常用坐标系变换

• 1.点旋转
• 2.坐标系旋转
• 3.常用坐标系变换

1.点旋转

• 绕z轴旋转 v'=r(z)v
  \[r(z)= \left\{ \begin{matrix} cos(\theta) & -sin(\theta) & 0 \\ sin(\theta) & cos(\theta) & 0 \\ 0 & 0 & 1 \end{matrix} \right\} \]
  
• 绕x轴旋转 v'=r(x)v
  \[r(x)= \left\{ \begin{matrix} 1 & 0 & 0 \\ 0 & cos(\theta) & -sin(\theta) \\ 0 & sin(\theta) & cos(\theta) \end{matrix} \right\} \]

• 绕y轴旋转 v'=r(y)v
  \[r(x)= \left\{ \begin{matrix} cos(\theta) & 0 & sin(\theta) \\ 0 & 1 & 0 \\ -sin(\theta) & 0 & cos(\theta) \end{matrix} \right\} \]

  2.坐标系旋转(同一个点在不同坐标系下的表示)

• 绕z轴旋转 vⁿ =\(C^{new}_{raw}(z)\)v raw坐标系绕z轴旋转-->new系
  \[C^{new}_{raw}(z)= \left\{ \begin{matrix} cos(\theta) & sin(\theta) & 0 \\ -sin(\theta) & cos(\theta) & 0 \\ 0 & 0 & 1 \end{matrix} \right\} \]
  
• 绕x轴旋转
  \[C^{new}_{raw}(x)= \left\{ \begin{matrix} 1 & 0 & 0 \\ 0 & cos(\theta) & sin(\theta) \\ 0 & -sin(\theta) & cos(\theta) \end{matrix} \right\} \]

• 绕y轴旋转
  \[C^{new}_{raw}(y)= \left\{ \begin{matrix} cos(\theta) & 0 & -sin(\theta) \\ 0 & 1 & 0 \\ sin(\theta) & 0 & cos(\theta) \end{matrix} \right\} \]

  3.常用坐标系变换

• 从地球系e转换到地心惯性系i:
  e绕Z轴逆时针转\(\theta\)=-W_et--->i
  \[C^{i}_{e}(z)= \left\{ \begin{matrix} cos(\theta) & sin(\theta) & 0 \\ -sin(\theta) & cos(\theta) & 0 \\ 0 & 0 & 1 \end{matrix} \right] = \left\{ \begin{matrix} cos(wt) & -sin(wt) & 0 \\ sin(wt) & cos(wt) & 0 \\ 0 & 0 & 1 \end{matrix} \right\} \]
  
• 从地理系g转换到地球系e:
  e——绕z轴(\(\lambda\)+90)——>t系——绕x轴(90-L)——>g系
  e<——绕z轴(-(\(\lambda\)+90))——t系<——绕x轴(-(90-L))<——g系
  \[C^{e}_{g} = C^{e}_{t}*C^{t}_{g}= \left\{ \begin{matrix} cos(\lambda+90) & -sin(\lambda+90) & 0 \\ sin(\lambda+90) & cos(\lambda+90) & 0 \\ 0 & 0 & 1 \end{matrix} \right] * \left\{ \begin{matrix} 1 & 0 & 0 \\ 0 & cos(90-L) & -sin(90-L) \\ 0 & sin(90-L) & cos(90-L) \end{matrix} \right\} \]
  \[ = \left\{ \begin{matrix} -sin(\lambda) & -sin(L)cos(\lambda) & cos(L)cos(\lambda) \\ cos(\lambda) & -sin(L)sin(\lambda) & cos(L)sin(\lambda) \\ 0 & cos(L) & sin(L) \end{matrix} \right\} \]

• 从机体系b到地理系g：
  g系——绕z轴(\(\phi\))——>t1系——绕x轴(\(\theta\))——>t2系——绕y轴(\(\gamma\))——>b系
  g系<——绕z轴(-\(\phi\))——t1系<——绕x轴(-\(\theta\))——t2系<——绕y轴(-\(\gamma\))——b系
  \[C^{g}_{b} = C^{g}_{t1}*C^{t1}_{t2}*C^{t2}_{b}\]
  \[ = \left\{ \begin{matrix} cos(\phi) & -sin(\phi) & 0 \\ sin(\phi) & cos(\phi) & 0 \\ 0 & 0 & 1 \end{matrix} \right] \left\{ \begin{matrix} 1 & 0 & 0 \\ 0 & cos(\theta) & -sin(\theta) \\ 0 & sin(\theta) & cos(\theta) \end{matrix} \right] \left\{ \begin{matrix} cos(\gamma) & 0 & sin(\gamma) \\ 0 & 1 & 0 \\ -sin(\gamma) & 0 & cos(\gamma) \end{matrix} \right\} \]
  对应：\(\theta\)--i,\(\gamma\)--j,\(\phi\)--z
  \[ \left\{ \begin{matrix} cj*ck-si*sj*sk & -ci*sk & sj*ck+si*cj*sk \\ cj*sk+si*sj*ck & ci*ck & sj*sk-si*cj*ck \\ -ci*sj & si & ci*cj \end{matrix} \right\} \]

  附录matlab code

  syms a b c;% i j k
  ci = cos(a);
  si =sin(a);
  cj = cos(b);
  sj =sin(b);
  ck = cos(c);
  sk =sin(c);
  %z
  t1=[ck -sk 0;
  sk ck 0;
  0 0 1];
  %x
  t2 = [1 0 0;
  0 ci -si;
  0 si ci];
  %y
  t3=[cj 0 sj;
  0 1 0;
  -sj 0 cj];
  t3t = [cj 0 -sj;
  0 1 0;
  sj 0 cj];
  t2t = [1 0 0;
  0 ci si;
  0 -si ci];
  t1t = [ck sk 0;
  -sk ck 0;
  0 0 1];

cgb=t1*t2*t3