Proper motion control for lunar prolate polar orbits

Abstract

A method implemented by a computer for maintaining a spacecraft near an orbit includes a step (3) of detecting that a distance from the spacecraft to the orbit is greater than a spacecraft threshold, and, in response thereto, a step (5) of linearizing the dynamics of the spacecraft from the current time over a certain time horizon with respect to a high-fidelity reference trajectory to generate a state transition matrix (STM) for uncontrollable motion of the spacecraft within the time horizon. The STM includes non-expanding eigenvectors having a magnitude of 1 or less and expanding eigenvectors having a magnitude greater than 1. The method further includes a step (7) of determining a control action that changes the next state of the spacecraft to a linear combination of non-expanding eigenvectors of the STM, and a step (9) of generating a control command to an actuator of the spacecraft.

Description

[Technical Field]

[0001] The present disclosure relates generally to controlling the operation of spacecraft near unstable orbits around celestial bodies such as the Moon, and more particularly to methods and systems for generating fuel-efficient maneuvers that leverage natural motion to remain near unstable orbits for extended periods of time without intervention. [Background technology]

[0002] Launching and deploying space stations and spacecraft into space remains a challenging task, requiring precise analysis and investigation of orbital motion, celestial bodies, and other space objects. One example of such a space station is the Lunar Orbital Platform-Gateway (LOP-G), also known as the Gateway, a small space station in lunar orbit intended to serve as a solar-powered communications hub, a scientific laboratory, a short-term habitation module for government astronauts, and a staging area for planetary probes and other robots. The Gateway is deployable in a seven-day, highly elliptical, near-rectilinear halo orbit (NRHO) around the Moon and is intended to play a key role in facilitating missions in and beyond cislunar space. NRHOs are a type of halo orbit with slightly curved, therefore nearly straight, sides between flybys. The Gateway can be deployed in close proximity to an NRHO, which is a closed periodic orbit in the Earth-Moon Circular-Restricted Three-Body Problem (CR3BP) due to its favorable stability characteristics and visibility from Earth. One of the advantages of such an NRHO is that it minimizes the amount of communication disruption with Earth.

[0003] The Gateway can be deployed near the NRHO, but not on it. This is because the NRHO of the CR3BP does not take into account perturbations such as the gravitational pull of the Sun, solar radiation pressure (SRP), or lunar J2 influence. Instead of attempting to trace the NRHO of the CR3BP and using fuel to compensate for predictable perturbations, standard practice is to use multiple-shooting or collocation-based techniques to find a high-fidelity trajectory near the NRHO that accounts for all major predictable forces in cislunar space. This high-fidelity solution is still referred to as the NRHO, even though it is no longer closed, periodic, or stable. However, the benefit of this high-fidelity NRHO solution is that, in the absence of additional perturbing forces, the spacecraft can naturally trace its trajectory without consuming fuel, which is a key performance indicator for ensuring the long-term viability of the Gateway.

[0004] However, two factors hinder Gateway deployment based solely on high-fidelity solutions. First, navigation uncertainties and unpredictable disturbance forces in the sky prevent the spacecraft from being deployed precisely on the calculated trajectory. Second, and importantly, the trajectory is highly unstable compared to its ideal CR3BP counterpart. As a result, any small deviation from the solution can cause the spacecraft to rapidly deviate from the calculated trajectory, necessitating stabilizing control actions. These deviation issues also arise on other celestial bodies, where it is likely that a spacecraft will deviate from its planned trajectory for several reasons.

[0005] In recent years, several stationkeeping strategies have been developed for high-fidelity NRHO. However, while the Gateway itself can utilize such control strategies, these methods cannot be directly applied to approaching spacecraft for resupply missions, human transport missions, or inspection and maintenance missions, which would require long-term, constrained, and collision-free relative motion around the Gateway. Formation control for multiple spacecraft in halo orbits has been developed specifically for NRHO, but these methods propose control schemes based on periodic solutions in the CR3BP and rely on the computationally expensive process of generating high-fidelity solutions for each spacecraft in the formation.

[0006] Therefore, among other aspects, a need exists in the art for improved methods of controlling the operation of multiple spacecraft for long-term, limited, collision-free motion near NRHOs. Summary of the Invention

[0007] This disclosure relates to a control policy for reliable and fuel-efficient stationkeeping and constrained relative motion control for Gateway and approaching spacecraft that utilizes a single pre-calculated high-fidelity NRHO solution while ensuring a safe separation distance between the spacecraft. The high-fidelity NRHO may be referred to throughout this disclosure as a reference trajectory, a high-fidelity reference trajectory, a baseline, a baseline solution, and a baseline reference trajectory.

[0008] There were assumptions and realizations that aided in the development of some of the embodiments of the present disclosure. At least one realization included that the control policy should utilize a linear approximation of spacecraft dynamics in the vicinity of high-fidelity NRHO.

[0009] Although small deviations from conditions on a high-fidelity NRHO trajectory generally lead to rapid deviations, some exemplary embodiments recognize that at any given time instance, there exist several special conditions near the reference trajectory that result in desired deviation-free natural motion, where natural motion means motion without control (i.e., without using fuel / onboard power). These conditions form a desired space around the reference trajectory. When a spacecraft is controlled to this space rather than controlled by the orbit itself, the requirement is relaxed from staying strictly on orbit to staying “near” orbit, which trades off distance to orbit for improved fuel efficiency. That is, the proposed control policy aims to significantly improve fuel efficiency by maintaining the spacecraft within a desired space where deviation-free desired natural motion is possible, rather than consuming significant fuel / onboard power by continuously controlling the spacecraft to remain on the intended orbit.

[0010] Some example embodiments are also based on the recognition that the region (subspace) of such special states can be estimated using local mode decomposition of a receding horizon State Transition Matrix (STM) associated with a high-fidelity reference trajectory. In other words, the trajectory dynamics for the reference trajectory over a finite, user-determined horizon can be linearized, and the eigenvalue decomposition of the resulting linearized STM can be used. The state transition matrix is ​​used to solve for a general state-space representation of a linear system of the form:

number

[0011] At this point, it is important to understand that a (non-zero) vector v of dimension N is an eigenvector of a square N × N matrix A if it satisfies a linear equation of the form Av = λv for some scalar λ. In that case, λ is called the eigenvalue corresponding to the eigenvector v. Special states that exhibit desirable natural motion arise from eigenvectors of the STM that have eigenvalues ​​with magnitudes less than 1. This is because in such states, where the eigenvalues ​​have magnitudes less than 1, the solution provided by the STM approaches the baseline solution (i.e., converges toward the reference trajectory).

[0012] The natural motions due to initial conditions along the eigenvectors of the STM are called local eigenmotions. The natural motions due to initial conditions along the extended (eigenvalue > 1) and non-extended (eigenvalue <= 1) eigenvectors of the STM are called extended and non-extended local eigenmotions, respectively.

[0013] Another realization of some exemplary embodiments included that these eigenvectors are directions that do not have fixed magnitudes. This means that the spacecraft can move in these directions at various distances from the reference trajectory and exhibit the desired type of proper motion. Some exemplary embodiments utilize the distance to the reference trajectory as a trigger condition for determining when to apply control, because the spacecraft can be controlled to move from a current undesired deviation state to a desired state in the non-extended subspace, regardless of the spacecraft's distance from the reference trajectory.

[0014] Some exemplary embodiments recognize that whenever a deviation from a vicinity of a reference trajectory is detected, proper motion-based control solves a nonlinear optimization problem to find one or more fuel-efficient maneuvers (also referred to as control actions) that move the spacecraft into a set of states that result in the desired natural motion. The nonlinear optimization is a finite-horizon optimization of a spacecraft dynamics model, a set of spacecraft motion objectives, and constraints on the spacecraft propulsion system and motion, with the ability to anticipate future events and take appropriate control actions.

[0015] One or more fuel-efficient maneuvers can be achieved by optimizing the spacecraft's motion according to a set of objectives over a finite future time horizon using predictions obtained according to a spacecraft model subject to constraints. These constraints may correspond to, for example, physical limitations of the spacecraft, safety constraints on the spacecraft's motion, and performance constraints on the spacecraft's trajectory. In some non-limiting examples, constraints on the spacecraft's propulsion system may include constraints on thruster inputs that define the thruster's rotation range. A spacecraft control strategy is acceptable if the motion generated by the spacecraft for such control strategy satisfies all constraints.

[0016] In theory, it is possible to use nonlinear optimal control to obtain optimal stationkeeping maneuvers over the entire duration of a spacecraft mission, rather than over a finite time horizon. However, this poses a very large optimization problem, resulting in a very high computational load, which cannot be realized with the computationally constrained hardware on the spacecraft. Therefore, some exemplary embodiments are based on the recognition that using local proper motion-based control allows the spacecraft to use natural motion for long periods of time, thereby reducing the computational load of solving the optimization problem and speeding up the solution of the optimization problem.

[0017] A further realization is that by using trigger-based control, the spacecraft can be controlled less frequently because it can follow its natural motion for an extended period of time before crossing a trigger threshold, at which point a near-desired condition exists in the subspace in which the spacecraft can be maneuvered, and it begins to deviate from the reference trajectory. Because the spacecraft is controlled less frequently, the number of times its thrusters are fired is significantly reduced, which also translates into improved fuel efficiency over a period of time.

[0018] In some exemplary embodiments, the distance from the reference trajectory is an adjustable parameter that enables control of multiple spacecraft relative to one another without having to calculate a reference trajectory independently for each spacecraft. Multiple spacecraft, the planets they orbit, and their orbits form a multi-body celestial system. In other words, the same subspace associated with a single reference trajectory is used, but the distances from the reference trajectory are different, which is sufficient to ensure differences in the trajectories of the multiple spacecraft. Thus, in a scenario where a host spacecraft is approached by a visitor spacecraft for some reason, the same reference trajectory for the host spacecraft can be utilized. Therefore, because the exemplary embodiments utilize the distance between the spacecraft and the baseline solution (reference trajectory) as a trigger for the control policy, the approach is applicable to controlling any number of spacecraft in addition to a gateway.

