Acta Astronautica 54 (2003) 69–75
www.elsevier.com/locate/actaastro
Academy transactions note
Minimum-time spacecraft maneuver using
sliding-mode control
Y.W. Jan, J.C. Chiou
∗Department of Electrical and Control Engineering, National Chiao-Tung University, Hsinchu 30010, Taiwan, ROC Received 27 March 2003; accepted 31 March 2003
Abstract
The purpose of this paper is to present a sliding mode control method that can be used to perform a spacecraft large angle maneuver with minimum time. An algorithm of minimum-time SMC is developed to provide the robust tracking control. The simulation results are compared with previous developed control schemes, eigenaxis quaternion regulator, to demonstrate the superiority of the proposed sliding mode control algorithm.
c
2003 Elsevier Ltd. All rights reserved.
1. Introduction
The major tasks in operating a remote-sensing satel-lite are to perform rapid multitarget acquisition, point-ing, and tracking. The conventional satellite control systems for these tasks are based on a sequence of rotational maneuvers about each control axis to con-duct the required three-axis large angle maneuver. The time of such successive rotations is longer (by a fac-tor of 2 or 3) than that of a single maneuver about the eigenaxis that has been known and studied in the last two decades. Among the research, Vadali and Junk-ins [1] proposed the so-called open-loop schemes for large-angle maneuvers. The open-loop schemes, how-ever, are sensitive to spacecraft parameter uncertainty. Wie et al. [2] chose a linear quaternion feedback reg-ulator with open-loop decoupling control torque for
Endorsed by Jeng-Shing Chern, AIAA, IAA Member
(Section 2).
∗Corresponding author. Fax: +886-35-715-998.
E-mail address:[email protected](J.C. Chiou).
gyroscopic forces to ensure eigenaxis rotations. How-ever, in their approach, the constraints for reaction wheel torque were not considered. The sliding mode control (SMC) is a robust control technique [3] that has applied to the spacecraft attitude-tracking prob-lem by Dwyer and Sira-Ramirez [4], Dwyer and Kim [5], Chen and Lo [6], using the Rodriques parameters as the attitude measurement. Wie et al. [7] proposed a PID saturation control logic to provide a rest-to-rest eigenaxis rotation under slew rate constraint. How-ever, the attitude tracking with minimum time was not considered in their research.
In order to improve the aforementioned drawbacks, an SMC for spacecraft minimum-time tracking ma-neuver is proposed in this paper. An algorithm of minimum-time SMC is developed to provide the de-sired tracking trajectories, which are based on an eige-naxis rotation with maximum torques. The eigeeige-naxis rotation with maximum torques eventually provides the minimum-time maneuver. The condition of satura-tion on wheel torque has been imposed on the control algorithm. Meanwhile, the proposed SMC controller 0094-5765/03/$ - see front matter c 2003 Elsevier Ltd. All rights reserved.
provides the robust control for removing the eEects of model uncertainty.
2. Spacecraft model description
A rigid spacecraft rotating under the inFuence of body-Gxed torquing devices is derived in this section. Reorientation of the spacecraft is accomplished us-ing three reaction wheels that are aligned along three body-Gxed control axes. The kinematic and dynamic equations are described by [8]
˙Iq = Q*! (1) and II ˙*! =*L − d * Hw dt − * ! × ( II*! +H*w); (2)
respectively, where *L is the external torque,
*
! = ( !1 !2 !3)T is the angular rate and
Iq = [ q0 q1 q2 q3]T denotes the quaternion. The
elements of Iq are constrained by
Iq = q2
0+ q21+ q22+ q23= 1: (3)
Furthermore, Q is an orthogonal matrix of quaternion that can be expressed by the following equation:
Q =12 −q1 −q2 −q3 q0 −q3 q2 q3 q0 −q1 −q2 q1 q0 : (4)
The angular momentum of reaction wheels isH*w=
IIw(! +* ), where II* w= diag( Iw11 Iw22 Iw33)
de-notes moment of inertia matrix of the three reaction wheels, and = * 1ˆb1+ 2ˆb2+ 3ˆb3is the relative
spin rate. In terms of the wheel angular momentum,
*
Hw, Eq. (2) can be rewritten as
II ˙*! =*L − IIw( ˙*! + ˙ * ) −! × ( II + II* w)*! − *! × IIw * : (5) The control torque, T , generated by reaction wheels* can be expressed as
*
T = IIw( ˙*! +* ):˙ (6)
3. Attitude tracking by SMC
The design procedure of SMC generally contains two fundamental steps.
