WA7 11:15

AN INVERSE DYNkMIC METHOD

YIELDING FLEXIBLE MANIPULATOR STATE TRAJECTORIES

"3ont,-So(, Kwon and Wayne J. Book

George Vt Wcodruff School of Mechrnical En3inecring

Ge _rgia Institute of Techr, ology

Atlanta, GA 30332

7

) J

ABSTRACT

An inverse dynamic equation for .2 fl.a'ible

manipulator is derived in a state form. By dividing the

inverse system into the causal part and the antic mst_' part,

we can calculate torque in the time domain for a certain

end point trajectory, as well as tra]ectoK_ c f all state

variable_ The open loop control of the inverse dynamic

method shows an excellent re:ult in simulation. For

practical applications, a control strategy adapting feedback

tracking control to the invert ; dynamic feedforward control

is illustrated, and its good experimental result is presented.

1. Introduction

Flexible manipulators have been widely discussed

in the literat.are recemly. Most of these papers have

mentiofied the flexible manipulator's advantages such as

low mass to payload ratio, use of small actuators, and

high travelling speed. However, the inherent flexibility

of the manipulatm makes its actual industrial application

difficult. In spite of its flexibility, a flexible manipulator

should have the capability to follow a given end point

trajectory for practical applications. This paper proposes

a simple inverse dynamic method which can make the

end point of a flexible manipulator accurately follow a

given trajectory without overshoot or residual vibration.

The feedback control regulating metEod, which

adds damping effect to the flexible structure is one of

typical methods used to suppress the structural vibration

of the manipulator in literature. By using joint and

strain feedback control, Hastings and Book [7]

demonstrated good results in regulating the vibration.

Even though the feedback control is able to dampen the

residual vibration during settling time, their experiment

showed the undershoot, the overshoot, and the flexible

vibration in case of a step response. For a step input

command, these vibrating phenomena are inevitable with

the feedback control scheme because the feedback

control signal contains high frequency components,

which excite the system natural frequencies. Instead of

a step response, a smooth nominal trajectory can be

used as the joint reference command of tracking control.

However, the strain feedback tracking control requires

the desired trajectory values of flexible mode variables

which match the joint trajectory value. Because of

lacking flexible mode values, assigning zero values to the

desired values of flexible mode states has been

acceptable for the nominal trajectory tracking control to

suppress the vibration during motion. In other words,

reference commands are given to the flexible

manipalator to follow the trajectcry like a rigid

manipulator. Even though the increa_d damping by the

feedback control regulates the vibration, such unrealistic

commands cannot avoid the generation of the vibration.

To avoid the above feedback control problems,

De Luca and Siciliano [6] suggested a joint based

inversion control scheme. This method showed good

tracking result for a certain joint trajectory. However, it

ctmidn't be extended for an end point trajectory

following control because of nonminimnm phase system

characteristics. Oosting and Dickerson [8] demonstrated

a calculation method of the torque to follow a smooth

trajectory for a simple lumped parameter two link

flexible manipulator.

To make the end point of a flexible manipulator

follow a given trajectory, Bayo [2] proposed a new

approach. For a given end point acceleration profile,

the required torque was calculated by solving the inverse

dynamic equation in the frequency domain through the

inverse fast Fourier transform. The inverse dynamic

system considering the end point acceleration as input

and the joint torque as output is a noncausal system

since the output (torque) must begin before the input

(end point acceleration) begins. Therefore, this inverse

dynamic method provides a noncausal solution torque

which acts before the tip moves, and after the tip stops.

In spite of excellent results, his method has a drawback

because it requires heavy computation for the

transformation of a dynamic model and an input

trajectory from the time domain to the frequency

domain and the inverse transformation of the output to

the time domain. To reduce such computation burden,

Bayo and Moulin [4] recently introduced the convolution

integral method to solve the inverse dynamic equation.

Asada and Ma [1] derived an inverse dynamic

equation by using assumed mode functions for a general

n-link ease. As the transfer function of a flexible

manipulator between the input torque and the output

end point position is a non-minimum phase system, it

has some positive real value zeros. These zeros become

positive poles of the inverse dynamic system transfer

function. They cause the inverse system to be unstable

P/

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] .....

r: L" r

186
b

if the inverse system output is restricted to causal

solutions. In the reference [1], they used rigid body

torque to show nonlinear effect without solving the

inverse dynamic equation.

