This paper studies higher order sliding-modecontrol laws to deal with some spacecraft-attitude-tracking problems. Quasi-continuous second- and third-order sliding controllers and differentiators are applied to quaternion-based spacecraftattitude- tracking maneuvers. A class of linear sliding manifolds is selected as a function of angular velocities and quaternion errors. The second method of Lyapunov is used to show that tracking is achieved globally. An example of multiaxial attitude-tracking maneuvers is presented, and simulation results are included to verify and compare the practical usefulness of the various controllers

Pukdeboon, C.

Thein, M.W.

Zinober, A.S.I.

English

C. Pukdeboon

A.S.I. Zinober

M.-W.L. Thein

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Quasi-Continuous Higher Order Sliding-Mode Controllers for Spacecraft-Attitude-Tracking Maneuvers

White Rose Research Online

Quasi-continuous higher-order sliding-mode controllers for spacecraft-attitude-tracking manoeuvres

This is a repository copy of Quasi-continuous higher-order sliding-mode controllers for spacecraft-attitude-tracking manoeuvres.White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/10727/Article:Pukdeboon, C., Zinober, A.S.I. and Thein, M.W. (2010) Quasi-continuous higher-order sliding-mode controllers for spacecraft-attitude-tracking manoeuvres. IEEE Transactions on Industrial Electronics, 57 (4). pp. 1436-1444. ISSN 0278-0046 https://doi.org/10.1109/TIE.2009.2030215eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. Quasi-Continuous Higher-Order Sliding Mode Controller Designs forSpacecraft Attitude Tracking ManoeuvresC. PukdeboonDepartment of Applied Mathematics, University of Sheffield, Sheffield, UKA. S. I. ZinoberDepartment of Applied Mathematics, University of Sheffield, Sheffield, UKM.-W. L. TheinDepartment of Mechanical Engineering, University of New Hampshire, Durham, NH 03824, USAAbstract— This paper studies high-order sliding mode controllaws to deal with some spacecraft attitude tracking problems.Second and third order quasi-continuous sliding control areapplied to quaternion-based spacecraft attitude tracking ma-noeuvres. A class of linear sliding manifolds is selected as afunction of angular velocities and quaternion errors. The secondmethod of Lyapunov theory is used to show that tracking isachieved globally. An example of multiaxial attitude trackingmanoeuvres is presented and simulation results are included toverify and compare the usefulness of the various controllers.I. INTRODUCTIONIn general spacecraft motion is governed by the so-calledkinematics equations and dynamics equations [1]. Thesemathematical descriptions are highly nonlinear and thuslinear feedback control techniques are not suitable for theglobal controller design.First-order sliding mode control has been considered asa useful scheme for spacecraft attitude control. Vadeli [2]designed a variable structure attitude control law based onquaternion kinematics. A similar approach was later pro-posed in [3] where sliding mode controller was designedfor spacecraft tracking problems. This was illustrated byan example of multiaxis attitude tracking manoeuvres. Anadaptation of the sliding mode control technique was de-rived and applied to a quaternion-based spacecraft attitudetracking manoeuvres. This modified version presented in[4] is the smoothing model-reference sliding mode control(SMRSMC). This technique improves the transient responseand reduces the chatter phenomenon. In [5] the (additive)quaternion-based tracking of spacecraft manoeuvres usedsliding mode control in the sense of optimal control. Mc-Duffie and Shtessel [6] designed a de-coupled sliding modecontroller and observer for spacecraft attitude control.From the previous literature we conclude that sliding modecontrol can be used for quaternion-based spacecraft