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< / div > < div id = "content" >
< h1 class = "title" > Effect of Uncertainty on the payload’ s dynamics on the isolation platform dynamics< / h1 >
< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#org4c816e0" > 1. Simple Introductory Example< / a >
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< ul >
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< li > < a href = "#orgaf607f3" > 1.1. Equations of motion< / a > < / li >
< li > < a href = "#org08d6c58" > 1.2. Initialization of the payload dynamics< / a > < / li >
< li > < a href = "#orgd871eaf" > 1.3. Initialization of the isolation platform< / a > < / li >
< li > < a href = "#org28efeb7" > 1.4. Comparison< / a > < / li >
< li > < a href = "#org95c8f67" > 1.5. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org7c75d19" > 2. Generalization to arbitrary dynamics< / a >
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< ul >
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< li > < a href = "#org5080024" > 2.1. Introduction< / a > < / li >
< li > < a href = "#org1b0c4a4" > 2.2. Equations of motion< / a > < / li >
< li > < a href = "#org0505562" > 2.3. Impedance \(G^\prime(s)\) of a mass-spring-damper payload< / a > < / li >
< li > < a href = "#orge0f3a40" > 2.4. First Analytical analysis< / a > < / li >
< li > < a href = "#orgc1a00ca" > 2.5. Impedance of the Payload and Dynamical Uncertainty< / a > < / li >
< li > < a href = "#org456b213" > 2.6. Equivalent Inverse Multiplicative Uncertainty< / a > < / li >
< li > < a href = "#org1d2455c" > 2.7. Effect of the Isolation platform Stiffness< / a > < / li >
< li > < a href = "#org1321ab2" > 2.8. Reduce the Uncertainty on the plant< / a >
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< ul >
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< li > < a href = "#org7519de0" > 2.8.1. Effect of the platform’ s stiffness \(k\)< / a > < / li >
< li > < a href = "#org2cfe073" > 2.8.2. Effect of the platform’ s damping \(c\)< / a > < / li >
< li > < a href = "#org20f04de" > 2.8.3. Effect of the platform’ s mass \(m\)< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgff05731" > 2.9. Conclusion< / a > < / li >
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< / ul >
< / li >
< / ul >
< / div >
< / div >
< p >
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 >
< ul class = "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 >
< ul class = "org-ul" >
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< li > In section < a href = "#org752f885" > 1< / a > , simple mass-spring-damper systems are chosen to model both the isolation platform and the payload< / li >
< li > In section < a href = "#org1cfcec8" > 2< / a > , we consider arbitrary payload dynamics with multiplicative input uncertainty to study the unmodelled dynamics of the payload< / li >
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< / ul >
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< div id = "outline-container-org4c816e0" class = "outline-2" >
< h2 id = "org4c816e0" > < span class = "section-number-2" > 1< / span > Simple Introductory Example< / h2 >
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< div class = "outline-text-2" id = "text-1" >
< p >
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< a id = "org752f885" > < / a >
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< / p >
< p >
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Let’ s consider the system shown in Figure < a href = "#orge9db1fc" > 1< / a > consisting of:
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< / p >
< ul class = "org-ul" >
< li > An < b > isolation platform< / b > represented by a mass \(m\), a stiffness \(k\) and a dashpot \(c\) and an actuator \(F\)< / li >
< li > A < b > payload< / b > represented by a mass \(m^\prime\), a stiffness \(k^\prime\) and a dashpot \(c^\prime\)< / li >
< / ul >
< p >
The goal is to stabilize \(x\) using \(F\) in spite of uncertainty on the payload mechanical properties.
