Model Predictive control of spacecraft

The present invention relates comprehensively to controlling the movement of a spacecraft, and more specifically to controlling the movement using model predictive control (MPC) over a backward horizon.

Spacecraft in orbit are subject to various disturbances that affect their location, that is, their ability to maintain the desired orbit and position in the desired orbit. To offset these forces, spacecraft are generally equipped with thrusters for orbital maneuvers. Existing methods of handling trackkeeping requirements use impact propulsion systems manually commanded by ground control centers.

In addition to orbital perturbation, the spacecraft is provided by on-board momentum exchange devices such as reaction wheels or control moment gyroscopes that allow the spacecraft to maintain its desired orientation relative to the Earth or stars. It is disturbed by the external torque that is generally absorbed. The stored angular momentum is periodically unloaded via the onboard thrusters to prevent saturation of the momentum exchange device and subsequent loss of desired spacecraft attitude. This is also a manually commanded process from the ground control center.

The process of determining and instructing the onboard thrusters from the ground control center is a tedious manual process, eg, space in a specific orbit, eg geostationary orbit, as required for spacecraft colocation. Not easily scaled to an increasing number of ships and their tight orbital retention windows. Also, such manual control results in an open-loop strategy, which cannot automatically correct the errors introduced in the modeling or implementation of the desired orbital maintenance maneuver and momentum management maneuver. Therefore, the accuracy of positioning and positioning of the spacecraft is limited.

Orbital retention and momentum unloading are generally achieved by different sets of thrusters, which include mass being a driving consideration in spacecraft design and increased complexity and cost. Not desirable due to. Combining orbital stationkeeping and momentum unloading problems with the same set of thrusters results in multiple goals, and it is difficult to coordinate these goals to achieve them at the same time. It becomes. For this, see, for example, the method described in Patent Document 1 that simplifies control by using the maximum available values for thruster torque and force.

<p><patcit num="1"><text>U.S. Pat. No. 8282043</text></patcit></p>

<p> An object of some embodiments of the present invention is to provide a system and method for simultaneous control of spacecraft orbital position and accumulated on-board momentum using a single set of thrusters. Another object of some embodiments is to provide a way to achieve the simultaneous control using Model Predictive Control (MPC) over backward horizon. A further objective of some embodiments is to avoid control manually commanded from the ground and provide autonomous control that can be performed in the onboard control system, resulting in tighter and more accurate. Orbital maintenance and momentum unloading.</p>

<p> Some embodiments of the present invention provide a single set of thrusters to orbital maneuver and orbital maneuver by adjusting the required thrust between available thrusters for both maneuvers, taking into account the overall thrust limit. It is based on the recognition that it can be used simultaneously for momentum unloading maneuvers. For example, tight and acceptable trajectory retention windows, strict constraints on available thrust, and trajectory retention such as adjustments required between trajectory control for trajectory retention and attitude control for momentum unloading. Momentum management requirements impose constraints on the conditions and inputs that the controller must meet.</p><p> Further recognition is that well-defined models, cost functions, and model predictive control (MPC) with constraints may be beneficial in producing fuel-efficient maneuvers that extend the effective life of the spacecraft. It means that there is. For example, the MPC cost function can have the dual purpose of simultaneously controlling the orbital position of a spacecraft and the accumulated on-board momentum using a single set of thrusters. In addition, the optimization of its cost function can be conditioned on conditions and inputs for adjusted trajectory and attitude control.</p><p> In addition, MPC is an autonomous closed loop control that can be implemented in the onboard control system. Some embodiments are quadratic programming (QP) that optionally utilizes a linearized orbital mechanics equation for the spacecraft's nominal operating conditions and a spacecraft predictive model based on the linearized stance dynamics equations. By formulating MPC as: quadratic program), the computational complexity of MPC can be further reduced.</p><p> Also, one embodiment of controlling a spacecraft in geostationary earth orbit (GEO) utilizes the orbital plane coupling of a linearized orbital (CWH) equation to produce a fuel-efficient maneuver. Achieve. This embodiment includes a disturbance prediction model using the associated non-Kepler disturbance force analysis formula as the MPC prediction model.</p><p> Therefore, one embodiment of the present invention discloses a method of controlling the movement of a spacecraft according to a model of the spacecraft. The method uses a control input that simultaneously controls the thruster and spacecraft momentum exchange devices using cost function optimization over a backward horizon subject to constraints on the spacecraft's attitude and spacecraft's input to the thruster. The cost function includes components that control the attitude of the spacecraft and the momentum stored by the momentum exchange device, and a command to simultaneously control the thruster and the momentum exchange device according to at least a portion of the control input. To generate and include. The steps of this method are performed by the spacecraft processor.</p><p> Another embodiment discloses a control system that controls the operation of a spacecraft according to a spacecraft model, the control system comprising at least one processor that executes a module of the control system. The module provides control inputs that simultaneously control the thruster and spacecraft momentum exchange devices using cost function optimizations across backward horizon subject to constraints on the spacecraft's attitude and constraints on the spacecraft's input to the thruster. The desired control input module, the cost function, includes a control input module that includes components that control the momentum stored by the spacecraft's attitude and momentum exchange device, and thrusters and momentum exchange devices according to at least a portion of the control input. It is a force torque map module that generates commands to control at the same time, and the generated commands are a command to a momentum exchange device that unloads the stored momentum, and a momentum exchange while maintaining or changing the attitude of the spacecraft. Includes a force torque map module, which includes a force compensating for the torque generated by the device unloading the stored momentum and commands to individual thrusters to generate the torque.</p><p> Yet another embodiment is a spacecraft, a set of thrusters that change the attitude of the spacecraft, a set of momentum exchange devices that absorb the disturbance torque acting on the spacecraft, and thrusters and momentum exchange devices. A control system that simultaneously controls the operation of a spacecraft, comprising a control system comprising at least one processor that executes a module of the control system. The module provides control inputs that simultaneously control the spacecraft thrusters and spacecraft momentum exchange devices using cost function optimizations across the backward horizon subject to spacecraft attitude constraints and spacecraft input constraints. The desired control input module, the cost function, includes a control input module, which includes components that control the attitude of the spacecraft and the momentum stored by the momentum exchange device.</p>

