+
+By analyzing the uncontrollable output directions, an engineer can decide on whether it is acceptable to keep certain output combinations uncontrolled, or if additional actuators are needed.
+
+
+### Limitation Imposed by Time Delays {#limitation-imposed-by-time-delays}
+
+Time delays pose limitation also in MIMO systems. Let \\(\theta\_{ij}\\) denote the time delay in the \\(ij\\)'th element of \\(G(s)\\). Then a **lower bound on the time delay for output** \\(i\\) is given by the smallest delay in row \\(i\\) of \\(G(s)\\), that is
+\\[ \theta\_i^{\min} = \min\_j \theta\_{ij} \\]
+
+For MIMO systems, we have the surprising result that an increase time delay may sometimes improve the achievable performance. The time delay may indeed increase the decoupling between the outputs.
+
+
+### Limitations Imposed by RHP-Zeros {#limitations-imposed-by-rhp-zeros}
+
+The limitations imposed by RHP-zeros on MIMO systems are similar to those for SISO system, although they only apply in particular directions.
+
+The limitations of a RHP-zero located at \\(z\\) may be derived from the bound:
+\\[ \hnorm{w\_P S(s)} = \max\_{\w} \abs{w\_P(j\w)} \maxsv(S(j\w)) \ge \abs{w\_P(z)} \\]
+
+All the results derived for SISO systems generalize if we consider the "worst" direction corresponding to the maximum singular value \\(\maxsv(S)\\).
+For instance, if we choose \\(w\_P(s)\\) to require tight control at low frequencies, the bandwidth must satisfy \\(w\_B^\* < z/2\\).
+
+In MIMO systems, one can often **move the deteriorating effect of a RHP-zero to a given output** which may be less important to control well.
+This is possible because, although the interpolation constraint \\(y\_z^H T(z) = 0\\) imposes a certain relationship between the elements within each column of \\(T(s)\\), the columns of \\(T(s)\\) may still be selected independently.
+
+Requiring a decoupled response from \\(r\\) to \\(y\\) generally leads to the introduction of additional RHP-zero in \\(T(s)\\) which are not present in \\(G(s)\\).
+Moving the effect of the RHP-zero to a particular output generally add some interaction. Also, moving to RHP-zero in a direction where \\(y\_z\\) is small usually introduces more interaction than in a direction where \\(y\_z\\) is large.
+
+For example, if we have a RHP-zero with \\(y\_z = [0.03,\ -0.04,\ 0.9,\ 0.43]^T\\), then one may in theory move the bad effect of the RHP-zero to any of the outputs. However, in practice, it will be difficult to avoid the effect of the RHP-zero on output 3, because the zero direction is mainly in that output. Trying to move it somewhere else will give large interactions and poor performance.
+
+
+### Limitation Imposed by Unstable (RHP) Poles {#limitation-imposed-by-unstable--rhp--poles}
+
+For unstable plants, feedback is needed for stabilization. More precisely, the presence of an unstable pole \\(p\\) requires for internal stability \\(T(p) y\_p = y\_p\\) where \\(y\_p\\) is the output pole direction.
+
+
+
+
+The transfer function \\(KS\\) from plant output to plant inputs must satisfy for any RHP-pole \\(p\\)
+\\[ \hnorm{KS} \ge \normtwo{u\_p^H G\_s(p)^{-1}} \\]
+where \\(u\_p\\) is the input pole direction, and \\(G\_s\\) is the "stable version" of \\(G\\) with its RHP-poles mirrored in the LHP.
+
+
+
+
+From the bound \\(\hnorm{w\_T(s) T(s)} \ge \abs{w\_T(p)}\\), we find that a RHP-pole \\(p\\) imposes restrictions on \\(\maxsv(T)\\) which are identical to those derived on \\(\abs{T}\\) for SISO systems.
+Thus, we need to react sufficiently fast and we must require that \\(\maxsv(T(j\w))\\) is about 1 or larger up to the frequency \\(2 \abs{p}\\).
+
+
+
+
+Consider a scalar disturbance \\(d\\) and let the vector \\(g\_d\\) represents its effect on the outputs (\\(y = g\_d d\\)).
+The disturbance direction is defined as
+
+\begin{equation}
+ y\_d = \frac{1}{\normtwo{g\_d}} g\_d
+\end{equation}
+
+For a plant with multiple disturbances, \\(g\_d\\) is a column of the matrix \\(G\_d\\).
+
+
+
+
+\begin{equation}
+ \gamma\_d (G) = \maxsv(G) \maxsv(G^\dagger y\_d)
+\end{equation}
+
+where \\(G^\dagger\\) is the pseudo inverse of \\(G\\)
+
+
+
+Let assume \\(r=0\\) and that the system has been scaled. With feedback control \\(e = S g\_d d\\) and the performance objective is
+\\[ \normtwo{S g\_d} = \maxsv(S g\_d) < 1 \ \forall\w \quad \Leftrightarrow \quad \hnorm{S g\_d} < 1 \\]
+
+We derive bounds in terms of the singular values of \\(S\\):
+\\[ \minsv(S) \normtwo{g\_d} \le \normtwo{S g\_d} \le \maxsv(S) \normtwo{g\_d} \\]
+
+
+
+
+For acceptable performance **we must at least require that**
+\\[ \maxsv(I+L) > \normtwo{g\_d} \\]
+
+And **we may require that**
+\\[ \minsv(I+L) > \normtwo{g\_d} \\]
+
+
+
+For a square plant, the input needed for perfect disturbance rejection is \\(u = -G^{-1} G\_d d\\).
+
+For a single disturbance, as the worst-cast disturbance is \\(\abs{d(\w)} = 1\\), we get that input saturation is avoided (\\(\\|u\\|\_{\text{max}} \le 1\\)) if all elements in the vector \\(G^{-1} g\_d\\) are less than 1 in magnitude:
+\\[ \\|G^{-1} g\_d\\|\_{\text{max}} < 1, \ \forall\w \\]
+
+It is first recommended to **consider one disturbance at a time** by plotting as a function of frequency the individual elements of \\(G^{-1} G\_d\\). This will yields more information about which particular input is most likely to saturate and which disturbance is the most problematic.
+
+
+#### Inputs for Acceptable Control {#inputs-for-acceptable-control}
+
+We here consider the question: is it possible to achieve \\(\\|e\\|<1\\) while using inputs with \\(\\|u\\| \le 1\\)?
+
+For SISO systems, we have to required \\(\abs{G} > \abs{g\_d} - 1\\) at frequencies where \\(\abs{g\_d} > 1\\).
+We would like to generalize this result to MIMO systems.
+
+
+
+
+Each singular value \\(\sigma\_i\\) of \\(G\\) must approximately satisfy:
+
+\begin{equation}
+ \sigma\_i(G) \ge \abs{u\_i^H g\_d} - 1 \text{ where } \abs{u\_i^H g\_d} > 1
+\end{equation}
+
+with \\(u\_i\\) the \\(i\\)'th output singular vector of \\(G\\).
+
+\\(u\_i^H g\_d\\) may be interpreted as the projection of \\(g\_d\\) onto the \\(i\\)'th output singular vector of the plant.
+
+
+
+If all the elements of \\(E\_O\\) and \\(E\_I\\) are non-zero, then we have **full block (unstructured) uncertainty**.
+
+In many cases, the source of uncertainty is in the individual input or output channels, and we have that \\(E\_I\\) and \\(E\_O\\) are **diagonal matrices**. For example \\(E\_I = \text{diag}\\{\epsilon\_1, \epsilon\_2, \dots\\}\\) where \\(\epsilon\_i\\) is the **relative uncertainty in input channel** \\(i\\).
+
+Diagonal input uncertainty is **always** present in real systems and the magnitude of \\(\epsilon\_i\\) is typically \\(0.1\\) or larger.
+
+
+#### Effect of Uncertainty on Feedforward Control {#effect-of-uncertainty-on-feedforward-control}
+
+Consider a feedforward controller \\(u = K\_r r\\) for the case with no disturbance (\\(d = 0\\)). We assume that \\(G\\) is inversible and we select \\(K\_r = G^{-1}\\) to achieve perfect control (\\(e = 0\\)).
+However, for the actual plant \\(G^\prime\\) (with uncertainty), the actual control error \\(e^\prime = y^\prime - r = G^\prime G^{-1} r - r\\) is not null and we get:
+
+- For output uncertainty: \\(e^\prime = E\_O r\\)
+- For input uncertainty: \\(e^\prime = G E\_I G^{-1} r\\)
+
+For output uncertainty, we have an identical result as for SISO systems: the worst case relative control error \\(\normtwo{e^\prime}/\normtwo{r}\\) is equal to the magnitude of the relative output uncertainty \\(\maxsv(E\_O)\\).
+However, for input uncertainty, the sensitivity may be much larger because the elements in the matrix \\(G E\_I G^{-1}\\) can be much larger than the elements in \\(E\_I\\).
+
+
+
+
+For diagonal input uncertainty, the elements of \\(G E\_I G^{-1}\\) are directly related to the RGA:
+\\[ \left[ G E\_I G^{-1} \right]\_{ii} = \sum\_{j=1}^n \lambda\_{ij}(G) \epsilon\_j \\]
+
+
+
+**Feedforward** Let the nominal transfer function with feedforward control be \\(y = T\_r r\\) where \\(T\_r = G K\_r\\) and \\(K\_r = G^{-1}\\).
+With model error \\(T\_r^\prime = G^\prime K\_r\\) and the change in response is \\(y^\prime - y = (T\_r^\prime - T\_r) r = (G^\prime - G)G^{-1} T\_r r = E\_O T\_r r\\).
+Thus, the control error caused by the uncertainty is equal to the relative output uncertainty.
+
+**Feedback control** The output is \\(y = T r\\).
+The change in response is \\(y^\prime - y = (T^\prime - T)r = S^\prime E\_O T r = S^\prime E\_O y\\).
+With feedback control, **the effect of the uncertainty is reduced** by a factor \\(S^\prime\\) compared to that with feedforward control.
+
+
+#### Uncertainty and the Sensitivity Peak {#uncertainty-and-the-sensitivity-peak}
+
+Consider a controller \\(K(s) = l(s)G^{-1}(s)\\) which results in a nominally decoupled response with sensitivity \\(S = s \cdot I\\) and complementary sensitivity \\(T = t \cdot I\\) where \\(t(s) = 1 - s(s)\\).
+Suppose the plant has diagonal input uncertainty of relative magnitude \\(\abs{w\_I(j\w)}\\) in each input channel.
+Then there exists a combination of input uncertainties such that at each frequency:
+\\[ \maxsv(S^\prime) \ge \maxsv(S) \left( 1 + \frac{\abs{w\_I t}}{1+\abs{w\_I t}} \\|\Lambda(G)\\|\_{i\infty} \right) \\]
+where \\(\\| \Lambda(G) \\|\_{i\infty}\\) is the maximum row sum of the RGA and \\(\maxsv(S) = \abs{s}\\).
+
+We can see that with an inverse based controller, the worst case sensitivity will be much larger than the nominal sensitivity at frequencies where the plant has large RGA elements.
+
+
+
+
+These statements apply to the frequency range around crossover.
+By "small", we mean smaller than 2 and by "large" we mean larger than 10.
+
+- Condition number \\(\gamma(G)\\) or \\(\gamma(K)\\) small: robust performance to both diagonal and full-block input uncertainty
+- Minimized condition number \\(\gamma\_I^\* (G)\\) or \\(\gamma\_O^\*(K)\\) small: robust performance to diagonal input uncertainty
+- \\(\text{RGA}(G)\\) has large elements: inverse based controller is not robust to diagonal input uncertainty.
+ Since diagonal input uncertainty is unavoidable in practice, the rule is never to use a decoupling controller for a plant with large RGA-elements.
+ **Plant with large RGA elements are fundamentally difficult to control**.
+
+
+
+
+The matrix \\(G\\) becomes singular if we make a relative change \\(-1/\lambda\_{ij}\\) in its \\(ij\\)'th elements, that is, if a single element in \\(G\\) is perturbed from \\(g\_{ij}\\) to \\(g\_{pij} = g\_{ij}(1-\frac{1}{\lambda\_{ij}})\\)
+
+
+
+To account for model uncertainty, we will assume that the dynamic behavior of a plant is described not by a single linear time invariant model but by a **set \\(\Pi\\) of possible linear time invariant models**, sometimes denoted the "**uncertainty set**".
+
+We adopt the following notation:
+
+- \\(\Pi\\) - a set of possible perturbed plant models
+- \\(G(s) \in \Pi\\) - nominal plant model
+- \\(G\_p(s) \in \Pi\\) - particular perturbed plant models
+
+We will use a "**norm-bounded uncertainty description**" where the set \\(\Pi\\) is generated by allowing \\(\hinf\\) norm-bounded stable perturbations to the nominal plant \\(G(s)\\).
+We let \\(E\\) denote a perturbation which is not normalized, and let \\(\Delta\\) denote a normalized perturbation with its \\(\hinf\\) norm less than 1.
+
+
+### Representing Uncertainty {#representing-uncertainty}
+
+Uncertainty in the plant model may have **several origins**:
+
+1. There are always parameters in the linear model which are only known approximatively
+2. Parameters in the model may vary due to **non-linearities** or changes in the operating conditions
+3. Measurement devices have imperfections
+4. At high frequencies, even the structure and the model order is unknown, and the uncertainty will always exceed \\(\SI{100}{\percent}\\) at some frequency
+5. Even when a very detailed model is available, we may choose to work with a simpler nominal model and **represent the neglected dynamics as "uncertainty"**
+6. The controller implemented may differ from the one obtained by solving the synthesis problem.
+ One may include uncertainty to allow for controller order reduction and implementation inaccuracies
+
+The various sources of model uncertainty may be grouped into two main classes:
+
+1. **Parametric uncertainty**. The structure of the model is known, but some parameters are uncertain
+2. **Neglected and unmodelled dynamics uncertainty**. The model is in error because of missing dynamics, usually at high frequencies
+
+
+
+
+Parametric uncertainty will be quantified by assuming that **each uncertain parameters is bounded within some region** \\([\alpha\_{\min}, \alpha\_{\text{max}}]\\). That is, we have parameter sets of the form
+
+\begin{equation}
+ \alpha\_p = \bar{\alpha}(1 + r\_\alpha \Delta); \quad r\_\alpha = \frac{\alpha\_{\text{max}} - \alpha\_{\min}}{\alpha\_{\text{max}} + \alpha\_{\min}}
+\end{equation}
+
+where \\(\bar{\alpha}\\) is the mean parameter value, \\(r\_\alpha\\) is the relative uncertainty in the parameter, and \\(\Delta\\) is any real scalar satisfying \\(\abs{\Delta} \le 1\\).
+
+
+
+There is also a third class of uncertainty (which is a combination of the other two) called **Lumped uncertainty**.
+Here the uncertainty description represents one or several sources of parametric and/or unmodelled dynamics uncertainty combined into a single lumped perturbation of a chosen structure.
+The frequency domain is also well suited for describing lumped uncertainty.
+
+
+
+
+In most cases, we prefer to lump the uncertainty into a multiplicative uncertainty of the form
+\\[ G\_p(s) = G(s)(1 + w\_I(s)\Delta\_I(s)); \quad \abs{\Delta\_I(j\w)} \le 1 \, \forall\w \\]
+which may be represented by the diagram in Fig. [fig:input_uncertainty_set](#fig:input_uncertainty_set).
+
+
+
+
+\\[ G\_p(s) = k\_p G\_0(s); \quad k\_{\min} \le k\_p \le k\_{\text{max}} \\]
+where \\(k\_p\\) is an uncertain gain and \\(G\_0(s)\\) is a transfer function with no uncertainty.
+By writing \\(k\_p = \bar{k}(1 + r\_k \Delta)\\) where \\(r\_k\\) is the relative magnitude of the gain uncertainty and \\(\bar{k}\\) is the average gain, be may write
+\\[ G\_p = \underbrace{\bar{k}G\_0(s)}\_{G(s)} (1 + r\_k \Delta), \quad \abs{\Delta} \le 1 \\]
+where \\(\Delta\\) is a real scalar and \\(G(s)\\) is the nominal plant.
+
+
+
+
+\\[ G\_p(s) = \frac{1}{\tau\_p s + 1}G\_0(s); \quad \tau\_{\min} \le \tau\_p \le \tau\_{\text{max}} \\]
+By writing \\(\tau\_p = \bar{\tau}(1 + r\_\tau \Delta)\\), with \\(\abs{\Delta} \le 1\\), the model set can be rewritten as
+\\[ G\_p(s) = \frac{G\_0}{1+\bar{\tau} s + r\_\tau \bar{\tau} s \Delta} = \underbrace{\frac{G\_0}{1+\bar{\tau}s}}\_{G(s)} \frac{1}{1 + w\_{iI}(s) \Delta} \\]
+with \\(\displaystyle w\_{iI}(s) = \frac{r\_\tau \bar{\tau} s}{1 + \bar{\tau} s}\\).
+
+
+
+
+The disc-shaped regions may be generated by additive complex norm-bounded perturbations around a nominal plant \\(G\\)
+
+\begin{equation}
+ \begin{aligned}
+ \Pi\_A: \ G\_p(s) &= G(s) + w\_A(s) \Delta\_A(s) \\\\\\
+ & \text{with }\abs{\Delta\_A(j\w)} \le 1 \, \forall\w
+ \end{aligned}
+\end{equation}
+
+At each frequency, all possible \\(\Delta(j\w)\\) "generates" a disc-shaped region with radius 1 centered at 0, so \\(G(j\w) + w\_A(j\w)\Delta\_A(j\w)\\) generates at each frequency a disc-shapes region of radius \\(\abs{w\_A(j\w)}\\) centered at \\(G(j\w)\\) as shown in Fig. [fig:uncertainty_disc_generated](#fig:uncertainty_disc_generated).
+
+
+
+
+The disc-shaped region may alternatively be represented by a **multiplicative uncertainty**
+
+\begin{equation}
+ \begin{aligned}
+ \Pi\_I: \ G\_p(s) &= G(s)(1 + w\_I(s)\Delta\_I(s)); \\\\\\
+ & \text{with }\abs{\Delta\_I(j\w)} \le 1 \, \forall\w
+ \end{aligned}
+\end{equation}
+
+
+
+
+We want to represent the following set using multiplicative uncertainty with a rational weight \\(w\_I(s)\\)
+\\[ \Pi: \quad G\_p(s) = \frac{k}{\tau s + 1} e^{-\theta s}, \quad 2 \le k, \theta, \tau \le 3 \\]
+To simplify subsequent controller design, we select a delay-free nominal model
+\\[ G(s) = \frac{\bar{k}}{\bar{\tau} s + 1} = \frac{2.5}{2.5 s + 1} \\]
+
+To obtain \\(l\_I(\w)\\), we consider three values (2, 2.5 and 3) for each of the three parameters (\\(k, \theta, \tau\\)).
+The corresponding relative errors \\(\abs{\frac{G\_p-G}{G}}\\) are shown as functions of frequency for the \\(3^3 = 27\\) resulting \\(G\_p\\) (Fig. [fig:uncertainty_weight](#fig:uncertainty_weight)).
+To derive \\(w\_I(s)\\), we then try to find a simple weight so that \\(\abs{w\_I(j\w)}\\) lies above all the dotted lines.
+
+
+
+
+To represent unmodelled dynamics, we usually use a simple **multiplicative weight** of the form
+
+\begin{equation}
+ w\_I(s) = \frac{\tau s + r\_0}{(\tau/r\_\infty) s + 1}
+\end{equation}
+
+where \\(r\_0\\) is the relative uncertainty at steady-state, \\(1/\tau\\) is the frequency at which the relative uncertainty reaches \\(\SI{100}{\percent}\\), and \\(r\_\infty\\) is the magnitude of the weight at high frequency.
+
+
+
+
+The requirement of robust stability for the case with multiplicative uncertainty gives an **upper bound on the complementary sensitivity**
+
+\begin{equation}
+ \text{RS} \quad \Leftrightarrow \quad \abs{T} < 1/\abs{w\_I}, \ \forall\w
+\end{equation}
+
+
+
+
+The requirement for robust stability for the case with inverse multiplicative uncertainty gives an **upper bound on the sensitivity**
+
+\begin{equation}
+ \text{RS} \quad \Leftrightarrow \quad \abs{S} < 1/\abs{w\_{iI}}, \ \forall\w
+\end{equation}
+
+We see that we need tight control and have to make \\(S\\) small at frequencies where the uncertainty is large and \\(w\_{iI}\\) exceeds 1 in magnitude.
+
+
+
+
+The condition for nominal performance when considering performance in terms of the **weighted sensitivity** function is
+
+\begin{equation}
+ \begin{aligned}
+ \text{NP} &\Leftrightarrow \abs{w\_P S} < 1 \ \forall\omega \\\\\\
+ &\Leftrightarrow \abs{w\_P} < \abs{1 + L} \ \forall\omega
+ \end{aligned}
+\end{equation}
+
+
+
+
+For robust performance, we require the performance condition to be satisfied for **all** possible plants:
+
+\begin{equation}
+ \begin{aligned}
+ \text{RP}\ &\overset{\text{def}}{\Leftrightarrow}\ \abs{w\_P S} < 1 \quad \forall S\_p, \forall \omega\\\\\\
+ \ &\Leftrightarrow\ \abs{w\_P} < \abs{1 + L\_p} \quad \forall L\_p, \forall \omega
+ \end{aligned}
+\end{equation}
+
+
+
+
+Finally, we obtain the following condition for Robust Performance:
+
+\begin{equation}
+ \text{RP} \ \Leftrightarrow\ \max\_{\omega} \left(\abs{w\_P S} + \abs{w\_I T} \right) < 1
+\end{equation}
+
+
+
+
+\begin{subequations}
+ \begin{align}
+ \text{NP} & \Leftrightarrow |w\_P S| < 1,\ \forall \omega \\\\\\
+ \text{RS} & \Leftrightarrow |w\_I T| < 1,\ \forall \omega \\\\\\
+ \text{RP} & \Leftrightarrow |w\_P S| + |w\_I T| < 1,\ \forall \omega
+ \end{align}
+\end{subequations}
+
+
+
+In addition, for SISO systems, if we satisfy both RS and NP, then we have at each frequency:
+\\[ |w\_P S| + |w\_I T| < 2 \cdot \max \\{|w\_P S|, |w\_I T|\\} < 2 \\]
+It then follows that, within a factor at most 2, we will automatically get RP when NP and RS are satisfied.
+This, RP is not a "big issue" for SISO systems.
+
+To satisfy RS we generally want \\(T\\) small, whereas to satisfy \\(NP\\) we generally want \\(S\\) small.
+However, we cannot make both \\(S\\) and \\(T\\) small at the same frequency because of the identity \\(S + T = 1\\).
+This has implications for RP:
+
+\begin{align\*}
+ |w\_P S| + |w\_I T| &\ge \text{min}\\{|w\_P|, |w\_I|\\}(|S| + |T|) \\\\\\
+ &\ge \text{min}\\{|w\_P|, |w\_I|\\}(|S + T|) \\\\\\
+ &\ge \text{min}\\{|w\_P|, |w\_I|\\}
+\end{align\*}
+
+This means that we cannot have both \\(|w\_P|>1\\) (i.e. good performance) and \\(|w\_I|>1\\) (i.e. more than \\(\si{100}{\%}\\) uncertainty) at the same frequency.
+
+
+### Examples of Parametric Uncertainty {#examples-of-parametric-uncertainty}
+
+
+#### Parametric Pole Uncertainty {#parametric-pole-uncertainty}
+
+Consider the following set of plants:
+\\[ G\_p(s) = \frac{1}{s - a\_p} G\_0(s); \quad a\_\text{min} \le a\_p \le a\_{\text{max}} \\]
+
+If \\(a\_\text{min}\\) and \\(a\_\text{max}\\) have different signs, then this means that the plant can change from stable to unstable with the pole crossing through the origin.
+
+This set of plants can be written as
+\\[ G\_p(s) = \frac{G\_0(s)}{s - \bar{a}(1 + r\_a \Delta)}; \quad -1 \le \Delta \le 1 \\]
+which can be exactly described by inverse multiplicative uncertainty:
+\\[ G(s) = \frac{G\_0(s)}{(s - \bar{a})}; \quad w\_{iI}(s) = \frac{r\_a \bar{a}}{s - \bar{a}} \\]
+
+The magnitude of \\(w\_{iI}(s)\\) is equal to \\(r\_a\\) at low frequency and goes to \\(0\\) at high frequencies.
+
+
+##### Time constant form {#time-constant-form}
+
+It is also interesting to consider another form of pole uncertainty, namely that associated with the time constant:
+\\[ G\_p(s) = \frac{1}{\tau\_p s + 1} G\_0(s); \quad \tau\_\text{min} \le \tau\_p \le \tau\_\text{max} \\]
+
+The corresponding uncertainty weight is
+\\[ w\_{iI}(s) = \frac{r\_\tau \bar{\tau} s}{1 + \bar{\tau} s} \\]
+
+This results in uncertainty in the pole location, but here the uncertainty affects the model at high frequency.
+
+
+#### Parametric Zero Uncertainty {#parametric-zero-uncertainty}
+
+Consider zero uncertainty in the "time constant" form as in:
+\\[ G\_p(s) = (1 + \tau\_p s)G\_0(s); \quad \tau\_\text{min} \le \tau\_p \le \tau\_\text{max} \\]
+
+This set of plants may be written as multiplicative uncertainty with:
+\\[ w\_I(s) = \frac{r\_\tau \bar{\tau} s}{1 + \bar{\tau} s} \\]
+The magnitude \\(|w\_I(j\omega)|\\) is small at low frequencies and approaches \\(r\_\tau\\) at high frequencies.
+For cases with \\(r\_\tau > 1\\) we allow the zero to cross from the LHP to the RHP.
+
+
+#### Parametric State-Space Uncertainty {#parametric-state-space-uncertainty}
+
+A general procedure for handling parametric uncertainty which is more suited for numerical calculations, is parametric state-space uncertainty.
+Consider an uncertain state-space model:
+
+\begin{align\*}
+ \dot{x} &= A\_p x + B\_p u \\\\\\
+ y &= C\_p x + D\_p u
+\end{align\*}
+
+Assume that the underlying cause for the uncertainty is uncertainty in some real parameters \\(\delta\_1, \delta\_2, \dots\\) and assume that the state space matrices depends linearly on these parameters:
+
+\begin{align\*}
+ A\_p = A + \sum \delta\_i A\_i; \quad & B\_p = B + \sum \delta\_i B\_i \\\\\\
+ C\_p = C + \sum \delta\_i C\_i; \quad & D\_p = D + \sum \delta\_i D\_i
+\end{align\*}
+
+where \\(A\\), \\(B\\), \\(C\\) and \\(D\\) model the nominal system.
+
+We can collect the perturbations \\(\delta\_i\\) in a large diagonal matrix \\(\Delta\\) with the real \\(\delta\_i\\)'s along its diagonal:
+\\[ A\_p = A + \sum \delta\_i A\_i = A + W\_2 \Delta W\_1 \\]
+
+In the transfer function form:
+
+\begin{align\*}
+ (s I - A\_p)^{-1} &= (sI - A - W\_2 \Delta W\_1)^{-1} \\\\\\
+ &= (I - \Phi(s) W\_2 \Delta W\_1)^{-1} \Phi(s)
+\end{align\*}
+
+with \\(\Phi(s) \triangleq (sI - A)^{-1}\\).
+
+This is illustrated in the block diagram of Fig. [fig:uncertainty_state_a_matrix](#fig:uncertainty_state_a_matrix), which is in the form of an inverse additive perturbation.
+
+
+
+{{< figure src="/ox-hugo/skogestad07_uncertainty_state_a_matrix.png" caption="Figure 25: Uncertainty in state space A-matrix" >}}
+
+
+### Conclusion {#conclusion}
+
+Model uncertainty for SISO systems can be represented in the frequency domain using complex norm-bounded perturbations \\(\hnorm{\Delta} \le 1\\).
+
+Requirements of robust stability for the case of multiplicative complex uncertainty imposes an upper bound on the allowed complementary sensitivity, \\(\abs{w\_I T} < 1, \ \forall\w\\).
+
+Similarly, the inverse multiplicative uncertainty imposes an upper bound on the sensitivity, \\(\abs{w\_{iI} S} < 1, \ \forall\w\\).
+
+We also derived a condition for robust performance with multiplicative uncertainty, \\(\abs{w\_P S} + \abs{w\_I T} < 1, \ \forall\w\\).
+
+
+## Robust Stability and Performance Analysis {#robust-stability-and-performance-analysis}
+
+
+
+
+### General Control Configuration with Uncertainty {#general-control-configuration-with-uncertainty}
+
+The starting point for our robustness analysis is a system representation in which the uncertain perturbations are "pulled out" into a **block diagonal matrix**
+\\[ \Delta = \text{diag} \\{\Delta\_i\\} = \begin{bmatrix}\Delta\_1 \\ & \ddots \\ & & \Delta\_i \\ & & & \ddots \end{bmatrix} \\]
+where each \\(\Delta\_i\\) represents a **specific source of uncertainty**, e.g. input uncertainty \\(\Delta\_I\\) or parametric uncertainty \\(\delta\_i\\).
+
+If we also pull out the controller \\(K\\), we get the generalized plant \\(P\\) as shown in Fig. [fig:general_control_delta](#fig:general_control_delta). This form is useful for controller synthesis.
+
+
+
+{{< figure src="/ox-hugo/skogestad07_general_control_delta.png" caption="Figure 26: General control configuration used for controller synthesis" >}}
+
+If the controller is given and we want to analyze the uncertain system, we use the \\(N\Delta\text{-structure}\\) in Fig. [fig:general_control_Ndelta](#fig:general_control_Ndelta).
+
+
+
+{{< figure src="/ox-hugo/skogestad07_general_control_Ndelta.png" caption="Figure 27: \\(N\Delta\text{-structure}\\) for robust performance analysis" >}}
+
+\\(N\\) is related to \\(P\\) and \\(K\\) by a **lower LFT**
+
+\begin{align\*}
+ N &= F\_l(P, K) \\\\\\
+ &\triangleq P\_{11} + P\_{12} K (I - P\_{22}K)^{-1} P\_{21}
+\end{align\*}
+
+Similarly, the uncertain closed-loop transfer function from \\(w\\) to \\(z\\), is related to \\(N\\) and \\(\Delta\\) by an **upper LFT**
+
+\begin{align\*}
+ F &= F\_u(N, \Delta) \\\\\\
+ &\triangleq N\_{22} + N\_{21} \Delta (I - N\_{11} \Delta)^{-1} N\_{12}
+\end{align\*}
+
+To analyze robust stability of \\(F\\), we can rearrange the system into the \\(M\Delta\text{-structure}\\) shown in Fig. [fig:general_control_Mdelta_bis](#fig:general_control_Mdelta_bis) where \\(M = N\_{11}\\) is the transfer function from the output to the input of the perturbations.
+
+
+
+{{< figure src="/ox-hugo/skogestad07_general_control_Mdelta_bis.png" caption="Figure 28: \\(M\Delta\text{-structure}\\) for robust stability analysis" >}}
+
+
+### Representing Uncertainty {#representing-uncertainty}
+
+Each individual perturbation is assumed to be **stable and normalized**:
+\\[ \maxsv(\Delta\_i(j\w)) \le 1 \quad \forall\w \\]
+
+As the maximum singular value of a block diagonal matrix is equal to the largest of the maximum singular values of the individual blocks, it then follows for \\(\Delta = \text{diag}\\{\Delta\_i\\}\\) that
+\\[ \maxsv(\Delta\_i(j\w)) \le 1 \quad \forall\w, \forall i \quad \Leftrightarrow \quad \tcmbox{\hnorm{\Delta} \le 1} \\]
+
+
+#### Differences Between SISO and MIMO Systems {#differences-between-siso-and-mimo-systems}
+
+The main difference between SISO and MIMO systems is the concept of directions which is only relevant in the latter.
+As a consequence, MIMO systems may experience **much larger sensitivity to uncertainty** than SISO systems.
+
+
+#### Parametric Uncertainty {#parametric-uncertainty}
+
+The representation of parametric uncertainty for MIMO systems is the same as for SISO systems.
+However, the inclusion of parametric uncertainty may be more significant for MIMO plants because it offers a simple method of representing uncertain transfer function elements.
+
+
+#### Unstructured Uncertainty {#unstructured-uncertainty}
+
+Unstructured perturbations are often used to get a simple uncertainty model.
+We here define unstructured uncertainty as the use of a "full" complex perturbation matrix \\(\Delta\\), usually with dimensions compatible with those of the plant, where at each frequency any \\(\Delta(j\w)\\) satisfying \\(\maxsv(\Delta(j\w)) < 1\\) is allowed.
+
+Three common forms of **feedforward unstructured uncertainty** are shown Fig. [fig:feedforward_uncertainty](#fig:feedforward_uncertainty): additive uncertainty, multiplicative input uncertainty and multiplicative output uncertainty.
+
+
+
+
+\begin{alignat\*}{3}
+ &\Pi\_A: \quad &&G\_p = G + E\_A; \quad& &E\_a = w\_A \Delta\_a \\\\\\
+ &\Pi\_I: \quad &&G\_p = G(I + E\_I); \quad& &E\_I = w\_I \Delta\_I \\\\\\
+ &\Pi\_O: \quad &&G\_p = (I + E\_O)G; \quad& &E\_O = w\_O \Delta\_O
+\end{alignat\*}
+
+
+
Table 4:
+ Common feedforward unstructured uncertainty
+
+
+
+\begin{alignat\*}{3}
+ &\Pi\_{iA}: \quad &&G\_p = G(I - E\_{iA} G)^{-1}; & & \quad E\_{ia} = w\_{iA} \Delta\_{ia} \\\\\\
+ &\Pi\_{iI}: \quad &&G\_p = G(I - E\_{iI})^{-1}; & & \quad E\_{iI} = w\_{iI} \Delta\_{iI} \\\\\\
+ &\Pi\_{iO}: \quad &&G\_p = (I - E\_{iO})^{-1}G; & & \quad E\_{iO} = w\_{iO} \Delta\_{iO}
+\end{alignat\*}
+
+
+
Table 5:
+ Common feedback unstructured uncertainty
+
+
+Since **output uncertainty is frequently less restrictive than input uncertainty in terms of control performance**, we first attempt to lump the uncertainty at the output. For example, a set of plant \\(\Pi\\) may be represented by multiplicative output uncertainty with a scalar weight \\(w\_O(s)\\) using
+\\[ G\_p = (I + w\_O \Delta\_O) G, \quad \hnorm{\Delta\_O} \le 1 \\]
+where
+\\[ l\_O(\w) = \max\_{G\_p \in \Pi} \maxsv\left( (G\_p - G)G^{-1} \right); \ \abs{w\_O(j\w)} \ge l\_O(\w), \, \forall\w \\]
+
+If the resulting uncertainty weight is reasonable and the analysis shows that robust stability and performance may be achieve, then this lumping of uncertainty at the output is fine.
+If this is not the case, then one may try to lump the uncertainty at the input instead, using multiplicative input uncertainty with a scalar weight,
+\\[ G\_p = G(I + w\_I \Delta\_I), \quad \hnorm{\Delta\_I} \le 1 \\]
+where
+\\[ l\_I(\w) = \max\_{G\_p \in \Pi} \maxsv\left( G^{-1}(G\_p - G) \right); \ \abs{w\_I(j\w)} \ge l\_I(\w), \, \forall\w \\]
+
+However, in many cases, this approach of lumping uncertainty either at the output or the input does **not** work well because **it usually introduces additional plants** that were not present in the original set.
+
+
+##### Conclusion {#conclusion}
+
+Ideally, we would like to lump several sources of uncertainty into a single perturbation to get a simple uncertainty description.
+Often an unstructured multiplicative output perturbation is used.
+However, we should be careful about doing this, at least for plants with a large condition number.
+In such cases we may have to represent the uncertainty as it occurs physically (at the input, in the elements, etc.) thereby generating several perturbations.
+
+
+#### Diagonal Uncertainty {#diagonal-uncertainty}
+
+By "diagonal uncertainty" we mean that the perturbation is a complex diagonal matrix
+\\[ \Delta(s) = \text{diag}\\{\delta\_i(s)\\}; \quad \abs{\delta\_i(j\w)} \le 1, \ \forall\w, \, \forall i \\]
+
+Diagonal uncertainty usually arises from a consideration of uncertainty or neglected dynamics in the **individual input or output channels**.
+This type of diagonal uncertainty is **always present**.
+
+
+
+
+Let us consider uncertainty in the input channels. With each input \\(u\_i\\), there is a physical system (amplifier, actuator, etc.) which based on the controller output signal \\(u\_i\\), generates a physical plant input \\(m\_i\\)
+\\[ m\_i = h\_i(s) u\_i \\]
+The scalar transfer function \\(h\_i(s)\\) is often absorbed into the plant model \\(G(s)\\).
+We can represent its uncertainty as multiplicative uncertainty
+\\[ h\_{pi}(s) = h\_i(s)(1 + w\_{Ii}(s)\delta\_i(s)); \quad \abs{\delta\_i(j\w)} \le 1, \, \forall\w \\]
+which after combining all input channels results in diagonal input uncertainty for the plant
+
+\begin{align\*}
+ G\_p(s) = G(I + W\_I \Delta\_I) \text{ with } &\Delta\_I = \diag{\delta\_i} \\\\\\
+ &W\_I = \diag{w\_{Ii}}
+\end{align\*}
+
+
+
+**Diagonal input uncertainty should always be considered because**:
+
+- it is **always** present and a system which is sensitive to this uncertainty will not work in practice
+- it often **restrict achievable performance** with multivariable control
+
+
+### Obtaining \\(P\\), \\(N\\) and \\(M\\) {#obtaining--p----n--and--m}
+
+Let's consider the feedback system with multiplicative input uncertainty \\(\Delta\_I\\) shown Fig. [fig:input_uncertainty_set_feedback_weight](#fig:input_uncertainty_set_feedback_weight).
+\\(W\_I\\) is a normalization weight for the uncertainty and \\(W\_P\\) is a performance weight.
+
+
+
+{{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback_weight.png" caption="Figure 29: System with multiplicative input uncertainty and performance measured at the output" >}}
+
+We want to derive the generalized plant \\(P\\) which has inputs \\([u\_\Delta,\ w,\ u]^T\\) and outputs \\([y\_\Delta,\ z,\ v]^T\\).
+
+By breaking the loop before and after \\(K\\) and \\(\Delta\_I\\), we get
+\\[ P = \begin{bmatrix}
+ 0 & 0 & W\_I \\\\\\
+ W\_P G & W\_P & W\_P G \\\\\\
+ -G & -I & -G
+\end{bmatrix} \\]
+
+Next, we want to derive the matrix \\(N\\). We fist partition \\(P\\) to be compatible with \\(K\\):
+
+\begin{align\*}
+ P\_{11} = \begin{bmatrix}0&0\\GW\_P&W\_P\end{bmatrix},\quad & P\_{12} = \begin{bmatrix}W\_I\\GW\_P\end{bmatrix} \\\\\\
+ P\_{21} = \begin{bmatrix}G&-1\end{bmatrix}, \quad & P\_{22} = -G \\\\\\
+\end{align\*}
+
+and then we find \\(N\\) using \\(N = F\_l(P, K)\\).
+
+
+### Definitions of Robust Stability and Robust Performance {#definitions-of-robust-stability-and-robust-performance}
+
+The next step is to check whether we have stability and acceptable performance for all plant in the set:
+
+1. **Robust stability analysis**: with a given controller \\(K\\) we determine whether the system remains stable for all plants in the uncertainty set
+2. **Robust performance analysis**: is RS is satisfied, we determine how "large" the transfer function from exogenous inputs \\(w\\) to outputs \\(z\\) may be for all plants in the uncertainty set
+
+We have \\(z = F(\Delta) \cdot w\\) with
+
+\begin{align\*}
+ F &= F\_u(N, \Delta) \\\\\\
+ &\triangleq N\_{22} + N\_{21}\Delta(I - N\_{11}\Delta)^{-1} N\_{12}
+\end{align\*}
+
+We here use \\(\hinf\\) norm to define performance and require for RP that \\(\hnorm{F(\Delta)} \le 1\\) for all allowed \\(\Delta\\).
+A typical choice is \\(F = w\_P S\_P\\) where \\(w\_P\\) is the performance weight and \\(S\_P\\) represents the set of perturbed sensitivity functions.
+
+
+
+
+In terms of the \\(N\Delta\text{-structure}\\), our requirements for stability and performance can be summarized as follows:
+
+\begin{align\*}
+\text{NS} &\ \stackrel{\text{def}}{\Longleftrightarrow} \ N \text{ is internally stable} \\\\\\
+\text{NP} &\ \stackrel{\text{def}}{\Longleftrightarrow} \ \text{NS and } \hnorm{N\_{22}} < 1 \\\\\\
+\text{RS} &\ \stackrel{\text{def}}{\Longleftrightarrow} \ \text{NS and } F = F\_u(N, \Delta) \text{ is stable } \forall\Delta \\\\\\
+\text{RP} &\ \stackrel{\text{def}}{\Longleftrightarrow} \ \text{NS and } \hnorm{F} < 1, \quad \forall \Delta, \,\hnorm{\Delta} \le 1 \\\\\\
+\end{align\*}
+
+
+
+
+Assume that the nominal system \\(M(s)\\) and the perturbations \\(\Delta(s)\\) are stable.
+Consider the convex set of perturbations \\(\Delta\\), such that if \\(\Delta^\prime\\) is an allowed perturbation then so is \\(c\Delta^\prime\\) where c is any **real** scalar such that \\(\abs{c} \le 1\\).
+Then the \\(M\Delta\text{-structure}\\) is stable for all allowed perturbations **if and only if** the Nyquist plot of \\(\det\left( I - M\Delta(s) \right)\\) does not encircle the origin, \\(\forall\Delta\\):
+
+\begin{equation}
+ \det( I - M\Delta(j\w)) \ne 0, \quad \forall\w, \, \forall\Delta
+\end{equation}
+
+
+
+
+Assume that the nominal system \\(M(s)\\) and the perturbations \\(\Delta(s)\\) are stable.
+Consider the class of perturbations, \\(\Delta\\), such that if \\(\Delta^\prime\\) is an allowed perturbation, then so is \\(c\Delta^\prime\\) where c is any **complex** scalar such that \\(\abs{c} \le 1\\).
+Then the \\(M\Delta\text{-structure}\\) is stable for all allowed perturbations **if and only if**:
+
+\begin{equation}
+\begin{aligned}
+ &\rho(M\Delta(j\w)) < 1, \quad \forall\w, \, \forall\Delta\\\\\\
+ \Leftrightarrow \quad &\max\_{\Delta} \rho(M\Delta(j\w)) < 1, \quad \forall\w
+\end{aligned}
+\end{equation}
+
+
+
+
+Assume that the nominal system \\(M(s)\\) is stable and that the perturbations \\(\Delta(s)\\) are stable.
+Then the \\(M\Delta\text{-system}\\) is stable for all perturbations \\(\Delta\\) satisfying \\(\hnorm{\Delta} \le 1\\) if and only if
+
+\begin{equation}
+ \maxsv(M(j\w)) < 1 \ \forall\w \quad \Leftrightarrow \quad \hnorm{M} < 1
+\end{equation}
+
+
+
+
+This applies for any \\(D\\), and therefore the "most improved" (least conservative) RS-condition is obtained by minimizing at each frequency the scaled singular value and we have
+\\[ \text{RS if } \min\_{D(\w) \in \mathcal{D}} \maxsv(D(\w)M(j\w)D(\w)^{-1}) < 1, \ \forall\w \\]
+where \\(\mathcal{D}\\) is the set of block-diagonal matrices whose structure is compatible to that of \\(\Delta\\), i.e, \\(\Delta D = D \Delta\\).
+
+
+
+The above definition of \\(\mu\\) involves varying \\(\maxsv(\Delta)\\). However, we prefer to normalize \\(\Delta\\) such that \\(\maxsv(\Delta)\le1\\). We can do that by scaling \\(\Delta\\) by a factor \\(k\_m\\), and looking for the smallest \\(k\_m\\) which makes the matrix \\(I - k\_m M \Delta\\) singular. \\(\mu\\) is then the reciprocal of this small \\(k\_m\\): \\(\mu = 1/k\_m\\). This results in the following alternative definition of \\(\mu\\).
+
+
+
+
+Let \\(M\\) be a given complex matrix and let \\(\Delta = \diag{\Delta\_i}\\) denote a set of complex matrices with \\(\maxsv(\Delta) \le 1\\) and with a given block-diagonal structure.
+The real non-negative function \\(\mu(M)\\), called the structured singular value, is defined by
+
+\begin{align\*}
+ \mu(M) \triangleq &(\min\\{ k\_m | \det(I - k\_m M \Delta) = 0\\\\\\
+ &\text{for structured } \Delta, \maxsv(\Delta) \le 1 \\} )^{-1}
+\end{align\*}
+
+If no such structured \\(\Delta\\) exists then \\(\mu(M) = 0\\)
+
+
+
+
+Assume that the nominal system \\(M\\) and the perturbations \\(\Delta\\) are stable.
+Then the \\(M\Delta\text{-system}\\) is stable for all allowed perturbations with \\(\maxsv(\Delta)\le 1, \ \forall\w\\) if on only if
+
+\begin{equation}
+ \mu(M(j\w)) < 1, \ \forall \omega
+\end{equation}
+
+
+
+
+Let \\(\Delta = \diag{\Delta\_1, \Delta\_2}\\) and assume we have fixed \\(\norm{\Delta\_1} \le 1\\) and we want to find how large \\(\Delta\_2\\) can be before we get instability.
+The solution is to select
+\\[ K\_m = \begin{bmatrix}I & 0 \\ 0 & k\_m I\end{bmatrix} \\]
+and look at each frequency for the smallest value of \\(k\_m\\) which makes \\(\det(I - K\_m M \Delta) = 0\\) and we have that skewed-\\(\mu\\) is
+\\[ \mu^s(M) \triangleq 1/k\_m \\]
+
+
+
+
+Rearrange the uncertain system into the \\(N\Delta\text{-structure}\\). Assume nominal stability such that \\(N\\) is stable.