< / p >
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< div id = "orge9db1fc" class = "figure" >
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< p > < img src = "figs/2dof_system_stiffness_uncertainty_payload.png" alt = "2dof_system_stiffness_uncertainty_payload.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Two degrees-of-freedom system< / p >
< / div >
< / div >
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< div id = "outline-container-orgaf607f3" class = "outline-3" >
< h3 id = "orgaf607f3" > < span class = "section-number-3" > 1.1< / span > Equations of motion< / h3 >
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< div class = "outline-text-3" id = "text-1-1" >
< p >
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If we write the equation of motion of the system in Figure < a href = "#orge9db1fc" > 1< / a > , we obtain:
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< / p >
\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}
< p >
After eliminating \(x^\prime\), we obtain:
< / p >
\begin{equation}
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\label{orgfd11867}
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\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}
< / div >
< / div >
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< div id = "outline-container-org08d6c58" class = "outline-3" >
< h3 id = "org08d6c58" > < span class = "section-number-3" > 1.2< / span > Initialization of the payload dynamics< / h3 >
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< div class = "outline-text-3" id = "text-1-2" >
< p >
Let the payload have:
< / p >
< ul class = "org-ul" >
< li > a nominal mass of \(m^\prime = 50\ [kg]\)< / li >
< li > a nominal stiffness of \(k^\prime = 5 \cdot 10^6\ [N/m]\)< / li >
< li > a nominal damping of \(c^\prime = 3 \cdot 10^3\ [N/(m/s)]\)< / li >
< / ul >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > mpi = 50;
kpi = 5e6;
cpi = 3e3;
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kpi = (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 50)< span class = "org-type" > ^< / span > 2< span class = "org-type" > *< / span > mpi;
cpi = 0.2< span class = "org-type" > *< / span > sqrt(kpi< span class = "org-type" > *< / span > mpi);
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< / pre >
< / div >
< p >
Let’ s also consider some uncertainty in those parameters:
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > mp = ureal(< span class = "org-string" > 'm'< / span > , mpi, < span class = "org-string" > 'Range'< / span > , [1, 100]);
cp = ureal(< span class = "org-string" > 'c'< / span > , cpi, < span class = "org-string" > 'Percentage'< / span > , 30);
kp = ureal(< span class = "org-string" > 'k'< / span > , kpi, < span class = "org-string" > 'Percentage'< / span > , 30);
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< / pre >
< / div >
< p >
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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 < a href = "#orgfaa9b30" > 2< / a > .
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< / p >
< p >
One can see that the payload has a resonance frequency of \(\omega_0^\prime = 250\ Hz\).
< / p >
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< div id = "orgfaa9b30" class = "figure" >
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< p > < img src = "figs/nominal_payload_compliance_dynamics.png" alt = "nominal_payload_compliance_dynamics.png" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > Nominal compliance of the payload (< a href = "./figs/nominal_payload_compliance_dynamics.png" > png< / a > , < a href = "./figs/nominal_payload_compliance_dynamics.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgd871eaf" class = "outline-3" >
< h3 id = "orgd871eaf" > < span class = "section-number-3" > 1.3< / span > Initialization of the isolation platform< / h3 >
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< div class = "outline-text-3" id = "text-1-3" >
< p >
Let’ s first fix the mass of the isolation platform:
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > m = 10;
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< / pre >
< / div >
< p >
And we generate three isolation platforms:
< / p >
< ul class = "org-ul" >
< li > A soft one with \(\omega_0 = 0.1 \omega_0^\prime = 5\ Hz\)< / li >
< li > A medium stiff one with \(\omega_0 = \omega_0^\prime = 50\ Hz\)< / li >
< li > A stiff one with \(\omega_0 = 10 \omega_0^\prime = 500\ Hz\)< / li >
< / ul >
< / div >
< / div >
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< div id = "outline-container-org28efeb7" class = "outline-3" >
< h3 id = "org28efeb7" > < span class = "section-number-3" > 1.4< / span > Comparison< / h3 >
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< div class = "outline-text-3" id = "text-1-4" >
< p >
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The obtained dynamics from \(F\) to \(x\) for the three isolation platform are shown in Figure < a href = "#org57a4827" > 3< / a > .
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< / p >
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< div id = "org57a4827" class = "figure" >
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< p > < img src = "figs/plant_dynamics_uncertainty_payload_variability.png" alt = "plant_dynamics_uncertainty_payload_variability.png" / >
< / p >
< p > < span class = "figure-number" > Figure 3: < / span > Obtained plant for the three isolation platforms considered (< a href = "./figs/plant_dynamics_uncertainty_payload_variability.png" > png< / a > , < a href = "./figs/plant_dynamics_uncertainty_payload_variability.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org95c8f67" class = "outline-3" >
< h3 id = "org95c8f67" > < span class = "section-number-3" > 1.5< / span > Conclusion< / h3 >
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< div class = "outline-text-3" id = "text-1-5" >
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< div class = "important" id = "orgae2c60b" >
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< p >
The stiff platform dynamics does not seems to depend on the dynamics of the payload.
