SuPLE: Robot Learning with Lyapunov Rewards

The reward function is an essential component in robot learning. Reward directly affects the sample and computational complexity of learning, and the quality of a solution. The design of informative rewards requires domain knowledge, which is not always available. We use the properties of the dynamics to produce system-appropriate reward without adding external assumptions. Specifically, we explore an approach to utilize the Lyapunov exponents of the system dynamics to generate a system-immanent reward. We demonstrate that the `Sum of the Positive Lyapunov Exponents' (SuPLE) is a strong candidate for the design of such a reward. We develop a computational framework for the derivation of this reward, and demonstrate its effectiveness on classical benchmarks for sample-based stabilization of various dynamical systems. It eliminates the need to start the training trajectories at arbitrary states, also known as auxiliary exploration. While the latter is a common practice in simulated robot learning, it is unpractical to consider to use it in real robotic systems, since they typically start from natural rest states such as a pendulum at the bottom, a robot on the ground, etc. and can not be easily initialized at arbitrary states. Comparing the performance of SuPLE to commonly-used reward functions, we observe that the latter fail to find a solution without auxiliary exploration, even for the task of swinging up the double pendulum and keeping it stable at the upright position, a prototypical scenario for multi-linked robots. SuPLE-induced rewards for robot learning offer a novel route for effective robot learning in typical as opposed to highly specialized or fine-tuned scenarios. Our code is publicly available for reproducibility and further research.