The paper presents the simulation-based variant of the LQR-tree feedback-motion-planning approach. The algorithm generates a control policy that stabilizes a nonlinear dynamic system from a bounded set of initial conditions to a goal. This policy is represented by a tree of feedback-stabilized trajectories. The algorithm explores the bounded set with random state samples and, where needed, adds new trajectories to the tree using motion planning. Simultaneously, the algorithm approximates the funnel of a trajectory, which is the set of states that can be stabilized to the goal by the trajectory’s feedback policy. Generating a control policy that stabilizes the bounded set to the goal is equivalent to adding trajectories to the tree until their funnels cover the set. In previous work, funnels are approximated with sums-of-squares verification. Here, funnels are approximated by sampling and falsification by simulation, which allows the application to a broader range of systems and a straightforward enforcement of input and state constraints. A theoretical analysis shows that, in the long run, the algorithm tends to improve the coverage of the bounded set as well as the funnel approximations. Focusing on the practical application of the method, a detailed example implementation is given that is used to generate policies for two example systems. Simulation results support the theoretical findings, while experiments demonstrate the algorithm’s state-constraints capability, and applicability to highly-dynamic systems.

In this work, we investigate the state observer problem for linear mechanical systems with a single unilateral constraint, for which neither the impact time instants nor the contact distance is explicitly measured. We propose to attack the observer problem by transforming and approximating the original continuous‐time system by a discrete linear complementarity system (LCS) through the use of the Schatzman–Paoli scheme. From there, we derive a deadbeat observer in the form of a linear complementarity problem. Sufficient conditions guaranteeing the uniqueness of its solution then serve as observability conditions. In addition, the discrete adaptation of an existing passivity‐based observer design for LCSs can be applied. A key point in using a time discretization is that the discretization acts as a regularization, that is, the impacts take place over multiple time steps (here two time steps). This makes it possible to render the estimation error dynamics asymptotically stable. Furthermore, the so‐called peaking phenomenon appears as singularity within the time discretization approach, posing a challenge for robust observer design.

A state observer that only uses the collision time information has recently been developed for linear time-invariant multibody systems with unilateral constraints. The observer is based on synchronization and makes use of switched geometric unilateral constraints, which generate a unidirectional coupling in a master-slave setup. In presence of uncertainties, such as model inaccuracies or disturbances, an exact reconstruction of the observed state is not possible. As a first step in assessing the robustness of the proposed observer, we present an experimental verification of the observer’s performance. Furthermore, we account for dry friction in the observer design.

One of the main difficulties in the state observer design for impulsive mechanical systems is the so-called peaking phenomenon: even for an arbitrarily small pre-impact estimation error, a slight mismatch between the impact time instants of the observer and the observed system can lead to large post-impact estimation error. Therefore, Lyapunov's stability theorems cannot be directly applied. For linear mechanical systems with unilateral constraints, we propose to take a Nonsmooth Dynamics perspective on the problem, which allows to sidestep the main difficulties by transforming and approximating the original continuous-time system by a discrete linear complementarity system through the use of the Schatzman-Paoli scheme. The discretization acts as a regularization, i.e. the impacts take place over two consecutive time steps. Furthermore, it involves force and impact laws on position-level with the favorable property of maximal monotonicity. Finally, a passivity-based observer design for discrete linear complementarity systems can be applied.