< / p >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org7c75d19" class = "outline-2" >
< h2 id = "org7c75d19" > < span class = "section-number-2" > 2< / span > Generalization to arbitrary dynamics< / h2 >
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< div class = "outline-text-2" id = "text-2" >
< p >
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< a id = "org1cfcec8" > < / a >
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< / p >
< / div >
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< div id = "outline-container-org5080024" class = "outline-3" >
< h3 id = "org5080024" > < span class = "section-number-3" > 2.1< / span > Introduction< / h3 >
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< div class = "outline-text-3" id = "text-2-1" >
< p >
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Let’ s now consider a general payload described by its < b > impedance< / b > \(G^\prime(s) = \frac{F^\prime}{x}\) as shown in Figure < a href = "#org8c5577f" > 4< / a > .
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< / p >
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< div class = "note" id = "org70c39e9" >
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< p >
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Note here that we use the term < i > impedance< / i > , however, the mechanical impedance is usually defined as the ratio of the force over the velocity \(F^\prime/\dot{x}\). We should refer to < i > resistance< / i > instead of < i > impedance< / i > .
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< / p >
< / div >
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< div id = "org8c5577f" class = "figure" >
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< p > < img src = "figs/general_payload_impedance.png" alt = "general_payload_impedance.png" / >
< / p >
< p > < span class = "figure-number" > Figure 4: < / span > General support< / p >
< / div >
< p >
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Now let’ s consider the system consisting of a mass-spring-system (the isolation platform) supporting the general payload as shown in Figure < a href = "#orgdc750e6" > 5< / a > .
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< / p >
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< div id = "orgdc750e6" class = "figure" >
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< p > < img src = "figs/general_payload_with_isolator.png" alt = "general_payload_with_isolator.png" / >
< / p >
< p > < span class = "figure-number" > Figure 5: < / span > Mass-Spring-Damper (isolation platform) supporting a general payload< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org1b0c4a4" class = "outline-3" >
< h3 id = "org1b0c4a4" > < span class = "section-number-3" > 2.2< / span > Equations of motion< / h3 >
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< div class = "outline-text-3" id = "text-2-2" >
< p >
We have to following equations of motion:
< / p >
\begin{align}
ms^2 x & = F - (cs + k) x - F^\prime \\
F^\prime & = G^\prime(s) x
\end{align}
< p >
And by eliminating \(F^\prime\), we find the plant dynamics \(G(s) = \frac{x}{F}\).
< / p >
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< div class = "important" id = "org7e9adf1" >
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\begin{equation}
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\label{org11f0d43}
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\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 >
< ul class = "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 >
< / ul >
< / div >
< / div >
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< div id = "outline-container-org0505562" class = "outline-3" >
< h3 id = "org0505562" > < span class = "section-number-3" > 2.3< / span > Impedance \(G^\prime(s)\) of a mass-spring-damper payload< / h3 >
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< div class = "outline-text-3" id = "text-2-3" >
< p >
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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 < a href = "#orge9db1fc" > 1< / a > :
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< / p >
\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*}
< p >
By eliminating \(x^\prime\) of the equations, we obtain:
< / p >
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< div class = "important" id = "orgb96f8e8" >
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\begin{equation}
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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}
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\end{equation}
< p >
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The impedance of a 1dof mass-spring-damper system is described by Eq. \eqref{eq:impedance_mass_spring_damper}.
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< / p >
< / div >
< p >
And we obtain
< / p >
\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*}
< p >
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Which is the same transfer function that was obtained in section < a href = "#org752f885" > 1< / a > (Eq. \eqref{eq:plant_simple_system}).
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< / p >
< p >
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The impedance of the mass-spring-damper system is shown in Figure < a href = "#orgb67a7fe" > 6< / a > .