[0019] In some exemplary embodiments, particular eigenvector direction states are combined into linear combinations to achieve mixed states with natural motions, which are combinations of corresponding eigenmotions. For example, controlling a spacecraft to an eigenvector combination having an eigenvalue strictly less than one will cause the natural motion of the spacecraft to converge toward the reference trajectory. In contrast, including eigenvector directions with a magnitude near one in the linear combination will cause the natural motion to oscillate at the current distance from the reference trajectory. By selecting a particular combination of eigenvector directions for the spacecraft to transition to, an operator can shape the natural motion of the spacecraft about the reference trajectory to move the spacecraft closer to or further from the reference trajectory as desired.

[0020] According to one non-limiting embodiment, the spacecraft is powered by eight thrusters, each mounted in a manner aligned with the spacecraft's center of mass to generate a force to change the spacecraft's position without generating a torque to rotate the spacecraft.

[0021] Toward these ends, some illustrative embodiments provide controllers, methods, and programs for maintaining spacecraft near desired orbits. Some illustrative embodiments provide a key solution to controlling the operation of multiple spacecraft for long-duration, constrained, collision-free motion near NRHOs for missions performing satellite servicing, active debris mitigation, in-space manufacturing, space station resupply, and planetary sample return.

[0022] Accordingly, one embodiment discloses a controller for maintaining a spacecraft near orbit. The controller includes a memory having instructions stored thereon and a processor, the processor executing the instructions to cause the controller to detect that a distance from the spacecraft to the orbit is greater than a spacecraft threshold. The processor is further configured, in response to detecting that the distance from the spacecraft to the orbit is greater than a spacecraft threshold, to linearize dynamics of the spacecraft in its current state relative to a high-fidelity reference trajectory over a time horizon to generate a state transition matrix (STM) for uncontrollable motion of the spacecraft within the time horizon. The STM includes non-extended eigenvectors having a magnitude less than or equal to one and extended eigenvectors having a magnitude greater than one. The processor is further configured to determine a control action that changes a next state of the spacecraft to a linear combination of the non-extended eigenvectors of the state transition matrix. The processor is further configured to generate control commands to actuators of the spacecraft that cause a modification of the next state of the spacecraft along a direction corresponding to at least one of the non-extended eigenvectors of the state transition matrix.

[0023] Another embodiment discloses a computer-implemented method for maintaining a spacecraft near-orbit, the method including detecting that a distance from the spacecraft to the orbit is greater than a spacecraft threshold. In response to detecting that the distance from the spacecraft to the orbit is greater than the spacecraft threshold, dynamics of the spacecraft in its current state are linearized with respect to a high-fidelity reference trajectory over a time horizon to generate a state transition matrix for uncontrolled motion of the spacecraft within the time horizon. The state transition matrix includes non-extended eigenvectors having a magnitude less than or equal to one and extended eigenvectors having a magnitude greater than one. The method further includes determining a control action to change a next state of the spacecraft to a linear combination of the non-extended eigenvectors of the state transition matrix, and generating control commands to actuators of the spacecraft that cause a modification of the next state of the spacecraft along a direction corresponding to at least one of the non-extended eigenvectors of the state transition matrix.

[0024] Yet another embodiment discloses a non-transitory computer-readable storage medium having embodied thereon a program executable by a processor for performing a method for maintaining a spacecraft near-orbit, the method including detecting that a distance from the spacecraft to the orbit is greater than a spacecraft threshold. In response to detecting that the distance from the spacecraft to the orbit is greater than the spacecraft threshold, dynamics of the spacecraft in its current state are linearized with respect to a high-fidelity reference trajectory over a time horizon to generate a state transition matrix for uncontrolled motion of the spacecraft within the time horizon. The state transition matrix includes non-extended eigenvectors having a magnitude less than or equal to one and extended eigenvectors having a magnitude greater than one. The method further includes determining a control action to change a next state of the spacecraft to a linear combination of the non-extended eigenvectors of the state transition matrix, and generating control commands to actuators of the spacecraft that cause a modification of the next state of the spacecraft along a direction corresponding to at least one of the non-extended eigenvectors of the state transition matrix.

[0025] The presently disclosed embodiments are further described with reference to the accompanying drawings, in which: The drawings shown are not necessarily to scale, emphasis instead generally being placed upon illustrating the principles of the presently disclosed embodiments. [Brief explanation of the drawings]

[0026] [Figure 1A] FIG. 2 is a block diagram illustrating several method steps for proper motion maneuver design that provides fuel-efficient stationkeeping and limited relative motion control for a spacecraft, according to some illustrative embodiments. [Figure 1B] 1 is a flowchart of some steps for determining a state transition matrix for uncontrolled motion of a spacecraft, according to some illustrative embodiments. [Figure 1C]1 is another flowchart of some steps for determining a state transition matrix for uncontrolled motion of a spacecraft, according to some illustrative embodiments. [Figure 1D] FIG. 1 is a block diagram illustrating a method for controlling the operation of a spacecraft to remain within certain boundaries of a reference trajectory, according to some illustrative embodiments. [Figure 1E] FIG. 2 is a block diagram illustrating some components of a controller for maintaining a spacecraft near orbit in accordance with some illustrative embodiments. [Figure 2A] FIG. 2 is a schematic diagram illustrating a high-fidelity reference trajectory, also referred to as a baseline solution, according to some exemplary embodiments. [Figure 2B] FIG. 1 is a schematic diagram illustrating a maneuver that transitions a spacecraft into non-extended local proper motion at apolumn if a trigger condition is met, according to some illustrative embodiments. [Figure 3] 10 is a graph illustrating exemplary distance from baseline of proper motion, according to some exemplary embodiments. [Figure 4] FIG. 10 illustrates an algorithm associated with local proper motion control, according to some exemplary embodiments. [Figure 5] 1 is a graph illustrating deviations from a baseline trajectory for two spacecraft operating under local proper motion control, according to some exemplary embodiments. [Figure 6A] 1 is a schematic diagram of some conventional parameters illustrating aspects used to implement methods and systems, according to some exemplary embodiments. [Figure 6B] 1 is a schematic diagram of some conventional parameters illustrating aspects used to implement methods and systems, according to some exemplary embodiments. [Figure 6C] 1 is a schematic diagram of some conventional parameters illustrating aspects used to implement methods and systems, according to some exemplary embodiments. [Figure 6D]1 is a schematic diagram of some conventional parameters illustrating aspects used to implement methods and systems, according to some exemplary embodiments. [Figure 7A] FIG. 1 is a block diagram illustrating some components for implementing generated control commands, according to some exemplary embodiments. [Figure 7B] 1 is a schematic diagram illustrating aspects of a thruster configuration, according to some illustrative embodiments. FIG. [Figure 8] FIG. 1 is a schematic diagram illustrating some components used to implement methods and systems, according to some exemplary embodiments. [Figure 9] FIG. 1 is a schematic diagram illustrating, by way of non-limiting example, a computing device for implementing some techniques of the methods and systems, according to some illustrative embodiments. DETAILED DESCRIPTION OF THE INVENTION

[0027] While the above-identified drawing figures set forth embodiments disclosed herein, other embodiments are contemplated, as noted in the present discussion. The present disclosure presents exemplary embodiments by way of representation and not limitation. Those skilled in the art may devise numerous other modifications and embodiments that fall within the scope and spirit of the principles of the embodiments disclosed herein.

[0028] Deploying spacecraft and space stations in orbits around celestial bodies is a challenging task, requiring robust, low-cost strategies for stationkeeping and relative motion tailored to these specialized orbits. One example of such a deployment is when an approaching spacecraft needs to be deployed around the Lunar Orbital Platform-Gateway (LOP-G), also known as the Gateway, and is scheduled to be deployed in a long-ellipsoidal, seven-day lunar polar orbit (NRHO) around the Moon. In the context of such a spacecraft deployment, it is desirable to have as much uncontrolled natural motion as possible to reduce fuel consumption and the need for frequent onboard energy replenishment. Several factors can hinder or prevent attempts to deploy a spacecraft in a region near the NRHO. Therefore, any small deviation from the solution can cause the spacecraft to rapidly deviate from the calculated trajectory, necessitating stabilizing control actions.

[0029] Exemplary embodiments disclosed herein provide a control approach that utilizes eigenvectors of a state transition matrix (STM) associated with a high-fidelity NRHO solution in an ephemeris model to design long-term stationkeeping and constrained relative motion. The proposed strategy effectively utilizes the spacecraft's natural motion so that control actions are infrequent and fuel-efficient. This ensures that the spacecraft can exhibit long-term, constrained, collision-free relative motion around the Gateway. Furthermore, because the proposed strategy does not involve any periodic solutions or compute a high-fidelity solution for each spacecraft, the approach on which some exemplary embodiments are based is fast, computationally efficient, inexpensive, and scalable, resulting in improved spacecraft energy efficiency.

[0030] In this regard, exemplary embodiments utilize a single pre-calculated high-fidelity NRHO solution while ensuring a safe separation distance between spacecraft. Some exemplary embodiments exploit several special conditions in the vicinity of the reference trajectory that result in a desired natural motion without deviation. These conditions form a desired space around the reference trajectory. When a spacecraft is controlled to this space rather than to the orbit itself, the requirement is relaxed from staying strictly on orbit to staying “near” orbit, which trades off distance to orbit for improved fuel efficiency. To determine these special conditions, some exemplary embodiments apply local mode decomposition of the receding horizon state transition matrix (STM) associated with the high-fidelity reference trajectory. To prevent unnecessary or undesired spacecraft control that could result in undue energy consumption, some exemplary embodiments utilize trigger conditions to determine when a control law / policy should be executed. The trigger conditions are based on the spacecraft’s distance from the reference trajectory. Using trigger-based control can reduce the frequency at which a spacecraft executes control. This is because the spacecraft can follow natural motion for an extended period of time before crossing a trigger threshold, the point at which a near-desired condition exists in the subspace in which the spacecraft can be maneuvered, and beginning to deviate from the reference trajectory.

