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Proceedings of the American Control Conference Philadelphia, Pennsylvania June 1998

HIGH PERFORMANCE SWING VELOCITY TRACKING CONTROL OF HYDRAULIC EXCAVATORS Bin Yao +,

Jiao Zhang ++,

Douglas Koehler ++,

John Litherland

++

+ School of Mechanical Engineering Purdue University, West Lafayette, IN 47907 [email protected]

++ Advanced Hydraulics Group, Caterpillar Inc. Joliet , Illinois 60434-0504

Abstract

T h e objective of the swing motion control of a n industrial hydraulic excavator is t o regulate the supplied flow rate to hydraulic swing motors such that the excavator follows the swing velocity command given by the h u m a n driver with a smooth acceleration/deceleration in spite of various uncertainties. Dificulties in the design of a high performance controller are: ( i ) T h e swing inertia i s time-varying and unknown due t o the movement of the linkage and the unknown payload in the bucket; (ii) Hydraulic parameters such as bulk modulus exhibits large variations during actual operation; (iii) T h e system is subjected t o mild extent of uncertain nonlinearities (e.g., leakage flows of swing motors) during normal operations o n a flat ground; and (iv) T h e syst e m m a y experience large extent of uncertain nonlinearities when operating on a n uneven ground; the swing torque due to gravity forces o n a slope surface i s usually very large and changes quite rapidly. All these dificulties are addressed in the paper and a high performance adaptive robust control algorithm that can deliver the required performance is presented. Simulation results are presented t o illustrate the proposed method. 1 INTRODUCTION Hydraulic systems have been used in industry in a wide number of applications by virtue of their small size-to-power ratios and the ability t o apply very large forces and torques. However, hydraulic systems also have a number of characteristics which complicate the development of high performance closed-loop controllers; dynamics of hydraulic systems are highly nonlinear [l]and have large extent of model uncertainties. Advanced control techniques have not been developed to address these issues. This leads to the urgent need for advancing hydraulics technologies by combining the high power of hydraulic actuation with the versatility of electronic control. In fact, hydraulic industry has been actively interacting with universities for help. The recent strategic partnership developed between Caterpillar Inc., the largest earth moving equipment manufacturer, and several major research universities including Purdue University is one such good example. In the past, much of the work in the control of hydraulic sys0-7803-4530-4/98 $10.00 0 1998 AACC

tems uses linear control theory [2, 3, 41. In [ 5 ] , Alleyne and Hedrick applied the nonlinear adaptive control t o the force control of an active suspension driven by a double-rod cylinder. They demonstrated that nonlinear control schemes can achieve a much better performance than conventional linear controllers. They considered the parametric uncertainties of the cylinder only. In [6], the adaptive robust control (ARC) approach proposed by Yao and Tomizuka in [7, 8, 9, 101 was generalized t o provide a rigorous theoretic framework for the high performance robust control of a one DOF electro-hydraulic servo-system by taking into account the particular nonlinearities and model uncertainties of the electro-hydraulic servosystems. The effects of parametric uncertainties coming from the inertia load and the cylinder and the uncertain nonlinearities such as friction forces were considered. The non-smoothness of nonlinearities associated with hydraulic dynamics (e.g., the nonlinear function describing the relationship between the flow rate and the valve opening) was carefully examined. A physical intuition based modification was provided to overcome the difficulty in carrying out the backstepping design via ARC Lyapunov function [9] caused by the non-smooth nonlinearities of the hydraulic dynamics. This paper focuses on the high performance swing velocity tracking control of an industrial hydraulic excavator. An industrial hydraulic excavator consists of a base, a rotating structure driven by hydraulic motors to provide swing motions, and a robot-arm-like three-DOF linkage (or boom, stick, and bucket called in hydraulic industry). The linkage is mounted on the rotating structure and is driven by three independent hydraulic cylinders to provide the necessary motion for carrying and dumping the load in the bucket. The objective of the swing motion control is to regulate the supplied flow rate to the swing motors such that the resulting swing motion follows the swing velocity command given by the human driver with a smooth acceleration/deceleration in spite of various uncertainties. However, several difficulties exist in the development of a high performance swing motion controller. First, the swing inertia is time-varying and unknown due to the movement of the linkage and the unknown payload. This makes the swing motion control the-

