Abstract
This paper presents an extension of impedance control of robots based on fractional calculus. In classical impedance control, the end-effector reactions are proportional to the end-effector position errors through the stiffness matrix K, while damping is proportional to the first-order time-derivative of the end-effector coordinate errors through the damping matrix D. In the proposed approach, a half-derivative damping is added, proportional to the half-order time-derivative of the end-effector coordinate errors through the half-derivative damping matrix HD. The discrete-time digital implementation of the half-order derivative alters the steady-state behavior, in which only the stiffness term should be present. Consequently, a compensation method is proposed, and its effectiveness is validated by multibody simulation on a 3-PUU parallel robot. The proposed approach can be considered the extension to MIMO robotic systems of the PDD1/2 control scheme for SISO mechatronic systems, with potential benefits in the transient response performance.
Highlights
In a wide range of robotic applications, for example, assembly of electronic boards or handling of objects to be placed on horizontal pallets, the full mobility (6-DOF) of the end-effector is not necessary, since the tasks can be proficiently performed by means of a3-DOF translational motion or by a 4-DOF motion with three translations and one rotation around a vertical axis (Schoenflies motion [1])
Considering parallel kinematics machines (PKMs), translational motion can be obtained by parallel Cartesian robots [6,7,8], characterized by three legs that are planar serial mechanisms moved by three orthogonal linear actuators perpendicularly to their planes, or by other closed-loop schemes which are not purely translational in general, but become purely translational in case of specific orientations of the joint axes [1,9,10,11]
The stiffness and damping matrices for the KD and KDHD impedance controls have been obtained starting from the nondimensional values of Table 2, imposing kKD = kKDHD = 1·103 N/m and using Equations (15) to (22); the two control laws have been compared in the presence of a trapezoidal reference trajectory, xd, characterized by four phases: 1
Summary
In a wide range of robotic applications, for example, assembly of electronic boards or handling of objects to be placed on horizontal pallets, the full mobility (6-DOF) of the end-effector is not necessary, since the tasks can be proficiently performed by means of a. The impedance behavior of the end-effector is usually defined by means of the stiffness and damping matrices, which respectively represent the zero-order and the first-order terms of a three-dimensional PD control in the external coordinates, expressed in the principal reference frame. The stiffness/damping behavior imposed by the impedance control is linear, but a half-order term, based on the fractional derivative of order 1/2 of the position error, is added to the zero-order and first-order terms of classical impedance control. This impedance control generalizes to a three-dimensional system the fractionalorder PDD1/2 control scheme developed, so far, for single axes [30].
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