Orbit Propagation Using Gauss-Legendre Collocation

Transcription

1 Orbit Propagation Using Gauss-Legendre Collocation Brandon A. Jones University of Colorado, Boulder, CO, 839, USA This paper discusses the application of Gauss-Legendre collocation for orbit propagation. Previous presentations of collocation propagators for astrodynamics used a fixed-step implementation, thereby limiting their effectiveness for eccentric orbits. The presented formulation uses a variable-step implementation, thereby improving its use for scenarios where the necessary time step varies over time. Additionally, low- and high-fidelity force models are combined to reduce the overall computation time for the integration. The method was compared to other commonly employed integration tools for both two-body and higher-fidelity propagation, and matched or reduced the number of high-fidelity force function calls required for accurate propagation. These tests only considered a serial implementation of the Gauss-Legendre propagator, but parallel methods and their implications for computationally efficient orbit propagation are also discussed. Nomenclature α β ǫ θ j ξ j τ A a i,j b b i c c i d i e F f() h k i l j () N M m p s t u w j y Empirically determined scale parameter Step size predictor safety factor Matrix of changes in initial state per unit time for each node Integration tolerance Integrator interpolation weights Gaussian quadrature node Scaled time in integration interval Integration matrix Coefficients in the integration matrix s-vector of integration weights Integration weights s-vector of integration nodes Integration nodes Intermediate values in initial step size calculation Integration error Matrix of forces at each integrator stage Force model Integrator step size (s) Force-model output for integrator state i Lagrange interpolating polynomial of order j Number of Gaussian quadrature nodes Maximum number of iterations in nonlinear solver Number of iterations required for the nonlinear solver Dimension of the integrated state vector Number of integration nodes (or stages) Time (s) Generic polynomial Gaussian quadrature weight p-vector for the integrated state Research Associate, Colorado Center for Astrodynamics Research, UCB 431, AIAA Member. 1 of 16

2 I. Introduction This paper examines the use of Gauss-Legendre (GL) collocation for satellite orbit propagation. Such methods have several stability and computation qualities of benefit to satellite orbit propagation, some of which are demonstrated in this research. The presented method includes a variable-step formulation to improve its efficiency for orbits with non-zero eccentricities, and compares its performance to several of the more common orbit propagation methods currently employed by the astrodynamics community. Although not commonly used, applications of collocation methods in astrodynamics have increased in recent years. Barrio, et al. 1 compares the use of Chebyshev collocation methods and an eighth-order embedded Runge-Kutta (RK) technique for orbit propagation. Chapter 7 of Beutler 2 describes the application of collocation methods to orbit propagation, and defines a general method based on the location of the zeroes of the underlying polynomial. This discussion also includes tests using polynomials defined by equally-spaced nodes and a brief comparison between Chebyshev and Legendre collocation methods in a small number of toy problems. Those results demonstrate that Legendre collocation outperforms Chebyshev collocation by about an order of magnitude in accuracy. However, these comparisons use polynomials of equal degree without compensating for the order of the resulting integration method. More recent research considers a Chebyshev-polynomial-based method that is very similar to collocation, and demonstrates a parallel implementation to further reduce overall CPU time. 3 Applications of collocation integrators also exist for boundary value problems, specifically to aid in trajectory optimization. Previous uses of the Lobatto IIIA methods include trajectory optimization for Earth-Moon trajectories. 4,5 These same Lobatto IIIA methods were used for trajectory design for solar sail 6,7 and low-thrust trajectories 8,9 in the N-body problem for Lunar pole coverage. The two papers by Bradley, et al. 1 and Jones and Anderson 11 recently discussed and presented results for orbit propagation using other forms of collocation. Jones and Anderson 11 described a fixed-step implementation of integration using GL nodes, and provided sample results based on tests with a small number of nodes ( 4). As discussed in Bradley, et al., 1 methods such as those based on GL or Chebyshev nodes exhibit oversampling towards the ends of the integration interval. This oversampling results in an insensitivity in the step size with increases in the number of nodes. Hence, the maximum step size possible, which is desired for long-term propagation, increases slowly with heightend computation cost. However, no demonstration of this limitation could be found in the literature for astrodynamics applications. To reduce this oversampling, Bradley, et al. 1 presented the BandLimited Collocation using Implicit Runge-Kutta (BLC-IRK) method that provides a more uniform distribution of points and combines low- and high-fidelity force models to reduce computation time. Both of these papers demonstrated that collocation methods may be used to provide an efficient means of orbit propagation, but were not optimal for scenarios with high eccentricities. Hence, a variable-step method for collocation integrators is desired. This paper presents an implementation of the GL integrator that allows for more efficient propagation of eccentric orbits when compared to the previous demonstrations of collocation methods. The variable-step implementation uses the rate of convergence for the implicit method to determine a step size appropriate for the next integration step. To further reduce computation time, the method uses a combination of high- and low-fidelity force models to reduce the number of evaluations of the most computationally expensive portions of the software. This is similar to the BLC-IRK method previously described. The paper then compares this method to other commonly used ordinary differential equation (ODE) solvers in astrodynamics. Through this research, we hope to quantify the maximum number of nodes useful in GL collocation for astrodynamics, and begin developing a variable-step method with applications to other collocation methods. The paper is organized as follows. Section II provides a general discussion of integration using GL nodes in the form of an implicit Runge-Kutta(IRK) scheme. Section III then describes the implementation considered in this paper. Results for two-body propagation are presented, followed by higher-fidelity propagation that includes perturbations in the gravity field. Finally, the paper is summarized and conclusions are presented in Section VI. II. Gauss-Legendre Methods Research into integration using the GL nodes accelerated in the 195s, and was partially motivated by the discovery of features of the method that are desirable in solving the initial value problem. Identification of the GL methods began with a reinterpretation of the common implicit trapezoid rule as a collocation 2 of 16

