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Choice of sensors and control architecture Predictability and limitations of the system dynamics Two-Sensor control architecture Vibration isolation using a Stewart platform Experimental comparison of Force sensor and Inertial Sensor and associated control architecture for vibration isolation
Figure 1: Hexapod for active vibration isolation" />
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<h1 class="post-title">Sensors and control of a space-based six-axis vibration isolation system</h1>
</header>
<div class="post-toc" id="post-toc">
<h2 class="post-toc-title">Contents</h2>
<div class="post-toc-content">
<nav id="TableOfContents"></nav>
</div>
</div>
<div class="post-content">
<dl>
<dt>Tags</dt>
<dd><a href="/zettels/stewart_platforms/">Stewart Platforms</a>, <a href="/zettels/vibration_isolation/">Vibration Isolation</a>, <a href="/zettels/cubic_architecture/">Cubic Architecture</a></dd>
<dt>Reference</dt>
<dd><sup id="f9698a1741fe7492aa9b7b42c7724670"><a href="#hauge04_sensor_contr_space_based_six" title="Hauge \&amp; Campbell, Sensors and Control of a Space-Based Six-Axis Vibration Isolation System, {Journal of Sound and Vibration}, v(3-5), 913-931 (2004).">(Hauge &amp; Campbell, 2004)</a></sup></dd>
<dt>Author(s)</dt>
<dd>Hauge, G., &amp; Campbell, M.</dd>
<dt>Year</dt>
<dd>2004</dd>
</dl>
<p><strong>Discusses</strong>:</p>
<ul>
<li>Choice of sensors and control architecture</li>
<li>Predictability and limitations of the system dynamics</li>
<li>Two-Sensor control architecture</li>
<li>Vibration isolation using a Stewart platform</li>
<li>Experimental comparison of Force sensor and Inertial Sensor and associated control architecture for vibration isolation</li>
</ul>
<p><a id="org666133a"></a></p>
<figure>
<img src="/ox-hugo/hauge04_stewart_platform.png"
alt="Figure 1: Hexapod for active vibration isolation"/> <figcaption>
<p>Figure 1: Hexapod for active vibration isolation</p>
</figcaption>
</figure>
<p><strong>Stewart platform</strong> (Figure <a href="#org666133a">1</a>):</p>
<ul>
<li>Low corner frequency</li>
<li>Large actuator stroke (\(\pm5mm\))</li>
<li>Sensors in each strut (Figure <a href="#org4d96564">2</a>):
<ul>
<li>three-axis load cell</li>
<li>base and payload geophone in parallel with the struts</li>
<li>LVDT</li>
</ul>
</li>
</ul>
<p><a id="org4d96564"></a></p>
<figure>
<img src="/ox-hugo/hauge05_struts.png"
alt="Figure 2: Strut"/> <figcaption>
<p>Figure 2: Strut</p>
</figcaption>
</figure>
<blockquote>
<p>Force sensors typically work well because they are not as sensitive to payload and base dynamics, but are limited in performance by a low-frequency zero pair resulting from the cross-axial stiffness.</p>
</blockquote>
<p><strong>Performance Objective</strong> (frequency domain metric):</p>
<ul>
<li>The transmissibility should be close to 1 between 0-1.5Hz
\(-3dB &lt; |T(\omega)| &lt; 3db\)</li>
<li>The transmissibility should be below -20dB in the 5-20Hz range
\(|T(\omega)| &lt; -20db\)</li>
</ul>
<p>With \(|T(\omega)|\) is the Frobenius norm of the transmissibility matrix and is used to obtain a scalar performance metric.</p>
<p><strong>Challenge</strong>:</p>
<ul>
<li>small frequency separation between the two requirements</li>
</ul>
<p><strong>Robustness</strong>:</p>
<ul>
<li>minimization of the transmissibility amplification (Bode&rsquo;s &ldquo;pop&rdquo;) outside the performance region</li>
</ul>
<p><strong>Model</strong>:</p>
<ul>
<li>single strut axis as the cubic Stewart platform can be decomposed into 6 single-axis systems</li>
</ul>
<p><a id="org74432f8"></a></p>
<figure>
<img src="/ox-hugo/hauge04_strut_model.png"
alt="Figure 3: Strut model"/> <figcaption>
<p>Figure 3: Strut model</p>
</figcaption>
</figure>
<p><strong>Zero Pair when using a Force Sensor</strong>:</p>
<ul>
<li>The frequency of the zero pair corresponds to the resonance frequency of the payload mass and the &ldquo;parasitic&rdquo; stiffness (sum of the cross-axial, suspension, wiring stiffnesses)</li>
<li>This zero pair is usually not predictable nor repeatable</li>
<li>In this Stewart platform, this zero pair uncertainty is due to the internal wiring of the struts</li>
</ul>
<p><strong>Control</strong>:</p>
<ul>
<li>Single-axis controllers =&gt; combine them into a full six-axis controller =&gt; evaluate the full controller in terms of stability and robustness</li>
<li>Sensitivity weighted LQG controller (SWLQG) =&gt; address robustness in flexible dynamic systems</li>
<li>Three type of controller:
<ul>
<li>Force feedback (cell-based)</li>
<li>Inertial feedback (geophone-based)</li>
