Speaker
Description
Precise setpoint control of formations of differential-drive robots poses significant challenges. Here, formation control refers to the cooperative control of the formation's centroid and the relative postures of the individual robots. Even for a single nonholonomic robot, controller design is challenging as no continuous, static state-feedback law can asymptotically stabilize a given setpoint. This also complicates deriving stabilizing terminal ingredients for model predictive controllers tailored to nonholonomic systems. Moreover, an MPC without terminal ingredients subject to conventional quadratic costs provably fails for certain initial conditions. This arises from the geometry of nonholonomic systems such that the MPC cost must align with the so-called sub-Riemannian distance -- an actual measure of the effort required to drive the robot to its origin.
Using these insights, a stabilizing MPC controller can be designed using a mixed-exponents cost function, which, close to the setpoint, penalizes deviations in hard-to-control directions more, i.e., in those directions following from interleaving direct input vector fields. From a differential-geometric perspective, the mixed-exponent pseudo-norm bounds the sub-Riemannian distance to the origin and follows from the nonholonomic system's homogeneous approximation, a first-order approximation retaining controllability. This tailored MPC controller is probably capable of stabilizing several nonholonomic vehicles as well as formations of robots, also in real-world experiments.
However, up to this point, only formation setpoints in which all robots shall have the same goal orientation have been considered. Stabilizing setpoints with distinct robot orientations is an open question. In such cases, robots do not share mutual hard-to-control directions in a common frame of reference but still have mutual input directions. Consequently, close to the setpoint, a robot cannot correct its relative error by solely using direct input vector fields without simultaneously altering the formation's center in hard-to-control directions. As a result, the formation is not capable of approaching its desired setpoint by driving along well-controllable directions, which has been the intrinsic key characteristic of the tailored mixed-exponent cost function. Using polar coordinates offers a promising solution to retain this characteristic to also stabilize formation setpoints with distinct robot orientations. Particularly, close to the setpoint, a single robot can maneuver without altering the formation's center in hard-to-control directions. As a downside, this approach requires special attention to constraint satisfaction, as the system's inputs are no longer linear combinations of the robots' motor velocities. Beyond differential-drive robots, the approach's potential to govern generic setpoints of other nonholonomic vehicles, such as car-like robots, is investigated.
