Rigid Body Motions
Reference
- Modern Robotics
- Modern Robotics (YouTube)
- https://ingmec.ual.es/~jlblanco/papers/jlblanco2010geometry3D_techrep.pdf
Rotation Matrix
SO(3)
The special orthogonal group: The special orghogonal group
, also known as the group of rotation matrices, is the set of all SO(3) real matrices 3 \times 3 that satisfy (i) R and (ii) RR^T = I \det{R} = 1
Definition of a group
A group consists of a set of elements and an operation on two elements such that, for all
- closure:
is also in the groupAB - associativity:
(AB)C = A(BC) - identity element existence: There exists an element
in the group (the identity matrix forI ) such thatSO(n) AI=IA=A - inverse element existence: There exists an element
in the group such thatA^{-1} AA^{-1} = A^{-1}A = I
Definition of Lie group
A Lie group is a (non-empty) subset
is a groupG is a manifold inG \mathbb{R}^N - Both the group product operation and its inverse are smooth functions
Reference: https://ingmec.ual.es/~jlblanco/papers/jlblanco2010geometry3D_techrep.pdf (P.40)
What is Manifold??
An
Intuitively, this means that in an infinitely small vicinity of a point
Reference https://ingmec.ual.es/~jlblanco/papers/jlblanco2010geometry3D_techrep.pdf (P.40)
Exponential Coordinate Representation of Rotation (3.2.3; P.77)
The exponential coordinates parameterize a rotation matrix in terms of:
- a rotation axis (as a unit vector
)\hat{\omega} \in \mathbb{R}^3 - an angle of rotation
about the axis\theta \in \mathbb{R}
A rotation with this is represented as
There are two views to explain the connection with a rotation matrix
- If a frame was rotated by
about\theta , its final orientation (relative to the original frame) would be expressed by\hat{\omega} R - If a frame followed angular velocity
for\hat{\omega} units of time, its final orientation (relative to the original frame) would be expressed by\theta R
Let's think about rotating a vector
We can see it as
Then we use 3
skew-symmetric matrix
Given a vector
It's called skew-symmetric because
The set of all 3
This is a linear differential equation (
Summary
Rodrigues' formula