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<title>Control of the Nano-Active-Stabilization-System</title>
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<a accesskey="H" href="./index.html"> HOME </a>
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</div><div id="content">
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<h1 class="title">Control of the NASS</h1>
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||||
<h1 class="title">Control of the Nano-Active-Stabilization-System</h1>
|
||||
<div id="table-of-contents">
|
||||
<h2>Table of Contents</h2>
|
||||
<div id="text-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#org15699e9">1. Control Configuration - Introduction</a></li>
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||||
<li><a href="#orgfb6ef96">2. Tracking Control - Basic Architectures</a>
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||||
<ul>
|
||||
<li><a href="#org970ab39">2.1. Control in the frame of the Legs</a></li>
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||||
<li><a href="#org82193fb">2.2. Control in the Cartesian frame</a></li>
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||||
</ul>
|
||||
</li>
|
||||
<li><a href="#orgbfd345b">3. Active Damping Architecture - Collocated Control</a>
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||||
<ul>
|
||||
<li><a href="#org3546873">3.1. Integral Force Feedback</a></li>
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||||
<li><a href="#org722b371">3.2. Direct Relative Velocity Feedback</a></li>
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||||
</ul>
|
||||
</li>
|
||||
<li><a href="#orgaee8981">4. HAC-LAC Architectures</a>
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||||
<ul>
|
||||
<li><a href="#orgd9c84f0">4.1. HAC-LAC using IFF and Tracking control in the frame of the Legs</a></li>
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||||
<li><a href="#orgeb80da1">4.2. HAC-LAC using IFF and Tracking control in the Cartesian frame</a></li>
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||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org1b20de0">5. Cascade Architectures</a>
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||||
<ul>
|
||||
<li><a href="#org3e5154f">5.1. Cascade Control with HAC-LAC Inner Loop and Primary Controller in the task space</a></li>
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||||
<li><a href="#org4353aca">5.2. Cascade Control with HAC-LAC Inner Loop and Primary Controller in the joint space</a></li>
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||||
</ul>
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||||
</li>
|
||||
<li><a href="#org49649c2">6. Sensor Fusion Architectures</a></li>
|
||||
<li><a href="#org8b94438">7. \(\mathcal{H}_\infty\) Architectures</a></li>
|
||||
<li><a href="#org7311546">8. Force Control</a></li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The system consist of the following inputs and outputs (Figure <a href="#org2d9f6d0">1</a>):
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>\(\bm{\tau}\): Forces applied in each leg</li>
|
||||
<li>\(\bm{\tau}_m\): Force sensor located in each leg</li>
|
||||
<li>\(\bm{\mathcal{X}}\): Measurement of the payload position with respect to the granite</li>
|
||||
<li>\(d\bm{\mathcal{L}}\): Measurement of the (small) relative motion of each leg</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="org2d9f6d0" class="figure">
|
||||
<p><img src="figs/control_architecture_plant.png" alt="control_architecture_plant.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Block diagram with the inputs and outputs of the system</p>
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||||
</div>
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||||
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||||
<p>
|
||||
In order to position the Sample with respect to the granite, we must use the measurement \(\bm{\mathcal{X}}\) in the control loop.
|
||||
The wanted position of the sample with respect to the granite is represented by \(\bm{r}_\mathcal{X}\).
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||||
From \(\bm{r}_\mathcal{X}\) and \(\bm{\mathcal{X}}\), we can compute the required small change of pose of the nano-hexapod’s top platform expressed in the frame of the nano-hexapod’s base as shown in Figure <a href="#orgc4acef7">2</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
This can we considered as:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>the position error \(\bm{\epsilon}_{\mathcal{X}_n}\) expressed in a frame attach to the base of the nano-hexapod</li>
|
||||
<li>the wanted (small) pose displacement \(\bm{r}_{d\mathcal{X}_n}\) of the nano-hexapod mobile platform with respect to its base</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div id="orgc4acef7" class="figure">
|
||||
<p><img src="figs/control_architecture_pos_error.png" alt="control_architecture_pos_error.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>Block diagram corresponding to the computation of the position error in the frame of the nano-hexapod</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
In this document, we see how the different outputs of the system can be used to control of position \(\bm{\mathcal{X}}\).
