Juggling dynamics

Figure 1 shows two five-ball juggling patterns above plots that define their sensitivities to deviations in throw speed and angle. On the left is the most common pattern, the cascade, in which each hand catches and throws balls to equal heights across the body’s centerline. Hand motions in the cascade are left–right antisymmetric—that is, 180° out of phase. It is an amusing exercise to prove what all jugglers know—that only an odd number of balls can be juggled in the cascade. It’s impossible to juggle an even number of balls in a cascade without breaking the antisymmetry—for example, by throwing with both hands simultaneously or by throwing balls to different heights.

The shower pattern (shown on the right), by contrast, lacks symmetry and can be performed with either even or odd numbers of balls. Although the pattern is dynamically simpler, it’s also less stable—and thus more difficult—than the cascade. To appreciate why, consider the sensitivity plots (figure 1b) for both cascade and shower.

Both panels show the results of simulated parabolic trajectories under gravity. Figure 2a illustrates the ideal positions: The trajectories define throws that leave the exact center of the throwing hand and land precisely in the center of the catching hand. The hands move in ellipses, centered 50 cm apart and 180° out of phase, with horizontal radius 10 cm and vertical radius 5 cm. In simulations we add Gaussian deviations in initial speed v₀ and angle θ₀ to the throws and record how rates of catch success depend on throw deviations. Those rates are determined by counting the number of failures from 15 trials. Each trial comprises 50 throws for each combination of v₀ and θ₀ in the plots. Any throw in a trial that misses a catching hand 10 cm in diameter counts as a failure.

The take-home message of figure 1 is that the success region is larger for the cascade than for the shower. Cascade patterns can evidently tolerate greater variability in throw speed than can the shower, with balls thrown to the same height. For that reason, balls in a shower are typically thrown higher than those in a cascade. The higher throws provide more time between catches, but at the expense of requiring tighter control over the angle, as we’ll see.

For throws to the same height, the two patterns are indistinguishable in sensitivity to throw angle. That feature makes sense in terms of the dynamics of the problem. The trajectories of both patterns are parabolic and nearly identical, so changing the throw angle produces the same displacement of a catch from its expected location in both.

In the shower pattern, each ball travels first through a parabola and then through a quick shuffle, whereas in the cascade, each ball must travel through two parabolas to return to its starting point. So hands must move nearly twice as rapidly in the shower as in the cascade, which makes catches in the shower much more sensitive to timing.

The different effects of throw angle and speed can be quantified by plotting sensitivities of throws to independent deviations in the two parameters, shown in figure 2b for a five-ball cascade. Notice that throws to greater heights permit less angle deviation but more speed deviation than throws to lower heights. That phenomenon occurs because errors in throw angle produce variations in catch location that worsen as throws become higher.

Initial speed, on the other hand, chiefly affects the timing of the catch. And because greater heights are produced at lower hand frequencies, higher throws provide more tolerance to time variation. So for high, nearly vertical throws, it’s speed, not angle, that largely determines the time between throw and catch. The effects—particularly for higher throws—are that throw angle determines the location of a catch, whereas throw speed determines its timing. This distinguishes juggling from darts and other throwing games in which targeting accuracy does not depend on the relative timing between two hands.

The distinction between angle and speed is complicated by so-called siteswaps, juggling patterns defined by the order in which successive balls are manipulated, often resulting in throws to varying heights. When balls are thrown many meters high, throw angles must be accurate to within 0.1 degree. That’s more than an order of magnitude tighter than is achievable by world-class athletes, and the time between successive catches is shorter than human reaction times. For an analysis of sensitivity plots for several siteswaps, see the online supplement.

It’s unclear how jugglers achieve accuracy and response times beyond apparent physiological limits. But two experiments offer clues to how people can successfully juggle despite these limitations. In 2004 researchers presented evidence that both human and monkey brains can compute trajectories using an internal representation of equations of motion. According to that picture, jugglers keep track of the locations of their balls—and so effectively extend their reaction times—using dynamical prediction. Just as an outfielder predicts where a fly ball will land, a juggler predicts trajectories from how balls are thrown.