In this paper, we introduce a new simple inverse

dynamic method, which relieves the calculation burden

considerably. With this method, we can calculate the

required torque in the time domain. In addition to that,

we can generate very natural trajectories of flexible

mode coordinates, which match the given end point

trajectory dynamically. These trajectories can be and are

used as reference commands of flexible coordinates for

feedback tracking control.

In following sections, we first describe a direct

dynamic model of a single link manipulator by using

assumed mode methods. And we derive an inverse

dynamic system equation from the dynamic model in a

state space form. By dividing the inverse system into the

causal part and the anticausal part, we can integrate the

inverse dynamic differential equation of each part

separately in the time domain. Next, this inverse

dynamic method is implemented on the single link

flexible manipulator in Fig. 1 through simulations. Its

results are compared with the output of other typical

control methods. Finally, the controller design with the

inverse dynamic method is presented, and its

experimental results are discussed.

2. Modeling

A single link flexible manipulator having planar

motions is described as shown in Fig. 1. The rotating

inertia of the servo motor, the tachometer, and the

clamping hub are modeled as the hub inertia Ih. The

payload is modeled as the end mass Me. Even though

flexible beam damping exists, it is ignored in modeling.

Ib, Io __

0.C, Notot L-47

Ib _guaqe .- L*4" *

Tot q'_e

Fig 1. A single link flexible manipulator

To derive equations of motion of the

manipulator, we describe the position of a point on the

beam with a combination of a virtual rigid body motion

and flexible deflections using a Bernoulli-Euler beam

model. The virtual rigid body motion is represented by

the motion of the moving coordinate which is attached

to the beam. The flexible deflections are described by

finite series of assumed modes with respect to that

moving reference frame. Defining the rigid body motion

is important because different mode shape functions

have to be used according to the choice of the rigid body

coordinate. Several authors [7,9] used the rigid body

coordinate which is attached at the base hub like a) of

Fig. 2, and used the clamp-free boundary condition

mode functions. Other authors [5] defined the rigid

body mode coordinate to pass through the center of

mass of the beam, and used pin-free mode functions.

And others [1] let the rigid body coordinate pass through

the end point, and used the pin-pin mode functions. All

of these definitions for the rigid body mode can be valid

because appropriate mode functions,which satisfy the

geometric boundary conditions can be chosen for each

case.

a) ¢Ialp-£zet

b) pll,lleO

/

Fig 2. The kinematic descriptions of flexible manipulator

coordinates

In this paper, the rigid body mode coordinate

which passes through the end point of the beam is

selected, and the pin-pin mode functions are used to

describe deflections. In order to obtain a accurate

model with a small number of modes, we have

considered more accurate boundary conditions such as

the joint hub inertia and the end mass when we derive

the mode shape functions. For the inverse dynamic

model, two mode functions of pin, hub inertia-pin,end

mass boundary conditions are used. The reason why this

rigid body coordinate is selected is that the end point

position of the beam can be expressed by the rigid body

mode variable, and this simple representation of the end

point position makes the derivation of the inverse

dynamics equation easy.

The velocity of any point of the beam can be

expressed as follows.

• - xJ*wl where w - w(x,t)

187
, ' L

, (X(I,+Vb)2 where w < x

The kinetic energy of a beam is obtained _ t_e

integration the velocxty product term. Its s-cond term

generates centrifJga_and coriolis forces in t ae d.narric

equation. In this single link case, t'_e ,econd

,2

q, velocity produc: te,"m can be _,ssumed to be

negligible to provide a Knearized model except in the

case of very high rotational speed.

As we can consider x as Co(X) and q,(O as qo(O ,

let's introduce another variable y which represents the

total displacement of the point on the beam as shown in

Fig. 2.

y - x_, ÷ w - xq,(n + _ ¢,(x)q+(O

t-I

Then i_ - _ - {_ _A+} z

|-0

Thus the kinetic energy and the elastic potential energy

will written as

" _fo'i (+,,tyd_ + S',(i +',(O)qy

L. £-0 2 l-O

2 _-o 2 i-o

2 J0 i-o

And the virtual work is expressed as

8 W = _'Syt(x'O,t) - _'(_ ¢,(O)Sq+

_..0

i..O

"rtlerefore, the generalized force Q+ is expressed as

follows:

Q, - ¢,(0)"

By using Lagrange s eqaatieas of motion, the dynamic

equat:on of a fle.6ble mar, ipulator is obtained with

generalized coordinates.

d( _.__.] _1" + OV _ Q,, i-O,l,..,n

_Laq,) aq, Oq,

[MI# + [lOq - [BI_

where

[MI- _ ..., [KI-

,o[:]+It

K_

i-0,1,..n

M# -

' I

. _, +',(o)+_(o)

+ M,¢_o¢_t)

/ !

