Proper Motion Control for Lunar Elliptical Polar Orbit

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 generally relates to controlling the operation of a spacecraft in the vicinity of an unstable orbit around a celestial body such as the moon, and more particularly to methods and systems for creating fuel-efficient maneuvers that utilize natural motion to remain in the vicinity of an unstable orbit for an extended period without intervention.

Background Art

[0002] Launching and deploying space stations and spacecraft into space has remained a difficult challenge to date and requires accurate analysis and investigation of orbital motion, celestial bodies, and other space objects. An example of such a space station is the Lunar Orbital Platform-Gateway (LOP-G), also referred to as the Gateway, a small space station in lunar orbit intended to function as a solar-powered communication hub, science laboratory, short-term habitation module for government agency astronauts, and staging area for planetary probes and other robots. The Gateway can be deployed in a near-rectilinear halo orbit (NRHO) around the moon with a 7-day lunar semi-elliptical polar orbit, and is intended to play an important role in facilitating missions inside and outside cis-lunar space. The NRHO is a type of halo orbit with a slightly curved, and thus nearly straight, side between close approaches to the orbiting body. The Gateway can be deployed in close proximity to the NRHO, which is a closed periodic trajectory in the 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 the amount of interruption of communication with Earth is minimal.

[0003] The gateway can be deployed near NRHO rather than on NRHO. The reason is that the NRHO of CR3BP does not take into account perturbations such as the gravitational attraction of the sun, solar radiation pressure (SRP), or the lunar J2 effect. Instead of attempting to follow the NRHO of CR3BP and using fuel to compensate for predictable perturbations, the standard conventional practice is to obtain high-fidelity trajectories near NRHO that account for all the major predictable forces within cis-lunar space by means of multiple shooting or collocation-based techniques. This high-fidelity solution is no longer closed, not periodic, and not stable, but is also referred to as NRHO. However, the advantage of this high-fidelity NRHO solution is that in the absence of additional perturbing forces, the spacecraft can naturally follow the trajectory without consuming fuel, which is an important performance metric for ensuring the long-term viability of the gateway.

[0004] However, there are two factors that prevent the deployment of the gateway based solely on the high-fidelity solution. First, navigation uncertainties and unpredictable disturbing forces in cislunar space prevent the spacecraft from being accurately deployed on the calculated trajectory. Second, and importantly, the trajectory is very unstable compared to the ideal CR3BP counterpart. As a result, any small deviation from the solution can cause the spacecraft to rapidly deviate from the calculated trajectory, necessitating stabilization control actions. These deviation problems also occur at other celestial bodies, and for several reasons, the spacecraft is likely to deviate from the planned trajectory.

[0005] In recent years, several station-keeping 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 supply missions, human transportation missions, or inspection and maintenance missions, etc., which would require long-term, restricted, collision-free relative motion around the gateway. Formation control for multiple spacecraft in halo orbits has been particularly developed for NRHO, but these methods propose control schemes based on periodic solutions in the CR3BP and rely on a computationally expensive process to generate high-fidelity solutions for each spacecraft in the formation.

[0006] Therefore, there is a need in the art to improve methods for controlling the operation of multiple spacecraft for long-term, restricted, collision-free motion, particularly in the vicinity of NRHO, in several existing scenarios.

Summary of the Invention

[0007] The present disclosure relates to control policies for reliable and fuel-efficient station-keeping and restricted relative motion control for gateways and approaching spacecraft, which utilize a single pre-calculated high-fidelity NRHO solution while ensuring a safe separation distance between spacecraft. High-fidelity NRHO may be referred to throughout the present disclosure as a reference trajectory, a high-fidelity reference trajectory, a baseline, a baseline solution, and a baseline reference trajectory.

[0008] To develop some of the embodiments of the present disclosure, there were assumptions and understandings that helped with those developments. At least one understanding included that the control policy utilizes a linear approximation of spacecraft dynamics in the vicinity of high-fidelity NRHO.

[0009] Small deviations from states on a high-fidelity NRHO trajectory generally lead to rapid departures, but some exemplary embodiments are based on the recognition that there are some special states near the reference trajectory that result in a desired natural motion without departure at any given time instance, where natural motion means motion without control (i.e., without using fuel / onboard power). These states form a desired space around the reference trajectory. When the spacecraft is controlled to this space rather than to the orbit itself, the requirement for staying "near" the orbit is relaxed from staying exactly on the orbit, which sacrifices the distance to the 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 a desired natural motion without departure is possible, rather than continuously controlling the spacecraft to stay on the desired orbit, which would consume a large amount of fuel / onboard power.

[0010] Also, some exemplary embodiments are based on the recognition that such a region (subspace) of special states can be estimated using a local mode decomposition of the backwards horizon state transition matrix (STM) associated with the high-fidelity reference trajectory. In other words, the orbit dynamics for the reference trajectory over a finite horizon determined by the user can be linearized, and the resulting eigenvalue decomposition of the linearized STM can be used. The state transition matrix is used to find the solution to the general state-space representation of a linear system expressed in the following form.

Number

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

[0012] The natural motion resulting from initial conditions along the eigenvectors of the STM is called local natural motion. The natural motion resulting from initial conditions along the extended (eigenvalue > 1) eigenvectors and non-extended (eigenvalue <= 1) eigenvectors of the STM are called extended local natural motion and non-extended local natural motion, respectively.

[0013] Another recognition of some exemplary embodiments included that these eigenvectors are directions that do not have a fixed magnitude. This means that the spacecraft can move to various distances from the reference trajectory in these directions and exhibit the desired type of natural motion. Some exemplary embodiments utilize the distance to the reference trajectory as a trigger condition for determining when to apply control. The reason is that regardless of the distance of the spacecraft from the reference trajectory, the spacecraft can be controlled to transition from the current undesirable deviation state to a desired state in the non-extended subspace.

[0014] The recognition of some exemplary embodiments is that, whenever a deviation from the reference trajectory is detected, the natural motion-based control solves a non-linear optimization problem to transfer the spacecraft to a set of states that result in a desired natural motion, finding one or more fuel-efficient maneuvers (also referred to as control actions). The non-linear optimization is a finite-horizon optimization of the spacecraft's dynamics model, a set of objectives of the spacecraft's motion, and the spacecraft's propulsion system and constraints on the motion, having the ability to anticipate future events and take appropriate control actions.

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

[0016] In theory, it is possible to use non-linear optimal control to obtain an optimal station-keeping maneuver over the entire duration of the spacecraft mission rather than a finite-time horizon. However, it causes a very large optimization problem and results in a very high computational load, so it cannot be realized with hardware having limited computational resources within the spacecraft. Therefore, some exemplary embodiments are based on the recognition that by using local natural motion-based control, the spacecraft can use natural motion over a long period, thereby reducing the computational load for solving the optimization problem and speeding up the solution of the optimization problem.

[0017] Further recognition is that by using trigger-based control, the control frequency of the spacecraft can be reduced because the spacecraft can follow natural motion over a long period until it begins to deviate from the reference trajectory beyond a trigger threshold, which is the point at which a state close to the desired state exists in the subspace where the spacecraft can be maneuvered. Since the control frequency of the spacecraft is reduced, the number of times the thruster is activated is significantly reduced, which also leads to an improvement in fuel efficiency over a certain period.

[0018] In some exemplary embodiments, the distance from the reference trajectory is an adjustable parameter that enables control of multiple spacecraft relative to each other without the need to calculate the reference trajectory independently for each spacecraft. The 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 multiple spacecraft. Thus, in a scenario where for some reason the host spacecraft is approached by a visitor spacecraft, the same one as the reference trajectory of the host spacecraft can be utilized. Therefore, since the exemplary embodiments utilize the distance between the spacecraft and the baseline solution (reference trajectory) as a trigger for the control policy, this approach is applicable to controlling any number of spacecraft in addition to the gateway.

