旋转运动学

旋转的描述：矩阵欧拉角角轴四元数 Lie Group

一、绕全局直角坐标系旋转

\begin{aligned} ^{B} r & = x \hat{i} + y \hat{j} + z \hat{k}^{G} r = X \hat{I} + Y \hat{J} + Z \hat{K} \\ ^{G} r & = Q^{B} r \Rightarrow (\begin{array}{c} X \\ Y \\ Z \end{array}) = (\begin{array}{c} \hat{I} \\ \hat{J} \\ \hat{K} \end{array}) (\begin{array}{c} \hat{i} & \hat{j} & \hat{k} \end{array}) (\begin{array}{c} x \\ y \\ z \end{array}) \end{aligned}

\begin{array}{r} Q_{Z, α} = (\begin{array}{c} \cos α & - \sin α & 0 \\ \sin α & \cos α & 0 \\ 0 & 0 & 1 \end{array}) \\ Q_{Y, β} = (\begin{array}{c} \cos β & 0 & \sin β \\ 0 & 1 & 0 \\ - \sin β & 0 & \cos β \end{array}) \\ Q_{X, γ} = (\begin{array}{c} 1 & 0 & 0 \\ 0 & \cos γ & - \sin γ \\ 0 & \sin γ & \cos γ \end{array}) \end{array}

实际推导

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二、绕全局直角坐标系连续旋转

\begin{array}{r} ^{G} Q_{B} = Q_{X, γ} Q_{Y, β} Q_{Z, α} \end{array}

\begin{array}{r} (\begin{array}{ccc} \cos α \cos β & - \cos β \sin α & \sin β \\ \cos γ \sin α + \cos α \sin β \sin γ & \cos α \cos γ - \sin α \sin β \sin γ & - \cos β \sin γ \\ \sin α \sin γ - \cos α \cos γ \sin β & \cos α \sin γ + \cos γ \sin α \sin β & \cos β \cos γ \end{array}) \end{array}

MATLAB

syms a b c 
Q1=[cos(a),-sin(a),0
    sin(a),cos(a),0
    0,0,1
    ]

Q2=[cos(b),0,sin(b)
    0,1,0
    -sin(b),0,cos(b)]

Q3=[1,0,0
    0,cos(c),-sin(c)
    0,sin(c),cos(c)]

Q3*Q2*Q1

\begin{array}{r} α = \frac{2 π}{n} Q_{X, α}^{m} = (\begin{array}{c} 1 & 0 & 0 \\ 0 & \cos (m \frac{2 π}{n}) & - \sin (m \frac{2 π}{n}) \\ 0 & \sin (m \frac{2 π}{n}) & \cos (m \frac{2 π}{n}) \end{array}) \end{array}

unknown

syms  m n
assume(m, 'positive')  
assumeAlso(m, 'integer') 
assume(n, 'integer')    
k=2*pi/n
Q=[1,0,0
    0,cos(k),-sin(k)
    0,sin(k),cos(k)]
simplify(Q^m)

三、全局翻滚角、俯仰角、偏航角