Step 1: Choose the sliding surface such that the control goal can be achieved. The four quaternions sliding surfaces are chosen as
Is = K( Iq − Iqr) + ( ˙Iq − ˙Iqr) = I0 (7)
and their time derivatives are given by
˙Is = K( ˙Iq − ˙Iqr) + ( LIq − LIqr) = I0; (8)
where Is=[ s0 s1 s2 s3]T; I0=[ 0 0 0 0 ]T;
K is a constant and positive deGnite diagonal matrix, and Iqris the reference quaternion trajectory. Note that
for the trajectories conGned to the surface of Eq. (7),
˙Iqe= −K Iqe; (9)
where Iqe= Iq− Iqr. The solution of Eq. (9) can be easily
obtained as
Iqe(t) = e−KtIqe(0): (10)
This solution shows that the selected sliding surfaces are exponential stable for the reason that K is positive deGnite.
Step 2: Design the control law such that the reach-ing and slidreach-ing conditions on the slidreach-ing surfaces are satisGed. With this in mind, the spacecraft dynamic equations can be expressed in terms of quaternion by taking the time derivative of Eq. (1) and substituting Eq. (5) to obtain
LIq = ˙Q*! + Q II−1(M −* *T ); (11)
where
*
M =*L −*! × ( II + IIw)*! −*! × IIw :*
The control law for the sliding phase can be found by enforcing ˙Is = I0. Substituting Eq. (8) into Eq. (11), which gives the control law for the reaction wheel torques,
*
T =M + IIQ* ∗{[K( ˙Iq − ˙Iqr) − LIqr+ ˙Q!]−D · sgn(Is)};
(12) where Q∗ is the pseudo inverse of Q deGned as
Q∗= (QTQ)−1QT= 2 −q1 q0 q3 −q2 −q2 −q3 q0 q1 −q3 q2 −q1 q0 :
Matrix D in Eq. (12) is a constant and positive deGnite diagonal matrix. The discontinuous term has the ef-fect of moving the trajectory back towards the sliding surface when deviations occur resulting from external disturbances or system modeling errors.
In order to demonstrate the stability of the proposed control design, Lyapunov’s direct method is used to show that the control law given in Eq. (12) is asymp-totically stable for the reaching phase motion. A posi-tive deGnite candidate Lyapunov function is given by V =1
2IsTIs. The requirement for ensuring the reaching
and sliding conditions on the sliding surfaces is ˙V = IsT[K( ˙Iq − ˙Iq
r) − LIqr+ ˙Q! + Q II−1(M −* T )] ¡ 0:*
(13) Substituting Eq. (12) into Eq. (13) results in a stability constraint of
0 ¿−IsT{(I − QQ∗)[K( ˙Iq − ˙Iqr) − LIqr] − ˙qqT˙Iq} IsTQQ∗D sgn(Is) − 1:
(14) The controller design parameters are the values of K and D matrices of Eq. (12). Typical uncertainty arising in spacecrafts during maneuver phase is mainly mass property. SpeciGc constraints on D for the uncertainty of mass property are considered as follows.
The true spacecraft inertia is denoted by
II = II0+ O II; (15)
where II0 is the measured value of spacecraft inertia,
and O II is the measurement error. It is assumed that O II is suPciently small such that the zeroth-order term from the binomial expansion of ( II0+ O II)−1is valid.
This assumption leads to the following relations: II−1II
0≈ I;
II−1
0 O II ≈ 0: (16)
Replacing II with II0 in control law equation
(12) results in
*
T =M + II* 0Q∗{[K( ˙Iq − ˙Iqr) − LIqr+ ˙Q!]−D sgn(Is)}:
(17) Again, performing the calculations analogous to Eqs. (13) and (14), but using Eqs. (16) and (17) yields the
constraint relation
0 ¿−IsT{(I − QQIs∗T)[K( ˙Iq − ˙IqQQ∗D sgn(Is)r) − LIqr] − ˙qqT˙Iq}− 1
(18) for ensuring closed-loop stability. This constraint is identical to that of the nominal case, where the space-craft inertia is known exactly.