[0019] In some exemplary embodiments, special eigenvector direction states are combined in a linear combination to realize a mixed state with natural motion, which is a combination of corresponding eigenmotions. For example, controlling the spacecraft with a combination of eigenvectors having eigenvalues strictly less than 1 causes the natural motion of the spacecraft to converge towards the reference trajectory. In contrast, including eigenvector directions having magnitudes near 1 in the linear combination causes natural motion that oscillates at the current distance from the reference trajectory. By selecting a specific combination of eigenvector directions as the spacecraft transitions, the operator can shape the natural motion of the spacecraft near the reference trajectory and bring the spacecraft closer to or farther from the reference trajectory as desired.

[0020] According to one non-limiting embodiment, the spacecraft is driven by eight thrusters, each of which is mounted in a manner aligned with the center of mass of the spacecraft so as to generate a force for changing the position of the spacecraft without generating a torque for rotating the spacecraft.

[0021] For these purposes, some exemplary embodiments provide a controller, method, and program for maintaining a spacecraft near a desired orbit. Some exemplary embodiments provide important solutions for controlling the operation of multiple spacecraft for long-term, limited, collision-free motion near NRHO for missions performing satellite services, 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 an orbit. The controller includes a memory storing instructions and a processor, and the processor is configured to execute 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 to linearize the dynamics of the spacecraft in the current state over a time horizon with respect to a high-fidelity reference trajectory in response to detecting that the distance from the spacecraft to the orbit is greater than the spacecraft threshold, and 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 processor is further configured to determine a control action that changes the next state of the spacecraft to a linear combination of the non-expanding eigenvectors of the state transition matrix. The processor is further configured to generate a control command to an actuator of the spacecraft that causes a correction of the next state of the spacecraft along a direction corresponding to at least one of the non-expanding eigenvectors of the state transition matrix.

[0023] Another embodiment discloses a computer-implemented method for maintaining a spacecraft near an 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, the dynamics of the spacecraft in its current state are linearized over a time horizon with respect to a high-fidelity reference trajectory, and a state transition matrix is generated for uncontrollable motion of the spacecraft within the time horizon. The state transition matrix includes non-expanding eigenvectors having a magnitude of 1 or less and expanding eigenvectors having a magnitude greater than 1. The method further includes determining a control action to change a next state of the spacecraft to a linear combination of the non-expanding eigenvectors of the state transition matrix, and generating a control command to the actuator of the spacecraft that causes a correction of the next state of the spacecraft along a direction corresponding to at least one of the non-expanding eigenvectors of the state transition matrix.

[0024] Yet another embodiment discloses a non-transitory computer-readable storage medium embodying a program executable by a processor for implementing a method for maintaining a spacecraft near an orbit. The method includes 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, the dynamics of the spacecraft in its current state are linearized over a time horizon with respect to a high-fidelity reference trajectory, and a state transition matrix is generated for uncontrollable motion of the spacecraft within the time horizon. The state transition matrix includes non-expanding eigenvectors having a magnitude of 1 or less and expanding eigenvectors having a magnitude greater than 1. The method further includes determining a control action that changes a next state of the spacecraft to a linear combination of the non-expanding eigenvectors of the state transition matrix, and generating a control command for an actuator of the spacecraft that causes a correction of the next state of the spacecraft along a direction corresponding to at least one of the non-expanding eigenvectors of the state transition matrix.

[0025] The embodiments disclosed herein will be further described with reference to the accompanying drawings. The drawings shown are not necessarily drawn to scale; instead, emphasis is placed on generally illustrating the principles of the embodiments disclosed herein. BRIEF DESCRIPTION OF THE DRAWINGS

[0026]

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[0027] The drawings identified above describe the embodiments disclosed herein, but as discussed herein, other embodiments are contemplated. This disclosure presents exemplary embodiments by way of illustration 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 and requires reliable and low-cost strategies for station-keeping and relative motion tailored to such special orbits. An example of such a deployment is when approaching spacecraft need to be deployed around a lunar orbit platform - gateway (LOP-G), also known as a gateway, and are planned to be deployed in a 7-day lunar near-rectilinear halo orbit (NRHO) that orbits the moon. In the context of such spacecraft deployments, it is desirable to have as much uncontrollable natural motion as possible to reduce frequent replenishment of fuel consumption and onboard energy. There are several factors that impede or prevent attempts to deploy spacecraft in the region near NRHO. Therefore, even a small arbitrary deviation from the solution causes the spacecraft to rapidly deviate from the calculated trajectory, so stabilization control actions are required.

[0029] The exemplary embodiments disclosed herein provide a control approach that utilizes the eigenvectors of the state transition matrix (STM) associated with a high-fidelity NRHO solution in a celestial ephemeris model to design long-term station-keeping and restricted relative motion. The proposed strategy effectively utilizes the natural motion of the spacecraft such that control actions are infrequent and fuel-efficient. This ensures that the spacecraft can exhibit restricted, collision-free relative motion around the gateway over a long period. Furthermore, since the proposed strategy does not include any periodic solutions or calculate high-fidelity solutions for each spacecraft, the approach on which some exemplary embodiments are based is fast, computationally efficient, inexpensive, scalable, and results in improved energy efficiency of the spacecraft.

[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 leverage some special states in the vicinity of a reference trajectory that result in a desirable natural motion without deviation. These states form a desirable space around the reference trajectory. When the spacecraft is controlled to this space rather than to the orbit itself, the requirements for staying "near" the orbit are relaxed compared to staying precisely on the orbit, which sacrifices the distance to the orbit for improved fuel efficiency. To determine these special states, some exemplary embodiments are directed to the local mode decomposition of the backwards horizon state transition matrix (STM) associated with the high-fidelity reference trajectory. To prevent unnecessary or undesirable control of the spacecraft that may lead to excessive energy consumption, some exemplary embodiments utilize trigger conditions for determining when to execute the control law / policy. The trigger conditions are based on the distance of the spacecraft from the reference trajectory. By using trigger-based control, the frequency of control execution by the spacecraft can be reduced. This is because the spacecraft can follow a natural motion over a long period until it begins to deviate from the reference trajectory beyond a trigger threshold at which a desirable state exists in the partial space where the spacecraft can be maneuvered.

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

[0032] Thus, some exemplary embodiments of the present disclosure provide important solutions for controlling the operations of multiple spacecraft for long-term, limited, collision-free motion in the vicinity of NRHO for missions performing satellite services, active debris mitigation, in-space manufacturing, space station resupply, and planetary sample return. Further, proximity operations are an important process for achieving mission objectives and an important 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 services, 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. Some exemplary embodiments are described in relation to lunar orbits, but it can be contemplated that the proposed control strategy is applicable to any multi-object celestial system. Thus, the exemplary embodiments described herein should not be limited to the lunar celestial system for illustrative purposes only. The scope of the proposed control strategy encompasses situations and systems related to any multi-object celestial system.

[0034] FIG. 1A is a block diagram showing some method steps for a natural motion maneuver design that results in fuel-efficient station-keeping 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 operations of a spacecraft over a finite time horizon.

[0035] In some exemplary embodiments, the spacecraft may be directed to orbit a central body such as a celestial body. The spacecraft can do so by itself performing an orbital motion or by being deployed near an orbiting space station orbiting the central body. In this regard, the spacecraft may be equipped with a function to check or obtain the current state of the spacecraft within a specified period. The state of the spacecraft may include data related to the position and translational velocity of the spacecraft, as well as one or a combination of perturbations that act on the multi-body celestial system of which the spacecraft is a part. A check may be performed as to whether the distance of the spacecraft to the orbit is greater than a spacecraft threshold. If the distance is greater than or in some cases equal to the spacecraft threshold, it corresponds to a situation where the spacecraft has deviated or is starting to deviate in any direction from the desired orbit. Therefore, it is desirable to perform prompt control actions to return the spacecraft to a path on the desired orbit or a path along the desired orbit. Thus, the condition (3) that it is detected that the distance of the spacecraft to the orbit is greater than the spacecraft threshold can be used as a trigger for executing a control policy to prevent the spacecraft from deviating from the desired orbit.