[0031] Another advantage of the control approach provided by various exemplary embodiments is that the proposed strategy allows for the control (collision avoidance) of multiple spacecraft relative to one another without the need to calculate a reference trajectory for each spacecraft separately: the same subspace of special states associated with a single reference trajectory can be used for each spacecraft at different distances from the single reference trajectory.

[0032] As such, some exemplary embodiments of the present disclosure provide an important solution for controlling the operation of multiple spacecraft for long-term, constrained, collision-free motion near NRHOs for missions performing satellite servicing, active debris mitigation, in-space manufacturing, space station resupply, and planetary sample return. Furthermore, proximity operations are a key process for achieving mission objectives and a key technology for space exploration. Indeed, the systems and methods of the present disclosure can be applied for several purposes, including, but not limited to, satellite servicing, orbital debris removal, in-space manufacturing, space station resupply, and planetary science sample return missions.

[0033] These and several other advantages will be apparent from the following detailed description of exemplary embodiments of the proposed control strategy. Although some exemplary embodiments are described in relation to lunar orbits, it can be contemplated that the proposed control strategy is applicable to any multi-body celestial system. Thus, the exemplary embodiments described herein should not be limited to the lunar celestial system, which is referenced for illustrative purposes only. The scope of the proposed control strategy encompasses situations and systems related to any multi-body celestial system.

[0034] 1A is a block diagram illustrating several method steps for proper motion maneuver design that provides fuel-efficient stationkeeping and limited relative motion control, according to some exemplary embodiments. In particular, the described method steps relate to any system and method for controlling the operation of a spacecraft over a finite time horizon.

[0035] In some exemplary embodiments, a spacecraft may be intended to orbit a central body, such as a celestial body. The spacecraft may do so by orbiting itself or by deployment to a nearby space station orbiting the central body. In this regard, the spacecraft may be capable of ascertaining or obtaining its current state within a specified time period. The spacecraft state may include data related to one or a combination of the spacecraft's position and translational velocity, and perturbations acting on a multi-body celestial system of which the spacecraft is a part. A check may be performed as to whether the spacecraft's distance to orbit is greater than a spacecraft threshold. If the distance is greater than or, as the case may be, equal to the spacecraft threshold, this corresponds to a situation in which the spacecraft is deviating, or, as the case may be, beginning to deviate in any direction from the desired orbit. Therefore, it is desirable to execute a prompt control action to return the spacecraft to a path on or along the desired orbit. In this manner, the condition (3) in which the spacecraft's distance to orbit is detected to be greater than the spacecraft threshold can be utilized as a trigger to execute a control policy to prevent the spacecraft from deviating from the desired orbit.

[0036] Upon detecting that the trigger condition in step 3 of FIG. 1A is satisfied, the controller may linearize the spacecraft dynamics in its current state against a high-fidelity reference trajectory over a time horizon to generate a state transition matrix (STM) for the uncontrolled motion of the spacecraft within that time horizon in step 5 of FIG. 1A. In some exemplary embodiments, the high-fidelity reference trajectory may be pre-computed and stored in storage, such as a database, accessible to the controller. In some other exemplary embodiments, the reference trajectory may be dynamically calculated at run time. The linearization of the spacecraft dynamics may be performed over discrete time instances, and the state transition matrix may be obtained directly.

[0037] In some exemplary embodiments, the time horizon is a finite time horizon, and the designated time period during which the current state of the spacecraft is ascertained may overlap the finite time horizon. In some exemplary embodiments, the designated time period may be outside the finite time horizon.

[0038] Following linearization of the spacecraft's orbital dynamics about a high-fidelity reference trajectory, an eigenvalue decomposition of the resulting linearized STM is performed to identify a region (subspace) of special states in the vicinity of the reference trajectory. The special states may be states that result in desired natural motion without deviation. Throughout this disclosure, natural motion or uncontrollable motion may refer to spacecraft motion without control (i.e., without using fuel or onboard power). The resulting STM includes non-extended eigenvectors with magnitudes less than or equal to one and extended eigenvectors with magnitudes greater than one. In some exemplary embodiments, the special states that indicate desired natural motion arise from eigenvectors of the STM that have eigenvalues ​​with magnitudes less than one.

[0039] Step 7 of Figure 1A involves determining a desired control action to achieve a state that is a linear combination of the non-extended eigenvectors of the state transition matrix. If the spacecraft follows its calculated trajectory and a deviation from the desired trajectory is detected in the next state, the controller needs to execute a corrective maneuver for the next state. In this regard, the controller determines a control action that changes the next state corresponding to a linear combination of the non-extended eigenvectors of the STM.

[0040] Step 9 of FIG. 1A includes generating one or more control commands to activate or deactivate one or more thrusters of the vehicle for a specified time period based on the one or more control commands to achieve a desired corrective maneuver. The control commands are intended to cause a modification of the spacecraft's next state along a direction corresponding to at least one of the non-extended eigenvectors of the STM. The generated control commands are output to an interface of the spacecraft for further action in step 11. In some exemplary embodiments, the controller may be embodied on the spacecraft itself, such that in step 11 the controller provides control commands directly to the spacecraft's actuators (propellers and / or thrusters) to achieve the desired corrective maneuver. In some exemplary embodiments, the controller may communicate with the spacecraft remotely, such that the control commands may be transmitted to the spacecraft, and an onboard CPU of the spacecraft may process the control commands accordingly to achieve the desired corrective maneuver.

[0041] FIG. 1B illustrates a flowchart 5a of several steps for determining a state transition matrix for uncontrolled motion of a spacecraft, according to some exemplary embodiments. A dynamic model of a multi-body celestial system may be obtained (151). The multi-body celestial system may include a spacecraft and at least one celestial body. The dynamic model may be expressed as analytical equations. Because the dynamic model may be nonlinear, it may be linearized in the next step at 153 to obtain a linear model of the multi-body celestial system (155). This may represent continuous-time analytical equations for linearization, but discrete-time matrices may be required to obtain the state transition matrix. As part of the steps leading to the state transition matrix, flowchart 5a includes propagating the linearized model over a specified period of time (157). For example, the specified period may span the entire baseline period. In this regard, a differential equation (such as Differential Equation 11, described below) may be solved (157a) over the specified period of time to obtain a discrete-time state transition matrix 159.

[0042] 1C illustrates another flowchart 5b of steps for determining a state transition matrix for uncontrolled motion of a spacecraft, according to some exemplary embodiments. At 161, a dynamical model of a multi-body celestial system expressed as analytical equations may be obtained. A baseline solution, also referred to as a reference trajectory, may also be obtained (163). Discrete-time updates of perturbations of the baseline solution may be calculated (165). For example, let r(t) represent a solution, which may be the baseline solution in some exemplary embodiments. Discrete-time updates of perturbations of r(t) may be calculated.

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[0043]

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[0044] In this way, the STM can be obtained numerically with fewer calculations, thereby improving fuel efficiency.

[0045] 1D is a block diagram illustrating a method for controlling spacecraft motion to remain within certain bounds of a reference trajectory, according to some embodiments of the present disclosure, for example, the method controlling spacecraft motion using control inputs determined using a model of a joint multi-body celestial body system based on optimizing a cost function having an objective function that minimizes fuel / onboard power consumption.

[0046] The first step 110 of FIG. 1D involves determining the current state of the spacecraft, which may be determined using sensors or other aspects such as hardware or software. Additionally or alternatively, the current state of the spacecraft may be obtained from communications with a ground command center located on Earth or another spacecraft located in space, such as GPS, relative distance measurements, star trackers, horizon sensors, etc. In some exemplary embodiments, the current state of the spacecraft may be determined based on previous control inputs determined in a previous iteration, optimized with a previous cost function using a previous model of the spacecraft. As used herein, state includes one or a combination of the spacecraft's position and translational velocity and perturbations acting on a multi-body celestial body. Continuing to refer to FIG. 1D, the state determined in step 110 may be an absolute state relative to a central body around which the spacecraft is orbiting.

[0047] Step 130 of Figure 1D determines a current control action / input for controlling the spacecraft in a current time period using the current model of joint multi-body celestial body dynamics. In this regard, the current model of joint multi-body celestial body dynamics may be fetched from a database or other storage accessible to the controller implementing the method of Figure 1D. In some exemplary embodiments, the current model of joint multi-body celestial body dynamics may be generated based on dynamics data of the spacecraft, the central body around which the spacecraft is orbiting, and / or other bodies and objects around the spacecraft.

[0048] 1D , the method determines a sequence of future thruster force inputs over a period of time into the future using the current model of the joint multi-body celestial system dynamics, at least as long as new state measurements are acquired such that the predicted future spacecraft states and inputs satisfy the constraints on spacecraft motion and the constraints on the control inputs. For example, the sequence of future thruster force inputs may include forces that result in a new future state in which the spacecraft is no longer deviating from the desired orbit.

[0049] Step 136 of Figure 1D uses the thruster profile as an input to the spacecraft. Using this profile, future inputs can be applied to the spacecraft.

[0050] 1D, the next state of the spacecraft is determined based on the current state of the spacecraft determined in step 110 and the current control inputs to the spacecraft determined in step 130, and the controller waits until a new state measurement is received in step 140. Control of the step then returns to step 110 for the next iteration, and the process is repeated.

