Robust Hybrid Supervisory Control for a 3-DOF Spacecraft in Close-Proximity Operations⁎

Proceedings, 2018 IFAC Workshop on Networked & Autonomous Proceedings, Proceedings, 2018 2018 IFAC IFAC Workshop Workshop on on Networked Networked & & Autonomous Autonomous Air & Space Systems Proceedings, 2018 IFAC IFAC Workshop Workshop on on Networked Networked & & Autonomous Autonomous Proceedings, 2018 Air & Space Systems Air & Space Systems Available online at www.sciencedirect.com June NM Air & 13-15, Space 2018. Systems Proceedings, 2018Santa IFAC Fe, Workshop Air & Space Systems June 13-15, 2018. Santa Fe, NM USA USAon Networked & Autonomous June 13-15, 2018. Santa Fe, NM USA June 13-15, 2018. Santa Fe, NM USA Air & Space Systems June 13-15, 2018. Santa Fe, NM USA June 13-15, 2018. Santa Fe, NM USA

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IFAC PapersOnLine 51-12 (2018) 88–93

Robust Robust Hybrid Hybrid Supervisory Supervisory Control Control for for Robust Hybrid Supervisory Control for 3-DOF Spacecraft in Close-Proximity Robust Hybrid Supervisory Control for 3-DOF Spacecraft in Close-Proximity 3-DOF Spacecraft in Close-Proximity  Operations 3-DOF Spacecraft in Close-Proximity Operations Operations  Operations ∗∗ Giulia Zucchini ∗∗∗ Bharani P. Malladi ∗∗ ∗∗

