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Formation Control of Spacecraft under orbital perturbation
4th 4th International International Conference Conference on on Advances Advances in in Control Control and and 4th International Conference on Advances in Control and Optimization of Dynamical Dynamical Systems Optimization of Systems 4th International Conference on Advances in Control and Optimization of Dynamical Systems Available online at www.sciencedirect.com February 2016. NIT India February 1-5, 1-5, of 2016. NIT Tiruchirappalli, Tiruchirappalli, India Optimization Dynamical Systems February 1-5, 2016. NIT Tiruchirappalli, India February 1-5, 2016. NIT Tiruchirappalli, India
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IFAC-PapersOnLine 49-1 (2016) 130–135
Formation Control of Spacecraft Formation Control of Spacecraft Formation Control of Spacecraft Formation Control of Spacecraft orbital perturbation orbital perturbation orbital perturbation orbital perturbation
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∗ ∗∗ ∗ Prof. Arpita Sinha ∗∗ Shashank Agarwal Shashank Agarwal ∗ ∗ Prof. Arpita Sinha ∗∗ ∗∗ Shashank Agarwal ∗ Prof. Arpita Sinha ∗∗ Shashank Agarwal Prof. Arpita Sinha ∗ ∗ Systems and Control Department, Indian Institute of Technology Institute of Technology ∗ ∗ Systems and Control Department, Indian Indian ∗ SystemsMumbai and Department, Bombay, 411014 (email :[email protected]) Systems and Control Control Department, Indian Institute Institute of of Technology Technology Bombay, Mumbai 411014 (email :[email protected]) ∗∗ Bombay, Mumbai 411014 (email :[email protected]) ∗∗ Systems and Control Department, Indian Institute of Bombay, Mumbai 411014 (email :[email protected]) Systems and Control Department, Indian Institute ∗∗ ∗∗ Systems and Control Department, Indian Institute of ∗∗ Technology,Bombay, Mumbai 411014 (email: [email protected]) Systems and Mumbai Control Department, Indian Institute of of Technology,Bombay, 411014 (email: [email protected]) Technology,Bombay, Mumbai 411014 (email: [email protected]) Technology,Bombay, Mumbai 411014 (email: [email protected]) Abstract: The novel spacecraft flying aa substitute aa Abstract: The novel concept concept of of multiple multiple spacecraft formation formation flying as as substitute for for Abstract: The multiple formation as for single large vehicle vehicle will concept enhance of future spacespacecraft mission performance. performance. The benefits benefits of aa spacecraft spacecraft Abstract: The novel novel concept of multiple spacecraft formation flying flying as aa substitute substitute for aa single large will enhance future space mission The of single vehicle will future space performance. The of formation cost effective synthetic aperture radar for observations, of the single large largeinclude vehiclemore will enhance enhance future space mission mission performance. The benefits benefitsflexibility of aa spacecraft spacecraft formation include more cost effective synthetic aperture radar for observations, flexibility of the formation include more cost effective synthetic aperture radar for observations, flexibility of the satellites altering their roles, reduction of cost owing to the reduction of mass launched into orbit formationaltering includetheir moreroles, cost effective synthetic aperture radar for observations, flexibility oforbit the satellites reduction of cost owing to the reduction of mass launched into satellites altering their roles, reduction of cost owing to the reduction of mass launched into orbit etc. A in the control to formation maintenance satellites altering challenge their roles, of of cost owingdesign to theis of aamass launched into orbit etc. A significant significant challenge inreduction the domain domain of control design isreduction to contrive contrive formation maintenance etc. challenge the domain control is aa formation maintenance controller that will will enablein the member spacecrafts to maintain maintain desired relative orbit with with etc. A A significant significant challenge inthe themember domain of of control design design is to to contrive contrive formation maintenance controller that enable spacecrafts to aaa desired relative orbit controller that will enable the member spacecrafts to maintain desired relative orbit with optimal propellant expenditure while maintaining the desired formation. This paper examines controllerpropellant that will expenditure enable the member spacecraftsthetodesired maintain a desiredThis relative orbit with optimal while maintaining formation. paper examines optimal propellant expenditure while maintaining the desired formation. This paper examines a low earth orbit formation control methodology, with the aim of evaluating formation from propellant expenditure while maintaining with the desired Thisformation paper examines aoptimal low earth orbit formation control methodology, the aimformation. of evaluating from aa a orbit control methodology, the of evaluating formation from aa propellant thrust and dynamics standpoint. feedback controller has a low low earth earthbudget, orbit formation formation control methodology, with the aim aim A of State evaluating formation from propellant budget, thrust level level and error error dynamicswith standpoint. A State feedback controller has propellant budget, thrust level and error dynamics standpoint. A State feedback controller has been applied on J perturbed Clohessy-Wiltshire dynamics, and the system is checked for its propellant budget, level Clohessy-Wiltshire and error dynamicsdynamics, standpoint. A State feedback controller been applied on J22thrust perturbed and the system is checked forhas its been applied on stability and performance. been applied on J J222 perturbed perturbed Clohessy-Wiltshire Clohessy-Wiltshire dynamics, dynamics, and and the the system system is is checked checked for for its its stability and performance. stability and performance. stability and performance. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Perturbation. Keywords: Formation Formation flying, flying, State State feedback feedback control, control, Optimal Optimal control, control, J J22 Perturbation. Keywords: Formation flying, State feedback control, Optimal control, J Perturbation. Keywords: Formation flying, State feedback control, Optimal control, J222 Perturbation. 1. INTRODUCTION Lyapunov type controller 1. INTRODUCTION Lyapunov type controller based based on on feedback feedback linearizalineariza1. Lyapunov type tion approach. 1. INTRODUCTION INTRODUCTION Lyapunov type controller controller based based on on feedback feedback linearizalinearization approach. approach. •• tion Leader-Follower:The leader/follower(L/F) archition approach. Leader-Follower:The leader/follower(L/F) archi• Leader-Follower:The leader/follower(L/F) archiThe major reason for formation of satellites is the desire to tecture is the most studied formation flying conThe major reason for formation of satellites is the desire to • Leader-Follower:The leader/follower(L/F) architecture is the most studied formation flying conThe major reason for formation of satellites is the desire to tecture distribute the functionality of large satellites. The ability is the most studied formation flying control(FFC) architecture, also termed as chief/deputy The majorthe reason for formation of satellites is the desire to distribute functionality of large satellites. The ability tecture is the most studied formation flying control(FFC) architecture, also termed as chief/deputy distribute the of satellites. The ability trol(FFC) architecture, also termed as chief/deputy of small satellites to fly in precise aa or master/slave. Extensive literature exits on this distribute the functionality functionality of large largeformation satellites. will Themake ability of small satellites to fly in precise formation will make trol(FFC) architecture, also termed as chief/deputy or master/slave. Extensive literature exits on this of small satellites to fly in precise formation will make a or master/slave. Extensive literature exits on this wide array of new applications possible, including nextsubject. The best part of these approaches is that of small satellites toapplications fly in precise formation will make a wide array of new possible, including nextor master/slave. Extensive literature exits on this subject. The best part of these approaches is that wide array of new applications possible, including nextsubject. The best part of these approaches is that generation internet, space-based radar and ultra powerful sufficient conditions for stability are available for genwide array of new applications possible, including nextgeneration internet, space-based radar and ultra powerful subject. The best part of these approaches is that sufficient conditions for stability are available for gengeneration internet, space-based radar and powerful sufficient conditions for are for space telescopes. There is also an economic aspect to this; eral L/F formation, these stability conditions are generation internet, space-based radar and ultra ultra powerful space telescopes. There is also an economic aspect to this; sufficient conditions and for stability stability are available available for gengeneral L/F formation, and these stability conditions are space telescopes. There is also an economic aspect to this; eral L/F formation, and these stability conditions are often it is more expensive to place one big satellite with all broadly classified as mesh stability. space telescopes. There is to also an economic aspect with to this; often it is more expensive place one big satellite all eral L/F formation, and these stability conditions are broadly classified as mesh stability. often it is more expensive to place one big satellite with all broadly classified as mesh stability. the functions built-in into orbit than several smaller ones • Virtual structure:In the virtual structure architecoften it is more expensive to place one big satellite with all the functions built-in into orbit than several smaller ones broadly classified as mesh stability. • Virtual structure:In the virtual structure architecthe functions built-in into than smaller ones •• Virtual structure:In the structure architecof the the same collective collective weights. Therefore, as the the number of ture, the spacecraft as points embedded in the functions built-in weights. into orbit orbit than several several smaller ones of same Therefore, as number of Virtual structure:Inbehave the virtual virtual structure architecture, the spacecraft behave as points embedded in of the same collective weights. Therefore, as the number of ture, the spacecraft behave as points embedded in missions that use spacecraft flying in formation, proposed a virtual virtual rigid body.behave