[0051] 1E is a block diagram illustrating some components of a controller that implements at least some steps of the methods of FIGS. 1A and 1D , according to some embodiments of the present disclosure. The method of FIG. 1D may include a control system or controller 101 having at least one processor 113 for executing the modules of the controller. The controller 101 may be in communication with the processor 113 and a memory 119. The memory may store instructions and data including a cost function 121, a joint multi-body celestial body model 123, and constraints 129.

[0052] 1D may determine control inputs 107 via processor 113 using joint multi-body celestial body model 123, subject to constraints 129. In some exemplary embodiments, determined control inputs 107 may be transmitted to spacecraft 102. To that end, controller 101 may be included in or operatively connected to an output interface, which is configured to issue control commands 107 to thrusters 103 of spacecraft 102. Furthermore, spacecraft 102 may have thrusters 103 and sensors 108, among other components. Current state 106 of spacecraft 102 may be obtained from sensors 108 and communicated to processor 113.

[0053] 1E , in some exemplary embodiments, processor 113 may determine at least one of cost function 121, joint multi-body celestial body model 123, and constraints 129 during control. For example, controller 101 may execute a method such as that of FIG. 1D to iteratively control the operation of spacecraft 102 using control inputs of step 130 of FIG. 1D determined using joint multi-body celestial body model 123 based on optimizing cost function 121. It is also contemplated that the method of FIG. 1D may be executed by controller 101 based on a previous iterative operation of spacecraft 102, i.e., from a previous iterative control action having previous control inputs determined in a previous iteration that were optimized by a previous cost function using a previous model of spacecraft 102.

[0054] 2A is a schematic diagram illustrating a high-fidelity reference trajectory, according to some exemplary embodiments. The exemplary high-fidelity reference trajectory shown in FIG. 2A may also be referred to as a baseline NRHO solution 201. The baseline NRHO solution 201 (represented in the Earth-Moon rotating coordinate system) may consist of 60 revolutions around the Moon over a period of 394 days. With reference to FIG. 2A, the Moon is indicated by the label 203, and the Earth is indicated by the label 205.

[0055] Several concepts related to spacecraft control are discussed below. A lunar long-recurve polar orbit (NRHO) is a periodic trajectory around the L1 and L2 Lagrangian points of the Earth-Moon circular restricted three-body problem (CR3BP). Due to their favorable stability characteristics and relatively low stationkeeping costs, NRHOs around the L2 point, where the synchronic resonance is 9:2 and the perilunar radius is approximately 3150 km, have been selected for Gateway deployment. However, NRHOs do not exist in reality because CR3BP ignores effects such as solar radiation pressure (SRP), gravitational forces due to celestial bodies other than the Earth and Moon, and the lunar J2 zonal spherical harmonics. Ignoring these higher-order effects during mission design would lead to unacceptably high fuel consumption. Therefore, a high-fidelity astrodynamic model using ephemeris data is used in practice to generate a solution that most closely resembles an NRHO in CR3BP. This high-fidelity solution is nonperiodic and consists of a finite number of orbits around the Moon. The astrodynamics model considered in some exemplary embodiments accounts for all major predictable forces acting on a spacecraft in cislunar space. Any major predictable force has a magnitude greater than the magnitude of the largest unpredictable force. In the model considered here, the largest unpredictable force affecting a spacecraft is determined to be an indirect disturbance caused by navigation errors affecting the spacecraft's controller. Navigation errors are quantified under the assumption that spacecraft states are estimated using measurements from the Deep Space Network (DSN). Under these assumptions, the major predictable forces acting on a spacecraft in the region of space occupied by the NRHO of the CR3BP are the SRP, the lunar J2 zonal spherical harmonics, and gravity due to the Earth, Moon, and Sun.

[0056]

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[0057]

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[0058]

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[0059] The state of the spacecraft at t2, obtained after the impulse at t', can then be determined as follows:

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[0060]

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[0061]

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[0062]

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[0063]

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[0064]

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[0065]

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[0066] Local proper motion control Some exemplary embodiments are called receding horizon STM, [0, t max] is used to determine the non-extended local proper motions corresponding to those intervals.

[0067] 2B is a schematic diagram illustrating a maneuver that transitions a spacecraft to a non-extended local proper motion at apolumn if a trigger condition is met, according to some exemplary embodiments. The spacecraft may have a planned trajectory having an initial spacecraft path 251. For the reasons described above, the spacecraft may tend to deviate from the initial spacecraft path 251 to a deviation path 253.

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[0068] In principle, the STM could be calculated over the entire baseline period, and the non-extended proper motion could be chosen so that a constrained relative motion trajectory over the entire baseline period is obtained in one shot. However, this approach is computationally infeasible because the condition number of the STM increases as the period over which it applies increases. In practice, the longest period for which the STM is reliable is generally the time required for 12 revolutions around the Moon, approximately 78 days.

[0069]

number

[0070]

number

[0071]

number

[0072] Note that the resulting linear combination does not include components along directions with two or more eigenvalues. The basis for selecting states in the linear combination is based on the criterion that it should provide a desired behavior (motion) that does not cause the spacecraft to deviate from the reference trajectory / baseline solution. The basis for determining how many of the states should be in the linear combination is based on an optimization problem. For example, an objective function for minimizing fuel consumption may be a major factor in determining which components of the states should be in the resulting linear combination. Therefore, among many special states, only those that the optimization problem determines have the lowest fuel consumption cost can be selected. In this regard, in some exemplary embodiments, it may be contemplated that the resulting combination may include at least one of the special states. In scenarios where the resulting combination includes multiple states, the resulting state has a direction that is a hybrid of the directions of the selected eigenvectors. Because the eigenvectors that result in the desired stable motion only provide several major directions, it may be necessary in some cases to select a combination of those major directions.

[0073]

number

[0074]

number

[0075] 4 is a diagram illustrating an algorithm associated with local proper motion control, according to some exemplary embodiments. Referring to FIG. 4, the proposed control approach results in an automated routine summarized by algorithm 400 of FIG.

[0076]

number

[0077]

number

[0078]

number

[0079]

number

[0080] Continuing with FIG. 4, Trigger evaluates to true if the trigger condition (14) is met.

[0081]

number

[0082]

number

[0083] Still referring to Figure 4, Δν j (j=1,...,M) is the period [0,t max ]. The cumulative cost of M maneuvers is denoted by Δν.

[0084] According to some exemplary embodiments, a practical implementation of the proposed control strategy can be demonstrated by augmenting algorithm 400 of FIG. 4 with a Kalman filter that estimates state using simulated range and range-rate measurements from a Deep Space Network (DSN). The measurements can be received at a rate of 6 hours, which is reasonable for the DSN and sufficient to provide accurate state estimation. The errors in the range and range-rate measurements follow a standard normal distribution, with standard deviations of 10 m and 1 mm s, respectively.-1 is.

[0085] The proposed approach is demonstrated for the case where two spacecraft perform constrained relative motion near the baseline solution. In particular, the Gateway undergoes tight stationkeeping near the baseline while the approaching spacecraft is made to perform collision-free relative motion near the Gateway.

[0086] The calculated trajectory for the Gateway, referred to as Solution 1, uses ρ = 10, Δt = 6 hours, Δr = 0.5 km, and N = 4, while the trajectory for the approaching spacecraft, referred to as Solution 2, is calculated with ρ = 2, Δt = 12 hours, Δr = 50 km, and N = 6. Specific parameters in Algorithm 9 affect the properties of the constrained motion solutions. In particular, the selection of ρ and Δr in (14) for the two solutions helps ensure they remain collision-free. These parameters are adjusted so that the resulting solutions are both rare and fuel-efficient.

[0087] It is worth noting that the effects of navigation uncertainty are more pronounced for tight stationkeeping maneuvers near the baseline. When maneuvers are initiated near the baseline, the final state (located near the baseline and aligned with the desired eigenvector) is more susceptible to being corrupted by navigation uncertainty due to its small magnitude measured relative to the baseline. As a result, the spacecraft may be maneuvered into a state that is not properly aligned with the desired eigenvector, which will cause the spacecraft to deviate prematurely. Therefore, tight stationkeeping may require more maneuvers per year in some cases. This pitfall is avoided while generating Solution 1 by selecting a small value for Δr0 and a large value for ρ. This allows the spacecraft to slowly offset from the baseline over a 150-day period, settling at a distance of approximately 9 km from the baseline where maneuvers are triggered less frequently.

[0088] FIG. 5 is a graph illustrating the deviation from the baseline trajectory for two spacecraft operating under local proper motion control, according to some exemplary embodiments. The annual stationkeeping performance of Solution 1 (502) and Solution 2 (504) is shown in Table 1 below, and the distance of those solutions from the baseline as a function of time is shown in FIG. 5. With the proposed strategy, the annual stationkeeping cost of the Gateway is comparable to or better than the cost of the current state of the art for stationkeeping on the NRHO. Because the approaching spacecraft maintains a greater distance from the baseline than the Gateway, the approaching spacecraft requires significantly more fuel. Therefore, each maneuver to transition to the baseline's non-extended local proper motion is more expensive. [Table 1]

[0089] Another notable benefit of the proposed approach is the relatively small number of maneuvers required to maintain the annual constrained motion. The dark areas on Solutions 1 and 2 in Figure 5 highlight the significantly smaller annual control duty cycle. Furthermore, Solutions 1 and 2 pose no risk of collision between the Gateway and approaching spacecraft. The distance between the solutions is never less than 8 km. Features

[0090] It is contemplated that a controller can maintain a spacecraft near a desired orbit using the proposed approach. To detect whether a trigger condition is met, the controller can determine a current state of the spacecraft within a specified time period. To determine the current state of the spacecraft, a processor can obtain measurements corresponding to one or a combination of a position and translational velocity of the spacecraft and a perturbation, the perturbation acting on a multi-body celestial system defined by the spacecraft and at least one celestial body. To determine the state transition matrix, the processor can obtain a linear model of a dynamical system defined by the spacecraft and the at least one celestial body. The processor can also propagate the linear model of the dynamical system over a time period defined by a first time instance during which the spacecraft is in a current state and a second time instance during which the spacecraft is in a next state.