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Giulia Zucchini ∗∗ Bharani P. Malladi ∗∗ ∗∗ ∗∗ ∗∗∗ ∗∗∗ ∗∗ Ricardo Sanfelice Butcher Giulia G. Zucchini Bharani P.A. ∗∗∗ Eric ∗∗ Ricardo G. Sanfelice Eric A.Malladi Butcher ∗ ∗∗ ∗∗ ∗∗∗ ∗∗∗ Giulia Zucchini Bharani P. Malladi Ricardo G. Sanfelice ∗∗∗ Eric A. Butcher ∗∗ ∗ Ricardo G. Sanfelice Eric A. Butcher ∗∗ ∗ ∗ Department of Electrical, Electronic and Information Engineering Department of Electrical, Electronic and Engineering ∗ ∗ Department of Electrical, Electronic and Information Information Engineering (DEI), of Bologna, 40136 (DEI), University University of Bologna, Bologna, Bologna, 40136 IT IT (e-mail: (e-mail: ∗ Department of Electrical, Electronic and Information Engineering [email protected]) (DEI), University of Bologna, Bologna, 40136 IT (e-mail: [email protected]) ∗∗ (DEI), University of Bologna, Bologna, 40136 IT (e-mail: ∗∗ of Aerospace and Mechanical Engineering, [email protected]) ∗∗ Department and Mechanical Engineering, University University of of ∗∗ Department of Aerospace ∗∗ [email protected]) Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ 85721 USA (e-mail: [email protected], Arizona, Tucson, AZ 85721 USA (e-mail: [email protected], ∗∗ Department of Aerospace Engineering, University of Arizona, Tucson, [email protected]) 85721and USAMechanical (e-mail: [email protected], [email protected]) ∗∗∗ Arizona, Tucson, AZ 85721 USA (e-mail: [email protected], ∗∗∗ Department of Computer Engineering, University of California [email protected]) ∗∗∗ Engineering, University of California ∗∗∗ Department of Computer ∗∗∗ [email protected]) Santa Cruz, Santa Cruz, CA 95064 USA (e-mail: [email protected]) Department of Computer Engineering, University of California Santa Cruz, CA USA [email protected]) ∗∗∗ Department of Cruz, Computer Engineering, University of California Santa Cruz, Santa Santa Cruz, CA 95064 95064 USA (e-mail: (e-mail: [email protected]) Santa Cruz, Santa Cruz, CA 95064 USA (e-mail: [email protected]) Abstract: Abstract: In In this this paper paper we we propose propose a a hybrid hybrid control control strategy strategy to to solve solve the the problem problem of of rendezvous, rendezvous, Abstract:operations, In this paper we docking propose aof control strategy to solve rendezvous, proximity operations, and docking ofhybrid an autonomous autonomous spacecraft inthe 3D.problem Due to toofthe the different proximity and an spacecraft in 3D. Due different Abstract:operations, In this paper we docking propose aofhybrid control strategy toimplemented solve the problem ofthe rendezvous, proximity and an autonomous spacecraft in 3D. Due to constraints and tasks to perform, a hybrid systems approach is to solve the problem constraints and tasks to perform, a hybrid systems approach is implemented to solve the different problem proximity operations, and docking of an systems autonomous spacecraft indistance; 3D. Due to the different in three phases: 1) rendezvous; 2) rendezvous with smaller relative 3) docking phase; constraints and tasks to perform, a hybrid approach is implemented to solve the in three phases: 1) rendezvous; 2) rendezvous with smaller relative distance; 3) dockingproblem phase; constraints and phase; tasks towith perform, a and hybrid systems approach isInimplemented to 3) solve the problem and 4) docked range angle measurements. this approach, we implement aa in three phases: 1) rendezvous; 2) rendezvous with smaller relative distance; docking phase; and 4) docked phase; with range and angle measurements. In this approach, we implement in phases: 1) rendezvous; 2) and rendezvous with controllers smaller relative distance; 3)we docking phase; andthree 4) docked with range measurements. approach, implement a supervisor that phase; robustly coordinates theangle individual tothis accomplish the whole mission. supervisor that robustly coordinates the individual controllersInto accomplish the whole mission. and 4) docked phase; with range and angle measurements. Intothis approach, wewhole implement a supervisor that robustly coordinates the individual controllers accomplish the mission. We also present the designs of these individual controllers that solve the appropriate control We also present the designs of these individual controllers that solve the appropriate control supervisor that robustly coordinates the individual controllers to accomplish the perturbed whole mission. problems for the individual phases. Numerical results for both the nominal and case We also present the designs of these individual controllers that solve the appropriate control problems for the individual phases. Numerical results for both the nominal and perturbed case We also the present designsphases. of theseforindividual controllers that the appropriate control validate control close-proximity mission. problems forhybrid the the individual results for both thesolve nominal and perturbed case validate the hybrid control strategy strategy Numerical for the the spacecraft spacecraft close-proximity mission. problems forhybrid the individual phases. Numerical results for both the nominal and perturbed case validate the control strategy for the spacecraft close-proximity mission. © 2018, IFAC (International Control)close-proximity Hosting by Elsevier Ltd. All rights reserved. validate the hybrid controlFederation strategy of forAutomatic the spacecraft mission. Keywords: Keywords: Hybrid Hybrid systems, systems, Spacecraft Spacecraft close-proximity close-proximity missions, missions, Supervisory Supervisory control, control, Keywords: Robustness RobustnessHybrid systems, Spacecraft close-proximity missions, Supervisory control, Keywords: Hybrid systems, Spacecraft close-proximity missions, Supervisory control, Robustness Robustness 1. dimensional 1. INTRODUCTION INTRODUCTION dimensional spacecraft spacecraft modeled modeled using using the the CWH CWH equations. equations. 1. INTRODUCTION Similar to the strategy presented in Malladi et (2016), dimensional spacecraft modeled using the CWH equations. Similar to the strategy presented in Malladi et al. al. (2016), 1. INTRODUCTION dimensional spacecraft modeled using the CWH equations. this problem consists of the following four main phases: 1) Similar to the strategy presented in Malladi et al. (2016), In recent years there has been an increasing necessity to problem consists of the following four main phases: 1) In recent years there has been an increasing necessity to this Similar to the strategy presented in Malladi et al. (2016), problem consists of the following four main phases: 1) rendezvous with large relative distance; 2) rendezvous with In recent there of hasrelative been an increasing necessity to this control theyears dynamics satellite motion for closerendezvous with large relative distance; 2) rendezvous with control the dynamics of relative satellite motion for closethis problem consists of the following four main phases: 1) In recent years there has been an increasing necessity to rendezvous with large relative distance; 2) rendezvous with smaller relative distance; 3) docking phase; and 4) docked control themissions. dynamics of relative satellite motion proximity Often the motion between twofor or closemore smaller relative distance; 3) docking phase; and 4) docked proximity missions. Often the motion between two or more rendezvous withdistance; large relative distance; 2) rendezvous with control the dynamics of relative satellite motion for closephase. We consider that the range and angle measurements proximity missions. Often the motion between two or more smaller relative 3) docking phase; and 4) docked satellites is modeled assuming a circular chief orbit and satellites is modeled assuming a circular chief orbit and phase. We consider that the range andphase; angle measurements relative distance; 3)while docking and 4) docked proximityis missions. Often the motion between two or more are in phase the and satellites modeled assuming circular orbit and smaller phase. We consider the range andstate angleconstraints measurements a linearized about chief’s motion. This are available available in each eachthat phase while the state constraints and a deputy deputy orbit orbit linearized abouta the the chief’schief motion. This phase. We consider that the range and angle measurements satellites is modeled assuming a circular chief orbit and are available in each phase while the state constraints and tasks to perform are different access phases. Precisely, aresults deputy linearized about the chief’s motion. This the in orbit the Clohessy-Wiltshire-Hill (CWH) equations the tasks to perform are different access phases. Precisely, results in the Clohessy-Wiltshire-Hill (CWH) equations are available in each phase while the state constraints and a deputy orbit