ISARC-2002-081.DOC- 1 – 

ADAPTIVE CONTROL OF A CONSTRUCTION MANIPULATOR 

by 

V. Gradetsky a

, M. Rachkov b, M. Pushkin a

a Institute for Problems in Mechanics, Russian Academy of Sciences, Vernadsky ave. 101 

Moscow 117526, Russia, Phone: +7-095-4344149, gradet@ipmnet.ru 

b

 Moscow State Industrial University, ul. Avtozavodskaya 16, Moscow, 109280, Russia, 

Phone: +7-095-3684883, rachk-v@mail.msiu.ru 

ABSTRACT: The paper describes an advanced control technique that uses a self-adjusting 

controller. The technique allows changing control parameters of the manipulator according 

to different technological forces and disturbances acting to the end-effector of the 

manipulator while performing surface treatment construction operations such as cleaning 

and polishing. The system is asymptotically stable for the phase error and can be 

implemented on a base of existed industrial robots. 

KEYWORDS: adaptive control; construction manipulator, parametrical error; reference 

model; self-adjusting controller. 

1. INTRODUCTION 

Automation of some surface treatment 

construction operations such as cleaning and 

polishing by means of manipulators demands 

adaptive control that allows keeping required 

technological forces of a tool at any disturbances 

from surface unevenness. 

There is a program that allow for efficient 

management of relevant operational parameters 

related to surface finish quality [1]. This program 

is intended for on-board computers in graders or 

pavers. 

One of approaches in manipulator control is 

based on a calculation method by incomplete 

information about external parameters [2]. 

However, the known algorithm of control is 

characterized by complexity of calculations and 

is usually used for manipulators with two or three 

degrees of freedom. 

Optimal positioning of a pneumatic manipulator 

with minimum control energy consumption is 

solved in [3]. This system can place a tool with 

high accuracy taking into account information 

about current position of the manipulator but 

some disturbances can act to the tool from treated 

surface unevenness while operating. 

The proposed technique solves a task of 

qualitative control of the manipulator tool with a 

reference model where a wide range of possible 

dynamic properties can be changed. The 

technique allows changing control parameters of 

the manipulator according to different 

technological forces and disturbances acting to 

the end-effector of the manipulator while 

performing surface treatment construction 

operations such as cleaning and polishing by 

means of a self-adjusting manipulator controller. 

2. SYSTEM WITH A REFERENCE MODEL 

Let's consider the approach based on the selfadjusting system with a reference model on the 

basis of information about object parameters 

that is received while functioning. The active 

change of regulator parameters should be 

carried out on a base of this information. 
ISARC-2002-081.DOC- 2 – 

The self-adjusting system with the reference 

model can be described by the block diagram 

presented in Figure 1. 

The system motion is described by the linear norder differential equation with constant 

beforehand-unknown factors, which can vary in 

a wide range scale 

( ),

( ) [ ( )] ( ) 1

( ) ( )

f t

x t a a a t x t

n

i

n i

i i i

n

=

+ ∑ + ∆ + = =

− δ

 (1) 

where ∆ai - constant beforehand-unknown 

factors, a (t) δ i - factors, generated by the selfadjusting circuit. 

The reference model is described by the 

equation: 

 ∑=

− + =

n

i

n i

i

n y t a y t f t 1

( ) ( ) ( ) ( ) ( ) , (2) 

where y(t) - output signal of the reference 

model, f (t) - control signal. It is supposed that 

the reference model is asymptotically steady. 

The purpose the system with the reference 

model is to design such a self-adjustment circuit, 

at which x(t)→ y(t), t →∞ by generated 

a (t) δ i . The equations of the object and the 

reference model in the matrix form are 

[ ]

( ), ( ) ,

( ) ( ) ( , , ) ( )

0 0 f t x t x

x t Ax t A A t x y x t

+ =

& = + ∆ +δ +

 (2) 

( ) ( ) ( ), ( ) , 0 0 y& t = Ay t + f t y t = y (3) 

where n x(t), y(t)∈R - phase coordinate 

vectors of the object and the reference model, 

A - a real constant matrix (Gurvitz matrix) that 

can be written as 

























=

a a a an

A

...

0 0 0 ... 1

.................

