Dsc-17a-2 Smooth and Time-optimal Trajectory Planning for Industrial Manipulators along Specified Paths

This paper presents a method for determining smooth and time-optimal path-constrained trajectories for robotic manipulators. The desired smoothness of the trajectory is imposed through limits on the actuator jerks. The third derivative of the path parameter with respect to time, the pseudo-jerk, is the controlled input. The limits on the actuator torques translate into state-dependent limits on the pseudo-acceleration. The time-optimal control objective is cast as an optimization problem by using cubic splines to parameterize the state space trajectory. The optimization problem is solved using the exible tolerance method. INTRODUCTION The need for increased productivity in path-following industrial robotic applications has been addressed in the literature by determining path-constrained time-optimal motions (PCTOM) while accounting for actuator torque limits; see (Bobrow, 1985)[1], (Pfei er, 1987)[9], (Shiller, 1992)[13]. In these formulations, the joint actuator torques are the controlled inputs and the open loop control schemes result in bang-bang or bang-singular-bang controls (Chen, 1989)[2]. PCTOM trajectories compute the maximum velocity achievable at the robot tip while still following the prescribed path. However, implementation of PCTOM in physical manipulators has drawbacks, such as joint oscillations due to nite joint sti ness and overshoot of the nominal torque limits due to unmodelled actuator dynamics. The resultant extra strain on the robot actuators could cause them to fail frequently (Li, 1997)[6], reducing the productivity of the entire workcell. At the trajectory planning level, a number of di erent techniques have been devised to address the problem of discontinuous actuator torques. A modi ed cost function, such as time-joint torques (Pfei er, 1987)[9] or time-square of joint torques (Shiller, 1994)[11], smoothes the controls and helps to improve the tracking accuracy, at the expense of motion time. Another way of smoothing the controls is to parameterize the path by using functions that are at least C continuous. Cubic splines used for path parameterization with time as the cost function (Lin, 1983)[7] result in trajectories that have continuous joint accelerations. However, the limits on the joint variables are very conservative, since they need be constant over the entire space and, therefore, are chosen as the lowest maximum. Incorporating the actuator dynamics in this problem formulation (Tarkiainen, 1993)[14] transforms the actuator voltages into the limited controlled inputs. The optimal trajectory is bang-bang in the new controls and the actuator torques are no longer limited. Also, the case of singular controls is not considered since they can be avoided by an appropriate selection of the path (Shiller, 1992)[13] or by convexifying the set of admissible controls (Shiller, 1994)[10]. In this paper, a method is presented for determining time-optimal trajectories subject to actuator torque and jerk limits. The resulting trajectories will be called smooth path-constrained time-optimal motions (SPCTOM) to distinguish them from the path-constrained time-optimal motions (PCTOM), which do not consider jerk limits. The actuator jerk limits are imposed in order to plan feasible optimal motions for industrial applications, which, more and more, use newer light direct drive manipulators. Moreover, in industrial applications, an accurate robot model is likely not readily available and the controller strategy cannot be modi ed by the user. In such cases, unlimited jerks can cause severe vibrations in the arm possibly leads to the failure of the actuators themselves. The SPCTOM trajectories lter the jerks from the trajectory at the planning level, thus leaving more authority to the tracking controller to compensate for disturbances. Potentially, they could also compensate for larger modelling errors. Motion planning problems such as obstacle avoidance are beyond the scope of this paper, since the path is assumed to be preimposed. Path-constrained motions are speci c to contour following applications. Typically, they are also used in point-to-point motion planning in cluttered environments (Shiller, 1989)[12], when the geometric constraints (obstacles, joint limits) are satis ed by the path planner and the dynamic constraints are left to the trajectory planner. PROBLEM FORMULATION The basic problem The problem of smooth path-constrained time-optimal motion (SPCTOM) planning can be stated as follows: min _ T2 J = Z tf 0 1dt, (1) subject to the manipulator dynamics: M(q) q+ _ qC(q) _ q +G(q) = T, (2) the boundary conditions: q(0) = q0 ; q(tf ) = qf ; _ q(0) = _ q(tf ) = 0 ;  q(0) =  q(tf ) = 0, (3) the path constraints: r = r(s), (4) the actuator torque limits: Tmin T Tmax, (5) and the actuator jerk limits: _ Tmin _ T _ Tmax, (6) where n is the number of degrees of freedom of the manipulator. Furthermore, q 2 R is the vector of joint positions, T 2 R is the vector of actuator torques, _ T 2 R is the vector of actuator jerks, M(q) 2 R n is the inertia matrix of the manipulator, C(q) 2 R n n is a third order tensor representing the coe cients of the centrifugal and Coriolis forces, G(q) 2 R is the vector of gravity terms, and r 2 R is a C continuous curve parametrized by s, which may be, for example, the arc length. To simplify the dynamics, viscous and static friction terms have been neglected. In the above formulation, the actuator jerks represent the bounded controls. However, these controls would later have to be integrated in order to derive the actual system inputs in terms of desired actuator torques. Since the Lagrangian form of the robot dynamics incorporates only the actuator torques, the third order dynamics is required. Di erentiation of (2) with respect to time results in: M(q) ... q + _ M(q) q+  qC(q) _ q + _ q _ C(q) _ q + _ qC(q) q + _ G(q) = _ T. (7) Equation (7) is taken as the dynamics of the system, with _ representing the n-dimensional bounded controls. System dynamics for path-constrained motions The dynamic system described by Equation (7) has 3n degrees of freedom. However, the path constraints (4) parameterize the tip position by a single parameter s and reduce the order of the system to 3. By expressing the joint positions, velocities, accelerations, and jerks as functions of the path parameter s, the actuator torque and jerk limits are transformed into limits on  s, the pseudo-acceleration, and limits on the ... s , the pseudo-jerk, respectively: Tmin A(s) s+ B(s) _ s 2 + C(s) Tmax (8) _ Tmin a(s) ... s + b(s) _ s s+ c(s) _ s + d(s) _ s _ Tmax, (9) w h er e: A (s ) = M q 0 , (1 0 ) B (s ) = M q 00 + q 0 T C q 0 , (1 1 )