J. Y. S. Luh

Formulation and optimization of cubic polynomial joint trajectories for industrial robots

Because of physical constraints, the optimum control of industrial robots is a difficult problem. An alternative approach is to divide the problem into two parts: optimum path planning for off-line processing followed by on-line path tracking. The path tracking can be achieved by adopting the existing approach. The path planning is done at the joint level. Cubic spline functions are used for constructing joint trajectories for industrial robots. The motion of the robot is specified by a sequence of Cartesian knots, i.e., positions and orientations of the hand. For an N -joint robot, these Cartesian knots are transformed into N sets of joint displacements, with one set for each joint. Piecewise cubic polynomials are used to fit the sequence of joint displacements for each of the N joints. Because of the use of the cubic spline function idea, there are only n - 2 equations to be solved for each joint, where n is the number of selected knots. The problem is proved to be uniquely solvable. An algorithm is developed to schedule the time intervals between each pair of adjacent knots such that the total traveling time is minimized subject to the physical constraints on joint velocities, accelerations, and jerks. Fortran programs have been written to implement: 1) the procedure for constructing the cubic polynomial joint trajectories; and 2) the algorithm for minimizing the traveling time. Results are illustrated by means of a numerical example.

Optimum Path Planning for Mechanical Manipulators

To assure a successful completion of an assigned task without interruption, such as the collision with fixtures, the hand of a mechanical manipulator often travels along a preplanned path. An advantage of requiring the path to be composed of straight-line segments in Cartesian coordinates is to provide a capability for controlled interaction with objects on a moving conveyor. This paper presents a method of obtaining a time schedule of velocities and accelerations along the path that the manipulator may adopt to obtain a minimum traveling time, under the constraints of composite Cartesian limit on linear and angular velocities and accelerations. Because of the involvement of a linear performance index and a large number of nonlinear inequality constraints, which are generated from physical limitations, the “method of approximate programming (MAP)” is applied. Depending on the initial choice of a feasible solution, the iterated feasible solution, however, does not converge to the optimum feasible point, but is often entrapped at some other point of the boundary of the constraint set. To overcome the obstacle, MAP is modified so that the feasible solution of each of the iterated linear programming problems is shifted to the boundaries corresponding to the original, linear inequality constraints. To reduce the computing time, a “direct approximate programming algorithm (DAPA)” is developed, implemented and shown to converge to optimum feasible solution for the path planning problem. Programs in FORTRAN language have been written for both the modified MAP and DAPA, and are illustrated by a numerical example for the purpose of comparison.

An anatomy of industrial robots and their controls

In the United States, commercially available industrial robots perform very well in limited areas of industrial tasks such as arc welding, paint spraying, etc. These tasks mainly involve synchronization but no task interaction. A close examination of the basic structure and controls of the robots reveals their resulting limitations which lead to unnatural specifications and inefficient performance of task interactions. It is our opinion that, to expand the range of robot tasks to include labor intensive jobs such as product assembly, sensors of multiple purposes must be added onto the robots and integrated into their control systems. Computer command language must be developed to enable nonexpert users to operate the robots, and a work-method must be available for analyzing robot time-motion so that the robots can be programmed to achieve best efficiency with least production cost.

Minimum distance collision-free path planning for industrial robots with a prismatic joint

A collision-free path is a path which an industrial robot can physically take while traveling from one location to another in an environment containing obstacles. Usually the obstacles are expanded to compensate for the body width of the robot. For robots with a prismatic joint, which allows only a translational motion along its axis, additional problems created by the long boom are handled by means of pseudoobstacles which are generated by real obstacles edges and faces. The environment is then modified by the inclusion of pseudoobstacles which contribute to the forbidden regions. This process allows the robot itself again to be represented by a point specifying the location of its end effector in space. An algorithm for determining the shortest distance collision-free path given a sequence of edges to be traversed has been developed for the case of stationary obstacles.

Formulation and optimization of cubic polynomial joint trajectories for mechanical manipulators

Because of physical constraints, the optimum control of industrial robots is a difficult problem. An alternative approach is to divide the problem into two parts: optimum path planning for off-line processing followed by on-line path tracking. The path tracking can be achieved by adopting the existing approach. The path planning is done at the joint level. Cubic spline functions are used for constructing joint trajectories for mechanical manipulators. The motion of the manipulator is specified by a sequence of Cartesian knots, i.e. positions and orientations of the hand. For a 6-joint manipulator, these Cartesian knots are transformed into six sets of joint values, with each set corresponding to a joint. Piece-wise cubic polynomials are used to fit the sequence of joint values for each of the six joints. The problem is proved to be uniquely solvable. Furthermore, an algorithm is developed to schedule the time periods between each pair of adjacent knots such that the total traveling time is minimized subject to the physical constraints on joint velocities, accelerations, and jerks. FORTRAN programs have been written to implement (1) the procedure for constructing the cubic polynomial joint trajectories and (2) the algorithm for minimizing the traveling time. Results are given as an illustration.

