Spacecraft Constrained Maneuver Planning for Moving DebrisAvoidance Using Positively Invariant Constraint Admissible Sets

Avishai Weiss, Morgan Baldwin, R. Scott Erwin and Ilya Kolmanovsky

Abstract— This paper extends our prior work onspacecraft debris avoidance based on positively invariantsets to the case of moving debris. In this approach, a time-varying connectivity graph is constructed between a setof forced equilibria, forming a virtual net that is centeredaround a nominal orbital position of the spacecraft. Theconnectivity between two equilibria is determined basedon contractive sets in order to guarantee that collision-free transitions between equilibria can be effected withina giving time period. A graph search algorithm is imple-mented to find the shortest path around the debris, orthrough the debris’ path once it has moved on. Simulationresults are presented that illustrate this approach.

I. INTRODUCTION

Given the rapidly growing amount of debris inEarth orbit, spacecraft maneuvers must satisfy debrisavoidance requirements. While obstacle avoidance hasbeen studied extensively in robotics applications [3],[4], the related spacecraft problems have several uniquefeatures and benefit from the development of spe-cialized approaches and algorithms. In particular, thespace environment is relatively uncluttered. Spacecraftdynamics are different from those of typical robots.Maneuver efficiency with respect to time and fuelconsumption is important. The states of the spacecraftand the debris can only be estimated, often with asignificant estimation error. Finally, computational al-gorithms must be suitable for implementation on-boardwhere computing power is very limited.

In [1], we have introduced an on-board maneuverplanning approach based on the use of constraint-admissible positively invariant sets to determine con-nectivity between a set of forced and unforced space-craft equilibria forming a virtual net in the vicinity of

A. Weiss is a Graduate Student, Department of AerospaceEngineering, The University of Michigan, Ann Arbor, MI.

M. Baldwin is a Research Aerospace Engineer, Space VehiclesDirectorate, Air Force Research Laboratory, Kirtland Air ForceBase, NM.

R. Scott Erwin is a Principal Research Aerospace Engineer,Space Vehicles Directorate, Air Force Research Laboratory, Kirt-land Air Force Base, NM.

I. Kolmanovsky is a Professor, Department of Aerospace Engi-neering, The University of Michigan, Ann Arbor, MI.

the spacecraft. Two equilibria are connected if a choiceof a Linear Quadratic (LQ) feedback gain can be madethat results in a transition between the equilibria andwhich avoids the debris collision while satisfying theimposed thrust limits. The connectivity graph for all theequilibria in the net is constructed and graph searchalgorithms are employed to optimize an equilibriahopping sequence that avoids the debris collisions.

Unlike the open-loop trajectory optimization ap-proaches, we do not rely on precise assignment ofspacecraft position to the time instants along the trajec-tory, but instead switch to the next set-point and con-troller gain once appropriate conditions are satisfied.While this approach can be conservative, it facilitatesfault-tolerant and disturbance-tolerant execution of themaneuvers. Furthermore, by employing disturbance-invariant sets [14], we can assure robustness to unmea-sured (but set-bounded) disturbances and uncertainties.This extension to handling unmeasured disturbancesand uncertainties using techniques of [14] is not pur-sued in the present paper.

To facilitate the on-board construction of the connec-tivity graph, a fast growth distance computation proce-dure between two ellipsoidal sets has been proposed in[1]. In this approach, using the Karush-Kuhn-Tuckerconditions, the growth distance computations are re-duced to a root finding problem for the scalar valueof the Lagrange multiplier. Then a predictor-correctordynamic Newton-Raphson algorithm is used to updatethe Lagrange multiplier thereby rapidly estimating thegrowth distance from different equilibria in the virtualnet to the debris.

In [2], we incorporated limited thrust requirementsinto the computation of thrust limit on the growthdistance. Even though the computation of thrust limitson the growth distance can be performed offline forthe nominal operating conditions, fast computationalprocedures are beneficial in case of thruster failures,degradations, and restrictions on thrust directions (e.g.,caused by the presence of other spacecraft nearby), allof which can lead to changing constraints on thrustduring spacecraft missions. If the thrust limits are pre-scribed in the form of 2-norm bounds, the optimization

problem involved in computing the thrust limit on thegrowth distance is non-convex (unlike computing thegrowth distance itself). Consequently, we used poly-hedral norm bounds on thrust, that lent themselves toexplicit solutions.

