From de392e5c40f31cf91c9a93d86fa0917052a24068 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Fri, 14 Feb 2020 14:11:57 +0100 Subject: [PATCH] Add schematic to explain the dynamics --- docs/dynamics-study.html | 45 +++++++++++++---- docs/figs/1dof_actuator_external_forces.pdf | Bin 0 -> 31018 bytes docs/figs/1dof_actuator_external_forces.png | Bin 0 -> 4418 bytes docs/figs/2dof_actuator_external_forces.pdf | Bin 0 -> 33690 bytes docs/figs/2dof_actuator_external_forces.png | Bin 0 -> 6291 bytes org/dynamics-study.org | 52 ++++++++++++++++++++ 6 files changed, 87 insertions(+), 10 deletions(-) create mode 100644 docs/figs/1dof_actuator_external_forces.pdf create mode 100644 docs/figs/1dof_actuator_external_forces.png create mode 100644 docs/figs/2dof_actuator_external_forces.pdf create mode 100644 docs/figs/2dof_actuator_external_forces.png diff --git a/docs/dynamics-study.html b/docs/dynamics-study.html index 84372cd..4f2e730 100644 --- a/docs/dynamics-study.html +++ b/docs/dynamics-study.html @@ -4,7 +4,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Stewart Platform - Dynamics Study @@ -272,13 +272,13 @@ for the JavaScript code in this tag.
  • 2. Comparison of the static transfer function and the Compliance matrix
  • @@ -382,6 +382,17 @@ The comparison of the two transfer functions is shown in Figure Figure 1: Comparison of the transfer functions from \(\bm{\mathcal{F}}\) to \(\mathcal{\bm{X}}\) and from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\) (png, pdf)

    + +

    +This can be understood from figure 2 where \(\mathcal{F}_{x}\) and \(\mathcal{F}_{x,\text{ext}}\) have clearly the same effect on \(\mathcal{X}_{x}\). +

    + + +
    +

    1dof_actuator_external_forces.png +

    +

    Figure 2: Schematic representation of the stewart platform on a rigid support

    +
    @@ -426,20 +437,34 @@ Gd.OutputName = {'Edx',

    -The comparison between the obtained transfer functions is shown in Figure 2. +The comparison between the obtained transfer functions is shown in Figure 3.

    comparison_Fext_F_flexible_base.png

    -

    Figure 2: Comparison of the transfer functions from \(\bm{\mathcal{F}}\) to \(\mathcal{\bm{X}}\) and from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\) (png, pdf)

    +

    Figure 3: Comparison of the transfer functions from \(\bm{\mathcal{F}}\) to \(\mathcal{\bm{X}}\) and from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\) (png, pdf)

    +
    + +

    +The addition of a flexible support can be schematically represented in Figure 4. +We see that \(\mathcal{F}_{x}\) applies a force both on \(m\) and \(m^{\prime}\) whereas \(\mathcal{F}_{x,\text{ext}}\) only applies a force on \(m\). +And thus \(\mathcal{F}_{x}\) and \(\mathcal{F}_{x,\text{ext}}\) have clearly not the same effect on \(\mathcal{X}_{x}\). +

    + + +
    +

    2dof_actuator_external_forces.png +

    +

    Figure 4: Schematic representation of the stewart platform on top of a flexible support

    -
    -

    1.3 Conclusion

    + +
    +

    1.3 Conclusion

    @@ -677,8 +702,8 @@ And now at the Compliance matrix.

    -
    -

    2.2 Conclusion

    +
    +

    2.2 Conclusion

    @@ -692,7 +717,7 @@ The low frequency transfer function matrix from \(\mathcal{\bm{F}}\) to \(\mathc

