[足式机器人]Part3 机构运动学与动力学分析与建模 Ch00-4(1) 刚体的速度与角速度

本文仅供学习使用，总结很多本现有讲述运动学或动力学书籍后的总结，从矢量的角度进行分析，方法比较传统，但更易理解，并且现有的看似抽象方法，两者本质上并无不同。

2024年底本人学位论文发表后方可摘抄
若有帮助请引用
本文参考：
.

食用方法
求解逻辑：速度与加速度都是在知道角速度与角加速度的前提下——旋转运动更重要
所求得的速度表达-需要考虑是否为刚体相对固定点！
旋转矩阵？转换矩阵？有什么意义和性质？——与角速度与角加速度的关系
务必自己推导全部公式，并理解每个符号的含义

4. 刚体的速度与角速度

对于运动坐标系下任意一点$P_{\mathrm{i}}$而言，有：
$$\begin{array}{rl}& R_{\mathrm{P}}^F=R_{\mathrm{M}}^F+\left[Q_{\mathrm{M}}^F\right]R_{{\mathrm{P}}_{\mathrm{i}}}^M\\ & \Rightarrow v_{\mathrm{P}}^F=v_{\mathrm{M}}^F+\left[{\dot{Q}}_{\mathrm{M}}^F\right]R_{{\mathrm{P}}_{\mathrm{i}}}^M+\left[Q_{\mathrm{M}}^F\right]{\dot{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M={\omega }^F\times R_{\mathrm{P}}^F={\tilde{\omega }}^F{R}_{\mathrm{P}}^F={\tilde{\omega }}^F\left(R_{\mathrm{M}}^F+\left[Q_{\mathrm{M}}^F\right]R_{{\mathrm{P}}_{\mathrm{i}}}^M\right)\\ & \Rightarrow \left[{\dot{Q}}_{\mathrm{M}}^F\right]R_{{\mathrm{P}}_{\mathrm{i}}}^M+\left[Q_{\mathrm{M}}^F\right]{\dot{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M={\tilde{\omega }}^F\left[Q_{\mathrm{M}}^F\right]R_{{\mathrm{P}}_{\mathrm{i}}}^M\\ & \Rightarrow v_{{\mathrm{P}}_{\mathrm{i}}}^M=\left({\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}{\tilde{\omega }}^F\left[Q_{\mathrm{M}}^F\right]-{\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[{\dot{Q}}_{\mathrm{M}}^F\right]\right)R_{{\mathrm{P}}_{\mathrm{i}}}^M=\left(\left({\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}{\omega }^F\right)-{\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[{\dot{Q}}_{\mathrm{M}}^F\right]\right)R_{{\mathrm{P}}_{\mathrm{i}}}^M=\left({\tilde{\omega }}^M-{\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[{\dot{Q}}_{\mathrm{M}}^F\right]\right)R_{{\mathrm{P}}_{\mathrm{i}}}^M\end{array}$$

因此，当$P_{\mathrm{i}}$为刚体上的固定点时，有：$v_{{\mathrm{P}}_{\mathrm{i}}}^M=0$，进而可知（下式仅当$P_{\mathrm{i}}$为刚体上的固定点时成立）：
$$\begin{array}{rl}{\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}{\tilde{\omega }}^F\left[Q_{\mathrm{M}}^F\right]-{\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[{\dot{Q}}_{\mathrm{M}}^F\right]=0& \Rightarrow {\tilde{\omega }}^F=\left[{\dot{Q}}_{\mathrm{M}}^F\right]{\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\\ {\tilde{\omega }}^M-{\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[{\dot{Q}}_{\mathrm{M}}^F\right]=0& \Rightarrow {\tilde{\omega }}^M={\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[{\dot{Q}}_{\mathrm{M}}^F\right]\end{array}$$

观察上式，角速度本身是一个向量，与刚体实际绕哪根确定的轴旋转并没有关系，而是需要一个向量作为其回转方向——而真正影响到速度的表达，则与所在的观测点有关——在地球上观测月球绕地球回转，而在太阳上观测月球则作更为复杂的运动——与所选的$R$有关

