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Thomas Dehaeze
2022-03-15 16:40:48 +01:00
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content/article/chen00_ident_decoup_contr_flexur_joint_hexap.md
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title = "Identification and decoupling control of flexure jointed hexapods"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
Reference
: ([Chen and McInroy 2000](#org1c74a9c))
: (<a href="#citeproc_bib_item_1">Chen and McInroy 2000</a>)
Author(s)
: Chen, Y., & McInroy, J.
: Chen, Y., &amp; McInroy, J.
Year
: 2000
@@ -31,7 +31,7 @@ Year
## Introduction {#introduction}
Typical decoupling algorithm ([Decoupled Control]({{< relref "decoupled_control" >}})) impose two constraints:
Typical decoupling algorithm ([Decoupled Control]({{< relref "decoupled_control.md" >}})) impose two constraints:
- the payload mass/inertia matrix is diagonal
- the geometry of the platform and attachment of the payload must be carefully chosen
@@ -43,11 +43,11 @@ The algorithm derived herein removes these constraints, thus greatly expanding t
## Dynamic Model of Flexure Jointed Hexapods {#dynamic-model-of-flexure-jointed-hexapods}
The derivation of the dynamic model is done in ([McInroy 1999](#orgebf33dd)) ([Notes]({{< relref "mcinroy99_dynam" >}})).
The derivation of the dynamic model is done in (<a href="#citeproc_bib_item_2">McInroy 1999</a>) ([Notes]({{< relref "mcinroy99_dynam.md" >}})).
<a id="orga594879"></a>
<a id="figure--fig:chen00-flexure-hexapod"></a>
{{< figure src="/ox-hugo/chen00_flexure_hexapod.png" caption="Figure 1: A flexured joint Hexapod. {P} is a cartesian coordiante frame located at (and rigidly connected to) the payload's center of mass. {B} is a frame attached to the (possibly moving) base, and {U} is a universal inertial frame of reference" >}}
{{< figure src="/ox-hugo/chen00_flexure_hexapod.png" caption="<span class=\"figure-number\">Figure 1: </span>A flexured joint Hexapod. {P} is a cartesian coordiante frame located at (and rigidly connected to) the payload's center of mass. {B} is a frame attached to the (possibly moving) base, and {U} is a universal inertial frame of reference" >}}
In the joint space, the dynamics of a flexure jointed hexapod are written as:
@@ -56,9 +56,9 @@ In the joint space, the dynamics of a flexure jointed hexapod are written as:
\end{equation}
\begin{aligned}
& \left( {}^U\_P\bm{R} {}^P\bm{M}\_x {}^B\_P\bm{R}^T \bm{J}^{-1} \right) \ddot{\vec{l}} + \\\\\\
& {}^U\_B\bm{R} \bm{J}^T \bm{B} \dot{\vec{l}} + {}^U\_B\bm{R}\bm{J}^T \bm{K}(\vec{l} - \vec{l}\_r) = \\\\\\
& {}^U\_B\bm{R} \bm{J}^T \vec{f}\_m + \vec{\mathcal{F}}\_e + \vec{\mathcal{F}} + \vec{\mathcal{C}} - \\\\\\
& \left( {}^U\_P\bm{R} {}^P\bm{M}\_x {}^B\_P\bm{R}^T \bm{J}^{-1} \right) \ddot{\vec{l}} + \\\\
& {}^U\_B\bm{R} \bm{J}^T \bm{B} \dot{\vec{l}} + {}^U\_B\bm{R}\bm{J}^T \bm{K}(\vec{l} - \vec{l}\_r) = \\\\
& {}^U\_B\bm{R} \bm{J}^T \vec{f}\_m + \vec{\mathcal{F}}\_e + \vec{\mathcal{F}} + \vec{\mathcal{C}} - \\\\
& \left( {}^U\_B\bm{R} \bm{J}^T \bm{M}\_s + {}^U\_P\bm{R} {}^P\bm{M}\_x {}^U\_P\bm{R}^T \bm{J}\_c \bm{J}\_B^{-1} \right) \ddot{\vec{q}}\_s
\end{aligned}
@@ -79,7 +79,7 @@ where:
- \\(\vec{\mathcal{G}}\\) is a vector containing all gravity terms
\begin{aligned}
\bm{M}\_p & \ddot{\vec{p}}\_s + \bm{B} \dot{\vec{p}}\_s + \bm{K} \vec{p}\_s = \vec{f}\_m + \\\\\\
\bm{M}\_p & \ddot{\vec{p}}\_s + \bm{B} \dot{\vec{p}}\_s + \bm{K} \vec{p}\_s = \vec{f}\_m + \\\\
& \bm{M}\_q \ddot{\vec{q}}\_s + \bm{B} \dot{\vec{q}}\_s + \bm{J}^{-T} {}^U\_B\bm{R}^T \vec{\mathcal{F}}\_e
\end{aligned}
@@ -100,9 +100,9 @@ where
## Experimental Results {#experimental-results}
## Bibliography {#bibliography}
<a id="org1c74a9c"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. <https://doi.org/10.1109/robot.2000.844878>.
<a id="orgebf33dd"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. <https://doi.org/10.1109/cca.1999.806694>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In <i>Proceedings 2000 Icra. Millennium Conference. Ieee International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00ch37065)</i>, nil. doi:<a href="https://doi.org/10.1109/robot.2000.844878">10.1109/robot.2000.844878</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In <i>Proceedings of the 1999 Ieee International Conference on Control Applications (Cat. No.99ch36328)</i>, nil. doi:<a href="https://doi.org/10.1109/cca.1999.806694">10.1109/cca.1999.806694</a>.</div>
</div>