In this document we will consider an <b>isolation platform</b> (e.g. the nano-hexapod) with a <b>payload</b> on top (e.g. the the sample to be positioned).
</p>
<p>
The goal is to study:
</p>
<ulclass="org-ul">
<li>how does the dynamics of the payload influence the dynamics of the isolation platform</li>
<li>similarly: how does the uncertainty on the payload’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="#org971d11c">1</a>, simple mass-spring-damper systems are chosen to model both the isolation platform and the payload</li>
<li>In section <ahref="#org7065358">2</a>, we consider arbitrary payload dynamics with multiplicative input uncertainty to study the unmodelled dynamics of the payload</li>
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 <ahref="#org3c89797">2</a>.
</p>
<p>
One can see that the payload has a resonance frequency of \(\omega_0^\prime = 250\ Hz\).
<p><spanclass="figure-number">Figure 2: </span>Nominal compliance of the payload (<ahref="./figs/nominal_payload_compliance_dynamics.png">png</a>, <ahref="./figs/nominal_payload_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_payload_variability.png">png</a>, <ahref="./figs/plant_dynamics_uncertainty_payload_variability.pdf">pdf</a>)</p>
Let’s now consider a general payload described by its <b>impedance</b> \(G^\prime(s) = \frac{x}{F^\prime}\) as shown in Figure <ahref="#orgb54b79a">4</a>.
</p>
<divclass="note">
<p>
Note here that we use the term <i>impedance</i>, however, the mechanical impedance is usually defined as the ratio of the velocity over the force \(\dot{x}/F^\prime\). We should refer to <i>resistance</i> instead of <i>impedance</i>.
Now let’s consider the system consisting of a mass-spring-system (the isolation platform) supporting the general payload as shown in Figure <ahref="#orga07f362">5</a>.
And by eliminating \(F^\prime\), we find the plant dynamics \(G(s) = \frac{x}{F}\).
</p>
<divclass="important">
\begin{equation}
\label{org8b9a6a7}
\frac{x}{F} = \frac{1}{ms^2 + cs + k + G^\prime(s)}
\end{equation}
</div>
<p>
We can learn few things about the obtained transfer function:
</p>
<ulclass="org-ul">
<li>the zeros of \(x/F\) will be the poles of \(G^\prime(s)\).</li>
<li>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</li>
<h3id="orge217a33"><spanclass="section-number-3">2.3</span> Impedance \(G^\prime(s)\) of a mass-spring-damper payload</h3>
<divclass="outline-text-3"id="text-2-3">
<p>
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="#orgaa77a57">1</a>:
<p><spanclass="figure-number">Figure 6: </span>Example of the impedance of a mass-spring-damper system (<ahref="./figs/example_impedance_mass_spring_damper.png">png</a>, <ahref="./figs/example_impedance_mass_spring_damper.pdf">pdf</a>)</p>
<li>\(\lim_{\omega \to \infty} |G^\prime(j\omega)| = c^\prime s + k\)</li>
</ul>
<p>
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:
\[ |G^\prime(s)| \ll |ms^2 + cs + k| \]
</p>
<p>
Let’s take the case of a <b>stiff payload</b> (\(\omega_0^\prime \gg \omega_0\)).
</p>
<p>
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 <b>payload mass should be small with respect to the isolation platform mass</b>.
</p>
<p>
Above \(\omega_0\):
\[ |G^\prime(j\omega)| \ll m \omega^2 \]
</p>
<p>
Until \(\omega_0^\prime\), we have \(m^\prime \ll m\) which is the same condition as before.
Above \(\omega_0^\prime\), we obtain \(|jc^\prime \omega + k| \ll m \omega^2\).
</p>
<divclass="important">
<p>
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.
\[ w_I(s) = \frac{\tau s + r_0}{(\tau/r_\infty) s + 1} \]
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>
<p>
The parameters are defined below.
</p>
<divclass="org-src-container">
<preclass="src src-matlab">r0 = 0.5;
tau = 1<spanclass="org-type">/</span>(50<spanclass="org-type">*</span>2<spanclass="org-type">*</span><spanclass="org-constant">pi</span>);
rinf = 10;
wI = (tau<spanclass="org-type">*</span>s <spanclass="org-type">+</span> r0)<spanclass="org-type">/</span>((tau<spanclass="org-type">/</span>rinf)<spanclass="org-type">*</span>s <spanclass="org-type">+</span> 1);
<p><spanclass="figure-number">Figure 8: </span>Uncertainty of the payload’s impedance (<ahref="./figs/payload_impedance_uncertainty.png">png</a>, <ahref="./figs/payload_impedance_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 10: </span>Obtained plant for the three isolators (<ahref="./figs/plant_uncertainty_impedance_payload.png">png</a>, <ahref="./figs/plant_uncertainty_impedance_payload.pdf">pdf</a>)</p>
<h3id="org1466bd9"><spanclass="section-number-3">2.8</span> Reduce the Uncertainty on the plant</h3>
<divclass="outline-text-3"id="text-2-8">
<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>Norm of the inverse multiplicative uncertainty weight for various values of the the isolation platform’s stiffness (<ahref="./figs/inverse_multiplicative_uncertainty_payload_few_k.png">png</a>, <ahref="./figs/inverse_multiplicative_uncertainty_payload_few_k.pdf">pdf</a>)</p>
</div>
<p>
The norm of the uncertainty weight \(|w_iI(j\omega)|\) is displayed as a function of \(\omega\) and \(k\) in Figure <ahref="#org362ed76">12</a>.
<p><spanclass="figure-number">Figure 12: </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_payload_uncertainty_norm_k.png">png</a>, <ahref="./figs/inverse_multiplicative_payload_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="#org27fe0c1">13</a>.
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.
<p><spanclass="figure-number">Figure 13: </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_payload_uncertainty_k_normalized_frequency.png">png</a>, <ahref="./figs/inverse_multiplicative_payload_uncertainty_k_normalized_frequency.pdf">pdf</a>)</p>
<h4id="org4c45fb5"><spanclass="section-number-4">2.8.2</span> Effect of the platform’s damping \(c\)</h4>
<divclass="outline-text-4"id="text-2-8-2">
<p>
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 <ahref="#org51df34a">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 platform’s damping ratio \(\xi\) (<ahref="./figs/inverse_multiplicative_payload_uncertainty_c.png">png</a>, <ahref="./figs/inverse_multiplicative_payload_uncertainty_c.pdf">pdf</a>)</p>
<h4id="org9086831"><spanclass="section-number-4">2.8.3</span> Effect of the platform’s mass \(m\)</h4>
<divclass="outline-text-4"id="text-2-8-3">
<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="#orgd260e86">15</a>).
<p><spanclass="figure-number">Figure 15: </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_payload_uncertainty_m.png">png</a>, <ahref="./figs/inverse_multiplicative_payload_uncertainty_m.pdf">pdf</a>)</p>
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.
</p>
<p>
The best thing to do is to have a stiff isolation platform.
</p>
<p>
If a soft isolation platform is to be used, it is first a good idea to damp the isolation platform as shown in Figure <ahref="#org51df34a">14</a>.
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.