818

oretical challenging; the approach in [6]can handle constant unknown inertia only. 5econd, the system has other parametric uncertainties (e.g.., bulk modulus) and large extent of uncertain nonlinearities such as the swing torque due to gravity forces and leakage flows. Third, the nominal swing motor operating pressuro should not exceed the swing motor line relief pressure. Partly due to these difficulties, current industrial hydraulic excavators have not been able to employ any kind of closed-loop control algorithm; instead, open-loop control is a common practice and the severe practical problems such as bouncing cannot be solved. All these difficulties will be addressed in the E aper. Thus the paper serves for two purposes: one is to advance the design of high performance adaptive robust controllers by addressing unsolved theoretical problems due to the time-varying and unknown swing inertia, and the other ir to construct simple and practical nonlinear robust controllers to solve practical problems to provide quality products.

2 PROBLEM FORMULATION A N D DYNAMIC MODELS For this initial investigat on, we consider the two major components of the swing cirmit only: (i) Rotating Structure of the Excavator; and (ii) Gwing Motors. Other related components such as valves ;and the pump are neglected and it is assumed that the pump and the valves can supply the designed flow rate. The analytical model of each individual element is given below. 2.1 D y n a m i c Model f Excavator Swin Motion The excavator swing m&on can be describef by

2.2 D y n a m i c M o d e l of Swing Motors The swing motor dynamics can be described by

%@I = Q Iz @ z=

- Qlossi(W,P1,Pz) - Qtoaarl(W,P1)

-Qz+ D m W

+

Qloasi(W,PlrPZ)

-Q1oaoe2(~>~2)

(3)

where p l and p z are the inlet and outlet pressures of the swing motor respectively, Q 1 is the supplied flow rate at the inlet; positive for the flow in and negative for the flow out, Q2 is the supplied flow rate at the outllet; positive for the flow out and negative for the flow in, Qiossi represents the internal or cross-port leakage flow; a simplified model is given by Qiossi = Ci,(pi - p z ) , and Q , i o s s e j , j = 1,2, represent the external leakage flows; a simplified model is given by Q i o s s e j = C,,(pj - p r e f ) where p,,f is the tank reference pressure.

3 SWING VELOCITY CONTROL STRATEGY The proposed control strategy consists of two parts: (i) Online desired swing velocity trajectory generation, and (ii) Velocity tracking control algorithm.

3.1 On-line Desired Velocity Trajectory G e n e r a t i o n This block is to provide a feasible desired velocity trajectory that the swing circuit can track. The purpose is to achieve a smooth acceleration/deceleration and pressure limiting. A by-product of this on-line algorithm is that we can use trajectory initialization t o reduce transient response of the velocity tracking controller.

I

q t ) &= D m p . P ~- iw

- T~,,., - T.

+ T~

(1)

where w represents the swing velocity, I is the swing inertia, PL := p l - pa is the load pressure built up in the swing motors, D, is a constant coefficient, Iw is the torque due to the change of swing iiertia, TI,,, = b F,sign(w) is the torque losses due to frict on of the swing motor and gears, in which b represents the coefficient of the viscous friction forces including swing motor torque losses and F, is the Columb friction force, T, represmts the reaction torque of the contacting environment (e.i;., the reaction force of a wall), and Tg is the torque due t o gravity forces on a slope ground.

+

The detailed expressions of I ( t ) and Tgare given in [ll],from which the followin ger era1 statements can be concluded. The swing inertia fconrlists of three parts given by I = I1

+ Tz(4J(t))+ 13(4J(t), m L )

(2)

As seen from (l),the load pressure PL is proportional to the inertia load I h . So for pressure limiting, the maximal allowable desired acceleration ~ J Mshould be smaller enough such that IWM

IDmpPLmas

(4)

where P L is the ~ maximal ~ ~ allowable working pressure. Since I is unknown, a conservative bound can be calculated by G~ =

DmpPLmaz Imaz

(5)

where I,,, is the largest possible inertia load that the system is expected to encounter.

For driver's comfort, it may be desirable to limit the rate of pressure change. By differentiating (l), it is obvious that @ is proportional to I$. Thus, the rate of pressure change can be indirectly regulated by limiting the maximal allowable jerk W M to GM =

where PL,,, rate.

