tdehaeze/digital-brain
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Update Content - 2024-12-17

Browse Source
This commit is contained in:
Thomas Dehaeze
2024-12-17 16:38:55 +01:00
parent 59f9b8ae81
commit 91ca6068b5
13 changed files with 89 additions and 89 deletions
Download Patch File Download Diff File Expand all files Collapse all files

8
content/article/mcinroy02_model_desig_flexur_joint_stewar.md
View File

@@ -115,7 +115,7 @@ where:
- \\(\mathcal{F}\_e\\) represents a vector of exogenous generalized forces applied at the center of mass
- \\(g\\) is the gravity vector
By combining <eq:strut_dynamics_vec>, <eq:payload_dynamics> and <eq:generalized_force>, a single equation describing the dynamics of a flexure jointed hexapod can be found:
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{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}
@@ -200,7 +200,7 @@ In order to get dominance at low frequencies, the hexapod must be designed so th
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.
By satisfying <eq:cond_stiff>, \\(f\_p\\) can be aligned with the strut for frequencies much below the spherical joint's resonance mode:
By satisfying \eqref{eq:cond\_stiff}, \\(f\_p\\) can be aligned with the strut for frequencies much below the spherical joint's resonance mode:
\\[ \omega \ll \sqrt{\frac{k\_r}{m\_s l^2}} \rightarrow x\_{\text{gain}\_\omega} \approx \frac{k}{k\_r/l^2} \gg 1 \\]
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 \\]
@@ -225,14 +225,14 @@ In this case, it is reasonable to use:
<div class="important">
By designing the flexure jointed hexapod and its controller so that both <eq:cond_stiff> and <eq:cond_bandwidth> are met, the dynamics of the hexapod can be greatly reduced in complexity.
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}
Equation <eq:eom_fjh> is not suitable for control analysis and design because \\(\ddot{\mathcal{X}}\\) is implicitly a function of \\(\ddot{q}\_u\\).
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\\).
This section will derive this implicit relationship.
Let denote:

4
content/article/mcinroy99_dynam.md
View File

@@ -53,7 +53,7 @@ In order to provide low frequency passive vibration isolation, the hard actuator
<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 <a href="#orgffe7e8f">2</a>
Definition of quantities on <a href="#org1f8da5d">2</a>
</div>
| **Symbol** | **Meaning** |
@@ -142,7 +142,7 @@ where:
- \\(\mathcal{F}\_e\\) represents a vector of exogenous generalized forces applied at the center of mass
- \\(g\\) is the gravity vector
By combining <eq:strut_dynamics_vec>, <eq:payload_dynamics> and <eq:generalized_force>, a single equation describing the dynamics of a flexure jointed hexapod can be found:
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\\\\