Feature motion control of nearly linear halo orbits

Abstract

A computer-implemented method for maintaining a spacecraft in the vicinity of an orbit, the method comprising the steps of: detecting (3) that the distance from the spacecraft to the orbit is greater than a spacecraft threshold, and in response, linearizing (5) the dynamics of the spacecraft from a current time within a temporal field of view with respect to a high-fidelity reference trajectory, to generate a state transition matrix (STM) for uncontrolled motion of the spacecraft within the temporal field of view. The STM includes a non-extended feature vector having a size less than or equal to 1 and an extended feature vector having a size greater than 1. The method further comprises determining (7) a control action that changes an upcoming state of the spacecraft to a linear combination of non-extended feature vectors of the STM, and generating (9) a control command to an actuator of the spacecraft.

Description

Technical Field

[0001] The present disclosure relates generally to controlling spacecraft operations near an unstable orbit around a celestial body such as the Moon, and more particularly to methods and systems for producing fuel efficient maneuvers that exploit natural motion to remain near an unstable orbit for extended periods of time without intervention. Background Art

[0002] To date, launching and deploying space stations and spacecraft to outer space remains a challenging task, requiring precise analysis and research 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), which is a small space station in lunar orbit and is intended to be used as a solar communications hub, a scientific laboratory, a short-term living cabin for astronauts of a government agency, and a waiting area for lunar rovers and other robots. The Gateway can be deployed in a highly elliptical seven-day near-rectilinear halo orbit (NRHO) around the moon, and is intended to play an important role in promoting missions in lunar space and beyond. The NRHO is a halo orbit that has a slight bend between close passes with an orbiting celestial body, so it is close to a straight edge. Due to its good stability and visibility from the earth, the Gateway can be deployed close to the NRHO, a closed periodic trajectory in the Earth-Moon Circular Restricted Three-Body Problem (CR3BP). One of the advantages of this NRHO is the minimal amount of communication interruption with the earth.

[0003] Since CR3BP's NRHO does not account for perturbations such as solar gravity, solar radiation pressure (SRP), or the lunar J2 effect, the Gateway can be deployed near the NRHO instead of on it. Instead of attempting to follow CR3BP's NRHO and using fuel to compensate for predictable perturbations, standard practice is to solve, via multi-shooting or combination-based techniques, a high-fidelity trajectory near the NRHO that accounts for all major predictable forces in lunar and terrestrial space. This high-fidelity solution is also called the NRHO, although it is no longer closed, periodic, or stable. However, the benefit of this high-fidelity NRHO solution is that in the absence of any additional perturbing forces, the spacecraft can naturally follow the trajectory without consuming any fuel, which is a key performance metric to ensure the long-term availability of the Gateway.

[0004] However, two factors hinder the deployment of a portal based solely on a high-fidelity solution. First, navigation uncertainties and unpredictable perturbations in celestial space prevent the spacecraft from being deployed exactly on the calculated trajectory. Second, and critically, its trajectory is highly unstable compared to its ideal CR3BP counterpart. Thus, arbitrarily small deviations from the solution can cause the spacecraft to rapidly deviate from the calculated trajectory, necessitating stabilizing control actions. These deviation issues also occur for other celestial bodies, and there is a high probability that a spacecraft will deviate from the planned trajectory for a variety of reasons.

[0005] In recent years, several station-keeping strategies have been developed for high-fidelity NRHO. However, while the Gateway itself can exploit such control strategies, these approaches are not directly applicable to visiting spacecraft, which are used, for example, for cargo resupply, crew transportation, or inspection and maintenance missions that may require long-term bounded, collision-free relative motion around the Gateway. While formation control for multiple spacecraft in halo orbit has been developed (particularly for NRHO), these approaches propose control schemes based on periodic solutions in CR3BP and rely on a computationally expensive process of generating high-fidelity solutions for each spacecraft in the formation.

[0006] Therefore, there is a particular need in the art for an improved method to control the operation of multiple spacecraft for long-term bounded, collision-free motion near an NRHO. Summary of the invention

[0007] The present disclosure relates to a control strategy for reliable, fuel-efficient station keeping and bounded relative motion control of a gateway and visiting spacecraft that utilizes a single precomputed high-fidelity NRHO solution while ensuring a safe separation distance between the spacecraft. Throughout the disclosure, the high-fidelity NRHO may be referred to as a reference trajectory, a high-fidelity reference trajectory, a baseline, a baseline solution, and a baseline reference trajectory.

[0008] In order to develop some embodiments of the present disclosure, there are assumptions and realizations that contribute to its development, including at least one realization that the control strategy should utilize a linear approximation of the dynamics of the spacecraft near the high-fidelity NRHO.

[0009] Although small deviations from states on a high-fidelity NRHO trajectory would normally result in rapid divergence, some exemplary embodiments are based on the recognition that, at any given instance in time, there are some special states near the reference trajectory that produce desired non-deviation natural motion, where natural motion means motion without control (i.e., without using fuel / onboard power). These states form a desired space around the reference trajectory. If the spacecraft is controlled in this space, rather than to the orbit itself, the requirement to stay exactly on the orbit is relaxed to staying "near" the orbit, which sacrifices distance to the orbit for improved fuel efficiency. That is, the proposed control strategy aims to keep the spacecraft within the desired space where the desired non-deviation natural motion may occur, resulting in a significant increase in fuel efficiency, rather than continuously controlling the spacecraft to stay on the predetermined orbit, which results in significant fuel / onboard power consumption.

[0010] Some exemplary embodiments are also based on the recognition that the region (subspace) of such special states can be estimated using a local modal decomposition of a rolling horizon state transfer matrix (STM) associated with a high-fidelity reference trajectory. In other words, the orbital dynamics with respect to the reference trajectory within a user-determined finite time horizon can be linearized and the eigenvalue decomposition of the resulting linearized STM can be used. The state transfer matrix is ​​used to find a solution to a general state space representation of a linear system described in the following form: x(t0)=x0, where x(t) is the state of the system, u(t) is the input signal, A(t) and B(t) are matrix functions, and x0 is the initial condition at t0. The state transition matrix Φ(t,τ) of this system can be used to give the solution The first term is called the zero input response and represents how the state of the system will evolve in the absence of any input. The second term is called the zero state response and defines how the input affects the system. Thus, the STM provides the state of the system.

[0011] At this point, it is important to understand that an N-dimensional (non-zero) vector v is an eigenvector of a square N×N matrix A if it satisfies a linear equation of the form Av=λv for some scalar λ. Then λ is called the eigenvalue corresponding to the eigenvector v. The special state that exhibits the expected natural motion arises from the eigenvectors of the STM whose eigenvalues ​​have a magnitude less than 1. This is because for such states where the magnitude of the eigenvalues ​​is less than 1, the STM provides a solution that is closer to the baseline solution (i.e., converges towards the reference trajectory).

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

[0013] Another insight included with some exemplary embodiments is that these eigenvectors are directions that do not have a fixed magnitude. This means that the spacecraft can move in these directions at various distances from the reference trajectory and exhibit the desired characteristic motion types. Some exemplary embodiments utilize the distance to the reference trajectory as a trigger condition to determine when to apply control because at any spacecraft distance from the reference trajectory, the spacecraft can be controlled to move from the current undesirable deviation state to a desired state in the non-expanded subspace.

[0014] Some exemplary embodiments recognize that whenever a deviation from a vicinity of a reference trajectory is detected, the characteristic motion-based control solves a nonlinear optimization problem to find a set of one or more fuel-efficient maneuvers (also referred to as control actions) that transfer the spacecraft to a state that produces the desired natural motion. The nonlinear optimization is a finite-horizon optimization of a dynamic model of the spacecraft, a set of motion goals for the spacecraft, and constraints on the spacecraft propulsion system and motion, with the ability to predict future events to take appropriate control actions.

[0015] One or more fuel efficient maneuvers may be achieved by optimizing the operation of the spacecraft according to a set of objectives over a future finite time horizon, where predictions are obtained according to a model of the spacecraft subject to constraints. The constraints may represent, for example, physical limitations on the spacecraft, safety limitations on the operation of the spacecraft, and performance limitations on the trajectory of the spacecraft. In some non-limiting examples, constraints on the propulsion system of the spacecraft may include constraints on inputs to thrusters that define a range of rotation of the thrusters. A control strategy for the spacecraft is admissible when the motion produced by the spacecraft for such a control strategy satisfies all constraints.

[0016] In theory, nonlinear optimal control can be used to obtain an optimal position-keeping maneuver for the duration of the entire spacecraft mission rather than within a finite time horizon. However, this creates a very large optimization problem and results in a very high computational burden, which may not be feasible in the computationally constrained hardware in the spacecraft. Thus, some exemplary embodiments are based on the recognition that by using control based on local eigenmotions, the spacecraft can use natural motions for a long time, which reduces the computational burden of the optimization problem and speeds up the solution of the optimization problem.

[0017] A further insight is that the use of trigger-based control allows the spacecraft to be controlled infrequently, as the spacecraft can follow the natural motion for an extended period of time until it begins to deviate from the reference trajectory beyond a trigger threshold, at which point there is a nearby desired state in subspace that can be steered to. Since the spacecraft is controlled infrequently, the number of times the thrusters are activated is greatly reduced, which again results in improved fuel efficiency over a period of time.

[0018] In some exemplary embodiments, the distance from the reference trajectory is an adjustable parameter that allows multiple spacecraft to be controlled relative to each other without having to independently calculate the reference trajectory of each spacecraft. Multiple spacecraft, the planets around which the multiple spacecraft orbit, and the orbits of the multiple spacecraft constitute a multi-target celestial system. In other words, the same subspace associated with a single reference trajectory is used, but at different distances from the reference trajectory, which is sufficient to ensure the difference in the trajectories of the multiple spacecraft. Therefore, in a scenario where a visiting spacecraft visits the main spacecraft for some reason, the same reference trajectory of the main spacecraft can be used. In this way, since the exemplary embodiments use the distance between the spacecraft and the baseline solution (reference trajectory) as a trigger for the control strategy, the method is applicable to controlling any number of spacecraft other than a portal.

[0019] In some exemplary embodiments, particular eigenvector direction states are combined into linear combinations to achieve a hybrid state using natural motion that is a combination of corresponding eigenmotions. For example, controlling the spacecraft to a combination of eigenvectors with eigenvalues ​​strictly less than 1 produces a natural motion of the spacecraft that converges toward a reference trajectory. Conversely, including eigenvector directions with magnitudes close to 1 in the linear combination produces a natural motion that oscillates at the current distance from the reference trajectory. Selecting a particular combination of eigenvector directions to which the spacecraft is to be transitioned allows the operator to form a natural motion of the spacecraft near the reference trajectory, moving the spacecraft closer to or further away from the reference trajectory as desired.

[0020] According to one non-limiting embodiment, the spacecraft is actuated by eight thrusters, each thruster being mounted in alignment with the center of mass of the spacecraft so that it generates a force to change the position of the spacecraft while not generating a torque to rotate the spacecraft.

[0021] To this end, some exemplary embodiments provide controllers, methods, and procedures for maintaining a spacecraft near a desired orbit. Some exemplary embodiments provide important solutions to control the operation of multiple spacecraft for long-term bounded collision-free motion near the NRHO for missions performing satellite servicing, active debris mitigation, in-space manufacturing, space station resupply, and planetary sample return.