A second clue lies in the well-known but poorly understood phenomenon of muscle memory: A practiced sequence of movements can be recalled and repeated (see Physics Today, November 2018, page 16). Human physiologists have long known that the brain’s motor cortex contains somatotopic maps such that when particular locations are briefly stimulated electrically, specific muscles contract. In 2002 neuroscientist Michael Graziano and his colleagues showed that by prolonging the duration of stimulation, they could produce—at least in monkeys—complex, coordinated motions involving multiple muscles. They concluded that the brain appears to have developed a clear-cut and hierarchical way to encode complex tasks.

Complex tasks like juggling can be successfully performed without understanding the physiology behind motor control, although a deeper understanding is both intriguing and useful. Feedback-controlled robots can juggle a small number of balls by tracking them at 60 Hz. Physiology has apparently found ways to do the same, but with many more balls tracked at the slower rate of just 5 Hz. Unraveling the secrets behind how our nervous systems pull that off may pave the way for more dexterous robots.

By: Jonah Botvinick-Greenhouse and Troy Shinbrot

Figure 1 in our February 2020 Quick Study demonstrates that when hand frequency is permitted to vary, some juggling patterns can be highly sensitive to deviations in throw speed but not to throw angle. Sensitivity to throw angle can be assessed by maintaining a constant hand frequency and comparing different siteswaps—periodic juggling patterns such as those shown in figure S1. Just as a musical score defines how long each note is played, the digits in a siteswap define, roughly speaking, how long it takes for a ball to be caught after it has been thrown. More specifically, a siteswap digit defines the number of throws that pass before an object is thrown again. For this reason, even numbers represent throws back to the same hand, whereas odd numbers cross to the opposite hand.

In a five-ball siteswap, the order in which the throws are made is swapped at constant hand frequency. That is, the sequence 555555. . . , in which each height and period is identical, can be changed to 753753. . . by throwing one ball higher (7), so that it arrives at the catch location two catches later than a standard throw (5) would have. The timing change frees the hand that would have caught that ball, so it can now catch a lower throw (3) two catches earlier than a standard (5). Thus two throws of a height corresponding to a five-ball cascade are swapped for heights of a seven- and a three-ball cascade at a five-ball hand frequency, as shown in the center panel of figure S1.

Similarly, if a ball is thrown higher still (9), even more throws can be made before it lands and is thrown again. The result, shown in the right panel of figure S1, is a 933933. . . siteswap. But that is only one of numerous options; the siteswaps 97531, 94444, and 942 are some alternatives. Readers interested in the mathematics of siteswaps can find a comprehensive overview and demonstrations at Allen Knutson’s YouTube channel, Mathematics of Juggling.

As shown beneath each panel of figure S1, the siteswaps with progressively higher throws reduce the tolerance of successful catches to angle deviation, while leaving their dependence on throw speed unchanged. All siteswaps are simulated at the same frequency, so unlike figure 2 in the main text, the timing of throws and catches is identical. A small throw-angle variation in a high throw will, however, cause the ball to land far from its intended target. Consequently, the higher the throw, the more sensitive the juggling routine is to angle variations.

Figure S1 shows the intrinsic sensitivity of three juggling patterns. From left to right, directing the dynamics toward the most stable regime of motion becomes increasingly difficult (see Physics Today, November 2018, page 16). People can juggle up to five balls blindfolded, as mentioned in the main text, so steering toward stable dynamics seems to be at work.

The stable regions (yellow) in figure S1 are nevertheless smaller than can be practically achieved without additional control: The 933 pattern, for example, would require throw angles to vary by less than 0.1°, more than an order of magnitude tighter than is achievable by world-class jugglers. And yet juggling of even more complex patterns can be achieved with eyes open. But because the available response times are so short, it is unclear what control schemes are actually used.

Jonah Botvinick-Greenhouse is a junior mathematics major at Amherst College in Massachusetts, and Troy Shinbrot is a professor in the department of biomedical engineering at Rutgers University in New Jersey.