3 solution using the quadratic, shifted Legendre polynomial. 12 Butcher 13 later defined an IRK method using the shifted Legendre polynomials of any degree, 13 which corresponded to the first general definition of integration using the GL nodes. The GL collocation methods are optimal in terms of the number of nodes for a given order of the integration method. Specifically, a method using a Legendre polynomials of order s has an integration order of 2s. 14,15 Additionally, such methods are symmetric (due to the distribution of nodes in [,1]), A-stable, 16 B-stable, 17 and symplectic. 18 Since this integrator is A-stable, it may be used for generating a solution to a system of ODEs that are stiff. Unfortunately, such stability properties may be lost with implementation. 19 Also, when compared to other symplectic integrators, the GL methods tend to be less efficient. 2,14 However, the combination of these properties increases the versatility of the GL methods when compared to other algorithms used in special perturbations propagation, thereby motivating their study for use in astrodynamics. This paper considers this collocation integrator, but reformulated as an IRK method. Hence, this section first presents a very basic overview of the RK solver. Then, the basics of collocation are described, along with the methods required to generate the necessary RK schemes for any number of integration nodes. II.A. Runge-Kutta Methods Runge-Kutta methods provide a single-step solution to the initial value problem. 21,22 In a generalized form, a s-stage RK method may be written as s = f t n +hc i,y n +h a ij k j, i = 1,...,s (1) k i y n+1 = y n +h j=1 s b j k j (2) j=1 where the integration state vector y n at time t n is of length p, f(t,y) is the ODE describing the dynamic model and returns a p-vector, h is the step size, and the a ij, c i, and b j coefficients are specific to a given RK scheme. The coefficients are often expressed in terms of the Butcher table c A s s b T where the integration (or Runge-Kutta) matrix A is comprised of the terms a ij, and the vectors c and b include the c i and b j coefficients, respectively. RK methods are typically classified by their order and the number of stages s. Efficient methods minimize s for a given order, and higher order methods allow for larger h. When a ij = for j i, this yields an explicit method in which each of the k i terms may be determined sequentially. When this is not the case, i.e. the Runge-Kutta scheme is implicit, a nonlinear solver must be used to generate k i. The most common methods of RK-based orbit propagation use explicit schemes, which are easier to implement and benefit from decades of development. One important advantage of the explicit RK methods is the development of variable-step techniques to autonomously determine the step size required. Such methods usually select a time step that constrains the maximum local error of the solution. The most efficient of these methods are the embedded schemes that produce one solution and require only one additional evaluation of f(t,y) to generate a solution of a higher or lower order. By comparing these two solutions, the local integration error is approximated and the step size may be predicted. The most commonly used of these methods are those developed by Dormand and Prince. 23,24 In this study, the developed GL integrator is compared to the thirteen-stage DOPRI 8(7), or DP 8(7), method, 24 which yields an eighth order solution and uses a seventh order solution for step size control. Comparisons with the seven-stage DOPRI 5(4) (DP 5(4)) scheme 23 are also provided. The commonly employed explicit methods fail to utilize the multi-core processing capabilities of modern computers in the sense that evaluations of the force model function cannot be computed in parallel. This is increasingly important since, for high fidelity orbit propagation, most of the computation time is spent in the force model. While concepts exist to define an explicit RK scheme that allows for the parallel computation of the s stages, such methods result in a reduced order scheme [25, p. 171]. However, implicit methods allow for parallelization in the generation of the solution to the nonlinear system. With the use iterative solvers, the evaluations of f(t, y) may be executed in parallel for a single iteration. Additionally, such techniques 3 of 16

