Add definition of K ronde

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Thomas Dehaeze 2021-02-05 15:45:49 +01:00
parent dc72858a1f
commit 192841352e
2 changed files with 330 additions and 292 deletions

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@ -1929,6 +1929,14 @@ The stiffness matrix in the frame $\{K\}$ can be expressed as:
where:
- $J_{\{K\}}$ is the Jacobian transformation from the struts to the frame $\{K\}$
- $\mathcal{K}$ is a diagonal matrix with the strut stiffnesses on the diagonal
\begin{equation}
\mathcal{K} = \begin{bmatrix}
k_1 & & & 0 \\
& k_2 & & \\
& & \ddots & \\
0 & & & k_n
\end{bmatrix}
\end{equation}
The Jacobian for a planar manipulator, evaluated in a frame $\{K\}$, can be expressed as follows:
\begin{equation} \label{eq:jacobian_planar}
@ -2203,8 +2211,17 @@ For a fully parallel manipulator, the stiffness matrix $K_{\{K\}}$ expressed in
K_{\{K\}} = J_{\{K\}}^T \mathcal{K} J_{\{K\}}
\end{equation}
where:
- $K_{\{K\}}$ is the Jacobian transformation from the struts to the frame $\{K\}$
- $\mathcal{K}$ is a diagonal matrix with the strut stiffnesses on the diagonal
- $J_{\{K\}}$ is the Jacobian transformation from the struts to the frame $\{K\}$
- $\mathcal{K}$ is a diagonal matrix with the strut stiffnesses on the diagonal:
\begin{equation}
\mathcal{K} = \begin{bmatrix}
k_1 & & & 0 \\
& k_2 & & \\
& & \ddots & \\
0 & & & k_n
\end{bmatrix}
\end{equation}
The analytical expression of $J_{\{K\}}$ is:
\begin{equation}
@ -2286,7 +2303,7 @@ Taking the transpose and re-arranging:
k_i ({}^Mb_i \times \hat{s}_i) \hat{s}_i^T = k_i ({}^MO_K \times \hat{s}_i) \hat{s}_i^T
\end{equation}
As the vector cross product also can be expressed as the product of a skew-symmetric matrix and a vector, we obtain:
As the vector cross product also can be expressed as the product of a skew-symmetric matrix and a vehttps://rwth.zoom.us/j/92311133102?pwd=UTAzS21YYkUwT2pMZDBLazlGNzdvdz09tor, we obtain:
\begin{equation}
k_i ({}^Mb_i \times \hat{s}_i) \hat{s}_i^T = {}^M\bm{O}_{K} ( k_i \hat{s}_i \hat{s}_i^T )
\end{equation}