Brief PaperA phase-plane approach to time-optimal control of single-DOF mechanical systems with friction☆
Introduction
Most mechanical systems are subject to the effect of friction to some extent. Nonetheless, the effect of Coulomb friction was ignored or precompensated in most prior works on the time-optimal control of mechanical systems (Van Willigenburg & Loop, 1991). This is mainly because the conventional Pontryagin's maximum principle (PMP) cannot be applied directly to time-optimal control of mechanical systems with friction, since their dynamic behavior is governed by second-order differential equations with discontinuous right-hand sides. In this context, a version of PMP for optimal multiprocess (Clarke & Vinter, 1989) was employed in Kim and Ha (2001) to time-optimal control of single-degree-of-freedom (DOF) mechanical systems with friction.
In general, the direct application of the PMP leads to a nonlinear two-point boundary-value problem (TPBVP), which cannot be solved analytically. Therefore, it should be solved via sophisticated iterative numerical techniques such as the quasi-linearization and the gradient projection techniques (Kirk, 1970). Usually, these methods require good estimates of the initial trajectory, or may show very poor convergence property (Kirk, 1970). On the other hand, it is well known that the so-called phase-space technique is useful to solve the time-optimal control problem for a specific class of second-order systems (Shin & Mckay, 1985; (Bobrow, Dubosky, & Gibson, 1985).
In this paper, we attempt to solve the time-optimal control problem for single-DOF mechanical systems with friction, while taking into account not only the control input constraint but also the state constraint. Specifically, we take a phase-plane analysis instead of resorting to the PMP. Thereby, The exact time-optimal solution can be obtained simply by solving a set of first-order differential equations with continuous right-hand sides. Hence, it can be constructed numerically via the direct application of the well-known Euler or Runge–Kutta methods. Finally, we present some simulation results to demonstrate the practical use of the time-optimal solution.
Section snippets
Problem statement
The dynamic behavior of a single-DOF mechanical system with friction is governed bywhere m>0 is the mass. Here, denote position, velocity, and control input, respectively. The above system is subject to the following constraints:In what follows, the trajectory of the system in (1) with (respectively, ) passing through a point (x0,v0)∈R2 is called a maximum acceleration trajectory
Construction of switching curves
We begin with the introduction of some notations and definitions needed in our development. Note that the state variables x and v of the system in (1) satisfy the following system:Define the variable y byUsing the variables x and y, we then can write the system in (9) asNote that the right-hand side of the system in (11) is continuous, while that of the system in (9) is not. In this context, our main result will be
Time-optimal trajectory
Recall that the initial state (x0,v0) in the (x,v)-phase plane corresponds to the point (x0,y0) in the (x,y)-phase plane where . We show in the following that there exist some points (x0,y0) in , from which the origin is not reachable without violating the control input constraint in (2) and the state constraint in (3). To see this, we define a subset of aswhere
Simulation results
In this section, we apply the algorithms developed in the preceding section to the time-optimal control of a single-DOF robotic manipulator whose dynamics are given by (1) and (4) with the mass and the friction modelwhere , , , . Here, the function Fs in (14) represents Tustin's static friction model (Armstrong-Hélouvry et al., 1994). In our simulation work, the functions ū, u̱, , and in , that determine the
Conclusion
In this paper, we have solved the time-optimal control problem of single-DOF mechanical systems with friction, while taking into account not only the velocity-dependent control input constraint but also the state constraint. Since the friction effect is neither ignored nor precompensated, the admissible range of the control input can be maximized so as to achieve faster response.
Dong-Soo Choi received the B.S., M.S., and Ph.D. degrees in electrical engineering & computer science from Seoul National University, Seoul, Korea, in 1996, 1998, and 2002, respectively. Currently, he is working as a Senior Research Engineer in the R&D Center, JUSTEK, Inc., Seoul. His current fields of interest includes nonlinear control theory and its applications to high-precision motion control of linear servo motors.
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Dong-Soo Choi received the B.S., M.S., and Ph.D. degrees in electrical engineering & computer science from Seoul National University, Seoul, Korea, in 1996, 1998, and 2002, respectively. Currently, he is working as a Senior Research Engineer in the R&D Center, JUSTEK, Inc., Seoul. His current fields of interest includes nonlinear control theory and its applications to high-precision motion control of linear servo motors.
Seung-Jean Kim received the Ph.D. degree in electrical engineering and computer science from Seoul National University, Seoul, Korea, in 2000. Currently, he is a Postdoctoral Researcher in the Information Systems Laboratory at Stanford University, U.S.A. His current research interests include system and control theory, circuit optimization, and interior point methods for convex optimization.
In-Joong Ha (M-87) received the Ph.D. degree in computer, information, and control engineering (CICE) from the University of Michigan, Ann Arbor, in 1985. He is presently a Professor in the School of Electrical Engineering and Computer Science, Seoul National University, Seoul, Korea. From 1985 to 1986, he worked as a senior research engineer in the General Motors Research Laboratories, Troy, MI. From 1982 to 1985, he was a Research Assistant at the Center for Research on Integrated Manufacturing, University of Michigan, Ann Arbor. From 1973 to 1981, he worked in the area of missile guidance and control at the Agency of Defense Development in Korea. His current research interest includes nonlinear system theory and its applications to servo motors, robots, high-density storage systems, and communication networks. Dr. Ha was the recipient of the 1985 Outstanding Achievement Award in the CICE program. From 2000 to 2002, he served as an associate editor of AUTOMATICA for nonlinear systems.
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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Kenko Uchida under the direction of Editor Tamer Basar.