Linear Continuous Sliding Mode-based Attitude Controller with Modified Rodrigues Parameters Feedback
LLinear Continuous Sliding Mode-based AttitudeController with Modified Rodrigues ParametersFeedback
Harry Septanto Satellite Technology CenterNational Institute of Aeronautics and Space (LAPAN)
Bogor, Indonesia Ministry of Research and Technology/National Agency for Research and Innovation (BRIN)
Jakarta, [email protected]
Djoko Suprijanto
Faculty of Mathematics and Natural SciencesInstitute of Technology Bandung
Bandung, [email protected]
Abstract —This paper studies an attitude control system designbased on modified Rodrigues parameters feedback. It employsa linear continuous sliding mode controller. The sliding modecontroller is able to bring the existence of the sliding motionasymptotically. Besides, the attitude control system equilibriumpoint is proved to have an asymptotic stability guarantee throughfurther analysis. This stability analysis is conducted since thesliding mode existence on the designed sliding surface does notimply the stability guarantee of the system’s equilibrium. Thispaper ends with some numerical examples that confirm theeffectiveness of the designed attitude control system.
Index Terms —attitude control, modified Rodrigues parame-ters, linear continuous sliding mode, asymptotic stability guar-antee.
I. I
NTRODUCTION
Attitude control system design is a challenging problem.It contains nonlinear dynamics and kinematics models. Be-sides, the complexity increases since there is some kine-matics representation with each own constraints. Researchefforts have been conducted to design the attitude controlsystems using various kinematics representations. For instance,Ozgoren [1] reported a comparison of the attitude controlsystems design based on Euler angles, quaternion, Euler angle-axis pairs, and orientation matrices (or rotation matrices [2]).Another kinematics representation that has been also widelystudied is modified Rodrigues parameters (MRP), e.g., [3],[4], [5], to name a few. MRP representation is not uniquenor global [2], i.e., a physical orientation is not representedby a single-value in MRP as well as MRP represents notall physical orientations. Nevertheless, MRP can representorientation between − and that cannot be conductedby Euler angles. Besides, its feedback control does not exhibitan unwinding phenomenon that may occur with quaternionrepresentation [6]. Hence, this work is focused on the attitude Final version. Published in 2020 International Conference on Radar, An-tenna, Microwave, Electronics, and Telecommunications (ICRAMET) control system design based on MRP kinematics representa-tion.Many approaches can be conducted in designing an attitudecontrol system, .e.g., variable structure or sliding mode controlapproach. There are many types of sliding mode controller,e.g., discontinuous or switching [7], terminal sliding mode[8] [9], super-twisting [10], saturation function-approximation[11] [12], and linear continuous-type of controller [13] [14][15]. The last type of sliding mode is less complex forimplementation since it is a continuous system. Besides, itmight chattering-free. The chattering occurs because switchingof the control of the switching-type will excite the unmodeleddynamics [16]. Boiko and Fridman [17] showed the chatteringof the so-called continuous sliding mode control systems.However, the analysis covered only the terminal sliding modeand super-twisting-type. This paper presents the resultingstudy of the sliding mode controller that is a linear continuous-type employed in the attitude control system using the MRPkinematics state feedback.The paper is organized as follows. The next section de-scribes the methodology of this research. Section III presentsthe main result that consist of a theorem, numerical simula-tions, and discussion. We end the paper by concluding remarksin Section IV. II. M
