Partial
Feedback
Linearization
of
Underactuated
Mechanical
Systems
*
Mark W. Spong
Coordinated Science Labor at ory
University of Illinois at UrbanaChampaign
1308 W. Main St., Urbana, IL 61801
Abstract
things
as
actuator dynamics in the model description.
In this paper we discuss .the partial feedback lin
earization control of underactuated mechanical sys
tems.
We consider an n degree of freedom system
having m actuated, or active, degrees of freedom and
B
=
n

m unactuated, or passive, degrees of freedom.
It is known that the portion of the dynamics corre
sponding to the active degrees of freedom may be lin
earized by nonlinear feedback. In this paper we show,
alternatively, that the portion of the dynamics corre
sponding to the passive degrees of freedom may be lin
earized by nonlinear feedback under a condition that
we call Strong Inertial Coupling. We derive and ana
lyze the resulting zero dynamics which are crucial to
an understanding of the response of the overall system.
Simulation results are presented showing the perfor
mance of two link underactuated robots under partial
feedback linearization control.
1
Introduction
Underactuated mechanical systems are mechanical
systems with fewer actuators than degreesoffreedom
and arise in several ways, from intentional design
in
the brachiation robot of Fukuda
[13]
or the Acrobot
[2],
in mobile robot systems when a manipulator arm is
attached to a mobile platform, a space platform, or an
undersea vehicle,
[8],
or because of the mathematical
model used for control design
when joint flexibility
is included in the model
[14].
In the latter sense, then,
all mechanical systems are underactuated if one wishes
to control flexible modes that are not directly actuated
(the noncollocation problem), or even to include such
Our main interest in this paper is the control of
gymnast type robots like the Acrobot
[2],
and the
threelink gymnast robot in
[17].
We will show that
the method of partial feedback linearization
[7]
and
the recent method of integrator backstepping
[9]
pro
vide effective design tools for controlling such robots
to perform various motions.
It has long been known
[16]
that fully actuated
robots are feedback linearizable by nonlinear feedback.
For underactuated robots it is known that the portion
of the dynamics corresponding to the actuated (or ac
tive) degrees of freedom may be linearized by nonlinear
feedback[5]. The remaining portion of the dynamics
after such partial feedback linearization is nonlinear
and represents internal dynamics. In this paper we
show that, under a condition which we call Strong In
ertial Coupling, it is alternatively possible to linearize
the portion of the dynamics corresponding to non
actuated (or passive) degrees of freedom. .This some
what surprising result is quite interesting and, roughly
speaking, means for a system with m actuators, that m
of the equations of motion may be linearized whether
or not they are directly actuated. We will show how
these results may be used to control underactuated
robots performing gymnastic type motions.
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 Fall '08
 Smith,R
 Zero dynamics, partial feedback linearization, Acrobot

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