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Thomas Dehaeze
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content/article/mcinroy02_model_desig_flexur_joint_stewar.md
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@@ -9,7 +9,7 @@ Tags
Reference
: ([McInroy 2002](#orgf7c9a88))
: ([McInroy 2002](#org48d21a1))
Author(s)
: McInroy, J.
@@ -17,7 +17,7 @@ Author(s)
Year
: 2002
This short paper is very similar to ([McInroy 1999](#org4526c4b)).
This short paper is very similar to ([McInroy 1999](#org287d886)).
> 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](#org4526c4b)).
## Flexure Jointed Hexapod Dynamics {#flexure-jointed-hexapod-dynamics}
<a id="orgd884ef4"></a>
<a id="org66e9285"></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" >}}
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](#orgd884ef4)).
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](#org66e9285)).
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](#orgd884ef4) with:
The model of the strut are shown in Figure [1](#org66e9285) with:
- \\(m\_{s\_i}\\) moving strut mass
- \\(k\_i\\) spring constant
@@ -132,16 +132,16 @@ 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="org54c99c4"></a>
<a id="org343afcb"></a>
{{< figure src="/ox-hugo/mcinroy02_model_strut_joint.png" caption="Figure 2: A simplified dynamic model of a strut and its joint" >}}
Figure [2](#org54c99c4) depicts a strut, along with the corresponding force diagram.
Figure [2](#org343afcb) 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](#org54c99c4) (b), Newton's second law yields:
From Figure [2](#org343afcb) (b), Newton's second law yields:
\begin{equation}
f\_p = \begin{bmatrix}
@@ -188,7 +188,7 @@ The first part depends on the mechanical terms and the frequency of the movement
x\_{\text{gain}\_\omega} = \frac{|-m\_s \omega^2 + k|}{|-m\_s \omega^2 + \frac{k\_r}{l^2}|}
\end{equation}
<div class="bred">
<div class="important">
<div></div>
In order to get dominance at low frequencies, the hexapod must be designed so that:
@@ -206,7 +206,7 @@ By satisfying \eqref{eq:cond_stiff}, \\(f\_p\\) can be aligned with the strut fo
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="bred">
<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:
@@ -225,7 +225,7 @@ In this case, it is reasonable to use:
\text{control bandwidth} \ll \sqrt{\frac{k}{m\_s}}
\end{equation}
<div class="bred">
<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.
@@ -271,6 +271,6 @@ By using the vector triple identity \\(a \cdot (b \times c) = b \cdot (c \times
## Bibliography {#bibliography}
<a id="org4526c4b"></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="org287d886"></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="orgf7c9a88"></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>.
<a id="org48d21a1"></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>.