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Research Papers

Active Joint Stiffness Regulation to Achieve Isotropic Compliance in the Euclidean Space

[+] Author and Article Information
Nicola P. Belfiore1

Matteo Verotti

Department of Mechanical and Aerospace Engineering,  Sapienza University of Rome, via Eudossiana, 18, Rome 00184 Italy

Paolo Di Giamberardino

Department of Computer and System Sciences,  Antonio Ruberti Sapienza University of Rome, via Eudossiana, 18, Rome 00184 Italydigiamberardino@dis.uniroma1.it

Imre J. Rudas

Rector  Óbuda University, Bécsi út 96/B, Budapest H-1034, Hungaryrudas@uni-obuda.hu

1

Corresponding author.

J. Mechanisms Robotics 4(4), 041010 (Sep 17, 2012) (11 pages) doi:10.1115/1.4007307 History: Received September 16, 2011; Revised June 29, 2012; Published September 17, 2012; Online September 17, 2012

Abstract

This paper is dedicated to the relationship between the external force applied on a point of a robot end-effector and its consequent displacement in static conditions. Both the force and the displacement are herein considered in the Euclidean space $E(3)$. This fact represents a significant simplification of the approach, since it avoids some problems related to the absence of a natural positive definite metric on the Special Euclidean Group $SE(3)$. On the other hand, such restriction allows the method to find closed-form solutions to a large class of problems in robot statics. The peculiar goal of this investigation consists of setting up a procedure which guarantees at least one pose at which any force applied (in $E(3)$) to an end-effector point is always parallel to its consequent displacement (also in $E(3)$). This property, which will be referred to as isotropic compliance in $E(3)$, makes the robot tip static behavior uniform with respect to all directions, namely, isotropic, although not homogeneous, since it holds only in some poses. Achieving isotropic compliance in $E(3)$ is a task more general than the classical problem of finding a pose with unit condition number, which does not include the case of different elements in the diagonal joint stiffness matrix. For this reason, the object of the present investigation could not be furtherer simplified to the classical kinetostatic problem in terms of the jacobian matrix alone. The paper reveals how the force–displacement parallelism can be achieved by using a method based on a simple proportional-derivative (PD) controller strategy. The method can be applied when the passive and active stiffness act, on the joints, either in parallel or in series, and the magnitude of the displacement response can be chosen by imposing appropriate values for the overall joints compliance. Results show that for the three analyzed examples, namely, the RR, RRP, and RRR manipulators, with arbitrary lengths of the links, there is, at least, one pose for which the sought property is achieved.

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Figures

Figure 1

A schematic representation of the mechanical layout of a compliant RR in the neutral configuration

Figure 2

A schematic representation of the mechanical layout of a serial RR manipulator with elastic joints

Figure 3

The reduced quasi-static control scheme for the case of parallel action of the passive and active stiffness

Figure 4

A rearrangement of the reduced scheme reported in Fig. 3

Figure 5

Compliance ellipsoid (scaled) for RR manipulator with l2=0.4 m (a) and l2=0.8 m (b)

Figure 6

Compliance ellipsoid (scaled) for RRP manipulator with l2=0.55 m (a) and l2=0.85 m (b)

Figure 7

Compliance ellipsoid (scaled) for RRR manipulator with l2=0.7 m (a) and l2=0.9 m (b)

Figure 8

The elementary case of the actuator acting in parallel to the passive stiffness

Figure 9

A simplified scheme of a single actuator with secondary mass m and its corresponding link (primary mass M)

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