ETHODOLOGY
A. Mathematical Preliminaries
The time derivative of function f is denoted by ˙ f . A squarematrix G > and G < mean a positive definite and anegative matrix, respectively. I is the × identity matrix, I = .A vector of an angular velocity ( rad/s ) are defined asfollows: (cid:126)ω lb = ω blbT F b = ω llbT F l = ω dlbT F d , where a r X i v : . [ ee ss . S Y ] F e b blb , ω llb , ω dlb ∈ R and F b , F l , F d ∈ R × ; ω blb denotes the an-gular velocity of the satellite’s body frame ( F b ) with respect tothe inertial reference frame ( F l ) that is expressed in the bodyframe ( F b ); ω llb denotes the angular velocity of the satellite’sbody frame ( F b ) with respect to the inertial reference frame( F l ) that is expressed in the inertial reference frame ( F l ); and ω dlb denotes the angular velocity of the satellite’s body frame( F b ) with respect to the inertial reference frame ( F l ) that isexpressed in the target or desired frame ( F d ). Note that theframe variables F b , F l and F d are matrices whose rows consistof the vector basis of R . B. Rigid Body Dynamics and Kinematics
The rigid body satellite dynamics with a × symmetricmatrix of inertia calculated about its center of mass J > (cid:0) Kg · m (cid:1) , J ∈ R × , and the control torque expressedin the body frame τ ( N · m ) , τ ∈ R , is defined in (1).The attitude kinematics represented by modified Rodriguesparameters (MRP) is shown in (2)-(3) [5], where [ σ db ] × is askew-symmetric matrix as defined in (4). Note that σ db ∈ R denotes the attitude of the satellite’s body frame ( F b ) withrespect to the desired frame ( F d ), where σ db = σ lb − σ ld . Itis also called the attitude error. J ˙ ω blb = − ω blb × Jω blb + τ (1) ˙ σ db = G ( σ db ) ω bdb (2)where G ( σ db ) = 12 (cid:18) − σ dbT σ db I − [ σ db ] × + σ db σ dbT (cid:19) (3)and [ σ db ] × = − σ db σ db σ db − σ db − σ db σ db , σ db = σ db σ db σ db (4) Definition 1:
Consider the rigid body dynamics (1) andkinematics (2)-(4). τ is an MRP-based feedback sliding modecontroller if it is able to bring the system’s states to reach thesliding surface asymptotically and to asymptotically stabilizethe system’s equilibrium point. Remarks 1:
Some other authors stated that a finite-timestable of the sliding surface is required for a sliding modeexistence, as stated in [7] and [18], to name a few. However,this work refers to that the sliding surface’s asymptoticallystable condition is necessary for a sliding mode existence. Thisnecessary condition may be found in [19] and [13], to namea few.
C. Problem Statement
Considering the rigid body dynamics (1) and the MRProtation kinematics (2)-(4), design an attitude control systemusing MRP feedback and a linear continuous-type slidingmode controller for every constant desired attitude σ ld andzero desired angular velocity ω bld from any initial angularvelocity ω blb (0) and attitude σ lb (0) .III. M AIN R ESULT
The main result of this paper is presented by Theorem 1.
Theorem 1:
Consider the rigid body dynamics (1) and theMRP rotation kinematics (2)-(3) with the initial angular veloc-ity ω blb ( t = 0) ∈ R , the initial attitude error σ db ( t = 0) ∈ R ,the constant desired attitude σ ld ∈ R , and the zero desiredangular velocity ω bld = (cid:2) (cid:3) T .Then we have τ = u eq + u N is a sliding mode controllerwith the sliding surface (5), where u eq and u N are presentedin equation (6) and (7), respectively, for certain k , k ∈ R ,such that k k > and L > , L ∈ R × . S = (cid:40)(cid:20) ω blb σ db (cid:21) : k ω blb + k σ db = ξ, ξ = (cid:2) (cid:3) T (cid:41) (5) u eq = (cid:0) ω blb × Jω blb (cid:1) − k k JG ( σ db ) ω blb , ∀ (cid:20) ω blb σ db (cid:21) ∈ S (6) u N = − k JLξ, ∀ ξ (cid:54) = (cid:2) (cid:3) T (7) Proof:
