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content/article/mcinroy02_model_desig_flexur_joint_stewar.md
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@@ -9,7 +9,7 @@ Tags
Reference
: <sup id="8bfe2d2dce902a584fa016e86a899044"><a class="reference-link" href="#mcinroy02_model_desig_flexur_joint_stewar" title="McInroy, Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes, {IEEE/ASME Transactions on Mechatronics}, v(1), 95-99 (2002).">(McInroy, 2002)</a></sup>
: ([McInroy 2002](#org8c59af1))
Author(s)
: McInroy, J.
@@ -17,8 +17,7 @@ Author(s)
Year
: 2002
This short paper is very similar to <sup id="5da427f78c552aa92cd64c2a6df961f1"><a class="reference-link" href="#mcinroy99_dynam" title="McInroy, Dynamic modeling of flexure jointed hexapods for control purposes, nil, in in: {Proceedings of the 1999 IEEE International Conference on
Control Applications (Cat. No.99CH36328)}, edited by (1999)">(McInroy, 1999)</a></sup>.
This short paper is very similar to ([McInroy 1999](#org69da46e)).
> This paper develops guidelines for designing the flexure joints to facilitate closed-loop control.
@@ -37,15 +36,15 @@ This short paper is very similar to <sup id="5da427f78c552aa92cd64c2a6df961f1"><
## Flexure Jointed Hexapod Dynamics {#flexure-jointed-hexapod-dynamics}
<a id="org1e5260a"></a>
<a id="orgccb4fed"></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](#org1e5260a)).
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](#orgccb4fed)).
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](#org1e5260a) with:
The model of the strut are shown in Figure [1](#orgccb4fed) with:
- \\(m\_{s\_i}\\) moving strut mass
- \\(k\_i\\) spring constant
@@ -118,10 +117,9 @@ where:
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:
\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)
\end{aligned}
\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}
\end{equation}
Joint (\\(l\\)) and Cartesian (\\(\mathcal{X}\\)) terms are still mixed.
In the next section, a connection between the two will be found to complete the formulation
@@ -134,16 +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="orgbd4aaf0"></a>
<a id="orgaffe5ce"></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](#orgbd4aaf0) depicts a strut, along with the corresponding force diagram.
Figure [2](#orgaffe5ce) 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](#orgbd4aaf0) (b), Newton's second law yields:
From Figure [2](#orgaffe5ce) (b), Newton's second law yields:
\begin{equation}
f\_p = \begin{bmatrix}
@@ -190,11 +188,16 @@ 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}
> In order to get dominance at low frequencies, the hexapod must be designed so that:
>
> \begin{equation}
> \frac{k\_r}{l^2} \ll k \label{eq:cond\_stiff}
> \end{equation}
<div class="important">
<div></div>
In order to get dominance at low frequencies, the hexapod must be designed so that:
\begin{equation}
\frac{k\_r}{l^2} \ll k \label{eq:cond\_stiff}
\end{equation}
</div>
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.
@@ -203,11 +206,16 @@ 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 \\]
> 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:
>
> \begin{equation}
> \text{control bandwidth} \ll \sqrt{\frac{k\_r}{m\_s l^2}} \label{eq:cond\_bandwidth}
> \end{equation}
<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:
\begin{equation}
\text{control bandwidth} \ll \sqrt{\frac{k\_r}{m\_s l^2}} \label{eq:cond\_bandwidth}
\end{equation}
</div>
The control bandwidth can be increase for hexapods that are designed so that \\(x\_{\text{gain}\_\omega} \gg 1\\) for \\(\omega \ll \sqrt{k/m\_s}\\).
This can be achieve, for instance, by adding damping.
@@ -217,12 +225,52 @@ In this case, it is reasonable to use:
\text{control bandwidth} \ll \sqrt{\frac{k}{m\_s}}
\end{equation}
> 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.
<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.
</div>
## Relationships between joint and cartesian space {#relationships-between-joint-and-cartesian-space}
# Bibliography
<a class="bibtex-entry" id="mcinroy02_model_desig_flexur_joint_stewar">McInroy, J., *Modeling and design of flexure jointed stewart platforms for control purposes*, IEEE/ASME Transactions on Mechatronics, *7(1)*, 95–99 (2002). http://dx.doi.org/10.1109/3516.990892</a> [↩](#8bfe2d2dce902a584fa016e86a899044)
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\\).
<a class="bibtex-entry" id="mcinroy99_dynam">McInroy, J., *Dynamic modeling of flexure jointed hexapods for control purposes*, In , Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328) (pp. ) (1999). : .</a> [↩](#5da427f78c552aa92cd64c2a6df961f1)
This section will derive this implicit relationship.
Let denote:
- \\(\mathcal{X}\_B\\) the pose of {B} with respect to {U}
- \\({}^B\mathcal{X}\_P\\) the pose of {P} with respect to {B}
- \\({}^Uq\_i = {}^U\_BR {}^Bq\_i + {}^UP\_{BORG}\\) the position of the ith base attachment point, expressed in the universal frame {U}
- \\(P\_{BORG}\\) the position of the origin of frame {B}
Note that although \\({}^Bq\_i\\) is fixed, \\({}^Uq\_i\\) varies due to base motion.
Differentiating twice and converting derivatives of rotation matrices into angular velocity cross products yields:
\begin{equation}
{}^U\dot{q}\_i = \omega\_B \times {}^U\_BR {}^Bq\_i + \underbrace{{}^U\_BR {}^B\dot{q}\_i}\_{= 0} + v\_B
\end{equation}
\begin{equation}
{}^U\ddot{q}\_i = \dot{\omega}\_B \times {}^U\_BR {}^Bq\_i + \omega\_B \times \omega\_B \times {}^U\_BR {}^Bq\_i + \dot{v}\_B
\end{equation}
where:
- \\(\omega\_B\\) denotes the angular velocity of {B} with respect to {U}
- \\(v\_B = {}^U\dot{P}\_{BORG}\\) denotes the linear velocity of the origin of {B} with respect to {U}
By using the vector triple identity \\(a \cdot (b \times c) = b \cdot (c \times a)\\) and putting the equation in a matrix form:
\begin{equation}
{}^U \hat{u}\_i^T {}^U\ddot{q}\_i = \left[ {}^U\hat{u}\_i^T \left( {}^U\_BR {}^Bq\_i \times {}^U\hat{u}\_i \right)^T \right] \ddot{\mathcal{X}}\_B + {}^U\hat{u}\_i^T \left( \omega\_B \times \left[ \omega\_B \times {}^U\_BR {}^Bq\_i \right] \right)
\end{equation}
## Bibliography {#bibliography}
<a id="org69da46e"></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="org8c59af1"></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>.