Title: Robust Tracking with Model Mismatch for Fast and Safe Planning: an SOS Optimization Approach,
Abstract: In the pursuit of real-time motion planning, a commonly adopted practice is
to compute a trajectory by running a planning algorithm on a simplified,
low-dimensional dynamical model, and then employ a feedback tracking controller
that tracks such a trajectory by accounting for the full, high-dimensional
system dynamics. While this strategy of planning with model mismatch generally
yields fast computation times, there are no guarantees of dynamic feasibility,
which hampers application to safety-critical systems. Building upon recent work
that addressed this problem through the lens of Hamilton-Jacobi (HJ)
reachability, we devise an algorithmic framework whereby one computes, offline,
for a pair of "planner" (i.e., low-dimensional) and "tracking" (i.e.,
high-dimensional) models, a feedback tracking controller and associated
tracking bound. This bound is then used as a safety margin when generating
motion plans via the low-dimensional model. Specifically, we harness the
computational tool of sum-of-squares (SOS) programming to design a bilinear
optimization algorithm for the computation of the feedback tracking controller
and associated tracking bound. The algorithm is demonstrated via numerical
experiments, with an emphasis on investigating the trade-off between the
increased computational scalability afforded by SOS and its intrinsic
conservativeness. Collectively, our results enable scaling the appealing
strategy of planning with model mismatch to systems that are beyond the reach
of HJ analysis, while maintaining safety guarantees.