In this paper, we address the problem of avoiding amoving debris. Our connectivity graph construction uti-lizes the sets, centered about the equilibria in the virtualnet, that are guaranteed to contract to specified smallersubsets in N steps. The connectivity of two equilibriain the net ensures that over N steps the transition tothe neighborhood of the target equilibrium (i.e., smallersubset) can occur without the debris colliding with thespacecraft. The graph vertices are formed based onequilibria and the control time intervals of N steps.A graph search algorithm is employed to determine afeasible debris avoidance trajectory.

The related literature on spacecraft trajectory opti-mization with obstacle/debris avoidance is surveyed in[1]. Previous research addresses topics in spacecrafttrajectory optimization [6], collision avoidance strate-gies based on risk assessment [7], the use of artificialpotential functions [8], [9], and the use of conventionaland mixed integer linear programming techniques [10],[11], [12], [13].

The paper is organized as follows. In Section II wediscuss the nonlinear and the linearized models usedto represent spacecraft relative motion dynamics. InSection III we review our approach to constructing thevirtual net based on a set of forced and unforced equi-libria, and using this virtual net for debris avoidance.The procedure to compute the thrust limit on the growthdistance is presented. Simulation results are reported inSection IV. Finally, concluding remarks are made inSection V.

II. RELATIVE MOTION MODEL

The spacecraft relative motion model represents thespacecraft dynamics in the (non-inertial) Hill’s framewith the origin at a target location on a nominalcircular orbit. For small distances, the linearization ofthe relative motion model given by the Hill-Clohessy-Wiltshire (CWH) equations is used [15].

A. Nonlinear equations of motion

The relative position vector of the spacecraft withrespect to a target location on a circular orbit isexpressed as

δ~r = xı+ yı+ zk,

where x, y and z are the components of the positionvector of the spacecraft relative to the target location

and ı, , k are the unit vectors of the Hill’s frame. TheHill’s frame has its x-axis along the orbital radius, y-axis along the orbital track, and z-axis is orthogonal toorbital plane.

The position vector of the spacecraft with respectto the center of the Earth can be expressed as ~R =~R0+δ~r = (R0+x)ı+y+zk, where R0 is the nominalorbital radius. The nonlinear equations of motion forthe spacecraft (relative to an inertial frame) can beexpressed in vector form as

~R = −µ~R

R3+

1

mc

~F , (1)

where ~F is the vector of external forces applied to thespacecraft, R = |~R|, mc is the mass of the spacecraft,µ is the gravitational constant, and

~R = (x− 2ny − n2x)ı+ (y + 2nx− n2y)+ (z)k.

In these equations, n =√

µR3

0denotes the mean motion

on the nominal orbit. Similar equations can be used todescribe the motion of the debris.

B. Linearized HCW equations in discrete-time

For δr << R, the linearized Hill-Clohessy-Wiltshire(HCW) equations [15] approximate the relative motionof the spacecraft on a circular orbit as

x− 3n2x− 2ny =Fxmc

,

y + 2nx =Fymc

,

z + n2z =Fzmc

,

(2)

where Fx, Fy, Fz are components of the external forcevector (excluding gravity) acting on the spacecraft. Thelinearized dynamics account for differences in gravitybetween the spacecraft and nominal orbital location,and for relative motion effects. The spacecraft relativemotion dynamics in the orbital plane (x and y) andin the out-of-orbital plane (z) are decoupled. The in-plane dynamics are Lyapunov unstable, while the out-of-plane dynamics are Lyapunov stable. The in-planedynamics are completely controllable from Fy inputbut are not controllable from Fx input. The out-of-plane dynamics are controllable from Fz input. Thesedynamics are clearly different from typical groundrobots.

Assuming a sampling period of ∆T sec, we canconvert the model (2) to a discrete-time form

X(t+ 1) = AX(t) +BU(t), (3)

where X(t) = [x(t), y(t), z(t), x(t), y(t), z(t)]T

is the state vector at the time instant t ∈ Z+,U(t) = [Fx(t), Fy(t), Fz(t)]

T is the control vectorof thrust forces at the time instant t ∈ Z+, and A =exp(Ac∆T ), B =

∫ ∆T0 exp(Ac(∆T − τ))dτBc are the

discretized matrices obtained based on the continuous-time system realization (Ac, Bc) in (2). Alternatively,the control vector U can represent an instantaneouschange in the velocity of the spacecraft, ∆v, inducedby thrust, with an appropriately re-defined B-matrix,

B∆v = eAc∆T

0 0 00 0 00 0 01 0 00 1 00 0 1

.

