SPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

SPATIAL IMPEDANCE CONTROL IN COORDINATED

MANIPULATION

Andreu Fargas-Marquès Ramon Costa-Castelló

Luis Basañez

Institut d’Organització i Control de Sistemes Industrials

Universitat Politècnica de Catalunya

Barcelona-SPAIN

ImplementationExperimental Setup

Equation Discretization

Impedance FormulationError Definition

Potential Energy

Complete Formulation

Introduction

SPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

Experimental Results

Conclusions

Introduction

Internal forces sources– inaccurate modeling– lack of synchronization


Goal– Keep internal forces under control– Position controlled Manipulators– Decoupled controllers

Coordinated Manipulation may apply undesired forces over manipulated object

Introduction Previous Works

– Impedance over n(Kosuge et al

1997),(Ferreti et al 2000)

– Impedance over nx SO(3)

(Natale et al, 1999)


Impedance Control– Impedance control is a control

strategy to manage the interaction with an unstructured environment.

Proposed approach– Impedance over SE(3)




Energy Study


Introduction



Conclusions

Impedance Formulation Classical Formulation

– Mass + Spring + Damping

System Behaviour:– Motion Equation

– Energy formulation


f Kx Bv Ma

xK x2

1v Mv

2

1 TT

UTEt

wii=100 kg (kg.m), bii=266 N.s/m (N.s/rad), kii=711N/m (N/rad)

SE(3) Formulation – End effector position and orientation can be seen as an

SE(3) element.

– Homogenous matrix (4x4)

Position

Rotation Matrix

Impedance FormulationSPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

)3(10

pRH hh SE

)3( R h SO

3hp

Error Definition


To be consistent with SE(3), E should be defined in terms of the group operation.

– Exploits group structure

– Inertial Frame Independent

101 dr

Tdr

Td

rd

ppRRRHHE



Spatial Impedance defined over the error (E). Energies:

– Kinetic Energy

– Potential Energy v Wv

2

1 e

TeT

Kee 2

1 TU

(E) loge

6T

SE(3)

γβαzyx eeeeee


Newton Law over SE(3)



exteev fKe Bv vWe

extev fvWe

wii=100 kg (kg.m), bii=266 N.s/m (N.s/rad), kii=711N/m (N/rad)


Internal Forces

Complete Scheme

Steady State


mmm e K f

mmext

mmext

eK eK 0f

eK - eK f




Energy Study


Introduction



Conclusions

Implementation Control Scheme


Steps– Impedance Generator Impedance equations

discretization– PD controller design

Experimental Setup

1 JR3 Force Sensors


Crimson Elan SGI Workstation

2 Stäubli RX-90 Robots


Problem:– Impedance equations over SE(3) are nonlinear– Robots are seen as discrete time systems (T=48 ms)

Proposed Approach– Online-Integration Method– Integrator: P-2/PC-3/C-3 - R.M. Howe [Howe, 1991]


3)(2161

Te


Ex:– wii=100 kg (kg.m), bii=266 N.s/m (N.s/rad), kii=711N/m (N/rad).

– fext= 1 N (N.m)





Energy Study


Introduction



Conclusions

Experimental ResultsSPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

Individual Manipulation- Impedance Generator


1 Robot Impedance Behaviour– wii=100 kg (kg.m), bii=266 N.s/m (N.s/rad), kii=711N/m (N/rad).


Experimental ResultsSPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

COORDINATED MANIPULATION

Experimental Results COORDINATED MANIPULATION

– wii=1000 kg (kg.m), bii=4000 N.s/m (N.s/rad), kii=4000 N/m (N/rad).


X axis trajectory Multi axis trajectory




Energy Study


Introduction



Conclusions

Conclusions Impedance behaviour defined over SE(3). On-line Integration Methodology to compute trajectory

in Real Time. Validated in a real Experimental Setup. (Preliminary

Results)

Future works: – Both arms under impedance behaviour.– Saturation Effects Study– Discretization Equations– Adaptative Behaviours– Haptic Manipulation


SPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

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