DmpPLmas Imas

(6)

is the maximal allowable pressure changing

where I1 is the swing inertia due to the upper structure, which is constant., 1 2 is the swing inertia due to the boom, stick , and bucket, which is a function of the link position q E R3 and thus time-tarying, and 13 is the swing inertia due to the inertia load in the bucket, which is a function of the link position q and t ne mass of the inertia load m L ; m L

Let winput(t)be the swing velocity commanded by the driver. The desired swing velocity trajectory wd(t) is generated through a second-order filter with an acceleration and jerk limit, i.e.,

is normally unknown.

with the constraint that Iwdl 5 W M and lt3dI 5 3 ~the; detailed trajectory generation algorithm is given in [ll].The initial values wd(0) and w d (0) will be chosen later t o minimize the transient response. 'with a break frequency of ut and a damping ration of d ( t ) - W : ( W d ( t ) - Winput(t))

(7)

where e, = W - W d is the velocity tracking error. The detailed design is given in [ll]and the resulting control function is

3.2 Velocity Tracking Controller The closed-loor, velocity tracking control law is synthesized based on the &ysical models in section 2 . From ( 2 ) , the swing inertia can be separated into two parts given by I =IC(@))

+

(8)

Ic,(dt))mL

I(t);t = D m p P ~ (i, PL = & [ Q L

+ ~ , , P , ) w-

- Dmw - Q l o s s 1 +

+ dZ, I = I , + I + P ,

04

8;

where PL = p l and Q L = Q1 or PL = -p2 and Q L = depending on the port t o be controlled (inlet for p l and outlet for p 2 ) . Flossand Q l o s s are the estimates of torque losses and flow losses respectively, and the lumped physical parameters PI, pz, P 3 , and p 4 are defined as

For controller design purpose, we also define P 5 and P5 06

&

PLds

where I,(&)) and I , p ( q ( t ) ) can be calculated based on the link configuration q ( t ) ;the analytical expressions for the calculation are given in [ l l ] . The physical model ( 1 ) and ( 3 ) are then rewritten in the following form for controller design:

= PIP3 = m L % = 0104 = m L % Q i o s s

P6

+

= PLdo PLda = [?loss -PZ = - lDC m ps i e 2

PLd PLda

+

iwd

+i w d - kwIc(t)ew]

(13)

where irepresentSAtheestimate of 0 by substituting the parameter estimate /3 for ,B in the expression of 0 , k , > 0 and C s l > 0 are positive scalars. It can be seen from (13) that PLd consists of model compensation (the first four terms of &a) with a desired compensation structure [ 1 2 ] , a simple proportional feedback control law with a time-varying gain k , I c ( t ) , and a nonlinear robust feedback term PLds. The purpose of using a time-varying linear gain is to achieve a constant bandwidth over a large operating range since the swing inertia I changes with the link configuration.

Step 2: Since PL is not the actual control in ut, the second Q L so that step is to design a desired flow rate f L d the actual load pressure PL tracks the esired load pressure PLd; this step is based on the swing motor equation (the second equation of (9)) and the design is accomplished by dissipating an total-energy-like positive function given by

gr

v = VI + ijwpIe:

as (11)

which represent the coupling effect between the unknown inertia load and the unknown motor dynamics.

(14)

where e p = PL - PLd is the pressure tracking error and wp is a weighting factor. Using the newly proposed adaptive robust control techniques [lo, 7, 13, 81, the resulting control law for QL is Q L ( W , P L , P , W d r L j d r W d r q , q ,= q) QLdr + Q L d s

where bY

Assumptions: (A1 ) Valves and the pump can provide the designed flow rate Q L ; (A2 ) Velocity and pressure sensors are available, i.e., w 1 p 1 , p 2 and p s are available; (A3 ) Physical quantities P ( t ) are bounded with known bounds, i.e., Pmzn 5 p(t) 5 Pmax, where Pmtn = [Plmzn , . . . ,P6mznIT and = [Plmox, . . . ,PSmax]' are known; (A4 ) Positions, velocities, and accelerations of boom, stick, and bucket (i.e., q ( t ) ,q(t),and q ( t ) )are available for the calculation of known parts of the swing inertia and their derivatives, i.e., I , and ICs in (8) and their derivatives I, , &, I,, and &p can be calculated.