linearized about the chief’s motion. This the tasks to perform are different access phases. Precisely, we contribute to the problem by results the Wiltshire Clohessy-Wiltshire-Hill (CWH)which equations Clohessyin and (1960); Hill (1878), is a we contribute to the problem by Clohessy (1960); Hill (1878), which is aa the tasks to perform are different access phases. Precisely, results time-invariant in and the Wiltshire Clohessy-Wiltshire-Hill (CWH) equations Clohessy and Wiltshire (1960); Hill (1878), which is we contribute to the problem by linear model. Such missions include both linear time-invariant model. Such missions include both we••contribute Characterizing family of to thethe problem Characterizing the familyby of individual individual controllers controllers in in Clohessy and (1960); Hill (1878), which both is a linear time-invariant model. missions include formation flyingWiltshire missions and Such rendezvous in 3-dimensional • the Characterizing the family of individual controllers in 3D case and the required properties they formation flying missions and rendezvous in 3-dimensional the 3D case and the required properties they should should linear time-invariant model. Such missions include both formation flying missionsclosed-loop and rendezvous in 3-dimensional space, where guidance, control, and naviga• the Characterizing the family of individual controllers in induce to the closed-loop system to solve the problem space, where guidance, closed-loop control, and naviga3D case and the required properties they should induce to the closed-loop system to solve the problem formation flying missions and rendezvous in 3-dimensional space, where guidance, closed-loop control, and account navigation algorithms must be designed taking into the 3D each case and the required properties theyproblem should within phase of operation. tion algorithms must be designed taking into account induce to the closed-loop system to solve the space,algorithms where guidance, closed-loop control, and navigawithin each phase of tion mustand be designed account mission requirements the natural orbital dynamics of induce to the closed-loop system to solve the problem mission requirements and the naturaltaking orbitalinto dynamics of each of operation. operation. Designing supervisor that robustly coordinates the •• within Designing aa phase supervisor that robustly coordinates the tion system. algorithms mustand be the designed taking into account mission requirements natural orbital dynamics of the Feedback control solutions for such missions within eachacontrollers phase of operation. • Designing supervisor that robustly coordinates the individual so as to provide a solution to the system. Feedback control solutions for such missions individual controllers so as to provide a solution to mission requirements and the natural orbital dynamics of the control solutions for such missions maysystem. involve Feedback LQR control Kluever (1999), time-varying • Designing acontrollers supervisorsothat robustly coordinates the the problem. may involve LQR control Kluever (1999), time-varying individual as to provide a solution to the problem. the system. Feedback control solutions for such missions may involve LQR control Kluever (1999), time-varying gain control Nazari and Butcher (2016), output tracking individual controllers so as todesigns providethat a solution to • Providing specific controller approprigain control Nazari and Butcher (2016), output tracking the problem. • Providing specific controller designs that approprimay control involve LQR control Kluever (1999), time-varying gain and Butcher (2016), output tracking schemes that successfully reject Lee et the problem. schemes thatNazari successfully reject disturbances disturbances Lee et al. al. • Providing specific controller designs that appropriately solve solve the the control control problems problems for for individual individual phases phases ately gain control and Butcher (2016), output tracking schemes thatNazari successfully reject strategies disturbances Lee et (2014), model predictive control Vazquez et al. • Providing controller designs that appropriately solve specific the control problems for individual phases and validate them numerically. (2014), model predictive control strategies Vazquez et al. and validate them numerically. schemes that successfully reject disturbances Lee et al. (2014), predictive Vazquez (2011); model Di Cairano et al. control (2012);strategies Weiss et al. (2015)etand al. ately solve the control problems for individual phases (2011); Di Cairano et al. (2012); Weiss et al. (2015) and and validate them numerically. (2014), control model predictive control Vazquez al. The and (2011); Di Cairano et al.Malladi (2012);strategies Weiss et al. (2015)etand remainder of them the paper paper is organized organized as as follows. follows. The The hybrid strategies Malladi et al. (2016). (2016). validate numerically. The remainder of the is hybrid control strategies et al. (2011); control Di Cairano et al.Malladi (2012); et Weiss et al. (2015) and notation The remainder of the paper ispaper organized as needed follows.backThe hybrid strategies al. (2016). used throughout the and the notation used throughout the paper and the needed backIn thiscontrol paper,strategies we extend the hybrid control strategy The remainder ofonthe paper ispaper organized as needed follows. The hybrid Malladi et al. (2016). ground material hybrid controllers is presented inbackSecIn this paper, we extend the hybrid control strategy notation used throughout the and the material on hybridthe controllers is presented inbackSecIn paper, proximity we extendoperations, the hybridand control strategy for this rendezvous, proximity operations, and docking of an an ground notation used throughout paper and the needed for rendezvous, docking of tion problem of is in 3 ground material on hybrid controllers is presented in Section 2. 2. The The problem of interest interest is formalized formalized in Section Section 3 In this paper, we extend the hybridand control strategy for rendezvous, proximity docking autonomous spacecraft in Malladi to 3material on hybrid controllers is presented in Secautonomous spacecraft in operations, Malladi et et al. al. (2016) (2016) to ofaa an 3- ground tion 2. The problem of interest is formalized in Section 3 and a general hybrid feedback control solution is presented and a general hybrid feedback control solution is presented for rendezvous, proximity dockingto ofa an autonomous spacecraft in operations, Malladi et and al. (2016) 3- tion 2. The problem of interest is formalized in Section 3  in Section 4. Section 5 presents specific designs for each and a general hybrid feedback control solution is presented Research by R. G. Sanfelicein partially supported NSF Grants  Section 4. Section 5 presents specific designs for each autonomous spacecraft Malladi et al. by (2016) to a no. 3- in  G. supported by NSF no. Research by by R. R. G. Sanfelice Sanfelice partially partially supported by NSF Grants Grants no. and a general feedback control solution is presented  Research controller and numerical simulations for both the nominal  in Section 4. hybrid Section 5 presents specific designs for each ECS-1150306 and CNS-1544396, and by AFOSR Grant FA9550-16-1Research by R. G. Sanfelice partially supported by NSF Grants no. Research by R. G. Sanfelice partially supported by NSF Grants no. ECS-1150306 and CNS-1544396, and by AFOSR Grant FA9550-16-1controller and numerical simulations for both the nominal ECS-1150306 and CNS-1544396, and by AFOSR Grant FA9550-16-1 in Section 4. Section 5 presents specific designs for each controller and numerical simulations for both the nominal case as well as the more general case in which we consider 0015. Research by E. A. Butcher and B. Malladi partially supported Research by R. G. Sanfelice partially supported by NSF Grants no. ECS-1150306 and CNS-1544396, and by AFOSR Grant FA9550-16-1ECS-1150306 and CNS-1544396, and by AFOSR Grant FA9550-16-1case as well as the more general case in which we consider 0015. 0015. Research Research by by E. E. A. A. Butcher Butcher and and B. B. Malladi Malladi partially partially supported supported controller and numerical simulations for both the nominal case as well as the more general case in which we consider by NSF Grant CMMI-1657637. ECS-1150306 and CNS-1544396, and by AFOSR Grant FA9550-16-10015. Research by E. A. Butcher and B. Malladi partially supported 0015. Research by E. A. Butcher and B. Malladi partially supported by Grant by NSF NSF Grant CMMI-1657637. CMMI-1657637. case as well as the more general case in which we consider 0015. Research by E. A. Butcher and B. Malladi partially supported by NSF NSF Grant CMMI-1657637. CMMI-1657637. by Grant