[0022] Thus, one embodiment discloses a controller for maintaining a spacecraft near an orbit. The controller includes a memory having instructions stored thereon and a processor, the processor being 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. In response to detecting that the distance from the spacecraft to the orbit is greater than the spacecraft threshold, the processor is further configured to linearize the dynamics of the spacecraft in a current state within a time horizon relative to a high-fidelity reference trajectory to generate a state transfer matrix (STM) for uncontrolled motion of the spacecraft within the time horizon. The STM includes a non-extended eigenvector having a magnitude less than or equal to 1 and an extended eigenvector having a magnitude greater than 1. The processor is further configured to determine a control action that changes an upcoming state of the spacecraft to a linear combination of the non-extended eigenvectors of the state transfer matrix. The processor is further configured to generate a control command to an actuator of the spacecraft, the control command resulting in a correction to the upcoming state of the spacecraft in a direction corresponding to at least one of the non-extended eigenvectors of the state transfer matrix.

[0023] Another embodiment discloses a computer-implemented method for maintaining a spacecraft in the vicinity of an orbit, the method comprising the step of 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 in a current state relative to a high-fidelity reference trajectory within a time horizon to produce a state transfer matrix for uncontrolled motion of the spacecraft within the time horizon. The state transfer matrix includes non-extended eigenvectors having a magnitude less than or equal to 1 and extended eigenvectors having a magnitude greater than 1. The method also includes determining a control action that changes an upcoming state of the spacecraft to a linear combination of the non-extended eigenvectors of the state transfer matrix, and generating control commands to actuators of the spacecraft, the control commands resulting in a correction to the upcoming state of the spacecraft in a direction corresponding to at least one of the non-extended eigenvectors of the state transfer matrix.

[0024] Yet another embodiment discloses a non-transitory computer-readable storage medium having a program implemented thereon, the program executable by a processor for performing a method of 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, linearizing the dynamics of the spacecraft in a current state within a time horizon relative to a high-fidelity reference trajectory to generate a state transfer matrix for uncontrolled motion of the spacecraft within the time horizon. The state transfer matrix includes non-extended eigenvectors having a magnitude less than or equal to 1 and extended eigenvectors having a magnitude greater than 1. The method also includes determining a control action that changes an upcoming state of the spacecraft to a linear combination of the non-extended eigenvectors of the state transfer matrix, and generating control commands to actuators of the spacecraft, the control commands resulting in a correction to the upcoming state of the spacecraft in a direction corresponding to at least one of the non-extended eigenvectors of the state transfer matrix.

[0025] The presently disclosed embodiments will be further explained with reference to the accompanying drawings. The drawings shown are not necessarily drawn to scale, emphasis instead generally being placed upon illustrating the principles of the presently disclosed embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] [ Figure 1A ]

[0027] Figure 1A is a block diagram illustrating some method steps for producing a characteristic motion maneuver design for fuel efficient position keeping and bounded relative motion control of a spacecraft according to some exemplary embodiments;

[0028] [ Figure 1B ]

[0029] Figure 1B A flow chart illustrating some steps for determining a state transition matrix for uncontrolled motion of a spacecraft according to some exemplary embodiments;

[0030] [ Figure 1C ]

[0031] Figure 1C Another flow chart illustrating some steps for determining a state transition matrix for uncontrolled motion of a spacecraft according to some exemplary embodiments;

[0032] [ Figure 1D ]

[0033] Figure 1D is a block diagram illustrating a method for controlling the operation of a spacecraft staying within some bounds of a reference trajectory, according to some exemplary embodiments;

[0034] [ Figure 1E ]

[0035] Figure 1E is a block diagram illustrating some components of a controller for maintaining a spacecraft near orbit, according to some exemplary embodiments;

[0036] [ Figure 2A ]

[0037] Figure 2A is a schematic diagram illustrating a representation of a high-fidelity reference trajectory (also referred to as a baseline solution) according to some exemplary embodiments;

[0038] [ Figure 2B ]

[0039] Figure 2B is a schematic diagram showing a representation of a maneuver of a non-extended local feature motion to transfer a spacecraft to aphelion when a trigger condition is met in accordance with some exemplary embodiments;

[0040] [ Figure 3 ]

[0041] Figure 3 is a graph illustrating distance from a baseline for exemplary feature motions according to some exemplary embodiments;

[0042] [ Figure 4 ]

[0043] Figure 4 illustrating algorithms associated with local feature motion control according to some exemplary embodiments;

[0044] [ Figure 5 ]

[0045] Figure 5 is a graph illustrating position deviations from a baseline trajectory for two spacecraft operating under local feature motion control according to some exemplary embodiments;

[0046] [ Fig. 6A ]

[0047] Fig. 6A is a schematic diagram illustrating some general parameters for implementing aspects of methods and systems according to some exemplary embodiments;

[0048] [ Figure 6B ]

[0049] Figure 6B is a schematic diagram illustrating some general parameters for implementing aspects of methods and systems according to some exemplary embodiments;

[0050] [ Figure 6C ]

[0051] Figure 6Cis a schematic diagram illustrating some general parameters for implementing aspects of methods and systems according to some exemplary embodiments;

[0052] [ Fig.6D ]

[0053] Fig.6D is a schematic diagram illustrating some general parameters for implementing aspects of methods and systems according to some exemplary embodiments;

[0054] [ Fig. 7A ]

[0055] Fig. 7A is a block diagram illustrating some components for implementing generated control commands according to some exemplary embodiments;

[0056] [ Figure 7B ]

[0057] Figure 7B is a schematic diagram illustrating aspects of a thruster configuration according to some exemplary embodiments;

[0058] [ Figure 8 ]

[0059] Figure 8 is a schematic diagram showing some components for implementing methods and systems according to some exemplary embodiments; and

[0060] [ Fig. 9 ]

[0061] Fig. 9 is a schematic diagram illustrating, by way of non-limiting example, a computing device for implementing some techniques of methods and systems according to some exemplary embodiments.

[0062] Although the above-mentioned drawings illustrate the embodiments of the present disclosure, other embodiments are also conceivable as mentioned in the discussion. The present disclosure presents the embodiments by way of representation and not limitation. Those skilled in the art may devise many other variations and embodiments that fall within the scope and spirit of the principles of the embodiments of the present disclosure. DETAILED DESCRIPTION

[0063] Deploying spacecraft and space stations into orbits around celestial bodies is a challenging task that requires reliable, low-cost strategies for station keeping and relative motion tailored for such special orbits. An example of such a deployment is when a visiting spacecraft needs to be deployed around the Lunar Orbital Platform-Gateway (LOP-G), also known as the Gateway, and is planned to be deployed in a highly elliptical seven-day near-rectilinear halo orbit (NRHO) around the Moon. In the context of deploying such a spacecraft, it is desirable to perform as much natural, uncontrolled motion as possible in order to reduce fuel consumption and frequent replenishment of onboard energy. Several factors prevent or hinder attempts to deploy spacecraft into the vicinity of the NRHO. Therefore, arbitrarily small deviations from the solution can cause the spacecraft to quickly deviate from the calculated trajectory, requiring stabilizing control actions.

[0064] Exemplary embodiments disclosed herein provide a control method that exploits the eigenvectors of the state transfer matrix (STM) associated with a high-fidelity NRHO solution in an ephemeris model to design long-term position keeping and bounded relative motion. The proposed strategy effectively exploits the natural motion of the spacecraft, making control actions infrequent and fuel efficient. This ensures that the spacecraft can exhibit long-term bounded, collision-free relative motion around the portal. Furthermore, since the proposed strategy does not involve any periodic solutions or computing high-fidelity solutions for each spacecraft, the method on which some exemplary embodiments are based is computationally efficient, inexpensive, and scalable, while being fast and leading to improved energy efficiency of the spacecraft.

[0065] In this regard, exemplary embodiments utilize a single pre-computed high-fidelity NRHO solution while ensuring safe separation distances between spacecraft. Some exemplary embodiments utilize some special states near a reference trajectory that produce desired non-deviant natural motion. These states form a desired space around the reference trajectory. If the spacecraft is controlled to this space, rather than to the orbit itself, the requirement is relaxed from staying precisely on the orbit to staying "near" the orbit, which sacrifices the distance to the orbit to improve fuel efficiency. To determine these special states, some exemplary embodiments involve local modal decomposition of a rolling time domain state transfer matrix (STM) associated with a high-fidelity reference trajectory. In order to prevent unnecessary or undesirable spacecraft control that may result in excessive energy consumption, some exemplary embodiments utilize trigger conditions to determine when to execute control laws / strategies. The trigger conditions are based on the distance of the spacecraft from the reference trajectory. The use of trigger-based control allows the spacecraft to perform control infrequently. This is because the spacecraft can follow the natural motion for an extended period of time until it begins to deviate from the reference trajectory beyond a trigger threshold, at which time there is a nearby desired state in the subspace that can be manipulated.

[0066] Another advantage of the control method provided by the various exemplary embodiments is that the proposed strategy allows controlling multiple spacecraft relative to each other (collision avoidance) without the need to calculate a reference trajectory for each of them separately. The same subspace of special states associated with a single reference trajectory can be used for each spacecraft at a different distance from the single reference trajectory.

[0067] In this way, some exemplary embodiments of the present disclosure provide important solutions to control the operation of multiple spacecraft for long-term bounded collision-free motion near the NRHO for missions of satellite servicing, active debris mitigation, in-space manufacturing, space station resupply, and planetary sample return. In addition, close operations are an important process for achieving mission objectives and are a key technology for space exploration. In fact, the systems and methods of the present disclosure can be applied to a variety of purposes, such as but not limited to satellite servicing, orbital debris removal, in-space manufacturing, space station resupply, and planetary science sample return missions.

[0068] These and many other advantages will become apparent from the following detailed description of exemplary embodiments of the proposed control strategy. Although some exemplary embodiments are described with reference to lunar orbit, it is contemplated that the proposed control strategy is applicable to any multi-target celestial body system. Therefore, the exemplary embodiments described herein are not limited to the lunar celestial body system mentioned for exemplary purposes only. The scope of the proposed control strategy includes situations and systems related to any multi-target celestial body system.

[0069] Figure 1A is a block diagram illustrating some method steps for producing a characteristic motion maneuver design for fuel efficient position keeping and bounded relative motion control according to some exemplary embodiments. In particular, the method steps described relate to any system and method for controlling the operation of a spacecraft within a finite time horizon.

[0070] In some exemplary embodiments, a spacecraft may be directed to orbit a central body such as a celestial body. The spacecraft may achieve this by orbiting itself or by being deployed around a space station orbiting the central body. In this regard, the spacecraft may be equipped with the ability to determine or obtain the current state of the spacecraft within a specified time period. The state of the spacecraft may include data related to one or a combination of the position of the spacecraft, the translational velocity, and the perturbations acting on the multi-target 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 may deviate from or in some cases begin to deviate from the desired orbit in any direction. Therefore, it is expected that an immediate control action can be performed to manipulate the spacecraft back to the desired orbit or along the path of the desired orbit. Therefore, the condition 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 strategy to prevent the spacecraft from deviating from the desired orbit.

[0071] Upon detection of satisfaction Figure 1A When the trigger condition in step 3 is Figure 1A The controller in step 5 may linearize the dynamics of the spacecraft in the current state within the time horizon relative to the high-fidelity reference trajectory to produce a state transfer matrix (STM) of the uncontrolled motion of the spacecraft within the time horizon. In some exemplary embodiments, the high-fidelity reference trajectory may be pre-computed and stored in a storage unit such as a database accessible to the controller. In some other exemplary embodiments, the reference trajectory may be dynamically calculated at runtime. The linearization of the dynamics of the spacecraft may be performed at discrete time instances, and the state transfer matrix may be directly obtained.

