- Email: [email protected]
Sub-optimal cooperative collision avoidance maneuvers of multiple active spacecraft via discrete-time generating functions
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Sub-optimal cooperative collision avoidance maneuvers of multiple active spacecraft via discrete-time generating functions
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a
b
Kwangwon Lee , Hyeongjun Park , Chandeok Park
a,c,∗
, Sang-Young Park
a
a
NARA Space Technology, Seoul 03628, Republic of Korea b Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Curces, NM 88003, USA c Yonsei University Observatory, Yonsei University, Seoul 03722, Republic of Korea
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i n f o
Article history: Received 21 August 2018 Received in revised form 10 July 2019 Accepted 15 July 2019 Available online xxxx Keywords: Collision avoidance Generating function Hamiltonian system Optimal feedback control Discrete-time
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This study presents real-time sub-optimal control for cooperative collision-free transfers of multiple active (actuated) spacecraft in proximity operations. The constrained optimal control problem for collisionfree transfers of multiple active spacecraft is decentralized and approximated as an unconstrained optimal control problem for single active spacecraft to mitigate the complexity and difficulty. The new penalty function is proposed by considering relative velocities for cooperative maneuvers between multiple active spacecraft, and is integrated with the quadratic cost function for optimal tracking by continuous-thrust control instead of the inequality constraints for avoiding collision. Then, the infinite-horizon control law applicable to each of multiple active spacecraft is obtained as an algebraic function of the states of both reference solutions and obstacles by employing discrete-time generating functions. Unlike conventional methods based on shooting, the proposed approach does not require repetitive process and initial guesses regardless of the number of active spacecraft. Illustrative examples demonstrate the effectiveness of the proposed approach with the new penalty function especially in simultaneous collision avoidance maneuvers of multiple active spacecraft. © 2019 Elsevier Masson SAS. All rights reserved.
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1. Introduction
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Numerous studies about control algorithms for space missions with multiple spacecraft have been actively proposed for over ten years since multiple spacecraft can deal with various mission objectives in service missions and deep space missions. For instance, missions that may utilize multiple spacecraft with flexibility and efficiency include implementing a virtual telescope [1], on-orbit assembly [2], and measuring gravitational and magnetic fields [3,4]. Among control algorithms for operating multiple spacecraft, the collision avoidance algorithm is among the most important and essential to achieve successful missions since it is directly connected with operational safety. Various algorithms have been proposed for collision avoidance maneuvers of spacecraft. Some studies used the penalty function or artificial potential function (APF) based on the Lyapunov theory, see, e.g., [2,5–11] and the references therein. They derived feedback control laws as an analytic form by employing the appropriate potential function that increases as closed to obstacles.
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*
Corresponding author at: Yonsei University Observatory, Yonsei University, Seoul 03722, Republic of Korea. E-mail address: [email protected] (C. Park). https://doi.org/10.1016/j.ast.2019.07.031 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.
They considered not only collision avoidance maneuvers but also attitude motion [2,5,8–11], communication range [5,8], underactuated system [7], parametric uncertainties [6,11], and estimation of unknown disturbances [10] as practical applications. Using the APF has an advantage that it is easy to implement and to combine with other nonlinear control approaches based on the Lyapunov theory, but the APF-based approaches do not consider the optimality of control input. There were some attempts to combine the APF-based approach with the optimal control theory by tracking optimal solution or using inverse optimal approach. Nonetheless, they do not take into account the optimality of control input under constraints for collision avoidance exactly [12–17]. Previous studies for implementing optimization-based collisionfree transfers of multiple active spacecraft were usually based on model predictive control (MPC) approaches [18–23]. Distributed MPC methods have been also proposed for the structural flexibility, less computation cost, and lower communication burden [24,25]. In the MPC framework, an optimal control problem is solved subject to constraints over a receding horizon based on the direct optimization. A control sequence is calculated at each time step, and only the first element is applied to the system. The procedure is repeated at the next time step. Thus, MPC can effectively deal with state/control equality/inequality constraints for collision
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avoidance directly. However, this advantage comes at the expense of repetitive procedure and initial guesses at each time step, which is generally required in the direct optimization approach. To reduce the computational burden which is usually inherent in direct optimizations, explicit MPC approaches have been proposed by implementing the direct optimization in the off-line calculation [26,27]. The explicit MPC approaches require the direct optimization only once, but, at the same time, large memory may be required to store the solution. Moreover, they require trajectories of obstacles to be known a priori. Considering that a spacecraft in multiple-spacecraft proximity missions is relatively small with limited hardware resources (e.g., onboard computers with low computing power and limited data memory) and that more spacecraft lead to more stringent constraints and calculations, the iterative procedure in the optimization-based approaches might be inefficient and/or indefinite for operating multiple spacecraft in real time. However, the small spacecraft also need to consider the optimality in fuel consumption because the limited space forces to load only a small amount of fuel. These contradictory requirements imply that it is required to develop a collision avoidance algorithm considering both optimality in fuel consumption and efficiency in computation. In the last years, some approaches that exploit the generating functions have been used for solving various types of optimal control problems subject to nonlinear dynamics and constraints [28–32]. In these approaches, based on the Hamilton-Jacobi theory, a sub-optimal control law truncated at a desired order can be obtained as an explicit function of current states by employing generating functions without repetitive procedure and initial guesses. Especially, the recently proposed approach employing the discrete-time generating function derived the sub-optimal control law for collision-free transfers of single spacecraft [32]. By incorporating the penalty function into the cost function for optimal tracking, and by assigning the states of obstacles and reference solution to independent variables, the control law was derived as an algebraic form of the states of obstacles and reference solution. As the control law presented in [32] is an explicit function of the states of obstacles, it seems to be straightforward or even trivial to extend its applicability into collision-free transfers of multiple active spacecraft. However, the control law proposed in [32], in itself, does not work properly for multiple active spacecraft in simultaneous maneuver, and yields infeasible and/or inefficient trajectories. These undesired behaviors are mainly caused by the penalty function presented in [32], which does not consider the relative velocities of obstacles from the spacecraft at present. Thus, this technical issue must be resolved for maneuvering multiple active spacecraft robustly and efficiently, which is in significant contrast to collision avoidance maneuvers for a single spacecraft. Motivated by both efficiency and limitation of the previous algorithm in [32] for maneuvering a single spacecraft, this study aims to expand the previous algorithm into cooperative collisionfree transfers of multiple active spacecraft as real-time implementations. To achieve multiple spacecraft operations in an optimal fashion without conflicts between collision avoidance maneuvers, a penalty function is newly designed by employing relative velocities between spacecraft for cooperatively avoiding each other. Then, the sub-optimal control law applicable to all the active spacecraft is developed only once for collision-free transfers in an explicit form. As the overall process does not require repetitive procedure and initial guesses, the proposed approach shows an extremely low computational burden barely affected by the number of active spacecraft. In the rest of this paper, the proposed approach is first introduced in detail. Then, multiple spacecraft operation scenarios are described. Compared with the previous approach presented in [32], the effectiveness of the proposed approach with the newly de-
signed penalty function for cooperative collision-free transfers is clearly shown in the presented examples in which collision avoidance maneuvers of multiple active spacecraft interfere with each other. The performance of the proposed approach such as computational time and control effort is analyzed and verified in detail by comparing with conventional approaches and applying to scenarios with even up to twelve active spacecraft. We also verify and analyze the performance under different levels of obstacle-sensing ability of spacecraft; the levels are quantified as the number of the nearest obstacles which are able to be simultaneously recognized. Finally, concluding remarks are followed.