An approximate minimal time closed-loop controller for processes with bounded control amplitudes and rates

This paper presents a method for designing an approximate minimal time closed-loop controller for linear systems with bounded control amplitudes and rates. The method is based on obtaining an approximate functional expression (explicitly in terms of the state variables) that describes the minimal time isochrones of the system. This expression is obtained by a series of least-squares fits to the computed system states on the various isochrones. The computation of the system states on the isochrones and the determination of the approximate expressions are achieved off-line. For on-line operation, it is only required to store a limited number of coefficients of the approximate expressions, and to compute the closed-loop control function by some algebraic manipulation. Consequently, the on-line computer storage requirement as well as the on-line computation requirement is relatively small. Thus, the method is feasible for high-order practical systems. To evaluate its usefulness in applications, the scheme is used to design a fourth-order Ranger Block III Attitude Control System. The results are compared to those obtained by applying the minimal time open-loop control.

NEWTON – EULER FORMULATION OF MANIPULATOR DYNAMICS FOR COMPUTER CONTROL

This paper presents a new approach to obtaining the equations by the method of Newton-Euler formulation which is independent of the type of manipulator-configuration. The method involves the successive transformation of velocities and accelerations from the base of the manipulator out to the gripper, link by link, using the relationships of moving coordinate systems. Forces are then transformed back from the gripper to the base to obtain the joint torques. Using this formulation, the amount of computation increases linearly as the number of joints increases while the conventional Lagrangian formulation causes the increase proportional to the quatic of the number of joints. With a program written in floating point language, it is possible to achieve an average execution time of 4.5 milli-seconds on a PDP 11/45 computer for a Stanford manipulator.

Joint torque control by a direct feedback for industrial robots

Two joints of an industrial robot have been redesigned and fabricated to include torque sensing capability by means of strain gauges. The resulting control systems reduced the effective frictional torques of the joints from 1072 oz.-inches to 33.5 oz-inches. The stability of the closed-loop systems was analyzed by means of the describing function for limit cycles exhibited in the system, which can be removed by an insertion of phased-lead series compensating networks.

Construction of the minimal time control for processes with bounded control amplitudes and rates

Minimal time control of linear time-varying processes with bounded control amplitudes and rates is discussed. Based on the maximum principle, necessary conditions are given which include a specific modified adjoint solution. Two additional conditions that are associated with this modified adjoint solution are derived. These conditions are then utilized to develop a systematic method of constructing the extremal control as an explicit time function. The construction scheme consists of a set of decision rules in the form of inequalities. These decision rules are implemented on a digital computer. The computer program basically checks out the decision rules and determines the appropriate set of prestored parameters, which describes each extremal control. To obtain the minimal time control from the extremal controls, a point-by-point search from the grid on a hypercylindrical surface is required. As an illustration, the computer program is applied to the Ranger Block III attitude control system.

Mininum-time along the path for a mechanical arm

I HTROWCT I ON The t r a v e l i n g p a t h of a mechanical manipulator i s canposed o f s t r a i g h t l i n e s e g m n t s i n C a r t e s i a n coordinates connected by smooth arcs. The manipu-l a t o r r e s t s a t i t s i n i t i a l p o s i t i o n b e f o r e it moves, and s t o p s a t i t s f i n a l p o s i t i o n. The j o i n t s o f t h e manipulator are driven by e l e c t r i c a l motors and c o n t r o l l e d by servo systems which are designed for the worst case. In actual operation, the manipula-tor t h u s p e r f o r m below i t s c a p a b i l i t y. I n a d d i-tion, conventional controls are designeA +such a way that the manipulator stops at the end, ne seg-It i s seen t h a t t h e e l i m i n a t i o n o f t h e s t o p a t t h e end o f each l i n e segnent and the use o f t h e maximum v e l o c i t y under physical constraints w i l l allow the manipulator t o move more r a p i d l y and hence the travel ing t i m e i s reduced. on the arc of the actual path from the corner of two adjacent straight line segments must be kept s m a l l enough so that the actual path does not s i g n i f i c a n t l y d e v i a t e from the planned path. Otherwise i t may have a r i s k o f r u n n i n g i n t o obstacles. These …