[0072] In some exemplary embodiments, the time horizon is a finite time horizon, and the specified time period for determining the current state of the spacecraft may overlap with the finite time horizon. In some exemplary embodiments, the specified time period may be outside the finite time horizon.

[0073] After linearization of the orbital dynamics of the spacecraft around a high-fidelity reference trajectory, the resulting linearized STM is subjected to eigenvalue decomposition to identify regions (subspaces) of special states near the reference trajectory. Special states can be states that produce desired non-deviation natural motion. Throughout the disclosure, natural motion or uncontrolled motion can refer to the motion of a spacecraft without control (i.e., without the use of fuel or onboard power). The resulting STM includes non-extended eigenvectors with a size less than or equal to 1 and extended eigenvectors with a size greater than 1. In some exemplary embodiments, special states that exhibit desired natural motion originate from eigenvectors of the STM whose eigenvalues ​​are less than 1.

[0074] Figure 1A Step 7 of includes determining the desired control action to achieve the state as a linear combination of the unextended eigenvectors of the state transfer matrix. If the spacecraft is to follow its calculated trajectory, and detects that it will deviate from the desired trajectory in the upcoming state, then the controller needs to perform a corrective maneuver for the upcoming state. At this point, the controller determines the control action that changes the upcoming state corresponding to the linear combination of the unextended eigenvectors of the STM.

[0075] Figure 1A Step 9 of includes generating one or more control commands to activate or deactivate one or more thrusters of the spacecraft within a specified time period based on the control commands to achieve the desired corrective maneuver. The control commands are intended to cause a correction to the upcoming 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 an interface of the spacecraft in step 11 for further action. In some exemplary embodiments, the controller can be implemented on the spacecraft itself, so that in step 11 the controller can directly provide control commands to the actuators (propellers and / or thrusters) of the spacecraft to achieve the desired corrective maneuver. In some exemplary embodiments, the controller can communicate remotely with the spacecraft, so the control commands can be transmitted to the spacecraft, and the onboard CPU of the spacecraft can process the control commands accordingly to achieve the desired corrective maneuver.

[0076] Figure 1B Flowchart showing some steps for determining a state transition matrix for uncontrolled motion of a spacecraft according to some exemplary embodiments Figure 5 a. A dynamic model of a multi-target celestial system may be obtained 151. The multi-target celestial system may include a spacecraft and at least one celestial body. The dynamic model may be expressed as an analytical equation. Since the dynamic model may be nonlinear, it may be linearized in a next step 153 to obtain a linear model of the multi-target celestial system 155. This may present a continuous time analytical equation for linearization, and a discrete time matrix may be required to obtain a state transfer matrix. As part of the step of obtaining the state transfer matrix, the process Figure 5 a includes propagating 157 the linearized model over a defined time period. For example, the defined time period may span the entire duration of the baseline. In this regard, a differential equation (e.g., differential equation 11 described below) may be solved 157a over the defined time period to obtain a discrete time state transfer matrix 159.

[0077] Figure 1C Another flow chart showing some steps for determining a state transition matrix for uncontrolled motion of a spacecraft according to some exemplary embodiments Figure 5b. At 161, a dynamical model of a multi-target celestial system represented as an analytical equation may be obtained. A baseline solution (also referred to as a reference trajectory) may also be obtained 163. A discrete-time update of the baseline solution to the perturbation may be calculated 165. For example, let r(t) denote a solution, which may be a baseline solution in some exemplary embodiments. A discrete-time update of r(t) to the perturbation may be calculated, i.e., given a time t h and perturbation state Where δr(t) represents the relative state with respect to r, and the goal is to find the local linear mapping (state transfer matrix) Φ(t,t+t h ), so that

[0078] process Figure 5 b also includes defining 167 the basic basis vectors for the STM. In order to numerically compute Φ(t,t+t h ), limiting e i is the basic basis vector, where e i The i-th element of is equal to 1, and ε>0 is set as a small parameter. Next, the numerical expression defined by the baseline solution r(t) and the basic basis vectors is propagated 169 over a limited time period. For example, the expression r(t)+εe i and r(t)-εe i can be propagated numerically from t to t+t h , and at time t+t h The solutions can be called and Then, the STMΦ(t,t+t h )'s i-th column approximates 171:

[0079]

[0080] In this way, the STM can be numerically obtained with fewer computations, resulting in improved fuel efficiency.

[0081] Figure 1D is a block diagram illustrating a method for controlling the operation of a spacecraft staying within some limits of a reference trajectory according to some embodiments of the present disclosure. For example, the method controls the operation of the spacecraft using control inputs determined using a model of a joint multi-objective celestial system based on optimization of a cost function with an objective function of minimizing fuel / onboard power consumption.

[0082] Figure 1DAn initial step 110 of includes determining a current state of the spacecraft, wherein the current state of the spacecraft may be determined using sensors or other aspects such as hardware or software. Additionally, or alternatively, the current state of the spacecraft may be obtained from communications with a ground command center located on Earth or another spacecraft located in outer space (e.g., GPS, relative range measurements, star trackers, horizon sensors, etc.). In some exemplary embodiments, the current spacecraft state may be determined based on a previous control input determined for a previous iteration optimized using a previous model of the spacecraft using a previous cost function. As used herein, a state includes one or a combination of the spacecraft's position, translational velocity, and perturbations acting on the multi-target celestial system. Still referring to Figure 1D , the state determined in step 110 may be an absolute state relative to a central body about which the spacecraft orbits.

[0083] Figure 1D Step 130 uses the current model of the joint multi-objective celestial system dynamics to determine the current control action / input for controlling the spacecraft during the current time period. In this regard, the current model of the joint multi-objective celestial system dynamics can be obtained from the current control action / input for controlling the spacecraft during the current time period. Figure 1D In some exemplary embodiments, the current model of the joint multi-target celestial system dynamics may be generated based on the dynamics data of the spacecraft, the central body around which the spacecraft orbits, and / or other targets and objects around the spacecraft.

[0084] exist Figure 1D In step 132, the method uses the current model of the joint multi-objective celestial system dynamics to determine a sequence of future inputs of thruster forces for a fixed amount of time in the future from the current moment, as long as at least new state measurements are obtained such that the predicted future spacecraft states and inputs satisfy constraints on the operation of the spacecraft and constraints on the control inputs. For example, the sequence of future inputs of thruster forces may include forces that result in a new future state in which the spacecraft no longer deviates from the desired orbit.

[0085] Figure 1D Step 136 uses the thruster properties as input to the spacecraft. Using the properties, future input can be applied to the spacecraft.

[0086] exist Figure 1D In step 140, the next state of the spacecraft is determined based on the current state of the spacecraft determined in step 110 and the current control input of the spacecraft determined in step 130, and the controller waits until a new state measurement is received in step 140. Control of the step then returns to step 110 for the next iteration, and the process repeats.

[0087] Figure 1E The present invention is to illustrate the implementation of some embodiments of the present invention. Figure 1A and Figure 1D A block diagram of some components of a controller that performs at least some steps of a method. Figure 1D The method 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 a memory 119. The memory may have instructions and data stored thereon including a cost function 121, a joint multi-objective celestial system model 123, and constraints 129.

[0088] also, Figure 1D The method may use a model of a joint multi-objective celestial system 123 subject to constraints 129 to determine control inputs 107 via a processor 113. In some exemplary embodiments, the determined control inputs 107 may be sent to a spacecraft 102. To this end, the controller 101 may include or be operably connected to an output interface configured to submit control commands 107 to thrusters 103 of the spacecraft 102. In addition, the spacecraft 102 may have thrusters 103 and sensors 108, among other components. The current state 106 of the spacecraft 102 may be obtained from the sensors 108 and transmitted to the processor 113.

[0089] Still refer to Figure 1E In some exemplary embodiments, the processor 113 may determine at least one of the cost function 121, the joint multi-objective celestial system model 123, and the constraint 129 during control. For example, the controller 101 may execute a method (e.g., Figure 1D ), based on the optimization of the cost function 121, using the model 123 of the joint multi-objective celestial system Figure 1D The operation of the spacecraft 102 is iteratively controlled based on the control input of step 130. It is contemplated that Figure 1D The method may also be performed by controller 101 based on a previous iterative operation of spacecraft 102 (ie, according to a previous iterative control operation having a previous control input determined for the previous iteration, the previous iteration being optimized by a previous cost function using a previous model of spacecraft 102).

[0090] Figure 2A is a schematic diagram illustrating a representation of a high-fidelity reference trajectory, according to some exemplary embodiments. Figure 2A The exemplary high-fidelity reference trajectory shown in FIG. 200 may also be referred to as a baseline NRHO solution 201. The baseline NRHO solution 201 (expressed in the Earth-Moon rotating coordinate system) may consist of 60 revolutions around the Moon in 394 days. Figure 2A, the moon is shown as labeled 203 , and the earth is shown as labeled 205 .

[0091] In the following, some concepts related to spacecraft control will be described. The near-rectilinear halo orbit (NRHO) is a periodic orbit around the L1 and L2 Lagrangian points of the Earth-Moon Circular Restricted Three-Body (CR3BP). Due to its good stability and relatively low position-keeping cost, the NRHO around the L2 point with a 9:2 rendezvous resonance and a near-moon radius of about 3150km is selected to deploy the portal. However, the NRHO does not exist in reality because the CR3BP ignores the solar radiation pressure (SRP), the gravity generated by celestial bodies outside the Earth and the Moon, and effects such as the J2 band harmonics of the Moon. Ignoring these high-order effects during mission design will result in unacceptably high fuel consumption. Therefore, a high-fidelity astrodynamic model with ephemeris data is used in practice to produce the solution closest to the NRHO in CR3BP. This high-fidelity solution is non-periodic and consists of a finite number of revolutions around the Moon. The astrodynamic model considered in some exemplary embodiments considers all the main predictable forces acting on the spacecraft in the lunar-terrestrial space. The magnitude of any major predictable force is greater than the magnitude of the largest unpredictable force. For the model considered here, the largest unpredictable force affecting the spacecraft is determined to be the indirect disturbance caused by navigation errors fed back to the spacecraft's controllers. The navigation errors are quantified under the assumption that the spacecraft state is estimated using measurements from the Deep Space Network (DSN). Under these assumptions, the major predictable forces acting on the spacecraft in the region of space occupied by the NRHO of CR3BP are the SRP, the lunar J2 zonal harmonics, and the gravitational forces due to the Earth, Moon, and Sun.

[0092] Notation :Let the set of natural numbers be represented by And the set of real numbers is represented by All vectors are column vectors. The real vector belongs to represents a set. A vector is represented as a list of comma-separated elements enclosed in [·]. A vector with two elements [a,b] containing real numbers a and b such that a≤b also represents a closed interval on the real line. Similarly, (a,b) represents an open interval. A vector constructed by vertically stacking two vectors c and d can be concisely represented as [c T d T ] T . A vector of length n containing zeros is represented by 0 ndenoted by θ[k]. The kth element of a vector θ of length n≥k is denoted by θ[k]. Given a matrix θ with n≥k rows and m≥k columns, row k is denoted by θ[k,:] and column k is denoted by θ[:,k]. If κ is a vector consisting of elements from {1,…,n}, then θ[κ] is a vector of elements of θ indexed by elements in the vector κ. Similarly, θ[:,κ] and θ[κ,:] are matrices consisting of columns and rows of θ, respectively, indexed by elements in the vector κ. The identity matrix is ​​denoted by I and its dimension is inferred by the context. Unless otherwise stated, the eigenvalues ​​of a matrix are denoted by λ.