|
||||
</p>
|
||||
|
||||
<div id="outline-container-org15699e9" class="outline-2">
|
||||
<h2 id="org15699e9"><span class="section-number-2">1</span> Control Configuration - Introduction</h2>
|
||||
<div class="outline-text-2" id="text-1">
|
||||
<p>
|
||||
In this section, we discuss the control configuration for the NASS.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
From <a class='org-ref-reference' href="#skogestad07_multiv_feedb_contr">skogestad07_multiv_feedb_contr</a>:
|
||||
</p>
|
||||
<blockquote>
|
||||
<p>
|
||||
We define the <b>control configuration</b> to be the restrictions imposed on the overall controller \(K\) by decomposing it into a set of <b>local controllers</b> with predetermined links and with a possibly predetermined design sequence where subcontrollers are designed locally.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Some elements used to build up a specific control configuration are:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li><b>Cascade controllers</b>. The output from one controller is the input to another</li>
|
||||
<li><b>Decentralized controllers</b>. The control system consists of independent feedback controllers which interconnect a subset of the output measurements with a subset of the manipulated inputs.
|
||||
These subsets should not be used by any other controller</li>
|
||||
<li><b>Feedforward elements</b>. Link measured disturbances and manipulated inputs</li>
|
||||
<li><b>Decoupling elements</b>. Link one set of manipulated inputs with another set of manipulated inputs.
|
||||
They are used to improve the performance of decentralized control systems.</li>
|
||||
</ul>
|
||||
</blockquote>
|
||||
|
||||
<p>
|
||||
Decoupling elements will be used to convert quantities from the joint space to the task space and vice-versa.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Decentralized controllers will be largely used both for Tracking control (Section <a href="#org251e3c9">2</a>) and for Active Damping techniques (Section <a href="#org1b3cc21">3</a>)
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Combining both can be done in an HAC-LAC topology presented in Section <a href="#org31fa800">4</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The use of decentralized controllers is proposed in Section <a href="#orga038762">5</a>.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgfb6ef96" class="outline-2">
|
||||
<h2 id="orgfb6ef96"><span class="section-number-2">2</span> Tracking Control - Basic Architectures</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
<p>
|
||||
<a id="org251e3c9"></a>
|
||||
</p>
|
||||
<p>
|
||||
In this section, we suppose that we want to track some reference position \(\bm{r}_{\mathcal{X}_n}\) corresponding to the pose of the nano-hexapod’s mobile platform with respect to its fixed base.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
To do so, we have to the use the leg’s length measurement \(d\bm{\mathcal{L}}\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
However, thanks to the forward and inverse kinematics, the controller can either be designed in the task space or in the joint space.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
These to configuration are described in the next two sections.
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org970ab39" class="outline-3">
|
||||
<h3 id="org970ab39"><span class="section-number-3">2.1</span> Control in the frame of the Legs</h3>
|
||||
<div class="outline-text-3" id="text-2-1">
|
||||
<p>
|
||||
<a id="org8583193"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
From the wanted small change in pose of the nano-hexapod’s mobile platform \(\bm{r}_{d\mathcal{X}_n}\), we can use the Inverse Kinematics of the nano-hexapod to compute the corresponding small change of the leg length of the nano-hexapod \(\bm{r}_{d\mathcal{L}}\).
|
||||
Then, this is subtracted by the measurement of the leg relative displacement \(d\bm{\mathcal{L}}\) to obtain to displacement error of each leg \(\bm{\epsilon}_{d\mathcal{L}}\).
|
||||
Finally, a diagonal (Decentralized) controller \(\bm{K}_\mathcal{L}\) can be used.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org3211e10" class="figure">
|
||||
<p><img src="figs/control_architecture_leg_frame.png" alt="control_architecture_leg_frame.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 3: </span>Control in the frame of the legs</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org82193fb" class="outline-3">
|
||||
<h3 id="org82193fb"><span class="section-number-3">2.2</span> Control in the Cartesian frame</h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
<p>
|
||||
<a id="orgbd7e263"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
From the relative displacement of each leg \(d\bm{\mathcal{L}}\), the pose of the nano-hexapod’s mobile platform \(\bm{\mathcal{X}_n}\) is estimated.