+Then
+
+\begin{align\*}
+\text{RP} \ &\stackrel{\text{def}}{\Longleftrightarrow} \ \hnorm{F} = \hnorm{F\_u(N, \Delta)} < 1, \ \forall \hnorm{\Delta} < 1 \\\\\\
+ &\Longleftrightarrow \ \mu\_{\hat{\Delta}}(N(j\w)) < 1, \ \forall\w
+\end{align\*}
+
+where \\(\mu\\) is computed with respect to the structure
+\\[ \hat{\Delta} = \begin{bmatrix}\Delta & 0 \\ 0 & \Delta\_P\end{bmatrix} \\]
+and \\(\Delta\_P\\) is a full complex perturbation with the same dimensions as \\(F^T\\).
+
+
+
+
+Rearrange the uncertain system into the \\(N\Delta\text{-structure}\\) where the block-diagonal perturbation satisfy \\(\hnorm{\Delta} \le 1\\).
+Introduce
+\\[ F = F\_u(N, \Delta) = N\_{22} + N\_{21}\Delta(I - N\_{11} \Delta)^{-1} N\_{12} \\]
+Let the performance requirement be \\(\hnorm{F} \le 1\\).
+
+\begin{align\*}
+ \text{NS} \ &\Leftrightarrow \ N \text{ (internally) stable} \\\\\\
+ \text{NP} \ &\Leftrightarrow \ \text{NS and } \maxsv(N\_{22}) = \mu\_{\Delta\_P} < 1, \ \forall\w \\\\\\
+ \text{RS} \ &\Leftrightarrow \ \text{NS and } \mu\_\Delta(N\_{11}) < 1, \ \forall\w \\\\\\
+ \text{RP} \ &\Leftrightarrow \ \text{NS and } \mu\_{\tilde{\Delta}}(N) < 1, \ \forall\w, \ \tilde{\Delta} = \begin{bmatrix}\Delta & 0 \\ 0 & \Delta\_P\end{bmatrix}
+\end{align\*}
+
+
+
+The starting point is the upper bound on \\(\mu\\) in terms of the scaled singular value
+
+\begin{equation}
+ \mu(N) \le \min\_{D \in \mathcal{D}} \maxsv(D N D^{-1})
+\end{equation}
+
+The idea is to find the controller that minimizes the peak value over frequency of this upper bound, namely
+
+\begin{equation}
+ \min\_{K} \left( \min\_{D \in \mathcal{D}} \hnorm{D N(K) D^{-1} } \right)
+\end{equation}
+
+by alternating between minimizing \\(\hnorm{DN(K)D^{-1}}\\) with respect to either \\(K\\) or \\(D\\) (while holding the other fixed).
+
+To start the iterations, one selects an initial stable rational transfer matrix \\(D(s)\\) with appropriate structure.
+The identity matrix is often a good initial choice for \\(D\\) provided the system has been reasonably scaled for performance.
+
+
+
+
+1. **K-step**. Synthesize an \\(\hinf\\) controller for the scaled problem, \\(\min\_{K} \hnorm{DN(K)D^{-1}}\\) with fixed \\(D(s)\\)
+2. **D-step**. Find \\(D(j\w)\\) to minimize at each frequency \\(\maxsv(DND^{-1}(j\w))\\) with fixed \\(N\\)
+3. Fit the magnitude of each element of \\(D(j\w)\\) to a stable and minimum phase transfer function \\(D(s)\\) and go to step 1
+
+
+
+The order of the controller resulting from each iteration is equal to the number of the states in the plant \\(G(s)\\) plus the number of states in the weights plus twice the number of state in \\(D(s)\\).
+The obtain \\(\mu\text{-optimal}\\) controller will usually be of **high order** and will have a flat \\(\mu\text{-curve}\\) until some high frequency.
+
+The DK-iteration depends heavily on optimal solutions for steps 1 and 2, and also on good fits in step 3.
+We usually **prefers to have a low-order fit** in step 3 as it will reduces the order of the \\(\hinf\\) problem which usually improves the numerical properties of the optimization.
+In some cases, the iterations converge slowly, the \\(\mu\text{-value}\\) can even increase. This may be caused by numerical problems and one may consider going back to the initial problem and **rescaling the inputs and outputs**.
+
+
+#### Adjusting the Performance Weight {#adjusting-the-performance-weight}
+
+If \\(\mu\\) at a given frequency is different from 1, then the interpretation is that at this frequency we can tolerate \\(1/\mu\\) times more uncertainty and still satisfy our performance objective with a margin of \\(1/\mu\\).
+In \\(\mu\text{-synthesis}\\), the designer will usually adjust some parameter in the performance or uncertainty weights until the weight of the peak \\(\mu\text{-value}\\) is close to 1.
+
+Sometimes, uncertainty is fixed and we effectively optimize worst-cast performance by adjusting a parameter in the performance weight.
+Consider the performance weight
+\\[ w\_p(s) = \frac{s/M + \w\_B^\*}{s + \w\_B^\* A} \\]
+where we want to keep \\(M\\) constant and find the high achievable bandwidth frequency \\(\w\_B^\*\\).
+The optimization problem becomes
+\\[ \text{max} \abs{\w\_B^\*} \quad \text{such that} \quad \mu(N) < 1, \ \forall\w \\]
+where \\(N\\), the interconnection matrix for the RP-problem, depends on \\(\w\_B^\*\\). This may be implemented as an **outer loop around the DK-iteration**.
+
+
+#### Fixed Structure Controller {#fixed-structure-controller}
+
+Sometimes it is desirable to find a low-order controller with a given structure.
+This may be achievable by numerical optimization where \\(\mu\\) is minimized with respect to the controller parameters.
+This problem here is that the optimization is not generally convex in the parameters.
+Sometimes it helps to switch the optimization between minimizing the peak of \\(\mu\\) and minimizing the integral square deviation of \\(\mu\\) away from \\(k\\) (i.e. \\(\normtwo{\mu(j\w) - k}\\)) where \\(k\\) is usually close to 1.
+The latter is an attempt to "flatten out" \\(\mu\\).
+
+
+#### Example: \\(\mu\text{-synthesis}\\) with DK-iteration {#example--mu-text-synthesis--with-dk-iteration}
+
+For simplicity, we will consider again the case of multiplicative uncertainty and performance defined in terms of weighted sensitivity.
+The uncertainty weight \\(w\_I I\\) and performance weight \\(w\_P I\\) are shown graphically in Fig. [fig:weights_distillation](#fig:weights_distillation).
+
+
+
+{{< figure src="/ox-hugo/skogestad07_weights_distillation.png" caption="Figure 33: Uncertainty and performance weights" >}}
+
+The objective is to minimize the peak value of \\(\mu\_{\tilde{\Delta}}(N)\\), \\(\tilde{\Delta} = \text{diag}\\{\Delta\_I, \Delta\_P\\}\\).
+\\(\Delta\_I\\) is a diagonal \\(2 \times 2\\) matrix representing the diagonal input uncertainty and \\(\Delta\_P\\) is a full \\(2 \times 2\\) matrix representing the performance specifications.
+
+First, the generalized plant \\(P\\) is constructed which includes the plant model, the uncertainty weight and the performance weight.
+Then the block structure is defined, it consists of two \\(1 \times 1\\) blocks to represent \\(\Delta\_I\\) and a \\(2 \times 2\\) block to represent \\(\Delta\_P\\).
+The scaling matrix \\(D\\) for \\(DND^{-1}\\) then has the structure \\(D = \text{diag}\\{d\_1, d\_2, d\_3I\_2\\}\\). We select \\(d\_3 = 1\\) and as initial scalings we select \\(d\_1^0 = d\_2^0 = 1\\).
+\\(P\\) is then scaled with the matrix \\(\text{diag}\\{D, I\_2\\}\\) where \\(I\_2\\) is associated with the inputs and outputs from the controller (we do not want to scale the controller).
+
+- Iteration No. 1.
+ Step 1: with the initial scalings, the \\(\mathcal{H}\_\infty\\) synthesis produced a 6 state controller (2 states from the plant model and 2 from each of the weights).
+ Step 2: the upper \\(\mu\text{-bound}\\) is shown in Fig. [fig:dk_iter_mu](#fig:dk_iter_mu).
+ Step 3: the frequency dependent \\(d\_1(\omega)\\) and \\(d\_2(\omega)\\) from step 2 are fitted using a 4th order transfer function shown in Fig. [fig:dk_iter_d_scale](#fig:dk_iter_d_scale)
+- Iteration No. 2.
+ Step 1: with the 8 state scalings \\(D^1(s)\\), the \\(\mathcal{H}\_\infty\\) synthesis gives a 22 state controller.
+ Step 2: This controller gives a peak value of \\(\mu\\) of \\(1.02\\).
+ Step 3: the scalings are only slightly changed
+- Iteration No. 3.
+ Step 1: The \\(\mathcal{H}\_\infty\\) norm is only slightly reduced. We thus decide the stop the iterations.
+
+
+
+{{< figure src="/ox-hugo/skogestad07_dk_iter_mu.png" caption="Figure 34: Change in \\(\mu\\) during DK-iteration" >}}
+
+
+
+{{< figure src="/ox-hugo/skogestad07_dk_iter_d_scale.png" caption="Figure 35: Change in D-scale \\(d\_1\\) during DK-iteration" >}}
+
+The final \\(\mu\text{-curves}\\) for NP, RS and RP with the controller \\(K\_3\\) are shown in Fig. [fig:mu_plot_optimal_k3](#fig:mu_plot_optimal_k3).
+The objectives of RS and NP are easily satisfied.
+The peak value of \\(\mu\\) is just slightly over 1, so the performance specification \\(\bar{\sigma}(w\_P S\_p) < 1\\) is almost satisfied for all possible plants.
+
+
+
+{{< figure src="/ox-hugo/skogestad07_mu_plot_optimal_k3.png" caption="Figure 36: \\(mu\text{-plots}\\) with \\(\mu\\) \"optimal\" controller \\(K\_3\\)" >}}
+
+To confirm that, 6 perturbed plants are used to compute the perturbed sensitivity functions shown in Fig. [fig:perturb_s_k3](#fig:perturb_s_k3).
+
+
+
+{{< figure src="/ox-hugo/skogestad07_perturb_s_k3.png" caption="Figure 37: Perturbed sensitivity functions \\(\bar{\sigma}(S^\prime)\\) using \\(\mu\\) \"optimal\" controller \\(K\_3\\). Lower solid line: nominal plant. Upper solid line: worst-case plant" >}}
+
+
+### Further Remarks on \\(\mu\\) {#further-remarks-on--mu}
+
+For complex perturbations, the scaled singular value \\(\maxsv(DND^{-1})\\) is a tight upper bound on \\(\mu(N)\\) in most cases, and minimizing the upper bound \\(\hnorm{DND^{-1}}\\) form the basis for the DK-iteration.
+
+The use of constant D-scales (\\(D\\) is not allowed to vary with frequency), provides a necessary and sufficient condition for robustness to arbitrary fast time varying linear uncertainty. While such perturbations are unlikely in a practical situation, we know that this controller will work very well even for rapid changes in the plant. Moreover, the use of constant D-scales make the computation of \\(\mu\\) straightforward and solvable using LMIs.
+
+
+### Conclusion {#conclusion}
+
+We have discussed how to represent uncertainty and how to analyze its effect on stability (RS) and performance (RP) using the **structured singular value** \\(\mu\\).
+
+To analyze robust stability of an uncertain system, we make use of the \\(M\Delta\text{-structure}\\) where \\(M\\) represents the transfer function for the "new" feedback part generated by the uncertainty.
+From the small gain theorem
+\\[ \tcmbox{RS \quad \Leftarrow \quad \maxsv(M) < 1, \ \forall\w} \\]
+which is tight (necessary and sufficient) for the special case where at each frequency any complex \\(\Delta\\) satisfying \\(\maxsv(\Delta) \le 1\\) is allowed.
+More generally, the **tight condition is**
+\\[ \tcmbox{RP \quad \Leftrightarrow \quad \mu(M) < 1, \ \forall\w} \\]
+where \\(\mu(M)\\) is the **structured singular value**. The calculation of \\(\mu\\) makes use of the fact that \\(\Delta\\) has a given block-diagonal structure, where certain blocks may also be real (e.g. to handle parametric uncertainty).
+
+We defined robust performance as \\(\hnorm{F\_l(N, \Delta)} < 1\\) for all allowed \\(\Delta\\).
+Since we used the \\(\hinf\\) norm in both the representation of uncertainty and the definition of performance, we found that RP could be viewed as a special case of RS, and we derived
+\\[ \tcmbox{RS \quad \Leftrightarrow \quad \mu(N) < 1, \ \forall\w} \\]
+where \\(\mu\\) is computed with respect to the **block-diagonal structure** \\(\diag{\Delta, \Delta\_P}\\).
+Here \\(\Delta\\) represents the uncertainty and \\(\Delta\_P\\) is a fictitious full uncertainty block representing the \\(\hinf\\) performance bound.
+
+There are **two main approaches to getting a robust design**:
+
+1. We aim to make the system robust to some **"general" class of uncertainty** which we do not explicitly model.
+ For SISO systems, the classical gain and phase margins and the peaks of \\(S\\) and \\(T\\) provide useful robustness measures.
+ For MIMO systems, normalized coprime factor uncertainty provides a good general class of uncertainty, and the associated Glover-McFlarlane \\(\hinf\\) loop-shaping design procedure has proved itself very useful in applications
+2. We **explicitly model and quantify the uncertainty** in the plant and aim to make the system robust to this specific uncertainty.
+ Potentially, it yields better designs, but it may require a much larger effort in terms of uncertainty modelling, especially if parametric uncertainty is consider.
+ Analysis and in particular, synthesis using \\(\mu\\) can be very involved
+
+In applications, it is therefore recommended to **start with the first approach**, at least for design.
+The robust stability and performance is then analyzed in simulations and using the structured singular value, for example, by considering first simple sources of uncertainty such as multiplicative input uncertainty.
+One then iterates between design and analysis until a satisfactory solution is obtained.
+If resulting control performance is not satisfactory, one may switch to the second approach.
+
+
+##### Practical \\(\mu\text{-synthesis}\\) in practice: {#practical--mu-text-synthesis--in-practice}
+
+1. Because of the effort involved in deriving detailed uncertainty descriptions, and the subsequent complexity in synthesizing controllers, the rule is to **start simple** with a crude uncertainty description, and then to see whether the performance specifications can be met. Only if they can't, one should consider more detailed uncertainty descriptions such as parametric uncertainty
+2. The use of \\(\mu\\) implies a worst-case analysis, so one should be **careful about including too many sources of uncertainty**, noise and disturbances - otherwise it becomes very unlikely for the worst case to occur, and the resulting analysis and design may be **unnecessarily conservative**
+3. There is always uncertainty with respect to the inputs and outputs, so **it is generally sage to include diagonal input and output uncertainty**. The relative multiplicative form is very convenient in this case
+4. \\(\mu\\) is most commonly used for analysis. If \\(\mu\\) is used for synthesis, then we recommend that you keep the uncertainty fixed and adjust the parameters in the performance weight until \\(\mu\\) is close to 1
+
+
+## Controller Design {#controller-design}
+
+
+
+
+### Trade-offs in MIMO Feedback Design {#trade-offs-in-mimo-feedback-design}
+
+The shaping of multivariable transfer functions is based on the idea that a satisfactory definition of gain for a matrix transfer function is given by the **singular values**.
+By multivariable transfer function shaping, therefore, we mean the shaping of the singular values of appropriate specified transfer functions such as the loop transfer function of one or more closed-loop transfer functions.
+
+The classical loop-shaping ideas can be further generalized to MIMO systems by considering the singular values.
+
+Consider the one degree-of-freedom system as shown in Fig. [fig:classical_feedback_small](#fig:classical_feedback_small).
+We have the following important relationships:
+
+\begin{subequations}
+ \begin{align}
+ y(s) &= T(s) r(s) + S(s) d(s) - T(s) n(s) \\\\\\
+ u(s) &= K(s) S(s) \big(r(s) - n(s) - d(s) \big)
+ \end{align}
+\end{subequations}
+
+
+
+{{< figure src="/ox-hugo/skogestad07_classical_feedback_small.png" caption="Figure 38: One degree-of-freedom feedback configuration" >}}
+
+
+
+
+1. For disturbance rejection make \\(\maxsv(S)\\) small
+2. For noise attenuation make \\(\maxsv(T)\\) small
+3. For reference tracking make \\(\maxsv(T) \approx \minsv(T) \approx 1\\)
+4. For control energy reduction make \\(\maxsv(KS)\\) small
+5. For robust stability in presence of an additive perturbation (\\(G\_p = G + \Delta\\)) make \\(\maxsv(KS)\\) small
+6. For robust stability in presence of a multiplicative output perturbation (\\(G\_p = (I + \Delta) G\\)) make \\(\maxsv(T)\\) small
+
+
+
+In classical loop shaping, it is the magnitude of the open-loop transfer function \\(L = GK\\) which is shaped, whereas the above requirements are all in terms of closed-loop transfer functions.
+However, we have that
+\\[ \minsv(L) - 1 \le \frac{1}{\maxsv(S)} \le \minsv(L) + 1 \\]
+from which we see that \\(\maxsv(S) \approx 1/\minsv(L)\\) at frequencies where \\(\minsv(L)\\) is much larger than \\(1\\).
+Furthermore, from \\(T = L(I+L)^{-1}\\) it follows that \\(\maxsv(T) \approx \maxsv(L)\\) at frequencies where \\(\maxsv(L)\\) is much smaller than \\(1\\).
+
+Thus, over specified frequency ranges, it is relatively easy to approximate the closed-loop requirements by open-loop objectives.
+
+
+
+
+1. For disturbance rejection make \\(\minsv(GK)\\) large
+2. For noise attenuation make \\(\maxsv(GK)\\) small
+3. For reference tracking make \\(\minsv(GK)\\) large
+4. For control energy reduction make \\(\maxsv(K)\\) small
+5. For robust stability in presence of an additive perturbation make \\(\maxsv(K)\\) small
+6. For robust stability in presence of an multiplicative output perturbation make \\(\maxsv(GK)\\) small
+
+
+
+Shaping the singular values of \\(GK\\) by selecting \\(K\\) is relatively easy task, but to do this in a way which also guarantees closed-loop stability is in general difficult as **closed-loop stability cannot be determined from open-loop singular values**.
+
+For SISO systems, closed-loop stability is closely related to the open-loop roll-off rate from high to low gain at the crossover (which is in practice less than \\(\SI{40}{\decibel\per dec}\\)).
+An immediate consequence of this is that there is a lower limit to the difference between \\(\omega\_h\\) and \\(\omega\_l\\).
+
+For MIMO systems, a similar gain/phase relationship holds in the crossover frequency region, but this is in terms of roll-off rate of the magnitude of the eigenvalues of \\(GK\\) and not the singular values.
+The stability constraint is therefore more difficult to handle.
+
+
+### LQG Control {#lqg-control}
+
+LQG control was developed and successfully applied for aerospace problems where accurate plants are available.
+For other control problems, it was not easy, and the **assumption of white noise disturbance** is not always relevant.
+As a result, LQG designs were sometimes not robust enough to be used in practice.
+
+It is assumed that the plant dynamics are linear and known, and that the measurement noise and disturbance signals are stochastic with known statistical properties:
+
+\begin{align\*}
+ \dot{x} &= A x + B u + w\_d \\\\\\
+ y &= C x + D u + w\_n
+\end{align\*}
+
+with \\(w\_d\\) and \\(w\_n\\) are the disturbance and measurement noise which are assumed to be uncorrelated zero-mean Gaussian stochastic processes with constant power spectral density matrices \\(W\\) and \\(V\\) respectively.
+
+
+
+
+The LQG control problem is to find the optimal control \\(u(t)\\) that minimize:
+\\[J = E \bigg\\{ \lim\_{T\rightarrow \infty} \frac{1}{T} \int\_0^T \big[ x^T Q x + u^T R u \big] dt \bigg\\} \\]
+Where \\(Q\\) and \\(R\\) are appropriately chosen constant **weighting matrices** (design parameters) such that \\(Q = Q^T \ge 0\\) and \\(R = R^T > 0\\).
+
+
+
+The next step is to find an **optimal estimate** \\(\hat{x}\\) of the state \\(x\\) so that \\(E \big\\{ [x-\hat{x}]^T [x-\hat{x}] \big\\}\\) is minimized.
+The optimal state estimate is given by a **Kalman filter**.
+
+The solution to the LQG problem is then found by replacing \\(x\\) by \\(\hat{x}\\) to give \\(u(t) = -K\_r \hat{x}\\).
+
+We therefore see that the LQG problem and its solution can be separated into two distinct parts as illustrated in Fig. [fig:lqg_separation](#fig:lqg_separation): the optimal state feedback and the optimal state estimator (the Kalman filter).
+
+
+
+{{< figure src="/ox-hugo/skogestad07_lqg_separation.png" caption="Figure 40: The separation theorem" >}}
+
+
+
+
+The LQR problem, where all the states are known is to find the input signal \\(u(t)\\) that takes the system \\(\dot{x} = Ax+Bu\\) to the zero state (\\(x=0\\)) by minimizing the deterministic cost
+
+\begin{equation}
+ J\_r = \int\_0^\infty \big( x(t)^T Q x(t) + u(t)^T R u(t) \big) dt
+\end{equation}
+
+The optimal solution is \\(u=-K\_r x(t)\\) with
+
+\begin{equation}
+ K\_r = R^{-1} B^T X
+\end{equation}
+
+and \\(X\\) is the unique positive-semi definite solution of the algebraic Riccati equation:
+
+\begin{equation}
+ A^T X + X A - XBR^{-1}B^TX + Q = 0
+\end{equation}
+
+
+
+
+The Kalman filter has the structure of an ordinary state-estimator, as shown on Fig. [fig:lqg_kalman_filter](#fig:lqg_kalman_filter), with:
+
+\begin{equation}
+ \dot{\hat{x}} = A\hat{x} + Bu + K\_f(y-C\hat{x})
+\end{equation}
+
+The optimal choice of \\(K\_f\\), which minimize \\(E \big\\{ [x-\hat{x}]^T [x-\hat{x}] \big\\}\\) is given by
+
+\begin{equation}
+ K\_f = Y C^T V^{-1}
+\end{equation}
+
+Where \\(Y\\) is the unique positive-semi definite solution of the algebraic Riccati equation
+
+\begin{equation}
+ YA^T + AY - YC^TV^{-1}CY + W = 0
+\end{equation}
+
+
+
+For the LQG-controller, as shown on Fig. [fig:lqg_kalman_filter](#fig:lqg_kalman_filter), it is not easy to see where to position the reference input \\(r\\) and how integral action may be included, if desired. Indeed, the standard LQG design procedure does not give a controller with integral action. One strategy is illustrated in Fig. [fig:lqg_integral](#fig:lqg_integral). Here, the control error \\(r-y\\) is integrated and the regulator \\(K\_r\\) is designed for the plant augmented with these integral states.
+
+
+
+{{< figure src="/ox-hugo/skogestad07_lqg_integral.png" caption="Figure 42: LQG controller with integral action and reference input" >}}
+
+For an LQG-controller system with a combined Kalman filter and LQR control law, there are **no guaranteed stability margins**, and there exist LQG combinations with arbitrary small gain margins.
+However, there are procedures for improving robustness properties of LQG control such as **Loop Transfer Recovery** (LTR).
+
+These procedure are somehow difficult to use in practice.
+Their main limitation is that they can only be applied to minimum phase plants.
+
+
+### \\(\htwo\\) and \\(\hinf\\) Control {#htwo--and--hinf--control}
+
+
+
+
+#### General Control Problem Formulation {#general-control-problem-formulation}
+
+
+There are many ways in which feedback design problems can be cast as \\(\htwo\\) and \\(\hinf\\) optimization problems.
+It is very useful therefore to have a **standard problem formulation** into which any particular problem may be manipulated.
+
+Such a general formulation is afforded by the general configuration shown in Fig. [fig:general_control](#fig:general_control).
+
+
+
+{{< figure src="/ox-hugo/skogestad07_general_control.png" caption="Figure 43: General control configuration" >}}
+
+The system is described by
+
+\begin{subequations}
+ \begin{align}
+ \colvec{z\\v} &= P(s) \colvec{w\\u} = \begin{bmatrix}P\_{11}(s) & P\_{12}(s)\\ P\_{21}(s) & P\_{22}(s) \end{bmatrix} \colvec{w\\u}\\\\\\
+ u &= K(s) v
+ \end{align}
+\end{subequations}
+
+With a state space realization of the generalized plant \\(P\\) given by
+
+\begin{equation}
+ P = \left[
+ \begin{array}{c|cc}
+ A & B\_1 & B\_2 \\\\\\
+ \hline
+ C\_1 & D\_{11} & D\_{12} \\\\\\
+ C\_2 & D\_{21} & D\_{22}
+ \end{array}
+ \right]
+\end{equation}
+
+The closed loop transfer function from \\(w\\) to \\(z\\) is given by the linear fractional transformation:
+
+\begin{align\*}
+ z &= F\_l(P, K) w \\\\\\
+ &= [P\_{11} + P\_{12}K(I-P\_{22}K)^{-1} P\_{21}] w
+\end{align\*}
+
+\\(\htwo\\) and \\(\hinf\\) control involve the minimization of the \\(\htwo\\) and \\(\hinf\\) norms of \\(F\_l(P, K)\\).
+
+The most general and widely used algorithms for \\(\htwo\\) and \\(\hinf\\) control problems are based on the state space formulation and requires the solution of two Riccati equations.
+
+The following **assumptions** are typically made in \\(\htwo\\) and \\(\hinf\\) problems:
+
+1. \\((A, B\_2, C\_2)\\) is stabilizable and detectable.
+ This is required for the existence of stabilizing controllers
+2. \\(D\_{12}\\) and \\(D\_{21}\\) have full rank.
+ This is sufficient to ensure that the controllers are proper
+3. \\(\begin{bmatrix} A-j\w I & B\_2 \\ C\_1 & D\_{12} \end{bmatrix}\\) and \\(\begin{bmatrix} A-j\w I & B\_1 \\ C\_2 & D\_{21} \end{bmatrix}\\) have respectively full column and full row rank for all \\(\w\\).
+ This ensures that the controller does not cancel poles or zeros in the imaginary axis which would result in closed-loop instability
+4. \\(D\_{11} = 0\\) and \\(D\_{22} = 0\\) is a conventional requirement for \\(\htwo\\) control.
+ This is not required for \\(\hinf\\) control but this significantly simplify algorithm formulas
+5. \\(D\_{12}^T C\_1 = 0\\) and \\(B\_1 D\_{12}^T = 0\\) is common in \\(\htwo\\) control.
+ \\(D\_{12}^T C\_1 = 0\\) means that there is no cross terms in the cost function and \\(B\_1 D\_{12}^T = 0\\) that the process noise and measurement noise are uncorrelated
+6. \\((A, B\_1)\\) is stabilizable and \\((A, C\_1)\\) is detectable
+
+If the Matlab Robust Control Toolbox complains, then it probably means that the control problem is not well formulated and that some assumptions are not met.
+
+\\(\hinf\\) algorithms, in general, find a **sub-optimal controller**. That is, for a specified \\(\gamma\\) a stabilizing controller is found for which \\(\hnorm{F\_l(P,K)}<\gamma\\).
+This contrasts with \\(\htwo\\) theory, in which the optimal controller is **unique** and can be found from the solution of two Riccati equations.
+
+
+#### \\(\htwo\\) Optimal Control {#htwo--optimal-control}
+
+
+
+
+The standard \\(\htwo\\) optimal control problem is to find a stabilizing controller \\(K\\) which minimizes
+\\[\normtwo{F(s)} = \sqrt{\frac{1}{2\pi}\int\_{-\infty}^{\infty} tr[F(j\w)F(j\w)^H]d\w }\\]
+With \\(F = F\_l(P, K)\\).
+
+
+
+The \\(\htwo\\) norm can be given different **deterministic interpretations**.
+It also has the following **stochastic interpretation**.
+
+Suppose in the general control configuration that the exogenous input \\(w\\) is white noise of unit density. That is
+\\[ E\\{w(t)w(\tau)^T\\} = I \delta(t-\tau) \\]
+
+Expected power in the error signal \\(z\\) is then given by
+
+\begin{align\*}
+ P\_z &= E\bigg\\{ \lim\_{T\rightarrow\infty} \frac{1}{2T} \int\_{-T}^{T} z(t)^T z(t) dt \bigg\\} \\\\\\
+ &= \text{tr}\ E\\{z(t)z(t)^H\\}\\\\\\
+ &= \frac{1}{2\pi} \int\_{-\infty}^{\infty}\text{tr}\left[F(j\omega)F(j\omega)^H\right]d\omega\\\\\\
+ &= \normtwo{F}^2 = \normtwo{F\_l(P,K)}^2
+\end{align\*}
+
+
+
+
+Thus, by minimizing the \\(\htwo\\) norm, the error power of the generalized system, due to a unit intensity white noise input, is minimized.
+We are **minimizing the Root Mean Square value** of \\(z\\).
+
+
+
+In practice, it is usually not necessary to obtain an optimal controller for the \\(\hinf\\) problem, and it is simpler to design a **sub-optimal** one.
+
+Let \\(\gamma\_\text{min}\\) be the minimum value of \\(\hnorm{F\_l(P,K)}\\) over all stabilizing controllers \\(K\\).
+Then the \\(\hinf\\) **sub-optimal control problem** is: given a \\(\gamma > \gamma\_\text{min}\\), find all stabilizing controllers \\(K\\) such that
+
+\begin{equation}
+ \hnorm{F\_l(P, K)} < \gamma
+\end{equation}
+
+By reducing \\(\gamma\\) in an iterative way, an optimal solution is approached.
+
+**General \\(\hinf\\) algorithm**.
+For the general control configuration and with assumptions described above, there exists a stabilizing controller \\(K(s)\\) such that \\(\hnorm{F\_l(P,K)}<\gamma\\) if and only if
+
+1. \\(X\_\infty \ge 0\\) is a solution to the algebraic Riccati equation \\(A^T X\_\infty + X\_\infty A + C\_1^T C\_1 + X\_\infty (\gamma^{-2} B\_1 B\_1^T - B\_2 B\_2^T)X\_\infty = 0\\) such that \\(\text{Re } \lambda\_i \left[ A + (\gamma^{-2}B\_1B\_1^T - B\_2B\_2^T)X\_\infty \right] < 0, \ \forall i\\)
+2. \\(Y\_\infty \ge 0\\) is a solution to the algebraic Riccati equation \\(A Y\_\infty + Y\_\infty A^T + B\_1 B\_1^T + Y\_\infty (\gamma^{-2} C\_1^T C\_1 - C\_2^T C\_2)Y\_\infty = 0\\) such that \\(\text{Re } \lambda\_i \left[ A + Y\_\infty(\gamma^{-2}C\_1^TC\_1 - C\_2^TC\_2)Y\_\infty \right] < 0, \ \forall i\\)
+3. \\(\rho(X\_\infty Y\_\infty) < \gamma^2\\)
+
+All such controllers are then given by \\(K = F\_l(K\_c, Q)\\) where
+
+\begin{align\*}
+ K\_c(s) &= \left[ \begin{array}{c|cc}
+ A\_\infty & -Z\_\infty L\_\infty & Z\_\infty B\_2 \\ \hline
+ F\_\infty & 0 & I \\\\\\
+ -C\_2 & I & 0
+ \end{array} \right], \ L\_\infty = -Y\_\infty C\_2^T \\\\\\
+ F\_\infty &= -B\_2^T X\_\infty, \ Z\_\infty = (I - \gamma^2 Y\_\infty X\_\infty)^{-1} \\\\\\
+ A\_\infty &= A + \gamma^{-2} B\_1 B\_1^T X\_\infty + B\_2F\_\infty + Z\_\infty L\_\infty C\_2
+\end{align\*}
+
+and \\(Q(s)\\) is any stable proper transfer function such that \\(\hnorm{Q} < \gamma\\).
+
+For \\(Q(s) = 0\\), we get
+
+\begin{equation}
+ K(s) = K\_{c11}(s) = -Z\_\infty L\_\infty (s I - A\_\infty)^{-1} F\_\infty
+\end{equation}
+
+This is called the **central controller** and has the same number of states as the generalized plant \\(P(s)\\).
+
+The central controller can be separated into a state estimator (observer) of the form
+\\[ \dot{\hat{x}} = A\hat{x} + B\_1 \gamma^{-2} B\_1^T X\_\infty \hat{x} + B\_2 u + Z\_\infty L\_\infty (C\_2 \hat{x} - y) \\]
+and a state feedback \\(u = F\_\infty \hat{x}\\).
+
+
+
+
+If we desire a controller that achieves \\(\gamma\_\text{min}\\), to within specified tolerance, then we can perform a **bisection** on \\(\gamma\\) until its value is sufficiently accurate.
+The above conditions provide a test for each value of \\(\gamma\\) to determine if \\(\gamma<\gamma\_\text{min}\\) or \\(\gamma>\gamma\_\text{min}\\).
+
+
+
+In the **shaping approach**, \\(\hinf\\) optimization is used to shape the singular values of specified transfer functions over frequency.
+The maximum singular values are relatively easy to shape by forcing them to lie below user defined bounds, thereby ensuring desirable bandwidth and roll-off rates.
+
+In the **signal-based approach**, we seek to minimize the energy in certain error signal given a set of exogenous input signals.
+
+A difficulty that sometimes arises with \\(\hinf\\) control is the **selection of weights** such that the \\(\hinf\\) optimal controller provides a good trade-off between conflicting objectives in various frequency ranges.
+Thus, for practical designs it is sometimes recommended to perform only a few iterations of the \\(\hinf\\) algorithm.
+The justification for this is that the initial design, after one iteration, is similar to an \\(\htwo\\) design which does trade-off over various frequency ranges.
+Therefore stopping the iterations before the optimal value is achieved gives the design an \\(\htwo\\) flavor which may be desirable.
+
+
+#### Mixed-Sensitivity \\(\hinf\\) Control {#mixed-sensitivity--hinf--control}
+
+Mixed-sensitivity is the name given to transfer function shaping problems in which the sensitivity function \\(S = (I + GK)^{-1}\\) is shaped along with one or more other closed-loop transfer functions such as \\(KS\\) or \\(T = I - S\\).
+
+Suppose that we have a regulation problem in which we want to reject a disturbance \\(d\\) entering at the plant output and it is assumed that the measurement noise is relatively insignificant.
+It makes sense to shape the closed-loop transfer functions \\(S\\) and \\(KS\\).
+Recall that \\(S\\) is the transfer function between \\(d\\) and the output, and \\(KS\\) the transfer function from \\(d\\) and the control signal.
+It is important to include \\(KS\\) as a mechanism for **limiting the size and bandwidth of the controller**, and hence the energy used.
+The size of \\(KS\\) is also important for robust stability with respect to uncertainty modeled as additive plant perturbations.
+
+The disturbance \\(d\\) is typically a low frequency signal, and therefore it will be successfully rejected if the maximum singular value of \\(S\\) is made small over the same low frequency range.
+To do this, we could select a scalar low pass filter \\(w\_1(s)\\) with a bandwidth equal to that of the disturbance, and then find a stabilizing controller that minimizes \\(\hnorm{w\_1 S}\\).
+This cost function alone is not very practical, it focuses on just one closed-loop transfer function and the controller may have infinite gain.
+It is far more useful in practice to minimize
+
+\begin{equation}
+ \hnorm{\begin{matrix} w\_1 S \\ w\_2 KS \end{matrix}}
+\end{equation}
+
+where \\(w\_2(s)\\) is a scalar high pass filter with a crossover frequency approximately equal to that of the desired closed-loop bandwidth.
+
+In general, the scalar weighting functions \\(w\_1(s)\\) and \\(w\_2(s)\\) can be replaced by matrices \\(W\_1(s)\\) and \\(W\_2(s)\\).
+This can be useful for **systems with channels of quite different bandwidths**.
+In that case, **diagonal weights are recommended** as anything more complicated is usually not worth the effort.
+
+To see how this mixed sensitivity problem can be formulated in the general setting, we can imagine the disturbance \\(d\\) as a single exogenous input and define and error signal \\(z = [z\_1^T\ z\_2^T]^T\\), where \\(z\_1 = W\_1 y\\) and \\(z\_2 = -W\_2 u\\) as illustrated in Fig. [fig:mixed_sensitivity_dist_rejection](#fig:mixed_sensitivity_dist_rejection).
+We can then see that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\) as required.
+The elements of the generalized plant are
+
+\begin{equation\*}
+ \begin{array}{ll}
+ P\_{11} = \begin{bmatrix}W\_1\\0\end{bmatrix} & P\_{12} = \begin{bmatrix}W\_1G\\-W\_2\end{bmatrix} \\\\\\
+ P\_{21} = -I & P\_{22} = -G
+ \end{array}
+\end{equation\*}
+
+
+
+{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_dist_rejection.png" caption="Figure 44: \\(S/KS\\) mixed-sensitivity optimization in standard form (regulation)" >}}
+
+Another interpretation can be put on the \\(S/KS\\) mixed-sensitivity optimization as shown in the standard control configuration of Fig. [fig:mixed_sensitivity_ref_tracking](#fig:mixed_sensitivity_ref_tracking).
+Here we consider a tracking problem.
+The exogenous input is a reference command \\(r\\), and the error signals are \\(z\_1 = -W\_1 e = W\_1 (r-y)\\) and \\(z\_2 = W\_2 u\\).
+As the regulation problem of Fig. [fig:mixed_sensitivity_dist_rejection](#fig:mixed_sensitivity_dist_rejection), we have that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\).
+
+
+
+{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_ref_tracking.png" caption="Figure 45: \\(S/KS\\) mixed-sensitivity optimization in standard form (tracking)" >}}
+
+Another useful mixed sensitivity optimization problem, is to find a stabilizing controller which minimizes
+
+\begin{equation}
+ \hnorm{\begin{matrix} W\_1 S \\ W\_2 T \end{matrix}}
+\end{equation}
+
+The ability to shape \\(T\\) is desirable for tracking problems and noise attenuation.
+It is also important for robust stability with respect to multiplicative perturbations at the plant output.
+
+The \\(S/T\\) mixed-sensitivity minimization problem can be put into the standard control configuration as shown in Fig. [fig:mixed_sensitivity_s_t](#fig:mixed_sensitivity_s_t).
+
+The elements of the generalized plant are
+
+\begin{equation\*}
+ \begin{array}{ll}
+ P\_{11} = \begin{bmatrix}W\_1\\0\end{bmatrix} & P\_{12} = \begin{bmatrix}-W\_1G\\W\_2G\end{bmatrix} \\\\\\
+ P\_{21} = -I & P\_{22} = -G \\\\\\
+ \end{array}
+\end{equation\*}
+
+
+
+{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_s_t.png" caption="Figure 46: \\(S/T\\) mixed-sensitivity optimization in standard form" >}}
+
+The shaping of closed-loop transfer functions as described above with the stacked cost functions becomes difficult with more than two functions whereas with two, the process is relatively easy.
+The bandwidth requirements on each are usually complementary and simple, stable low-pass and high-pass filters are sufficient to carry out the required shaping and trade-offs.
+
+The weights \\(W\_i\\) in mixed-sensitivity \\(\hinf\\) optimal control must all be stable.
+Therefore, if we wish, for example, to emphasize the minimization of \\(S\\) at low frequency by weighting with a term including integral action, we would have to approximate \\(\frac{1}{s}\\) by \\(\frac{1}{s + \epsilon}\\) where \\(\epsilon \ll 1\\).
+Similarly, one might be interested in weighting \\(KS\\) with a non-proper weight to ensure that \\(K\\) is small outside of the system bandwidth.
+The trick is to replace a non proper term such as \\((1 + \tau\_1 s)\\) by \\(\frac{1 + \tau\_1 s}{1 + \tau\_2 s}\\) where \\(\tau\_2 \ll \tau\_1\\).
+
+
+#### Signal-Based \\(\hinf\\) Control {#signal-based--hinf--control}
+
+The signal-based approach to controller design is very general and is appropriate for multivariable problems in which several objectives must be taken into account simultaneously.
+In this approach, we define the plant, possibly the model uncertainty, the **class of external signals affecting the system** and the **norm of the error signals we want to keep small**.
+
+
+
+
+The focus of attention has moved to the size of signals and away from the size and bandwidth of selected closed-loop transfer functions.
+
+
+
+When we consider a system's response to persistent sinusoidal signals of varying frequency, or when we consider the induced 2-norm between the exogenous input signals and the error signals, we are required to minimize the \\(\hinf\\) norm.
+In the absence of model uncertainty, there does not appear to be an overwhelming case for using the \\(\hinf\\) norm rather than the more traditional \\(\htwo\\) norm.
+However, when uncertainty is addressed, as it always should be, \\(\hinf\\) is clearly the more **natural approach** using component uncertainty models as in Fig. [fig:input_uncertainty_hinf](#fig:input_uncertainty_hinf).
+
+A typical problem using the signal-based approach to \\(\hinf\\) control is illustrated in the interconnection diagram of Fig. [fig:hinf_signal_based](#fig:hinf_signal_based).
+\\(G\\) and \\(G\_d\\) are nominal models of the plant and disturbance dynamics, and \\(K\\) is the controller to be designed.
+The weights \\(W\_d\\), \\(W\_r\\), and \\(W\_n\\) may be constant or dynamic and describe the relative importance and/or the frequency content of the disturbance, set points and noise signals.
+The weight \\(W\_\text{ref}\\) is a desired closed-loop transfer function between the weighted set point \\(r\_s\\) and the actual output \\(y\\).
+The weights \\(W\_e\\) and \\(W\_u\\) reflect the desired frequency content of the error \\((y-y\_\text{ref})\\) and the control signals \\(u\\), respectively.
+The problem can be cast as a standard \\(\hinf\\) optimization in the general control configuration by defining
+
+\begin{equation\*}
+ w = \begin{bmatrix}d\\r\\n\end{bmatrix},\ z = \begin{bmatrix}z\_1\\z\_2\end{bmatrix}, \ v = \begin{bmatrix}r\_s\\y\_m\end{bmatrix},\ u = u
+\end{equation\*}
+
+
+
+{{< figure src="/ox-hugo/skogestad07_hinf_signal_based.png" caption="Figure 48: A signal-based \\(\hinf\\) control problem" >}}
+
+Suppose we now introduce a multiplicative dynamic uncertainty model at the input to the plant as shown in Fig. [fig:hinf_signal_based_uncertainty](#fig:hinf_signal_based_uncertainty).
+The problem we now want to solve is: find a stabilizing controller \\(K\\) such that the \\(\hinf\\) norm of the transfer function between \\(w\\) and \\(z\\) is less that 1 for all \\(\Delta\\) where \\(\hnorm{\Delta} < 1\\).
+We have assumed in this statement that the **signal weights have normalized the 2-norm of the exogenous input signals to unity**.
+This problem is a non-standard \\(\hinf\\) optimization.
+It is a robust performance problem for which the \\(\mu\text{-synthesis}\\) procedure can be applied where we require the structured singular value:
+\\[ \mu(M(j\omega)) < 1, \quad \forall\omega \\]
+
+
+
+{{< figure src="/ox-hugo/skogestad07_hinf_signal_based_uncertainty.png" caption="Figure 49: A signal-based \\(\hinf\\) control problem with input multiplicative uncertainty" >}}
+
+However, whilst the structured singular value is a useful analysis tool for assessing designs, \\(\mu\text{-synthesis}\\) is sometimes difficult to use and often too complex for the practical problems.
+
+
+### \\(\hinf\\) Loop-Shaping Design {#hinf--loop-shaping-design}
+
+The loop-shaping design procedure described in this section is based on \\(\hinf\\) robust stabilization combined with classical loop shaping.
+It is essentially a **two stage design process**:
+
+- First the open-loop plant is augmented by pre and post compensators to give a desired shape to the singular values of the open-loop frequency response
+- Then the resulting shaped plant is robustly stabilized with respect to coprime factor uncertainty using \\(\hinf\\) optimization
+
+An important advantage is that no problem-dependent uncertainty modelling, or weight selection, is required in this second step.
+
+
+#### Robust Stabilization {#robust-stabilization}
+
+For multivariable systems, **classical gain and phase margins are unreliable indicators of robust stability** when defined for each channel (or loop), taken one at a time, because simultaneous perturbations in more than one loop are not then catered for.
+
+It is now common practice to model uncertainty by stable **norm-bounded** dynamic (complex) **matrix perturbations**.
+With a single perturbation, the associated robustness tests is in terms of the maximum singular values of various closed-loop transfer functions.
+Use of a single stable perturbation restricts the plant and perturbed plant models to either have the same number of unstable poles or the same number of RHP zeros.
+
+To overcome this, **two stable perturbations** can be used, one on each of the factors in a **coprime factorization** of the plant.
+Although this uncertainty description seems unrealistic and less intuitive than the others, it is in fact quite general, and for our purposes it leads to a very useful \\(\hinf\\) robust stabilization problem.
+
+Let's consider the stabilization of a plant \\(G\\) which has a normalized left coprime factorization
+
+\begin{equation}
+ G = M^{-1} N
+\end{equation}
+
+where we have dropped the subscripts from \\(M\\) and \\(N\\) for simplicity.
+
+A perturbed plant model \\(G\_p\\) can then we written has
+
+\begin{equation}
+ G\_p = (M + \Delta\_M)^{-1} (N + \Delta\_N)
+\end{equation}
+
+where \\(\Delta\_M\\), \\(\Delta\_N\\) are stable unknown transfer functions which represent the uncertainty in the nominal plant \\(G\\).
+
+The objective of robust stabilization is to stabilize not only the nominal model \\(G\\), but a family of perturbed plants defined by
+
+\begin{equation}
+ G\_p = \\{ (M + \Delta\_M)^{-1} (N + \Delta\_N) \ :\ \hnorm{\Delta\_N\ \Delta\_M} < \epsilon \\}
+\end{equation}
+
+where \\(\epsilon > 0\\) is then the **stability margin**.
+
+For the perturbed feedback system of Fig. [fig:coprime_uncertainty_bis](#fig:coprime_uncertainty_bis), the stability property is robust if and only if the nominal feedback system is stable and
+\\[ \gamma \triangleq \hnorm{\begin{bmatrix}K \\ I\end{bmatrix} (I - GK)^{-1} M^{-1}} \le \frac{1}{\epsilon} \\]
+
+Notice that \\(\gamma\\) is the \\(\hinf\\) norm from \\(\phi\\) to \\(\begin{bmatrix}u\\y\end{bmatrix}\\) and \\((I-GK)^{-1}\\) is the sensitivity function for this positive feedback arrangement.