[0093] Assume that the high-fidelity dynamic system is represented by the following formula

[0094]

[0095] It describes the equations of motion of the spacecraft in the non-inertial rotating coordinate system (hereinafter referred to as the rotating coordinate system) considered in CR3BP. The origin of this coordinate system is located at the Earth-Moon barycenter, the x-axis points to the mass center of the Moon, and the z-axis is along the angular momentum vector of the Earth-Moon system in CR3BP. This coordinate system is usually chosen for analyzing NRHO because it is relevant to observations and communications from Earth. In equation (1), corresponds to the time derivative of the spacecraft state θ, g is a function of time, and u(t) corresponds to the control input representing the rate of change of the state θ. The right side of the model in (1) describes the main predictable external forces acting on the spacecraft as described above. The state vector It consists of the spacecraft position vector r and velocity vector v in the rotating coordinate system, and u represents the control input vector.

[0096] Control Model: The control method developed in the next section uses impulse thrusters to perform maneuvers, where the thrust pulses are modeled as causing instantaneous changes in the spacecraft velocity. Consider the motion of the spacecraft in the time interval [t1, t2], and a thrust pulse is applied at t′∈(t1, t2). The control input u(t) can be expressed as

[0097]

[0098] where δ(t) is the Dirac delta function. The right side of (1) can be rewritten as

[0099] g(t,θ(t),u(t))=f(t,θ(t))+u(t), (3)

[0100] for t∈[t1,t2], where f includes the external forces acting on the spacecraft as mentioned above. Therefore, the (uncontrolled) natural motion of the spacecraft is given by

[0101]

[0102] The spacecraft state at t2 obtained after the pulse at t′ can be determined as follows

[0103]

[0104] in

[0105]

[0106] Although propagation of spacecraft dynamics with impulse thrust (6) was used in subsequent developments with minor modifications, the control method can be applied to thrust pulses of finite duration. Note that finite duration thrust pulses of the order of minutes can be well approximated by thrust pulses due to the sufficiently slow time scale of the system dynamics (4).

[0107] Baseline solution and linearization: The high-fidelity NRHO solution of (4) is expressed as the baseline solution The baseline solution is an uncontrolled natural motion trajectory located near the NRHO of CR3BP and is estimated using a multi-targeting approach.

[0108] Reference Figure 2A , baseline solution By cross The orbit of the moon (shown as 203) revolutions, where t∈[0,t max ], and t = 0 corresponds to the Julian calendar epoch of January 13, 2023. The farthest point from the moon in each revolution is called the aphelion, and the corresponding time is particularly important for the control strategy in Section 3. Baseline solution The apogee approach time of the Kth revolution is given by Given, where and

[0109] In addition to the baseline, the following sections utilize a linear time-varying system to represent the uncontrolled natural motion of the spacecraft as:

[0110]

[0111] in,

[0112] A(t) is the linearized dynamics of the system in continuous time and is expressed as

[0113]

[0114] as well as

[0115]

[0116] It is Nearby approximate solutions of (4).

[0117] The proposed control strategy aims to keep the spacecraft in a region of the state space close to the baseline (i.e., the region where the spacecraft’s deviation from the baseline is much smaller than the scale of the baseline) where the spacecraft can be solved with the baseline. Together, the linear approximation (7) computed a priori is valid. Therefore, the linear model, although imperfect, is useful for predicting the future evolution of the nonlinear dynamics (4) and determining appropriate control actions.

[0118] The finite-time behavior of the solutions of (4) near the baseline solution is particularly useful and can be analyzed using the discrete-time counterpart of (7). In some exemplary embodiments, if for some t1∈[0,t max ], the spacecraft is Nearby At t2>t1, the state (given by It can be approximated as

[0119]

[0120] where Φ(t2,t1) is the linear system propagated on the time interval [t1,t2] with the initial condition Φ(t1,t1)=I.

[0121]

[0122] The state transfer matrix (STM) for the time interval [t1, t2] is obtained, and χ(t2, t1) is the residual given by:

[0123]

[0124] This implementation is Figure 1B The process Figure 5 For the eigenvector v of STMΦ(t2,t1), there exists a complex number λ such that

[0125] Φ(t2,t1)ν=λν. (13)

[0126] If |λ|≤1, then ν is called a non-extended eigenvector, and if |λ|>1, then v is called an extended eigenvector. STM provides information about Useful local information about the solution near (4). The natural motions generated by the initial conditions along the extended and non-extended eigenvectors of the STM are called extended local eigenmotions and non-extended local eigenmotions, respectively. Ideally, the non-extended local eigenmotion associated with Φ(t2,t1) does not deviate from the baseline within the time interval [t1,t2], while the extended local eigenmotion deviates from the baseline within the same time interval. In practice, due to the fact that the linear estimate in (10) is only an approximation of the propagating non-linear dynamics (4), the non-extended local eigenmotion may still deviate within t < t2. Since the linear model is only valid near the baseline, the performance of the non-extended local eigenmotion may deteriorate when the initial condition is far from the baseline solution When this occurs, the performance of the non-extended local eigenmotion may degrade.

[0127] To generate the STM, the orbital dynamics with respect to the reference trajectory / baseline solution within a user-defined finite horizon can be linearized, and the eigenvalue decomposition of the resulting linearized STM can be used. In some exemplary embodiments, the STM is obtained as a direct result of the linearization process. Consider an n-dimensional dynamic system described by

[0128]

[0129] Let r(t) represent the solution. The discrete-time update of r(t) for perturbations can be calculated, i.e., given time t h and perturbation state The goal is to find the local linear mapping (state transition matrix) Φ(t,t + t h ), such that To numerically calculate Φ(t,t + t h ), let e i be the basic basis vector, where the i-th element of e i is equal to 1, and let ε > 0 be a small parameter. Numerically propagate r(t)+εe i and r(t)-εe i from t to t + t h , and the solutions at t + t h are called and Then, the i-th column of the STM Φ(t,t + t h ) can be approximated by the central difference formula:

[0130]

[0131] Local feature motion control : Some exemplary embodiments utilize the span of adjacent time intervals [0,t max] computed STM (called rolling horizon STM) to determine the non-extended local feature motion corresponding to these intervals.

[0132] Figure 2B 2 is a schematic diagram showing a representation of a maneuver that transfers a spacecraft to a non-extended local feature motion at apogee 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 reasons discussed previously, the spacecraft may tend to deviate from the initial spacecraft path 251 to a deviation path 253. Whenever the spacecraft is predicted to deviate from the baseline solution When a specified threshold is exceeded (trigger 255), it is steered to an initial condition of one of the non-extended local feature motions 259 in a fuel efficient manner via one or more control maneuvers 257.

[0133] In principle, the STM can be computed for the entire duration of the baseline, and the non-extended characteristic motions can be picked to obtain bounded relative motion trajectories for the entire duration of the baseline in one go. However, this approach is computationally infeasible because the condition number of the STM increases with the duration for which it is held. In practice, the longest duration for which an STM is reliable is typically the time required for 12 revolutions around the Moon, which is about 78 days.

[0134] In this regard, some exemplary embodiments consider maneuvers at aphelion, but not necessarily every time the spacecraft visits aphelion. The reasons are as follows. For some G≤12 and k≤KG, assuming that by placing the spacecraft precisely at The spacecraft is initialized on the desired local characteristic motion about the baseline. Numerical simulations of (4) show that the spacecraft exhibits a satisfactorily bounded motion relative to the baseline for at least the subsequent G revolutions around the Moon.

[0135] Figure 3 is a graph showing the distance from a baseline of exemplary feature motions according to some exemplary embodiments. Figure 3 As mentioned above, STM is numerically reliable (here means less than 10 8 The longest duration of the condition number is around the moon The time required for one revolution. The estimated eigenvalues ​​of STM are It can be observed that this STM has two extended eigenvectors and four non-extended eigenvectors. Due to the very large eigenvalues, a small perturbation to the spacecraft on the baseline NRHO solution may cause it to deviate rapidly. This is in sharp contrast to the good stability of the NRHO in the CR3BP model. In addition, it can be observed that for any k≤KH The estimated STM spectrum is qualitatively similar to the spectrum of the initial H revolutions. Any vector in the span of the four non-expanded eigenvectors of will cause a non-expanded local eigenmotion for any k ≤ KH. Let these be the set The feature vector of the index is k≤KH express.

[0136] Still refer to Figure 3 , showing the The distance of some of the moving spacecraft from the baseline in the local characteristic motion. As shown, as the number of days increases, the local characteristic motion 302 continues to deviate from the baseline solution, while the local characteristic motions 304 and 306 converge in some cases. The magnitudes of the characteristic motions 304 and 306 are both less than 1. These local characteristic motions 304 and 306 correspond to motions near the baseline solution based on special states from the eigenvectors. There can be multiple special states such that when the controller proposes control to move the spacecraft from its current position to a desired position, the controller inherently chooses to keep the spacecraft within a span of states with eigenvalues ​​less than 1. To this end, such a span of states can be produced by any linear combination of states that have eigenvalues ​​less than 1 in both the local characteristic motions 304 and 306.

[0137] It should be noted that the resulting linear combination does not include any components along directions having eigenvalues ​​greater than one. The basis for the selection of states in the linear combination is based on the criterion that they should provide the desired behavior (motion) that does not cause the spacecraft to deviate from the reference trajectory / baseline solution. The basis for determining how many states should be in the linear combination is based on an optimization problem. For example, an objective function that minimizes fuel consumption can be a major factor in determining what state components should be present in the resulting linear combination. In this way, among many special states, only the special state for which the optimization problem determines the cost of fuel consumption to be the smallest can be selected. In this regard, it is conceivable that in some exemplary embodiments, the resulting combination may include at least one of the special states. In the case where the resulting combination includes multiple states, the direction of the resulting state is a mixture of the directions of the selected eigenvectors. The eigenvectors that result in the desired stable motion may only provide some main directions, so in some cases it is necessary to select a combination of these main directions.

[0138] The spacecraft maneuver occurs in a short time interval (called the control horizon) just before the spacecraft reaches the apogee. When the trigger condition is met, the maneuver that returns the spacecraft state to the non-extended local characteristic motion is initiated. For k≤KH, assume that the spacecraft is The position deviation is Δr kThe trigger condition used in this work takes into account the change in the distance between the spacecraft and the baseline solution since t = 0 and since the most recent visit to the apogee, and is given by

[0139] Δr k >min{ρΔr0,min{2Δr k-1 ,Δr0}}, (14)

[0140] where ρ is a tuning parameter that helps determine the nature of the bounded motion trajectories produced near the baseline.

[0141] If for some k <K,Δr k If the trigger condition (14) is met, the following discrete-time optimal control problem is solved to calculate the transfer of the spacecraft from Fuel-efficient manipulation of non-extended local feature motions initiated

[0142] minimize

[0143]

[0144] Subject to the following constraints, for the time grid

[0145]

[0146]

[0147]

[0148] For j=0,…,N-1, The operation is performed by Start applying N-1 thrust pulses at intervals Δt Under the assumption of an impulse input, the propagation equations of motion in (6) are used to constrain (16). Boundary conditions (17) stipulate that the state at the end of the control horizon is located at a position that will result in non-extended local feature motion. In addition, in practical implementations, the controller will not have access to the true state of the spacecraft. It receives the state estimator from (18) Represents an estimate.

[0149] Figure 4 An algorithm associated with local feature motion control according to some exemplary embodiments is shown. Figure 4 , the proposed control method is obtained by Figure 4 The algorithm 400 summarizes the automatic procedure.