对转换矩阵${\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}$而言，有：
$$\begin{array}{rl}& {\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[Q_{\mathrm{M}}^F\right]=E\\ \Rightarrow & {\left[{\dot{Q}}_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[Q_{\mathrm{M}}^F\right]+{\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[{\dot{Q}}_{\mathrm{M}}^F\right]=0\\ \Rightarrow & {\left[{\dot{Q}}_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[Q_{\mathrm{M}}^F\right]+{\left[{\left[{\dot{Q}}_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[Q_{\mathrm{M}}^F\right]\right]}^{\mathrm{T}}=0\end{array}$$
即，${\left[{\dot{Q}}_{\mathrm{M}}^F\right]}^{\mathrm{T}}\left[Q_{\mathrm{M}}^F\right]$为反(斜)对称矩阵。

因此，对于矩阵 ${\tilde{\omega }}^F$与 ${\tilde{\omega }}^M$具有如下转换关系（下式仅当$P_{\mathrm{i}}$为刚体上的固定点时成立）：
$$\begin{array}{rl}{\tilde{\omega }}^M& ={\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}{\tilde{\omega }}^F\left[Q_{\mathrm{M}}^F\right]\\ {\tilde{\omega }}^F& =\left[Q_{\mathrm{M}}^F\right]{\tilde{\omega }}^M{\left[Q_{\mathrm{M}}^F\right]}^{\mathrm{T}}\end{array}$$

进而可将上式中的项term $\left[{\dot{Q}}_{\mathrm{M}}^F\right]R_{{\mathrm{P}}_{\mathrm{i}}}^M$改写为（下式仅当$P_{\mathrm{i}}$为刚体上的固定点时成立）：
$$\left[{\dot{Q}}_{\mathrm{M}}^F\right]R_{{\mathrm{P}}_{\mathrm{i}}}^M=\{\begin{array}{l}{\tilde{\omega }}^F\left[Q_{\mathrm{M}}^F\right]R_{{\mathrm{P}}_{\mathrm{i}}}^M={\tilde{\omega }}^F{R}_{{\mathrm{P}}_{\mathrm{i}}}^F={\omega }^F\times R_{{\mathrm{P}}_{\mathrm{i}}}^F\\ \left[Q_{\mathrm{M}}^F\right]{\tilde{\omega }}^M{R}_{{\mathrm{P}}_{\mathrm{i}}}^M=\left[Q_{\mathrm{M}}^F\right]\left({\omega }^M\times R_{{\mathrm{P}}_{\mathrm{i}}}^M\right)\end{array}$$

4.1 角速度的表达

4.1.1 欧拉参数表示角速度

结合定义矩阵： B 3 × 4 = [ − q 2 q 1 − q 4 q 3 − q 3 q 4 q 1 − q 2 − q 4 − q 3 q 2 q 1 ] B_{3\times 4}=\left[ \begin{array}{cccc} -q_2& q_1& -q_4& q_3\\ -q_3& q_4& q_1& -q_2\\ -q_4& -q_3& q_2& q_1\\ \end{array} \right] B3×4​= ​−q2​−q3​−q4​​q1​q4​−q3​​−q4​q1​q2​​q3​−q2​q1​​ ​ B ˉ 3 × 4 = [ − q 2 q 1 q 4 − q 3 − q 3 − q 4 q 1 q 2 − q 4 q 3 − q 2 q 1 ] \bar{B}_{3\times 4}=\left[ \begin{array}{cccc} -q_2& q_1& q_4& -q_3\\ -q_3& -q_4& q_1& q_2\\ -q_4& q_3& -q_2& q_1\\ \end{array} \right] Bˉ3×4​= ​−q2​−q3​−q4​​q1​−q4​q3​​q4​q1​−q2​​−q3​q2​q1​​ ​, 带入同样的式子可得：

