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with Hyper Degrees of Freedom

Hiromi Mochiyama

School of Information Science,

Japan Advanced Institute of Science and Technology

January 16, 1998

Abstract

Thisthesisprovidesatheoreticalframeworktocontrolamanipulatorwithhyp erdegrees

of freedom. The term \Hyper Degrees Of Freedom" (HDOF in short) is an emblematic

word toexpressstrong necessityof muchmorekinematic degreesoffreedomfor amanip-

ulator nowadays. An HDOFmanipulator has ability to achieve various kindof tasks. In

order to make full use of its ability, shape control is prop osed here, that is, not only the

tip of a manipulator,but also itswhole shapeis controlledpositively.

Before discussing shap e control,we denerigorouslyshapecorresp ondence between an

HDOFmanipulatorandaspatialcurveusedforprescribingthedesiredshape. Itisdened

by using the solution of some nonlinear optimization problems termed the shap e inverse

problem. We give not only the existence theorem of its solution, but also a theorem on

the existenceregionwhichallowsustoconvertcontrolproblems app earedlaterintomore

tractable ones.

ShaperegulationcontrolisconsideredrsttobringanHDOFmanipulatorontoagiven

time invariant curve. The idea of estimating the desired curve parameter enable us to

nd the shape regulation lawwith curveparameter estimation law by Lyapunov design.

This crucial ideaof curveparameter estimation isalso eectivefor the shape tracking,

where a time-varying curve is given to prescribe the desired shape. Two shap e track-

ing control laws are derived by utilizing familiar tracking control laws for conventional

manipulator tracking.

Furthermore, it is shown that joint velo city signals are not essential to achieve shape

tracking,that is, shape trackingusing only joint anglesis attained. Basedon the idea of

conceptual duality,we derive anobserver that does not directly estimate the jointangle

velo cities, but estimate the velo city of the shape. After properly tuned, the modied

shapetracking controllerand shapevelo city observerassure local asymptotic stabilityof

the closed-lo op system.

We also give an example to show that new tasks which have never done b efore can

be accomplished by shape control. The useful motion control for sophisticated obstacle

c

for going into a narrow space. Two controllers achieving this motion are shown as the

counterpartsof twoshap e trackingcontrollersderivedbefore.

We also propose the way to nd essentials of an HDOF manipulators by increase in

DOF. Conditions of the kinematic structure are derived from a geometrically natural

requirementthatitsdirectkinematicstendstoFrenet-SerretformulabyincreaseinDOF.

The results inthis thesis giveacompletely new vision for robotic manipulatorcontrol.

Key Words: hyp er degrees of freedom manipulators, spatial curves,

dynamics-based shape control, Lyapunovstability theory, de-

sired curve parameter estimation