ANTI-SATURATION DYNAMIC SURFACE CONTROL FOR SPACECRAFT TERMINAL SAFE APPROACH BASED ON COMMAND FILTER. Guanqun Wu, Shenmin Song and Jingguang Sun

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1 International Journal of Innovative Computing, Information and Control ICIC International 2018 ISSN Volume 14, Number 1, February 2018 pp ANTI-SATURATION DYNAMIC SURFACE CONTROL FOR SPACECRAFT TERMINAL SAFE APPROACH BASED ON COMMAND FILTER Guanqun Wu, Shenmin Song and Jingguang Sun Center for Control Theory and Guidane Tehnology Harbin Institute of Tehnology No. 92, West Dazhi Street, Harbin , P. R. China { wgqwuguanqun; sunjingguanghit }@163.om; Corresponding author: songshenmin@hit.edu.n Reeived June 2017; revised Otober 2017 Abstrat. This paper studies anti-saturation ontrol shemes of the spaeraft terminal safe approah. Based on the spaeraft relative motion model of terminal approah and spherial ollision avoidane potential funtion, anti-saturation ontroller and adaptive anti-saturation ontroller are designed for the situations of known and unknown upper bound of external disturbanes respetively using dynami surfae ontrol DSC and auxiliary system. The designed ontrollers not only take advantage of the first-order ommand filter to avoid the differential of the virtual ontrol signals, but also introdue the ompensating signals to remove the effet of the error aused by the ommand filter. Under the proposed ontrol strategies, the bounded and uniformly ultimate bounded of the system states are proved via Lyapunov stability theory, and the haser an approah the desired position without ollision with the target. The numerial simulations are onduted to demonstrate that the haser spaeraft using the designed ontrollers an realize the terminal approah to the target safely, whih further illustrate the effetiveness of the proposed ontrollers. Keywords: Terminal safe approah, Collision avoidane, Dynami surfae ontrol, Command filter, Input saturation 1. Introdution. With the inreasing of spae strategy position, the spaeraft proximity relative motion ontrol has beome a researh hotspot in spae tehnology, whih is being widely applied in many spae missions, suh as rendezvous and doking, on-orbit servies and spae attak-defense [1,2]. Considering the atual requirements as the external disturbanes, input onstraint and ollision area, ontrolling the haser to approah the target safely and aurately beomes the key to omplete spae missions. Consequently, the robust ontrol of spaeraft proximity relative motion onsidering safety onstraints has beome an essential tehnology in aerospae [3]. Reently, experts and sholars at home and abroad have onduted a lot of researh with respet to spaeraft relative motion ontrol and obtained many important ahievements [4-8]. On the basis of Tshauner-Hempel equations, a parametri Lyapunov differential equation method was proposed in [4] to solve the spaeraft rendezvous problem. In order to drive the haser to approah the target, a modified adaptive ontroller with unertain parameters was presented in [5], and the system states were proved asymptoti onvergent with the ontroller. In [6], an integral sliding mode ontroller was proposed using linear quadrati optimal ontrol theory to realize spaeraft hovering around ellipti orbits by traking fuel-optimal trajetories. Contraposing spaeraft formation with unertainties and disturbanes, to improve the ontrol preision of the system, an input-output linearization minimum sliding mode error feedbak ontroller was designed in [7]. Using 33

2 34 G. WU, S. SONG AND J. SUN bak-stepping method, a robust adaptive ontroller was designed in [8] for spaeraft rendezvous and doking, with whih the system states were globally uniformly ultimately bounded, and the unertainties of the relative dynamis were ompensated by using radial basis funtion neural networks. Collision may happen in the proess of the spaeraft relative motion. Thus, to ensure the safety of the haser spaeraft, muh literature onsiders ollision avoidane in the design of relative motion ontrol sheme [9-16]. For autonomous rendezvous and doking with a non-ooperative target, a new guidane ontrol method using fuzzy logi ombined with potential funtion was presented in [9] to ensure safe approahing. Aiming at spaeraft formation maintaining, new sliding mode ontrollers that used the speial potential funtions to realize avoiding obstales were presented in [10]. For proximity operations of spaeraft formation flying near ellipti referene orbits, a traking ontroller using Riati proedure was provided in [11], also ollision avoidane problem was settled via using Gaussian-like funtion. In [12], a nonlinear optimal ontrol sheme was put forward using the optimal sliding mode ontrol, whih was ombined with a quadrati funtion to keep away from moving obstale. To realize aurate formation, finite-time ontrollers are designed for multiple Euler-Lagrange systems with avoiding obstales using null-spae-based and fast terminal sliding mode in [13]. As the in-depth study on the model preditive ontrol MPC, many ontrol strategies based on MPC were put forward for rendezvous and doking with obstale avoidane [14-16]. Robust ontrol shemes based on expliit MPC and the linear quadrati MPC with dynamially reonfigurable onstraints were proposed in [14,15], under whih spaeraft rendezvous onsidering obstale avoidane and a line-of-sight one onstraint an aomplish suessfully. In [16], linear quadrati MPC and nonlinear MPC were applied for spaeraft rendezvous and doking, and multiple obstales were avoided by using the ontrollers. In the atual ontrol system, the ontrol fore provided by the atuators is limited. The atuator saturation an weaken system ontrol performane, even may result in system instability if the input saturation is not taken into onsideration in the ontroller design. Thus, the problem of input saturation should be solved in the proess of ontroller design [17-22]. The auxiliary system was introdued in [17] to analyze the effet of input saturation, with whih adaptive traking ontrollers were designed for MIMO systems with input saturation. Using the mean-value theorem, an adaptive fuzzy traking ontroller was onstruted in [18], whih introdued a pieewise smooth funtion to approximate the saturation funtion. In [19], fousing on unertain nonstrit-feedbak systems, an adaptive fuzzy ontrol sheme was proposed, and the problem of input saturation was solved by adopting an auxiliary design system. In [20], an auxiliary system was employed to deal with the influene of input onstraints, ombining with whih robust bak-stepping ontroller was presented for unertain nonlinear systems with input onstraints. Conentrating on spaeraft irular orbit rendezvous system subjet to input saturation, a new robust gain sheduling ontroller was provided in [21]. For spaeraft rendezvous and proximity operations, a saturated adaptive bak-stepping ontroller was put forward in [22], whih adopted auxiliary design system for input saturation. The ontrol strategies above an settle the problem of input onstraint effetively, but there are a smaller number of ontrol shemes with safe onstraint onsidering input saturation at the same time. In order to further researh the spaeraft terminal safe approah ontrol sheme onsidering input saturation, and being inspired by above researh, anti-saturation ontrollers are designed using DSC and auxiliary system for the situations of known and unknown upper bound of external disturbanes respetively, whih use spherial ollision avoidane

