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content/article/mcinroy02_model_desig_flexur_joint_stewar.md
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+++
title = "Modeling and design of flexure jointed stewart platforms for control purposes"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = false
+++
@@ -9,7 +9,7 @@ Tags
Reference
: ([McInroy 2002](#org2871bf9))
: (<a href="#citeproc_bib_item_2">McInroy 2002</a>)
Author(s)
: McInroy, J.
@@ -17,7 +17,7 @@ Author(s)
Year
: 2002
This short paper is very similar to ([McInroy 1999](#org1d169f9)).
This short paper is very similar to (<a href="#citeproc_bib_item_1">McInroy 1999</a>).
> This paper develops guidelines for designing the flexure joints to facilitate closed-loop control.
@@ -36,15 +36,15 @@ This short paper is very similar to ([McInroy 1999](#org1d169f9)).
## Flexure Jointed Hexapod Dynamics {#flexure-jointed-hexapod-dynamics}
<a id="org4ea1e8b"></a>
<a id="figure--fig:mcinroy02-leg-model"></a>
{{< figure src="/ox-hugo/mcinroy02_leg_model.png" caption="Figure 1: The dynamics of the ith strut. A parallel spring, damper, and actautor drives the moving mass of the strut and a payload" >}}
{{< figure src="/ox-hugo/mcinroy02_leg_model.png" caption="<span class=\"figure-number\">Figure 1: </span>The dynamics of the ith strut. A parallel spring, damper, and actautor drives the moving mass of the strut and a payload" >}}
The strut can be modeled as consisting of a parallel arrangement of an actuator force, a spring and some damping driving a mass (Figure [1](#org4ea1e8b)).
The strut can be modeled as consisting of a parallel arrangement of an actuator force, a spring and some damping driving a mass (Figure [1](#figure--fig:mcinroy02-leg-model)).
Thus, **the strut does not output force directly, but rather outputs a mechanically filtered force**.
The model of the strut are shown in Figure [1](#org4ea1e8b) with:
The model of the strut are shown in Figure [1](#figure--fig:mcinroy02-leg-model) with:
- \\(m\_{s\_i}\\) moving strut mass
- \\(k\_i\\) spring constant
@@ -78,10 +78,10 @@ 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}
@@ -107,7 +107,7 @@ 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:
@@ -115,10 +115,10 @@ 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{equation}
{}^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) \label{eq:eom\_fjh}
{}^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) \label{eq:eom\_fjh}
\end{equation}
Joint (\\(l\\)) and Cartesian (\\(\mathcal{X}\\)) terms are still mixed.
@@ -132,21 +132,21 @@ Many prior hexapod dynamic formulations assume that the strut exerts force only
The flexure joints Hexapods transmit forces (or torques) proportional to the deflection of the joints.
This section establishes design guidelines for the spherical flexure joint to guarantee that the dynamics remain tractable for control.
<a id="org5bc5fa8"></a>
<a id="figure--fig:mcinroy02-model-strut-joint"></a>
{{< figure src="/ox-hugo/mcinroy02_model_strut_joint.png" caption="Figure 2: A simplified dynamic model of a strut and its joint" >}}
{{< figure src="/ox-hugo/mcinroy02_model_strut_joint.png" caption="<span class=\"figure-number\">Figure 2: </span>A simplified dynamic model of a strut and its joint" >}}
Figure [2](#org5bc5fa8) depicts a strut, along with the corresponding force diagram.
Figure [2](#figure--fig:mcinroy02-model-strut-joint) depicts a strut, along with the corresponding force diagram.
The force diagram is obtained using standard finite element assumptions (\\(\sin \theta \approx \theta\\)).
Damping terms are neglected.
\\(k\_r\\) denotes the rotational stiffness of the spherical joint.