Spacecraft statesembedded are coupled coupled of the same collective weights.flying Therefore, as the number of missions that use spacecraft in formation, proposed ture, the spacecraft as points in a rigid body. Spacecraft states are missions that use spacecraft flying in formation, proposed a virtual rigid body. Spacecraft states are coupled or under development, increases, one can imagine assembly through the template fitting step(Lamy-1993) considmissions that use spacecraft flying incan formation, proposed or under development, increases, one imagine assembly a virtual rigid body. Spacecraft states are coupled through the template fitting step(Lamy-1993) considor under development, increases, can imagine through the fitting lines of standardized thus drastically lowering ers Earth-orbiting formations. or under development,spacecraft, increases, one one can imagine assembly assembly lines of standardized spacecraft, thus drastically lowering through the template template fitting step(Lamy-1993) step(Lamy-1993) considconsiders Earth-orbiting formations. lines of standardized spacecraft, thus drastically lowering ers Earth-orbiting formations. the cost cost of building building them. them. Thesethus standardized spacecraft •• ers Behavioral:In behavioral architecture the output lines of standardized spacecraft, drasticallyspacecraft lowering the of These standardized Earth-orbiting formations. Behavioral:In behavioral architecture the output the cost of them. These standardized spacecraft •• Behavioral:In behavioral architecture the output will be fully equipped with proper instruments for their of multiple multiple controllers controllers designed for achieving achieving difthe cost of building building them. These standardized spacecraft will be fully equipped with proper instruments for their Behavioral:In behavioral architecture the output of designed for difwill be fully equipped with proper instruments for their of multiple controllers designed for achieving difmission. ferent and competing behavior is combined. Anderwill be fully equipped with proper instruments for their mission. of multiple controllers designed for achieving different and competing behavior is combined. Andermission. ferent and competing behavior is combined. AnderThe concept of multiple satellite formation flying is drasson et. al(1998) provides an excellent example of a mission. The concept of multiple satellite formation flying is drasferent and competing behavior is combined. Anderson et. al(1998) provides an excellent example of a The concept of satellite formation flying is son et. al(1998) provides an excellent example of a tically transforming Earth and Space Space science. This techBehavioral FFC algorithm. They consider velocityThe concept of multiple multiple satellite formation flying is drasdrastically transforming Earth and science. This techson et. al(1998) provides an excellent example of a Behavioral FFC algorithm. They consider velocitytically transforming Earth and Space science. This techBehavioral FFC algorithm. They consider velocitynological revolution heralds novel technique in spacecraft commandedFFC aircraft with collision collision avoidance, obstacle tically transforming Earth and Space science. This technological revolution heralds novel technique in spacecraft Behavioral algorithm. They consider velocitycommanded aircraft with avoidance, obstacle nological revolution heralds novel in commanded aircraft with collision avoidance, obstacle guidance, navigation, and control, and the manner inforavoidance, move to goal formation maintenance nological revolution heralds novel technique technique in spacecraft spacecraft guidance, navigation, and control, and the manner inforcommanded aircraft with and collision avoidance, obstacle avoidance, move to goal and formation maintenance guidance, navigation, and control, and the manner inforavoidance, move to goal and formation maintenance mation is shared between space borne vehicles and ground. behaviors. guidance, navigation, and control, and the manner information is shared between space borne vehicles and ground. avoidance, move to goal and formation maintenance behaviors. mation is between borne and The formation by NASA’s Earth Observing-1 •• behaviors. Cyclic:- In aa cyclic FFC algorithm, the spacecraft are mation is shared sharedconstituted between space space borne vehicles vehicles and ground. ground. The formation constituted by NASA’s Earth Observing-1 behaviors. In FFC algorithm, the spacecraft are The formation constituted by NASA’s Earth Observing-1 •• Cyclic:Cyclic:In aa cyclic cyclic FFC algorithm, the spacecraft are and Landsat-7 carry instruments to create high-resolution connected in aa cyclic or ring structure. The spacecraft The formation carry constituted by NASA’s Earth Observing-1 and Landsat-7 instruments to create high-resolution Cyclic:In cyclic FFC algorithm, the spacecraft are connected in cyclic or ring structure. The spacecraft and Landsat-7 carry instruments to create high-resolution connected in a cyclic or ring structure. images for the study of climatic trends in the Earth’s The spacecraft share information with their neighbors in the cyclic and Landsat-7 carry instruments totrends create high-resolution images for the study of climatic in the Earth’s connected in a cyclic or ring structure. The spacecraft share information with their neighbors in the cyclic images for share information with their neighbors in cyclic environment. topology and generate control law based the images for the the study study of of climatic climatic trends trends in in the the Earth’s Earth’s environment. share information with the their neighbors in the theon cyclic topology and generate the control law based on the environment. topology and generate the control law based on the The literature on the subject is divided into five architeclocal information. environment. The literature on the subject is divided into five architectopology and generate the control law based on the local information. The literature on the subject is divided into five architeclocal information. tures: The literature on the subject is divided into five architec- Existing tures: local information. literature is quite rich with respect to formation tures: Existing literature is quite rich with respect to formation tures: Existing literature is quite rich with respect to formation • Multiple-input, multi-output:The research on FFC flying, though the inclusion of orbital perturbation is still • Multiple-input, multi-output:The research on FFC Existing literature is quite rich with respect to formation flying, though the inclusion of orbital perturbation is still • Multiple-input, multi-output:The research on FFC flying, though the inclusion of orbital perturbation is still is not restricted to spacecraft formations only. Exnot completely explored. The paper aims at examining low • Multiple-input, multi-output:The researchonly. on FFC is not restricted to spacecraft formations Exflying, though the inclusion of orbital perturbation is still not completely explored. The paper aims at examining low is not restricted to spacecraft formations only. Exnot completely explored. The paper aims at examining low tensive literature exists on achieving formation in a earth orbit formation control under orbital perturbations. is not restricted to spacecraft formations only. in Extensive literature exists on achieving formation a not completely explored. The paper aims at examining low earth orbit formation control under orbital perturbations. tensive literature exists on achieving formation in aa earth orbit formation control under orbital perturbations. group of robots and UAV’s and similar autonomous A State feedback controller has been applied on J pertensive literature exists on achieving formation in 2 group of robots and UAV’s and similar autonomous earth orbit formation control under orbital perturbations. A State feedback controller has been applied on J per2 group of robots and UAV’s and autonomous A vehicles. recent (Zhang-2008) the group of A robots andpaper UAV’s and similar similar presents autonomous vehicles. A recent paper (Zhang-2008) presents the A State State feedback feedback controller controller has has been been applied applied on on J J222 perpervehicles. A recent paper (Zhang-2008) presents the vehicles. A recent paper (Zhang-2008) presents the
Copyright © 2016, 2016 IFAC IFAC 130Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 130 Copyright 2016 responsibility IFAC 130Control. Peer review© of International Federation of Automatic Copyright ©under 2016 IFAC 130 10.1016/j.ifacol.2016.03.041
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turbed Clohessy-Wiltshire dynamics. Major contribution from current work is : • Formulation and implementation of control algorithm that achieves precise formation maintenance, since the formation must be maintained in the face of external disturbances, robustness of the controllers is an essential feature. • Evaluation of the performance of this control law with respect to its fuel consumption , thrust required and transient behaviour. The flow of paper is as follows, next session briefly introduces the translational dynamics of spacecraft. Section further introduces the J2 perturbation as the unmodeled force and discusses the ways to incorporate it in the translational dynamics. Section 3 and 4 explain the control law and proves the system’s stability under J2 perturbation. Simulation based verification of the control law is presented in section 5 followed by conclusion in section 6.
Fig. 1. Rotating Euler-Hill frame, centered at the leader spacecraft. This figure also shows the follower spacecraft, whose position vectors in the rotating and inertial reference frames are denoted by ρ and rf , respectively (Vadali 2009).
2. BACKGROUND
x ¨ − 2ny˙ − 3n2 x = ux
2.1 Kepler’s law and Energy and Momentum associated with Spacecraft
(3)
y¨ + 2nx˙ = uy
(4)
2
(5)
z¨ + n z = uz
Kepler’s laws of motion describe satellite’s motion with respect to the earth(general two body motion)Vadali(2009), µ.r (1) r¨ + 3 = 0 r where, r = [X, Y, Z]T is the position vector of the spacecraft in the ECI frame and µ is the gravitational constant of earth.
where, (x, y, z) are the coordinates of spacecraft in Hill’s frame or LVLH frame and (ux , uy , uz ) is the control input and n is f˙, where f is the true anomaly of the leader satellites orbit.
2.2 The translational dynamics of spacecraft
HCW equation(3-5) is the linear version of original dynamics and does not account for the oblateness of the earth surface. This oblateness leads to important consequences in practical spacecraft orbit design.The most pronounced effect on Low Earth orbits is caused by the second harmonic of the Earth potential, which reflects the oblateness of Earth Samuel(2002).
