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We describe the design of a juggling robot that is able to vertically bounce a completely unconstrained ball without any sensing. The robot consists of a linear motor actuating a machined aluminum paddle. The curvature of this paddle keeps the ball from falling off while the apex height of the ball is stabilized by decelerating the paddle at impact. We analyze the mapping of perturbations of the nominal trajectory over a single bounce to determine the design parameters that stabilize the system. The first robot prototype confirms the results from the stability analysis and exhibits substantial robustness to perturbations in the horizontal degree of freedoms. We then measure the performance of the robot and characterize the noise introduced into the system as white noise. This allows us to refine the design parameters by minimizing the H2 norm of an input-output representation of the system. Finally, we design an H2 optimal controller for the apex height using impact time measurements as feedback and show that the closed-loop performance is only marginally better than what is achieved with open-loop control.

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Abstract-We present an algorithm that probabilistically covers a bounded region of the state space of a nonlinear system with a sparse tree of feedback stabilized trajectories leading to a goal state. The generated tree serves as a lookup table control policy to get any reachable initial condition within that region to the goal. The approach combines motion planning with reasoning about the set of states around a trajectory for which the feedback policy of the trajectory is able to stabilize the system. The key idea is to use a random sample from the bounded region for both motion planning and approximation of the stabilizable sets by falsification; this keeps the number of samples and simulations needed to generate covering policies reasonably low. We simulate the nonlinear system to falsify the stabilizable sets, which allows enforcing input and state constraints. Compared to the algebraic verification using sums of squares optimization in our previous work, the simulationbased approximation of the stabilizable set is less exact, but considerably easier to implement and can be applied to a broader range of nonlinear systems. We show simulation results obtained with model systems and study the performance and robustness of the generated policies.

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.