From 568a37f2201ad201e302487f62c6345242430b31 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Wed, 29 Jan 2020 14:55:54 +0100 Subject: [PATCH] Remove duplicated explainations --- static-analysis.org | 39 --------------------------------------- 1 file changed, 39 deletions(-) diff --git a/static-analysis.org b/static-analysis.org index 9069f8e..8fc2fdc 100644 --- a/static-analysis.org +++ b/static-analysis.org @@ -20,45 +20,6 @@ #+PROPERTY: header-args:matlab+ :output-dir figs :END: -* Jacobian -** Relation to platform parameters -A Jacobian is defined by: -- the orientations of the struts $\hat{s}_i$ expressed in a frame $\{A\}$ linked to the fixed platform. -- the vectors from $O_B$ to $b_i$ expressed in the frame $\{A\}$ - -Then, the choice of $O_B$ changes the Jacobian. - -** Jacobian for displacement -\[ \dot{q} = J \dot{X} \] -With: -- $q = [q_1\ q_2\ q_3\ q_4\ q_5\ q_6]$ vector of linear displacement of actuated joints -- $X = [x\ y\ z\ \theta_x\ \theta_y\ \theta_z]$ position and orientation of $O_B$ expressed in the frame $\{A\}$ - -For very small displacements $\delta q$ and $\delta X$, we have $\delta q = J \delta X$. - -** Jacobian for forces -\[ F = J^T \tau \] -With: -- $\tau = [\tau_1\ \tau_2\ \tau_3\ \tau_4\ \tau_5\ \tau_6]$ vector of actuator forces -- $F = [f_x\ f_y\ f_z\ n_x\ n_y\ n_z]$ force and torque acting on point $O_B$ - -* Stiffness matrix $K$ - -\[ K = J^T \text{diag}(k_i) J \] - -If all the struts have the same stiffness $k$, then $K = k J^T J$ - -$K$ only depends of the geometry of the stewart platform: it depends on the Jacobian, that is on the orientations of the struts, position of the joints and choice of frame $\{B\}$. - -\[ F = K X \] - -With $F$ forces and torques applied to the moving platform at the origin of $\{B\}$ and $X$ the translations and rotations of $\{B\}$ with respect to $\{A\}$. - -\[ C = K^{-1} \] - -The compliance element $C_{ij}$ is then the stiffness -\[ X_i = C_{ij} F_j \] - * Coupling What causes the coupling from $F_i$ to $X_i$ ?