4 typically reduce the number of stages required for a given order, but at the expense of solving the nonlinear system. Before the proliferation of multi-core processing, theoretical studies of parallel methods looked at the effective number of evaluations, i.e., the number of function evaluations assuming a given number of processors [25, p. 171]. When comparing the GL propagator to other methods, we present some results in terms of the number of effective stages. For this case, it is assumed that the number of available processors is greater than or equal to s. Of course, this does not capture the inherent overhead in the generation of and communication between the threads for a parallel implementation. Many of the operations required in the RK solver may be expressed in terms of linear algebra operations. For an explicit RK solver where each of the k i are found in sequence, a linear system formulation is of little benefit. The same is not the case for implicit methods. Such techniques must iterate on the solution, which allows for evaluating the series in Eq. (1) for all i before any calls to the force models. Let where Then, Eq. (1) may be written as where i indicates the i-th row of. The final solution is then = FA T (3) ] F = [k 1 k. (4) s p s F i = k i = f (t n +hc i,y n +h i ) (5) y n+1 = y n +hfb. (6) With this formulation, the sums in Eqs. (1) and (2) may be performed using Basic Linear Algebra Subprograms, or BLAS, library functions. BLAS routines are more computationally efficient than typical for-loop implementations, and are especially advantageous as the number of stages s or the size of the state p increases. II.B. Collocation Methods Collocation methods approximate the continuous solution to an ODE using a polynomial basis that exactly equals the solution at a set of nodes. In these methods, the first derivatives of the polynomial equal the slopes of the continuous solution y(t) at a given set of nodes c i. Given c i [,1] where i = 1,...,s, a collocation solution satisfies u(t n ) = y n (7) u(t n +hc i ) = ẏ(t n +hc i,u(t n +hc i )) (8) which then results in y n+1 = u(t n + h). It is noted that the resulting polynomial u(t) provides a continuous approximation of the system, and the formulation may easily be extended to a vector y n. Practical implementations of collocation methods generate a solution that satisfies Eq. (8) to a finite accuracy. Gauss Nodes Lobatto Nodes Chebyshev Nodes Figure 1. Node locations for various collocation polynomials as the number of nodes increases. Figure 1 illustrates the location of nodes c i for three ofthe most commonly used collocationmethods. For these polynomials, the concentration of c i increases towards the endpoints of the time interval. This results 4 of 16

