We employ a simplified linear mathematical model to describe the ballbot's dynamics. Rudolf Kalman, a Hungarian-American mathematical system theorist, invented in 1960 an elegant method for deriving control policies for such systems, which he called the linear quadratic regulator. This approach considers the measurements of the system's internal states to be proportional to the values of the states themselves. Further, it assumes that the states change over time at a rate proportional to the values of the states plus a proportional contribution of any control actions that might occur, such as motor torques. Kalman's technique cleverly minimizes an integral function over time that includes a quadratic measure of the states plus a quadratic measure of the control actions. Its solution yields a final set of constants, which, when multiplied by each of the internal states, gives a recommended, or optimal, control action for the ballbot to take at each moment in time. These calculations run several hundred times a second in the ballbot's main computer.
When the ballbot's goal is to stand still, its control policy tries to simultaneously drive the body's position and speed, as well as its tilt and tilt rate, to zero in each direction, while minimizing the actions needed to do so. When its objective is to go from one place to another, the control policy automatically institutes a retrograde ball rotation to establish a body tilt, allowing it to accelerate forward. As the goal position is approached, the ball automatically speeds up to reverse the tilt and bring the ballbot to rest [see box above].
WE HAVE BEGUN to experiment with the ballbot, interacting with it over a wireless radio link. We plan to add a pair of arms, as well as a head that pans and tilts, with a binocular vision system and many other sensors, in an effort to develop the machine into a capable robot with a significant degree of autonomy. Our goals are to understand how well such robots can perform around people in everyday settings and to compare quantitatively its performance, safety and navigation abilities with those of traditional, statically stable robots. Our hypothesis is that the latter may turn out to be an evolutionary dead end when it comes to operating in such environments.
We are not alone in betting on the notion of dynamically stable robots. Other research groups have produced two-wheeled robots that are dynamically stable in the pitch direction but statically stable in the roll orientation. Although these robots are not omnidirectional like a ballbot is, they show promise for agile mobility—especially outdoors.
It may turn out that dynamically stable biped robots, perhaps in humanoid form, will have the long-term edge—particularly for their ability to deal with stairways. Research teams worldwide are working intensively to develop these complex and often expensive machines. Meanwhile it would seem that ballbots will serve as interesting and effective platforms for studying how mobile robots can interact dynamically and gracefully with humans in the places where people live.