[0036] When it is detected that the trigger condition in step 3 of FIG. 1A is satisfied, the controller may linearize the dynamics of the current state of the spacecraft with respect to a high-fidelity reference trajectory over a certain time horizon in step 5 of FIG. 1A to generate a state transition matrix (STM) for the uncontrollable motion of the spacecraft within the time horizon. In some exemplary embodiments, the high-fidelity reference trajectory may be pre-calculated and stored in a storage such as a database accessible to the controller. In some other exemplary embodiments, the reference trajectory may be calculated dynamically at runtime. The linearization of the dynamics of the spacecraft 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 specified period during which the current state of the spacecraft is checked may overlap with the finite time horizon. In some exemplary embodiments, the specified period may be outside the finite time horizon.

[0038] Following the linearization of the spacecraft's orbital dynamics for the 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 state can be a state that results in a desirable natural motion without deviation. Throughout this disclosure, natural motion or uncontrollable motion can mean the motion of the spacecraft without control (i.e., without using fuel or onboard power). The resulting STM includes non-expanding eigenvectors having magnitudes less than or equal to 1 and expanding eigenvectors having magnitudes greater than 1. In some exemplary embodiments, the special state indicating the desirable natural motion results from the eigenvectors of the STM having eigenvalues with magnitudes less than 1.

[0039] Step 7 in FIG. 1A includes determining the desired control action to achieve a state that is a linear combination of the non-expanding 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 the control action that changes the next state to correspond to a linear combination of the non-expanding eigenvectors of the STM.

[0040] Step 9 in FIG. 1A includes generating one or more control commands to activate or not activate one or more thrusters of the vehicle over a specified period based on one or more control commands to achieve a desired corrective maneuver. The control commands are intended to cause a correction of the next state of the spacecraft along a direction corresponding to at least one of the non-extended eigenvectors of the STM. The generated control commands are output to the interface of the spacecraft in step 11 for further action. In some exemplary embodiments, since the controller may be embodied on the spacecraft itself, in step 11, the controller may directly provide the control commands to the actuators (propellers and / or thrusters) of the spacecraft to achieve the desired corrective maneuver. In some exemplary embodiments, since the controller may communicate remotely with the spacecraft, the control commands may be transmitted to the spacecraft, and the on-board CPU of the spacecraft may process the control commands accordingly to achieve the desired corrective maneuver.

[0041] FIG. 1B is a diagram showing a flowchart 5a of some steps for determining a state transition matrix for uncontrollable motion of a spacecraft according to some exemplary embodiments. A multi-body celestial system mechanical model can be obtained (151). The multi-body celestial system may include a spacecraft and at least one celestial body. The mechanical model can be expressed as an analytical equation. Since the mechanical model can be non-linear, in the next step at 153, it can be linearized to obtain a linear model of the multi-body celestial system (155). This can show the continuous-time analytical equation for linearization, but a discrete-time matrix 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 (157). For example, the specified period may be over the entire baseline period. In this regard, a differential equation (such as differential equation 11 described later) is solved over the specified period (157a) to obtain a discrete-time state transition matrix 159.

[0042] FIG. 1C is a diagram showing another flowchart 5b of some steps for determining a state transition matrix for uncontrollable motions of a spacecraft according to some exemplary embodiments. At 161, a mechanical model of a multi-body celestial system expressed as an analytical equation can be obtained. A baseline solution, also referred to as a reference trajectory, can also be obtained (163). Discrete-time up-to-date information on the perturbation of the baseline solution can be calculated (165). For example, r(t) represents a solution, which can be a baseline solution in some exemplary embodiments. Discrete-time up-to-date information on the perturbation of r(t) can be calculated.

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[0043]

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

[0045] FIG. 1D is a block diagram showing a method for controlling the operation of a spacecraft to stay within some boundaries of a reference trajectory according to some embodiments of the present disclosure. For example, the method controls the operation of the spacecraft using a control input determined using a model of a joint multi-body celestial system based on the optimization of a cost function having an objective function that minimizes fuel / payload power consumption.

[0046] The first step 110 of FIG. 1D involves determining the current state of the spacecraft, which can 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 communication with a ground command center located on Earth or another spacecraft located in space, such as GPS, relative distance measurement, star tracker, horizon sensor, etc. In some exemplary embodiments, the current state of the spacecraft may be determined based on previous control inputs determined in previous iterations that were optimized using a previous cost function using a previous model of the spacecraft. The state, as used herein, includes the position and translational velocity of the spacecraft, and one or a combination of perturbations acting on the multi-body celestial system. Continuing to refer to FIG. 1D, the state determined in step 110 can be the absolute state with respect to the central body around which the spacecraft is orbiting.

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

[0048] In step 132 of FIG. 1D, the method uses the current model of the joint multi-body celestial system dynamics to determine a sequence of future inputs of thruster forces over a certain period of time from the current time, as long as the method obtains at least new state measurement values such that the predicted future state of the spacecraft and the inputs satisfy the constraints on the operation of the spacecraft and the constraints on the control inputs. For example, the sequence of future inputs of thruster forces may include forces that bring the spacecraft to a new future state where it no longer deviates from the desired orbit.

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

[0050] In step 140 of FIG. 1D, based on the current state of the spacecraft determined in step 110 and the determined current control input to the spacecraft in step 130, the next state of the spacecraft is determined. In step 140, the controller waits until new state measurement values are received. Then, the control of the steps returns to step 110 for the next iteration, and the process is repeated.

[0051] FIG. 1E is a block diagram showing some components of a controller that implements at least some of the 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 communicate with the processor 113 and the memory 119. Instructions and data including a cost function 121, a joint multi-body celestial system model 123, and constraints 129 may be stored in the memory.

[0052] Furthermore, the method of FIG. 1D can determine the control input 107 via the processor 113 using the joint multi-body celestial system model 123 that is subject to the constraint 129. In some exemplary embodiments, the determined control input 107 can be transmitted to the spacecraft 102. To that end, the controller 101 can be included in or operatively connected to an output interface, and the output interface is configured to input the control command 107 to the thruster 103 of the spacecraft 102. Furthermore, the spacecraft 102 can have, among other components, thrusters 103 and sensors 108. The current state 106 of the spacecraft 102 can be obtained from the sensors 108 and communicated to the processor 113.

[0053] Continuing to refer to FIG. 1E, in some exemplary embodiments, the processor 113 can determine at least one of the cost function 121, the joint multi-body celestial system model 123, and the constraint 129 during control. For example, the controller 101 can execute a method such as the method of FIG. 1D to repeatedly control the operation of the spacecraft 102 using the control input of step 130 of FIG. 1D determined using the joint multi-body celestial system model 123 based on the optimization of the cost function 121. Also, it is contemplated that the method of FIG. 1D may be executed by the controller 101 from a previous iterative control operation having a previous control input determined in a previous iteration that was optimized by a previous cost function using a previous model of the spacecraft 102, i.e., based on a previous iterative operation of the spacecraft 102.

[0054] FIG. 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 can also be referred to as the baseline NRHO solution 201. The baseline NRHO solution 201 (represented in the Earth-Moon rotating coordinate system) can be composed of 60 revolutions around the Moon over 394 days. Referring to FIG. 2A, the Moon is shown with the designation 203 and the Earth is shown with the designation 205.

[0055] Next, several concepts related to space vehicle control will be described below. The Near-Rectilinear Halo Orbit (NRHO) is a periodic trajectory around the L1 and L2 Lagrange points of the Earth-Moon Circular Restricted Three-Body Problem (CR3BP). Due to their favorable stability characteristics and relatively low station-keeping costs, the NRHO around the L2 point with an association resonance of 9:2 and a periselene radius of approximately 3150 km has been selected for deploying a gateway. However, NRHO does not actually exist. This is because CR3BP ignores effects such as solar radiation pressure (SRP), gravity caused by celestial bodies other than the Earth and the Moon, and the lunar J2 zonal harmonic function. Ignoring these higher-order effects during mission design would lead to an unacceptably large amount of fuel consumption. Therefore, a high-fidelity astrodynamics model based on celestial ephemeris data is actually used to generate the solution closest to NRHO in CR3BP. This high-fidelity solution is aperiodic and consists of a finite number of revolutions around the Moon. The astrodynamics models considered in some exemplary embodiments account for all the major predictable forces acting on a space vehicle within cis-lunar space. Any major predictable force has a magnitude larger than the magnitude of the largest unpredictable force. In the models considered here, the largest unpredictable force affecting the space vehicle is determined to be an indirect disturbance caused by navigation errors that affect the space vehicle's controller. Navigation errors are quantified under the assumption that the state of the space vehicle is estimated using measurements from the Deep Space Network (DSN). Under these assumptions, the major predictable forces acting on a space vehicle within the region of space occupied by the NRHO of CR3BP are SRP, the lunar J2 zonal harmonic function, and the gravity caused by the Earth, the Moon, and the Sun.

