From 8c6768f372b89ba7b9b92227ef61dfb507ec868a Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Thu, 26 Mar 2020 17:25:43 +0100 Subject: [PATCH] Publish html --- docs/control.html | 145 +-- docs/control_active_damping.html | 261 ++--- docs/control_decentralized.html | 136 ++- docs/control_force.html | 302 ++++++ docs/control_requirements.html | 96 +- docs/figs/control_architecture_force.pdf | Bin 68814 -> 68814 bytes docs/figs/general_payload_impedance.pdf | Bin 0 -> 34229 bytes docs/figs/general_payload_impedance.png | Bin 0 -> 5619 bytes ...nt_dynamics_uncertainty_stiff_mid_soft.pdf | Bin 291365 -> 285013 bytes ...nt_dynamics_uncertainty_stiff_mid_soft.png | Bin 124029 -> 121817 bytes .../plant_uncertainty_stiffness_isolator.pdf | Bin 361960 -> 341106 bytes .../plant_uncertainty_stiffness_isolator.png | Bin 148771 -> 129692 bytes docs/uncertainty_payload.html | 992 ++++++++++++++++++ docs/uncertainty_support.html | 40 +- org/control_active_damping.org | 4 +- org/control_decentralized.org | 4 - org/control_requirements.org | 2 +- org/uncertainty_payload.org | 17 +- org/uncertainty_support.org | 45 +- 19 files changed, 1712 insertions(+), 332 deletions(-) create mode 100644 docs/control_force.html create mode 100644 docs/figs/general_payload_impedance.pdf create mode 100644 docs/figs/general_payload_impedance.png create mode 100644 docs/uncertainty_payload.html diff --git a/docs/control.html b/docs/control.html index 643d72e..3517c2a 100644 --- a/docs/control.html +++ b/docs/control.html @@ -4,7 +4,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Control of the Nano-Active-Stabilization-System @@ -202,28 +202,50 @@ + + + + + + + + +
+ UP + | + HOME +
+

Force Control applied on the Simscape Model

+ +

+Cascade Control where the inner loop goal is to control the total wrench (forces and torques) applied to the upper part of the nano-hexapod. +

+ +

+The force will induce an acceleration of the upper part, thus we can consider the new input \(\bm{r}_\mathcal{F}\) as an acceleration input. +

+ +

+Ideas: +

+ + + +
+

control_cascade_force_F.png +

+
+ + +
+

control_cascade_force_tau.png +

+
+
+
+

Author: Dehaeze Thomas

+

Created: 2020-03-26 jeu. 17:25

+
+ + diff --git a/docs/control_requirements.html b/docs/control_requirements.html index 40f906b..69d7fde 100644 --- a/docs/control_requirements.html +++ b/docs/control_requirements.html @@ -4,7 +4,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Control Requirements @@ -202,28 +202,50 @@ + + + + + + + + +
+ UP + | + HOME +
+

Effect of Uncertainty on the payload’s dynamics on the isolation platform dynamics

+ + +

+In this document we will consider an isolation platform (e.g. the nano-hexapod) with a payload on top (e.g. the the sample to be positioned). +

+ +

+The goal is to study: +

+ + +

+Two models are made to study these effects: +

+ + +
+

1 Simple Introductory Example

+
+

+ +

+

+Let’s consider the system shown in Figure 1 consisting of: +

+
    +
  • An isolation platform represented by a mass \(m\), a stiffness \(k\) and a dashpot \(c\) and an actuator \(F\)
    • +
  • A payload represented by a mass \(m^\prime\), a stiffness \(k^\prime\) and a dashpot \(c^\prime\)
    • +
+ +

+The goal is to stabilize \(x\) using \(F\) in spite of uncertainty on the payload mechanical properties. +

+ + +
+

2dof_system_stiffness_uncertainty_payload.png +

+

Figure 1: Two degrees-of-freedom system

+
+
+ +
+

1.1 Equations of motion

+
+

+If we write the equation of motion of the system in Figure 1, we obtain: +

+\begin{align} + ms^2 x &= F - (cs + k) x + (c^\prime s + k^\prime) (x^\prime - x) \\ + m^\prime s^2 x^\prime &= - (c^\prime s + k^\prime) (x^\prime - x) +\end{align} + +

+After eliminating \(x^\prime\), we obtain: +

+\begin{equation} +\label{orge5d69a3} + \frac{x}{F} = \frac{m^\prime s^2 + c^\prime s + k^\prime}{(ms^2 + cs + k)(m^\prime s^2 + c^\prime s + k^\prime) + m^\prime s^2(c^\prime s + k^\prime)} +\end{equation} +
+
+ +
+

