Bellman Value Decomposition for Task Logic in Safe Optimal Control
For safe optimal control in complex, high-dimensional tasks, this work provides a principled decomposition that automates reward tuning and improves performance.
This work decomposes the Bellman value for temporal logic tasks into a graph of Bellman equations, enabling automatic balancing of safety and liveness. The proposed VDPPO method achieves improved performance over baselines in high-dimensional simulated and hardware experiments.
Real-world tasks involve nuanced combinations of goal and safety specifications. In high dimensions, the challenge is exacerbated: formal automata become cumbersome, and the combination of sparse rewards tends to require laborious tuning. In this work, we consider the innate structure of the Bellman Value as a means to naturally organize the problem for improved automatic performance. Namely, we prove the Bellman Value for a complex task defined in temporal logic can be decomposed into a graph of Bellman Values, connected by a set of well-known Bellman equations (BEs): the Reach-Avoid BE, the Avoid BE, and a novel type, the Reach-Avoid-Loop BE. To solve the Value and optimal policy, we propose VDPPO, which embeds the decomposed Value graph into a two-layer neural net, bootstrapping the implicit dependencies. We conduct a variety of simulated and hardware experiments to test our method on complex, high-dimensional tasks involving heterogeneous teams and nonlinear dynamics. Ultimately, we find this approach greatly improves performance over existing baselines, balancing safety and liveness automatically.