<figref num="1A">It is a schematic diagram of the formulation of the problem by one embodiment of the present invention.</figref><figref num="1B">It is a schematic diagram of the formulation of the problem by one embodiment of the present invention.</figref><figref num="1C">It is a schematic diagram of the formulation of the problem by one embodiment of the present invention.</figref><figref num="2">It is a block diagram of the controller which controls the operation of the spacecraft by one Embodiment of this invention.</figref><figref num="3">It is a block diagram of the general structure of the controller of FIG. 1A according to one embodiment of the present invention.</figref><figref num="4">It is a block diagram of various modules of a controller according to one embodiment of the present invention.</figref><figref num="5">FIG. 6 is a block diagram of a method performed by a module of a controller according to one embodiment of the present invention.</figref><figref num="6">It is the schematic of the disturbance prediction problem by one Embodiment of this invention.</figref><figref num="7">FIG. 6 is a schematic view of an exemplary region of feasible force and torque values for a pair of thrusters according to one embodiment of the invention.</figref>

1A and 1B show a spacecraft 102 equipped with multiple actuators such as thrusters 150 and momentum exchange device 151. Examples of actuator types include reaction wheels (RW) and control moment gyroscopes (CMG). The spacecraft was designed to fly in extra-atmospheric space whose movements change physical quantities such as the position, speed, and orientation or orientation of the spacecraft in response to commands sent to the actuator. A vehicle, airship, or machine. When the actuator receives a command, it increases or decreases the speed of the spacecraft, thus giving the spacecraft a force to translate its position, and when the actuator receives a command, the spacecraft rotates. It also gives the spacecraft torque to change its orientation or orientation. As used herein, the motion of a spacecraft is determined by the motion of an actuator that determines the motion of the spacecraft that alters such physical quantities.

Spacecraft are around, between, or near the Earth 161, the Moon, and / or one or more gravitational bodies such as celestial planets, stars, asteroids, comets, etc. in extra-atmospheric space. Fly along an open orbital path or a closed orbital path 160. Usually, a desired position or target position 165 along the orbital path is given. The reference coordinate system 170 is mounted in the desired position. Here, the origin of this coordinate, that is, the coordinate of all zeros in the reference coordinate system, is always the coordinate of the desired position.

Spacecraft are subject to various disturbance forces 114. These disturbance forces are all forces that were not taken into account when determining the orbital path of the spacecraft. These disturbances act on the spacecraft to pull the spacecraft away from the desired position on the orbital path. These forces can include, but are not limited to, gravity, radiation pressure, atmospheric resistance, non-spherical centrosomes, and leaked fuels. Therefore, the spacecraft may be at a distance of 167 away from the target position.

Due to these disturbances, it is not always possible to keep the spacecraft in the desired position along its orbit. Therefore, instead, the spacecraft is desired to remain within the window 166 with the specified dimensions 604 around the desired position. To that end, the spacecraft is controlled to move along any path 606 contained within the window. In this example, the window 166 has a rectangular shape, but the shape of the window may change in different embodiments.