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< / p >
< ul class = "org-ul" >
< li > Before the resonance frequency \(\omega_0^\prime\), the impedance follows the mass line< / li >
< li > After the resonance, the impedance will follow the stiffness line (depending on the relative values of the stiffness and damping)< / li >
< li > At high frequency, it will follow the damping line< / li >
< / ul >
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< div id = "orgb67a7fe" class = "figure" >
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< p > < img src = "figs/example_impedance_mass_spring_damper.png" alt = "example_impedance_mass_spring_damper.png" / >
< / p >
< p > < span class = "figure-number" > Figure 6: < / span > Example of the impedance of a mass-spring-damper system (< a href = "./figs/example_impedance_mass_spring_damper.png" > png< / a > , < a href = "./figs/example_impedance_mass_spring_damper.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orge0f3a40" class = "outline-3" >
< h3 id = "orge0f3a40" > < span class = "section-number-3" > 2.4< / span > First Analytical analysis< / h3 >
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< div class = "outline-text-3" id = "text-2-4" >
< p >
To summarize, we consider:
< / p >
< ul class = "org-ul" >
< li > an Isolation platform represented by a mass \(m\), a damper \(c\) and a stiffness \(k\).
This system resonate at \(\omega_0 = \sqrt{\frac{k}{m}}\)< / li >
< li > A payload represented by a mass \(m^\prime\), a damper \(c^\prime\) and a stiffness \(k^\prime\).
The payload resonate at \(\omega_0^\prime = \sqrt{\frac{k^\prime}{m^\prime}}\)< / li >
< / ul >
< p >
The “ impedance” of the payload is represented by:
\[ G^\prime(s) = \frac{m^\prime s^2 (c^\prime s + k^\prime)}{m^\prime s^2 + c^\prime s + k^\prime} \]
< / p >
< p >
And the plant is:
\[ G(s) = \frac{x}{F} = \frac{1}{ms^2 + cs + k + G^\prime(s)} \]
< / p >
< p >
Let’ s write the asymptotic behavior of \(|G^\prime(j\omega)|\):
< / p >
< ul class = "org-ul" >
< li > \(\lim_{\omega \to 0} |G^\prime(j\omega)| = m^\prime s^2\)< / li >
< li > \(|G^\prime(j\omega_0)| = \frac{k^\prime \sqrt{1 + (2\xi^\prime)^2}}{2 \xi^\prime}\)< / li >
< 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 >
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< div class = "important" id = "orgc2589fd" >
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< 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.
< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgc1a00ca" class = "outline-3" >
< h3 id = "orgc1a00ca" > < span class = "section-number-3" > 2.5< / span > Impedance of the Payload and Dynamical Uncertainty< / h3 >
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< div class = "outline-text-3" id = "text-2-5" >
< p >
We model the payload by a mass-spring-damper model with some uncertainty.
< / p >
< p >
Let the payload have:
< / p >
< ul class = "org-ul" >
< li > a nominal mass of \(m^\prime = 50\ [kg]\)< / li >
< li > a nominal stiffness of \(k^\prime = 5 \cdot 10^6\ [N/m]\)< / li >
< li > a nominal damping of \(c^\prime = 3 \cdot 10^3\ [N/(m/s)]\)< / li >
< / ul >
< p >
The main resonance of the payload is then \(\omega^\prime = \sqrt{\frac{m^\prime}{k^\prime}} \approx 50\ Hz\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > m0 = 10;
c0 = 3e2;
k0 = 5e5;
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Gp0 = (m0< span class = "org-type" > *< / span > s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > *< / span > (c0< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > k0))< span class = "org-type" > /< / span > (m0< span class = "org-type" > *< / span > s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > c0< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > k0);
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< / pre >
< / div >
< p >
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Let’ s represent the uncertainty on the impedance of the payload by a multiplicative uncertainty (Figure < a href = "#org6a86e10" > 7< / a > ):
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\[ G^\prime(s) = G_0^\prime(s)(1 + w_I^\prime(s)\Delta_I(s)) \quad |\Delta_I(j\omega)| < 1 \ \ forall \ omega \ ]
< / p >
< p >
This could represent < b > unmodelled dynamics< / b > or unknown parameters of the payload.
