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