From 42a7ca9244b79f72cfdba8c7afb06d478d5a0c76 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Mon, 7 Apr 2025 14:36:23 +0200 Subject: [PATCH] Modify transpose notation --- nass-geometry.org | 38 +++++++++++++++++++------------------- nass-geometry.pdf | Bin 2725576 -> 2737247 bytes nass-geometry.tex | 32 ++++++++++++++++---------------- 3 files changed, 35 insertions(+), 35 deletions(-) diff --git a/nass-geometry.org b/nass-geometry.org index 23bf414..69e471f 100644 --- a/nass-geometry.org +++ b/nass-geometry.org @@ -364,8 +364,8 @@ ylim([1e-10, 2e-3]) #+end_src -** TODO [#B] Change review based on christophe's comments -SCHEDULED: <2025-04-04 Fri> +** DONE [#B] Change review based on christophe's comments +CLOSED: [2025-04-04 Fri 21:11] SCHEDULED: <2025-04-04 Fri> - [-] make sure that all papers are cited - [X] geng93_six_degree_of_freed_activ @@ -713,7 +713,7 @@ Compute: \begin{equation} - \sum_{i = 1}^{6} \hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T = 2 \bm{I}_3 + \sum_{i = 1}^{6} \hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^{\intercal} = 2 \bm{I}_3 \end{equation} @@ -731,12 +731,12 @@ This is wrong, check from matlab script \begin{equation} - \bm{K} = k \bm{J}^T \bm{J} = + \bm{K} = k \bm{J}^{\intercal} \bm{J} = k \left[ \begin{array}{c|c} - \Sigma_{i = 0}^{6} \hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T & \Sigma_{i = 0}^{6} \bm{\hat{s}}_i \cdot ({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i)^T \\ + \Sigma_{i = 0}^{6} \hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^{\intercal} & \Sigma_{i = 0}^{6} \bm{\hat{s}}_i \cdot ({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i)^{\intercal} \\ \hline - \Sigma_{i = 0}^{6} ({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i) \cdot \hat{\bm{s}}_i^T & \Sigma_{i = 0}^{6} ({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i) \cdot ({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i)^T\\ + \Sigma_{i = 0}^{6} ({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i) \cdot \hat{\bm{s}}_i^{\intercal} & \Sigma_{i = 0}^{6} ({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i) \cdot ({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i)^{\intercal}\\ \end{array} \right] \end{equation} @@ -1216,12 +1216,12 @@ The analysis is significantly simplified when considering small motions, as the \begin{equation}\label{eq:detail_kinematics_jacobian} \begin{bmatrix} \delta l_1 \\ \delta l_2 \\ \delta l_3 \\ \delta l_4 \\ \delta l_5 \\ \delta l_6 \end{bmatrix} = \underbrace{\begin{bmatrix} - {{}^A\hat{\bm{s}}_1}^T & ({}^A\bm{b}_1 \times {}^A\hat{\bm{s}}_1)^T \\ - {{}^A\hat{\bm{s}}_2}^T & ({}^A\bm{b}_2 \times {}^A\hat{\bm{s}}_2)^T \\ - {{}^A\hat{\bm{s}}_3}^T & ({}^A\bm{b}_3 \times {}^A\hat{\bm{s}}_3)^T \\ - {{}^A\hat{\bm{s}}_4}^T & ({}^A\bm{b}_4 \times {}^A\hat{\bm{s}}_4)^T \\ - {{}^A\hat{\bm{s}}_5}^T & ({}^A\bm{b}_5 \times {}^A\hat{\bm{s}}_5)^T \\ - {{}^A\hat{\bm{s}}_6}^T & ({}^A\bm{b}_6 \times {}^A\hat{\bm{s}}_6)^T + {{}^A\hat{\bm{s}}_1}^{\intercal} & ({}^A\bm{b}_1 \times {}^A\hat{\bm{s}}_1)^{\intercal} \\ + {{}^A\hat{\bm{s}}_2}^{\intercal} & ({}^A\bm{b}_2 \times {}^A\hat{\bm{s}}_2)^{\intercal} \\ + {{}^A\hat{\bm{s}}_3}^{\intercal} & ({}^A\bm{b}_3 \times {}^A\hat{\bm{s}}_3)^{\intercal} \\ + {{}^A\hat{\bm{s}}_4}^{\intercal} & ({}^A\bm{b}_4 \times {}^A\hat{\bm{s}}_4)^{\intercal} \\ + {{}^A\hat{\bm{s}}_5}^{\intercal} & ({}^A\bm{b}_5 \times {}^A\hat{\bm{s}}_5)^{\intercal} \\ + {{}^A\hat{\bm{s}}_6}^{\intercal} & ({}^A\bm{b}_6 \times {}^A\hat{\bm{s}}_6)^{\intercal} \end{bmatrix}}_{\bm{J}} \begin{bmatrix} \delta x \\ \delta y \\ \delta z \\ \delta \theta_x \\ \delta \theta_y \\ \delta \theta_z \end{bmatrix} \end{equation} @@ -1606,28 +1606,28 @@ The contribution of joints stiffness is not considered here, as the joints were However, theoretical frameworks for evaluating flexible joint contribution to the stiffness matrix have been established in the literature [[cite:&mcinroy00_desig_contr_flexur_joint_hexap;&mcinroy02_model_desig_flexur_joint_stewar]]. \begin{equation}\label{eq:detail_kinematics_stiffness_matrix} - \bm{K} = \bm{J}^T \bm{\mathcal{K}} \bm{J} + \bm{K} = \bm{J}^{\intercal} \bm{\mathcal{K}} \bm{J} \end{equation} It is assumed that the stiffness of all struts is the same: $\bm{\mathcal{K}} = k \cdot \mathbf{I}_6$. In that case, the obtained stiffness matrix linearly depends on the strut stiffness $k$, and is structured as shown in equation eqref:eq:detail_kinematics_stiffness_matrix_simplified. \begin{equation}\label{eq:detail_kinematics_stiffness_matrix_simplified} - \bm{K} = k \bm{J}^T \bm{J} = + \bm{K} = k \bm{J}^{\intercal} \bm{J} = k \left[ \begin{array}{c|c} - \Sigma_{i = 0}^{6} \hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T & \Sigma_{i = 0}^{6} \bm{\hat{s}}_i \cdot ({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i)^T \\ + \Sigma_{i = 0}^{6} \hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^{\intercal} & \Sigma_{i = 0}^{6} \bm{\hat{s}}_i \cdot ({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i)^{\intercal} \\ \hline - \Sigma_{i = 0}^{6} ({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i) \cdot \hat{\bm{s}}_i^T & \Sigma_{i = 0}^{6} ({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i) \cdot ({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i)^T\\ + \Sigma_{i = 0}^{6} ({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i) \cdot \hat{\bm{s}}_i^{\intercal} & \Sigma_{i = 0}^{6} ({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i) \cdot ({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i)^{\intercal}\\ \end{array} \right] \end{equation} **** Translation Stiffness -As shown by equation eqref:eq:detail_kinematics_stiffness_matrix_simplified, the translation stiffnesses (the $3 \times 3$ top left terms of the stiffness matrix) only depend on the orientation of the struts and not their location: $\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T$. +As shown by equation eqref:eq:detail_kinematics_stiffness_matrix_simplified, the translation stiffnesses (the $3 \times 3$ top left terms of the stiffness matrix) only depend on the orientation of the struts and not their location: $\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^{\intercal}$. In the extreme case where all struts are vertical ($s_i = [0\ 0\ 1]$), a vertical stiffness of $6k$ is achieved, but with null stiffness in the horizontal directions. -If two struts are aligned with the X axis, two struts with the Y axis, and two struts with the Z axis, then $\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T = 2 \bm{I}_3$, resulting in well-distributed stiffness along all directions. +If two struts are aligned with the X axis, two struts with the Y axis, and two struts with the Z axis, then $\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^{\intercal} = 2 \bm{I}_3$, resulting in well-distributed stiffness along all directions. This configuration corresponds to the cubic architecture presented in Section ref:sec:detail_kinematics_cubic. When the struts are oriented more vertically, as shown in Figure ref:fig:detail_kinematics_stewart_mobility_vert_struts, the vertical stiffness increases while the horizontal stiffness decreases. @@ -1658,7 +1658,7 @@ Under very specific conditions, the equations of motion in the Cartesian frame, These conditions are studied in Section ref:ssec:detail_kinematics_cubic_dynamic. \begin{equation}\label{eq:detail_kinematics_transfer_function_cart} - \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(s) = ( \bm{M} s^2 + \bm{J}^{T} \bm{\mathcal{C}} \bm{J} s + \bm{J}^{T} \bm{\mathcal{K}} \bm{J} )^{-1} + \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(s) = ( \bm{M} s^2 + \bm{J}^{\intercal} \bm{\mathcal{C}} \bm{J} s + \bm{J}^{\intercal} \bm{\mathcal{K}} \bm{J} )^{-1} \end{equation} In the frame of the struts, the equations of motion eqref:eq:detail_kinematics_transfer_function_struts are well decoupled at low frequency. diff --git a/nass-geometry.pdf b/nass-geometry.pdf index 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