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2. Infinite-horizon optimal collision-free transfers for multiple spacecraft
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Optimal transfers of multiple active spacecraft with powerlimited low propulsion systems can be defined as the following problem:
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Problem 1. Minimize
J=
1 2
t nsc f i =1 t
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T
ui ui dt
(1)
0
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subject to continuous-time nonlinear equations of motion in the affine form with initial and final conditions:
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x˙ i = f(xi , t ) + g(xi , t )ui ,
xi = [ri , vi ] T
xi (t 0 ) and xi (t f ) are given,
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(2)
i = 1, · · · , nsc
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Here, xi ∈ Rn×1 and ui ∈ Rm×1 are the state and control vectors of the i-th active spacecraft, respectively. ri and vi are the position and velocity vectors of the i-th active spacecraft, respectively. nsc is the number of active spacecraft. Since the active spacecraft might collide with other active spacecraft while transferring to the final conditions, the following inequality constraints must be additionally considered to avoid collision:
i r − r j ≥ dmin , i = 1, · · · , nsc − 1,
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i < j ≤ nsc
(3)
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Here, dmin is a desired minimum distance between the active spacecraft. Solving Problem 1 with Eq. (3) becomes more complicated and time-consuming as the number of the active spacecraft increases. To mitigate this complexity, Problem 1 is approximated as the following optimal tracking problem for a single active spacecraft by considering other active spacecraft as obstacles:
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Problem 2. Minimize
J=
1 2
+
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T
(x N − xr ) Q (x N − xr ) N −1 k =0
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(4)
1
[ (xk − xr ) T Q (xk − xr ) + ukT Ruk + (xk )]
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subject to discrete-time nonlinear equations of motion in the affine form with initial conditions:
xk+1 = fd (xk , tk ) + gd (xk , tk )uk , x0 is given
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xk = [rk , vk ]
T
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(5)
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Here, xk ∈ Rn×1 and uk ∈ Rm×1 are the state and control vectors of an active spacecraft at the k-th time step, respectively. rk and
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vk are the position and velocity vectors at the k-th time step, respectively. tk and t N are the k-th time step and the final time step, respectively. Q ∈ Rn×n and R ∈ Rm×m are weighting matrices for optimal tracking. (xk ) is the penalty function for indirectly implementing collision avoidance maneuvers; its value should increase sharply as the spacecraft becomes closer to obstacles. xr is the state vector of reference solution, which is defined as the unconstrained optimal transfer without collision avoidance, i.e. the solution of Problem 1 without the constraint, Eq. (3). This problem formulation is designed such that an active spacecraft detours obstacles while tracking the unconstrained optimal transfer as close as possible. Thus, if we obtain a control law for the above problem with the appropriate penalty function, then we can indirectly implement the collision-free transfers of multiple active spacecraft by applying the control law to each active spacecraft. The infinite-horizon optimal control law for the above problem can be derived with the recently developed approach employing the discrete-time generating function presented in [32] as the following lemma.
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Lemma 1 ([32]). Consider the relative states zk in the stationary frame of which origin is located at the current position of active spacecraft, and that Eqs. (4)-(5) in Problem 2 are reformulated with respect to zk as
1
J=
2
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(z N − zr )T Q (z N − zr ) N −1
+
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2
k =0
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1
T
(zk − zr ) Q
(zk − zr ) + ukT
Ruk + (zk )
zk+1 = fd (zk + xc , tk ) + gd (zk + xc , tk )uk − xc ,
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(zk ) =
no i =1
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where η and κ are positive weighting parameters, and no is the number of obstacles. αi , βi , and w i are defined as follows:
αi = α0 1 − exp −γ rf β0 if ro ,i T rf ≥ 0 and ro ,i < rf βi =
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0
otherwise
w i rk , hi =
hiT rk
(9) (10) (11)
where
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¯ (zk ) =
i =1
hi =
ro ,i × (rf × ro ,i )
r
o ,i
× (r
f
× r
o ,i
)
(12)
Here, r
= ro,i − rk and rf o ,i
= r f − rk where ro,i is the position vector of the i-th obstacle and r f is the terminal condition of the active spacecraft. α0 , β0 , and γ are positive weighting parameters. Then, the control law for collision-free transfers in Problem 2 can be derived by employing the discrete-time generating function of the second kind (F 2 ) obtained for arbitrarily defined receding horizon N ∗ as follows: uk = R
−1
2 αi β¯i + exp −ηrk − ro ,i 4 rk − ro ,i 1 + exp(κ w¯ i )
(14)
∂ F 2 (zk+1 , λ N ∗ ) gd (zk + xc , tk ) ∂ zk+1 T
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(13)
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where
(8)
no
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2 βi + exp −ηrk − ro ,i 4 rk − ro,i 1 + exp(κ w i )
αi
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As presented in [32], the penalty function in Eq. (8) is effective to implement collision-free transfers of a single active spacecraft. The first term in the penalty function allows the active spacecraft to detour the forbidden region around the obstacles by sharply increasing the value of penalty function as the spacecraft approaches the obstacles. The second term is an auxiliary term to avoid obstacles in advance by guiding the active spacecraft out of the plane that includes the active spacecraft and obstacles. w i is defined such that w i = 0 is the equation of the plane including ro ,i and is perpendicular to another plane including ro ,i and rf . Because the second term decreases as w i increases and the normal vector of the plane (hi ) satisfies hiT r f ≥ 0, the spacecraft can be guided to the side including the terminal condition by minimizing the penalty function. Both terms in the penalty function also converge to zero when the spacecraft becomes farther from the obstacles or approaches the terminal condition. However, Lemma 1 with the penalty function in Eq. (8) might lead to inappropriate and inefficient collision-free transfers when multiple active spacecraft perform collision avoidance maneuvers simultaneously, because it does not consider the relative velocities of obstacles from the spacecraft at present. Considering only the current positions may lead to maneuvering spacecraft to the direction where obstacles move. Thus, we propose an alternative approach utilizing the newly designed penalty function for cooperative collision avoidance maneuvers as the following proposition: Proposition 1. Consider replacing the penalty function (zk ) in ¯ (zk ): Lemma 1 with the following
Here, xc = [rk , 0] T , and zr = xr − xc . Consider the penalty function (zk ) as follows:
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Here, λ N ∗ is the terminal costates. zk+1 can be replaced with the function of zk and λ N ∗ as zk+1 = M (zk , λ N ∗ ) by using the linear transformation. λ N ∗ is replaced with zero by the transversality condition.
(7)
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zk = [rk , vk ] T
z0 is given
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(6)
3
β¯i =
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⎧ ⎪ ⎨β ⎪ ⎩
ro ,i rf T
0
r
o ,i
r
f
T if ro ,i rf
× r
o ,i
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(16)
(vo ,i × ro ,i ) × ro ,i o ,i
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(15)
otherwise
¯ i ) = h¯ T r w¯ i (rk , h i k (v
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0
¯i = h
≥ 0 and ro ,i < rf
) × r
o ,i
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(17)
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Here, vo ,i is the relative velocity of the obstacle from the spacecraft. Then, the control law for collision-free transfers in Lemma 1 can guide the T spacecraft to the direction rk satisfying rk vo ,i ≤ 0.
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¯ (zk )/∂ w¯ i ≤ 0 and that w¯ i is maximized when Proof. Note that ∂ ¯ i . Since h¯ T v o,i ≤ 0 and the control law derived rk is parallel with h i ¯ (zk ), the spacecraft can be guided to the by Lemma 1 minimizes T direction rk satisfying rk vo ,i ≤ 0. 2
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Remark 1. Proposition 1 implies that the newly designed penalty function considering the relative velocities of obstacles from the spacecraft guides spacecraft to the side where obstacles will not move. It allows the multiple spacecraft actuated by their own control laws to avoid each other cooperatively by predicting the movements based on the relative velocities between them.
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Remark 2. The proposed control law is obtained once at most without iterative procedure and initial guesses regardless of the number of active spacecraft [32]. Once the control law is derived, it is available to all the other active spacecraft for cooperative collision-free transfers of multiple active spacecraft, as it is an algebraic and explicit form of the time-varying parameters and the states of active spacecraft, reference solution, and obstacles. uk of each spacecraft is calculated by simple algebraic substitutions. The algorithm computing control input at arbitrary states and time with the obtained F 2 is summarized as the pseudo-code presented in Algorithm 2.1. For more detailed procedure obtaining F 2 , refer to [32,33].
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Algorithm 2.1 CtrlGF(tk , xk , xr , r f , ro,i , vo,i ).