[0091] Furthermore, the control action can be determined by determining the direction of modification of the next state by a linear combination of the directions of each non-extended eigenvector of the STM to realize a mixed state with the natural motion, where the natural motion is a combination of the corresponding eigenmotions of each non-extended eigenvector of the STM.

[0092] To generate the control commands, the processor may solve a nonlinear optimization problem to find fuel-efficient maneuvers that transition the spacecraft into a set of desired states that result in the desired natural motion. The nonlinear optimization problem is directed to a finite-horizon optimization of a model of the spacecraft, a set of spacecraft motion objectives, and constraints on the spacecraft propulsion system and spacecraft motion. The constraints on the spacecraft propulsion system and spacecraft motion include one or more physical limitations of the spacecraft, one or more safety limitations on the spacecraft operation, and one or more performance limitations on the spacecraft trajectory.

[0093] The spacecraft may perform a task of docking with a space station that maintains a position near orbit at a third distance less than the station threshold, and the controller is further configured to select a spacecraft threshold greater than the station threshold.

[0094] The next state of the spacecraft may include one or a combination of the position, orientation, translational and angular velocity of one or more of the spacecraft and its payload, and perturbations acting on a multi-body celestial body defined by the spacecraft, payload, and one or more celestial bodies. The perturbations acting on the multi-body celestial body are natural orbital forces, including solar and lunar gravitational perturbations, anisotropic gravitational perturbations due to non-sphericity of the central body, solar radiation pressure, and atmospheric drag.

[0095] The high-fidelity reference trajectory is an uncontrolled natural motion trajectory located near the lunar prolate polar orbit (NRHO) in the Earth-Moon circularly restricted three-body problem, and is estimated using a multiple shooting approach. The high-fidelity reference trajectory includes a finite number of non-periodic orbits around the Moon over a finite number of days.

[0096] The orbit may be one of a circular orbit, an elliptical orbit, a halo orbit, a lunar polar elliptical orbit, or a quasi-satellite orbit. The control commands are generated as a solution to a model predictive control policy that generates the control commands by optimizing a cost function over a receding horizon. The cost function includes a stabilization component for directing spacecraft motion to a next state, a component for an objective of spacecraft operation, and a performance component for optimizing spacecraft motion until the next state is achieved. In some exemplary embodiments, the cost function includes an objective function that minimizes spacecraft fuel consumption. The control commands are generated for each specified period of a plurality of specified periods in the time horizon or are generated iteratively over the receding time horizon. The control commands are output to an operation module of the controller, which communicates the control commands to a thruster command module, which receives the control commands as delta v commands, which converts the delta v commands into thruster commands and sends the thruster commands to a thruster processor of at least one thruster to activate or deactivate the at least one thruster for trajectory tracking control of the vehicle according to the converted delta v commands. definition

[0097] In accordance with aspects of the present disclosure and based on experimentation, the following definitions have been established, although these definitions are certainly not a complete definition of each phrase or term. The definitions provided are provided merely as examples based on what has been learned from experimentation, and other interpretations, definitions, and other aspects may be relevant. However, such definitions are provided for at least a basic preview of the phrases or terms presented.

[0098] Space rendezvous: A space rendezvous may be a set of orbital maneuvers in which two spacecraft (or a chaser spacecraft and a target (i.e., the target may be another spacecraft, a space station, a celestial body, or orbital debris)) arrive in the same orbit and come very close (e.g., within visual range).

[0099] Celestial system (astronomical reference frame): In astronomy, an astronomical coordinate system (or astronomical reference frame) is a system for specifying the positions of satellites, planets, stars, galaxies, and other celestial objects relative to a physical reference point available to a present observer (e.g., horizontal north for an observer present on the Earth's surface). A coordinate system can specify the location of an object in three-dimensional space, or simply plot its direction on the celestial sphere if the object's distance is unknown or irrelevant. Coordinate systems are realized as either spherical or Cartesian coordinate systems. Spherical coordinate systems projected onto the celestial sphere are similar to geographic coordinate systems used on the Earth's surface. The difference between them lies in the choice of base plane, which divides the celestial sphere into two equal hemispheres along great circles. A Cartesian coordinate system with appropriate units is simply the Cartesian equivalent of a spherical coordinate system with the same base (x, y) plane and principal (x-axis) direction. Each coordinate system is named according to its choice of base plane.

[0100] 6A, 6B, 6C, 6D, and 6E are schematic diagrams of some conventional parameters illustrating aspects used to implement methods and systems, according to some exemplary embodiments.

[0101] conic sectionsReferring to FIG. 6A, a conic section (also synonymously referred to as a conic section) is a curve formed by intersecting a plane with a right circular cone 602. FIG. 6A illustrates the angular orientation of the plane 600 relative to the cone 602, which determines whether the conic section is a circle 604, an ellipse 606, a parabola 608, or a hyperbola 610, 612. The circle 604 and the ellipse 606 arise when the intersection of the cone 602 and the plane 600 forms a boundary curve. The circle 604 is a special case of the ellipse 606 in which the plane 600 is perpendicular to the axis of the cone 602. If the plane 600 is parallel to the generatrix of the cone 602, the conic section is called a parabola 608. Finally, if the intersection forms a boundary curve and the plane 600 is not parallel to the generatrix of the cone 602, the figure is a hyperbola 610, 612. In the latter case, plane 600 intersects both halves of cone 602, forming two separate curves. All conic sections can be defined by their eccentricity. The type of conic section is also related to the semimajor axis and energy. Table 2 below shows the relationship between eccentricity, semimajor axis, and energy and the type of conic section. [Table 2] The satellite orbit can be any of four conic sections.

[0102] Referring to Figures 6B, 6C and 6D, to mathematically describe a conventional orbit, six quantities called orbital elements must be defined. The six orbital elements are: Orbit semi-major axis a Eccentricity e Inclination angle i Argument of periapsis ω Periapsis passing time T Ascending node ecliptic longitude is.

[0103] Figures 6B-6D show a conventional orbiting satellite 650 following an oblong-shaped path known as an ellipse 620, with the celestial body about which the orbit is centered, called the primary star, located at one of two points called foci 622, 624. Figure 6C shows the ellipse 620, which is defined to be a curve with the property that at each point on the ellipse, the sum of its distances from two fixed points, called foci 622, 624, is constant. The longest and shortest lines that can be drawn through the center of the ellipse are called the major and minor axes, respectively. The semimajor axis is one-half the major axis and represents the average distance from the satellite to its primary star. The eccentricity is the distance between the foci divided by the length of the major axis and is a number between 0 and 1. An eccentricity of zero indicates a circle.

[0104] Figure 6D shows the inclination angle i, which is the angular distance between the satellite's orbital plane and the equatorial plane of its star (or the ecliptic plane in the case of a heliocentric or sun-centered orbit). An inclination angle i of zero degrees indicates an orbit around the equatorial plane of the star in the same direction as the star's rotation, an orientation called prograde (or direct). An inclination angle i of 90 degrees indicates a polar orbit. An inclination angle i of 180 degrees indicates a retrograde equatorial orbit. A retrograde orbit is an orbit in which the satellite moves in the opposite direction to the rotation of its star.

[0105] Continuing with FIG. 6D, periapsis ω is the point in the orbit closest to the star (i.e., for an object moving in an elliptical orbit around another celestial body, the point of closest approach is the periapsis, and at this point in the orbit, the object moves with the greatest velocity, per Kepler's second law). The opposite of periapsis ω is the point of furthest approach, called apoapsis (i.e., for an object moving in an elliptical orbit around another celestial body, the point of furthest away is the apoapsis, and at this point in the orbit, the object moves with the least velocity, per Kepler's second law). Perihelion is the position of closest approach, i.e., the point at which the distance between the Sun and the planet is the shortest, and at this point in the orbit, the planet moves with its greatest velocity, per Kepler's second law. Aphelion is the point at which the distance between the Sun and the planet is the farthest, and at this point in the orbit, the planet moves with the least velocity, per Kepler's second law. Aphelion, specifically in orbits around the Sun, corresponds to the apoapsis of a general orbit. Perigee ω and apogee are usually corrected for the celestial body about which the orbit is centered, e.g., perihelion and aphelion for the Sun, perigee and apogee for the Earth, perijove and apojove for Jupiter, and perilunar and apojuvenile for the Moon. The argument of periapsis ω is the angular distance between the ascending node N1 and the periapse (see Figure 6D). The time of periapsis T is the time when the satellite passes through its periapse.

[0106] periapsis: In an elliptical orbit of a celestial body around the center of mass of a system, the point at which the distance between the celestial body and the center of mass is smallest. Symbolized by ω (also called the perifocus argument or pericenter argument), ω is one of the orbital elements of an orbiting celestial body. As a parameter, ω is the angle from the ascending node of the celestial body to periapsis, measured in the direction of motion. For certain types of orbits, terms such as perihelion (orbit around the Sun), perigee (orbit around the Earth), and periasterism (orbit around a star) may be substituted for the word periapsis (see orbital poles for details). A periapsis argument of 0° means that the orbiting body is closest to the center of mass when crossing the reference plane from south to north. A periapsis argument of 90° means that the orbiting body reaches periapsis at its farthest north from the reference plane. The longitude of the ascending node plus the periapsis argument equals the longitude of the periapsis. However, especially in discussions of binary stars and exoplanets, the terms "periapsis longitude" or "pericelestial longitude" are often used synonymously with "periapsis argument."

[0107] far point : In an elliptical orbit of a celestial body around the center of mass of the system, it is the point at which the distance between the celestial body and the center of mass is greatest.