[0056]

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[0059] Next, the state of the spacecraft at t, which is obtained after the impulse at t′, can be determined as follows. 2 : The state of the spacecraft at t can be determined as follows.

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[0060]

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[0061]

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[0063]

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[0064]

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[0065]

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[0066] Local Intrinsic Motion Control : Some exemplary embodiments are called the rear horizon STM, [0,t maxDetermine the non-extended local proper motion corresponding to those intervals using the STM calculated during a sequence of adjacent time intervals spanning them.

[0067] Figure 2B is a schematic diagram illustrating the maneuver of transferring a spacecraft to non-extended local proper motion at apocenter when a trigger condition is met, according to some exemplary embodiments. The spacecraft may have a planned trajectory with 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 can be calculated over the entire baseline period, and the non-extended proper motion can be selected such that a restricted 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 to which it applies increases. In fact, the longest period for which the STM is reliable is generally the time required for 12 revolutions around the moon, which is about 78 days.

[0069]

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[0070]

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[0071]

Number

[0072] Note that the resulting linear combination does not include components along directions having two or more eigenvalues. The basis for the selection of states in the linear combination is based on the criterion that it should provide the desired behavior (motion) without deviating the spacecraft from the reference trajectory / baseline solution. The basis for the linear combination 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 can be a major factor in determining which components of the states should be in the resulting linear combination. Thus, among many special states, only those determined by the optimization problem to have the minimum fuel consumption cost can be selected. In this regard, in some exemplary embodiments, it can be contemplated that the resulting combination may include at least one of the special states. In a scenario where the resulting combination includes multiple states, the resulting state has a direction that is a hybrid of the directions of the selected eigenvectors. Since the eigenvectors that provide the desired stable motion provide only some major directions, in some cases, it is essential to select a combination of those major directions.

[0073]

Number

[0074]

Number

[0075] FIG. 4 is a diagram showing an algorithm associated with local eigenmotion control according to some exemplary embodiments. Referring to FIG. 4, the proposed control approach results in an automatic routine summarized by algorithm 400 in FIG. 4.

[0076]

Number

[0077]

Number

[0078]

Number

[0079]

Number

[0080] Continuing to refer to FIG. 4, Trigger is evaluated as true when the trigger condition (14) is satisfied.

[0081]

Number

[0082]

Number

[0083] Continuing to refer to FIG. 4, Δν j (j = 1,..., M) is the cost of M maneuvers over the period [0, t max initiated in Algorithm 1. The cumulative cost of the M maneuvers is indicated by Δν.

[0084] According to some exemplary embodiments, an actual implementation of the proposed control strategy can be demonstrated by enhancing Algorithm 400 of FIG. 4 using a Kalman filter that estimates the 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 for accurate state estimation. The errors in the range and range rate measurements are due to a standard normal distribution, and the standard deviations are 10 m and 1 mm s, respectively-1 It is.

[0085] The proposed approach is demonstrated for the case where two spacecraft perform a restricted relative motion in the vicinity of the baseline solution. In particular, the gateway undergoes tight station-keeping near the baseline while the approaching spacecraft performs a collision-free relative motion near the gateway.

[0086] The trajectory calculated for the gateway, referred to as solution 1, uses ρ = 10, Δt = 6 hours, Δr 0 = 0.5 km and N = 4, while the trajectory of the approaching spacecraft, referred to as solution 2, is calculated with ρ = 2, Δt = 12 hours, Δr 0 = 50 km and N = 6. The specific parameters in algorithm 9 affect the nature of the restricted motion solution. In particular, the selection of ρ and Δr 0 in the two solutions helps to ensure that they remain collision-free. These parameters are adjusted such that the resulting solutions are sparse and fuel-efficient.

[0087] It is meaningful to note that the effect of navigation uncertainty is more prominent in the tight station-keeping maneuvers near the baseline. When the maneuver starts near the baseline, the final state (located near the baseline and aligned with the desired eigenvector) is more vulnerable to being damaged by navigation uncertainty because the magnitude measured with respect to the baseline is small. As a result, the spacecraft may be maneuvered into a state where it is not properly aligned with the desired eigenvector, which will cause the spacecraft to deviate prematurely. Therefore, tight station-keeping may, in some cases, require more annual maneuvers. This pitfall is Δr 0By selecting a small value of and a large value of ρ, it is avoided while generating Solution 1. As a result, the spacecraft can slowly offset from the baseline over a period of 150 days and settle at a distance of about 9 km from the baseline where the maneuvers are not triggered so frequently.

[0088] FIG. 5 is a graph showing the displacement from the baseline trajectory for two spacecraft operating under local proper motion control according to some exemplary embodiments. The annual station-keeping performance of Solution 1 (502) and Solution 2 (504) is shown in Table 1 below, and their distances from the baseline as a function of time are shown in FIG. 5. With the proposed strategy, the annual station-keeping cost of the gateway is comparable to or better than the state-of-the-art cost for station-keeping on NRHO. Since the approaching spacecraft maintains a greater distance from the baseline than the gateway, significantly more fuel is required for the approaching spacecraft. Therefore, each maneuver to transition to non-extended local proper motion of the baseline is more expensive.

Table 1

[0089] Another notable advantage of the proposed approach is that relatively few maneuvers are required to maintain annual limited motion. The dark portions on Solutions 1 and 2 in FIG. 5 highlight a significantly small annual control duty cycle. Further, Solutions 1 and 2 do not pose a risk of collision between the gateway and the approaching spacecraft. The distance between the solutions does not become less than 8 km. Feature

[0090] It is contemplated that the controller can use the proposed approach to maintain the spacecraft near the desired orbit. To detect whether the trigger condition is satisfied, the controller can determine the current state of the spacecraft within a specified period. To determine the current state of the spacecraft, the processor can obtain measurements corresponding to the position and translational velocity of the spacecraft and one or a combination of perturbations, where the perturbations act on the 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 the mechanical system defined by the spacecraft and at least one celestial body. Further, the processor can propagate the linear model of the mechanical system over a period defined by a first time instance when the spacecraft is in the current state and a second time instance when the spacecraft is in the next state.

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

[0092] To generate the control command, the processor can solve a non-linear optimization problem to find a fuel-efficient maneuver that transfers the spacecraft to a desired set of states that result in the desired natural motion. The non-linear optimization problem is directed towards the finite horizon optimization of the spacecraft model, a set of objectives of the spacecraft motion, and the constraints on the spacecraft propulsion system and the spacecraft motion. The constraints on the spacecraft propulsion system and the spacecraft motion include one or more physical limitations of the spacecraft, one or more safety limitations on the operation of the spacecraft, and one or more performance limitations on the trajectory of the spacecraft.

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

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

[0095] The high-fidelity reference trajectory is an uncontrollable natural motion trajectory located near the near-rectilinear halo orbit (NRHO) in the Earth-Moon circular restricted three-body problem and is estimated using a multiple shooting approach. The high-fidelity reference trajectory includes a finite number of non-periodic revolutions around the Moon over a finite number of days.

