Controlling the Cascade: Kinematic Planning for N-ball Toss Juggling

TL;DR

The paper addresses dexterous N-ball toss juggling as a dynamic manipulation challenge by decomposing an infinite-horizon task into sequential, short-horizon trajectory optimizations in both task-space and joint-space. It introduces explicit take-off and catch constraints, leverages a CasADi-based direct-shooting approach with piece-wise-jerk control, and evaluates several tracking controllers, including an inverse-dynamics strategy. The key contributions include a formalized planning and control framework, identification of essential trajectory constraints, and demonstration of stable juggling up to 17 balls on two anthropomorphic arms in simulation and controlled experiments. This work advances dynamic manipulation and informs design choices for multi-ball juggling on real hardware, highlighting practical limits and avenues for real-world deployment.

Abstract

Dynamic movements are ubiquitous in human motor behavior as they tend to be more efficient and can solve a broader range of skill domains than their quasi-static counterparts. For decades, robotic juggling tasks have been among the most frequently studied dynamic manipulation problems since the required dynamic dexterity can be scaled to arbitrarily high difficulty. However, successful approaches have been limited to basic juggling skills, indicating a lack of understanding of the required constraints for dexterous toss juggling. We present a detailed analysis of the toss juggling task, identifying the key challenges and formalizing it as a trajectory optimization problem. Building on our state-of-the-art, real-world toss juggling platform, we reach the theoretical limits of toss juggling in simulation, evaluate a resulting real-time controller in environments of varying difficulty and achieve robust toss juggling of up to 17 balls on two anthropomorphic manipulators.

Figures (7)

• Figure 1: Stable juggling of seven and more balls in a cascade pattern on an anthropomorphic two-arm robotic setup. Trajectories are planned in real-time and adapted to disturbances.
• Figure 2: (Left) Crossing throws in a five ball cascade (Middle) Non-crossing throws in a four-ball fountain pattern (Right) The handcycle with dwell time $T_d$ and vacant time $T_v$ and carry distance $d_d$.
• Figure 3: Pre-touch-down and post-take-off constraints: (Left) After take-off, the relative hand acceleration $\ddot{\mathbf{x}}-\mathbf{g}$ and the hand normal $\mathbf{e}_h$ are constrained to be collinear. (Right) Before touch-down, the hand velocity $\dot{\mathbf{x}}$ and the ball velocity $\dot{\mathbf{b}}$ are constrained to be collinear.
• Figure 4: Theoretical limit vs. achieved patterns: The maximum number of balls is limited by the horizontal distance they travel. Stable juggling close to the upper bound was achieved in task space for more than $500$ catches.
• Figure 5: Robustness to disturbances: Ball trajectories in cascade patterns were disturbed by isotropic white noise on the take-off velocity. Catches counted until $100$ averaged over $50$ runs, with and without ball collisions.

Theorems & Definitions (1)

• Definition I.1: Simple Toss Juggling