Regulated sliding mode control of satellite rotation

CoPll'rI&bt

~IPAC

GENERALIZED SOLUTIONS IN CONTROL PROBLEMS. 2004.

GENERAL THEORY, LUMPED PARAMETER SYSTEMS. P. 185--103

REGULATED SLIDING MODE CONTROL OF SATELLITE ROTATION Ali A. Moosavian, M. Reza Homaeinejad Department of Mechanical Engineering K.N .Toosi University of Technology Tehran, Iran, P.G. Box 16765-3381 E-mail: [email protected]

Abstract: Development of space platforms, structures and satellites increasingly requires servicing, construction, and maintenance operations on orbit. In this paper, a regulated sliding mode control law is developed and applied to space assemblies during a rotational maneuver, which results in an improved performance compared to the conventional algorithm. Although the sliding mode controller usually results in an acceptable performance, but the chattering frequency is often high which may excite flexible dynamics of space systems. Besides, a chattering control demand may cause practical difficulties for actuators, particularly for the on-off type as is the case in space. In order to fulfill stability requirements, robustness properties and chattering elimination, a regulating routine is proposed to determine proper positive values for the coefficient of sliding condition. To this end, first the rotation dynamics of a typical satellite described in body coordinates is derived in terms of its angular velocity. Explicit relationship between actuator torques and satellite orientation is obtained using Euler quasi-coordinates. Next, focusing on the chattering phenomenon that is of main concern in space due to excitation of higher modes, a new approach is proposed to alleviate (ideally eliminate) the chattering trend. The developed control law is applied on a given satellite during a rotational maneuver, while parametric uncertainties and noisy feedback signals are taken into considerations. Also, it is assumed that only micro reaction jet actuators are available, which can generate a constant positive or negative (on-off) force. Therefore, the exact demanded torque can not be applied on the system. The simulation results show that the regulated sliding condition can significantly alleviate the chattering trend, which in turn make the on-off actuators properly follow the control demand. These results reveal the merits of the new regulated sliding mode control law.

Sliding mode control, Satellite control, Rotation, Uncertainty, On-Off Actuator, Simulation

Keyword8:

1. INTRODUCTION

In order to develop control systems for space assemblies, it is essential to develop proper kinematics/dynamics model for the system 1 • This has been studied under the assumption of rigid elements, [2-4], and also elastic elements, [5-6]. Due to complicated nonlinearities in space systems, and maneuver time limitations, besides restrictions of energy consumption in space, there have been various studies on the nonlinear control problem of such systems with both rigid and flexible elements [7-10]. Systems with nonlinear dynamics are controlled in a different way compared to linear systems. Systems which include uncertainties such as parametric or structural uncertainties, also need appropriate strategies to be controlled. Two important approaches used for tackling against nonlinearities and uncertainties are Robust control and Adaptive control. One of the main approaches in robust control is Sliding Mode control, [11]. Sliding Mode control is usually accompanied by a phenomenon called "chattering", [12-13]. Chattering should be avoided to reduce the energy consumption of the control system, and prevent any potential damages on actuators, especially in case of on-off type. In addition, due to high frequency content of chattering, it can easily stimulate the flexible modes which in turn may cause instability. To alleviate the chattering phenomenon one can use saturation functions instead of switching operators, which degrade the control precision. Another approach is to choose the controller coefficients by an intelligent

method such as adding an adaptive, a fuzzy or a neural network subsystem, or even a combination of them to the main control strategy, [14]. In this paper, a regulated sliding mode control law for space platforms during a rotational maneuver is developed which results in considerable performance improvements. To fulfill stability requirements, a regulation procedure is proposed to determine proper positive values for the coefficient of sliding condition, which satisfies robustness properties, and chattering elimination characteristics. To this end, the rotation dynamics of a typical satellite described in body coordinates is derived in terms of its angular velocity. Using Euler quasi-coordinates, explicit relationship between actuator torques and satellite orientation is obtained for integration purposes. Then, the developed control law is applied on a given satellite during a rotational maneuver. To examine the new controller merits, and to consider practical limitations, it is assumed that only micro reaction jet actuators are available, which can generate a constant positive or negative (on-off) force. Therefore, the exact demanded torque can not be applied on the system. Also, parametric uncertainties and noisy feedback signals are taken into considerations. The simulation results reveal the merits of the new regulated sliding mode control law.

2. SYSTEM DYNAMICS Considering typical space systems such as satellites, as shown in Figure 1, it can be assumed that the main structure consists of rigid elements. For a flexible system an appropriate dynamics model can be

I For a through review of the dynamics and control of space multi-body systems (space free-flyers) one could see ref. [1].

