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Slerp(Spherical Linear Interpolation)

Interpolation betwee two orientations, oa and ob?

  • oa and ob can be represented by 
    • Euler-angles
    • 3x3 matrices
    • quaternions
  • How can them be interpolated?
  • Is the linear interpolation applicable?

 

When the axis-angle is known, oaob

 

Rotation between two orientations?

  • Rotation between two SO(3) matrices, ma and mb
    • mr=mbma1
  • Rotation between two quaternions, qa and qb
    • qr=qbqa1

 

Quaternion → Axis-Angle

 

Rotation Matrix → Axis-Angle

 

Interpolation mamb

 

 

Interpolation qaqb

 

Alternative Method

 

 

Quaternion Slerp

2D Complex Number Form

 

2D Spherical Interpolation

 

Spherical Linear Interpolation

  • SLERP [Shoemake 1985]
    • Linear Interpolation of two quaternions

 

Antipodal Equivalence Problem

 

  • In gerneral, the shorter pass is better
  • You should test both qa and qa , and choose one

 

Rotation Matrix vs. Unit Quaternion

  • Equivalent in many aspects
    • No singularity
    • Exp & Log
    • Special tangent space
  • Why quaternions?
    • Fewer parameters
    • Simpler algebra
    • Easy to fix numerical error
  • Why rotation matrices?
    • One-to-one correspondence
    • Handle rotation and translation in a uniform way
      • Eg) 4x4 homogeneous matrices

 

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📖 Contents 📖
Slerp(Spherical Linear Interpolation)Interpolation betwee two orientations, oa and ob?When the axis-angle is known, oaobRotation between two orientations?Quaternion → Axis-AngleRotation Matrix → Axis-AngleInterpolation mambInterpolation qaqbAlternative MethodQuaternion Slerp2D Complex Number Form2D Spherical InterpolationSpherical Linear InterpolationAntipodal Equivalence ProblemRotation Matrix vs. Unit Quaternion