1.2 Initialization of the payload dynamics

+
+

+Let the payload have: +

+
    +
  • a nominal mass of \(m^\prime = 50\ [kg]\)
    • +
  • a nominal stiffness of \(k^\prime = 5 \cdot 10^6\ [N/m]\)
    • +
  • a nominal damping of \(c^\prime = 3 \cdot 10^3\ [N/(m/s)]\)
    • +
+ +
+
mpi = 50;
+kpi = 5e6;
+cpi = 3e3;
+
+kpi = (2*pi*50)^2*mpi;
+cpi = 0.2*sqrt(kpi*mpi);
+
+
+ +

+Let’s also consider some uncertainty in those parameters: +

+
+
mp = ureal('m', mpi, 'Range', [1, 100]);
+cp = ureal('c', cpi, 'Percentage', 30);
+kp = ureal('k', kpi, 'Percentage', 30);
+
+
+ +

+The compliance of the payload without the isolation platform is \(\frac{1}{m^\prime s^2 + c^\prime s + k^\prime}\) and its bode plot is shown in Figure 2. +

+ +

+One can see that the payload has a resonance frequency of \(\omega_0^\prime = 250\ Hz\). +

+ + +
+

nominal_payload_compliance_dynamics.png +

+

Figure 2: Nominal compliance of the payload (png, pdf)

+
+
+
+
+

1.3 Initialization of the isolation platform

+
+

+Let’s first fix the mass of the isolation platform: +

+
+
m = 10;
+
+
+ +

+And we generate three isolation platforms: +

+
    +
  • A soft one with \(\omega_0 = 0.1 \omega_0^\prime = 5\ Hz\)
    • +
  • A medium stiff one with \(\omega_0 = \omega_0^\prime = 50\ Hz\)
    • +
  • A stiff one with \(\omega_0 = 10 \omega_0^\prime = 500\ Hz\)
    • +
+
+
+ +
+

1.4 Comparison

+
+

+The obtained dynamics from \(F\) to \(x\) for the three isolation platform are shown in Figure 3. +

+ + +
+

plant_dynamics_uncertainty_payload_variability.png +

+

Figure 3: Obtained plant for the three isolation platforms considered (png, pdf)

+
+
+
+ +
+

1.5 Conclusion

+
+
+

+The stiff platform dynamics does not seems to depend on the dynamics of the payload. +

+ +
+
+
+
+ +
+

2 Generalization to arbitrary dynamics

+
+

+ +

+
+
+

2.1 Introduction

+
+

+Let’s now consider a general payload described by its impedance \(G^\prime(s) = \frac{x}{F^\prime}\) as shown in Figure 4. +

+ +
+

+Note here that we use the term impedance, however, the mechanical impedance is usually defined as the ratio of the velocity over the force \(\dot{x}/F^\prime\). We should refer to resistance instead of impedance. +

+ +
+ + +
+

general_payload_impedance.png +

+

Figure 4: General support

+
+ +

+Now let’s consider the system consisting of a mass-spring-system (the isolation platform) supporting the general payload as shown in Figure 5. +

+ +
+

general_payload_with_isolator.png +

+

Figure 5: Mass-Spring-Damper (isolation platform) supporting a general payload

+
+
+
+ +
+

2.2 Equations of motion

+
+

+We have to following equations of motion: +

+\begin{align} + ms^2 x &= F - (cs + k) x - F^\prime \\ + F^\prime &= G^\prime(s) x +\end{align} + +

+And by eliminating \(F^\prime\), we find the plant dynamics \(G(s) = \frac{x}{F}\). +

+ +
+\begin{equation} +\label{org8b9a6a7} + \frac{x}{F} = \frac{1}{ms^2 + cs + k + G^\prime(s)} +\end{equation} + +
+ +

+We can learn few things about the obtained transfer function: +

+
    +
  • the zeros of \(x/F\) will be the poles of \(G^\prime(s)\).
    • +
  • if the impedance of the payload is small \(|G^\prime(s)| \ll |ms^2 + cs + k|\), then the payload will have small influence on the obtained dynamics
    • +
+
+
+ +
+

2.3 Impedance \(G^\prime(s)\) of a mass-spring-damper payload

+
+

+In order to verify that the formula is correct, let’s take the same mass-spring-damper system used in the system shown in Figure 1: +

+\begin{align*} + m^\prime s^2 x^\prime &= (x - x^\prime) (c^\prime s + k^\prime) \\ + F^\prime &= (x - x^\prime) (c^\prime s + k^\prime) +\end{align*} + +

+By eliminating \(x^\prime\) of the equations, we obtain: +

+
+\begin{equation} +\label{orgae0b162} + G^\prime(s) = \frac{F^\prime}{x} = \frac{m^\prime s^2 (c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime} +\end{equation} + +

+The impedance of a 1dof mass-spring-damper system is described by Eq. \eqref{orgae0b162}. +