+
+
+
+{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_bis.png" caption="Figure 50: \\(\hinf\\) robust stabilization problem" >}}
+
+The lowest achievable value of \\(\gamma\\) and the corresponding maximum stability margin \\(\epsilon\\) are given as
+
+\begin{equation}
+ \gamma\_\text{min} = \epsilon\_{\text{max}}^{-1} = \left\\{ 1 - \\|N \ M\\|\_H^2 \right\\}^{-\frac{1}{2}} = (1 + \rho(XZ))^{\frac{1}{2}}
+\end{equation}
+
+where \\(\\|\ \cdot\ \\|\_H\\) denotes Hankel norm, \\(\rho\\) denotes the spectral radius (maximum eigenvalue), and for a minimal state space realization of G, Z is the unique positive definite solution of the algebraic Riccati equation
+
+\begin{align\*}
+ (A - BS^{-1} D^TC)Z &+ Z(A - BS^{-1}D^TC)^T \\\\\\
+ &- ZC^TR^{-1}CZ + BS^{-1}B^T = 0
+\end{align\*}
+
+where
+\\[ R = I + D D^T, \quad S = I + D^T D \\]
+\\(X\\) is the unique positive definite solution of the following algebraic Riccati equation
+
+\begin{align\*}
+ (A - BS^{-1} D^T C)X &+ X(A - BS^{-1}D^TC)^T \\\\\\
+ &- XBS^{-1} B^T X + C^TR^{-1}C = 0
+\end{align\*}
+
+A controller which guarantees that
+\\[ \hnorm{ \begin{bmatrix}K\\I\end{bmatrix} (I-GK)^{-1} M^{-1} } \le \gamma \\]
+for a specified \\(\gamma > \gamma\_\text{min}\\), is given by
+
+\begin{subequations}
+ \begin{align}
+ K &\triangleq \left[ \begin{array}{c|c}
+ {\scriptstyle A + BF + \gamma^2L^{-T} Z C^T(C + DF)} & {\scriptstyle \gamma^2L^{-T} Z C^T} \\ \hline
+ {\scriptstyle B^T X} & {\scriptstyle -D^T}
+ \end{array} \right] \label{eq:control\_coprime\_factor} \\\\\\
+ F &= -S^{-1}(D^T C + B^T X)\\\\\\
+ L &= (1-\gamma^2) I + XZ
+ \end{align}
+\end{subequations}
+
+The Matlab function `coprimeunc` can be used to generate the controller in [eq:control_coprime_factor](#eq:control_coprime_factor).
+It is important to emphasize that since we can compute \\(\gamma\_\text{min}\\) from [eq:gamma_min_coprime](#eq:gamma_min_coprime) we get an explicit solution by solving just two Riccati equations and avoid the \\(\gamma\text{-iteration}\\) needed to solve the general \\(\mathcal{H}\_\infty\\) problem.
+
+
+#### A Systematic \\(\hinf\\) Loop-Shaping Design Procedure {#a-systematic--hinf--loop-shaping-design-procedure}
+
+
+Robust stabilization alone is not much used in practice because the designer is not able to specify any performance requirements.
+
+To do so, **pre and post compensators** are used to **shape the open-loop singular values** prior to robust stabilization of the "shaped" plant.
+
+If \\(W\_1\\) and \\(W\_2\\) are the pre and post compensators respectively, then the shaped plant \\(G\_s\\) is given by
+
+\begin{equation}
+ G\_s = W\_2 G W\_1
+\end{equation}
+
+as shown in Fig. [fig:shaped_plant](#fig:shaped_plant).
+
+
+
+{{< figure src="/ox-hugo/skogestad07_shaped_plant.png" caption="Figure 51: The shaped plant and controller" >}}
+
+The controller \\(K\_s\\) is synthesized by solving the robust stabilization problem for the shaped plant \\(G\_s\\) with a normalized left coprime factorization \\(G\_s = M\_s^{-1}N\_s\\).
+The feedback controller for the plant \\(G\\) is then \\(K = W\_1 K\_s W\_2\\).
+
+Systematic procedure for \\(\hinf\\) loop-shaping design:
+
+1. **Scale the plant outputs and inputs**.
+ This is very important for most design procedures.
+ In general, scaling improves the conditioning of the design problem, it enables meaningful analysis to be made of the robustness properties of the feedback system in the frequency domain, and for loop shaping it can simplify the selection of weights:
+ - The outputs are scaled such that equal magnitudes of cross-coupling into each of the outputs is equally undesirable
+ - Each input is scaled by a given percentage (say \\(\SI{10}{\%}\\)) of its expected range of operation.
+ That is, the inputs are scaled to reflect the relative actuator capabilities.
+2. **Order the inputs and outputs** so that the plant is as diagonal as possible.
+ The relative gain array can be useful here.
+ The purpose of this pseudo-diagonalization is to ease the design of the pre and post compensators which, for simplicity, will be chosen to be diagonal.
+ Next, we discuss the selection of weights to obtain the shaped plant \\(G\_s = W\_2 G W\_1\\) where \\(W\_1 = W\_p W\_a W\_g\\)
+3. **Select the elements of diagonal pre and post compensators** \\(W\_p\\) and \\(W\_2\\) so that the singular values of \\(W\_2 G W\_p\\) are desirable.
+ This would normally mean high gain at low frequencies, a slope of about \\(-1\\) at the desired bandwidth(s), with higher rates at high frequencies.
+ The weights should be chosen so that no unstable hidden modes are created in \\(G\_s\\)
+ - \\(W\_2\\) is usually chosen as a constant, reflecting the relative importance of the outputs to be controlled and the other measurements being fed back to the controller
+ - \\(W\_p\\) contains the dynamic shaping. Integral action, for low frequency performance; phase-advance for reducing the roll-off rates at crossover; and phase-lag to increase the roll-off rates at high frequencies should all be places in \\(W\_p\\) is desired
+4. _Optional_: Align the singular values at a desired bandwidth using a further constant weight \\(W\_a\\) cascaded with \\(W\_p\\)
+5. _Optional_: Introduce an additional gain matrix \\(W\_g\\) cascaded with \\(W\_a\\) to provide control over actuator range. \\(W\_g\\) is diagonal and is adjusted so that actuator rate limits are not exceeded for reference demands and typical disturbances on the scaled plant outputs
+6. **Robustly stabilize the shaped plant** \\(G\_s = W\_2 G W\_1\\) where \\(W\_1 = W\_p W\_a W\_g\\)
+ - First, calculate the maximum stability margin \\(\epsilon\_{\text{max}} = 1/\gamma\_\text{min}\\)
+ - If the margin is too small, \\(\epsilon\_{\text{max}} < 0.25\\), then go back to step 4 and modify the weights. Otherwise, select \\(\gamma > \gamma\_\text{min}\\), by about \\(\SI{10}{\%}\\), and synthesize a sub-optimal controller. There is usually no advantage to be gained by using the optimal controller
+ - When \\(\epsilon\_{\text{max}} > 0.25\\) (respectively \\(\gamma\_\text{min} < 4\\)) the design is usually successful. In this case, at least \\(\SI{25}{\%}\\) coprime factor uncertainty is allowed, and we also find that the shape of the open-loop singular values will not have changed much after robust stabilization
+ - A small value of \\(\epsilon\_{\text{max}}\\) indicates that the chosen singular value loop-shapes are incompatible with robust stability requirements
+7. **Analyze the design** and if not all the specification are met, make further modifications to the weights
+8. **Implement the controller**.
+ The configuration shown in Fig. [fig:shapping_practical_implementation](#fig:shapping_practical_implementation) has been found useful when compared with the conventional setup in Fig. [fig:classical_feedback_small](#fig:classical_feedback_small).
+ This is because the references do not directly excite the dynamics of \\(K\_s\\), which can result in large amounts of overshoot.
+ The constant prefilter ensure a steady-state gain of \\(1\\) between \\(r\\) and \\(y\\), assuming integral action in \\(W\_1\\) or \\(G\\)
+
+
+
+{{< figure src="/ox-hugo/skogestad07_shapping_practical_implementation.png" caption="Figure 52: A practical implementation of the loop-shaping controller" >}}
+
+We will conclude this section with a summary of the **advantages** offered by the above \\(\hinf\\) loop-shaping design procedure:
+
+- It is relatively easy to use, being based on classical loop-shaping ideas
+- There exists a closed formula for the \\(\hinf\\) optimal cost \\(\gamma\_\text{min}\\), which in turn corresponds to a maximum stability margin \\(\epsilon\_{\text{max}} = 1/\gamma\_\text{min}\\)
+- No \\(\gamma\text{-iteration}\\) is required in the solution
+- Except for special systems, ones with all-pass factors, there are no pole-zero cancellations between the plant and controller.
+ Pole-zeros cancellations are common in many \\(\hinf\\) control problems and are a problem when the plant has lightly damped modes
+
+
+#### Two Degrees-of-freedom Controllers {#two-degrees-of-freedom-controllers}
+
+Many control design problems possess two degrees-of-freedom:
+
+- on one hand, **measurement of feedback signals**
+- and on the other hand, **commands and reference**
+
+Sometimes, one degree-of-freedom is left out of the design, and the controller is driven by an error signal i.e. the difference between a command and the output.
+But in cases where stringent time-domain specifications are set on the output response, a one degree-of-freedom structure may not be sufficient.
+
+A general two degrees-of-freedom feedback control scheme is depicted in Fig. [fig:classical_feedback_2dof_simple](#fig:classical_feedback_2dof_simple).
+The commands and feedbacks enter the controller separately and are independently processed.
+
+
+
+{{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_simple.png" caption="Figure 53: General two degrees-of-freedom feedback control scheme" >}}
+
+The presented \\(\mathcal{H}\_\infty\\) loop-shaping design procedure in section [sec:hinf_loop_shaping_procedure](#sec:hinf_loop_shaping_procedure) is a one-degree-of-freedom design, although a **constant** pre-filter can be easily implemented for steady-state accuracy.
+However, this may not be sufficient and a dynamic two degrees-of-freedom design is required.
+
+The design problem is illustrated in Fig. [fig:coprime_uncertainty_hinf](#fig:coprime_uncertainty_hinf).
+The feedback part of the controller \\(K\_2\\) is designed to meet robust stability and disturbance rejection requirements.
+A prefilter is introduced to force the response of the closed-loop system to follow that of a specified model \\(T\_{\text{ref}}\\), often called the **reference model**.
+
+
+
+{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_hinf.png" caption="Figure 54: Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping design problem" >}}
+
+The design problem is to find the stabilizing controller \\(K = [K\_1,\ K\_2]\\) for the shaped plant \\(G\_s = G W\_1\\), with a normalized coprime factorization \\(G\_s = M\_s^{-1} N\_s\\), which minimizes the \\(\mathcal{H}\_\infty\\) norm of the transfer function between the signals \\([r^T\ \phi^T]^T\\) and \\([u\_s^T\ y^T\ e^T]^T\\) as defined in Fig. [fig:coprime_uncertainty_hinf](#fig:coprime_uncertainty_hinf).
+This problem is easily cast into the general configuration.
+
+The control signal to the shaped plant \\(u\_s\\) is given by:
+\\[ u\_s = \begin{bmatrix} K\_1 & K\_2 \end{bmatrix} \begin{bmatrix} \beta \\ y \end{bmatrix} \\]
+where \\(K\_1\\) is the prefilter, \\(K\_2\\) is the feedback controller, \\(\beta\\) is the scaled reference and \\(y\\) is the measured output.
+The purpose of the prefilter is to ensure that:
+\\[ \left\\| (I - G\_s K\_2)^{-1} G\_s K\_1 - T\_{\text{ref}} \right\\|\_\infty < \gamma \rho^2 \\]
+\\(T\_{\text{ref}}\\) is the desired closed-loop transfer function and \\(\rho\\) is a scalar parameter that the designer can increase to place more emphasis on model matching in the optimization at the expense of robustness.
+
+The main steps required to synthesize a two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller are:
+
+1. Design a one degree-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller (section [sec:hinf_loop_shaping_procedure](#sec:hinf_loop_shaping_procedure)) but without a post-compensator \\(W\_2\\)
+2. Select a desired closed-loop transfer function \\(T\_{\text{ref}}\\) between the commands and controller outputs
+3. Set the scalar parameter \\(\rho\\) to a small value greater than \\(1\\); something in the range \\(1\\) to \\(3\\) will usually suffice
+4. For the shaped \\(G\_s = G W\_1\\), the desired response \\(T\_{\text{ref}}\\), and the scalar parameter \\(\rho\\), solve the standard \\(\mathcal{H}\_\infty\\) optimization problem to a specified tolerance to get \\(K = [K\_1,\ K\_2]\\)
+5. Replace the prefilter \\(K\_1\\) by \\(K\_1 W\_i\\) to give exact model-matching at steady-state.
+6. Analyze and, if required, redesign making adjustments to \\(\rho\\) and possibly \\(W\_1\\) and \\(T\_{\text{ref}}\\)
+
+The final two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller is illustrated in Fig. [fig:hinf_synthesis_2dof](#fig:hinf_synthesis_2dof).
+
+
+
+{{< figure src="/ox-hugo/skogestad07_hinf_synthesis_2dof.png" caption="Figure 55: Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller" >}}
+
+
+#### Observer-Based Structure for \\(\hinf\\) Loop-Shaping Controllers {#observer-based-structure-for--hinf--loop-shaping-controllers}
+
+\\(\mathcal{H}\_\infty\\) designs exhibit an observer/state feedback structure in the controller.
+The clear structure of the \\(\mathcal{H}\_\infty\\) loop-shaping controllers has several advantages:
+
+- It is helpful in describing a controller's function
+- It lends itself to implementation in a gain-schedule scheme
+- If offers computational savings in digital implementations
+
+Let's assume that the shaped plant is strictly proper, with a stabilizable and detectable state space realization
+\\[ G\_s \triangleq \left[ \begin{array}{c|c}
+ A\_s & B\_s \\ \hline
+ C\_s & 0
+\end{array} \right] \\]
+
+The single degree-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller can be realized as an observer for the shaped plant plus a state feedback control law:
+
+\begin{align\*}
+ \dot{\hat{x}}\_s &= A\_s \hat{x}\_s + H\_s(C\_s \hat{x}\_s - y\_s) + B\_s u\_s \\\\\\
+ u\_s &= K\_s \hat{x}\_s
+\end{align\*}
+
+where \\(\hat{x}\_s\\) is the observer state, \\(u\_s\\) and \\(y\_s\\) are respectively the input and output of the shaped plant, and
+
+\begin{align\*}
+ H\_s &= -Z\_s C\_s^T \\\\\\
+ K\_s &= -B\_s^T [I - \gamma^{-2}I - \gamma^{-2} X\_s Z\_s]^{-1} X\_s
+\end{align\*}
+
+where \\(Z\_s\\) and \\(X\_s\\) are the appropriate solutions to the generalized algebraic Riccati equations for \\(G\_s\\).
+
+The same can be done for two degrees-of-freedom controllers.
+
+
+#### Implementation Issues {#implementation-issues}
+
+
+##### Discrete-time controllers {#discrete-time-controllers}
+
+For implementation purposes, discrete-time controllers are usually required.
+These can be obtained from a continuous-time design using a **bilinear transformation** from the \\(s\text{-domain}\\) to the \\(z\text{-domain}\\), but there can be advantages in being able to design directly in discrete time.
+
+
+##### Anti-windup {#anti-windup}
+
+In \\(\hinf\\) loop-shaping the pre compensator weight \\(W\_1\\) would normally include integral action in order to reject low frequency disturbances acting on the system.
+However, in the case of actuator saturation, the integrators continue to integrate their input and hence cause **windup** problems.
+An anti-windup scheme is therefore required on the weighting function \\(W\_1\\).
+The approach we recommend is to implement the weight \\(W\_1\\) in its self-conditioned or Hanus form.
+Let the weight \\(W\_1\\) have a realization
+\\[ W\_1 \triangleq \left[ \begin{array}{c|c}
+ A\_w & B\_w \\ \hline
+ C\_w & D\_w
+\end{array} \right] \\]
+and let \\(u\\) be the input to the plant actuators and \\(u\_s\\) the input to the shaped plant.
+Then \\(u = W\_1 u\_s\\).
+When implemented in Hanus form, the expression for \\(u\\) becomes
+\\[ u = \left[ \begin{array}{c|cc}
+ A\_w - B\_wD\_w^{-1}C\_w & 0 & B\_wD\_w^{-1} \\ \hline
+ C\_w & D\_w & 0
+\end{array} \right] \begin{bmatrix}
+ u\_s \\ u\_a
+\end{bmatrix} \\]
+where \\(u\_a\\) is the **actual plant input**, that is the measurement at the **output of the actuators** which therefore contains information about possible actuator saturation.
+
+The situation is illustrated in Fig. [fig:weight_anti_windup](#fig:weight_anti_windup), where the actuators are each modeled by a unit gain and a saturation.
+
+
+
+{{< figure src="/ox-hugo/skogestad07_weight_anti_windup.png" caption="Figure 56: Self-conditioned weight \\(W\_1\\)" >}}
+
+The Hanus form prevents windup by keeping the states of \\(W\_1\\) consistent with the actual plant input at all times.
+When there is no saturation, \\(u\_a=u\\), the dynamics of \\(W\_1\\) remains unaffected.
+But when \\(u\_a\neq u\\), the dynamics are inverted and driven by \\(u\_a\\) so that the states remain consistent with the actual plant input \\(u\_a\\).
+Notice that such an implementation requires \\(W\_1\\) to be invertible and minimum phase.
+
+
+##### Bumpless transfer {#bumpless-transfer}
+
+When multi-mode switched controller is designed, one should ensure **smooth transition from one controller to the other** (bumpless transfer).
+It was found useful to condition the reference models and the observers in each of the controllers.
+When on-line, the observer state evolves according to
+\\[ \dot{\hat{x}}\_s = A\_s \hat{x}\_s + H\_s (C\_s \hat{x}\_s - y\_s) + B\_s u\_s \\]
+but when off-line, the state equation becomes
+\\[ \dot{\hat{x}}\_s = A\_s \hat{x}\_s + H\_s (C\_s \hat{x}\_s - y\_s) + B\_s u\_{as} \\]
+where \\(u\_{as}\\) is the actual input to the shaped plant governed by the on-line controller.
+
+Doing so ensure that the inputs to the shaped plant for the off-line controller follows the actual shaped plant input \\(u\_{as}\\) given by the on-line controller.
+The observer based structure of the \\(\mathcal{H}\_\infty\\) loop-shaping controller is then helpful for such technique.
+
+
+### Conclusion {#conclusion}
+
+Several methods and techniques for controller design have been described.
+The emphasis has been on \\(\hinf\\) loop shaping which is easy to apply and works well in practice.
+It combines classical loop-shaping ideas with an effective method for robustly stabilizing the feedback loop.
+
+For complex problems, such as unstable plants with multiple gain crossover frequencies, it may not be easy to decide on a desired loop shape.
+In which case, we would suggest doing an initial LQG design (with simple weights) and using the resulting loop shape as the desired one for the \\(\hinf\\) loop shaping.
+
+And alternative to \\(\hinf\\) loop shaping is a standard \\(\hinf\\) design with a stacked cost function such as in \\(S/KS\\) mixed-sensitivity optimization.
+In this approach, \\(\hinf\\) optimization is used to shape two or sometimes three closed-loop transfer functions.
+However, with more functions, the shaping becomes increasingly difficult for the designer.
+
+In other design situations where there are several performance objectives, it may be more appropriate to follow a signal-based \\(\htwo\\) or \\(\hinf\\) approach.
+But again, the problem formulations become so complex that the designer has little direct influence on the design.
+
+After a design, the resulting controller should be analyzed with respect to robustness and tested using nonlinear simulations.
+For the study of robustness, we recommend \\(\mu\text{-analysis}\\). If the design is not robust, then the weights should be modified.
+Sometimes, one might consider synthesizing a \\(\mu\text{-optimal}\\) controller, but this complexity is rarely necessary in practice.
+Moreover, one should be careful about combining controller synthesis and analysis into a single step.
+
+
+## Controller Structure Design {#controller-structure-design}
+
+
+
+
+### Introduction {#introduction}
+
+In previous sections, we considered the general problem formulation in Fig. [fig:general_control_names_bis](#fig:general_control_names_bis) and stated that the controller design problem is to find a controller \\(K\\) which based on the information in \\(v\\), generates a control signal \\(u\\) which counteracts the influence of \\(w\\) on \\(z\\), thereby minimizing the closed loop norm from \\(w\\) to \\(z\\).
+
+
+
+{{< figure src="/ox-hugo/skogestad07_general_control_names_bis.png" caption="Figure 57: General Control Configuration" >}}
+
+In this chapter we are concerned with the **structural decisions** associated with the following selection tasks of control structure design:
+
+- **Controlled outputs**: What are the variables \\(z\\)?
+- **Manipulations and measurements**: What are the variable set \\(u\\) and \\(v\\)?
+- **Control configuration**: What is the structure of \\(K\\)?
+- **Controller type**: What algorithm is used for \\(K\\)?
+
+The distinction between the words under control _structure_ and control _configuration_ are significant.
+The _control structure_ refers to all structural decisions included in the design of a control system.
+On the other hand, the _control configuration_ refers only to the structuring of the controller \\(K\\) itself.
+
+Ideally, the tasks involved in designing a complete control system are performed sequentially; first a "top down" selection of controller outputs, measurements and inputs, and then a "bottom up" design of the control system in which the selection of the control configuration is the most important decision.
+However, in practice the tasks are closely related so the procedure may involve iteration.
+
+One important reason for decomposing the control system into a specific _control configuration_ is that it may **allow for simple tuning** of the sub-controllers **without the need for a detailed plant model** describing the dynamics and interactions in the process.
+Multivariable centralized controllers may always outperform decomposed (decentralized) controllers, bus this performance gain must be traded off against the cost of obtaining and maintaining a sufficiently detailed plant model.
+
+The number of possible control structure is usually very large.
+Fortunately, we can often from physical insight obtain a reasonable choice of controlled outputs, measurements and manipulated inputs.
+
+
+### Optimization and Control {#optimization-and-control}
+
+The selection of controlled outputs involves selecting the variables \\(y\\) to be controlled at given reference values \\(y \approx r\\).
+The reference value \\(r\\) is usually set at some higher layer in the control hierarchy which is often divided into two layers:
+
+- **Optimization layer**: computes the desired reference commands \\(r\\)
+- **Control layer**: implements these commands to achieve \\(y \approx r\\)
+
+Additional layers are possible, as is illustrated in Fig. [fig:control_system_hierarchy](#fig:control_system_hierarchy) which shows a typical control hierarchy for a chemical plant.
+
+
+
+{{< figure src="/ox-hugo/skogestad07_system_hierarchy.png" caption="Figure 58: Typical control system hierarchy in a chemical plant" >}}
+
+In general, the information flow in such a control hierarchy is based on the higher layer sending reference values (setpoints) to the layer below reporting back any problems achieving this (see Fig. [fig:optimize_control_b](#fig:optimize_control_b)).
+There is usually a time scale separation between the layers which means that the **setpoints**, as viewed from a given layer, are **updated only periodically**.
+
+The optimization tends to be performed open-loop with limited use of feedback. On the other hand, the control layer is mainly based on feedback information.
+The **optimization is often based on nonlinear steady-state models**, whereas we often use **linear dynamic models in the control layer**.
+
+From a theoretical point of view, the optimal performance is obtained with a **centralized optimizing controller**, which combines the two layers of optimizing and control (see Fig. [fig:optimize_control_c](#fig:optimize_control_c)).
+All control actions in such an ideal control system would be perfectly coordinated and the control system would use on-line dynamic optimization based on nonlinear dynamic model of the complete plant.
+However, this solution is normally not used for a number a reasons, included the cost of modeling, the difficulty of controller design, maintenance, robustness problems and the lack of computing power.
+
+
+
+
Table 6:
+ Alternative structures for optimization and control
+
+
+In the following, we let \\(y\\) denote the selected controller outputs in the control layer.
+Two distinct questions arise:
+
+1. What variables \\(y\\) should be selected?
+2. What is the optimal reference value \\(y\_\text{opt}\\)?
+
+For the first problem, we make the following assumptions:
+
+1. The overall goal can be quantified in terms of a **scalar cost function** \\(J\\) which we want to minimize
+2. For a given disturbance \\(d\\), there exists an optimal value \\(u\_\text{opt}(d)\\) and corresponding value \\(y\_\text{opt}(d)\\) which minimizes the cost function \\(J\\)
+3. The reference values \\(r\\) for the controlled outputs \\(y\\) should be constant, i.e. \\(r\\) should be independent of the disturbances \\(d\\)
+
+The system behavior is a function of the independent variables \\(u\\) and \\(d\\): \\(J = J(u, d)\\).
+For a given disturbance \\(d\\) the optimal value of the cost function is
+
+\begin{equation}
+ J\_\text{opt}(d) \triangleq J(u\_\text{opt}, d) = \min\_u J(u, d)
+\end{equation}
+
+In practice \\(u \neq u\_\text{opt}\\), and we have a loss which can be quantified by \\(L = J - J\_\text{opt}\\).
+A reasonable objective for selecting controlled outputs \\(y\\) is to minimize some norm of the loss, for instance the worst-case loss:
+
+\begin{equation}
+ \Phi \triangleq \max\_{d \in \mathcal{D}} |\underbrace{J(u, d) - J(u\_\text{opt}, d)}\_{L}|
+\end{equation}
+
+where \\(\mathcal{D}\\) is the set of possible disturbances.
+
+
+#### Direct Evaluation of Cost {#direct-evaluation-of-cost}
+
+The "brute force" approach for selecting controlled variables is to evaluate the loss for alternative sets of controlled variable.
+By solving the non linear equations, we evaluate directly the cost function \\(J\\) for various disturbances \\(d\\).
+The set of controlled outputs with smallest worst case or average value of \\(J\\) is then preferred.
+This approach may be time consuming because the solution of the nonlinear equations must be repeated for each candidate set of controlled outputs.
+
+
+#### Linear Analysis {#linear-analysis}
+
+Consider the loss \\(L = J(u,d) - J\_\text{opt}(d)\\) where \\(d\\) is a fixed disturbance.
+We make the following additional assumptions:
+
+
+- The cost function \\(J\\) is smooth (twice differentiable)
+- The optimization problem is unconstrained.
+If it is optimal to keep some variable at a constant, then we assume that this is implemented and consider the remaining unconstrained problem
+- The dynamics of the problem can be neglected, that is, **we consider the steady-state control and optimization**
+
+
+For a fixed \\(d\\) we may express \\(J(u, d)\\) in terms of a Taylor series expansion in \\(u\\) around the optimal point.
+By neglecting terms of third order and higher, we obtain:
+\\[ J(u, d) = J\_\text{opt}(d) + \frac{1}{2} (u - u\_\text{opt}(d))^T \left(\frac{\partial^2 J}{\partial u^2}\right)\_\text{opt} (u - u\_\text{opt}(d)) \\]
+This quantifies how \\(u-u\_\text{opt}\\) affects the cost function.
+For a fixed \\(d\\), we have: \\(y - y\_\text{opt} = G (u - u\_\text{opt})\\) where \\(G\\) is the steady state gain matrix.
+Thus, we get:
+\\[ J - J\_\text{opt} \approx \frac{1}{2} \big(G^{-1}(y-y\_\text{opt})\big)^T \left(\frac{\partial^2 J}{\partial u^2}\right)\_\text{opt} G^{-1} (y - y\_\text{opt}) \\]
+
+We conclude that we should select \\(y\\) such that:
+
+1. \\(G^{-1}\\) is small: the inputs have a large effect on \\(y\\)
+2. \\(e\_\text{opt} = r - y\_\text{opt}(d)\\) is small: its optimal value \\(y\_\text{opt}(d)\\) depends only weakly on the disturbances and other changes
+3. \\(e = y - r\\) is small: it is easy to keep the control error \\(e\\) small
+
+Note that \\(\bar{\sigma}(G^{-1}) = 1/\underline{\sigma}(G)\\) and so **we want the smallest singular value of the steady state gain matrix to be large**.
+
+As this depends of scaling, we should first **scale the outputs** such that the expected magnitude of \\(y\_i - y\_{i\_\text{opt}}\\) is similar in magnitude for each output, and **scale the inputs** such that the effect of a given deviation \\(u\_j - u\_{j\_\text{opt}}\\) on the cost function \\(J\\) is similar for each input.
+
+
+
+
+The use of the minimum singular value to select controlled outputs may be summarized in the following procedure:
+
+1. From a (nonlinear) model compute the optimal parameters (inputs and outputs) for various conditions (disturbances, operating points).
+ This yields a "look-up" table for optimal parameter values as a function of the operating conditions
+2. From this data, obtain for each candidate output the variation in its optimal value
+ \\[ v\_i = \frac{(y\_{i\_{\text{opt,max}}} - y\_{i\_{\text{opt,min}}})}{2} \\]
+3. Scale the candidate outputs such that for each output the sum of the magnitudes of \\(v\_i\\) and the control error (\\(e\_i\\), including measurement noise \\(n\_i\\)) is similar (e.g. \\(|v\_i| + |e\_i| = 1\\))
+4. Scale the inputs such that a unit deviation in each input from its optimal value has the same effect on the cost function \\(J\\)
+5. Select as candidates those sets of controlled outputs which corresponds to a large value of \\(\underline{\sigma}(G)\\).
+ \\(G\\) is the transfer function for the effect of the scaled inputs on the scaled outputs
+
+
+
+The controlled outputs are often measured, but we may also estimated their values based on other measured variables.
+We may also use other measurements to improve the control of the controlled outputs, for example, by use of cascade control.
+Thus, the selection of controlled and measured outputs are two separate issues.
+
+
+### Selection of Manipulations and Measurements {#selection-of-manipulations-and-measurements}
+
+We are here concerned with the variable sets \\(u\\) and \\(v\\) in Fig. [fig:general_control_names_bis](#fig:general_control_names_bis).
+Note that **the measurements** \\(v\\) used by the controller **are in general different from the controlled variables** \\(z\\) because we may not be able to measure all the controlled variables and we may want to measure and control additional variables in order to:
+
+- Stabilize the plant, or more generally change its dynamics
+- Improve local disturbance rejection
+
+
+##### Stabilization {#stabilization}
+
+We usually start of controller design by designing a lower-layer controller to stabilize the plant.
+The issue is then: which outputs and inputs should be used for stabilization?
+A reasonable objective is to minimize the required input usage of the stabilizing control system.
+
+
+##### Local disturbance rejection {#local-disturbance-rejection}
+
+For measurements, the rule is generally to select those which have a **strong relationship with the controlled outputs**, or which may **quickly detect a major disturbance**.
+
+The selected manipulations should have a **large effect on the controlled outputs** and should be located "close" (in terms of dynamic response) to the outputs and measurements.
+
+To evaluate the combinations of manipulations and measurements, one may perform an **input-output controllability analysis** for each combination (e.g. consider the minimum singular values, RHP-zeros, interactions, etc).
+A more involved approach would be to perform a achievable robust performance analysis.
+An even more involved (and exact) approach would be to synthesize controllers for optimal robust performance for each candidate combination.
+However, the number of combination has a combinatorial growth and the analysis may become very time-consuming.
+
+
+### RGA for Non-Square Plant {#rga-for-non-square-plant}
+
+A simple but effective tool for selecting inputs and outputs, which avoids to combinatorial problem is the **Relative Gain Array** (RGA) of the "big" transfer matrix \\(G\_\text{all}\\) with all candidates inputs and outputs included:
+
+\begin{equation}
+ \tcmbox{\Lambda = G\_{\text{all}} \times G\_{\text{all}}^{\dagger^T}}
+\end{equation}
+
+Essentially, one may consider not using those manipulations \\(u\\) corresponding to columns in the RGA where the sum of the elements is much smaller than 1.
+
+Similarly, one may consider not using those outputs \\(v\\) corresponding to rows in the RGA where the sum of the elements is much small than 1.
+
+
+### Control Configuration Elements {#control-configuration-elements}
+
+We now assume that the measurements, manipulations and controlled outputs are fixed.
+The available synthesis theories presented in this book result in a _multivariable controller_ \\(K\\) which connects all available measurements \\(v\\) with all available manipulations \\(u\\):
+\\[ u = K v \\]
+However, such a "big" controller may not be desirable.
+
+
+
+
+We define the **control configuration** to be the restrictions imposed on the overall controller \\(K\\) by decomposing it into a set of **local controllers** with predetermined links and with a possibly predetermined design sequence where subcontrollers are designed locally.
+
+
+
+The choice of control configuration leads to two different ways of partitioning the control system:
+
+- **Vertical decomposition**. This usually results from a sequential design of the control system
+- **Horizontal decomposition**. This usually involves a set of independent decentralized controllers
+
+Of course, a **performance loss** is inevitable if we decompose the control system.
+For example, if we select a poor configuration at the lower control layer, then this may pose fundamental limitations on the achievable performance (RHP zeros, strong interactions, etc).
+
+
+#### Cascade Control Systems {#cascade-control-systems}
+
+We here use SISO controllers of the form
+
+\begin{equation}
+ u\_i = K\_i(s) (r\_i - y\_i)
+\end{equation}
+
+where \\(K\_i(s)\\) is a scalar.
+Then when a SISO control loop is closed, we lose the input \\(u\_i\\) as a degree-of-freedom but the reference \\(r\_i\\) becomes a new degree-of-freedom.
+
+A cascade control structure results when either of the following two situations arise:
+
+- The reference \\(r\_i\\) is an output from another controller.
+ This is the **conventional cascade control** (Fig. [fig:cascade_extra_meas](#fig:cascade_extra_meas))
+- The "measurement" \\(y\_i\\) is an output from another controller.
+ This is referred to as **input resetting** (Fig. [fig:cascade_extra_input](#fig:cascade_extra_input))
+
+
+
+
Table 7:
+ Cascade Implementations
+
+
+
+With reference to the special (but common) case of cascade control shown in Fig. [fig:cascade_control](#fig:cascade_control), the use of extra measurements is useful under the following circumstances:
+
+- The disturbance \\(d\_2\\) is significant and \\(G\_1\\) is non-minimum phase.
+ If \\(G\_1\\) is minimum phase, the input-output controllability of \\(G\_2\\) and \\(G\_1 G\_2\\) are the same and there is no fundamental advantage in measuring \\(y\_2\\)
+- The plant \\(G\_2\\) has considerable uncertainty associated with it and the inner loop serves to remove the uncertainty.
+ The inner loop \\(L\_2 = G\_2 K\_2\\) removes the uncertainty if it is sufficiently fast and yields a transfer function \\((I + L\_2)^{-1} L\_2\\) close to \\(I\\) at frequencies where \\(K\_1\\) is active.
+
+
+
+
+Consider the system in Fig. [fig:cascade_control_two_layers](#fig:cascade_control_two_layers) with two manipulated inputs (\\(u\_2\\) and \\(u\_3\\)), one controlled output (\\(y\_1\\) which should be close to \\(r\_1\\)) and two measured variables (\\(y\_1\\) and \\(y\_2\\)).
+Input \\(u\_2\\) has a more direct effect on \\(y\_1\\) than does input \\(u\_3\\) (there is a large delay in \\(G\_3(s)\\)).
+Input \\(u\_2\\) should only be used for transient control as it is desirable that it remains close to \\(r\_3 = r\_{u\_2}\\).
+The extra measurement \\(y\_2\\) is closer than \\(y\_1\\) to the input \\(u\_2\\) and may be useful for detecting disturbances affecting \\(G\_1\\).
+
+Controller \\(K\_1\\) controls the primary output \\(y\_1\\) at its reference \\(r\_1\\) by adjusting the "input" \\(\hat{u}\_1\\), which is the reference value for \\(y\_2\\).
+Controller \\(K\_2\\) controls the secondary output \\(y\_2\\) using input \\(u\_2\\).
+Finally, controller \\(K\_3\\) manipulates \\(u\_3\\) slowly in order to reset input \\(u\_2\\) to its desired value \\(r\_3\\).
+We would probably tune the three controllers in the order \\(K\_2\\), \\(K\_3\\), and \\(K\_1\\).
+
+
+
+However, there are a **number of reason why cascade and decentralized control are used in practice**.
+The most important one is the **cost associated with obtaining good plant models**, which are a prerequisite for applying multivariable control.
+Since cascade and decentralized control systems depend more strongly on feedback rather than models as their source of information, it is usually more important (relative to centralized multivariable control) that the fast control loops be tuned to respond quickly.
+
+The cascade and decentralized control are often easier to understand, their tuning parameters have a direct and "localized" effect, and they tend to be **less sensitive to uncertainty**.
+
+The **main challenge** is then to find a control configuration which allows the controllers to be tuned independently based on a minimum of model information.
+To be able to tune the controllers independently, we must require that the loops interact only to a limited extent.
+For example, one desirable property is that the steady-state gain from \\(u\_i\\) to \\(y\_i\\) in an "inner" loop does not change too much as outer loops are closed.
+
+
+### Hierarchical and Partial Control {#hierarchical-and-partial-control}
+
+
+#### Partial Control {#partial-control}
+
+
+
+
+Partial control involves controlling only a subset of the outputs for which there is a control objective.
+
+
+
Table 8:
+ Applications of partial control
+
+
+The objectives for this hierarchical decomposition are:
+
+- to allow for simple or even on-line tuning of \\(K\_2\\)
+- to allow the use of longer sampling intervals for \\(K\_1\\)
+- to allow simple models when designing \\(K\_1\\)
+- to "stabilize" the plant using \\(K\_2\\) such that it is amenable to manual control
+
+
+
+
+The selection of \\(u\_2\\) and \\(y\_2\\) for use in the lower-layer control system can be done with the following criteria:
+
+- The lower-layer must quickly implement the setpoints computed by the higher layers, that is, the input-output controllability of the subsystem involving the use of \\(u\_2\\) to control \\(y\_2\\) should be good (consider \\(G\_{22}\\) and \\(G\_{d2}\\))
+- The control of \\(y\_2\\) using \\(u\_2\\) should provide local disturbance rejection, that is, it should minimize the effect of disturbances on \\(y\_1\\)
+- The control of \\(y\_2\\) using \\(u\_2\\) should not impose unnecessary control limitations (RHP-zero, ill-conditioning, etc.) on the remaining control problem which involves using \\(u\_1\\) to control \\(y\_1\\)
+
+
+
+
+A set of outputs \\(y\_1\\) may be left uncontrolled only if the effects of all disturbances (including \\(r\_2\\)) on \\(y\_1\\), as expressed by the elements in the corresponding partial disturbance gain matrix \\(P\_d\\) are less than \\(1\\) in magnitude at all frequencies.
+
+
+
+
+Scale the disturbances \\(d\\) to be of magnitude 1, and scale the outputs \\(y\_2\\) so that the expected control error \\(e\_2\\) (measurement noise) is of magnitude 1 for each outputs.
+Then to minimize the control error for the primary output, \\(J = \\|y\_1 - r\_1\\|\\), we should select sets of controlled outputs which minimizes \\(\\|[ P\_d \ P\_r]\\|\\).
+
+
+
+
+Intuitively, we would like to pair variables \\(u\_j\\) and \\(y\_i\\) so that \\(\lambda\_{ij}\\) is close to \\(1\\), because this means that the gain from \\(u\_j\\) to \\(y\_i\\) is unaffected by closing the other loops.
+More precisely, we would like to pair such that the rearranged system, with the pairings along the diagonal, has a RGA matrix close to identity.
+
+
+
+**Sufficient conditions in terms of RGA**.
+Suppose the plant \\(G(s)\\) is stable. If the RGA-matrix \\(\Lambda(G) = I\ \forall\omega\\) (which can only arise for a triangular plant \\(G(s)\\)), then stability of each of the individual loops implies stability of the entire system.
+
+In most cases, it is sufficient for overall stability to require that \\(G(j\omega)\\) is close to triangular (or \\(\Lambda(G) \approx I\\)) at crossover frequencies.
+This gives the "first pairing rule".
+
+
+
+
+To achieve stability with decentralized control, prefer pairings such that at frequencies \\(\omega\\) around crossover, the rearranged matrix \\(G(j\omega)\\) (with the paired elements along the diagonal) is close to triangular.
+This is equivalent to requiring \\(\Lambda(G(j\omega)) \approx I\\), i.e. the RGA-number \\(\\|\Lambda(G(j\omega)) - I\\|\_\text{sum}\\) should be small.
+
+
+
+An even stronger requirement is that the system remains stable as the gain in various loops are reduced: \\(0 \le \epsilon\_i \le 1\\).
+
+
+
+
+The plant \\(G(s)\\) (corresponding to a given pairing with the paired elements along its diagonal) is **Decentralized Integral Controllability** (DIC) if there exists a stabilizing decentralized controller with **integral action in each loop** such that each individual loop may be detuned independently by a factor \\(\epsilon\_1\\) (\\(0 \le \epsilon\_i \le 1\\)) without introducing instability.
+
+
+
+
+\begin{equation}
+\begin{aligned}
+ &\text{A stable (reordered) plant } G(s)\\\\\\
+ &\text{is DIC only if } \lambda\_{ii}(0) \ge 0 \text{ for all } i
+\end{aligned}
+\end{equation}
+
+
+
+
+For a stable plant, avoid pairings that corresponds to negative steady-state RGA-elements \\(\lambda\_{ij}(0) < 0\\)
+
+
+
+
+\begin{align\*}
+G(0) &= \begin{bmatrix}
+ 10.2 & 5.6 & 1.4 \\\\\\
+ 15.5 & -8.4 & -0.7 \\\\\\
+ 18.1 & 0.4 & 1.8
+\end{bmatrix} \\\\\\
+\Lambda(0) &= \begin{bmatrix}
+ 0.96 & 1.45 & -1.41 \\\\\\
+ 0.94 & -0.37 & 0.43 \\\\\\
+ -0.90 & -0.07 & 1.98
+\end{bmatrix}
+\end{align\*}
+
+For a \\(3 \times 3\\) plant there are 6 alternative pairings.
+From the steady state RGA, we see that there is only one positive element in columns 2, and only positive element in row 3, and therefore there is only on possible pairing if we require DIC:
+\\[ u\_1 \leftrightarrow y\_2,\ u\_2 \leftrightarrow y\_1,\ u\_3 \leftrightarrow y\_3 \\]
+
+
+
+
+\begin{align\*}
+G(s) &= \frac{-s + 1}{(5 s + 1)^2} \begin{bmatrix}
+ 1 & 4 & -26 \\\\\\
+ 6.2 & 1 & -26 \\\\\\
+ 1 & 1 & 1
+\end{bmatrix}\\\\\\
+\Lambda(G) &= \begin{bmatrix}
+ 1 & 5 & -5 \\\\\\
+ -5 & 1 & 5 \\\\\\
+ 5 & -5 & 1
+\end{bmatrix}
+\end{align\*}
+
+Only two of the six possible pairings gives positive steady-state RGA-elements: the diagonal pairing on all \\(\lambda\_{ii} = 1\\) or the pairing on all \\(\lambda\_{ii} = 5\\).
+Intuitively, one may expect pairing with \\(\lambda\_{ii} = 1\\) since it corresponds to pairing on RGA-elements equal to \\(1\\).
+However, the RGA matrix is far from identify, and the RGA-number \\(\\| \Lambda - I \\|\_\text{sum} = 30\\) for both alternative.
+Thus none of the two alternatives satisfy _Pairing Rule 1_, and decentralized control should not be used for this plant.
+
+
+
+At frequencies where feedback is effective (\\(\tilde{S} \approx 0\\)), \\(S \approx \tilde{S} \Gamma\\) which shows that \\(\Gamma\\) is important when evaluating performance with decentralized control.
+
+Note that the diagonal elements of the PRGA-matrix are equal to the diagonal elements of the RGA and that the off-diagonal elements of the PRGA depend on the relative scaling on the outputs which is not the case for the RGA.
+
+We will also use the related **Closed-Loop Disturbance Gain** (CLDG) matrix:
+
+\begin{equation}
+ \tcmbox{\tilde{G}\_d(s) \triangleq \Gamma(s)G\_d(s) = \tilde{G}(s) G^{-1}(s) G\_d(s)}
+\end{equation}
+
+which depends on both output and disturbance scaling.
+
+Suppose the system has been scaled such that:
+
+- Each disturbance magnitude is less than \\(1\\), \\(|d\_k| < 1\\)
+- Each reference change is less than the corresponding diagonal element in \\(R\\), \\(|r\_j| < R\_j\\)
+- For each output the acceptable control error is less than \\(1\\), \\(|e\_i| < 1\\)
+
+
+##### Single disturbance {#single-disturbance}
+
+Consider a single disturbance, in which case \\(G\_d\\) is a vector, and let \\(g\_{di}\\) denote the \\(i\text{'th}\\) element of \\(G\_d\\).
+Let \\(L\_i = g\_{ii} k\_i\\) denote the loop transfer function in loop \\(i\\).
+Consider frequencies where feedback is effective so \\(\tilde{S}\Gamma\\) is small.
+Then for **acceptable disturbance rejection** (\\(|e\_i| < 1\\)) we must with decentralized control required for each loop \\(i\\)
+
+\begin{equation}
+ \tcmbox{|1 + L\_i| > |\tilde{g}\_{di}| \quad \forall i}
+\end{equation}
+
+which is the same as the SISO-condition except that \\(G\_d\\) is replaced by the CLDG.
+In words, \\(\tilde{g}\_{di}\\) gives the "apparent" disturbance gain as seen from the loop \\(i\\) when the system is controlled using decentralized control.
+
+
+##### Single reference change {#single-reference-change}
+
+Consider a change in reference for output \\(j\\) of magnitude \\(R\_j\\).
+Consider frequencies where feedback is effective.
+Then for **acceptable reference tracking** (\\(|e\_i|<1\\)) we must require for each loop \\(i\\)
+
+\begin{equation}
+ \tcmbox{|1 + L\_i| > |\gamma\_{ij}| \cdot |R\_j| \quad \forall i}
+\end{equation}
+
+which is the same as the SISO-condition except for the PRGA-factor \\(|\gamma\_{ij}|\\).
+
+Consequently, for performance it is desirable to have small elements in \\(\Gamma\\), at least at frequencies where feedback is effective.
+However, at frequencies close to crossover, stability is the main issue and since the diagonal elements of the PRGA and RGA are equal, we usually prefer to have \\(\gamma\_{ii}\\) close to \\(1\\).
+
+
+#### Summary: Controllability Analysis for Decentralized Control {#summary-controllability-analysis-for-decentralized-control}
+
+When considering decentralized diagonal control of a plant, one should first check that the plant is controllable with any controller.