by NSF Grant CMMI-1657637. 2405-8963 © 2018 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © IFAC 88 Copyright © 2018 88 Copyright © under 2018 IFAC IFAC 88 Control. Peer review responsibility of International Federation of Automatic Copyright © © 2018 2018 IFAC IFAC 88 Copyright 88 10.1016/j.ifacol.2018.07.093 Copyright © 2018 IFAC 88

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bounded, and Gc (xc , uc ) is a nonempty subset of Rnc for all (xc , uc ) ∈ Dc , then Hc is said to be well-posed. Note that this interconnection is well-posed when its data satisfies the hybrid basic conditions. For more details on the definitions of hybrid time domain, hybrid arc, hybrid basic conditions, asymptotic stability and well-posedness of a hybrid system, see Goebel et al. (2012).

the presence of noise in measurements. Due to space limitations, additional details and the proof of the main result will be published elsewhere. 2. PRELIMINARIES 2.1 Notation

3. PROBLEM DESCRIPTION

The following notation and definitions are used throughout the paper. Rn denotes n-dimensional Euclidean space. R denotes the real numbers. Z denotes the integers. R≥0 denotes the nonnegative real numbers, i.e., R≥0 = [0, ∞). N denotes the natural numbers including 0, i.e., N = {0, 1, . . .}. B denotes the open unit ball in a Euclidean space. Given a set S, S denotes its closure. Given a vector x ∈ Rn , |x| denotes the Euclidean vector norm. Given a closed set S ⊂ Rn and a point x ∈ Rn , |x|S := inf y∈S |x − y|. Given subsets S1 , S2 , S3 subsets of Rn , S1 + S2 + S3 := {x1 + x2 + x3 : x1 ∈ S1 , x2 ∈ S2 , x3 ∈ S3 }. The equivalent notation [x y  z  ] , and (x, y, z) is used for vectors. S(+) denotes the set of positive definitive matrices. 0 denotes a 3 × 3 matrix with zeros and I denotes a 3 × 3 identity matrix.

We consider a model of the chaser spacecraft given by Clohessy-Wiltshire equations, namely, Fx x ¨ − 2ny˙ − 3n2 x = mc Fy (4) y¨ + 2nx˙ = mc Fz z¨ + n2 z = mc where (x, y, z) and (x, ˙ y, ˙ z) ˙ are the position and velocity of the chaser spacecraft with respect to the target spacecraft resolved into the target LVLH (local-vertical - localhorizontal) frame, respectively; Fx , Fy and Fz are the control forces in the x, y and z directions, respectively,  µ mc the mass of the chaser, and n := ro 3 , where µ is the gravitational parameter of the Earth and ro is the orbit radius of the target spacecraft. The target spacecraft is located at (x, y, z) = (0, 0, 0) and has mass mt .

2.2 Hybrid controllers In this paper, we consider stabilization problems for nonlinear control systems of the form P : η˙ = fP (η, u), y = hP (η) (η, u) ∈ CP × UP (1) where UP ⊂ RmP is a set defining the available input values, CP ⊂ RnP is a set where the plant state η ∈ RnP is allowed to evolve, fP : CP × UP → RnP is a function defining the continuous dynamics, and hP : CP → RnP is the output function. A hybrid controller Hc = (Cc , fc , Dc , Gc , hc ) takes the form (see Goebel et al. (2012))   yc = hc (uc , xc ) x˙ c = fc (uc , xc ) (uc , xc ) ∈ Cc (2) Hc :  x+ ∈ G (u , x ) (u , x ) ∈ D c c c c c c c

The state space representation of (4) is given by: η˙ = Aη + Bu 

(5)

˙ ∈ R is the state vector, where η := [x y z x˙ y˙ z]  3 u := [Fx Fy Fz ] ∈ R is the input vector, and   0 0 0 1 0 0 0 1 0  0 0 0    0 0 0 1 0 0 0 1   A :=  2 , B := 0 2n 0 3n 0 0  mc I  0 0 0 −2n 0 0 0 0 −n2 0 0 0 are the state and input matrices, respectively. The relative position betweenthe chaser and the target is represented by ρ(x, y, z) := x2 + y 2 + z 2 . In addition, let N n (0, σ 2 ) be the set of measurable functions in an n-dimensional Euclidean space with Gaussian distribution having zero mean and variance σ 2 .