+ +
+ +

+And we obtain +

+\begin{align*} + \frac{x}{F} &= \frac{1}{ms^2 + cs + k + G^\prime(s)} \\ + &= \frac{1}{ms^2 + cs + k + \frac{m^\prime s^2 (c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime}} \\ + &= \frac{m^\prime s^2 + c^\prime s + k^\prime}{(ms^2 + cs + k) (m^\prime s^2 + c^\prime s + k^\prime) + m^\prime s^2 (c^\prime s + k)} +\end{align*} + +

+Which is the same transfer function that was obtained in section 1 (Eq. \eqref{orge5d69a3}). +

+ +

+The impedance of the mass-spring-damper system is shown in Figure 6. +

+
    +
  • Before the resonance frequency \(\omega_0^\prime\), the impedance follows the mass line
    • +
  • After the resonance, the impedance will follow the stiffness line (depending on the relative values of the stiffness and damping)
    • +
  • At high frequency, it will follow the damping line
    • +
+ + +
+

example_impedance_mass_spring_damper.png +

+

Figure 6: Example of the impedance of a mass-spring-damper system (png, pdf)

+
+
+
+ +
+

2.4 First Analytical analysis

+
+

+To summarize, we consider: +

+
    +
  • an Isolation platform represented by a mass \(m\), a damper \(c\) and a stiffness \(k\). +This system resonate at \(\omega_0 = \sqrt{\frac{k}{m}}\)
    • +
  • A payload represented by a mass \(m^\prime\), a damper \(c^\prime\) and a stiffness \(k^\prime\). +The payload resonate at \(\omega_0^\prime = \sqrt{\frac{k^\prime}{m^\prime}}\)
    • +
+ +

+The “impedance” of the payload is represented by: +\[ G^\prime(s) = \frac{m^\prime s^2 (c^\prime s + k^\prime)}{m^\prime s^2 + c^\prime s + k^\prime} \] +

+ +

+And the plant is: +\[ G(s) = \frac{x}{F} = \frac{1}{ms^2 + cs + k + G^\prime(s)} \] +

+ +

+Let’s write the asymptotic behavior of \(|G^\prime(j\omega)|\): +

+
    +
  • \(\lim_{\omega \to 0} |G^\prime(j\omega)| = m^\prime s^2\)
    • +
  • \(|G^\prime(j\omega_0)| = \frac{k^\prime \sqrt{1 + (2\xi^\prime)^2}}{2 \xi^\prime}\)
    • +
  • \(\lim_{\omega \to \infty} |G^\prime(j\omega)| = c^\prime s + k\)
    • +
+ +

+Let’s find some conditions in order to have that the dynamics of the payload does not influence to much the dynamics of the plant: +\[ |G^\prime(s)| \ll |ms^2 + cs + k| \] +

+ +

+Let’s take the case of a stiff payload (\(\omega_0^\prime \gg \omega_0\)). +

+ +

+Below \(\omega_0\), the condition becomes: +\[ |G^\prime(s)| \ll k \Leftrightarrow m^\prime \omega_0^2 \ll k \Leftrightarrow m^\prime \ll m \] +The payload mass should be small with respect to the isolation platform mass. +

+ +

+Above \(\omega_0\): +\[ |G^\prime(j\omega)| \ll m \omega^2 \] +

+ +

+Until \(\omega_0^\prime\), we have \(m^\prime \ll m\) which is the same condition as before. +Above \(\omega_0^\prime\), we obtain \(|jc^\prime \omega + k| \ll m \omega^2\). +

+ +
+

+When using a soft isolation platform and a stiff payload such that the payload resonate above the first resonance of the isolation platform, the mass of the payload should be small compared to the isolation platform mass in order to not disturb the dynamics of the isolation platform. +

+ +
+
+
+ +
+

2.5 Impedance of the Payload and Dynamical Uncertainty

+
+

+We model the payload by a mass-spring-damper model with some uncertainty. +

+ +

+Let the payload have: +

+
    +
  • a nominal mass of \(m^\prime = 50\ [kg]\)
    • +
  • a nominal stiffness of \(k^\prime = 5 \cdot 10^6\ [N/m]\)
    • +
  • a nominal damping of \(c^\prime = 3 \cdot 10^3\ [N/(m/s)]\)
    • +
+ +

+The main resonance of the payload is then \(\omega^\prime = \sqrt{\frac{m^\prime}{k^\prime}} \approx 50\ Hz\). +

+ +
+
m0 = 10;
+c0 = 3e2;
+k0 = 5e5;
+
+Gp0 = (m0*s^2 * (c0*s + k0))/(m0*s^2 + c0*s + k0);
+
+
+ +

+Let’s represent the uncertainty on the impedance of the payload by a multiplicative uncertainty (Figure 7): +\[ G^\prime(s) = G_0^\prime(s)(1 + w_I^\prime(s)\Delta_I(s)) \quad |\Delta_I(j\omega)| < 1\ \forall \omega \] +

+ +

+This could represent unmodelled dynamics or unknown parameters of the payload. +

