Trajectory Optimization for High-Dimensional Nonlinear Systems Under STL Specifications

Abstract

Signal Temporal Logic (STL) has gained popularity in recent years as a specification language for cyber-physical systems, especially in robotics. Beyond being expressive and easy to understand, STL is appealing because the synthesis problemgenerating a trajectory that satisfies a given specification-can be formulated as a trajectory optimization problem. Unfortunately, the associated cost function is nonsmooth and non-convex. As a result, existing synthesis methods scale poorly to high-dimensional nonlinear systems. In this letter, we present a new trajectory optimization approach for STL synthesis based on Differential Dynamic Programming (DDP). It is well known that DDP scales well to extremely high-dimensional nonlinear systems like robotic quadrupeds and humanoids: we show that these advantages can be harnessed for STL synthesis. We prove the soundness of our proposed approach, demonstrate order-of-magnitude speed improvements over the state-of-the-art on several benchmark problems, and demonstrate the scalability of our approach to the full nonlinear dynamics of a 7 degree-of-freedom robot arm.

A More Scalable Mixed-Integer Encoding for Metric Temporal Logic

“…Remark 3. The fragment (12) is the same as that considered in [26], which presents a scalable but incomplete synthesis method. In contrast, our proposed encoding is sound and complete for all specifications, including those in this fragment.…”

“…The scalability limitations of this standard MICP formulation are well-known [9], [15], [17], [18], [26]. In particular, this encoding is associated with rapidly increasing solve times in the case of long time horizons (which increase the number of binary variables) and complex specifications (which increase the complexity of the constraint structure).…”

Temporal Logic Motion Planning with Convex Optimization via Graphs of Convex Sets

“…Finally, we acknowledge the recent trend of attempting to avoid the NP-hardness of temporal logic motion planning altogether by providing approximate solutions via non-convex optimization [24,25,26,27], learning [28,29], or control barrier functions [30,31]. While such approaches can be extremely efficient and may be practical for some applications, they offer limited or no completeness guarantees and rarely scale to very complex specifications like that shown in Fig.…”

Mixed-Integer Programming for Signal Temporal Logic with Fewer Binary Variables

“…S IGNAL Temporal Logic (STL) is a powerful means of expressing complex control objectives. STL combines boolean operators ("and", "or", "not") with temporal operators ("always", "eventually", "until") and is defined over continuous-valued signals, making it an appealing choice for dynamical systems ranging from mobile robots [1]- [3] and quadrotors [4] to high-DoF manipulators [5] and traffic networks [6], [7].…”

“…Much STL research in recent years has focused on avoiding MICP entirely, and instead designing efficient heuristic methods which are sound but not complete. Such methods include gradient-based optimization [2], [3], [16], [17], differential dynamic programming [5], control barrier functions [18] and neural-network-based methods [19]. These approaches tend to improve scalability, especially with respect to the specification time horizon, but offer limited formal guarantees and often struggle to handle more complex specifications.…”