attitudetracking manoeuvres. Floquet [7] presented the stabilizationof the angular velocity of rigid body via first-order andsecond-order sliding mode controllers but it has not beenapplied to spacecraft tracking problems. Higher-order slidingmode control has desired properties, such as robustness,similar to sliding mode control. It also may reduce chatteringand provides better accuracy than first order sliding. Hencewe will study spacecraft attitude tracking manoeuvres usinghigher-order sliding mode control.This paper is organized as follows. Section II presentsthe kinematics and dynamic equations of a rigid spacecraft.In Section III the sliding manifold and first-order slidingmode control are presented for attitude tracking manoeuvres.In Section IV the sliding manifold and the second-orderquasi-continuous controller [8] are presented. A first-orderdifferentiator [9] is applied to estimate the time derivativeof the sliding vector. Section V presents the design of third-order quasi-continuous controller. We add a precompensator(first-order lag) to the spacecraft model description to smooththe control signal, and use a second-order differentiator [7]to estimate the first and second time derivatives of thesliding vector. A numerical example of the multiaxial attitudetracking problem [4] is illustrated in Section VI to verifythe usefulness the third-order quasi continuous controller.Section VII is the conclusion.II. SPACECRAFT MODEL DESCRIPTIONWe consider the general case of a rigid spacecraft ro-tating under the influence of body-fixed torquing devices.According to [10], the kinematics equation and the dynamicsequation are given byq̇ =12T (Q)ωq̇4 = −12qTω (1)andJω̇ = −[ω×]Jω + u+ d (2)where Q = [qT q4]T is the quaternion with q =[q1 q2 q3]T , ω = [ω1 ω2 ω3]T is the angular velocityvector, andT (Q) = (q4I3 + [q×]) (3)where I3 is a 3 × 3 identity matrix and [q×] is a skew-symmetric matrix expressed by[q×] =0 −q3 q2q3 0 −q1−q2 q1 0 (4)In (2) u = [u1 u2 u3]T is the control vector, d =[d1 d2 d3]T represents bounded disturbances, and J is theinertia matrix. The kinematic equation (1) can be rewritten978-1-4244-2200-5/08/$25.00 ©2008 IEEE 215Authorized licensed use limited to: Sheffield University. Downloaded on March 29,2010 at 06:54:23 EDT from IEEE Xplore.  Restrictions apply. in a more compact form as [11]Q̇ =12E(Q)ω (5)whereE(Q) =[T (Q)−qT](6)Note that the elements of Q are restricted by‖Q‖ = 1 or q21+ q22+ q23+ q24= 1 (7)III. ATTITUDE TRACKING BY FIRST-ORDER SLIDINGCONTROLLERWe mention briefly the first order sliding approach [4]so that we can compare our improved results later. Thedevelopment of their controller is not presented here (forlack of space). The sliding vector iss = ωe +Kqe (8)and the sliding controller isu = [ω×]J0ω + J0υ̇d − J0K[12T (Q)ω − q̇d] + τ (9)where τ = [τ1 τ2 τ3]T and τi = −gisign(si). They alsoproposed an improved more complicated SMRSMC con-troller that improves the reaching phase transient dynamicsand avoids chattering.IV. ATTITUDE TRACKING BY SECOND-ORDERQUASI-CONTINUOUS CONTROLLERThe quasi-continuous controller presented in [8] is a classof higher-order sliding mode controller. Here the second-order quasi-continuous controller (QC2S) is developed toachieve robust attitude tracking.To avoid the singularity of T (Q) that will occur at q4 = 0,let the attitude of the spacecraft be restricted in the workspaceW [4] defined byW = {Q|Q = [qT q4]T , ‖q‖ ≤ β < 1, q4 ≥√1 − β2 > 0}(10)A. Sliding manifoldA class of linear sliding vectors is chosen as follows:s = ωe +Kqe (11)where K is a 3 × 3 symmetric positive-definition constantmatrix, ωe = ω− υd, qe = q− qd, and υd = 2T−1(Q)q̇d. In[4] a similar sliding manifold was introduced to apply to thesliding mode controller for the spacecraft attitude trackingmanoeuvres. It was proved that, by choosing Lyapunovfunction V = 12qTe Kqe with positive