Robustness of a Few Control Methods Used in Robotics

Copyright @ IFAC Control Systems Design, Bratislava, Slovak Republic, 2000

ROBUSTNESS OF A FEW CONTROL METHODS USED IN ROBOTICS

Vesna KrajCi, Ladislav Jurisica

Dipl.Ing. Vesna KrajCi*, MSc; Pro! dr. LadislavJurisica**, PhD. Department ofAutomation and Control Faculty ofElectrical Engineering and Information Technology Slovak University of Technology Ilkovicova 3,81219 Bratislava, Slovakia *Phone: 00421 760291 599; E-mail: [email protected] **Phone: 00421 760291351; Fax: 00421 765429521; E-mail: [email protected]

Abstract: The paper deals with proving of robustness for a few simple control methods often used in robotics. These methods are: two variants of variable-structure control, a control method with an auxiliary controller developed by Hsia (1989) and a fuzzy control. Robustness of the proposed continuous-path control methods is evaluated in the case of planar articulated robot moving along two common used trajectories: straight-line and circular. Computer simulations performed with load change, added external disturbance and white measurement noise proved satisfying robustness of all presented robot control methods. Copyright @2000IFAC Keywords: robotics, robustness, variable-structure control, robust control, fuzzy control, robot arm, computer simulation.

system being controlled. If the reference tool tip trajectory is smooth, the error of each ith motor shaft angle ~(t) and speed dej(t)/dt can be chosen as system state variables:

1. IN1RODUCTION

Significant robot uncertainties, such as unpredictable robot parameter perturbation from the norm, unmodelled robot dynamics, external disturbance acting on robot and white measurement noise are very common in robotics. Therefore, the robot control methods have to be robust, i.e. they have to ensure accurate control system in the presence of these uncertainties. In addition, several robust robot control methods are explained: two variations of variable-structure control, a control method with an auxiliary controller developed by Hsia (1989) and a fuzzy control.

e;(t) == (1) (t) - 8;(t),

I::; i ::; n,

(I)

ej(t) == BrI (t) - Bj(t),

I::; i ::; n.

(2)

According to the (Schilling, (990), the following common error can be defined for state variables of the ith robot motor, l::;i::;n:

The goal of this control method is elimination of motor errors crj(t):

2. VARIABLE-STRUCTURE CONTROL

The control signal in variable-structure system is changed abruptly on the basis of the state of the

(4)

601

3. CONTROL METHOD WITH AN AUXILIARY CONTROLLER

For detennining the control law for the ith robot motor which ensures that the system is operating in the sliding mode, the following Liapunov function can be used for I~i~n (Schilling, 1990):

v, ( .) = CT; (e"e,)' CT,(e"e,) L;

(5)

2

e"e,

The control law proposed in (Hsia, 1989) contains two control signals: (12) The first part U1 ;(t) is a signal from a standard servo PD controller which ensures asymptotic tracking of the trajectory.

The controlled system will be asymptotically stable, if the following condition is fulfilled for all motors I~i~ (Schilling, 1990):

The second part U2;(t) is a signal from an auxiliary controller (Hsia, 1989):

(6)

The last condition can be fulfilled, if the following selection of dQ";(t)/dt, by using equations (3) and (4), is made for all robot motors I~~ (Chen, ef aI., 1990; Morgan and Ozguner, 1985):

(13)

where Jrni denotes moment of inertia for the ith robot armature-controlled DC motor (1~~), K Ai is coefficient of the ith amplifier, K,.;=11Ru, R.u is the resistance of the armature winding and K; is the torque constant of the ith motor.

From equations (7) and (3), the following control law can be derived for the ith robot motor, l~~n: UR; (f)

= u1; (f) + u2; (f),

Ut/f)

= r; ·sgn{K

G

;

•

This is used for system parameters uncertainty and disturbance effects cancellation.

[A.; -e;(t) +e;(t)]},

(8)

From the equation (13) it can be seen that the ith auxiliary controller needs the signal of the ith motor shaft acceleration d 28;1dt2, which can be derived from measured motor shaft angle or speed. In practice, the following filter can be used for realization of the ith auxiliary controller, as shown on Fig. l.

u2; (f) = KG; -A.; -e;(f). To reduce the chattering of the control signal, the first part of the control law (8) which contains sign function: CT;(t)

-r ; -Sgn[CT; (f)] = r ; .!CT;(t)I'

(9)

The time constant of the ith filter T fi has to be very small to ensure very good system output.

is replaced by continuous control signal proposed in (Hashimoto, ef aI., 1987) (the 1st VSC method): (10)

or by a continuous approximation with a large slope, i.e. saturation function, which is suggested in (Myszkorowski, 1990) (the 2nd VSC method):

-r;, l

r;-saf[CT;(f)]= r;.;;,

ri'

CT; < -/i;! -/i;
,

(ll)

CT; > /i;

Fig. I.

where ~\ denotes the thickness of boundary layer for the ith robot motor, 1~~n.