0 1 0 ... 0

1 2 3

, (4) 

∆A - real constant matrix with beforehandunknown factors depending on the control 

object, δ A(t, x, y) - matrix of parameters 

changed by the self-adjustment circuit. The 

matrixes ∆Aand δ A can be expressed as 

























∆ ∆

∆ =

a an

A

...

0 ... 0

.............

0 ... 0

1

, 

 (5) 

























=

a an

A t x y

δ δ

δ

...

0 ... 0

................

0 ... 0

( , , )

1

. 

The control vector is f (t) (0,...,0, f (t)) n

T = , 

where the top index hereinafter means a 

transpose operation. For an error vector 

ε (t) = x(t) − y(t) , subtracting equations (3) 

from (2), we have 

[ ]

0 0 ( )

( ) ( ) ( , , ) ( ),

ε ε

ε ε δ

=

= + ∆ +

t

& t A t A A t x y x t

 (6) 

or in more convenient form 

0 0 ( )

( ) ( ) ( ) ( , , ),

ε ε

ε ε α

=

= +

t

& t A t X t t x t

 (7) 
ISARC-2002-081.DOC- 3 – 

where X (t) - matrix, at which the last row 

coincides with the vector x (t) T , and 

α (t, x, y) - vector of a parametrical error 

( ) ( , , ),..., ( , , )

( , , )

1 1 a a t x y a a t x y

t x y

n n

T

δ δ

α

= ∆ + ∆ +

= . 

 

 

The Lyapunov ́s second theorem is used for a 

synthesis of the self-adjustment circuit. Let's 

choose Lyapunov ́s function as the square-law 

form 

ε α ε ε α α T T V (t, , ) = Г + , (8) 

where the positively certain matrix Г satisfies 

to the matrix equation 

A Г ГA CE T + =− . 

 

The equation of the self-adjustment circuit is 

α& (t) =θ (t,ε , x, y) . 

 

 

A derivative of the V function by this equation 

should be not positive, i.e. 

≤0

dt

dV . 

The derivative of the function (8) by system (7) 

is equal 

0 2

2

+ + =− ≤

=− + + +

α α α α ε

ε α ε ε α

C

V C X Г ГX

T T

T T T

& &

&

, (9) 

if the self-adjustment circuit is described by the 

following equation 

(t) X (t)Г (t) T α& =− ε . (10) 

 

The algorithm (10) covers the majority of 

known algorithms of adaptation. If vectors 

x(t), y(t), ε (t) are measurable, the algorithm 

(10) can be implemented. Thus, the system (7), 

(10) is stable for ε (t) and α (t) . 

3. DEFINITION OF THE SELFADJUSTMENT CIRCUIT VECTOR 

In practice, there are objects, in which the 

matrix of factors depends on some number of 

parameters. Let the object is described by the 

equation 

n x&(t) = A(q)x(t)+δ Z(t) + f (t), x ∈R , (11) 

 

and the reference model is described by the 

equation 

n

n

f t R

y t A q y t f t y R

∈

= + ∈

( )

( ) ( ) ( ) ( ), , 0 & (12) 

where m q ∈ R - the vector of parameters, from 

which a matrix of object is depended, ( )

0 A q - 

Gurvitz matrix, δ Z(t) - vector generated by 

the self-adjustment circuit. The task consists in 

definition of structure and algorithm of change 

of the vector δ Z(t) to have 

ε (t) = x(t) − y(t) →0, t → ∞ . The vector 

of a phase error ε (t)satisfies to the equation 

[ ]

( )

( ) ( ) ( ) ( ) ( ) 0 0

Z t

t A q A q A q x t

δ

ε ε

+

& = + − +

 . (13) 

Decomposing A(q) in a row, we have 
ISARC-2002-081.DOC- 4 – 

( ) ( ) ( ) ( ) 1

0 O q

q

A q A q A q q

m

i i

i + ∆

∂

∂ − = ∑∆

=

, 

m q − q = ∆q, ∆q ∈ R 0 , (14) 

where values of derivatives in (13) are 

calculated for 0 q = q . 

Let's enter the vector δ Z(t) , generated by the 

self-adjustment circuit, as 

( ) ( ) ( )

( ) ( ) ( ),

0

1

x t

q

A q B t

Z t q t B t

q q

i

i

m

i

i i

=

=

∂

∂ =

δ = ∑δ

. (15) 

Here q (t) δ i - scalar functions that is 

determined below. 