5 in oversampling and high-frequency oscillations at the ends (Runge phenomenon), which reduces the overall efficiency of the integration method for high order schemes. An alternative method for orbit propagation reduces node density at the ends by using collocation with band-limited functions. 1 This is designed to sacrifice some accuracy for the sake of computation speed, symplecticity, and improved stability. Although several methods exist for using a polynomial basis to generate a solution to the initial value problem, all collocation methods may be expressed as an IRK method. 26 Given a set of nodes c defined by the collocation polynomial, the integrator is equivalent to a s-stage Runge-Kutta integration method with 26,14,15 ci 1 a ij = l j (γ)dγ, b i = l i (γ)dγ (9) where l j (γ) is the Lagrange interpolating polynomial l j (γ) = l j γ c l c j c l. (1) These equations may then be used to generate an IRK scheme when given the zeros, or nodes, of the collocation method, and they provide an analytic means for generating the Butcher table for a method of arbitrary order. Equation (9) may be evaluated using Gaussian quadrature, which provides a numerically stable method for generating the RK scheme. Software for generating the nodes ξ j [ 1,1] and the weights w j of the quadratures may be found online a, and these nodes correspond to the zeros of the Legendre polynomials. Given these quadratures, the nodes of the shifted Legendre polynomials c j [,1] are c i = (ξ i +1)/2. (11) Since the integrals in Eq. (9) are expressed in terms of the Legendre polynomials, evaluations of the integrals in this way is exact when using Gaussian quadrature of the same degree. Hence, if c j and w j are accurate to quadruple precision, evaluation of the integrals will be accurate to the same precision. The solution for the RK scheme is then and a ij = ci = ξ i +1 4 = ξ i +1 4 b i = = 1 2 = l i (γ)dγ (12) 1 l i (ξ)dξ (13) N w k l i (ξ k ) (14) k=1 l j (γ)dγ (15) 1 1 ( ) 1 l j 2 (ξ(ξ i +1)+ξ i 1) dξ (16) N ( ) 1 w k l j 2 (ξ k(ξ i +1)+ξ i 1). (17) k=1 For the current implementation of the GL integrator, schemes for s = 2,...,6 are generated in quadruple precision and stored in double precision for future use in orbit propagation. This minimizes round-off errors resulting from the terms in the integration coefficients. III. Variable-Step GL Propagator Implementation The GL propagator considered in this paper uses the variable-step implementation suggested by van der Houwen and Sommeijer 27 with some alterations to customize its use for orbit propagation. In their paper, a For example, the GAUSSQ routine may be found on netlib ( 5 of 16

6 van der Houwen and Sommeijer demonstrate that a variable-step implementation may be developed by approximating the local truncation error by the rate of convergence for the iterative process. This technique is implemented with the GL collocation method expressed as a IRK solver. Other methods of generating a variable step integrator for an IRK method exist, 19,28,29 but were not considered in this study. For easy identification of this integrator, the implementation described here is denoted VGL-s where s is the number of nodes in the scheme. III.A. Step-Size Selection As mentioned previously, generic solutions using an IRK scheme require an iterative solver for the nonlinear system in Eq. (5). The current implementation uses fixed point, or Picard, iteration. Given a solution for the forces F (m) at iteration m, (m) = F (m) A T, F (m+1) i ( = f t n +hc i,y n +h (m) i ). (18) This process is repeated with m = m+1 until convergence is achieved. As demonstrated by van der Houwen and Sommeijer, 27 the resulting solution of the RK solver y (m) n+1 = y n +hf (m) b (19) provides a solution of increasing order with each iteration m. Hence, y (m) may be considered an embedded solution of order m and compared to the lower-order solution y (m 1). This method requires no additional evaluations of the force model since it is already calculated in the process of generating y n+1. After m iterations, the local error is then e = 1 p y (m) i y (m 1) ( i p ), (2) max 1., i=1 y (m) i andthesolutionisconsideredconvergedife < ǫwhereǫisauserdefinedrelativetolerance. Ifthisconvergence is not achieved within a maximum number of iterations M, then the step size is rejected and the final value of e is used to generate a new step size. Such rejections can be expensive, and care must be taken to minimize their number. This may be accomplished by limiting the change in the step size or generating a more accurate step size prediction. A possible candidate for the later was presented by Gustafsson. 3 To set the maximum number of iterations, the current implementation uses int [ ] s M = 3 s > 5 (21) s s 5 which was found to allow for a sufficient number of iterations to converge while limiting the accumulation of floating-point error. After the iteration process is complete, regardless of the success of convergence, the predicted step size is h = h β min ( 6., max ( ( )) ǫ 1/(2s+1).1,. (22) e) where β is a safety factor commonly employed in such a step size predictor. The values 6. and.1 ensure that the time step does not increase or decrease by a given factor after a single step. If the nonlinear solver failed to converge, this new step size is used with the iterative process restarted. Implementations of Eq. (22) in common embedded RK methods set β [.8,.95]. However, Hairer, et al. [19, p. 124] recommends altering β to account for the rate of convergence when iterating. Similar to the formulation of Haier, et al., 19 this implementation uses ( ) min.9 3M 2M+m β =,1. e < ǫ. (23).8 e ǫ This selectionofβ yields avalue of.9ifthe maximumnumber ofiterationswererequired, and imposesafactor of.8 if the previous step failed to converge on the solution. Testing of this implementation demonstrated 6 of 16