3 ANTI-SATURATION DYNAMIC SURFACE CONTROL 35 potential funtion to deal with taboo area in the haser spaeraft motion. Compared with the above mentioned literature, the main ontributions of the paper are as follows. 1 The ollision avoidane onstraint is onsidered in terminal approah ontrol. And the spherial ollision avoidane potential funtion is introdued to make ollision avoidane problem in terminal approah ontrol be simplified into boundedness problem of reiproal of the potential funtion. 2 The proposed dynami surfae ontrollers use ommand filter to avoid the omputations of time derivatives of the virtual ontrol signals, and the ompensating signals are introdued to remove the effets of the error aused by the ommand filter, whih an improve the performane of the ontrollers. 3 By introduing the auxiliary system, input saturation is taken into onsideration in ontroller design to ope with physial onstraints of the atuators, whih makes the designed ontrollers have more pratial value in engineering. The rest of the paper is strutured in following manner. Setion 2 states a brief desription of the spaeraft terminal safe approah ontrol problem, inluding the target orbit oordinate system, spaeraft relative motion model of terminal approah, ollision avoidane potential funtion and related assumptions for ontroller design. In Setion 3, for the situations of known and unknown upper bound of external disturbanes, two anti-saturation dynami surfae ontrollers are proposed respetively, and the Lyapunov theory is used to prove and analyze the performanes of the ontrollers. Setion 4 gives numerial simulations to further verify the effetiveness of the proposed ontrollers. Finally, the onlusions of the paper are presented in Setion Problem Formulation Spaeraft relative motion model of terminal approah. The target orbit oordinate frame F t O t x t y t z t is shown in Figure 1. The origin of oordinate is the enter of target. The positive diretion of x t axis is the diretion that is from the enter point to the target position. The positive diretion of y t axis points to the target movement diretion. The positive diretion of z t axis is perpendiular to target orbital plane. Figure 1. Target orbit oordinate frame The dynamial equations of the haser and the target in earth entered inertial ECI frame F o OXY Z are given as follows: r = µr r 3 + f d m + u m 1

4 36 G. WU, S. SONG AND J. SUN r t = µr t r 3 t + f dt m t 2 where r and r t are defined as position vetors of the target and haser in ECI frame F o OXY Z respetively, so the orresponding aeleration vetors are r and r t. u is the ontrol fore of haser. µ is the gravitational onstant. f d and f dt are all perturbation effets on the target and haser. m and m t are the mass of haser and target. Relative position vetor in ECI frame an be defined as X r = r r t ; thus the relative aeleration vetor Ẍr an be obtained: X r = r r t = µr + µr t + u + f d f dt 3 r 3 rt 3 m m m t Converting the Ẍr into F t O t x t y t z t yields: µr r 3 + µr t r 3 t + u m + d m = Ẍr + ω t ω t X r + ω t X r + 2ω t X v 4 In Equation 4, X r and Ẍr represent relative position vetor and the relative aeleration vetor in F t, and u and disturbane d are denoted as u and f d m m t f dt in F t. In target orbit oordinate frame, define X r = [x y z] T and X v = Ẋr = [ẋ ẏ ż] T. The angular veloity of the target is defined as ω t = [0 0 θt ] T. The position vetor from earth s ore to the target and the haser an be defined as r t = [r t 0 0] T and r = [x + r t y z] T. Substituting the definition of the variables above into Equation 4, after simplifiation, spaeraft terminal approah relative motion model an be obtained as follows: Ẋ r = X v Ẋ v = AX v + BX r + C + d + u 5 m m where Ẋv = [ẍ ÿ z] T. A, B and C are defined as follows: A = 2 θ t B = µ r 3 I θ 2 t θ t 0 θ t θ2 t [ 1 C = µ rt 2 r t r ] T 8 where θ t = n t1+e t os θ t 2 1 e 2 t 3/2, θ t = 2n2 t et1+et os θt3 sin θ t, and n 1 e 2 t 3 t = µ. a 3 t 2.2. Desription of potential funtion and ontrol objetive. In the proess of aessing to the desired position, the haser will be lose to the target spaeraft within short distane, beause ollision may happen between the haser and the target. To ontrol the haser to aess to the desired position without ollision, taboo area where ollision an happen must be onsidered in the proess. Taboo area in the paper is set

