GCRB Extension

[dhan2325]

In the implementation of the algorithm presented by Garrigas-Casanovas and Rodriguez y Baena (dubbed the GCRB solver), we found that the expressions provided in the original paper for the constants in the quadratic expression (denoted with $c_i$) did not provide the desired results. Additionally, the paper only provides the formulation of these constants when the z-coordinate is used to constrain the free parameter in the inverse kinematics problem of the two-segment robot.

To derive the expressions from scratch for the 2-segment case in which the z-coordinate is parameterized, a quick sympy script was used to derive the expressions for these constants from scratch (which revealed that our implementation had a missing sign).

The paper presented by the original authors also mention that any of the x, y, and z coordinates could be the parameterized, and that this solution can be extended to a three-segment case. The sympy script provided in this repository could be extended to derive full formulations for each of the parameterizable coordinates and for the three-segment case as well.

It should also be noted that while in theory the sympy script could be run with substitutions in the actual solver implementation, this would greatly impact performance, so it is generally speaking worthwhile to use sympy to derive the expressions for the constants in the quadratic expression and then hard code functions that evaluate those constants in the solver implementation