186

substituted for the rigid model. Therefore, rotational dynamics of the main rigid body of a given satellite can be outained based on Euler equations. To do so, the angular momentum (H) about center of mass (C) can be expressed in the body attached coordinat.e (C%y.) as

{ll}%yz = [1]

{~: }

Eqs. (4) into Eqs. (3), besides required simplifications yields

Ql = ~9 (11 cos (} Sill'lp) + ~Ip (11 sin (} cos 1/1) -- Olp (1 1 sin 1/1) + (h _. 12 ) 4)2 .. x ( -2 sin 2(}cos t/J + fjJT/Jsin 0 cos 'ljJ

(1)

- O~ sin 1/1 cos 0 -

where [I] describes the mass moment of inertia matrix. Based on Euler equations described as d Mc = diHe (2)

Q2 = ~9(hcos(}cos1/1)

L

- ~Ip (hsinOsin1/1) .- Olp (12 cos 4') + (11 - h)

where I: Me defines the resultant moment of all external forces auout satellite center of mass. Assumiug the body attached coordinate as principal axes, Eq. (2) yields = Ilwl + (13 - 12 ) W2 W 3 M2 =, 12W2 + (11 .- 13 ) Wj:;)3 M3 = hW3 + (/2 _. 11 ) WI W 2

x

sin 20 sin 1/1

+ ~Ip sin 0 sin 1/1

(3)

where M,'s arc the resultant external moment about principal axes. To find a direct relationship between the control inputs (il,f, 's) and the satellite orientation in terms of a set of Euler angles, one can substitute the angular velocity components by Euler angle rates, see [15), as (w, = 4>sinOsintb

(;2

+ B~cos't/J cos 0 + Blpcos 1/1)

Ml {

91p sin 'I/!)

(5a)

where

+ 9cosw

NI = M I - QI, N 2 = M 2 - Q2, N 3 = M 3 - Q3, Al = 11 sin Osin t/J, A 2 = Ir cos'lj;, B I = 12 sin () cos 1/1, B 2 = -12 sin 'Ij; Cl = 13 cos 0, C 2 = h to,. = A I E 2 - A 2 B I

(5b) F.A)s. (5) can be written in the following form

(6) Fig. 1. The main body of a typical satellite (Surrey Space Centre UoSat-12).

where:

where 4J,0 and t/J describe yaw, pitch, and spin angle, respectively. Substituting 187

- A 2Q2 - B2 QI f1 Ll '

be nonsingular, too. Therefore. it is defined that

R _ B2 M 1

- A2 M 2 Ll ' f - B1Ql - A 1Q2 2 Ll ' R = A 1 M 2 - BlMl

= (I + Ll) iJ ILl,] I ~ D,] t =

B

1 -

2

-Q3

'1;1

li, - f,1 ~ Fa

1

13 = - - - - (-B 2 QI + A 2Q2) , 9) C2 Ll M3 Cl 1 R 3 = C - C ~ (B 2 M l + A 2 M 2 ). 2

2

(7) where f, describes the dynamics for each angle and R. are the inputs that must be determined appropriately. It should be noted that Eqs. (6) describe the system dynamics in a suitable format for developing sliding mode nonlinear controllers as will be discussed in the next section.

d

S,

)n.-I

X,

x!n.-I) - X~:·-l)

s, =

() {~=I = f() , x + ~b ~ '] X u] _ 1.... m .=1 J- •... m

(11)

(12)

where X~~·-l) is computed based on the error between z, and Xad . For instance, considering a system with two state arrays and two independent inputs, it can be obtained

(8) where u describes control input array. z is defined as state matrix that is composed of x.'s as state arrays

[x.i, ... x:·-f

= ( dt + Aa

Xt = x, - Xad where A,'s are controller parameters and in fact are time constants in a low pass filter sequence, [12], and i, describes the tracking error of x,. Eq. (11) can be written as

Suppose a multi input nonlinear system defined by:

x, =

(10)

i,

where is the estimated value of f. that can be obtained from dynamics model, and Ll,]'s can yield the error values of the input matrix estimation procedure. In fact, the exact value of ~] is not known but the upper bound limitation (Le. D,) ) can be substituted. Therefore, the distance from a sliding surface is defined as

