Configuration space
Describe robot configurations using degrees of freedom, joint spaces, and coordinates.
Lie I: groups
Treat rigid-body configuration spaces as Lie groups, then define tangent spaces, Exp and Log.
Rotation representations
Describe frames and rotation matrices in the plane and in 3-D. Use a rotation matrix to represent, re-express and move a frame.
Angular velocity
Derive angular velocity in the plane and in space, and write it as a skew-symmetric matrix in so(3).
Exponential coordinates
Turn a constant angular velocity into a finite rotation with the matrix exponential, Rodrigues’ formula and the matrix logarithm.
SO(2) as a manifold
Examine the simplest Lie group through its tangent space and exponential map.
Motion representations
Describe poses, twists and screw motions in 3-D and in the plane. Use the exponential map to turn a twist into motion.
Homogeneous coordinates
Represent points and transformations in projective space.
Quaternions
Build unit quaternions from complex numbers and relate them to the double cover of SO(3).
Lie II: calculus
Apply plus and minus operations, the adjoint and Jacobians. Connect them to uncertainty and the formulas in Modern Robotics.
Lie cheatsheet
Look up the elements, dimension and meaning of each Lie group and Lie algebra used for rigid-body motion.
References
- Motion in a plane (Khan Academy). If your vectors or circular motion are rusty, watch these short videos before the rotation and angular velocity pages. The unit’s projectile motion videos are not needed here.
- Vectors: adding and subtracting vectors and unit vector notation.
- Products of vectors: the dot product, cross product 1 and cross product 2.
- Circular motion: angular motion variables, arc length from angular displacement, angular velocity and speed and relating angular and regular motion variables.
- Centripetal acceleration: deriving it from angular velocity, a visual explanation and a calculus proof.
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