(15)

represents an adaptive-control-like term given

QLda

- p

h

fi * ( D m p p ~ - i W - ?,os* + P Z ) + 1% in which IC, > 0 is any feedback gain. &Lds is a robust IPLd

control law given by QLds

= - { c ~ z (nI*ll4yll+YZ/%l)'+C=a

-C*4Ic

+'YZl%il4p21))

+

(WI*/I@pll

cs6 ( I ~ P ~ ~ P eM p ) ~ }

(17)

Justification of Assumptions: Assumption A1 is necessary for any kind of control algorithm development. A2 is satisfied under current hardware setting. A3 is practically reasonable since we know the extent of model uncertainties the system is going to handle; for example, the lower bound of the load p1 will be zero and the upper bound will be the maximal load that the bucket can hold. Assumption A4 can be removed at the expense of a degraded performance.

in which 'I = d i a g { y l , y 2 , y s , y 4 , y s , y s } > 0 is the diagonal adaptation rate matrix, Csj are constants given by C ~= Z ~ p I m a x / ( 2 k w P 3 m i n ) , Cs3 = w p / h m i n , Cs4 = k p / ( 2ImazP3max), Cs5 = w p / ( d I m i n P ~ m i n & ,p )and

The design of tracking algorithm consists of two steps as follows:

where Proj represents the widely used projection mapping [14, 7 , l o ] , and the adaptation function is given by

A

#w1

= -(icpWd

= A +i,pw eP

+- I,,%,

Icpcd)

4p2

A

-*

(18)

The parameter adaptation law is given by

b = P T O j ( r 7 p ( W , P L rb l W d r h d , f i d T

7'4

Step 1: The swing motion is produced by the motor pressure PL as described by the first equation of (9). So the first step is to design a desired load pressure PLd for PL so that any feasible desired velocity trajectory W d ( t ) can be tracked when PL = PLd. This can be accom lished by dissipating an energy-like positive function given %y

rp

2

1

ew#wl

~

+wpep

[

F z c q

4, i')

~

(19)

(20)

Icpffzcq

It is proved in [ l l ]that the above control law (15) with the adaptation law (19) achieves

8 20

R1 Stability in spite of model uncertainties associated with the physical quantities

R2 A guaranteed trans ient performance and final tracking accuracy; the tracking error e, can be made arbitrarily small by increEiing feedback gains k, ,k,, and C,I , transient performance . and decreasing E ~ Guaranteed is especially impor;ant for practical applications (e.g., swing control) since execute time is normally short. Theoretically, this result is what a well-designed robust controller can achieve.

R3 Zero final tracking error without using infinite feedback gains when /? is constant. Practically, this result implies that even a low bandwidth controller (feedback gains are small) can have a very small tracking error, which is a nice feature for the swing velocity control since actual valve's bandwidth is low and sampling

ll

4 Sim lation Results

A simulation program has been developed to simulate the performance of the proposed swing control strategy. The following units are used to normalize different tities to minimize the eflkct of numerical calculation error: Pressure: bar, Flow Rate: Llmin, Swing Velocity: radls, Torque: k N , Mass: 1000kcg,Length: m. Physical parameters are obtained from the spccification of an industrial hydraulic excavator manufactured 1)y Caterpillar Inc.. Specifically, the parameters needed for the calculation of the inertia I are: I = I,+I,pmL, I , = 123+15.7cq1 -s,1 +206.2C~,-15.1C,1Sq1 1.5Si1 +3.66Cq12 - 0.17sqi; 115CqiCqiz-5.4CqiS,iz 25Ci12

+

+

+0.69Cq123

0.75c,12s,12+0.2s~,2

+ -

-o.01sq123

-o.34cq1s1r123 +10.8Cql~Cq123 -0.17Cq12sq123 I,p = 0.053 4+1.23C,2,,, - 0.04Cq123Sqle3 +0.0003S,2123, 1.65Cq12 +0.38C, 23-0.006Sq123+51.8C921 +51.8cq1cq12 3.3cqi +21.6CqlCq123

+

-o.1gcplsq123 +12.96C,212+5.93Cqi~Cqi~~ -0.09Cq12Sq123 +0.68C,2123 -0.02Cq123Sq123 0 . 0 0 0 1 7 S ~ 1 2 3where

+11.85Cq1Cq123 c q 1

= cos(91),

+

sin(q1 4721, qz q3).

+

+

sq1

cqlZ3

+

= Si?L(9l), Cql2 = cos(91 921, s q 1 2 = cosi:q1 92 931, and s q l Z 3 = ain(91

The gravit

+ +

the simulation.