From Figure [2](#org5bc5fa8) (b), Newton's second law yields:
From Figure [2](#figure--fig:mcinroy02-model-strut-joint) (b), Newton's second law yields:
\begin{equation}
f\_p = \begin{bmatrix}
-f\_m + m\_s \Delta \ddot{x} + k\Delta x \\\\\\
m\_s \Delta \ddot{y} + \frac{k\_r}{l^2} \Delta y \\\\\\
-f\_m + m\_s \Delta \ddot{x} + k\Delta x \\\\
m\_s \Delta \ddot{y} + \frac{k\_r}{l^2} \Delta y \\\\
m\_s \Delta \ddot{z} + \frac{k\_r}{l^2} \Delta z
\end{bmatrix}
\end{equation}
@@ -157,16 +157,16 @@ The force is aligned perfectly with the strut only if \\(m\_s = 0\\) and \\(k\_r
To examine the passive behavior, let \\(f\_m = 0\\) and consider a sinusoidal motion:
\begin{equation}
\begin{bmatrix} \Delta x \\ \Delta y \\ \Delta z \end{bmatrix} =
\begin{bmatrix} A\_x \cos \omega t \\ A\_y \cos \omega t \\ A\_z \cos \omega t \end{bmatrix}
\begin{bmatrix} \Delta x \\\ \Delta y \\\ \Delta z \end{bmatrix} =
\begin{bmatrix} A\_x \cos \omega t \\\ A\_y \cos \omega t \\\ A\_z \cos \omega t \end{bmatrix}
\end{equation}
This yields:
\begin{equation}
f\_p = \begin{bmatrix}
\Big( -m\_s \omega^2 + k \Big) A\_x \cos \omega t \\\\\\
\Big( -m\_s \omega^2 + \frac{k\_r}{l^2} \Big) A\_y \cos \omega t \\\\\\
\Big( -m\_s \omega^2 + k \Big) A\_x \cos \omega t \\\\
\Big( -m\_s \omega^2 + \frac{k\_r}{l^2} \Big) A\_y \cos \omega t \\\\
\Big( -m\_s \omega^2 + \frac{k\_r}{l^2} \Big) A\_z \cos \omega t
\end{bmatrix}
\end{equation}
@@ -189,7 +189,6 @@ The first part depends on the mechanical terms and the frequency of the movement
\end{equation}
<div class="important">
<div></div>
In order to get dominance at low frequencies, the hexapod must be designed so that:
@@ -201,13 +200,12 @@ In order to get dominance at low frequencies, the hexapod must be designed so th
This puts a limit on the rotational stiffness of the flexure joint and shows that as the strut is made softer (by decreasing \\(k\\)), the spherical flexure joint must be made proportionately softer.
By satisfying \eqref{eq:cond_stiff}, \\(f\_p\\) can be aligned with the strut for frequencies much below the spherical joint's resonance mode:
By satisfying <eq:cond_stiff>, \\(f\_p\\) can be aligned with the strut for frequencies much below the spherical joint's resonance mode:
\\[ \omega \ll \sqrt{\frac{k\_r}{m\_s l^2}} \rightarrow x\_{\text{gain}\_\omega} \approx \frac{k}{k\_r/l^2} \gg 1 \\]
At frequencies much above the strut's resonance mode, \\(f\_p\\) is not dominated by its \\(x\\) component:
\\[ \omega \gg \sqrt{\frac{k}{m\_s}} \rightarrow x\_{\text{gain}\_\omega} \approx 1 \\]
<div class="important">
<div></div>
To ensure that the control system acts only in the band of frequencies where dominance is retained, the control bandwidth can be selected so that:
@@ -226,16 +224,15 @@ In this case, it is reasonable to use:
\end{equation}
<div class="important">
<div></div>
By designing the flexure jointed hexapod and its controller so that both \eqref{eq:cond_stiff} and \eqref{eq:cond_bandwidth} are met, the dynamics of the hexapod can be greatly reduced in complexity.
By designing the flexure jointed hexapod and its controller so that both <eq:cond_stiff> and <eq:cond_bandwidth> are met, the dynamics of the hexapod can be greatly reduced in complexity.
</div>
## Relationships between joint and cartesian space {#relationships-between-joint-and-cartesian-space}
Equation \eqref{eq:eom_fjh} is not suitable for control analysis and design because \\(\ddot{\mathcal{X}}\\) is implicitly a function of \\(\ddot{q}\_u\\).
Equation <eq:eom_fjh> is not suitable for control analysis and design because \\(\ddot{\mathcal{X}}\\) is implicitly a function of \\(\ddot{q}\_u\\).
This section will derive this implicit relationship.
Let denote:
@@ -269,9 +266,9 @@ By using the vector triple identity \\(a \cdot (b \times c) = b \cdot (c \times
\end{equation}
## Bibliography {#bibliography}
<a id="org1d169f9"></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="org2871bf9"></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>