|
||||
It is then subtracted from reference pose of the nano-hexapod \(\bm{r}_{\mathcal{X}_n}\) to obtain the pose error \(\bm{\epsilon}_{\mathcal{X}_n}\).
|
||||
A diagonal controller \(\bm{K}_\mathcal{X}\) is used to generate forces and torques applied on the payload in a frame attached to the nano-hexapod’s base.
|
||||
These forces are then converted to forces applied in each of the nano-hexapod’s actuators by the use of the Jacobian \(\bm{J}^{-T}\).
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org81b6823" class="figure">
|
||||
<p><img src="figs/control_architecture_cartesian_frame.png" alt="control_architecture_cartesian_frame.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Control in the cartesian Frame (rotating frame attached to the nano-hexapod’s base)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbfd345b" class="outline-2">
|
||||
<h2 id="orgbfd345b"><span class="section-number-2">3</span> Active Damping Architecture - Collocated Control</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<p>
|
||||
<a id="org1b3cc21"></a>
|
||||
</p>
|
||||
<p>
|
||||
From <a class='org-ref-reference' href="#preumont18_vibrat_contr_activ_struc_fourt_edition">preumont18_vibrat_contr_activ_struc_fourt_edition</a>:
|
||||
</p>
|
||||
<blockquote>
|
||||
<p>
|
||||
Active damping is very effective in reducing the settling time of transient disturbances and the effect of steady state disturbances near the resonance frequencies of the system; however, away from the resonances, the active damping is completely ineffective and leaves the closed-loop response essentially unchanged.
|
||||
Such low-gain controllers are often called Low Authority Controllers (LAC), because they modify the poles of the system only slightly.
|
||||
</p>
|
||||
</blockquote>
|
||||
|
||||
<p>
|
||||
Two very well known active damping techniques are <b>Integral Force Feedback</b> and <b>Direct Velocity Feedback</b>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
These two active damping techniques are collocated control techniques.
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org3546873" class="outline-3">
|
||||
<h3 id="org3546873"><span class="section-number-3">3.1</span> Integral Force Feedback</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
<a id="orgb398117"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
In this active damping technique, the force sensors in each leg is used.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The controller \(\bm{K}_\text{IFF}\) is a diagonal matrix, each of its diagonal element consists of:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>an pure integrator</li>
|
||||
<li>a gain \(g\) that can be tuned to achieve a maximum damping</li>
|
||||
</ul>
|
||||
|
||||
\begin{equation}
|
||||
\bm{K}_\text{IFF}(s) = \frac{g}{s} \bm{I}_{6}
|
||||
\end{equation}
|
||||
|
||||
<p>
|
||||
A lead-lag can also be used instead of a pure integrator.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org19b5f2d" class="figure">
|
||||
<p><img src="figs/control_architecture_iff.png" alt="control_architecture_iff.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Integral Force Feedback</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org722b371" class="outline-3">
|
||||
<h3 id="org722b371"><span class="section-number-3">3.2</span> Direct Relative Velocity Feedback</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<p>
|
||||
<a id="orgfaf575b"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The controller \(\bm{K}_\text{DVF}\) is a diagonal matrix.
|
||||
Each diagonal element consists of:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>a derivative action up to some frequency \(\omega_0\)</li>
|
||||
<li>a gain \(g\) that can be tuned to achieve a maximum damping</li>
|
||||
</ul>
|
||||
|
||||
\begin{equation}
|
||||
\bm{K}_\text{DVF}(s) = \frac{g s}{\omega_0 + s} \bm{I}_{6}
|
||||
\end{equation}
|
||||
|
||||
|
||||
<div id="org402f972" class="figure">
|
||||
<p><img src="figs/control_architecture_dvf.png" alt="control_architecture_dvf.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Direct Velocity Feedback</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgaee8981" class="outline-2">
|
||||
<h2 id="orgaee8981"><span class="section-number-2">4</span> HAC-LAC Architectures</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<p>
|
||||
<a id="org31fa800"></a>
|
||||
</p>
|
||||
<p>
|
||||
Here we can combine Active Damping Techniques (Low authority control) with a tracking controller (high authority control).