7 that allowing too many iterations yields faster accumulation of floating point error for long integration time spans. Conversely, reducing this threshold results in small step sizes to satisfy the constrained number of iterations and reduces integrator efficiency. Thus, Eqs. (21) and (23) provided an adequate balance, and are effectively design parameters for this integration method. III.B. Initial Step Size Selection Selection of the initial step size for the VGL-s integrator influences integrator efficiency. For a step size that is too large, the method must reject integrator steps until an appropriate value has been determined. For a value too small, early steps through the integrator are smaller than necessary until the step size has settled on the appropriate value. In some cases, the user may already known an appropriate value. For those cases without such a value, a method for proper step size selection is desired. This selection outlines a method based on the one presented in Hairer, et al. [31, p. 169] with alterations empirically determined in this research. This method for step size selection is based on the assumption that local error Ch m+1 y (m+1) (t ), (24) i.e. the truncation error in the Taylor expansion of order m. Since, for the currently implemented integrator, only y (1) = f(t,x) is known, y (p+1) is approximated using a small Euler step and the first derivatives. The algorithm is then presented in Algorithm 1 the computed norms are based on Eq. (2), but with no reference. The parameter α is empirically determined in Section V.C, and is a function of the orbit. Algorithm 1 Initial Step Size Selection f 1 = f(t,xy ) d = y, d 1 = f 1 h =.1 (d /d 1 ) if d or d 1 < 1 5 then h = 1 6 end if y 1 = y +hf 1 f 2 = f(t +h,y 1 ) d 2 = f 2 f 1 /h h 1 = (.1/max(d 1,d 2 )) 1/(p+1) if max(d 1,d 2 ) 1 15 then h 1 = max(1 6,h 1 3 ) end if h = min(1 h 1,α h 2 ) III.C. Continuous Approximation The solution to the GL-based collocation method also provides a continuous approximation to the ODE solution. Specifically, interpolation using the the collocation polynomial provides a solution of order s [14, p. 326]. Specialized code may be written for each polynomial, however a generalize algorithm may be employed for any scheme where the node locations are the zeroes of the polynomial [14, p. 335]. For this method, s y(t) = y n +h θ j (t)f(t n +hc j,y n +h j ) (25) j=1 where the θ j (t) are coefficients defined by the integration scheme, t [t n,t n+1 ], and j is the solution at the nodes after convergence. The coefficients θ j (t) are generated by solving the Vandermonde-like system θ 1 (t) τ c 1 c 2 c s θ 2 (t) τ..... = 2 /2 (26)... c1 s 1 c2 s 1 cs s 1 θ s (t) τ s /s s s s 1 s 1 7 of 16

8 with τ = t t n h. (27) Assuming the Vandermonde matrix is non-singular (which is the case for the GL methods), the inverse is precomputed and later used to define a polynomial in τ. This yields a general method applicable to all collocation methods formulated as IRK. It is noted that such a method may be used for extrapolation ( τ > 1), which is also a common method used to initialize the iteration procedure in the next time step. III.D. Stage Initialization Several techniques exist for initializing the nonlinear solver, with each attempting to provide the most accuratestartingvaluetoreducethe totalnumberofiterations. Thesimplest choicecallsforsettingf () =, but this fails to incorporate any of the information already available for stage prediction. Other methods for stage prediction include: the previously mentioned extrapolation using the collocation polynomial, 14 additional evaluations of the RHS function in an explicit integration method, 32 equistage approximation, 33 or two-step, explicit methods. 34,35 However, presentations of all of these methods except the polynomial extrapolation assume a relatively small number of nodes. Since this paper considers up to 3 nodes, the current method is limited to the polynomial extrapolation. To generate a more accurate initialization of the stages, the polynomial extrapolation is combined with a lowfidelity propagation to reduce the overall computation time. To accomplish this, iterations of the nonlinear system of equations are first performed using dominant forces with relatively low computation time, e.g., the two-body equation and the J 2 gravity perturbation. After the solution has converged to within a given tolerance, then the high-fidelity force model is used until convergence to ǫ. This provides a warm start for the iterations, and reduces the overall computation cost. A discussion of the effects of the low-fidelity convergence tolerance may be found in Section V.D. Further research may also allow for improved efficiency in stage initialization. IV. Two-Body Propagation The VGL-s integrator is compared to several common propagators to assess computational efficiency and accuracy. As mentioned previously, the DP 8(7) and DP 5(4) propagators are used as a basis of comparison. The DP 5(4) includes a first-same-as-last (FSAL) implementation to reduce the number of force evaluations per integration step by one. A comparison with an eighth-order Gauss-Jackson propagator (GJ 8) is also included. Implementation of this integrator is based on Berry and Healy 36 with usually only one correction per time step, i.e., one force-model evaluation per integrator step. Hull, et al. 37 introduced a series of tests that consider two-body propagation with orbits of various eccentricities. The results of a given ODE solver may then be compared to the analytic result of Kepler s equation. Such tests were also used in the more extensive solver comparison by Montenbruck. 38 These tests are designated as D1 through D5, with eccentricities ranging from.1 to.9 in increments of.2. This section presents results of a similar test with the DP 5(4), DP 8(7), GJ 8, and VGL-s methods. Additionally, a D test is added with an eccentricity of.. The normalized initial conditions of the propagators are then defined such that the semimajor axis is 1., the inclination is 45, and the gravitational parameter is also 1.. The orbits are propagated until t f = 2, which corresponds to approximately 3.2 revolutions. Since only two-body dynamics are considered for this case, combining low- and high-fidelity descriptions of the satellilte forces is not possible. Hence, the stage initialization only uses extrapolation of the collocation polynomial and the reported number of function calls is the total calls to the two-body force model. These tests are provided here to compare the methods using already existing test cases in the community, and provide some insight into the effects of design parameters on the orbit propagation in a simplified system. Before comparing to the other integrators, Fig. 2 illustrates the variations in the integrator efficiency as a function of the maximum number of iterations M and the number of nodes s. Precision is defined as ( 6 ) 1/2 precision = log (y i (t f ) ŷ i (t f )) 2 (28) i=1 where ŷ i is the i-th component of the solution generated via Kepler s equation. The number of effective function calls designates the number of evaluations if computations for each node are run in parallel, and 8 of 16

