Off-line programming of six-axis robots for optimum five-dimensional tasks

• A novel approach to functional-redundancy resolution is introduced.
• A detailed derivation of the gradient and Hessian of the condition number is provided.
• An example shows the practicality of our approach to functional-redundancy resolution.

Five-degree-of-freedom (five-dof) tasks are of particular interest in industry, since machining, arc-welding and deburring operations all fall into this category. These tasks, normally conducted with industrial six-dof robots, render the robot functionally redundant. Upon exploiting this redundancy to minimize the condition number of the Jacobian matrix, it is expected that the accuracy of the performed task will be increased. Traditional methods for redundancy-resolution are normally used to solve the more frequent intrinsic redundancy; however, they are not applicable to functional redundancy. Five-dof tasks are formulated using an approach that leads to a system of six velocity-level kinematics relations in six unknowns, with a 6 × 6 Jacobian matrix, of nullity 1, which reflects the functional redundancy of the problem at hand. To resolve the foregoing redundancy, a method based on sequential quadratic programming (SQP) is proposed. A novel method to compute the gradient of the condition number is also discussed, as it is a key element for finding the posture of minimum condition number using a gradient method. An example then shows how the SQP algorithm can be applied to offline robot trajectory-planning for five-dof tasks. In this example, a comparison is also made between the quasi-Newton and the Newton–Raphson methods to find the posture of minimum condition number for the robot. This is an essential step in finding the trajectory of minimum condition number.