< / p >
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< div id = "org6a86e10" class = "figure" >
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< p > < img src = "figs/input_uncertainty_set.png" alt = "input_uncertainty_set.png" / >
< / p >
< p > < span class = "figure-number" > Figure 7: < / span > Input Multiplicative Uncertainty< / p >
< / div >
< p >
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.
< / p >
< p >
The parameters are defined below.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > r0 = 0.5;
tau = 1< span class = "org-type" > /< / span > (50< span class = "org-type" > *< / span > 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > );
rinf = 10;
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wI = (tau< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > r0)< span class = "org-type" > /< / span > ((tau< span class = "org-type" > /< / span > rinf)< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > 1);
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< / pre >
< / div >
< p >
We then generate a complex \(\Delta\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > DeltaI = ucomplex(< span class = "org-string" > 'A'< / span > ,0);
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< / pre >
< / div >
< p >
We generate the uncertain plant \(G^\prime(s)\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > Gp = Gp0< span class = "org-type" > *< / span > (1< span class = "org-type" > +< / span > wI< span class = "org-type" > *< / span > DeltaI);
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< / pre >
< / div >
< p >
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A set of uncertainty payload’ s impedance transfer functions is shown in Figure < a href = "#org62be1c1" > 8< / a > .
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< / p >
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< div id = "org62be1c1" class = "figure" >
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< p > < img src = "figs/payload_impedance_uncertainty.png" alt = "payload_impedance_uncertainty.png" / >
< / p >
< p > < span class = "figure-number" > Figure 8: < / span > Uncertainty of the payload’ s impedance (< a href = "./figs/payload_impedance_uncertainty.png" > png< / a > , < a href = "./figs/payload_impedance_uncertainty.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org456b213" class = "outline-3" >
< h3 id = "org456b213" > < span class = "section-number-3" > 2.6< / span > Equivalent Inverse Multiplicative Uncertainty< / h3 >
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< div class = "outline-text-3" id = "text-2-6" >
< p >
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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 >
\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*}
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< div class = "important" id = "org8137f47" >
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< p >
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We can the plant dynamics that as an inverse multiplicative uncertainty (Figure < a href = "#org4d0f12d" > 9< / a > ):
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< / p >
\begin{equation}
\frac{x}{F} = G_0(s) (1 + w_{iI}(s) \Delta(s))^{-1}
\end{equation}
< p >
with:
< / p >
< ul class = "org-ul" >
< li > \(G_0(s) = \frac{1}{ms^2 + cs + k + G_0^\prime(s)}\)< / li >
< li > \(w_{iI}(s) = \frac{G_0^\prime(s) w_I(s)}{ms^2 + cs + k + G_0^\prime(s)} = G_0(s) G_0^\prime(s) w_I(s)\)< / li >
< / ul >
< / div >
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< div id = "org4d0f12d" class = "figure" >
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< p > < img src = "figs/inverse_uncertainty_set.png" alt = "inverse_uncertainty_set.png" / >
< / p >
< p > < span class = "figure-number" > Figure 9: < / span > Inverse Multiplicative Uncertainty< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org1d2455c" class = "outline-3" >
< h3 id = "org1d2455c" > < span class = "section-number-3" > 2.7< / span > Effect of the Isolation platform Stiffness< / h3 >
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< div class = "outline-text-3" id = "text-2-7" >
< p >
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Let’ s first fix the mass of the isolation platform:
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > m = 20;
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< / pre >
< / div >
< p >
And we generate three isolation platforms:
< / p >
< ul class = "org-ul" >
< li > A soft one with \(\omega_0 = 5\ Hz\)< / li >
< li > A medium stiff one with \(\omega_0 = 50\ Hz\)< / li >
< li > A stiff one with \(\omega_0 = 500\ Hz\)< / li >
< / ul >
< p >
Soft Isolation Platform:
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > k_soft = m< span class = "org-type" > *< / span > (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 5)< span class = "org-type" > ^< / span > 2;