[0096] The orbit can be one of a circular orbit, an elliptical orbit, a halo orbit, a lunar long ellipse polar orbit, or a quasi-satellite orbit. The control command is generated as a solution to a model predictive control policy that generates the control command by optimizing a cost function over a receding horizon. The cost function includes a stabilization component for directing the movement of the spacecraft to the next state, a component for the purpose of the operation of the spacecraft, and a performance component for optimizing the movement of the spacecraft until the next state is achieved. In some exemplary embodiments, the cost function includes an objective function that minimizes the fuel consumption of the spacecraft. The control command is generated over each of a plurality of specified periods within a time horizon or is iteratively generated over a receding time horizon. The control command is output to an operation module of the controller, and the operation module communicates the control command to a thruster command module, and the thruster command module receives the control command as a delta-v command, and the thruster command module converts the delta-v command into a thruster command and transmits the thruster command to a thruster processor of at least one thruster to activate or not activate at least one thruster for vehicle trajectory tracking control according to the converted delta-v command. Definitions

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

[0098] Space rendezvous: Space rendezvous can be a set of orbital maneuvers in which two spacecraft (or a chaser spacecraft and a target, where the target can be another spacecraft, a space station, a celestial body, or orbital debris) reach the same orbit and approach to a very close distance (e.g., within visual range).

[0099] Celestial system (astronomical reference coordinate system): In astronomy, an astronomical coordinate system (or astronomical reference coordinate system) is a system for specifying the positions of satellites, planets, stars, galaxies, and other celestial bodies relative to a physical reference point available to an existing observer (e.g., the horizontal north direction of an observer existing on the Earth's surface). The coordinate system can specify the position of an object in three-dimensional space or, when the distance of the object is unknown or not a problem, simply depict its direction on the celestial sphere. The coordinate system is realized either as a spherical coordinate system or a rectangular coordinate system. The spherical coordinate system projected onto the celestial sphere is similar to the geographic coordinate system used on the Earth's surface. These differences lie in the choice of the fundamental plane that divides the celestial sphere into two equal hemispheres along a great circle. The rectangular coordinate system with appropriate units is simply the Cartesian equivalent of the spherical coordinate system having the same fundamental (x, y) plane and principal (x-axis) direction. Each coordinate system is named according to the choice of its fundamental plane.

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

[0101] Conic Section: Referring to FIG. 6A, a conic section (also synonymously referred to as a conic cut surface) is a curve formed by intersecting a plane with a right circular cone 602. FIG. 6A shows the angular direction of the plane 600 with respect to the cone 602 that determines whether the conic section is a circle 604, an ellipse 606, a parabola 608, or hyperbolas 610, 612. The circle 604 and the ellipse 606 occur 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 where the plane 600 is perpendicular to the axis of the cone 602. When the plane 600 is parallel to the generatrix of the cone 602, the conic section is called a parabola 608. Finally, when 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, the plane 600 intersects both halves of the cone 602, forming two separate curves. All conic sections can be defined by the eccentricity. The type of conic section is also related to the orbital semi-major axis and energy. Table 2 below shows the relationship between the eccentricity, orbital semi-major axis, energy, and the type of conic section.

Table 2

[0102] Referring to FIGS. 6B, 6C, and 6D, to mathematically describe a conventional orbit, six quantities called orbital elements must be defined. The six orbital elements are Orbital semi-major axis a Eccentricity e Inclination angle i Argument of periapsis ω Time of periapsis passage T Right ascension of the ascending node That is.

[0103] Figures 6B to 6D show a conventional orbiting satellite 650 following an oval-shaped path known as ellipse 620. The celestial body at the center of the orbit, called the primary star, is located at one of two points called foci 622, 624. FIG. 6C shows ellipse 620, which is defined as a curve having the property that the sum of the distances from two fixed points called foci 622, 624 is constant at each point on the ellipse. The longest and shortest lines that can be drawn through the center of the ellipse are called the major axis and the minor axis, respectively. The orbital semi-major axis is half of 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 value between 0 and 1. An eccentricity of zero means a circle.

[0104] FIG. 6D shows the inclination angle i, which is the angular distance between the orbital plane of the satellite and the equatorial plane of its primary star (or the ecliptic plane in the case of a heliocentric orbit, i.e., an orbit centered on the sun). An inclination angle i of 0 degrees indicates an orbit around the equatorial plane of the primary star in the same direction as the rotation of the primary star, which is called the prograde (or direct) direction. 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 direction opposite to the rotation of its primary star.

[0105] Continuing to refer to FIG. 6D, the periapsis ω is the point in the orbit that is closest to the primary star (i.e., in the case of 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 at its maximum speed according to Kepler's second law). The opposite of the periapsis ω is the apoapsis, which is the farthest point in the orbit (i.e., in the case of an object moving in an elliptical orbit around another celestial body, the farthest point is the apoapsis, and at this point in the orbit, the object moves at its minimum speed according to Kepler's second law). The perihelion is the position of closest approach, i.e., the point where the distance between the sun and the planet is the shortest, and at this point in the orbit, the planet moves at its maximum speed according to Kepler's second law. The aphelion is the point where the distance between the sun and the planet is the farthest, and at this point in the orbit, the planet moves at its minimum speed according to Kepler's second law. The aphelion specifically refers to the orbit around the sun and corresponds to the apoapsis of a general orbit. The periapsis ω and the apoapsis are usually corrected according to the celestial body at the center of the orbit. For example, in the case of the sun, they are the perihelion and the aphelion; in the case of the earth, they are the perigee and the apogee; in the case of Jupiter, they are the perijove and the apojove; in the case of the moon, they are the perilune and the apolune, etc. The argument of periapsis ω is the angular distance between the ascending node N 1 and the periapsis (see FIG. 6D). The time of periapsis passage T is the time when the satellite passes through its periapsis.

[0106] Periapsis: In an elliptical orbit of a celestial body around the center of mass of a system, it is the point where the distance between the celestial body and the center of mass is minimized. Represented by the symbol ω (also called the argument of perifocus or argument of pericenter), it is one of the orbital elements of the celestial body orbiting the orbit. As a parameter, ω is the angle from the ascending node of the celestial body to the pericenter, measured in the direction of motion. For certain types of orbits, words including perihelion (orbit centered on the sun), perigee (orbit centered on the earth), periastron (orbit centered on a star), etc. may be replaced with the word pericenter (see orbital poles for details). The argument of pericenter being 0° means that the celestial body orbiting the orbit approaches the central body most closely when crossing the reference plane from south to north. The argument of pericenter being 90° means that the celestial body orbiting the orbit reaches the pericenter at the place farthest from the reference plane on the north side. Adding the argument of pericenter to the longitude of the ascending node gives the longitude of the pericenter. However, especially in discussions of binary stars and exoplanets, the terms "longitude of pericenter" or "longitude of periastron" are often used synonymously with the "argument of pericenter".

[0107] Apoapsis : In an elliptical orbit of a celestial body around the center of mass of a system, it is the point where the distance between the celestial body and the center of mass is maximized.

[0108] Intersection Point : The point where the orbit intersects a plane, for example, the point where a satellite intersects the equatorial plane of the earth. When the satellite intersects the plane from south to north, the intersection point is the ascending node N 1 and when it intersects the plane from north to south, the intersection point is the descending node N 2 is. The longitude of the ascending node N 1 is the ecliptic longitude of this intersection point. The ecliptic longitude is similar to the longitude on the earth, measured counterclockwise from zero in degrees, and the zero longitude is in the direction of the vernal equinox Ω.

[0109] Orbit Type: The geosynchronous orbit (GEO) is a circular orbit around the Earth with a period of 24 hours. The geosynchronous orbit with an inclination angle of 0 degrees is called the geostationary orbit. A spacecraft in the geostationary orbit appears to be hanging stationary above a certain position on the Earth's equator. Therefore, this is ideal for certain types of communication satellites and meteorological satellites. A spacecraft in an inclined geosynchronous orbit appears to trace a regular figure-eight pattern in the sky with each orbit revolution. To obtain a geosynchronous orbit, a spacecraft is first launched into an elliptical orbit called the geosynchronous transfer orbit (GTO) with an apogee of 35,786 km (22,236 miles). Next, the orbit is made circular by firing the spacecraft's engine at the apogee.

