An algorithm is presented for generating near-time-optimal trajectories for an open-kinematic-chain manipulator moving in a cluttered workspace. This is the first algorithm to guarantee bounds on the closeness of an approximation to a time-optimal trajectory. The running time and space required are polynomial in the desired accuracy of the approximation. The user may also specify tolerances by which the trajectories must clear obstacles in the workspace to allow modeling of control errors.>

The authors develop simple, configuration-independent bounds for the Jacobian and the higher order derivatives of the forward kinematic map of a manipulator with revolute and prismatic joints. In doing so they obtain: a simple formula for the higher order derivatives of the forward kinematic map; an easily computable and meaningful metric on the group of rigid-body motions; and useful norms on the joint space of a manipulator. The bounds on the forward kinematic map are practical to implement and have proved to be very useful for trajectory interpolation, space generation, and interpolation of the robot dynamics.>

An algorithm for interpolating workspace trajectories in the joint space of a manipulator having revolute and prismatic joints is described. This algorithm relies on bounds on the derivatives of the kinematic map of such a manipulator derived by the authors (ibid., vol.3, p.1744-9). The algorithm generates interpolates such that user-specified tolerances on position, velocity and acceleration are maintained. This is the first joint space interpolation algorithm to guarantee workspace error bounds on velocity and acceleration. The algorithm includes the prismatic case, which is also novel, and it takes into account factors beyond kinematic errors, making it a practical addition to a robotic manipulator system.>

The authors develop simple configuration-independent bounds on the dynamics of a robot manipulator. In particular, the inertia and gravity tensors and their derivatives of all orders for open-kinematic-chain manipulators. The bounds are useful in dynamics interpolation bounding the Coriolis and centrifugal forces (since these depend on the derivatives of the inertia tensor), suboptimal control, and path planning with dynamical constraints. These bounds can be used for verifying that trajectories satisfy actuator constraints. The bounds on the derivatives of the inertia tensor bound Coriolis and centrifugal forces and determine whether or not they can be ignored along a trajectory. The bounds on the derivatives also permit reliable interpolation of feedforward terms of dynamic compensation in control. In addition, they may also be used to simplify convergence proofs in nonlinear control schemes where detailed estimates of the dynamics are required, and they are useful in a variety of planning and optimization problems to prove the correctness of algorithms.>