+The next step is to compute the RGA matrix as a function of frequency, and to determine if one can find a good set of input-output pairs bearing in mind the following:
+
+1. Prefer pairings which have the **RGA-matrix close to identity at frequencies around crossover**, i.e. the RGA-number \\(\\|\Lambda(j\omega)-I\\|\\) should be small
+2. Avoid a pairing \\(ij\\) with negative steady-state RGA elements \\(\lambda\_{ij}(G(0)\\)
+3. Prefer a pairing \\(ij\\) where \\(g\_{ij}(s)\\) puts minimal restrictions on the achievable bandwidth.
+ Specifically, the frequency \\(\omega\_{uij}\\) where \\(\angle g\_{ij}(j\omega\_{uij}) = \SI{-180}{\degree}\\) should be as large as possible
+ This rule favors parings on variables "close to each other"
+
+When a reasonable choice of pairings have been made, one should rearrange \\(G\\) to have the **paired elements along the diagonal** and perform a **controllability analysis**:
+
+
+- Compute the CLDG and PRGA, and plot these as a function of frequency
+- For systems with many loops, it is best to perform the analysis one loop at the time, that is, for each loop \\(i\\), plot \\(|\tilde{g}\_{dik}|\\) for each disturbance \\(k\\) and plot \\(|\gamma\_{ij}|\\) for each reference \\(j\\).
+For performance, we need \\(|1 + L\_i|\\) to be larger than each of these:
+
+\begin{equation}
+ |1 + L\_i| > \max\_{k,j}\\{|\tilde{g}\_{dik}|, |\gamma\_{ij}|\\}
+\end{equation}
+
+To achieve stability of the individual loops, one must analyze \\(g\_{ii}(s)\\) to ensure that the bandwidth required by [eq:decent_contr_one_loop](#eq:decent_contr_one_loop) is achievable.
+Note that RHP-zeros in the diagonal elements may limit achievable decentralized control, whereas they may not pose any problems for a multivariable controller.
+Since with decentralized control, we usually want to use simple controllers, the achievable bandwidth in each loop will be limited by the frequency where \\(\angle g\_{ii}\\) is \\(\SI{-180}{\degree}\\)
+- Check for constraints by considering the elements of \\(G^{-1} G\_d\\) and make sure that they do not exceed one in magnitude within the frequency range where control is needed.
+Equivalently, one may for each loop \\(i\\), plot \\(|g\_{ii}|\\) and the requirement is then that
+
+\begin{equation}
+ |g\_{ii}| > |\tilde{g}\_{dik}| \quad \forall k
+\end{equation}
+
+at frequencies where \\(|\tilde{g}\_{dik}|\\) is larger than \\(1\\).
+This provides a direct generalization of the requirement \\(|G| > |G\_d|\\) for SISO systems.
+
+
+If the plant is not controllable, then one may consider another choice of pairing and go back to Step 4.
+If one still cannot find any pairing which are controllable, then one should consider multivariable control.
+
+
+- If the chosen pairing is controllable, then [eq:decent_contr_one_loop](#eq:decent_contr_one_loop) tells us how large \\(|L\_i| = |g\_{ii} k\_i|\\) must be.
+This can be used as a basis for designing the controller \\(k\_i(s)\\) for loop \\(i\\)
+
+
+
+#### Sequential Design of Decentralized Controllers {#sequential-design-of-decentralized-controllers}
+
+Usually the local controllers \\(k\_i(s)\\) are designed locally and then all the loops are closed.
+One problem with this is that the **interactions** may cause the overall system \\(T\\) so be unstable, even though the local loops \\(\tilde{T}\\) are stable.
+This will not happen if the plant is **diagonally dominant**, such that we satisfy, for example \\(\maxsv(\tilde{T}) < 1/\mu(E)\\).
+
+The stability problem is avoided if the controllers are **designed sequentially** when, for example, the bandwidths of the loops are quite different.
+In this case, the outer loops are tuned with the inner loops in place, and each step may be considered as a SISO control problem.
+In particular, overall stability is determined by \\(m\\) SISO stability conditions.
+However, the issue of performance is more complicated because the closing of a loop may cause "disturbances" (interactions) into a previously designed loop.
+The engineer must then go back and redesign a loop that has been designed earlier.
+Thus sequential design may involve many iterations.
+
+
+#### Conclusion on Decentralized Control {#conclusion-on-decentralized-control}
+
+A number of **conditions for the stability**, e.g. [eq:decent_contr_cond_stability](#eq:decent_contr_cond_stability) and [eq:decent_contr_necessary_cond_stability](#eq:decent_contr_necessary_cond_stability), and **performance**, e.g. [eq:decent_contr_cond_perf_dist](#eq:decent_contr_cond_perf_dist) and [eq:decent_contr_cond_perf_ref](#eq:decent_contr_cond_perf_ref), of decentralized control systems have been derived.
+
+The conditions may be useful in **determining appropriate pairings of inputs and outputs** and the **sequence in which the decentralized controllers should be designed**.
+
+The conditions are also useful in an **input-output controllability analysis** for determining the viability of decentralized control.
+
+
+## Model Reduction {#model-reduction}
+
+
+
+
+### Introduction {#introduction}
+
+Modern controller design methods such as \\(\mathcal{H}\_\infty\\) and LQG, produce controllers of order at least equal to that of the plant, and usually higher because of the inclusion of weights.
+These control laws may be **too complex** with regards to **practical implementation** and simpler designs are then sought.
+For this purpose, one can **either reduce the order of the plant model prior to controller design, or reduce the controller in the final stage**.
+
+
+
+
+Given a high-order linear time-invariant stable model \\(G\\), find a low-order approximation \\(G\_a\\) such that the infinity (\\(\mathcal{H}\_\infty\\) or \\(\mathcal{L}\_\infty\\)) norm of the difference \\(\\|G - G\_a\\|\_\infty\\) is small.
+
+
+
+So far we have only been interested in the infinity (\\(\mathcal{H}\_\infty\\)) norm of stable systems.
+But the error \\(G-G\_a\\) may be unstable and **the definition of the infinity norm needs to be extended to unstable systems**.
+
+
+
+
+\\(\mathcal{L}\_\infty\\) defines the set of rational functions which have no poles on the imaginary axis, it includes \\(\mathcal{H}\_\infty\\), and its norm (like \\(\mathcal{H}\_\infty\\)) is given by
+
+\begin{equation}
+ \\|G\\|\_\infty = \sup\_\omega \maxsv(G(j\omega))
+\end{equation}
+
+
+
+If however, \\(A\\) is in **Jordan form**, then it is easy to **order the states** so that \\(x\_2\\) corresponds to **high frequency or fast modes**.
+
+
+##### Modal Truncation {#modal-truncation}
+
+For simplicity, assume that \\(A\\) has been diagonalized so that
+
+\begin{align\*}
+ A &= \begin{bmatrix}
+ \lambda\_1 & 0 & \dots & 0 \\\\\\
+ 0 & \lambda\_2 & \dots & 0 \\\\\\
+ \vdots & \vdots & \ddots & \vdots \\\\\\
+ 0 & 0 & \dots & \lambda\_n \\\\\\
+ \end{bmatrix},\quad B = \begin{bmatrix}
+ b\_1^T \\ b\_2^T \\ \vdots \\ b\_n^T
+ \end{bmatrix} \\\\\\
+ C &= \begin{bmatrix}
+ c\_1, c\_2, \dots, c\_n
+ \end{bmatrix}
+\end{align\*}
+
+Then, if the \\(\lambda\_i\\) are ordered so that \\(|\lambda\_1| < |\lambda\_2| < \dots\\), the fastest modes are removed from the model after truncation.
+The difference between \\(G\\) and \\(G\_a\\) following a k-th order model truncation is given by
+\\[ G - G\_a = \sum\_{i = k+1}^n \frac{c\_i b\_i^T}{s - \lambda\_i} \\]
+and therefore
+
+\begin{equation}
+ \\| G - G\_a \\|\_\infty \le \sum\_{i = k+1}^n \frac{\maxsv(c\_i b\_i^t)}{|\text{Re}(\lambda\_i)|}
+\end{equation}
+
+It is interesting to note that the error depends on the residues \\(c\_i b\_i^T\\) as well as the \\(\lambda\_i\\).
+The distance of \\(\lambda\_i\\) from the imaginary axis is therefore not a reliable indicator of whether the associated mode should be included in the reduced order model or not.
+
+An advantage of modal truncation is that the poles of the truncated model are a subset of the poles of the original model and therefore **retain any physical interpretation** they might have.
+
+
+#### Residualization {#residualization}
+
+In truncation, we discard all the states and dynamics associated with \\(x\_2\\).
+Suppose that instead of this, we simply set \\(\dot{x}\_2 = 0\\), i.e. we residualize \\(x\_2\\), in the state-space equations.
+One can then solve for \\(x\_2\\) in terms of \\(x\_1\\) and \\(u\\), and back substitution of \\(x\_2\\), then gives
+
+\begin{align\*}
+ \dot{x}\_1 &= (A\_{11} - A\_{12} A\_{22}^{-1} A\_{21}) x\_1 + (B\_1 - A\_{12} A\_{22}^{-1} B\_2) u \\\\\\
+ y &= (C\_1 - C\_2 A\_{22}^{-1} A\_{21}) x\_1 + (D - C\_2 A\_{22}^{-1} B\_2) u
+\end{align\*}
+
+And let assume \\(A\_{22}\\) is invertible and define
+
+\begin{alignat\*}{3}
+ &A\_r \triangleq A\_{11} - A\_{12}A\_{22}^{-1}A\_{21} & & \quad B\_r \triangleq B\_1 - A\_{12}A\_{22}^{-1}B\_2\\\\\\
+ &C\_r \triangleq C\_1 - C\_2A\_{22}^{-1}A\_{21} & & \quad D\_r \triangleq D - C\_2A\_{22}^{-1}B\_2
+\end{alignat\*}
+
+The reduced order model \\(G\_a(s) = (A\_r, B\_r, C\_r, D\_r)\\) is called a **residualization** of \\(G(s) = (A, B, C, D)\\).
+Usually \\((A, B, C, D)\\) will have been put into **Jordan form**, with the eigenvalues ordered so that \\(x\_2\\) contains the fast modes.
+
+Model reduction by residualization is then equivalent to singular perturbation approximation, where **the derivatives of the fastest states are allowed to approach zero** with some parameter \\(\epsilon\\).
+
+An important property of residualization is that **it preserves the steady-state gain of the system**:
+
+\begin{equation}
+ \tcmbox{G\_a(0) = G(0)}
+\end{equation}
+
+This should be no surprise since the residualization process sets derivatives to zero, which are zero anyway at steady-state.
+But it is in stark contrast to truncation which retains the system behavior at infinite frequency.
+This contrast between truncation and residualization follows from the simple bilinear relationship \\(s \to \frac{1}{s}\\) which relates the two.
+
+It is clear that **truncation is to be preferred when accuracy is required at high frequencies**, whereas **residualization is better for low frequency modelling**.
+
+Both methods depend to a large extent on the original realization and we have suggested to use of the Jordan form.
+A better realization, with many useful properties, is the **balanced realization**.
+
+
+### Balanced Realization {#balanced-realization}
+
+A balanced realization is an asymptotically stable minimal realization in which the **controllability and observability Gramiams are equal and diagonal**.
+
+Let \\((A,B,C,D)\\) be a minimal realization of a stable, rational transfer function \\(G(s)\\), then \\((A,B,C,D)\\) is called **balanced** if the solutions to be following Lyapunov equations
+
+\begin{subequations}
+ \begin{align}
+ AP + PA^T + BB^T &= 0 \\\\\\
+ A^TQ + QA + C^TC &= 0
+ \end{align}
+\end{subequations}
+
+are \\(P = Q = \text{diag}(\sigma\_1, \sigma\_2, \dots, \sigma\_n) \triangleq \Sigma\\), where \\(\sigma\_1 \ge \sigma\_2 \ge \dots \ge \sigma\_n > 0\\).
+\\(P\\) and \\(Q\\) are the **controllability and observability Gramiams**, also defined by
+
+\begin{subequations}
+ \begin{align}
+ P &\triangleq \int\_0^\infty e^{At} B B^T e^{A^Tt} dt \\\\\\
+ Q &\triangleq \int\_0^\infty e^{A^Tt} C^T C e^{At} dt
+ \end{align}
+\end{subequations}
+
+\\(\Sigma\\) is therefore simply referred to as the Gramiam of \\(G(s)\\).
+The \\(\sigma\_i\\) are the **ordered Hankel singular values** of \\(G(s)\\), more generally defined as \\(\sigma\_i \triangleq \lambda\_i^{\frac{1}{2}}(PQ)\\), \\(i = 1, \dots, n\\).
+Notice that \\(\sigma\_1 = \\|G\\|\_H\\) is the Hankel norm of \\(G(s)\\).
+
+In balanced realization the value of each \\(\sigma\_i\\) is associated with a state \\(x\_i\\) of the balanced system.
+
+
+
+
+The size of \\(\sigma\_i\\) is a relative measure of the contribution that \\(x\_i\\) makes to the input-output behavior of the system.
+
+
+
+After balancing a system, each state is just as controllable as it is observable, and a measure of a state's joint observability and controllability is given by its associated Hankel singular value.
+This property is fundamental to the model reduction methods in the remainder of this chapter which work by removing states having little effect on the system's input-output behavior.
+
+
+### Balanced Truncation and Balanced Residualization {#balanced-truncation-and-balanced-residualization}
+
+Let the balanced realization \\((A,B,C,D)\\) of \\(G(s)\\) and the corresponding \\(\Sigma\\) be partitioned compatibly as
+
+\begin{equation}
+ \begin{aligned}
+ A &= \begin{bmatrix}
+ A\_{11} & A\_{12} \\\\\\
+ A\_{21} & A\_{22}
+ \end{bmatrix}, \quad B = \begin{bmatrix}
+ B\_1 \\ B\_2
+ \end{bmatrix} \\\\\\
+ C &= \begin{bmatrix}
+ C\_1 & C\_2
+ \end{bmatrix}, \quad \Sigma = \begin{bmatrix}
+ \Sigma\_1 & 0 \\\\\\
+ 0 & \Sigma\_2
+ \end{bmatrix}
+ \end{aligned}
+\end{equation}
+
+where
+
+\begin{align\*}
+ \Sigma\_1 &= \text{diag}(\sigma\_1, \sigma\_2, \dots, \sigma\_k)\\\\\\
+ \Sigma\_2 &= \text{diag}(\sigma\_{k+1}, \sigma\_{k+2}, \dots, \sigma\_n),\ \sigma\_k > \sigma\_{k+1}
+\end{align\*}
+
+
+##### Balanced Truncation {#balanced-truncation}
+
+The reduced order model given by \\((A\_{11},B\_1,C\_1,D)\\) is called a **balanced truncation** of the full order system \\(G(s)\\).
+The idea of balancing truncation is thus to first make a balanced realization of the system and then to discard the states corresponding to small Hankel singular values.
+
+A balanced truncation is also a balanced realization, and the infinity norm of the error between \\(G(s)\\) and the reduced order system \\(G\_a(s)\\) is bounded by twice the sum of the last \\(n-k\\) Hankel singular values, i.e. twice the trace of \\(\Sigma\_2\\):
+
+\begin{equation}
+ \\|G(s) - G\_a(s)\\|\_\infty \le 2 \cdot \text{Tr}\big( \Sigma\_2 \big)
+\end{equation}
+
+For the case of repeated Hankel singular values, each repeated Hankel singular value is to be counted only once in calculating the sum.
+
+Useful algorithms that compute balanced truncations without first computing a balanced realization still require the computation of the observability and controllability Gramiam, which can be a problem if the system to be reduced is of very high order.
+
+
+##### Balanced Residualization {#balanced-residualization}
+
+In balanced truncation above, we discarded the least controllable and observable states corresponding to \\(\Sigma\_2\\).
+In balanced residualization, we simply set to zero the derivatives of all these states.
+
+
+##### Theorem {#theorem}
+
+Let \\(G(s)\\) be a stable rational transfer function with Hankel singular values \\(\sigma\_1 > \sigma\_2 > \dots > \sigma\_N\\) where each \\(\sigma\_i\\) has multiplicity \\(r\_i\\) and let \\(G\_a^k(s)\\) be obtained by truncating or residualizing the balanced realization of \\(G(s)\\) to the first \\((r\_1 + r\_2 + \dots + r\_k)\\) states.
+Then
+
+\begin{equation}
+ \\|G(s) - G\_a^k(s)\\|\_\infty \le 2(\sigma\_{k+1} + \sigma\_{k+2} + \dots + \sigma\_N)
+\end{equation}
+
+
+### Optimal Hankel Norm Approximation {#optimal-hankel-norm-approximation}
+
+In this approach to model reduction, the problem that is directly addressed is the following: given a stable model \\(G(s)\\) of order \\(n\\), find a reduced order model \\(G\_h^k(s)\\) of degree \\(k\\) such that the Hankel norm of the approximation error, \\(\\| G(s) - G\_h^k(s) \\|\_H\\), is minimized.
+
+
+
+
+The Hankel norm of any stable transfer function \\(E(s)\\) is defined as
+
+\begin{equation}
+ \\| E(s) \\|\_H \triangleq \rho^{\frac{1}{2}} (PQ)
+\end{equation}
+
+where \\(P\\) and \\(Q\\) are the controllability and observability Gramiams of \\(E(s)\\).
+
+
+
+
+Hence **residualization** is to be preferred whenever low frequency matching is desired.
+
+
+
+Residualization, unlike truncation and optimal Hankel norm approximation, preserves the steady-state gain of the system, and like truncation, it is simple and computationally inexpensive.
+It is observed that truncation and optimal Hankel norm approximation perform better at high frequencies, where residualization performs better at low and medium frequencies, i.e. up to the critical frequencies.
+
+Thus **for plant model reduction**, where models are not accurate at high frequencies to start with, **residualization would seem to be a better option**.
+Further, if the steady state gains are to be kept unchanged, truncated and optimal Hankel norm approximated systems require scaling, which may result in large errors.
+In such a case, too, residualization would be preferred choice.
+
+For **controller reduction**, we have shown in a two degrees-of-freedom example, the importance of scaling and steady-state gain matching.
+
+In general, steady-state gain matching may not be crucial, but the matching should usually be good near the desired closed-loop bandwidth.
+Balanced residualization has been seen to perform close to the full order system in this frequency range.
+Good approximation at high frequencies may also sometimes be desired.
+In such a case, using truncation or optimal Hankel norm approximation with appropriate frequency weightings may yield better results.
+
+# Bibliography
+Skogestad, S., & Postlethwaite, I., *Multivariable feedback control: analysis and design* (2007), : John Wiley. [↩](#ad6f62e369b7a8d31c21671886adec1f)
diff --git a/content/book/taghirad13_paral.md b/content/book/taghirad13_paral.md
new file mode 100644
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--- /dev/null
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++++
+title = "Parallel robots : mechanics and control"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Reference Books]({{< relref "reference_books" >}})
+
+Reference
+: (Taghirad, 2013)
+
+Author(s)
+: Taghirad, H.
+
+Year
+: 2013
+
+
+## Introduction {#introduction}
+
+
+
+This book is intended to give some analysis and design tools for the increase number of engineers and researchers who are interested in the design and implementation of parallel robots.
+A systematic approach is presented to analyze the kinematics, dynamics and control of parallel robots.
+To define the motion characteristics of such robots, it is necessary to represent 3D motion of the robot moving platform with respect to a fixed coordinate.
+This issue leads to the requirements for 3D representation of position, orientation and motion of bodies in space.
+In chapter [sec:motion_representation](#sec:motion_representation), such representation are introduced with emphasis on screw coordinates, which makes the representation of the general motion of the robot much easier to follow.
+
+Kinematic analysis refers to the study of robot motion geometry without considering the forces and torques that cause the motion.
+In this analysis (chapter [sec:kinematics](#sec:kinematics)), the relation between the geometrical parameters of the manipulator and the final motion of the moving platform is derived and analyzed.
+
+In Chapter [sec:jacobian](#sec:jacobian), the kinematics analysis of robot manipulators is further examined beyond static positioning.
+Jacobian analysis not only reveals the relation between the joint variable velocities of a parallel manipulator and the moving platform linear and angular velocities, but also constructs the transformation needed to find the actuator forces from the forces and moments acting on the moving platform.
+A systematic means to perform Jacobian analysis of parallel manipulators is given in this chapter.
+
+Dynamic analysis of parallel manipulators presents an inherent complexity due to their closed-loop structure and kinematic constraints.
+Nevertheless, dynamic modeling is quite important for the control, in particular because parallel manipulators are preferred in applications where precise positioning and suitable dynamic performance under high loads are the prime requirements.
+In Chapter [sec:dynamics](#sec:dynamics), the dynamic analysis of such robots is examined through three methods, namely the Newton-Euler principle of virtual work and Lagrange formations.
+Furthermore, a method is presented in this chapter to formulate the dynamic equation of parallel robots into closed form, by which the dynamic matrices are more tractable, and dynamics verification becomes feasible.
+
+The control of parallel robots is elaborated in the last two chapters, in which both the motion and force control are covered.
+
+
+## Motion Representation {#motion-representation}
+
+
+
+
+### Spatial Motion Representation {#spatial-motion-representation}
+
+Six independent parameters are sufficient to fully describe the spatial location of a rigid body.
+
+Consider a rigid body in a spatial motion as represented in Figure [fig:rigid_body_motion](#fig:rigid_body_motion).
+Let us define:
+
+- A **fixed reference coordinate system** \\((x, y, z)\\) denoted by frame \\(\\{\bm{A}\\}\\) whose origin is located at point \\(O\_A\\)
+- A **moving coordinate system** \\((u, v, z)\\) denoted by frame \\(\\{\bm{B}\\}\\) attached to the rigid body at point \\(O\_B\\)
+
+The absolute position of point \\(P\\) of the rigid body can be constructed from the relative position of that point with respect to the moving frame \\(\\{\bm{B}\\}\\), and the **position and orientation** of the moving frame \\(\\{\bm{B}\\}\\) with respect to the fixed frame \\(\\{\bm{A}\\}\\).
+
+
+
+{{< figure src="/ox-hugo/taghirad13_rigid_body_motion.png" caption="Figure 1: Representation of a rigid body spatial motion" >}}
+
+
+#### Position of a point {#position-of-a-point}
+
+The position of a point \\(P\\) with respect to a frame \\(\\{\bm{A}\\}\\) can be described by a \\(3 \times 1\\) position vector.
+The name of the frame is usually added as a leading superscript: \\({}^A\bm{P}\\) which reads as vector \\(\bm{P}\\) in frame \\(\\{\bm{A}\\}\\).
+
+\begin{equation}
+ \tcmbox{{}^A\bm{P} = \begin{bmatrix} P\_x\\ P\_y\\ P\_z \end{bmatrix}}
+\end{equation}
+
+
+#### Orientation of a Rigid Body {#orientation-of-a-rigid-body}
+
+The orientation of the whole rigid body is the same for all its points (by definition).
+Hence, representation of the orientation of a rigid body can be viewed as that for the orientation of a moving frame attached to the rigid body.
+It can be **represented in several different ways**: the rotation matrix, the screw axis representation and Euler angles are common descriptions.
+
+
+##### Rotation Matrix {#rotation-matrix}
+
+We consider a rigid body that has been exposed to a pure rotation.
+Its orientation has changed from a state represented by frame \\(\\{\bm{A}\\}\\) to its current orientation represented by frame \\(\\{\bm{B}\\}\\) (Figure [2](#org42e4742)).
+
+A \\(3 \times 3\\) rotation matrix \\({}^A\bm{R}\_B\\) is defined by
+
+\begin{equation}
+ \tcmbox{{}^A\bm{R}\_B = \left[ {}^A\hat{\bm{x}}\_B | {}^A\hat{\bm{y}}\_B | {}^A\hat{\bm{z}}\_B \right] = \begin{bmatrix}
+ u\_{x} & v\_{x} & z\_{x} \\\\\\
+ u\_{y} & v\_{y} & z\_{y} \\\\\\
+ u\_{z} & v\_{z} & z\_{z}
+ \end{bmatrix}}
+\end{equation}
+
+in which \\({}^A\hat{\bm{x}}\_B, {}^A\hat{\bm{y}}\_B\\) and \\({}^A\hat{\bm{z}}\_B\\) are the Cartesian unit vectors of frame \\(\\{\bm{B}\\}\\) represented in frame \\(\\{\bm{A}\\}\\).
+
+\begin{align\*}
+ {}^A\hat{\bm{x}}\_B &= {}^A\hat{u} = u\_x \hat{i} + u\_y \hat{j} + u\_z \hat{k} \\\\\\
+ {}^A\hat{\bm{y}}\_B &= {}^A\hat{v} = v\_x \hat{i} + v\_y \hat{j} + v\_z \hat{k} \\\\\\
+ {}^A\hat{\bm{z}}\_B &= {}^A\hat{w} = w\_x \hat{i} + w\_y \hat{j} + w\_z \hat{k}
+\end{align\*}
+
+The nine elements of the rotation matrix can be simply represented as the projections of the Cartesian unit vectors of frame \\(\\{\bm{B}\\}\\) on the unit vectors of frame \\(\\{\bm{A}\\}\\).
+
+
+
+{{< figure src="/ox-hugo/taghirad13_rotation_matrix.png" caption="Figure 2: Pure rotation of a rigid body" >}}
+
+The rotation matrix has a number of properties linking each of its nine elements:
+
+- **Orthonormality**: the rotation matrix is an orthonormal matrix
+- **Transposition**: \\({}^B\bm{R}\_A = {}^A\bm{R}\_B^{T}\\)
+- **Inverse**: \\({}^B\bm{R}\_A = {}^A\bm{R}\_B^{-1} = {}^A\bm{R}\_B^{T}\\)
+- **Pure Rotation Mapping**: Suppose that the point of a rigid body with respect to the moving frame \\(\\{\bm{B}\\}\\) is given and denoted by \\({}^B\bm{P}\\) and we wish to express the position of this point with respect to the fixed frame \\(\\{\bm{A}\\}\\). Consider that the rigid body has been exposed to a pure rotation (\\(\\{\bm{A}\\}\\) and \\(\\{\bm{B}\\}\\) are coincident at their origins). Then
+ \\[ \tcmbox{{}^A\bm{P} = {}^A\bm{R}\_B {}^B\bm{P}} \\]
+- **Determinant**: \\(\det({}^A\bm{R}\_B) = 1\\)
+- **Eigenvalues**: The eigenvalues of a rotation matrix \\({}^A\bm{R}\_B\\) are equal to \\(1\\), \\(e^{i\theta}\\) and \\(e^{-i\theta}\\) where \\(\theta\\) is calculated from \\(\theta = \cos^{-1}\frac{\text{tr}({}^A\bm{R}\_B) - 1}{2}\\).
+
+
+##### Screw Axis Representation {#screw-axis-representation}
+
+As seen previously, there exist an **invariant angle** \\(\theta\\) corresponding to the rotation matrix. This angle is an equivalent angle of rotation.
+The rotation is a spatial change of orientation about an axis which is called the **screw axis**.
+It can be shown that this screw axis is also an invariant of the rotation matrix, it is the eigenvector corresponding to the eigenvalue \\(\lambda = 1\\).
+
+The term screw axis for this axis of rotation has the benefit that a general motion of a rigid body, which is composed as a pure translation and a pure rotation, can be further represented by the same axis of rotation.
+
+The screw axis representation has the benefit of **using only four parameters** to describe a pure rotation.
+These parameters are the angle of rotation \\(\theta\\) and the axis of rotation which is a unit vector \\({}^A\hat{\bm{s}} = [s\_x, s\_y, s\_z]^T\\).
+
+
+
+{{< figure src="/ox-hugo/taghirad13_screw_axis_representation.png" caption="Figure 3: Pure rotation about a screw axis" >}}
+
+The Rodrigue's rotation formula for spatial rotation of a rigid body gives us the new position \\(\bm{P}\_2\\) of point \\(\bm{P}\_1\\) after a rotation represented by the screw axis \\(\hat{\bm{s}}\\) and the angle \\(\theta\\):
+
+\begin{equation}
+ \tcmbox{\bm{P}\_2 = \bm{P}\_1 \cos \theta + (\hat{\bm{s}} \times \bm{P}\_1)\sin\theta + (\bm{P}\_1 \cdot \hat{\bm{s}})\hat{\bm{s}}}
+\end{equation}
+
+
+##### Euler Angles {#euler-angles}
+
+Since rotation in space is a motion with three-degrees-of-freedom, a set of three independent parameters is sufficient to represent the orientation.
+
+In an Euler angle representation, three **successive** rotations about the coordinate system of either **fixed** or **moving** frame are used to describe the orientation of the rigid body.
+
+One type of Euler angle corresponds to rotations considered with respect to the fixed frame. The representation is called pitch-roll-yaw, or fixed X-Y-Z Euler angles.
+
+Three other types of Euler angles are consider with respect to a moving frame: they are denoted \\(w-v-u\\), \\(w-v-w\\) and \\(w-u-w\\) Euler angles.
+
+
+#### Pitch-Roll-Yaw Euler Angles {#pitch-roll-yaw-euler-angles}
+
+The pitch, roll and yaw angles are defined for a moving object in space as the rotations along the lateral, longitudinal and vertical axes attached to the moving object.
+
+
+
+{{< figure src="/ox-hugo/taghirad13_pitch-roll-yaw.png" caption="Figure 4: Definition of pitch, roll and yaw angles on an air plain" >}}
+
+Since all three rotations take place about the axes of a **fixed coordinate frame**, the resulting rotation matrix is obtained by multiplying the three basic rotation matrices as follows:
+\\[ \tcmbox{\bm{R}\_{PRY}(\alpha, \beta, \gamma) = \bm{R}\_z(\gamma) \bm{R}\_y(\beta) \bm{R}\_x(\alpha)} \\]
+
+To go from rotation matrix to Pitch-Roll-Yaw angles, the following set of equations can be used:
+
+\begin{align\*}
+ \alpha &= \atan2\left( \frac{r\_{32}}{\cos \beta}, \frac{r\_{33}}{\cos \beta} \right) \\\\\\
+ \beta &= \atan2\left( -r\_{31}, \pm \sqrt{r\_{11}^2 + r\_{21}^2} \right) \\\\\\
+ \gamma &= \atan2\left( \frac{r\_{21}}{\cos \beta}, \frac{r\_{11}}{\cos \beta} \right)
+\end{align\*}
+
+
+#### u-v-w Euler Angles {#u-v-w-euler-angles}
+
+Another way to describe the orientation of a moving object is to consider three successive rotations about the **coordinate axes of the moving frame**.
+Since the rotations do not occur about fixed axes, pre-multiplications of the individual rotation matrices fails to give the correct solution. It can be shown that the resulting matrix can be derived by post-multiplication of the individual rotation matrices as follows:
+\\[ {}^A\bm{R}\_B(\alpha, \beta, \gamma) = \bm{R}\_u(\alpha) \bm{R}\_v(\beta) \bm{R}\_w(\gamma) \\]
+
+The inverse solution for the u-v-w Euler angles is the following (for \\(\cos \beta \ne 0\\)):
+
+\begin{align\*}
+ \alpha &= \atan2\left( -\frac{r\_{23}}{\cos \beta}, \frac{r\_{33}}{\cos \beta} \right) \\\\\\
+ \beta &= \atan2\left( r\_{13}, \pm \sqrt{r\_{11}^2 + r\_{12}^2} \right) \\\\\\
+ \gamma &= \atan2\left( -\frac{r\_{12}}{\cos \beta}, \frac{r\_{11}}{\cos \beta} \right)
+\end{align\*}
+
+
+#### w-v-w Euler Angles {#w-v-w-euler-angles}
+
+Similarly:
+\\[ \bm{R}\_{wvw}(\alpha, \beta, \gamma) = \bm{R}\_w(\alpha) \bm{R}\_v(\beta) \bm{R}\_w(\gamma) \\]
+
+And for \\(\sin\beta\ne0\\):
+
+\begin{align\*}
+ \alpha &= \atan2\left( \frac{r\_{23}}{\sin\beta}, \frac{r\_{13}}{\sin\beta} \right) \\\\\\
+ \beta &= \atan2\left( \pm \sqrt{r\_{31}^2 + r\_{32}^2}, r\_{33} \right) \\\\\\
+ \gamma &= \atan2\left( \frac{r\_{32}}{\sin\beta}, -\frac{r\_{31}}{\sin\beta} \right)
+\end{align\*}
+
+
+#### w-u-w Euler Angles {#w-u-w-euler-angles}
+
+Here, the second rotation is about the \\(u\\) axis:
+\\[ \bm{R}\_{wuw}(\alpha, \beta, \gamma) = \bm{R}\_w(\alpha) \bm{R}\_u(\beta) \bm{R}\_w(\gamma) \\]
+
+And for \\(\sin\beta\ne0\\):
+
+\begin{align\*}
+ \alpha &= \atan2\left( \frac{r\_{13}}{\sin\beta}, -\frac{r\_{23}}{\sin\beta} \right) \\\\\\
+ \beta &= \atan2\left( \pm \sqrt{r\_{31}^2 + r\_{32}^2}, r\_{33} \right) \\\\\\
+ \gamma &= \atan2\left( \frac{r\_{31}}{\sin\beta}, \frac{r\_{32}}{\sin\beta} \right)
+\end{align\*}
+
+
+#### Notes about Euler Angles {#notes-about-euler-angles}
+
+If the Euler angle is given, a **unique rotation matrix** is determined for the orientation of the rigid body.
+However, the inverse map is not one-to-one, and at least two Euler angle sets can be found for each orientation.
+
+If the Euler angle is chosen for the representation of the orientation, extra care should be taken. From the continuity of the motion, a suitable solution may be chosen, such that no abrupt changes are seen in the variation of the Euler angles in a typical maneuver.
+
+The use of rotation matrix to represent the orientation of a rigid body is then generally preferred although there are nine parameters for that description.
+
+
+### Motion of a Rigid Body {#motion-of-a-rigid-body}
+
+Since the relative positions of a rigid body with respect to a moving frame \\(\\{\bm{B}\\}\\) attached to it is fixed for all time, it is sufficient to know the **position of the origin of the frame** \\(O\_B\\) and the **orientation of the frame** \\(\\{\bm{B}\\}\\) with respect to the fixed frame \\(\\{\bm{A}\\}\\), to represent the position of any point \\(P\\) in the space.
+
+Representation of the position of \\(O\_B\\) is uniquely given by the position vector, while orientation of the rigid body is represented in different forms.
+However, for all possible orientation representation, a rotation matrix \\({}^A\bm{R}\_B\\) can be derived.
+
+
+
+
+Therefore, the location or **pose** of a rigid body, can be **fully determined** by:
+
+1. The **position vector** of point \\(O\_B\\) with respect to frame \\(\\{\bm{A}\\}\\) which is denoted \\({}^A\bm{P}\_{O\_B}\\)
+2. The **orientation of the rigid body**, or the moving frame \\(\\{\bm{B}\\}\\) attached to it with respect to the fixed frame \\(\\{\bm{A}\\}\\), that is represented by \\({}^A\bm{R}\_B\\).
+
+
+
+
+If the pose of a rigid body \\(\\{{}^A\bm{R}\_B, {}^A\bm{P}\_{O\_B}\\}\\) is given, then the position of any point \\(P\\) of this rigid body with respect to \\(\\{\bm{A}\\}\\) is given by:
+
+\begin{equation}
+ {}^A\bm{P} = {}^A\bm{R}\_B {}^B\bm{P} + {}^A\bm{P}\_{O\_B}
+\end{equation}
+
+
+
+The first type is called **line vector**. Line vectors refer to a vector of which its value depends on the line of action, or the position of where it is applied. Examples are the position vector, linear velocity, force vector.
+
+On the contrary, there exist quantities likes orientation that **hold for the whole rigid body** and do not correspond to a point. They can be positioned freely throughout the whole rigid body, without any change in their quantity. These types of vectors are called **free vectors**.
+
+For line vectors, both orientation and translation of the moving frame contribute to their value.
+Homogeneous coordinate of such vectors is generated by appending \\(1\\) to the three components of that vector:
+
+\begin{equation}
+ \tcmbox{\bm{V} = \begin{bmatrix} v\_x \\ v\_y \\ v\_z \\ 1 \end{bmatrix}}
+\end{equation}
+
+For free vectors, only the orientation of the moving frame contributes to their value.
+The homogeneous coordinate is then
+
+\begin{equation}
+ \tcmbox{\bm{\omega} = \begin{bmatrix} \omega\_x \\ \omega\_y \\ \omega\_z \\ 0 \end{bmatrix}}
+\end{equation}
+
+
+#### Homogeneous Transformation Matrix {#homogeneous-transformation-matrix}
+
+
+
+
+The **homogeneous transformation matrix** is a \\(4\times4\\) matrix, defined for the purpose of transformation mapping of a vector in a homogeneous coordinate from one frame to another in a compact form.
+The matrix is composed of the rotation matrix \\({}^A\bm{R}\_B\\) representing the orientation and the position vector \\({}^A\bm{P}\_{O\_B}\\) representing the translation.
+It is partitioned as follows:
+
+\begin{equation}
+ {}^A\bm{T}\_B =
+ \left[ \begin{array}{ccc|c}
+ & & & \\\\\\
+ & {}^A\bm{R}\_B & & {}^A\bm{P}\_{O\_B} \\\\\\
+ & & & \\\\\\
+ \hline
+ 0 & 0 & 0 & 1 \\\\\\
+ \end{array} \right]
+\end{equation}
+
+
+
+
+Kinematic analysis refers to the study of the geometry of motion of a robot, without considering the forces an torques that cause the motion.
+In this analysis, the relation between the geometrical parameters of the manipulator with the final motion of the moving platform is derived and analyzed.
+
+
+
+If in addition to this condition, if the type and number of joints at each limb, and the number and location of the actuated joints are identical in all the limbs, such a parallel robot is called **symmetric**.
+
+There are three main cases for fully parallel manipulators.
+Planar robots with two translation and one rotational degree-of-freedom in the plane.
+Spatial orientation manipulators with three rotational degrees-of-freedom in space.
+And a general spatial robot with three translational and three rotational degrees-of-freedom in space.
+
+It is known that unlike serial manipulators, **inverse kinematic analysis of parallel robots is usually simple and straightforward**.
+In most cases, limb variable may be computed independently using the given pose of the moving platform, and the solution in most cases even for redundant manipulators is uniquely determined.
+However, **forward kinematics of parallel manipulators is generally very complicated**, and its solution usually involves systems of nonlinear equations, which are highly coupled and in general have no closed form and unique solution.
+
+
+### Loop Closure Method {#loop-closure-method}
+
+A typical parallel manipulator consists of two main bodies.
+Body \\(A\\) is arbitrary designated as fixed and is called the **base**, while body \\(B\\) is designated to be movable and is called the **moving platform**.
+These two bodies are coupled via \\(n\\) **limbs**, each attached to points \\(A\_i\\) and \\(B\_i\\) and called fixed and moving attachment points of the limb \\(i\\).
+
+At the **displacement** level, the **forward kinematic** problem permits the determination of the actual location or pose of the moving platform relative to the base from a set of joint-position readouts.
+
+At the **velocity** level, the **forward kinematic** problem refers to the determination of the translational and angular velocities of the moving platform relative to the base, from a set of joint-velocity readouts and for a known configuration.
+
+To describe the motion of the moving platform relative to the base, frame \\(\\{\bm{A}\\}\\) is attached to body \\(A\\) and frame \\(\\{\bm{B}\\}\\) to body \\(B\\).
+The pose of the moving platform relative to the base is thus defined by:
+
+- A position vector \\(\bm{p}\\) which denotes the position vector of the origin of \\(\\{\bm{B}\\}\\) with respect to frame \\(\\{\bm{A}\\}\\)
+- A \\(3\times3\\) rotation matrix \\(R\\) which denotes the rotation of \\(\\{\bm{B}\\}\\) with respect to \\(\\{\bm{A}\\}\\)
+
+Each limb of a parallel manipulator defines a kinematic loop passing through the origins of frames \\(\\{\bm{A}\\}\\) and \\(\\{\bm{B}\\}\\), and through the two limb attachment points \\(A\_i\\) and \\(B\_i\\).
+
+At the displacement level, the **closure of each kinematic loop** can be express in the vector form as
+\\[ \vec{AB} = \vec{AA\_i} + \vec{A\_iB\_i} - \vec{BB\_i} \quad \text{for } i = 1,2,\dots,n \\]
+in which \\(\vec{AA\_i}\\) and \\(\vec{BB\_i}\\) can be easily obtained from the geometry of the attachment points in the base and in the moving platform.
+
+Let us defined \\(\bm{a}\_i = \vec{AA\_i}\\) in the fixed frame \\(\\{\bm{A}\\}\\), and \\(\bm{b}\_i = \vec{BB\_i}\\) in the moving frame \\(\\{\bm{B}\\}\\).
+Furthermore, \\(\bm{q}\_i = \vec{A\_iB\_i}\\) is defined as the **limb variable**, which indicated the geometry of the limb.
+
+
+
+
+The **loop closure** can be written as the unknown pose variables \\(\bm{p}\\) and \\(\bm{R}\\), the position vectors describing the known geometry of the base and of the moving platform, \\(\bm{a}\_i\\) and \\(\bm{b}\_i\\), and the limb vector \\(\bm{q}\_i\\)
+
+\begin{equation}
+ \bm{p} = \bm{a}\_i + \bm{q}\_i - \bm{R} \bm{b}\_i \quad \text{for } i=1,2,\dots,n
+\end{equation}
+
+
+
+
+Since all the limbs are connected to the moving platform and to the base by spherical joints, **no twisting torque** can be transmitted and the force acting on each limb is directed along the longitudinal axis of the limb.
+
+
+
+
+The position of the attachment points on the fixed base are denoted by the vectors \\(\bm{a}\_i\\) and the position of moving attachment points are denoted by the vectors \\(\bm{b}\_i\\).
+The geometry of each limb is described by its length \\(l\_i\\) and its direction is denoted by a unit vector \\(\hat{\bm{s}}\_i\\).
+
+
+
+
+For **inverse kinematic analysis**, it is assumed that the position \\({}^A\bm{P}\\) and orientation of the moving platform \\({}^A\bm{R}\_B\\) are given and the problem is to obtain the joint variables \\(\bm{L} = \left[ l\_1, l\_2, l\_3, l\_4, l\_5, l\_6 \right]^T\\).
+
+
+
+
+In **forward kinematic analysis**, it is assumed that the vector of limb lengths \\(\bm{L}\\) is given and the problem is to find the position \\({}^A\bm{P}\\) and the orientation \\({}^A\bm{R}\_B\\).
+
+
+
+
+The Jacobian matrix not only reveals the **relation between the joint variable velocities of a parallel manipulator to the moving platform linear and angular velocities**, it also constructs the transformation needed to find the **actuator forces from the forces and moments acting on the moving platform**.
+
+
+
+
+The **angular velocity vector** \\(\bm{\Omega}\\) describes the instantaneous rotation of frame \\(\\{\bm{B}\\}\\) with respect to the fixed frame \\(\\{\bm{A}\\}\\).
+The direction of \\(\bm{\Omega}\\) indicates the instantaneous axis of rotation and its magnitude indicates the speed of rotation.
+
+
+
+
+\begin{equation}
+ {}^A\bm{v}\_P = {}^A\bm{v}\_{O\_B} + {}^A\bm{\Omega}^\times \ {}^A\bm{R}\_B {}^B\bm{P}
+\end{equation}
+
+
+
+
+The most general rigid-body displacement can be produced by a translation along a line followed by a rotation about the same line. Since this displacement is reminiscent of the displacement of a screw, it is called a **screw displacement**, and the line of axis is called the **screw axis**.
+
+
+
+\\(\bm{q}\\) and \\(\bm{\mathcal{X}}\\) are related through a system of **nonlinear algebraic equations** representing the **kinematic constraints imposed by the limbs**, which can be generally written as
+
+\begin{equation}
+ f(\bm{q}, \bm{\mathcal{X}}) = 0
+\end{equation}
+
+We can differentiate this equation with respect to time and obtain:
+
+\begin{equation}
+ \tcmbox{\bm{J}\_x \dot{\bm{\mathcal{X}}} = \bm{J}\_q \dot{\bm{q}}}
+\end{equation}
+
+where
+
+\begin{equation}
+ \bm{J}\_x = \pdv{f}{\bm{\mathcal{X}}} \quad \text{and} \quad \bm{J}\_q = -\pdv{f}{\bm{q}}
+\end{equation}
+
+
+
+
+The **general Jacobian matrix** is defined as:
+
+\begin{equation}
+ \dot{\bm{q}} = \bm{J} \dot{\bm{\mathcal{X}}}
+\end{equation}
+
+From equation [eq:jacobians](#eq:jacobians), we have:
+
+\begin{equation}
+ \bm{J} = {\bm{J}\_q}^{-1} \bm{J}\_x
+\end{equation}
+
+
+
+Kinematic loop closures are:
+\\[ \bm{p} = \bm{a}\_i + \bm{d}\_i - \bm{R} \bm{b}\_i \quad \text{for}\ i = 1, \ldots, m \\]
+
+with
+
+- \\(\bm{p}\\) the position vector of the moving platform w.r.t. frame \\(\\{\bm{A}\\}\\)
+- \\(\bm{R}\\) the rotation matrix of the moving platform
+- \\(\bm{a}\_i\\) the position vector of the \\(i\\)'th limb of the fixed platform w.r.t. frame \\(\\{\bm{A}\\}\\)
+- \\(\bm{b}\_i\\) the position vector of the \\(i\\)'th limb of the moving platform w.r.t. frame \\(\\{\bm{B}\\}\\)
+- \\(\bm{d}\_i\\) the limb vector
+
+By taking the time derivative, we obtain the following **Velocity Loop Closure**:
+
+\begin{equation}
+ \tcmbox{\dot{\bm{p}} = \dot{\bm{d}\_i} - \bm{\omega} \times \bm{R} \bm{b}\_i \quad \text{for}\ i = 1, \ldots, m}
+\end{equation}
+
+
+### Singularity Analysis of Parallel Manipulators {#singularity-analysis-of-parallel-manipulators}
+
+The singularities occur when:
+
+- \\(\bm{J}\_q\\) is rank deficient (Inverse kinematic singularity)
+- \\(\bm{J}\_x\\) is rank deficient (Forward kinematic singularity)
+
+
+#### Inverse Kinematic Singularity {#inverse-kinematic-singularity}
+
+Inverse kinematic singularity happens when \\(\bm{J}\_q\\) (\\(m \times m\\) matrix) is rank deficient (\\(\det \bm{J}\_q = 0\\)).