9 Function Calls Function Calls Effective Fcn. Calls Precision s = 5, M = 1 s = 1, M = 2 s = 15, M = 2 s = 2, M = 26 s = 25, M = 33 Effective Fcn. Calls Precision s = 5, M = 5 s = 1, M = 1 s = 15, M = 5 s = 2, M = 7 s = 25, M = 9 Figure 2. Work-precision plots for the D (left) and D4 (right) cases for different design parameter s and M. only represent a theoretical value for computation work load. These results indicate a coupling between the eccentricity, s, and M for the optimal method. Additionally, arbitrarily increasing the number of nodes, even though it increases the order of the integrator, may yield an increase in the total number of function evaluations. This is contrary to the explicit methods where higher order methods typically allow for longer step sizes and reduced computation times. As mentioned previously, this is due to the oversampling at the endpoints as s increases. Fcn. Calls Fcn. Calls Fcn. Calls 8 7 D D D Precision DP 5(4) DP 8(7) GJ 8 VGL-s D1 D3 D Precision VGL-s, eff. Figure 3. D-D5 work-precision diagrams for various integration methods. Figure 3 provides a comparison of the different integrators for the D through D5 test cases. The presented results come in the form of work-precision plots. Here, the VGL-s methods use 2 nodes for the D-D2 cases, 15 nodes for D3, and 1 nodes for D4 and D5. For highly accurate solutions, the VGL-s methods require the same number of function calls as the DP 8(7) methods for the D-D2 cases. The work required for the VGL-s method then increases faster as a function of eccentricity than DP 8(7). However, it still performs better than DP 5(4). In terms of effective stages, the VGL-s methods prove to be the most efficient for these two-body cases. 9 of 16