[0150] Figure 4The input of the algorithm 400 consists of the following: The moment corresponding to the far-moon point in the K+1-visit baseline (for k = 0, ..., K), the initial distance Δr0 of the spacecraft from the baseline, the number of downstream revolutions H for which the STM is calculated, the time interval before each visit (excluding the first) to the apogee in the baseline solution, Each time interval has N grid points evenly spaced by Δt (for k=1,…,K).

[0151] Still refer to Figure 4 , Propagate(ψ,T) returns the state after propagating the high-fidelity prediction model (4) with initial condition ψ over the time interval specified by the binary vector T. When the algorithm 400 works with a state estimator, the first argument ψ of Propagate is an estimate of the current state of the spacecraft.

[0152] Still refer to Figure 4 , PosDev(θ,ψ) returns the position deviation between states θ and ψ.

[0153] Still refer to Figure 4 , EigenMotion(Δr,k,H) returns a state with a position deviation Δr from the state of the kth apogee of the baseline, and the state is along the vector Positioning, where v i (for )for The non-expanded eigenvector of .

[0154] Still refer to Figure 4 , if the trigger condition (14) is met, Trigger evaluates to true.

[0155] Still refer to Figure 4 , For time interval (i.e., solve the discrete-time optimal control problem (8) for the maneuvers that occur just before the kth apogee visit of the spacecraft.) The constraints (17) are constructed from the non-expanded eigenvectors of the STM, and the initial conditions (18) are specified by ψ. OptCtrl returns the control work required for the manipulation and is estimated as The control input (for j=0,…,N-2) is the control solution of a matrix stacked in rows.

[0156] Still refer to Figure 4 , Use the control solution U returned by OptCrl with the initial condition ψ in the control horizon Upward propagation (1).

[0157] Still refer to Figure 4 , Δv j (for j = 1, ..., M) is the duration [0, t max ] is the cost of the M operations initiated in Algorithm 1. The cumulative cost of the M operations is given by Δv.

[0158] According to some exemplary embodiments, the actual implementation of the proposed control strategy can be extended by using a Kalman filter Figure 4 The algorithm 400 of FIG. 4 is used to demonstrate the Kalman filter using range rate measurements and simulated range from a deep space network (DSN) to estimate the state. The measurements can be received at a rate of 6 hr which is reasonable for the DSN and sufficient for accurate state estimation. The errors in the range and range rate measurements come from standard deviations of 10 m and 1 mm s, respectively. -1 The standard normal distribution of .

[0159] The proposed method is demonstrated for the case where two spacecraft perform bounded relative motion around a baseline solution. In particular, the portal is subject to strict position keeping with respect to the baseline while the visiting spacecraft is made to perform collision-free relative motion around the portal.

[0160] The trajectory calculated for the portal (referred to as Solution 1) uses ρ = 10, Δt = 6 hr, Δr0 = 0.5 km, and N = 4, while the trajectory for the visiting spacecraft (referred to as Solution 2) is calculated using ρ = 2, Δt = 12 hr, Δr0 = 50 km, and N = 6. Certain parameters in Algorithm 9 affect the properties of the bounded motion solutions. In particular, the choice of ρ in (14) and Δr0 for both solutions plays an important role in ensuring that they remain conflict-free. These parameters are adjusted so that the resulting solutions are infrequent and fuel efficient.

[0161] It is worth noting that the impact of navigation uncertainty is more prominent for strict station keeping maneuvers near the baseline. When the maneuver is initiated close to the baseline, the final state (whose position is close to the baseline and aligned with the desired eigenvector) is more easily corrupted by navigation uncertainty due to its small size relative to the baseline measurement. As a result, the spacecraft may be manipulated to a state that is not properly aligned with the desired eigenvector, which will cause the spacecraft to deviate prematurely. Therefore, strict station keeping may require more annual maneuvers. This hidden danger can be avoided by choosing a smaller value of Δr0 and a larger value of ρ when generating Solution 1. This allows the spacecraft to slowly deviate from the baseline over a duration of 150 days and stay at a distance of about 9 km from the baseline where maneuvers will not be triggered too frequently.

[0162] Figure 5is a graph showing the position deviation of two spacecraft operating under local feature motion control relative to a baseline trajectory according to some exemplary embodiments. The annual position keeping performance of Solution 1 (502) and Solution 2 (504) is shown in Table 1 below, and the distance of the solution from the baseline as a function of time is shown in Table 1. Figure 5 As shown. With the proposed strategy, the annual station keeping cost of the Gateway is comparable to or better than the existing techniques for station keeping on NRHO. Since the visiting spacecraft maintains a greater distance from the baseline than the Gateway, the fuel required for it is significantly higher. Therefore, each maneuver for a non-extended local feature motion to transfer to the baseline is more expensive.

[0163] Table 1: Annual station keeping performance

[0164]

[0165] Another significant advantage of the proposed method is that relatively few manipulations are required to maintain annual bounded motion. Figure 5 The black part of Solution 1 and Solution 2 highlights the very small annual control duty cycle. Furthermore, Solution 1 and Solution 2 do not pose a risk of collision between the Gateway and a visiting spacecraft. The distance between the solutions is never less than 8 km.

[0166] feature

[0167] It is contemplated that a controller may use the proposed method to keep a spacecraft near a desired orbit. To detect whether a trigger condition is satisfied, the controller may determine a current state of the spacecraft over a specified time period. To determine the current state of the spacecraft, the processor may obtain measurements corresponding to one or a combination of a position and a translational velocity of the spacecraft and a perturbation acting on a multi-target celestial body system defined by the spacecraft and at least one celestial body. To determine a state transition matrix, the processor may obtain a linear model of a dynamic system defined by the spacecraft and at least one celestial body. The processor may also propagate the linear model of the dynamic system over a time period defined by a first time instant when the spacecraft is in a current state and a second time instant when the spacecraft will be in an upcoming state.

[0168] Furthermore, the control action may be determined by determining the correction direction of the upcoming state by a linear combination of the directions of each non-extended eigenvector of the STM to achieve a hybrid state utilizing a natural motion that is a combination of the corresponding eigenmotions of each non-extended eigenvector of the STM.

[0169] To generate the control commands, the processor may solve a nonlinear optimization problem to find a fuel efficient maneuver that moves the spacecraft to a desired set of states that produces a desired natural motion. The nonlinear optimization problem is directed to a finite horizon optimization of a model of the spacecraft, a set of objectives for the spacecraft motion, and constraints on the spacecraft's propulsion system and the spacecraft's motion. The constraints on the spacecraft's propulsion system and the spacecraft's motion include one or more physical limitations on the spacecraft, one or more safety limitations on the spacecraft's operation, and one or more performance limitations on the spacecraft's trajectory.

[0170] The spacecraft may perform a docking mission with a space station that maintains its position near the orbit at a third distance less than the station threshold, wherein the controller is further configured to select a spacecraft threshold greater than the station threshold.

[0171] The upcoming state of the spacecraft may include the position, orientation, translational velocity, and angular velocity of one or more of the spacecraft and the spacecraft's payload, and one or a combination of perturbations acting on a multi-object celestial body system defined by the spacecraft, the payload, and the one or more celestial bodies. The perturbations acting on the multi-object celestial body system are natural orbital forces including solar and lunar gravitational perturbations, anisotropic gravitational perturbations due to the non-spherical shape of the central body, solar radiation pressure, and air drag.

[0172] The high-fidelity reference trajectory is an uncontrolled natural motion trajectory located near a near-rectilinear halo orbit (NRHO) in the constrained three-body Earth-Moon circle and estimated using the multiple shooting method. The high-fidelity reference trajectory includes a finite number of non-periodic revolutions around the Moon in a finite number of days.

[0173] The orbit may be one of a circular orbit, an elliptical orbit, a halo orbit, a near-rectilinear halo orbit, or a quasi-satellite orbit. The control command is generated as a solution of a model predictive control strategy that generates the control command by optimizing a cost function over a rolling horizon. The cost function includes a stability component for directing the motion of the spacecraft to an upcoming state, a component for a goal of spacecraft operation, and a performance component for optimizing the motion of the spacecraft until the upcoming 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 for each of a plurality of specified time periods within a time horizon, or iteratively within a rolling time horizon. The control command is output to an operation module of a controller so that the operation module transmits the control command to a thruster command module, the thruster command module receives the control command as a Δv command, and the thruster command module converts the Δv command into a thruster command, and sends the thruster command to a thruster processor of at least one thruster to activate or deactivate at least one thruster for trajectory tracking control of the aircraft according to the converted Δv command.

[0174] definition

[0175] According to various aspects of the present disclosure, and based on experiments, the following definitions have been established, and these definitions are certainly not complete definitions of each phrase or term. The definitions provided therein are provided only as examples based on knowledge gained from experiments, where other explanations, definitions and other aspects may also be applicable. However, such definitions have been provided for at least a basic preview of the phrases or terms presented.

[0176] Space rendezvous: A space rendezvous can be a collection of orbital maneuvers during which two spacecraft (or a pursuit spacecraft and a target (i.e., the target can be another spacecraft, a space station, a celestial body, or orbital debris)) arrive in the same orbit and approach to a very close range (e.g., within visual contact).

[0177] Celestial system (celestial reference system): In astronomy, a celestial coordinate system (or celestial reference system) is a system used to specify the positions of satellites, planets, stars, galaxies, and other celestial bodies relative to physical reference points available to an observer located at a particular location (e.g., the true horizon and the principal direction of north for an observer located on the surface of the Earth). A coordinate system can specify the position of an object in three-dimensional space, or just plot its direction on the celestial sphere (if the distance of the object is unknown or insignificant). Coordinate systems are implemented in spherical coordinates or rectangular coordinates. Spherical coordinates, projected on the celestial sphere, are similar to the geographic coordinate systems used on the surface of the Earth. The difference lies in their choice of fundamental planes, which divide the celestial sphere into two equal hemispheres along a great circle. In appropriate units, rectangular coordinates are just the Cartesian equivalent of spherical coordinates, and have the same fundamental (x, y) planes and principal (x-axis) directions. Each coordinate system is named after its choice of fundamental planes.

[0178] Fig. 6A , Figure 6B , Figure 6C , Fig.6D and FIG. 6E is a schematic diagram illustrating some general parameters for implementing aspects of methods and systems according to some exemplary embodiments.

[0179] Conic sections :Reference Fig. 6A A conic section (also interchangeably referred to as a conic section) is a curve formed by passing a plane through a right circular cone 602. Fig. 6AThe angular orientation of the plane 600 relative to the cone 602 is shown, which determines whether the conic section is a circle 604, an ellipse 606, a parabola 608, or a hyperbola 610, 612. The circle 604 and the ellipse 606 appear when the intersection of the cone 602 and the plane 600 is a bounded 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. If the plane 600 is parallel to the generatrix of the cone 602, the conic section is called a parabola 608. Finally, if the intersection is an unbounded 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 will intersect the two halves of the cone 602, resulting in two separate curves. All conic sections can be defined in terms of eccentricity. The type of conic section is also related to the semi-major axis and energy. Table 2 below shows the relationship between eccentricity, semi-major axis and energy and the type of conic section.

[0180] Table 2

[0181] Conic Section Eccentricity, e Semi-major axis energy round 0 = Radius <0 oval 0<e<1 >0 <0 parabola 1 endless 0 hyperbola >1 <0 >0

[0182] Satellite orbits can be any of four conic sections.