c_soft = 0.1< span class = "org-type" > *< / span > sqrt(m< span class = "org-type" > *< / span > k_soft);
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G_soft = 1< span class = "org-type" > /< / span > (m< span class = "org-type" > *< / span > s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > c_soft< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > k_soft < span class = "org-type" > +< / span > Gp);
G0_soft = 1< span class = "org-type" > /< / span > (m< span class = "org-type" > *< / span > s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > c_soft< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > k_soft < span class = "org-type" > +< / span > Gp0);
wiI_soft = Gp0< span class = "org-type" > *< / span > G0_soft< span class = "org-type" > *< / span > wI;
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< / pre >
< / div >
< p >
Mid Isolation Platform
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > k_mid = m< span class = "org-type" > *< / span > (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 50)< span class = "org-type" > ^< / span > 2;
c_mid = 0.1< span class = "org-type" > *< / span > sqrt(m< span class = "org-type" > *< / span > k_mid);
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G_mid = 1< span class = "org-type" > /< / span > (m< span class = "org-type" > *< / span > s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > c_mid< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > k_mid < span class = "org-type" > +< / span > Gp);
G0_mid = 1< span class = "org-type" > /< / span > (m< span class = "org-type" > *< / span > s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > c_mid< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > k_mid < span class = "org-type" > +< / span > Gp0);
wiI_mid = Gp0< span class = "org-type" > *< / span > G0_mid< span class = "org-type" > *< / span > wI;
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< / pre >
< / div >
< p >
Stiff Isolation Platform
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > k_stiff = m< span class = "org-type" > *< / span > (2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 500)< span class = "org-type" > ^< / span > 2;
c_stiff = 0.1< span class = "org-type" > *< / span > sqrt(m< span class = "org-type" > *< / span > k_stiff);
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G_stiff = 1< span class = "org-type" > /< / span > (m< span class = "org-type" > *< / span > s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > c_stiff< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > k_stiff < span class = "org-type" > +< / span > Gp);
G0_stiff = 1< span class = "org-type" > /< / span > (m< span class = "org-type" > *< / span > s< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > c_stiff< span class = "org-type" > *< / span > s < span class = "org-type" > +< / span > k_stiff < span class = "org-type" > +< / span > Gp0);
wiI_stiff = Gp0< span class = "org-type" > *< / span > G0_stiff< span class = "org-type" > *< / span > wI;
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< / pre >
< / div >
< p >
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The obtained transfer functions \(x/F\) for each of the three platforms are shown in Figure < a href = "#orgf9a8ac5" > 10< / a > .
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< / p >
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< div id = "orgf9a8ac5" class = "figure" >
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< p > < img src = "figs/plant_uncertainty_impedance_payload.png" alt = "plant_uncertainty_impedance_payload.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 10: < / span > Obtained plant for the three isolators (< a href = "./figs/plant_uncertainty_impedance_payload.png" > png< / a > , < a href = "./figs/plant_uncertainty_impedance_payload.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org1321ab2" class = "outline-3" >
< h3 id = "org1321ab2" > < span class = "section-number-3" > 2.8< / span > Reduce the Uncertainty on the plant< / h3 >
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< div class = "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:
\[ w_{iI}(s) = \frac{G_0^\prime(s) w_I(s)}{ms^2 + cs + k + G_0^\prime(s)} \]
< / p >
< p >
Let’ s study separately the effect of the platform’ s mass, damping and stiffness.
< / p >
< / div >
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< div id = "outline-container-org7519de0" class = "outline-4" >
< h4 id = "org7519de0" > < span class = "section-number-4" > 2.8.1< / span > Effect of the platform’ s stiffness \(k\)< / h4 >
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< div class = "outline-text-4" id = "text-2-8-1" >
< p >
Let’ s fix \(\xi = \frac{c}{2\sqrt{km}} = 0.1\), \(m = 100\ [kg]\) and see the evolution of \(|w_{iI}(j\omega)|\) with \(k\).
