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Aas 17-204 Validation of a Low-Thrust Mission Design (Preprint) AAS 17-204 VALIDATION OF A LOW-THRUST MISSION DESIGN TOOL USING OPERATIONAL NAVIGATION SOFTWARE Jacob A. Englander∗ Jeremy M. Knittely Ken Williamsz Dale Stanbridgex Donald H. Ellison{ Design of flight trajectories for missions employing solar-electric propulsion requires a suitably high-fidelity design tool. In this work, the Evolutionary Mission Trajectory Gen- erator (EMTG) is presented as a medium-high fidelity design tool that is suitable for mis- sion proposals. EMTG is validated against the high-heritage deep-space navigation tool MIRAGE, demonstrating both the accuracy of EMTG’s model and an operational mission design and navigation procedure using both tools. The validation is performed using a bench- mark mission to the Jupiter Trojans. INTRODUCTION The design of trajectories for interplanetary missions requires a highly specialized toolset. This is espe- cially true for missions employing solar-electric propulsion (SEP). Several medium-fidelity tools exist that are suitable for early mission formulation, including EMTG [1, 2], Mission Analysis Low-Thrust Optimiza- tion (MALTO) [3], Gravity Assisted Low-thrust Local Optimization Program (GALLOP) [4], and Parallel Global Multiobjective Optimizer (PaGMO) [5]. These tools allow a designer to rapidly determine the key parameters of a mission concept, including flight time, propellant mass, delivered dry mass, etc., but the tra- jectories generated by these tools only roughly resemble those that a spacecraft might actually fly. There are also several flight-fidelity low-thrust tools, including Copernicus [6] and Mystic [7]. In the mission proposal stage, medium-fidelity is often sufficiently accurate to be used for high-level mis- sion and systems trades. However, when it comes time to submit a proposal, it is a good idea to provide at least one trajectory that is designed to a higher level of fidelity. The flight-fidelity tools that produce these solutions, however, require a great deal of human analyst time that may not be available or affordable in the context of a proposal. In addition, the flight-fidelity tools are not always readily available to all teams wishing to submit a proposal. An intermediate level of design fidelity that is easy and inexpensive to use and also readily available to all potential users is therefore highly desirable. Such a “medium-high” fidelity tool should integrate the equations of motion rather than approximating them and its force model should include reason- ably high fidelity models of propulsion and power systems as well as significant orbit perturbations such as third-body gravity and solar radiation pressure (SRP). Such a tool is sufficient to ensure that a proposed trajectory is actually feasible and is not an artifact of the low-fidelity and medium-fidelity models available in most preliminary design tools. However, before such a medium-high fidelity tool can be trusted, it must be validated against a high-heritage navigation tool that might be used to actually fly the mission. ∗Aerospace Engineer, Navigation and Mission Design Branch, NASA Goddard Space Flight Center, Member AAS yAerospace Engineer, Navigation and Mission Design Branch, NASA Goddard Space Flight Center zFlight Director, Space Navigation and Flight Dynamics Practice, KinetX Aerospace, Simi Valley, California, 93065 xSenior Navigation Analyst, Space Navigation and Flight Dynamics Practice, KinetX Aerospace, Simi Valley, California, 93065 {Ph.D. Candidate, Department of Aerospace Engineering, University of Illinois at Urbana-Champaign 1 In this work, the EMTG [1, 2] is presented as a tool for medium-high fidelity trajectory design, that is suitable for mission proposals. The finite-burn low-thrust (FBLT) [8] force model is extended to include third-body perturbations and solar radiation pressure. The resulting medium-high fidelity design tool is then validated against the operational navigation tool Multiple Interferometric Ranging Analysis using GPS En- semble (MIRAGE). MIRAGE is a licensed version of the Double-Precision Trajectory and Orbit Deter- mination Program (DPTRAJ/ODP) originally developed at JPL to support high-fidelity orbit determination, trajectory propagation and maneuver planning [9]. MIRAGE has flight heritage from the 1970s to the present day and has been used to navigate many deep-space missions, including NEAR, Stardust, Genesis, and many other missions at JPL, and, more recently at KinetX, MESSENGER, New Horizons, and OSIRIS-REx. Most relevant to this work, MIRAGE was used to navigate the Deep Space 1 mission, which employed solar elec- tric propulsion. MIRAGE is therefore an ideal environment in which to check the accuracy of EMTG’s FBLT mode. THE FINITE-BURN LOW-THRUST TRAJECTORY MODEL The FBLT transcription divides a continuous-thrust spacecraft trajectory into Np phases, each consisting of N control decision segments. The spacecraft’s state (X) is propagated from either phase boundary to the center of the phase. A constraint is applied at the “match point” that ensures state continuity to some numerical tolerance. 