diff --git a/content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md b/content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md index b033f86..fbebb72 100644 --- a/content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md +++ b/content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md @@ -8,7 +8,7 @@ Tags : [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Reference Books]({{< relref "reference_books" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [HAC-HAC]({{< relref "hac_hac" >}}) Reference -: ([Preumont 2018](#org09bc150)) +: ([Preumont 2018](#org2443fdb)) Author(s) : Preumont, A. @@ -61,11 +61,11 @@ There are two radically different approached to disturbance rejection: feedback #### Feedback {#feedback} - + {{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="Figure 1: Principle of feedback control" >}} -The principle of feedback is represented on figure [1](#org3a6239d). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance. +The principle of feedback is represented on figure [1](#org17539d6). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance. The design problem consists of finding the appropriate compensator \\(K(s)\\) such that the closed-loop system is stable and behaves in the appropriate manner. In the control of lightly damped structures, feedback control is used for two distinct and complementary purposes: **active damping** and **model-based feedback**. @@ -87,12 +87,12 @@ The objective is to control a variable \\(y\\) to a desired value \\(r\\) in spi #### Feedforward {#feedforward} - + {{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="Figure 2: Principle of feedforward control" >}} The method relies on the availability of a **reference signal correlated to the primary disturbance**. -The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in figure [2](#org0c27bf0). +The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in figure [2](#orgb6b4033). The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized. @@ -123,11 +123,11 @@ The table [1](#table--tab:adv-dis-type-control) summarizes the main features of ### The Various Steps of the Design {#the-various-steps-of-the-design} - + {{< figure src="/ox-hugo/preumont18_design_steps.png" caption="Figure 3: The various steps of the design" >}} -The various steps of the design of a controlled structure are shown in figure [3](#org4371fad). +The various steps of the design of a controlled structure are shown in figure [3](#org0ed85b6). The **starting point** is: @@ -154,14 +154,14 @@ If the dynamics of the sensors and actuators may significantly affect the behavi ### Plant Description, Error and Control Budget {#plant-description-error-and-control-budget} -From the block diagram of the control system (figure [4](#orgffc7bae)): +From the block diagram of the control system (figure [4](#org90c3880)): \begin{align\*} y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\\\ z &= T\_{zw} w = [G\_{zw} + G\_{zu}H(I - G\_{yu}H)^{-1} G\_{yw}] w \end{align\*} - + {{< figure src="/ox-hugo/preumont18_general_plant.png" caption="Figure 4: Block diagram of the control System" >}} @@ -186,12 +186,12 @@ Even more interesting for the design is the **Cumulative Mean Square** response It is a monotonously decreasing function of frequency and describes the contribution of all frequencies above \\(\omega\\) to the mean-square value of \\(z\\). \\(\sigma\_z(0)\\) is then the global RMS response. -A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#org3ce8916). +A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#org3209437). It is useful to **identify the critical modes** in a design, at which the effort should be targeted. The diagram can also be used to **assess the control laws** and compare different actuator and sensor configuration. - + {{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="Figure 5: Error budget distribution in OL and CL for increasing gains" >}} @@ -398,11 +398,11 @@ With: D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \xi\_i \omega/\omega\_i} \end{equation} - + {{< figure src="/ox-hugo/preumont18_neglected_modes.png" caption="Figure 