< / p >
< p >
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This is first shown for few values of the stiffness \(k\) in figure < a href = "#orgd59ef39" > 11< / a >
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< / p >
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< div id = "orgd59ef39" class = "figure" >
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< p > < img src = "figs/inverse_multiplicative_uncertainty_payload_few_k.png" alt = "inverse_multiplicative_uncertainty_payload_few_k.png" / >
< / p >
< p > < span class = "figure-number" > Figure 11: < / span > Norm of the inverse multiplicative uncertainty weight for various values of the the isolation platform’ s stiffness (< a href = "./figs/inverse_multiplicative_uncertainty_payload_few_k.png" > png< / a > , < a href = "./figs/inverse_multiplicative_uncertainty_payload_few_k.pdf" > pdf< / a > )< / p >
< / div >
< p >
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The norm of the uncertainty weight \(|w_iI(j\omega)|\) is displayed as a function of \(\omega\) and \(k\) in Figure < a href = "#org20cbb63" > 12< / a > .
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< / p >
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< div id = "org20cbb63" class = "figure" >
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< p > < img src = "figs/inverse_multiplicative_payload_uncertainty_norm_k.png" alt = "inverse_multiplicative_payload_uncertainty_norm_k.png" / >
< / p >
< p > < span class = "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 \(k\) (< a href = "./figs/inverse_multiplicative_payload_uncertainty_norm_k.png" > png< / a > , < a href = "./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 >
< ul class = "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 >
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The obtain plot is shown in Figure < a href = "#orgab7b13f" > 13< / a > .
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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 >
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< div id = "orgab7b13f" class = "figure" >
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< p > < img src = "figs/inverse_multiplicative_payload_uncertainty_k_normalized_frequency.png" alt = "inverse_multiplicative_payload_uncertainty_k_normalized_frequency.png" / >
< / p >
< p > < span class = "figure-number" > Figure 13: < / span > Evolution of the norm of the uncertainty weight \(|w_{iI}(j\omega)|\) as a function of the frequency ratio \(\omega_0/\omega_0^\prime\) (< a href = "./figs/inverse_multiplicative_payload_uncertainty_k_normalized_frequency.png" > png< / a > , < a href = "./figs/inverse_multiplicative_payload_uncertainty_k_normalized_frequency.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org2cfe073" class = "outline-4" >
< h4 id = "org2cfe073" > < span class = "section-number-4" > 2.8.2< / span > Effect of the platform’ s damping \(c\)< / h4 >
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< div class = "outline-text-4" id = "text-2-8-2" >
< p >
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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 < a href = "#org96c279d" > 14< / a > ).
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< / p >
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< div id = "org96c279d" class = "figure" >
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< p > < img src = "figs/inverse_multiplicative_payload_uncertainty_c.png" alt = "inverse_multiplicative_payload_uncertainty_c.png" / >
< / p >
< p > < span class = "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 \(\xi\) (< a href = "./figs/inverse_multiplicative_payload_uncertainty_c.png" > png< / a > , < a href = "./figs/inverse_multiplicative_payload_uncertainty_c.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org20f04de" class = "outline-4" >
< h4 id = "org20f04de" > < span class = "section-number-4" > 2.8.3< / span > Effect of the platform’ s mass \(m\)< / h4 >
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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 < a href = "#org24afcde" > 15< / a > ).
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< p > < img src = "figs/inverse_multiplicative_payload_uncertainty_m.png" alt = "inverse_multiplicative_payload_uncertainty_m.png" / >
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< p > < span class = "figure-number" > Figure 15: < / span > Evolution of the norm of the uncertainty weight \(|w_{iI}(j\omega)|\) as a function of the payload mass \(m\) (< a href = "./figs/inverse_multiplicative_payload_uncertainty_m.png" > png< / a > , < a href = "./figs/inverse_multiplicative_payload_uncertainty_m.pdf" > pdf< / a > )< / p >
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< h3 id = "orgff05731" > < span class = "section-number-3" > 2.9< / span > Conclusion< / h3 >
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As was expected from Eq. \eqref{org11f0d43}, 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.
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The best thing to do is to have a stiff isolation platform.
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If a soft isolation platform is to be used, it is first a good idea to damp the isolation platform as shown in Figure < a href = "#org96c279d" > 14< / a > .
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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.
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< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2021-02-20 sam. 23:08< / p >
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