where uc ∈ Rmc denotes the input to the controller, yc ∈ Yc ⊂ Rrc denotes the controller output, xc ∈ Rnc is the controller state, the sets Cc and Dc define regions where the controller state can flow and jump, respectively, hc : Cc → Yc defines the output of the controller and fc : Cc → Rnc the flows, while Gc : Dc ⇒ Rnc is a map that defines how the controller state xc is updated at jumps. When Yc = UP and system (1) is controlled by Hc via the interconnection conditions uc = y, and u = yc , the resulting hybrid closed-loop system Hcl is given by   η˙ = fP (η, hc (hP (η), xc ))   =: F (x)  x˙ = fc (hP (η), xc )    c  (η, xc ) ∈ C, H: (3) + = η η   =: G(x)  +    xc ∈ Gc (hP (η), xc ) (η, xc ) ∈ D where, C := {(η, xc ) : (η, hc (hP (η), xc )) ∈ CP × UP , (hP (η), xc ) ∈ Cc }, D := {(η, xc ) : (hP (η), xc ) ∈ Dc }.

6

With these details, the problem to solve is the following. Problem 1: Given positive constants mc , mt , µ, ro , umax , ρmax > ρr > ρd , V , Vmax , σ1 , σ2 , σ3 , σ4 , tf > te , θ ∈ [0, π2 ), and (xp , yp , zp ) ∈ R3 , design a feedback controller that measures angle and range y = h(η) + v i.e.,   y arctan  x

    z (6) h(η) = arcsin   ρ(x, y, z)  ρ(x, y, z)

where arctan : R → [−π, π], arcsin : R → [0, 2π] are four-quadrant inverse tangent and inverse sine, re-

If Hc is such that Cc and Dc are closed, fc and hc are continuous, Gc is outer semicontinuous and locally 89

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spectively, v ∈ N (0, σn2 ), n ∈ {1, 2, 3, 4}; and assigns u such that for every initial condition η0 ∈  M0 := η ∈ R6 : ρ(x, y, z) ∈ [0, ρmax ], ρ(x, ˙ y, ˙ z) ˙ ∈ [0, V ] of the chaser with dynamics as in (5) under the constraints • The control signal t → u(t) satisfies the “maximum thrust” constraint sup max{|Fx (t)|, |Fy (t)|, |Fz (t)|} ≤ umax t≥0

namely, for each t ≥ 0,   u(t) ∈ UP := u ∈ R3 : max{|Fx |, |Fy |, |Fz |} ≤ umax ; (7)   • For each η ∈ M1 := η ∈ R6 : ρ(x, y, z) ∈ [ρr , ∞) , angle and range measurements are available as in (6), and v ∈ N (0, σ12 );   • For each η ∈ M2 := η ∈ R6 : ρ(x, y, z) ∈ [ρd , ρr ) , angle and range measurements are available as in (6), and v ∈ N 2 (0, σ22 );   • For each η ∈ Ma3 := η ∈ R6 : ρ(x, y, z) ∈ [0, ρd ) , angle and range measurements are available as in (6) and v ∈ N 2 (0, σ32 ); While, in addition, if η ∈ Ma3 ∩ Mb , where Mb3 (θ) :=  3       0  sin(θ/2) cos(θ/2) 0    x 0 sin(θ/2) − cos(θ/2)   0 y ≤  η ∈ R6 :  sin(θ/2) 0 cos(θ/2) 0     z sin(θ/2) 0 − cos(θ/2) 0 namely, the position state is in a 3-dimensional cone with aperture θ centered about the x axis, then the following constraint on closing/approaching velocity is satisfied:   ˙ y, ˙ z) ˙ ≤ Vmax η ∈ Mc3:= η ∈ R6 : ρ(x,  where ρ(x, ˙ y, ˙ z) ˙ := x˙ 2 + y˙ 2 + z˙ 2 .

When the chaser docks to the target (docked-phase), the chaser-target dynamics are given as in (5) with mc + mt in place of mc under the constraint (7) and with available position measurements relative to a partner at location (xp , yp , zp ). The constrained dynamics of the chaser-target are

η˙ = Aη + BR u (8) (η, u) ∈ CR × UP yb = hR (η) := h3 (η)

where

(3) η(t4f ) ∈ M4 , where  M4:= η ∈ R6 : (x, y,z) = (xp , yp , zp ),(x, ˙ y, ˙ z) ˙ = (0,0,0) ; namely, the docked chaser (or chaser-target) reach the partner location no later than t4f time units.  Remark 3.1. The values of the constants mc , mt , µ, ro , umax , and (xp , yp , zp ) are imposed by the vehicles and their environment. The constants ρmax , ρr , ρd , V , Vmax , θ, tf , and te are imposed by the mission and the desired performance. 4. GENERAL HYBRID FEEDBACK CONTROL STRATEGY Following Malladi et al. (2016), we extend the algorithm that supervises multiple hybrid controllers that are designed to cope with the individual constraints and to satisfy the desired temporal properties to 3-dimensional chaser proximity mission. Similar to Malladi et al. (2016), the supervising algorithm is modeled as a hybrid system, which we denote Hs , and is in charge of supervising the following individual hybrid controllers: • Hybrid controller for rendezvous from distances far from target (Phase I): this controller is denoted Hc,1 and its goal is to steer the chaser to a point in the interior of M, in particular, from points in the compact set M1 ∩ M0 . • Hybrid controller for rendezvous in close-proximity to target (Phase II): this controller is denoted Hc,2 and its goal is to steer the chaser to a point in the interior of Xlos ⊂ M2 ∪Ma3 , in particular, from points in M2 . • Hybrid controller for docking to target (Phase III): this controller is denoted Hc,3 and its goal is to steer the chaser to nearby η = 0 from points in M2 ∪ Ma3 . • Hybrid controller for relocation of target (Phase IV): this controller is denoted Hc,4 and its goal is to steer the chaser-target from nearby Mc3 to a neighborhood of the partner position (xp , yp , zp ). The operations described above are subject to the constraints stated in Problem 1. Similar to the 2-dimensional chaser close-proximity mission presented in Malladi et al. (2016), each of the hybrid controllers operates in specific regions of the state space. The tasks performed by the controllers Hc,3 and Hc,4 are practical, in the sense that the trajectories η are steered from and to neighborhoods of the desired sets respectively. With this problem formulation, the goals of the individual hybrid controllers are formalized next.

