Siddhartha Ganguly, George Rapakoulias, Panagiotis Tsiotras
We study stochastic density control between Gaussian-mixture endpoint distributions under Brownian prior dynamics. Since the direct Schrödinger bridge between Gaussian mixtures is generally not available in closed form, we introduce a lifted path-space construction in which each trajectory is augmented with a source--target component label. Consequently, the problem decomposes into Gaussian component-to-component Schrödinger bridges with explicit marginal, drift, and cost formulas, while the mixture-level assignment reduces to a finite-dimensional entropic coupling problem with a Sinkhorn scaling form. We then analyze the projection obtained by discarding or forgetting the label. By construction, the projected law satisfies the original Gaussian-mixture endpoint constraints, but its relative entropy generally differs from the lifted relative entropy by a nonnegative conditional label-information gap. This gap reveals a path-space obstruction: the lifted optimizer cannot, in general, be identified with the direct unlabeled Schrödinger bridge after projection. We also derive the posterior-averaged Markov drift associated with the projected marginal flow, prove a kinetic-energy upper bound, and identify a common path-potential condition under which the projection gap vanishes. Several numerical illustrations showing density and shape control are recorded for a self-contained exposition.
Lorenzo Ticozzi, Patricio A. Vela, Panagiotis Tsiotras
Spacecraft-mounted robotic manipulators can support long-term operability of orbiting assets by conducting tasks such as on-orbit assembly, repair, upgrade, and refueling. In this context, the integration of continuum/soft robotic mechanisms into space manipulator designs promises to reduce stowage volume and mass, as well as to improve operational safety through the inherent mechanical compliance of soft structures. This work presents a novel state reconstruction pipeline for a hybrid soft-rigid space manipulator, in which the distal link is a tendon-driven continuum robot. The estimation problem is formulated as a maximum a posteriori (MAP) inference problem over the hybrid manipulator's configuration, conditioned on discrete, noisy measurements from joint encoders and a vision-based sensing pipeline. The resulting soft-constrained optimization problem is transcribed as a factor graph, providing a compact representation for fusing heterogeneous measurements with the rigid-soft kinematic model. Furthermore, the adoption of the Geometric Variable Strain (GVS) approach yields a minimum-order parameterization of the configuration, a property expected to facilitate model-based feedback control design in future work. The proposed implementation is validated in simulation on a closed-loop trajectory where, for better realism, position measurements for the continuum link are obtained by means of a simulated visual pipeline leveraging synthetic photorealistic images. The state reconstruction accuracy and computational performance confirm the potential of the proposed framework for hybrid soft-rigid robotic applications in orbit.