So I’ve been trying to simulate a rope as a series of IK points with momentum join lines and motors. The first part is just adding the forces and deriving the new motor and momentum for each particle in the chain. Then i run IK on each particle in the chain to keep the lengths consistent. I believe at the IK step, the correcting motor generated at the IK step should be multiplied by the current motor in order to get the final motor, however when calculating the derivative, the formula calls for motor addition. What is the mathematical meaning behind adding motors as derivatives vs multiplying motors together?
It’s essentially the relationship of an additive group and a multiplicative group via the exponential/logarithm maps:
\log(ab) = \log(a) + \log(b) and \exp(a+b) = \exp(a)\exp(b).
The motors are elements of the Lie group of euclidean motions, while their derivatives (when they are properly transformed to be bivectors) are elements of the Lie algebra. The multiplication in any Lie group is related to the addition in the Lie algebra by the exponential/logarithm map.
Sec. 8 of the 2019 SIGGRAPH course notes on geometric algebra contains the details.