5 ANTI-SATURATION DYNAMIC SURFACE CONTROL 37 as sphere interior whose entre is the enter of target and the radius is R. The ollision avoidane potential funtion is presented as follows: hx r = 1 R 2 x 2 + y 2 + z 2 R 2 9 Assume hx r0 > 0, that is, the initial position of the haser is in safe area. In the proess of approahing to the desired position, if the inequality hx r > 0 holds all the time, the haser an realize terminal safe approah. On the ontrary, if hx r 0, the haser may have ollision with the target in the taboo area. Thus, if hx r > 0 holds from beginning to end, that is to say, the 1/hX r is bounded all the time, the haser an approah desired position around the target safely. Control objetive: Aiming at the situation of known and unknown upper bound of external disturbanes, two anti-saturation ontrollers should be designed respetively, under whih the haser an approah the expeted position safely subjet to input saturation onstraint, that is, the relative position of the haser X r an onvergene to the neighborhood of the desired loation X rd, also 1/hX r is bounded all the time. Remark 2.1. The motivation of employing the potential funtion is to onvert ollision avoidane problem into boundedness problem of reiproal of the potential funtion, and the designed ontroller ombined with the potential funtion an guarantee the haser omplete the spaeraft terminal approah without ollision happening Related lemmas and assumptions. To failitate the design and analysis of ontroller, related lemmas and assumptions are introdued as follows. Lemma 2.1. [23]. The ompensating signals ξ i i = 2,..., n are defined as follows ξ 1 = 1 ξ 1 ξ 1 + ξ 2 + x 2, a 1 ξ i = i ξ i ξ i 1 + ξ i+1 + x i+1, a i ξ n = n ξ n ξ n 1 10 where x i+1, and a i are output and input of first-order ommand filter respetively. When t ξ i is bounded, satisfying lim ξ i µ t 2k 0 where k 0 = 1/2 min i. Lemma 2.2. [17]. For any arbitrary number x and non-zero real number y, then the following inequality holds where α = x 1 tanh x/y α y 11 Assumption 2.1. The initial position and the desired position of the haser X r0, X rd are assumed to be set in safe area. Assumption 2.2. The external disturbane d in Equation 5 is assumed to be bounded, and satisfies the inequality d d m, where d m is a positive onstant. Remark 2.2. For further on-orbit operation, the desired position X rd should be in the safe area. Also, if the initial position X r0 is set in the taboo area, the haser may have ollided with the target. Thus, in order to design the ontrollers for the spaeraft terminal approah with ollision avoidane, Assumption 2.1 is reasonable. In pratie, the external disturbanes are unknown but bounded whih ontain atmospheri drag, radiation pressure, et. And [8,10,22] have presented assumptions similar to Assumption 2.2. Thus, Assumption 2.2 is reasonable.

6 38 G. WU, S. SONG AND J. SUN 3. Main Results. For the spaeraft terminal approah relative motion model Equations 5-8, for the situations of known and unknown upper bound of external disturbanes, two anti-saturation ontrollers are designed based on dynami surfae ontrol with signal ompensation and auxiliary system, by whih the haser an approah the desired position safely with input onstraint Anti-saturation ontroller design for the situation of known upper bound of external disturbanes. For the spaeraft terminal approah relative motion model Equations 5-8 with the known upper bound of external disturbanes, the detailed proesses of designing an anti-saturation ontroller are shown as follows. Step 1: Define the traking error variable z 1 as follows: where X rd is referene signal. Computing the first order derivative of Equation 12: Define the virtual ontrol α v1 as follows: z 1 = X r X rd 12 ż 1 = Ẋr Ẋrd = X v Ẋrd 13 α v1 = k 1 z 1 14 where k 1 is a positive onstant. To avoid the differential of the virtual ontrol signal, the first order ommand filter is introdued as follows: τ 1 α vd1 + α vd1 = α v1, α vd1 0 = α v where α v1, α vd1 are the input and output of the ommand filter respetively, and τ 1 is a positive onstant. To eliminate the effet of the error aused by the α vd1 α v1, the ompensating signal is introdued referring to Lemma 2.1. ξ 1 is defined as follows: ξ 1 = k 1 ξ 1 + ξ 2 + α vd1 α v1 16 Define the ompensated traking error signal z v1 as: Choose Lyapunov funtion v 1 as: Compute the first order derivative of v 1 : z v1 = z 1 ξ 1 17 v 1 = 1 2 zt v1z v1 18 v 1 = zv1żv1 T = zv1 T X v Ẋrd + zv1 T k 1 ξ 1 ξ 2 α vd1 α v1 = zv1 T z 2 + Ẋrd + α vd1 Ẋrd + zv1 T k 1 ξ 1 ξ 2 α vd1 α v1 = z T v1 z 2 + α v1 + α vd1 α v1 + z T v1 k 1 ξ 1 ξ 2 α vd1 α v1 = z T v1 z 2 k 1 z 1 + k 1 ξ 1 ξ 2 = z T v1 z 2 k 1 z v1 k 1 ξ 1 + k 1 ξ 1 ξ 2 = k 1 z T v1z v1 + z T v1 z 2 ξ 2 Step 2: Define the traking error variable z 2 as follows: 19 z 2 = X v Ẋrd α vd1 20