[0183] Reference Figure 6B , Figure 6C and Fig.6D , in order to describe regular orbits mathematically, six quantities (called orbital elements) must be defined. The six orbital elements are:

[0184] Semi-major axis, a

[0185] Eccentricity, e

[0186] Inclination angle, i

[0187] Pericenter argument, ω

[0188] After the pericentric period, T

[0189] Longitude of ascending node,

[0190] FIG. 6B to FIG. 6D A conventional orbiting satellite 650 is shown following an oval path known as an ellipse 620 , with the orbited object (known as the primary) located at one of two points known as foci 622 , 624 . Figure 6CAn ellipse 620 is shown defined as a curve with the following properties: For each point on the ellipse, the sum of its distances to two fixed points (called foci 622, 624) is constant. The longest and shortest straight lines that can be drawn through the center of the ellipse are called the major and minor axes, respectively. The semi-major axis is half the major axis and represents the average distance of a satellite from its host star. The eccentricity is the distance between the foci divided by the length of the major axis and is a number between 0 and 1. An eccentricity of zero represents a circle.

[0191] Fig.6D The inclination i is shown, which is the angular distance between the satellite's orbital plane and the equator of its host star (or the ecliptic plane in a heliocentric or sun-centered orbit). An inclination i of zero degrees represents an orbit around the host star's equator in the same direction as the host star's rotation (this direction is called co-rotating (or prograde)). An inclination i of 90 degrees represents a polar orbit. An inclination i of 180 degrees represents a retrograde equatorial orbit. A retrograde orbit is one in which the satellite moves in the opposite direction of its host star's rotation.

[0192] Still refer to Fig.6D , the periapsis ω is the point in the orbit that is closest to the primary star (i.e., for an object moving in an elliptical orbit around another celestial body, the closest point is the periapsis, and at this point in the orbit, according to Kepler's second law, the celestial body moves at its maximum speed). The opposite of the periapsis ω (i.e., the farthest point in the orbit) is called the apoapsis (i.e., for an object moving in an elliptical orbit around another celestial body, the point of maximum separation is the apoapsis, and at this point in the orbit, according to Kepler's second law, the object moves at its slowest speed). The perihelion is the position of closest approach (i.e., the shortest distance between the Sun and the planet), at which point in the orbit, according to Kepler's second law, the planet moves at its maximum speed. The aphelion is the maximum distance between the Sun and the planet, at which point in the orbit, according to Kepler's second law, the planet moves at its slowest speed, so the aphelion refers specifically to the orbit around the Sun and is equivalent to the apoapsis of an ordinary orbit. Among them, the periapsis ω and apoapsis are usually corrected to be applicable to the objects being orbited, such as the perihelion and aphelion of the sun, the perigee and apogee of the earth, the perijove and apojove of Jupiter, the perilunar and aphelion of the moon, etc. The angular distance of the periapsis ω is the angular distance between the ascending node N1 and the periapsis (see Fig.6D ). The periapsis period T is the time when the satellite passes through its periapsis.

[0193] Peripapsis: The point in an elliptical orbit of a body about the system's center of mass at which the distance between the body and the center of mass is at its minimum. (Also called the angular distance of periapsis or pericentricity), symbol ω, is one of the orbital elements of an orbiting body. Parametrically, ω is the angle from the ascending node of a body to its periapsis, measured in the direction of motion. For certain types of orbits, terms such as perihelion (for heliocentric orbits), perigee (for geocentric orbits), periastron (for orbits about fixed stars), etc. may be substituted for the term periapsis. (See apses for more information.) An angular distance of 0° for periapsis means that the orbiting body will be closest to the centroid at the same moment that it crosses the reference plane from south to north. An angular distance of 90° for periapsis means that the orbiting body will reach periapsis at its northernmost distance from the reference plane. The longitude of periapsis is obtained by adding the angular distance of periapsis to the longitude of the ascending node. However, especially when discussing binary stars and exoplanets, the term "longitude of periapsis" or "longitude of periapsis" is often used as a synonym for "angular distance of periapsis".

[0194] Apoapsis : The point on the elliptical orbit of a celestial body about the center of mass of a system where the distance between the celestial body and the center of mass is at its maximum.

[0195] Intersection : is the point where the orbit intersects the plane (e.g., where the satellite intersects the Earth's equatorial plane). If the satellite intersects the plane from south to north, then that node is the ascending node N1; if it moves from north to south, then it is the descending node N2. The longitude of the ascending node N1 is the ecliptic longitude of that node. Ecliptic longitude is similar to longitude on Earth and is measured in degrees counterclockwise from zero degrees, which is in the direction of the vernal equinox, Ω.

[0196] Track Type : Geosynchronous orbit (GEO): A circular orbit around the Earth with a period of 24 hours. A geosynchronous orbit with an inclination of zero degrees is called a geostationary orbit. Spacecraft in geostationary orbits appear to hang motionless in one position at the Earth's equator. Therefore, they are ideal for certain types of communication satellites and weather satellites. A spacecraft in an inclined geosynchronous orbit will appear to follow a regular figure-eight pattern in the sky with each orbit. To obtain a geosynchronous orbit, a spacecraft is first launched into an elliptical orbit with an apogee of 35,786 kilometers (22,236 miles), which is called a geosynchronous transfer orbit (GTO). The orbit is then re-entered by firing the spacecraft's engine at the apogee.

[0197] Polar Orbit (PO): is an orbit with an inclination of 90 degrees. Polar orbits are very useful for satellites performing mapping and / or surveillance operations because as the planet rotates, the spacecraft can visit almost every point on the planet's surface. Promenade Orbit: Orbiting satellites are subject to many gravitational influences. First, planets are not perfectly spherical and their mass is slightly unevenly distributed. These fluctuations have an effect on the trajectory of the spacecraft. In addition, the Sun, Moon, and planets have gravitational influences on orbiting satellites. With proper planning, an orbit can be designed that takes advantage of these influences to induce precession in the satellite's orbital plane. The resulting orbit is called a promenade orbit.

[0198] Sun-synchronous orbit (SSO): An orbit in which the orbital plane precesses with the same period as the planet's solar orbit. In such an orbit, the satellite passes through periapsis at approximately the same local time in each orbit. This is useful if the satellite carries instruments that rely on sunlight shining at a certain angle on the planet's surface. Occasional propulsion maneuvers may be required to adjust the orbit in order to maintain precise synchronous timing.

[0199] Lightning orbit: is a highly eccentric Earth orbit with a period of about 12 hours (2 revolutions per day). The orbit inclination is chosen so that the rate of change of the perigee is zero, so that the apogee and perigee can be maintained at fixed latitudes. This happens at inclinations of 63.4 degrees and 116.6 degrees. For these orbits, the angular distance of the perigee is usually in the Southern Hemisphere, so the satellite remains over the Northern Hemisphere near the apogee for about 11 hours per orbit. This orientation can provide good ground coverage in the high latitudes of the Northern Hemisphere.

[0200] Hohmann transfer orbits: are interplanetary orbits whose advantage is that they consume as little propellant as possible. A Hohmann transfer orbit to an outer planet (such as Mars) is accomplished by launching a spacecraft and accelerating it in the direction of the Earth's orbit around the Sun until it escapes the Earth's gravity and reaches a speed that places it in a solar orbit with aphelion equal to the orbit of the outer planet. After it reaches its destination, the spacecraft must decelerate so that the planet's gravity can capture it into the planet's orbit. For example, to send a spacecraft to an inner planet (such as Venus), the spacecraft is launched and accelerated (i.e. decelerated) in the opposite direction of the Earth's orbit around the Sun until it reaches a solar orbit with a perihelion equal to the orbit of the inner planet. It should be noted that the spacecraft continues to move in the same direction as the Earth, just at a slower speed. To reach a planet, the spacecraft is required to insert itself into the interplanetary orbit at the right time, so that the spacecraft arrives at the planet's orbit when the planet will be at the point where the spacecraft will intercept it. This task is equivalent to a quarterback "leading" his receiver so that the football and the receiver arrive at the same point at the same time. The time interval in which a spacecraft must be launched to complete its mission is called the launch window.

[0201] Nearly Rectilinear Halo Orbit (NRHO): This can be defined as an "almost stable" orbit, where the stability is measured using the stability exponent v.

[0202] CR3BP model Nearly rectilinear halo orbits are members of the broader L1 and L2 family of halo orbits, i.e., basic structures that exist in dynamic environments modeled after multiple gravitating bodies. L1 is the point 1 / 100 of the way from Earth to the Sun, or the first Lagrange point, where the centripetal force and the gravitational forces of Earth and Sun cancel each other out. It is one of five such points in the Earth-Sun system, and a space probe could, in principle, stay there forever, like balancing on a gravitational version of a pinhead. L2, on the other hand, is on the side of Earth away from the Sun, 1.6 million kilometers away. Both L1 and L2 are ideal places to look out over the universe, and L1 is also a great spot to look down on the Earth and Sun. However, they also have drawbacks: At L1, a spacecraft's signal would be drowned out by radiation from the Sun behind it. At L2, the Earth's shadow blocks the solar radiation that the probe needs to power its instruments. The solution is to put spacecraft into "halo orbits" around Lagrange points. A spacecraft in a halo orbit around L1 describes a giant lazy loop perpendicular to the Earth-Sun axis, which falls endlessly toward an equilibrium point. This fundamental behavior is also present in high-fidelity models, thus supporting potential long-duration mission scenarios for spacecraft (possibly manned) in orbits near the Moon. This type of trajectory was first identified in a simplified representation of the gravitational effects of the Earth-Moon system, namely the Circular Restricted Three-Body Problem (CR3BP). In the CR3BP model, near-rectilinear halo orbits (NRHOs) (i.e., orbits that can be defined as “almost stable”, where stability is measured using the stability exponent ν) are characterized by good stability, which suggests the possibility of maintaining NRHO-like motions for long durations while consuming small amounts of propellant resources. Some NRHOs also have good resonance properties, which can be used for mission design and are particularly helpful for avoiding lunar eclipses. However, for practical mission realization, transfers to such orbits as well as position-keeping strategies must be demonstrated in higher-fidelity ephemeris models. Position-keeping algorithms for libration point orbits have previously been explored in this dynamic range in the context of planar Lyapunov orbits and classical three-dimensional halo orbits. However, NRHOs are constructed in the ephemeris regime.

[0203] Position holding: In astrodynamics, orbital maneuvers by thruster burning required to keep a spacecraft in a particular designated orbit are called orbital 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 uniform sphere, the gravity from the Sun / Moon, solar radiation pressure, and air drag) must be counteracted. The deviation of the Earth's gravitational field from that of a uniform sphere and the gravity from the Sun / Moon usually perturb the orbital plane. For sun-synchronous orbits, the precession of the orbital plane caused by the Earth's oblateness is a desired feature as part of the mission design, but the inclination changes caused by the gravity of the Sun / Moon are undesirable. For geostationary spacecraft, the inclination changes caused by the gravity of the Sun and Moon must be counteracted by considerable fuel consumption, because the inclination should be kept small enough to enable the spacecraft to be tracked by non-steerable antennas. For spacecraft in low orbits, the effects of atmospheric drag must often be compensated. For some missions, this is simply to avoid reentry; for other missions (especially those whose orbits should be precisely synchronized with the Earth's rotation), this is necessary to avoid shortening the orbital period. Solar radiation pressure typically perturbs the eccentricity (i.e. the eccentricity vector), see Orbital Perturbation Analysis (Spacecraft). For some missions this must be actively countered with maneuvers. For geostationary spacecraft, the eccentricity must be kept small enough to track the spacecraft with non-steerable antennas. Furthermore, for Earth observation spacecraft that want very repetitive orbits with fixed ground tracks, the eccentricity vector should be kept as fixed as possible. A large part of this compensation can be done by using a frozen orbit design, but for fine control maneuvers the use of thrusters is required. For spacecraft in halo orbits around Lagrange points, position keeping is even more important, because such orbits are unstable; without active control with thruster burns, the smallest deviations in position / velocity will result in the spacecraft completely falling out of orbit.