[0108] intersection : The point where an orbit intersects a plane, for example, the point where a satellite intersects the Earth's equatorial plane. If the satellite intersects the plane from south to north, the point of intersection is the ascending node N1; if it intersects the plane from north to south, the point of intersection is the descending node N2. The longitude of the ascending node N1 is the umeltima of this node. The umeltima is similar to longitude on Earth and is measured in degrees counterclockwise from zero, with zero longitude being towards the vernal equinox Ω.

[0109] Orbital TypesA geosynchronous orbit (GEO) is a circular orbit around the Earth with a period of 24 hours. A geosynchronous orbit with an inclination of 0 degrees is called a geostationary orbit. A spacecraft in a geostationary orbit appears to be stationary and hanging above a point on the Earth's equator, making it ideal for some communications and weather satellites. A spacecraft in an inclined geosynchronous orbit appears to trace a regular figure-eight pattern in the sky with each orbit. To achieve a geosynchronous orbit, a spacecraft is first launched into an elliptical orbit with an apogee of 35,786 km (22,236 mi), called a geosynchronous transfer orbit (GTO). The orbit is then circularized by igniting the spacecraft's engines at the apogee.

[0110] Polar Orbit (PO): An orbit with an inclination angle of 90 degrees. Polar orbits are useful for satellites performing mapping and / or monitoring missions because they allow the spacecraft to access virtually every point on the planet's surface as the planet rotates. Walking Orbit: An orbit in which the orbiting satellite is subject to significant gravitational influences. First, planets are not perfectly spherical and have slightly uneven mass distributions. These instabilities affect the trajectory of a spacecraft. In addition, the sun, moon, and planets exert gravitational influences on orbiting satellites. With proper planning, it is possible to design an orbit that takes advantage of these influences to cause precession in the satellite's orbital plane. The resulting orbit is called a walking orbit.

[0111] Sun-synchronous orbit (SSO): A walking orbit whose orbital plane precesses with the same period as the planet's solar orbital period. In such an orbit, the satellite passes periapsis at the same local time every orbit. This is useful if the satellite carries instruments that depend on some angle of sunlight irradiating the planet's surface. To maintain precise synchronous timing, it may be necessary to perform occasional thrust maneuvers to adjust the orbit.

[0112] Molniya orbit: A highly eccentric Earth orbit with a period of approximately 12 hours (two rotations per day). The orbital inclination is chosen so that the rate of change of perigee is zero, thus maintaining both apogee and perigee over a fixed latitude. This condition occurs between inclinations of 63.4 and 116.6 degrees. For these orbits, the argument of perigee is typically located in the Southern Hemisphere, so the satellite remains over the Northern Hemisphere near the apogee for approximately 11 hours per orbit. This orientation allows for good ground coverage at high latitudes in the Northern Hemisphere.

[0113] Hohmann Transfer Orbit: An interplanetary trajectory whose advantage is that it consumes the least amount of propellant. A Hohmann transfer orbit to an outer planet, such as Mars, is achieved by launching a spacecraft in the direction of Earth's orbit around the Sun and accelerating it until it reaches a velocity that eliminates the Earth's gravitational influence and allows it to enter a solar orbit whose aphelion is identical to that of the outer planet. Once the spacecraft reaches its destination, it must decelerate so that the planet's gravity can capture the spacecraft into the planet's orbit. For example, to send a spacecraft to an inner planet, such as Venus, the spacecraft is launched in the opposite direction to Earth's orbit around the Sun and accelerates (i.e., decelerates) until it reaches a solar orbit whose perihelion is identical to that of the inner planet. Note that the spacecraft continues to move in the same direction as Earth, but slightly slower. To reach a planet, the spacecraft must be inserted into the interplanetary trajectory at the correct time so that it reaches the planet's orbit when the planet is positioned at the point where the spacecraft would intercept it. This task is similar to a quarterback "leading" his receiver so that the ball and the receiver reach the same point at the same time. The time interval within which a spacecraft must be launched to complete its mission is called the launch window.

[0114] Lunar Prolate Polar Orbit (NRHO): can be defined as a "nearly stable" orbit where the stability is measured using the stability index v.

[0115] CR3BP modelLunar polar elliptical orbits are members of the broader L1 and L2 family of halo orbits—structures that exist in a dynamic environment modeled in terms of multiple gravitational bodies. L1 is the first Lagrangian point, a point 1 / 100th the distance from Earth to the Sun, where the centripetal and gravitational forces of Earth and the Sun cancel each other out. It is one of five such points in the Earth-solar system, where a spacecraft could essentially remain suspended forever, balancing on a gravitational version of the head of a pin. Another point, L2, is 1.6 million kilometers away from Earth on the opposite side of the Sun. Both L1 and L2 are ideal vantage points for space exploration, with L1 offering a commanding vantage point of both Earth and the Sun. However, they have drawbacks. At L1, a spacecraft's signal is overwhelmed by radiation from the Sun behind it. At L2, the Earth's shadow blocks the sunlight the spacecraft needs to power its instruments. The solution is to place the spacecraft in a "halo orbit" around a Lagrangian point. A spacecraft in a halo orbit around L1 follows a large, slack loop, perpendicular to the Earth-Sun axis and descending endlessly toward equilibrium. Furthermore, the fundamental behavior adheres to higher-fidelity models, supporting potentially long-term mission scenarios for potentially manned spacecraft in near-lunar orbits. This type of trajectory is first identified in a simple representation of the gravitational effects in the Earth-Moon system, the circularly restricted three-body problem (CR3BP). In the CR3BP model, a non-repeated polar lunar orbit (NRHO) can be defined as a "nearly stable" orbit, where stability is measured using the stability index v. It is characterized by favorable stability properties that suggest the possibility of maintaining a motion similar to an NRHO for extended periods while consuming minimal propellant resources. Some NRHOs also have favorable resonance properties that can be exploited for mission design, particularly to help avoid eclipses. However, to realize a practical mission, such orbital transfer and stationkeeping strategies must be accounted for in higher fidelity ephemeris models.Stationkeeping algorithms for libration point orbits have previously been investigated within this dynamical framework, in the context of both planar Lyapunov orbits and conventional 3D halo orbits. However, NRHO is constructed within the framework of an ephemeris.

[0116] Station KeepingIn astrodynamics, orbital maneuvers performed by thruster burns necessary to maintain a spacecraft in a specific assigned orbit are called orbital stationkeeping. For many Earth satellites, the effects of non-Keplerian forces, namely the deviation of Earth's gravity from that of a homogeneous sphere, the gravity of the Sun / Moon, solar radiation pressure, and air resistance, must be countered. The deviation of Earth's gravity field from that of a homogeneous sphere and the gravity of the Sun / Moon generally perturb the orbital plane. For sun-synchronous orbits, precession of the orbital plane caused by Earth's oblateness is a desirable feature that is part of the mission design, but inclination changes caused by the gravity of the Sun / Moon are undesirable. For geostationary spacecraft, the inclination should be kept small enough to allow tracking by non-steerable antennas, so the inclination changes caused by the gravity of the Sun and Moon must be countered at the expense of a significant amount of fuel. Spacecraft in low Earth orbit often must compensate for the effects of atmospheric drag. For some missions, this is simply necessary to avoid re-entry. For other missions, typically those whose orbits must be precisely synchronized with the Earth's rotation, this is necessary to avoid orbital period shortening. Solar radiation pressure generally perturbs the eccentricity (i.e., the eccentricity vector). See Orbital Perturbation Analysis (Spacecraft). For some missions, this must be actively countered by maneuvers. For geostationary spacecraft, the eccentricity must be kept small enough to allow the spacecraft to be tracked by non-steerable antennas. Also, for Earth-observing spacecraft, where highly repeatable orbits with fixed ground tracks are desirable, the eccentricity vector should remain as fixed as possible. Most of this compensation can be achieved using frozen orbit designs, but thruster maneuvers are required for fine adjustments. Stationkeeping is even more important for spacecraft in halo orbits around Lagrangian points because such orbits are unstable, and without active control through thruster burns, even the smallest deviations in position / velocity would cause the spacecraft to deorbit completely.

[0117] Dominant disturbance force A dominant disturbance is one that causes a deviation greater than the navigation error of the Deep Space Network (DSN) positioning system. A dominant force can also be defined as a force whose influence is greater than the largest unpredictable force. The largest unpredictable force can also be attributed to position and velocity measurement errors.

[0118] perturbation : It can be the complex motion of a massive body subject to other forces besides the gravitational pull of a single other massive body. These other forces can include a third (fourth, fifth, etc.) body, resistance from the atmosphere, and the off-center gravitational pull of an oblate or otherwise distorted body. A perturbing force acts from the Sun on the Moon at two locations in its orbit. The dotted black arrow represents the direction and magnitude of gravity relative to the Earth. Applying it to both the Earth's position and the Moon's position does not perturb their relative positions to each other. When this is subtracted from the force on the Moon (solid black line), what remains is the perturbing force on the Moon relative to the Earth (double black arrow). The perturbing force changes the shape of the orbit because its direction and magnitude differ on each side of the orbit.

[0119] FIG. 7A is a block diagram illustrating some components for implementing generated control commands according to an embodiment of the present disclosure. The thruster controller module 710 may include at least one processor 720 communicatively coupled to a controller, such as controller 101 of FIG. 1E, via one or more interfaces. Furthermore, the processor 720 may be coupled to a thruster sub-controller 740 via buses 728 and 734. The processor 720 may receive control commands as delta-v-commands 701. The processor 720 may convert 724 the received delta-v commands 701 into thruster commands 726, which may be transmitted via bus 728 to a sub-controller 740 for a thruster 743, which may be connected to sensors 748. The sub-controller thruster 740 may include another processor 741 for processing the thruster commands and issuing control actions as commanded.

