Twists As Elements of A Lie Algebra
Consider the movement of a rigid body defined by the parameterized 4x4 homogeneous transform,
This notation does not distinguish between P = (X, Y, Z, 1), and P = (X, Y, Z), which is hopefully clear in context.
The velocity of this movement is defined by computing the velocity of the trajectories of the points in the body,
The dot denotes the derivative with respect to time, and because p is constant its derivative is zero.
Substitute the inverse transform for p into the velocity equation to obtain the velocity of P by operating on its trajectory P(t), that is
where
Recall that is the angular velocity matrix. The matrix is an element of the Lie algebra se(3) of the Lie group SE(3) of homogeneous transforms. The components of are the components of the twist screw, and for this reason is also often called a twist.
From the definition of the matrix, we can formulate the ordinary differential equation,
and ask for the movement that has a constant twist matrix . The solution is the matrix exponential
This formulation can be generalized such that given an initial configuration g(0) in SE(n), and a twist ξ in se(n), the homogeneous transformation to a new location and orientation can be computed with the formula,
where θ represents the parameters of the transformation.
Read more about this topic: Screw Theory
Famous quotes containing the words elements, lie and/or algebra:
“The Laws of Nature are just, but terrible. There is no weak mercy in them. Cause and consequence are inseparable and inevitable. The elements have no forbearance. The fire burns, the water drowns, the air consumes, the earth buries. And perhaps it would be well for our race if the punishment of crimes against the Laws of Man were as inevitable as the punishment of crimes against the Laws of Nature—were Man as unerring in his judgments as Nature.”
—Henry Wadsworth Longfellow (1807–1882)
“It is only for a little while, only occasionally, methinks, that we want a garden. Surely a good man need not be at the labor to level a hill for the sake of a prospect, or raise fruits and flowers, and construct floating islands, for the sake of a paradise. He enjoys better prospects than lie behind any hill. Where an angel travels it will be paradise all the way, but where Satan travels it will be burning marl and cinders.”
—Henry David Thoreau (1817–1862)
“Poetry has become the higher algebra of metaphors.”
—José Ortega Y Gasset (1883–1955)