+ + +
+

input_uncertainty_set.png +

+

Figure 7: Input Multiplicative Uncertainty

+
+ +

+We choose a simple uncertainty weight: +\[ w_I(s) = \frac{\tau s + r_0}{(\tau/r_\infty) s + 1} \] +where \(r_0\) is the relative uncertainty at steady-state, \(1/\tau\) is the frequency at which the relative uncertainty reaches \(100\ \%\), and \(r_\infty\) is the magnitude of the weight at high frequency. +

+ +

+The parameters are defined below. +

+
+
r0 = 0.5;
+tau = 1/(50*2*pi);
+rinf = 10;
+
+wI = (tau*s + r0)/((tau/rinf)*s + 1);
+
+
+ +

+We then generate a complex \(\Delta\). +

+
+
DeltaI = ucomplex('A',0);
+
+
+ +

+We generate the uncertain plant \(G^\prime(s)\). +

+
+
Gp = Gp0*(1+wI*DeltaI);
+
+
+ +

+A set of uncertainty payload’s impedance transfer functions is shown in Figure 8. +

+ + +
+

payload_impedance_uncertainty.png +

+

Figure 8: Uncertainty of the payload’s impedance (png, pdf)

+
+
+
+ +
+

2.6 Effect of the Isolation platform Stiffness

+
+

+Let’s first fix the mass of the isolation platform: +

+
+
m = 20;
+
+
+ +

+And we generate three isolation platforms: +

+
    +
  • A soft one with \(\omega_0 = 5\ Hz\)
    • +
  • A medium stiff one with \(\omega_0 = 50\ Hz\)
    • +
  • A stiff one with \(\omega_0 = 500\ Hz\)
    • +
+ +

+Soft Isolation Platform: +

+
+
k_soft = m*(2*pi*5)^2;
+c_soft = 0.1*sqrt(m*k_soft);
+
+G_soft = 1/(m*s^2 + c_soft*s + k_soft + Gp);
+
+
+ +

+Mid Isolation Platform +

+
+
k_mid = m*(2*pi*50)^2;
+c_mid = 0.1*sqrt(m*k_mid);
+
+G_mid = 1/(m*s^2 + c_mid*s + k_mid + Gp);
+
+
+ +

+Stiff Isolation Platform +

+
+
k_stiff = m*(2*pi*500)^2;
+c_stiff = 0.1*sqrt(m*k_stiff);
+
+G_stiff = 1/(m*s^2 + c_stiff*s + k_stiff + Gp);
+
+
+ +

+The obtained transfer functions \(x/F\) for each of the three platforms are shown in Figure 9. +

+ + +
+

plant_uncertainty_impedance_payload.png +

+

Figure 9: Obtained plant for the three isolators (png, pdf)

+
+
+
+ +
+

2.7 Equivalent Inverse Multiplicative Uncertainty

+
+

+Let’s express the uncertainty of the plant \(x/F\) as a function of the parameters as well as of the uncertainty on the platform’s compliance: +

+\begin{align*} + \frac{x}{F} &= \frac{1}{ms^2 + cs + k + G_0^\prime(s)(1 + w_I(s)\Delta(s))}\\ + &= \frac{1}{ms^2 + cs + k + G_0^\prime(s)} \cdot \frac{1}{1 + \frac{G_0^\prime(s) w_I(s)}{ms^2 + cs + k + G_0^\prime(s)} \Delta(s)}\\ +\end{align*} + +
+

+We can the plant dynamics that as an inverse multiplicative uncertainty (Figure 10): +

+\begin{equation} + \frac{x}{F} = G_0(s) (1 + w_{iI}(s) \Delta(s))^{-1} +\end{equation} +

+with: +

+
    +
  • \(G_0(s) = \frac{1}{ms^2 + cs + k + G_0^\prime(s)}\)
    • +
  • \(w_{iI}(s) = \frac{G_0^\prime(s) w_I(s)}{ms^2 + cs + k + G_0^\prime(s)} = G_0(s) G_0^\prime(s) w_I(s)\)
    • +
+ +
+ + +
+

inverse_uncertainty_set.png +

+

Figure 10: Inverse Multiplicative Uncertainty

+
+
+
+ +
+

2.8 Reduce the Uncertainty on the plant

+
+

+Now that we know the expression of the uncertainty on the plant, we can wonder what parameters of the isolation platform would lower the plant uncertainty, or at least bring the uncertainty to reasonable level. +

+ +

+The uncertainty of the plant is described by an inverse multiplicative uncertainty with the following weight: +\[ w_{iI}(s) = \frac{G_0^\prime(s) w_I(s)}{ms^2 + cs + k + G_0^\prime(s)} \] +

+ +

+Let’s study separately the effect of the platform’s mass, damping and stiffness. +

+
+ +
+

2.8.1 Effect of the platform’s stiffness \(k\)

+
+

+Let’s fix \(\xi = \frac{c}{2\sqrt{km}} = 0.1\), \(m = 100\ [kg]\) and see the evolution of \(|w_{iI}(j\omega)|\) with \(k\). +