The sliding mode controller will be designed tofollow the equivalent control method. First, we have to findthe u eq . At this step, we assume that the sliding mode isexist. Therefore, at the sliding surface, the system satisfiesthe condition (8) ξ = k ω blb + k σ db = (cid:2) (cid:3) T ⇒ ˙ ξ = k ˙ ω blb + k ˙ σ db = (cid:2) (cid:3) T (8)Substituting ˙ ω blb and ˙ σ db in (8) by equation (1) and (2),respectively, hence equation (9) is satisfied. (cid:2) (cid:3) T = k (cid:16) J − (cid:0) − ω blb × Jω blb (cid:1) + J − τ (cid:17) + k G ( σ db ) ω bdb ⇔ J − (cid:0) − ω blb × Jω blb (cid:1) + J − τ = − k k G ( σ db ) ω blb , where ω bld = (cid:2) (cid:3) T ⇔ τ = (cid:0) ω blb × Jω blb (cid:1) − k k JG ( σ db ) ω blb , ∀ k (cid:54) = 0 and k (cid:54) = 0 (9)At this point, we have the control torque that will only workwhen the states reach the sliding surface, u eq , as shown in (10). = u eq = (cid:0) ω blb × Jω blb (cid:1) − k k JG ( σ db ) ω blb , ∀ (cid:20) ω blb σ db (cid:21) ∈ S (10)Next, we have to determine the part of the control torquethat works to ensure the existence of the sliding mode, i.e., u N .This control torque is derived through the Lyapunov stabilitytheory using Lyapunov function candidate, a positive definitefunction V = ξ T ξ . The complete derivation is shown in (11). V = 12 ξ T ξ > , ∀ ξ (cid:54) = (cid:2) (cid:3) T ⇒ ˙ V = ξ T ˙ ξ = ξ T (cid:16) k ˙ ω blb + k ˙ σ db (cid:17) (11)Substituting by ˙ ω blb in (1) and ˙ σ db in (2)-(3), we obtain (12). ˙ V = ξ T (cid:16) k J − (cid:0) − ω blb × Jω blb (cid:1) + k J − ( u eq + u N ) + k G ( σ db ) ω blb (cid:17) ⇔ ˙ V = k ξ T J − u N (12)If we have u N as shown in (13), then (14) is satisfied.Hence, V is a Lyapunov function. u N = − k JLξ, ∀ L > (13) ˙ V = − ξ T Lξ < (14)This fact implies that the sliding surface is asymptoticallystable. In other words, it proves that the sliding mode exists.Nevertheless, it is not the end of the proof since we alsowant to make sure that the system’s states will also reachthe equilibrium point at t → ∞ .Let ¯ V = ξ T ξ + 2¯ k log e (cid:0) σ dbT σ db (cid:1) , ∀ ¯ k > isa candidate Lyapunov function. Recall G ( σ db ) in (3) andsince σ dbT σ dbT σ db = σ dbT σ db σ dbT and σ dbT [ σ db ] × = (cid:2) (cid:3) T , hence we have an MRP property shown in (15).Noting this fact, we can have the time derivation of ¯ V that isshown in (16). σ dbT G ( σ db ) = σ dbT (cid:16) − σ dbT σ db I − [ σ db ] × + σ db σ dbT (cid:17) = 14 σ dbT (cid:0) σ Tdb σ db (cid:1) (15) ˙¯ V = − ξ T Lξ + ¯ kω blbT σ db ⇔ ˙¯ V = − (cid:0) k ω blbT Lω blb + 2 k k ω blbT Lσ db + k σ dbT Lσ db (cid:1) + ¯ kω blbT σ db (16)Therefore, if k k L = ¯ kI , then ˙¯ V < . This fact are statedin (17). Fig. 1: Sliding motion; ξ = (cid:2) ξ ξ ξ (cid:3) T ; ξ − ; ξ − − ; ξ · · · .Fig. 2: Angular velocity of the satellite’s body frame with respect to the in-ertial reference frame expressed in the satellite body frame; ω blb = (cid:2) ω blb ω blb ω blb (cid:3) T ; ω blb − ; ω blb − − ; ω blb · · · . ˙¯ V = − k ω blbT Lω blb − k σ dbT Lσ db < , if k k L = ¯ kI > (17)Since, ¯ V > and ˙¯ V < for any non-zero value of thestate (cid:0) ω blb , σ db (cid:1) , hence the equilibrium point, (cid:0) ω blb , σ db (cid:1) = (cid:2) (cid:3) T , is asymptotically stable for any ini-tial angular velocity ω blb ( t = 0) , the initial attitude er-ror σ db ( t = 0) , and the desired angular velocity ω bld = (cid:2) (cid:3) T .In addition, since ¯ k > and L > , it implies more strictcondition regarding k and k , i.e., k k > . Note that since ¯ k is any positive value, hence L can be any positive definitematrix as well as k and k can be any non-zero scalar suchthat k k > . It completes the proof. Remarks 2:
