Minimum‐time control of systems with Coulomb friction: near global optima via mixed integer linear programming

Abstract

SUMMARYThis work presents a method of "nding near global optima to minimum-time trajectory generation problems for systems that would be linear if it were not for the presence of Coulomb friction. The required "nal state of the system is assumed to be maintainable by the system, and the input bounds are assumed to be large enough so that the role of maintaining zero acceleration during "nite time intervals of zero velocity (the role of static friction) can always be assumed by the input. Other than the previous work for generating minimum-time trajectories for robotic manipulators for which the path in joint space is already speci"ed, this work represents, to the best of our knowledge, the "rst approach for generating near global optima for minimum-time problems involving a non-linear class of dynamic systems. The reason the optima generated are near global optima instead of exactly global optima is due to a discrete-time approximation of the system (which is usually used anyway to simulate such a system numerically). The method closely resembles previous methods for generating minimum-time trajectories for linear systems, where the core operation is the solution of a Phase I linear programming problem. For the non-linear systems considered herein, the core operation is instead the solution of a mixed integer linear programming problem.

An iterative method for time optimal control of dynamic systems

“…Here, a double springmass system with coulomb friction, which was investigated in [3] using a linear programming method, is considered. A schematic image of system is illustrated in Fig.…”

“…The physical parameters, initial conditions and final conditions of system are taken from [3] and are noted in Tab. 1.…”

Optimal control of systems with discontinuous differential equations

“…From the third branch in (9), the problem of following the trajectory in the discontinuity should be expressed as…”

“…Lemma 1 is useful for showing that the smoothed right-hand side f σ in (17) also satisfies a one-sided Lipschitz condition, although the Lipschitz constant might not be the same as for (9). To show this, we do need to assume that f 1 and f 2 satisfy an ordinary ("two-sided") Lipschitz condition with constant L f and that ∇ψ is also Lipschitz with constant L ∇ψ .…”

Rest-to-Rest Motion of an Experimental Flexible Structure subject to Friction: Linear Programming Approach

“…A more recent approach poses the problem in a mixed integer linear programming setting, in order to accommodate for the friction sign change for the two-mass harmonic dynamics [14]. This is computationally expensive which precludes fine discretization of the maneuver time.…”