+ +

+This is first shown for few values of the stiffness \(k\) in figure 11 +

+ + +
+

inverse_multiplicative_uncertainty_payload_few_k.png +

+

Figure 11: Norm of the inverse multiplicative uncertainty weight for various values of the the isolation platform’s stiffness (png, pdf)

+
+ +

+The norm of the uncertainty weight \(|w_iI(j\omega)|\) is displayed as a function of \(\omega\) and \(k\) in Figure 12. +

+ + +
+

inverse_multiplicative_payload_uncertainty_norm_k.png +

+

Figure 12: Evolution of the norm of the uncertainty weight \(|w_{iI}(j\omega)|\) as a function of the platform’s stiffness \(k\) (png, pdf)

+
+ +

+Instead of plotting as a function of the platform’s stiffness, we can plot as a function of \(\omega_0/\omega_0^\prime\) where: +

+
    +
  • \(\omega_0\) is the resonance of the platform alone
    • +
  • \(\omega_0^\prime\) is the resonance of the support alone
    • +
+ +

+The obtain plot is shown in Figure 13. +In that case, we can see that with a platform’s resonance frequency 10 times higher than the resonance of the payload, we get less than \(1\%\) uncertainty until some fairly high frequency. +

+ + +
+

inverse_multiplicative_payload_uncertainty_k_normalized_frequency.png +

+

Figure 13: Evolution of the norm of the uncertainty weight \(|w_{iI}(j\omega)|\) as a function of the frequency ratio \(\omega_0/\omega_0^\prime\) (png, pdf)

+
+
+
+ +
+

2.8.2 Effect of the platform’s damping \(c\)

+
+

+Let’s fix \(k = 10^7\ [N/m]\), \(m = 100\ [kg]\) and see the evolution of \(|w_{iI}(j\omega)|\) with the isolation platform damping \(c\) (Figure 14). +

+ + +
+

inverse_multiplicative_payload_uncertainty_c.png +

+

Figure 14: Evolution of the norm of the uncertainty weight \(|w_{iI}(j\omega)|\) as a function of the platform’s damping ratio \(\xi\) (png, pdf)

+
+
+
+ +
+

2.8.3 Effect of the platform’s mass \(m\)

+
+

+Let’s fix \(k = 10^7\ [N/m]\), \(\xi = \frac{c}{2\sqrt{km}} = 0.1\) and see the evolution of \(|w_{iI}(j\omega)|\) with the payload mass \(m\) (Figure 15). +

+ + +
+

inverse_multiplicative_payload_uncertainty_m.png +

+

Figure 15: Evolution of the norm of the uncertainty weight \(|w_{iI}(j\omega)|\) as a function of the payload mass \(m\) (png, pdf)

+
+
+
+
+ +
+

2.9 Conclusion

+
+
+

+As was expected from Eq. \eqref{org8b9a6a7}, it is usually a good idea to maximize the mass, damping and stiffness of the isolation platform in order to be less sensible to the payload dynamics. +

+ +

+The best thing to do is to have a stiff isolation platform. +

+ +

+If a soft isolation platform is to be used, it is first a good idea to damp the isolation platform as shown in Figure 14. +This can make the uncertainty quite low until the first resonance of the payload. +In that case, maximizing the stiffness of the payload is a good idea. +

+ +
+
+
+
+
+
+

Author: Dehaeze Thomas

+

Created: 2020-03-26 jeu. 17:25

+
+ + diff --git a/docs/uncertainty_support.html b/docs/uncertainty_support.html index 80d4a16..493e293 100644 --- a/docs/uncertainty_support.html +++ b/docs/uncertainty_support.html @@ -4,7 +4,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Effect of Uncertainty on the support’s dynamics on the isolation platform dynamics @@ -271,17 +271,17 @@ for the JavaScript code in this tag. @@ -346,8 +346,8 @@ The goal is to stabilize \(x\) using \(F\) in spite of uncertainty on the suppor -
-

1.1 Equations of motion

+
+

1.1 Equations of motion

If we write the equation of motion of the system in Figure 1, we obtain: @@ -451,12 +451,12 @@ The obtained dynamics from \(F\) to \(x\) for the three isolation platform are s

-
-

1.5 Conclusion

+
+

1.5 Conclusion

-The soft platform dynamics does not seems to depend on the dynamics of the support. +The soft platform dynamics does not seems to depend on the dynamics of the support nor to be affect by the dynamic uncertainty of the support.

@@ -497,8 +497,8 @@ Now let’s consider the system consisting of a mass-spring-system (the isol
-
-

2.2 Equations of motion

+
+

2.2 Equations of motion

We have to following equations of motion: @@ -674,6 +674,7 @@ G_stiff = 1/(m*s8.