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3. Simulation results
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As demonstrations of sub-optimal cooperative collision avoidance maneuvers between active spacecraft by the proposed approach, it is applied to four scenarios. When deriving the control law in the scenarios, the horizon length is defined as two time steps, where the time step of discrete-time equations of motion is set as 1 second. F 2 is obtained as the second-order power series combined with the coefficients, which are an explicit form ¯ i , αi , and β¯i . The control input is computed by of rk , zr , ro ,i , h using Algorithm 2.1 at every second and then is applied to the continuous-time equations of motion for 1 second. It is assumed that the reference solution, xr , is obtained in advance by utilizing the state-transition matrix of the linearized Hamiltonian system for Problem 1 without the inequality constraints, Eq. (3).
In Scenario 1, two active spacecraft avoid each other while crossing each other in the two-dimensional double-integrator dynamics for 800 seconds. We present three cases to demonstrate the effectiveness of the revised penalty function. Cases 1 and 2 use the control law presented in our previous study [32] for a collision-free transfer, which is based on Eq. (8). In Case 1, one of the spacecraft uses the control law for a collision-free transfer but the other spacecraft simply follows the reference solution as an inactive spacecraft. In Case 2, both spacecraft use the control law as an active spacecraft. The proposed control law employing the newly designed penalty function, i.e., Eq. (14), for cooperative collision-free transfers is applied to both spacecraft in Case 3. In all cases, we use the same weighting parameters set as follows:
Q =
Q p × I 2×2 02×2
02×2 Q v × I 2×2
R = R 0 × I 2×2 ,
Q v = 1,
β0 = 1 × 10−2 ,
γ = 6 × 10−1 , η = 1, κ = 10
R 0 = 1,
α0 = 4 × 10−4 ,
T
u = [u x , u y , u z ] ,
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(18)
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⎤
⎡
x˙ 0 ⎢ y˙ ⎥ ⎢0 ⎢ ⎥ ⎢ ⎢ z˙ ⎥ ⎢0 ⎥ =⎢ ⎢ 2ω0 y˙ + ω2 x − μ(ρ0 + x)/ρ 3 + μ/ρ 2 ⎥ + ⎢ ⎢ ux 0 0 ⎥ ⎢ ⎣ −2ω0 x˙ + ω2 y − μ y /ρ 3 ⎦ ⎣ uy 0 uz −μ z/ρ 3 x = [x, y , z, x˙ , y˙ , z˙ ] T ,
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Here, I n×n and 0n×n are n by n identity and zero matrices, respectively. These settings are for implementing forbidden regions whose radii are 0.25 m around active spacecraft for preventing collision, which is empirically adjusted based on the systematic procedure presented in [32]. The control law, once derived, is applied to all the active spacecraft with the same weighting parameters. Figs. 1–2 show the trajectories of spacecraft with the forbidden regions around them. The results of the unconstrained transfer and Case 1 are described in the upper left and right panels, respectively. The results of Cases 2 and 3 are described in the lower left and right panels, respectively. The forbidden regions at 0, 400, and 800 seconds are described as the empty circles. The initial and final positions are stated in Table 1 and are presented as the small empty and filled circles, respectively. The initial velocities are set as zero. As shown in Figs. 1–2, when Spacecraft 1 is actuated by the previous control law in Case 1, it barely detours the inactive spacecraft (Spacecraft 2) to the direction where the inactive spacecraft moves. Moreover, both active spacecraft in Case 2 cannot avoid each other for a while and eventually collide with each other. These results of Cases 1-2 imply that considering only the relative positions in the penalty function may lead to inefficient or failed maneuvers for multiple spacecraft operations and that these kinds of malfunction can easily occur even in this simple example. In contrast, the spacecraft in Case 3 with the revised penalty function successfully avoid each other cooperatively; Spacecraft 1 detours Spacecraft 2 to the direction opposite to where Spacecraft 2 moves. This result clearly validates that the proposed approach with the penalty function considering relative velocities between spacecraft is effective for implementing cooperative collision avoidance maneuvers of multiple active spacecraft. Scenarios 2-3 deal with formation reconfiguration of multiple active spacecraft in the relative motion with respect to the chief in a circular low Earth orbit. The equations of the relative motion in continuous-time domain are expressed as follows:
⎡
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,
Q p = 4 × 10−2 ,
x˙ = f(x, t ) + g(x, t )u
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ρ = (ρ0 + x)2 + y 2 + z2
122
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦
123 124 125 126
(19)
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Fig. 1. Trajectories of the unconstrained transfer (upper left) and the collision-free transfers by Case 1 (upper right), Case 2 (lower left), and Case 3 (lower right) in Scenario 1.
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Fig. 2. Relative distance between active spacecraft from the unconstrained transfer (upper left) and the collision-free transfers by Case 1 (upper right), Case 2 (lower left), and Case 3 (lower right) in Scenario 1.
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Q =
Table 1 Boundary conditions in Scenario 1.
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Spacecraft No.
Initial position (m)
Final position (m)
1 2
[5.5, 5.5] T [5.5, 0.5] T
[0.5, 0.5] T [0.5, 5.5] T
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Here, ρ0 and ω0 are the orbital radius and frequency of the chief orbit, respectively, and μ is the gravitational constant of the Earth. In the procedure obtaining the second-order F 2 , the relative motion is defined as the linear system in discrete-time domain by approximating Eq. (19) as follows:
fd (xk , tk ) ≡ A k xk ,
gd (xk , tk ) ≡ B k
A k = I 6×6 + A = I 6×6 +
⎡
19 21 22
3ω02 A 21 = ⎣ 0 0
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33 34 35 36 37 38 39 40
03×3 A 21
⎤
0 0 0 0 ⎦, 0 −ω02
I 3×3 A 22
,
Bk = B =
⎡
0 A 22 = ⎣ −2ω0 0
2ω0 0 0
(20)
03×3 I 3×3
⎤
0 0⎦ 0
(21) (22)
A and B are obtained by linearizing f(x, t ) and g(x, t ) in Eq. (19), respectively. In Scenarios 2-3, dmin is given as 0.3 km by assuming the radius of forbidden region around active spacecraft as 0.15 km. In Scenario 2, four active spacecraft are initially located on the y-axis and are arranged on the x-axis after maneuvering on the x-y plane for 2500 seconds. The weighting parameters for Scenario 2 are set as follows:
31 32
Q =
Q p × I 2×2 02×2
02×2 Q v × I 2×2
R = R 0 × I 2×2 ,
,
Q p = 1 × 10−2 ,
Q v = 1,
β0 = 1 × 10−3 ,
γ = 1, η = 4, κ = 10
R 0 = 1,
α0 = 1 × 10−5 ,
(23)
As references for comparative analyses, we also apply the linear quadratic MPC (LQ-MPC) and the stabilizing control law using artificial potential function (LQR/APF) to each active spacecraft in Scenario 2. The LQ-MPC solves the following problem [19,21]:
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Problem 3. Minimize
J=
1 2
(x N − xr )T P (x N − xr ) +
N −1 k =0
1
[ (xk − xr ) T Q (xk − xr ) 2
(24)
+ ukT Ruk ] subject to the discrete-time linear equations of motion in affine form with initial conditions and linear constraints:
xk+1 = A k xk + B k uk , pˆ kT (rk − ro,i ) ≥ dmin,i ,
x0 is given,
x N is free
i = 1, · · · , n o
(25)
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Here, pˆ k is the unit normal vector of the hyperplane tangential to the forbidden region around obstacles. P is the solution of the following algebraic Ricatti equation.