$$\begin{array}{rl}{\tilde{\omega }}^F& =2\bar{B}{\dot{\bar{B}}}^{\mathrm{T}}\\ {\tilde{\omega }}^M& =2B{\dot{B}}^{\mathrm{T}}\end{array}$$
将上式展开，由四元数的归一化可知：${\dot{q}}_1{q}_1+{\dot{q}}_2{q}_2+{\dot{q}}_3{q}_3+{\dot{q}}_4{q}_4=0$，可得：
[ w 1 F w 2 F w 3 F ] = 2 [ q ˙ 4 q 3 − q ˙ 3 q 4 + q ˙ 2 q 1 − q ˙ 1 q 2 q ˙ 2 q 4 − q ˙ 1 q 3 + q ˙ 4 q 2 − q ˙ 3 q 1 q ˙ 3 q 2 − q ˙ 4 q 1 + q ˙ 1 q 4 − q ˙ 2 q 3 ] [ w 1 M w 2 M w 3 M ] = 2 [ q 4 q ˙ 3 − q 3 q ˙ 4 − q 2 q ˙ 1 + q 1 q ˙ 2 q 2 q ˙ 4 + q 1 q ˙ 3 − q 4 q ˙ 2 − q 3 q ˙ 1 q 3 q ˙ 2 − q 4 q ˙ 1 + q 1 q ˙ 4 − q 2 q ˙ 3 ] \begin{split} \left[ \begin{array}{c} {w_1}^F\\ {w_2}^F\\ {w_3}^F\\ \end{array} \right] &=2\left[ \begin{array}{c} \dot{q}_4q_3-\dot{q}_3q_4+\dot{q}_2q_1-\dot{q}_1q_2\\ \dot{q}_2q_4-\dot{q}_1q_3+\dot{q}_4q_2-\dot{q}_3q_1\\ \dot{q}_3q_2-\dot{q}_4q_1+\dot{q}_1q_4-\dot{q}_2q_3\\ \end{array} \right] \\ \left[ \begin{array}{c} {w_1}^M\\ {w_2}^M\\ {w_3}^M\\ \end{array} \right] &=2\left[ \begin{array}{c} q_4\dot{q}_3-q_3\dot{q}_4-q_2\dot{q}_1+q_1\dot{q}_2\\ q_2\dot{q}_4+q_1\dot{q}_3-q_4\dot{q}_2-q_3\dot{q}_1\\ q_3\dot{q}_2-q_4\dot{q}_1+q_1\dot{q}_4-q_2\dot{q}_3\\ \end{array} \right] \end{split} ​w1​Fw2​Fw3​F​ ​ ​w1​Mw2​Mw3​M​ ​​=2 ​q˙​4​q3​−q˙​3​q4​+q˙​2​q1​−q˙​1​q2​q˙​2​q4​−q˙​1​q3​+q˙​4​q2​−q˙​3​q1​q˙​3​q2​−q˙​4​q1​+q˙​1​q4​−q˙​2​q3​​ ​=2 ​q4​q˙​3​−q3​q˙​4​−q2​q˙​1​+q1​q˙​2​q2​q˙​4​+q1​q˙​3​−q4​q˙​2​−q3​q˙​1​q3​q˙​2​−q4​q˙​1​+q1​q˙​4​−q2​q˙​3​​ ​​
继续观察上式，将上式进行化简：
$$\begin{gathered} {\omega }^F=2B\dot{q}=-2\dot{B}q \\ {\omega }^M=2\bar{B}\dot{q}=-2\dot{\bar{B}}q \end{gathered}$$

进而可将$\left[{\dot{Q}}_{\mathrm{M}}^F\right]R_{{\mathrm{P}}_{\mathrm{i}}}^M=\left[Q_{\mathrm{M}}^F\right]\left({\omega }^M\times R_{{\mathrm{P}}_{\mathrm{i}}}^M\right)$改写为（下式仅当$P_{\mathrm{i}}$为刚体上的固定点时成立）：
$$\left[{\dot{Q}}_{\mathrm{M}}^F\right]R_{{\mathrm{P}}_{\mathrm{i}}}^M=\left[Q_{\mathrm{M}}^F\right]\left({\omega }^M\times R_{{\mathrm{P}}_{\mathrm{i}}}^M\right)=-\left[Q_{\mathrm{M}}^F\right]\left(R_{{\mathrm{P}}_{\mathrm{i}}}^M\times {\omega }^M\right)=-\left[Q_{\mathrm{M}}^F\right]{\tilde{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M\left(2\bar{B}\dot{q}\right)$$

因为所有表达方式都能转换成欧拉参数-四元数的形式，因此上式在计算过程中具有普适性。

进而可知：
$$\frac{\partial \left(\left[Q_{\mathrm{M}}^F\right]R_{{\mathrm{P}}_{\mathrm{i}}}^M\right)}{\partial q}=-\left[Q_{\mathrm{M}}^F\right]{\tilde{R}}_{{\mathrm{P}}_{\mathrm{i}}}^M\left(2\bar{B}\right)$$

