Implicit Runge-Kutta Methods for Orbit Propagation

Transcription

1 Implicit Runge-Kutta Methods for Orbit Propagation Jeffrey M. Aristoff and Aubrey B. Poore Numerica Corporation, 4850 Hahns Peak Drive, Suite 200, Loveland, Colorado, 80538, USA Accurate and efficient orbital propagators are critical for space situational awareness because they drive uncertainty propagation which is necessary for tracking, conjunction analysis, and maneuver detection. We have developed an adaptive, implicit Runge-Kuttabased method for orbit propagation that is superior to existing explicit methods, even before the algorithm is potentially parallelized. Specifically, we demonstrate a significant reduction in the computational cost of propagating objects in low-earth orbit, geosynchronous orbit, and highly elliptic orbit. The new propagator is applicable to all regimes of space, and additional features include its ability to estimate and control the truncation error, exploit analytic and semi-analytic methods, and provide accurate ephemeris data via built-in interpolation. Finally, we point out the relationship between collocation-based implicit Runge-Kutta and Modified Chebyshev-Picard Iteration. I. Introduction Accurate and timely surveillance of objects in the near-earth space environment is becoming increasingly important to national security and presents a unique and formidable challenge. Efficiently and accurately modeling trajectories of the vast number of objects in orbit around the Earth is difficult because the equations of motion are non-linear, the infrequency of observations may require modeling trajectories over long periods of time, and the number of objects to be modeled is on the order of 10 5 and growing owing to object breakups and improved sensors. While existing numerical methods for orbit propagation appear to meet the current needs of satellite operators, advances in numerical analysis, together with improved speed, memory, and architecture of modern computers (e.g., the availability of parallel processors), necessitate new and improved algorithms for orbit propagation, especially when future needs in space situational awareness are taken into consideration. In this paper, we demonstrate the construction, implementation, and performance of collocation-based implicit Runge-Kutta (IRK) methods for orbit propagation (i.e., the numerical solution of initial value problems arising in orbital mechanics). This includes collocation-based IRK methods such as Gauss-Legendre, Gauss-Chebyshev, and their Radau and Lobatto variants, 1 4 as well as band-limited collocation-based IRK methods. 5, 6 Particular attention is given to a class of collocation-based IRK methods known as Gauss- Legendre IRK (GL-IRK). Collocation-based IRK methods are ideally suited for orbit propagation for several reasons. First, unlike multistep methods, GL-IRK and other collocation-based IRK methods are A-stable at all orders, thus allowing for larger (and fewer) time steps to be taken without sacrificing stability. 1 4 As a result, less error is accumulated and the computational cost is reduced. Second, and perhaps more importantly, collocation-based IRK methods can exploit parallel computing architectures, which makes them well-poised to meet the future demands of space surveillance. It is worth noting that current state-of-the-art numerical integrators such as Dormand-Prince 8(7), Runge-Kutta-Nystrom 12(10), Gauss-Jackson (GJ), and Adams-Bashforth-Moulton (ABM) are explicit methods that are not A-stable nor parallelizable. 1 4, 7 The properties of these methods are summarized in Table 1. A survey of numerical integration methods used for orbit propagation can be found in Montenbruck & Gill 8 and Jones & Anderson. 9 Our implementation of IRK overcomes the disadvantages often associated with using implicit numerical methods: the need to solve a system of nonlinear equations at time step. We mitigate this problem by making 8, 10 use of available analytic and semi-analytic methods in astrodynamics. Such approximate solutions are Research Scientist, Numerica Corporation. Chief Scientist, Numerica Corporation. 1 of 19

2 DP8(7) RKN12(10) GJ & ABM GL-IRK Type of method Explicit RK Explicit RK Multistep Implicit RK Error control Yes Yes No/Yes Yes Non-conservative forces Yes No Yes Yes A-stable No No No Yes Parallelizable No No No Yes Table 1. Summary of the properties of numerical integration methods used for orbit propagation. While it is possible to implement error control with Gauss-Jackson (GJ), its standard implementation lacks this feature. 7 RKN12(10) is unable to handle cases wherein the force acting on the object depends on its velocity (e.g., when non-conservative forces are present). used to warm-start the iterations arising on each time step of IRK, thereby reducing the computational cost. Second, we have developed an efficient error-control mechanism that allows for the propagation to be entirely adaptive a. Time-adaptivity is achieved through the use of an error estimator which makes use of the propagated solution in order to select the optimal time step for a given tolerance, thereby controlling the accumulation of local error, and minimizing the cost of propagation per time step. This feature, among others, distinguishes our work from that of Bai & Junkins, 11 Bradley et al., 5 and Beylkin & Sandberg, 6 who studied the use of fixed-step collocation-based methods for orbit propagation. By virtue of being adaptive, our implementation of IRK is not restricted to the propagation of nearlycircular orbits. Highly elliptic orbits can also be efficiently propagated. Fixed-step propagators, on the other hand, are forced to take extremely small time steps when propagating highly elliptic orbits in order to maintain high accuracy. Alternatively, the equations of motion can be transformed (e.g., using the Sundman transform) so that fixed steps can be taken in an orbital anomaly, rather than in time, in an effort to distribute the numerical error evenly across the steps. Unfortunately, this approach results in the need to solve an additional ordinary differential equation, 12 and hence both options incur increased computational costs. This paper focuses on the propagation of objects in low-earth orbit (LEO), geosynchronous orbit (GEO), and highly elliptic orbit (HEO). The new propagator is found to be superior to existing propagators when medium to high accuracy is required. Specifically, we propagate objects in LEO, GEO, and HEO for multiple orbital periods. The performance is compared to that of DP8(7) and ABM, the latter being a method that is similar to GJ, albeit ABM has error control, whereas the standard implementation of GJ does not have error control. 7 In a serial computing environment, we demonstrate that the new propagator outperforms DP8(7) and ABM by 60% to 85% in LEO and in GEO, and by 30% to 55% in HEO. In a parallel computing environment, the new propagator outperforms DP8(7) and ABM by 92% to 99% in LEO and in GEO, and by 86% to 97% in HEO. Hence, the real impact of the new propagator is its ability to significantly reduce the computational cost of orbit propagation, even before the algorithm is potentially parallelized. The layout of this paper is the following. In Section II, we introduce implicit Runge-Kutta methods and describe the construction of collocation-based IRK methods. In Section III, we describe the implementation of adaptive IRK methods for efficient orbit propagation. In Section IV, we compare the performance of our adaptive collocation-based IRK propagator to that of DP8(7) and ABM. In Section V, we summarize our results. II. Mathematical Preliminaries The first-order ordinary differential equation (ODE) y (t) = f (t, y(t)), (1) together with the initial condition y(t 0 ) = y 0, (2) a An adaptive numerical method for solving initial value problems estimates the local (truncation) error arising after each time step and compares it to a user-specified tolerance to determine whether or not to accept the step, as well as to adjust the size of the subsequent time step. 2 of 19

