Mathematical Background
Long-Exposure Shot of the Robot Juggling an Oversize Table-Tennis Ball.
How do you build a robot that can juggle a ball without sensing? In a perfect world, it would not be hard to build: the robot would strike the ball with perfect motion and timing, the ball would be perfectly round, and it would bounce along an identical path to exactly the same height each time.
The world is not perfect, however. Balls that appear round to the eye are actually covered with tiny manufacturing defects, and even the most accurate mechanisms cannot repeat the same motion twice. The result of this imperfect world is that balls do not bounce along a perfect path – they spin, they bounce too high, too low or sideways. In other words, they deviate.
Dealing with Deviations
Because the Blind Juggler has no cameras, no microphones or any other sensor to measure the deviations it cannot actively correct the trajectory of the ball. We have to find a design of the robot that automatically compensates for the deviations. For example, if the ball bounces up too high, the design has to force the ball to bounce lower again. The goal of the mathematical analysis is to find such a design.
Designing a Stable System
In technical terms, a system that automatically compensates for deviations is called a stable system. We have two parameters at our disposal to stabilize the juggling system: 1) the motion and 2) the shape of the paddle. The goal of the mathematical analysis is find a stabilizing pair of parameters.
The principle of the analysis is simple: After calculating the ideal trajectory of the ball, we add small deviations to the ideal trajectory (see video below). We then use the physical laws of free fall and impact to model how the deviations develop over a series of bounces. If the deviation are getting larger, the system is not stable. Using this measure we now can search a range of motions and shapes to get a stable design. Below, we explain the parameters and values we found using the analysis.
The green disk represents the ideal trajectory: Bouncing in the center of the paddle and exactly 0.8 meters high. The red ball represents the perturbed trajectory: The ball starts quite a bit too high and too far to the right (0.1 meters to be precise). Over a couple of bounces, the error in the perturbed ball's trajectory becomes smaller until the two trajectories are almost identical. This is exactly what our calculations predicted. Note that there is no noise present in this simulation.
Higher Quality Video Available Here
1) The Motion
Fig. 2: Illustration of the Trajectory Stabilization in the Vertical Direction
The analysis shows that the paddle needs to be decelerating as it strikes the ball. We can intuitively understand why this makes sense: If the ball bounces too high, it takes longer to return to the paddle; since the paddle is slowing down, the paddle has a lower speed at impact and consequently, the ball bounces less high. The opposite is also true: when the ball is too low, it hits the paddle again earlier, so the ball is struck harder and the ball reaches a higher apex again. Therefore, the motion of the paddle forces the ball to maintain a consistent height (Click on Fig. 2 to see all those cases illustrated). In the robot section, you may find experimental results showing that the ball's apex height is indeed very consistent.
Fig. 3: Illustration of the Trajectory Stabilization in the Horizontal Directions
2) The Shape
We find that the ball's trajectory is stabilized in the horizontal directions by a slightly concave paddle shape. The slight curvature of the paddle keeps the ball in the paddle's center and prevents the ball from falling off. This is illustrated in Fig. 3: The further the ball hits the paddle away from the center, the stronger the ball is pushed inwards again. We find in our analysis that the flatter the plate, the higher we should be able to juggle. We manufactured two paddles of which one is a bit flatter than the other one: The prediction our mathematical model makes is indeed true: with first paddle, we can juggle up to about 1.4 meters before the ball falls off; with the second flatter paddle we can go up to 2.1 meters (see video section, under "High Bounces").
