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Thomas Dehaeze
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content/article/mcinroy99_dynam.md
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+++
title = "Dynamic modeling of flexure jointed hexapods for control purposes"
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
: ([McInroy 1999](#orgc5d256d))
: (<a href="#citeproc_bib_item_1">McInroy 1999</a>)
Author(s)
: McInroy, J.
@@ -16,7 +16,7 @@ Author(s)
Year
: 1999
This conference paper has been further published in a journal as a short note ([McInroy 2002](#orge25929e)).
This conference paper has been further published in a journal as a short note (<a href="#citeproc_bib_item_2">McInroy 2002</a>).
## Abstract {#abstract}
@@ -38,22 +38,22 @@ The actuators for FJHs can be divided into two categories:
1. soft (voice coil), which employs a spring flexure mount
2. hard (piezoceramic or magnetostrictive), which employs a compressive load spring.
<a id="org89aa8b3"></a>
<a id="figure--fig:mcinroy99-general-hexapod"></a>
{{< figure src="/ox-hugo/mcinroy99_general_hexapod.png" caption="Figure 1: A general Stewart Platform" >}}
{{< figure src="/ox-hugo/mcinroy99_general_hexapod.png" caption="<span class=\"figure-number\">Figure 1: </span>A general Stewart Platform" >}}
Since both actuator types employ force production in parallel with a spring, they can both be modeled as shown in Figure [2](#org0b2b1e5).
Since both actuator types employ force production in parallel with a spring, they can both be modeled as shown in Figure [2](#figure--fig:mcinroy99-strut-model).
In order to provide low frequency passive vibration isolation, the hard actuators are sometimes placed in series with additional passive springs.
<a id="org0b2b1e5"></a>
<a id="figure--fig:mcinroy99-strut-model"></a>
{{< figure src="/ox-hugo/mcinroy99_strut_model.png" caption="Figure 2: The dynamics of the i'th strut. A parallel spring, damper and actuator drives the moving mass of the strut and a payload" >}}
{{< figure src="/ox-hugo/mcinroy99_strut_model.png" caption="<span class=\"figure-number\">Figure 2: </span>The dynamics of the i'th strut. A parallel spring, damper and actuator drives the moving mass of the strut and a payload" >}}
<a id="table--tab:mcinroy99-strut-model"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:mcinroy99-strut-model">Table 1</a></span>:
Definition of quantities on Figure <a href="#org0b2b1e5">2</a>
Definition of quantities on Figure <a href="#org84f1a50">2</a>
</div>
| **Symbol** | **Meaning** |
@@ -70,11 +70,11 @@ In order to provide low frequency passive vibration isolation, the hard actuator
| \\(v\_i = p\_i - q\_i\\) | vector pointing from the bottom to the top |
| \\(\hat{u}\_i = v\_i/l\_i\\) | unit direction of the strut |
It is here supposed that \\(f\_{p\_i}\\) is predominantly in the strut direction (explained in ([McInroy 2002](#orge25929e))).
It is here supposed that \\(f\_{p\_i}\\) is predominantly in the strut direction (explained in (<a href="#citeproc_bib_item_2">McInroy 2002</a>)).
This is a good approximation unless the spherical joints and extremely stiff or massive, of high inertia struts are used.
This allows to reduce considerably the complexity of the model.
From Figure [2](#org0b2b1e5) (b), forces along the strut direction are summed to yield (projected along the strut direction, hence the \\(\hat{u}\_i^T\\) term):
From Figure [2](#figure--fig:mcinroy99-strut-model) (b), forces along the strut direction are summed to yield (projected along the strut direction, hence the \\(\hat{u}\_i^T\\) term):
\begin{equation}
m\_i \hat{u}\_i^T \ddot{p}\_i = f\_{m\_i} - f\_{p\_i} - m\_i \hat{u}\_i^Tg - k\_i(l\_i - l\_{r\_i}) - b\_i \dot{l}\_i
@@ -105,10 +105,10 @@ The payload is modeled as a rigid body:
\begin{equation}
\underbrace{\begin{bmatrix}
m I\_3 & 0\_{3\times 3} \\\\\\
m I\_3 & 0\_{3\times 3} \\\\
0\_{3\times 3} & {}^cI
\end{bmatrix}}\_{M\_x} \ddot{\mathcal{X}} + \underbrace{\begin{bmatrix}
0\_{3 \times 1} \\ \omega \times {}^cI\omega
0\_{3 \times 1} \\\ \omega \times {}^cI\omega
\end{bmatrix}}\_{c(\omega)} = \mathcal{F} \label{eq:payload\_dynamics}
\end{equation}
@@ -134,7 +134,7 @@ where \\(J\\) is the manipulator Jacobian and \\({}^U\_BR\\) is the rotation mat
The total generalized force acting on the payload is the sum of the strut, exogenous, and gravity forces:
\begin{equation}
\mathcal{F} = {}^UJ^T f\_p + \mathcal{F}\_e - \begin{bmatrix} mg \\ 0\_{3\times 1} \end{bmatrix} \label{eq:generalized\_force}
\mathcal{F} = {}^UJ^T f\_p + \mathcal{F}\_e - \begin{bmatrix} mg \\\ 0\_{3\times 1} \end{bmatrix} \label{eq:generalized\_force}
\end{equation}
where:
@@ -142,11 +142,11 @@ where:
- \\(\mathcal{F}\_e\\) represents a vector of exogenous generalized forces applied at the center of mass
- \\(g\\) is the gravity vector
By combining \eqref{eq:strut_dynamics_vec}, \eqref{eq:payload_dynamics} and \eqref{eq:generalized_force}, a single equation describing the dynamics of a flexure jointed hexapod can be found:
By combining <eq:strut_dynamics_vec>, <eq:payload_dynamics> and <eq:generalized_force>, a single equation describing the dynamics of a flexure jointed hexapod can be found:
\begin{aligned}
& {}^UJ^T [ f\_m - M\_s \ddot{l} - B \dot{l} - K(l - l\_r) - M\_s \ddot{q}\_u\\\\\\
& - M\_s g\_u + M\_s v\_2] + \mathcal{F}\_e - \begin{bmatrix} mg \\ 0\_{3\times 1} \end{bmatrix} = M\_x \ddot{\mathcal{X}} + c(\omega)
& {}^UJ^T [ f\_m - M\_s \ddot{l} - B \dot{l} - K(l - l\_r) - M\_s \ddot{q}\_u\\\\
& - M\_s g\_u + M\_s v\_2] + \mathcal{F}\_e - \begin{bmatrix} mg \\\ 0\_{3\times 1} \end{bmatrix} = M\_x \ddot{\mathcal{X}} + c(\omega)
\end{aligned}
Joint (\\(l\\)) and Cartesian (\\(\mathcal{X}\\)) terms are still mixed.
@@ -162,9 +162,9 @@ In the next section, a connection between the two will be found to complete the
## Control Example {#control-example}
## Bibliography {#bibliography}
<a id="orgc5d256d"></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>.
<a id="orge25929e"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. <https://doi.org/10.1109/3516.990892>.
<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>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 class="csl-entry"><a id="citeproc_bib_item_2"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” <i>Ieee/Asme Transactions on Mechatronics</i> 7 (1): 95–99. doi:<a href="https://doi.org/10.1109/3516.990892">10.1109/3516.990892</a>.</div>
</div>