3 define an initial value problem (IVP). Many standard problems in dynamics can be converted to this form. 2 The independent variable t (time) is permitted to take on any real value, and the dependent variable y R d (the state of the object) is a vector-valued function y : R R d, where d is the dimension of the state, and f : R R d R d. Henceforth, we assume existence and uniqueness of the solution on some interval [t 0, t 0 + h]. Runge-Kutta methods may be used to solve the IVP given by (1)-(2), that is, to find the state of the object at time t = t 0 + h. 1 4 An s-stage Runge-Kutta method is defined by its weights b = (b 1, b 2,... b s ), nodes c = (c 1, c 2,..., c s ), and the s by s integration matrix A whose elements are a ij. It is used to solve the IVP over a given time step b t 0 to t 0 + h by writing ( ξ i = f t 0 + c i h, y 0 + h ) a ij ξ j j=1 i = 1, 2,..., s, (3) where ξ i are the function evaluations at the intermediate times t 0 + c i h for i = 1, 2,..., s, and where the internal stage z i = y 0 + h s j=1 a ijξ j is the approximation of y at time t 0 + c i h. Once obtained, the internal stages are used to advance the solution from t 0 to t 0 + h according to y 1 = y 0 + h b i ξ i. (4) If a ij = 0 for i < j, then (3) is an explicit expression for the internal stages. Otherwise, as with implicit Runge-Kutta (IRK) methods, iterative techniques must be used to solve the nonlinear system of equations (3), and an initial guess is required. In fact, as we shall demonstrate in Section III, it is the availability of approximate analytical solutions to (1)-(2) that make the iterative solution of (4) tractable in orbital mechanics. Examples of fixed-point methods for solving (3) include the Jacobi method, the Gauss-Seidel method, and the Steffensen method. Newton and quasi-newton methods can also be used. 13 Note that the Runge-Kutta method (A, b, c) can be reformulated so that the s evaluations of f in (4) need not be performed. 4 A. Runge-Kutta Methods for Second-Order Systems Of particular importance in the modeling of the motion of objects subject to forces are equations of the form i=1 Y (t) = g ( t, Y (t), Y (t) ), (5) which can be written as a first-order ODE (1) where ( y(t) = Y (t) Y (t) ) (6) and Let the initial condition (2) be f (t, y(t)) = ( y 0 = Y (t) g ( t, Y (t), Y (t) ) ( Y 0 Y 0 ) ). (7). (8) In this case, the system of equations defining the internal stages of the IRK method (3) becomes ξ i = Y 0 + h s ( j=1 a ijξ j ξ i = g t 0 + c i h, Y 0 + h s j=1 a ijξ j, Y 0 + h s j=1 a ijξ j ) i = 1, 2,..., s, (9) and (4) becomes Y 1 = Y 0 + h b i ξ i, i=1 Y 1 = Y 0 + h b i ξ i. (10) i=1 b Typically, multiple time-steps must be taken to solve an IVP using Runge-Kutta methods. 3 of 19

4 Substituting the first formula of (9) into the second yields ξ i = g (t 0 + c i h, Y 0 + c i hy 0 + h 2 ā ij ξ j, Y 0 + h and substituting the first formula of (9) into (10) yields j=1 ) a ij ξ j, (11) j=1 Y 1 = Y 0 + hy 0 + h 2 bi ξ i, i=1 Y 1 = Y 0 + h b i ξ i. (12) i=1 where ā ij = a ik a kj, bi = k=1 b j a ji, and c i = j=1 a ij. (13) Provided that ā ij, b i, and c i are pre-computed, these substitutions reduce the storage required by the implicit Runge-Kutta methods as well as the number of iterations. 3 Similar substitutions can be done for higher-order ODEs to reformulate the associated IRK method. B. Construction of Collocation-Based Runge-Kutta Methods There is an extensive literature devoted to the construction of IRK methods. 1 4 Here, we describe the construction of a class of IRK methods known as collocation-based IRK methods, which have particularly strong stability properties. This framework allows us to readily construct collocation-based IRK methods such as Gauss-Legendre, Gauss-Chebyshev, and their Radau and Lobatto variants. Without loss of generality c, a collocation-based, s-stage IRK method consists of choosing s collocation points c 1, c 2,..., c s in [ 1, 1], and seeking a vector function u that obeys the initial condition (2) and satisfies the differential equation (1) at the collocation points. In other words, one seeks a function u such that j=1 u(t 0 ) = y 0 u (t 0 + c i h) = f (t 0 + c i h, u(t 0 + c i h)) i = 1, 2,..., s. (14) A collocation method consists of finding such a u and setting y 1 = u(t 0 + h). The collocation method for c 1, c 2,..., c s is equivalent to the s-stage IRK method with coefficients and where a ij = b i = ci L j (τ)dτ, i, j = 1, 2,..., s, (15) L i (τ)dτ, i = 1, 2,..., s, (16) L i (τ) = s l=1, l i τ c l c i c l (17) is the Lagrange interpolating polynomial. Band-limited collocation-based IRK (BLC-IRK) methods, 5 and methods wherein the interpolating function need not be polynomial, may be constructed by choosing a family of orthogonal basis functions and computing ci a ij = Ψ 1 lj Ψ l (τ)dτ, i, j = 1, 2,..., s, (18) 1 and b j = l=1 l=1 c The interval for construction of the IRK method need not be [ 1, 1]. 1 Ψ 1 lj Ψ l (τ)dτ, 1 j = 1, 2,..., s, (19) 4 of 19

