Rotation representation

星期四, 31. 八月 2017 03:28下午

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Outline

Roll,pitch,yaw

roll： 翻滚 沿着x轴旋转
pitch： 倾斜，坠落 沿着y轴旋转
yaw: 偏航 沿着z轴旋转

Quaternions

* Definition:

1. i² = j² = k² =-1
2. ij = k, jk = i, ki = j, ji = -k, kj = -i, ik = -j
3. q = s + x i + y j + z k

* Rules:

1. q1 q2 ≠ q2 q1
2. (q1 q2) q3 = q1 (q2 q3)
3. (q1 + q2) q3 = q1 q3 + q2 q3
4. q1 (q2 + q3) = q1 q2 + q1 q3
5. α (q1 + q2) = α q1 + α q2 (α is scalar)
6. (αq1) q2 = α (q1q2) = q1 (αq2) (α is scalar)
7. Norm: |q|²= s² + x² + y² + z² (|q|² != q*q= s² - x² - y² -z²+2s(xi+yj+zk))
8. Conjugate: q* = s-xi-yj-zk
9. Inverse: q^-1=q*/|q|²
10. unit quaternion: |q| = 1 –> q^-1= q*

* Quaternions and Rotations

• Rotations are represented by unit quaternions.
  |q|²= s² + x² + y² + z² = 1

• Let (unit) rotation axis be [u_x, u_y, u_z], and angle θ

  Corresponding quaternion is q = cos(θ/2) + sin(θ/2)u_xi+sin(θ/2)u_yj+sin(θ/2)u_zk
  Composition of rotations q₁ and q₂: q = q₂ q₁ ≠ q₁ q₂(not commute)

• Quaternion-xyzw:
  x = u_x * sin(θ / 2)
  y = u_y * sin(θ / 2)
  z = u_z * sin(θ / 2)
  w = cos(θ / 2)

• Let v be a (3-dim) vector and let q be a unit quaternion:
  the corresponding rotation transforms vector v to q v q^-1 ,where v = 0+xi+yj+zk = [i,j,k] * [x;y;z]
  q v q^-1 = [i,j,k] *(R [x;y;z]) (s化为0了)

• For q = a+bi+cj+dk, R can be expressed as below…
  
  Quaternions q and -q give the same rotation

* Geometric Explanation of Quaternions

• A quaternion is a point on the 4-D unit sphere.
• Interpolating rotations corresponds to curves on the 4-D sphere
  (两点+圆心总能作出一个圆，由该圆为大圆可以得到一个球，该大圆的法线方向为旋转轴，两点的弧对应的角度为绕该轴的旋转角，这些是最开始定义中，q = cos(θ/2) + sin(θ/2)u_xi+sin(θ/2)u_yj+sin(θ/2)u_zk 里面的u_x u_y u_z 和θ)

* Applications: Spherical Linear intERPolation(SLERP)

• Situation：Interpolate along the great circle on the 4-D unit sphere(changing from one rotation state to another)
• Solution: pick the shortest(SLERP)
  
  cos(θ)=q1q2 =s1s2+x1x2+y1y2+z1z2
  u: from 0–>1
  q_m = s_m + x_m i + y_m j + z_m k, m =1,2
  The above formula does not produce a unit quaternion and must be normalized, replace q by q / |q|
• Realization:
  
• Improvement:
  To be more smooth:combined with spline.

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From http://run.usc.edu/cs520-s12/quaternions/quaternions-cs520.pdf