+ L ¢,(/)¢/0

id-o,..n

The mass and stiffness matrices are represented in these

general forms which are valid for different definitions of

rigid body modes. For the viscous friction of a joint, the

damping term is included. The dynamic equation can be

divided into a rigid body motion part and a flexible

motion part as follows:

LD: to

where q, " qo ; rigid body coord.,

q!" _} mode coord.

For a state space form, we obtain the following dynamic

equation.

188
3f- -I£ M-t X+ -x _ (2)

y- (c_ + (,'+"J.,r

where X - lq,//_,,4/lr

" lqo,q,,'",#o,#v'"l r

From now on,we callthisdynamic equationthedirect

dynamic equationto distinguishitfrom the inverse

dynamic equationwhich isderivedatthenextsection.

3. Inverse dynamic equations

Base on the above the direct dynamic equations, we will

derive the inverse dynamic equation. Eqn. (1) can be

written in two parts.

[_.]+.+[M_,+:[_.],+,+ [o,+,+: [B,]'+ c_)

- [Bjl,

From Eqn. (3), torque is give by

- [B,]-'I[U.]+,+tM,_+/+ [O.],+. + [O_41 (:3,,)

Substitution of above Eqn. (3a) to Eqn. (4) gives the

following relations between the flexible coordinates qf

and the rigid body coordinate qr

[l_,]+:[o,j4/+txjq/-[B,,]#,+[e,214, <s)

where [MJ - {[Mbi-tB:tB,I-ttMj}

[Dj- {[o_-[B,_tsj-'[o,_}

tr,] - It:

[n,,]- ttn_tn,3-'toj-wjq

[B,zl - I [S:[n,1-'tM,,]-tM:r}

From Eqn. (4), the acceleration of flexible coordinates

are expressed as follows:

+f.-JM_-'tMy'+,-tM_-'[O,,:,+, _+_

-tM_J-'tou]+-tM++"tr+]q/+t+_-'te_"

Substitute this Eqn. (6) to Eqn. (3), then we will get the

following Eqn.

- [Culq:(C_)#:(Ful¢,+[Fa)4, (7)

where [m-t[S,.1-CMJCM_-'tS:}-'

(C,_l-[6qI-[UJ[M_I"[X_ l

(Ca]-[G]I[DJ-[MJ(M_]-t[D_]}

tFu]-t6_{WJ-tMJtM_-'tOJ'}

(Fa]-(G]I(M,,]-(MJ[Mj]-'IMJr}

If Eqn. (5) and Eqn. (7) are represented in a state space

form, the inverse dynamic equations will be

Let X_- {q_}r, q=,- 14,_,1

0

+[+o

t - [Cu,CaIX _ + [Fu,Fa]q_,

(e)

The inverse dynamic equation is obtained in this

simple form. By integrating this 1st order differential

equation, the required torque can be calculated for the

desired endpoint acceleration, and velocity, which are

expressed with the rigid body coordinate qr Even

though the inverse equation is derived in the simple

state space form, the inverse system matrix A i has

positive poles (which came from the positive zeros of the

transfer function of the direct dynamic system) as well as

negative poles. It seems to be impossible to integrate

that equation. However, if we relax the solution range

of that equation to include noncausal solutions, we can

obtain a unique stable solution.

To analyze the inverse system of a flexible

manipulator, first of all, let's define several terms: a

causal system is the system of which output (impulse

response) always occurs after a input (impulse) is given.

An anticausal system, on the contrary, always has the

output (backwards impulse response) before a input is

given (impulse). A noncausal system is defined as the

synthesized system of a causal system and an anticausal

system.

In order to grasp the meanings of the above

definitions, we need to understand the physical

phenomena during the motion of a flexible manipulator.

If a certain torque profile is applied to the manipulator,

there is a unique motion of the end point. On the other

hand, if this unique motion of the end point is given as

a desired motion, we should be able to get that previous

189
i i i i I IIIII I III I

-- torque profile by using the inverse dynamic equations.

In some cases, to make the end point follow a certain

trajectory profile, we have to preshalx (przbend) the

flexible manipulator. Therefore, the requi'ed torque,

which is necessa-y to pre.shape it, ha_ to be _pplied

before the end point starts to move. This _eaas that

the inverse system has anticausal characteristics.

