Abstract
This paper presents an extension of the concept of dynamically consistent Jacobian inverse from robotic manipulators (holonomic systems) to non-holonomic robotic systems, like mobile robots. This new inverse is derived within the framework of the endogenous configuration space approach, following a strict analogy with the original derivation of the dynamically consistent Jacobian inverse for holonomic systems. The analogy is founded on replacing a finite-dimensional configuration space of the manipulation robot by the space of control functions steering the non-holonomic system. Consequently, a curve in the space of control functions corresponds to the manipulator’s trajectory in the configuration space, whereas endogenous velocities and forces are defined as elements of the tangent and cotangent spaces to the control space. Three ways of introducing the dynamically consistent Jacobian inverse are proposed, referred to as the geometric method, the force method, and the optimization method. A crucial concept underlying all these methods is a Riemannian metric in the space of control functions of the non-holonomic system as well as in its operational space. It has been shown that, similarly as for holonomic systems, the dynamically consistent Jacobian inverse obtained prevents the transmission of certain internal forces acting in the system from the endogenous configuration space to the operational space. This property is illustrated with the example of the Pioneer 2DX mobile platform. Performance of the new Jacobian inverse is demonstrated in the context of motion planning of the rolling ball.
Highlights
The concept of a dynamically consistent Jacobian inverse for manipulation robots was founded by Khatib, see [11,12]; it is known as the inertia weighted Jacobian pseudoinverse [10,15]
The dynamically consistent Jacobian inverse serves as a tool for solving the inverse kinematics or the motion planning problem in the way that respects some basic requirements of the force transmission from the configuration to the operational space of the robot
In order to further demonstrate the performance of a non-holonomic system driven by the dynamically consistent Jacobian inverse, we shall solve an example motion planning problem for the rolling ball, using the dynamically consistent inverse Jq#0D,TC and the Jacobian pseudoinverse Jq#0P,T
Summary
The concept of a dynamically consistent Jacobian inverse for manipulation robots was founded by Khatib, see [11,12]; it is known as the inertia weighted Jacobian pseudoinverse [10,15]. An incorporation of the constraint consistent inertia matrix into a weighted Jacobian inverse for non-holonomic wheeled-based mobile manipulators has been proposed in [16,25]; an inherent flaw of such an approach in the context of general nonholonomic systems was revealed in [26] In this last reference, the endogenous configuration space approach was suggested as an alternative. For completeness and clarity of presentation, this paper begins with three methods of derivation of the dynamically consistent Jacobian inverse for holonomic robots All these methods rely on Riemannian metrics in the configuration space and in the operational space, defined by the inertia matrix of the robot dynamics. The inertia matrix computed along the system’s trajectory defines a Riemannian metric in the endogenous configuration space In this setting, the derivation of the dynamically consistent Jacobian inverse for a non-holonomic system is patterned on the derivation performed for the holonomic systems.
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