[0204] Dominant interference force : A dominant disturbance is a disturbance that causes a deviation greater than the navigation error of the Deep Space Network (DSN) positioning system. Furthermore, a dominant force can be defined as a force whose effect is greater than the maximum unpredictable force. Furthermore, the maximum unpredictable force can be due to position and velocity measurement errors.

[0205] Perturbation: Can be a complex motion of a massive object subject to forces other than the gravity of a single other large body. Other forces may include third (fourth, fifth, etc.) bodies, drag (e.g., from the atmosphere), and the eccentric gravity of oblate or other malformed bodies. The Sun's gravitational pull on the Moon is at two places in the Moon's orbit. The black dashed arrows represent the direction and magnitude of the gravitational pull on the Earth. Applying this to the positions of the Earth and Moon does not interfere with their relative positions. When this is subtracted from the force on the Moon (dark solid arrow), what remains is the gravitational pull on the Moon relative to the Earth (dark double arrow). Because the direction and magnitude of the gravitational pull are different on opposite sides of the orbit, it changes the shape of the orbit.

[0206] Fig. 7A 7 is a block diagram illustrating 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, which is communicatively coupled to a controller (e.g., Figure 1E Controller 101). In addition, processor 720 may be coupled to a sub-controller 740 of a thruster via buses 728 and 734. Processor 720 may receive control commands as Δv commands 701. Processor 720 may convert 724 the received Δv commands 701 into thruster commands 726, which may be sent via bus 728 to sub-controller 740 of a thruster 743, which may be connected to sensor 748. Sub-controller thruster 740 may include another processor 741 to process thruster commands and issue control actions as commanded.

[0207] Figure 7B is a schematic diagram showing aspects of a thruster configuration according to an embodiment of the present disclosure. In some exemplary embodiments, a spacecraft may be equipped with eight thrusters mounted at the corners of the spacecraft such that they are aligned and produce a net force acting on the center of mass of the spacecraft without producing any torque that would rotate the spacecraft. Fig. 7A The controller of subcontroller 740 can send signals to enable or deactivate thrusters to move the spacecraft along the commanded trajectory.

[0208] Reference Figure 7B , the chief 752 and the deputy 754 may be in orbit around a central body (eg, the Earth). Coordinate system F e is the Earth-centered inertial (ECI) coordinate system, e is an unforced particle, and it can be assumed that e is collocated with the center of the Earth. The center of mass of the secondary spacecraft is denoted by "d" and has the secondary spacecraft fixed coordinate system F d The center of mass of the main spacecraft is denoted by "c" and has the main spacecraft fixed coordinate system F c The angular velocity of the main spacecraft relative to the inertial coordinate system is ωc / e , and may be non-zero, i.e., the host spacecraft may be uncontrolled and tumble. The orbital coordinate system of the host spacecraft is a Hill coordinate system with radial, along-orbit, and across-orbit basis vectors. The vector Parallel to the position vector of the main satellite, points in the direction of the orbital angular momentum, and For simplicity, and considering that the reorientation of the slave spacecraft can be achieved faster than its orbital control by the reaction wheel attitude control system, the slave spacecraft is controlled and assumed to be aligned with the orbital coordinate system F of the primary spacecraft. o Alignment, i.e. ω d / o = 0. Both the primary and secondary spacecraft are assumed to be rigid, and all external forces acting on the spacecraft are assumed to act on the center of mass of their respective bodies.

[0209] Still refer to Figure 7B , the main spacecraft and the auxiliary spacecraft relative to the inertial coordinate system F e The translational motion equation of

[0210]

[0211]

[0212] where r c 、r d is the position vector of the main and secondary centroids relative to the center of the earth, m c 、m d are the primary and secondary masses, μ is the gravitational constant of the Earth, and f c 、f d denote the perturbations acting on the primary and secondary spacecraft, respectively. Usually, these perturbations include orbital perturbations and control. In this study, it is assumed that the primary spacecraft follows Keplerian motion, i.e., f c = 0, and we ignore the orbital perturbations to the secondary spacecraft.

[0213] Given the primary and secondary spacecraft, the position of the secondary spacecraft relative to the primary spacecraft is given by

[0214] ρ = r d -r c . (twenty one)

[0215] Still refer to Figure 7B , relative to the orbital coordinate system F of the main spacecraft o The relative position (21) is derived to obtain

[0216]

[0217] For the orbital coordinate system F relative to the main spacecraft o The relative velocity (22) is derived to obtain

[0218]

[0219] Substituting (20) into (23) yields the complete nonlinear relative motion equation.

[0220] ||ρ||<<||r c ,(twenty four)

[0221] The relative motion equation (23) can be linearized about the trajectory of the host spacecraft and expressed in the orbital coordinate system F of the host spacecraft. o Analyze in, get:

[0222]

[0223] in o ρ=[δx δy δz] T It is in F o The relative position of the analysis, r c =||r c ||,h=||r c ×r′ c || is the inertial specific angular momentum of the host spacecraft's orbit, and o f d =[u x u y u z ] T It is in F o The control inputs applied to the orbit of the secondary spacecraft are analyzed in .

[0224] Still refer to Figure 7B , when r c As and h vary along the trajectory, the equation of motion (25) yields a linear time-varying system

[0225]

[0226] in And u= o f d .

[0227] Figure 8810 is a block diagram showing some components of the system and method that can be used to implement some embodiments of the present disclosure. For example, the computer system 870 can be suitable for controlling the movement of a spacecraft or aircraft. 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. The storage device 818, the control interface 820, the display interface 822 and the external interface 824 can also be connected to the bus system 813. The external interface 824 can be connected to the expansion memory 850 and the aircraft tool parameter database 852 storing the spacecraft specifications, thruster specifications, dimensions, weight, etc. In addition, the external interface 824 can be connected to the initial orbit database 854 and other orbit databases 856 (i.e., unique orbit data), and the initial orbit database 854 stores the time, date and parameters about the orbit including altitude, inclination and eccentricity, etc. The bus system 813 can also connect the control interface 826, the output interface 827, the receiver 828 and the transmitter 830. In addition, the bus system can connect the GPS receiver module 832 to the GPS 834.

[0228] The bus system 813 may connect to the output thruster command module 858 to output thruster commands. In addition, the bus 859 may connect back to the orbit maintenance module 840, which includes a transfer orbit generator for generating one or more transfer orbits for the spacecraft, a feedback gain module 844 for calculating feedback gains, and a feedback controller 846. In addition, the orbit maintenance module 840 may also include a thruster command generator 848.

[0229] Still refer to Figure 8 , the computer 870 may be a server or desktop computer, laptop computer, mobile device or other computer device or system having one or more processors 810. The processor 810 may be a central processing unit adapted to access code in the form of a transfer trajectory generator 842 in a memory 812 or stored data 818 of the computer 870 (or an extended memory 850 or other data storage devices 852, 854, 856). In accordance with aspects related to the systems and methods of the present disclosure, external storage devices may be contemplated if further desired, depending on the specific design and aspects of the intended hardware and target implementation. For example, the computer 870 may be used to implement the steps of the systems and methods, wherein the memory 812 and / or the storage device 818 may store data.

[0230] Figure 8The data stored in the memory 812 may include executable modules, aircraft data, and historical space data. For example, the aircraft data may include the specifications, dimensions, weight, and performance data of the spacecraft under various conditions (including gravity and other perturbations, i.e., the complex motion of a massive object in space subject to forces other than the gravitational attraction of a single other massive object). In addition, the aircraft data may include data related to aspects related to the dynamics of the aircraft associated with one or more of a plurality of variables, the plurality of variables being (1) unusual orbital characteristics of a celestial body (i.e., a natural object located outside the Earth's atmosphere (e.g., the Moon, the Sun, an asteroid, a planet, or a star)); (2) unusual orbital motion of a celestial body; (3) an unusual proximity of a celestial body to an orbit around another celestial body; and (4) other known perturbations. The space data may include data related to celestial body systems and past missions to celestial bodies, as well as any other data related to space, spacecraft, and planned orbital designs for other celestial bodies in the universe. For example, data stored as space data may include data about satellites of a celestial body (e.g., characteristics of a celestial body that may be considered in developing an orbital design from an initial celestial body orbit to a similar target celestial body orbit).

[0231] Optionally, the stored data can be stored in Figure 8 The storage device 818 is connected to the external interface 824 connected to the expansion memory 850 (which is connected to the initial orbit data database 854, the other orbit data database 856 and the aircraft parameters, specifications, performance, etc. data database 852).

[0232] Still refer to Figure 8 Depending on the specific application, the processor 810 of the computer 870 may be two or more processors. For example, some steps may require separate processors to ensure a specific processing time or processing speed associated with the systems and methods of the present disclosure.

[0233] Following the historical spatial data stored in the memory 812, the receiver 828 or input interface can receive spatial data, which can be the latest spatial data obtained from the Earth mission control center or sensors associated with the spacecraft or some other location. The receiver 828 and transmitter 830 can provide a wireless location for receiving and sending data to, for example, the Earth mission control center or some other destination. The GPS receiver module 832 connected to the GPS 834 can be used for navigation related aspects. The computer 870 can include a control interface 820, a display interface 822, and optional external devices, control interfaces, displays, sensors, machines, etc. (not shown, see Figure 8) that are contemplated for use in connection with the systems and methods of the present disclosure. Figure 8 ).

[0234] Fig. 99 is a schematic diagram illustrating, by way of non-limiting example, a computing device 900 for implementing some techniques of methods and systems according to embodiments of the present disclosure. Computing device or apparatus 900 represents various forms of digital computers, such as laptops, desktops, workstations, personal digital assistants, servers, blade servers, mainframes, and other appropriate computers.

[0235] The computing device 900 may include a power supply 908, a processor 909, a memory 910, a storage device 911, all of which are connected to a bus 950. In addition, a high-speed interface 912, a low-speed interface 913, a high-speed expansion port 1214, and a low-speed connection port 915 may be connected to the bus 1250. In addition, a low-speed expansion port 916 is connected to the bus 950. Depending on the specific application, various component configurations that can be mounted on a common motherboard (non-limiting example) 930 are envisioned. In addition, an input interface 917 can be connected to an external receiver 906 and an output interface 918 via the bus 950. The receiver 919 can be connected to an external transmitter 907 and a transmitter 920 via the bus 950. The external memory 904, the external sensor 903, the machine 902, and the environment 901 can also be connected to the bus 950. In addition, one or more external input / output devices 905 can be connected to the bus 950. Network interface controller (NIC) 921 may be adapted to connect to network 922 via bus 950 , where data or other information and the like may be presented on a third party display device, third party imaging device, and / or third party printing device external to computer device 900 .

[0236] Still refer to Fig. 9 It is contemplated that the memory 910 may store instructions executable by the computer device 900, historical data, and any data that may be utilized by the methods and systems of the present disclosure. The memory 910 may include a random access memory (RAM), a read-only memory (ROM), a flash memory, or any other suitable storage system. The memory 910 may be one or more volatile memory units and / or one or more non-volatile memory units. The memory 910 may also be another form of computer-readable medium (e.g., a magnetic disk or optical disk).

[0237] Storage device 911 can be suitable for storing supplementary data and / or software modules used by computer device 900. For example, storage device 911 can store historical data and other relevant data as described above about the present disclosure. Additionally or alternatively, storage device 911 can store historical data similar to the data mentioned above about the present disclosure. Storage device 911 can include hard disk drive, optical disk drive, thumb drive, drive array or its arbitrary combination. In addition, storage device 911 can include computer readable medium (for example, floppy disk device, hard disk device, optical disk device or tape device, flash memory or other similar solid-state memory device or array of device (including storage area network or other configuration)). Instructions can be stored in information carriers. When executed by one or more processing devices (for example, processor 909), instructions execute one or more methods (for example, methods described above).