 1 0 , CR := M, mc + mt I  

rx (x) arctan  ry (y)   

 , rz (z) hR (η) =   arcsin  ρ(rx , ry , rz )  ρ(rx , ry , rz ) BR :=

Due to space limitations, the formal results about the specific controllers and the supervisor will be published elsewhere.

rx (x) = x−xp , ry (y) = y−yp , rz (z) = z−zp , v ∈ N 2 (0, σ42 )  and ρ(rx , ry , rz ) := rx (x)2 + ry (y)2 + rz (z)2 .

5. SPECIFIC DESIGNS AND SIMULATIONS

The following holds for the η-component t → η(t) of each solution to the closed-loop system: for some t2f < t3f < t4f such that t3f ≤ te , t4f ≤ tf , we have

5.1 An observer-based 3D LQR design of Hc,1 The controller Hc,1 is designed such that the inflated closed set A1 + δ1 B ⊂ M, where δ1 > 0, is finite-time attractive for the initial conditions starting from basin of attraction induced by Hc,1 in η space. A controller with linear continuous-time state feedback κ1 given by κ1 (η) := −K1 η, where K1 ∈ R6×6 , is obtained from a LQR

(1) η(t2f ) ∈ Ma3 ∩ Mb3 and ρ(x(t2f ), y(t2f ), z(t2f )) = ρd ; namely, the chaser reaches the cone first;   (2) η(t3f ) ∈ Mc3 = η ∈ R6 : η = 0 ; namely, the chaser docks on the target next, no later than t3f time units; 90

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controller design. A saturation on the controller is implemented to satisfy the maximum thrust constraint and the resulting closed-loop hybrid system, denoted H1 := (C1 , F1 , D1 , G1 ), has data given by F1 (η) = Aη + Bκ1 (η)

This construction is obtained by changing to a coordinate system (in polar coordinates) that is fixed to the target spacecraft with its origin moving at a constant angular rate n. The resulting hybrid feedback is such that, from points in C2 nearby α = 0, with  ∈ (0, π), it steers the chaser clockwise to −α∗ if α <  and counter-clockwise to α∗ if α > −. An additional LQR controller is implemented for the z component, with the state feedback given by κb2 (ηb ) and a saturation on both controllers is implemented to satisfy the maximum thrust constraint.

∀η ∈ C1

(9) where C1 := R , D1 := ∅ and arbitrary G1 (that is, no jumps). 6

5.2 A logic-based line-of-sight controller design of Hc,2 The hybrid controller Hc,2 is designed to render the inflated closed set A2 + δ2 B finite-time attractive for the solution components η, η2 starting from D12 . For this purpose, using the fact that initial conditions of the chaser belong to D12 , we exploit the ideas in Kluever (1999) (in particular, the change of coordinates), where a proportional-derivative control law that guides the chaser to dock with the target at a desired docking direction (α∗ ) and position (ρ∗ ) is proposed. We introduce a logic variable to handle the topological obstruction of stabilizing a set on a manifold. In fact, with a continuous state feedback law, there will be antipodal points to A2 (nearby α = 0) from where the chaser can move either left or right to reach the desired line of sight. While, alternatively, a discontinuous controller can be designed, such a discontinuous controller would not be robust to small measurement noise as previously shown in literature Sanfelice et al. (2006). We design a logic-based hybrid controller that steers the chaser can either clockwise or counter-clockwise to take shortest route εδ and reach a point in Xlos and be robust to small perturbations. With the proposed controller, the resulting closedloop hybrid system is denoted H2 := (C2 , F2 , D2 , G2 ), and has data given by   Aη + Bκ2 (hη) F2 (η, h) := ∀(η, h) ∈ C2 0   (10) η G2 (η, h) := ∀(η, h) ∈ D2 −h

5.3 A uniting local and “global” design of Hc,3 The hybrid controller Hc,3 steers the η components of ε the solutions from A2 + δ2 B to Xlos in finite time. This controller is designed to induce forward invariance and to satisfy the closing speed constraints for the chaser. We do this in two stages. First, a controller with output κ13 , thrusts the chaser towards the reference way-point  ε ηr := [xr 0 0 0 0 0] ∈ Xlos (in the y axis) within T3a ε seconds while guaranteeing forward invariance of Xlos ∪ εδ 2 Xlos . Second, a controller with output κ3 implements a  damping control law that guides the chaser from Xlos to the inflated set A3 +δ3 B within T3b , along the vertical axis and slowing down the vehicle so as to satisfy the closing speed constraint. The data of the resulting hybrid closedloop system, which is denoted H3 := (C3 , F3 , D3 , G3 ), is given by   Aη + Bκp3 (η) ∀(η, p) ∈ C3 F3 (η, p) := 0   (12) η G3 (η, p) := ∀(η, p) ∈ D3 3−p where p ∈ {1, 2} is a logic variable the denotes the subcontroller (κ13 or κ23 ) being used, C3 := ∪p∈{1,2} C3p × {p}, D3 := ∪p∈{1,2} D3p × {p}. The set C31 is taken to be a compact neighborhood of the reference way-point ηr that is contained in the basin of attraction of κ23 . The set D32 is taken as a compact neighborhood of ηr such that solutions using κ13 that start in D32 do not reach the boundary of C31 . Then, we define C32 = R6 \ D32 and D31 = R6 \ C31 . All the controllers are tuned in a way to satisfy the maximum thrust constraint.

