  Complementarity problems, a class of mathematical optimization problems with
orthogonality constraints, are widely used in many robotics tasks, such as
locomotion and manipulation, due to their ability to model non-smooth phenomena
(e.g., contact dynamics). In this paper, we propose a method to analyze the
stability of complementarity systems with neural network controllers. First, we
introduce a method to represent neural networks with rectified linear unit
(ReLU) activations as the solution to a linear complementarity problem. Then,
we show that systems with ReLU network controllers have an equivalent linear
complementarity system (LCS) description. Using the LCS representation, we turn
the stability verification problem into a linear matrix inequality (LMI)
feasibility problem. We demonstrate the approach on several examples, including
multi-contact problems and friction models with non-unique solutions.