7 ANTI-SATURATION DYNAMIC SURFACE CONTROL 39 Computing the first order derivative of Equation 20 yields: ż 2 = Ẋv Ẍrd α vd1 = AX v + BX r + C + d m + u m Ẍrd 1 τ 1 α v1 α vd1 21 The ompensating signal ξ 2 is introdued as follows: where k 2 is a positive onstant. Define the ompensated traking error signal z v2 as: Choose Lyapunov funtion v 2 as: ξ 2 = k 2 ξ 2 ξ 1 22 z v2 = z 2 ξ 2 23 v 2 = 1 2 zt v2z v2 24 Compute the first order derivative of v 2 : v 2 = zv2żv2 T = zv2 AX T v + BX r + C + dm + um Ẍrd α vd1 + zv2 T k 2 ξ 2 + ξ 1 25 To handle input saturation, the auxiliary system Equation 26 is introdued: k η = η η 1 zv2 T u η m 2 ut u η + u, η σ V 26 0, η < σ V where u = u u, u is the ideal ontrol input, u is the atual ontrol input, and σ V is a positive onstant. For the situations that the upper bound of external disturbanes is known, based on Equations 12-26, the ontroller is designed as Equation 27, where k 2, k h, k η, k h1 are positive onstants, and inequalities k 2 > 1k 2 η, k η > 1 hold. k 2 z 2 z 1 AX v BX r C + Ẍrd + α vd1 d m sign z v2 u = m m z v2 +k η η k h ḣxr z v2 2 h 2 X r + k h1 h 1 X r 27 Theorem 3.1. Consider the spaeraft terminal approah relative motion model Equations 5-8 with Assumption 2.1 and Assumption 2.2, for the situations of known upper bound of external disturbanes, the state of the system is regulated under the designed ontroller Equation 27 and auxiliary system Equation 26, and the following onlusion an be drawn. I The variables v 1, v 2, η, 1/hX r are bounded. II The ollision avoidane potential funtion hx r is positive, that is, the haser an avoid ollision with the target in the proess of motion. III The traking error variables z 1, z 2 an onverge to any small neighborhood, that is, the relative position of the haser X r an onverge to any small range of the desired position X rd. Proof: Choose the Lyapunov funtion V 1 as: V 1 = v 1 + v ηt η + k h 1 hx r 28

8 40 G. WU, S. SONG AND J. SUN Computing the derivative of V 1 yields: V 1 = v 1 + v 2 + η T η + k h ḣx r h 2 X r AX v + BX r + C + d m + u m = k 1 zv1z T v1 + zv1 T z 2 ξ 2 + zv2 T + u m Ẍrd α vd1 = k 1 zv1z T v1 + zv1z T v2 + zv2 T k 2 z 2 z 1 + zv2 T k 2 ξ 2 + ξ 1 + η T ḣx r d η + k h h 2 X r + zt v2 + u d m m m + z T v2 k 2 ξ 2 + ξ 1 + η T η + k h ḣx r h 2 X r sign z v2 + k η η m z v2 k h ḣxr z v2 2 h 2 X r + k h1 h 1 X r = k 1 zv1z T v1 + zv2 T z v1 k 2 z 2 z 1 + k 2 ξ 2 + ξ 1 + η T ḣx r d η + k h h 2 X r + zt v2 + u d m z v2 sign z v2 + k η η k h ḣxr m m z v2 2 h 2 X r + k h1 h 1 X r = k 1 zv1z T v1 k 2 zv2z T v2 k η η T 1 η k h k h1 hx r + zt v2 d d m sign z v2 m + k η zv2η T + zv2 T u z T v2 u + 1 m m 2 ut u + η T u Two inequalities an be derived: k η z T v2η + η T u = k η m 29 z T v2 u z T v2 u z v2,i η i + i=1 3 u i η i i=1 1 2 k ηz T v2z v k η + 1 η T η ut u Substituting Equations 30 and 31 into Equation 29, then Equation 29 an be rewritten as: V 1 k 1 zv1z T v1 k k η zv2z T v2 2 k η 1 η T 1 η k h k h1 2 hx r 31 + zt v2 d d m sign z v2 m k k η k 1 z T v1z v1 + z v2 d d m m k 1 z T v1z v1 ε 1 V 1 k k η z T v2z v2 1 2 k η 1 η T 1 η k h k h1 2 hx r 1 zv2z T v2 2 k η 1 η T 1 η k h k h1 2 hx r where ε 1 = min { 2k 1, 2 k k η, k η 1 2, kh1 }, k2 > 1 2 k η, k η > 1. 32