5 where Ψ 1 ij = [Ψ j (c i )] 1, i, j = 1, 2,..., s (20) are the components of the interpolation matrix Ψ 1. Collocation-based IRK methods are amenable to interpolation (and extrapolation). The solution of the IVP at the collocation points, together with the interpolation matrix Ψ 1, the elements of which are given by (20), can also be used to determine the state of the object at points that do not necessarily correspond to the collocation points. 5 Since Ψ 1 can be pre-computed, interpolation can be done efficiently. C. Other Collocation Methods (e.g., Modified Chebyshev-Picard Iteration) An alternative, but mathematically equivalent (in the sense that it produces the same discrete approximation) 14, 15 approach to collocation-based IRK is the following. Rather than working in terms of the values of the function f, one works in terms of the coefficients of a chosen family of basis functions R j for j = 1, 2,..., s (e.g., Chebyshev or Legendre polynomials) by expressing the approximate solution u(t) as u(t) = a j R j (t). (21) j=1 The vector coefficients a j are obtained by satisfying the ODE at s chosen collocation points and by satisfying the initial condition; the resulting system of nonlinear equations relating the coefficients a j and the function f is solved using iterative techniques. This approach is sometimes referred to as a collocation method, and has 11, been used in astrodynamics by several authors. We emphasize that it is mathematically equivalent 14, 15 to an s-stage collocation-based IRK method having the same collocation points and basis functions, since this fact does not appear to be well known. Since collocation methods are a subset of IRK methods, we prefer to describe the new propagator in the broader context of IRK. This choice also allows us to easily make use of a number of results from numerical analysis, most notably, the theory of order conditions. D. Order Conditions Let ŷ be the exact solution to the initial value problem (1) and (2) at t = t 0 +h, and let y 1 be an approximate numerical solution at t = t 0 + h. The approximate numerical solution is said to have order p if the local error satisfies y 1 ŷ = O(h p+1 ) (22) as h 0. For an s-stage Runge-Kutta method to obtain order p, it must satisfy the order conditions B(p), C(η), and D(ζ) with p η + ζ + 1 and p 2η + 2, where B(p) : C(η) : D(ζ) : i=1 j=1 i=1 b i c q 1 i a ij c q 1 j = 1 q = cq i q q = 1,..., p; (23) i = 1,..., s, q = 1,..., η; (24) b i c q 1 i a ij = b j q (1 cq j ) j = 1,..., s q = 1,..., ζ. (25) Note that for p = 1, B(p) reduces to b i = 1, (26) i=1 which is the condition required for any Runge-Kutta method to be consistent and convergent, 1, 2 a prerequisite for its use. Gauss-Legendre IRK methods achieve the highest possible order (p = 2s) and are deemed super-convergent. Their Radau IA and IIA variants achieve order p = 2s 1, and their Lobatto IIIA, IIIB, and IIIC variants achieve order p = 2s 2. 2, 3 Gauss-Chebyshev IRK methods are order p = s. Their Radau IA and IIA variants achieve order p = s 1, and their Lobatto IIIA, IIIB, and IIIC variants achieve order p = s 2. 2, 3 The order of a BLC-IRK method is between zero and 2s depending upon the choice of band-limit. 5 of 19

6 III. Implementation of Implicit Runge-Kutta for Orbit Propagation One of the main difficulties with the implementation of implicit Runge-Kutta (IRK) methods is the need to solve a nonlinear system of equations at each time step. These must be solved iteratively and in an efficient manner (see Subsection A), and a stopping criterion for the iterations must be established (see Subsection B). Moreover, for an IRK method to be practical, approaches to limit the number of iterations should be used that exploit approximate solutions to the initial value problem (see Subsection C). Another difficulty with the implementation of IRK arises from the requirement that the accumulation of error is monitored and controlled so that the accuracy of the numerical solution can be established (see Subsection D). The present implementation of IRK for orbit propagation addresses and circumvents each of these difficulties. A. Solution of the Nonlinear System via Fixed-Point Iteration Newton s method or one of its many variants can be used for solving the nonlinear system of equations, but this requires calculation (or approximation) of the Jacobian. In orbit propagation, evaluation of a single component of the Jacobian is more expensive than evaluation of a single component of the force, and there are more components to evaluate. Hence, this approach can be costly. Simplified Newton methods avoid having to calculate the Jacobian at each of the intermediate times, but reduce the convergence from quadratic to linear. 4 Alternatively, fixed-point methods can be used for solving the nonlinear system. Although fixed-point methods converge more slowly than Newton s method (but at the same rate as simplified Newton methods), fixed-point methods do not require the computation of a Jacobian, and are thus well-suited for orbit propagation. The downside of using fixed-point methods is that they generally degrade the superior stability properties of IRK methods. In particular, provided that the initial guess is sufficiently close to the solution, there is no limit on the size of the time step that can be taken when a method is A-stable and Newton s method is used. If a fixed-point, rather than Newton s method, is used, then there is a maximum step size above which the iterations diverge. While this may appear to limit the applicability of fixed-point methods to orbit propagation, we have studied the convergence of fixed-point methods in this context, and determined a bound on the maximum step size, below which the iterations will converge. For the case of unperturbed Keplerian dynamics, our proof of convergence indicates that steps sizes up to three-quarters of an orbital period are acceptable. Practically speaking, the propagator we develop monitors the convergence of the iterations during each step of the propagation, and reduces the step size if necessary. There are a number of different fixed-point iteration methods that may be used to solve the nonlinear system (11), the simplest being the Jacobi method (see Algorithm 1). Note that the s evaluations of the force model and the s internal stage updates can each be done in parallel. An efficient and robust convergence criteria for stopping the iterations will be presented in Subsection B. Algorithm 1: Jacobi Method Jacobi method for solving the nonlinear system (11) arising in orbit propagation. inputs : Initial guess for the reformulated internal stages z i and z i, second-order integration matrices and nodes (a ij, ā ij, c i, c i ) for i, j = 1, 2,..., s, time step h, current state (y 0, y 0 ), and iteration tolerance. outputs: Solution for the reformulated internal stages and corresponding force-model evaluations. begin while stopping criterion for the iterations has not been met (see Subsection B) do for i 1 to s do Evaluate the force model at node i: g(t 0 + c i h, y 0 + z i, y 0 + z i ) g i. for i 1 to s do Update the i th internal stage: c i hy 0 + h2 s j=1 āijg j z i and h s j=1 a ijg j z i. An alternative to the Jacobi fixed-point method, known as the Gauss-Seidel method (see Algorithm 2), combines the two for-loops of the Jacobi method into a single step. In practice, the Gauss-Seidel method converges more quickly because the newly calculated function evaluations are immediately incorporated into the internal stage updates. However, neither the s evaluations of the force model nor the s internal stage updates can be done in parallel, so the computational advantages are only obtained in a serial computing environment or in a parallel computing environment in which the number of processors does not exceed the number of orbits to be propagated multiplied by the number of internal stages. In the latter situation, 6 of 19