3. CONTROL LAW SYNTHESIS

x,(n) •

1, ... ,m J = 1, ... , m

(9)

s, =

and f, represents the dynamics of the tth state as a function of state vector z (not essentially a linear function), b,] is the corresponding element of input matrix "B" that describes the gain function of the lh input on subsystem "t" ,n. is the order of corresponding differential equation. and "m" is the number of independent inputs. The control aim can be expressed as making the state vector z follow the desired time dependent vector Zd. In the presence of modeling uncertainties, it is assumed that all parametric uncertainties appear in the input matrix B, and it is nonsingular in the state space domain. If iJ describes the estimated value of B. it is assumed to

(ft + Aa) .x,

= x, - (Xad - A,x,)

t=I.2

(13a)

which yields X r • = x... - A,X, ~ = 1,2 (13b) Therefore, in a general case, the control inputs must be determined such that satisfy the following sliding condition

1d

2

2dt s,

~ -1], Is.1 1],

>0

(14)

where 1], 's are controller parameters and chosen as positive values which reflect how the states are converged to their sliding 188

surfaces. The higher values of '7, shows that the corresponding state reaches its sliding surface faster. Assuming that K,'s are positive values that must be determined so that the sliding condition (14) is satisfied,. then the control law is obtained

noise effect will be decreased. Second, after reaching the state trend to the sliding surface, chattering may happen. Unfortunately, the amplitude of this phenomenon increases with a high value of '7,. Therefore, the noise rejection characteristics conflicts the chattering effect reduction. However, following the proposed procedure even starting with high initial values for '70. , it will compensate between the two effects, and alleviates the chattering soon, which will be shown in simulations in the next section. In figure 2 the switching threshold can be chosen based on the initial value of '7,'S and the required approaching speed of the system state vector toward the sliding surface. If the initisl value of '7, 's is large enough to complete the control activity, then the ERP activation margin (L e. switching threshold) can be chosen in a narrow bond. But if those initial values are not large enough, the threshold must be chosen wide enough to activate the ERP. Therefore, if the initisl values are chosen large enough heuristically, the trend of approaching toward the sliding surface can be tuned in the desired way. In the next part, the switching threshold will be discussed in more details.

as: u

= B- 1 (:z:~ - j -

KSgn(s))

(15)

where K,'s can be calculated using Filippov's Construction of Equivalent Dynamics, [12], by calculation 8, = 0 (for z = 1, ... , n), which yields

(1 + D.. ) K,

+L

D"K,

= F.+

,~,

n

L D"

,=1

IX~J

- 1,1 + '7,

1=

1, ... , n

(16) In fact, Eqs.(16) define a set of "n" equations with "n" unknowns (Le. K,'s). To solve these equations, one should choose '7, 'so AY. explained before, '7, is a factor that indicates the speed of the corresponding state in approaching to its sliding surface. Therefore, if one can determine '7, in such a way that the speed of the corresponding state becomes lower based on the absolute vslue of distance from sliding surface, and becomes zero on the surface, then the performance will be as desired and chattering will be alleviated if not vanished. Therefore, rather than the conventionsl heuristic method to choose '7, , here it is proposed to select that as

4. SIMULATION RESULTS. In this section, to compare the performance of the two conventionsl and the new regulated sliding mode controllers, a 3-DOF rotating satellite about a fixed position of its center of mass is simulated. Figure 3 shows the essential mathematical calculations to implement the conventional sliding mode control and the proposed algorithm, in a Simulink format. The task is that the state matrix follows the desired one, Le. 11/1 8 t/JI, during a 3-DOF rotating maneuver. The satellite specifications are given in table (1).

I

'7, = '70.1 1 - e l',' (17) In practice, one can follow the procedure shown in Figure 2, to use Eq. (17) for selection of '7, , which will be refereed as Etta Regulating Procedure (ERP). To select the initial value, '70. ' any large value can be chosen. In fact, in the absence of any uncertainties, there are two aspects in the conventional heuristic method of choosing '7,. First, regarding the noise effects, the controller must be capable enough to tackle against separation of the state trend from corresponding sliding surface. Therefore, with a reasonable high value of '7. the

Table(I). The satellite specifications. 189

Value 550 < kg· rn' > 550 < kg· rn' > 450 < kg· rn' > 1050 < kg >

Specification t/J 9

1/1 Total Mass

Parametric uncertainties and measurement faults for the two controllers have been considered as follows: (1) 5% uncertain variations in the dynamic parameters, Le. mass and mass moments of inertia. (2) 5% deviations in measurements. Obtained results show that when the state vector of the system reach to the sliding surface, using the new ER? controller, the effect of parametric uncertainties will become negligible as shown in Fig(4). The model for the actuators has been shown in the Fig. 5 . In this model the constant value can be be determined by calculation of required moment obtained from inverse dynamics. Since the maneuver takes place in a relatively long time interval, L e. 720 s, the time constants for switching between "on" and "off" actions of the actuators can be neglected. According to this model as shown in Fig. 5, if the demanding controlling torque is positive/negative, the actuator will deliver its constant value in positive/negative direction, respectively.

