In this document we will consider an <b>isolation platform</b> (e.g. the nano-hexapod) on top of a <b>flexible support</b> (e.g. the micro-station).
</p>
<p>
The goal is to study:
</p>
<ulclass="org-ul">
<li>how does the dynamics of the support influence the dynamics of the plant to control</li>
<li>similarly: how does the uncertainty on the support’s dynamics will be transferred to uncertainty on the plant</li>
<li>what design choice should be made in order to minimize the resulting uncertainty on the plant</li>
</ul>
<p>
Two models are made to study these effects:
</p>
<ulclass="org-ul">
<li>In section <ahref="#org232d01f">1</a>, simple mass-spring-damper systems are chosen to model both the isolation platform and the flexible support</li>
<li>In section <ahref="#orgb01b074">2</a>, we consider arbitrary support dynamics with multiplicative input uncertainty to study the unmodelled dynamics of the support</li>
The compliance of the support without the isolation platform is \(\frac{1}{m^\prime s^2 + c^\prime s + k^\prime}\) and its bode plot is shown in Figure <ahref="#orgf0e5d13">2</a>.
</p>
<p>
One can see that support has a resonance frequency of \(\omega_0^\prime = 50\ Hz\).
<p><spanclass="figure-number">Figure 2: </span>Nominal compliance of the support (<ahref="./figs/nominal_support_compliance_dynamics.png">png</a>, <ahref="./figs/nominal_support_compliance_dynamics.pdf">pdf</a>)</p>
<p><spanclass="figure-number">Figure 3: </span>Obtained plant for the three isolation platforms considered (<ahref="./figs/plant_dynamics_uncertainty_stiff_mid_soft.png">png</a>, <ahref="./figs/plant_dynamics_uncertainty_stiff_mid_soft.pdf">pdf</a>)</p>
Let’s now consider a general support described by its <b>compliance</b> \(G^\prime(s) = \frac{x^\prime}{F^\prime}\) as shown in Figure <ahref="#orgaa4cf23">4</a>.
Now let’s consider the system consisting of a mass-spring-system (the isolation platform) on top of a general support as shown in Figure <ahref="#org524a33a">5</a>.
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 <ahref="#org41bc770">1</a>:
<h3id="orgc20cabb"><spanclass="section-number-3">2.3</span> Compliance of the Support</h3>
<divclass="outline-text-3"id="text-2-3">
<p>
We model the support by a mass-spring-damper model with some uncertainty.
</p>
<p>
The nominal compliance of the support is corresponding to the compliance of a mass-spring-damper system with a mass of \(1000\ kg\) and a stiffness of \(10^8\ [N/m]\).
The main resonance of the support is then \(\omega^\prime = \sqrt{\frac{m^\prime}{k^\prime}} \approx 50\ Hz\).
\[ w_I(s) = \frac{\tau s + r_0}{(\tau/r_\infty) s + 1} \]
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.
<p><spanclass="figure-number">Figure 7: </span>Uncertainty of the support’s compliance (<ahref="./figs/compliance_support_uncertainty.png">png</a>, <ahref="./figs/compliance_support_uncertainty.pdf">pdf</a>)</p>
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:
<p><spanclass="figure-number">Figure 9: </span>Obtained plant for the three isolators (<ahref="./figs/plant_uncertainty_stiffness_isolator.png">png</a>, <ahref="./figs/plant_uncertainty_stiffness_isolator.pdf">pdf</a>)</p>
The obtain result is very similar to the one obtain in section <ahref="#org232d01f">1</a>, except for the stiff isolation that experience lot’s of uncertainty at high frequency.
This is due to the fact that with the current model, at high frequency, the support’s compliance uncertainty is much higher than the previous model.
<h3id="org6967854"><spanclass="section-number-3">2.6</span> Reduce the Uncertainty on the plant</h3>
<divclass="outline-text-3"id="text-2-6">
<p>
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.
</p>
<p>
The uncertainty of the plant is described by an inverse multiplicative uncertainty with the following weight:
<p><spanclass="figure-number">Figure 11: </span>Evolution of the norm of the uncertainty weight \(|w_{iI}(j\omega)|\) as a function of the platform’s stiffness \(k\) (<ahref="./figs/inverse_multiplicative_uncertainty_norm_k.png">png</a>, <ahref="./figs/inverse_multiplicative_uncertainty_norm_k.pdf">pdf</a>)</p>
</div>
<p>
Instead of plotting as a function of the platform’s stiffness, we can plot as a function of \(\omega_0/\omega_0^\prime\) where:
</p>
<ulclass="org-ul">
<li>\(\omega_0\) is the resonance of the platform alone</li>
<li>\(\omega_0^\prime\) is the resonance of the support alone</li>
</ul>
<p>
The obtain plot is shown in Figure <ahref="#org9adcd50">12</a>.
In that case, we can see that with a platform’s resonance frequency 10 times lower than the resonance of the support, we get less than \(1\%\) uncertainty.
<p><spanclass="figure-number">Figure 12: </span>Evolution of the norm of the uncertainty weight \(|w_{iI}(j\omega)|\) as a function of the frequency ratio \(\omega_0/\omega_0^\prime\) (<ahref="./figs/inverse_multiplicative_uncertainty_k_normalized_frequency.png">png</a>, <ahref="./figs/inverse_multiplicative_uncertainty_k_normalized_frequency.pdf">pdf</a>)</p>
<h4id="orgd9a82cb"><spanclass="section-number-4">2.6.2</span> Effect of the platform’s damping \(c\)</h4>
<divclass="outline-text-4"id="text-2-6-2">
<p>
Let’s fix \(k = 10^7\ [N/m]\), \(m = 100\ [kg]\) and see the evolution of \(|w_{iI}(j\omega)|\) with the isolator damping \(c\) (Figure <ahref="#org983fa6b">13</a>).
<p><spanclass="figure-number">Figure 13: </span>Evolution of the norm of the uncertainty weight \(|w_{iI}(j\omega)|\) as a function of the platform’s damping ratio \(\xi\) (<ahref="./figs/inverse_multiplicative_uncertainty_norm_c.png">png</a>, <ahref="./figs/inverse_multiplicative_uncertainty_norm_c.pdf">pdf</a>)</p>
<h4id="orgd2fc303"><spanclass="section-number-4">2.6.3</span> Effect of the platform’s mass \(m\)</h4>
<divclass="outline-text-4"id="text-2-6-3">
<p>
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 <ahref="#orgf899c7a">14</a>).
<p><spanclass="figure-number">Figure 14: </span>Evolution of the norm of the uncertainty weight \(|w_{iI}(j\omega)|\) as a function of the payload mass \(m\) (<ahref="./figs/inverse_multiplicative_uncertainty_norm_m.png">png</a>, <ahref="./figs/inverse_multiplicative_uncertainty_norm_m.pdf">pdf</a>)</p>
If the goal is to have an acceptable (\(<10\%\))uncertaintyontheplantuntilthehighestfrequency,twodesignchoicefortheisolationplatformarepossible:
</p>
<ulclass="org-ul">
<li>a very soft isolation platform \(\omega_0 \ll \omega_0^\prime\)</li>
<li>a very stiff isolation platform \(\omega_0 \gg \omega_0^\prime\)</li>
</ul>
<p>
If a very soft isolation platform is used, the uncertainty due to the support’s compliance is filtered out and never reaches problematic values.
</p>
<p>
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).