Taking into account expressions (14) and (15), 

equation (13) is possible to write as 

∑=

+ ∆

∂

∂

+ ∆ +

= +

m

i i

i i x t O q x t

q

A

q q

t A q t

1

0

( ) ( ) ( ) ( ).

( ) ( ) ( )

δ

ε& ε

 (16) 

Let's assume, that the matrix A(q) depends on 

the parameter q linearly. Then, last member in 

the equation (16) is absent and it will have a 

form 

 

 

∑ ∂

∂ ( ) = ( ) ( )+ ( ) ( ) 0

x t

q

A

t A q t t

i

βi ε& ε , 

(t) q q (t) β i =∆ i +δ i . (17) 

The further synthesis of the self-adjustment 

circuit uses Lyapunov ́s function. It can be 

chosen as 

∑=

> =

= +

m

i

T

i

T

t Г Г Г

V t Г

1

2

( ), 0,

( , , )

β

ε β ε ε

, (18) 

where the matrix function Г generally looks 

like Г = Г(t, x, y). 

A complete derivative of this function, taking 

into account equations (17), is 

 

∑ + +

= + + +

( )( 2 ( ))

( ( ) ( ) ) 0 0

t B Г ГB t

A q Г ГA q Г

dt

dV

i i

T T

i i

T T

β ε ε β

ε ε

&

&

 . (19) 

Since a matrix Г is symmetrical, we have 

i

T T Bi Гε =ε ГB . Let's consider the following 

algorithm of adjustment of parameters q (t) δ i 

(0) , (0) 0

( )

( ( )) ( )

= ∆ =

















∂

∂ =−

= = − =

i i i

T

i

T

T

i i i

q q

Г

q

A

x t

q t t B Г

dt

d

β δ

ε

δ β ε &

 . (20) 

Let matrix Г(t, x, y) satisfies to conditions 

( ) ,

( , , )

2

3

2

2

2

1

p A Г ГA Г C p

C p p Г t x y C p

T T

T

+ + ≤ −

≤ ≤

& (21) 

in which ( )

0 A = A q , the matrix A(q) linearly 

depends on parameters, ( )

0 A q - Gurvitz matrix. 

Then it is possible to show, that the system (17), 

(20) is stable for ε (t) and (t) β i , and also is 

asymptotically stable for the phase error. If the 

matrix A(q) depends on parameters not 

linearly, the system may be not stable. 
ISARC-2002-081.DOC- 5 – 

However, described self-adjustment circuit has 

the following property. If the control signal is 

restricted, the phase error is also limited, and 

the error can be so small as desired, if the 

parametrical error between the object and the 

reference model is small enough. It results from 

general stability theorems. 

4. CONCLUSIONS 

The advanced control technique that uses the 

self-adjusting controller is developed. The 

technique allows changing control parameters 

of the manipulator according to different 

technological forces and disturbances acting to 

the end-effector of the manipulator while 

performing surface treatment construction 

operations such as cleaning and polishing. 

The system is asymptotically stable for the 

phase error in a linear case. In non-linear case, 

the phase error is also limited, and the error can 

be so small as desired, if the parametrical error 

between the object and the reference model is 

small enough. 

The described control technique can be 

implemented on a base of industrial robots RM01 or PUMA 560 that has six rotary joints and 

can perform different construction operations 

with complex trajectories of the tool. 

5. REFERENCES 

1. Skibniewski, M., Michalowski, S., Sobczyk, 

A.: “Predictive control systems in 

construction equipment”, 18th ISARC 

Proceedings, 2001, Krakow, Poland, pp. 

277-282. 

2. Fu, K.S., Gonsalez, R.C., Lee G.S.G.: 

“Robotics: Control, Sensing, Vision and 

Intelligence”, McGraw-Hill Book Company, 

New York, 1987, 624 p. 

3. Rachkov, M., Marques, L., Almeida, A.T.: 

“Optimal Control of Pneumatic 

Manipulators for Construction Application”, 

18th ISARC Proceedings, 2001, Krakow, 

Poland, pp. 253-258. 

Additional 

signal 

Reference

model 

Manipulator 

Regulator 

Self-adjusting circuit 

Disturbances from a 

construction surface

Parameter 

changing 

f(t) 

y(t)

x(t)

Figure 1. Self-adjusting system with the reference model