|
||||
Usually, the low authority controller is designed first, and the high authority controller is designed based on the damped plant.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
From <a class='org-ref-reference' href="#preumont18_vibrat_contr_activ_struc_fourt_edition">preumont18_vibrat_contr_activ_struc_fourt_edition</a>:
|
||||
</p>
|
||||
<blockquote>
|
||||
<p>
|
||||
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure <a href="#org1b2c5c7">7</a>.
|
||||
The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure.
|
||||
This approach has the following advantages:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>The active damping extends outside the bandwidth of the HAC and reduces the settling time of the modes which are outsite the bandwidth</li>
|
||||
<li>The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin)</li>
|
||||
<li>The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin)</li>
|
||||
</ul>
|
||||
</blockquote>
|
||||
|
||||
|
||||
<div id="org1b2c5c7" class="figure">
|
||||
<p><img src="figs/control_architecture_hac_lac.png" alt="control_architecture_hac_lac.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>HAC-LAC Control Architecture</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
If there is only one input to the system, the HAC-LAC topology can be represented as depicted in Figure <a href="#org91828a2">8</a>.
|
||||
Usually, the Low Authority Controller is first design, and then the High Authority Controller is designed based on the damped plant.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org91828a2" class="figure">
|
||||
<p><img src="figs/control_architecture_hac_lac_one_input.png" alt="control_architecture_hac_lac_one_input.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>HAC-LAC Architecture with a system having only one input</p>
|
||||
</div>
|
||||
</div>
|
||||
<div id="outline-container-orgd9c84f0" class="outline-3">
|
||||
<h3 id="orgd9c84f0"><span class="section-number-3">4.1</span> HAC-LAC using IFF and Tracking control in the frame of the Legs</h3>
|
||||
<div class="outline-text-3" id="text-4-1">
|
||||
|
||||
<div id="orgd235561" class="figure">
|
||||
<p><img src="figs/control_architecture_hac_iff_L.png" alt="control_architecture_hac_iff_L.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>IFF + Control in the frame of the legs</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgeb80da1" class="outline-3">
|
||||
<h3 id="orgeb80da1"><span class="section-number-3">4.2</span> HAC-LAC using IFF and Tracking control in the Cartesian frame</h3>
|
||||
<div class="outline-text-3" id="text-4-2">
|
||||
|
||||
<div id="orgb89bca0" class="figure">
|
||||
<p><img src="figs/control_architecture_hac_iff_X.png" alt="control_architecture_hac_iff_X.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>IFF + Control in the cartesian frame</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1b20de0" class="outline-2">
|
||||
<h2 id="org1b20de0"><span class="section-number-2">5</span> Cascade Architectures</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
<p>
|
||||
<a id="orga038762"></a>
|
||||
</p>
|
||||
<p>
|
||||
The principle of Cascade control is shown in Figure <a href="#org03ef231">11</a> and explained as follow:
|
||||
</p>
|
||||
<blockquote>
|
||||
<p>
|
||||
To follow <b>two objectives</b> with different properties in one control system, usually a <b>hierarchy</b> of two feedback loops is used in practice.
|
||||
This kind of control topology is called <b>cascade control</b>, which is used when there are <b>several measurements and one prime control variable</b>.
|
||||
Cascade control is implemented by <b>nesting</b> the control loops, as shown in Figure <a href="#org03ef231">11</a>.
|
||||
The output control loop is called the <b>primary loop</b>, while the inner loop is called the secondary loop and is used to fulfill a secondary objective in the closed-loop system. – <a class='org-ref-reference' href="#taghirad13_paral">taghirad13_paral</a>
|
||||
</p>
|
||||
</blockquote>
|
||||
|
||||
|
||||
<div id="org03ef231" class="figure">
|
||||
<p><img src="figs/control_architecture_cascade_control.png" alt="control_architecture_cascade_control.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 11: </span>Cascade Control Architecture</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
This control topology seems adapted for the NASS, as indeed we have more inputs than outputs
|
||||
</p>
|
||||
|
||||
<p>
|
||||
In the NASS’s case:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>The primary objective is to position the sample with respect to the granite, thus the outer loop (and primary controller) should corresponds to a motion control loop</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
The inner loop can be composed of the system controlled with the HAC-LAC topology.