4.1.2 轴角参数表示角速度

将 [ θ v 1 v 2 v 3 ] = [ 2 a r c cos ⁡ ( q 1 ) q 2 sin ⁡ θ 2 q 3 sin ⁡ θ 2 q 4 sin ⁡ θ 2 ] \left[ \begin{array}{c} \theta\\ v_1\\ v_2\\ v_3\\ \end{array} \right] =\left[ \begin{array}{c} 2\mathrm{arc}\cos \left( q_1 \right)\\ \frac{q_2}{\sin \frac{\theta}{2}}\\ \frac{q_3}{\sin \frac{\theta}{2}}\\ \frac{q_4}{\sin \frac{\theta}{2}}\\ \end{array} \right] ​θv1​v2​v3​​ ​= ​2arccos(q1​)sin2θ​q2​​sin2θ​q3​​sin2θ​q4​​​ ​带入 [ w 1 F w 2 F w 3 F ] = 2 [ q ˙ 4 q 3 − q ˙ 3 q 4 + q ˙ 2 q 1 − q ˙ 1 q 2 q ˙ 2 q 4 − q ˙ 1 q 3 + q ˙ 4 q 2 − q ˙ 3 q 1 q ˙ 3 q 2 − q ˙ 4 q 1 + q ˙ 1 q 4 − q ˙ 2 q 3 ] , [ w 1 M w 2 M w 3 M ] = 2 [ q 4 q ˙ 3 − q 3 q ˙ 4 − q 2 q ˙ 1 + q 1 q ˙ 2 q 2 q ˙ 4 + q 1 q ˙ 3 − q 4 q ˙ 2 − q 3 q ˙ 1 q 3 q ˙ 2 − q 4 q ˙ 1 + q 1 q ˙ 4 − q 2 q ˙ 3 ] \left[ \begin{array}{c} {w_1}^F\\ {w_2}^F\\ {w_3}^F\\ \end{array} \right] =2\left[ \begin{array}{c} \dot{q}_4q_3-\dot{q}_3q_4+\dot{q}_2q_1-\dot{q}_1q_2\\ \dot{q}_2q_4-\dot{q}_1q_3+\dot{q}_4q_2-\dot{q}_3q_1\\ \dot{q}_3q_2-\dot{q}_4q_1+\dot{q}_1q_4-\dot{q}_2q_3\\ \end{array} \right] , \left[ \begin{array}{c} {w_1}^M\\ {w_2}^M\\ {w_3}^M\\ \end{array} \right] =2\left[ \begin{array}{c} q_4\dot{q}_3-q_3\dot{q}_4-q_2\dot{q}_1+q_1\dot{q}_2\\ q_2\dot{q}_4+q_1\dot{q}_3-q_4\dot{q}_2-q_3\dot{q}_1\\ q_3\dot{q}_2-q_4\dot{q}_1+q_1\dot{q}_4-q_2\dot{q}_3\\ \end{array} \right] ​w1​Fw2​Fw3​F​ ​=2 ​q˙​4​q3​−q˙​3​q4​+q˙​2​q1​−q˙​1​q2​q˙​2​q4​−q˙​1​q3​+q˙​4​q2​−q˙​3​q1​q˙​3​q2​−q˙​4​q1​+q˙​1​q4​−q˙​2​q3​​ ​, ​w1​Mw2​Mw3​M​ ​=2 ​q4​q˙​3​−q3​q˙​4​−q2​q˙​1​+q1​q˙​2​q2​q˙​4​+q1​q˙​3​−q4​q˙​2​−q3​q˙​1​q3​q˙​2​−q4​q˙​1​+q1​q˙​4​−q2​q˙​3​​ ​可得：