7 parallelization at the level of the orbits can still take place, but finer parallelization, at a the level of the internal stages, cannot. Algorithm 2: Gauss-Seidel Method Gauss-Seidel method for solving the nonlinear system (11) arising in orbit propagation. inputs : Initial guess for the reformulated internal stages z i and z i and corresponding force-model evaluations, second-order integration matrices and nodes (a ij, ā ij, c i, c i ) for i, j = 1, 2,..., s, time step h, current state (y 0, y 0 ), and iteration tolerance. outputs: Solution for the reformulated internal stages and corresponding force-model evaluations. begin while stopping criterion for the iterations has not been met (see Subsection B) do for i 1 to s do Update the i th internal stage: c i hy 0 + h2 s j=1 āijg j z i and h s j=1 a ijg j z i. Evaluate the force model at node i: g(t 0 + c i h, y 0 + z i, y 0 + z i ) g i. B. Stopping Criterion for the Iterations Rather than performing a fixed number of iterations, we seek a stopping criterion that minimizes the number of iterations necessary for convergence to a given accuracy. Following Hairer & Wanner, 4 denote the set of internal stages after m iterations as V m = ( z 1 z 2... z s z 1 z 2... z s ), (27) and let V m = V m+1 V m denote the iteration error. Since convergence is linear, we have V m+1 Θ V m, (28) where is a suitable matrix norm that is consistent with the way in which the local error is estimated (see Subsection D). Additionally, since we are dealing with a second order system, one must decide whether to monitor the error Z or Z (or some combination d ). Provided that the iterations are converging (see Subsection A), the convergence rate Θ < 1. Let V be the exact solution to (11). Applying the triangle inequality to yields V m+1 V = (V m+1 V m+2 ) + (V m+2 V m+3 ) +... (29) V m+1 V An estimate of the convergence rate Θ can be made using Θ 1 Θ Vm. (30) Θ = V m / V m 1, (31) or if V m 1 is unavailable (as is true after only one iteration), Θ from the previous integration step can be used, provided that the previous and current time steps are comparable. We therefore stop the iterations when Θ 1 Θ Vm κ tol, (32) and accept V m+1 as an approximation to V, where tol is the local error tolerance and κ is a numerical pre-factor. If Θ 1 or the number of iterations reaches the maximum allowable number of iterations, we halt the iterations and reduce the time step by a factor of two. d Note that the physical dimensions of these matrices differ. 7 of 19

8 C. Solution of the Nonlinear System Using Multiple Force Models Physical systems are typically described by a hierarchy of mathematical models of increasing complexity. In orbital mechanics, for example, the complexity of the force model depends upon the degree and order of the gravity model, and how drag, solar radiation, and third-body effects and other perturbations are accounted for. It is therefore reasonable to ask whether low- and medium-fidelity force models can be used in conjunction with high-fidelity force models to efficiently solve differential equations of the form (11). The answer depends upon the underlying numerical methods. Implicit numerical methods can exploit multiple force models, whereas explicit methods cannot. In particular, the iterative technique used for solving the nonlinear system of equations that arise on each step of an implicit Runge-Kutta (11), whether it be fixed-point or Newton-step, requires a guess for the internal stages to start the iterations. Rather than using a single, high-fidelity force model, we use a tiered approach wherein multiple force models are evaluated sequentially (see Algorithm 3). This approach significantly reduces the number of high-fidelity force-model evaluations, and is similar to the approach used by Beylkin & Sandberg 6 and Bradley et. al. 5 Algorithm 3: Sequentially Evaluate Force Models Sequentially evaluate force models in succession together with IRK to solve the nonlinear system (11) arising in orbit propagation. inputs : Initial guess for the reformulated internal stages z i and z i and corresponding force-model evaluations, second-order integration matrices and nodes (a ij, ā ij, c i, c i ) for i, j = 1, 2,..., s, time step h, current state (y 0, y 0 ), and iteration tolerance. outputs: Solution for the reformulated internal stages and corresponding force-model evaluations. begin Use Algorithm 1 or 2 with g = g (low) where g (low) is a low-fidelity force model, and pass the result as input to the following step. Use Algorithm 1 or 2 with g = g (med) where g (med) is a medium-fidelity force model, and pass the result as input to the following step. Use Algorithm 1 or 2 with g = g (high) where g (high) is a high-fidelity force model. D. Error Estimation and Control Numerical error is intrinsic to orbit propagation (unless the propagation is done analytically). It arises due to the finite precision of computations involving floating-point values (i.e., round-off error), and due to the difference between the exact mathematical solution and the approximation to the exact mathematical solution (i.e., truncation error). Owing to round-off error, the final few digits of a floating-point computation cannot be considered to be accurate. Round-off error can be mitigated to some extent, 4 but it is truncation error that is typically the dominant contributor to numerical error in orbit propagation. Recall that the local (truncation) error is related to the order of the numerical method (see Section II). Fortunately, the local error can be estimated and controlled. Note that the orbital propagators presented by Bradley et al. 5 and Bai & Junkins 11 do not estimate nor control the error. Modern methods for solving initial value problems (IVPs) estimate and control the error by adjusting the step size accordingly. 19 For higher order methods such as those needed for orbit propagation, 8, 9 rigorous error bounds become unpractical, and it is therefore necessary to consider the first non-zero term in the Taylor expansion of the error. Even so, one cannot simply compute the global error (without access to the true solution), that is, the error of the computed solution after several steps. Rather, the local error committed at each step of the method can be estimated and used to adjust the step size, keeping in mind that local errors are accumulated and transported to the final state. Ideally, local error control results in global error control, and the method takes the largest time steps possible without committing too much error. Hence, the method does just the right amount of work solving the IVP. Let tol = atol + rtol y 0 be the local error tolerance, where atol is a user-specified absolute tolerance, and rtol is a user-specified relative tolerance. On each step of the propagation, we estimate the local error and require that it fall below tol. This approach is known as error per step (EPS) control e. The local error e Conversely, error per unit step (EPUS) control requires that the local error fall below tol h for the step to be accepted. 8 of 19

