Trajectory optimization in the implicit representation

Develop trajectory optimization directly in the implicit representation of a constraint submanifold under induced Riemannian metrics, with decision variables constrained to the level set rather than represented in a parameter domain.

Background

The paper implements trajectory optimization only for the explicit representation, using minimal coordinates supplied by a parameterization and lifting the resulting trajectory into configuration space. The authors identify direct optimization in the implicit representation as an unresolved direction, in which decision variables would lie on the constraint level set itself while still using the induced metric.