602

Auxiliary controller with filter for the ith robot motor.

4. FUZZY CONTROL METHOD

5. SIMULATION RESULTS

The fuzzy controller used in the ith joint servo control loop has two discrete inputs: motor position error signal ej(k) and change of motor position error signal dej(k):

The proposed control methods have been tested by computer simulations in programming language C++ in the case of moving the tool of a three-axis electric driven articulated planar robot (Schilling, 1990) along the two common used trajectories:

e;(k) = Br; (k) - B;(k),

1:5: i :5: n,

(14)

de;(k) = e;(k) - e;(k -I),

1:5: i :5: n.

(15)

The fuzzy controller operates in nonincremental mode, i.e. the current output value does not depend on the previous output values. Seven linguistic subsets have been defined for both inputs: large negative (LN), medium negative (MN), small negative (SN), zero (Z), small positive (SP), medium positive (MP) and large positive (LP). The distribution of membership functions for both inputs is shown in Fig. 2. As may be seen from that figure, the controller input membership functions have a triangular form and only two adjacent functions overlap. Because of simplicity and good interpolative features, the controller output value is calculated according to the centre of gravity principle (Kosko, 1992). In case of using regular and unimodal controller output fuzzy subsets, they can be substituted in the fuzzy rule-table with their centroid. During the creation of seven by seven fuzzy ruletable, the limits of motor torques have to be taken into account. The same fuzzy algorithm is used in all servo loops but with a proper scaling of the fuzzy controller inputs and outputs: I K•. = - - , ,

e·

K

d.,

1 =-de.

'max

(16)

'

Fig. 2.

straight-line: between the start point A (0.4 [m], 0.1 [m)) and the end point B (0.1 [m], 0.4 [m)),

b)

circular: clockwise along the circle (S (0.2[m], 0.2[m)), r=O.l[m)), with change of tool roll angle for 1£/2 [rad].

Lengths of robot segments are: a l =O.3 [m], a 2=O.2 [m] and the distance between the tool tip and the working plane is d 3=O.1 [m] (i.e. length of the third segment). Masses of the segments are: m\ = I [kg], m2=O.7 [kg], m 3=O.3 [kg]. The computer simulations have been performed by using realistic dynamic model of robot with viscous, dynamic and static joint and motor frictional forces (Schilling, 1990) (l:5:i:5:n):

The following friction coefficients have been used in simulations: viscous motor friction coefficients bvml = bvm2=bvm3 =O.00003855 [kg om 2/s]; viscous joint friction coefficients hv 1=O.5 [kg'm2/s], bv2 =O.25 [kg'm2/s], bv3=O.2 [kg'm 2/s]; dynamic joint friction 2 coefficients bdl =O.2 [kg om 2], b d2 =O.l [kg om ], 2 bd3 =O.05 [kg'm ]; static joint friction coefficients bs1 =2 [kg-m2], bs2 =1 [kg om2], bs3 =O.5 [kg·m2]; small constants &1 =&2=&3=0.1.

'max

where emu.x is maximal error of the ith motor shaft angle, demu.x is maximal change of this error, Uimax is maximal value ofith fuzzy controller output, 1:5:i:5:n.

-0.6

a)

-0.3

-0.\

o

0.\

0.3

0.6

e;!e-, de/dean

The distributions of the ith fuzzy controller input membership functions.

Other robot motor parameters are: resistances of armature winding R.l=~=Ru=8.2 [0], inductances of armature winding La1 =L a2 =L a3 =O.0000165 [H), torque constants K]=K2=K 3=O.0394 [Nom/A], moments of inertia Jm1 =Jm2=Jm3=0.OOO00268 [kg'm2], maximal armature currents Iam1 =I am2=Iam3 =O.745 [A], maximal output controller [V], amplifier voltage U Rm1 =URrn2 =U Rm3 =1 0 coefficients K A1 =KA2 =KA3 =2.4, gear ratios Nr]=291, Nr2=388, Nr3 =582. Parameters of controllers for the 1st and the 2nd VSC method are adjusted according to these principles: maximal allowed tracking error (0.5 [mm)), no chattering and minimum of energy. The parameters are the following: Kc]=Kc2=Kc 3=O.01, ".1=".2=".3=13 [I/s], 8]=~~=O.07; Y]=Y2=Y3=8.109 (1st VSC method) and Yl=Y2=Y3=4.185 (2nd VSC method) for straight-line; Yl= Y2=Y3=5.541 (1st VSC

method) and YI==Y2==Y3=3.664 (2nd YSC method) for circular trajectory.