+

plant_uncertainty_stiffness_isolator.png

@@ -699,10 +700,10 @@ Let’s express the uncertainty of the plant \(x/F\) as a function of the pa &= \frac{1}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)} \cdot \frac{1}{1 + \frac{ms^2(cs + k)G_0^\prime(s) w_I(s)}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)} \Delta(s)}\\ \end{align*} -

-We can rewrite that as an inverse multiplicative uncertainty (Figure 9): -

+

+We can the plant dynamics that as an inverse multiplicative uncertainty (Figure 9): +

\begin{equation} \frac{x}{F} = G_0(s) (1 + w_{iI}(s) \Delta(s))^{-1} \end{equation} @@ -827,9 +828,10 @@ Let’s fix \(k = 10^7\ [N/m]\), \(\xi = \frac{c}{2\sqrt{km}} = 0.1\) and se
-
-

2.7 Conclusion

+
+

2.7 Conclusion

+

If the goal is to have an acceptable (\(<10\%\)) uncertainty on the plant until the highest frequency, two design choice for the isolation platform are possible:

@@ -847,13 +849,15 @@ If a very stiff isolation platform is used, the uncertainty will be high around It will then be high around \(\omega_0\) and probably be higher than one. Thus, if a stiff isolation platform is used, the recommendation is to have the largest possible resonance frequency, as the control bandwidth will be limited by the first resonance of the isolation platform (if not already limited by the resonance of the support).

