Trajectory Optimization via Schr\"{o}dinger Bridge Sampling

Abstract

We take a new look at the relation between finite-horizon trajectory optimization and Schr\"odinger bridge sampling. Viewed as inference, KL-regularized trajectory optimization is solved by sampling from a Gibbs--Boltzmann distribution whose energy is the trajectory cost, and the adjoint Schr\"odinger bridge sampler (ASBS) is a simulation-free diffusion sampler designed for exactly such unnormalized targets. Hard equality path and terminal constraints, by contrast, confine the admissible decision variables to a measure-zero feasibility manifold, on which the target must be redefined intrinsically. In particular: $(a)$ we analyze two complementary parametrizations; a rollout parametrization, in which the dynamics are eliminated and only the remaining constraints shape the manifold, and a double-shooting parametrization, in which states and controls are sampled jointly and the dynamics themselves become part of the manifold; $(b)$ we establish regularity conditions under which both admissible sets are smooth embedded manifolds; $(c)$ under compactness and path-connectedness assumptions, we sample from the resulting intrinsic Gibbs measures via Riemannian ASBS, treating strict inequalities through exponential slack variables. Experiments, including contact-rich locomotion and manipulation, demonstrate the effectiveness of both regimes.