[0238] Still refer to Fig. 9 The system can be optionally linked to a display interface or user interface (HMI) 923 via a bus 950, which are suitable for connecting the system to a display device 925 and a keyboard 924, wherein the display device 925 may include a computer monitor, camera, television, projector or mobile device, etc.

[0239] The computer device 900 may include a user input interface 917 suitable for a printer interface (not shown), and may also be connected via a bus 950 and suitable for connecting to a printing device (not shown), wherein the printing device may include a liquid inkjet printer, a solid ink printer, a large-scale commercial printer, a thermal printer, a UV printer or a dye sublimation printer, etc.

[0240] Still refer to Fig. 9 , the high-speed interface 912 manages bandwidth-intensive operations of the computing device 900, while the low-speed interface 913 manages lower bandwidth-intensive operations. This functional allocation is only an example. In some implementations, the high-speed interface 912 can be connected to the memory 910, the user interface (HMI) 923, the keyboard 924 and the display 925 (for example, through a graphics processor or accelerator), and the high-speed expansion port 914 (which can accept various expansion cards (not shown) via the bus 950). In an implementation, the low-speed interface 913 is connected to the storage device 911 and the low-speed expansion port 915 via the bus 950. The low-speed expansion port 915, which may include various communication ports (for example, USB, Bluetooth, Ethernet, wireless Ethernet), can be connected to one or more input / output devices 905 and other devices (for example, keyboard 924, pointing device (not shown), scanner (not shown) or network devices such as switches or routers) through a network adapter.

[0241] As shown, computing device 900 can be implemented in a variety of different forms. For example, computing device 900 can be implemented as a standard server 926, or multiple implementations in a group of such servers. In addition, computing device 900 can be implemented as a personal computer such as a laptop computer 927. Computing device 900 can also be implemented as a part of a rack server system 928. Alternatively, the components of computing device 900 can be combined with other components in a mobile device (not shown). Each such device can contain one or more computing devices and mobile computing devices, and the entire system can be composed of multiple computing devices that communicate with each other.

[0242] This description provides only exemplary embodiments and is not intended to limit the scope, applicability or configuration of the present disclosure. Instead, the following description of the exemplary embodiments will provide a description of implementing one or more exemplary embodiments for those skilled in the art. It is contemplated that various changes may be made to the function and arrangement of the elements without departing from the spirit and scope of the disclosed subject matter as set forth in the appended claims.

[0243] Specific details are given in the following description to provide a comprehensive understanding of the embodiments. However, it will be appreciated by those skilled in the art that these embodiments can be practiced without these specific details. For example, the systems, processes and other elements in the disclosed subject matter can be shown as components in block diagram form to avoid obscuring the embodiments with unnecessary details. In other cases, in order to avoid obscuring the embodiments, known processes, structures and techniques can be shown without unnecessary details. In addition, the same reference numerals and names in the various drawings represent the same elements.

[0244] In addition, each embodiment can be described as a process, which is depicted as a flow chart, a job chart, a data flow chart, a structure chart or a block diagram. Although a flow chart can describe an operation as a sequential process, many operations can be performed in parallel or concurrently. In addition, the order of the operation can be rearranged. The process can terminate when its operation is completed, but there can be additional steps that are not discussed or included in the figure. In addition, not all operations in any specifically described process can occur in all embodiments. The process can correspond to a method, a function, a process, a subroutine, a subprogram, etc. When the process corresponds to a function, the termination of the function can correspond to the function returning to the calling function or the main function.

[0245] In addition, embodiments of the disclosed subject matter may be implemented at least in part manually or automatically. Manual or automatic implementation may be performed or at least assisted by the use of machines, hardware, software, firmware, middleware, microcode, hardware description languages, or any combination thereof. When implemented in software, firmware, middleware, or microcode, program code or code segments that perform the necessary tasks may be stored in a machine-readable medium. A processor may perform the necessary tasks.

[0246] The above-mentioned embodiments of the present disclosure can be implemented in any of a variety of ways. For example, the embodiments can be implemented using hardware, software, or a combination thereof. When implemented in software, the software code can be executed on any suitable processor or set of processors, whether it is set in a single computer or distributed in multiple computers. Such a processor can be implemented as an integrated circuit and have one or more processors in an integrated circuit assembly. However, the processor can be implemented using a circuit in any suitable format.

[0247] In addition, the various methods or processes outlined herein may be encoded as software that can be executed on one or more processors using any of a variety of operating systems or platforms. In addition, such software may be written using any of a variety of suitable programming languages ​​and / or programming or scripting tools, and may also be compiled into executable machine language code or intermediate code that is executed on a framework or virtual machine. Typically, in various embodiments, the functionality of the program modules may be combined or distributed as desired.

[0248] In addition, the embodiments of the present disclosure can be implemented as methods, and examples thereof have been provided. The actions performed as a part of the method can be ordered in any suitable manner. Therefore, an embodiment can be constructed, wherein the actions are performed in a sequence different from that shown, which can include performing some actions concurrently, even if these actions are shown as sequential actions in the illustrative embodiments. In addition, the use of ordinal numbers such as first and second in the claims to modify the claim elements themselves does not represent any priority, precedence or order of a claim element relative to another claim element, nor does it represent the time sequence in which the actions of the method are performed, but is only used as a mark to distinguish a claim element with a specific name from another element with the same name (except the use of ordinal numbers), thereby distinguishing the claim elements.

[0249] Although the present disclosure has been described with reference to certain preferred embodiments, it should be understood that various other variations and modifications may be made within the spirit and scope of the present disclosure. Therefore, the aspects of the appended claims are to cover all such variations and modifications that fall within the true spirit and scope of the present disclosure.

Claims

1. A controller for maintaining a spacecraft near an orbit, the controller comprising: a memory configured to store executable instructions; as well as a processor configured to execute the executable instructions to cause the controller to: 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 within a time horizon from a current time relative to a high-fidelity reference trajectory to generate a state transfer matrix STM for uncontrolled motion of the spacecraft within the time horizon, wherein the state transfer matrix includes non-extended eigenvectors having a magnitude less than or equal to 1 and extended eigenvectors having a magnitude greater than 1; determining a control action that changes an upcoming state of the spacecraft to a linear combination of the non-extended eigenvectors of the state transfer matrix; and Control commands are generated for actuators of the spacecraft, the control commands resulting in a correction to the upcoming state of the spacecraft in a direction corresponding to at least one of the non-extended eigenvectors of the state transfer matrix.

2. The controller according to claim 1, wherein: To detect that the distance is greater than the spacecraft threshold, the processor is configured to: determining the current status of the spacecraft during a specified time period; and determining a distance between a position controlled by the current state of the spacecraft and the orbit; and The determined distance between the position controlled by the current state of the spacecraft and the orbit is compared to the spacecraft threshold.

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

4. The controller according to claim 1, wherein: In order to determine the state transition matrix, the processor is further configured to: obtaining a linear model of a dynamic system defined by the spacecraft and at least one celestial body; computing a discrete-time update of a perturbation of the high-fidelity reference trajectory; and A numerical expression defined by the basic basis vectors of the STM and the high-fidelity reference trajectory is propagated over a defined period of time.

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

6. The controller according to claim 1, wherein: In order to determine the control action, the processor is also configured to determine the direction of the correction to the upcoming state by a linear combination of the directions of each non-extended eigenvector of the STM to achieve a mixed state using a natural motion that is a combination of the corresponding eigenmotions of each non-extended eigenvector of the STM.

7. The controller according to claim 1, wherein: To generate the control commands, the processor is further configured to solve a nonlinear optimization problem to find a fuel efficient maneuver that transfers the spacecraft to a set of desired states that produces a desired natural motion.

8. The controller according to claim 7, wherein: The nonlinear optimization problem is a finite-horizon optimization of a model of the spacecraft, a set of objectives for the motion of the spacecraft, and constraints on the propulsion system of the spacecraft and the motion of the spacecraft.

9. The controller according to claim 8, wherein: Constraints on the propulsion system of the spacecraft and the motion of the spacecraft include one or more physical limitations on 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.

10. The controller according to claim 1, wherein: The spacecraft is configured to perform a docking mission to a space station that maintains the space station's position near the orbit at a distance less than a station threshold, wherein the controller is further configured to select the spacecraft threshold greater than the station threshold.

11. The controller according to claim 1, wherein: The upcoming state of the spacecraft includes the position, orientation, and translational and angular velocity of one or more of the spacecraft and the spacecraft's payload, and one or a combination of perturbations acting on a multi-object celestial system defined by the spacecraft, the payload, and one or more celestial bodies.

12. The controller according to claim 11, wherein: The perturbations acting on the multi-object celestial body system are natural orbital forces including the gravitational perturbations of the sun and the moon, the anisotropic gravitational perturbations caused by the non-spherical shape of the central body, the solar radiation pressure and the air resistance.

13. The controller according to claim 1, wherein: The high-fidelity reference trajectory is an uncontrolled natural motion trajectory, which is located near the near-rectilinear halo orbit NRHO in the Earth-Moon circular restricted three-body problem and is estimated using a multiple shooting method.

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

15. The controller according to claim 1, wherein: The orbit is one of a circular orbit, an elliptical orbit, a halo orbit, a nearly linear halo orbit or a quasi-satellite orbit.

16. The controller according to claim 1, wherein: The control commands are generated as solutions to a model predictive control strategy that generates the control commands by optimizing a cost function over a rolling horizon.

17. The controller according to claim 16, wherein: The cost function includes a stabilization component for directing the motion of the spacecraft to the upcoming state, a component for an operational goal of the spacecraft, and a performance component for optimizing the motion of the spacecraft until the upcoming state is achieved.

18. The controller according to claim 1, wherein: The control command is generated for each of a plurality of specified time periods within the time horizon, or iteratively within a rolling time horizon.

19. A computer-implemented method for maintaining a spacecraft in the vicinity of an orbit, the method comprising the steps of: 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 within a time horizon from a current time relative to a high-fidelity reference trajectory to generate a state transfer matrix STM of uncontrolled motion of the spacecraft within the time horizon, wherein the state transfer matrix includes non-extended eigenvectors having a size less than or equal to 1 and extended eigenvectors having a size greater than 1; determining a control action that changes an upcoming state of the spacecraft to a linear combination of the non-extended eigenvectors of the state transfer matrix; and Control commands are generated to actuators of the spacecraft, the control commands resulting in a correction to the upcoming state of the spacecraft in a direction corresponding to at least one of the non-extended eigenvectors of the state transfer matrix.

20. A non-transitory computer-readable storage medium having a program embodied thereon, the program being executable by a processor for performing a method for maintaining a spacecraft near an orbit, the method comprising the steps of: 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 within a time horizon from a current time relative to a high-fidelity reference trajectory to generate a state transfer matrix STM for uncontrolled motion of the spacecraft within the time horizon, wherein the state transfer matrix includes non-extended eigenvectors having a size less than or equal to 1 and extended eigenvectors having a size greater than 1; determining a control action that changes an upcoming state of the spacecraft to a linear combination of the non-extended eigenvectors of the state transfer matrix; and Control commands are generated to actuators of the spacecraft, the control commands resulting in a correction to the upcoming state of the spacecraft in a direction corresponding to at least one of the non-extended eigenvectors of the state transfer matrix.

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