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A T P + P A − P B R −1 B T P + Q = 0
(26)
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,
R = I 2×2
(27)
where A and B are stated in Eqs. (21)-(22). The length of receding horizon for LQ-MPC is defined as 40 steps, and pˆ k is set to be rotated 1 degrees per time step to the side not including vo ,i . To obtain an open-loop solution for the receding horizon, we use the function quadprog, a built-in solver for quadratic programming in MATLAB. Q and R are set for LQ-MPC as follows:
u = − R −1 B T P (x − xr ) −
∂ V (x, ro,i ) R −1 B T 2 ∂x
(28)
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⎧ 0 if d s < r − ro,i ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ (d2s − r − ro,i ) ⎪ ⎨ (r − ro,i ) −χ ∂ V (x, ro,i ) 2 2 = 2 (r − ro,i − dmin ) ⎪ ∂x ⎪ ⎪ ⎪ ⎪ if dmin < r − ro,i < ds ⎪ ⎪ ⎩ not defined if r − ro,i < dmin
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where χ is a positive constant, and d s is the distance sensing obstacles. χ and d s are set as 2 × 10−5 and 0.5 km, respectively. Q and R in Eq. (28) are set as follows:
02×2 I 2×2
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(30)
Q and R in Eqs. (27) and (30) are adjusted to arrive at the final desired position within similar level of position errors to that of our approach. As the same with the proposed approach, LQ-MPC and LQR/APF compute the control input at every second and the input is applied to the continuous-time equations of motion for 1 second. Figs. 3–4 present the resultant transfer trajectories and the relative distances between active spacecraft in Scenario 2. The initial and final positions of active spacecraft are stated in Table 2 and are denoted as the same as those in Scenario 1. The results of the unconstrained transfer and proposed approach are described in the upper left and upper right panels, respectively, and the results of LQ-MPC and LQR/APF are presented in the lower left and lower right panels, respectively. Table 3 states rela the minimum tive distance between the active spacecraft (r − ro,i min ), the total u), the maximum relafuel consumption of all spacecraft (
t =t
tive distance from target at final time (r − r f maxf ), and the mean elapsed time for obtaining control input at each time step (dt ctrl ) in Scenario 2. GF in Table 3 stands for the proposed generating function approach. As presented in Figs. 3–4 and Table 3, the control efforts of avoiding collision from the proposed approach is larger than those from LQ-MPC and smaller than those from LQR/APF. LQMPC shows the best performance in fuel consumption, but dt ctrl of LQ-MPC is about 160 times larger than that of the proposed approach. Note that this difference in dt ctrl could become more critical as the number of active spacecraft increases. As the most complicated multiple spacecraft maneuver, Scenario 3 considers as many as twelve active spacecraft to switch their locations for 2500 seconds. The weighting parameters are set as the same as those in Scenario 2 with 3 by 3 identity and zero matrices. The boundary conditions of spacecraft are presented in Table 4. As representing the limited obstacle-sensing ability of spacecraft, we assume that each spacecraft estimates the relative states of only adjacent spacecraft. This limitation can be expressed by the directed graph G = ( V G , E G ) where V G and E G are a set of vertices and directed edges, respectively. The vertices are defined as active spacecraft as follows:
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Fig. 3. Trajectories of the unconstrained transfer (upper left) and the collision-free transfers by the proposed approach (upper right), LQ-MPC (lower left), and LQR/APF (lower right) in Scenario 2.
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Fig. 4. Relative distances between the active spacecraft from the unconstrained transfer (upper left) and the collision-free transfers by the proposed approach (upper right), LQ-MPC (lower left), and LQR/APF (lower right) in Scenario 2.
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Table 5 Minimum relative distances, fuel consumption, final position errors, and mean elapsed time for deriving u in Scenario 3.
Table 2 Boundary conditions in scenario 2.
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Final position (km)
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Unconstrained transfer GF LQ-MPC LQR/APF
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min
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Table 4 Boundary Conditions in Scenarios 3.
max
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Unconstrained transfer GF (s = 1) GF (s = 3) GF (s = 5) GF (s = 7) GF (s = 9) GF (s = 11)
Initial position (km)
Final position (km)
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[2, 0, 0] T [1, 0, 0] T [−1, 0, 0] T [−2, 0, 0] T [0, 2, 0] T [0, 1, 0] T [0, −1, 0] T [0, −2, 0] T [0, 0, 2] T [0, 0, 1] T [0, 0, −1] T [0, 0, −2] T
[0, 1, 0] T [0, 2, 0] T [0, −2, 0] T [0, 1, 0] T [0, 0, 1] T [0, 0, 2] T [0, 0, −2] T [0, 0, −1] T [1, 0, 0] T [2, 0, 0] T [−2, 0, 0] T [−1, 0, 0] T
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min
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(km)
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max
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v 1 , · · · , v nsc ∈ V G
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It is defined that v j can acquire the relative states of v i if the directed edge e i j = ( v i , v j ) exists. Then, the number of spacecraft simultaneously recognized in v j can be denoted as the indegree of v j , which is the number of head ends to v j . The indegree of all vertices is set to be the same value s as follows:
∀v ∈ V G ,
Spacecraft No.
r − ro,i
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−
deg ( v ) = s
(32)
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as the number of obstacles grows. This tendency seems to result from the reason that only the number of terms in the penalty function increases linearly with no effect on complexity as the number of obstacles increases, because the same penalty function is added sequentially. Fig. 5 presents the resultant trajectories of unconstrained transfer at 0, 800, 1600, 2500 seconds, and Fig. 6 describes those of the collision-free transfer when s = 7 at the same epochs. The left and right panels in Fig. 7 present the relative distances between active spacecraft of unconstrained transfer and collision-free transfer when s = 7, respectively. Note that the proposed approach can easily implement suboptimal collision-free transfers while considering even more than ten obstacles simultaneously, and that the mean elapsed time of generating function approach is within two milliseconds. This extremely low computational burden allows us to utilize the proposed approach in real-time regardless of the number of obstacles while online MPC-based approaches such as LQ-MPC might have computational challenges for avoidance many obstacles at the
same time. For reference, we measured computation time by CPU time usage. The simulations were performed on a computer with Intel® Core™ i7-4790K CPU @ 4.00 GHz and the controller code was implemented in MATLAB.
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4. Conclusion
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We proposed an alternative approach for optimal collision-free transfers of multiple active spacecraft in real time by utilizing the canonical transformation for the discrete-time Hamiltonian system. The new penalty function considering the relative velocities was presented to implement efficient and safe collision avoidance between multiple active spacecraft. Then, the sub-optimal collisionfree control law available to all active spacecraft was derived as an explicit function of the states of active spacecraft, reference solution, and obstacles without repetitive process and initial guesses. The proposed approach with the revised penalty function was validated by numerical simulations changing positions and formation
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of multiple (as many as twelve) active spacecraft. Regardless of the number of active spacecraft and the number of obstacles simultaneously recognized, all the active spacecraft successfully accomplished collision-free transfers with extremely low computational burden by using the same control law obtained only once. Considering the relatively lower computational burden and reasonable control effort, we expect the proposed approach to be suitable for operating a large number of spacecraft for swarm, formation flying, and constellation.
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Declaration of Competing Interest
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None declared.
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Acknowledgements
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This work was supported by Global Surveillance Research Center (GSRC) program funded by the Defense Acquisition Program Administration (DAPA) and Agency for Defense Development (ADD).
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Sub-optimal cooperative collision avoidance maneuvers of multiple active spacecraft via discrete-time generating functions
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a
b
Kwangwon Lee , Hyeongjun Park , Chandeok Park
a,c,∗
, Sang-Young Park
a
a
NARA Space Technology, Seoul 03628, Republic of Korea b Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Curces, NM 88003, USA c Yonsei University Observatory, Yonsei University, Seoul 03722, Republic of Korea
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Article history: Received 21 August 2018 Received in revised form 10 July 2019 Accepted 15 July 2019 Available online xxxx Keywords: Collision avoidance Generating function Hamiltonian system Optimal feedback control Discrete-time
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This study presents real-time sub-optimal control for cooperative collision-free transfers of multiple active (actuated) spacecraft in proximity operations. The constrained optimal control problem for collisionfree transfers of multiple active spacecraft is decentralized and approximated as an unconstrained optimal control problem for single active spacecraft to mitigate the complexity and difficulty. The new penalty function is proposed by considering relative velocities for cooperative maneuvers between multiple active spacecraft, and is integrated with the quadratic cost function for optimal tracking by continuous-thrust control instead of the inequality constraints for avoiding collision. Then, the infinite-horizon control law applicable to each of multiple active spacecraft is obtained as an algebraic function of the states of both reference solutions and obstacles by employing discrete-time generating functions. Unlike conventional methods based on shooting, the proposed approach does not require repetitive process and initial guesses regardless of the number of active spacecraft. Illustrative examples demonstrate the effectiveness of the proposed approach with the new penalty function especially in simultaneous collision avoidance maneuvers of multiple active spacecraft. © 2019 Elsevier Masson SAS. All rights reserved.