III. DEBRIS AVOIDANCE BASED ON A VIRTUAL

NET

Our approach to debris avoidance is based on utiliz-ing constraint-admissible positively invariant sets [14],[16] centered around the spacecraft forced and unforcedrelative motion equilibria. A finite set of these equilibriaused for constructing debris avoidance maneuvers isreferred to as a virtual net. Given an estimate of thedebris position, we build a connectivity graph thatidentifies the equilibria in the virtual net between whichthe spacecraft can move, with guaranteed collision-freemotion and within the available thrust authority. Wethen employ graph search to determine an efficient pathbetween the equilibria that ensures debris avoidance.

A. Virtual Net

The virtual net comprises a finite set of equilibria,Xe(r), corresponding to a finite set of prescribed space-craft positions r ∈ N = {r1, r2, . . . , rn} ⊂ R3,

Xe(rk) =

[rk0

]=

rkxrkyrkz000

, k = 1, · · · , n, (4)

whose velocity states are zero, and where n is thenumber of equilibria in the virtual net. See Figure 1.We assume that for all r ∈ N , the corresponding valuesof control necessary to support the specified equilibriain steady-state satisfy the imposed thrust limits.

Debris

Fig. 1: The virtual net for debris avoidance. Dotscorrespond to positions at equilibria, Xe(r), on a virtualnet. The ellipsoid represents the debris position anduncertainty.

B. LQ Controller with Gain Switching

A conventional Linear-Quadratic (LQ) feedback isused to control the spacecraft to a commanded equilib-rium in (4),

U = K(X −Xe(r)) + Γr = KX +H(K)r, (5)

where

Γ =

−3n2mc 0 00 0 00 0 n2mc

,H(K) = Γ−K

[I3

03

],

and where I3 denotes the 3 × 3 identity matrix and03 denotes the 3 × 3 zero matrix. This LQ controllerprovides an asymptotically stable closed-loop systembut does not enforce the constraints.

To provide greater flexibility in handling constraints,a multimode controller architecture is employed [16].Specifically, we assume that a finite set of LQ gainsK ∈ K = {K1, · · · ,Km} is available to control thespacecraft. By using a large control weight in the LQcost functional, motions with low fuel consumptionyet large excursions can be generated; using a largecontrol weight in the LQ cost, motions with shorttransition time can be generated [17]. We assume that apreference ordering has been defined and the gains arearranged in the order of descending preference, fromK1 being the highest preference gain to Km being thelowest preference gain.

C. Positively Invariant Sets

The ellipsoidal set

C(r,K) = {X ∈ R6 :

1

2(X −Xe(r))

TP (K)(X −Xe(r)) ≤ 1} ⊂ R6, (6)

where A(K)TPA(K) − P < 0, A(K) = (A + BK),P = P (K) > 0, is positively invariant. Positiveinvariance implies that any trajectory of the closed-loopsystem that starts in C(r,K) is guaranteed to stay inC(r,K) as long as the same LQ gain K is used and thesame set-point command r is maintained. To achievethe positive invariance, the matrix P can be chosenas the solution of the discrete-time Riccati equation orof the above Lyapunov equation for the closed-loopasymptotically stable system. We note that, becausethe system is linear, the positive invariance of C(r,K)implies the positive invariance of the scaled set

C(r,K, ρ) = {X ∈ R6 :

1

2(X −Xe(r))

TP (K)(X −Xe(r)) ≤ ρ2}, ρ ≥ 0.

(7)

Geometrically, the set C(r,K, ρ) corresponds to anellipsoid scaled by the value of ρ and centered aroundXe(r), r ∈ N .

D. Debris Representation

We use a set, O(z,Q), centered around the positionz ∈ R3, to over-bound the current position of thedebris, i.e.,

O(z,Q) = {X ∈ R6 : (SX − z)TQ(SX − z) ≤ 1},(8)

where Q = QT > 0 and

S =

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 0

. (9)

The set O(z,Q) can account for the debris and space-craft physical sizes and also for the uncertainties in theestimation of the debris/spacecraft position. Note thatthe set O(z,Q) has an ellipsoidal shape in the positiondirections and it is unbounded in the velocity directions.Ellipsoidal sets rather than polyhedral sets are usedto over-bound the debris since ellipsoidal bounds aretypically produced by position estimation algorithms,such as the Extended Kalman Filter (EKF).