The trajectory of the satellite cannot be chosen arbitrarily, but is constrained by the laws of physics. One of the big challenges is to find appropriate paths for all the satellites in a formation, so that the desired functionalities are achieved, both with a view to fuel efficiency and to fulfil the predefined mission. As discussed in section 2.1 , Kepler’s laws govern the motion of any spacecraft in inertial frame. Consider two spacecraft, where one is the leader and the other is the follower. The most common linear passive relative orbits of the followers with respect to leaders are the solutions to the Hill-Clohessy-Wiltshire Equations. These equations were introduced in Clohessy(1960). Though it’s worth noticing HCW equations do not consider the eccentricity of the orbit, and the same has not been considered for the work carried out in this paper. Let the subscripts i = l, f denote the leader and the follower satellite respectively. The position vector from the leader to the follower satellite can be expressed as
2.3 Perturbation due to non-spherical Earth
Incorporating J2 Perturbation in HCW Equations In Samuel(2002), the authors show that the equations of motion relative to circular non-Keplerian reference orbit and including the J2 term are well approximated by the linear system:3 ¯ x ¨i = 2ncy˙ i + (5c2 − 2)n2 xi + KJ2 Cos(2kt) 4 1 ¯ y¨i = −2ncx˙ i + KJ2 Sin(2kt) 2 z¨i = −q 2 zi + 2lqCos(qt + φ)
(6)
where, iref and rref are parameters of the reference orbit, ρ = rf − rl
(2)
= xer + yeθ + zez
where x, y and z are the components of ρ in the Hill frame. Using Kepler’s law and following the derivation as in Clohessy(1960) under linearized approximation, the equations will be given by: 131
3J R2
Re is the nominal radius of the earth, s = 8r22 e (1 + ref √ 3n2 J R2 3 cos(2iref )), c = 1 + s, KJ2 = rref2 e sin2 (iref ), k¯ = c+
3J2 Re2 2 2rref
cos2 iref , q is approximately equal to cn, and φ,
l are time varying functions of the difference in orbit inclination (see Samuel(2002) and Yeh(2002) for the details).
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Fig. 2. (a)Interaction Topology where dji denoted the desired relative separation between each agent (b) SC maintaining constant separation along z axis 3. PROBLEM FORMULATION AND CONTROL LAW DERIVATION 3.1 Problem Statement The problem addressed in this paper, is to maintain n spacecraft along z coordinate of LVLH frame in the low earth orbit(LEO). Main reason of selecting formation about z coordinate is the inherent stability of out of plane formation keeping, which requires very little thrust activity when compared to in-plane dynamics. The paper focus on formation maintenance and not the attitude control of spacecraft, which can be the scope for future work. • Complete system consists of a reference spacecraft at the origin of LVLH frame which is controlled by the ground station, all other spacecraft are autonomous and the exchange topology is given in figure (2). • Control law for spacecraft uses the information of its own state as well as the relative states of its neighbours. Own state information is obtained from GPS, while relative states are obtained through communication. • Paper examines the formation about z coordinate of LVLH frame. 3.2 State Feedback Control Law Control law in this paper considered the very general interaction topology, where the communication graph always consists of a spanning tree. The control law for spacecraft i be given by: ui = (k1 (Xj − Xi ) + k2 (X˙ j − X˙ i )) + k3 X˙ i + k4 Xi (7)
Fig. 3. Interaction Topology considered. where, Ln is the laplacian matrix of the interaction graph of spacecraft. And, 2 0 2n 0 3n 0 0 A1 = 0 0 0 , B1 = −2n 0 0 0 0 0 0 0 −n2 (8) can be simplified to
ξ ξ˙ = (Σ1 ⊕ Σ2 ) ζ ζ˙
where,
(9)
1 I 0 3 3 Σ1 = 2 A1 B 1 1 I 0 n n Σ2 = 2 −(k1 Ln + k4 In ) −(k2 Ln + k3 In ) it can be easily proved that eigenvalue of Σ1 ⊕ Σ1 will always be negative (for detailed proof refer Schaub(2004), Schur(1960), Shashank(2011)). Hence for the closed loop system (9) at steady state, ξ Limt→∞ =0 (10) ζ
which mean position and velocity of all the spacecraft will go to zero at steady state. This shows that all the spacecraft will eventually come to a rendezvous point. Though, as discussed this paper discuses about the formation control of spacecraft. To achieve the desired formation pattern and separation between the spacecraft, the control law(7) is modified as discussed in next subsection.
j∈Ni
where, k1 , k2 , k3 , k4 ∈ R+ , ui = (uix , uiy , uiz ) is the control 3.3 Formation Control Law input, Xi = (xi , yi , zi ) is the position of spacecraft i and Ni denotes the neighbors of spacecraft i. Thus, the control The control law presented (7) is complete but it always has law for spacecraft i depends on the position and velocity zero as equilibrium point, now in the context of formation of itself and its neighbors. T flying, a desired distance need to be maintained between T Let, ξ = [X1 X2 . . . . . . Xn ] and ζ = X˙ 1 X˙ 2 . . . . . . X˙ n . all the spacecraft. To achieve this purpose, control law is being modified as given: Now, for n agents, applying above control law to (6−8): u = (k (X −X −di−1 )+k (X˙ ˙ ˙ 03n I3n i 1 i−1 i 2 i−1 − Xi ))+k3 Xi +k4 (Xi ) i ξ˙ ξ (11) I ⊗ A − = n 1 ζwhere, for i = 1, i−1 denotes the reference spacecraft, di is ζ˙ (k1 Ln + k4 In ) ⊗ I3 In ⊗ B1 − (k2 Ln + k3 In ) ⊗ I3 i (8) the absolute desired distance for any spacecraft and dji = 132
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djj −dii is the desired relative distance between spacecraft j and i as mentioned in fig.(2). As discussed in the problem definition, the reference spacecraft is controlled by the ground station, and will always remain at the origin of the LVLH frame as in fig.(3). In this case, first spacecraft is the reference spacecraft which is controlled by ground station, T which implies d11 = 0. Using ξ = [X1 X2 . . . . . . Xn ] and T ζ = X˙ 1 X˙ 2 . . . . . . X˙ n and applying above control law to the HCW dynamics (3),
Theorem 1. Consider n spacecraft with dynamics (3) and the control law (11), if k1 , k2 , k3 , k4 ∈ R + , where R+ ξ is set of all positive real numbers then → Ddesired ζ asymptotically which ensures that the spacecraft achieve the desired formation.
V˙ = F V + D
In the previous subsection general control law for formation flying has been introduced. The control law is derived considering HCW dynamics as the dynamics given in LVLH frame. Including J2 perturbation into the dynamics equation (6) can be written further in the form of
with,
(12)
d11 .. . 0n×n 0n×n dnn H= −k1 Ln 0n×n 0 . . . 0
using above equation in eqn.(13), U˙ = F U now F is a Hurwitz matrix, which implies
⇒ Limt→∞ (V (t) + F
−1
(14)
D) = 0
V → F −1 D as t → ∞ (15) ξ which implies → F −1 D. ζ Above analysis shows that by using control law (11), the separation between any two spacecraft will settle to F −1 D. To ensure the desired formation separation, a different T need to be passed to the value of d = d11 d12 . . . dn−1 n T 1 2 ˜ controller. Let, ddesired = d1 d˜2 . . . d˜nn denotes the required separation of any spacecraft from reference and d as defined above is the separation that need to be passed to the control law. Then, d can be such that calculated ddesired D = F Ddesired where, Ddesired = . 0 From equation (14) Limt→∞ (V (t) + F −1 F Ddesired ) = 0
3.4 Performance of the Control Law Under J2 Perturbation
¨ i = AXi + B X˙ i + d(t) + ui X ¯ i+B ¯ X˙ i + ∆(t) + ui = AXi + B X˙ i + AX
ξ ,V = , F = (In ⊗ Σ1 + Σ2 ⊗ I3 ) and D = H.F being a ζ nonsingular matrix (none of the eigenvalue of F have zero real part, as proved in previous section), further solving equation (12), (13) V˙ = F (V + F −1 D) Assuming U = V + F −1 D, and differentiating, U˙ = V˙
Limt→∞ U (t) = 0
Proof: As follows from above analysis.