9 estimate err is used to adjust the subsequent step according to h next = h max(0.5, min(2, (0.35 tol/err) 1/(p+1) )), (33) where p is the order of the IRK method (see Section II). The parameters 0.5 and 2 are chosen so that the ratio h next /h is neither too large nor too small. The parameter 0.35 is chosen to increase the probability that the next step is accepted, with the probability depending on the order of the IRK method. We note that at the culmination of each step, we ensure that the step size to be taken for the following step does not exceed the final time. If so, the step size is reduced. Our approach to error control is summarized in Algorithm 4. Algorithm 4: Error Estimation and Control Estimate and control the local error. inputs : Initial guess for the reformulated internal stages z i and z i and corresponding force-model evaluations, second-order integration matrices and nodes (a ij, ā ij, c i, c i ) for i, j = 1, 2,..., s, time step h, current state (y 0, y 0 ), iteration tolerance, and local error tolerance. outputs: Solution for the reformulated internal stages and corresponding force-model evaluations accurate to within the local error tolerance. begin while step has not been accepted do Use Algorithm 3 to compute the s-stage IRK step and the error estimate err. if err tol then Accept the step. Use (33) to increase or decrease the time step. else Reject the step. Use (33) to decrease the time step. We choose to estimate and control the error in propagating an orbit by using Earth-centered-inertial coordinates (ECI). The reason for this choice is two-fold. First, the components of the internal stages naturally separate into positions and velocities in ECI. Then, one simply decides whether to control the error in position, the error in velocity, or both. Second is the fact that the geopotential calculation and its derivatives (the component of the force model evaluation that is most expensive) are naturally expressed in Earth-centered-Earth-fixed coordinates, which convert to ECI through a trivial rotation. 8 Use of an alternative coordinate system for propagation would incur an additional cost due to the necessary change of variable at each node. Furthermore, one would not be able to exploit the second-order formulation of implicit Runge-Kutta to reduce the computational cost of integration. IV. Performance of the New Orbital Propagator In this section, we test the performance of the new IRK-based propagator in six realistic orbit propagation scenarios. Both the efficiency and the accuracy of the propagations will be quantified. Although any IRK scheme can be used within the new propagator (including those not based on collocation) we have chosen to use the Gauss-Legendre IRK (GL-IRK) family of methods in the present study because: GL-IRK methods are A-stable (and B-stable) and can therefore be applied to stiff problems (e.g., satellite re-entry, highly elliptic orbits), GL-IRK methods are parallelizable and can therefore exploit advanced computing architectures (each evaluation of the force model can be done independently and on a different processor), GL-IRK methods are super-convergent, they achieve the highest order of convergence possible for a Runge-Kutta method (methods based on Chebyshev collocation are not super-convergent), GL-IRK methods are symmetric and thus preserve time-reversibility of a dynamical system (if applicable), and GL-IRK methods are symplectic and thus preserve the first integral of a Hamiltonian system (if applicable). 9 of 19