[0110] Polar orbit (PO): An orbit with an inclination angle of 90 degrees. The polar orbit is useful for satellites performing mapping and / or surveillance operations because the spacecraft can access virtually all points on the planet's surface while the planet rotates. Walking orbit: An orbit in which a satellite orbiting the orbit is subject to a very large number of gravitational effects. First, the planet is not a perfect sphere and has a slightly non-uniform mass distribution. These instabilities affect the spacecraft's trajectory. In addition, the sun, moon, and planets exert gravitational effects on the satellite orbiting the orbit. If properly planned, it is possible to design the orbit to cause a precession motion in the satellite's orbital plane by taking advantage of these effects. The orbit thus obtained is called a walking orbit.

[0111] Sun-synchronous orbit (SSO): A walking orbit in which the orbital plane precesses with the same period as the planet's solar orbital period. In such an orbit, the satellite passes the perigee at the same local time with each orbit revolution. This is useful when the satellite is equipped with equipment that depends on some angle of sunlight illumination on the planet's surface. To maintain accurate synchronization timing, it may be necessary to perform occasional propulsion maneuvers to adjust the orbit.

[0112] Molniya orbit: An eccentric Earth orbit with a period of about 12 hours (two revolutions per day). The orbital inclination angle is selected so that the rate of change of the perigee is zero, and thus both the apogee and perigee can be maintained over a fixed latitude. This state occurs between inclination angles of 63.4 degrees and 116.6 degrees. In the case of these orbits, since the argument of perigee is typically placed in the Southern Hemisphere, the satellite will remain over the Northern Hemisphere near the apogee for about 11 hours per orbit revolution. This orientation allows for sufficient coverage of the ground in the high-latitude regions of the Northern Hemisphere.

[0113] Hohmann transfer orbit: An interplanetary trajectory, the advantage of which is that the amount of propellant consumed is minimized. A Hohmann transfer orbit to an outer planet such as Mars is obtained by launching the spacecraft in the direction of the Earth's revolution around the Sun, getting out of the influence of the Earth's gravity, and accelerating until the speed at which the aphelion enters the same solar orbit as the outer planet's orbit is reached. When the spacecraft reaches its destination, it must decelerate so that the planet's gravity can capture the spacecraft into the planetary orbit. For example, to send a spacecraft towards an inner planet such as Venus, the spacecraft is launched in the direction opposite to the Earth's revolution around the Sun and accelerated (i.e., decelerated) until the perihelion reaches the same solar orbit as the inner planet's orbit. Note that the spacecraft continues to move in the same direction as the Earth but slightly slower. To reach the planet, it is necessary to insert the spacecraft into the interplanetary trajectory at the right time so that the spacecraft reaches the planetary orbit when the planet is located at the point where the spacecraft will 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 during which the spacecraft must be launched to complete its mission is called the launch window.

[0114] Near-Rectilinear Halo Orbit (NRHO): It can be defined as an "almost stable" orbit whose stability is measured using the stability index v.

[0115] CR3BP Model: The lunar near-rectilinear halo orbit is a member of a wider set of the L1 and L2 families of halo orbits, i.e., it is an underlying structure that exists in a dynamic environment modeled in terms of multiple gravitational bodies. L1 is a point at 1 / 100 of the distance from the Earth to the Sun, i.e., the first Lagrange point, where the centripetal and gravitational forces of the Earth and the Sun cancel each other out. This is one of five such points within the Earth-Solar system where a spacecraft can in principle remain forever as if balanced on the gravitational version of the head of a pin. Another point, L2, is located 1.6 million kilometers away from the Earth on the opposite side of the Sun. Both L1 and L2 are ideal locations for surveying the universe, and L1 also offers a clear view with respect to the Earth and the Sun. However, they have drawbacks. At L1, the signals of the spacecraft are overwhelmed by the radiation from the Sun behind it. At L2, the Earth's shadow blocks the sunlight that the probe needs to power its instruments. The solution is to place the spacecraft in a "halo orbit" around the Lagrange point. A spacecraft in a halo orbit around L1 traces a huge, slack loop that is perpendicular to the Earth-Sun axis and descends towards the equilibrium point without end. Also, the basic behavior adheres to a more faithful model, thus supporting long-term possible mission scenarios for, in some cases, manned spacecraft in orbits near the Moon. This type of trajectory is first identified in a simplified representation of the gravitational effects in the Earth-Moon system, i.e., in the Circular Restricted Three-Body Problem (CR3BP). In the CR3BP model, the lunar near-rectilinear halo orbit (NRHO) can be defined as an "almost stable" orbit whose stability is measured using the stability index v, and is characterized by favorable stability properties that suggest the possibility of maintaining a motion similar to NRHO over the long term while consuming only a small amount of propellant resources. Also, some NRHOs have favorable resonance properties that can be exploited for mission design and are particularly useful for avoiding eclipses. However, in order to actually realize a mission, the transfer to such orbits and the station-keeping strategy must be clarified in a more faithful ephemeris model.The station-keeping algorithm for the libration point orbits has previously been investigated in the context of both planar Lyapunov orbits and conventional three-dimensional halo orbits within this dynamic framework. However, NRHOs are constructed within the framework of the celestial ephemeris.

[0116] Station Keeping: In astrodynamics, the orbital maneuvers performed by thruster firings necessary to maintain a spacecraft on a specified assigned orbit are called orbit station-keeping. For many Earth satellites, the effects of non-Keplerian forces, i.e., the deviation of the Earth's gravity from that of a homogeneous sphere, gravity from the Sun / Moon, solar radiation pressure, and air drag must be compensated for. The deviation of the Earth's gravitational field from that of a homogeneous sphere and the gravity from the Sun / Moon generally cause perturbations in the orbital plane. In the case of a sun-synchronous orbit, the precession motion of the orbital plane caused by the Earth's oblateness is a desirable feature that is part of the mission design, but the inclination angle change caused by the Sun / Moon gravity is not desirable. For a geostationary spacecraft, the inclination angle change caused by the Sun and Moon gravity must be compensated for by expending a fairly large amount of fuel because the inclination angle should be kept small enough to be tracked by non-maneuverable antennas. For spacecraft in low Earth orbit, the effects of atmospheric drag often must be compensated for. In some missions, this is necessary simply to avoid re-entry. In other missions, typically those where the orbit should be precisely synchronized with the Earth's rotation, this is necessary to avoid shortening the orbital period. Solar radiation pressure generally causes perturbations in the eccentricity (i.e., the eccentricity vector). See Orbital Perturbation Analysis (Spacecraft). In some missions, this must be actively compensated for by maneuvers. For a geostationary spacecraft, the eccentricity must be kept small enough to be tracked by non-maneuverable antennas. Also, for an Earth observation spacecraft with a very repetitive orbit having a fixed ground track, the eccentricity vector should be kept as fixed as possible. Most of this compensation can be done using frozen orbit design, but thruster maneuvers are necessary for fine-tuning. For a spacecraft on a halo orbit around a Lagrange point, station-keeping is even more important. Because such orbits are unstable and without active control by thruster firings, the slightest deviation in position / velocity would cause the spacecraft to completely leave the orbit.

[0117] Dominant Disturbing Force : The dominant disturbance causes a deviation greater than the navigation error of the deep space network (DSN) positioning system. Also, the dominant force can be defined as a force greater than the most unpredictable force with the greatest impact. Also, the most unpredictable force can be due to position and velocity measurement errors.

[0118] Perturbation : It can be the complex motion of a massive object that is subject to forces other than the gravitational force of a single other massive object. The other forces can include the gravitational force of a third (fourth, fifth, etc.) object, resistance from the atmosphere, and the off-center gravitational force of an oblate or otherwise distorted object. Perturbing forces act on the moon from the sun at two locations within its orbit. The black dotted arrows represent the direction and magnitude of the gravitational force on the earth. Applying this to both the position of the earth and the position of the moon does not disrupt their relative positions to each other. When this is subtracted from the force on the moon (black solid line), what remains is the perturbing force on the moon in comparison to the earth (black double arrow). Since the perturbing force has different directions and magnitudes on both sides of the orbit, it changes the shape of the orbit.