-

, ,,, ,

- 1st VSC method - - 2nd VSC method

---- Bsia'B method

- - funy method

Parameters of Hsia's controllers are defined according to demand of maximal allowed tracking error (0.5 [mm]) and no overshooting as follows: acceleration coefficients K RaJ =KIW=KRa3 =
,, ,. 1

, ,, , ,, /1

/

I

,., .-.--

.--

~:::-.

The parameters of fuzzy controllers are also chosen according to the principle of maximal allowed tracking error (0.5 [mm]) and minimum of energy. The parameters are the following: Kel=Ke2=Ke 3=O.1, ~el=O.5, ~e2=O.5, ~3=2; Kul =35.l5, Ku2=25, K u3 =30 for straight-line trajectory; Kul =43, Ku2 =32, Ku3=20 for circular trajectory.

-=--=-=-:.:-=--.:::;-;o.;;-,-,,-~-

0,1

Fig. 4.

,

I " J __ "~ /

1

8.0

/

0.2

0.6

O.~

0.3 0.4 load [Iq]

Total error on straight-line trajectory in the case of load change.

25

The robustness of the proposed robot control methods is evaluated by changing robot parameters (load change) M 1i and adding sinusoidal external disturbances Mw and white measurement noise Om, as shown in Fig. 3 for the ith motor (1$i$n).

20

In computations of total energy along the trajectory a feedback of electrical energy into the network or into a storage battery is not considered (Desoyer, 1994):

tilt wc method 2nd WC method Hllia'lI method fuzzy method

-

- -- ---

9.t..O~""""'O.L.l"""""""""'0J..l.2~"""""'0':'".':'"3""""'''-'-'::'O'''''.''''''''''''''''''''O~.5~~'="'O.6 IDad [Iq]

Fig. 5. where T. denotes trajectory traverse time, Uai is armature voltage and lai is armature current for the ith motor. The trajectory traverse times, with defined limits of joint accelerations and velocities, are the following: T,=3.33 [s] for straight-line and T,=? [s] for circular trajectory.

Total energy on straight-line trajectory in the case of load change.

3.5 -

- -----

3.0

1

25 .

Oni (s)

Oi(S)

+J+

/ /

/

/ / / /

/ /

",2.0

M(s) +

/

lilt VSC method 2nd VSC method Bllia'lI method fuzzy method

/

e

/

'"

/

" 1.5

Oti(s)

, ,

:3oS 1.0 0.5

I

-r

---

-'

e--.:~-=:'-"':::=-----------=------'

Mm (s) 0'%.0

0.2

0.4

0.6

0.8

load

Fig. 3.

Load, disturbance and noise ofith motor.

Fig. 6.

The results of computer simulations, i.e. total trajectory tracking errors and total energy of all motors, are shown on Figs. 4-15.

604

1.0

[k.]

1.2

1..

UI

Total error on circular trajectory in the case of load change.

40

1.6

-

- 1st VSC method - - 2nd VSC method - --- Bma'lI method - - fuzzy method

35 ~30

, I I

I

::!..

I

~::

--

c:l

11

/" /"

/"

3 15 .s 10

-

- -- ---

i /"/f ) I

-

I

/"

1st VSC Jnethod 2nd VSC method Bm'lI method fuzzy method

IS 9.0

0.2

OA

0.6 0.6 1.0 load [ke]

1.2

1.4

0.002

1.6

5

S

40

j

- -

I

1st. VSC method - - 2nd VSC method ---- Baie.'& method - - fUZzy methocl

f,

,,"',', ,

!4 ,

...

~SO ~

I

~

,, I

..ea

I /

CIl

/ /

3a .s

~20

,

I

11

I

, ,, I

:3

-

- 1st VSC :method - - 2nd VSC method ---- BBi.'. method - - fuzzy method

.s 10

1

(1.001

0.002

0.004

disturbance


Fig. 10. Total error on circular trajectory in the case of external disturbance.

Fig. 7. Total energy on circular trajectory in the case of load change.

6

0.004 0.006 0.008 disturbance [Nm]

[Nm)

0.005

0.~,=,oo~..u...JOo'-.~OO'='2~..L.L..lOo'-.~004~L.L..L..uOo'-.~OO'='6~~Oo'-.~OO':!:8=-'-'"~Oo'-.'='OI0

0.000

disturbance

[Nm]

Fig. 11. Total energy on circular trajectory in the case of external disturbance.