+ +

Author: Dehaeze Thomas

-

Created: 2020-03-25 mer. 19:20

+

Created: 2020-03-26 jeu. 17:24

diff --git a/org/control_active_damping.org b/org/control_active_damping.org index 0fd0179..42d84c7 100644 --- a/org/control_active_damping.org +++ b/org/control_active_damping.org @@ -87,7 +87,7 @@ After that, a tomography experiment is simulation without any active damping tec <> #+end_src -#+begin_src matlab :tangle no +#+begin_src matlab simulinkproject('../'); #+end_src @@ -2985,7 +2985,7 @@ Inertial Control should not be used. #+end_src #+begin_src matlab - cd('../'); + simulinkproject('../'); #+end_src ** Load the plants diff --git a/org/control_decentralized.org b/org/control_decentralized.org index eeca927..83e6c0c 100644 --- a/org/control_decentralized.org +++ b/org/control_decentralized.org @@ -99,10 +99,6 @@ The signals are: \draw[->] (Ex.east) -- (J.west) node[above left]{$\bm{r}_{\mathcal{X}_n}$}; \draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{\mathcal{L}}$}; \draw[<-] (Ex.west)node[above left]{$\bm{r}_{\mathcal{X}}$} -- ++(-1, 0); - - % \draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{\mathcal{L}}$}; - % \draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_\mathcal{L}$}; - % \draw[->] (G.east) node[above right]{$\bm{\mathcal{L}}$} -| ($(G.east)+(1, -1)$) -| (subr.south); \end{tikzpicture} #+end_src diff --git a/org/control_requirements.org b/org/control_requirements.org index 8e45186..7aec795 100644 --- a/org/control_requirements.org +++ b/org/control_requirements.org @@ -554,7 +554,7 @@ Instead of a pure integrator, let's use a low pass filter with a cut-off frequen | | $d_\mu$ | $F_d$ | $w$ | |-----+------------------------------------+-----------------------------------+--------------------------------------| | IFF | Better filtering of the vibrations | More sensitive to External forces | | -| DVF | inverse | inverse | Little bit better at low frequencies | +| DVF | Opposite | Opposite | Little bit better at low frequencies | ** Control using $x$ *** Analytical analysis diff --git a/org/uncertainty_payload.org b/org/uncertainty_payload.org index bbf0da3..688b7f1 100644 --- a/org/uncertainty_payload.org +++ b/org/uncertainty_payload.org @@ -350,13 +350,13 @@ The obtained dynamics from $F$ to $x$ for the three isolation platform are shown * Generalization to arbitrary dynamics <> ** Introduction -Let's now consider a general payload described by its *impedance* $G^\prime(s) = \frac{x}{F^\prime}$ as shown in Figure [[fig:general_payload_impdeance]]. +Let's now consider a general payload described by its *impedance* $G^\prime(s) = \frac{x}{F^\prime}$ as shown in Figure [[fig:general_payload_impedance]]. #+begin_note Note here that we use the term /impedance/, however, the mechanical impedance is usually defined as the ratio of the velocity over the force $\dot{x}/F^\prime$. We should refer to /resistance/ instead of /impedance/. #+end_note -#+begin_src latex :file general_payload_impdeance.pdf +#+begin_src latex :file general_payload_impedance.pdf \begin{tikzpicture} \def\massw{2.2} % Width of the masses \def\massh{0.8} % Height of the masses @@ -379,10 +379,10 @@ Note here that we use the term /impedance/, however, the mechanical impedance is \end{tikzpicture} #+end_src -#+name: fig:general_support_compliance +#+name: fig:general_payload_impedance #+caption: General support #+RESULTS: -[[file:figs/general_payload_impdeance.png]] +[[file:figs/general_payload_impedance.png]] Now let's consider the system consisting of a mass-spring-system (the isolation platform) supporting the general payload as shown in Figure [[fig:general_payload_with_isolator]]. #+begin_src latex :file general_payload_with_isolator.pdf @@ -441,8 +441,9 @@ We have to following equations of motion: And by eliminating $F^\prime$, we find the plant dynamics $G(s) = \frac{x}{F}$. #+begin_important +#+name: eq:plant_dynamics_general_payload \begin{equation} - \frac{x}{F} = \frac{1}{ms^2 + cs + k + G^\prime(s)} \label{eq:plant_dynamics_general_payload} + \frac{x}{F} = \frac{1}{ms^2 + cs + k + G^\prime(s)} \end{equation} #+end_important @@ -467,8 +468,9 @@ In order to verify that the formula is correct, let's take the same mass-spring- By eliminating $x^\prime$ of the equations, we obtain: #+begin_important +#+name: eq:impedance_mass_spring_damper \begin{equation} - G^\prime(s) = \frac{F^\prime}{x} = \frac{m^\prime s^2 (c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime} \label{eq:impedance_mass_spring_damper} + G^\prime(s) = \frac{F^\prime}{x} = \frac{m^\prime s^2 (c^\prime s + k)}{m^\prime s^2 + c^\prime s + k^\prime} \end{equation} The impedance of a 1dof mass-spring-damper system is described by Eq. [[eq:impedance_mass_spring_damper]]. @@ -667,7 +669,8 @@ Stiff Isolation Platform G_stiff = 1/(m*s^2 + c_stiff*s + k_stiff + Gp); #+end_src -The obtained transfer functions $x/F$ for each of the three platforms are shown in Figure [[fig:plant_uncertainty_stiffness_isolator]]. +The obtained transfer functions $x/F$ for each of the three platforms are shown in Figure [[fig:plant_uncertainty_impedance_payload]]. + #+begin_src matlab :exports none freqs = logspace(0, 3, 1000); diff --git a/org/uncertainty_support.org b/org/uncertainty_support.org index 3a1d10d..31dfc84 100644 --- a/org/uncertainty_support.org +++ b/org/uncertainty_support.org @@ -272,7 +272,7 @@ The obtained dynamics from $F$ to $x$ for the three isolation platform are shown set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]'); - title('Soft Platform'); + title('$\omega_0 \ll \omega_0^\prime$'); hold off; ax4 = subplot(2,3,4); @@ -293,7 +293,7 @@ The obtained dynamics from $F$ to $x$ for the three isolation platform are shown end set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); - title('Medium Stiff Platform'); + title('$\omega_0 \approx \omega_0^\prime$'); hold off; ax5 = subplot(2,3,5); @@ -314,7 +314,7 @@ The obtained dynamics from $F$ to $x$ for the three isolation platform are shown end set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); - title('Stiff