If larger errors in the inertia matrix exist, then Eq. (16) is not valid and constraint equation (18) should be used with a worst-case estimate of the inertia er-rors. We assume that the disturbance torque due to the inertia errors on the spacecraft*L is bounded. This value is unique for a given maneuver. The derivation for the constraint on D to ensure stability is similar to that of Eqs. (13) and (14), where *L is unknown and therefore not included in control law (12). The time derivative of Lyapunov function V results in a maneuver-dependent constraint similar to Eq. (14)
0 ¿− IsT{(I − QQ∗)[K( ˙Iq − ˙Iqr) − LIqr] − ˙qqT˙Iq + Q II−1*L} IsTQQ∗D sgn( Is) − 1
(19) for ensuring stability.
4. Tracking reference of eigenaxis maneuver with maximum torque
It has been recognized that minimization of the maneuver time around the eigenaxis might be the solution to Gnd the most fast tracking reference. An eigenaxis rotation results in the shortest angu-lar path and, therefore, also in minimum time when the maximum torque is applied. In this section, a minimum-time maneuver trajectory is derived through the concept of applying maximum torque during the eigenaxis maneuver. Firstly, we can express Eq. (12) as
*
T =M +* N*slew− IIQ∗D sgn(Is): (20)
The gyroscopic torque, disturbance torque M and* the torque needed for on-sliding-phase control, IIQ∗D sgn(Is), is assumed to be relatively small
com-pared to the slew torque. This assumption is true if the satellite is three-axes stabilized with low reaction
wheel momentum before the eigenaxis rotation com-mences. Note that using a suitable momentum dump-ing method can ensure this assumption. If we ignore gyroscopic and disturbance torqueM and accept the* fact that the torque needed for on-sliding phase is small. Then by assuming II = diag( Ix Iy Iz), the
minimum-time maneuver trajectory can be obtained from the following equations:
II ˙I!r=N*slew; (21) ˙Iqr=12 −qr1 −qr2 −qr3 qr0 −qr3 qr2 qr3 qr0 −qr1 −qr2 qr1 qr0 * !r: (22)
For an eigenaxis slewing, we have Nslew;i Nslew;j = Iiqei Ijqej; (23) where if i or j = x; i or j = y; i or j = z; then qei or qej= qe1; qei or qej= qe2; qei or qej= qe3:
The unique constant ratio properties in Eq. (23) during an eigenaxis rotation can be used to determine the available reaction wheel torque for a minimum time slew. With Eq. (23), the minimum-time slew torque can be written as * Nslew= + min i Nsat−i Iiqei diag(II)qvec; t ∈ (0; th); − min i Nsat−i Iiqei diag(II)qvec; t ∈ (th; 2th); (24) where th is the time to reach halfway mark during
slew maneuver, *!r = ( !r1 !r2 !r3)T; qvec =
[ qe1(0) qe2(0) qe3(0) ]T;N*sat−i is the saturated
wheel torque in body axis i, and ∈ (0; 1) is the total saturated torque. To satisfy the torque constraint, we
have ¡ 1. This is to enable a minimum-time slew while providing a small additional torque for satellite’s rotation around eigenaxis. This small additional torque is used to accommodate gyroscopic and disturbance torque M and to counteract any perturbations due to* modeling errors and external disturbances.
While the slew torque N*slew in Eq. (24) used to
generate the minimum-time tracking in open loop, the halfway mark is determined using feedback from the error quaternion. The largest error quaternion vector components [ qerr1(t) qerr2(t) qerr3(t) ] are
com-pared to its precomputed values at the halfway mark (qhalf) to determine if the maneuver time reaches th:
max i |qerri(t)| − qhalf = ¿ 0 ∀t ¡ th; ¡ 0 ∀t ¿ th; (25) where Iqerr= qerr0 qerr1 qerr2 qerr3 = qr0 qr1 qr2 qr3 −qr1 qr0 qr3 −qr2 −qr2 −qr3 qr0 qr1 −qr3 qr2 −qr1 qr0 q0 q1 q2 q3 ; (26) qhalf=max|sin( =2)|i|qerri(0)| sin
4 : (27) 5. Simulation results
In order to demonstrate the superiority of the proposed minimum-time SMC algorithm, we further adapt the eigenaxis quaternion regulator:
*
T =M + Kq* vec+ D!:* (28)
It has been shown in Ref. [2] that an eigenaxis rotation will occur when K = k diag( II) and D = d diag( II), where k and d are positive constants.