E. Debris Avoidance Approach

Consider now ri ∈ N , representing a possibleposition on the net that the spacecraft can move to asa part of the debris avoidance maneuver. Suppose firstthat the debris is stationary and that the current stateof the spacecraft is X(t0) at the time instant t0 ∈ Z+.If there exists a ρ ≥ 0 and Kj ∈ K such that

X(t0) ∈ C(ri,Kj , ρ) and O(z,Q) ∩ C(ri,Kj , ρ) = ∅,(10)

the spacecraft can move to the position ri ∈ N byengaging the control law with r(t) = ri and K(t) =Kj , t ≥ t0, and without hitting the debris confinedto O(z,Q). This idea underlies our approach to debrisavoidance, in which we maintain the spacecraft trajec-tories within the tube formed by positively invariantsets that do not overlap with the debris.

To avoid a non-stationary debris, in [1], [2] weassume that its path can be covered by a union of afinite number of sets of ellipsoidal shape,

D =

l=nd⋃l=1

O(zl, Ql), (11)

where the center of the lth set is denoted by zl ∈ R3

and the lth set shape is defined by Ql = QTl > 0. Then

the debris path avoidance condition is

X(t0) ∈ C(ri,Kj , ρ) andO(zl, Ql) ∩ C(ri,Kj , ρ) = ∅, l = 1, . . . , nd.

(12)The same approach, with larger nd, can be used tohandle multiple non-stationary debris. Note, however,that this approach is conservative as it does not accountfor the debris progressions along their paths versustime.

In this paper, we introduce the notion of time intothe problem; whereas a transition between ri and rjmight not be feasible at time t1, based on the motionof a debris, it might become feasible at time t2.To accommodate moving debris, we introduce a setCk(r,K, ρ), 0 ≤ k ≤ N, defined by the followingrelation,

A(K)k(Ck(r,K, ρ)− {Xe(r)}

)=

(C(r,K, ρ)− {Xe(r)}

),

(13)

Note that if X(0) ∈ Ck(r,K, ρ), then X(1) ∈Ck−1(r,K, ρ), X(2) ∈ Ck−2(r,K, ρ), · · · , X(k) ∈C0(r,K, ρ) = C(r,K, ρ).

F. Connectivity Graph and Graph Search

We now introduce a notion of connectivity betweentwo vertices of the virtual net, ri ∈ N and rj ∈ N at aspecified time t0. This notion is based on the fact thatthe time to transition from any state in CN (r,K, ρ) toC(r,K, ρ) is less or equal than N steps. Suppose thatthe debris path D(t0 : t0 +N ·H) has been predictedover the N ·H discrete steps from the time instant t0,where

D(tk : tr) =

t=tr⋃t=tk

O(z(t), Q(t)).

The node ri ∈ N is connected to ri ∈ N at the timeinstant tk = t0 + kN if there exists K ∈ K such that

D(tk : tk +N) ∩ C(ri,K, ρ) = {∅}. (14)

The node ri ∈ N is connected to node rj ∈ N at timetk if there exists K ∈ K such that

D(tk : tk +N) ∩ CN (rj ,K, ρ) = {∅} (15)

and

C(ri,K, ρ) ⊂ CN (rj ,K, ρ). (16)

The connectivity implies that a spacecraft locatedclose to an equilibrium corresponding to ri, Xe(ri),can transition to an equilibrium Xe(rj) between thetime instants tk and tk + N while avoiding collisionwith the debris. We note that if ri is connected to rjthis does not imply that, in turn, rj is connected to ri.We also note that connectivity depends on the existenceof an appropriate control gain from the set of gains Kbut does not need to hold for all gains. Furthermore,since connectivity depends on the predicted motion ofthe debris, connectivity is time-varying.

The on-line motion planning with debris avoidanceis performed according to the following procedure:

Step 1: Determine the debris location, shape andpredict the debris path D(t0 : t0 +N ·H)

Step 2: Construct graph connectivity matrices cor-responding to tk, k = 0, 1, · · · , H . In thegraph connectivity matrix, if two vertices, riand rj , are not connected at tk, the corre-sponding matrix element is zero; if they areconnected the corresponding matrix elementis 1. In parallel, build the control gain selec-tivity matrix, which identifies the index of thehighest preference gain K for which ri andrj are connected. This gain will be applied ifthe edge connecting ri and rj is traversed.