+
+The corresponding configurations are located at the boundary of the manipulator workspace or on the internal boundaries between sub-regions of the workspace.
+
+In such cases, there exist nonzero vectors \\(\dot{\bm{q}}\\) which correspond to a null Cartesian twist vector \\(\dot{\bm{\mathcal{X}}}\\). In other words, infinitesimal motion of the moving platform along certain directions cannot be accomplished.
+**The manipulator looses one or more degrees-of-freedom**.
+
+
+#### Forward Kinematic Singularity {#forward-kinematic-singularity}
+
+Forward kinematic singularity happens when \\(\bm{J}\_x\\) (\\(m \times n\\) matrix) is rank deficient (\\(\det({\bm{J}\_x}^T \bm{J}\_x) = 0\\)).
+If the manipulator is not redundantly actuated (\\(m=n\\)), then the Jacobian matrix \\(\bm{J}\_x\\) is square and the forward kinematic singularity happens when \\(\det \bm{J}\_x = 0\\).
+
+The degeneracy occur inside the manipulator's Cartesian workspace and corresponds to the set of configurations for which two different branches of forward kinematic problem meet.
+
+There exist nonzero cartesian twist vectors \\(\dot{\bm{\mathcal{X}}}\\) that are mapped into a vanishing actuator velocity vector \\(\dot{\bm{\mathcal{q}}}\\).
+
+The corresponding configuration will be one in which an infinitesimal motion of the platform is possible even if the actuator are locked. **The manipulator gains one or several degrees-of-freedom and its stiffness vanishes in the corresponding direction(s)**.
+
+
+### Jacobian Analysis of the Stewart-Gough Platform {#jacobian-analysis-of-the-stewart-gough-platform}
+
+
+#### Velocity Loop Closure {#velocity-loop-closure}
+
+The input joint rate is denoted by \\(\dot{\bm{\mathcal{L}}} = [ \dot{l}\_1, \dot{l}\_2, \dot{l}\_3, \dot{l}\_4, \dot{l}\_5, \dot{l}\_6 ]^T\\), and the output twist vector is denoted by \\(\dot{\bm{\mathcal{X}}} = [^A\bm{v}\_p, {}^A\bm{\omega}]^T\\).
+
+The jacobian matrix can be derived by formulating a velocity loop closure equation of each limb.
+The loop closure equations for each limb are:
+
+\begin{equation}
+ {}^A\bm{P} + {}^A\bm{R}\_B {}^B\bm{b}\_i = l\_i {}^A\hat{\bm{s}}\_i + {}^A\bm{a}\_i
+\end{equation}
+
+By differentiate this with respect to time:
+
+\begin{equation}
+ {}^A\bm{v}\_p + {}^A \dot{\bm{R}}\_B {}^B\bm{b}\_i = \dot{l}\_i {}^A\hat{\bm{s}}\_i + l\_i {}^A\dot{\hat{\bm{s}}}\_i
+\end{equation}
+
+Moreover, we have:
+
+- \\({}^A\dot{\bm{R}}\_B {}^B\bm{b}\_i = {}^A\bm{\omega} \times {}^A\bm{R}\_B {}^B\bm{b}\_i = {}^A\bm{\omega} \times {}^A\bm{b}\_i\\) in which \\({}^A\bm{\omega}\\) denotes the angular velocity of the moving platform expressed in the fixed frame \\(\\{\bm{A}\\}\\).
+- \\(l\_i {}^A\dot{\hat{\bm{s}}}\_i = l\_i \left( {}^A\bm{\omega}\_i \times \hat{\bm{s}}\_i \right)\\) in which \\({}^A\bm{\omega}\_i\\) is the angular velocity of limb \\(i\\) express in fixed frame \\(\\{\bm{A}\\}\\).
+
+Then, the velocity loop closure [eq:loop_closure_limb_diff](#eq:loop_closure_limb_diff) simplifies to
+\\[ {}^A\bm{v}\_p + {}^A\bm{\omega} \times {}^A\bm{b}\_i = \dot{l}\_i {}^A\hat{\bm{s}}\_i + l\_i ({}^A\bm{\omega}\_i \times \hat{\bm{s}}\_i) \\]
+
+By dot multiply both side of the equation by \\(\hat{\bm{s}}\_i\\):
+\\[ \hat{\bm{s}}\_i {}^A\bm{v}\_p + ({}^A\bm{b}\_i \times \hat{\bm{s}}\_i) {}^A\bm{\omega} = \dot{l}\_i \\]
+
+We then omit the superscript \\(A\\) and we can rearrange the 6 equations into a matrix form
+
+\begin{equation}
+ \tcmbox{\dot{\bm{\mathcal{L}}} = \bm{J} \dot{\bm{\mathcal{X}}}}
+\end{equation}
+
+
+
+
+\begin{equation}
+ \bm{J} = \begin{bmatrix}
+ {\hat{\bm{s}}\_1}^T & (\bm{b}\_1 \times \hat{\bm{s}}\_1)^T \\\\\\
+ {\hat{\bm{s}}\_2}^T & (\bm{b}\_2 \times \hat{\bm{s}}\_2)^T \\\\\\
+ {\hat{\bm{s}}\_3}^T & (\bm{b}\_3 \times \hat{\bm{s}}\_3)^T \\\\\\
+ {\hat{\bm{s}}\_4}^T & (\bm{b}\_4 \times \hat{\bm{s}}\_4)^T \\\\\\
+ {\hat{\bm{s}}\_5}^T & (\bm{b}\_5 \times \hat{\bm{s}}\_5)^T \\\\\\
+ {\hat{\bm{s}}\_6}^T & (\bm{b}\_6 \times \hat{\bm{s}}\_6)^T
+ \end{bmatrix}
+\end{equation}
+
+\\(\bm{J}\\) then **depends only** on:
+
+- \\(\hat{\bm{s}}\_i\\) the orientation of the limbs
+- \\(\bm{b}\_i\\) the position of the joints with respect to \\(O\_B\\) and express in \\(\\{\bm{A}\\}\\).
+
+
+
+From equation [eq:jacobians](#eq:jacobians), it is clear that for the Stewart-Gough platform, \\(\bm{J}\_q = \bm{I}\\) and \\(\bm{J}\_x = \bm{J}\\).
+Hence the manipulator has **no inverse kinematic singularities** within the manipulator workspace, but **may possess forward kinematic singularity** when \\(\bm{J}\\) becomes rank deficient. This may occur when
+\\[ \det \bm{J} = 0 \\]
+
+
+### Static Forces in Parallel Manipulators {#static-forces-in-parallel-manipulators}
+
+The relation between the forces/moments applied to the environment at the point of contact and the actuator forces/torques is determined and analyzed in the study of static force analysis.
+
+It is assumed that the manipulator is at a static equilibrium, and that the actuator forces required to produce the desired contact forces are determined.
+
+Two methods are usually applied: the **free-body diagram** and the **principle of virtual work**.
+
+In the **free-body diagram approach**, the actuator forces are determined to produce desired contact forces/moments as well as the internal and interacting forces/torques applied at the limbs. The analysis of such forces is essential in the design of a manipulator to determine the stresses and deflection of each link and joint.
+
+However, if only the actuator forces are desired to be determined, the **principle of virtual work** is more efficient and computationally less expensive.
+
+
+#### Virtual Work Approach {#virtual-work-approach}
+
+A virtual displacement for a parallel manipulator refers to an infinitesimal change in the general displacement of the moving platform as a result of any arbitrary infinitesimal changes in the joint variables at a given instant of time.
+
+The virtual displacement of the joints can be written as \\(\delta \bm{q} = [\delta q\_1, \delta q\_2, \cdots, \delta q\_m]^T\\) and \\(\delta \bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta\_x, \delta \theta\_y, \delta \theta\_z ]^T\\) denotes the virtual displacement of a contacting point of the moving platform.
+\\([\delta \theta\_x, \delta \theta\_y, \delta \theta\_z ]^T = \delta \theta \hat{\bm{s}}\\) are the orientation variables represented by screw coordinates.
+
+Let the vector of actuator forces be denoted by \\(\bm{\tau} = [\tau\_1, \tau\_2, \cdots, \tau\_m]^T\\), and the external forces/torque acting on the contact point of the moving platform denoted by a wrench in a screw coordinate as \\(\bm{\mathcal{F}} = [\bm{f}, \bm{n}]^T\\) in which \\(\bm{f} = [f\_x, f\_y, f\_z]^T\\) denotes the external forces, and \\(\bm{n} = [n\_x, n\_y, n\_z]^T\\) denotes the external torque action on the moving platform at the point of contact to the environment.
+
+We assume that the frictional forces acting on the joints are negligible, and also that the gravitational forces of the limb links are much smaller than the interacting force of the moving platform to the environment.
+The principle of virtual work states that the total virtual work, \\(\delta W\\), done by all actuators and external forces is equal to zero:
+
+\begin{equation}
+ \tcmbox{\delta W = \bm{\tau}^T \delta \bm{q} - \bm{\mathcal{F}}^T \delta \bm{\mathcal{X}} = 0}
+\end{equation}
+
+Furthermore, from the definition of the Jacobian, the virtual displacements \\(\delta \bm{q}\\) and \\(\delta \bm{\mathcal{X}}\\) are related by the Jacobian:
+\\[ \delta \bm{q} = \bm{J} \cdot \delta \bm{\mathcal{X}} \\]
+
+We then have \\(\left( \bm{\tau}^T \bm{J} - \bm{\mathcal{F}}^T \right) \delta \bm{\mathcal{X}} = 0\\) that holds for any arbitrary virtual displacement \\(\delta \bm{\mathcal{X}}\\), hence
+\\[ \bm{\tau}^T \bm{J} - \bm{\mathcal{F}}^T = 0 \\]
+
+
+
+
+We obtain that the **Jacobian matrix** constructs the **transformation needed to find the actuator forces** \\(\bm{\tau}\\) **from the wrench acting on the moving platform** \\(\bm{\mathcal{F}}\\):
+
+\begin{equation}
+ \bm{\mathcal{F}} = \bm{J}^T \bm{\tau}
+\end{equation}
+
+
+
+
+
+{{< figure src="/ox-hugo/taghirad13_stewart_static_forces.png" caption="Figure 8: Free-body diagram of forces and moments action on the moving platform and each limb of the Stewart-Gough platform" >}}
+
+Consider an external wrench generated by the manipulator and applied to the environment at the point \\(O\_B\\) denoted by \\(\bm{\mathcal{F}} = [\bm{f} \ \bm{n}]^T\\).
+
+It is assumed that no external forces are applied to the limbs except the actuator forces.
+Therefore, the static force can be assumed to be **along the limb axis** \\(\hat{\bm{s}}\_i\\), and the limb is subject to a tension/compression force \\(f\_i\\).
+
+At static equilibrium, the summation of all acting forces on the moving platform shall be zero, therefore
+
+\begin{equation}
+ -\bm{f} + \sum\_{i=1}^6 f\_i \hat{\bm{s}}\_i = 0
+\end{equation}
+
+in which \\(-\bm{f}\\) if the **external force** applied to the moving platform from the environment.
+
+The summation of moments contributed by all forces acting on the moving platform about \\(O\_B\\) is as follows:
+
+\begin{equation}
+ -\bm{n} + \sum\_{i=1}^6 b\_i \times f\_i \hat{\bm{s}}\_i = 0
+\end{equation}
+
+in which \\(-n\\) is the **external moment** applied to the moving platform by the environment, and \\(b\_i\\) is the position vector from the point \\(O\_B\\) to the attached point \\(B\_i\\) on the moving platform.
+
+Writing the two equations together in a matrix form results in
+
+\begin{equation}
+ \begin{bmatrix}
+ \hat{\bm{s}}\_1 & \hat{\bm{s}}\_2 & \cdots & \hat{\bm{s}}\_6 \\\\\\
+ \bm{b}\_1 \times \hat{\bm{s}}\_1 & \bm{b}\_2 \times \hat{\bm{s}}\_2 & \cdots & \bm{b}\_6 \times \hat{\bm{s}}\_6
+ \end{bmatrix} \cdot \begin{bmatrix}
+ f\_1 \\ f\_2 \\ \vdots \\ f\_6
+ \end{bmatrix} = \begin{bmatrix}
+ \bm{f} \\ \bm{n}
+ \end{bmatrix}
+\end{equation}
+
+There we can recognize the transpose of the Jacobian matrix:
+
+\begin{equation}
+ \tcmbox{\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}}
+\end{equation}
+
+in which \\(\bm{\tau} = [f\_1, f\_2, \cdots, f\_6]^T\\) is the vector of actuator forces, and \\(\bm{\mathcal{F}} = [\bm{f}, \bm{n}]^T\\) is the 6D wrench applied by the manipulator to the environment.
+
+
+### Stiffness Analysis of Parallel Manipulators {#stiffness-analysis-of-parallel-manipulators}
+
+Here, we focus on the deflections of the manipulator moving platform that are the result of the applied wrench to the environment.
+The amount of these deflections are a function of the applied wrench as well as the manipulator **structural stiffness**.
+Thus, the stiffness of a manipulator has a direct impact on its overall positioning accuracy if the manipulator is in contact with a stiff environment.
+
+
+#### Stiffness and Compliance Matrices {#stiffness-and-compliance-matrices}
+
+The relation between the applied actuator force \\(\tau\_i\\) and the corresponding small deflection \\(\Delta q\_i\\) along the applied force axis can be approximated as a **linear function**:
+
+\begin{equation}
+ \tcmbox{\tau\_i = k\_i \cdot \Delta q\_i}
+\end{equation}
+
+in which \\(k\_i\\) denotes the **stiffness constant of the actuator**.
+
+Re-writing the equation [eq:stiffness_actuator](#eq:stiffness_actuator) for all limbs in a matrix form result in
+
+\begin{equation}
+ \tcmbox{\bm{\tau} = \mathcal{K} \cdot \Delta \bm{q}}
+\end{equation}
+
+in which \\(\bm{\tau}\\) is the vector of actuator forces, and \\(\Delta \bm{q}\\) corresponds to the actuator deflections.
+\\(\mathcal{K} = \text{diag}\left[ k\_1 \ k\_2 \dots k\_m \right]\\) is an \\(m \times m\\) diagonal matrix composed of the actuator stiffness constants.
+
+Writing the Jacobian relation given in equation [eq:jacobian_disp](#eq:jacobian_disp) for infinitesimal deflection read
+
+\begin{equation}
+ \Delta \bm{q} = \bm{J} \cdot \Delta \bm{\mathcal{X}}
+\end{equation}
+
+in which \\(\Delta \bm{\mathcal{X}} = [\Delta x\ \Delta y\ \Delta z\ \Delta\theta x\ \Delta\theta y\ \Delta\theta z]\\) is the infinitesimal linear and angular deflection of the moving platform.
+
+Furthermore, rewriting the Jacobian as the projection of actuator forces to the moving platform [eq:jacobian_forces](#eq:jacobian_forces) gives
+
+\begin{equation}
+ \bm{\mathcal{F}} = \bm{J}^T \bm{\tau}
+\end{equation}
+
+Hence, by substituting [eq:stiffness_matrix_relation](#eq:stiffness_matrix_relation) and [eq:jacobian_disp_inf](#eq:jacobian_disp_inf) in [eq:jacobian_force_inf](#eq:jacobian_force_inf), we obtain:
+
+\begin{equation}
+ \tcmbox{\bm{\mathcal{F}} = \underbrace{\bm{J}^T \mathcal{K} \bm{J}}\_{\bm{K}} \cdot \Delta \bm{\mathcal{X}}}
+\end{equation}
+
+Equation [eq:stiffness_jacobian](#eq:stiffness_jacobian) implies that the moving platform output wrench is related to its deflection by the **stiffness matrix** \\(K\\).
+
+
+
+
+\begin{equation}
+ \bm{K} = \bm{J}^T \mathcal{K} \bm{J}
+\end{equation}
+
+
+
+One way to **characterize these transformation** is to compare the amplitude and direction of the moving platform velocity generated by a **unit** actuator joint velocity.
+To achieve this goal, we confine the actuator joint velocity vector on a m-dimensional unit sphere
+\\[ \dot{\bm{q}}^T \dot{\bm{q}} = 1 \\]
+and compare the resulting moving platform velocity in n-dimensional space:
+\\[ \dot{\bm{\mathcal{X}}}^T \bm{J}^T \bm{J} \dot{\bm{\mathcal{X}}} = 1 \\]
+
+Similarly, we can confine the exerted moving platform wrench \\(\bm{\mathcal{F}}^T \bm{\mathcal{F}} = 1\\) and compare the required actuator forces: \\(\bm{\tau}^T \bm{J} \bm{J}^T \bm{\tau} = 1\\).
+
+Consider the case of fully parallel manipulators.
+Then \\(\bm{J}\bm{J}^T\\) and \\(\bm{J}^T\bm{J}\\) transformations are represented by \\(n \times n\\) matrices.
+Geometrically, these transformations represent a **hyper-ellipsoid** in n-dimensional space, whose principal axes are the **eigenvectors** of \\(\bm{J}^T\bm{J}\\) and \\(\bm{J}\bm{J}^T\\) respectively.
+Furthermore, the lengths of the principal axes are equal to the reciprocals of the square roots of the eigenvalues of \\(\bm{J}\bm{J}^T\\) and \\(\bm{J}^T\bm{J}\\), which are also equal to the reciprocals of the **singular values** of \\(\bm{J}\\).
+
+The shape of this hyper-ellipsoid in space indicates the characteristics of the transformation.
+As this hyper-ellipsoid is closer to a hyper-sphere, the transformation becomes more uniform in different directions.
+Since Jacobian matrix is **configuration dependent**, the shape of the hyper-ellipsoid is also configuration dependent, and as the moving platform moves from one pose to the other, the shape of the hyper-ellipsoid changes accordingly.
+
+A measure of the dexterity of the manipulator is the reciprocal of the Jacobian matrix condition number defined as:
+
+\begin{equation}
+ \frac{1}{\kappa} = \frac{\sigma\_{\text{min}}}{\sigma\_{\text{max}}}
+\end{equation}
+
+in which \\(\sigma\_{\text{min}}\\) and \\(\sigma\_{\text{max}}\\) are the smallest and the largest singular values of the Jacobian matrix.
+
+
+#### Stiffness Analysis of the Stewart-Gough Platform {#stiffness-analysis-of-the-stewart-gough-platform}
+
+In this section, we restrict our analysis to a 3-6 structure (Figure [fig:stewart36](#fig:stewart36)) in which there exist six distinct attachment points \\(A\_i\\) on the fixed base and three moving attachment point \\(B\_i\\).
+
+
+
+{{< figure src="/ox-hugo/taghirad13_stewart36.png" caption="Figure 9: Schematic of a 3-6 Stewart-Gough platform" >}}
+
+Denote the vector of actuated joint forces by \\(\bm{\tau} = [f\_1 \ f\_2 \ f\_3 \ f\_4 \ f\_5 \ f\_6 ]\\), and the corresponding vector of infinitesimal displacements of actuated limbs denoted by \\(\Delta \bm{L} = [\Delta l\_1 \ \Delta l\_2 \ \Delta l\_3 \ \Delta l\_4 \ \Delta l\_5 \ \Delta l\_6 ]\\).
+The relation between \\(\Delta \bm{L}\\) and \\(\bm{\tau}\\) is described by a diagonal \\(6 \times 6\\) matrix \\(\mathcal{K}\\):
+\\[ \bm{\tau} = \mathcal{K} \cdot \Delta \bm{L} \\]
+
+Also, from the definition of the Jacobian, we have:
+\\[ \Delta \bm{L} = \bm{J} \cdot \Delta \bm{\mathcal{X}} \\]
+in which \\(\Delta \bm{\mathcal{X}} = [\Delta\_x \ \Delta\_y \ \Delta\_z \ \Delta \theta\_x \ \Delta \theta\_y \ \Delta \theta\_z ]\\) is the vector of infinitesimal linear and angular motions of the moving platform.
+
+Also, the vector of the moving platform output wrench denoted by \\(\bm{F} = [f\_x \ f\_y \ f\_z \ n\_x \ n\_y \ n\_z ]\\) is related to the vector of actuated joint forces \\(\bm{\tau}\\) by:
+\\[ \bm{F} = \bm{J}^T \cdot \bm{\tau} \\]
+
+By substitution, we obtain:
+\\[ \bm{\mathcal{F}} = \bm{K} \cdot \Delta \bm{\mathcal{X}} \\]
+in which
+\\[ \bm{K} = \bm{J}^T \mathcal{K} \bm{J} \\]
+where \\(K\\) is called the **stiffness matrix** of the Stewart-Gough manipulator.
+
+For a given configuration of the moving platform, the eigenvalue of the stiffness matrix represents the stiffness of the manipulator in the corresponding eigenvector direction.
+Furthermore, the reciprocal of the stiffness matrix condition number may be used to represent the dexterity of the manipulator.
+
+The maximum stiffness of the manipulator can be analyzed by the maximum singular value of the Jacobian matrix.
+The largest axis of the stiffness transformation hyper-ellipsoid is given by this value at each configuration.
+
+
+## Dynamics {#dynamics}
+
+
+
+
+### Introduction {#introduction}
+
+The dynamic analysis of parallel manipulators presents an inherent complexity due to their closed-loop structure.
+Several approaches have been proposed.
+
+Traditional **Newton-Euler** formulation is used for dynamic analysis of general parallel manipulators.
+In this formulation, the equation of motion of each limb and the moving platform must be derived, which inevitably leads to a large number of equations and less computational efficiency.
+On the other hand, all the reaction forces can be computed, which is very useful in the design of a parallel manipulator.
+
+The **Lagrangian** formulation eliminates all the unwanted reaction forces at the outset, and therefore, is quite efficient.
+However, because of the constraints imposed by the closed-loop structure, deriving explicit equations of motions in terms of a set of generalized coordinates becomes a prohibitive task.
+
+A third approach is to use the **principle of virtual work**, in which the computation of the constraint forces are bypassed.
+In this method, inertial forces and moments are computed using linear and angular accelerations of the bodies.
+Then, the whole manipulator is considered to be in static equilibrium by using the d'Alembert's principle, and the principle of virtual work is applied to derive the input forces and torques.
+
+Different objectives require different forms of formulations, there are three key issues pursued to derive dynamic formulation of parallel manipulators:
+
+1. **Calculation of internal forces** either active or passive for the design process of the manipulator
+2. **Study on dynamical properties** of the manipulator for controller design
+3. Utilization of dynamic specifications in an inverse dynamics controller or any **model-based control topology**
+
+The first item is the main advantage of the Newton-Euler formulation, the second and third items are the benefits of using the Lagrange or virtual work approaches.
+
+The dynamic equations in an **explicit form** can be written as:
+
+\begin{equation}
+ \tcmbox{\bm{M}(\bm{\mathcal{X}}) \ddot{\bm{\mathcal{X}}} + \bm{C}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}}) \dot{\bm{\mathcal{X}}} + \bm{G}(\bm{\mathcal{X}}) = \bm{\mathcal{F}}}
+\end{equation}
+
+in which:
+
+- \\(\bm{\mathcal{X}}\\) is a vector of the generalized coordinates
+- \\(\bm{M}(\bm{\mathcal{X}})\\) denotes the system **mass matrix**
+- \\(\bm{C}(\bm{X},\dot{\bm{X}})\\) denotes the **Coriolis and centrifugal matrix**
+- \\(\bm{G}(\bm{X})\\) denotes the **gravity vector**
+- \\(\bm{\mathcal{F}}\\) denotes the generalized force
+
+Deriving explicit dynamic equations for parallel manipulators is a very prohibitive task because of the closed-loop nature of the manipulator.
+
+
+### Dynamics of the Rigid Bodies {#dynamics-of-the-rigid-bodies}
+
+
+#### Acceleration of Rigid Bodies {#acceleration-of-rigid-bodies}
+
+
+
+
+The acceleration analysis consists of studying the variations of linear velocity of a point and angular velocity of a rigid body with respect to time.
+
+
+
+
+**Angular acceleration vector**, denoted by the symbol \\(\dot{\bm{\Omega}}\\), describes the instantaneous change of the angular velocity of frame \\(\\{\bm{B}\\}\\), denoted by \\(\bm{\Omega}\\), with respect to the fixed frame \\(\\{\bm{A}\\}\\):
+
+\begin{equation}
+ \begin{aligned}
+ \dot{\bm{\Omega}} = \frac{d \bm{\Omega}}{dt} &= \ddot{\theta} \hat{\bm{s}} + \dot{\theta} \dot{\hat{\bm{s}}} \\\\\\
+ &= \ddot{\theta} \hat{\bm{s}} + \dot{\theta} (\bm{\Omega} \times \hat{\bm{s}}) \\\\\\
+ &= \ddot{\theta} \hat{\bm{s}}
+ \end{aligned}
+\end{equation}
+
+where \\(\\{\theta, \hat{\bm{s}}\\}\\) are the screw parameters representing the rotation of the rigid body.
+
+
+
+Now consider the general motion of a rigid body, in which a moving frame \\(\\{\bm{B}\\}\\) is attached to the rigid body and the problem is to find the absolute acceleration of point \\(P\\) with respect to the fixed frame \\(\\{\bm{A}\\}\\).
+The rigid body performs a general motion, which is a combination of a translation, denoted by the velocity vector \\({}^A\bm{v}\_{O\_B}\\), and an instantaneous angular rotation denoted by \\(\bm{\Omega}\\) (see Figure [fig:general_motion](#fig:general_motion)).
+To determine acceleration of point \\(P\\), we start with the relation between absolute and relative velocities of point \\(P\\):
+
+\begin{equation}
+ {}^A\bm{v}\_P = {}^A\bm{v}\_{O\_B} + {}^A\bm{R}\_B{}^B\bm{v}\_P + {}^A\bm{\Omega}^\times {}^A\bm{R}\_B {}^B\bm{P}
+\end{equation}
+
+In order to derive acceleration of point \\(P\\), we differentiate both sides with respect to time and we obtain
+
+\begin{equation}
+ \begin{split}
+ {}^A\bm{a}\_p\ &= {}^A\bm{a}\_{O\_B} \quad \text{(linear acc. of } \\{\bm{B}\\} \text{)} \\\\\\
+ &+ {}^A\bm{R}\_B{}^B\bm{a}\_p \quad \text{(relative acc. of } P \text{ w.r.t. } \\{\bm{B}\\} \text{)} \\\\\\
+ &+ {}^A\dot{\bm{\Omega}}^\times {}^A\bm{R}\_B {}^B\bm{P} \quad \text{(angular acceleration of } \\{\bm{B}\\} \text{)} \\\\\\
+ &+ {}^A\bm{\Omega}^\times ({}^A\bm{\Omega}^\times {}^A\bm{R}\_B {}^B\bm{P}) \quad \text{(centrifugal)} \\\\\\
+ &+ 2{}^A\bm{\Omega}^\times {}^A\bm{R}\_B {}^B\bm{v}\_P \quad \text{(Coriolis)}
+ \end{split}
+\end{equation}
+
+For the case where \\(P\\) is a point embedded in the rigid body, \\({}^B\bm{v}\_P = 0\\) and \\({}^B\bm{a}\_P = 0\\) and we obtain:
+
+\begin{equation}
+ \begin{split}
+ {}^A\bm{a}\_P = {}^A\bm{a}\_{O\_B} &+ {}^A\dot{\bm{\Omega}}^\times {}^A\bm{R}\_B {}^B\bm{P} \\\\\\
+ &+ {}^A\bm{\Omega}^\times ({}^A\bm{\Omega}^\times {}^A\bm{R}\_B {}^B\bm{P})
+ \end{split}
+\end{equation}
+
+
+#### Mass Properties {#mass-properties}
+
+In this section, the properties of mass, namely **center of mass**, **moments of inertia** and its characteristics and the required transformations are described.
+
+
+
+{{< figure src="/ox-hugo/taghirad13_mass_property_rigid_body.png" caption="Figure 10: Mass properties of a rigid body" >}}
+
+
+##### Center of Mass {#center-of-mass}
+
+Consider a reference frame \\(\\{\bm{A}\\}\\) in which the mass distribution of a material body is measured, and let \\(\bm{p}\\) denote the position vector of a differential mass \\(\rho dV\\) with respect to a reference frame.
+
+The **center of mass** of a rigid body is defined as the point \\(C\\) which satisfied the following condition
+
+\begin{equation}
+ \bm{p}\_c = \frac{1}{m} \int\_V \bm{p} \rho dV
+\end{equation}
+
+in which the mass of the material body \\(\\{\bm{B}\\}\\) with density \\(\rho\\) and volume \\(V\\) is defined as
+
+\begin{equation}
+ m = \int\_V \rho dV
+\end{equation}
+
+
+##### Moments of Inertia {#moments-of-inertia}
+
+As opposed to the mass, which introduces inertia to linear accelerations, moment of inertia is the property of mass which introduces **inertia to angular accelerations**.
+Basically, for rotational motion, **the distribution of mass with respect to the axis of rotation introduces resistance to the angular acceleration**.
+
+Moments of inertia \\(\bm{I}\\) about \\(\bm{A}\\) is defined by the second moment of the mass with respect to a reference frame of rotation as:
+
+\begin{equation}
+ {}^A\bm{I} = \begin{bmatrix}
+ I\_{XX} & I\_{XY} & I\_{XZ} \\\\\\
+ I\_{YX} & I\_{YY} & I\_{YZ} \\\\\\
+ I\_{ZX} & I\_{ZY} & I\_{ZZ}
+ \end{bmatrix}
+\end{equation}
+
+in which
+
+\begin{equation\*}
+ \begin{aligned}
+ I\_{XX} &= \int\_V (y^2 + z^2) \rho dV, \quad I\_{XY} = I\_{YX} = -\int\_V xy \rho dV \\\\\\
+ I\_{YY} &= \int\_V (x^2 + z^2) \rho dV, \quad I\_{YZ} = I\_{ZY} = -\int\_V yz \rho dV \\\\\\
+ I\_{ZZ} &= \int\_V (x^2 + y^2) \rho dV, \quad I\_{XZ} = I\_{ZX} = -\int\_V xz \rho dV
+ \end{aligned}
+\end{equation\*}
+
+
+##### Principal Axes {#principal-axes}
+
+As seen in equation [eq:moment_inertia](#eq:moment_inertia), the inertia matrix elements are a function of mass distribution of the rigid body with respect to the frame \\(\\{\bm{A}\\}\\).
+Hence, it is possible to find **orientations of frame** \\(\\{\bm{A}\\}\\) in which the product of inertia terms vanish and inertia matrix becomes **diagonal**:
+
+\begin{equation}
+ {}^A\bm{I} = \begin{bmatrix}
+ I\_{XX} & 0 & 0 \\\\\\
+ 0 & I\_{YY} & 0 \\\\\\
+ 0 & 0 & I\_{ZZ}
+ \end{bmatrix}
+\end{equation}
+
+Such axes are called the **principal axes of inertia**, and diagonal terms are called the **principal moments of inertia**, which represent the maximum, minimum and intermediate values of the moments of inertia for a particular chosen origin \\(\\{\bm{A}\\}\\).
+
+It can be shown that the principal moments of inertial and principal axes are invariant parameters and can be determined from an eigen value decomposition of the inertia matrix in any configuration of the reference frame \\(\\{\bm{A}\\}\\).
+
+
+##### Inertia Matrix Transformations {#inertia-matrix-transformations}
+
+The moment of inertia is usually given for **frames passing through the center of mass of the rigid body**.
+**The inertia matrix changes under change of the reference frame**.
+
+Consider frame \\(\\{\bm{C}\\}\\) **parallel** to \\(\\{\bm{A}\\}\\) and attached to the center of mass of a rigid body and let \\(\bm{p}\_c = [x\_c, y\_c, z\_c]^T\\) denote the vector of the position of the center of mass with respect to frame \\(\\{\bm{A}\\}\\).
+The relation between the inertia matrix about \\(A\\) and that about \\(C\\) is given by the following relation:
+
+\begin{equation}
+ \tcmbox{{}^A\bm{I} = {}^C\bm{I} + m(\bm{p}\_c^T \bm{p}\_c \bm{I}\_{3 \times 3} - \bm{p}\_c \bm{p}\_c^T)}
+\end{equation}
+
+in which \\(m\\) denotes the mass of the rigid body and \\(\bm{I}\_{3 \times 3}\\) denotes the identity matrix.
+
+On the other hand, if the reference frame \\(\\{B\\}\\) has **pure rotation** with respect to the frame attached to the center of mass \\(\\{A\\}\\):
+
+\begin{equation}
+ {}^A\bm{I} = {}^A\bm{R}\_C {}^C\bm{I} {}^A\bm{R}\_C^T
+\end{equation}
+
+
+#### Momentum and Kinetic Energy {#momentum-and-kinetic-energy}
+
+
+##### Linear Momentum {#linear-momentum}
+
+Linear momentum of a material body, shown in Figure [fig:angular_momentum_rigid_body](#fig:angular_momentum_rigid_body), with respect to a reference frame \\(\\{\bm{A}\\}\\) is defined as
+
+\begin{equation}
+ {}^A\bm{G} = \int\_V \frac{d\bm{p}}{dt} \rho dV
+\end{equation}
+
+For any mass element \\(\rho dV\\), the position vector \\(\bm{p}\\) can be written as
+\\[ p = p\_c + r \\]
+
+And because \\(\int\_V r \rho dV = 0\\), we have by substitution
+\\[ {}^A\bm{G} = \frac{d\bm{p}\_c}{dt} \int\_V \rho dV \\]
+and thus
+
+\begin{equation}
+ \tcmbox{{}^A\bm{G} = m \cdot {}^A\bm{v}\_C}
+\end{equation}
+
+in which \\({}^A\bm{v}\_C\\) denotes the velocity of the center of mass with respect to the frame \\(\\{\bm{A}\\}\\).
+
+This result implies that the **total linear momentum** of differential masses is equal to the linear momentum of a **point mass** \\(m\\) located at the **center of mass**.
+This highlights the important of the center of mass in dynamic formulation of rigid bodies.
+
+
+
+{{< figure src="/ox-hugo/taghirad13_angular_momentum_rigid_body.png" caption="Figure 11: The components of the angular momentum of a rigid body about \\(A\\)" >}}
+
+
+##### Angular Momentum {#angular-momentum}
+
+Consider the solid body represented in Figure [fig:angular_momentum_rigid_body](#fig:angular_momentum_rigid_body).
+Angular momentum of the differential masses \\(\rho dV\\) about a reference point \\(A\\), expressed in the reference frame \\(\\{\bm{A}\\}\\) is defined as
+\\[ {}^A\bm{H} = \int\_V \left(\bm{p} \times \frac{d\bm{p}}{dt} \right) \rho dV \\]
+in which \\(d\bm{p}/dt\\) denotes the velocity of differential mass with respect to the reference frame \\(\\{\bm{A}\\}\\).
+
+By substituting \\(\bm{p} = \bm{p}\_c + \bm{r}\\) in the previous equations, be obtain:
+\\[ {}^A\bm{H} = \bm{p}\_c \times m \bm{v}\_c + \int\_V \bm{r} \times (\bm{\Omega} \times \bm{r}) \rho dV \\]
+
+Therefore, angular momentum of the rigid body about point \\(A\\) is reduced to
+
+\begin{equation}
+ \tcmbox{{}^A\bm{H} = \bm{p}\_c \times \bm{G}\_c + {}^C\bm{H}}
+\end{equation}
+
+in which
+\\[ {}^C\bm{H} = \int\_V \bm{r} \times (\bm{\Omega} \times \bm{r}) \rho dV = {}^C\bm{I} \cdot \bm{\Omega} \\]
+
+Equation [eq:angular_momentum](#eq:angular_momentum) reveals that angular momentum of a rigid body about a point \\(A\\) can be written as \\(\bm{p}\_c \times \bm{G}\_c\\), which is the contribution of linear momentum of the rigid body about point \\(A\\), and \\({}^C\bm{H}\\) which is the angular momentum of the rigid body about the center of mass.
+
+This also highlights the important of the center of mass in the dynamic analysis of rigid bodies.
+If the center of mass is taken as the reference point, the relation describing angular momentum [eq:angular_momentum](#eq:angular_momentum) is very analogous to that of linear momentum [eq:linear_momentum](#eq:linear_momentum).
+
+
+##### Kinetic Energy {#kinetic-energy}
+
+The Kinetic energy of a rigid body is defined as
+
+\begin{equation}
+ \tcmbox{\bm{K} = \frac{1}{2} \int\_V \bm{v} \cdot \bm{v} \rho dV}
+\end{equation}
+
+The velocity of a differential mass \\(\rho dV\\) can be represented by linear velocity of the center of mass and angular velocity of the rigid body as
+\\[ \bm{v} = \bm{v}\_p + \bm{\Omega} \times \bm{r} \\]
+
+By substitution, the kinetic energy of the rigid body may be obtained by:
+
+\begin{equation}
+ \tcmbox{\bm{K} = \frac{1}{2} \bm{v}\_c \times \bm{G}\_c + \frac{1}{2} \bm{\Omega} \cdot {}^C\bm{H}}
+\end{equation}
+
+in which \\(\bm{G}\_C\\) is the linear momentum of the rigid body and \\({}^C\bm{H}\\) is the angular momentum of the rigid body about the center of mass.
+
+This equation reveals that kinetic energy of a moving body can be represented as the **kinetic energy of a point mass located as the center of mass**, in addition to the **kinetic energy of a body rotating about the center of mass**.
+
+
+#### Newton-Euler Laws {#newton-euler-laws}
+
+The Newton and Euler laws can be written for three different cases where the angular motion:
+
+1. is about a fixed point in space
+2. is represented about the center of mass
+3. is represented about an arbitrary moving point in space
+
+We only examine the case in which all rotations are represented about the center of mass.
+
+Consider a rigid body under general motion, that is, a combination of translation and rotation.
+
+
+
+
+The Newton's law relates the change of linear momentum of the rigid body to the resulting external forces applied to it
+\\[ \sum \bm{f}\_\text{ext} = \frac{d\bm{G}\_c}{dt} \\]
+
+
+
+
+
+
+The Euler's law relates the change of angular momentum of a rigid body about the center of mass, to the summation of all external moments applied to the rigid body about center of mass
+\\[ \sum {}^c\bm{n}\_\text{ext} = \frac{d}{dt}({}^c\bm{H}) \\]
+
+
+
+Furthermore, all the external forces and moments applied to the links and to the moving platform must be carefully determined.
+Gravitational forces acting on the center of masses, frictional forces and moments acting on the joints, and any possible disturbance force or moment applied to the links and to the moving platform would be identified.
+The most important external forces or moments applied on the manipulator are the one applied by the actuators, denoted by \\(\bm{\tau} = [\tau\_{1}, \tau\_{2}, \dots, \tau\_{m}]^{T}\\).
+The forces and moments shall be derived from the set of Newton-Euler laws, which are written separately for each link and the moving platform.
+
+Finally, by elimination of these constraints forces and moments on the Newton-Euler equations written for the moving platform, the dynamic equations **relating the actuator forces and moments** \\(\bm{\tau}\\) to the **motion variables of the moving platform** \\(\bm{\mathcal{X}}\\), \\(\dot{\bm{\mathcal{X}}}\\) and \\(\ddot{\bm{\mathcal{X}}}\\) are derived.
+
+
+#### Dynamic Formulation of the Stewart-Gough Platform {#dynamic-formulation-of-the-stewart-gough-platform}
+
+
+##### Acceleration Analysis {#acceleration-analysis}
+
+In acceleration analysis, it is intended to **derive expressions for linear and angular acceleration of the limbs**, namely \\(\ddot{l}\_{i}\\) and \\(\dot{\bm{\omega}}\_{i}\\) as a function of the moving platform acceleration \\(\ddot{\bm{\mathcal{X}}} = [\dot{\bm{v}}\_{p}, \dot{\bm{\omega}}]^{T}\\).
+To obtain such a relation, let us rewrite the **velocity loop closure**:
+
+\begin{equation}
+ \bm{v}\_{p} + \bm{\omega} \times \bm{b}\_{i} = \dot{l}\_{i} \hat{\bm{s}}\_{i} + l\_{i}(\bm{\omega}\_{i} \times \hat{\bm{s}}\_{i})
+\end{equation}
+
+Since there is no actuation torque about \\(\hat{\bm{s}}\_{i}\\), the limb angular velocity and acceleration vectors (\\(\bm{\omega}\_i\\) and \\(\dot{\bm{\omega}}\_i\\)) are normal to \\(\hat{\bm{s}}\_{i}\\) provided that the following assumption are considered for the platform:
+
+- both end joints of the limb are spherical
+- the limbs are symmetric with respect to their axes
+- the effects of friction in spherical joints are neglected
+
+Considering these assumptions, it can be concluded that the limbs cannot spin about their axes: \\(\bm{\omega}\_{i} \cdot \hat{\bm{s}}\_{i} = 0\\) and \\((\hat{\bm{s}}\_i \times (\bm{\omega}\_i \times \hat{\bm{s}}\_i)) = \bm{\omega}\_i\\).
+
+To obtain the angular velocity of the limbs \\(\bm{\omega}\_{i}\\), we cross multiply \\(\hat{\bm{s}}\_{i}\\) to both sides of the previous equation:
+
+\begin{equation}
+ \bm{\omega}\_{i} = \frac{1}{l\_i} ( \hat{\bm{s}}\_i \times \bm{v}\_{b\_i} )
+\end{equation}
+
+With \\(\bm{v}\_{b\_{i}}\\) an **intermediate variable** corresponding to the velocity of point \\(\bm{b}\_{i}\\):
+
+\begin{equation}
+ \bm{v}\_{b\_{i}} = \bm{v}\_{p} + \bm{\omega} \times \bm{b}\_{i}
+\end{equation}
+
+As illustrated in Figure [fig:free_body_diagram_stewart](#fig:free_body_diagram_stewart), the piston-cylinder structure of the limbs is decomposed into two separate parts, the masses of which are denoted by \\(m\_{i\_1}\\) and \\(m\_{i\_2}\\).
+The position vector of these two center of masses can be determined by the following equations:
+
+\begin{align}
+ \bm{p}\_{i\_1} &= \bm{a}\_{i} + c\_{i\_1} \hat{\bm{s}}\_{i} \\\\\\
+ \bm{p}\_{i\_2} &= \bm{a}\_{i} + ( l\_i - c\_{i\_2}) \hat{\bm{s}}\_{i}
+\end{align}
+
+
+
+{{< figure src="/ox-hugo/taghirad13_free_body_diagram_stewart.png" caption="Figure 12: Free-body diagram of the limbs and the moving platform of a general Stewart-Gough manipulator" >}}
+
+By differentiating the previous equations and doing some manipulations, we obtain:
+
+\begin{align}
+ \ddot{l}\_i &= \bm{a}\_{b\_i} \times \hat{\bm{s}}\_i + l\_i (\bm{\omega\_i} \cdot \bm{\omega\_i}) \\\\\\
+ \dot{\bm{\omega}}\_i &= \frac{1}{l\_i} (\hat{\bm{s}}\_i \times \bm{a}\_{b\_i} - 2 \dot{l}\_i \bm{\omega}\_i) \\\\\\
+ \bm{a}\_{i\_1} &= c\_{i\_1} ( \dot{\bm{\omega}}\_i \times \hat{\bm{s}}\_{i} + \bm{\omega}\_i \times (\bm{\omega}\_i \times \hat{\bm{s}}\_i)) \\\\\\
+ \bm{a}\_{i\_2} &= ( l\_i - c\_{i\_2}) (\dot{\bm{\omega}}\_i \times \hat{\bm{s}}\_{i} - (\bm{\omega}\_i \cdot \bm{\omega}\_i) \hat{\bm{s}}\_i) + 2 \dot{l}\_i (\bm{\omega}\_i \times \hat{\bm{s}}\_i) + \ddot{l}\_i \hat{\bm{s}}\_i
+\end{align}
+
+with
+
+\begin{equation}
+ \bm{a}\_{b\_i} = \bm{a}\_p + \dot{\bm{\omega}} \times \bm{b}\_i + \bm{\omega} \times (\bm{\omega} \times \bm{b}\_i)
+\end{equation}
+
+
+##### Dynamic Formulation of the Limbs {#dynamic-formulation-of-the-limbs}
+
+To derive the dynamic formulation of the Stewart-Gough platform, the manipulator is decomposed into a moving platform and six identical limbs.
+We assume that each limb consists of two parts, the cylinder and the piston, where the velocities and the accelerations of their centers of masses are determined.
+We also assume that the centers of masses of the cylinder and the piston are located at a distance of \\(c\_{i\_1}\\) and \\(c\_{i\_2}\\) above their foot points, and their masses are denoted by \\(m\_{i\_1}\\) and \\(m\_{i\_2}\\).
+Moreover, consider that the pistons are symmetric about their axes, and their centers of masses lie at their midlengths.
+
+The free-body diagrams of the limbs and the moving platforms is given in Figure [fig:free_body_diagram_stewart](#fig:free_body_diagram_stewart).
+The reaction forces at fixed points \\(A\_i\\) are denoted by \\(\bm{f}\_{a\_i}\\), the internal force at moving points \\(B\_i\\) are dentoed by \\(\bm{f}\_{b\_i}\\), and the internal forces and moments between cylinders and pistons are denoted by \\(\bm{f}\_{c\_i}\\) and \\(\bm{M\_{c\_i}}\\) respectively.
+
+Assume that the only existing external disturbance wrench is applied on the moving platform and is denoted by \\(\bm{\mathcal{F}}\_d = [\bm{F}\_d, \bm{n}\_d]^T\\).