where h ∈ {−1, 1} is the logic state variable of the controller,  ∈ (0, π) is a controller parameter, C2 := {(η, h) ∈ R6 × {−1, 1} : h(α − α∗ ) ≥ −}, D2 := {(η, h) ∈ R6 ×{−1, 1} : h(α−α∗ ) ≤ −}. The continuous-time state feedback κ2 is given by   a κ2 (ηa ) κ2 (η) := b κ2 (ηb ) where ηa := (x, y, x, ˙ y) ˙ ∈ R4 and ηb := (z, z) ˙ ∈ R2 . For the PD controller done on the xy system we can express      cos(α) − sin(α) aρ ax a κ2 (ηa ) := = (11) sin(α) cos(α) ay aα

5.4 A 3D LQR design of Hc,4 In Phase IV, the controller Hc,4 has to steer the docked chaser-target from points in D34 to M4 + δ4 B, δ4 > 0, in finite time. A controller with linear continuous-time state feedback κ4 given by κ4 (η) := −K4 (η − ηp ) is designed using the LQR method. The gain K4 is designed to satisfy the maximum thrust constraint and with this controller, the resulting closed-loop hybrid system, denoted H4 := (C4 , F4 , D4 , G4 ), has data given by

aρ =uρ + nρ , aα = uα + nα uρ =−k1 ρ˙e − k2 ρe , uα = −ρ(k3 α˙e + k4 αe )

˙ + α)] ˙ cos(α) + x(2n ˙ + α) ˙ sin(α) nρ =−[3n2 x + y(2n nα =[3n2 x + y(2n ˙ + α)] ˙ sin(α) + x(2n ˙ + α) ˙ cos(α) + vρ α˙ with k1 , k2 , k3 , k4 positive constants, ρe = ρ−ρ∗ , αe = α − hα∗ 1 , ρ˙ = v˙ ρ , and vρ = x˙ cos(α) + y˙ sin(α).

F4 (η) = Aη + BR κ4 (η)

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Once again to avoid the discontinuities associated with angle calculations, we embed the angle error on a unit circle, i.e. the error  sin(α ) calculation is performed as αe = atan2 cos(αe ) .

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(13)

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5.5 Simulation results for the nominal case 

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µ = 3.986 × ro = 7100000m, We use n = mc = 500Kg and mt = 2000Kg in the simulations. In the problem definition provided in the previous paper Malladi et al. (2016), which we also use here for this invited session, the chaser starts at a distance of no more than ρmax = 10Km away from the target. Once docked, the chaser-target has to reach a relocation position with range ρ(x, y, z) = 20Km, which is 10Km away from the partner spacecraft in worst-case time of tf = 12hr. In Phase I-IV both range ρ and angle α measurements are available and hence we consider that the states η ∈ R6 can be easily reconstructed.

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With these mission parameters, simulations for the entire closed-loop system are performed for the chaser starting from η ∈ M0 ∩ M1 , which corresponds to various initial conditions in the 10Km radius with a initial velocity ρ(x(0, ˙ 0), y(0, ˙ 0), z(0, ˙ 0)) ∈ [0, 0.707m/sec]. At this step we are assuming to have all measurements known and two LQR-based controllers are implemented for the xy and z system, respectively, with the following of weight  choice  20 × 104 0 matrices: Q1a = 0.015 × I4×4 , R1a = , 0 11 × 104 −2 3 Q1b = 1.5×10 ×I2×2 , and R1b = 99×10 . The trajectories of the chaser during Phase I are shown in Figure 1, and the chaser completes the desired maneuver in this phase in T1 ≈ 1, 7hr. Due to the interesting chaser motion,

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Fig. 3. Trajectories of the chaser during Phase III In the last phase (Phase IV), the specific motion is provided by the controller Hc,4 and the goal is to reach a desired partner position for the two system given by  ηp = [0km 20km 0km 0km/sec 0km/sec 0km/sec] . For that phase the 3D LQR controller has weight  matrices:  138 0 −1 4 Qa = 6×10 ×I4×4 , Ra = 11×10 ×I2×2 , Qb = 0 10 and Rb = 30 × 106 . The motion of the chaser with mass mc + mt is presented in Figure 4 and this maneuver is completed by the chaser in T4 ≈ 1.7hr.