[0119] Figure 7A is a block diagram showing some components for implementing the 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 the controller 101 of FIG. 1E via one or more interfaces. Further, the processor 720 may be coupled to the sub-control device 740 of the thruster via buses 728 and 734. The processor 720 may receive the control command as a delta v-command 701. The processor 720 may convert the received delta-v command 701 into a thruster command 726 (724), and the thruster command 726 may be transmitted via the bus 728 to the sub-control device 740 of the thruster 743, and the thruster 743 may be connected to the sensor 748. The sub-control device thruster 740 may include another processor 741 for processing the thruster command and issuing a control action as commanded.

[0120] Figure 7B is a schematic diagram showing an aspect of the thruster configuration according to an embodiment of the present disclosure. In some exemplary embodiments, the spacecraft may include eight thrusters, which are mounted at the corners of the spacecraft so as to produce a pure force acting on the center of mass of the spacecraft without generating any torque that would rotate the spacecraft, when aligned. A controller, such as the sub-control device 740 in FIG. 7A, may transmit signals for activating or deactivating the thrusters in order to move the spacecraft along a commanded trajectory.

[0121]

Number

[0122] Continuing to refer to FIG. 7B, the translational equations of motion of the main and auxiliary machines with respect to the inertial frame F e are shown as follows.

Number

[0123] For the main spacecraft and the auxiliary spacecraft, the position of the auxiliary machine with respect to the main machine is shown as follows.

Number

[0124] Continuing to refer to FIG. 7B, the derivative of the relative position (21) with respect to the host's orbital frame F o is adopted, and the following is obtained.

Equation

[0125] The host's orbital frame F o is adopted, and the following is obtained.

Equation

[0126]

Equation

[0127] Continuing to refer to FIG. 7B, since r c and h vary along the orbit, the equation of motion (25) becomes a linear time-varying system.

Equation

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

[0129] The bus system 813 can be connected to the output thruster command module 858 for outputting thruster commands. Further, the bus 859 can be connected to the orbit maintenance module 840, and the orbit maintenance module 840 includes a transfer orbit generation unit for generating one or more transfer orbits of the spacecraft, a feedback gain module 844 for calculating a feedback gain, and a feedback controller 846. Further, the orbit maintenance module 840 can also include a thruster command generation unit 848.

[0130] Continuing to refer to FIG. 8, computer 870 can be a server or a desktop, laptop, mobile, or other computer device or system having one or more processors 810. Processor 810 can be a central processing unit adapted to access code in the form of a transfer orbit generation unit 842 in memory 812 or storage data 818 of computer 870 (or in extended memory 850 or other data storage 852, 854, 856). According to aspects related to the systems and methods of the present disclosure, an external storage device is contemplated if further required depending on the intended hardware and specific design and aspects of achieving the objectives. For example, computer 870 can be used to implement steps of the systems and methods where memory 812 and / or storage device 818 can store data.

[0131] The data stored in memory 812 of FIG. 8 can include executable modules, vehicle data, and history space data. For example, vehicle data can include spacecraft specifications, dimensions, weight, performance data under varying conditions including gravity, and other perturbations, i.e., the complex motion of a massive object subject to forces other than the gravitational pull of a single other massive object in space. Further, vehicle data can include data related to aspects related to vehicle dynamics associated with one or more of the following multivariables: (1) abnormal orbital characteristics of celestial bodies, i.e., natural objects located outside the Earth's atmosphere such as the moon, sun, asteroid, planet, or star; (2) abnormal orbital motion of celestial bodies; (3) orbits of celestial bodies that are abnormally close to another celestial body; and (4) other known perturbations. Space data can include data related to celestial systems, past missions to celestial bodies, and any other data related to the planning of orbital designs for space, spacecraft, and other celestial bodies in the universe. For example, space data can include data related to the moon of a celestial body, such as the characteristics of the celestial body that can be considered when developing an orbital design from an initial celestial orbit to a similar target celestial orbit.

[0132] Optionally, the stored data can be stored in an external interface 824 connected to the storage device 818 and the extended memory 850. The extended memory 850 is connected to the initial orbit data database 854, the other orbit data database 856, and the database 852 of data such as vehicle parameters, specifications, and performance in FIG. 8.

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

[0134] The receiver 828 or the input interface can receive the stored historical space data stored in the memory 812, the latest space data that may be obtained from either a sensor associated with the Earth mission control center or the spacecraft, or some other location. The receiver 828 and the transmitter 830 can provide a wireless location for receiving data and transmitting it, for example, to the Earth mission control center or some other destination. The GPS receiver module 832 connected to the GPS 834 can be used in aspects related to navigation. 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] FIG. 9 is a schematic diagram showing, as a non-limiting example, a computing device 900 that can be used to implement some of the techniques of the methods and systems according to embodiments of the present disclosure. The computing device or device 900 corresponds to 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 can include a power supply 908, a processor 909, a memory 910, and a storage device 911, all connected to a bus 950. Further, a high-speed interface 912, a low-speed interface 913, a high-speed expansion port 1214, and a low-speed connection port 915 can be connected to a bus 1250. Additionally, a low-speed expansion port 916 is connected to the bus 950. Various component configurations that can be mounted on a common motherboard (such as 930 by way of non-limiting example) are contemplated depending on the particular application. Further, an input interface 917 can be connected to an external receiver 906 and an output interface 918 via the bus 950. A receiver 919 can be connected to an external transmitter 907 and a transmitter 920 via the bus 950. Also, an external memory 904, an external sensor 903, a machine 902, and an environment 901 can be connected to the bus 950. Further, one or more external input / output devices 905 can be connected to the bus 950. A network interface controller (NIC) 921 can be adapted to connect to a network 922 through the bus 950, and data or other data can be rendered, inter alia, on a third-party display device, a third-party imaging device, and / or a third-party printing device external to the computing device 900.

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

[0138] The storage device 911 can be adapted to store supplementary data and / or software modules used by the computer device 900. For example, the storage device 911 can store historical data and other related data as described above with respect to the present disclosure. Additionally, or alternatively, the storage device 911 can store historical data such as data as mentioned above with respect to the present disclosure. The storage device 911 can include a hard drive, an optical drive, a thumb drive, an array of drives, or any combination thereof. Further, the storage device 911 can include a computer-readable medium such as a floppy (registered trademark) 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 an array of devices including devices in a storage area network or other configuration. Instructions can be stored on an information carrier. When executed by one or more processing devices (e.g., the processor 909), the instructions execute one or more methods such as those described above.

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

[0140] The computer device 900 can also be adapted to a printer interface (not shown) connected via bus 950 and can include a user input interface 917 adapted to connect to a printing device (not shown). The printing device can include, among other things, 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 to refer to FIG. 9, the high-speed interface 912 manages bandwidth-intensive operations for the computing device 900, and the low-speed interface 913 manages lower-bandwidth-intensive operations. Such a function assignment is just an example. In some implementations, the high-speed interface 912 can be coupled to a high-speed expansion port 914 that can receive memory 910, a user interface (HMI) 923, a keyboard 924 and a display 925 (e.g., via a graphics processor or accelerator), and various expansion cards (not shown) via bus 950. In one implementation, the low-speed interface 913 is coupled to the storage device 911 and the low-speed expansion port 915 via bus 950. The low-speed expansion port 915, which may include various communication ports (e.g., USB, Bluetooth®, Ethernet®, wireless Ethernet®), can be coupled 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, for example via a network adapter.

[0142] As shown in the figures, computing device 900 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. Additionally, it may be implemented in a personal computer such as a laptop computer 927. Also, it may be implemented as part of a rack server system 928. Alternatively, components from computing device 900 may be combined with other components within a mobile device (not shown). Each of such devices may include one or more of a computing device and a mobile computing device, and the overall system may be composed of multiple computing devices that communicate with each other.

[0143] The description provides only exemplary embodiments and is not intended to limit the scope, applicability, or configuration of the present disclosure. Rather, the following description of exemplary embodiments provides those skilled in the art with a feasible description for implementing one or more exemplary embodiments. What is contemplated are various changes that can be made in the functions and configurations of the elements without departing from the spirit and scope of the disclosed subject matter as recited in the claims.