of regulation procedure for selection of 11" based on Eq. (17), the obtained results will be compared. Figure 7 shows a typical result of the ER? algorithm, Le. the regulated value of 113' Application of the ER? for other subsystem parameters yields similar results. In order to provide quick response of the control system to reach to the sliding surface, a margin that is in direct proportion to the absolute value of the s, has been considered. In this case, this margin has been chosen as Isl ~ 0.0850. Next, the performance of the two conventional and the new regulated sliding mode controllers is compared and discussed. The effect of initial choice of the controller parameters, i.e. 11,'S and A/s, on the performance of the two controllers in terms of orientation error (1/1 angle) is shown in Fig. (8). These results are related to an 80% deterioration in initial choice of 11, 's for both controllers. It is seen that the ER? shows almost no sensitivity to the initial choice of the control parameters, while the performance of the conventional controller does certainly depend on these choices. It should be mentioned that the results for other Euler angles show similar effect. Figure 9 compares tracking errors for the Euler angles (
The conventional controller coefficients are chosen as Al = A2 = A3 = 2.5 'h = 112 = 113 = 0.0450 It should be mentioned that the conventional heuristic method of choosing controller coefficients for various trials does not results in much difference. The initial error of each Euler angle is chosen about 20· . Figure 6 shows corresponding desired values of Euler angles of the satellite for the rotating maneuver. These values have been chosen such that the main performance characteristics of the controllers can be compared. Now, applying the new idea 190

5. CONCLUSION

[2] Z. Vafa and S. Dubowsky, "On The Dynamics of Manipulators in Spaoe Using The Virtual Manipulator Approach," Proc. of IEEE Int. ConI. on Robotiu and Automation, pp. 579585,1987. [3] S. All A. MOOllavian and E. Papadopoul08, "011 the Kinematics of Multiple Manipulator Space Free-Flyers and Their Computation," Journal 0/ Robotic S"ltel'1U, 15(4), 207-216, 1998. [4] S. All A. Moosavian and E. Papadopoul08, "Explicit Dynamics of Space Free-Flyers v,ith Multiple Manipulators via SPACEMAPL," Journal 0/ Advanced Robotic.t, 18(2), pp 223244,2004. [5] J. L. Kuang, B. J. Kim, H. W. Lee, and D. K. Sung, "The Attitude Stability Analysis of a Rigid Body with Multi-Elastic Appendages and Multi-Liquid-Filled Cavities Using The Chateau Method," Journal 0/ the A,tronautieal Space Sciences, 15(1), pp. 209-220, 1998. [6] H. W. Mah, V. J. Modi, Y. Morita, and H. Yokota, "Dynamics During Slewing and Translational Maneuven of the Space Station Based MRMS," Journal 0/ the A.tronautical Space Sciencu, 38(4), pp. 557-579, 1990. [7J S. Ali A. MOOllavian and E. Papadopoul08, "On the Control of Space Free-Flyers Using Multiple Impedance Control," Proc. IEEE Int. ConI. Robotiu Automation, NM, USA, April 21-27 1997. [8] S. All A. Moosavian and R. Rastegari, "Force Tracking in Multiple hnpedance Control of Spaoe Free-Flyers," Proc. o/Int. ConI. on Inulligent Robot. and S"atel'1U (IROS), Japan,

In this paper, a regulated sliding mode control law was developed and applied to space platforms during a rotational maneuver. Since on-off reaction jet actuators in space can not deliver the exact demanding moment to the system so the control becomes more complicated. The sliding mode controller is a reliable algorithm with acceptable performance, while the chattering frequency is often high and may excite flexible dynamics, which is of main concern for space systems due to minimum weight design as a dominant trend. Therefore, it is not easy to find proper values for the coefficient of sliding condition (17,'S) that can provide an acceptable performance during a usual maneuver. To alleviate chattering phenomenon, a new procedure was proposed to change 17. 's and smoothly converge to appropriate positive values. This satisfies robustness properties, and chattering elimination characteristics. To this end, the rotation dynamics of a typical satellite described in body coordinates was derived in terms of its angular velocity. Using Euler quasi-coordinates, explicit relationship between actuator torques and satellite orientation was obtained for integration purposes. Then, control input functions were obtained bnsed on a multi input sliding mode control law. Next, focusing on the chattering phenomenon to fulfill energy limitations in space, a new approach was proposed to alleviate (ideally eliminate) the chattering trend. Finally, the developed control law was applied on a given satellite during a rotational maneuver. The simulation results revealed that the new proposed algorithm yields an improved performance compared to those of the conventional routine.