(16)
V (t) → Ddesired (17) ξ which implies → Ddesired . This guarantees all the ζ spacecraft to maintain the required separation between each of them. The analysis presented above can briefly be written in the form of a theorem as stated next. 133
(18)
where, ui is the input applied as given by control law x 3n2 0 0 0 2n 0 (7), X = y , A = 0 0 0 , B = −2n 0 0 , z 0 0 0 0 0 −n2 3 2 2 ¯ K Cos(2 kt) J 0 5s n 0 4 2 1 0 , ∆(t) = A¯ = 0 0 ¯ K Sin(2 kt) J 2 2 2 0 0 −s n 2 2lqCos(qt + φ) √ 1 + s − 1) 0 0 2n( ¯ = −2n(√1 + s − 1) B 0 0, 0 0 0 Now,for n agents above equation (18) can be written by T T taking ζ = [X1 X2 · · · Xn ] and ξ = X˙ 1 X˙ 2 · · · X˙ n as the system states, and is given by: I3n 03n ζ˙ ζ ζ T = (Σ1 ⊕Σ2 ) + ¯ ¯ ⊗ In ξ +[0n 1n ] ∆(t) ξ A ⊗ In B ξ˙ (19) where, Σ1 and Σ2 are the matrices as explained in previous section. All the eigenvalues of (Σ1 ⊕ Σ2 ) are negative and hence it is Hurwitz. 4. STABILITY ANALYSIS UNDER J2 PERTURBATION As can be seen from the above section, system in this case consists of both vanishing as well as non-vanishing perturbation part. In this paper both the parts will be analyzed separately and then clubbed together to see their effect as a whole on the system. Let, V (t, X) corresponds to the Lyapunov function for the perturbed system (19) in [0, ∞) × D, where D = {X ∈ R2n | X < r}, where T r > 0 and X = [ζ ξ] . Let, (Σ1 ⊕ Σ2 ) = Q, then Q is a Hurwitz matrix. Equation (19) can be written in the form of X˙ = QX + χX + g(t) (20) 0 In T where, χ = ¯ ¯ n and g(t) = [0n 1n ] ∆(t) . Let AIn BI
χX = γ X , ∀t ≥ 0, ∀X ∈ D . Neither the stability of origin as an equilibrium point can no longer be studied, nor the solution of perturbed system be expected to approach
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origin as t → ∞. The best scenario can be that X(t) will be ultimately bounded by a small bound, if the perturbation term g(t) is small in some sense. As can be seen from the above equation, g(t) is always bounded and let the bound on the value of g(t) is given by | g(t) |≤ β for all t ≥ 0. Now, let V (t, X) = X T P X where, P is the solution of P Q + QT P = −I The solution of above equation will always result a positive definite value of P . Implies V (t, X) > 0. Also, V (t, X) satisfies conditions as given in (Khalil 2002) (21) λmin (P ) X 22 ≤ V (X) ≤ λmax (P ) X 22 ∂V QX = X T (P Q + QT P )X = −X T X (22) ∂X where λmax and λmax corresponds to the maximum and minimum eigenvalues of matrix P . Now, Q being Hurwitz X = 0 is a exponential stable equilibrium point of the system X˙ = QX. The derivative of V (X) along the trajectories of (20) satisfies
Fig. 4. Simulations for three spacecraft under formation control law with HCW dynamics
∂V ∂V ∂V QX + χX + g(t) V˙ (X) = ∂X ∂X ∂X ≤ −(1 − 2λmax (P )γ) X 2 +2λmax (P )g(t) X Now, using (21),and √ taking W (t) = V (t, x(t)) and use ˙ = V˙ /2 V , when V = 0, to obtain, the fact W 2λmax (P ) 1 1 λmax (P ) ˙ − γ W +g(t) W ≤− λmin (P ) 2 λmax (P ) λmin (P ) (23) Now, by comparison lemma (Khalil 2002), W (t) satisfies the inequality t λmax (P ) φ(t, τ )g(τ )dτ (24) Fig. 5. Control/Thrust profile along X,Y and Z Coordinate W (t) ≤ φ(t, t0 )W (t0 )+ under formation control law λmin (P ) t0 where using the fact that γ is a constant,function φ(t, t0 ) separation of 5 units. The interaction topology is as deis given by fined in figure(3). As the satellites maintain a separa λmax (P ) 1 tion of 5 units along z axis they should be at the point − φ(t, t0 ) = exp − (t − t0 ) (0, 0, 0); (0, 0, 5); (0, 0, 10). To obtain this desired configu2λmax (P ) λmin (P ) (25) ration some different values of D are need to be passed min (P ) to the controller by calculating F ∗ Ddesired . For this case which implies, for φ(t, t0 ) to be bounded, γ < − 2λλmax (P )2 . with the values of gains k1 = .0001, k2 = .01, k3 = .01 and This bound on γ can be ensured by choosing proper gain k4 = .0001, the value of D is calculated and is given by values of control law i.e k1 , k2 , k3 , k4 which will affect the T eigenvalues of Q, and hence the eigenvalues of P . So, by [0 0 . . . 0 10.058 ∗ k1 0 0 15.116 ∗ k1 ] choosing proper values of gains the bound over the value The desired formation can be seen along the z axis of of γ can be ensured. Using the bound on γ, equation (25) LVLH frame as shown in figure (4). implies It can be seen from figure (4) that the controller sucφ(t, t0 ) = exp[−σ(t − t0 )] = ρ (26) cessfully performs the desired maneuver. Next, it is interwhere σ > 0. Using (26) in (24), it can be seen that X(t) is esting to observe thrust profile or the acceleration profile always bounded even in the presence of perturbation g(t). of this control action in plots in fig(5). As can observe Also to limit the bound to the ball of radius r ∈ D, from the plots control acceleration reduces to zero in the steady state. Though author would like to emphasize on λmax (P ) λmax (P ) X(t) ≤ max ρX(t0 ), supt≥t0 g(t) the point that the control acceleration reduces to zero, λmin (P ) λmin (P ) as the inherent dynamics along z axis for HCW equation (27) has stabilizing effect and hence require no acceleration component, once the equilibrium is achieved. 5. SIMULATION RESULTS The fuel consumption measure ∆V as is shown in figure 5 is calculated by integrating the resultant control For simulation, three satellites are considered having dy- acceleration over the time period of simulation Tsim , The namics as given by HCW equations in LVLH frame. The fuel needed for the maneuver is related to the total velocity initial positions of satellite are (15, 5, 10); (12, 7, 12); (5, 15, 20) change, as given in Gurfil 2007. Km. The desired formation is along the z axis with a 134
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Table 1. Maximum Control Effort by SC’s along X Y and Z coordinates under J2 perturbation Coordinate X Y Z
SC1 18.23 0.4123 1.145
SC2 19.67 0.6412 0.4567
SC3 5.009 2.5734 0.5845
Fig. 7. Control/Thrust profile along X,Y and Z Coordinate with J2 perturbation under formation control law REFERENCES Fig. 6. Simulations for three spacecraft under formation control law under the effect of J2 perturbation Performance of Control law with J2 Perturbation: To check the performance or the robustness of the controller, controller is implemented with the J2 perturbation well. Investigations carried out so far amply prove that such a control law is robust enough to perform its task of trajectory following even in the face of J2 perturbation. Same initial conditions and desired positions are given to spacecrafts while simulating the result with the inclusion of perturbation. Therefore, one can conclude that relative trajectory design using HCW equations is useful even in the presence of J2 perturbation. Plots (4 and 6) give the comparison between controller implemented with and without J2 . Maximum amount of control effort by controller considering J2 perturbation is given in Table (1). Simulations investigates the effect of perturbation on the amount of fuel consumption by the controller over the period of 6400 Sec. Plots (7) show the the fuel budget of the implemented controller while considering the effect of perturbation with HCW equations for SC1. 6. CONCLUSION Paper has briefly outlined the concept of multiple spacecraft formation flying and has investigated in some detail the dynamics and control of satellite formations. The stated aim was accomplished by implementing state feedback control law with HCW equations incorporating J2 perturbation. The stability of controller was studied by means of Lyapunov theory.Numerical simulations conducted in Matlab/Simulink are presented for control verification. An important outcome of this portion of the work is to analyze the performance of controller while in the presence of J2 perturbation. Though as mentioned earlier as well, current work still ignores the eccentricity of the orbit, and incorporating the same will remain the future interest for authors. 135
Zhang Zhi-guo, Li Jun-Feng (2008).“Orbit and Attitude Control of Spacecraft Formation Flying,” Applied Mathematics and Mechanics, England, Vol 29(1),pp. 43-50. A. Lamy and S. Pascal (1993), “Station keeping strategies for constellations of satellites:” in the Proceedings of AAS NASA International Symp. on Spaceflight dynamics, Toulouse, France,pp. 819-833. M.R. Anderson and A.C. Robbins (1998), “Formation flight as a cooperative game,” in An adaptive control approach to satellite formation flying with relative distance comtraints AlAA Guidance, Navigation, and Control Conference, AIAA-984124, Boston, MA, pp. 244-251. Kyle T. Alfriend, Srinivas R. Vadali, Pini Gurfil, Jonathan P. How, Louis S.Breger.(2009) “Spacecraft Formation Flying” Elsevier Astrodynamics Series, London. Clohessy, W. and Wiltshire, R.(1960),“Terminal Guidance System for Satellite Rendezvous,” Journal of the Astronautical Sciences, Vol. 27, No. 9, pp. 653678. Samuel A. Schweighart and Raymond J. Sedwick.(2002) “High-Fidelity Linearized J2 Model for Satellite Formation Flight,” Journal of Guidance, Control, and Dynamics, Vol. 25, No. 6, pp. 546-554 . Alan J. Laub (2004),“Matrix Analysis for Scientists and Engineers”, SIAM Publication, 139-147. F. R. Gantmacher (1960),“The Theory of Matrices”, Volume 1, AMS Chelsea Publishing, pp. 42-47. Yeh, H.-H., Nelson, E., and Sparks, A.(2002), “Formation Flying: Accommodating Nonlinearity and Eccentricity Perturbations”, Journal of Guidance, Control and Dynamics, Vol. 25, No. 2, pp. 376-386. Khalil, H. K.(2002), “Nonlinear Systems”, 3rd edition, prenticeHall, Upper Saddle River, NJ, pp. 339-372. Pini Gurfil, David Mishne,(2007)“Cyclic Spacecraft Formations: Relative Motion Control Using Line-of-Sight Measurements Only”, Journal of Guidance, Control and Dynamics, Vol. 30, No. 1, pp. 214-226. Agarwal, Shashank,(2011) “Formation control of spacecraft”, M.Tech. thesis, Systems and Control Department, Indian Institute of Technology, Bombay, Mumbai India.