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1. Introduction
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Numerous studies about control algorithms for space missions with multiple spacecraft have been actively proposed for over ten years since multiple spacecraft can deal with various mission objectives in service missions and deep space missions. For instance, missions that may utilize multiple spacecraft with flexibility and efficiency include implementing a virtual telescope [1], on-orbit assembly [2], and measuring gravitational and magnetic fields [3,4]. Among control algorithms for operating multiple spacecraft, the collision avoidance algorithm is among the most important and essential to achieve successful missions since it is directly connected with operational safety. Various algorithms have been proposed for collision avoidance maneuvers of spacecraft. Some studies used the penalty function or artificial potential function (APF) based on the Lyapunov theory, see, e.g., [2,5–11] and the references therein. They derived feedback control laws as an analytic form by employing the appropriate potential function that increases as closed to obstacles.
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*
Corresponding author at: Yonsei University Observatory, Yonsei University, Seoul 03722, Republic of Korea. E-mail address: [email protected] (C. Park). https://doi.org/10.1016/j.ast.2019.07.031 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.
They considered not only collision avoidance maneuvers but also attitude motion [2,5,8–11], communication range [5,8], underactuated system [7], parametric uncertainties [6,11], and estimation of unknown disturbances [10] as practical applications. Using the APF has an advantage that it is easy to implement and to combine with other nonlinear control approaches based on the Lyapunov theory, but the APF-based approaches do not consider the optimality of control input. There were some attempts to combine the APF-based approach with the optimal control theory by tracking optimal solution or using inverse optimal approach. Nonetheless, they do not take into account the optimality of control input under constraints for collision avoidance exactly [12–17]. Previous studies for implementing optimization-based collisionfree transfers of multiple active spacecraft were usually based on model predictive control (MPC) approaches [18–23]. Distributed MPC methods have been also proposed for the structural flexibility, less computation cost, and lower communication burden [24,25]. In the MPC framework, an optimal control problem is solved subject to constraints over a receding horizon based on the direct optimization. A control sequence is calculated at each time step, and only the first element is applied to the system. The procedure is repeated at the next time step. Thus, MPC can effectively deal with state/control equality/inequality constraints for collision
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avoidance directly. However, this advantage comes at the expense of repetitive procedure and initial guesses at each time step, which is generally required in the direct optimization approach. To reduce the computational burden which is usually inherent in direct optimizations, explicit MPC approaches have been proposed by implementing the direct optimization in the off-line calculation [26,27]. The explicit MPC approaches require the direct optimization only once, but, at the same time, large memory may be required to store the solution. Moreover, they require trajectories of obstacles to be known a priori. Considering that a spacecraft in multiple-spacecraft proximity missions is relatively small with limited hardware resources (e.g., onboard computers with low computing power and limited data memory) and that more spacecraft lead to more stringent constraints and calculations, the iterative procedure in the optimization-based approaches might be inefficient and/or indefinite for operating multiple spacecraft in real time. However, the small spacecraft also need to consider the optimality in fuel consumption because the limited space forces to load only a small amount of fuel. These contradictory requirements imply that it is required to develop a collision avoidance algorithm considering both optimality in fuel consumption and efficiency in computation. In the last years, some approaches that exploit the generating functions have been used for solving various types of optimal control problems subject to nonlinear dynamics and constraints [28–32]. In these approaches, based on the Hamilton-Jacobi theory, a sub-optimal control law truncated at a desired order can be obtained as an explicit function of current states by employing generating functions without repetitive procedure and initial guesses. Especially, the recently proposed approach employing the discrete-time generating function derived the sub-optimal control law for collision-free transfers of single spacecraft [32]. By incorporating the penalty function into the cost function for optimal tracking, and by assigning the states of obstacles and reference solution to independent variables, the control law was derived as an algebraic form of the states of obstacles and reference solution. As the control law presented in [32] is an explicit function of the states of obstacles, it seems to be straightforward or even trivial to extend its applicability into collision-free transfers of multiple active spacecraft. However, the control law proposed in [32], in itself, does not work properly for multiple active spacecraft in simultaneous maneuver, and yields infeasible and/or inefficient trajectories. These undesired behaviors are mainly caused by the penalty function presented in [32], which does not consider the relative velocities of obstacles from the spacecraft at present. Thus, this technical issue must be resolved for maneuvering multiple active spacecraft robustly and efficiently, which is in significant contrast to collision avoidance maneuvers for a single spacecraft. Motivated by both efficiency and limitation of the previous algorithm in [32] for maneuvering a single spacecraft, this study aims to expand the previous algorithm into cooperative collisionfree transfers of multiple active spacecraft as real-time implementations. To achieve multiple spacecraft operations in an optimal fashion without conflicts between collision avoidance maneuvers, a penalty function is newly designed by employing relative velocities between spacecraft for cooperatively avoiding each other. Then, the sub-optimal control law applicable to all the active spacecraft is developed only once for collision-free transfers in an explicit form. As the overall process does not require repetitive procedure and initial guesses, the proposed approach shows an extremely low computational burden barely affected by the number of active spacecraft. In the rest of this paper, the proposed approach is first introduced in detail. Then, multiple spacecraft operation scenarios are described. Compared with the previous approach presented in [32], the effectiveness of the proposed approach with the newly de-
signed penalty function for cooperative collision-free transfers is clearly shown in the presented examples in which collision avoidance maneuvers of multiple active spacecraft interfere with each other. The performance of the proposed approach such as computational time and control effort is analyzed and verified in detail by comparing with conventional approaches and applying to scenarios with even up to twelve active spacecraft. We also verify and analyze the performance under different levels of obstacle-sensing ability of spacecraft; the levels are quantified as the number of the nearest obstacles which are able to be simultaneously recognized. Finally, concluding remarks are followed.
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2. Infinite-horizon optimal collision-free transfers for multiple spacecraft
79 80 81
Optimal transfers of multiple active spacecraft with powerlimited low propulsion systems can be defined as the following problem:
82 83 84 85 86
Problem 1. Minimize
J=
1 2
t nsc f i =1 t
87 88
T
ui ui dt
(1)
0
89 90 91
subject to continuous-time nonlinear equations of motion in the affine form with initial and final conditions:
92 93 94
x˙ i = f(xi , t ) + g(xi , t )ui ,
xi = [ri , vi ] T
xi (t 0 ) and xi (t f ) are given,
95
(2)
i = 1, · · · , nsc
97
Here, xi ∈ Rn×1 and ui ∈ Rm×1 are the state and control vectors of the i-th active spacecraft, respectively. ri and vi are the position and velocity vectors of the i-th active spacecraft, respectively. nsc is the number of active spacecraft. Since the active spacecraft might collide with other active spacecraft while transferring to the final conditions, the following inequality constraints must be additionally considered to avoid collision:
i r − r j ≥ dmin , i = 1, · · · , nsc − 1,
96
i < j ≤ nsc
(3)
98 99 100 101 102 103 104 105 106 107 108
Here, dmin is a desired minimum distance between the active spacecraft. Solving Problem 1 with Eq. (3) becomes more complicated and time-consuming as the number of the active spacecraft increases. To mitigate this complexity, Problem 1 is approximated as the following optimal tracking problem for a single active spacecraft by considering other active spacecraft as obstacles:
109 110 111 112 113 114 115 116 117
Problem 2. Minimize
J=
1 2
+
118
T
(x N − xr ) Q (x N − xr ) N −1 k =0
119 120
(4)
1
[ (xk − xr ) T Q (xk − xr ) + ukT Ruk + (xk )]
122
2
123
subject to discrete-time nonlinear equations of motion in the affine form with initial conditions:
xk+1 = fd (xk , tk ) + gd (xk , tk )uk , x0 is given
121
xk = [rk , vk ]
T
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(5)
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Here, xk ∈ Rn×1 and uk ∈ Rm×1 are the state and control vectors of an active spacecraft at the k-th time step, respectively. rk and
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vk are the position and velocity vectors at the k-th time step, respectively. tk and t N are the k-th time step and the final time step, respectively. Q ∈ Rn×n and R ∈ Rm×m are weighting matrices for optimal tracking. (xk ) is the penalty function for indirectly implementing collision avoidance maneuvers; its value should increase sharply as the spacecraft becomes closer to obstacles. xr is the state vector of reference solution, which is defined as the unconstrained optimal transfer without collision avoidance, i.e. the solution of Problem 1 without the constraint, Eq. (3). This problem formulation is designed such that an active spacecraft detours obstacles while tracking the unconstrained optimal transfer as close as possible. Thus, if we obtain a control law for the above problem with the appropriate penalty function, then we can indirectly implement the collision-free transfers of multiple active spacecraft by applying the control law to each active spacecraft. The infinite-horizon optimal control law for the above problem can be derived with the recently developed approach employing the discrete-time generating function presented in [32] as the following lemma.