9 ANTI-SATURATION DYNAMIC SURFACE CONTROL 41 Equation 32 shows that V 1 0, so V 1 is not inreasing, whih means that v 1, v 2, η, 1/hX r are bounded. Beause initial value hx r0 is positive and 1/hX r is bounded, it an be obtained that hx r 0 holds during the hange of the system states, whih indiates that ollision avoidane potential funtion hx r is not equal to zero in the proess of approahing the target, that is to say, the haser moves in safe area all the time and an realize spaeraft terminal approah safely. Therefore, onlusions I and II have been proved. Aording to Equation 32, inequality an be obtained as: V 1 t V 1 0e ε 1t From Equation 33, it an be obtained that v 1, v 2 V 1 0e ε 1t. When t, v 1, v 2 0, further z v1, z v2 0. From Lemma 2.1 it an be seen that ξ 1 and ξ 2 are bounded. So when ε 1 is seleted large enough, the traking error variables z 1, z 2 an onverge to any small neighborhood. Therefore, onlusion III has been proved. The proof of Theorem 3.1 has been ompleted. Remark 3.1. The use of ommand filter may ause the filtering errors whih an lead to great traking error. Therefore, the thesis introdues the ompensating signals Equation 16 and Equation 22 to remove the effet of the error aused by the ommand filter so that the traking error an be diminished. Remark 3.2. Generally, the upper bound of external disturbanes of the system is unknown beause of the omplex external environment; thus the ontroller Equation 27 results in failure. To make the ontrol sheme more valuable to pratie, an adaptive ontroller is designed as the following Adaptive anti-saturation ontroller design for the situation of unknown upper bound of external disturbanes. To deal with the spaeraft terminal approah with unknown upper bound of external disturbanes, the paper further designs an adaptive anti-saturation ontroller. Based on Equations derived using DSC, ommand filter and the ompensating signal, a novel ontroller Equation 34 is designed whih adopts adaptive law Equations 35 and 36 to estimate the upper bound of unknown external disturbanes, where k 2, k 3, k h, k η, γ d are positive onstants, and inequalities k 2 > 1k 2 η, k η > 1 hold, ˆdm is the estimation value of upper bound of external disturbanes d m. k 2 z 2 z 1 AX v BX r C + Ẍrd + α vd1 u = m ˆd m tanh z v2 /p 2 V m + k z v2 ηη k h ḣxr z v2 2 h 2 X r + h 1 34 X r ˆd m = γ d z v d dt p2 V = 3k 3 α ˆdm p 2 V 36 Theorem 3.2. Consider the spaeraft terminal approah relative motion model Equations 5-8 with Assumption 2.1 and Assumption 2.2, for the situations of unknown upper bound of external disturbanes, the state of the system is regulated under the designed adaptive ontroller Equation 34, adaptive law Equations 35 and 36 and auxiliary system Equation 26, the following onlusion an be drawn. I The ollision avoidane potential funtion hx r is positive, that is, the haser an avoid ollision with the target in the proess of motion. 33

10 42 G. WU, S. SONG AND J. SUN II The variables v 1, v 2, η, 1/hX r are uniformly ultimately bounded. III The traking error variables z 1, z 2 an onverge to any small neighborhood, that is, the relative position of the haser X r an onverge to any small range of the desired position X rd. Proof: Choose the Lyapunov funtion andidate as V 2 = v 1 + v ηt η + 1 2γ d m d2 m + 1 m k 3 p 2 V + k h 1 hx r where d m is the estimation error d m = d m ˆd m of upper bound of external disturbanes. Computing the derivative of V 2 yields: V 2 = v 1 + v 2 + η T η + 1 dm dm + 1 d ḣx r γ d m m k 3 dt p2 V + k h h 2 X r = k 1 zv1z T v1 + zv1 T z 2 ξ 2 + zv2 T AX v + BX r + C + d + u + u m m m Ẍrd α vd1 + z T v2 k 2 ξ 2 + ξ 1 + η T η + 1 γ d m dm dm + 1 m d dt p2 V + k h ḣx r h 2 X r = k 1 zv1z T v1 + zv1z T v2 + zv2 T k 2 z 2 z 1 + zv2 T k 2 ξ 2 + ξ 1 d + zv2 T + u ˆd m tanh z v2 /p 2 z v2 V + kη η k h ḣxr m m m z v2 2 h 2 X r + h 1 X r + η T η + 1 dm dm + 1 d ḣx r γ d m m k 3 dt p2 V + k h h 2 X r = k 1 zv1z T v1 k 2 zv2z T v2 k η η T 1 η k h hx r + zt v2 d m ˆd m tanh z v2 /p 2 V + k h zv2 T + k η zv2η T + zv2 T u z T v2 u + 1 m m 2 ut u + η T u + 1 dm dm + 1 d γ d m m k 3 dt p2 V Beause Equations 30 and 31 hold, Equation 38 an be further rewritten as: V 2 k 1 zv1z T v1 k k η zv2z T v2 2 k η 1 η T 1 η k h 2 hx r + zt v2 d m ˆd m tanh z v2 /p 2 V + 1 dm dm d γ d m m k 3 dt p2 V Aording to Lemma 2.2 and Assumption 2.2, the following inequality an be derived: 1 z m v2d T ˆd m zv2 T tanh z v2 /p 2 V 1 3 z v2,i d i + m ˆd m p 2 V α z v2,i /p 2 V i=1 1 3 z v2,i d m + m ˆd 40 m p 2 V α z v2,i /p 2 V i=1 1 3α ˆdm p 2 V + m d 3 m z v2,i i=