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IFAC-PapersOnLine 49-1 (2016) 130–135
Formation Control of Spacecraft Formation Control of Spacecraft Formation Control of Spacecraft Formation Control of Spacecraft orbital perturbation orbital perturbation orbital perturbation orbital perturbation
under under under under
∗ ∗∗ ∗ Prof. Arpita Sinha ∗∗ Shashank Agarwal Shashank Agarwal ∗ ∗ Prof. Arpita Sinha ∗∗ ∗∗ Shashank Agarwal ∗ Prof. Arpita Sinha ∗∗ Shashank Agarwal Prof. Arpita Sinha ∗ ∗ Systems and Control Department, Indian Institute of Technology Institute of Technology ∗ ∗ Systems and Control Department, Indian Indian ∗ SystemsMumbai and Department, Bombay, 411014 (email :[email protected]) Systems and Control Control Department, Indian Institute Institute of of Technology Technology Bombay, Mumbai 411014 (email :[email protected]) ∗∗ Bombay, Mumbai 411014 (email :[email protected]) ∗∗ Systems and Control Department, Indian Institute of Bombay, Mumbai 411014 (email :[email protected]) Systems and Control Department, Indian Institute ∗∗ ∗∗ Systems and Control Department, Indian Institute of ∗∗ Technology,Bombay, Mumbai 411014 (email: [email protected]) Systems and Mumbai Control Department, Indian Institute of of Technology,Bombay, 411014 (email: [email protected]) Technology,Bombay, Mumbai 411014 (email: [email protected]) Technology,Bombay, Mumbai 411014 (email: [email protected]) Abstract: The novel spacecraft flying aa substitute aa Abstract: The novel concept concept of of multiple multiple spacecraft formation formation flying as as substitute for for Abstract: The multiple formation as for single large vehicle vehicle will concept enhance of future spacespacecraft mission performance. performance. The benefits benefits of aa spacecraft spacecraft Abstract: The novel novel concept of multiple spacecraft formation flying flying as aa substitute substitute for aa single large will enhance future space mission The of single vehicle will future space performance. The of formation cost effective synthetic aperture radar for observations, of the single large largeinclude vehiclemore will enhance enhance future space mission mission performance. The benefits benefitsflexibility of aa spacecraft spacecraft formation include more cost effective synthetic aperture radar for observations, flexibility of the formation include more cost effective synthetic aperture radar for observations, flexibility of the satellites altering their roles, reduction of cost owing to the reduction of mass launched into orbit formationaltering includetheir moreroles, cost effective synthetic aperture radar for observations, flexibility oforbit the satellites reduction of cost owing to the reduction of mass launched into satellites altering their roles, reduction of cost owing to the reduction of mass launched into orbit etc. A in the control to formation maintenance satellites altering challenge their roles, of of cost owingdesign to theis of aamass launched into orbit etc. A significant significant challenge inreduction the domain domain of control design isreduction to contrive contrive formation maintenance etc. challenge the domain control is aa formation maintenance controller that will will enablein the member spacecrafts to maintain maintain desired relative orbit with with etc. A A significant significant challenge inthe themember domain of of control design design is to to contrive contrive formation maintenance controller that enable spacecrafts to aaa desired relative orbit controller that will enable the member spacecrafts to maintain desired relative orbit with optimal propellant expenditure while maintaining the desired formation. This paper examines controllerpropellant that will expenditure enable the member spacecraftsthetodesired maintain a desiredThis relative orbit with optimal while maintaining formation. paper examines optimal propellant expenditure while maintaining the desired formation. This paper examines a low earth orbit formation control methodology, with the aim of evaluating formation from propellant expenditure while maintaining with the desired Thisformation paper examines aoptimal low earth orbit formation control methodology, the aimformation. of evaluating from aa a orbit control methodology, the of evaluating formation from aa propellant thrust and dynamics standpoint. feedback controller has a low low earth earthbudget, orbit formation formation control methodology, with the aim aim A of State evaluating formation from propellant budget, thrust level level and error error dynamicswith standpoint. A State feedback controller has propellant budget, thrust level and error dynamics standpoint. A State feedback controller has been applied on J perturbed Clohessy-Wiltshire dynamics, and the system is checked for its propellant budget, level Clohessy-Wiltshire and error dynamicsdynamics, standpoint. A State feedback controller been applied on J22thrust perturbed and the system is checked forhas its been applied on stability and performance. been applied on J J222 perturbed perturbed Clohessy-Wiltshire Clohessy-Wiltshire dynamics, dynamics, and and the the system system is is checked checked for for its its stability and performance. stability and performance. stability and performance. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Perturbation. Keywords: Formation Formation flying, flying, State State feedback feedback control, control, Optimal Optimal control, control, J J22 Perturbation. Keywords: Formation flying, State feedback control, Optimal control, J Perturbation. Keywords: Formation flying, State feedback control, Optimal control, J222 Perturbation. 1. INTRODUCTION Lyapunov type controller 1. INTRODUCTION Lyapunov type controller based based on on feedback feedback linearizalineariza1. Lyapunov type tion approach. 1. INTRODUCTION INTRODUCTION Lyapunov type controller controller based based on on feedback feedback linearizalinearization approach. approach. •• tion Leader-Follower:The leader/follower(L/F) archition approach. Leader-Follower:The leader/follower(L/F) archi• Leader-Follower:The leader/follower(L/F) archiThe major reason for formation of satellites is the desire to tecture is the most studied formation flying conThe major reason for formation of satellites is the desire to • Leader-Follower:The leader/follower(L/F) architecture is the most studied formation flying conThe major reason for formation of satellites is the desire to tecture distribute the functionality of large satellites. The ability is the most studied formation flying control(FFC) architecture, also termed as chief/deputy The majorthe reason for formation of satellites is the desire to distribute functionality of large satellites. The ability tecture is the most studied formation flying control(FFC) architecture, also termed as chief/deputy distribute the of satellites. The ability trol(FFC) architecture, also termed as chief/deputy of small satellites to fly in precise aa or master/slave. Extensive literature exits on this distribute the functionality functionality of large largeformation satellites. will Themake ability of small satellites to fly in precise formation will make trol(FFC) architecture, also termed as chief/deputy or master/slave. Extensive literature exits on this of small satellites to fly in precise formation will make a or master/slave. Extensive literature exits on this wide array of new applications possible, including nextsubject. The best part of these approaches is that of small satellites toapplications fly in precise formation will make a wide array of new possible, including nextor master/slave. Extensive literature exits on this subject. The best part of these approaches is that wide array of new applications possible, including nextsubject. The best part of these approaches is that generation internet, space-based radar and ultra powerful sufficient conditions for stability are available for genwide array of new applications possible, including nextgeneration internet, space-based radar and ultra powerful subject. The best part of these approaches is that sufficient conditions for stability are available for gengeneration internet, space-based radar and powerful sufficient conditions for are for space telescopes. There is also an economic aspect to this; eral L/F formation, these stability conditions are generation internet, space-based radar and ultra ultra powerful space telescopes. There is also an economic aspect to this; sufficient conditions and for stability stability are available available for gengeneral L/F formation, and these stability conditions are space telescopes. There is also an economic aspect to this; eral L/F formation, and these stability conditions are often it is more expensive to place one big satellite with all broadly classified as mesh stability. space telescopes. There is to also an economic aspect with to this; often it is more expensive place one big satellite all eral L/F formation, and these stability conditions are broadly classified as mesh stability. often it is more expensive to place one big satellite with all broadly classified as mesh stability. the functions built-in into orbit than several smaller ones • Virtual structure:In the virtual structure architecoften it is more expensive to place one big satellite with all the functions built-in into orbit than several smaller ones broadly classified as mesh stability. • Virtual structure:In the virtual structure architecthe functions built-in into than smaller ones •• Virtual structure:In the structure architecof the the same collective collective weights. Therefore, as the the number of ture, the spacecraft as points embedded in the functions built-in weights. into orbit orbit than several several smaller ones of same Therefore, as number of Virtual structure:Inbehave the virtual virtual structure architecture, the spacecraft behave as points embedded in of the same collective weights. Therefore, as the number of ture, the