[ w 1 F w 2 F w 3 F ] = [ 2 ( v ˙ 3 v 2 − v ˙ 2 v 3 ) sin ⁡ 2 θ 2 + v ˙ 1 sin ⁡ θ + θ ˙ v 1 2 ( v ˙ 1 v 3 − v ˙ 3 v 1 ) sin ⁡ 2 θ 2 + v ˙ 2 sin ⁡ θ + θ ˙ v 2 2 ( v ˙ 2 v 1 − v ˙ 1 v 2 ) sin ⁡ 2 θ 2 + v ˙ 3 sin ⁡ θ + θ ˙ v 3 ] [ w 1 M w 2 M w 3 M ] = [ 2 ( v 3 v ˙ 2 − v 2 v ˙ 3 ) sin ⁡ 2 θ 2 + v ˙ 1 sin ⁡ θ + θ ˙ v 1 2 ( v 1 v ˙ 3 − v 3 v ˙ 1 ) sin ⁡ 2 θ 2 + v ˙ 2 sin ⁡ θ + θ ˙ v 2 2 ( v 2 v ˙ 1 − v 1 v ˙ 2 ) sin ⁡ 2 θ 2 + v ˙ 3 sin ⁡ θ + θ ˙ v 3 ] \begin{split} \left[ \begin{array}{c} {w_1}^F\\ {w_2}^F\\ {w_3}^F\\ \end{array} \right] &=\left[ \begin{array}{c} 2\left( \dot{v}_3v_2-\dot{v}_2v_3 \right) \sin ^2\frac{\theta}{2}+\dot{v}_1\sin \theta +\dot{\theta}v_1\\ 2\left( \dot{v}_1v_3-\dot{v}_3v_1 \right) \sin ^2\frac{\theta}{2}+\dot{v}_2\sin \theta +\dot{\theta}v_2\\ 2\left( \dot{v}_2v_1-\dot{v}_1v_2 \right) \sin ^2\frac{\theta}{2}+\dot{v}_3\sin \theta +\dot{\theta}v_3\\ \end{array} \right] \\ \left[ \begin{array}{c} {w_1}^M\\ {w_2}^M\\ {w_3}^M\\ \end{array} \right] &=\left[ \begin{array}{c} 2\left( v_3\dot{v}_2-v_2\dot{v}_3 \right) \sin ^2\frac{\theta}{2}+\dot{v}_1\sin \theta +\dot{\theta}v_1\\ 2\left( v_1\dot{v}_3-v_3\dot{v}_1 \right) \sin ^2\frac{\theta}{2}+\dot{v}_2\sin \theta +\dot{\theta}v_2\\ 2\left( v_2\dot{v}_1-v_1\dot{v}_2 \right) \sin ^2\frac{\theta}{2}+\dot{v}_3\sin \theta +\dot{\theta}v_3\\ \end{array} \right] \end{split} ​w1​Fw2​Fw3​F​ ​ ​w1​Mw2​Mw3​M​ ​​= ​2(v˙3​v2​−v˙2​v3​)sin22θ​+v˙1​sinθ+θ˙v1​2(v˙1​v3​−v˙3​v1​)sin22θ​+v˙2​sinθ+θ˙v2​2(v˙2​v1​−v˙1​v2​)sin22θ​+v˙3​sinθ+θ˙v3​​ ​= ​2(v3​v˙2​−v2​v˙3​)sin22θ​+v˙1​sinθ+θ˙v1​2(v1​v˙3​−v3​v˙1​)sin22θ​+v˙2​sinθ+θ˙v2​2(v2​v˙1​−v1​v˙2​)sin22θ​+v˙3​sinθ+θ˙v3​​ ​​