_ Furthermore, wh.'.n the flexible manipulator :top._ some

torque should be applied to release the f exible

deflection after the exd lxint stops. Thi: me*ms t'lat the

inverse system hzs a causal system characteristics, too.

_ Thus, such _nverse system can be called a nou:ausal

system, which is composed of a causal system and

anticausal scstem. With this intuitive motivation, we

divide the inverse dynamic system into the caus'.d part

--and the anticausal part by using a similarity

transformation as follows.

[Tj:Orthogonal rrans/orma_n matrix

xj - [rll',

- [L.r.Jle,,e,.I T

(9)

'¢ . '_¢ 4- TiC

Such a coordinate change decouples the inverse

system into two subsystems of Eqn. (9). The new

variable P_ represents the coordinate of the causal

-- system, an_ the Pac does that of the anticausal system.

In Eqn. (9), [1] matrix doesn't have to be divided

equally. However, it has been done to make the causal

system and the anticausal system symmetric. If both

-- systems are symmetric and the acceleration profile is

skew symmetric, we don't have to integrate both parts.

As the output of one system is the same as the reflected

output of the other, the total output torque can be

-- obtained by adding one output itself and its reflection.

In general cases, both systems should be

integrated separately. For a given end point trajectory

_ q_ the causal part torque is obtained by integrating the

causal part inverse dynamic equations forward from the

initial time of the trajectory. However, the anticausal

system equations must be integrated backwards from the

_ final time of the trajectory. The interpretation of this

separate integration method to a convolution integral

gives good explanations about the backward integration.

The total torque, which is the output of these equations,

;s obt.ained by adding the both part t)rqu-_s as shown in

Fig. 5.

As additioaal outputs, we cou:d build the

reference trzject_ry ,_f the who!e flzxible mode

coordinates of the direct dynamic s:'stem from a rigid

body mode trajectory. As we can expect in Eqn. (9), the

space: of the full state vector of the direct dynamic

system can be di_ded into three stbspaces: the rigid

body coordinate subspace, the causal part flexible

coordinate subspaee, and the anticausal part flexible

coordinate subspace. These subspaces are linearly

independent and orthogonal to another. The relations

of these spaces are illustrated in Fig. 3, and described by

the following Eqn. (10) when two flexible modes are

used.

X -

10! I000

00 000

00 100

÷

01" _000

001 010

001 001

X_

x - n,q, ÷ tt_re_

x- n,a, • u,r,e, • tttr.),., (10)

Itgid Body Cau|al

Coozd. flexible

Sublpace Coozd.

Sublpice

Anticausal

I Pac I rlexLble

, Coozd.

Sub|pace

Fig 3. Dimensional analysis of state variables of flexible

manipulator dynamic equations

From the end point trajectory, the rigid body coordinate

trajectory qr is given, and the flexible coordinate

trajectories of P_ an a P_ are calculated from the

integration of Eqa. (c). IlShus, the trajectories of the

whole states X can be obtained by using Eqn. (10).

These trajectory values can be used as reference

commands for feedback tracking control.

190
4. Trajectory Generation

Among the desired trajectories of the manipulator

end point, the acceleration, and velocity profile are used

as input of the inverse dynamic equations. Theoretically,

the inverse dynamic equation can give a torque profile

for an arbitrary acceleration profile. However, as Bayo

mentioned in the ref. [3], it is important to apply the

acceleration profile which doesn't produce high

frequency content of the torque. In this paper, several

constraints are applied for a desired acceleration profile

to get fast motion and no residual vibration. First, the

acceleration profile should not have too much high

frequency component. If the acceleration changes

sharply, the calculated torque profile will contain high

peak impulses. This impulse component of the torque

may excite natural frequencies of the flexible

manipulator. Moreover, it can saturate the actuator, or

it may be beyond the actuator bandwidth. Second, the

maximum acceleration limit should be chosen not to

saturate the actuator. Third, traveling time should be

minimum by using the full capacity of the actuator. This

third constraint makes the acceleration profile close to

a bang bang type which will result in unwanted high

frequency problems. Therefore, we have to select the

acceleration profile by compromising the profile

smoothness and the use the full capacity of the actuator.

Considering the above constraints, the

acceleration profile of Fig. 4 was adopted. It is

composed of four 3rd-order polynomial parts and two

constant acceleration parts.

,I Accelerotion

500 maK occelerotion

_250

0

u

0.0 /_02

-250.