Fig. 8. Total error on straight-line trajectory in the case of external disturbance.

1.0

20

,

i 10~------------------~

, ,,

/"

,

~O.8

! .......

~

-

0.8

o

Q 11

11

... 0 .•

-

.3

- -

hit VSC :method 2nd VSC :method - -- - HIli.'. method - - fUZzy method

-

.s 0.2

0.~':'Ou..u..uO='.-='00~1~.1..L..L;O:-'".002~:"-'-'-~O~.0:':04~"""""''=0~.O'=05=,",",u..u.O='".-='OOll di.turbance [Nm]

- -----

ht VSC :method 2nd VSC method HIIia'. method fuzzy method

0.0 O~~""""'-'-7-1"""''''''''''..u...J':.':2'''''''''~L.L..L.''-;3~''''''''''''''''''~'-'-'-~~5

noise

Fig. 9. Total energy on straight-line trajectory in the case of external disturbance.

r/~

Fig. 12. Total error on straight-line trajectory in the case of white measurement noise.

605

20

From the Figs. 4-7 it can be seen that maximal load of 0.55 [kg] for straight-line and 1.5 [kg] for circular trajectory rapidly causes a big increasing of total trajectory tracking error in all control methods, while the total energies are very similar.

---- /'

./

~10t----=--------~--------------

=

The results of acting the sinusoidal external disturbance ~i=MAdi·sin(2·t)+MAdi·sin(20·t) (MAdi changes from zero to maximal amplitude shown on x-axis) on all robot motors are shown in Figs. 8-11. The signals of total energies are again very similar for all simulated robot control methods.

Cll

:3o

-

.... 5

- -

llrl VSC method 2nd VSC method

---- Hlli&'1I method

- - - fuzzy

~ethod

°0l..LLL..L..l.JL.L.U...L.I...1..J...LL-U..L.'":J2...L.1...1.L.J...L.w..L5~.1..U-L.L.WLJ.4-U..L.w...L~5

noise

Et.]

The white motor shaft speed measurement noise (Figs. 12-15) (from 0% to 5% of the speed) has the greatest influence on the 2nd VSC method and the smallest influence on fuzzy and Hsia's controller.

Fig. 13. Total energy on straight-line trajectory in the case of white measurement noise.

6. CONCLUSION 0.8

Computer simulations performed with load change, added external disturbance and white measurement noise proved satisfying robustness of all presented robot control methods, with very similar signals of total energy.

- --

-

- -----

REFERENCES

lilt VSC method 2nd VSC method HIIia'. method fuzzy method

Chen, Y-F., T. Mita and S. Wakui (1990). A New and Simple Algorithm for Sliding Mode Trajectory Control of the Robot Arm. IEEE Transactions on Automatic Control, 35, 828829. Desoyer, K. (1994). Geometry, Kinematics and Kinetics of Industrial Robots. In: Proceedings of the International Summer School CIM and Robotics, Krems. Hashimoto, H., K. Maruyama and F. Harashima (1987). A Microprocessor-Based Robot Manipulator Control with Sliding Mode. IEEE Transactions on Industrial Electronics, IE-34, 11-18. Hsia, T. C. (1989). A New Technique for Robust Control of Servo Systems. IEEE Transactions on Industrial Electronics, IE-36, 1-7. Kosko, B. (1992). Neural Networks and Fuzzy Systems. Prentice-Hall, New Jersey. A Morgan, R G. and U. Qzguner (1985). Decentralized Variable Structure Control Algorithm for Robotic Manipulators. IEEE Journal of Robotics and Automation, RA-I, 57-65. Myszkorowski, P. (1990). Comments on "A New and Simple Algorithm for Sliding Mode Trajectory Control of the Robot Arm". IEEE Transactions on Automatic Control, 37, 1088. Schilling, R 1. (1990). Fundamentals of Robotics: Analysis and Control. Prentice-Hall, New Jersey.

D. 0 O~l..LLL.L.LO~1.1..U-""""'1..J....I..:!2""""'.L.J...L.w..L3~"""""'L.U...l..LL"""""..L.U-'-:5

noise

~J

Fig. 14. Total error on circular trajectory in the case of white measurement noise.

30 25

-

5

- -

llrl VSC method 2nd VSC method - - - - Blli&'1I method - - fuzzy method

°O~L..L..l.JL.L.U...L.I...1..J...LL-U..L.'":J2L.L.LO..J...LL.w..L5~.1..U-L.L.WLJ.4-U..L.w...L~5

noise

~J

Fig. 15. Total energy on circular trajectory in the case of white measurement noise.

606