Abstract:

Robotic manipulators are kinematically redundant when they have more degrees of freedom than those strictly necessary to complete a task. For example, a planar manipulator with three degrees of freedom becomes redundant if its main task or objective is to control the position of its gripper in the plane, neglecting its orientation. Kinematic redundancy can be exploited to achieve secondary goals besides the main task, for instance: to avoid obstacles or singularities, to maximize dexterity, to minimize energy, etc. This grants redundant manipulators greater flexibility, which is possible because, for redundant manipulators, a given task can be solved by infinitely many different joint displacements. This is in contrast to non-redundant manipulators, for which a given task can be achieved by only a finite number of joint displacements.

The problem of redundancy resolution consists in deciding a criterion to pick one of these infinitely many possible solutions of joint displacements that achieve the desired main task. Typically, when teaching redundancy resolution in undergraduate and master’s studies, this problem is addressed at velocity level, i.e., the velocity equation of the manipulator is pseudo-inverted to solve the joint velocities as functions of the task velocities, and then these joint velocities are numerically integrated in order to obtain the joint displacements that need to be applied to the manipulator to complete the task. This velocity-based approach is widely adopted mainly because of its simplicity; however, it exhibits deficiencies that can be addressed by alternative position-based approaches such as the feasibility maps, the vanishing self-motion manifolds, or the more recent differentiable manifold formulation.

Regardless of the approach followed, teaching and learning the kinematics of redundant manipulators is not easy because of the infinitude of joint displacements that solve a given task. When following more traditional velocity-based methods, one has to deal with pseudo-inversion of Jacobian matrices and projection of the gradients of secondary goals on the null space of these Jacobians. When using more recent position-based methods, it is necessary to map and explore the manifolds that contain these infinitely many solutions. In both cases, the problem can be understood much more easily if graphical representations of the involved computations are available. For this reason, in this paper we present an interactive and graphical simulation addressed to facilitate the teaching and learning of the redundancy resolution problem of redundant robotic manipulators.

The presented simulator consists of a simulated planar parallel robot with three degrees of freedom whose main task is to control the position of its gripper in the plane, which makes this robot redundant. For a position of the gripper specified by the student, the simulator represents graphically the self-motion manifolds formed by all the joint displacements that allow the robot to reach the said position. For a desired trajectory of the gripper defined by the student, the simulator represents the associated feasibility maps and illustrates the transformations of self-motion manifolds along the trajectory. Also, the students can try velocity-based solutions to redundancy resolution and analyze how they relate to position-based approaches, facilitating the comprehension of redundancy resolution from both position and velocity-based perspectives.

Keywords:

Simulation, interactive, robotic manipulator, redundancy resolution, manifold.