Impedance_Control_An_Approach_to_Manipulation_Part_I_II

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Impedance_Control_An_Approach_to_Manipulation_I_II_III.zip （3个子文件）

Impedance_Control_An_Approach_to_Manipulation_Part_III—Applications.pdf 863KB

Impedance_Control_An_Approach_to_Manipulation_Part_I—Theory.pdf 808KB

Impedance control - An approach to manipulation_ II - Implementation.pdf 968KB

Neville Mogan

Associate Professor,

Department

Mechanical Engineering

and Laboratory for Manufacturing

and Productivity,

Massachusetts Institute of Technology,

Cambridge, Mass. 02139

Impedance

Control:

Approach

to Manipulation:

Part SI—Implementation

This three-part paper presents an approach to the control of dynamic interaction

between a manipulator and its environment. Part I presented the theoretical

reasoning behind impedance control. In Part II the implementation of impedance

control is considered. A feedback control algorithm for imposing a desired car-

tesian impedance on the end-point of a nonlinear manipulator is presented. This

algorithm completely eliminates the need to solve the "inverse kinematics problem"

in robot motion control. The modulation of end-point impedance without using

feedback control is also considered, and it is shown that apparently "redundant"

actuators and degrees of freedom such vs exist in the primate musculoskeletal

system may be used to modulate end-point impedance and may play an essential

functional role in the control of dynamic interaction.

Introduction

Most successful applications of industrial robots to date

have been based on position control, in which the robot is

treated essentially as an isolated system. However, many

practical tasks to be performed by an industrial robot or an

amputee with a prosthesis fundamentally require dynamic

interaction. The work presented in this three-part paper is an

attempt to define a unified approach to manipulation which is

sufficiently general to control manipulation under these

circumstances.

In Part I this approach was developed by starting with the

reasonable postulate that no controller can make the

manipulator appear to the environment as anything other

than a physical system. An important consequence of

dynamic interaction between two physical systems such as a

manipulator and its environment is that one must physically

complement the other: Along any degree of freedom, if one is

an impedance, the other must be an admittance and vice

versa.

One of the difficulties of controlling manipulation stems

from the fact that while the bulk of existing control theory

applies to linear systems, manipulation is a fundamentally

nonlinear problem. The familiar concepts of impedance and

admittance are usually applied to linear systems and regarded

as equivalent and interchangeable. As shown in Part I, for a

nonlinear system, the distinction between the two is fun-

damental.

Now, for almost all manipulatory tasks the environment at

least contains inertias and kinematic constraints, physical

systems which accept force inputs and which determine their

motion in response and are properly described as admittances.

When a manipulator is mechanically coupled to such an

Contributed by the Dynamic Systems and Control Division for publication in

the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL.

Manuscript

received by the Dynamic Systems and Control Division, June 1983.

environment, to ensure physical compatibility with the en-

vironmental admittance, something has to give, and the

manipulator should assume the behavior of an impedance.

Thus a general strategy for controlling a manipulator is to

control its motion (as in conventional robot control) and in

addition give it a "disturbance response" for deviations from

that motion which has the form of an impedance. The

dynamic interaction between manipulator and environment

may then be modulated, regulated, and controlled by

changing that impedance.

This second part of the paper presents some techniques for

controlling the impedance of a general nonlinear multiaxis

manipulator.

Implementation of Impedance Control

A distinction between impedance control and the more

conventional approaches to manipulator control is that the

controller attempts to implement a dynamic relation between

manipulator variables such as end-point position and force

rather than just control these variables alone. This change in

perspective results in a simplification of several control

problems.

Most of our work to date [3, 6, 13, 14, 16] has focused on

controlling the impedance of a manipulator as seen at its

"port of interaction" with the environment, its end effector.