[0120] 7B is a schematic diagram illustrating aspects of a thruster configuration, according to embodiments of the present disclosure. In some exemplary embodiments, the spacecraft may be equipped with eight thrusters mounted at the corners of the spacecraft such that they are aligned to produce a net force acting on the center of mass of the spacecraft without producing any torque that would cause the spacecraft to rotate. A controller, such as subcontroller 740 of FIG. 7A, may send signals to activate and deactivate the thrusters to move the spacecraft along a commanded trajectory.

[0121]

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[0122] Continuing to refer to FIG. 7B, inertial frame F e The translational motion equations for the main and secondary engines are given by:

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[0123] For the primary and secondary spacecraft, the position of the secondary spacecraft relative to the primary spacecraft is given by:

number

[0124] Continuing to refer to FIG. 7B, the main engine track frame F o Taking the derivative of the relative position (21) with respect to

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[0125] Main engine track frame F o Taking the derivative of the relative velocity (22) with respect to

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[0126]

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[0127] Still referring to Figure 7B, r c Since and h vary along the trajectory, the equations of motion (25) become a linear time-varying system.

number

[0128] FIG. 8 is a block diagram illustrating some components that may be used to implement systems and methods according to some embodiments of the present disclosure. For example, a computer system 870 may be adapted for use in controlling the movement of a spacecraft or vehicle. A CPU or processor 810 may be connected to a memory 812, an input / output device 814, and a communication interface 816 via a bus system 813. Also connected to the bus system 813 may be a storage device 818, a control interface 820, a display interface 822, and an external interface 824. The external interface 824 may be connected to an expansion memory 850, a vehicle parameter database 852 that stores spacecraft specifications, thruster specifications, size, weight, etc., and an initial orbit database 854 that stores parameters including time, date, orbit-related altitude, inclination, eccentricity, etc., and another orbit database 856 (i.e., specific orbit data). The bus system 813 may also connect a control interface 826, an output interface 827, a receiver 828, and a transmitter 830. Additionally, the bus system may connect the GPS receiver module 832 to the GPS 834 .

[0129] Bus system 813 may connect output thruster command module 858 for outputting thruster commands. Further, bus 859 may connect to orbit keeping module 840, which includes a transfer orbit generator for generating one or more transfer orbits for the spacecraft, a feedback gain module 844 for calculating feedback gains, and a feedback controller 846. Further, orbit keeping module 840 may also include a thruster command generator 848.

[0130] Continuing to refer to FIG. 8 , computer 870 can be a server or desktop, laptop, mobile, or other computing device or system having one or more processors 810. Processor 810 may be a central processing unit adapted to access code in the form of transition trajectory generator 842 in memory 812 or stored data 818 of computer 870 (or in expansion memory 850 or other data storage 852, 854, 856). In accordance with aspects related to the systems and methods of the present disclosure, external storage devices are contemplated as further needed depending on the specific design and aspects of the intended hardware and implementation. For example, computer 870 can be used to implement system and method steps in which memory 812 and / or storage 818 can store data.

[0131] The data stored in memory 812 of FIG. 8 can include executable modules, vehicle data, and historical spatial data. For example, vehicle data can include spacecraft specifications, dimensions, weight, performance data under varied conditions, including gravity, and other perturbations, i.e., the complex motion of a massive object subject to forces other than the gravitational attraction of a single other massive object in space. Furthermore, vehicle data can include data related to aspects of vehicle dynamics associated with multiple variables, i.e., one or more of the following: (1) anomalous orbital characteristics of a celestial body, i.e., a natural object located outside the Earth's atmosphere, such as the Moon, the Sun, an asteroid, a planet, or a star; (2) anomalous orbital motion of a celestial body; (3) a near-anomalous orbit of a celestial body around another celestial body; and (4) other known perturbations. Spatial data can include data related to celestial systems, past missions to celestial bodies, and any other data related to the planning of space, spacecraft, and orbital designs for other celestial bodies in space. For example, spatial data can include data related to the moon of a celestial body, such as the properties of the celestial body that can be considered when developing an orbital design from an initial celestial body orbit to a similar target celestial body orbit.

[0132] Optionally, the stored data can be stored in storage device 818, external interface 824 connected to expansion memory 850, which is connected to initial trajectory data database 854, other trajectory data database 856, and database 852 of data such as vehicle parameters, specifications, performance, etc. of FIG.

[0133] 8, processor 810 of computer 870 may be more than one processor depending on the particular application. For example, some steps may require separate processors to ensure a particular processing time or processing speed associated with the systems and methods of the present disclosure.

[0134] The receiver 828 or input interface can receive spatial data, which may be current spatial data obtained from either an Earth mission control center or sensors associated with the spacecraft, or some other location, after stored historical spatial data stored in memory 812. The receiver 828 and transmitter 830 can receive data and provide a radio location for transmission, for example, to an Earth mission control center or some other destination. A GPS receiver module 832 connected to a GPS 834 can be used for navigation-related aspects. The computer 870 can include a control interface 820, a display interface 822, and optionally external devices, control interfaces, displays, sensors, machines, etc. (not shown, see FIG. 8) contemplated for use in connection with the systems and methods of the present disclosure.

[0135] 9 is a schematic diagram illustrating, by way of non-limiting example, a computing device 900 that may be used to implement some techniques of the methods and systems according to embodiments of the present disclosure. The computing device or device 900 represents various forms of digital computers, such as laptops, desktops, workstations, personal digital assistants, servers, blade servers, mainframes, and other suitable computers.

[0136] The computing device 900 may include a power supply 908, a processor 909, a memory 910, and a storage device 911, all connected to a bus 950. Additionally, a high-speed interface 912, a low-speed interface 913, a high-speed expansion port 1214, and a low-speed connection port 915 may be connected to the bus 950. Additionally, a low-speed expansion port 916 is connected to the bus 950. Various component configurations are contemplated that may be mounted on a common motherboard (930, as a non-limiting example) depending on the particular application. Furthermore, the input interface 917 may be connected to an external receiver 906 and an output interface 918 via the bus 950. The receiver 919 may be connected to an external transmitter 907 and a transmitter 920 via the bus 950. Additionally, an external memory 904, an external sensor 903, a machine 902, and an environment 901 may be connected to the bus 950. Additionally, one or more external input / output devices 905 may be connected to the bus 950. A network interface controller (NIC) 921 may be adapted to connect to a network 922 through a bus 950, and data or other data may be rendered on, among other things, a third-party display device, a third-party imaging device, and / or a third-party printing device external to the computing device 900.

[0137] 9 , it is contemplated that memory 910 may store instructions executable by computing device 900, historical data, and any data that may be utilized by the methods and systems of the present disclosure. Memory 910 may include random access memory (RAM), read only memory (ROM), flash memory, or any other suitable memory system. Memory 910 may be a volatile memory unit and / or a non-volatile memory unit. Memory 910 may also be another form of computer-readable medium, such as a magnetic disk or an optical disk.

[0138] The storage device 911 may be adapted to store supplemental data and / or software modules used by the computing device 900. For example, the storage device 911 may store historical data and other relevant data, as described above with respect to this disclosure. Additionally or alternatively, the storage device 911 may store historical data, such as the data referred to above with respect to this disclosure. The storage device 911 may include a hard drive, an optical drive, a thumb drive, an array of drives, or any combination thereof. Furthermore, the storage device 911 may include an array of devices, including computer-readable media such as a floppy disk device, a hard disk device, an optical disk device, or a tape device, a flash memory or other similar solid-state memory device, or a device in a storage area network or other configuration. The instructions may be stored on an information carrier. When executed by one or more processing devices (e.g., the processor 909), the instructions perform one or more methods, such as those described above.

[0139] Continuing with reference to FIG. 9, the system can be linked via a bus 950 to a display interface or user interface (HMI) 923, optionally adapted to connect the system to a display device 925 and a keyboard 924, which can include, among other things, a computer monitor, a camera, a television, a projector, or a mobile device.

[0140] The computing device 900 may be adapted with a printer interface (not shown), which may also be connected via a bus 950, and may include a user input interface 917 adapted to connect to a printing device (not shown), which may include, among others, a liquid inkjet printer, a solid ink printer, a large scale commercial printer, a thermal printer, a UV printer, or a dye sublimation printer.

[0141] Continuing with reference to FIG. 9 , high-speed interface 912 manages bandwidth-intensive operations for computing device 900, and low-speed interface 913 manages less bandwidth-intensive operations. This allocation of functionality is merely an example. In some implementations, high-speed interface 912 may be coupled to memory 910, a user interface (HMI) 923, a keyboard 924 and a display 925 (e.g., via a graphics processor or accelerator), and a high-speed expansion port 914 that may accept various expansion cards (not shown) via a bus 950. In one implementation, low-speed interface 913 is coupled to storage device 911 and a low-speed expansion port 915 via a bus 950. The low-speed expansion port 915, which may include various communication ports (e.g., USB, Bluetooth, Ethernet, wireless Ethernet), may be coupled, for example, via a network adapter, to one or more input / output devices 905 and other devices, such as a keyboard 924, a pointing device (not shown), a scanner (not shown), or a networking device such as a switch or router.

[0142] Computing device 900, as shown in the figure, may be implemented in several different forms. For example, it may be implemented as a standard server 926 or multiple times within a group of such servers. In addition, it may be implemented in a personal computer, such as a laptop computer 927. It may also be implemented as part of a rack server system 928. Alternatively, components from computing device 900 may be combined with other components in a mobile device (not shown). Each such device may include one or more of a computing device and a mobile computing device, and the entire system may be made up of multiple computing devices communicating with each other.

[0143] The description provides exemplary embodiments only and is not intended to limit the scope, applicability, or configuration of the present disclosure. Rather, the following description of exemplary embodiments will provide those skilled in the art with an enabling description for implementing one or more exemplary embodiments. Contemplated are various changes that may be made in the function and arrangement of elements without departing from the spirit and scope of the disclosed subject matter as set forth in the claims.

