We present the design of the Blind Juggler: a robot that is able to juggle an unconstrained ball without feedback at heights of up to 2 m. The robot actuates a parabolic aluminum paddle with a linear motor. We achieve open-loop stability of the ball trajectory with two design parameters: 1) the curvature of the parabolic paddle and 2) the acceleration of the paddle at impact. We derive a linear map of perturbations of the nominal ball trajectory over a single bounce and obtain local stability of the trajectory by tuning the eigenvalues of this mapping with the two design parameters. We consider nine ball states in this analysis, including ball spin. Experimental data provide the impact states of the ball and paddle. From these data, we can identify system parameters and infer the process noise introduced into the system. We then combine the experimental noise power spectral densities with a model of the system and optimize the design parameters such that the impact of the process noise on juggling performance is minimized. Theoretical as well as experimental results of the optimization are discussed.

We present the analysis and design process of the Pendulum Juggler, a robot that juggles a ball with an actuated paddle mounted to a swinging pendulum. The unconstrained ball travels up to 1 m horizontally, reaching an apex height of 1.1 m between impacts. Using a perturbation analysis, we show that the ball trajectory is locally stable without feedback. The pendulum consists of a four-bar linkage, optimized to keep the paddle perpendicular to the impact velocity of the ball. We verify the stability in experiments with a prototype of the Pendulum Juggler, demonstrating sustained juggling at amplitudes of up to 25°.

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 simulation-based 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.

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 H2norm of an input-output representation of the system. Finally, we design an H2optimal 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.