P(t).

torque Tg is given by

T~

=

+ =

+ m ~ ( 0 . 2 3 + 7 . 2 C , l+3.6Cq12 +0.82Cq123 -0.01Sq~~3)]sin(6')+(0.2+

B is the swing angle. The parameters needed for the calculation of the torque loss Z, in (1) and the flow losses in (3) are obtained from the swing motor specifications and are given by b , = 28.4, F, = 8.652, C,; = 0.0584, c., = 0.05572 . The swing motors constants are D,, = 0.8185 and Dm = 491.1 respectively. 0 . 0 3 m L ) c o s ( ~ ) } g s i n ( a ) where

The designed flow rate is assumed to be supplied by a set of valves with a supplied pressure of P, = 255bar and a tank pressure of p r , f = lbar. To verify the ideal performance of the proposed algorithm, the valves are assumed to have a relative high break frequcency of w, = 50Hz and a damping ratio of Cv = 0.71 so thai; the effect of valve dynamics may

Although simulation has been carried out for various operating conditions, only the following typicid operation is shown here. During the first 5 seconds of the simulation, the driver's commanded velocity is assumed to be winput = 0.5rad/s,which is near the full speed of the excavator. After that, the driver's commanded velocity is zero. To see the effect of fast changing link configuration, the movement of boom, stick, and bucket is assumed tro be described by = [30" 15" (1-COS(?t)), 120"-30"(1-MIS( $ t ) ) , 90° -

+

45"(1 - cos(;t))lT. In the simulation, the estimated torque loss; and flow loss are assumed t? be zero- to test t,he robustness o,f the algorithm, i.e., b, = 0, F, = 0, Cim = 0, and Cem = 0. The initial load estimate is zero, i.e., The actual values of the load and effective bulk modulus are assumed to be mL = 3000kg and /?e 2625bar while their initial estimates of m~ = 0 and /?e = 4pe are used to simulate the large variations of these parameters. Other controller paas follows: (i) bounds rameters used are categorized ., Physical are /?,in = [0, -100,27.6,-4421,0,-15030]T, and ,&,, = [3.4,100,110.5,4421,375.8,15030]T; (ii) Velocity trajectory = 18.85~Ct =I ' WA4 = 0'2177, planning parameters are and GM = 5.44;( 4 Velocity tracking contro1.k parameters , c8l = 1.63e7, are k, = 62.8,k, = 62.8,w, = 4 x E, = 100 a n d r = di~g{442,2xlO~,l.lx10~,:2.7x10~,4.26x

104,109}. The desired velocity trajectory profile and the actual velocity are shown in Fig.1. As shown, the desired velcicity trajectory approaches to the driver's command smoothly and the actual velocity follows the desired velocity very closely to achieve a smooth acceleration and deceleration. This verifies the velocity tracking capability of the proposed ARC strategy in spite of various model uncertainties. Furthermore, the working pressures shown in Fig.:! verify that the working pressures did not exceed the line relief pressure. All these show that the proposed strategy achieves the objectives set forth. To see the effect of parameter adaptation, we re-run the simulation but without using parameter adaptation. In such a case, the control law is equivalent to a deterministic robust control (DRC) law [7,81. The tracking errors e, are shown in Fig. 3 for both ARC and DRC. It can be seen that ARC achieves a better tracking performance than DRC due to the use of parameter adaptation. To see the effect of uneven ground, the above simulations are re-run with a ground slope of (Y = 7". Due to the large swing velocity between 1 to 6 seconds, the gravity torque Tg changes quite rapidly. Also, its magnitude is so large that it dominates model uncertainties-a perfect example where model uncertainties are mainly due to uncertain nonlinearities. Traditionally, in such a case, adaptive control will normally perform poorly and even become unstable sometimes. However, the tracking errors shown in Fig. 4 reveal that the proposed ARC still has an excellent velocity tracking capability and achieves a better performance than DRC

in spite of the unavoidable wrong parameter adaptation due to the appearance of large uncertain nonlinearities. Overall, the proposed strategy can handle the variation of changing gravity load well. 5 Conclusions

This paper considered the high performance swing velocity tracking control of an industria hydraulic excavator. An ARC control strategy has been ro osed t o handle various model uncertainties. Special$, &e scheme considered the effect of the unknown. and time-varying swing inertia, large variations of hydraulic parameters, the unknown leakage flows of.the swing motors, and the fast chan in and unknown swin gravity tor ue on an uneven groun%. &nulation results slowed that %e proposed strategy can deliver the expected swing velocit tracking with a smooth acceleration and deceleration wide keeping working ressures of swing motors below the line relief pressure. Impgmentation of a simplified version of the proposed a1 orithm on an actual industrial hydraulic excavator manufacfured by Caterpillar Inc. is underway.

