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Add schematic to explain the dynamics

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Thomas Dehaeze
2020-02-14 14:11:57 +01:00
parent b5b3a756a4
commit de392e5c40
6 changed files with 87 additions and 10 deletions
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@@ -159,6 +159,29 @@ The comparison of the two transfer functions is shown in Figure [[fig:comparison
#+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_fixed_base.png][png]], [[./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.