+
+
+
+
+\begin{equation}
+\begin{aligned}
+ \bm{f}\_{b\_i} &= \frac{1}{l\_i}(I\_{xx\_i} + l\_i^2 m\_{c\_e}) \hat{\bm{s}}\_i \times \dot{\bm{\omega}}\_i \\\\\\
+ &\quad + \frac{2}{l\_i} m\_{i\_2} c\_{i\_2} \dot{l}\_i \hat{\bm{s}}\_i \times \bm{\omega}\_i \\\\\\
+ &\quad - m\_{g\_e} \hat{\bm{s}}\_i \times (\hat{\bm{s}}\_i \times \bm{g})
+\end{aligned}
+\end{equation}
+
+in which \\(m\_{c\_e}\\) is defined as
+
+\begin{equation}
+ m\_{c\_e} = \frac{1}{l\_i^2} \left( m\_{i\_1} c\_{i\_1}^2 + m\_{i\_2} c\_{i\_2}^2 \right)
+\end{equation}
+
+
+
+
+\begin{align}
+ & m(\bm{a}\_p - \bm{g}) - \bm{F}\_d - \sum\_{i=1}^{6} \bm{f}\_{b\_i} = \bm{0} \label{eq:dyn\_form\_implicit\_trans}\\\\\\
+ & {}^A\bm{I}\_P \dot{\bm{\omega}} + \bm{\omega} \times {}^A\bm{I}\_P\bm{\omega} - \bm{n}\_d - \sum\_{i=1}^{6} \bm{b}\_{i} \times \bm{f}\_{b\_i} = \bm{0} \label{eq:dyn\_form\_implicit\_rot}
+\end{align}
+
+
+
+
+\begin{equation}
+ \begin{aligned}
+ \bm{M}(\bm{\mathcal{X}}) &= \bm{M}\_p + \sum\_{i=1}^6 \bm{M}\_{li} \\\\\\
+ \bm{C}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}}) &= \bm{C}\_p + \sum\_{i=1}^6 \bm{C}\_{li} \\\\\\
+ \bm{G}(\bm{\mathcal{X}}) &= \bm{G}\_p + \sum\_{i=1}^6 \bm{G}\_{li} \\\\\\
+ \bm{\mathcal{F}}(\bm{\mathcal{X}}) &= \bm{\mathcal{F}}\_d + \sum\_{i=1}^6 \bm{\mathcal{F}}\_{\tau\_i}
+ \end{aligned}
+\end{equation}
+
+
+
+To combine the corresponding matrices, an to generate the whole manipulator dynamics, it is necessary to find intermediate Jacobian matrices \\(\bm{J}\_i\\), given in [eq:jacobian_intermediate](#eq:jacobian_intermediate), and then compute compatible matrices for the limbs given in [eq:closed_form_stewart_manipulator](#eq:closed_form_stewart_manipulator).
+Now that all the terms required to **computed to actuator forces required to generate such a trajectory** is computed, let us define \\(\bm{\mathcal{F}}\\) as the resulting Cartesian wrench applied to the moving platform.
+This wrench can be calculated from the summation of all inertial and external forces **excluding the actuator torques** \\(\bm{\tau}\\) in the closed-form dynamic formulation [eq:closed_form_dynamic_stewart_wanted](#eq:closed_form_dynamic_stewart_wanted).
+
+By this definition, \\(\bm{\mathcal{F}}\\) can be viewed as the projector of the actuator forces acting on the manipulator, mapped to the Cartesian space.
+Since there is no redundancy in actuation in the Stewart-Gough manipulator, the Jacobian matrix \\(\bm{J}\\), squared and actuator forces can be uniquely determined from this wrench, by \\(\bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}}\\), provided \\(\bm{J}\\) is non-singular.
+Therefore, actuator forces \\(\bm{\tau}\\) are computed in the simulation from
+
+\begin{equation}
+ \bm{\tau} = \bm{J}^{-T} \left( \bm{M}(\bm{\mathcal{X}})\ddot{\bm{\mathcal{X}}} + \bm{C}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}})\dot{\bm{\mathcal{X}}} + \bm{G}(\bm{\mathcal{X}}) - \bm{\mathcal{F}}\_d \right)
+\end{equation}
+
+
+
+{{< figure src="/ox-hugo/taghirad13_stewart_inverse_dynamics.png" caption="Figure 14: Flowchart of inverse dynamics implementation sequence" >}}
+
+
+### Virtual Work Formulation {#virtual-work-formulation}
+
+
+### Lagrange Formulation {#lagrange-formulation}
+
+\begin{equation}
+ K(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}}) = \frac{1}{2} \dot{\bm{\mathcal{X}}}^T \bm{M}(\bm{\mathcal{X}}) \dot{\bm{\mathcal{X}}}
+\end{equation}
+
+\begin{equation}
+ \bm{G}(\bm{\mathcal{X}}) = \frac{\partial P(\bm{\mathcal{X}})}{\partial \bm{\mathcal{X}}}
+\end{equation}
+
+\begin{equation}
+ \bm{C}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}}) = \frac{1}{2} (\dot{\bm{M}} + \bm{U}^T - \bm{U})
+\end{equation}
+
+
+## Motion Control {#motion-control}
+
+
+
+
+### Introduction {#introduction}
+
+Parallel robots are designed for two different types of applications.
+
+In the first type, the moving platform of the robot accurately follows a **desired position and orientation** path in a specific time frame, while **no interacting forces** need to be applied to the environment.
+
+The second type of application include situations where the robot moving platform is in **contact with a stiff environment** (e.g. precision machining).
+In such application, the contact force describe the state of interaction more effectively than the position and orientation of the moving platform.
+The problem of **force control** can be described as to derive the actuator forces for such a manipulator required to generate a prescribed desired wrench (force and torque) at the manipulator moving platform, while the manipulator is performing its motion.
+
+Although a multiple degrees-of-freedom robotic manipulator can usually be represented by a MIMO and nonlinear model, many industrial controllers for such robots consist of a number of linear controller designed to **control individual joint motions**.
+One of the reasons why such decentralization can perform well in practice is the use of large gear reductions in robot actuators, which significantly reduces the coupling and non linear behavior of robot dynamics.
+
+However, using advanced techniques in nonlinear and MIMO control permits to overcome limitations of the SISO approach.
+
+
+### Controller Topology {#controller-topology}
+
+
+
+
+
+
+In motion control of parallel manipulator, it is assumed that the controller computes the **required actuator forces** or torques to cause the robot motion to follow a desired position and orientation trajectory.
+
+
+
+Consider the general closed-form dynamics formulation of a parallel robot
+
+\begin{equation}
+ \tcmbox{\bm{M}(\bm{\mathcal{X}})\ddot{\bm{\mathcal{X}}} + \bm{C}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}})\dot{\bm{\mathcal{X}}} + \bm{G}(\bm{\mathcal{X}}) = \bm{\mathcal{F}}}
+\end{equation}
+
+where
+
+- \\(\bm{M}(\bm{\mathcal{X}})\\) denotes the mass matrix
+- \\(\bm{C}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}})\\) denotes the Coriolis and centrifugal matrix
+- \\(\bm{G}(\bm{\mathcal{X}})\\) denotes the gravity vector
+- \\(\bm{\mathcal{F}}\\) denotes the generalized forces applied to the moving platform center of mass
+
+The generalized forces can be decomposed as follow
+\\[ \bm{\mathcal{F}} = \bm{J}^T \bm{\tau} + \bm{\mathcal{F}}\_d \\]
+with
+
+- \\(\bm{J}\\) is the Jacobian
+- \\(\bm{\tau}\\) are the actuator forces
+- \\(\bm{\mathcal{F}}\_d\\) are any external wrenches
+
+
+
+
+**Control topology** is referred to the **structure of the control system** used to compute the actuator forces/torques from the measurements, and the required pre and post processing.
+
+
+
+To perform such motion in closed loop, it is necessary to **measure the output motion** \\(\bm{\mathcal{X}}\\) of the manipulator by an instrumentation system.
+Such instrumentation usually consists of two subsystems: the first subsystem may use accurate accelerometers, or global positioning systems to calculate the position of a point on the moving platform; and a second subsystem may use inertial or laser gyros to determine orientation of the moving platform.
+
+Figure [fig:general_topology_motion_feedback](#fig:general_topology_motion_feedback) shows the general topology of a motion controller using direct measurement of the motion variable \\(\bm{\mathcal{X}}\\), as feedback in the closed-loop system.
+In such a structure, the measured position and orientation of the manipulator is compared to its desired value to generate the **motion error vector** \\(\bm{e}\_\mathcal{X}\\).
+The controller uses this error information to generate suitable commands for the actuators to minimize the tracking error.
+
+
+
+{{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback.png" caption="Figure 15: The general topology of motion feedback control: motion variable \\(\bm{\mathcal{X}}\\) is measured" >}}
+
+However, it is usually much **easier to measure the active joint variable** rather than measuring the final position and orientation of the moving platform.
+The relation between the **joint variable** \\(\bm{q}\\) and **motion variable** of the moving platform \\(\bm{\mathcal{X}}\\) is dealt with the **forward and inverse kinematics**.
+The relation between the **differential motion variables** \\(\dot{\bm{q}}\\) and \\(\dot{\bm{\mathcal{X}}}\\) is studied through the **Jacobian analysis**.
+
+It is then possible to use the forward kinematic analysis to calculate \\(\bm{\mathcal{X}}\\) from the measured joint variables \\(\bm{q}\\), and one may use the control topology depicted in Figure [fig:general_topology_motion_feedback_bis](#fig:general_topology_motion_feedback_bis) to implement such a controller.
+
+
+
+{{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback_bis.png" caption="Figure 16: The general topology of motion feedback control: the active joint variable \\(\bm{q}\\) is measured" >}}
+
+In this topology, the forward kinematic analysis of the manipulator has to be performed to implement the feedback loop.
+As described earlier, this is a **complex task** for parallel manipulators.
+It is even more complex when a solution has to be found in real time.
+
+However, as shown herein before, the inverse kinematic analysis of parallel manipulators is much easier to carry out.
+To overcome the implementation problem of the control topology in Figure [fig:general_topology_motion_feedback_bis](#fig:general_topology_motion_feedback_bis), another control topology is usually implemented for parallel manipulators.
+
+In this topology, depicted in Figure [fig:general_topology_motion_feedback_ter](#fig:general_topology_motion_feedback_ter), the desired motion trajectory of the robot \\(\bm{\mathcal{X}}\_d\\) is used in an **inverse kinematic analysis** to find the corresponding desired values for joint variable \\(\bm{q}\_d\\).
+Hence, the controller is designed based on the **joint space error** \\(\bm{e}\_q\\).
+
+
+
+{{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback_ter.png" caption="Figure 17: The general topology of motion feedback control: the active joint variable \\(\bm{q}\\) is measured, and the inverse kinematic analysis is used" >}}
+
+Therefore, the **structure and characteristics** of the controller in this topology is totally **different** from that given in the first two topologies.
+
+The **input and output** of the controller depicted in Figure [fig:general_topology_motion_feedback_ter](#fig:general_topology_motion_feedback_ter) are **both in the joint space**.
+However, this is not the case in the previous topologies where the input to the controller is the motion error in task space, while its output is in the joint space.
+
+For the topology in Figure [fig:general_topology_motion_feedback_ter](#fig:general_topology_motion_feedback_ter), **independent controllers** for each joint may be suitable.
+
+To generate a **direct input to output relation in the task space**, consider the topology depicted in Figure [fig:general_topology_motion_feedback_quater](#fig:general_topology_motion_feedback_quater).
+A force distribution block is added which maps the generated wrench in the task space \\(\bm{\mathcal{F}}\\), to its corresponding actuator forces/torque \\(\bm{\tau}\\).
+
+
+
+{{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback_quater.png" caption="Figure 18: The general topology of motion feedback control in task space: the motion variable \\(\bm{\mathcal{X}}\\) is measured, and the controller output generates wrench in task space" >}}
+
+For a fully parallel manipulator such as the Stewart-Gough platform, this mapping can be constructed from the **Jacobian** transpose of the manipulator:
+\\[ \bm{\mathcal{F}} = \bm{J}^T \bm{\tau}; \quad \bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}} \\]
+
+
+### Motion Control in Task Space {#motion-control-in-task-space}
+
+
+
+
+#### Decentralized PD Control {#decentralized-pd-control}
+
+In the control structure in Figure [fig:decentralized_pd_control_task_space](#fig:decentralized_pd_control_task_space), a number of linear PD controllers are used in a feedback structure on each error component.
+The decentralized controller consists of **six disjoint linear controllers** acting on each error component \\(\bm{e}\_x = [e\_x,\ e\_y,\ e\_z,\ e\_{\theta\_x},\ e\_{\theta\_y},\ e\_{\theta\_z}]\\).
+The PD controller is denoted by \\(\bm{K}\_d s + \bm{K}\_p\\), in which \\(\bm{K}\_d\\) and \\(\bm{K}\_p\\) are \\(6 \times 6\\) **diagonal matrices** denoting the derivative and proportional controller gains for each error term.
+
+
+
+{{< figure src="/ox-hugo/taghirad13_decentralized_pd_control_task_space.png" caption="Figure 19: Decentralized PD controller implemented in task space" >}}
+
+Hence, by this structure, each tracking error component is **treated separately**.
+The output of the controller is denoted by \\(\bm{\mathcal{F}} = [F\_x\ F\_y\ F\_z\ \tau\_x\ \tau\_y\ \tau\_z]\\).
+
+In practice, the calculated output wrench is transformed into actuator forces through the force distribution block.
+This mapping is implemented through inverse of the manipulator **Jacobian** transpose by \\(\bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}}\\).
+
+Different alternatives of linear controllers can be used instead of the PD controller used in this structure, however PD controller is the simplest form which can preserve the manipulator stability while providing suitable tracking performance.
+
+The proposed decentralized PD controller is very simple in structure and therefore easily implementable.
+The design of such a controller needs no detailed information on the manipulator dynamics.
+The controller gains are generally tuned experimentally based on physical realization of the controller by trial and error.
+
+
+#### Feed Forward Control {#feed-forward-control}
+
+A feedforward wrench denoted by \\(\bm{\mathcal{F}}\_{ff}\\) may be added to the decentralized PD controller structure as depicted in Figure [fig:feedforward_control_task_space](#fig:feedforward_control_task_space).
+This term is generated from the dynamic model of the manipulator in the task space, represented in a closed form by the following equation:
+\\[ \bm{\mathcal{F}}\_{ff} = \bm{\hat{M}}(\bm{\mathcal{X}}\_d)\ddot{\bm{\mathcal{X}}}\_d + \bm{\hat{C}}(\bm{\mathcal{X}}\_d, \dot{\bm{\mathcal{X}}}\_d)\dot{\bm{\mathcal{X}}}\_d + \bm{\hat{G}}(\bm{\mathcal{X}}\_d) \\]
+
+
+
+{{< figure src="/ox-hugo/taghirad13_feedforward_control_task_space.png" caption="Figure 20: Feed forward wrench added to the decentralized PD controller in task space" >}}
+
+The desired trajectory in task space \\(\bm{\mathcal{X}}\_d\\), and its derivatives \\(\dot{\bm{\mathcal{X}}}\_d\\), \\(\ddot{\bm{\mathcal{X}}}\_d\\) are the required inputs for the feedforward block.
+This term is called feedforward since no online information of the output motion trajectory \\(\bm{\mathcal{X}}\\) is needed for its computation.
+
+In order to generate this term, dynamic formulation of the robots and its kinematic and dynamic parameters are needed.
+In practice, exact knowledge of dynamic matrices are not available, and therefore, **estimate** of these matrices are used in practice, denoted by \\(\hat{\bm{M}}\\), \\(\hat{\bm{C}}\\) and \\(\hat{\bm{G}}\\).
+
+The information required to generate the feedforward wrench \\(\bm{\mathcal{F}}\_{ff}\\) is usually available beforehand and can be derived offline.
+The closed-loop dynamic formulation for the manipulator becomes:
+
+\begin{equation}
+ \begin{aligned}
+ \bm{M}(\bm{\mathcal{X}})\ddot{\bm{\mathcal{X}}} &+ \bm{C}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}})\dot{\bm{\mathcal{X}}} + \bm{G}(\bm{\mathcal{X}}) \\\\\\
+ &= \bm{\mathcal{F}} + \bm{\mathcal{F}}\_d \\\\\\
+ &= \bm{\mathcal{F}}\_{pd} + \bm{\mathcal{F}}\_{ff} + \bm{\mathcal{F}}\_d \\\\\\
+ &= \bm{K}\_d \dot{\bm{e}}\_x + \bm{K}\_p \bm{e}\_x + \bm{\mathcal{F}}\_{ff} + \bm{\hat{M}}\ddot{\bm{\mathcal{X}}}\_d + \bm{\hat{C}}\dot{\bm{\mathcal{X}}}\_d + \bm{\hat{G}}
+ \end{aligned}
+\end{equation}
+
+If the knowledge of the dynamic matrices is complete, we may assume that \\(\hat{\bm{M}} = \bm{M}\\), \\(\hat{\bm{C}} = \bm{C}\\) and \\(\hat{\bm{G}} = \bm{G}\\).
+Furthermore, if we consider that the controller performs well such that \\(\bm{\mathcal{X}}(t) \simeq \bm{\mathcal{X}}\_d(t)\\) and \\(\dot{\bm{\mathcal{X}}}(t) \simeq \dot{\bm{\mathcal{X}}}\_d(t)\\), the simplified closed-loop dynamics become:
+
+\begin{equation}
+ \begin{aligned}
+ \bm{M} (\ddot{\bm{\mathcal{X}}}\_d - \ddot{\bm{\mathcal{X}}}) + \bm{K}\_d \dot{\bm{e}}\_x + \bm{K}\_p \bm{e}\_x + \bm{\mathcal{F}}\_d &= 0 \\\\\\
+ \bm{M} \ddot{\bm{e}}\_x + \bm{K}\_d \dot{\bm{e}}\_x + \bm{K}\_p \bm{e}\_x + \bm{\mathcal{F}}\_d &= 0
+ \end{aligned}
+\end{equation}
+
+This equation implies that, if the mentioned assumptions hold, the **error dynamics** satisfies a set of **second-order system** in the presence of disturbance.
+By choosing appropriate gains for PD controller, the transient and steady-state performance of tracking error can be designed so as to satisfy the application requirements.
+
+Note that except the mass matrix, the error dynamic terms are all **configuration independent**, and therefore, it is much **easier to tune the PD controller gains** to work well within the whole workspace of the robot.
+
+However, this method faces a number of **limitations** in practice.
+The most important limitation of this control technique is the **stringent assumption of a complete knowledge requirement of the dynamic matrices**.
+In practice, derivation of these matrices is a prohibitive task.
+
+Finally, because of the dependency of the mass matrix to the configuration of the robot, the error dynamics are not completely decoupled.
+This means that correction in one error component may be considered as a disturbance effect to the other components.
+To overcome these limitations, inverse dynamic approach is given in the following section.
+
+
+#### Inverse Dynamics Control {#inverse-dynamics-control}
+
+
+
+
+In inverse dynamics control (IDC), nonlinear dynamics of the model is used to add a **corrective term** to the decentralized PD controller.
+By this means, **nonlinear and coupling behavior of the robotic manipulator is significantly attenuated**, and therefore, the performance of linear controller is greatly improved.
+
+
+
+
+
+{{< figure src="/ox-hugo/taghirad13_inverse_dynamics_control_task_space.png" caption="Figure 21: General configuration of inverse dynamics control implemented in task space" >}}
+
+The controller output wrench applied to the manipulator may be derived as follows:
+
+\begin{subequations}
+\begin{align}
+ \bm{\mathcal{F}} &= \hat{\bm{M}}(\bm{\mathcal{X}}) \bm{a} + \bm{\mathcal{F}}\_{fl} \\\\\\
+ &= \hat{\bm{M}}(\bm{\mathcal{X}}) \bm{a} + \hat{\bm{C}}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}}) \dot{\bm{\mathcal{X}}} + \hat{\bm{G}}(\bm{\mathcal{X}}) \\\\\\
+ \bm{a} &= \ddot{\bm{\mathcal{X}}}\_d + \bm{K}\_d \dot{\bm{e}}\_x + \bm{K}\_p \bm{e}\_x
+\end{align}
+\end{subequations}
+
+The closed-loop dynamic formulation for the manipulator becomes:
+
+\begin{equation}
+ \begin{aligned}
+ \bm{M}(\bm{\mathcal{X}})\ddot{\bm{\mathcal{X}}} &+ \bm{C}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}})\dot{\bm{\mathcal{X}}} + \bm{G}(\bm{\mathcal{X}}) \\\\\\
+ &= \bm{\mathcal{F}} + \bm{\mathcal{F}}\_d \\\\\\
+ &= \hat{\bm{M}}(\bm{\mathcal{X}}) \left(\ddot{\bm{\mathcal{X}}}\_d + \bm{K}\_d \dot{\bm{e}}\_x + \bm{K}\_p \bm{e}\_x \right) \\\\\\
+ &\quad + \hat{\bm{C}}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}})\dot{\bm{\mathcal{X}}} + \hat{\bm{G}}(\bm{\mathcal{X}}) + \bm{\mathcal{F}}\_d \\\\\\
+ \end{aligned}
+\end{equation}
+
+If the knowledge of the dynamic matrices is complete, the closed-loop dynamic formulation simplifies to:
+
+\begin{equation}
+ \hat{\bm{M}}(\bm{\mathcal{X}}) \left(\ddot{\bm{e}}\_d + \bm{K}\_d \dot{\bm{e}}\_x + \bm{K}\_p \bm{e}\_x \right) + \bm{\mathcal{F}}\_d = 0
+\end{equation}
+
+This control technique is very popular in practice because of the fact that this technique can significantly **linearize and decouple dynamic formulation of the closed-loop error dynamics**.
+Furthermore, the error dynamic terms are all **configuration independent**, and therefore, it is much easier to tune the PD controller gains for suitable performance in the whole workspace of the robot.
+
+However, note that for a good performance, and **accurate model of the system is required**, and the overall procedure is **not robust to modeling uncertainty**.
+Furthermore, this technique is computationally intensive in terms of the online computations needed to carry out the closed-loop control structure.
+
+
+#### Partial Linearization IDC {#partial-linearization-idc}
+
+Inverse dynamics control has several features making it very attractive in practice.
+However, to apply this method, complete knowledge of the dynamic formulation matrices is required.
+This requirement has the main drawbacks that the dynamic formulation of the parallel manipulator is a complicated step to be carried out.
+
+To implement all the terms in IDC structure, not only the structure and components of such matrices must be carefully determined, but also the kinematics and inertial parameters of the robot are needed to be identified and calibrated.
+This step requires the use of high-precision calibration equipment which are not usually accessible.
+Finally, if all the terms and parameters are well known, implementation of full inverse dynamic linearization is computationally intensive.
+
+These are the reasons why, in practice, IDC control is extended to different forms where the above-mentioned stringent requirements are reduced.
+
+To develop the simplest possible implementable IDC, let us recall dynamic formulation complexities:
+
+- the manipulator mass matrix \\(\bm{M}(\bm{\mathcal{X}})\\) is derived from kinetic energy of the manipulator (Eq. [eq:kinetic_energy](#eq:kinetic_energy))
+- the gravity vector \\(\bm{G}(\bm{\mathcal{X}})\\) is derived from potential energy (Eq. [eq:gravity_vectory](#eq:gravity_vectory))
+- the Coriolis and centrifugal matrix \\(\bm{C}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}})\\) is derived from Eq. [eq:gravity_vectory](#eq:gravity_vectory)
+
+The computation of the Coriolis and centrifugal matrix is more intensive than that of the mass matrix.
+Gravity vector is more easily computable.
+
+However, it is shown that certain properties hold for mass matrix, gravity vector and Coriolis and centrifugal matrix, which might be directly used in the control techniques developed for parallel manipulators.
+One of the most important properties of dynamic matrices is the skew-symmetric property of the matrix \\(\dot{\bm{M}} - 2 \bm{C}\\) .
+
+Consider dynamic formulation of parallel robot given in Eq. [eq:closed_form_dynamic_formulation](#eq:closed_form_dynamic_formulation), in which the skew-symmetric property of dynamic matrices is satisfied.
+The simplest form of IDC control effort \\(\bm{\mathcal{F}}\\) consists of:
+\\[ \bm{\mathcal{F}} = \bm{\mathcal{F}}\_{pd} + \bm{\mathcal{F}}\_{fl} \\]
+in which the first term \\(\bm{\mathcal{F}}\_{pd}\\) is generated by the simplified PD form on the motion error:
+\\[ \bm{\mathcal{F}}\_{pd} = \bm{K}\_d \dot{\bm{e}}\_x + \bm{K}\_p + \bm{e}\_x \\]
+
+The second term \\(\bm{\mathcal{F}}\_{fl}\\) is considered to be only the gravity vector of the manipulator \\(\bm{G}(\bm{\mathcal{X}})\\), at any configuration, and the computationally intensive Coriolis and centrifugal term is not used:
+\\[ \bm{\mathcal{F}}\_{fl} = \bm{G}(\bm{\mathcal{X}}) \\]
+
+Note that for an appreciable tracking performance with no static error at steady state, it is required to have complete knowledge of **only** the gravity term.
+By this means, computations required in this control technique are significantly less than that of the general IDC.
+
+Despite the simple structure of such a controller, the resulting control technique is very well performed, especially at steady state.
+We can show that this control topology achieves asymptotic tracking for a constant desired trajectory motion, that is, \\(\dot{\bm{\mathcal{X}}}\_d = 0\\).
+
+This reveals the fact that even if the mass matrix and Coriolis and centrifugal matrix are not used in the feedback, and the closed-loop dynamics is not completely linearized, the PD control structure with gravity compensation can still lead to asymptotic tracking.
+However, to suitable transient performance, more information of the system dynamics must be used in the linearization technique given in IDC.
+
+
+### Robust and Adaptative Control {#robust-and-adaptative-control}
+
+Inverse dynamics control faces the stringent requirement that for a good performance, an **accurate model** of the system is required, and the overall procedure is **not robust** to modeling uncertainty.
+Furthermore, this technique is computationally intensive in terms of online computation needed to carry out the closed-loop control structure.
+The proposed modified inverse dynamics control, while being beneficial in terms of computational cost, is not suitable in terms of a closed-loop transient performance.
+
+Another approach to modify IDC is to consider a complete linearization, but **assume that complete knowledge of dynamic formulation matrices is not available**.
+To **compensate for the lack of knowledge**, two advanced control methods, namely **robust** and **adaptive control** are proposed:
+
+- In the **robust approach**, a **fixed controller** is designed to satisfy the control objectives for the **worst possible case of modeling uncertainty** and disturbance wrenches.
+- In the **adaptive approach**, the estimates of dynamic formulation matrices are **updated** such that the difference between the true values of these matrices to their estimates converges to zero.
+
+A global understanding of the trade-offs involved in each method is needed to employ either of them in practice.
+
+
+#### Robust Inverse Dynamics Control {#robust-inverse-dynamics-control}
+
+Various sources of uncertainties such as unmodelled dynamics, unknown parameters, calibration error, unknown disturbance wrenches, and varying payloads may exist, and are not seen in dynamic model of the manipulator.
+
+To consider these modeling uncertainty in the closed-loop performance of the manipulator, recall the general closed-form dynamic formulation of the manipulator given in Eq. [eq:closed_form_dynamic_formulation](#eq:closed_form_dynamic_formulation), and modify the inverse dynamics control input \\(\bm{\mathcal{F}}\\) as
+
+\begin{align\*}
+ \bm{\mathcal{F}} &= \hat{\bm{M}}(\bm{\mathcal{X}}) \bm{a}\_r + \hat{\bm{C}}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}}) \dot{\bm{\mathcal{X}}} + \hat{\bm{G}}(\bm{\mathcal{X}})\\\\\\
+ \bm{a}\_r &= \ddot{\bm{\mathcal{X}}}\_d + \bm{K}\_d \dot{\bm{e}}\_x + \bm{K}\_p \bm{e}\_x + \bm{\delta}\_a
+\end{align\*}
+
+in which \\(\bm{a}\_r\\) is the robustified control input.
+
+Comparing this equation to the usual IDC, a robustifying term \\(\bm{\delta}\_a\\) is added to compensate for modeling uncertainties.
+
+Note that, as defined earlier, the notation \\(\hat{(.)}\\) represents the estimated value of \\((.)\\) and \\(\tilde{(.)}\\) is defined as the error mismatch between the estimated value and the true value as \\(\tilde{(.)} = \hat{(.)}- (.)\\).
+
+In a similar manner \\(\tilde{(.)}\\) notation may be applied to the motion variables as
+\\[ \tilde{\bm{\mathcal{X}}} = \bm{\mathcal{X}} - \bm{\mathcal{X}}\_d = - \bm{e}\_x \\]
+
+The closed-loop dynamic formulation of the manipulator can be written as:
+\\[ \ddot{\bm{\mathcal{X}}} = \bm{a}\_r + \bm{\eta}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}}, \bm{a}\_r) \\]
+in which
+\\[ \bm{\eta} = \bm{M}^{-1} \left( \tilde{\bm{M}} \bm{a}\_r + \tilde{\bm{C}} \dot{\bm{\mathcal{X}}} + \tilde{\bm{G}} \right) \\]
+is a measure of modeling uncertainty.
+
+
+
+{{< figure src="/ox-hugo/taghirad13_robust_inverse_dynamics_task_space.png" caption="Figure 22: General configuration of robust inverse dynamics control implemented in the task space" >}}
+
+
+#### Adaptive Inverse Dynamics Control {#adaptive-inverse-dynamics-control}
+
+
+
+{{< figure src="/ox-hugo/taghirad13_adaptative_inverse_control_task_space.png" caption="Figure 23: General configuration of adaptative inverse dynamics control implemented in task space" >}}
+
+
+### Motion Control in Joint Space {#motion-control-in-joint-space}
+
+Although the motion control schemes developed in section [sec:control_task_space](#sec:control_task_space) are very effective for tracking performance, they suffer from an implementation constraint that the motion variable \\(\bm{\mathcal{X}}\\) must be measured in practice.
+
+If this measurement is available without any doubt, such topologies are among the best routines to be implemented in practice.
+However, as explained in Section [sec:control_topology](#sec:control_topology), in many practical situations measurement of the motion variable \\(\bm{\mathcal{X}}\\) is difficult or expensive, and usually just the active joint variables \\(\bm{q}\\) are measured.
+In such cases, the controllers developed in the joint space may be recommended for practical implementation.
+
+To generate a direct input to output relation in the joint space, consider the topology depicted in Figure [fig:general_topology_motion_feedback_bis](#fig:general_topology_motion_feedback_bis).
+In this topology, the controller input is the joint variable error vector \\(\bm{e}\_q = \bm{q}\_d - \bm{q}\\), and the controller output is directly the actuator force vector \\(\bm{\tau}\\), and hence there exists a **one-to-one correspondence between the controller input to its output**.
+
+The general form of dynamic formulation of parallel robot is usually given in the task space.
+For motion control in joint space, we need to transform the dynamic formulation in the joint space, by which the actuator forces \\(\bm{\tau}\\) are directly related to the active joint variables \\(\bm{q}\\).
+
+
+#### Dynamic Formation in the Joint Space {#dynamic-formation-in-the-joint-space}
+
+The relation between the task space variables to their counterparts in the joint space can be derived by forward and inverse kinematics relations.
+Although both analyses involve solution to a set of non-linear equations, for parallel manipulators, inverse kinematic solution proves to be much easier to obtain than that of forward kinematic solution.
+
+This relation in **differential kinematics** is much simpler and can be completely determined by the Jacobian matrix:
+\\[ \tcmbox{\dot{\bm{q}} = \bm{J} \dot{\bm{\mathcal{X}}} \Longrightarrow \dot{\bm{\mathcal{X}}} = \bm{J}^{-1} \dot{\bm{q}}} \\]
+
+The acceleration variables are then:
+\\[ \ddot{\bm{q}} = \dot{\bm{J}} \dot{\bm{\mathcal{X}}} + \bm{J} \ddot{\mathcal{X}} \Longrightarrow \ddot{X} = \bm{J}^{-1} \ddot{\bm{q}} - \bm{J}^{-1} \dot{\bm{J}} \dot{\bm{\mathcal{X}}} \\]
+
+Furthermore, the relation between the actuator force vector \\(\bm{\tau}\\) to the corresponding task space wrench is given by:
+\\[ \tcmbox{\bm{\mathcal{F}} = \bm{J}^T \bm{\tau} \Longrightarrow \bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}}} \\]
+
+Substituting \\(\dot{\bm{\mathcal{X}}}\\) and \\(\ddot{\bm{\mathcal{X}}}\\) from the above equations into the dynamic formulation of the parallel robot gives:
+
+\begin{equation\*}
+ \begin{aligned}
+ & \left( \bm{J}^{-T} \bm{M} \bm{J}^{-1} \right) \ddot{\bm{q}} \\\\\\
+ & \quad + \bm{J}^{-T} \left( \bm{C} - \bm{M} \bm{J}^{-1} \dot{\bm{J}} \right) \bm{J}^{-1} \dot{\bm{q}} \\\\\\
+ & \quad + \bm{J}^{-T} \bm{G} + \bm{J}^{-T} \bm{\mathcal{F}}\_d = \bm{\tau}
+ \end{aligned}
+\end{equation\*}
+
+
+
+
+\begin{equation}
+ \bm{M}\_q \ddot{\bm{q}} + \bm{C}\_q \dot{\bm{q}} + \bm{G}\_q + \bm{\tau}\_d = \bm{\tau} \label{eq:dynamics\_joint\_space}
+\end{equation}
+
+with:
+
+\begin{subequations}
+\begin{align}
+ \bm{M}\_q =& \bm{J}^{-T} \bm{M} \bm{J}^{-1} \\\\\\
+ \bm{D}\_q =& \bm{J}^{-T} \left( \bm{C} - \bm{M} \bm{J}^{-1} \dot{\bm{J}} \right) \bm{J}^{-1} \\\\\\
+ \bm{G}\_q =& \bm{J}^{-T} \bm{G} \\\\\\
+ \bm{\tau}\_q =& \bm{J}^{-T} \bm{\mathcal{F}}\_d
+\end{align}
+\end{subequations}
+
+
+
+Note that the dynamic matrices are **not** explicitly represented in terms of the joint variable vector \\(\bm{q}\\).
+In fact, to fully derive these matrices, the Jacobian matrices must be computed and are generally derived as a function of the motion variables \\(\bm{\mathcal{X}}\\).
+Furthermore, the main dynamic matrices are all functions of the motion variable \\(\bm{\mathcal{X}}\\).
+Hence, in practice, to find the dynamic matrices represented in the joint space, **forward kinematics** should be solved to find the motion variable \\(\bm{\mathcal{X}}\\) for any given joint motion vector \\(\bm{q}\\).
+
+Since in parallel robots the forward kinematic analysis is computationally intensive, there exist inherent difficulties in finding the dynamic matrices in the joint space as an explicit function of \\(\bm{q}\\).
+In this case it is possible to solve forward kinematics in an online manner, it is recommended to use the control topology depicted in [fig:general_topology_motion_feedback_bis](#fig:general_topology_motion_feedback_bis), and implement control law design in the task space.
+
+However, one implementable alternative to calculate the dynamic matrices represented in the joint space is to use the **desired motion trajectory** \\(\bm{\mathcal{X}}\_d\\) instead of the true value of motion vector \\(\bm{\mathcal{X}}\\) in the calculations.
+This approximation significantly reduces the computational cost, with the penalty of having mismatch between the estimated values of these matrices to their true values.
+
+
+#### Decentralized PD Control {#decentralized-pd-control}
+
+The first control strategy introduced in the joint space consists of the simplest form of feedback control in such manipulators.
+In this control structure, depicted in Figure [fig:decentralized_pd_control_joint_space](#fig:decentralized_pd_control_joint_space), a number of PD controllers are used in a feedback structure on each error component.
+
+The PD controller is denoted by \\(\bm{K}\_d s + \bm{K}\_p\\), where \\(\bm{K}\_d\\) and \\(\bm{K}\_p\\) are \\(n \times n\\) **diagonal** matrices denoting the derivative and proportional controller gains, respectively.
+
+
+
+{{< figure src="/ox-hugo/taghirad13_decentralized_pd_control_joint_space.png" caption="Figure 24: Decentralized PD controller implemented in joint space" >}}
+
+By this structure, each tracking error component is **treated separately** by its disjoint PD controller.
+The proposed decentralized PD controller is very simple in structure, and therefore very easy to be implemented on the manipulator.
+The design of such a controller **needs no detailed information on the manipulator dynamic formulation and parameters**.
+However, the tracking performance of such a controller is relatively poor, and **static tracking errors** might be unavoidable.
+Also, the performance of the closed-loop system is configuration dependent.
+
+In practice, the gains are tuned experimentally and obtained as a trade-off between transient behavior and steady-state errors at different configurations.
+As the dynamics of the system in the joint space is configuration dependent, finding suitable controller gains to result in required performance in all configurations is a difficult task.
+
+The performance of the controller to attenuate measurement noise and external disturbance wrenches are also poor in practice.
+To remedy these shortcomings, some modifications have been proposed to this structure and further described.
+
+
+#### Feedforward Control {#feedforward-control}
+
+The tracking performance of the simple PD controller implemented in the joint space is usually not sufficient at different configurations.
+To improve the tracking performance, a feedforward actuator force denoted by \\(\bm{\tau}\_{ff}\\) may be added to the structure of the controller as depicted in Figure [fig:feedforward_pd_control_joint_space](#fig:feedforward_pd_control_joint_space).
+
+
+
+{{< figure src="/ox-hugo/taghirad13_feedforward_pd_control_joint_space.png" caption="Figure 25: Feed forward actuator force added to the decentralized PD controller in joint space" >}}
+
+The feedforward term is generated from the dynamic formulation of the manipulator.
+The desired trajectory in the task space \\(\bm{\mathcal{X}}\_d\\) and its derivatives \\(\dot{\bm{\mathcal{X}}}\_d\\), \\(\ddot{\bm{\mathcal{X}}}\_d\\) are thus required.
+
+In practice, exact knowledge of dynamic matrices are not available, and therefore, estimates of these matrices are used in this derivation denoted by \\(\hat{\bm{M}}\\), \\(\hat{\bm{C}}\\) and \\(\hat{\bm{G}}\\).
+
+The information required to generate the feedforward actuator force \\(\bm{\tau}\_{ff}\\) is usually available beforehand, and in such a case, the feedforward term corresponding to a given trajectory can be **determined off-line**, while the computation of the decentralized feedback term would be executed online.
+
+If complete information of the dynamic matrices is available, and if we assume that the system is performing well, meaning that \\(\bm{\mathcal{X}}(t) \simeq \bm{\mathcal{X}}\_d(t)\\) and \\(\dot{\bm{\mathcal{X}}}(t) \simeq \dot{\bm{\mathcal{X}}}\_d(t)\\), we can write the closed loop dynamics as follow:
+\\[ \bm{M}\_q \ddot{\bm{e}}\_q + \bm{K}\_d \dot{\bm{e}}\_q + \bm{K}\_p \bm{e}\_q = \bm{\tau}\_d \\]
+
+The error dynamics satisfy a set of second-order differential equations in the presence of disturbance.
+Therefore, by choosing appropriate gains of the PD controller, the transient and steady-state performance of the tracking error can be suitably designed.
+
+Note that except for the mass matrix, the error dynamics terms are all **configuration independent**, and therefore, it is much easier to tune the PD controller gains to work well in the whole workspace of the robot in such a structure.
+
+However, this method suffers from a number of **limitations** in practice.
+The most important limitation is the **stringent assumption of the complete information requirement of dynamics matrices**.
+Furthermore, even is all the assumption hold, because of the configuration dependence of the mass matrix, the error dynamics is still not completely decoupled.
+This means that correction in one component may be considered as a disturbance effect to the other components.
+To overcome these limitations, the inverse dynamic approach has been developed and is given in the following section.
+
+
+#### Inverse Dynamics Control {#inverse-dynamics-control}
+
+As seen in the previous section, the tracking performance of a decentralized PD controller implemented in the joint space is not uniform at different configurations.
+To compensate for such effects, a feedforward torque is added to the structure of the controller, by which the shortcomings of the decentralized controller is partially remedied.
+However, the closed-loop performance still faces a number of limitations, which cannot be completely remedied because of the inherent conditions on feedforward structure of that proposed controller.
+To overcome these limitations, in this section, a control technique based on **inverse dynamic feedback** of the manipulator in the joint space is presented.
+
+
+
+
+In the inverse dynamics control (IDC) strategy, the **nonlinear dynamics of the model is used to add a corrective term to the decentralized PD controller**.
+By this means, the **nonlinear and coupling characteristics** of robotic manipulator is significantly **attenuated**, and therefore, the performance of linear controller is significantly improved.
+
+
+
+
+
+{{< figure src="/ox-hugo/taghirad13_inverse_dynamics_control_joint_space.png" caption="Figure 26: General configuration of inverse dynamics control implemented in joint space" >}}
+
+The controller output torque applied to the manipulator may be calculated by:
+
+\begin{subequations}
+\begin{align}
+ \bm{\tau} &= \hat{\bm{M}}\_q \bm{a}\_q + \bm{\tau}\_{fl} \\\\\\
+ \bm{\tau}\_{fl} &= \hat{\bm{C}}\_q \dot{\bm{q}} + \hat{\bm{G}}\_q \\\\\\
+ \bm{a}\_q &= \ddot{\bm{q}}\_d + \bm{K}\_d \dot{\bm{e}}\_q + \bm{K}\_p \bm{e}\_q
+\end{align}
+\end{subequations}
+
+If the knowledge of dynamic matrices is complete, the closed-loop dynamic formulation is simplified to:
+\\[ \hat{\bm{M}}\_q \left( \ddot{\bm{e}}\_q + \bm{K}\_d \dot{\bm{e}}\_q + \bm{K}\_p \bm{e}\_q \right) + \bm{\tau}\_d = 0 \\]
+
+This equation implies that if there exist complete knowledge of the dynamic matrices, the tracking error dynamic equation satisfies a set of second-order systems in the presence of disturbance.
+Consider the case where no disturbance wrench is applied to the manipulator, as the mass matrix \\(\bm{M}\_q\\) is positive definite at all non-singular configurations, it can be inverted, and the error dynamics simplifies to:
+\\[ \ddot{\bm{e}}\_q + \bm{K}\_d \dot{\bm{e}}\_q + \bm{K}\_p \bm{e}\_q = 0 \\]
+
+This control technique is very popular in practice because of the fact that it can significantly **linearize** and **decouple** the dynamic formulation of the closed-loop system for error dynamics components.
+Furthermore, the error dynamic terms are all **configuration independent**, and therefore, it is **much easier to tune the PD controller gains** to perform well in the whole workspace of the robot.
+
+However, note that for a good performance, an **accurate model of the system is required**, and the overall procedure is **not robust** to model uncertainties.
+
+
+### Summary of Motion Control Techniques {#summary-of-motion-control-techniques}
+
+In this section, a number of control techniques have been developed for parallel robots.
+Based on the dynamic formulation given in Section [sec:dynamics](#sec:dynamics), many **model-based** control techniques have been developed for implementation in the task space as well as in the joint space.
+These control techniques are presented from the simplest form of decentralized PD control to more advanced robust and adaptive inverse dynamics control.
+
+A summary of these techniques is given below.
+
+
+##### Dynamic Formulations {#dynamic-formulations}
+
+The dynamic formulation of a parallel robot may be directly represented as a function of motion variable \\(\bm{\mathcal{X}}\\) in the task space as follows:
+
+\begin{equation\*}
+ \bm{M}(\bm{\mathcal{X}})\ddot{\bm{\mathcal{X}}} + \bm{C}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}})\dot{\bm{\mathcal{X}}} + \bm{G}(\bm{\mathcal{X}}) = \bm{\mathcal{F}} + \bm{\mathcal{F}}\_d
+\end{equation\*}
+
+The dynamic formulation may be represented as a function of actuator motion variable \\(\bm{q}\\) as
+
+\begin{equation\*}
+ \bm{M}\_q \ddot{\bm{q}} + \bm{C}\_q \dot{\bm{q}} + \bm{G}\_q = \bm{\tau} + \bm{\tau}\_d
+\end{equation\*}
+
+in which these two formulations are closely related to each other by the following relations:
+
+\begin{equation\*}
+ \begin{aligned}
+ \bm{M}\_q &= \bm{J}^{-T} \bm{M} \bm{J}^{-1} \\\\\\
+ \bm{C}\_q &= \bm{J}^{-T} \left( \bm{C} - \bm{M}\bm{J}^{-1}\dot{\bm{J}} \right) \bm{J}^{-1} \\\\\\
+ \bm{D}\_q &= \bm{J}^{-T} \bm{G} \\\\\\
+ \bm{\tau}\_q &= \bm{J}^{-T} \bm{\mathcal{F}}
+ \end{aligned}
+\end{equation\*}
+
+
+##### Decentralized PD Control {#decentralized-pd-control}
+
+The simplest controller for a parallel robot can be considered as a decentralized PD controller being implemented individually on each error component.
+If such a structure is implemented in the task space, the control effort is calculated by
+
+\begin{equation\*}
+ \bm{\mathcal{F}} = \bm{K}\_d \dot{\bm{e}}\_x + \bm{K}\_p \bm{e}\_x
+\end{equation\*}
+
+and the actuator effort can be generally determined through a force distribution scheme.
+
+For a completely parallel manipulator, the actuator forces can be generated by \\(\bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}}\\) at non-singular configurations.
+
+Decentralized PD control can be directly implemented in the joint space by the following equation:
+
+\begin{equation\*}
+ \bm{\tau} = \bm{K}\_d \dot{\bm{e}}\_q + \bm{K}\_p \bm{e}\_q
+\end{equation\*}
+
+
+##### Feed Forward Control {#feed-forward-control}
+
+The reduce the performance limitations of simple PD control, the control effort may be enforced with a feed forward wrench given by
+
+\begin{equation\*}
+ \bm{\mathcal{F}} = \bm{\mathcal{F}}\_{pd} + \bm{\mathcal{F}}\_{ff}
+\end{equation\*}
+
+in which
+
+\begin{equation\*}
+ \begin{aligned}
+ \bm{\mathcal{F}}\_{ff} &= \bm{K}\_d \dot{\bm{e}}\_x + \bm{K}\_p \bm{e}\_x \\\\\\
+ &+ \hat{\bm{M}}(\bm{\mathcal{X}}\_d)\ddot{\bm{\mathcal{X}}}\_d + \hat{\bm{C}}(\bm{\mathcal{X}}\_d, \dot{\bm{\mathcal{X}}}\_d)\dot{\bm{\mathcal{X}}}\_d + \hat{\bm{G}}(\bm{\mathcal{X}}\_d)
+ \end{aligned}
+\end{equation\*}
+
+where \\(\hat{\bm{M}}\\), \\(\hat{\bm{C}}\\) and \\(\hat{\bm{G}}\\) are estimation of the dynamic matrices.