整理为：
$$\begin{array}{rl}{\omega }^F& =2v^F\times {\dot{v}}^F{\sin }^2\frac{\theta }{2}+{\dot{v}}^F\sin \theta +\dot{\theta }{v}^F\\ {\omega }^M& =2{\dot{v}}^F\times v^F{\sin }^2\frac{\theta }{2}+{\dot{v}}^F\sin \theta +\dot{\theta }{v}^F\end{array}$$

4.1.3 欧拉角表示角速度

对于ZYX欧拉角而言，有：
{ [ Q M F ] = [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] [ Q F 2 F 1 ( j ⃗ F , β ) ] [ Q F 1 F ( i ⃗ F , α ) ] ω ⃗ ~ F = [ Q ˙ M F ] [ Q M F ] T ω ⃗ ~ F = { [ Q ˙ F 3 ( M ) F 2 ( k ⃗ F , γ ) ] [ Q F 2 F 1 ( j ⃗ F , β ) ] [ Q F 1 F ( i ⃗ F , α ) ] ⋅ [ Q F 1 F ( i ⃗ F , α ) ] T [ Q F 2 F 1 ( j ⃗ F , β ) ] T [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] T + [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] [ Q ˙ F 2 F 1 ( j ⃗ F , β ) ] [ Q F 1 F ( i ⃗ F , α ) ] ⋅ [ Q F 1 F ( i ⃗ F , α ) ] T [ Q F 2 F 1 ( j ⃗ F , β ) ] T [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] T + [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] [ Q F 2 F 1 ( j ⃗ F , β ) ] [ Q ˙ F 1 F ( i ⃗ F , α ) ] ⋅ [ Q F 1 F ( i ⃗ F , α ) ] T [ Q F 2 F 1 ( j ⃗ F , β ) ] T [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] T ω ⃗ ~ F = { [ Q ˙ F 3 ( M ) F 2 ( k ⃗ F , γ ) ] [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] T + [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] [ Q ˙ F 2 F 1 ( j ⃗ F , β ) ] [ Q F 2 F 1 ( j ⃗ F , β ) ] T [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] T + [ [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] [ Q F 2 F 1 ( j ⃗ F , β ) ] ] [ Q ˙ F 1 F ( i ⃗ F , α ) ] ⋅ [ Q F 1 F ( i ⃗ F , α ) ] T [ [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] [ Q F 2 F 1 ( j ⃗ F , β ) ] ] T ω ⃗ ~ F = ω ⃗ ~ F 3 ( M ) F 2 + [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] ω ⃗ ~ F 2 F 1 ~ + [ [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] [ Q F 2 F 1 ( j ⃗ F , β ) ] ] ω ⃗ ~ F 1 F ~ ⇒ ω ⃗ F = ω ⃗ F 3 ( M ) F 2 + [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] ω ⃗ F 2 F 1 + [ [ Q F 3 ( M ) F 2 ( k ⃗ F , γ ) ] [ Q F 2 F 1 ( j ⃗ F , β ) ] ] ω ⃗ F 1 F ⇒ ω ⃗ F = [ 0 0 γ ˙ ] + [ cos ⁡ γ − sin ⁡ γ 0 sin ⁡ γ cos ⁡ γ 0 0 0 1 ] [ 0 β ˙ 0 ] + [ cos ⁡ γ − sin ⁡ γ 0 sin ⁡ γ cos ⁡ γ 0 0 0 1 ] [ cos ⁡ β 0 sin ⁡ β 0 1 0 − sin ⁡ β 0 cos ⁡ β ] [ α ˙ 0 0 ] ⇒ ω ⃗ F = [ cos ⁡ β cos ⁡ γ − sin ⁡ γ 0 cos ⁡ β sin ⁡ γ cos ⁡ γ 0 − sin ⁡ β 0 1 ] [ α ˙ β ˙ γ ˙ ] \begin{split} &\begin{cases} \left[ Q_{\mathrm{M}}^{F} \right] =\left[ Q_{\mathrm{F}_3\left( M \right)}^{F_2}\left( \vec{k}^F,\gamma \right) \right] \left[ Q_{\mathrm{F}_2}^{F_1}\left( \vec{j}^F,\beta \right) \right] \left[ Q_{\mathrm{F}_1}^{F}\left( \vec{i}^F,\alpha \right) \right]\\ \tilde{\vec{\omega}}^F=\left[ \dot{Q}_{\mathrm{M}}^{F} \right] \left[ Q_{\mathrm{M}}^{F} \right] ^{\mathrm{T}}\\ \end{cases} \\ \tilde{\vec{\omega}}^F&=\begin{cases} \left[ \dot{Q}_{\mathrm{F}_3\left( M \right)}^{F_2}\left( \vec{k}^F,\gamma \right) \right] \left[ Q_{\mathrm{F}_2}^{F_1}\left( \vec{j}^F,\beta \right) \right] \left[ Q_{\mathrm{F}_1}^{F}\left( \vec{i}^F,\alpha \right) \right] \cdot \left[ Q_{\mathrm{F}_1}^{F}\left( \vec{i}^F,\alpha \right) \right] ^{\mathrm{T}}\left[ Q_{\mathrm{F}_2}^{F_1}\left( \vec{j}^F,\beta \right) \right] ^{\mathrm{T}}\left[ Q_{\mathrm{F}_3\left( M \right)}^{F_2}\left( \vec{k}^F,\gamma \right) \right] ^{\mathrm{T}}+\\ \left[ Q_{\mathrm{F}_3\left( M \right)}^{F_2}\left( \vec{k}^F,\gamma \right) \right] \left[ \dot{Q}_{\mathrm{F}_2}^{F_1}\left( \vec{j}^F,\beta \right) \right] \left[ Q_{\mathrm{F}_1}^{F}\left( \vec{i}^F,\alpha \right) \right] \cdot \left[ Q_{\mathrm{F}_1}^{F}\left( \vec{i}^F,\alpha \right) \right] ^{\mathrm{T}}\left[ Q_{\mathrm{F}_2}^{F_1}\left( \vec{j}^F,\beta \right) \right] ^{\mathrm{T}}\left[ Q_{\mathrm{F}_3\left( M \right)}^{F_2}\left( \vec{k}^F,\gamma \right) \right] ^{\mathrm{T}}+\\ \left[ Q_{\mathrm{F}_3\left( M \right)}^{F_2}\left( \vec{k}^F,\gamma \right) \right] \left[ Q_{\mathrm{F}_2}^{F_1}\left( \vec{j}^F,\beta \right) \right] \left[ \dot{Q}_{\mathrm{F}_1}^{F}\left( \vec{i}^F,\alpha \right) \right] \cdot \left[ Q_{\mathrm{F}_1}^{F}\left( \vec{i}^F,\alpha \right) \right] ^{\mathrm{T}}\left[ Q_{\mathrm{F}_2}^{F_1}\left( \vec{j}^F,\beta \right) \right] ^{\mathrm{T}}\left[ Q_{\mathrm{F}_3\left( M \right)}^{F_2}\left( \vec{k}^F,\gamma \right) \right] ^{\mathrm{T}}\\ \end{cases} \\ \tilde{\vec{\omega}}^F&=\begin{cases} \left[ \dot{Q}_{\mathrm{F}_3\left( M \right)}^{F_2}\left( \vec{k}^F,\gamma \right) \right] \left[ Q_{\mathrm{F}_3\left( M \right)}^{F_2}\left( \vec{k}^F,\gamma \right) \right] ^{\mathrm{T}}+\left[ Q_{\mathrm{F}_3\left( M \right)}^{F_2}\left( \vec{k}^F,\gamma \right) \right] \left[ \dot{Q}_{\mathrm{F}_2}^{F_1}\left( \vec{j}^F,\beta \right) \right] \left[ Q_{\mathrm{F}_2}^{F_1}\left( \vec{j}^F,\beta \right) \right] ^{\mathrm{T}}\left[ Q_{\mathrm{F}_3\left( M \right)}^{F_2}\left( \vec{k}^F,\gamma \right) \right] ^{\mathrm{T}}\\ +\left[ \left[ Q_{\mathrm{F}_3\left( M \right)}^{F_2}\left( \vec{k}^F,\gamma \right) \right] \left[ Q_{\mathrm{F}_2}^{F_1}\left( \vec{j}^F,\beta \right) \right] \right] \left[ \dot{Q}_{\mathrm{F}_1}^{F}\left( \vec{i}^F,\alpha \right) \right] \cdot \left[ Q_{\mathrm{F}_1}^{F}\left( \vec{i}^F,\alpha \right) \right] ^{\mathrm{T}}\left[ \left[ Q_{\mathrm{F}_3\left( M \right)}^{F_2}\left( \vec{k}^F,\gamma \right) \right] \left[ Q_{\mathrm{F}_2}^{F_1}\left( \vec{j}^F,\beta \right) \right] \right] ^{\mathrm{T}}\\ \end{cases} \\ \tilde{\vec{\omega}}^F&=\tilde{\vec{\omega}}_{\mathrm{F}_3\left( M \right)}^{F_2}+\widetilde{\left[ Q_{\mathrm{F}_3\left( M \right)}^{F_2}\left( \vec{k}^F,\gamma \right) \right] \tilde{\vec{\omega}}_{\mathrm{F}_2}^{F_1}}+\widetilde{\left[ \left[ Q_{\mathrm{F}_3\left( M \right)}^{F_2}\left( \vec{k}^F,\gamma \right) \right] \left[ Q_{\mathrm{F}_2}^{F_1}\left( \vec{j}^F,\beta \right) \right] \right] \tilde{\vec{\omega}}_{\mathrm{F}_1}^{F}} \\ \Rightarrow \vec{\omega}^F&=\vec{\omega}_{\mathrm{F}_3\left( M \right)}^{F_2}+\left[ Q_{\mathrm{F}_3\left( M \right)}^{F_2}\left( \vec{k}^F,\gamma \right) \right] \vec{\omega}_{\mathrm{F}_2}^{F_1}+\left[ \left[ Q_{\mathrm{F}_3\left( M \right)}^{F_2}\left( \vec{k}^F,\gamma \right) \right] \left[ Q_{\mathrm{F}_2}^{F_1}\left( \vec{j}^F,\beta \right) \right] \right] \vec{\omega}_{\mathrm{F}_1}^{F} \\ \Rightarrow \vec{\omega}^F&=\left[ \begin{array}{c} 0\\ 0\\ \dot{\gamma}\\ \end{array} \right] +\left[ \begin{matrix} \cos \gamma& -\sin \gamma& 0\\ \sin \gamma& \cos \gamma& 0\\ 0& 0& 1\\ \end{matrix} \right] \left[ \begin{array}{c} 0\\ \dot{\beta}\\ 0\\ \end{array} \right] +\left[ \begin{matrix} \cos \gamma& -\sin \gamma& 0\\ \sin \gamma& \cos \gamma& 0\\ 0& 0& 1\\ \end{matrix} \right] \left[ \begin{matrix} \cos \beta& 0& \sin \beta\\ 0& 1& 0\\ -\sin \beta& 0& \cos \beta\\ \end{matrix} \right] \left[ \begin{array}{c} \dot{\alpha}\\ 0\\ 0\\ \end{array} \right] \\ \Rightarrow \vec{\omega}^F&=\left[ \begin{matrix} \cos \beta \cos \gamma& -\sin \gamma& 0\\ \cos \beta \sin \gamma& \cos \gamma& 0\\ -\sin \beta& 0& 1\\ \end{matrix} \right] \left[ \begin{array}{c} \dot{\alpha}\\ \dot{\beta}\\ \dot{\gamma}\\ \end{array} \right] \end{split} ω ~Fω ~Fω ~F⇒ω F⇒ω F⇒ω F​⎩ ⎨ ⎧​[QMF​]=[QF3​(M)F2​​(k F,γ)][QF2​F1​​(j ​F,β)][QF1​F​(i F,α)]ω ~F=[Q˙​MF​][QMF​]T​=⎩ ⎨ ⎧​[Q˙​F3​(M)F2​​(k F,γ)][QF2​F1​​(j ​F,β)][QF1​F​(i F,α)]⋅[QF1​F​(i F,α)]T[QF2​F1​​(j ​F,β)]T[QF3​(M)F2​​(k F,γ)]T+[QF3​(M)F2​​(k F,γ)][Q˙​F2​F1​​(j ​F,β)][QF1​F​(i F,α)]⋅[QF1​F​(i F,α)]T[QF2​F1​​(j ​F,β)]T[QF3​(M)F2​​(k F,γ)]T+[QF3​(M)F2​​(k F,γ)][QF2​F1​​(j ​F,β)][Q˙​F1​F​(i F,α)]⋅[QF1​F​(i F,α)]T[QF2​F1​​(j ​F,β)]T[QF3​(M)F2​​(k F,γ)]T​=⎩ ⎨ ⎧​[Q˙​F3​(M)F2​​(k F,γ)][QF3​(M)F2​​(k F,γ)]T+[QF3​(M)F2​​(k F,γ)][Q˙​F2​F1​​(j ​F,β)][QF2​F1​​(j ​F,β)]T[QF3​(M)F2​​(k F,γ)]T+[[QF3​(M)F2​​(k F,γ)][QF2​F1​​(j ​F,β)]][Q˙​F1​F​(i F,α)]⋅[QF1​F​(i F,α)]T[[QF3​(M)F2​​(k F,γ)][QF2​F1​​(j ​F,β)]]T​=ω ~F3​(M)F2​​+[QF3​(M)F2​​(k F,γ)]ω ~F2​F1​​ ​+[[QF3​(M)F2​​(k F,γ)][QF2​F1​​(j ​F,β)]]ω ~F1​F​ ​=ω F3​(M)F2​​+[QF3​(M)F2​​(k F,γ)]ω F2​F1​​+[[QF3​(M)F2​​(k F,γ)][QF2​F1​​(j ​F,β)]]ω F1​F​= ​00γ˙​​ ​+ ​cosγsinγ0​−sinγcosγ0​001​ ​ ​0β˙​0​ ​+ ​cosγsinγ0​−sinγcosγ0​001​ ​ ​cosβ0−sinβ​010​sinβ0cosβ​ ​ ​α˙00​ ​= ​cosβcosγcosβsinγ−sinβ​−sinγcosγ0​001​ ​ ​α˙β˙​γ˙​​ ​​文章来源地址https://www.toymoban.com/news/detail-783815.html