Engineering G2.5 Neural networks for optimal robot trajectory planning

This case study discusses the interpolation of minimum-jerk robot joint trajectories through an arbitrary number of knots using a hard-wired neural network. Minimum-jerk joint trajectories are desirable for their similarity to human joint movements and their amenability to accurate tracking. The resultant trajectories are numerical functions of time. The interpolation problem is formulated as a constrained quadratic minimization problem over a continuous joint angle domain and a discrete time domain. Time is discretized according to the robot controller rate. The outputs of the neural network specify the joint angles (one neuron for each discrete value of time) and the Lagrange multipliers (one neuron for each trajectory constraint). An annealing method is used to prevent the network from getting stuck in a local minimum. We show via simulation that this trajectory planning method can be used to improve the performance of other trajectory optimization schemes. G2.5.1 Project overview G2.5.1.1 Robot trajectory planning The industrial robot is a highly nonlinear, coupled multivariable system with nonlinear constraints. For this reason, robot control algorithms are often divided into two stages: path planning and path tracking (Craig 1989). Path planning is often done without much consideration for the robot dynamics, and with simplified constraints. This reduces the computational expense of the path planning algorithm. The output of the path planning algorithm is then input to a path tracking algorithm. There are algorithms for the robot control problem which do not separate path planning and path tracking. These algorithms take source and destination Cartesian points as inputs, and determine optimal joint torques. Shiller and Dubowsky (1989) provide a concise review of such algorithms. While such methods are attractive in that they provide optimal solutions to some robot control problems, they result in impractically complicated algorithms and a large computational expense. A simpler approach to the robot control problem is to generate a suboptimal joint trajectory, and then track the trajectory with a controller. This approach ignores most of the dynamics of the robot. So the resultant trajectories do not take full advantage of the robot’s capabilities, but are computationally much easier to obtain. In this approach, a number of knot points are chosen along the desired Cartesian path. The number of knots chosen is a tradeoff between exactness and computational expense. The Cartesian knots are then mapped into joint knots using inverse kinematics. Finally, for each robot joint, an analytic interpolating curve is fit to the joint knots. Some of the initial and final derivatives of the curve are constrained to zero so as to ensure that the robot begins and ends its motion smoothly. ‘Smoothness’ is a concept which combines the ideas of derivative continuity and derivative magnitudes. The most popular type of interpolation is algebraic splines (Lin and Chang 1983, Lin et al 1983, Thompson and Patel 1987). Higher-order splines result in continuity of higher-order derivatives, which reduces wear and tear on the robot (Craig 1989) but this is at the expense of large oscillations of the c © 1997 IOP Publishing Ltd and Oxford University Press Handbook of Neural Computation release 97/1 G2.5:1 Neural networks for optimal robot trajectory planning trajectory. Trigonometric splines can be used to provide a less oscillatory interpolating curve (Simon and Isik 1993). G2.5.1.2 Motivation for a neural solution Consider a sequence of knots through which an interpolating curve is required to pass. A human could create an interpolating curve, but in a different way than a computer algorithm would. Computer algorithms can calculate analytic functions which pass through given knots. A human can draw a smooth curve through a given set of knots, but without performing any mathematical calculations. In contrast with the computer algorithm, the interpolating curve drawn by the human would not be an analytic function of time. In addition, the human would not satisfy the constraints exactly, but only approximately. For example, if the human was requested to maintain a zero slope at the endpoints, the resulting slope would not be zero, but would be very small. Such a result would be satisfactory for most robot path planning applications. These facts indicate that an artificial neural network may be able to do well at interpolation. Of course, artificial neural networks are still quite far from any biological neural networks. Further motivation for seeking a neural solution to the robot trajectory optimization problem is obtained from the possibility of implementation in parallel hardware. This would give the advantage of quick solutions to E1.2 large problems which would not otherwise be practical using more conventional optimization methods. The robot path planning problem can be viewed as an optimization problem: given a desired set of knots and endpoint constraints, find the ‘best’ interpolating curve such that the knot errors and endpoint derivatives are not too ‘large’. Several researchers have solved continuous optimization problems using neural networks (Zhao and Mendel 1988, Jeffrey and Rosner 1986a, b, Jang et al 1988). Platt and Barr (1988) formulate a neural network which can calculate a minimum of a general function subject to inequality or equality constraints. Their network has the important property of local stability for the problem considered in this section. Due to its stability and generality, this is the network which is used to determine a minimum-jerk robot joint path through a given set of knots. In order to plan an optimal robot trajectory, the measure of optimality must be defined. Human arm movements satisfy some optimality criterion, and this would seem to be a desirable criterion to adopt when planning trajectories for robot arms. Flash and Hogan (1985) suggest that human arm movements minimize a measure of Cartesian jerk, while Flanagan and Ostry (1990) present evidence that a function of joint jerk is minimized. Uno et al (1989) and Kawato et al (1990) argue that the objective function is a measure of the derivative of the joint torques, and propose a neural network to learn such a trajectory. In this section, a joint jerk objective function is used. While this choice ignores the dynamics of the robot, it reduces the error of the path tracker (Kyriakopoulos and Saridis 1988) and thus is suitable for robotics applications. G2.5.2 Design process