spacecraft behave as points embedded in missions that use spacecraft flying in formation, proposed a virtual virtual rigid body.behave Spacecraft statesembedded are coupled coupled of the same collective weights.flying Therefore, as the number of missions that use spacecraft in formation, proposed ture, the spacecraft as points in a rigid body. Spacecraft states are missions that use spacecraft flying in formation, proposed a virtual rigid body. Spacecraft states are coupled or under development, increases, one can imagine assembly through the template fitting step(Lamy-1993) considmissions that use spacecraft flying incan formation, proposed or under development, increases, one imagine assembly a virtual rigid body. Spacecraft states are coupled through the template fitting step(Lamy-1993) considor under development, increases, can imagine through the fitting lines of standardized thus drastically lowering ers Earth-orbiting formations. or under development,spacecraft, increases, one one can imagine assembly assembly lines of standardized spacecraft, thus drastically lowering through the template template fitting step(Lamy-1993) step(Lamy-1993) considconsiders Earth-orbiting formations. lines of standardized spacecraft, thus drastically lowering ers Earth-orbiting formations. the cost cost of building building them. them. Thesethus standardized spacecraft •• ers Behavioral:In behavioral architecture the output lines of standardized spacecraft, drasticallyspacecraft lowering the of These standardized Earth-orbiting formations. Behavioral:In behavioral architecture the output the cost of them. These standardized spacecraft •• Behavioral:In behavioral architecture the output will be fully equipped with proper instruments for their of multiple multiple controllers controllers designed for achieving achieving difthe cost of building building them. These standardized spacecraft will be fully equipped with proper instruments for their Behavioral:In behavioral architecture the output of designed for difwill be fully equipped with proper instruments for their of multiple controllers designed for achieving difmission. ferent and competing behavior is combined. Anderwill be fully equipped with proper instruments for their mission. of multiple controllers designed for achieving different and competing behavior is combined. Andermission. ferent and competing behavior is combined. AnderThe concept of multiple satellite formation flying is drasson et. al(1998) provides an excellent example of a mission. The concept of multiple satellite formation flying is drasferent and competing behavior is combined. Anderson et. al(1998) provides an excellent example of a The concept of satellite formation flying is son et. al(1998) provides an excellent example of a tically transforming Earth and Space Space science. This techBehavioral FFC algorithm. They consider velocityThe concept of multiple multiple satellite formation flying is drasdrastically transforming Earth and science. This techson et. al(1998) provides an excellent example of a Behavioral FFC algorithm. They consider velocitytically transforming Earth and Space science. This techBehavioral FFC algorithm. They consider velocitynological revolution heralds novel technique in spacecraft commandedFFC aircraft with collision collision avoidance, obstacle tically transforming Earth and Space science. This technological revolution heralds novel technique in spacecraft Behavioral algorithm. They consider velocitycommanded aircraft with avoidance, obstacle nological revolution heralds novel in commanded aircraft with collision avoidance, obstacle guidance, navigation, and control, and the manner inforavoidance, move to goal formation maintenance nological revolution heralds novel technique technique in spacecraft spacecraft guidance, navigation, and control, and the manner inforcommanded aircraft with and collision avoidance, obstacle avoidance, move to goal and formation maintenance guidance, navigation, and control, and the manner inforavoidance, move to goal and formation maintenance mation is shared between space borne vehicles and ground. behaviors. guidance, navigation, and control, and the manner information is shared between space borne vehicles and ground. avoidance, move to goal and formation maintenance behaviors. mation is between borne and The formation by NASA’s Earth Observing-1 •• behaviors. Cyclic:- In aa cyclic FFC algorithm, the spacecraft are mation is shared sharedconstituted between space space borne vehicles vehicles and ground. ground. The formation constituted by NASA’s Earth Observing-1 behaviors. In FFC algorithm, the spacecraft are The formation constituted by NASA’s Earth Observing-1 •• Cyclic:Cyclic:In aa cyclic cyclic FFC algorithm, the spacecraft are and Landsat-7 carry instruments to create high-resolution connected in aa cyclic or ring structure. The spacecraft The formation carry constituted by NASA’s Earth Observing-1 and Landsat-7 instruments to create high-resolution Cyclic:In cyclic FFC algorithm, the spacecraft are connected in cyclic or ring structure. The spacecraft and Landsat-7 carry instruments to create high-resolution connected in a cyclic or ring structure. images for the study of climatic trends in the Earth’s The spacecraft share information with their neighbors in the cyclic and Landsat-7 carry instruments totrends create high-resolution images for the study of climatic in the Earth’s connected in a cyclic or ring structure. The spacecraft share information with their neighbors in the cyclic images for share information with their neighbors in cyclic environment. topology and generate control law based the images for the the study study of of climatic climatic trends trends in in the the Earth’s Earth’s environment. share information with the their neighbors in the theon cyclic topology and generate the control law based on the environment. topology and generate the control law based on the The literature on the subject is divided into five architeclocal information. environment. The literature on the subject is divided into five architectopology and generate the control law based on the local information. The literature on the subject is divided into five architeclocal information. tures: The literature on the subject is divided into five architec- Existing tures: local information. literature is quite rich with respect to formation tures: Existing literature is quite rich with respect to formation tures: Existing literature is quite rich with respect to formation • Multiple-input, multi-output:The research on FFC flying, though the inclusion of orbital perturbation is still • Multiple-input, multi-output:The research on FFC Existing literature is quite rich with respect to formation flying, though the inclusion of orbital perturbation is still • Multiple-input, multi-output:The research on FFC flying, though the inclusion of orbital perturbation is still is not restricted to spacecraft formations only. Exnot completely explored. The paper aims at examining low • Multiple-input, multi-output:The researchonly. on FFC is not restricted to spacecraft formations Exflying, though the inclusion of orbital perturbation is still not completely explored. The paper aims at examining low is not restricted to spacecraft formations only. Exnot completely explored. The paper aims at examining low tensive literature exists on achieving formation in a earth orbit formation control under orbital perturbations. is not restricted to spacecraft formations only. in Extensive literature exists on achieving formation a not completely explored. The paper aims at examining low earth orbit formation control under orbital perturbations. tensive literature exists on achieving formation in aa earth orbit formation control under orbital perturbations. group of robots and UAV’s and similar autonomous A State feedback controller has been applied on J pertensive literature exists on achieving formation in 2 group of robots and UAV’s and similar autonomous earth orbit formation control under orbital perturbations. A State feedback controller has been applied on J per2 group of robots and UAV’s and autonomous A vehicles. recent (Zhang-2008) the group of A robots andpaper UAV’s and similar similar presents autonomous vehicles. A recent paper (Zhang-2008) presents the A State State feedback feedback controller controller has has been been applied applied on on J J222 perpervehicles. A recent paper (Zhang-2008) presents the vehicles. A recent paper (Zhang-2008) presents the
Copyright © 2016, 2016 IFAC IFAC 130Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 130 Copyright 2016 responsibility IFAC 130Control. Peer review© of International Federation of Automatic Copyright ©under 2016 IFAC 130 10.1016/j.ifacol.2016.03.041
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turbed Clohessy-Wiltshire dynamics. Major contribution from current work is : • Formulation and implementation of control algorithm that achieves precise formation maintenance, since the formation must be maintained in the face of external disturbances, robustness of the controllers is an essential feature. • Evaluation of the performance of this control law with respect to its fuel consumption , thrust required and transient behaviour. The flow of paper is as follows, next session briefly introduces the translational dynamics of spacecraft. Section further introduces the J2 perturbation as the unmodeled force and discusses the ways to incorporate it in the translational dynamics. Section 3 and 4 explain the control law and proves the system’s stability under J2 perturbation. Simulation based verification of the control law is presented in section 5 followed by conclusion in section 6.
Fig. 1. Rotating Euler-Hill frame, centered at the leader spacecraft. This figure also shows the follower spacecraft, whose position vectors in the rotating and inertial reference frames are denoted by ρ and rf , respectively (Vadali 2009).