[0144] In the following description, specific details are provided for a complete understanding of the embodiments. However, what can be understood by those skilled in the art is that the embodiments can be implemented without these specific details. For example, systems, processes, and other elements in the disclosed subject matter may be shown as components in the form of block diagrams so as not to obscure the embodiments with unnecessary details. In other examples, well-known processes, structures, and technologies may be shown without unnecessary details to avoid obscuring the embodiments. Further, like reference numerals and names in the various drawings indicate like elements.

[0145] Also, individual embodiments may be described as a process shown as a flowchart, a flow diagram, a data flow diagram, a structure diagram, or a block diagram. A flowchart can describe operations as a sequential process, but many of the operations can be executed in parallel or simultaneously. Additionally, the order of the operations may be rearranged. A process may end when its operations are completed, but may have additional steps not discussed or included in the figure. Further, not all operations in any particular process described will 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 can correspond to the 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. Manual or automatic implementations may be executed or at least assisted through the use of a machine, hardware, software, firmware, middleware, microcode, a hardware description language, or any combination thereof. When implemented in software, firmware, middleware, or microcode, the program code or code segments for performing the necessary tasks may be stored on a machine-readable medium. The necessary tasks may be executed by a processor.

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

[0148] Also, the various methods or processes outlined in this specification may be encoded 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 several suitable programming languages and / or programming or scripting tools, and may also be compiled as executable machine language code or intermediate code to be executed on a framework or virtual machine. Typically, the functionality of program modules may be combined or distributed as desired in various embodiments.

[0149] Also, embodiments of the present disclosure may be embodied as a method for which an example has been provided. The acts performed as part of the method may be ordered in any suitable manner. Accordingly, embodiments may be constructed in which some acts shown as consecutive acts in exemplary embodiments are performed simultaneously, including embodiments in which acts are performed in a different order than that shown by way of example. Further, the use of terms such as "first," "second," etc. in the claims to denote order for elements of the claims does not itself imply any priority, precedence, or order of one claim element over another claim element, or imply any temporal order in which acts of a method are to be performed, but are used merely as labels to distinguish one claim element having a given name (except where terms denoting order are used) from another element having the same name for purposes of distinguishing among the claim elements.

[0150] Although the present disclosure has been described with reference to specific preferred embodiments, it should be understood that various other adaptations and modifications can be made within the spirit and scope of the present disclosure. Accordingly, it is the aspect of the claims to embrace all such variations and modifications as fall within the true spirit and scope of the present disclosure.

Claims

1. A controller for maintaining a spacecraft near an orbit, comprising: a memory configured to store executable instructions; and a processor, wherein the processor executes the executable instructions to cause the controller to: detect 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, linearize the dynamics of the spacecraft from a current time over a 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 state transition matrix including non-expanding eigenvectors having a magnitude of 1 or less and expanding eigenvectors having a magnitude greater than 1, and the processor further executes the executable instructions to cause the controller to: determine a control action to change a next state of the spacecraft to a linear combination of the non-expanding eigenvectors of the state transition matrix; generate a control command for an actuator of the spacecraft to cause a correction of the next state of the spacecraft along a direction corresponding to at least one of the non-expanding eigenvectors of the state transition matrix.

2. To detect that the distance is greater than the spacecraft threshold, the processor: determines a current state of the spacecraft within a specified period; determines a distance between a location defined by the current state of the spacecraft and the orbit; and is configured to compare the determined distance between the location defined by the current state of the spacecraft and the orbit with the spacecraft threshold. The controller according to claim 1.

3. The specified period is outside the time horizon, and to determine the current state of the spacecraft, the processor is further configured to obtain measurement values corresponding to one or a combination of a position and a translational velocity of the spacecraft and perturbations, the perturbations acting on a multi-body celestial system defined by the spacecraft and at least one celestial body. The controller according to claim 2.

4. To determine the state transition matrix, the processor further: Obtain a linear model of the dynamical system defined by the spacecraft and at least one celestial body. Calculate the discrete-time up-to-date information of the perturbation of the high-fidelity reference trajectory. The controller according to claim 1, configured to propagate a mathematical formula defined by the high-fidelity reference trajectory and the basic basis vectors of the STM over a specified period.

5. The processor is further configured to obtain different spacecraft thresholds for a visiting spacecraft to perform eigenvector-based control of the motion of the visiting spacecraft using the high-fidelity reference trajectory, according to claim 1.

6. To determine the control action, the processor is further configured to determine the direction of the correction of the next state by a linear combination in the direction of each non-expanding eigenvector of the STM, so as to realize a mixed state with natural motion, where the natural motion is a combination of the corresponding eigenmotions of each non-expanding eigenvector of the STM, according to claim 1.

7. To generate the control command, the processor is further configured to solve a non-linear optimization problem to find a fuel-efficient maneuver that transfers the spacecraft to a set of desired states that produce a desired natural motion, according to claim 1.

8. The non-linear optimization problem is directed to finite-horizon optimization of the model of the spacecraft, a set of objectives of the motion of the spacecraft, the propulsion system of the spacecraft, and the constraints on the motion of the spacecraft, according to claim 7.

9. The constraints on the propulsion system of the spacecraft and the motion of the spacecraft include one or more physical limitations of the spacecraft, one or more safety limitations on the operation of the spacecraft, and one or more performance limitations on the trajectory of the spacecraft, according to claim 8.

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

11. The next state of the spacecraft includes one or more of the position, orientation, translational velocity, and angular velocity of the spacecraft and the payload of the spacecraft, and one or a combination of perturbations, and the perturbation acts on a multi-body celestial system defined by the spacecraft, the payload, and one or more celestial bodies. The controller according to claim 1.

12. The perturbation acting on the multi-body celestial system is a natural orbital force including gravitational perturbations of the sun and the moon, anisotropic gravitational perturbations due to the non-sphericity of the central body, solar radiation pressure, and air resistance. The controller according to claim 11.

13. The high-fidelity reference trajectory is an uncontrollable natural motion trajectory located near the near-rectilinear halo orbit (NRHO) in the Earth-moon circular restricted three-body problem, and is estimated using a multiple shooting approach. The controller according to claim 1.

14. The high-fidelity reference trajectory according to claim 13 includes a finite number of non-periodic revolutions around the moon over a finite number of days.

15. The orbit is one of a circular orbit, an elliptical orbit, a halo orbit, a near-rectilinear halo orbit, or a quasi-satellite orbit. The controller according to claim 1.

16. The control command is generated as a solution to a model predictive control policy that generates the control command by optimizing a cost function over a receding horizon. The controller according to claim 1.

17. The cost function includes a stabilization component for directing the motion of the spacecraft to the next state, a component for the purpose of the operation of the spacecraft, and a performance component for optimizing the motion of the spacecraft until the next state is achieved. The controller according to claim 16.

18. The control command is generated over each of a plurality of specified periods within the time horizon or is repeatedly generated over a receding time horizon. The controller according to claim 1.

19. A method implemented by a computer for maintaining a spacecraft near an orbit, comprising: 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, 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 the uncontrollable motion of the spacecraft within the time horizon, the state transition matrix including non-expanded eigenvectors having a magnitude of 1 or less and expanded eigenvectors having a magnitude greater than 1, the method further comprising determining a control action to change the next state of the spacecraft to a linear combination of the non-expanded eigenvectors of the state transition matrix; generating a control command to the actuator of the spacecraft that causes a correction of the next state of the spacecraft along a direction corresponding to at least one of the non-expanded eigenvectors of the state transition matrix. A method implemented by a computer.

20. A non-transitory computer-readable storage medium embodying a processor-executable program for performing a method for maintaining a spacecraft near an orbit, the method comprising 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, 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 the uncontrollable motion of the spacecraft within the time horizon, the state transition matrix including non-expanded eigenvectors having a magnitude of 1 or less and expanded eigenvectors having a magnitude greater than 1, the method further comprising determining a control action to change the next state of the spacecraft to a linear combination of the non-expanded eigenvectors of the state transition matrix; generating a control command to the actuator of the spacecraft that causes a correction of the next state of the spacecraft along a direction corresponding to at least one of the non-expanded eigenvectors of the state transition matrix. A non-transitory computer-readable storage medium.

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