Model Predictive Control of Vehicle Maneuvers with Guaranteed Completion Time and Robust Feasibility Arthur Richards 1 and Jonathan P. How 2 Abstract A formulation for Model Predictive Control is pre- sented for application to vehicle maneuvering problems in which the target regions need not contain equilib- rium points. Examples include a spacecraft rendezvous approach to a radial separation from the target and a UAV required to fly through several waypoints. Pre- vious forms of MPC are not applicable to this class of problems because they are tailored to the control of plants about steady-state conditions. Mixed-Integer Linear Programming is used to solve the trajectory op- timizations, allowing the inclusion of non-convex avoid- ance constraints. Analytical proofs are given to show that the problem will always be completed in finite time and that, subject to initial feasibility, the opti- mization solved at each step will always be feasible in the presence of a bounded disturbance. The formula- tion is demonstrated in several simulations, including both aircraft and spacecraft, with extension to multiple vehicle problems. 1 Introduction This paper extends Model Predictive Control (MPC) to applications in vehicle maneuvering problems. MPC is a feedback control scheme in which a trajectory op- timization is solved at each time step [5]. The first control input of the optimal sequence is applied and the optimization is repeated at each subsequent step. Because the on-line optimization explicitly includes the operating constraints, MPC can operate closer to con- straint boundaries than traditional control schemes [5]. The resulting efficiency gains have made MPC popu- lar for process control. In the field of aerospace con- trol, MPC has been used to stabilize an aerodynamic system [13] and for spacecraft maneuvers [14]. Other work [1] has used the combination of MILP and MPC to stabilize general hybrid systems around equilibrium points. The first innovation of this work is the formulation of MPC with general terminal constraints. When MPC is used for steady-state control, terminal constraints are applied to ensure stability. It has been shown [6] that stability depends on the terminal state lying in an invariant set. That is, a feasible feedback control law may be identified that would keep the state within that 1 Space Systems Laboratory, Massachusetts Institute of Tech- nology, Cambridge MA 02139 arthurr@mit.edu 2 Associate Professor, MIT jhow@mit.edu Fig. 1: Control System Overview set once it had been entered. For maneuvering prob- lems, it is natural that the terminal set be the target region for the maneuver. However, it is restrictive to enforce this set to be invariant. For example, a space- craft rendezvous may require the chaser vehicle to be at a certain position relative to the target, moving to- wards it with a certain velocity. All of these quantities are specified with some tolerance, forming a target re- gion in the state space. When the chaser enters that region, a physical connection would be made and the maneuver is complete. However, since the target ve- locity is non-zero, the region is not invariant. The for- mulation presented in this paper replaces the notion of stability with completion : given an arbitrary target set in the state space, the system will reach the set in finite time. Fig. 1 shows an overview of the control scheme pro- posed in this work. As well as the dynamic states x and inputs u, a finite state machine is included in the system model, with discrete state y and input v. The state y = 0 implies that the maneuver has been com- pleted, y = 1 otherwise. In the model, constraints on v couple the continuous and discrete systems such that the transition y → 0 can only occur when the vehicle states x are in the prescribed target region. MPC acts upon the combined, hybrid system to drive the com- bined state (x,y) to the terminal set y = 0 in finite time. The method can be extended to include multiple vehicles and multiple targets, allowing the scheme to be used for UAV maneuvers combining trajectory control and waypoint assignment [10]. The new formulation extends existing MPC methods. Previous work [16] has shown that allowing the horizon length to vary leads to finite-time completion. For the