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Lemma 1 ([32]). Consider the relative states zk in the stationary frame of which origin is located at the current position of active spacecraft, and that Eqs. (4)-(5) in Problem 2 are reformulated with respect to zk as
1
J=
2
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(z N − zr )T Q (z N − zr ) N −1
+
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2
k =0
30
1
T
(zk − zr ) Q
(zk − zr ) + ukT
Ruk + (zk )
zk+1 = fd (zk + xc , tk ) + gd (zk + xc , tk )uk − xc ,
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40
(zk ) =
no i =1
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where η and κ are positive weighting parameters, and no is the number of obstacles. αi , βi , and w i are defined as follows:
αi = α0 1 − exp −γ rf β0 if ro ,i T rf ≥ 0 and ro ,i < rf βi =
56 57 58 59 60 61 62 63 64 65 66
0
otherwise
w i rk , hi =
hiT rk
(9) (10) (11)
where
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¯ (zk ) =
i =1
hi =
ro ,i × (rf × ro ,i )
r
o ,i
× (r
f
× r
o ,i
)
(12)
Here, r
= ro,i − rk and rf o ,i
= r f − rk where ro,i is the position vector of the i-th obstacle and r f is the terminal condition of the active spacecraft. α0 , β0 , and γ are positive weighting parameters. Then, the control law for collision-free transfers in Problem 2 can be derived by employing the discrete-time generating function of the second kind (F 2 ) obtained for arbitrarily defined receding horizon N ∗ as follows: uk = R
−1
2 αi β¯i + exp −ηrk − ro ,i 4 rk − ro ,i 1 + exp(κ w¯ i )
(14)
∂ F 2 (zk+1 , λ N ∗ ) gd (zk + xc , tk ) ∂ zk+1 T
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(13)
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where
(8)
no
42
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97
2 βi + exp −ηrk − ro ,i 4 rk − ro,i 1 + exp(κ w i )
αi
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As presented in [32], the penalty function in Eq. (8) is effective to implement collision-free transfers of a single active spacecraft. The first term in the penalty function allows the active spacecraft to detour the forbidden region around the obstacles by sharply increasing the value of penalty function as the spacecraft approaches the obstacles. The second term is an auxiliary term to avoid obstacles in advance by guiding the active spacecraft out of the plane that includes the active spacecraft and obstacles. w i is defined such that w i = 0 is the equation of the plane including ro ,i and is perpendicular to another plane including ro ,i and rf . Because the second term decreases as w i increases and the normal vector of the plane (hi ) satisfies hiT r f ≥ 0, the spacecraft can be guided to the side including the terminal condition by minimizing the penalty function. Both terms in the penalty function also converge to zero when the spacecraft becomes farther from the obstacles or approaches the terminal condition. However, Lemma 1 with the penalty function in Eq. (8) might lead to inappropriate and inefficient collision-free transfers when multiple active spacecraft perform collision avoidance maneuvers simultaneously, because it does not consider the relative velocities of obstacles from the spacecraft at present. Considering only the current positions may lead to maneuvering spacecraft to the direction where obstacles move. Thus, we propose an alternative approach utilizing the newly designed penalty function for cooperative collision avoidance maneuvers as the following proposition: Proposition 1. Consider replacing the penalty function (zk ) in ¯ (zk ): Lemma 1 with the following
Here, xc = [rk , 0] T , and zr = xr − xc . Consider the penalty function (zk ) as follows:
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Here, λ N ∗ is the terminal costates. zk+1 can be replaced with the function of zk and λ N ∗ as zk+1 = M (zk , λ N ∗ ) by using the linear transformation. λ N ∗ is replaced with zero by the transversality condition.
(7)
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zk = [rk , vk ] T
z0 is given
34 35
(6)
3
β¯i =
106
⎧ ⎪ ⎨β ⎪ ⎩
ro ,i rf T
0
r
o ,i
r
f
T if ro ,i rf
× r
o ,i
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(16)
(vo ,i × ro ,i ) × ro ,i o ,i
108
(15)
otherwise
¯ i ) = h¯ T r w¯ i (rk , h i k (v
107
110
0
¯i = h
≥ 0 and ro ,i < rf
) × r
o ,i
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(17)
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Here, vo ,i is the relative velocity of the obstacle from the spacecraft. Then, the control law for collision-free transfers in Lemma 1 can guide the T spacecraft to the direction rk satisfying rk vo ,i ≤ 0.
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¯ (zk )/∂ w¯ i ≤ 0 and that w¯ i is maximized when Proof. Note that ∂ ¯ i . Since h¯ T v o,i ≤ 0 and the control law derived rk is parallel with h i ¯ (zk ), the spacecraft can be guided to the by Lemma 1 minimizes T direction rk satisfying rk vo ,i ≤ 0. 2
121
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Remark 1. Proposition 1 implies that the newly designed penalty function considering the relative velocities of obstacles from the spacecraft guides spacecraft to the side where obstacles will not move. It allows the multiple spacecraft actuated by their own control laws to avoid each other cooperatively by predicting the movements based on the relative velocities between them.
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Remark 2. The proposed control law is obtained once at most without iterative procedure and initial guesses regardless of the number of active spacecraft [32]. Once the control law is derived, it is available to all the other active spacecraft for cooperative collision-free transfers of multiple active spacecraft, as it is an algebraic and explicit form of the time-varying parameters and the states of active spacecraft, reference solution, and obstacles. uk of each spacecraft is calculated by simple algebraic substitutions. The algorithm computing control input at arbitrary states and time with the obtained F 2 is summarized as the pseudo-code presented in Algorithm 2.1. For more detailed procedure obtaining F 2 , refer to [32,33].
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Algorithm 2.1 CtrlGF(tk , xk , xr , r f , ro,i , vo,i ).
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3. Simulation results
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As demonstrations of sub-optimal cooperative collision avoidance maneuvers between active spacecraft by the proposed approach, it is applied to four scenarios. When deriving the control law in the scenarios, the horizon length is defined as two time steps, where the time step of discrete-time equations of motion is set as 1 second. F 2 is obtained as the second-order power series combined with the coefficients, which are an explicit form ¯ i , αi , and β¯i . The control input is computed by of rk , zr , ro ,i , h using Algorithm 2.1 at every second and then is applied to the continuous-time equations of motion for 1 second. It is assumed that the reference solution, xr , is obtained in advance by utilizing the state-transition matrix of the linearized Hamiltonian system for Problem 1 without the inequality constraints, Eq. (3).