The designed sliding mode controller τ is acontinuous but not linear feedback. ”Linear continuous” termused to name the type of the sliding mode is based on thecontrol structure of the u N . ig. 3: Attitude of the satellite’s body frame with respect to the desired frame (attitudeerror); σ db = (cid:2) σ db σ db σ db (cid:3) T ; σ db − ; σ db −− ; σ db · · · .Fig. 4: Attitude of the satellite’s body frame with respect to the inertial reference frame; σ lb = (cid:2) σ lb σ lb σ lb (cid:3) T ; σ lb − ; σ lb − − ; σ lb · · · .Fig. 5: Control signal u N ; u N = (cid:2) u N u N u N (cid:3) T ; u N − ; u N −− ; u N · · · . Fig. 6: Control signal u eq ; u eq = (cid:2) u eq u eq u eq (cid:3) T ; u eq − ; u eq −− ; u eq · · · . Some numerical examples are presented to figure out howeffective the designed sliding mode-based attitude controlsystem is. The simulations are conducted using a variable-step solver. The rigid body satellite’s moment of inertia, thecontroller parameters, and the initial conditions are presentedin (18), (19), and (20), respectively. The value of the momentof inertia J is adopted from [20], while the controller param-eters are arbitrary chosen. The desired attitude σ ld value istaken from the simulation setting in [4]. J = .
49 0 .
054 0 . .
054 1 .
51 00 . . (18) k = k = 0 . , L = 0 . I (19) ω blb (0) = (cid:2) − . (cid:3) T ,σ lb (0) = (cid:2) (cid:3) T ,σ ld = (cid:2) . − . − . (cid:3) T (20)Fig. 1-Fig. 6 show some dynamics characteristics relatingto the simulation setting. Fig. 1 verifies that the sliding motionexists. Meanwhile, Fig. 5 and Fig. 6 confirm that the controlsignal u N and u eq , respectively, are also able to bring thetrajectories converge to the equilibrium (Fig. 2 and Fig. 3),i.e., reach the desired states (Fig. 4).IV. C ONCLUDING R EMARKS
The designed attitude control system using a linearcontinuous-type sliding mode controller with the attitude statefeedback in modified Rodrigues parameters (MRP) representa-tion has been presented. It guarantees the sliding mode to existasymptotically. Furthermore, the equilibrium point of the con-trol system has an asymptotic stability guarantee. This stabilityanalysis is conducted since the sliding mode existence on theesigned sliding surface does not imply the stability guaranteeof the system’s equilibrium. Numerical examples verify thatthe sliding motion exists and the trajectories converge to thedesired states.Additional future work would concentrate on investigatingthe robustness properties of this control system. Besides,the attitude controller’s stability analysis in the discrete-timedomain for its digital implementation would also be the futureworks. A
CKNOWLEDGMENTS
This research is supported by the Ministry of Researchand Technology/ National Agency for Research and Inno-vation (BRIN), Jakarta, Republic of Indonesia. The authorsalso acknowledge the Satellite Technology Center, NationalInstitute of Aeronautics and Space (LAPAN) for providing theresearch facilities. HS is the main contributor of this paper withdetail contributions as follows: HS – idea, proof derivation,simulation, discussion, paper preparation; DS – discussion,reviewing, editing. R
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