Determining trajectories of redundant actuators jointly tracking reference trajectory

A method for controlling concurrently a first actuator and a second actuator of a machine, comprising:

• determining an inequality constraint on an operation of the first actuator as an error tolerance on tracking a reference trajectory, wherein the error tolerance is less than an operational ramie of the second actuator;
• solving a constraint optimization problem subject to the inequality constraint for determining a first trajectory of the first actuator tracking the reference trajectory with the error tolerance, such that the first trajectory is optimized according to a metric of performance;
• determining a second trajectory of the second actuator based on a difference between the first trajectory and the reference trajectory;
• determining a time sequence synchronizing an operation of the first actuator according to the first trajectory with an operation of the second actuator according to the second trajectory;
• and
• controlling concurrently the first actuator and the second actuator according to the first trajectory, the second trajectory, and the time sequence, wherein the determining the first trajectory comprises:
• determining a cost function representing an operation of the first actuator tracking the reference trajectory;
• and
• optimizing, according to the metric of performance, the cost function subject to the inequality constraint imposed by the error tolerance and an equality constraint, wherein the inequality constraint includes a position constraint, a velocity constraint, an acceleration constraint, and a torque constraint of the second actuator, and wherein the equality constraint includes a kinematic constraint on a sum of position contributions resulting from operations of the first and the second actuators.

A trajectory generator for determining trajectories of redundant actuators of a machine comprising a processor for executing a set of modules, the set of modules comprising:

• a path sampling module for sampling a reference trajectory to produce a set of points;
• a constrained optimization module for determining, based on the set of points, a first trajectory of a first actuator, such that the a curvature and a length of the first trajectory is minimized without violation of an operational constraint of a second actuator, and for determining a second trajectory of the second actuator based on the first and the reference trajectories, wherein the operational constraint is satisfied when a shortest distance between a point on the first trajectory and the reference trajectory is greater than an operational range of the second actuator, wherein the constrained optimization module determines a cost function representing an operation of the first actuator tracking the reference trajectory and solves a constraint optimization problem of operations of the first and the second actuators optimizing the cost function subject to an inequality constraint on the operation of the first actuator and an equality constraint on the operations of the first and the second actuators, wherein the inequality constraint includes a constraint on an operation ramie of the second actuator, and constraints on a position, a velocity, an acceleration, and a torque of the second actuator, and wherein the equality constraint includes a kinematic constraint on a sum of position contributions resulting from operations of the first and the second actuators;
• and
• a time assignment module for assigning time stamps for points on the first and the second trajectories, such that motions along the first and the second trajectories are synchronized without violation of operational constraints of the redundant actuators.