10 V. Higher Fidelity Propagation This section presents the efficiency of the VGL-s methods for higher fidelity propagation. As explained below, the forces describing the satellite motion are expanded for a collection of more realistic scenarios. At this point, the combination of low- and high-fidelity propagations is introduced to reduce total computation time. V.A. Test Description In the remaining sections, the tests of the VGL-s methods consider three orbits: a low-earth orbit (LEO), a geosynchronous orbit (GEO), and a Molniya orbit. Initial conditions for these cases may be found in Table 1. Perturbing accelerations include the Cartesian formulation of the spherical harmonic model 39 using the 7 7 GGM2C coefficients, 4 and Sun/Moon third-body perturbations. The positions of the third bodies are determined via the Jet Propulsion Laboratory planetary and lunar ephemerides model DE Each orbit is propagated for three orbit periods with a start time of March 28, 21 :: UTC. Orbit Semimajor Axis (km) Eccentricity Table 1. Orbit Initial Conditions Inclination (deg) Right Ascension of Ascending Node (deg) Argument of Perigee (deg) True Anomaly (deg) LEO Molniya 26, GEO 42, Since no analytic solution exists for orbit propagation using special perturbations, a reference trajectory must be defined to assess propagator accuracy. For this purpose, we use a solution based on the GJ 8 propagatorwith a step size of1s. This step size wasselected to minimize h without causing any accumulated floating point error that creates bias in the precision results. Reported global errors are then the RSS difference between the solution for a given integrator and this trajectory at the final time. V.B. VGL-s Setup To accommodate the increase in propagator fidelity, the VGL-s integrator is modified slightly in comparison with the two-body tests. Specifically, the low-fidelity iteration method previously described is employed for stage initialization. As demonstrated later, this reduces the number of evaluations for the more expensive forces. Here, the low-fidelity model includes the gravitational force of the primary body and the J 2 gravity term, and the reported number of function calls indicates the number of high-fidelity evaluations. Other design parameters for the higher fidelity tests may be found in Table 2. Section III.B discussed a method for initial step size determination, but did not state a specific value for α. Tests demonstrated that the optimal value varies with the orbit, with a brief treatment of the effects of initial step size selection provided in Section V.C. Design parameters used to automatically determine the step size in future tests may also be found in Table 2. A more robust method for determining an initial step size is a subject of future work. Table 2. Orbit Initial Step Size Design Parameters Orbit s α LEO 2 1. Molniya 2 33,333. GEO of 16

11 V.C. Initial Step-Size Selection For these more realistic propagation tests, the initial step size proved to effect the overall performance of the integrator. Although it is not demonstrated here, the step size selection for the embedded RK methods tends to have a relatively small effect on overall efficiency. The results of this section demonstrate some effects on initial step size selection on accuracy and efficiency. Figures 4 and 5 illustrate the effects of step size selection on efficiency and accuracy for the LEO and GEO cases. As seen in the plots on the left, selection of a step size that is too big or small can take a long time to settle on the optimal value. The sharp drop at the end of the plots correspond to rapid reductions in step size to match the final time of the integration. The LEO case exhibits two values where step sizes settle. These are largely determined by the design parameters in Eq. (22), and tests that included different values verified this hypothesis. For the 1, s case, this introduces global error into the solution as indicated by the plot on the right. For the GEO case, any initial step sizes larger than approximately 12. s are rejected due to the lack of convergence in the nonlinear solver. The plots on the right indicate that, as expected, small initial step sizes fail to improve results and are limited by the accumulation of floating-point error Time Step (s) Time (orbit periods) Auto Precision (meters) Initial Step Size (s) Number of Function Calls Figure 4. Evolution of the step size over time for different initial step sizes (left) and the effects of the initial step size on global integration accuracy (right) for the LEO satellite Time Step (s) Time (orbit periods) Auto Precision (meters) Initial Step Size (s) Number of Function Calls Figure 5. Evolution of the step size over time for different initial step sizes (left) and the effects of the initial step size on global integration accuracy (right) for the GEO satellite. Figure 6 provides the results of the same test for the Molniya orbit, and clearly indicate the evoluation of the time step over the orbit period. In this case, rejected step sizes for the first step prevent any extended settling time, i.e., the solution fails to converge for initial step sizes that are too large. Sharp spikes in the left plot correspond to rejected steps, some of which appear periodic. The exact cause is unknown and will be further investigated. V.D. Low-Fidelity Convergence As mentioned previously, this implementation of the VGL-s propagator uses a combination of low- and highfidelity force models to reduce the overall integrator computation time. Fig. 7 illustrates the effect of the 11 of 16

12 Time Step (s) Time (orbit periods) Auto Precision (meters) Initial Step Size (s) Number of Function Calls Figure 6. Evolution of the step size over time for different initial step sizes (left) and the effects of the initial step size on global integration accuracy (right) for the Molniya satellite. low-fidelity calls on the total number of high-fidelity force model evaluations. Effectively, including one lowfidelity call can reduce the number of more expensive evaluations by approximately 2%. This corresponds to the elimination of 1-2 high-fidelity iterations. These results also indicate that more than one iteration does not improve the propagator efficiency, and may actually increase the number of high-fidelity function calls. This results from differences in the low- and high-fidelity solutions, which grow over time. As demonstrated in the previous section, the VGL-s method allows for time steps on the order of hundreds of seconds. Errors in the propagation using a low-fidelity model grow over time and are largest for nodes towards the end of the integration interval. Hence, further iterations cause the solution to begin converging on a solution different from that of the high-fidelity propagator, and additional iterations may be required to correct this error. Molniya LEO Number of Low-Fidelity Iterations Figure 7. Effect of low-fidelity calls on overall work for the LEO and Molniya orbits. 12 of 16