6: Fourier spectrum of the excitation \\(F\\) and dynamic amplitification \\(D\_i\\) of mode \\(i\\) and \\(k\\) such that \\(\omega\_i < \omega\_b\\) and \\(\omega\_k \gg \omega\_b\\)" >}} -If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on figure [6](#org4219cc6)). +If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on figure [6](#org4377aea)). And \\(G(\omega)\\) can be rewritten on terms of the **low frequency modes only**: \\[ G(\omega) \approx \sum\_{i=1}^m \frac{\phi\_i \phi\_i^T}{\mu\_i \omega\_i^2} D\_i(\omega) + R \\] @@ -441,9 +441,9 @@ The open-loop FRF of a collocated system corresponds to a diagonal component of If we assumes that the collocated system is undamped and is attached to the DoF \\(k\\), the open-loop FRF is purely real: \\[ G\_{kk}(\omega) = \sum\_{i=1}^m \frac{\phi\_i^2(k)}{\mu\_i (\omega\_i^2 - \omega^2)} + R\_{kk} \\] -\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#orgd4f4723)). +\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#orgd6e521d)). - + {{< figure src="/ox-hugo/preumont18_collocated_control_frf.png" caption="Figure 7: Open-Loop FRF of an undamped structure with collocated actuator/sensor pair" >}} @@ -457,9 +457,9 @@ For lightly damped structure, the poles and zeros are just moved a little bit in -If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [8](#org5748d4e). +If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [8](#org3cc8875). - + {{< figure src="/ox-hugo/preumont18_collocated_zero.png" caption="Figure 8: Structure with collocated actuator and sensor" >}} @@ -474,9 +474,9 @@ The open-loop poles are independant of the actuator and sensor configuration whi -By looking at figure [7](#orgd4f4723), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right. +By looking at figure [7](#orgd6e521d), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right. - + {{< figure src="/ox-hugo/preumont18_alternating_p_z.png" caption="Figure 9: Bode plot of a lighly damped structure with collocated actuator and sensor" >}} @@ -486,7 +486,7 @@ The open-loop transfer function of a lighly damped structure with a collocated a G(s) = G\_0 \frac{\Pi\_i(s^2/z\_i^2 + 2 \xi\_i s/z\_i + 1)}{\Pi\_j(s^2/\omega\_j^2 + 2 \xi\_j s /\omega\_j + 1)} \end{equation} -The corresponding Bode plot is represented in figure [9](#org0dd5dad). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\). +The corresponding Bode plot is represented in figure [9](#org145a286). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\). In this way, the phase diagram is always contained between \\(\SI{0}{\degree}\\) and \\(\SI{-180}{\degree}\\) as a consequence of the interlacing property. @@ -508,12 +508,12 @@ Two broad categories of actuators can be distinguish: A voice coil transducer is an energy transformer which converts electrical power into mechanical power and vice versa. -The system consists of (see figure [10](#org576a7f9)): +The system consists of (see figure [10](#org589d929)): - A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap - A coil which is free to move axially - + {{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="Figure 10: Physical principle of a voice coil transducer" >}} @@ -551,9 +551,9 @@ Thus, at any time, there is an equilibrium between the electrical power absorbed #### Proof-Mass Actuator {#proof-mass-actuator} -A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) (figure [11](#orge7047bc)). +A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) (figure [11](#org29d3a41)). - + {{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="Figure 11: Proof-mass actuator" >}} @@ -583,9 +583,9 @@ with: -Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#orgf19efd8)). +Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#orgfadd8ec)). - + {{< figure src="/ox-hugo/preumont18_proof_mass_tf.png" caption="Figure 12: Bode plot \\(F/i\\) of the proof-mass actuator" >}} @@ -610,7 +610,7 @@ By using the two equations, we obtain: Above the corner frequency, the gain of the geophone is equal to the transducer constant \\(T\\). - + {{< figure src="/ox-hugo/preumont18_geophone.png" caption="Figure 