[0144] In the following description, specific details are given for a thorough understanding of the embodiments. However, it will be understood by those skilled in the art that the embodiments may be practiced without these specific details. For example, systems, processes, and other elements in the disclosed subject matter may be shown as components in block diagram form so as not to obscure the embodiments in unnecessary detail. In other instances, well-known processes, structures, and techniques may be shown without unnecessary detail to avoid obscuring the embodiments. Furthermore, like reference numbers and names in the various drawings indicate like elements.

[0145] Also, particular embodiments may be described as a process that is depicted as a flowchart, a flow diagram, a data flow diagram, a structure diagram, or a block diagram. While a flowchart may describe operations as a sequential process, many of the operations can be performed in parallel or simultaneously. Additionally, the order of operations may be rearranged. A process may terminate when its operations are completed, or may have additional steps not discussed or included in the diagram. Moreover, not all operations in any specifically described process may occur in all embodiments. A process may correspond to a method, a function, a procedure, a subroutine, a subprogram, etc. When a process corresponds to a function, the end of the function may correspond to a return of the function to the calling function or the main function.

[0146] Furthermore, embodiments of the disclosed subject matter may be implemented, at least in part, either manually or automatically. The manual or automatic implementation may be performed, or at least assisted, through the use of a machine, hardware, software, firmware, middleware, microcode, hardware description languages, or any combination thereof. When implemented in software, firmware, middleware, or microcode, the program code or code segments to perform the necessary tasks may be stored on a machine-readable medium. The necessary tasks may be performed by a processor.

[0147] The above-described embodiments of the present disclosure can be implemented in any of numerous ways. For example, the embodiments may be implemented using hardware, software, or a combination thereof. If implemented in software, the software code may be executed on any suitable processor or collection of processors, whether provided on a single computer or distributed across multiple computers. Such a processor may be implemented as an integrated circuit with one or more processors within an integrated circuit component. However, the processor may be implemented using any suitable form of circuitry.

[0148] Also, the various methods or processes outlined herein may be coded as software executable on one or more processors using any one of a variety of operating systems or platforms. Additionally, such software may be written using any of a number of suitable programming languages ​​and / or programming or scripting tools, and may be compiled as executable machine language code or intermediate code that runs on a framework or virtual machine. Typically, the functionality of the program modules may be combined or distributed as desired in various embodiments.

[0149] Also, embodiments of the present disclosure may be embodied as methods, of which an example is provided. Acts performed as part of a method may be ordered in any suitable manner. Thus, embodiments may be constructed in which acts are performed in an order other than that illustrated, including simultaneously performing some acts shown in an exemplary embodiment as sequential acts. Furthermore, the use of order terms such as "first," "second," etc. in the claims to modify claim elements does not, by itself, imply any priority, precedence, or order of one claim element relative to another claim element, or any chronological order in which method actions are performed, but is merely used as a label to distinguish one claim element having a certain name from another element having the same name (except when order terms are used) to distinguish between claim elements.

[0150] Although the present disclosure has been described with reference to certain preferred embodiments, it is to be understood that various other adaptations and modifications can be made within the spirit and scope of the disclosure. It is therefore the object of the appended claims to cover all such variations and modifications as come within the true spirit and scope of the disclosure.

Claims

1. 1. A controller for maintaining a spacecraft near orbit, comprising: a memory configured to store executable instructions; a processor that executes the executable instructions to cause the controller to: detecting a distance from the spacecraft to the orbit that is greater than a spacecraft threshold; in response to detecting that the distance from the spacecraft to the orbit is greater than the spacecraft threshold; and linearizing the dynamics of the spacecraft from a current time against a high-fidelity reference trajectory over a time horizon to generate a State Transition Matrix (STM) for uncontrollable motion of the spacecraft within the time horizon, the State Transition Matrix including non-extended eigenvectors having a magnitude less than or equal to one and extended eigenvectors having a magnitude greater than one, the processor further executing the executable instructions to cause the controller to: determining a control action that changes a next state of the spacecraft to a linear combination of the non-augmented eigenvectors of the state transition matrix; generating control commands to actuators of the spacecraft that cause a modification of the next state of the spacecraft along a direction corresponding to at least one of the non-extended eigenvectors of the state transition matrix.

2. To detect that the distance is greater than the spacecraft threshold, the processor: determining a current state of the spacecraft within a specified time period; determining a distance between a location defined by the current state of the spacecraft and the orbit; configured to compare the determined distance between the location defined by the current state of the spacecraft and the orbit to a spacecraft threshold; the specified period is outside the time horizon; 2. The controller of claim 1, wherein to determine the current state of the spacecraft, the processor is further configured to obtain measurements corresponding to one or a combination of a position and translational velocity of the spacecraft and a perturbation, the perturbation acting on a multi-body celestial body defined by the spacecraft and at least one celestial body.

3. To determine the state transition matrix, the processor further comprises: obtaining a linear model of a dynamic system defined by the spacecraft and at least one celestial body; calculating discrete-time updates of perturbations of the high-fidelity reference trajectory; The controller of claim 1 , configured to propagate a mathematical formula defined by the high-fidelity reference trajectory and the fundamental basis vectors of the STM over a defined period of time.

4. 2. The controller of claim 1, wherein the processor is further configured to obtain different spacecraft thresholds for a visitor spacecraft to use the high-fidelity reference trajectory to perform eigenvector-based control of the visitor spacecraft's motion.

5. 2. The controller of claim 1, wherein to determine the control action, the processor is further configured to determine the direction of the modification of the next state by a linear combination of directions of each non-extended eigenvector of the STM to achieve a mixed state with natural motion, the natural motion being a combination of corresponding eigenmotions of each non-extended eigenvector of the STM.

6. 2. The controller of claim 1, wherein to generate the control commands, the processor is further configured to solve a nonlinear optimization problem to find a fuel-efficient maneuver that puts the spacecraft into a desired set of states that results in a desired natural motion.

7. 7. The controller of claim 6, wherein the nonlinear optimization problem is directed to a finite horizon optimization of a model of the spacecraft, a set of objectives for motion of the spacecraft, and constraints on a propulsion system of the spacecraft and the motion of the spacecraft, the constraints on the propulsion system of the spacecraft and the motion of the spacecraft including one or more physical limits of the spacecraft, one or more safety limits on operation of the spacecraft, and one or more performance limits on a trajectory of the spacecraft.

8. 2. The controller of claim 1, wherein the spacecraft is configured to perform a task of docking with a space station that maintains a position near the orbit at a distance less than a station threshold, and the controller is further configured to select the spacecraft threshold greater than the station threshold.

9. the next state of the spacecraft includes one or a combination of a position, an orientation, a translational velocity, an angular velocity, and a perturbation of one or more of the spacecraft and the payload of the spacecraft, the perturbation acting on a multi-body celestial body system defined by the spacecraft, the payload, and one or more celestial bodies; 2. The controller of claim 1, wherein the perturbations acting on the multi-body celestial body are natural orbital forces including gravitational perturbations of the sun and moon, anisotropic gravitational perturbations due to non-sphericity of a central body, solar radiation pressure, and air resistance.

10. the high-fidelity reference trajectory is an uncontrolled natural motion trajectory located near a lunar near-rectilinear halo orbit (NRHO) in the Earth-Moon circularly restricted three-body problem, and is estimated using a multiple shooting approach; The controller of claim 1 , wherein the high fidelity reference trajectory comprises a finite number of non-periodic revolutions around the moon over a finite number of days.

11. The controller of claim 1 , wherein the orbit is one of a circular orbit, an elliptical orbit, a halo orbit, a lunar polar elliptical orbit, or a quasi-satellite orbit.

12. 2. The controller of claim 1, wherein the control commands are generated as a solution to a model predictive control policy that generates the control commands by optimizing a cost function over a receding horizon, the cost function including a stabilization component for directing motion of the spacecraft to the next state, a component for an objective of operation of the spacecraft, and a performance component for optimizing the motion of the spacecraft until the next state is achieved.

13. The controller of claim 1 , wherein the control commands are generated for each designated period of a plurality of designated periods within the time horizon or are generated repeatedly over a receding time horizon.

14. 1. A computer-implemented method for maintaining a spacecraft in near-orbit, comprising: detecting a distance from the spacecraft to the orbit that is greater than a spacecraft threshold; in response to detecting that the distance from the spacecraft to the orbit is greater than the spacecraft threshold; and linearizing the dynamics of the spacecraft from a current time against a high-fidelity reference trajectory over a time horizon to generate a state transition matrix (STM) for uncontrollable motion of the spacecraft within the time horizon, the state transition matrix including non-extended eigenvectors having a magnitude less than or equal to one and extended eigenvectors having a magnitude greater than one, the method further comprising: determining a control action that changes a next state of the spacecraft to a linear combination of the non-augmented eigenvectors of the state transition matrix; generating control commands to actuators of the spacecraft that cause a modification of the next state of the spacecraft along a direction corresponding to at least one of the non-extended eigenvectors of the state transition matrix.

15. 1. A non-transitory computer-readable storage medium having embodied thereon a program executable by a processor for performing a method for maintaining a spacecraft near orbit, the method comprising: detecting a distance from the spacecraft to the orbit that is greater than a spacecraft threshold; in response to detecting that the distance from the spacecraft to the orbit is greater than the spacecraft threshold; and linearizing the dynamics of the spacecraft from a current time against a high-fidelity reference trajectory over a time horizon to generate a state transition matrix (STM) for uncontrollable motion of the spacecraft within the time horizon, the state transition matrix including non-extended eigenvectors having a magnitude less than or equal to one and extended eigenvectors having a magnitude greater than one, the method further comprising: determining a control action that changes a next state of the spacecraft to a linear combination of the non-augmented eigenvectors of the state transition matrix; generating control commands to actuators of the spacecraft that cause a modification of the next state of the spacecraft along a direction corresponding to at least one of the non-extended eigenvectors of the state transition matrix.

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