Spacecraft are often also required to maintain the desired orientation. Spacecraft fixed, for example, aligned with a desired frame of reference, such as an inertia frame of reference 171 fixed to a distant star 172, or a frame of reference 173 oriented to always point to the Earth. A reference coordinate system 174 is required. However, depending on the shape of the spacecraft, various disturbance forces 114 can act non-uniformly on the spacecraft, which produces disturbance torque that causes the spacecraft to rotate away from its desired orientation. .. To compensate for these disturbance torques, a momentum exchange device 151, such as a reaction wheel, is used to absorb the disturbance torque, which allows the spacecraft to maintain its desired orientation.

The momentum stored in the momentum exchange devices is unloaded, for example, by reducing the amount of spin on the reaction wheel so that the momentum exchange devices do not saturate and thereby lose their ability to compensate for disturbance torque. .. Unnecessary torque is applied to the spacecraft by unloading the momentum exchange device. Such unnecessary torque is also compensated by the thrusters.

FIG. 1C shows Euler angles 175 between the spacecraft fixed frame of reference 174 and the desired frame of reference 171. For example, some embodiments control the spacecraft so that these Euler angles remain within the limit 181 during the momentum unloading process.

FIG. 2 shows a block diagram of the control system 101 that controls the operation of the spacecraft 102. The control system receives a target motion, eg, a desired motion 103 of the spacecraft, such as a desired flight trajectory or some target points of the physical quantities, and controls the spacecraft via a control input 104. These control inputs can include commands to change the parameters of the spacecraft's operation, or can include actual values of parameters such as voltage, pressure, torque, and force that affect the spacecraft's motion. As a result, the physical quantity 105 of the spacecraft is generated. In addition, the disturbance force and disturbance torque 114 affect the motion of the spacecraft.

An object of some embodiments of the present invention is to request a command 104 to the thruster 150 and the momentum exchange device 151 so that the spacecraft stays in the box 166 and at the same time unloads the stored excess momentum. .. This is done by implementing an automated control system 101 using a model of spacecraft 112. For example, some embodiments use the optimization of the cost function 116 over the receding horizon subject to constraints 115 on the spacecraft's attitude and input to the thruster, using the spacecraft's thrusters and spacecraft. The control input that simultaneously controls the momentum exchange device is obtained, and the appropriate control input command 104 is generated. The attitude of the spacecraft includes one or a combination of the absolute or relative positions and orientations of the spacecraft. In some embodiments, the cost function includes components that control the attitude of the spacecraft and components that unload the momentum stored by the momentum exchange device.

The control system 101 receives information 106 about the spacecraft's motion from sensors, hardware, or software that are directly or remotely connected to the spacecraft. Information 106 includes the state of the spacecraft. The spacecraft uses this state to select control input 104. Information 106 can include some or all of the physical quantities of motion 105, and can also include additional information about the spacecraft. The physical quantity 105, the control input 104, or a combination thereof can be required to stay within some predetermined range according to the constraint 115 on the operation of the spacecraft.

Control system 101 achieves simultaneous control using model predictive control (MPC) over backward horizon. MPC is based on iterative finite horizon optimization based on spacecraft models, a set of objectives for spacecraft motion, and constraints on spacecraft propulsion systems and spacecraft propulsion motion, to keep track of future events. Has the ability to anticipate and, as a result, take appropriate control actions. It uses the predictions obtained according to the spacecraft model subject to constraints, by optimizing the spacecraft's behavior according to the above set of objectives over a future finite time horizon, and then the current time slot. It is achieved only by implementing control over. For example, constraints can represent the physical limits of a spacecraft, the safety limits for spacecraft operation, and the performance limits for a spacecraft's flight trajectory. Spacecraft control strategies are acceptable when the motion generated by the spacecraft for such control strategies meets all constraints. For example, at time t, the spacecraft's current state is sampled and an acceptable cost minimization control strategy is sought for a relatively short time horizon in the future. Specifically, online or real-time calculation seeks a cost minimization control strategy up to time t + T. After the first step of this control is performed, the state is measured or estimated again and the calculation is repeated starting from the current state at that time to obtain new control and new predicted state flight trajectory. Be done. Predictive horizon shifts forward, so MPC is also called backward horizon control.

FIG. 3 shows the overall structure of the control system 101 according to one embodiment of the present invention. The control system 101 comprises at least one processor 130 running a module of the controller. Processor 130 is operatively connected to memory 120, which stores spacecraft model 112 and constraint 115. An object of some embodiments of the present invention is to obtain a control input 104 using a model of spacecraft 112 subject to constraint 115. The memory can also store the cost function 116. In one embodiment, the processor seeks and / or updates at least one of a cost function, constraint, and model during control.