10 BLC-IRK methods share many of the same properties as GL-IRK (and reduce to GL-IRK in a suitable limit). Our computational experience is that BLC-IRK methods can be up to 25% more efficient if large (fixed) time steps are taken and the optimal step-size is known a-priori so that the error need not be estimated. However, the challenge with using BLC-IRK methods is to develop an adaptive step-size control that is comparable in performance to GL-IRK methods. This remains an open research problem. We compare the performance of the new propagator to that of Dormand-Prince 8(7) (DP8(7)) and Adams-Bashforth-Moulton (ABM). DP8(7) is an explicit Runge-Kutta method, and ABM is a multistep method. Both methods are adaptive-step, high-order methods, and, like the new propagator, have the ability to estimate and control the local error. Both methods were designed with the goal of reducing the number of function calls, and, like the new propagator, both methods adaptively select the initial time step f. Hence, DP8(7) and ABM are good candidates to which we can compare the performance of IRK. Moreover, ABM is similar to Gauss-Jackson, what is used by Space Command, and we have a robust implementation of ABM (courtesy of MATLAB). An implementation of DP8(7) is available on the MATLAB file exchange. Review of the code reveals it to be a robust implementation as well, though not to the high standard achieved by ABM. In order to make a fair comparison between the performance of the three methods, DP8(7) and ABM are modified so that the same criterion for step acceptance is used as with the new propagator: on each step of a given method, we require that the relative error in the 2-norm of the position of the object fall below a user-specified relative tolerance g. Hence, we are using error per step (EPS) control. To quantify the computational cost of orbit propagation, we count the number of evaluations of the high-fidelity force model required to achieve a given global accuracy. We compute truth by propagating the initial orbital state using a high-accuracy 50-stage GL-IRK method. Recall that evaluation of the high-fidelity force-model is the dominant cost of orbit propagation, and that explicit methods can only use a single (high-fidelity) force model. The initial conditions of the six orbit propagation scenarios are listed in Table 2, along with the duration of the propagations. A degree and order 32 gravity model (EGM2008) is used in LEO and in HEO, and a degree and order 8 gravity model is used in GEO. For the sake of presentation, we do not consider drag or solar radiation in the first five scenarios. In the sixth scenario (satellite re-entry), a Harris-Priester drag model is used to demonstrate that the new propagator can handle cases wherein there is significant orbital decay. Unperturbed Keplerian dynamics is used for the low-fidelity force model, and J 2 gravity is used for the medium-fidelity force model. Since the computational cost of evaluating the gravitational force using a degree and order 32 gravity model is roughly 500 times greater than that using J 2 -gravity h, we can safely neglect the cost incurred during evaluation of the low- and medium-fidelity force models in LEO and in HEO. J 2 -gravity is roughly 32 times faster to evaluate than a degree and order 8 gravity model. Hence, we can safely neglect the cost incurred during evaluation of the low- and medium-fidelity force models in GEO. Scenario # Orbit Type a (km) e i ( ) Ω ( ) ω ( ) M ( ) orbital periods 1 LEO LEO GEO GEO HEO Re-Entry Table 2. Initial orbital elements for the orbit propagation scenarios. The first five are taken from Vinti, 10 the sixth from Peat. 20 For each scenario, we consider the relationship between the user-specified local accuracy, the global accuracy, and the computational cost. The relationship between the local and global error is shown first, followed by the relationship between the computational cost and the global accuracy. A quadratic-leastsquares-fit to each of the data sets is shown. As the local accuracy is progressively increased (i.e. the local f The new propagator also adaptively determines the number of IRK stages to use for a given propagation. g Henceforth, error refers to the relative error in the 2-norm of the position of the object. h The cost of evaluating a degree and order N gravity model using spherical harmonics is O(N 2 ), and J 2 -gravity is approximately twice as fast as a degree and order 2 gravity model. 10 of 19

11 error tolerance is decreased), the global accuracy increases (i.e. the global error decreases), as does the computational cost (i.e. the number of function calls). Effective error control is evidenced by the quadraticleast-squares-fit through the global vs. local accuracy data being nearly linear and the correlation being positive. Note that the global accuracy is always less than the local accuracy, in some cases by an order of magnitude or more. For a short and a long propagation to achieve the same global accuracy, the long propagation requires a greater local accuracy than does the short propagation. This fact is inherent to numerical propagation. The results shown pertain to a serial computing environment. Since 5-15 IRK stages are used in the scenarios, an additional speedup of 5-15 times would be observed in a parallel computing environment i. For reference, 7 digits of accuracy correspond to approximately one-meter accuracy for the object in LEO, and 7.5 digits of accuracy correspond to approximately one-meter accuracy for the object in GEO. For the object in HEO, 7 digits of accuracy correspond to approximately one-meter accuracy at perigee. Scenario 1. An object in low-altitude LEO is propagated for 3 orbital periods, or approximately 4.5 hours, using adaptive step-size control. The performance of the new IRK-based propagator is summarized in Figure 1, and compared to that of DP8(7) and ABM. All three methods effectively control the error, and IRK is 70-80% more efficient than DP8(7) and ABM over the range of accuracies considered. Scenario 2. An object in low-altitude LEO is propagated for 30 orbital periods, or approximately 45 hours, using adaptive step-size control. The performance of the new IRK-based propagator is summarized in Figure 2, and compared to that of DP8(7) and ABM. All three methods effectively control the error, and IRK is 60-70% more efficient than DP8(7) and ABM over the range of accuracies considered. Scenario 3. A object in GEO is propagated for 3 orbital periods, or approximately 3 days, using adaptive step-size control. The performance of the new IRK-based propagator is summarized in Figure 3, and compared to that of DP8(7) and ABM. All three methods effectively control the error, though the IRK solution is ten times more accurate than that of DP8(7) and ABM for the same local accuracy. Moreover, IRK is 80-90% more efficient than DP8(7) and 60-65% more efficient than ABM over the range of accuracies considered. Scenario 4. An object in GEO is propagated for 30 orbital periods, or approximately 30 days, using adaptive step-size control. The performance of the new IRK-based propagator is summarized in Figure 4, and compared to that of DP8(7) and ABM. All three methods effectively control the error. Again, IRK is much more accurate (globally) for a given local accuracy, and 90-95% more efficient than DP8(7) and 75-85% more efficient than ABM over the range of accuracies considered. Scenario 5. An object in HEO is propagated for 3 orbital periods, or approximately 36 hours, using adaptive step-size control. The performance of the new IRK-based propagator is summarized in Figure 5, and compared to that of DP8(7) and ABM. All three methods adequately control the error, and IRK is 50-70% more efficient than DP8(7) and 30-55% more efficient than ABM over the range of accuracies considered. Scenario 6. A satellite that re-entered the atmosphere, known as ROSAT, 20 is propagated for 3 orbital periods, or approximately 4.5 hours, using adaptive step-size control. The performance of the new IRK-based propagator is summarized in Figure 6, and compared to that of DP8(7) and ABM. Using the published weight of ROSAT (2400 kg) we estimate the the area-to-mass ratio to be m 2 kg 1. This results in a nearly 1 km decay in the altitude of the spacecraft per orbital period. All three methods adequately control the error, and IRK is 65-75% more efficient than DP8(7) and ABM over the range of accuracies considered. V. Conclusion The use of implicit numerical methods for orbit propagation represents a paradigm shift in astrodynamics. Standard algorithms are based on explicit and multistep numerical methods. 7, 8 As such, they solve an initial i This estimate assumes that the communication time between the processors is negligible. 11 of 19

