In this paper, we describe a new procedure for constructing robot trajectories. The procedure uses the theory of B-splines. The motion of the robot, as specified by a time sequence of position and orientation knots of the end effector, is first transformed into sets of joint displacements. B-splines are then used to fit these sequences for each joint. The procedure consists of a simple recursive algorithm and allows constraints to be imposed on the magnitudes of the joint velocities and accelerations as well as on their initial and final values. A trajectory constructed by this method has the property that a local modification can be made quickly and easily without recomputing the entire trajectory. The method has been used for generating trajectories for a PUMA 560 robot.

Formulation of Joint Trajectories for Industrial Robots Using B-Splines

The B-spline functions are used to generate the constrained minimum-time path for a robot manipulator. The interpolated path via B-spline functions is determined via a unique set of virtual knots so that the robot path can pass every intermediate knot and satisfy the boundary conditions. The local control property of the B-spline functions can also be utilized to simplify the involved computational complexities. The flexible polyhedron search algorithm is used to generate the near minimum-time path with velocity, acceleration, and jerk constraints on every intermediate point. This approach can simplify the formulation and procedures for generating the near minimum-time cubic B-spline path. The use of this method to generate the constrained minimum-time joint trajectories for a PUMA 560 robot is discussed as an example. >

Constrained minimum-time path planning for robot manipulators via virtual knots of the cubic B-spline functions

Interpolation of a robot joint trajectory is realized using trigonometric splines This method is based on the assumption that joint-space knots have been generated from cartesian knots by an inverse kinematics algorithm The use of trigonometric splines results in smooth joint trajectories and is amenable to real-time obstacle avoidance Some of the spline parameters can be chosen to minimize an objective function (eg jerk or energy) If the objective function is minimum jerk, a closed-form solution is obtained This paper introduces a trajectory interpolation algorithm, discusses a method for trajectory optimization, and includes simulation examples

A trigonometric trajectory generator for robotic arms

Robot motion is commonly specified as a Cartesian trajectory of its end-effector. For executing the end-effector trajectory on-line, a look-ahead of a few points on the trajectory is used to genera...

On-line polynomial trajectories for robot manipulators

Trajectory planning is a classic topic in robotics. Although cubic splines are widely used to interpolate the trajectory in the literature, they have drawbacks in satisfying more boundary conditions for high dynamic performance, say, the control of acceleration at the end points. On the other hand, if quartic splines are employed, more conditions are needed to determine the coefficients of the fourth-order polynomials for the trajectory. In this paper, we present a method to interpolate the trajectory by combining third-order and fourth-order polynomials. We use two quartic polynomials for the first and last segments of the trajectory, and cubic ones for other segments. The trajectory can be uniquely determined by a set of path points for boundary conditions including accelerations at end points. We also investigate the planning of optimal trajectories in the case of variant intervals between adjacent path points. In the optimization, we use multiple objectives including accelerations and jerks. How to determine succinctly the polynomial coefficients is described. Examples are provided for illustration, and comparisons with previous planning method are reported. Good results of application in humanoid robot motion planning for obstacle stepping-over are obtained

On robotic trajectory planning using polynomial interpolations

Methods of joint trajectory generation for mechanical manipulators using X-spline and quartic spline functions are presented. A sequence of Cartesian knots, i.e. positions and orientations of the hand, are selected from the desired Cartesian path. Joint values corresponding to these selected knots are solved from the kinematic equation. Approximate joint paths are constructed by interpolating these joint positions. At any instant, two or more future knots need to be known in advance in order to determine the current joint trajectories. Approximate errors for Cartesian paths are discussed. An illustrative example is used for comparing these trajectories to other existent polynomial-type trajectories.

P. R. Chang

Papers

Joint trajectories of mechanical manipulators for Cartesian path approximation