11 ANTI-SATURATION DYNAMIC SURFACE CONTROL 43 Substituting Equation 40 into Equation 39 yields V 2 k 1 zv1z T v1 k k η zv2z T v2 2 k η α ˆdm p 2 V + m d 3 m z v2,i 1 dm ˆdm 3α ˆdm p 2 V γ i=1 d m m = k 1 zv1z T v1 k k η zv2z T v2 2 k η d m z v2,i m d m z v2 1 i=1 = k 1 z T v1z v1 0 η T η k h 1 hx r η T η k h 1 hx r k k η zv2z T v2 2 k η 1 η T 1 η k h 2 hx r 41 Equation 41 shows that V 2 0, so V 2 is not inreasing, whih means that v 1, v 2, η, d m, 1/hX r are bounded. Therefore, the estimated value of the adaptive parameter ˆd m is bounded. That is to say, there exists a positive onstant d m > 0 satisfying ˆd m d m. Beause initial value hx r0 is positive and 1/hX r is bounded, it an be onluded that hx r 0 holds during the hange of the system states, whih indiates that ollision avoidane potential funtion hx r is not equal to zero in the proess of approahing the target, that is to say, the haser moves in safe area all the time. In order to further analyze the performane of the ontroller Equation 34, Lyapunov funtion V 3 is hosen as V 3 = v 1 + v ηt η + 1 d m 2γ d m ˆd 2 2 m + p 2 1 V + k h 42 m k 3 hx r Compute the derivative of Equation 42: V 3 = k 1 zv1z T v1 + zv1z T v2 + zv2 T k 2 z 2 z 1 + zv2 T k 2 ξ 2 + ξ 1 + η T η + 1 d m γ d m ˆd m ˆdm + zv2 T d + u ˆd m tanh z v2 /p 2 V + kη η m m m z v2 k h h 1 X zv2z T r + 2 d v2 m k 3 dt p2 V = k 1 zv1z T v1 k 2 zv2z T v2 k η η T 1 η k h hx r + zt v2 d m ˆd m tanh z v2 /p 2 V + k η zv2η T + zv2 T u z T v2 u + 1 m m 2 ut u + η T u + 1 γ d m d m ˆd m ˆdm + 2 d m k 3 k k η zv2z T v2 dt p2 V 1 k 1 zv1z T v1 2 k η 1 η T 1 η k h 2 hx r + zt v2 d m ˆd m tanh z v2 /p 2 V + 1 d m γ d m ˆd m ˆdm + 2 d m k 3 dt p2 V 43

12 44 G. WU, S. SONG AND J. SUN Aording to Lemma 2.2, Assumption 2.2 and ˆd m d m, the inequality an be derived: 1 z m v2d T ˆd m zv2 T tanh z v2 /p 2 V = 1 3 z v2,i d i m ˆd m tanh z v2,i /p 2 V i=1 1 3 z v2,i d m + m ˆd m p 2 V α z v2,i /p 2 V i=1 3α 1 ˆdm p 2 V + d m m ˆd 3 m z v2,i Substituting Equation 44 into Equation 43 yields: V 3 k 1 zv1z T v1 k k η zv2z T v2 i=1 1 2 k η d m 2γ d m ˆd 2 1 m p 2 V + 1 d m m 2γ d m ˆd 2 m ε 2 V 3 + C 1 η T η k h 1 hx r where ε 2 = min { 2k 1, 2 k k η, k η 1 2, k 3 2, 1 }, k 2 > 1 2 k η, k η > 1, C 1 = 1 2γ d m d m ˆd 2 m is bounded. From Equation 45, it an be known that v1, v 2, η, 1/hX r are uniformly ultimately bounded. Therefore, onlusion II has been proved. Aording to Equation 45, inequality an be derived: V 3 0 C 1 e ε 2t + C 1 0 V 3 t 46 ε 2 ε 2 From Equation 46, it an be known v 1, v 2 V 3 0 C 1 ε 2 e ε2t + C 1 ε 2. When t, v 1, v 2 C 1 2C ε 2, further it an be got lim z v1 = 1 2C t ε 2, lim z v2 = 1 t ε 2. From Lemma 2.1, it an be seen that ξ 1 and ξ 2 are bounded, so when ε 2 is seleted large enough, the traking error variables z 1, z 2 an onverge to any small neighborhood. Therefore, onlusion III has been proved. The proof of Theorem 3.2 has been ompleted. Remark 3.3. The auxiliary systems Equation 26 used in ontrol shemes an handle input saturation problem for the ases that the input saturation is symmetrial and asymmetri. Remark 3.4. Considering the anti-saturation ontroller Equation 27 and adaptive antisaturation ontroller Equation 34, if error z v2 onverges to zero infinitely, singularity problem may appear in the system. In the simulation analysis, the ontroller Equation 27 and the ontroller Equation 34 are modified to Equation 47 and Equation 48 respetively to avoid the phenomenon: k 2 z 2 z 1 AX v BX r C + Ẍrd + α vd1 d m sign z v2 u = m m z v2 +k η η k h ḣxr z v2 2 h 2 X r + k h1 h 1 X r

