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

SPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

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)

SPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

Impedance Control– Impedance control is a control

strategy to manage the interaction with an unstructured environment.

Proposed approach– Impedance over SE(3)

ImplementationExperimental Setup

Equation Discretization

Impedance FormulationError Definition

Energy Study

Complete Formulation

Introduction

SPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

Experimental Results

Conclusions

Impedance Formulation Classical Formulation

– Mass + Spring + Damping

System Behaviour:– Motion Equation

– Energy formulation

SPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

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

SPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

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

Complete Formulation

SPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

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

Complete Formulation

Newton Law over SE(3)

Complete Formulation

SPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

exteev fKe Bv vWe

extev fvWe

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

Complete Formulation

Internal Forces

Complete Scheme

Steady State

SPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

mmm e K f

mmext

mmext

eK eK 0f

eK - eK f

ImplementationExperimental Setup

Equation Discretization

Impedance FormulationError Definition

Energy Study

Complete Formulation

Introduction

SPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

Experimental Results

Conclusions

Implementation Control Scheme

SPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

Steps– Impedance Generator Impedance equations

discretization– PD controller design

Experimental Setup

1 JR3 Force Sensors

SPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

Crimson Elan SGI Workstation

2 Stäubli RX-90 Robots

Equation Discretization

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]

SPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

3)(2161

Te

Equation Discretization

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

– fext= 1 N (N.m)

SPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

ImplementationExperimental Setup

Equation Discretization

Impedance FormulationError Definition

Energy Study

Complete Formulation

Introduction

SPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

Experimental Results

Conclusions

Experimental ResultsSPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

Individual Manipulation- Impedance Generator

Experimental ResultsSPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

Individual Manipulation- Impedance Generator

Experimental Results

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

SPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

Experimental ResultsSPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

COORDINATED MANIPULATION

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).

SPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

X axis trajectory Multi axis trajectory

ImplementationExperimental Setup

Equation Discretization

Impedance FormulationError Definition

Energy Study

Complete Formulation

Introduction

SPATIAL IMPEDANCE CONTROL IN COORDINATED MANIPULATION

Experimental Results

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