+
+This controller can be implemented in joint space as follows
+
+\begin{equation\*}
+ \begin{aligned}
+ \bm{\tau} &= \bm{\tau}\_{pd} + \bm{\tau}\_{ff} \\\\\\
+ &= \bm{K}\_d \dot{\bm{e}}\_q + \bm{K}\_p \bm{e}\_q + \bm{J}^{-T} \bm{\mathcal{F}}\_{ff}
+ \end{aligned}
+\end{equation\*}
+
+
+##### Inverse Dynamics Control {#inverse-dynamics-control}
+
+In the inverse dynamics control, the nonlinear dynamics of the model is used to add a corrective term to the decentralized PD controller.
+If such a structure is implemented in the task space, the control effort is calculated by
+
+\begin{equation\*}
+ \begin{aligned}
+ \bm{\mathcal{F}} &= \hat{\bm{M}}(\bm{\mathcal{X}})\bm{a} + \hat{\bm{C}}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}})\dot{\bm{\mathcal{X}}} + \hat{\bm{G}}(\bm{\mathcal{X}}) \\\\\\
+ \bm{a} &= \ddot{\bm{\mathcal{X}}}\_d + \bm{K}\_d \dot{\bm{e}}\_x + \bm{K}\_p \bm{e}\_x
+ \end{aligned}
+\end{equation\*}
+
+In general, the tracking error dynamics can be represented by
+
+\begin{equation\*}
+ \ddot{\bm{e}}\_x + \bm{K}\_d \dot{\bm{e}}\_x + \bm{K}\_p \bm{e}\_x + \hat{\bm{M}}^{-1} \left[ \tilde{\bm{M}} \ddot{\bm{\mathcal{X}}} + \tilde{\bm{C}} \dot{\bm{\mathcal{X}}} + \tilde{\bm{G}} + \bm{\mathcal{F}}\_d \right] = 0
+\end{equation\*}
+
+This controller can be implemented in the joint space as follows:
+
+\begin{equation\*}
+ \begin{aligned}
+ \bm{\tau} &= \hat{\bm{M}}\_q \bm{a}\_q + \hat{\bm{C}}\_q \dot{\bm{q}} + \hat{\bm{G}}\_q \\\\\\
+ \bm{a}\_q &= \ddot{\bm{q}}\_d + \bm{K}\_d \dot{\bm{e}}\_q + \bm{K}\_p \bm{e}\_q
+ \end{aligned}
+\end{equation\*}
+
+by which the tracking error dynamics is summarized as
+
+\begin{equation\*}
+ \ddot{\bm{e}}\_q + \bm{K}\_d \dot{\bm{e}}\_q + \bm{K}\_p \bm{e}\_q + \hat{\bm{M}}\_{q}^{-1} \left[ \tilde{\bm{M}}\_q \ddot{\bm{q}} + \tilde{\bm{C}}\_q \dot{\bm{q}} + \tilde{\bm{G}}\_q + \bm{\mathcal{\tau}}\_d \right] = 0
+\end{equation\*}
+
+
+##### Partial Linearization IDC {#partial-linearization-idc}
+
+To reduce the computational cost of the inverse dynamic control, it is possible to use partial linearization of dynamic formulation, just by gravity compensation, while keeping asymptotic tracking stability of the closed-loop system.
+In which a case, the control input wrench in the task space is simplified to
+
+\begin{equation\*}
+ \bm{\mathcal{F}} = \bm{K}\_d \dot{\bm{e}}\_x + \bm{K}\_p \bm{e}\_x + \hat{\bm{G}}(\bm{\mathcal{X}})
+\end{equation\*}
+
+The following Lyapunov function may be used to analyze the stability of tracking dynamics of the closed-loop system:
+
+\begin{equation\*}
+ \dot{V} = \dot{\bm{\mathcal{X}}}^T \bm{M} \ddot{\bm{\mathcal{X}}} + \frac{1}{2} \dot{\bm{\mathcal{X}}}^T \dot{\bm{M}} \dot{\bm{\mathcal{X}}} + \bm{e}\_x^T \bm{K}\_p \bm{e}\_x
+\end{equation\*}
+
+Stability analysis of the closed-loop system in this case reveals the fact that this simplified version of inverse dynamics control can lead to asymptotic tracking for constant desired trajectories.
+
+
+##### Robust Inverse Dynamics Control {#robust-inverse-dynamics-control}
+
+To accommodate modeling uncertainties in inverse dynamic control, the following robust control scheme in the task space is developed:
+
+\begin{equation\*}
+ \begin{aligned}
+ \bm{\mathcal{F}} &= \hat{\bm{M}}(\bm{\mathcal{X}}) \bm{a}\_r + \hat{\bm{C}}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}})\dot{\bm{\mathcal{X}}} + \hat{\bm{G}}(\bm{\mathcal{X}}) \\\\\\
+ \bm{a}\_r &= \ddot{\bm{\mathcal{X}}}\_d + \bm{K}\_d \dot{\bm{e}}\_x + \bm{K}\_p \bm{e}\_x + \bm{\delta}\_a
+ \end{aligned}
+\end{equation\*}
+
+in which the robustifying corrective term \\(\bm{\delta}\_a\\) is found through a Lyapunov stability analysis of tracking error dynamics.
+The tracking error dynamics can be represented by the following linear and nonlinear components:
+
+\begin{equation\*}
+ \bm{A} = \begin{bmatrix}
+ \bm{0} & \bm{I} \\\\\\
+ -\bm{K}\_p & -\bm{K}\_d
+ \end{bmatrix}, \quad \bm{B} = \begin{bmatrix}
+ \bm{0} \\ \bm{I}
+ \end{bmatrix}
+\end{equation\*}
+
+The corrective term \\(\bm{\delta}\_a\\) can be found as
+
+\begin{equation\*}
+ \bm{\delta}\_a = \left\\{ \begin{matrix}
+ - \rho \frac{v}{\\|v\\|} & \text{if} \\|v\\| > \epsilon \\\\\\
+ - \rho \frac{v}{\epsilon} & \text{if} \\|v\\| \le \epsilon
+ \end{matrix} \right.
+\end{equation\*}
+
+in which \\(v\\) is defined by \\(v = \bm{B}^T \bm{P} \bm{\epsilon}\\), where \\(\bm{P}\\) is the solution to the matrix Lyapunov equation and \\(\epsilon\\) is a smoothing threshold.
+It is shown that by adding this corrective term to the regular inverse dynamics control, the closed-loop system achieves uniform ultimate bounded tracking errors.
+
+
+##### Adaptive Inverse Dynamics Control {#adaptive-inverse-dynamics-control}
+
+In the adaptive version of the inverse dynamics control, full feedback linearization is considered through adaptive update of dynamic formulation matrices.
+The error dynamics in this case is
+
+\begin{equation\*}
+ \dot{\bm{\epsilon}} = \bm{A} \bm{\epsilon} + \bm{B} \bm{\Phi} \tilde{\bm{\theta}}
+\end{equation\*}
+
+in which
+
+\begin{equation\*}
+ \begin{aligned}
+ \bm{A} &= \begin{bmatrix}
+ \bm{0} & \bm{I} \\ -bm{K}\_p & -\bm{K}\_d
+ \end{bmatrix}, \quad \bm{B} = \begin{bmatrix}
+ \bm{0} \\ \bm{I}
+ \end{bmatrix} \\\\\\
+ \bm{\Phi} &= \hat{\bm{M}}^{-1} \bm{\Upsilon}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}}, \ddot{\bm{\mathcal{X}}})
+ \end{aligned}
+\end{equation\*}
+
+Based on the Lyapunov stability analysis, by using the following Lyapunov function
+
+\begin{equation\*}
+ V = \bm{\epsilon}^T \bm{P} \bm{\epsilon} + \tilde{\bm{\theta}}^T \bm{\Gamma} \bm{\theta}
+\end{equation\*}
+
+the following parameter adaptation law is derived for updates
+
+\begin{equation\*}
+ \dot{\hat{\bm{\theta}}} = - \bm{\Gamma}^{-1} \bm{\Phi}^T \bm{B}^T \bm{P} \bm{\epsilon}
+\end{equation\*}
+
+By this means, the closed-loop system achieves asymptotic tracking performance, while the parameter estimation errors remain bounded.
+
+
+### Motion Control of the Stewart-Gough Platform {#motion-control-of-the-stewart-gough-platform}
+
+
+#### Control in the Task space {#control-in-the-task-space}
+
+For the Stewart-Gough platform, the motion variable in the task space is a six-dimensional vector
+\\[ \bm{\mathcal{X}} = \begin{bmatrix} \bm{x}\_p \\ \bm{\theta} \end{bmatrix} \\]
+with:
+
+- \\(\bm{x}\_p = [x\_p\ y\_p\ z\_p]^T\\) is the position vector of the motion platform center of mass
+- \\(\bm{\theta} = \theta [s\_x\ s\_y\ s\_z]^T = [\theta\_x\ \theta\_y\ \theta\_z]^T\\) is the moving platform orientation expressed by screw coordinates
+
+Therefore, the tracking error is defined as \\(\bm{e} = [e\_x\ e\_y\ e\_z\ e\_{\theta\_x}\ e\_{\theta\_y}\ e\_{\theta\_z}]^T\\).
+
+The decentralized controller consists of six disjoint proportional derivative controllers acting on each error component and is denoted by \\(\bm{K}\_d s + \bm{K}\_p\\).
+
+The output of the controller is denoted by \\(\bm{\mathcal{F}} = [F\_x\ F\_y\ F\_z\ \tau\_x\ \tau\_y\ \tau\_z]^T\\).
+Note that since the output of the controller is defined in the task space, each wrench component directly manipulates the corresponding tracking error component, and therefore, the overall tracking performance of the manipulator is suitable is high controller gains are used.
+
+In practice, the calculated output wrench is transformed into actuator forces through the force distribution block corresponding to the inverse of Jacobian transpose.
+
+
+#### Control in the Joint space {#control-in-the-joint-space}
+
+The joint variable \\(\bm{q}(t)\\) is a six-dimensional vector consisting of the limb lengths denoted by \\(\bm{q} = [l\_1\ l\_2\ l\_3\ l\_4\ l\_5\ l\_6]^T\\).
+Therefore, the tracking error is defined as \\(\bm{e}\_q = \bm{q}\_d - \bm{q}\\), in which is the desired motion variable in the joint space \\(\bm{q}\_d\\) is determined by the solution of inverse kinematics, and \\(\bm{q}\\) is given by direct measurement of the limb lengths.
+
+The decentralized controller, therefore, consists of six disjoint PD controllers acting on each error component.
+The output of the controller directly generates the actuator torques denoted by \\(\tau\\).
+
+In simulation, it is observe that the orientation error significantly increase in the joint space control scheme.
+The main reason is that the controller gains directly penalize the position error of the limb lengths, and not the orientation errors, and therefore, there is no direct controller action to be suitably tuned to reduce the orientation error.
+
+Comparing the closed-loop performance of the PD controllers designed in the joint space to those designed in the task space, it can be concluded that **tuning of the PD gains for a suitable performance is much easier in task space designs**.
+Furthermore, a very small error signature in the joint space may be accumulated to produce relatively larger tracking errors in the task space.
+Hence, it is recommended to design and implement controllers in the task space, if the required motion variables can be directly measured or the forward kinematic solution can be calculated in an online routine.
+
+
+## Force Control {#force-control}
+
+
+
+
+### Introduction {#introduction}
+
+In many applications, it may occur that the robot moving platform is in contact with a **stiff** environment and **specific interacting wrench is required**.
+In such applications, the contact wrench describes the state of interaction more effectively than the position and orientation of the moving platform.
+
+
+
+
+The problem of force control can be described as to derive the actuator forces required to generate a prescribed desired wrench at the manipulator moving platform, when the manipulator is carrying out its desired motion.
+
+
+
+If a pure **motion control** scheme is used for manipulator, in case it contacts an environment, the robot does not sense the presence of the environment, and its driving forces become harshly high to reduce the tracking errors.
+In such a case, the robot may break the object it is in contact or will break its internal structure.
+Additional sensors should be included in the manipulator in order for it to be able to feel the interaction and to control the interacting forces.
+Different wrench sensors are developed for such applications, and it is possible to use joint torque or link force measurement units to determine the projection of the interacting forces in the joint space.
+
+The use of wrench sensors either in the task space or in the joint space open horizons to use different force control topologies for the manipulators.
+Using such sensors does not imply that regular motion sensors used in motion control schemes are not necessary.
+The use of motion sensors and the usual corresponding control topologies are usually necessary, since the motion of the manipulator is one of the outputs to be controlled.
+Depending on the type and configuration of the wrench sensors, different force control topologies are developed.
+
+
+### Controller Topology {#controller-topology}
+
+For a force control scheme, the desired interacting wrench of the moving platform and the environment may be of interest.
+This quantity may be denoted by \\(\bm{\mathcal{F}}\_d\\), which has the same dimension and structure of the manipulator wrench \\(\bm{\mathcal{F}}\\).
+To carry out such a control task in a closed-loop structure, it is necessary to **measure the output wrench** of the manipulator through an instrumentation system.
+
+Although there are many commercial six-degrees-of-freedom wrench sensors available in the market, they are usually more expensive than single joint force measurement units.
+Another alternative for force measurement is **direct measurement of the actuator forces**.
+Many commercial linear actuators are available in the market in which **embedded force measurement** is considered in their design.
+Therefore, it might be preferable in some applications to use direct actuator force measurements to carry out the feedback control.
+
+
+#### Cascade Control {#cascade-control}
+
+In a general force control scheme, the **prime objective** is tracking of the interacting wrench between the moving platform and the environment.
+However, note that the motion control of the robot when the robot is in interaction with the environment is also another **less-important objective** and when the contact of the robot moving platform is released, motion control becomes the prime objective.
+
+
+
+
+To follow **two objectives** with different properties in one control system, usually a **hierarchy** of two feedback loops is used in practice.
+This kind of control topology is called **cascade control**, which is used when there are **several measurements and one prime control variable**.
+Cascade control is implemented by **nesting** the control loops, as shown in Figure [fig:cascade_control](#fig:cascade_control).
+The output control loop is called the **primary loop**, while the inner loop is called the secondary loop and is used to fulfill a secondary objective in the closed-loop system.
+
+
+
+To improve the performance of the control system for a particular objective, it is important to **choose the right variables for internal and external feedback loops**, and to **design suitable controllers for each feedback system**.
+Although these differ in different topologies described in the following sections, some **general rules** are applied to design a well performing cascade control system.
+
+A general idea in cascade control design is the **ideal case**, in which the inner loop is designed so tight that the secondary (inner) loop behaves as a **perfect servo**, and responds very quickly to the internal control command.
+This idea is effectively used in many applications, wherein a nearly-perfect actuator to respond to the requested commands is designed by using an inner control feedback.
+
+
+
+
+The design criteria for the inner loop is to have a high control gain such that the time response of the secondary variable is at least 5 times more than that of the primary variable, and such that it can overcome the effect of **disturbances** and **unmodelled dynamics** in the **internal feedback structure**.
+
+
+
+Consider first the cascade control topology shown in Figure [fig:taghira13_cascade_force_outer_loop](#fig:taghira13_cascade_force_outer_loop) in which the measured variables are both in the **task space**.
+The inner loop is constructed by position feedback while the outer loop is based on force feedback.
+As seen in Figure [fig:taghira13_cascade_force_outer_loop](#fig:taghira13_cascade_force_outer_loop), the force controller block is fed to the motion controller, and this might be seen as the **generated desired motion trajectory for the inner loop**.
+
+The output of motion controller is also designed in the task space, and to convert it to implementable actuator force \\(\bm{\tau}\\), the force distribution block is considered in this topology.
+
+
+
+{{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop.png" caption="Figure 28: Cascade topology of force feedback control: position in inner loop and force in outer loop. Moving platform wrench \\(\bm{\mathcal{F}}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured in the task space" >}}
+
+Other alternatives for force control topology may be suggested based on the variations of position and force measurements.
+If the force is measured in the joint space, the topology suggested in Figure [fig:taghira13_cascade_force_outer_loop_tau](#fig:taghira13_cascade_force_outer_loop_tau) can be used.
+In this topology, the measured actuator force vector \\(\bm{\tau}\\) is mapped into its corresponding wrench in the task space by the Jacobian transpose mapping \\(\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}\\).
+
+
+
+{{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop_tau.png" caption="Figure 29: Cascade topology of force feedback control: position in inner loop and force in outer loop. Actuator forces \\(\bm{\tau}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured" >}}
+
+Consider the case where the force and motion variables are both measured in the **joint space**.
+Figure [fig:taghira13_cascade_force_outer_loop_tau_q](#fig:taghira13_cascade_force_outer_loop_tau_q) suggests the force control topology in the joint space, in which the inner loop is based on measured motion variable in the joint space, and the outer loop uses the measured actuator force vector.
+In this topology, it is advised that the force controller is designed in the **task** space, and the Jacobian transpose mapping is used to project the measured actuator force vector into its corresponding wrench in the task space.
+However, as the inner loop is constructed in the joint space, the desired motion variable \\(\bm{\mathcal{X}}\_d\\) is mapped into joint space using **inverse kinematic** solution.
+
+Therefore, the structure and characteristics of the position controller in this topology is totally different from that given in the first two topologies.
+
+
+
+{{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop_tau_q.png" caption="Figure 30: Cascade topology of force feedback control: position in inner loop and force in outer loop. Actuator forces \\(\bm{\tau}\\) and joint motion variable \\(\bm{q}\\) are measured in the joint space" >}}
+
+
+#### Force Feedback in Inner Loop {#force-feedback-in-inner-loop}
+
+Consider the force control scheme in which the **motion-force relation is the prime objective**.
+In such a case, force tracking is not the primary objective, and it is advised that the outer loop of cascade control structure consists of a motion control feedback.
+
+Since different type of measurement units may be used in parallel robots, different control topologies may be constructed to implement such cascade controllers.
+
+Figure [fig:taghira13_cascade_force_inner_loop_F](#fig:taghira13_cascade_force_inner_loop_F) illustrates the cascade control topology for the system in which the measured variables are both in the task space (\\(\bm{\mathcal{F}}\\) and \\(\bm{\mathcal{X}}\\)).
+The inner loop is loop is constructed by force feedback while the outer loop is based on position feedback.
+By this means, when the manipulator is not in contact with a stiff environment, position tracking is guaranteed through the primary controller.
+However, when there is interacting wrench \\(\bm{\mathcal{F}}\_e\\) applied to the moving platform, this structure controls the force-motion relation.
+This configuration may be seen as if the **outer loop generates a desired force trajectory for the inner loop**.
+
+
+
+{{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_F.png" caption="Figure 31: Cascade topology of force feedback control: force in inner loop and position in outer loop. Moving platform wrench \\(\bm{\mathcal{F}}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured in the task space" >}}
+
+Other alternatives for control topology may be suggested based on the variations of position and force measurements.
+If the force is measured in the joint space, control topology shown in Figure [fig:taghira13_cascade_force_inner_loop_tau](#fig:taghira13_cascade_force_inner_loop_tau) can be used.
+In such case, the Jacobian transpose is used to map the actuator force to its corresponding wrench in the task space.
+
+
+
+{{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_tau.png" caption="Figure 32: Cascade topology of force feedback control: force in inner loop and position in outer loop. Actuator forces \\(\bm{\tau}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured" >}}
+
+If the force and motion variables are both measured in the **joint** space, the control topology shown in Figure [fig:taghira13_cascade_force_inner_loop_tau_q](#fig:taghira13_cascade_force_inner_loop_tau_q) is suggested.
+The inner loop is based on the measured actuator force vector in the joint space \\(\bm{\tau}\\), and the outer loop is based on the measured actuated joint position vector \\(\bm{q}\\).
+In this topology, the desired motion in the task space is mapped into the joint space using **inverse kinematic** solution, and **both the position and force feedback controllers are designed in the joint space**.
+Thus, independent controllers for each joint may be suitable for this topology.
+
+
+
+{{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_tau_q.png" caption="Figure 33: Cascade topology of force feedback control: force in inner loop and position in outer loop. Actuator forces \\(\bm{\tau}\\) and joint motion variable \\(\bm{q}\\) are measured in the joint space" >}}
+
+
+### Stiffness Control {#stiffness-control}
+
+
+#### Single-Degree-of-Freedom Stiffness Control {#single-degree-of-freedom-stiffness-control}
+
+
+#### General Stiffness Control {#general-stiffness-control}
+
+
+#### Stiffness Contorl of the Stewart-Gough Platform {#stiffness-contorl-of-the-stewart-gough-platform}
+
+
+### Direct Force Control {#direct-force-control}
+
+
+
+{{< figure src="/ox-hugo/taghira13_direct_force_control.png" caption="Figure 34: Direct force control scheme, force feedback in the outer loop and motion feedback in the inner loop" >}}
+
+
+### Impedance Control {#impedance-control}
+
+For the stiffness control and direct force control schemes, it is observed that when the manipulator-moving platform is in contact with a stiff environment, the motion variable \\(\bm{\mathcal{X}}\\) and the interacting force variable \\(\bm{\mathcal{F}}\\) are two dynamically **dependent** quantities.
+
+In stiffness control, it is aimed to adjust the **static** relation between these two quantities.
+In this scheme, no force measurement is required, however, careful design on the desired motion trajectory and PD controller gains is needed to tune the stiffness property of the interaction at steady stage.
+
+In force control schemes, on the other hand, the force tracking is the prime objective, and force measurement is a stringent requirement to implement such schemes.
+
+The main reason that the motion and force variables are not being gable to be controlled independently is that for an n-degrees-of-freedom manipulator, only n-independent control inputs are available, and therefore, only n-independent variables can be controlled, while the force and motion quantities count to \\(2n\\) independent variables.
+
+
+
+
+The key idea behind **impedance control** schemes, is to **tune the dynamic relation between the force and the motion variables**, and not a hierarchy of tracking objectives in force and in position variables.
+In this scheme, contrary to stiffness control schemes, **both force and position variables are measured** and used in the control structure.
+
+
+
+Impedance is usually denoted by \\(\bm{Z}(s)\\) and it may be represented by writing its magnitude and phase in the form of \\(\abs{\bm{Z}(s)}\\) and \\(\angle{\bm{Z}(s)}\\).
+The magnitude of the complex impedance \\(\abs{\bm{Z}}\\) is the ratio of the effort amplitude to that of the flow, while the phase \\(\angle{\bm{Z}}\\) is the phase shift by which the flow is ahead of the effort.
+
+
+
+
+Mechanical Impedance is defined as the ratio of the Laplace transform of the mechanical effort to the Laplace transform of the mechanical flow:
+
+\begin{equation}
+ \bm{Z}(s) = \frac{\bm{F}(s)}{\bm{v}(s)}
+\end{equation}
+
+in which effort in mechanical systems is represented by force \\(\bm{F}\\) and flow is represented by velocity \\(\bm{v}\\).
+
+
+
+Nevertheless, note that Laplace transform is only applicable for **linear time invariant** systems, and for a parallel manipulator the dynamic formulation of which is nonlinear, the concept of mechanical impedance may be extended to the differential equation relating the mechanical wrench \\(\bm{\mathcal{F}}\\) to motion twist \\(\dot{\bm{\mathcal{X}}}\\).
+
+
+
+
+Consider an RLC circuit depicted in Figure [fig:taghirad13_impedance_control_rlc](#fig:taghirad13_impedance_control_rlc).
+The differential equation relating voltage \\(v\\) to the current \\(i\\) is given by
+\\[ v = L\frac{di}{dt} + Ri + \int\_0^t \frac{1}{C} i(\tau)d\tau \\]
+in which \\(L\\) denote the inductance, \\(R\\) the resistance and \\(C\\) the capacitance.
+
+The impedance of the system may be found from the Laplace transform of the above equation:
+
+\begin{equation}
+ Z(s) = \frac{v(s)}{i(s)} = Ls + R + \frac{1}{Cs} \label{eq:rlc\_impedance}
+\end{equation}
+
+
+
+
+Consider the mass-spring-damper system depicted in Figure [fig:taghirad13_impedance_control_rlc](#fig:taghirad13_impedance_control_rlc).
+The governing dynamic formulation for this system is given by
+\\[ m \ddot{x} + c \dot{x} + k x = f \\]
+in which \\(m\\) denote the body mass, \\(c\\) the damper viscous coefficient and \\(k\\) the spring stiffness.
+
+The impedance of the system may be found from the Laplace transform of the above equation:
+
+\begin{equation}
+ Z(s) = \frac{f(s)}{v(s)} = ms + c + \frac{k}{s} \label{eq:mass\_spring\_damper\_impedance}
+\end{equation}
+
+
+
+
+An impedance \\(\bm{Z}(s)\\) is called
+
+- **Inductive** if \\(\abs{\bm{Z}(0)} = 0\\)
+- **Resistive** if \\(\abs{\bm{Z}(0)} = R\\)
+- **Capacitive** if \\(\lim\_{s\to 0} \abs{\bm{K}(s)} = \infty\\)
+
+
+
+In the impedance control scheme, **regulation of the motion-force dynamic relation is the prime objective**, and since force tracking is not the primary objective, it is advised to used a cascade control structure with motion control feedback in the outer loop and force feedback in the inner loop.
+
+Therefore, when the manipulator is not in contact with a stiff environment, position tracking is guaranteed by a primary controller.
+However, when there is an interacting wrench \\(\bm{\mathcal{F}}\_e\\) applied to the moving platform, this structure may be designed to control the force-motion dynamic relation.
+
+As a possible impedance control scheme, consider the closed-loop system depicted in Figure [fig:taghira13_impedance_control](#fig:taghira13_impedance_control), in which the position feedback is considered in the outer loop, while force feedback is used in the inner loop.
+This structure is advised when a desired impedance relation between the force and motion variables is required that consists of desired inductive, resistive, and capacitive impedances.
+As shown in Figure [fig:taghira13_impedance_control](#fig:taghira13_impedance_control), the motion-tracking error is directly determined from motion measurement by \\(\bm{e}\_x = \bm{\mathcal{X}}\_d - \bm{\mathcal{X}}\\) in the outer loop and the motion controller is designed to satisfy the required impedance.
+
+Moreover, direct force-tracking objective is not assigned in this control scheme, and therefore the desired force trajectory \\(\bm{\mathcal{F}}\_d\\) is absent in this scheme.
+However, an auxiliary force trajectory \\(\bm{\mathcal{F}}\_a\\) is generated from the motion control law and is used as the reference for the force tracking.
+By this means, no prescribed force trajectory is tracked, while the **motion control scheme would advise a force trajectory for the robot to ensure the desired impedance regulation**.
+
+
+
+{{< figure src="/ox-hugo/taghira13_impedance_control.png" caption="Figure 36: Impedance control scheme; motion feedback in the outer loop and force feedback in the inner loop" >}}
+
+The required wrench \\(\bm{\mathcal{F}}\\) in the impedance control scheme, is based on inverse dynamics control and consists of three main parts.
+In the inner loop, the force control scheme is based on a feedback linearization part in addition to a mass matrix adjustment, while in the outer loop usually a linear motion controller is considered based on the desired impedance requirements.
+
+Although many different impedance structures may be considered as the basis of the control law, in Figure [fig:taghira13_impedance_control](#fig:taghira13_impedance_control), a linear impedance relation between the force and motion variables is generated that consists of desired inductive \\(\bm{M}\_d\\), resistive \\(\bm{C}\_d\\) and capacitive impedances \\(\bm{K}\_d\\).
+
+According to Figure [fig:taghira13_impedance_control](#fig:taghira13_impedance_control), the controller output wrench \\(\bm{\mathcal{F}}\\), applied to the manipulator may be formulated as
+\\[ \bm{\mathcal{F}} = \hat{\bm{M}} \bm{M}\_d^{-1} \bm{e}\_F + \bm{\mathcal{F}}\_{fl} \\]
+with:
+
+\begin{align\*}
+ \bm{e}\_F &= \bm{\mathcal{F}}\_a - \bm{\mathcal{F}}\_m \\\\\\
+ \bm{\mathcal{F}}\_a &= \bm{M}\_d \ddot{\bm{\mathcal{X}}}\_{d} + \bm{C}\_{d} \dot{\bm{e}}\_{x} + \bm{K}\_{d} \bm{e}\_{x}
+\end{align\*}
+
+\\(\bm{M}\_d\\) denotes the desired inductive impedance, \\(\bm{C}\_d\\) the desired resistive impedance and \\(\bm{K}\_d\\) is desired capacitive impedance matrices.
+
+The feedback linearizing term is given by:
+\\[ \bm{\mathcal{F}}\_{fl} = \hat{\bm{C}}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}}) \dot{\bm{\mathcal{X}}} + \hat{\bm{G}}(\bm{\mathcal{X}}) + \bm{\mathcal{F}}\_m \\]
+
+If the information on the dynamic matrices is complete, and if the force measurements are noise free (\\(\bm{\mathcal{F}}\_m = \bm{\mathcal{F}}\_e\\)), the closed-loop dynamic formulation simplifies to:
+\\[ \bm{M}\_d \ddot{\bm{e}}\_x + \bm{C}\_d \dot{\bm{e}}\_x + \bm{K}\_d \bm{e}\_x = \bm{\mathcal{F}}\_e \\]
+
+And thus the closed-loop error dynamic equation satisfies a set of second-order systems with the desired impedance coefficients in relation to the interacting force.
+In other words, **the control structure guarantees that the force and motion relation follows a desired impedance**.
+Therefore, by choosing appropriate impedance matrices, the transient performance of the force-motion relation can be shaped so as to have a fast but stable interaction.
+By this means, what is controlled is the dynamic relation between the force and motion variables, and not directly the position.
+However, if the robot is moving freely in space and has no interaction with the environment, \\(\bm{\mathcal{F}}\_e = 0\\), the closed-loop system will provide a suitable motion tracking thanks to the motion controller in the outer loop.
+
+The impedance control scheme is very popular in practice, wherein tuning the force and motion relation in a robot manipulator interacting with a stiff environment is the prime objective.
+However, note that for a good performance, an accurate model of the system is required, and the obtained force and motion dynamics are not robust to modeling uncertainty.
+
+# Bibliography
+Taghirad, H., *Parallel robots : mechanics and control* (2013), Boca Raton, FL: CRC Press. [↩](#1a9683d2a60e031fa9b8ed16f4944c9f)
diff --git a/content/paper/_index.md b/content/paper/_index.md
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++++
+title = "Papers"
+author = ["Thomas Dehaeze"]
+type = "paper"
+draft = false
++++
+
+Here is the list of papers I took note about.
diff --git a/content/paper/alkhatib03_activ_struc_vibrat_contr.md b/content/paper/alkhatib03_activ_struc_vibrat_contr.md
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++++
+title = "Active structural vibration control: a review"
+author = ["Thomas Dehaeze"]
+draft = false
+tags = ["tag1", "tag2"]
+categories = ["cat1", "cat2"]
++++
+
+Tags
+:
+
+
+Reference
+: (Rabih Alkhatib \& Golnaraghi, 2003)
+
+Author(s)
+: Alkhatib, R., & Golnaraghi, M. F.
+
+Year
+: 2003
+
+
+## Process of designing an active vibration control system {#process-of-designing-an-active-vibration-control-system}
+
+1. Analyze the structure to be controled
+2. Obtain an idealized mathematical model with FEM or experimental modal analysis
+3. Reduce the model order is necessary
+4. Analyze the resulting model: dynamics properties, types of disturbances, ...
+5. Quantify sensors and actuators requirements. Decide on their types and location
+6. Analyze the impact of the sensors and actuators on the overall dynamic characteristics
+7. Specify performance criteria and stability tradeoffs
+8. Device of the type of control algorythm to be employed and design a controller to meet the specifications
+9. Simulate the resulting controlled system on a computer
+10. If the controller does not meet the requirements, adjust the specifications or modify the type of controller
+11. Choose hardware and software and integrate the components on a pilot plant
+12. Formulate experiments and perform system identification and model updating
+13. Implement controller and carry out system test to evaluate the performance
+
+
+## Feedback control {#feedback-control}
+
+
+### Active damping {#active-damping}
+
+The objective is to reduce the resonance peaks of the closed loop transfer function.
+
+\\[T(s) = \frac{G(s)H(s)}{1+G(s)H(s)}\\]
+
+Then \\(T(s) \approx G(s)\\) except near the resonance peaks where the amplitude is reduced.
+
+This method can be realized without a model of the structure with **guaranteed stability**, granted that the actuators and sensors are **collocated**.
+
+
+### Model based feedback {#model-based-feedback}
+
+Objective: keep a control variable (position, velocity, ...) to a desired value in spite of external disturbances \\(d(s)\\).
+
+We have \\[\frac{y(s)}{d(s)} = \frac{1}{1+G(s)H(s)}\\] so we need large values of \\(G(s)H(s)\\) in the frequency range where the disturbance has considerable effect.
+
+To do so, we need a mathematical model of the system, then the control bandwidth and effectiveness are restricted by the accuracy of the model.
+Unmodeled structural dynamics may destabilize the system.
+
+
+## Feedforward Control {#feedforward-control}
+
+We need a signal that is correlated to the disturbance. Then feedforward can improve performance over simple feedback control.
+
+An adaptive filter manipulates the signal correlated to the disturbance and the output is applied to the system by the actuator.
+The filter coefficients are adapted in such a way that an error signal is minimized.
+The idea is to generate a secondary disturbance, which destructively interferes with the effect of the primary distance at the location of the error sensor.
+However, there is no guarantee that the global response is also reduced at other locations.
+
+The method is considered to be a **local technique**, in contrast to feedback which is global.
+
+Contrary to active damping which can only reduce the vibration near the resonance, **feedforward control can be effective for any frequency**.
+The major restriction to the application of feedforward adaptive filtering is the accessibility of a reference signal correlated to the disturbance.
+
+
+
+
Table 1:
+ Comparison of control strategies
+
+
+
+\\[ S(s) + T(s) = 1 \\]
+
+
+
+
+- **Command following**: \\(S=0\\) and \\(T=1\\) => large gains
+- **Disturbance rejection**: \\(S=0\\) => large gains
+- **Sensor noise attenuation**: \\(T\\) small where the noise is concentrated
+- **Control Sensitivity minimization**: \\(K S\\) small
+- **Robustness to modeling errors**: \\(T\\) small in the frequency range of the expected model undertainties
+
+
+
+
+\\(\mathcal{H}\_\infty\\) control is a design technique with a state-space computation solution that utilizes frequency-dependent weighting functions to tune the controller's performance and robustness characteristics.
+
+
+
+
+\\[T\_{zw} = F\_l (P, K) = P\_{11} + P\_{12}K (I-P\_{22})^{-1} P\_{21}\\]
+
+
+
+
+- **Low frequency gain**: set to the inverse of the desired steady state tracking error
+- **High frequency gain**: set to limit overshoot (\\(0.1\\) to \\(0.5\\) is a good compromise between overshoot and response speed)
+- **Crossover frequency**: chosen to limit the maximum closed-loop time constant (\\(\omega\_c \approx 1/\tau\\))
+
+
+
+
+A high pass weight is usualy used on \\(T\\) because the noise energy is mostly concentrated at high frequencies. It should have the following characteristics:
+
+- The **crossover frequency** is chosen to **limit the closed-loop bandwidth**
+- The **high frequency gain** is set high to proide **sensor noise rejection** and high frequency gain attenuation
+
+
+
+
+\\[ X(s) = \underbrace{\frac{cs + k}{ms^2 + cs + k}}\_{T\_{wx}(s)} W(s) + \underbrace{\frac{1}{ms^2 + cs + k}}\_{T\_{Fx}(s)} F(s) \\]
+
+
+
Table 1:
+ Active isolation techniques
+
+
Table 2:
+ Sensor fusion configurations
+
+
Table 1:
+ Parallel VS serial manipulators
+
+
+A Gaussian random process is usually described by its autocorrelation function or its Power Spectral Density.
+
+The autocorrelation function of a random process \\(\mathcal{X}\\) is
+
+\begin{equation}
+ R\_{\mathcal{X}}(\tau) = E[\mathcal{X}(t)\mathcal{X}(t + \tau)]
+\end{equation}
+
+where \\(E\\) is the expected value operator.
+
+The variance of the process is equal to \\(R\_\mathcal{X}(0)\\) and is the expected value of the varying part squared:
+
+\begin{equation}
+ \text{Var} \mathcal{X} = E \left[ (\mathcal{X} - E[\mathcal{X}])^2 \right]
+\end{equation}
+
+The standard deviation \\(\sigma\\) is the square root of the variance:
+
+\begin{equation}
+ \sigma\_\mathcal{X} = \sqrt{\text{Var} \mathcal{X}}
+\end{equation}
+
+The standard deviation is also the Root Mean Square (RMS) value of a zero-mean random process.
+
+The Power Spectral Density \\(S\_\mathcal{X}(f)\\) of a random process represents the distribution of power (or variance) across frequency \\(f\\).
+
+For example, if the random process under consideration was measured in volts, the power spectral density would have the units of \\(V^2/\text{Hz}\\).
+
+The Power Spectral Density can be obtained from the autocorrelation function from the Wiener-Khinchin relation:
+
+\begin{equation}
+ S\_{\mathcal{X}} = 2 \mathcal{F}\\{ R\_\mathcal{X}(\tau) \\} = 2 \int\_{-\infty}^{\infty} R\_\mathcal{X}(\tau) e^{-2j\pi f \tau} d\tau
+\end{equation}
+
+If the power Spectral Density is known, the variance of the generating process can be found from the area under the curve:
+
+\begin{equation}
+ \sigma\_\mathcal{X}^2 = E[\mathcal{X}^2(t)] = R\_\mathcal{X}(0) = \int\_0^\infty S\_\mathcal{X}(f) df
+\end{equation}
+
+Rather than plotting the frequency distribution of power, it is often convenient to plot the frequency distribution of the standard deviation, which is referred to as the spectral density.
+It is related to the power spectral density by a square root:
+
+\begin{equation}
+ \text{spectral density} = \sqrt{S\_\mathcal{X}(f)}
+\end{equation}
+
+The units of \\(\sqrt{S\_\mathcal{X}(f)}\\) are \\(\text{units}/\sqrt{Hz}\\).
+
+The spectral density if preferred in the electronics literature as the RMS value of a noise process can be determined directly from the noise density and effective bandwidth.
+
+
+### Resolution {#resolution}
+
+The random noise of a position sensor causes an uncertainty in the measured position.
+If the distance between two measured locations is smaller than the uncertainty, it is possible to mistake one point for the other.
+
+To characterize the resolution, we use the probability that the measured value is within a certain error bound.
+
+If the measurement noise is approximately Gaussian, the resolution can be quantified by the standard deviation \\(\sigma\\) (RMS value).
+
+The empirical rule states that there is a \\(99.7\%\\) probability that a sample of a Gaussian random process lie within \\(\pm 3 \sigma\\).
+This if we define the resolution as \\(\delta = 6 \sigma\\), we will referred to as the \\(6\sigma\text{-resolution}\\).
+
+Another important parameter that must be specified when quoting resolution is the sensor bandwidth.
+There is usually a trade-off between bandwidth and resolution (figure [3](#orgd8c6776)).
+
+
+
+{{< figure src="/ox-hugo/fleming13_tradeoff_res_bandwidth.png" caption="Figure 3: The resolution versus banwidth of a position sensor." >}}
+
+Many type of sensor have a limited full-scale-range (FSR) and tend to have an approximated proportional relationship between the resolution and range.
+As a result, it is convenient to consider the ratio of resolution to the FSR, or equivalently, the dynamic range (DNR).
+A convenient method for reporting this ratio is in parts-per-million (ppm):
+
+\begin{equation}
+ \text{DNR}\_{\text{ppm}} = 10^6 \frac{\text{full scale range}}{6\sigma\text{-resolution}}
+\end{equation}
+
+
+## Comparison and summary {#comparison-and-summary}
+
+
+
+
Table 1:
+ Summary of position sensor characteristics. The dynamic range (DNR) and resolution are approximations based on a full-scale range of \(100\,\mu m\) and a first order bandwidth of \(1\,kHz\)
+
+
Table 1:
+ Typical characteristics of sensors used for isolation in hexapod systems
+
+ Table 1:
+ Zero/Low and High stiffness actuators
+
+
+| **Categories** | **Pros** | **Cons** |
+|----------------|---------------------------|-----------------------------|
+| Zero stiffness | No vibration transmission | Large and Heavy |
+| Low stiffness | High vibration isolation | Typically for low load |
+| High Stiffness | High control bandwidth | High vibration transmission |
+
+
+## Time Delay of Piezoelectric Electronics {#time-delay-of-piezoelectric-electronics}
+
+In this paper, the piezoelectric actuator/electronics adds a time delay which is much higher than the time delay added by the voice coil/electronics.
+
+
+## Definition of low-stiffness and high-stiffness actuator {#definition-of-low-stiffness-and-high-stiffness-actuator}
+
+- **Low Stiffness** actuator is defined as the ones where the transmissibility stays below 0dB at all frequency
+- **High Stiffness** actuator is defined as the ones where the transmissibility goes above 0dB at some frequency
+
+{{< figure src="/ox-hugo/ito16_low_high_stiffness_actuators.png" caption="Figure 1: Definition of low-stiffness and high-stiffness actuator" >}}
+
+
+## Low-Stiffness / High-Stiffness characteristics {#low-stiffness-high-stiffness-characteristics}
+
+- The low stiffness actuators achieve smooth transition from active isolation to passive isolation.
+- The high stiffness actuators can have a gap between the passive and active isolation vibration where the vibrations are amplified in a certain frequency band.
+
+
+## Controller Design {#controller-design}
+
+{{< figure src="/ox-hugo/ito16_transmissibility.png" caption="Figure 2: Obtained transmissibility" >}}
+
+
+## Discussion {#discussion}
+
+The stiffness requirement for low-stiffness actuators can be rephrased in the frequency domain as: "the cross-over frequency of the sensitivity function of the feedback system must be larger than \\(\sqrt{2} \omega\_r\\) with \\(\omega\_r\\) is the resonant frequency of the uncontrolled system".
+
+In practice, this is difficult to achieve with piezoelectric actuators as their first resonant frequency \\(\omega\_r\\) is **too close to other resonant frequencies to ensure close-loop stability**.
+In contrast, the frequency band between the first and the other resonances of Lorentz actuators can be broad by design making them more suitable to construct a low-stiffness actuators.
+
+# Bibliography
+Ito, S., & Schitter, G., *Comparison and classification of high-precision actuators based on stiffness influencing vibration isolation*, IEEE/ASME Transactions on Mechatronics, *21(2)*, 1169–1178 (2016). http://dx.doi.org/10.1109/tmech.2015.2478658 [↩](#aad53368e29e8a519e2f63857044fa46)
+
+
+## Backlinks {#backlinks}
+
+- [Actuators]({{< relref "actuators" >}})
diff --git a/content/paper/jiao18_dynam_model_exper_analy_stewar.md b/content/paper/jiao18_dynam_model_exper_analy_stewar.md
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index 0000000..0c618fe
--- /dev/null
+++ b/content/paper/jiao18_dynam_model_exper_analy_stewar.md
@@ -0,0 +1,20 @@
++++
+title = "Dynamic modeling and experimental analyses of stewart platform with flexible hinges"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
+
+Reference
+: (Jian Jiao {\it et al.}, 2018)
+
+Author(s)
+: Jiao, J., Wu, Y., Yu, K., & Zhao, R.
+
+Year
+: 2018
+
+# Bibliography
+Jiao, J., Wu, Y., Yu, K., & Zhao, R., *Dynamic modeling and experimental analyses of stewart platform with flexible hinges*, Journal of Vibration and Control, *25(1)*, 151–171 (2018). http://dx.doi.org/10.1177/1077546318772474 [↩](#ee917739f88877d6c2758c1c36565deb)
diff --git a/content/paper/legnani12_new_isotr_decoup_paral_manip.md b/content/paper/legnani12_new_isotr_decoup_paral_manip.md
new file mode 100644
index 0000000..f722f5b
--- /dev/null
+++ b/content/paper/legnani12_new_isotr_decoup_paral_manip.md
@@ -0,0 +1,34 @@
++++
+title = "A new isotropic and decoupled 6-dof parallel manipulator"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms" >}})
+
+Reference
+: (Legnani {\it et al.}, 2012)
+
+Author(s)
+: Legnani, G., Fassi, I., Giberti, H., Cinquemani, S., & Tosi, D.
+
+Year
+: 2012
+
+- Concepts of isotropy and decoupling for parallel manipulators
+- **isotropy**: the kinetostatic properties (same applicable force, same possible velocity, same stiffness) are identical in all directions (e.g. cubic configuration for Stewart platform)
+- **decoupling**: each DoF of the end effector can be controlled by a **single** actuator (not the case for the Stewart platform)
+
+Example of generated isotropic manipulator (not decoupled).
+
+
+
+{{< figure src="/ox-hugo/legnani12_isotropy_gen.png" caption="Figure 1: Location of the leg axes using an isotropy generator" >}}
+
+
+
+{{< figure src="/ox-hugo/legnani12_generated_isotropy.png" caption="Figure 2: Isotropic configuration" >}}
+
+# Bibliography
+Legnani, G., Fassi, I., Giberti, H., Cinquemani, S., & Tosi, D., *A new isotropic and decoupled 6-dof parallel manipulator*, Mechanism and Machine Theory, *58(nil)*, 64–81 (2012). http://dx.doi.org/10.1016/j.mechmachtheory.2012.07.008 [↩](#17295cbc2858c65ecc60d51b450233e3)
diff --git a/content/paper/li01_simul_fault_vibrat_isolat_point.md b/content/paper/li01_simul_fault_vibrat_isolat_point.md
new file mode 100644
index 0000000..c847f09
--- /dev/null
+++ b/content/paper/li01_simul_fault_vibrat_isolat_point.md
@@ -0,0 +1,233 @@
++++
+title = "Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Cubic Architecture]({{< relref "cubic_architecture" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}), [Multivariable Control]({{< relref "multivariable_control" >}})
+
+Reference
+: @phdthesis{li01_simul_fault_vibrat_isolat_point,
+ author = {Li, Xiaochun},
+ school = {University of Wyoming},
+ title = {Simultaneous, Fault-tolerant Vibration Isolation and
+ Pointing Control of Flexure Jointed Hexapods},
+ year = 2001,
+ tags = {parallel robot},
+ }
+
+Author(s)
+: Li, X.
+
+Year
+: 2001
+
+
+## Introduction {#introduction}
+
+**Stewart Platform**:
+
+- Cubic (mutually orthogonal)
+- Flexure Joints => eliminate friction and backlash but add complexity to the dynamics
+
+
+
+{{< figure src="/ox-hugo/li01_stewart_platform.png" caption="Figure 1: Flexure jointed Stewart platform used for analysis and control" >}}
+
+**Goal**:
+
+- Precise pointing in two axes (sub micro-radians)
+- simultaneously, providing both passive and active vibration isolation in six axes
+
+**Jacobian Analysis**:
+\\[ \delta \mathcal{L} = J \delta \mathcal{X} \\]
+The origin of \\(\\{P\\}\\) is taken as the center of mass of the payload.