A substantial body of literature has been published on

methods for implementing a planned end effector cartesian

path [5, 27, 28, 32, 34, 35]. The approach is widely used in the

control of industrial manipulators and there is some evidence

of a comparable strategy of motion control in biological

systems [1, 24]. Following the lead from this prior work we

have investigated ways of presenting the environment with a

dynamic behavior which is simple when expressed in

workspace (e.g., Cartesian) coordinates.

8/Vol. 107, MARCH 1985

Transactions of the ASME

Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 08/02/2013 Terms of Use: http://asme.org/terms

The lowest-order term

in any

impedance

is the

static

relation between output force

and

input displacement,

stiffness.

If, in

common with much

of the

current work

robot control,

assume actuators capable

generating

commanded forces

(or

torques), Tact, sensors capable

observing actuator position

(or

angle),

6, and a

purely

kinematic relation (i.e.,

structural elastic effects) between

actuator position

and

end-point position

L(6),

it is

straightforward

design

feedback control

law to im-

plement

actuator coordinates

desired relation between

end-point (interface) force, Fint,

and

position,

Defining

the desired equilibrium position

for the

end-point

in the

absence

environmental forces (the virtual position)

as X

, a

general form

for

the desired force-position relation is:

Fint=ATXo-X]-

(1)

Compute the Jacobian, J(0):

= 3(6)d6

(2)

From the principal

virtual work:

Tact

J'(0)Fint

(3)

The required relation

actuator coordinates

is:

Tact

J'(0)A-[X

-L(0)]

(4)

No restriction

linearity

has

been placed

on the

relation

K[X

-X],

and the

displacement

of X

from

need

not be

small. Note that

this equation

the

inverse Jacobian

is not

required.

Inverting

the

kinematic equations

of a

manipulator

determine

the

time-history

actuator (joint) positions

required

produce

desired time-history

end-point

positions

has

been described

as one of the

most difficult

problems

robot control [28].

For

some manipulators (e.g.,

those with nonintersecting wrist joint axes) no explicit (closed-

form) algebraic solution may

possible.

However, if

K[X

chosen

so as to

make

the

end-point sufficiently

stiff,

then

controller which implements equation

(4)

will

accomplish Cartesian end-point position control

and the

need

to solve

the

"inverse kinematics problem"

has

been com-

pletely eliminated. Only

the

forward kinematic equations

for

'Throughout this paper, "position" will refer

both location

and

orien-

tation,

and

"force" will refer

both force and moment.

the manipulator need

computed. This

is a

direct con-

sequence

of the

care which

was

taken

ensure that

the

desired behavior

was

compatible with

the

fundamental

mechanics

manipulation

and was

expressed

as an im-

pedance.

Another important term

in the

manipulator impedance

the relation between force

and

velocity. Again, given

the

above assumptions,

it is

straightforward

define

feedback

law

implement

actuator coordinates

desired relation

between end-point force

and

end-point velocity such

as:

(5)

(6)

Fint = fi[V

-V]

From the manipulator kinematics:

V = J(0)a>

The required relation

actuator coordinates

is:

Tact

J'(0)fl[V

J(0)co]

(7)

Again note that

the

relation

B[\

need

not be

linear

and

that inversion

the Jacobian

not

required.

The dynamic behavior

to be

imposed

on the

manipulator

should

as simple as possible,

but no

simpler. The foregoing

equations take

account

of the

inertial, frictional,

gravitational dynamics

of the

manipulator. Under some

circumstances this

may be

reasonable,

but in

many situations

these effects cannot

neglected.

ensure dynamic

feasibility

the

choice

of the

impedance

to be

imposed should

be based

on the

dominant dynamic behavior

of the

manipulator.

The

choice

is a

tradeoff between keeping

the

complexity

of the

controller within manageable limits while

ensuring that imposed behavior adequately reflects

the

real

dynamic behavior

of the

controlled system.

As a

result

depends both

on the

manipulator itself

and on the en-

vironment

which

operates.