definite K, the trackingerror qe converges to zero.B. Control lawIn this section we study QC2S and the first-order real-timedifferentiator for spacecraft attitude tracking manoeuvres. Inorder to use the second-order quasi-continuous controllerwe need to know the time derivative of the sliding vector(ṡ). Because it is very complicated to find ṡ theoreticallyfor this nonlinear system, we use the first-order Levantdifferentiator [9] for the estimation of ṡ. A first-order real-time differentiator has the formż0 = −λ1|z0 − s|1/2sign(z0 − s) + z1ż1 = −λ2sign(z0 − s) (12)where z0, z1 are real-time estimations of s and ṡ respectively.The second-order quasi-continuous SM controller [8] isdesigned asu = −αz1 + |z0|1/2signz0|ż1| + |z0|1/2. (13)Now we design the second-order quasi-continuous controllersuch that the reaching and sliding conditions are satisfied.We show that tracking is achieved globally (by using theLyapunov second method) following the approach of [4].Since J is symmetric and positive definite, the candidateLyapunov function is chosen asVs =12sTJs ≥ 0 (14)and Vs = 0 only when s = 0. Taking the first derivative ofVs and using (1), (2) and (11), we haveV̇s = sT {−[ω×]Jω + u+ d− Jυ̇d+JK(q̇ − q̇d) + J̇s}. (15)Let J = J0 + ∆J where J0 and ∆J denote the nominaland uncertain part of the inertia matrix. Using (1) then (15)becomesV̇s = sT {−[ω×]∆Jω − ∆Jυ̇d + ∆JK[12T (Q)ω−q̇d] + u+ d+ J̇s− [ω×]J0ω − J0v̇d+J0K[12T (Q)ω − q̇d]}. (16)Suppose that the external disturbances d and uncertain pa-rameters ∆J and J̇ are all bounded and that these bounds areknown. Let δ = {−[ω×]∆Jω − ∆Jυ̇d + ∆JK[12T (Q)ω −q̇d]+d+J̇s} and γ = {−[ω×]J0ω−J0υ̇d+J0K[12T (Q)ω−q̇d]}. Then (16) becomesV̇s = sT [δ + u+ γ]=3∑i=1si(δi + ui + γi). (17)By setting the controller asu = −kṡ+ |s|1/2signs|ṡ| + |s|1/2(18)216Authorized licensed use limited to: Sheffield University. Downloaded on March 29,2010 at 06:54:23 EDT from IEEE Xplore.  Restrictions apply. and letting Ψi = δi + γi, we haveV̇s =3∑i=1si[Ψi − ki(ṡi + |si|1/2sgn(si)|ṡi| + |si|1/2)]=3∑i=1siΨi −3∑i=1kisisgn(si)(ṡisgn(si) + |si|1/2|ṡi| + |si|1/2)=3∑i=1|si|ki[Ψisgn(si)ki−ṡisgn(si) + |si|1/2|ṡi| + |si|1/2](19)To guarantee the reaching and sliding on the manifold, werequireṡisgn(si) + |si|1/2|ṡi| + |si|1/2≥Ψisgn(si)ki(20)Sinceṡisgn(si) + |si|1/2|ṡi| + |si|1/2≤ 1, (20) can be written aski ≥ Ψisgn(si). (21)The upper bound of |Ψi| can be found and denoted as|Ψi| < Ψmaxi (Q,ω, qd, q̇d, q̈d). (22)Obviously, if we choose the gain ki as ki ≥Ψmaxi (Q,ω, qd, q̇d, q̈d) then V̇s < 0. This guarantees thereaching and sliding on the manifold. Note that the bounds(22) are functions of the states so simulation studies areneeded to assess their magnitudes and Lyapunov functionVs exists when condition (21) is satisfied.V. THIRD-ORDER QUASI-CONTINUOUS CONTROLLERWe next consider the third order quasi-continuous (QC3S)controller to achieve to the spacecraft attitude tracking ma-noeuvres. Because it is a third-order sliding mode controllerwhich normally provides very accurate outputs, we expecthigher accuracy of the tracking results. Moreover, we add afirst-order lag to the spacecraft model description to smooththe control signals.Because it is very complicated to find ṡ and s̈ from thissystem, we use the second order Levant differentiator [9] forthe estimations of ṡ and s̈.A second-order real-time differentiator [9] isż0 = v0v0 = −λ1|z0 − s|2/3sign(z0 − s) + z1ż1 = v1v1 = −λ2|z1 − v0|1/2sign(z1 − v0) + z2ż2 = −λ3sign(z2 − v1) (23)where z0, z1 and z2 are real-time estimations of s, ṡ and s̈respectively.The third-order quasi continuous SM controller isu = −α[z2 + 2(|z1| + |z0|2/3)−1/2(z1 + |z0|2/3sign(z0))|z2| + 2(|z1| + |z0|2/3)1/2](24)To guarantee the reaching and sliding on the manifold weselect the Lyapunov function Vs =12sTJs and follow thesame process as for the proof of QC2S. We select the controllaw asu = −k[s̈+ 2(|ṡ| + |s|2/3)−1/2(ṡ+ |s|2/3sign(s))|s̈| + 2(|ṡ| + |s|2/3)1/2](25)Substitute this controller into (17) and letting ψi = δi + γi,we haveV̇s = −3∑i=1siki[s̈i + 2(|ṡi| + |s|2/3i )−1/2(ṡi +|s|2/3i sign(si))]/[|s̈i| + 2(|ṡi| +|si|2/3)1/2] +3∑i=1siψi (26)For the first term of (26) we take sgn(si) outside the bracketandV̇s =3∑i=1siψi −3∑i=1kisisgn(si)[s̈isgn(si) +2(ṡisgn(si) + |si|2/3)−1/2(ṡisgn(si) +|si|2/3)]/[|s̈i| + 2(|ṡi| + |si|2/3)1/2]=3∑i=1ki|si|{ψisgn(si)ki−s̈isgn(si) + 2ṡisgn(si) + |si|2/3(|ṡi| + |si|2/3)1/2|s̈i| + 2(|ṡi| + |si|2/3)1/2} (27)To guarantee the reaching and sliding on the manifold, werequires̈isgn(si) + 2ṡisgn(si) + |si|2/3(|ṡi| + |si|2/3)1/2|s̈i| + 2(|ṡi| + |si|2/3)1/2≥ψisgn(si)ki. (28)Sinceṡisgn(si) + |si|2/3(|ṡi| + |si|2/3)1/2≤ (|ṡi| + |si|2/3)1/2ands̈isgn(si) ≤ |s̈i|consequently we haves̈isgn(si) + 2(ṡisgn(si) + |si|2/3)−1/2(ṡisgn(si) + |si|2/3)|s̈i| + 2(|ṡi| + |si|2/3)1/2≤ 1(29)Also the condition (28) can be written aski ≥ ψisgn(si). (30)The upper bound of |ψi| can be found and denoted as|ψi| < ψmaxi (Q,ω, qd, q̇d, q̈d). (31)Obviously, if we choose the gain ki as ki ≥ψmaxi (Q,ω, qd, q̇d, q̈d) then V̇s < 0. This guarantees thereaching and sliding on the manifold.217Authorized licensed use limited to: Sheffield University. Downloaded on March 29,2010 at 06:54:23 EDT from IEEE Xplore.  Restrictions apply. VI. MULTIAXIAL ATTITUDE TRACKING MANOEUVRESHere an example is presented with numerical simulationto validate and compare the various controllers; SMRSMC[4], QC2S and QC3S. The nominal part J0 and the uncertainpart ∆J of the inertia matrix areJ0 =1200 0 00 2200 00 0 3100and∆J =0 100 −200100 0 300−200 300 0The initial conditions are Q(0) = [0 0.5 0.5 0.7071]T ,and ω(0) = [−0.0005 0.0008 0.001]T . Suppose that theexternal disturbance di = 0 and the workspace W is definedby β2 = 0.75. The desired multiaxial attitude trackingmanoeuvres areqd(t) =0.5 cos[(π/50)t]0.5 sin[(π/50)t]−0.5 sin[(π/50)t]and the magnitude constraints on the controllers are |ui| ≤60(N ·m) for i = 1, 2, 3. For QC2S the positive scalar λ isselected as λ = 1.2 while for QC3S λ = 0.19. The slidingmanifold is chosen as (11) with K = λI3 and the gains inthe control laws are selected as gi = 60 for i = 1, 2, 3.Simulation results for the attitude tracking are shownin Figs. 1- 13. Figs. 1 and 3 show that the SMRSMCscheme gives good tracking output and the settling time isapproximately 60 s. The sliding vector remains on the slidingmanifold after 5 s. The actual control torques in Fig. 4 arevery smooth. Regarding accuracy the bound on |s| is 0.00047(at steady state) with O(h) = 0.005 for h = 0.005.As shown in Figs. 5 and 7, QC2S provides good trackingresults. The settling time is approximately 35 s. In Fig. 6the sliding vectors are driven to the sliding manifold andremain on the sliding manifold after 35 s. The actual controltorques presented in Fig. 8 are limited by 60 N-m and butchattering appears in this system. Regarding accuracy thebound on |s| is 0.00092 (at steady state) with O(h) = 0.005for h = 0.005.QC3S gives good tracking output(Figs. 9 and 11). Thesetting time is approximately 60 s. The sliding vector remainson the sliding manifold after 10 s. The calculated controltorques shown in Fig. 13 are limited to 60 N-m. For the actualcontrol torques applied to the spacecraft, Fig. 12 shows thatthe applied control torques are limited to 60 N-m for thefirst 15 s and then limited by 20 N-m, and are relativelysmooth.Regarding accuracy the bound on |s| is 0.000012 (atsteady state) with O(h2) = 0.000025 for h = 0.005.Although QC2S gives a small settling time, it has severechatter which is impractical for application to spacecraftattitude tracking. A smoothing scheme could reduce thechattering. Both SMRSMC scheme and and QC3S providesrelatively smooth control torque signals. For accuracy QC3Sobviously provides much more accurate tracking output. ItFig. 