2. BACKGROUND
x ¨ − 2ny˙ − 3n2 x = ux
2.1 Kepler’s law and Energy and Momentum associated with Spacecraft
(3)
y¨ + 2nx˙ = uy
(4)
2
(5)
z¨ + n z = uz
Kepler’s laws of motion describe satellite’s motion with respect to the earth(general two body motion)Vadali(2009), µ.r (1) r¨ + 3 = 0 r where, r = [X, Y, Z]T is the position vector of the spacecraft in the ECI frame and µ is the gravitational constant of earth.
where, (x, y, z) are the coordinates of spacecraft in Hill’s frame or LVLH frame and (ux , uy , uz ) is the control input and n is f˙, where f is the true anomaly of the leader satellites orbit.
2.2 The translational dynamics of spacecraft
HCW equation(3-5) is the linear version of original dynamics and does not account for the oblateness of the earth surface. This oblateness leads to important consequences in practical spacecraft orbit design.The most pronounced effect on Low Earth orbits is caused by the second harmonic of the Earth potential, which reflects the oblateness of Earth Samuel(2002).
The trajectory of the satellite cannot be chosen arbitrarily, but is constrained by the laws of physics. One of the big challenges is to find appropriate paths for all the satellites in a formation, so that the desired functionalities are achieved, both with a view to fuel efficiency and to fulfil the predefined mission. As discussed in section 2.1 , Kepler’s laws govern the motion of any spacecraft in inertial frame. Consider two spacecraft, where one is the leader and the other is the follower. The most common linear passive relative orbits of the followers with respect to leaders are the solutions to the Hill-Clohessy-Wiltshire Equations. These equations were introduced in Clohessy(1960). Though it’s worth noticing HCW equations do not consider the eccentricity of the orbit, and the same has not been considered for the work carried out in this paper. Let the subscripts i = l, f denote the leader and the follower satellite respectively. The position vector from the leader to the follower satellite can be expressed as
2.3 Perturbation due to non-spherical Earth
Incorporating J2 Perturbation in HCW Equations In Samuel(2002), the authors show that the equations of motion relative to circular non-Keplerian reference orbit and including the J2 term are well approximated by the linear system:3 ¯ x ¨i = 2ncy˙ i + (5c2 − 2)n2 xi + KJ2 Cos(2kt) 4 1 ¯ y¨i = −2ncx˙ i + KJ2 Sin(2kt) 2 z¨i = −q 2 zi + 2lqCos(qt + φ)
(6)
where, iref and rref are parameters of the reference orbit, ρ = rf − rl
(2)
= xer + yeθ + zez
where x, y and z are the components of ρ in the Hill frame. Using Kepler’s law and following the derivation as in Clohessy(1960) under linearized approximation, the equations will be given by: 131
3J R2
Re is the nominal radius of the earth, s = 8r22 e (1 + ref √ 3n2 J R2 3 cos(2iref )), c = 1 + s, KJ2 = rref2 e sin2 (iref ), k¯ = c+
3J2 Re2 2 2rref
cos2 iref , q is approximately equal to cn, and φ,
l are time varying functions of the difference in orbit inclination (see Samuel(2002) and Yeh(2002) for the details).
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Fig. 2. (a)Interaction Topology where dji denoted the desired relative separation between each agent (b) SC maintaining constant separation along z axis 3. PROBLEM FORMULATION AND CONTROL LAW DERIVATION 3.1 Problem Statement The problem addressed in this paper, is to maintain n spacecraft along z coordinate of LVLH frame in the low earth orbit(LEO). Main reason of selecting formation about z coordinate is the inherent stability of out of plane formation keeping, which requires very little thrust activity when compared to in-plane dynamics. The paper focus on formation maintenance and not the attitude control of spacecraft, which can be the scope for future work. • Complete system consists of a reference spacecraft at the origin of LVLH frame which is controlled by the ground station, all other spacecraft are autonomous and the exchange topology is given in figure (2). • Control law for spacecraft uses the information of its own state as well as the relative states of its neighbours. Own state information is obtained from GPS, while relative states are obtained through communication. • Paper examines the formation about z coordinate of LVLH frame. 3.2 State Feedback Control Law Control law in this paper considered the very general interaction topology, where the communication graph always consists of a spanning tree. The control law for spacecraft i be given by: ui = (k1 (Xj − Xi ) + k2 (X˙ j − X˙ i )) + k3 X˙ i + k4 Xi (7)
Fig. 3. Interaction Topology considered. where, Ln is the laplacian matrix of the interaction graph of spacecraft. And, 2 0 2n 0 3n 0 0 A1 = 0 0 0 , B1 = −2n 0 0 0 0 0 0 0 −n2 (8) can be simplified to
ξ ξ˙ = (Σ1 ⊕ Σ2 ) ζ ζ˙
where,
(9)
1 I 0 3 3 Σ1 = 2 A1 B 1 1 I 0 n n Σ2 = 2 −(k1 Ln + k4 In ) −(k2 Ln + k3 In ) it can be easily proved that eigenvalue of Σ1 ⊕ Σ1 will always be negative (for detailed proof refer Schaub(2004), Schur(1960), Shashank(2011)). Hence for the closed loop system (9) at steady state, ξ Limt→∞ =0 (10) ζ
which mean position and velocity of all the spacecraft will go to zero at steady state. This shows that all the spacecraft will eventually come to a rendezvous point. Though, as discussed this paper discuses about the formation control of spacecraft. To achieve the desired formation pattern and separation between the spacecraft, the control law(7) is modified as discussed in next subsection.
j∈Ni
where, k1 , k2 , k3 , k4 ∈ R+ , ui = (uix , uiy , uiz ) is the control 3.3 Formation Control Law input, Xi = (xi , yi , zi ) is the position of spacecraft i and Ni denotes the neighbors of spacecraft i. Thus, the control The control law presented (7) is complete but it always has law for spacecraft i depends on the position and velocity zero as equilibrium point, now in the context of formation of itself and its neighbors. T flying, a desired distance need to be maintained between T Let, ξ = [X1 X2 . . . . . . Xn ] and ζ = X˙ 1 X˙ 2 . . . . . . X˙ n . all the spacecraft. To achieve this purpose, control law is being modified as given: Now, for n agents, applying above control law to (6−8): u = (k (X −X −di−1 )+k (X˙ ˙ ˙ 03n I3n i 1 i−1 i 2 i−1 − Xi ))+k3 Xi +k4 (Xi ) i ξ˙ ξ (11) I ⊗ A − = n 1 ζwhere, for i = 1, i−1 denotes the reference spacecraft, di is ζ˙ (k1 Ln + k4 In ) ⊗ I3 In ⊗ B1 − (k2 Ln + k3 In ) ⊗ I3 i (8) the absolute desired distance for any spacecraft and dji = 132
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djj −dii is the desired relative distance between spacecraft j and i as mentioned in fig.(2). As discussed in the problem definition, the reference spacecraft is controlled by the ground station, and will always remain at the origin of the LVLH frame as in fig.(3). In this case, first spacecraft is the reference spacecraft which is controlled by ground station, T which implies d11 = 0. Using ξ = [X1 X2 . . . . . . Xn ] and T ζ = X˙ 1 X˙ 2 . . . . . . X˙ n and applying above control law to the HCW dynamics (3),
Theorem 1. Consider n spacecraft with dynamics (3) and the control law (11), if k1 , k2 , k3 , k4 ∈ R + , where R+ ξ is set of all positive real numbers then → Ddesired ζ asymptotically which ensures that the spacecraft achieve the desired formation.
V˙ = F V + D
In the previous subsection general control law for formation flying has been introduced. The control law is derived considering HCW dynamics as the dynamics given in LVLH frame. Including J2 perturbation into the dynamics equation (6) can be written further in the form of
with,
(12)
d11 .. . 0n×n 0n×n dnn H= −k1 Ln 0n×n 0 . . . 0
using above equation in eqn.(13), U˙ = F U now F is a Hurwitz matrix, which implies
⇒ Limt→∞ (V (t) + F
−1
(14)
D) = 0
V → F −1 D as t → ∞ (15) ξ which implies → F −1 D. ζ Above analysis shows that by using control law (11), the separation between any two spacecraft will settle to F −1 D. To ensure the desired formation separation, a different T need to be passed to the value of d = d11 d12 . . . dn−1 n T 1 2 ˜ controller. Let, ddesired = d1 d˜2 . . . d˜nn denotes the required separation of any spacecraft from reference and d as defined above is the separation that need to be passed to the control law. Then, d can be such that calculated ddesired D = F Ddesired where, Ddesired = . 0 From equation (14) Limt→∞ (V (t) + F −1 F Ddesired ) = 0
3.4 Performance of the Control Law Under J2 Perturbation
¨ i = AXi + B X˙ i + d(t) + ui X ¯ i+B ¯ X˙ i + ∆(t) + ui = AXi + B X˙ i + AX
ξ ,V = , F = (In ⊗ Σ1 + Σ2 ⊗ I3 ) and D = H.F being a ζ nonsingular matrix (none of the eigenvalue of F have zero real part, as proved in previous section), further solving equation (12), (13) V˙ = F (V + F −1 D) Assuming U = V + F −1 D, and differentiating, U˙ = V˙
Limt→∞ U (t) = 0
Proof: As follows from above analysis.