For

example,

manipulator

intended

for

underwater applications will operate

in a

predominantly viscous environment

and it may be

reasonable

to ignore inertial effects.

contrast,

manipulator intended

for operation

in a

free-fall orbit will encounter

predominantly inertial environment.

For

terrestrial

ap-

plications (which have been

the

main concern

of our

work)

both gravitational

and

inertial effects

are

important,

and the

dominant dynamic behavior

that

of a

mass driven

motion-dependent forces, second order

displacement along

each degree

freedom.

•Ac]

Fext

Fint

K[.]

£[•]

U-)

3(6)

admittance

impedance

modulation

command

set

flow source

effort source

external force

interface force

end-point position

commanded (virtual)

position

end-point velocity

commanded (virtual)

velocity

force/displacement rela-

tion

force/velocity relation

actuator position

angle

actuator velocity

linkage kinematic equa-

tions

Jacobian

F(-)

1(6)

C(0,co)

K(co)

S(6)

G(0,«)

Tact

Tint

Y(6)

W(6)

= inertia tensor

end-point

coordinates

= mass

= inertia

= time

= noninertial impedance

= environmental inertia

tensor

= inertia tensor

actuator

coordinates

= inertial coupling torques

= velocity-dependent torques

= position-dependent

torques

= accelerative coupling terms

= actuator force

torque

= interface torques

= mobility tensor

actuator

coordinates

= mobility tensor

in end-

point coordinates

H(.)

T,Ti,T

#1.02

1P2

-^1

J-^2

Ek*

X,X

= generalized momentum

actuator coordinates

= generalized momentum

end-point coordinates

= Hamiltonian

= torque

= absolute j oint angle

= relative joint angle

= link lengths

net

stiffness

due to

elbow

muscles

net

stiffness

due to

shoulder muscles

net

stiffness

due to two-

joint muscles

= stiffness tensor

in end-

point coordinates

= stiffness tensor

joint

coordinates

= potential energy

= kinetic energy

= kinetic coenergy

= eigenvalues

Journal

Dynamic Systems, Measurement,

and

Control

MARCH 1985, Vol.

107/9

Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 08/02/2013 Terms of Use: http://asme.org/terms

BONOS REPRESENT VECTOR QUANTITIES

F„1'S,

=^| 1 y -. 0 _] S

^jc)

' \

EXTERNAL

' Z • fc)

SOURCE I ' '

ENVIRONMENTAL | CONTROLLED MANIPULATOR

ADMITTANCE |

Fig.

1 A bond graph equivalent network representation of an im-

pedance-controlled manipulator interacting with an environmental

admittance. Each bond represents a vector of power flows along

multiple degrees of freedom.

When the manipulator is decoupled from its environment

the term due to the environmental admittance disappears, and

in principle the manipulator alone need exhibit no mass-like

behavior. In practice, the uncoupled manipulator still has

inertia (albeit nonlinear and configuration-dependent). This

means that the controlled system, both with the manipulator

coupled to and uncoupled from its environment, can be

represented by an admittance coupled to an impedance as

shown in Fig. 1.

No physically realisable strategy can eliminate the inertial

effects of a manipulator but the apparent inertia seen at the

end effector can be modified. The approach we have taken to

deal with inertial manipulator behavior is to "mask" the true

nonlinear inertial dynamics of the manipulator and impose

simpler dynamics, those of a rigid body. Most manipulatory

tasks are fundamentally described in relative coordinates, that

is,

in terms of displacements and rotations with respect to a

workpiece, tool or fixture whose location in the workspace is

not known in advance with certainty. As a result, task

planning and execution will be simplified if the end-point

inertial behavior is modified to be that of a rigid body with an

inertia tensor which remains invariant under translation and

rotation of the coordinate axes. This is achieved if:

This is the inertia tensor of a rigid body such as a cube of

uniform density. This inertia tensor eliminates the angular

velocity product terms in the Euler equations for the motion

of a rigid body [8] and ensures that if the system is at rest the

applied force and the resulting acceleration vectors are

colinear.