A typical remote sensing spacecraft system is used as an example. Fig. 1 depicts the remote sensing spacecraft on orbit conGguration. The
Fig. 1. Experimental spacecraft on-orbit conGguration.
Table 1
Initial and Gnal satellite orientation in Euler angles (1–2–3 sequence)
Boundary Roll Pitch Yaw
condition (deg) (deg) (deg)
Initial value 0 0 0 Final value 30 45 0 Table 2 Controller parameters SMC W = diag(1; 1; 1; 1) A = diag(0:001; 0:001; 0:001; 0:001), $ = 0:002; = 0:9 Quaternion K = diag(3:64; 6:58; 6:72), regulator D = diag(25:5; 46; 47)
large angle maneuver of interest is speciGed in terms of Euler angles as shown in Table 1, with II = diag[ 182 329 336 ] kg m2. The maneuver on
yaw is null for the most Earth-pointing remote sens-ing cases. The conGguration of reaction wheels in the simulated spacecraft has been deliberately arranged to accommodate the capability of fast attitude maneuver. According to the wheel conGguration, the maximum wheel torque Nsat= [ 0:56 0:52 0:24] N m,
max-imum wheel speed max = 5400 rpm, and MOI of
wheel Iw= 0:041 kg m2are used in the simulation.
In this paper, we assume the nominal inertia ma-trix is perturbed by 10% and an initial orientation er-ror of 2◦. Control parameters for all simulation cases
are listed in Table 2. Because of the existence of
Fig. 2. Euler angles for large-angle slew.
nonideality in the practical implementation of sgn(Is), the control law in Eq. (12) generally suEers from the chattering problem. To alleviate such undesirable performance, the sign function is modiGed as sat(si; $) = 1 si¿ $; si |si| 6 $ −1 si¡ − $; i = 0; 1; 2; 3: (29)
The simulation has been carried out by using Matlab/Simulink software with 100 s simulation time. Fig. 2 shows a large angular rotation with the cor-responding roll, pitch and yaw angles during the
Fig. 3. Reaction wheels torques during large-angle slew.
maneuvers. The proposed SMC algorithm precedes the quaternion regulator with respect to the slew time and pointing accuracy. Compared to the pro-posed SMC algorithm, the quaternion regulator is not inherently time optimal. In Fig. 3, the corre-sponding reaction wheel torques for two cases have been obtained. At the beginning of the slewing, we observe that the quaternion regulator produced a maximum constant wheel torque constraint of Nsat=[ 0:56 0:52 0:24 ] N m. The proposed SMC
algorithm satisGes the torque constraint by using
Fig. 4. Stability constraint residual for the fast-SMC maneuver (must be negative for stability as described in Eq. (14)).
0:468 N m on pitch axis with = 0:9 as the upper limit. Therefore, they provide 0:052 N m for the ad-ditional small torques. The maximum wheel speed is much lower than the speed limit of 5400 rpm given by hardware capability. Fig.4demonstrates the satis-faction of stability constraint, given in Eq. (14) that must be negative. This result veriGed the control law design of SMC where closed-loop stability has been ensured.
6. Conclusions
An algorithm of minimum-time SMC to perform fast large-angle maneuvers is proposed for a three-axis reaction wheel control of spacecraft. An eigenaxis rotation with maximum torques has been used to de-termine the tracking reference for the proposed SMC tracking control. The maneuver time of the proposed minimum-time SMC control is much faster (by a factor of 2 or 3) than the conventional approach that is based on a sequence of rotational maneuvers about each control axis to conduct the required three-axis large-angle maneuver. Compared to the quaternion regulator obtained from simulation optimization, the minimum-time SMC has been demonstrated to im-prove the slew time performance for the experimental target rotation of a typical small remote sensing satel-lite. Moreover, robustness against inertia modeling errors is ensured by tracking the reference maneuver.
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