Goal-conditioned planning benefits from learned low-dimensional representations of rich observations. While compact latent representations typically learned from variational autoencoders or inverse dynamics enable goal-conditioned decision making, they ignore state reachability, hampering their performance. In this paper, we learn a representation that associates reachable states together for effective planning and goal-conditioned policy learning. We first learn a latent representation with multi-step inverse dynamics (to remove distracting information), and then transform this representation to associate reachable states together in $\ell_2$ space. Our proposals are rigorously tested in various simulation testbeds. Numerical results in reward-based settings show significant improvements in sampling efficiency. Further, in reward-free settings this approach yields layered state abstractions that enable computationally efficient hierarchical planning for reaching ad hoc goals with zero additional samples.

The arising application of neural networks (NN) in robotic systems has driven the development of safety verification methods for neural network dynamical systems (NNDS). Recursive techniques for reachability analysis of dynamical systems in closed-loop with a NN controller, planner or perception can over-approximate the reachable sets of the NNDS by bounding the outputs of the NN and propagating these NN output bounds forward. However, this recursive reachability analysis may suffer from compounding errors, rapidly becoming overly conservative over a longer horizon. In this work, we prove that an alternative one-shot reachability analysis framework which directly verifies the unrolled NNDS can significantly mitigate the compounding errors, enabling the use of the rolling horizon as a design parameter for verification purposes. We characterize the performance gap between the recursive and one-shot frameworks for NNDS with general computational graphs. The applicability of one-shot analysis is demonstrated through numerical examples on a cart-pole system.

We propose a sampling-based approach to learn Lyapunov functions for a class of discrete-time autonomous hybrid systems that admit a mixed-integer representation. Such systems include autonomous piecewise affine systems, closed-loop dynamics of linear systems with model predictive controllers, piecewise affine/linear complementarity/mixed-logical dynamical systems in feedback with a ReLU neural network controller, etc. The proposed method comprises an alternation between a learner and a verifier to search for a Lyapunov function from a family of parameterized Lyapunov function candidates. In each iteration, the learner uses a collection of state samples to select a Lyapunov function candidate through a convex program in the parameter space. The verifier then solves a nonconvex mixed-integer quadratic program in the state space to either validate the proposed Lyapunov function candidate or reject it with a counterexample, i.e., a state where the Lyapunov condition fails. This counterexample is then added to the sample set of the learner to refine the set of Lyapunov function candidates in the next iteration. By designing the learner and the verifier according to the analytic center cutting-plane method from convex optimization, we show that when the set of Lyapunov functions is full-dimensional in the parameter space, our method finds a Lyapunov function in a finite number of steps. We demonstrate our stability analysis method on closed-loop MPC dynamical systems and a ReLU neural network controlled PWA system.