Step 3: Perform graph search to determine a se-quence r[tk] ∈ N and control gains K[k] ∈K, k = 1, · · · , lp, such that r[t1] satisfiesthe initial constraints, r[lp] satisfies the finalconstraints, and the path length lp (or anothercost function such as the expected fuel con-sumption) is minimized.

Per the above algorithm, a graph search is utilizedto determine the minimum number of equilibrium hopsaround a debris starting at t0.

Remark 1: The condition (15) is conservative. It canbe replaced by a less conservative condition,

D(tk : tk +m) ∩ CN−m(rj ,K, ρ) = {∅},

m = 0, 1, · · · , N,

at a price of more demanding computations.Remark 2: The condition (15) is checked com-

putationally using the fast growth distance algorithmdescribed in [1], [2]. The intersection is empty if CNcan be grown before it touches D(tk : tk + N). Thisfast growth distance algorithm is essential to be able torapidly construct the connectivity matrices.

Remark 3: In our simulations, the path search isperformed using the standard Dijkstra’s algorithm. It isapplied to a lifted graph with vertices being the pairs(ri, tk).

IV. SIMULATION RESULTS

Simulations are now provided that illustrate ourdebris avoidance approach, and contrast it with ourprior results in [1], [2]. We consider a nominal circularorbit of 850 km and discretize the HCW equations witha sampling period, ∆T = 30 sec. We construct anapproximately 2 km cubed virtual net. We let R =2 × 107I3 and Q = diag(100, 100, 100, 107, 107, 107),K = K1.

We first consider the case of a non-stationary de-bris where we treat its motion as the union of staticdebris along the path as described in [2]. A unionof ellipsoidal sets over-bounds the debris’ motion,⋃l=nd

l=1 O(zi, Qi), where zi are generated by samplingthe relative motion of the debris with the initial condi-tion [0 0.5 0 0 0.0006 0]T, and with Qi = 200I3, i =1 . . . nd. The spacecraft’s initial condition is X(0) =Xe(r0), where r0 = [0 1 0]T. The target equilibriumnode is Xe(0). We impose a maximum thrust constraintof 10 N in each axis. Dijkstra’s algorithm is used to findthe shortest length path from the initial node to the finalnode. The simulation results in Figure 2 demonstratethe the spacecraft is able to avoid the closed debrispath by hopping over it.

We now repeat this simulations using the approachdescribed in this paper which takes into account the de-bris’ motion as a function of time. Figure 2 shows thatthe graph search algorithm is able to find a path whichpasses through the debris’ path but avoids collision dueto the debris’ location elsewhere at the specific timeinstant at which the spacecraft path crosses the debris’path.

Finally, Figure 4 demonstrates another maneuver ofavoiding a debris, following a different path. Note thatthe thrust does not converge to zero since the targetspacecraft position is not an in-track equilibrium.

V. CONCLUSION

This paper described an approach to spacecraft ma-neuver planning that uses contractive constraint admis-

(a) Debris Avoidance Path.

0 500 1000 1500

0.2

0.4

0.6

0.8

1

Time, sec

Thr

ust,

N

(b) Norm of Thrust Profile.

Fig. 2: (a) Debris avoidance path for a non-stationarydebris using the union method described in [2]. Thegreen x marks the initial node. The blue x marks thefinal node. The red ellipsoids represent the debris path.The blue line is the path the spacecraft takes in orderto avoid the debris. The blue ellipsoids represent themaximally grown (see [2]) invariant sets, C, along thepath. (b) The time history of thrust magnitude.

sible sets in order to avoid collisions with a movingdebris. Future work will explore cost functions otherthan the path length and include set-bounded uncer-tainty into the motion planning.

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1

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Fig. 3: (a) Debris avoidance path for a non-stationarydebris using the contractive set approach described inthis paper. The green x marks the initial node. The bluex marks the final node. The red ellipsoids represent thedebris path. The blue line is the path the spacecrafttakes in order to avoid the debris. The blue ellipsoidsrepresent the invariant sets, CN , along the path. (b) Thetime history of thrust magnitude.

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0 5000 10000 150000

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