+
+**Decoupling**:
+If we refine the (force) inputs and (displacement) outputs as shown in Figure [2](#org2d875d1) or in Figure [3](#org3e247bd), we obtain a decoupled plant provided that:
+
+1. the payload mass/inertia matrix must be diagonal (the CoM is coincident with the origin of frame \\(\\{P\\}\\))
+2. the geometry of the hexapod and the attachment of the payload to the hexapod must be carefully chosen
+
+> For instance, if the hexapod has a mutually orthogonal geometry (cubic configuration), the payload's center of mass must coincide with the center of the cube formed by the orthogonal struts.
+
+
+
+{{< figure src="/ox-hugo/li01_decoupling_conf.png" caption="Figure 2: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
+
+
+
+{{< figure src="/ox-hugo/li01_decoupling_conf_bis.png" caption="Figure 3: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
+
+
+## Simultaneous Vibration Isolation and Pointing Control {#simultaneous-vibration-isolation-and-pointing-control}
+
+Basic idea:
+
+- acceleration feedback is used to provide high-frequency vibration isolation
+- cartesian pointing feedback can be used to provide low-frequency pointing
+
+The compensation is divided in frequency because:
+
+- pointing sensors often have low bandwidth
+- acceleration sensors often have a poor low frequency response
+
+The control bandwidth is divided as follows:
+
+- low-frequency disturbances as attenuated and tracking is accomplished by feedback from low bandwidth pointing sensors
+- mid-frequency disturbances are attenuated by feedback from band-pass sensors like accelerometer or load cells
+- high-frequency disturbances are attenuated by passive isolation techniques
+
+
+### Vibration Isolation {#vibration-isolation}
+
+The system is decoupled into six independent SISO subsystems using the architecture shown in Figure [4](#org3c42849).
+
+
+
+{{< figure src="/ox-hugo/li01_vibration_isolation_control.png" caption="Figure 4: Figure caption" >}}
+
+One of the subsystem plant transfer function is shown in Figure [4](#org3c42849)
+
+
+
+{{< figure src="/ox-hugo/li01_vibration_control_plant.png" caption="Figure 5: Plant transfer function of one of the SISO subsystem for Vibration Control" >}}
+
+Each compensator is designed using simple loop-shaping techniques.
+
+The unity control bandwidth of the isolation loop is designed to be from **5Hz to 50Hz**.
+
+> Despite a reasonably good match between the modeled and the measured transfer functions, the model based decoupling algorithm does not produce the expected decoupling.
+> Only about 20 dB separation is achieve between the diagonal and off-diagonal responses.
+
+
+### Pointing Control {#pointing-control}
+
+A block diagram of the pointing control system is shown in Figure [6](#org3c3e6ad).
+
+
+
+{{< figure src="/ox-hugo/li01_pointing_control.png" caption="Figure 6: Figure caption" >}}
+
+The plant is decoupled into two independent SISO subsystems.
+The compensators are design with inverse-dynamics methods.
+
+The unity control bandwidth of the pointing loop is designed to be from **0Hz to 20Hz**.
+
+A feedforward control is added as shown in Figure [7](#orgc8fa614).
+
+
+
+{{< figure src="/ox-hugo/li01_feedforward_control.png" caption="Figure 7: Feedforward control" >}}
+
+
+### Simultaneous Control {#simultaneous-control}
+
+The simultaneous vibration isolation and pointing control is approached in two ways:
+
+1. design and implement the vibration isolation control first, identify the pointing plant when the isolation loops are closed, then implement the pointing compensators
+2. the reverse design order
+
+Figure [8](#org987b709) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
+
+
+
+{{< figure src="/ox-hugo/li01_parallel_control.png" caption="Figure 8: A parallel scheme" >}}
+
+The transfer function matrix for the pointing loop after the vibration isolation is closed is still decoupled. The same happens when closing the pointing loop first and looking at the transfer function matrix of the vibration isolation.
+
+The effect of the isolation loop on the pointing loop is large around the natural frequency of the plant as shown in Figure [9](#orgb070c43).
+
+
+
+{{< figure src="/ox-hugo/li01_effect_isolation_loop_closed.png" caption="Figure 9: \\(\theta\_x/\theta\_{x\_d}\\) transfer function with the isolation loop closed (simulation)" >}}
+
+The effect of pointing control on the isolation plant has not much effect.
+
+> The interaction between loops may affect the transfer functions of the **first** closed loop, and thus affect its relative stability.
+
+The dynamic interaction effect:
+
+- only happens in the unity bandwidth of the loop transmission of the first closed loop.
+- affect the closed loop transmission of the loop first closed (see Figures [10](#org0d64bc7) and [11](#orgb43f022))
+
+As shown in Figure [10](#org0d64bc7), the peak resonance of the pointing loop increase after the isolation loop is closed.
+The resonances happen at both crossovers of the isolation loop (15Hz and 50Hz) and they may show of loss of robustness.
+
+
+
+{{< figure src="/ox-hugo/li01_closed_loop_pointing.png" caption="Figure 10: Closed-loop transfer functions \\(\theta\_y/\theta\_{y\_d}\\) of the pointing loop before and after the vibration isolation loop is closed" >}}
+
+The same happens when first closing the vibration isolation loop and after the pointing loop (Figure [11](#orgb43f022)).
+The first peak resonance of the vibration isolation loop at 15Hz is increased when closing the pointing loop.
+
+
+
+{{< figure src="/ox-hugo/li01_closed_loop_vibration.png" caption="Figure 11: Closed-loop transfer functions of the vibration isolation loop before and after the pointing control loop is closed" >}}
+
+> The isolation loop adds a second resonance peak at its high-frequency crossover in the pointing closed-loop transfer function, which may cause instability.
+> Thus, it is recommended to design and implement the isolation control system first, and then identify the pointing plant with the isolation loop closed.
+
+
+### Experimental results {#experimental-results}
+
+Two hexapods are stacked (Figure [12](#org12b1e53)):
+
+- the bottom hexapod is used to generate disturbances matching candidate applications
+- the top hexapod provide simultaneous vibration isolation and pointing control
+
+
+
+{{< figure src="/ox-hugo/li01_test_bench.png" caption="Figure 12: Stacked Hexapods" >}}
+
+Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in figure [13](#org4b99c02).
+
+
+
+{{< figure src="/ox-hugo/li01_vibration_isolation_control_results.png" caption="Figure 13: Vibration isolation control: open-loop (solid) vs. closed-loop (dashed)" >}}
+
+The simultaneous control is of dual use:
+
+- it provide simultaneous pointing and isolation control
+- it can also be used to expand the bandwidth of the isolation control to low frequencies because the pointing loops suppress pointing errors due to both base vibrations and tracking
+
+The results of simultaneous control is shown in Figure [14](#orged11c63) where the bandwidth of the isolation control is expanded to very low frequency.
+
+
+
+{{< figure src="/ox-hugo/li01_simultaneous_control_results.png" caption="Figure 14: Simultaneous control: open-loop (solid) vs. closed-loop (dashed)" >}}
+
+
+## Future research areas {#future-research-areas}
+
+Proposed future research areas include:
+
+- **Include base dynamics in the control**:
+ The base dynamics is here neglected since the movements of the base are very small.
+ The base dynamics could be measured by mounting accelerometers at the bottom of each strut or by using force sensors.
+ It then could be included in the feedforward path.
+- **Robust control and MIMO design**
+- **New decoupling method**:
+ The proposed decoupling algorithm do not produce the expected decoupling, despite a reasonably good match between the modeled and the measured transfer functions.
+ Incomplete decoupling increases the difficulty in designing the controller.
+ New decoupling methods are needed.
+ These methods must be static in order to be implemented practically on precision hexapods
+- **Identification**:
+ Many advanced control methods require a more accurate model or identified plant.
+ A closed-loop identification method is propose to solve some problems with the current identification methods used.
+- **Other possible sensors**:
+ Many sensors can be used to expand the utility of the Stewart platform:
+ - **3-axis load cells** to investigate the Coriolis and centripetal terms and new decoupling methods
+ - **LVDT** to provide differential position of the hexapod payload with respect to the base
+ - **Geophones** to provide payload and base velocity information
+
+# Bibliography
+Li, X., *Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods* (Doctoral dissertation) (2001). University of Wyoming, . [↩](#f885df380638b868e509fbbf75912d1e)
diff --git a/content/paper/li01_simul_vibrat_isolat_point_contr.md b/content/paper/li01_simul_vibrat_isolat_point_contr.md
new file mode 100644
index 0000000..dc73154
--- /dev/null
+++ b/content/paper/li01_simul_vibrat_isolat_point_contr.md
@@ -0,0 +1,23 @@
++++
+title = "Simultaneous vibration isolation and pointing control of flexure jointed hexapods"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
+
+Reference
+: (Xiaochun Li {\it et al.}, 2001)
+
+Author(s)
+: Li, X., Hamann, J. C., & McInroy, J. E.
+
+Year
+: 2001
+
+- if the hexapod is designed such that the payload mass/inertia matrix (\\(M\_x\\)) and \\(J^T J\\) are diagonal, the dynamics from \\(u\\) to \\(y\\) are decoupled.
+
+# Bibliography
+Li, X., Hamann, J. C., & McInroy, J. E., *Simultaneous vibration isolation and pointing control of flexure jointed hexapods*, In , Smart Structures and Materials 2001: Smart Structures and Integrated Systems (pp. ) (2001). : . [↩](#e3df2691f750617c3995644d056d553a)
diff --git a/content/paper/oomen18_advan_motion_contr_precis_mechat.md b/content/paper/oomen18_advan_motion_contr_precis_mechat.md
new file mode 100644
index 0000000..dbe51f7
--- /dev/null
+++ b/content/paper/oomen18_advan_motion_contr_precis_mechat.md
@@ -0,0 +1,20 @@
++++
+title = "Advanced motion control for precision mechatronics: control, identification, and learning of complex systems"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Motion Control]({{< relref "motion_control" >}})
+
+Reference
+: (Tom Oomen, 2018)
+
+Author(s)
+: Oomen, T.
+
+Year
+: 2018
+
+# Bibliography
+Oomen, T., *Advanced motion control for precision mechatronics: control, identification, and learning of complex systems*, IEEJ Journal of Industry Applications, *7(2)*, 127–140 (2018). http://dx.doi.org/10.1541/ieejjia.7.127 [↩](#73fd325bd20a6ef8972145e535f38198)
diff --git a/content/paper/poel10_explor_activ_hard_mount_vibrat.md b/content/paper/poel10_explor_activ_hard_mount_vibrat.md
new file mode 100644
index 0000000..d20a786
--- /dev/null
+++ b/content/paper/poel10_explor_activ_hard_mount_vibrat.md
@@ -0,0 +1,42 @@
++++
+title = "An exploration of active hard mount vibration isolation for precision equipment"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Vibration Isolation]({{< relref "vibration_isolation" >}})
+
+Reference
+: @phdthesis{poel10_explor_activ_hard_mount_vibrat,
+ author = {van der Poel, Gerrit Wijnand},
+ doi = {10.3990/1.9789036530163},
+ isbn = {978-90-365-3016-3},
+ school = {University of Twente},
+ title = {An Exploration of Active Hard Mount Vibration Isolation for
+ Precision Equipment},
+ url = {https://doi.org/10.3990/1.9789036530163},
+ year = 2010,
+ year = 2010,
+ tags = {parallel robot},
+ }
+
+Author(s)
+: van der Poel, G. W.
+
+Year
+: 2010
+
+# Bibliography
+van der Poel, G. W., *An exploration of active hard mount vibration isolation for precision equipment* (Doctoral dissertation) (2010). University of Twente, . [↩](#bcab548922e0e1ad6a2c310f63879596)
diff --git a/content/paper/preumont02_force_feedb_versus_accel_feedb.md b/content/paper/preumont02_force_feedb_versus_accel_feedb.md
new file mode 100644
index 0000000..6427946
--- /dev/null
+++ b/content/paper/preumont02_force_feedb_versus_accel_feedb.md
@@ -0,0 +1,49 @@
++++
+title = "Force feedback versus acceleration feedback in active vibration isolation"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Vibration Isolation]({{< relref "vibration_isolation" >}})
+
+Reference
+: (Preumont {\it et al.}, 2002)
+
+Author(s)
+: Preumont, A., A. Francois, Bossens, F., & Abu-Hanieh, A.
+
+Year
+: 2002
+
+Summary:
+
+- Compares the force feedback and acceleration feedback for active damping
+- The use of a force sensor always give alternating poles and zeros in the open-loop transfer function between for force actuator and the force sensor which **guarantees the stability of the closed loop**
+- Acceleration feedback produces alternating poles and zeros only when the flexible structure is stiff compared to the isolation system
+
+The force applied to a **rigid body** is proportional to its acceleration, thus sensing the total interface force gives a measured of the absolute acceleration of the solid body.
+Thus force feedback and acceleration feedback are equivalent for solid bodies.
+When there is a flexible payload, the two sensing options are not longer equivalent.
+
+- For light payload (Figure [1](#org7b4f6ee)), the acceleration feedback gives larger damping on the higher mode.
+- For heavy payload (Figure [2](#org361b58f)), the acceleration feedback do not give alternating poles and zeros and thus for high control gains, the system becomes unstable
+
+
+
+{{< figure src="/ox-hugo/preumont02_force_acc_fb_light.png" caption="Figure 1: Root locus for **light** flexible payload, (a) Force feedback, (b) acceleration feedback" >}}
+
+
+
+{{< figure src="/ox-hugo/preumont02_force_acc_fb_heavy.png" caption="Figure 2: Root locus for **heavy** flexible payload, (a) Force feedback, (b) acceleration feedback" >}}
+
+Guaranteed stability of the force feedback:
+
+> If two arbitrary flexible, undamped structures are connected with a single-axis soft isolator with force feedback, the poles and zeros of the open-loop transfer function from the force actuator to the force sensor alternate on the imaginary axis.
+
+The same is true for the transfer function from the force actuator to the relative displacement of the actuator.
+
+> According to physical interpretation of the zeros, they represent the resonances of the subsystem constrained by the sensor and the actuator.
+
+# Bibliography
+Preumont, A., A. Fran\ccois, Bossens, F., & Abu-Hanieh, A., *Force feedback versus acceleration feedback in active vibration isolation*, Journal of Sound and Vibration, *257(4)*, 605–613 (2002). http://dx.doi.org/10.1006/jsvi.2002.5047 [↩](#525e1e237b885f81fae3c25a3036ba6f)
diff --git a/content/paper/preumont07_six_axis_singl_stage_activ.md b/content/paper/preumont07_six_axis_singl_stage_activ.md
new file mode 100644
index 0000000..d0894e4
--- /dev/null
+++ b/content/paper/preumont07_six_axis_singl_stage_activ.md
@@ -0,0 +1,49 @@
++++
+title = "A six-axis single-stage active vibration isolator based on stewart platform"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
+
+Reference
+: (Preumont {\it et al.}, 2007)
+
+Author(s)
+: Preumont, A., Horodinca, M., Romanescu, I., Marneffe, B. d., Avraam, M., Deraemaeker, A., Bossens, F., …
+
+Year
+: 2007
+
+Summary:
+
+- **Cubic** Stewart platform (Figure [3](#org32a4f7c))
+ - Provides uniform control capability
+ - Uniform stiffness in all directions
+ - minimizes the cross-coupling among actuators and sensors of different legs
+- Flexible joints (Figure [2](#orgf807976))
+- Piezoelectric force sensors
+- Voice coil actuators
+- Decentralized feedback control approach for vibration isolation
+- Effect of parasitic stiffness of the flexible joints on the IFF performance (Figure [1](#org744bdc9))
+- The Stewart platform has 6 suspension modes at different frequencies.
+ Thus the gain of the IFF controller cannot be optimal for all the modes.
+ It is better if all the modes of the platform are near to each other.
+- Discusses the design of the legs in order to maximize the natural frequency of the local modes.
+- To estimate the isolation performance of the Stewart platform, a scalar indicator is defined as the Frobenius norm of the transmissibility matrix
+
+
+
+{{< figure src="/ox-hugo/preumont07_iff_effect_stiffness.png" caption="Figure 1: Root locus with IFF with no parasitic stiffness and with parasitic stiffness" >}}
+
+
+
+{{< figure src="/ox-hugo/preumont07_flexible_joints.png" caption="Figure 2: Flexible joints used for the Stewart platform" >}}
+
+
+
+{{< figure src="/ox-hugo/preumont07_stewart_platform.png" caption="Figure 3: Stewart platform" >}}
+
+# Bibliography
+Preumont, A., Horodinca, M., Romanescu, I., Marneffe, B. d., Avraam, M., Deraemaeker, A., Bossens, F., …, *A six-axis single-stage active vibration isolator based on stewart platform*, Journal of Sound and Vibration, *300(3-5)*, 644–661 (2007). http://dx.doi.org/10.1016/j.jsv.2006.07.050 [↩](#8096d5b2df73551d836ef96b7ca7efa4)
diff --git a/content/paper/saxena12_advan_inter_model_contr_techn.md b/content/paper/saxena12_advan_inter_model_contr_techn.md
new file mode 100644
index 0000000..c7e45ac
--- /dev/null
+++ b/content/paper/saxena12_advan_inter_model_contr_techn.md
@@ -0,0 +1,88 @@
++++
+title = "Advances in internal model control technique: a review and future prospects"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Complementary Filters]({{< relref "complementary_filters" >}})
+
+Reference
+: (Sahaj Saxena \& YogeshV Hote, 2012)
+
+Author(s)
+: Saxena, S., & Hote, Y.
+
+Year
+: 2012
+
+
+## Proposed Filter \\(F(s)\\) {#proposed-filter--fs}
+
+\begin{align\*}
+ F(s) &= \frac{1}{(\lambda s + 1)^n} \\\\\\
+ F(s) &= \frac{n \lambda + 1}{(\lambda s + 1)^n}
+\end{align\*}
+
+
+## Internal Model Control {#internal-model-control}
+
+Central concept in IMC: control can be acheive only if the control system involves, either implicitly or explicitly, some representation of the process to be controlled.
+
+
+### Basic IMC structure {#basic-imc-structure}
+
+IMC can be considered as a special case of classical feedback structure with plant \\(G(s)\\) and controller \\(C(s)\\).
+
+The plan model \\(G\_M(s)\\) is added and substracted into the feedback path of feedback controller.
+
+The structure can then be modified and we obtain a new controller \\(Q(s)\\).
+
+IMC is related to the classical controller through:
+
+\begin{align\*}
+Q(s) = \frac{C(s)}{1+G\_M(s)C(s)} \\\\\\
+C(s) = \frac{Q(s)}{1-G\_M(s)Q(s)}
+\end{align\*}
+
+Internal model control system is characterized by a control device consisting of the controller \\(Q(s)\\) and a predictive model \\(G\_M(s)\\) of the process (internal model).
+The internal model loop uses the difference between the outputs of the process \\(G(s)\\) to be controlled and the internal model.
+This difference \\(E(s)\\) represents the effect of disturbance and mismatch of the model.
+
+
+### Features of IMC Structure {#features-of-imc-structure}
+
+Three properties:
+
+- **Dual stability**: assume that, if the plant model is perfect (\\(G\_M(s) = G(s)\\)) and disturbance is absent, the system becomes open-loop and the closed-loop stability is characterized by the stability of \\(G(s)\\) and \\(Q(s)\\)
+- **Perfect control**: assume that, if the controller is equal to the model inverse (\\(Q(s) = G\_M^{-1}\\)) and \\(G(s) = G\_M(s)\\) with \\(G(s)\\) stable, then the system is perfectly controlled.
+- **Zero Offset**: assume that, if the steady state gain of the controller is equal to the inverse of model gain, then offset free control is obtained for constant step of ramp type inputs and disturbances. As expected, the equivalent classical controller leads to integral action.
+
+Issues:
+
+- the plant model is never perfect
+- inverting the model can cause instability
+- control signal may have large magnitude
+
+
+## Design procedure for IMC Compensator {#design-procedure-for-imc-compensator}
+
+1. factorize the plant model as \\(G\_M(s) = G\_{M-}(s)G\_{M+}(s)\\) where \\(G\_{M-}(s)\\) is invertible and minimum phase and \\(G\_{M+}(s)\\) is non-invertible and contains all non-minimum phase elements (delays, RHP zeros). Then, the controller is the inverse of the invertible portion of the plant model: \\(Q\_1(s) = G\_{M-}^{-1}(s)\\).
+2. Filter selection: to make the controller proper and robust against the plant-model mismatch, a low pass filter of the form \\(F(s) = \frac{n \lambda}{(\lambda s + 1)^n}\\) is augmented with the inverted model \\(Q\_1(s)\\): \\(Q(s) = Q\_1(s) F(s)\\). \\(\lambda\\) is a tuning parameter which has an inverse relationship with the speed of closed loop response, \\(n\\) is selected such that \\(Q(s)\\) becomes proper.
+
+
+## Issues in IMC {#issues-in-imc}
+
+
+### Filter selection and tuning guidelines {#filter-selection-and-tuning-guidelines}
+
+
+## Some advantages and future prospects {#some-advantages-and-future-prospects}
+
+
+## Conclusion {#conclusion}
+
+The interesting feature regarding IMC is that the design scheme is identical to the open-loop control design procedure and the implementation of IMC results in a feedback system, thereby copying the disturbances and parameter uncertainties, while open-loop control is not.
+
+# Bibliography
+Saxena, S., & Hote, Y., *Advances in internal model control technique: a review and future prospects*, IETE Technical Review, *29(6)*, 461 (2012). http://dx.doi.org/10.4103/0256-4602.105001 [↩](#14f767d8ba71d58fa8a3ec876628d750)
diff --git a/content/paper/schellekens98_desig_precis.md b/content/paper/schellekens98_desig_precis.md
new file mode 100644
index 0000000..37091c9
--- /dev/null
+++ b/content/paper/schellekens98_desig_precis.md
@@ -0,0 +1,20 @@
++++
+title = "Design for precision: current status and trends"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Precision Engineering]({{< relref "precision_engineering" >}})
+
+Reference
+: (Schellekens {\it et al.}, 1998)
+
+Author(s)
+: Schellekens, P., Rosielle, N., Vermeulen, H., Vermeulen, M., Wetzels, S., & Pril, W.
+
+Year
+: 1998
+
+# Bibliography
+Schellekens, P., Rosielle, N., Vermeulen, H., Vermeulen, M., Wetzels, S., & Pril, W., *Design for precision: current status and trends*, Cirp Annals, *(2)*, 557–586 (1998). http://dx.doi.org/10.1016/s0007-8506(07)63243-0 [↩](#89f7d8f4c31f79f83e3666017687f525)
diff --git a/content/paper/sebastian12_nanop_with_multip_sensor.md b/content/paper/sebastian12_nanop_with_multip_sensor.md
new file mode 100644
index 0000000..42e2c6f
--- /dev/null
+++ b/content/paper/sebastian12_nanop_with_multip_sensor.md
@@ -0,0 +1,20 @@
++++
+title = "Nanopositioning with multiple sensors: a case study in data storage"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Sensor Fusion]({{< relref "sensor_fusion" >}})
+
+Reference
+: (Abu Sebastian \& Angeliki Pantazi, 2012)
+
+Author(s)
+: Sebastian, A., & Pantazi, A.
+
+Year
+: 2012
+
+# Bibliography
+Sebastian, A., & Pantazi, A., *Nanopositioning with multiple sensors: a case study in data storage*, IEEE Transactions on Control Systems Technology, *20(2)*, 382–394 (2012). http://dx.doi.org/10.1109/tcst.2011.2177982 [↩](#eb5a15a8c900d93de0b9bab520e1b6da)
diff --git a/content/paper/spanos95_soft_activ_vibrat_isolat.md b/content/paper/spanos95_soft_activ_vibrat_isolat.md
new file mode 100644
index 0000000..9f531f9
--- /dev/null
+++ b/content/paper/spanos95_soft_activ_vibrat_isolat.md
@@ -0,0 +1,62 @@
++++
+title = "A soft 6-axis active vibration isolator"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
+
+Reference
+: (Spanos {\it et al.}, 1995)
+
+Author(s)
+: Spanos, J., Rahman, Z., & Blackwood, G.
+
+Year
+: 1995
+
+**Stewart Platform** (Figure [1](#org4317d08)):
+
+- Voice Coil
+- Flexible joints (cross-blades)
+- Force Sensors
+- Cubic Configuration
+
+
+
+{{< figure src="/ox-hugo/spanos95_stewart_platform.png" caption="Figure 1: Stewart Platform" >}}
+
+Total mass of the paylaod: 30kg
+Center of gravity is 9cm above the geometry center of the mount (cube's center?).
+
+Limitation of the **Decentralized Force Feedback**:
+
+- high frequency pole due to internal resonances of the struts
+- low frequency zero due to the rotational stiffness of the flexible joints
+
+After redesign of the struts:
+
+- high frequency pole at 4.7kHz
+- low frequency zero at 2.6Hz but non-minimum phase (not explained).
+ Small viscous damping material in the cross blade flexures made the zero minimum phase again.
+
+
+
+{{< figure src="/ox-hugo/spanos95_iff_plant.png" caption="Figure 2: Experimentally measured transfer function from voice coil drive voltage to collocated load cell output voltage" >}}
+
+The controller used consisted of:
+
+- second order low pass filter to gain stabilize the plant at high frequencies and provide steep roll-off
+- first order lead filter to provide adequate phase margin at the high frequency crossover
+- first order lag filter to provide adequate phase margin at the low frequency crossover
+- a first order high pass filter to attenuate the excess gain resulting from the low frequency zero
+
+The results in terms of transmissibility are shown in Figure [3](#orgf128817).
+
+
+
+{{< figure src="/ox-hugo/spanos95_results.png" caption="Figure 3: Experimentally measured Frobenius norm of the 6-axis transmissibility" >}}
+
+# Bibliography
+Spanos, J., Rahman, Z., & Blackwood, G., *A soft 6-axis active vibration isolator*, In , Proceedings of 1995 American Control Conference - ACC'95 (pp. ) (1995). : . [↩](#a48f6708d087625a42ca2375407a2bc4)
diff --git a/content/paper/stankevic17_inter_charac_rotat_stages_x_ray_nanot.md b/content/paper/stankevic17_inter_charac_rotat_stages_x_ray_nanot.md
new file mode 100644
index 0000000..e419720
--- /dev/null
+++ b/content/paper/stankevic17_inter_charac_rotat_stages_x_ray_nanot.md
@@ -0,0 +1,33 @@
++++
+title = "Interferometric characterization of rotation stages for x-ray nanotomography"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}}), [Positioning Stations]({{< relref "positioning_stations" >}})
+
+Reference
+: (Tomas Stankevic {\it et al.}, 2017)
+
+Author(s)
+: Stankevic, T., Engblom, C., Langlois, F., Alves, F., Lestrade, A., Jobert, N., Cauchon, G., …
+
+Year
+: 2017
+
+- Similar Station than the NASS
+- Similar Metrology with fiber based interferometers and cylindrical reference mirror
+
+
+
+{{< figure src="/ox-hugo/stankevic17_station.png" caption="Figure 1: Positioning Station" >}}
+
+- **Thermal expansion**: Stabilized down to \\(5mK/h\\) using passive water flow through the baseplate below the sample stage and in the interferometry reference frame.
+- **Controller**: Two Independant PID loops
+- Repeatable errors => feedforward (Look Up Table)
+- Non-repeatable errors => feedback
+- Result: 40nm runout error
+
+# Bibliography
+Stankevic, T., Engblom, C., Langlois, F., Alves, F., Lestrade, A., Jobert, N., Cauchon, G., …, *Interferometric characterization of rotation stages for x-ray nanotomography*, Review of Scientific Instruments, *88(5)*, 053703 (2017). http://dx.doi.org/10.1063/1.4983405 [↩](#abb1be5f48179255f7d8c45b1784bcf8)
diff --git a/content/paper/tang18_decen_vibrat_contr_voice_coil.md b/content/paper/tang18_decen_vibrat_contr_voice_coil.md
new file mode 100644
index 0000000..7086d0b
--- /dev/null
+++ b/content/paper/tang18_decen_vibrat_contr_voice_coil.md
@@ -0,0 +1,21 @@
++++
+title = "Decentralized vibration control of a voice coil motor-based stewart parallel mechanism: simulation and experiments"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms" >}})
+
+Reference
+: (Jie Tang {\it et al.}, 2018)
+
+Author(s)
+: Tang, J., Cao, D., & Yu, T.
+
+Year
+: 2018
+
+# Bibliography
+Tang, J., Cao, D., & Yu, T., *Decentralized vibration control of a voice coil motor-based stewart parallel mechanism: simulation and experiments*, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, *233(1)*, 132–145 (2018). http://dx.doi.org/10.1177/0954406218756941 [↩](#85f81ff678aabc195636437548e4234a)
diff --git a/content/paper/tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip.md b/content/paper/tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip.md
new file mode 100644
index 0000000..54ed4ad
--- /dev/null
+++ b/content/paper/tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip.md
@@ -0,0 +1,50 @@
++++
+title = "Sensor fusion for active vibration isolation in precision equipment"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Sensor Fusion]({{< relref "sensor_fusion" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
+
+Reference
+: (Tjepkema {\it et al.}, 2012)
+
+Author(s)
+: Tjepkema, D., Dijk, J. v., & Soemers, H.
+
+Year
+: 2012
+
+
+## Relative motion Control {#relative-motion-control}
+
+Control law: \\(f = -G(x-w)\\)
+
+\\[ \frac{x}{w} = \frac{k+G}{ms^2 + k+G} \\]
+\\[ \frac{x}{F} = \frac{1}{ms^2 + k+G} \\]
+
+
+## Force Control {#force-control}
+
+Control law: \\(f = -G F\_a = -G \left(f-k(x-w)\right)\\)
+
+\\[ \frac{x}{w} = \frac{k}{(1+G)ms^2 + k} \\]
+\\[ \frac{x}{F} = \frac{1+G}{(1+G)ms^2 + k} \\]
+
+
+## Inertial Control {#inertial-control}
+
+Control law: \\(f = -Gx\\)
+
+\\[ \frac{x}{w} = \frac{k}{ms^2 + k+G} \\]
+\\[ \frac{x}{F} = \frac{1}{ms^2 + k+G} \\]
+
+
+## Design constraints and control bandwidth {#design-constraints-and-control-bandwidth}
+
+Heavier sensor => lower noise but it is harder to maintain collocation with the actuator => that limits the bandwidth.
+There is a compromise between sensor noise and the influence of the sensor size on the system's design and on the control bandwidth.
+
+# Bibliography
+Tjepkema, D., Dijk, J. v., & Soemers, H., *Sensor fusion for active vibration isolation in precision equipment*, Journal of Sound and Vibration, *331(4)*, 735–749 (2012). http://dx.doi.org/10.1016/j.jsv.2011.09.022 [↩](#ef30bc07c91e9d46a42198757dc610de)
diff --git a/content/paper/wang12_autom_marker_full_field_hard.md b/content/paper/wang12_autom_marker_full_field_hard.md
new file mode 100644
index 0000000..bdf06bc
--- /dev/null
+++ b/content/paper/wang12_autom_marker_full_field_hard.md
@@ -0,0 +1,29 @@
++++
+title = "Automated markerless full field hard x-ray microscopic tomography at sub-50 nm 3-dimension spatial resolution"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}})
+
+Reference
+: (Jun Wang {\it et al.}, 2012)
+
+Author(s)
+: Wang, J., Chen, Y. K., Yuan, Q., Tkachuk, A., Erdonmez, C., Hornberger, B., & Feser, M.
+
+Year
+: 2012
+
+**Introduction of Markers**:
+That limits the type of samples that is studied
+
+There is a need for markerless nano-tomography
+=> the key requirement is the precision and stability of the positioning stages.
+
+**Passive rotational run-out error system**:
+It uses calibrated metrology disc and capacitive sensors
+
+# Bibliography
+Wang, J., Chen, Y. K., Yuan, Q., Tkachuk, A., Erdonmez, C., Hornberger, B., & Feser, M., *Automated markerless full field hard x-ray microscopic tomography at sub-50 nm 3-dimension spatial resolution*, Applied Physics Letters, *100(14)*, 143107 (2012). http://dx.doi.org/10.1063/1.3701579 [↩](#1bccbe15e35ed02229afbc6528c5057e)
diff --git a/content/paper/wang16_inves_activ_vibrat_isolat_stewar.md b/content/paper/wang16_inves_activ_vibrat_isolat_stewar.md
new file mode 100644
index 0000000..6bbeb70
--- /dev/null
+++ b/content/paper/wang16_inves_activ_vibrat_isolat_stewar.md
@@ -0,0 +1,57 @@
++++
+title = "Investigation on active vibration isolation of a stewart platform with piezoelectric actuators"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
+
+Reference
+: (Wang {\it et al.}, 2016)
+
+Author(s)
+: Wang, C., Xie, X., Chen, Y., & Zhang, Z.
+
+Year
+: 2016
+
+**Model of the Stewart platform**:
+
+- Struts are treated as flexible beams
+- Payload and the base are treated as flexible plates
+- The FRF synthesis method permits to derive FRFs of the Stewart platform
+
+The model is compared with a Finite Element model and is shown to give the same results.
+The proposed model is thus effective.
+
+
+
+{{< figure src="/ox-hugo/wang16_stewart_platform.png" caption="Figure 1: Stewart Platform" >}}
+
+**Control**:
+Combines:
+
+- the FxLMS-based adaptive inverse control => suppress transmission of periodic vibrations
+- direct feedback of integrated forces => dampen vibration of inherent modes and thus reduce random vibrations
+
+Force Feedback (Figure [2](#org4b1fbd9)).
+
+- the force sensor is mounted **between the base and the strut**
+
+
+
+{{< figure src="/ox-hugo/wang16_force_feedback.png" caption="Figure 2: Feedback of integrated forces in the platform" >}}
+
+Sorts of HAC-LAC control:
+
+- LAC: Decentralized integral force feedback
+- HAC: Inertial control using accelerometers. Use of the Jacobian to decouple the motion and then Fx-LMS based adaptive control is used
+
+**Experimental validation**:
+
+- All 6 transfer function from actuator force to force sensors are almost the same (gain offset)
+- Effectiveness of control methods are shown
+
+# Bibliography
+Wang, C., Xie, X., Chen, Y., & Zhang, Z., *Investigation on active vibration isolation of a stewart platform with piezoelectric actuators*, Journal of Sound and Vibration, *383()*, 1–19 (2016). http://dx.doi.org/10.1016/j.jsv.2016.07.021 [↩](#db95fac7cd46c14e2b4f38e8ca4158fe)
diff --git a/content/paper/yang19_dynam_model_decoup_contr_flexib.md b/content/paper/yang19_dynam_model_decoup_contr_flexib.md
new file mode 100644
index 0000000..23a1942
--- /dev/null
+++ b/content/paper/yang19_dynam_model_decoup_contr_flexib.md
@@ -0,0 +1,137 @@
++++
+title = "Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}), [Cubic Architecture]({{< relref "cubic_architecture" >}})
+
+Reference
+: (Yang {\it et al.}, 2019)
+
+Author(s)
+: Yang, X., Wu, H., Chen, B., Kang, S., & Cheng, S.
+
+Year
+: 2019
+
+**Discusses**:
+
+- flexible-rigid model of Stewart platform
+- the impact of joint stiffness is compensated using a displacement sensor and a force sensor
+- then the MIMO system is decoupled in modal space and 6 SISO controllers are applied for vibration isolation using force sensors
+
+The joint stiffness impose a limitation on the control performance using force sensors as it adds a zero at low frequency in the dynamics.
+Thus, this stiffness is taken into account in the dynamics and compensated for.
+
+**Stewart platform** (Figure [1](#org936d8f9)):
+
+- piezoelectric actuators
+- flexible joints (Figure [2](#orgd8c916a))
+- force sensors (used for vibration isolation)
+- displacement sensors (used to decouple the dynamics)
+- cubic (even though not said explicitly)
+
+
+
+{{< figure src="/ox-hugo/yang19_stewart_platform.png" caption="Figure 1: Stewart Platform" >}}
+
+
+
+{{< figure src="/ox-hugo/yang19_flexible_joints.png" caption="Figure 2: Flexible Joints" >}}
+
+The stiffness of the flexible joints (Figure [2](#orgd8c916a)) are computed with an FEM model and shown in Table [1](#table--tab:yang19-stiffness-flexible-joints).
+
+
+
+
Table 1:
+ Stiffness of flexible joints obtained by FEM
+
+ Table 1:
+ Characteristics of Environmental Units
+
+
+| | Temperature (\\(\pm\ ^oC\\)) | Pressure (\\(\pm\ hPa\\)) | Humidity \\(\pm\ \% RH\\) | Wavelength Accuracy (\\(\pm\ \text{ppm}\\)) |
+|-----------|------------------------------|---------------------------|---------------------------|---------------------------------------------|
+| Attocube | 0.1 | 1 | 2 | 0.5 |
+| Renishaw | 0.2 | 1 | 6 | 1 |
+| Picoscale | 0.2 | 2 | 2 | 1 |
+
+Figure [3](#orgce716a4) is taken from
+(Yoon-Soo Jang \& Seung-Woo Kim, 2017).
+
+
+
+{{< figure src="/ox-hugo/position_sensor_interferometer_precision.png" caption="Figure 3: Expected precision of interferometer as a function of measured distance" >}}
+
+
+### Fiber Optic Displacement Sensor {#fiber-optic-displacement-sensor}
+
+| | |
+|----------|----------------------------------------------------|
+| Unipulse | [link](https://www.unipulse.com/product/atw200-2/) |
+
+# Bibliography
+Collette, C., Janssens, S., Mokrani, B., Fueyo-Roza, L., Artoos, K., Esposito, M., Fernandez-Carmona, P., …, *Comparison of new absolute displacement sensors*, In , International Conference on Noise and Vibration Engineering (ISMA) (pp. ) (2012). : . [↩](#0b0b67de6dddc4d28031ab2d3b28cd3d)
+
+Fleming, A. J., *A review of nanometer resolution position sensors: operation and performance*, Sensors and Actuators A: Physical, *190(nil)*, 106–126 (2013). http://dx.doi.org/10.1016/j.sna.2012.10.016 [↩](#3fb5b61524290e36d639a4fac65703d0)
+
+Collette, C., Artoos, K., Guinchard, M., Janssens, S., Carmona Fernandez, P., & Hauviller, C., *Review of sensors for low frequency seismic vibration measurement* (2011). [↩](#642a18d86de4e062c6afb0f5f20501c4)
+
+Jang, Y., & Kim, S., *Compensation of the refractive index of air in laser interferometer for distance measurement: a review*, International Journal of Precision Engineering and Manufacturing, *18(12)*, 1881–1890 (2017). http://dx.doi.org/10.1007/s12541-017-0217-y [↩](#7658b1219a4458a62ae8c6f51b767542)
+
+
+## Backlinks {#backlinks}
+
+- [A review of nanometer resolution position sensors: operation and performance]({{< relref "fleming13_review_nanom_resol_posit_sensor" >}})
+- [Measurement technologies for precision positioning]({{< relref "gao15_measur_techn_precis_posit" >}})
diff --git a/content/zettels/positioning_stations.md b/content/zettels/positioning_stations.md
new file mode 100644
index 0000000..06850b2
--- /dev/null
+++ b/content/zettels/positioning_stations.md
@@ -0,0 +1,17 @@
++++
+title = "Positioning Stations"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+<./biblio/references.bib>
+
+
+## Backlinks {#backlinks}
+
+- [Interferometric characterization of rotation stages for x-ray nanotomography]({{< relref "stankevic17_inter_charac_rotat_stages_x_ray_nanot" >}})
+- [Position control in lithographic equipment]({{< relref "butler11_posit_contr_lithog_equip" >}})
+- [An instrument for 3d x-ray nano-imaging]({{< relref "holler12_instr_x_ray_nano_imagin" >}})
diff --git a/content/zettels/precision_engineering.md b/content/zettels/precision_engineering.md
new file mode 100644
index 0000000..5880820
--- /dev/null
+++ b/content/zettels/precision_engineering.md
@@ -0,0 +1,16 @@
++++
+title = "Precision Engineering"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+<./biblio/references.bib>
+
+
+## Backlinks {#backlinks}
+
+- [Design for precision: current status and trends]({{< relref "schellekens98_desig_precis" >}})
+- [Basics of precision engineering - 1st edition]({{< relref "leach18_basic_precis_engin_edition" >}})
diff --git a/content/zettels/reference_books.md b/content/zettels/reference_books.md
new file mode 100644
index 0000000..dace8c7
--- /dev/null
+++ b/content/zettels/reference_books.md
@@ -0,0 +1,20 @@
++++
+title = "Reference Books"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+<./biblio/references.bib>
+
+
+## Backlinks {#backlinks}
+
+- [Multivariable feedback control: analysis and design]({{< relref "skogestad07_multiv_feedb_contr" >}})
+- [The design of high performance mechatronics - 2nd revised edition]({{< relref "schmidt14_desig_high_perfor_mechat_revis_edition" >}})
+- [Parallel robots : mechanics and control]({{< relref "taghirad13_paral" >}})
+- [Modal testing: theory, practice and application]({{< relref "ewins00_modal" >}})
+- [The art of electronics - third edition]({{< relref "horowitz15_art_of_elect_third_edition" >}})
+- [Vibration Control of Active Structures - Fourth Edition]({{< relref "preumont18_vibrat_contr_activ_struc_fourt_edition" >}})
diff --git a/content/zettels/sensor_fusion.md b/content/zettels/sensor_fusion.md
new file mode 100644
index 0000000..25136d7
--- /dev/null
+++ b/content/zettels/sensor_fusion.md
@@ -0,0 +1,19 @@
++++
+title = "Sensor Fusion"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+<./biblio/references.bib>
+
+
+## Backlinks {#backlinks}
+
+- [Sensor fusion for active vibration isolation in precision equipment]({{< relref "tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip" >}})
+- [Vibration control of flexible structures using fusion of inertial sensors and hyper-stable actuator-sensor pairs]({{< relref "collette14_vibrat" >}})
+- [Sensor fusion methods for high performance active vibration isolation systems]({{< relref "collette15_sensor_fusion_method_high_perfor" >}})
+- [Nanopositioning system with force feedback for high-performance tracking and vibration control]({{< relref "fleming10_nanop_system_with_force_feedb" >}})
+- [Nanopositioning with multiple sensors: a case study in data storage]({{< relref "sebastian12_nanop_with_multip_sensor" >}})
diff --git a/content/zettels/stewart_platforms.md b/content/zettels/stewart_platforms.md
new file mode 100644
index 0000000..4061a45
--- /dev/null
+++ b/content/zettels/stewart_platforms.md
@@ -0,0 +1,34 @@
++++
+title = "Stewart Platforms"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+<./biblio/references.bib>
+
+
+## Backlinks {#backlinks}
+
+- [Six dof active vibration control using stewart platform with non-cubic configuration]({{< relref "zhang11_six_dof" >}})
+- [Decentralized vibration control of a voice coil motor-based stewart parallel mechanism: simulation and experiments]({{< relref "tang18_decen_vibrat_contr_voice_coil" >}})
+- [Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation]({{< relref "yang19_dynam_model_decoup_contr_flexib" >}})
+- [Parallel robots : mechanics and control]({{< relref "taghirad13_paral" >}})
+- [Investigation on active vibration isolation of a stewart platform with piezoelectric actuators]({{< relref "wang16_inves_activ_vibrat_isolat_stewar" >}})
+- [Identification and decoupling control of flexure jointed hexapods]({{< relref "chen00_ident_decoup_contr_flexur_joint_hexap" >}})
+- [The stewart platform manipulator: a review]({{< relref "dasgupta00_stewar_platf_manip" >}})
+- [Modeling and control of vibration in mechanical systems]({{< relref "du10_model_contr_vibrat_mechan_system" >}})
+- [Studies on stewart platform manipulator: a review]({{< relref "furqan17_studies_stewar_platf_manip" >}})
+- [Nanometre-cutting machine using a stewart-platform parallel mechanism]({{< relref "furutani04_nanom_cuttin_machin_using_stewar" >}})
+- [An intelligent control system for multiple degree-of-freedom vibration isolation]({{< relref "geng95_intel_contr_system_multip_degree" >}})
+- [Active isolation and damping of vibrations via stewart platform]({{< relref "hanieh03_activ_stewar" >}})
+- [Sensors and control of a space-based six-axis vibration isolation system]({{< relref "hauge04_sensor_contr_space_based_six" >}})
+- [Dynamic modeling and experimental analyses of stewart platform with flexible hinges]({{< relref "jiao18_dynam_model_exper_analy_stewar" >}})
+- [Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods]({{< relref "li01_simul_fault_vibrat_isolat_point" >}})
+- [A new isotropic and decoupled 6-dof parallel manipulator]({{< relref "legnani12_new_isotr_decoup_paral_manip" >}})
+- [Simultaneous vibration isolation and pointing control of flexure jointed hexapods]({{< relref "li01_simul_vibrat_isolat_point_contr" >}})
+- [A six-axis single-stage active vibration isolator based on stewart platform]({{< relref "preumont07_six_axis_singl_stage_activ" >}})
+- [Vibration Control of Active Structures - Fourth Edition]({{< relref "preumont18_vibrat_contr_activ_struc_fourt_edition" >}})
+- [A soft 6-axis active vibration isolator]({{< relref "spanos95_soft_activ_vibrat_isolat" >}})
diff --git a/content/zettels/system_identification.md b/content/zettels/system_identification.md
new file mode 100644
index 0000000..3a26238
--- /dev/null
+++ b/content/zettels/system_identification.md
@@ -0,0 +1,15 @@
++++
+title = "System Identification"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+<./biblio/references.bib>
+
+
+## Backlinks {#backlinks}
+
+- [Modal testing: theory, practice and application]({{< relref "ewins00_modal" >}})
diff --git a/content/zettels/test.md b/content/zettels/test.md
new file mode 100644
index 0000000..93e6581
--- /dev/null
+++ b/content/zettels/test.md
@@ -0,0 +1,103 @@
++++
+title = "Test File"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+> This is a quote!
+
+```matlab
+a = 2;
+figure;
+```
+
+
+
+
+This is an important part of the text.
+
+