-500

ill ) Vi4,,¢,_ _

,%

go :, o,,__,_o,

De, 06

Time ( sec )

¢)

•.J it

clio: p-O.2

.,-I , ,_

04 04 Oa '0

Fig 4. Desired End Point Trajectory

a)Acceleration, b)Velocity, c)Position

The above first and third constraints can be

compromised by adjusting the parameter p. If p =0, the

profile will be the bang bang type. And, if p--0.25, the

prof'de will be very smooth by being connected with 4

polynomials. In simulation and experiment, the case

p=Ol isused.

5. Simulation of the inverse dynamic open loop control

and alternative control methods

In this section, we present simulation results to

illustrate the performance of the inverse dynamic

method. First of all, the total torque profile is

calculated from the causal part torque and the anticausal

part torque for a given end point acceleration profile as

shown in Fig. 5. And the desired trajectories of the

output such as joint angle, joint velocity and strains are

generated as in Fig. 6.

2O

15

10

-- 5

l=

;, o_

=_ -I0-

o

o,,

-15-

-20-

/_ ...... cte|&l tq

/ _ -- 411ClCtUi&l gq

'rime ( lee ]

Fig 5. Calculation of torque with the inverse dynamic

method

(a) (b)

, _ j,_ rm_-u, T -- __ ¢.aa.IN _ I,,_

}. i =::

,2L . ..................

Fig 6. Trajectory generation by using the inverse

dynamic method: a) Joint angle, b) Strain at base, and

midpoint

The calculated torque was applied to the ideal flexible

manipulator model from which the inverse dynamic

model was derived. As the results are shown in Fig. 7,

the end point follows the desired trajectory exactly, and

no undershoot, no overshoot' nor noticeable residual

vibration occurs. However, the strain shows oscillations

after following the trajectory. It seems to be due to

numerical integration errors.

191 •,-,- " _" OUAL _
that a simple joint feedback PD controller performs

excellent tracking if it is combined with the inverse

dynamic feedforward control, and if the natural

trajectory of a joint angle is provided.

Joint Angle

t.O

0.8

O.E

0.4

_ 02

OC

-o_io

.... Oesired

...... 5;mulotion (operdoop)

-- Experiment (closedloop)

.k)_,t Amlkl 0 IL_t rid

, [hd Pmnt Pos,lion 4_)-

0.2 04 0.6 0.8 1.0 1.2 14 16 1.8 20 2.2

Time ( see )

00004

0 0003

_" 00002

0000_

00000

-00001

-0 0002

-0 0003

-0 0004

0 0 0 2 0 4 0 6 0 B

Stroln

.... Des,red

SimuIot,on(oPenloop)

-- [aper,ment (¢losedSoop)

10 12 14 16 18 2.0 22

Time(see)

Fig 10. Experimental results of the combined control of

the inverse dynamic feedforward control and the joint

tracking feedback control: a) Joint angle, b) Strain at

base

7. Conclusion

The proposed inverse dynamic method provides

the simple way to generate the required torque profile

in the time domain. Due to the simple computation

method, the implementation to the interactive control

became easy. Generation of the natural flexible

coordinate trajectories is quite useful for feedback

control. And the characteristics of flexible coordinates

were newly interpreted with causal and anticausal

concepts. The feedforward control using the inverse

dynamic method shows good tracking performance with

a simple joint feedback controller. In other words, this

inverse method can relieve the use of many sensors, or

feedback calculation burden such as observer

reconstruction.

Although several advantages of this method are

mentioned, this inverse dynamic method is limited to

linear systems. In order to be applied to a multi-link

flexible manipulator, this method should be combined

with a nonlinear compensation technique.

Acknowledgment

The Authors would like to acknowledge the

support of this research by the Nado_d Aeronautics and

Space Administration under the Grant No. NAG 1-623.

References

I. Asada, H. and Ma, Z, "Inverse Dynamics of Flexible

Robots", Proc. of American Control Conference, pp.2352-

2359, 1989.

2. Bayo, E., "A Finite Element Approach to Control the

End-Point Motion of a Single-Link Flexible Robot',

Journal of Robotic Systems, Vol.4, No.l, pp.63-75, 1987.

3. Bayo,E. and Paden, B., "On Trajectory generation for

Flexible Robots', Journal of Robotic Systems, Vol.4, No.?-,

pp.229-235, 1987.

4. Bayo, E. and Moulin, H., "An Efficient Computation

of the Inverse Dynamics of Flexible Manipulators in the

Tune Domain', Proc. of IEEE Conf. on Robotics and

Automation, pp.710-715, 1989.