Fig. 1. Trajectories of the chaser during Phase I we also perform multiple simulations when Hc,2 is used, for initial position (x(0, 0), y(0, 0), z(0, 0)) ∈ D12 , where D12 := {η ∈ R4 : ρ(x, y, z) ∈ [0, ρr ]}, ρr = 700m, and initial velocity ρ(x(0, ˙ 0), y(0, ˙ 0), z(0, ˙ 0)) ∈ [0, 0.64m/s]. With ρ∗ = 100m, α∗ = 179deg, and  = 10deg, the motion of the chaser with both h = 1 and h = −1 are shown in Figure 2, which highlights the capabilities conferred by the logic variable in the hybrid controller. For the PD controller κ2a the gains are chosen as: k1 = 40, k2 = 0.1, k3 = 25, and k4 = 0.047; instead, for the LQR controller,  138 0 the weight matrices are: Q = and R = 30 × 106 . 0 10 The trajectories of the chaser during Phase II, shown in Figure 2, are completed in this phase in T2 ≈ 1hr. We also show the chaser evolution during the approach/closing stage (Phase III) and highlight the specific motion provided by our controller Hc,3 . Multiple simulations from (x(0, 0), y(0, 0), z(0, 0)) ∈ A2 + δ2 B, where A2 = {η ∈ R6 : ρ = 150m, α = h 179deg} and δ2 = 10m, are presented in the Figure 3. The reference way-point, where

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The total time constraint is satisfied: T1 + T2 + T3 + T4 ≈ 5.2hr < tf . An overview of the nominal motion of the chaser is given in Figure 5. 10

chaser reaches the desired neighborhood of the target while maintaining the input constraint u∞ ≤ 0.02m/sec2 . The total worst case time to reach for the chaser rendezvous, docking and chaser-target rendezvous maneuver is T1 + T2 + T3 + T4 ≈ 8.88hr < tf , which is within specifications. 2

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REFERENCES

Fig. 5. Overview of the nominal motion

Clohessy, W.H. and Wiltshire, R.S. (1960). Terminal guidance system for satellite rendezvous. Journal of the Aerospace Sciences, 27(9), 653–658. Di Cairano, S., Park, H., and Kolmanovsky, I. (2012). Model predictive control approach for guidance of spacecraft rendezvous and proximity maneuvering. International Journal of Robust and Nonlinear Control, 22(12), 1398–1427. Goebel, R., Sanfelice, R.G., and Teel, A.R. (2012). Hybrid Dynamical Systems: Modeling, Stability, and Robustness. Princeton University Press, New Jersey. Hill, G. (1878). Researches in the lunar theory. American Journal of Mathematics, 1, 5–26. Kluever, C.A. (1999). Feedback control for spacecraft rendezvous and docking. Journal of Guidance, Control, and Dynamics, 22(4), 609–611. Lee, D., Bang, H., Butcher, E.A., and Sanyal, A.K. (2014). Nonlinear output tracking and disturbance rejection for autonomous close range rendezvous and docking of spacecraft. Transactions of the Japan Society for Aeronautical and Space Sciences, 57, 225–237. Malladi, B.P., Sanfelice, R.G., Butcher, E., and Wang, J. (2016). Robust hybrid supervisory control for rendezvous and docking of a spacecraft. In Proceedings of the Conference on Decision and Control, 3325 – 3330. Nazari, M. and Butcher, E.A. (2016). Fuel efficient periodic gain control strategies for spacecraft relative motion in elliptic chief orbits. International Journal of Dynamics and Control, 4, 104–122. Sanfelice, R.G., Messina, M.J., Tuna, S.E., and Teel, A.R. (2006). Robust hybrid controllers for continuous-time systems with applications to obstacle avoidance and regulation to disconnected set of points. In Proc. 25th American Control Conference, 3352–3357. Vazquez, R., Gavilan, F., and Camacho, E.F. (2011). Trajectory planning for spacecraft rendezvous with on/off thrusters. IFAC Proceedings Volumes, 44(1), 8473–8478. Weiss, A., Baldwin, M., Erwin, R.S., and Kolmanovsky, I. (2015). Model predictive control for spacecraft rendezvous and docking: Strategies for handling constraints and case studies. IEEE Transactions on Control Systems Technology, 23(4), 1638–1647.

5.6 Simulation results with noise A more complete overview of the simulations can be done by considering noise added into the system. A small zero-mean Gaussian residual noise (considering the best performance of a chosen filter) is added to the position and velocity components in every phase. The measurement noises added to each phase are shown in Table 1. Table 1. Variance of residual error, Phase I-IV Phase I II III IV

xy system Residual error (σres )2 (σpos )2 = (0.5m)2 (σvel )2 = (5 × 10−5 m/sec)2 (σpos )2 = (0.5m)2 (σvel )2 = (5 × 10−5 m/sec)2 (σpos )2 = (0.5 × 10−3 m)2 (σvel )2 = (5 × 10−5 m/sec)2 (σpos )2 = (0.5m)2 (σvel )2 = (5 × 10−5 m/sec)2

z system Residual error (σres )2 (σpos )2 = (0.5 × 10−3 m)2 (σvel )2 = (5 × 10−8 m/sec)2 (σpos )2 = (0.5 × 10−3 m)2 (σvel )2 = (5 × 10−8 m/sec)2 (σpos )2 = (0.5 × 10−7 m)2 (σvel )2 = (5 × 10−9 m/sec)2 (σpos )2 = (0.5 × 10−3 m)2 (σvel )2 = (5 × 10−8 m/sec)2

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Fig. 6. Full simulation of the chaser with noise The robustness of the controllers for small level of noise is shown in the simulations result, Figure 6, where the

2 Simulation files available at: https://github.com/ HybridSystemsLab/HybridRendezvousAndDocking3DOF

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