13 ANTI-SATURATION DYNAMIC SURFACE CONTROL 45 k 2 z 2 z 1 AX v BX r C + Ẍrd + α vd1 u = m ˆd m tanh z v2 /p 2 V m + k z v2 ηη k h ḣxr z v2 2 h 2 X r + h 1 X r + 48 where is a very small positive onstant. 4. Simulation Analysis. In order to validate the effetiveness of the designed ontrollers, simulations are onduted in this setion for the situations that the upper bound of the external disturbanes is known and unknown respetively. In the proess of simulations, to make these parameters more reasonable, the values of the basis parameters refer to the values of the orresponding parameters in [10]. The detailed names and values of parameters are shown in Table 1. Table 1. Parts of parameters for simulation Parameter Value The maximum input fore 5N The mass of the haser m = 120 kg The orbital element of the target: Semi-major axis a = km The orbital element of the target: Eentriity e = 0.02 The orbital element of the target: Longitude asending node Ω = 50 π/180 rad The orbital element of the target: Inlination i = 40 π/180 rad The orbital element of the target: Argument of perigee ω = 45 π/180 rad The orbital element of the target: Initial true anomaly f = 10 rad The initial relative speed vetor and the desired relative speed vetor are assumed as X v0 = [0 0 0] T m/s and X vd = [0 0 0] T m/s. To verify the effetiveness and robustness of the ontroller, simulations for two ases are onduted. Case1: The radius of the sphere in ollision avoidane potential and the external dis- turbanes are set as R = 10 m and d = sin πt/125 3 os πt/200 N. 8 + sin πt/ sin πt/ sin πt/ os πt/200 The initial relative position and the desired target position vetors are set as X r0 = [ ] T m and X rd = [ ] T m. Case2: The radius of the sphere in ollision avoidane potential and the external dis- turbanes are set as R = 15 m and d = sin πt/125 3 os πt/200 N. 8 + sin πt/ sin πt/ sin πt/ os πt/200 The initial relative position and the desired target position vetors are set as X r0 = [ ] T m and X rd = [ ] T m Simulation analysis of the anti-saturation ontroller. For the situation of known upper bound of external disturbanes, the parameters of the designed anti-saturation ontroller Equation 27 are seleted as: k 1 = 0.021, k 2 = 12, τ 1 = 0.08, k h = 40, k h1 = 0.8, k η = 1.02, d m = Simulation results of Case1 and Case2 using antisaturation ontroller Equation 27 are shown in Figures 2-4. The traking error urves of z 1 and z 2 are given in Figures 2a and 2b respetively. The urves of all ases indiate that the position and veloity of haser an onverge in short time though there are external disturbanes and input saturation, whih satisfies the requirement of ontrol auray. The ontrol fore urves of the system are provided in

14 46 G. WU, S. SONG AND J. SUN a The urves of traking error z 1 b The urves of traking error z 2 Figure 2. The urves of traking error under the ontroller Equation 27 Figure 3. The ontrol fore urves of the system under the ontroller Equation 27 Figure 3, from whih it an be seen that the ontrol fores are bounded in the whole ontrol proess without obvious hattering. Figure 4 shows the urve of potential funtion and the motion trajetory of haser in the target orbit oordinate frame whih desribes the proess of terminal approah more intuitively. Based on the results shown in Figures 4a- 4d, it an be known that there is no ollision happening when the haser approahes the target. Comparing the results of Case1 and the results of Case2 shows that the ontroller Equation 27 an realize the terminal safe approah with bigger differene of the external disturbanes and taboo areas, whih indiate the effetiveness and robustness of the ontroller Equation 27. To verify the effetiveness of ollision avoidane of the ontroller Equation 27 R = 15 m, let k h = 0, that is, the ollision avoidane is not onsidered. The orresponding simulation results are shown in Figure 5. Based on the results, it an be seen that the

15 ANTI-SATURATION DYNAMIC SURFACE CONTROL 47 a The urve of potential funtion for Case1 b The motion trajetory for Case1 The urve of potential funtion for Case2 d The motion trajetory for Case2 Figure 4. The urves of potential funtion and motion trajetory of haser under the ontroller Equation 27 a The urve of potential funtion b The motion trajetory of haser Figure 5. Simulation results using the ontroller Equation 27 with k h = 0

16 48 G. WU, S. SONG AND J. SUN haser ollides with the target in the proess of terminal approah if the ollision avoidane is not onsidered in ontroller design. Compared with Case2, it further indiates that the ontroller Equation 27 is useful for ollision avoidane. Aording to the above analysis, for the situation that the upper bound of the external disturbanes is known, the results presented in Figures 2-5 indiate that the antisaturation ontroller Equation 27 is effetive for the haser to realize the terminal approah without ollision Simulation analysis of the adaptive anti-saturation ontroller. For the situation of unknown upper bound of external disturbanes, the parameters of the designed adaptive anti-saturation ontroller Equation 34 are seleted as: k 1 = 0.033, k 2 = 18, k 3 = , τ 1 = 0.05, k h = 40, k η = 1.8, γ d = 0.4. Simulation results of Case1 and Case2 using adaptive ontroller Equation 34 are shown in Figures 6-9. a The urves of traking error z 1 b The urves of traking error z 2 Figure 6. The urves of traking error under the ontroller Equation 34 In Figure 6a and Figure 6b, the traking error urves of z 1 and z 2 are given respetively for two ases. From the simulation urves, it an be seen that the position and the veloity of haser an onverge in short time satisfying the requirement of ontrol auray regardless of external disturbane, parameter unertainty and input saturation of the system. The ontrol fore urves of the system presented in Figure 7 show that the ontrol fores are bounded in the whole ontrol proess without hattering whih satisfy the requirement of input onstraint, and the ontrol fores are better than the ontrol fores in Figure 3. The urves of adaptive parameters shown in Figure 8 manifest that the estimated values ˆd m approah a steady value after a period of time, whih indiates that the adaptive shemes are effetive to estimate the value of the upper bound of unknown external disturbanes d m. Figure 9 gives the urve of potential funtion and the motion trajetory of haser in the target orbit oordinate frame, whih shows that the value of the potential funtion is positive and the haser an aomplish terminal approah being away from the taboo area for two ases. Analyzing the results above, it an be seen that the ontroller Equation 34 an realize the terminal safe approah for the two ases with differene of the external disturbanes and taboo areas, whih indiate the effetiveness and robustness of the ontroller Equation 34. To verify the effetiveness of ollision avoidane of the adaptive ontroller Equation 34 R = 15 m, let k h = 0, the orresponding simulation results are shown in Figure 10. From the results, it an be seen that the haser ollides with the target during terminal