12 value problem by calculating the state of a system at a later time from the state of the system at the current time. Implicit numerical methods, on the other hand, solve an initial value problem by calculating the state of a system at a later time from the state of the system at the current time, together with the state of the system at future times. Hence, an initial approximation for the solution is required, and the resulting system of nonlinear equations must be solved iteratively. What makes implicit Runge-Kutta (IRK) methods practical for orbit propagation is the availability of analytic and semi-analytic approximations to the solution that can be computed efficiently and used to warm-start the iterations, thereby lowering the computational cost of orbit propagation. IRK methods can also be super-convergent, meaning that larger (and fewer) time steps can be taken than their explicit counterparts. What is more, IRK methods are parallelizable. Explicit Runge-Kutta and multistep methods are not. Even before parallelization, the new adaptive-step IRK-based orbital propagator is found to be significantly more efficient in our test scenarios than adaptive-step explicit and multistep methods often used for orbital propagation, specifically, Dormand-Prince 8(7) and Adams-Bashforth-Moulton. Table 3 lists the computational savings obtained in LEO (Scenarios 1 2), GEO (Scenarios 3 4), HEO (Scenario 5), and Satellite Re-Entry (Scenario 6) when medium- to high-accuracy propagations are performed. Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 Scenario 6 LEO LEO GEO GEO HEO Re-Entry SCE 70-80% 60-70% 60-65% 75-85% 30-55% 65-75% PCE 94-99% 92-98% 92-98% 95-99% 86-97% 93-98% Table 3. Summary of computational savings in a serial computing environment (SCE) or a parallel computing environment (PCE). The computational savings tend to increase as the accuracy of the propagation increases. Not all orbital propagators estimate and control the truncation error by adapting the step size. Instead, fixed steps (in time or in an orbital anomaly) are taken. 5, 7, 8, 11 By taking fixed steps, one is assuming that (1) the nonlinearity in the underlying dynamics is approximately uniform over each step and (2) the step size yields a truncation error that is less than the error in the force models (but not too much less or else unnecessary work is done to propagate the orbit). Unfortunately, one or both of these assumptions can be violated. The first assumption breaks down when propagating low-altitude orbits, highly-elliptic orbits, or orbits over long enough time intervals. The second assumption requires a-priori knowledge of the truncation error, which would need to be tabulated offline for a given set of force models and a large number of orbit propagation scenarios. Our approach is to use an adaptive-step orbital propagator that uses online error estimation and control wherein the acceptable level of truncation error can be tuned to the error intrinsic to the force models. Therefore, the error estimates are more accurate, and the propagator does just the right amount of work for a given propagation. A large class of methods for propagating an orbital state and its uncertainty (e.g. the unscented Kalman filter, 21, 22 particle filters, 23, 24 Gaussian sum filters ) require the propagation of an ensemble of particles or states through the nonlinear dynamics. Hence, orbit propagation is a prerequisite for uncertainty propagation using these methods. In this paper, we demonstrated the use of implicit Runge-Kutta methods for accurate and efficient orbit propagation. In a companion paper, 29 we demonstrate the use of implicit Runge-Kutta methods for accurate and efficient uncertainty propagation. Acknowledgments The authors thank G. Beylkin and J. T. Horwood for helpful comments on earlier versions of this paper. This work was funded, in part, by a Phase II STTR from the Air Force Office of Scientific Research (FA C-0034) and a grant from the Air Force Office of Scientific Research (FA ). References 1 Iserles, A., A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, Cambridge, UK, Butcher, J. C., Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, West Sussex, England, of 19

13 3 Hairer, E., Norsett, S. P., and Wanner, G., Solving Ordinary Differential Equations I: Nonstiff Problems, Springer Series in Computational Mathematics, Springer, 2nd ed., Hairer, E. and Wanner, G., Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics, Springer, 2nd ed., Bradley, B. K., Jones, B. A., Beylkin, G., and Axelrad, P., A new numerical integration technique in astrodynamics, Proceedings of the 22nd Annual AAS/AIAA Spaceflight Mechanics Meeting, AAS , Charleston, SC, Jan Feb , pp Beylkin, G. and Sandberg, K., ODE Solvers Using Bandlimited Approximations, arxiv: v1 [math.na], Berry, M. M. and Healy, L. M., Implementation of Gauss-Jackson integration for orbit propagation, Journal of the Astronautical Sciences, Vol. 52, No. 3, 2004, pp Montenbruck, O. and Gill, E., Satellite Orbits: Models, Methods, and Applications, Springer, Berlin, Jones, B. A. and Anderson, R. L., A survey of symplectic and collocation integration methods for orbit propagation, Proceedings of the 22nd Annual AAS/AIAA Spaceflight Mechanics Meeting, AAS , Charleston, SC, Jan Feb , pp Vinti, J. P., Orbital and Celestial Mechanics, Progress in Astronautics and Aeronautics, edited by G. J. Der and N. L. Bonavito, Vol. 177,, Cambridge, MA, Bai, X. and Junkins, J., Modified Chebyshev-Picard iteration methods for orbit propagation, J. Astronautical Sci., Vol. 3, 2011, pp Berry, M. and Healy, L., The generalized Sundman transformation for propagation of high-eccentricity elliptical orbits, Proceedings of the 12th AAS/AIAA Space Flight Mechanics Meeting, San Antonio, TX, January 2002, pp. 1 20, Paper AAS Kelley, C. T., Iterative methods for linear and nonlinear equations, Frontiers in Applied Mathematics, Vol. 16, SIAM, 1995, pp Wright, K., Some relationships between implicit Runge-Kutta, collocation and Lanczos methods and their stability properties, BIT, Vol. 10, 1970, pp Hulme, B. L., One-step piecewise polynomial Galerkin methods for initial value problems, Mathematics of Computation, Vol. 26, No. 118, 1972, pp Feagin, T. and Nacozy, P., Matrix formulation of the Picard method for parallel computation, Celestial Mechanics and Dynamical Astronomy, Vol. 29, 1983, pp Fukushima, T., Vector integration of the dynamical motions by the Picard-Chebyshev method, The Astronomical Journal, Vol. 113, 1997, pp Barrio, R., Palacios, M., and Elipe, A., Chebyshev collocation methods for fast orbit determination, Applied Mathematics and Computation, Vol. 99, 1999, pp Shampine, L. F., Error estimation and control for ODEs, J. Sci. Comp., Vol. 25, 2005, pp Peat, C., Heavens Above, July Julier, S. J. and Uhlmann, J. K., Method and Apparatus for Fusing Signals with Partially Known Independent Error Components, U.S. Patent Number 6,829,568 B2, Issued on 7 December Julier, S. J. and Uhlmann, J. K., Unscented filtering and nonlinear estimation, Proceedings of the IEEE, Vol. 92, 2004, pp A. Doucet, N.F. Freitas, N. G. and Smith, A., Sequential Monte Carlo Methods in Practice, Statistics for Engineering and Information Sciences, Springer, Ristic, B., Arulampalam, S., and Gordon, N., Beyond the Kalman Filter: Particle Filters for Tracking Applications, Artech House, Boston, Ito, K. and Xiong, K., Gaussian filters for nonlinear filtering problems, IEEE Transactions on Automatic Control, Vol. 45, No. 5, 2000, pp Horwood, J. T. and Poore, A. B., Adaptive Gaussian sum filters for space surveillance, IEEE Transactions on Automatic Control, Vol. 56, No. 8, 2011, pp Horwood, J. T., Aragon, N. D., and Poore, A. B., Gaussian sum filters for space surveillance: theory and simulations, Journal of Guidance, Control, and Dynamics, Vol. 34, No. 6, 2011, pp DeMars, K. J., Jah, M. K., Cheng, Y., and Bishop, R. H., Methods for splitting Gaussian distributions and applications within the AEGIS filter, Proceedings of the 22nd AAS/AIAA Space Flight Mechanics Meeting, Charleston, SC, February 2012, Paper AAS Aristoff, J. M., Horwood, J. T., and Poore, A. B., Implicit Runge-Kutta methods for uncertainty propagation, Proceedings of the 2012 AMOS Conference, September of 19