(16)
V (t) → Ddesired (17) ξ which implies → Ddesired . This guarantees all the ζ spacecraft to maintain the required separation between each of them. The analysis presented above can briefly be written in the form of a theorem as stated next. 133
(18)
where, ui is the input applied as given by control law x 3n2 0 0 0 2n 0 (7), X = y , A = 0 0 0 , B = −2n 0 0 , z 0 0 0 0 0 −n2 3 2 2 ¯ K Cos(2 kt) J 0 5s n 0 4 2 1 0 , ∆(t) = A¯ = 0 0 ¯ K Sin(2 kt) J 2 2 2 0 0 −s n 2 2lqCos(qt + φ) √ 1 + s − 1) 0 0 2n( ¯ = −2n(√1 + s − 1) B 0 0, 0 0 0 Now,for n agents above equation (18) can be written by T T taking ζ = [X1 X2 · · · Xn ] and ξ = X˙ 1 X˙ 2 · · · X˙ n as the system states, and is given by: I3n 03n ζ˙ ζ ζ T = (Σ1 ⊕Σ2 ) + ¯ ¯ ⊗ In ξ +[0n 1n ] ∆(t) ξ A ⊗ In B ξ˙ (19) where, Σ1 and Σ2 are the matrices as explained in previous section. All the eigenvalues of (Σ1 ⊕ Σ2 ) are negative and hence it is Hurwitz. 4. STABILITY ANALYSIS UNDER J2 PERTURBATION As can be seen from the above section, system in this case consists of both vanishing as well as non-vanishing perturbation part. In this paper both the parts will be analyzed separately and then clubbed together to see their effect as a whole on the system. Let, V (t, X) corresponds to the Lyapunov function for the perturbed system (19) in [0, ∞) × D, where D = {X ∈ R2n | X < r}, where T r > 0 and X = [ζ ξ] . Let, (Σ1 ⊕ Σ2 ) = Q, then Q is a Hurwitz matrix. Equation (19) can be written in the form of X˙ = QX + χX + g(t) (20) 0 In T where, χ = ¯ ¯ n and g(t) = [0n 1n ] ∆(t) . Let AIn BI
χX = γ X , ∀t ≥ 0, ∀X ∈ D . Neither the stability of origin as an equilibrium point can no longer be studied, nor the solution of perturbed system be expected to approach
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origin as t → ∞. The best scenario can be that X(t) will be ultimately bounded by a small bound, if the perturbation term g(t) is small in some sense. As can be seen from the above equation, g(t) is always bounded and let the bound on the value of g(t) is given by | g(t) |≤ β for all t ≥ 0. Now, let V (t, X) = X T P X where, P is the solution of P Q + QT P = −I The solution of above equation will always result a positive definite value of P . Implies V (t, X) > 0. Also, V (t, X) satisfies conditions as given in (Khalil 2002) (21) λmin (P ) X 22 ≤ V (X) ≤ λmax (P ) X 22 ∂V QX = X T (P Q + QT P )X = −X T X (22) ∂X where λmax and λmax corresponds to the maximum and minimum eigenvalues of matrix P . Now, Q being Hurwitz X = 0 is a exponential stable equilibrium point of the system X˙ = QX. The derivative of V (X) along the trajectories of (20) satisfies
Fig. 4. Simulations for three spacecraft under formation control law with HCW dynamics
∂V ∂V ∂V QX + χX + g(t) V˙ (X) = ∂X ∂X ∂X ≤ −(1 − 2λmax (P )γ) X 2 +2λmax (P )g(t) X Now, using (21),and √ taking W (t) = V (t, x(t)) and use ˙ = V˙ /2 V , when V = 0, to obtain, the fact W 2λmax (P ) 1 1 λmax (P ) ˙ − γ W +g(t) W ≤− λmin (P ) 2 λmax (P ) λmin (P ) (23) Now, by comparison lemma (Khalil 2002), W (t) satisfies the inequality t λmax (P ) φ(t, τ )g(τ )dτ (24) Fig. 5. Control/Thrust profile along X,Y and Z Coordinate W (t) ≤ φ(t, t0 )W (t0 )+ under formation control law λmin (P ) t0 where using the fact that γ is a constant,function φ(t, t0 ) separation of 5 units. The interaction topology is as deis given by fined in figure(3). As the satellites maintain a separa λmax (P ) 1 tion of 5 units along z axis they should be at the point − φ(t, t0 ) = exp − (t − t0 ) (0, 0, 0); (0, 0, 5); (0, 0, 10). To obtain this desired configu2λmax (P ) λmin (P ) (25) ration some different values of D are need to be passed min (P ) to the controller by calculating F ∗ Ddesired . For this case which implies, for φ(t, t0 ) to be bounded, γ < − 2λλmax (P )2 . with the values of gains k1 = .0001, k2 = .01, k3 = .01 and This bound on γ can be ensured by choosing proper gain k4 = .0001, the value of D is calculated and is given by values of control law i.e k1 , k2 , k3 , k4 which will affect the T eigenvalues of Q, and hence the eigenvalues of P . So, by [0 0 . . . 0 10.058 ∗ k1 0 0 15.116 ∗ k1 ] choosing proper values of gains the bound over the value The desired formation can be seen along the z axis of of γ can be ensured. Using the bound on γ, equation (25) LVLH frame as shown in figure (4). implies It can be seen from figure (4) that the controller sucφ(t, t0 ) = exp[−σ(t − t0 )] = ρ (26) cessfully performs the desired maneuver. Next, it is interwhere σ > 0. Using (26) in (24), it can be seen that X(t) is esting to observe thrust profile or the acceleration profile always bounded even in the presence of perturbation g(t). of this control action in plots in fig(5). As can observe Also to limit the bound to the ball of radius r ∈ D, from the plots control acceleration reduces to zero in the steady state. Though author would like to emphasize on λmax (P ) λmax (P ) X(t) ≤ max ρX(t0 ), supt≥t0 g(t) the point that the control acceleration reduces to zero, λmin (P ) λmin (P ) as the inherent dynamics along z axis for HCW equation (27) has stabilizing effect and hence require no acceleration component, once the equilibrium is achieved. 5. SIMULATION RESULTS The fuel consumption measure ∆V as is shown in figure 5 is calculated by integrating the resultant control For simulation, three satellites are considered having dy- acceleration over the time period of simulation Tsim , The namics as given by HCW equations in LVLH frame. The fuel needed for the maneuver is related to the total velocity initial positions of satellite are (15, 5, 10); (12, 7, 12); (5, 15, 20) change, as given in Gurfil 2007. Km. The desired formation is along the z axis with a 134
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Table 1. Maximum Control Effort by SC’s along X Y and Z coordinates under J2 perturbation Coordinate X Y Z
SC1 18.23 0.4123 1.145
SC2 19.67 0.6412 0.4567
SC3 5.009 2.5734 0.5845
Fig. 7. Control/Thrust profile along X,Y and Z Coordinate with J2 perturbation under formation control law REFERENCES Fig. 6. Simulations for three spacecraft under formation control law under the effect of J2 perturbation Performance of Control law with J2 Perturbation: To check the performance or the robustness of the controller, controller is implemented with the J2 perturbation well. Investigations carried out so far amply prove that such a control law is robust enough to perform its task of trajectory following even in the face of J2 perturbation. Same initial conditions and desired positions are given to spacecrafts while simulating the result with the inclusion of perturbation. Therefore, one can conclude that relative trajectory design using HCW equations is useful even in the presence of J2 perturbation. Plots (4 and 6) give the comparison between controller implemented with and without J2 . Maximum amount of control effort by controller considering J2 perturbation is given in Table (1). Simulations investigates the effect of perturbation on the amount of fuel consumption by the controller over the period of 6400 Sec. Plots (7) show the the fuel budget of the implemented controller while considering the effect of perturbation with HCW equations for SC1. 6. CONCLUSION Paper has briefly outlined the concept of multiple spacecraft formation flying and has investigated in some detail the dynamics and control of satellite formations. The stated aim was accomplished by implementing state feedback control law with HCW equations incorporating J2 perturbation. The stability of controller was studied by means of Lyapunov theory.Numerical simulations conducted in Matlab/Simulink are presented for control verification. An important outcome of this portion of the work is to analyze the performance of controller while in the presence of J2 perturbation. Though as mentioned earlier as well, current work still ignores the eccentricity of the orbit, and incorporating the same will remain the future interest for authors. 135
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