1. Attitude tracking response (quaternions) - SMRSMCFig. 2. Sliding functions - SMRSMCFig. 3. Two norm of attitude tracking errors - SMRSMCgives outstanding accuracy (better than O(h2)) while theaccuracy of SMRSMC satisfies O(h). In view of thesesimulation results, QC3S seem to be the most useful controldesign for practical spacecraft tracking, although its imple-mentation is more complicated..218Authorized licensed use limited to: Sheffield University. Downloaded on March 29,2010 at 06:54:23 EDT from IEEE Xplore.  Restrictions apply. Fig. 4. Control torques - SMRSMCFig. 5. Attitude tracking response(quaternions) - QC2SFig. 6. Sliding functions - QC2SVII. CONCLUSIONSQC3S has been successfully applied to spacecraft attitudetracking manoeuvres. An example of spacecraft multiaxialattitude tracking manoeuvres has been presented. Moreover,it reduces the undesirable chattering effect induced in theconventional sliding mode control and QC2S, and providesvery good accuracy of the tracking results. A class of linearFig. 7. Two norm of attitude tracking errors - QC2SFig. 8. Actual control torques - QC2SFig. 9. Attitude tracking response (quaternions) - QC3Ssliding manifold is chosen as a function of angular velocitiesand quaternion errors. The second method of Lyapunovtheory introduced to prove sliding system stability for allthe controller designs.REFERENCES[1] Dywer, T. A. W., III, and Sira-Ramirez, H., ”Variable-structure controlof spacecraft attitude manoeuvres”, Journal of Guidance, Control, andDynamics, vol. 11, No.3, 1988, pp 262-270.219Authorized licensed use limited to: Sheffield University. Downloaded on March 29,2010 at 06:54:23 EDT from IEEE Xplore.  Restrictions apply. Fig. 10. Sliding functions - QC3SFig. 11. Two norm of attitude tracking errors - QC3SFig. 12. Applied control torques - QC3S[2] Vadali, S. R., ”Variable sytucture control of spacecraft large manoeu-vres”, Journal of Guidance, Control, and Dynamics, vol. 29, No.2,1986, pp 235-239.[3] Chen, Y. -P., and Lo, S.-C., ”Sliding-Mode Controller Design forSpacecraft Attitude Tracking manoeuvres”, IEEE transactions onaerospace and electronic systems, vol. 29, No. 4, 1993, pp 1328-1333.[4] Lo, S.-C., and Chen, Y. -P., ”Smooth Sliding-Mode Control for Space-craft Attitude Tracking manoeuvres”, Journal of Guidance, Control,and Dynamics, vol. 18, No. 6, 1995, pp 1345-1349.[5] Crassidis, J. L., Vadali, S. R., and Markley F. L., ”Optimal variable-Fig. 13. Calculated control torques - QC3S.structure control tracking of spacecraft manoeuvres”, Journal of Guid-ance, Control, and Dynamics, vol. 23, No. 3, 2000, pp 564-566.[6] McDuffie, J. H., and Shtessel, Y. B., ”De-coupled sliding modecontroller and observer for satellite attitude control”, In Proceedingsof Annual Southeastern Symposium on System Theory, 1997.[7] Floquet, T., Perruquetti, W., and Barbot J. -P., ”Angular VelocityStabilization of a Ridgid Body Via VSS Control”, Journal of DynamicSystems, Measurement, and Control, vol. 122, 2000, pp 669-673.[8] Levant, A., ”Quasi-Continuous High-Order Sliding-Mode Con-trollers”, IEEE transactions on automatic control, vol. 50, No.11,2005, pp 1812-1816.[9] Levant, A.,”Higher-order sliding modes, differentiation and output-feedback control”, Int. J. Control, vol. 76, 2003, pp 924-941.[10] Wertz J. R., Spacecraft attitude determination and control, KluwerAcademic, Dordrecht, London 1978.[11] Kane T. R., Linkins, P. W., and Levinson, D. A., Spacecraft Dynamics,McGraw-Hill, New York, 1983.220Authorized licensed use limited to: Sheffield University. Downloaded on March 29,2010 at 06:54:23 EDT from IEEE Xplore.  Restrictions apply. 

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Quasi-continuous higher-order sliding-mode controllers for spacecraft-attitude-tracking manoeuvres

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