14 Global Accuracy (in digits) Local Accuracy (in digits) Function Calls Global Accuracy (in digits) Figure 1. Performance of Gauss-Legendre IRK vs. DP8(7) and ABM in Scenario 1 (LEO, 3 orbital periods). (a) Digits of accuracy in the propagated state versus the local accuracy. (b) Computational cost of orbit propagation, measured by the number of high-fidelity force-model evaluations, versus the global accuracy in a serial computing environment (an additional speedup of 5-15 times would be observed for IRK in a parallel computing environment). Note that 7 digits of accuracy corresponds to approximately one-meter accuracy. 14 of 19

15 Global Accuracy (in digits) Local Accuracy (in digits) x Function Calls Global Accuracy (in digits) Figure 2. Performance of Gauss-Legendre IRK vs. DP8(7) and ABM in Scenario 2 (LEO, 30 orbital periods). (a) Digits of accuracy in the propagated state versus the local accuracy. (b) Computational cost of orbit propagation, measured by the number of high-fidelity force-model evaluations, versus the global accuracy in a serial computing environment (an additional speedup of 5-15 times would be observed for IRK in a parallel computing environment). Note that 7 digits of accuracy corresponds to approximately one-meter accuracy. 15 of 19

16 Global Accuracy (in digits) Local Accuracy (in digits) Function Calls Global Accuracy (in digits) Figure 3. Performance of Gauss-Legendre IRK vs. DP8(7) and ABM in Scenario 3 (GEO, 3 orbital periods). (a) Digits of accuracy in the propagated state versus the local accuracy. (b) Computational cost of orbit propagation, measured by the number of high-fidelity force-model evaluations, versus the global accuracy in a serial computing environment (an additional speedup of 5-15 times would be observed for IRK in a parallel computing environment). Note that 7.5 digits of accuracy corresponds to approximately one-meter accuracy. 16 of 19

17 Global Accuracy (in digits) Local Accuracy (in digits) 2.5 x Function Calls Global Accuracy (in digits) Figure 4. Performance of Gauss-Legendre IRK vs. DP8(7) and ABM in Scenario 4 (GEO, 30 orbital periods). (a) Digits of accuracy in the propagated state versus the local accuracy. (b) Computational cost of orbit propagation, measured by the number of high-fidelity force-model evaluations, versus the global accuracy in a serial computing environment (an additional speedup of 5-15 times would be observed for IRK in a parallel computing environment). Note that 7.5 digits of accuracy corresponds to approximately one-meter accuracy. 17 of 19

18 Global Accuracy (in digits) Local Accuracy (in digits) Function Calls Global Accuracy (in digits) Figure 5. Performance of Gauss-Legendre IRK vs. DP8(7) and ABM in Scenario 5 (HEO, 3 orbital periods). (a) Digits of accuracy in the propagated state versus the local accuracy. (b) Computational cost of orbit propagation, measured by the number of high-fidelity force-model evaluations, versus the global accuracy in a serial computing environment (an additional speedup of 5-15 times would be observed for IRK in a parallel computing environment). Note that 7.5 digits of accuracy corresponds to approximately one-meter accuracy at perigee. 18 of 19

19 Global Accuracy (in digits) Local Accuracy (in digits) Function Calls Global Accuracy (in digits) Figure 6. Performance of Gauss-Legendre IRK vs. DP8(7) and ABM in Scenario 6 (Satellite Re-Entry, 3 orbital periods). (a) Digits of accuracy in the propagated state versus the local accuracy. (b) Computational cost of orbit propagation, measured by the number of high-fidelity force-model evaluations, versus the global accuracy in a serial computing environment (an additional speedup of 5-15 times would be observed for IRK in a parallel computing environment). Note that 7 digits of accuracy corresponds to approximately one-meter accuracy. 19 of 19