13: Model of a geophone based on a voice coil transducer" >}} @@ -619,9 +619,9 @@ Designing geophones with very low corner frequency is in general difficult. Acti ### General Electromechanical Transducer {#general-electromechanical-transducer} -The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#org6266d1c). +The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#org0f65711). - + {{< figure src="/ox-hugo/preumont18_electro_mechanical_transducer.png" caption="Figure 14: Electrical analog representation of an electromechanical transducer" >}} @@ -646,7 +646,7 @@ With: Equation \eqref{eq:gen_trans_e} shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals. Thus, if \\(Z\_ei\\) can be measured and substracted from \\(e\\), a signal proportional to the velocity is obtained. -To do so, the bridge circuit as shown on figure [15](#org37d6035) can be used. +To do so, the bridge circuit as shown on figure [15](#orgf6f982f) can be used. We can show that @@ -656,7 +656,7 @@ We can show that which is indeed a linear function of the velocity \\(v\\) at the mechanical terminals. - + {{< figure src="/ox-hugo/preumont18_bridge_circuit.png" caption="Figure 15: Bridge circuit for self-sensing actuation" >}} @@ -664,9 +664,9 @@ which is indeed a linear function of the velocity \\(v\\) at the mechanical term ### Smart Materials {#smart-materials} Smart materials have the ability to respond significantly to stimuli of different physical nature. -Figure [16](#org4f66ebd) lists various effects that are observed in materials in response to various inputs. +Figure [16](#org4c1156c) lists various effects that are observed in materials in response to various inputs. - + {{< figure src="/ox-hugo/preumont18_smart_materials.png" caption="Figure 16: Stimulus response relations indicating various effects in materials. The smart materials corresponds to the non-diagonal cells" >}} @@ -761,7 +761,7 @@ It measures the efficiency of the conversion of the mechanical energy into elect -If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#org9f89282)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating \eqref{eq:piezo_eq_matrix_bis} over the volume of the transducer: +If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#org9ace96d)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating \eqref{eq:piezo_eq_matrix_bis} over the volume of the transducer: \begin{equation} \begin{bmatrix}Q\\\Delta\end{bmatrix} @@ -782,7 +782,7 @@ where - \\(C = \epsilon^T A n^2/l\\) is the capacitance of the transducer with no external load (\\(f = 0\\)) - \\(K\_a = A/s^El\\) is the stiffness with short-circuited electrodes (\\(V = 0\\)) - + {{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="Figure 17: Piezoelectric linear transducer" >}} @@ -802,7 +802,7 @@ Equation \eqref{eq:piezo_stack_eq} can be inverted to obtain #### Energy Stored in the Piezoelectric Transducer {#energy-stored-in-the-piezoelectric-transducer} -Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org1d36e1c). +Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#orgb03700a). The total power delivered to the transducer is the sum of electric power \\(V i\\) and the mechanical power \\(f \dot{\Delta}\\). The net work of the transducer is @@ -810,7 +810,7 @@ The total power delivered to the transducer is the sum of electric power \\(V i\ dW = V i dt + f \dot{\Delta} dt = V dQ + f d\Delta \end{equation} - + {{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="Figure 18: Discrete Piezoelectric Transducer" >}} @@ -844,10 +844,10 @@ The ratio between the remaining stored energy and the initial stored energy is #### Admittance of the Piezoelectric Transducer {#admittance-of-the-piezoelectric-transducer} -Consider the system of figure [19](#org645378f), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\). +Consider the system of figure [19](#orga70a814), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\). The force acting on the mass is negative of that acting on the transducer, \\(f = -M \ddot{x}\\). - + {{< figure src="/ox-hugo/preumont18_piezo_stack_admittance.png" caption="Figure 19: Elementary dynamical