Platform'); + title('$\omega_0 \gg \omega_0^\prime$'); hold off; ax6 = subplot(2,3,6); @@ -344,7 +344,7 @@ The obtained dynamics from $F$ to $x$ for the three isolation platform are shown ** Conclusion #+begin_important - The soft platform dynamics does not seems to depend on the dynamics of the support. + The soft platform dynamics does not seems to depend on the dynamics of the support nor to be affect by the dynamic uncertainty of the support. #+end_important * Generalization to arbitrary dynamics @@ -615,6 +615,7 @@ Stiff Isolation Platform #+end_src The obtained transfer functions $x/F$ for each of the three platforms are shown in Figure [[fig:plant_uncertainty_stiffness_isolator]]. + #+begin_src matlab :exports none freqs = logspace(0, 3, 1000); @@ -633,9 +634,9 @@ The obtained transfer functions $x/F$ for each of the three platforms are shown set(gca, 'XTickLabel',[]); ylabel('Magnitude [dB]'); hold off; - title('Soft Isolator'); + title('$\omega_0 \ll \omega_0^\prime$'); - ax2 = subplot(2,3,4); + ax4 = subplot(2,3,4); hold on; for i = 1:length(Gs_soft) plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_soft(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]); @@ -646,10 +647,7 @@ The obtained transfer functions $x/F$ for each of the three platforms are shown ylabel('Phase [deg]'); hold off; - linkaxes([ax1,ax2],'x'); - xlim([freqs(1), freqs(end)]); - - ax1 = subplot(2,3,2); + ax2 = subplot(2,3,2); hold on; for i = 1:length(Gs_mid) plot(freqs, abs(squeeze(freqresp(Gs_mid(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]); @@ -657,9 +655,9 @@ The obtained transfer functions $x/F$ for each of the three platforms are shown set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); hold off; - title('Medium Stiff Isolator'); + title('$\omega_0 \approx \omega_0^\prime$'); - ax2 = subplot(2,3,5); + ax5 = subplot(2,3,5); hold on; for i = 1:length(Gs_mid) plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_mid(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]); @@ -670,10 +668,7 @@ The obtained transfer functions $x/F$ for each of the three platforms are shown xlabel('Frequency [Hz]'); hold off; - linkaxes([ax1,ax2],'x'); - xlim([freqs(1), freqs(end)]); - - ax1 = subplot(2,3,3); + ax3 = subplot(2,3,3); hold on; for i = 1:length(Gs_stiff) plot(freqs, abs(squeeze(freqresp(Gs_stiff(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]); @@ -681,9 +676,9 @@ The obtained transfer functions $x/F$ for each of the three platforms are shown set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); hold off; - title('Stiff Isolator'); + title('$\omega_0 \gg \omega_0^\prime$'); - ax2 = subplot(2,3,6); + ax6 = subplot(2,3,6); hold on; for i = 1:length(Gs_stiff) plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_stiff(:, :, i), freqs, 'Hz'))), '-', 'color', [0, 0, 0, 0.2]); @@ -693,8 +688,10 @@ The obtained transfer functions $x/F$ for each of the three platforms are shown ylim([-180 180]); hold off; - linkaxes([ax1,ax2],'x'); + linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x'); xlim([freqs(1), freqs(end)]); + + linkaxes([ax1,ax2,ax3],'y'); #+end_src #+header: :tangle no :exports results :results none :noweb yes @@ -717,8 +714,8 @@ Let's express the uncertainty of the plant $x/F$ as a function of the parameters &= \frac{1}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)} \cdot \frac{1}{1 + \frac{ms^2(cs + k)G_0^\prime(s) w_I(s)}{ms^2 + cs + k + ms^2(cs + k)G_0^\prime(s)} \Delta(s)}\\ \end{align*} -We can rewrite that as an inverse multiplicative uncertainty (Figure [[fig:inverse_uncertainty_set]]): #+begin_important +We can the plant dynamics that as an inverse multiplicative uncertainty (Figure [[fig:inverse_uncertainty_set]]): \begin{equation} \frac{x}{F} = G_0(s) (1 + w_{iI}(s) \Delta(s))^{-1} \end{equation} @@ -906,7 +903,7 @@ Let's fix $k = 10^7\ [N/m]$, $m = 100\ [kg]$ and see the evolution of $|w_{iI}(j surf(freqs, xi, wiI_c_soft', 'FaceColor', 'interp', 'EdgeColor', 'none') set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Damping Ratio'); - title('Soft Platform'); + title('$\omega_0 \ll \omega_0^\prime$'); view([0 0 1]); set(gca,'ColorScale','log') colorbar('location', 'west'); @@ -916,7 +913,7 @@ Let's fix $k = 10^7\ [N/m]$, $m = 100\ [kg]$ and see the evolution of $|w_{iI}(j surf(freqs, xi, wiI_c_mid', 'FaceColor', 'interp', 'EdgeColor', 'none') set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); - title('Medium Stiff Platform'); + title('$\omega_0 \approx \omega_0^\prime$'); view([0 0 1]); set(gca,'ColorScale','log') caxis([1e-3, 1]); @@ -924,7 +921,7 @@ Let's fix $k = 10^7\ [N/m]$, $m = 100\ [kg]$ and see the evolution of $|w_{iI}(j ax3 = subplot(1, 3, 3); surf(freqs, xi, wiI_c_stiff', 'FaceColor', 'interp', 'EdgeColor', 'none') set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - title('Stiff Platform'); + title('$\omega_0 \gg \omega_0^\prime$'); view([0 0 1]); set(gca,'ColorScale','log') caxis([1e-3 1e0]); @@ -983,6 +980,7 @@ Let's fix $k = 10^7\ [N/m]$, $\xi = \frac{c}{2\sqrt{km}} = 0.1$ and see the evol [[file:figs/inverse_multiplicative_uncertainty_norm_m.png]] ** Conclusion +#+begin_important If the goal is to have an acceptable ($<10\%$) uncertainty on the plant until the highest frequency, two design choice for the isolation platform are possible: - a very soft isolation platform $\omega_0 \ll \omega_0^\prime$ - a very stiff isolation platform $\omega_0 \gg \omega_0^\prime$ @@ -992,6 +990,7 @@ If a very soft isolation platform is used, the uncertainty due to the support's If a very stiff isolation platform is used, the uncertainty will be high around $\omega_0^\prime$ and may reach unacceptable value. It will then be high around $\omega_0$ and probably be higher than one. Thus, if a stiff isolation platform is used, the recommendation is to have the largest possible resonance frequency, as the control bandwidth will be limited by the first resonance of the isolation platform (if not already limited by the resonance of the support). +#+end_important * Numerical Analysis for the NASS :noexport: <>