Beyond the Cascade: Juggling Vanilla Siteswap Patterns

Abstract

Being widespread in human motor behavior, dynamic movements demonstrate higher efficiency and greater capacity to address a broader range of skill domains compared to their quasi-static counterparts. Among the frequently studied dynamic manipulation problems, robotic juggling tasks stand out due to their inherent ability to scale their difficulty levels to arbitrary extents, making them an excellent subject for investigation. In this study, we explore juggling patterns with mixed throw heights, following the vanilla siteswap juggling notation, which jugglers widely adopted to describe toss juggling patterns. This requires extending our previous analysis of the simpler cascade juggling task by a throw-height sequence planner and further constraints on the end effector trajectory. These are not necessary for cascade patterns but are vital to achieving patterns with mixed throw heights. Using a simulated environment, we demonstrate successful juggling of most common 3-9 ball siteswap patterns up to 9 ball height, transitions between these patterns, and random sequences covering all possible vanilla siteswap patterns with throws between 2 and 9 ball height. https://kai-ploeger.com/beyond-cascades

Figures (7)

• Figure 1: At a given throw frequency, throw heights are restricted to a discrete set. Successful juggling patterns of up to height 9 reach $4.53m$ tall, as measured from catch height, and include up to $9$ balls simultaneously.
• Figure 2: The three ball siteswap graph of height 4: Juggling sequences are planned on this type of graph. Each transition represents a throw of specified height and each loop in the graph is a unique pattern.
• Figure 3: Every planned trajectory starts and ends at takeoff (throw) and is discretized into 24 time steps. During the dwell time, a ball rests within the hand. Hands move toward the incoming ball during the vacant time. Post-takeoff and pre-touchdown, ball and hand movements must be collinear to ensure clean contact switches (\ref{['sec:hand_trajectory_planning']}). During the vacant time, premature contacts with low incoming balls need to be avoided (\ref{['sec:premature_contacts']}), and during dwell time balls need to be kept from rolling out of the hand (\ref{['sec:contact_management']}).
• Figure 4: To achieve clean contact switches balls are kept close to the symmetry line $\mathbf{e}_h$ of the funnel-shaped hand after takeoff (throw: left) and prior to touchdown (catch: right).
• Figure 5: (a) In the frictionless case, balls can orbit in the hand. Through sufficient friction, these orbits dissipate, allowing for (b) the preservation of the ball's resting position, by imposing the constraint $\measuredangle(\mathbf{e}_h, \mathbf{g} - \ddot{\mathbf{x}}) > \ang{90}+\alpha$, which governs the direction of the gravity-compensated hand acceleration $\ddot{\mathbf{x}} - \mathbf{g}$ based on the hand orientation $\mathbf{e}_h$ and slope angle $\alpha$.