13 V.E. Propagation Efficiency Figure 8 illustrates the number of high-fidelity force model evaluations required to achieve a given accuracy for the LEO case. For a serial implementation, the GJ 8 provides the most computationally efficient method. This is not surprising given that multistep methods tend to require fewer function evaluations then Runge- Kutta methods for scenarios that only require a constant step size. However, the theoretical number of parallel calls to the force model yields the most efficient method. The VGL-2 method does provide a more efficient solver than the embedded Runge-Kutta methods and approaches the efficiency of the GJ 8 method for high-precision integration. Global Integration Error (meters) VGL-2 VGL-2, effective DP54 DP87 GJ Number of Function Calls Figure 8. Work-Precision Plot for the LEO case. Global Integration Error (meters) VGL-2 VGL-2, effective DP54 DP87 GJ Number of Function Calls Figure 9. Work-Precision Plot for the GEO case. The work-precision plot for the GEO case is provided in Fig. 9. For this case, the VGL-2 method performance is still better than that of the DP 8(7) method for accuracies greater than approximately m. However, there is a larger disparity between the GJ 8 and the VGL-2 method at this altitude. Finally, Figure 1 illustrates the efficiency of the VGL-1 method when propagating a Molniya orbit. In this case, which is expected, the GJ 8 is a poor method for orbit propagation. This results from the unnecessary force model evaluations near apoapsis. When comparing the DP 8(7) and the VGL-1 results, thedp8(7)appearstorequirefewerfunctionevaluationsexceptnearaprecisionof1 4 m. Forthiseccentric 13 of 16

14 Global Integration Error (meters) VGL-1 VGL-1, effective DP54 DP87 GJ Number of Function Calls Figure 1. Work-Precision Plot for the Molniya case. case, the number of effective force evaluations greatly reduces the computation time for this eccentric orbit when compared to all other methods. VI. Conclusion This paper presented a variable-step implementation of integration using Gauss-Legendre nodes for orbit propagation, dubbed VGL-s. In combination with the variable-step implementation, a combination of lowand high-fidelity force models allowed for a further reduction in computation load. For the circular orbits considered, the Gauss-Legendre method required fewer function calls than the Dormand-Prince (DP) 8(7) and 5(4) embedded Runge-Kutta integrators, but still needed more evaluations than an eighth-order Gauss- Jackson method (GJ 8). For the Molniya orbit case, the Gauss-Legendre propagator required fewer function calls than the DP 5(4) and GJ 8 propagators, and matched the DP 8(7) method for high precision cases. Assuming force evaluations at each node within an interval may be performed in parallel for a given iteration, the VGL-s method would require less computation time than all of the sequential methods. However, this only represents a theoretical assessment of the computation time required for a parallel implementation. The tests considered also demonstrated the limited improvements in step size as nodes are added to the Gauss-Legendre method. Such a barrier results from the oversampling that occurs near the beginning and end of the integration interval as the number of nodes increase. This result helps provide a baseline for future endeavors that attempt to surpass this limitation. Several elements were identified that necessitate further research to improve robustness of the VGL-s integrator. In addition to the elements already described, an investigation of the quality of the continuous solution will aid in determining the capabilities of the method. As mentioned previously, collocation methods like the VGL-s integrator allows for a continuos approximation of the orbit, and this may be used to provide information on the propagated orbit at intermediate steps. Finally, a full exploration of the computation times when implemented with parallelization will be addressed with future work. Acknowledgements This research was partially funded by the Air Force Research Laboratories. The author thanks Vladimir Rokhlin of the Yale University Department of Mathematics who wrote the original version of the Fortran software used to calculated the Gaussian quadrature nodes and weights. This software was provided through Gregory Beylkin of the University of Colorado Department of Applied Mathematics, and then converted to quadruple precision for this application. 14 of 16

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