17 ANTI-SATURATION DYNAMIC SURFACE CONTROL 49 Figure 7. The ontrol fore urves of the system under the ontroller Equation 34 Figure 8. The urves of adaptive parameters under the ontroller Equation 34 approah if the ollision avoidane is not taken into aount, whih further indiate that the ontroller Equation 34 is effetive to avoid ollision in terminal approah. All in all, for the situation that upper bound of the external disturbanes is unknown, the results presented in Figures 6-10 indiate that the adaptive anti-saturation ontroller Equation 34 is effetive for the haser to realize the terminal safe approah. 5. Conlusions. The anti-saturation ontrol strategies for spaeraft terminal safe approah are researhed in this paper based on DSC, auxiliary system and spherial ollision

18 50 G. WU, S. SONG AND J. SUN a The urve of potential funtion for Case1 b The motion trajetory for Case1 The urve of potential funtion for Case2 d The motion trajetory Case2 Figure 9. The urves of potential funtion and motion trajetory of haser under the ontroller Equation 34 a The urve of potential funtion b The motion trajetory of haser Figure 10. Simulation results using the ontroller Equation 34 with k h = 0

19 ANTI-SATURATION DYNAMIC SURFACE CONTROL 51 avoidane potential funtion. Aording to the theoretial proofs and numerial simulations, onlusions are drawn as follows. 1 The safety problem of terminal approah ontrol is simplified into boundedness problem of the reiproal of the potential funtion by introduing the spherial ollision avoidane potential funtion. 2 For the situations of known and unknown upper bound of external disturbanes, two novel anti-saturation ontrol shemes for spaeraft terminal safe approah are put forward. The first-order ommand filter is used to avoid the differential of the virtual ontrol signals, and the effet of the error aused by the ommand filter an be removed by introduing the ompensating signals. 3 Under the designed ontrollers, the states of system are bounded and uniformly ultimately bounded respetively, and the traking errors an onverge to any small neighborhood. Also the haser spaeraft an approah the expeted position safely. The numerial simulation results further indiate the effetiveness of the designed ontrollers. Aknowledgment. The authors would like to aknowledge the support provided by the China Aerospae Siene and Tehnology Innovation Foundation CAST. No. JZ , the State Key Program of National Natural Siene Foundation of China and the Major Program of Natural Siene Foundation of China REFERENCES [1] C. Guariniello and D. A. DeLaurentis, Maintenane and reyling in spae: Funtional dependeny analysis of on-orbit serviing satellites team for modular spaeraft, AIAA SPACE 2013 Conferene and Exposition, pp.1-16, [2] E. Capello, E. Punta, F. Dabbene et al., Sliding-mode ontrol strategies for rendezvous and doking maneuvers, Journal of Guidane, Control, and Dynamis, vol.40, no.6, pp , [3] S. Di Cairano, H. Park and I. Kolmanovsky, Model preditive ontrol approah for guidane of spaeraft rendezvous and proximity maneuvering, International Journal of Robust and Nonlinear Control, vol.22, no.12, pp , [4] B. Zhou, Z. L. Lin and G. R. Duan, Lyapunov differential equation approah to elliptial orbital rendezvous with onstrained ontrols, Journal of Guidane, Control, and Dynamis, vol.34, no.2, pp , [5] S. N. Wu, Z. G. Wu, G. Radie et al., Adaptive ontrol for spaeraft relative translation with parametri unertainty, Aerospae Siene and Tehnology, vol.31, no.1, pp.53-58, [6] X. Huang, Y. Yan, Y. Zhou et al., Sliding mode ontrol for Lorentz-augmented spaeraft hovering around ellipti orbits, Ata Astronautia, vol.103, pp , [7] L. Cao and X. Q. Chen, Input-output linearization minimum sliding-mode error feedbak ontrol for spaeraft formation with large perturbations, Pro. of the Institution of Mehanial Engineers, Part G: Journal of Aerospae Engineering, vol.229, no.2, pp , [8] K. W. Xia and W. Huo, Robust adaptive bak-stepping neural networks ontrol for spaeraft rendezvous and doking with unertainties, Nonlinear Dynamis, vol.84, no.3, pp , [9] D. W. Zhang, S. M. Song and R. Pei, Safe guidane for autonomous rendezvous and doking with a non-ooperative target, AIAA Guidane, Navigation, and Control Conferene, pp.1-19, [10] Q. L. Hu, H. Y. Dong, Y. M. Zhang et al., Traking ontrol of spaeraft formation flying with ollision avoidane, Aerospae Siene and Tehnology, vol.42, pp , [11] L. Palaios, M. Ceriotti and G. Radie, Close proximity formation flying via linear quadrati traking ontroller and artifiial potential funtion, Advanes in Spae Researh, vol.56, no.10, pp , [12] L. C. Feng, Q. Ni, Y. Z. Bai et al., Optimal sliding mode ontrol for spaeraft rendezvous with ollision avoidane, IEEE Congress on Evolutionary Computation CEC, pp , [13] J. Chen, M. G. Gan, J. Huang et al., Formation ontrol of multiple Euler-Lagrange systems via nullspae-based behavioral ontrol, Siene China Information Sienes, vol.59, no.1, pp.1-11, [14] M. Leomanni, E. Rogers and S. B. Gabriel, Expliit model preditive ontrol approah for lowthrust spaeraft proximity operations, Journal of Guidane, Control, and Dynamis, vol.37, no.6, pp , 2014.

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