5. Cannon, R. and Schmitz, E., "Initial Experiments on

End Point Control of a Flexible One-Link Robot',

International Journal of Robotics Research, Vol.3, No.3,

ppA9-54, 1984.

6. De Laca, A. and Siciliano, B., "Joint-Based Control of

a Nonlinear Model of a Flexible Arm', Proc. of

American Control Conference, pp.935-940, Atlanta, June

1988.

7. Hastings, G. and Book, W. J., "Experiments in the

Optimal Control of A Flexible Manipulator', Proc. of

American Control Conference, pp.728-729, Boston, 1985.

8. Oosting, K. and Dickerson, S. L., "Simulation of a

High-Speed Lightweight Arm', Proc. of IEEE Int. Conf.

on Robotics and Automation, pp.494-496, Philadelphia,

1988.

9. Wang, D. and Vidyasagar, M., "Modelling and Control

of Flexible Beam Using the Stable Factorization

Approach', Proc. of ASME W'mter Annual Meeting,

Robotics: Theory and Application, Dec., 1986

193
I I II IIIII III I IIIIIIIIII I I

I

L4 oe I_fl_o{t_ie_ ia 1_ 1o z2

emm

A

4a

Fig 7. Simulation of the open loop contro wit1 the

inverse dynamic method

To demor_trate the excellence of this method, we

compare the resu!t with those of other typical me:hod._.

In Fig. 8 (curve .',), the result of a collocated joi[:t PD

controller fo r a step input command is demonstrated.

As we can expect, the feedback of position error

generates very high peak torque at the begimting, and it

excites the system natural frequencies. Hence, it

requires relatively long settling time. It also shows the

undershoot and the overshoot of the end point position.

6O

5O

4O

vfo

20

I0

End Point Position

/7

_ Desired •

Step mput

_ N_inal troj_

_ _ _ _inverse m t_ai_

0.2 0.4 0.6 0.8 t .0 1.2 14 t 6 t.8 20 z,2

Time (see)

Fig 8. Comparison of typical control methods: a) Joint

feedback control with a step input b) Full state

feedback tracking control with nominal trajectory c)

Full state feedback tracking with the inverse dynamic

method trajectory

As a typical alternative method, a nominal joint

trajectory tracking control can be used to improve the

response instead of a step input. This tracking control

is demonstrated with the very smooth nominal position

trajectory which is the same as the end point position

trajectory. Desired flexible coordinate values were set

to zero, q!- 0, q,- 0 . The feedback gain was

selected by LQ method, and the full state feedback was

used. Even though the response is better than that of

step input joint feedback case, it still has overshoot and

requires settling time. This poor tracking response is

due to the unnatural commands which assigns zero

values to the flexible coordinate commands. If we give

192

the natural flexible mode trajecteries, which was

obtained from the inverse dynamic mefhod, we can

i, nticipate better response. As it is shown in Fig. 8

(curve c), by assigning tl,e ratur.1 flexible mode

t:ajectory, good trazking perfornlanc¢ was achieved.

6. Experiment

Although the open loop control wi'h the inverse

dynaLqiC method showed the good re:ults with an ideal

model, it produced large positioning errors with the real

system, which has ,'iscous a.ad coulomb friction at the

jnint. Therefore, a joint feedback control loop was

added to give robustness to the feedforward control of

the inverse dynamic method. A friction compensation

also was added to cancel the friction effect as shown in

Fig. 9.

The above combined control scheme of the

inverse dynamic feedforward control and the feedback

control was implemented to the experimental single link

flexible manipulator, which has 47 inch long arm and a

0.1 lb end mass. It was driven by an Inland D.C. servo

motor with a current amplifier. For a fast real time

control, A Micro Vax II was used with 12 bit A/D and

D/A boards. The off line calculation of a trajectory and

a torque profile was also performed in the Micro Vax

interactively.

b flexible •

"*nt_"l*t°_ i 1

I

Fig 9. The control scheme of the experiment

By applying the precalculated torque and

compensate the friction, and using the feedback of the

tracking error at the joint, the excellent result of Fig. 10

was obtained. The flexible manipulator could stop

without any overshoot or any residual vibration after it

moved 40 inches (48.76 degrees) within less than 1

second. In the strain signal, there exists rough jerk

which could be vanished by using more smooth

acceleration profile.

Unfortunately, as the end point position sensor was not

attached to the system, the end point position couldn't

be measured directly. However, the end point tracking

performance can be estimated by joint angle tracking

result, and the strain on the beam base tracking result.

In experiment, only joint angle, and joint velocity signals

were used for feedback. The experimental results show

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