    Author: Dehaeze Thomas

    -

    Created: 2020-02-13 jeu. 15:36

    +

    Created: 2020-02-14 ven. 14:11

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[[./figs/comparison_Fext_F_fixed_base.pdf][pdf]]) [[file:figs/comparison_Fext_F_fixed_base.png]] +This can be understood from figure [[fig:1dof_actuator_external_forces]] where $\mathcal{F}_{x}$ and $\mathcal{F}_{x,\text{ext}}$ have clearly the same effect on $\mathcal{X}_{x}$. + +#+begin_src latex :file 1dof_actuator_external_forces.pdf + \begin{tikzpicture} + \draw[ground] (-1, 0) -- (1, 0); + + \draw[spring] (-0.6, 0) -- (-0.6, 1.5) node[midway, left=0.1]{$k$}; + \draw[actuator] ( 0.6, 0) -- ( 0.6, 1.5) node[midway, left=0.1](F){$\mathcal{F}_{x}$}; + + \draw[fill=white] (-1, 1.5) rectangle (1, 2) node[pos=0.5]{$m$}; + + \draw[dashed] (1, 2) -- ++(0.5, 0); + \draw[->] (1.5, 2) -- ++(0, 0.5) node[right]{$\mathcal{X}_{x}$}; + + \draw[->] (0, 2) node[]{$\bullet$} -- (0, 2.8) node[below right]{$\mathcal{F}_{x,\text{ext}}$}; + \end{tikzpicture} +#+end_src + +#+name: fig:1dof_actuator_external_forces +#+caption: Schematic representation of the stewart platform on a rigid support +#+RESULTS: +[[file:figs/1dof_actuator_external_forces.png]] + ** Comparison with a flexible support We now add a flexible support under the Stewart platform. #+begin_src matlab @@ -229,6 +252,35 @@ The comparison between the obtained transfer functions is shown in Figure [[fig: #+caption: Comparison of the transfer functions from $\bm{\mathcal{F}}$ to $\mathcal{\bm{X}}$ and from $\bm{\mathcal{F}}_{\text{ext}}$ to $\mathcal{\bm{X}}$ ([[./figs/comparison_Fext_F_flexible_base.png][png]], [[./figs/comparison_Fext_F_flexible_base.pdf][pdf]]) [[file:figs/comparison_Fext_F_flexible_base.png]] +The addition of a flexible support can be schematically represented in Figure [[fig:2dof_actuator_external_forces]]. +We see that $\mathcal{F}_{x}$ applies a force both on $m$ and $m^{\prime}$ whereas $\mathcal{F}_{x,\text{ext}}$ only applies a force on $m$. +And thus $\mathcal{F}_{x}$ and $\mathcal{F}_{x,\text{ext}}$ have clearly *not* the same effect on $\mathcal{X}_{x}$. + +#+begin_src latex :file 2dof_actuator_external_forces.pdf + \begin{tikzpicture} + \draw[ground] (-1, 0) -- (1, 0); + + \draw[spring] (0, 0) -- (0, 1.5) node[midway, left=0.1]{$k^{\prime}$}; + \draw[fill=white] (-1, 1.5) rectangle (1, 2) node[pos=0.5]{$m^{\prime}$}; + + \draw[spring] (-0.6, 2) -- (-0.6, 3.5) node[midway, left=0.1]{$k$}; + \draw[actuator] ( 0.6, 2) -- ( 0.6, 3.5) node[midway, left=0.1](F){$\mathcal{F}_{x}$}; + + \draw[fill=white] (-1, 3.5) rectangle (1, 4) node[pos=0.5]{$m$}; + + \draw[dashed] (1, 4) -- ++(0.5, 0); + \draw[->] (1.5, 4) -- ++(0, 0.5) node[right]{$\mathcal{X}_{x}$}; + + \draw[->] (0, 4) node[]{$\bullet$} -- (0, 4.8) node[below right]{$\mathcal{F}_{x,\text{ext}}$}; + \end{tikzpicture} +#+end_src + +#+name: fig:2dof_actuator_external_forces +#+caption: Schematic representation of the stewart platform on top of a flexible support +#+RESULTS: +[[file:figs/2dof_actuator_external_forces.png]] + + ** Conclusion #+begin_important The transfer function from forces/torques applied by the actuators on the payload $\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}$ to the pose of the mobile platform $\bm{\mathcal{X}}$ is the same as the transfer function from external forces/torques to $\bm{\mathcal{X}}$ as long as the Stewart platform's base is fixed.