In Scenario 1, two active spacecraft avoid each other while crossing each other in the two-dimensional double-integrator dynamics for 800 seconds. We present three cases to demonstrate the effectiveness of the revised penalty function. Cases 1 and 2 use the control law presented in our previous study [32] for a collision-free transfer, which is based on Eq. (8). In Case 1, one of the spacecraft uses the control law for a collision-free transfer but the other spacecraft simply follows the reference solution as an inactive spacecraft. In Case 2, both spacecraft use the control law as an active spacecraft. The proposed control law employing the newly designed penalty function, i.e., Eq. (14), for cooperative collision-free transfers is applied to both spacecraft in Case 3. In all cases, we use the same weighting parameters set as follows:
Q =
Q p × I 2×2 02×2
02×2 Q v × I 2×2
R = R 0 × I 2×2 ,
Q v = 1,
β0 = 1 × 10−2 ,
γ = 6 × 10−1 , η = 1, κ = 10
R 0 = 1,
α0 = 4 × 10−4 ,
T
u = [u x , u y , u z ] ,
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(18)
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⎤
⎡
x˙ 0 ⎢ y˙ ⎥ ⎢0 ⎢ ⎥ ⎢ ⎢ z˙ ⎥ ⎢0 ⎥ =⎢ ⎢ 2ω0 y˙ + ω2 x − μ(ρ0 + x)/ρ 3 + μ/ρ 2 ⎥ + ⎢ ⎢ ux 0 0 ⎥ ⎢ ⎣ −2ω0 x˙ + ω2 y − μ y /ρ 3 ⎦ ⎣ uy 0 uz −μ z/ρ 3 x = [x, y , z, x˙ , y˙ , z˙ ] T ,
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82
Here, I n×n and 0n×n are n by n identity and zero matrices, respectively. These settings are for implementing forbidden regions whose radii are 0.25 m around active spacecraft for preventing collision, which is empirically adjusted based on the systematic procedure presented in [32]. The control law, once derived, is applied to all the active spacecraft with the same weighting parameters. Figs. 1–2 show the trajectories of spacecraft with the forbidden regions around them. The results of the unconstrained transfer and Case 1 are described in the upper left and right panels, respectively. The results of Cases 2 and 3 are described in the lower left and right panels, respectively. The forbidden regions at 0, 400, and 800 seconds are described as the empty circles. The initial and final positions are stated in Table 1 and are presented as the small empty and filled circles, respectively. The initial velocities are set as zero. As shown in Figs. 1–2, when Spacecraft 1 is actuated by the previous control law in Case 1, it barely detours the inactive spacecraft (Spacecraft 2) to the direction where the inactive spacecraft moves. Moreover, both active spacecraft in Case 2 cannot avoid each other for a while and eventually collide with each other. These results of Cases 1-2 imply that considering only the relative positions in the penalty function may lead to inefficient or failed maneuvers for multiple spacecraft operations and that these kinds of malfunction can easily occur even in this simple example. In contrast, the spacecraft in Case 3 with the revised penalty function successfully avoid each other cooperatively; Spacecraft 1 detours Spacecraft 2 to the direction opposite to where Spacecraft 2 moves. This result clearly validates that the proposed approach with the penalty function considering relative velocities between spacecraft is effective for implementing cooperative collision avoidance maneuvers of multiple active spacecraft. Scenarios 2-3 deal with formation reconfiguration of multiple active spacecraft in the relative motion with respect to the chief in a circular low Earth orbit. The equations of the relative motion in continuous-time domain are expressed as follows:
⎡
68
81
,
Q p = 4 × 10−2 ,
x˙ = f(x, t ) + g(x, t )u
67
ρ = (ρ0 + x)2 + y 2 + z2
122
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦
123 124 125 126
(19)
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Fig. 1. Trajectories of the unconstrained transfer (upper left) and the collision-free transfers by Case 1 (upper right), Case 2 (lower left), and Case 3 (lower right) in Scenario 1.
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Fig. 2. Relative distance between active spacecraft from the unconstrained transfer (upper left) and the collision-free transfers by Case 1 (upper right), Case 2 (lower left), and Case 3 (lower right) in Scenario 1.
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Table 1 Boundary conditions in Scenario 1.
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Spacecraft No.
Initial position (m)
Final position (m)
1 2
[5.5, 5.5] T [5.5, 0.5] T
[0.5, 0.5] T [0.5, 5.5] T
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Here, ρ0 and ω0 are the orbital radius and frequency of the chief orbit, respectively, and μ is the gravitational constant of the Earth. In the procedure obtaining the second-order F 2 , the relative motion is defined as the linear system in discrete-time domain by approximating Eq. (19) as follows:
fd (xk , tk ) ≡ A k xk ,
gd (xk , tk ) ≡ B k
A k = I 6×6 + A = I 6×6 +
⎡
19 21 22
3ω02 A 21 = ⎣ 0 0
23 24 25 26 27 28 29 30
33 34 35 36 37 38 39 40
03×3 A 21
⎤
0 0 0 0 ⎦, 0 −ω02
I 3×3 A 22
,
Bk = B =
⎡
0 A 22 = ⎣ −2ω0 0
2ω0 0 0
(20)
03×3 I 3×3
⎤
0 0⎦ 0
(21) (22)
A and B are obtained by linearizing f(x, t ) and g(x, t ) in Eq. (19), respectively. In Scenarios 2-3, dmin is given as 0.3 km by assuming the radius of forbidden region around active spacecraft as 0.15 km. In Scenario 2, four active spacecraft are initially located on the y-axis and are arranged on the x-axis after maneuvering on the x-y plane for 2500 seconds. The weighting parameters for Scenario 2 are set as follows:
31 32
Q =
Q p × I 2×2 02×2
02×2 Q v × I 2×2
R = R 0 × I 2×2 ,
,
Q p = 1 × 10−2 ,
Q v = 1,
β0 = 1 × 10−3 ,
γ = 1, η = 4, κ = 10
R 0 = 1,
α0 = 1 × 10−5 ,
(23)
As references for comparative analyses, we also apply the linear quadratic MPC (LQ-MPC) and the stabilizing control law using artificial potential function (LQR/APF) to each active spacecraft in Scenario 2. The LQ-MPC solves the following problem [19,21]:
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Problem 3. Minimize
J=
1 2
(x N − xr )T P (x N − xr ) +
N −1 k =0
1
[ (xk − xr ) T Q (xk − xr ) 2
(24)
+ ukT Ruk ] subject to the discrete-time linear equations of motion in affine form with initial conditions and linear constraints:
xk+1 = A k xk + B k uk , pˆ kT (rk − ro,i ) ≥ dmin,i ,
x0 is given,
x N is free
i = 1, · · · , n o
(25)
54 55 56 57
Here, pˆ k is the unit normal vector of the hyperplane tangential to the forbidden region around obstacles. P is the solution of the following algebraic Ricatti equation.