model of the piezoelectric transducer" >}} @@ -866,9 +866,9 @@ And one can see that \frac{z^2 - p^2}{z^2} = k^2 \end{equation} -Equation \eqref{eq:distance_p_z} constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [20](#org68045c5)). +Equation \eqref{eq:distance_p_z} constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [20](#orgc632b09)). - + {{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="Figure 20: Typical admittance FRF of the transducer" >}} @@ -1566,7 +1566,7 @@ Their design requires a model of the structure, and there is usually a trade-off When collocated actuator/sensor pairs can be used, stability can be achieved using positivity concepts, but in many situations, collocated pairs are not feasible for HAC. -The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#orgb182ef8). +The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#orgfaf8470). The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure. This approach has the following advantages: @@ -1574,7 +1574,7 @@ This approach has the following advantages: - The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin) - The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin) - + {{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="Figure 21: Principle of the dual-loop HAC/LAC control" >}} @@ -1818,4 +1818,4 @@ This approach has the following advantages: ## Bibliography {#bibliography} -Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. . +Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. . diff --git a/content/zettels/singular_value_decomposition.md b/content/zettels/singular_value_decomposition.md new file mode 100644 index 0000000..323cc8f --- /dev/null +++ b/content/zettels/singular_value_decomposition.md @@ -0,0 +1,67 @@ ++++ +title = "Singular Value Decomposition" +author = ["Thomas Dehaeze"] +draft = false ++++ + +Tags +: + + +## SVD of a MIMO system {#svd-of-a-mimo-system} + +We are interested by the physical interpretation of the SVD when applied to the frequency response of a MIMO system \\(G(s)\\) with \\(m\\) inputs and \\(l\\) outputs. + +\begin{equation} +G = U \Sigma V^H +\end{equation} + +\\(\Sigma\\) +: is an \\(l \times m\\) matrix with \\(k = \min\\{l, m\\}\\) non-negative **singular values** \\(\sigma\_i\\), arranged in descending order along its main diagonal, the other entries are zero. + +\\(U\\) +: is an \\(l \times l\\) unitary matrix. The columns of \\(U\\), denoted \\(u\_i\\), represent the **output directions** of the plant. They are orthonormal. + +\\(V\\) +: is an \\(m \times m\\) unitary matrix. The columns of \\(V\\), denoted \\(v\_i\\), represent the **input directions** of the plant. They are orthonormal. + +The input and output directions are related through the singular values: + +\begin{equation} + G v\_i = \sigma\_i u\_i +\end{equation} + +So, if we consider an input in the direction \\(v\_i\\), then the output is in the direction \\(u\_i\\). +Furthermore, since \\(\normtwo{v\_i}=1\\) and \\(\normtwo{u\_i}=1\\), we see that **the singular value \\(\sigma\_i\\) directly gives the gain of the matrix \\(G\\) in this direction**. + +The **largest gain** for any input is equal to the **maximum singular value**: +\\[\maxsv(G) \equiv \sigma\_1(G) = \max\_{d\neq 0}\frac{\normtwo{Gd}}{\normtwo{d}} = \frac{\normtwo{Gv\_1}}{\normtwo{v\_1}} \\] +The **smallest gain** for any input direction is equal to the **minimum singular value**: +\\[\minsv(G) \equiv \sigma\_k(G) = \min\_{d\neq 0}\frac{\normtwo{Gd}}{\normtwo{d}} = \frac{\normtwo{Gv\_k}}{\normtwo{v\_k}} \\] + +We define \\(u\_1 = \bar{u}\\), \\(v\_1 = \bar{v}\\), \\(u\_k=\ubar{u}\\) and \\(v\_k = \ubar{v}\\). Then is follows that: +\\[ G\bar{v} = \maxsv \bar{u} ; \quad G\ubar{v} = \minsv \ubar{u} \\] + + +## SVD to pseudo inverse rectangular matrices {#svd-to-pseudo-inverse-rectangular-matrices} + +This is taken from [Preumont's book](preumont18_vibrat_contr_activ_struc_fourt_edition.md). + +The **Singular Value Decomposition** (SVD) is a generalization of the eigenvalue decomposition of a rectangular matrix: +\\[ J = U \Sigma V^T = \sum\_{i=1}^r \sigma\_i u\_i v\_i^T \\] +With: + +- \\(U\\) and \\(V\\) orthogonal matrices. The columns \\(u\_i\\) and \\(v\_i\\) of \\(U\\) and \\(V\\) are the eigenvectors of the square matrices \\(JJ^T\\) and \\(J^TJ\\) respectively +- \\(\Sigma\\) a rectangular diagonal matrix of dimension \\(m \times n\\) containing the square root of the common non-zero eigenvalues of \\(JJ^T\\) and \\(J^TJ\\) +- \\(r\\) is the number of non-zero singular values of \\(J\\) + +The pseudo-inverse of \\(J\\) is: +\\[ J^+ = V\Sigma^+U^T = \sum\_{i=1}^r \frac{1}{\sigma\_i} v\_i u\_i^T \\] + +The conditioning of the Jacobian is measured by the **condition number**: +\\[ c(J) = \frac{\sigma\_{max}}{\sigma\_{min}} \\] + +When \\(c(J)\\) becomes large, the most straightforward way to handle the ill-conditioning is to truncate the smallest singular value out of the sum. +This will have usually little impact of the fitting error while reducing considerably the actuator inputs \\(v\\). + +<./biblio/references.bib> diff --git a/content/zettels/stewart_platforms.md b/content/zettels/stewart_platforms.md index 3e70df7..99108cd 100644 --- a/content/zettels/stewart_platforms.md +++ b/content/zettels/stewart_platforms.md @@ -40,6 +40,9 @@ Tags | Newport | [link](https://www.newport.com/search/?q1=hexapod%3Arelevance%3Acompatibility%3AMETRIC%3AisObsolete%3Afalse%3A-excludeCountries%3AFR%3AnpCategory%3Ahexapods&ajax&text=hexapod) | USA | | Symetrie | [link](https://symetrie.fr/en/hexapods-en/positioning-hexapods/) | France | | CSA Engineering | [link](https://www.csaengineering.com/products-services/hexapod-positioning-systems/hexapod-models.html) | USA | +| Aerotech | [link](https://www.aerotech.com/product-catalog/hexapods.aspx) | USA | +| SmarAct | [link](https://www.smaract.com/smarpod) | Germany | +| Gridbots | [link](https://www.gridbots.com/hexamove.html) | India | ## Stewart Platforms at ESRF {#stewart-platforms-at-esrf} @@ -56,36 +59,36 @@ Tags Papers by J.E. McInroy: -- ([O’Brien et al. 1998](#org6990e82)) -- ([McInroy, O’Brien, and Neat 1999](#org2eaf165)) -- ([McInroy 1999](#org19d227f)) -- ([McInroy and Hamann 2000](#org0de51ce)) -- ([Chen and McInroy 2000](#orgfa121ee)) -- ([McInroy 2002](#org1ad3b68)) -- ([Li, Hamann, and McInroy 2001](#orga0bd81f)) -- ([Lin and McInroy 2003](#org01eae42)) -- ([Jafari and McInroy 2003](#org9490dc9)) -- ([Chen and McInroy 2004](#org3bd3a36)) +- ([O’Brien et al. 1998](#orgb1b8013)) +- ([McInroy, O’Brien, and Neat 1999](#org06dafcc)) +- ([McInroy 1999](#orga04f28d)) +- ([McInroy and Hamann 2000](#org61fde10)) +- ([Chen and McInroy 2000](#org4b76086)) +- ([McInroy 2002](#orgc3f19a5)) +- ([Li, Hamann, and McInroy 2001](#org469fdd4)) +- ([Lin and McInroy 2003](#org52d73bd)) +- ([Jafari and McInroy 2003](#orga8e7146)) +- ([Chen and McInroy 2004](#org97f0e30)) ## Bibliography {#bibliography} -Chen, Y., and J.E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” _IEEE Transactions on Control Systems Technology_ 12 (3):413–21. . +Chen, Y., and J.E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” _IEEE Transactions on Control Systems Technology_ 12 (3):413–21. . -Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. . +Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. . -Jafari, F., and J.E. McInroy. 2003. “Orthogonal Gough-Stewart Platforms for Micromanipulation.” _IEEE Transactions on Robotics and Automation_ 19 (4). Institute of Electrical and Electronics Engineers (IEEE):595–603. . +Jafari, F., and J.E. McInroy. 2003. “Orthogonal Gough-Stewart Platforms for Micromanipulation.” _IEEE Transactions on Robotics and Automation_ 19 (4). Institute of Electrical and Electronics Engineers (IEEE):595–603. . -Lin, Haomin, and J.E. McInroy. 2003. “Adaptive Sinusoidal Disturbance Cancellation for Precise Pointing of Stewart Platforms.” _IEEE Transactions on Control Systems Technology_ 11 (2):267–72. . +Lin, Haomin, and J.E. McInroy. 2003. “Adaptive Sinusoidal Disturbance Cancellation for Precise Pointing of Stewart Platforms.” _IEEE Transactions on Control Systems Technology_ 11 (2):267–72. . -Li, Xiaochun, Jerry C. Hamann, and John E. 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