To represent the dominant second-order behavior of the

manipulator the noninertial interface forces are assumed to

depend only on displacement, velocity and time:

Fint

F(X,V)-MdV/dt (9)

If the noninertial behavior to be imposed is nodic, it may be

written in terms of a displacement from a commanded (time-

varying) position X

Fint

F(X

-X,Y

-\)-MdV/dt (10)

Although there may be cases in which coupled nonlinear

viscoelastic behavior is useful, for simplicity the position- and

velocity-dependent terms may be separated:

Fint

K[X

-X]+B[V

-V]~MdY/dt (11)

All of the parameters in this expression are implicitly assumed

to be functions of the set of control commands (c) and of

time.

This set of assumptions defines a target behavior which

includes inertial effects. The first two terms are the position-

and velocity-dependent impedances of equations (1) and (5).

If the environment is a simple rigid body acted on by un-

predictable (or merely unpredicted) forces, its dynamic

equations are:

MedV/dt

Fext

¥mt (12)

and the coupled equations of motion for the complete system

of figure

are:

(Me+M)dV/dt=K[X

-X]+B[\

- V] +Fext (13)

Note that in this case both the coupled and uncoupled

equations for the system have the same second-order form.

To implement the target behavior of equation (11), one

approach we have used is to express the desired Cartesian

coordinate impedance in actuator coordinates (the kinematic

transformations between actuator coordinates and end-point

coordinates provides sufficient information to do this) and

then use a model of the manipulator dynamics to derive the

required controller equations. Assuming that the kinematic,

inertial, gravitational, and frictional effects provide an

adequate model of the manipulator dynamics as follows:

I(8)dco/dt

+ C(6,a>)+V(co) +

S(d)

Tact+Tint (14)

an expression for the required actuator torque as a function of

actuator position and velocity and end-point force can be

derived by straightforward substitution (see Appendix I):

Tact =

1(6)3

(6)M^

K[X

-L(6)]+S(6)

+ 1(6)5'

(d)M-

B[\

-3(d)u] + V(o)

+ I(ff)J~

(e)M-

Fmt-J

(9)Fmt

1(6)3

(6)G(6,u)

C(6,w) (15)

This equation expresses the required behavior to be

provided by the controller as a nonlinear impedance in ac-

tuator coordinates. It may be viewed as a nonlinear feedback

law relating actuator torques to observations of actuator

position, velocity and interface force. The input (command)

variables are the desired cartesian position (and velocity) and

the terms of the desired (possibly nonlinear) cartesian

dynamic behavior characterized by M, /?[•] and

AT*].

The feasibility of this approach to cartesian impedance

control has been investigated [6,16] by implementing this

nonlinear control law to impose cartesian end-point dynamics

on a servo-controlled, planar, two-link mechanism (similar to

the nonlinear linkage in a SCARA

robot). A simple analysis

estimating the computation required to implement this

controller on a six-degree-of-freedom manipulator indicated

that the computational burden is comparable to "exact"

approaches to generating forward-path manipulator com-

mands such as the recursive LaGrangian [17] and Newton-

Euler [21] methods or the configuration space method [18].

If the interface forces and torques in equations (11) and (15)

are eliminated and the position- and velocity-dependent terms

reduced to linear diagonal forms, this implementation of

impedance control resembles the resolved acceleration

method [22]. However, unlike the resolved acceleration

method, the impedance control algorithm presented above is

based on desired end-point behaviour which may be chosen

rationally using approaches such as the optimisation

technique presented in Part III. Furthermore, the impedance

control algorithm includes terms for coping with external

"disturbances." Without the external "disturbance" terms

(which have no counterpart in the resolved acceleration

algorithm) the manipulator is not capable of. controlled

Selective Compliance Assembly Robot Arm [23].

10/Vol.

107, MARCH 1985

Transactions of the ASME

Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 08/02/2013 Terms of Use: http://asme.org/terms

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