58 59
A T P + P A − P B R −1 B T P + Q = 0
(26)
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,
R = I 2×2
(27)
where A and B are stated in Eqs. (21)-(22). The length of receding horizon for LQ-MPC is defined as 40 steps, and pˆ k is set to be rotated 1 degrees per time step to the side not including vo ,i . To obtain an open-loop solution for the receding horizon, we use the function quadprog, a built-in solver for quadratic programming in MATLAB. Q and R are set for LQ-MPC as follows:
u = − R −1 B T P (x − xr ) −
∂ V (x, ro,i ) R −1 B T 2 ∂x
(28)
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⎧ 0 if d s < r − ro,i ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ (d2s − r − ro,i ) ⎪ ⎨ (r − ro,i ) −χ ∂ V (x, ro,i ) 2 2 = 2 (r − ro,i − dmin ) ⎪ ∂x ⎪ ⎪ ⎪ ⎪ if dmin < r − ro,i < ds ⎪ ⎪ ⎩ not defined if r − ro,i < dmin
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(29)
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where χ is a positive constant, and d s is the distance sensing obstacles. χ and d s are set as 2 × 10−5 and 0.5 km, respectively. Q and R in Eq. (28) are set as follows:
02×2 I 2×2
75 77
ventionally used to avoid obstacles presented in [13]:
Q =
69
76
∂ V (x, ro,i )/∂ x is the gradient of artificial potential function con-
5 × 10−5 × I 2×2 02×2
68
74
no 1 i =1
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73
x˙ = Ax + Bu
18 20
02×2 I 2×2
The stabilizing control law using artificial potential function can be derived as an analytic form for continuous-time linear dynamical system via the inverse optimal approach as follows [13]:
7 8
1.5 × 10−4 × I 2×2 02×2
89 90 91 92
,
R = I 2×2
(30)
Q and R in Eqs. (27) and (30) are adjusted to arrive at the final desired position within similar level of position errors to that of our approach. As the same with the proposed approach, LQ-MPC and LQR/APF compute the control input at every second and the input is applied to the continuous-time equations of motion for 1 second. Figs. 3–4 present the resultant transfer trajectories and the relative distances between active spacecraft in Scenario 2. The initial and final positions of active spacecraft are stated in Table 2 and are denoted as the same as those in Scenario 1. The results of the unconstrained transfer and proposed approach are described in the upper left and upper right panels, respectively, and the results of LQ-MPC and LQR/APF are presented in the lower left and lower right panels, respectively. Table 3 states rela the minimum tive distance between the active spacecraft (r − ro,i min ), the total u), the maximum relafuel consumption of all spacecraft (
t =t
tive distance from target at final time (r − r f maxf ), and the mean elapsed time for obtaining control input at each time step (dt ctrl ) in Scenario 2. GF in Table 3 stands for the proposed generating function approach. As presented in Figs. 3–4 and Table 3, the control efforts of avoiding collision from the proposed approach is larger than those from LQ-MPC and smaller than those from LQR/APF. LQMPC shows the best performance in fuel consumption, but dt ctrl of LQ-MPC is about 160 times larger than that of the proposed approach. Note that this difference in dt ctrl could become more critical as the number of active spacecraft increases. As the most complicated multiple spacecraft maneuver, Scenario 3 considers as many as twelve active spacecraft to switch their locations for 2500 seconds. The weighting parameters are set as the same as those in Scenario 2 with 3 by 3 identity and zero matrices. The boundary conditions of spacecraft are presented in Table 4. As representing the limited obstacle-sensing ability of spacecraft, we assume that each spacecraft estimates the relative states of only adjacent spacecraft. This limitation can be expressed by the directed graph G = ( V G , E G ) where V G and E G are a set of vertices and directed edges, respectively. The vertices are defined as active spacecraft as follows:
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Fig. 4. Relative distances between the active spacecraft from the unconstrained transfer (upper left) and the collision-free transfers by the proposed approach (upper right), LQ-MPC (lower left), and LQR/APF (lower right) in Scenario 2.
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Table 5 Minimum relative distances, fuel consumption, final position errors, and mean elapsed time for deriving u in Scenario 3.
Table 2 Boundary conditions in scenario 2.
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Spacecraft No.
Initial position (km)
Final position (km)
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[0, 2] T [0, 1] T [0, −1] T [0, −2] T
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Table 3 Minimum relative distances, fuel consumption, final position errors, and mean elapsed time for deriving u in Scenario 2.
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Algorithm
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Unconstrained transfer GF LQ-MPC LQR/APF
r − ro,i
(km)
(km/s)
min
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u
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r − r f t =t f (km) 0.0014 0.0018 0.0006 0.0014
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Table 4 Boundary Conditions in Scenarios 3.
max
dt ctrl (s) 0.0002 0.0324 0.0001
Unconstrained transfer GF (s = 1) GF (s = 3) GF (s = 5) GF (s = 7) GF (s = 9) GF (s = 11)
Initial position (km)
Final position (km)
1 2 3 4 5 6 7 8 9 10 11 12
[2, 0, 0] T [1, 0, 0] T [−1, 0, 0] T [−2, 0, 0] T [0, 2, 0] T [0, 1, 0] T [0, −1, 0] T [0, −2, 0] T [0, 0, 2] T [0, 0, 1] T [0, 0, −1] T [0, 0, −2] T
[0, 1, 0] T [0, 2, 0] T [0, −2, 0] T [0, 1, 0] T [0, 0, 1] T [0, 0, 2] T [0, 0, −2] T [0, 0, −1] T [1, 0, 0] T [2, 0, 0] T [−2, 0, 0] T [−1, 0, 0] T
0.0113 0.4058 0.4014 0.4033 0.4036 0.4036 0.4036
min
u
r − r f t =t f
(km/s)
(km)
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0.0014 0.0019 0.0019 0.0019 0.0019 0.0019 0.0020
max
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0.0002 0.0004 0.0006 0.0008 0.0010 0.0013
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v 1 , · · · , v nsc ∈ V G
(31)
It is defined that v j can acquire the relative states of v i if the directed edge e i j = ( v i , v j ) exists. Then, the number of spacecraft simultaneously recognized in v j can be denoted as the indegree of v j , which is the number of head ends to v j . The indegree of all vertices is set to be the same value s as follows:
∀v ∈ V G ,
Spacecraft No.
r − ro,i
Algorithm
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−
deg ( v ) = s
(32)
The directed edges are connected to each spacecraft from the closest s spacecraft at every time. To analyze the performances under the different level of obstacle-sensing ability of spacecraft, the proposed approach is applied to the examples with the different s set as 1, 3, 5, 7, 9, and 11. The six control laws are derived by the proposed approach with the different no set to be s. Regardless of the indegrees, it is confirmed that all the control law can successfully accomplish collision-free transfers as presented in Table 5. In detail, the fuel consumption decreases as the indegree increases, and converges at s = 7. Considering that the unconstrained transfer shows six close encounters between spacecraft, it appears to be reasonable to converge at s = 7. dt ctrl also tends to increase almost linearly
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Fig. 5. Unconstrained optimal trajectories at 0 (upper left), 800 (upper right), 1600 (lower left), and 2500 seconds (lower right) in Scenario 3.
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Fig. 7. Relative distances between the active spacecraft from the unconstrained transfer (left) and the collision-free transfers by the proposed approach (right) in Scenario 3.
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as the number of obstacles grows. This tendency seems to result from the reason that only the number of terms in the penalty function increases linearly with no effect on complexity as the number of obstacles increases, because the same penalty function is added sequentially. Fig. 5 presents the resultant trajectories of unconstrained transfer at 0, 800, 1600, 2500 seconds, and Fig. 6 describes those of the collision-free transfer when s = 7 at the same epochs. The left and right panels in Fig. 7 present the relative distances between active spacecraft of unconstrained transfer and collision-free transfer when s = 7, respectively. Note that the proposed approach can easily implement suboptimal collision-free transfers while considering even more than ten obstacles simultaneously, and that the mean elapsed time of generating function approach is within two milliseconds. This extremely low computational burden allows us to utilize the proposed approach in real-time regardless of the number of obstacles while online MPC-based approaches such as LQ-MPC might have computational challenges for avoidance many obstacles at the
same time. For reference, we measured computation time by CPU time usage. The simulations were performed on a computer with Intel® Core™ i7-4790K CPU @ 4.00 GHz and the controller code was implemented in MATLAB.
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4. Conclusion
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We proposed an alternative approach for optimal collision-free transfers of multiple active spacecraft in real time by utilizing the canonical transformation for the discrete-time Hamiltonian system. The new penalty function considering the relative velocities was presented to implement efficient and safe collision avoidance between multiple active spacecraft. Then, the sub-optimal collisionfree control law available to all active spacecraft was derived as an explicit function of the states of active spacecraft, reference solution, and obstacles without repetitive process and initial guesses. The proposed approach with the revised penalty function was validated by numerical simulations changing positions and formation
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of multiple (as many as twelve) active spacecraft. Regardless of the number of active spacecraft and the number of obstacles simultaneously recognized, all the active spacecraft successfully accomplished collision-free transfers with extremely low computational burden by using the same control law obtained only once. Considering the relatively lower computational burden and reasonable control effort, we expect the proposed approach to be suitable for operating a large number of spacecraft for swarm, formation flying, and constellation.
10 11
Declaration of Competing Interest
12 13
None declared.
14 15
Acknowledgements
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This work was supported by Global Surveillance Research Center (GSRC) program funded by the Defense Acquisition Program Administration (DAPA) and Agency for Defense Development (ADD).
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