November,2000.

[9] S. All A. MOOllavian and R. Rastegari, "Disturbance Rejection Analysis of Multiple hnpedance Control for Space Free-Flying Robots," Proc. 0/ the IEEE/RSJ Int. ConI. on Intdligent Robot. and So,ptcnl (IROS), Switzerland, 2002. [10] R. A. Hull, C. Ham , and R. W. Johnson, "Systematic Design of Attitude Control Systems for a Satellite in a Circular Orbit with Guaranteed Perlonnance and Stabili· ty," Proc. Of 14th AlAA/USU Conference on

Small SatelliUl, 2000. (11) John Y. Hung, Weibing Gao, James C. Hung, "Variable Structure Control: A Sur~,' IEEE ThmIactioTII on IndUltnal Dectronicl, Vol. 40, No. 1, 1993. [12] Ajay Thukral and Mano Innocenti, "A Sliding Mode Missile Pitch Autopilot Synthesis for High Angle of Attac1c Maneuvering," IEEE 7hJTllachon on Control S".tcnI TechnolDgJI, MAY 1998. [13] Jean-Jacques E. Slotine and Weiping Li, • Applied Nonlinear Control " Prentice Hall, 1991.

REFERENCES [IJ S. All A. MOO8avian, "Dynamics and Control of Free-Flying Robots in Space: A Surv~," Proc. Of the 5th IFAC/EURON Symposium on Intelligent Autonomous Vehicles IAV2004, Lisbon, Portugal, July 5-7, 2004. 191

{14J H.J. Zimmennann, "F'uZZJI Set Theo'1l and 11.3 Application.," Kluwer Academic Publishers, Boston, 1991. {IS\ Hughes, P. C., "Spacecraft Attitude Dtlnamic., " Wiley, 1986.

an

c.not.ot Fig. 2. The Procedure for choosing 1];.

r+ .....

...-

1-+ .....

4

.n.

JU

.. .. .

1/H...

L..+

r-i'

..... 01

r+" r-. I._' r-. n

r+ Q'

n

f-t

Cl

1ft

naoa JU

....... .XIl

RUD1

1IIPIIfl

r-."

f-- .........

..

.........

r- ..... rrr--

.XI_

" - I..... IUDJl

1IIPIIff

..

IU

~.1

I-t

n '"In

~u

I-t u I-t Cl

on.

f-t

f-- ..... 4.XI_

t--

..

_ott po. 1/0....

ICfIIUIIU

~u

01

Qf vuuat.or

r-.

JU JU

onUOT

onJU

f-- .1 Ir- 01

r---

~ ..

f-- ....

Q' S. . ptea

r+

Ql ......- .

sw-.-

I-t " - 1 'XI

n

u

r+ onUOT

.....1

DVlSTS

11

~QI

..

r+

Ir- ..

s....,.t_

o.tvott

ICfIIUIIU

"_1

JU

f-tn

r-.

1/».... "_I

u

"'"la - ,

U-u.ra

~

ICfIIUIIU

n

sw-,n_

Ras. .,.t....

-+ .ftUOT QJ .....r.tor

Fig. 3. Control system calculations according to the described procedure.

192

....

"'r--~·---"---"--""-~--~

.1

ut

'f

Du

H ,of

Fig. 4. Variation of 'I' cbannel dynamics due to parametric uncertainties.

Fig. 8. The effect of initial cbolce of controller parameten on the performance, in terms of orientation error (Dashed Line: Conventional; Solid Line: The new proposed controller).

Aduotot C.....III.m..1

Fig. 5. The reaction jet actuaton model.

Fig. 6. Desired values for Euler Angles during tbe maneuver.

""'r,..".--.-,nr",rr---------~

,,+

·.oot lo.OtIt

I ! ·-foa l

~-~-.--~-__"';:::___::::____=~=___'

,i ..... °0-

_-~_

Fig. 7. The ERP result for determination of

rh. 193

Fig. 9. Euler angles tracking error; (Dashed Line: Conventional; Solid Line: The new proposed controller).