On the Design of Model Predictive Setpoint Controllers for Nonholonomic Multibody Systems

ABSTRACT Systems subject to constraints that are not integrable to position‐level constraints pose significant challenges for setpoint control. In particular, conventional linearization techniques may fail to yield controllable approximations, since certain directions of motion are only accessible through higher‐order interactions of the control inputs. For systems subject to nonholonomic kinematic constraints, recent model predictive control (MPC) schemes have successfully addressed this issue by incorporating the underlying sub‐Riemannian geometry into the stage cost design, thereby enabling asymptotically stable setpoint control without terminal ingredients. Many practical systems, however, are naturally described as multibody systems with inertia, gyroscopic effects, and external forces. Underactuated multibody systems that do not exhibit restoring potential forces at the desired equilibrium possess non‐integrable constraints at the acceleration level, extending the notion of nonholonomy beyond the conventional kinematic setting. The present work investigates how recent geometry‐conforming MPC schemes for nonholonomic kinematic systems can be extended to underactuated multibody systems subject to such general non‐integrable constraints. The key idea is to construct a homogeneous approximation of the system dynamics in suitably chosen coordinates and to exploit recent MPC results for homogeneous control systems to derive tailored mixed‐exponents stage costs. The resulting framework provides a blueprint for the design of geometry‐conforming MPC controllers for this class of systems and serves as a first step towards a more systematic design methodology. The effectiveness of the proposed methodology is demonstrated by means of two simulation examples, namely an underactuated surface vessel and an omnidirectional mobile robot equipped with an unactuated joint.