Solid, Actual Dotted Desired ” -010

’ 3

4

5

6

7

8

Figure 1: Swing Velocity Tracking On An Even Ground 2

W

I

I

,

I

I

I

I

,

7

8

150

1W

H. E. Merritt, Hydraulic control systems. New York: Wiley,

2

1967. P. M. FitzSimons and J. J. Palazzolo, ‘‘Part i. Modeling of [2] a one-degree-of-freedom active hydraulic mount; part ii: Control,” ASME J. Dynamzc Systems, Measurement, and Control, vol. 118, no. 4, pp. 439-448, 1996. [3] A R. Plummer and N. D. Vaughan, “Robust adaptive control for hydraulic servosystems,” ASME J. Dynamrc System, Measurement and Control, vol. 118, p p 237-244, 1996. C. E. Jeronymo and T. Muto, “Application of unified predic[?I tive control (upc) for a n electro-hydraulic servo system,” JSME Int. J., Seraes C, vol. 38, no. 4, pp. 727-734, 1995. [5] A. Alleyne and 3. K. Hedrick, “Nonlinear adaptive control of active suspension,” IEEE Trans. on Control Systems Technology, vol 3, no. 1, pp. 94-101, 1995. B. Yao, G. T. C. Chiu, and J. T. Reedy, “Nonlinear adap[6] tive robust control of one-dof electro-hydraulic servo systems,” in ASME Internatronal Mechanical Engzneering Congress and Expositzon (ZMECE’97), FPST-Vo1.4, (Dallas), pp. 191-197, 1997. [7] B. Yao and M. Tomizuka, “Smooth robust adaptive sliding mode control of robot manipulators with guaranteed transient performance,’’ in Proc. of Amerzcan Control Conference, 1994 The full paper appeared in ASME Journal of Dynamzc Systems, Measurement and Control, Vol. 118, No.4, pp764-775, 1996. [8] B. Yao and M. Tomizuka, “Adaptive robust control of SISO nonlinear systems in a semi-strict feedback form,” Automataca, VOI. 33, no. 5, p p 893-900, 1997. (Part of t h e paper appeared in Proc of 1995 American Control Conference, pp2500-2505). [9] B. Yao and M. Tomizuka, “Adaptive robust control of MIMO nonlinear systems,” 1997. Submitted t o Aatomataca (under revision). Parts of the paper were presented in the IEEE Conf. on Decision and Control, pp2346-2351, 1995, and t h e IFAC World Congress, Vol. F , ~ ~ 3 3 5 - 3 4 1996. 0, [lo] B. Yao, “High performance adaptive robust control of nonlinear systems: a general framework and new schemes,” in Proc. of IEEE Conference on Deczsion and Control, pp. 2489-2494, 1997. [ll] B. Yao, “Hydraulic excavator (hex) swing control strategy.” Project Report, Aug 1997. [12] B. Yao and M. Tomizuka, “Comparative experiments of robust and adaptive control with new robust adaptive controllers for robot manipulators,” in Proc. of IEEE Conf. on Deciszon and Control, pp. 1290-1295, 1994. [13] B. Yao, M. Al-Majed, and M. Tomizuka, “High performance robust motion control of machine tools: An adaptive robust control approach and comparative experiments,” IEEE/ASME Trans. on Mechatronzcs, vol. 2, no. 2, pp. 63-76, 1997. (Part of t h e paper also appeared in Proc. of 1997 Amerzcan Control Conference ).

” 2

Time (s)

References [l]



1

E

o

Solid Load Pressure

-50 -1M)

-150

0

1

2

3

4

Time (s)

5

6

Figure 2: Working Pressures of Swing Motors loxioJ 8.

-81

0

. .

. I

3

,’. , .



1

:

,

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

‘: I



2

3

” 7 8



4

Time (sec)

5

6

Figure 3: Velocity Tracking Error On An Even Ground ,axlo* 14-

r’

:

\

12-

2 4.,

[14] S. Sastry and M. Bodson, Adaptzve Control: Stabilzty, Convergence and Robustness. Englewood Cliffs, NJ 07632, USA: Prentice Hall, Inc., 1989.

-0 4

,

1”

,

.

”2 ”

...,



Solid: ARC

.

Dashed:.DRC,

... :

3’

4

5

8

7

8

Time (sec)

Figure 4: Velocity Tracking Error On A Slope Ground ( a = 7”)

822