2 y y 3 rB − rF 6 7 y y y 6 y y 7 c = X − X = 6 r_ − r_ 7 = 0 (1) B F 6 B F 7 4 5 y y mB − mF Control point Match point (planet, satellite, free point in space) Numerically integrated segment with constant thrust direction Segment boundary Derivative information propagation † X X† F B Forward propagation Backward propagation Mission clock time flow Figure 1: An FBLT phase with N = 10 control segments. Partial derivatives of this constraint are computed using augmented state transition matrices (STM) (Eq. (2)) that are computed using the FBLT physics engine. These STMs are also useful for performing navigation 2 tasks such as filtering and orbit determination (OD). 2 @r @r @r @r 3 @r0 @r_ 0 @m0 @u0 6 @r_ @r_ @r_ @r_ 7 6 @r0 @r_ 0 @m0 @u0 7 Φ(t; t0) = 6 7 (2) 6 @m @m @m @m 7 4 @r0 @r_ 0 @m0 @u0 5 03×3 03×3 03×1 I3×3 The FBLT transcription models the effects of perturbing forces from sources other than the central body’s gravity and the spacecraft’s thruster, such as other gravitating bodies and solar radiation pressure SRP, by directly incorporating them into the equations of motion. This is known as Cowell’s method of special perturbations. The differential equations of motion governing a continuously thrusting spacecraft in orbit around a central body subject to gravitational forces from massive bodies and solar radiation pressure, using a spherical cannonball model [10], are: 0 G ¨m¨*+ mcb r X r − ri ri CrAsKφ rs= DT ¨ r = − 2 − G mi 3 + 3 + 2 + u (3) r r kr − rik r c r r m i i s= s= m_ = −kuk D m_ max (4) where G is the universal gravitational constant, m is the mass of the spacecraft, mcb is the mass of the central th gravitational body to which the inertial reference frame is fixed, mi is the mass of the i gravitational body, r is the position vector of the spacecraft with respect to the central body and ri is the position vector of the th i body with respect to the central body. The vectors ri and r are obtained from ephemeris calls. The vector rs= is the position vector of the spacecraft with respect to the sun and rs/i is the position vector of the spacecraft with respect to the ith gravitational body, both expressed in the coordinates of the inertial frame attached to the central body, i.e. rs= = r − r (5) Regarding the variables associated with radiation pressure, Cr is the coefficient of reflectivity on the in- terval [0; 2], As is the spacecraft area exposed to sunlight, K is the total percentage of the sun seen by the spacecraft ([0; 1]), φ is the solar flux and c is the speed of light in a vacuum. The thruster variable u is a throttle scaling vector whose three components are selected by the optimizer on the range [−1; 1], D is the thruster duty cycle (modeled continuously), T is the instantaneous thrust of the electric motor and m_ max is the maximum instantaneous mass flow rate of the propellant through the motor. Note that in Equation (3) the mass of the spacecraft is neglected in the first term since mcb >> m. The vector geometry of this problem is illustrated in Figure 2. The diagram on the left depicts the general case, where the central body need not be the sun, whereas the right hand diagram shows how the configuration simplifies for a heliocentric trajectory. 3 Figure 2: Vector diagrams for the general central body case and the simplified heliocentric case. For the situation shown on the left in Figure 2 the sun is included for two reasons, despite the fact that it is not the central body. The first reason is that for most mission designs, the sun would almost certainly be included in the problem dynamics (the summation in Equation (3)) due to its gravitational dominance in the solar system. The second reason is that for accurate solar-electric propulsion modeling, the position of the spacecraft with respect to the sun must be tracked at all times throughout the optimization since many of the quantities in Eq. (3) and (4) are dependent on the solar flux at any instant in time, i.e. T = T (rs= (r))m _ max =m _ max(rs= (r)) (6) SPACECRAFT POWER SYSTEM MODELING For a spacecraft using a solar electric propulsion (SEP) system, the maximum amount of thrust that the system can produce depends on the power available, which depends on its distance from the sun. This must be modeled in the low-thrust global optimization even at the preliminary design stage. Typically trajectory design engineers are supplied with polynomial representations of the thrust and mass flow rate of a given thruster, e.g., 4 3 2 T = eT Peff + dT Peff + cT Peff + bT Peff + aT (7) 4 3 2 m_ max = emPeff + dmPeff + cmPeff + bmPeff + am (8) where Peff is the power available to each thruster, Peff = P=Nactive (9) and Nactive is the number of thrusters firing at any point in time.
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