<li>Combined force/velocity feedback (load cell/geophone based)</li>
</ul>
</li>
</ul>
<blockquote>
<p>The use of multivariable and robust control on the full 6x6 hexapod does not improve performance over single-axis designs.</p>
</blockquote>
<p><a id="table--tab:hauge05-comp-load-cell-geophone"></a></p>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:hauge05-comp-load-cell-geophone">Table 1</a></span>:
Typical characteristics of sensors used for isolation in hexapod systems
</div>
<table>
<thead>
<tr>
<th></th>
<th><strong>Load cell</strong></th>
<th><strong>Geophone</strong></th>
</tr>
</thead>
<tbody>
<tr>
<td>Type</td>
<td>Relative</td>
<td>Inertial</td>
</tr>
<tr>
<td>Relationship with voice coil</td>
<td>Collocated and Dual</td>
<td>Non-Collocated and non-Dual</td>
</tr>
<tr>
<td>Open loop transfer function</td>
<td>(+) Alternating poles/zeros</td>
<td>(-) Large phase drop</td>
</tr>
<tr>
<td>Limitation from low-frequency zero pair</td>
<td>(-) Yes</td>
<td>(+) No</td>
</tr>
<tr>
<td>Sensitive to payload/base dynamics</td>
<td>(+) No</td>
<td>(-) Yes</td>
</tr>
<tr>
<td>Best frequency range</td>
<td>High (low-freq zero limitation)</td>
<td>Low (high-freq toll-off limitation)</td>
</tr>
</tbody>
</table>
<p><strong>Ability of a sensor-actuator pair to improve performance</strong>:
General system with input \(u\), performance \(z\), output \(y\) disturbance \(u\).</p>
<p>Given a sensor \(u\) and actuator \(y\) and a controller \(u = -K(s) y\), the closed loop disturbance to performance transfer function can be written as:</p>
<p>\[ \left[ \frac{z}{w} \right]_\text{CL} = \frac{G(s)_{zw} + K(G(s)_{zw} G(s)_{yu} - G(s)_{zu} G(s)_{yw})}{1 + K G(s)_{yu}} \]</p>
<p>In order to obtain a significant performance improvement is to use a high gain controller, <em>provided</em> the term \(G(s)_{zw} + K(G(s)_{zw} G(s)_{yu} - G(s)_{zu} G(s)_{yw})\) is small.</p>
<p>We can compare the transfer function from \(w\) to \(z\) with and without a high gain controller.
And we find that for \(u\) and \(y\) to be an acceptable pair for high gain control:
\[ \left| \frac{G(j\omega)_{zw} G(j\omega)_{yu} - G(j\omega)_{zu} G(j\omega)_{yw}}{K G(j\omega)_{yu}} \right| \ll |G_{zw}(j\omega)| \]</p>
<p><strong>Controllers</strong>:</p>
<p><strong>Force feedback</strong>:</p>
<ul>
<li>Performance limited by the low frequency zero-pair</li>
<li>It is desirable to separate the zero-pair and first most are separated by at least a decade in frequency</li>
<li>This can be achieve by reducing the cross-axis stiffness</li>
<li>If the low frequency zero pair is inverted, robustness is lost</li>
<li>Thus, the force feedback controller should be designed to have combined performance and robustness at frequencies at least a decade above the zero pair</li>
<li>The presented controller as a high pass filter at to reduce the gain below the zero-pair, a lag at low frequency to improve phase margin, and a low pass filter for roll off</li>
</ul>
<p><strong>Inertial feedback</strong>:</p>
<ul>
<li>Non-Collocated =&gt; multiple phase drops that limit the bandwidth of the controller</li>
<li>Good performance, but the transmissibility &ldquo;pops&rdquo; due to low phase margin and thus this indicates robustness problems</li>
</ul>
<p><strong>Combined force/velocity feedback</strong>:</p>
<ul>
<li>Use the low frequency performance advantages of geophone sensor with the high robustness advantages of the load cell sensor</li>
<li>A Single-Input-Multiple-Outputs (SIMO) controller is found using LQG</li>
<li>The performance requirements are met</li>
<li>Good robustness</li>
</ul>
<p><a id="orgca6905f"></a></p>
<figure>
<img src="/ox-hugo/hauge04_obtained_transmissibility.png"
alt="Figure 4: Experimental open loop (solid) and closed loop six-axis transmissibility using the geophone only controller (dotted), and combined geophone/load cell controller (dashed)"/> <figcaption>
<p>Figure 4: Experimental open loop (solid) and closed loop six-axis transmissibility using the geophone only controller (dotted), and combined geophone/load cell controller (dashed)</p>
</figcaption>
</figure>
<h1 id="bibliography">Bibliography</h1>
<p><a id="hauge04_sensor_contr_space_based_six"></a>Hauge, G., &amp; Campbell, M., <em>Sensors and control of a space-based six-axis vibration isolation system</em>, Journal of Sound and Vibration, <em>269(3-5)</em>, 913–931 (2004). <a href="http://dx.doi.org/10.1016/s0022-460x(03)00206-2">http://dx.doi.org/10.1016/s0022-460x(03)00206-2</a> <a href="#f9698a1741fe7492aa9b7b42c7724670">↩</a></p>
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