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content/article/mcinroy99_dynam.md
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@@ -4,12 +4,15 @@ author = ["Thomas Dehaeze"]
draft = false
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### Backlinks {#backlinks}
- [Identification and decoupling control of flexure jointed hexapods]({{< relref "chen00_ident_decoup_contr_flexur_joint_hexap" >}})
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference
: <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>
: ([McInroy 1999](#org5efb28a))
Author(s)
: McInroy, J.
@@ -17,7 +20,7 @@ Author(s)
Year
: 1999
This conference paper has been further published in a journal as a short note <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>.
This conference paper has been further published in a journal as a short note ([McInroy 2002](#org4990a96)).
## Abstract {#abstract}
@@ -39,22 +42,22 @@ 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="orgb4329bb"></a>
<a id="org5279430"></a>
{{< figure src="/ox-hugo/mcinroy99_general_hexapod.png" caption="Figure 1: 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](#org4a04030).
Since both actuator types employ force production in parallel with a spring, they can both be modeled as shown in Figure [2](#org6b356c7).
In order to provide low frequency passive vibration isolation, the hard actuators are sometimes placed in series with additional passive springs.
<a id="org4a04030"></a>
<a id="org6b356c7"></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" >}}
<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="#org4a04030">2</a>
Definition of quantities on Figure <a href="#org6b356c7">2</a>
</div>
| **Symbol** | **Meaning** |
@@ -71,11 +74,11 @@ 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 <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>).
It is here supposed that \\(f\_{p\_i}\\) is predominantly in the strut direction (explained in ([McInroy 2002](#org4990a96))).
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](#org4a04030) (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](#org6b356c7) (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
@@ -162,12 +165,9 @@ In the next section, a connection between the two will be found to complete the
## Control Example {#control-example}
# Bibliography
<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)
<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)
## Bibliography {#bibliography}
<a id="org5efb28a"></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>.
## Backlinks {#backlinks}
- [Identification and decoupling control of flexure jointed hexapods]({{< relref "chen00_ident_decoup_contr_flexur_joint_hexap" >}})
<a id="org4990a96"></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>.