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org3e5154f" class="outline-3">
|
||||
<h3 id="org3e5154f"><span class="section-number-3">5.1</span> Cascade Control with HAC-LAC Inner Loop and Primary Controller in the task space</h3>
|
||||
<div class="outline-text-3" id="text-5-1">
|
||||
|
||||
<div id="orgff7dfc6" class="figure">
|
||||
<p><img src="figs/control_architecture_cascade_L.png" alt="control_architecture_cascade_L.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 12: </span>Cascaded Control consisting of (from inner to outer loop): IFF, Linearization Loop, Tracking Control in the frame of the Legs</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org4353aca" class="outline-3">
|
||||
<h3 id="org4353aca"><span class="section-number-3">5.2</span> Cascade Control with HAC-LAC Inner Loop and Primary Controller in the joint space</h3>
|
||||
<div class="outline-text-3" id="text-5-2">
|
||||
|
||||
<div id="org4bc4c4c" class="figure">
|
||||
<p><img src="figs/control_architecture_cascade_X.png" alt="control_architecture_cascade_X.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 13: </span>Cascaded Control consisting of (from inner to outer loop): IFF, Linearization Loop, Tracking Control in the Cartesian Frame</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org49649c2" class="outline-2">
|
||||
<h2 id="org49649c2"><span class="section-number-2">6</span> Sensor Fusion Architectures</h2>
|
||||
<div class="outline-text-2" id="text-6">
|
||||
<p>
|
||||
<a id="org6e0650a"></a>
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8b94438" class="outline-2">
|
||||
<h2 id="org8b94438"><span class="section-number-2">7</span> \(\mathcal{H}_\infty\) Architectures</h2>
|
||||
<div class="outline-text-2" id="text-7">
|
||||
<p>
|
||||
<a id="org0440352"></a>
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7311546" class="outline-2">
|
||||
<h2 id="org7311546"><span class="section-number-2">8</span> Force Control</h2>
|
||||
<div class="outline-text-2" id="text-8">
|
||||
<p>
|
||||
Signals:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>\(\bm{r}_\mathcal{F}\) is the wanted total force/torque to be applied to the payload</li>
|
||||
<li>\(\bm{\epsilon}_\mathcal{F}\) is the force/torque errors that should be applied to the payload</li>
|
||||
<li>\(\bm{\tau}\) is the force applied in each actuator</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<div class="figure">
|
||||
<p><img src="figs/control_architecture_force.png" alt="control_architecture_force.png" />
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>
|
||||
|
||||
<h1 class='org-ref-bib-h1'>Bibliography</h1>
|
||||
<ul class='org-ref-bib'><li><a id="skogestad07_multiv_feedb_contr">[skogestad07_multiv_feedb_contr]</a> <a name="skogestad07_multiv_feedb_contr"></a>Skogestad & Postlethwaite, Multivariable Feedback Control: Analysis and Design, John Wiley (2007).</li>
|
||||
<li><a id="preumont18_vibrat_contr_activ_struc_fourt_edition">[preumont18_vibrat_contr_activ_struc_fourt_edition]</a> <a name="preumont18_vibrat_contr_activ_struc_fourt_edition"></a>Andre Preumont, Vibration Control of Active Structures - Fourth Edition, Springer International Publishing (2018).</li>
|
||||
<li><a id="taghirad13_paral">[taghirad13_paral]</a> <a name="taghirad13_paral"></a>Taghirad, Parallel robots : mechanics and control, CRC Press (2013).</li>
|
||||
</ul>
|
||||
</p>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Dehaeze Thomas</p>
|
||||
<p class="date">Created: 2020-03-06 ven. 15:09</p>
|
||||
<p class="date">Created: 2020-03-20 ven. 18:52</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
Reference in New Issue
Block a user