FIG. 4 shows a block diagram of various modules of the control system 101 according to one embodiment of the present invention. In some embodiments, the spacecraft model is model 112, such as spacecraft orbital mechanics that controls the translational motion of the spacecraft, and spacecraft attitude dynamics and spacecraft attitude kinetics that control the attitude motion of the spacecraft. Includes a nominal model 202 that defines the relationships between the parameters of. Model 112 also includes disturbance model 203, which defines the disturbance force 114 acting on the spacecraft. In some embodiments, these disturbance forces are whether the spacecraft is located in a predetermined position, eg, the desired position 165, for various time steps of control, regardless of the actual position of the spacecraft. Is required. Those embodiments are based on the recognition that such approximations simplify the computational complexity of disturbances without significantly reducing accuracy. The disturbance module 203 allows the MPC to utilize natural dynamics to compensate for the disturbance force, thus reducing fuel consumption while satisfying the spacecraft's kinetic objectives.

Some of the spacecraft's physical quantities need to remain within the desired range defined by the spacecraft's operational constraints 205. For example, such a physical quantity is a spacecraft attitude that includes a position constraint derived from the requirement to keep the spacecraft within window 166 and a orientation constraint derived from the requirement to keep Euler angles 175 within limits 181. Can include.

Some embodiments of the present invention are based on the additional recognition that constraints 206 on control inputs are required to meet operating limits of spacecraft actuators, such as limits on thrust magnitude. In some embodiments, the control input constraint 206 allows a single set of thrusters 150 to generate both trajectory control force and attitude control torque, taking into account the limits of the magnitude of the overall thrust. It is formulated to do. In some embodiments, constraint 206 is used in combination with at least some constraints 205 that control the spacecraft.

In some embodiments, the control input 104 is determined based on the optimization of the cost function 209 subject to the constraint 205 on the spacecraft operation and the constraint 206 on the control input. In some embodiments, this cost function is a component for spacecraft position 291, a spacecraft attitude component 292, a stored momentum component 293, and a spacecraft motion objective component. Includes a combination of multiple components, including 294, and component 295, which ensures the stability of spacecraft operation.

For example, spacecraft position component 291 penalizes the spacecraft's larger displacement 167 from the desired position 165, so control inputs are applied to the spacecraft as a result of optimization of cost function 209. Then, a control input that reduces the displacement 167 is obtained to help achieve the purpose of staying in the window 166.

Spacecraft attitude component 292 penalizes the greater magnitude of spacecraft Euler angles 175 between the spacecraft fixed frame of reference 174 and the desired frame of reference, eg 171, and therefore costs. As a result of the optimization of function 209, when a control input is applied to the spacecraft, control inputs that reduce Euler angles 175 are obtained to help achieve the goal of maintaining the desired orientation of the spacecraft. It has become like.

The stored momentum component 293 penalizes the greater magnitude of the stored momentum, so it was stored when the control input was applied to the spacecraft as a result of the optimization of the cost function 209. A control input that unloads momentum is now available, for example, a high reaction wheel spin amount is penalized, so that the optimization produces a control input that reduces the reaction wheel spin amount. To do.

The objective component 294 of the spacecraft's operation can also include a penalty for the amount of fuel used by the thruster, for example, so that the optimization of the cost function 209 results in a control input that uses less fuel. , More than the magnitude of the speed at which the spacecraft operates so that the optimization of the cost function results in a control input that causes the spacecraft to operate faster, i.e., a control input that achieves its purpose in a shorter period of time. It can also include penalties for small things.

Stability component 295 is required to provide control inputs that ensure the stability of spacecraft operation as a result of optimization of cost function 209. In one embodiment, if the desired orbit 160 is circular, the stability component of the cost function is at the end of the MPC horizon by using the solution of the Discrete Algebraic Riccati Equation (DARE). Penalize the position of the spacecraft. In other embodiments, the desired trajectory is not circular. For example, the desired orbit may be elliptical or otherwise non-circular and periodic. In that case, the stability component penalizes the position of the spacecraft at the end of the MPC horizon by using the solution of the Periodic Differential Riccati Equation (PDRE). Note that the solution of PDRE is not constant, so the current penalty of cost function 209 is chosen to correspond to the solution of PDRE at the time corresponding to the time at the end of the MPC horizon.

In some embodiments, for each of the components 291 to 294 of the cost function 209, the optimization of the cost function achieves the goals of the various individual components with priorities corresponding to their relative weights. Weighted to generate control inputs.

For example, in one embodiment, the weights are chosen so that the component 294, which penalizes the fuel used by the thrusters, is given the greatest weight. As a result, this embodiment produces a spacecraft operation that prioritizes the use of the smallest possible amount of fuel at the expense of higher average displacement 167. In different embodiments, the component 291 that penalizes the displacement 167 from the desired position 165 is given the greatest weight. As a result, this embodiment produces a spacecraft operation that prioritizes maintaining a small average displacement of 167 at the expense of using more fuel. In some embodiments, the stability component 295 is defined by its weight according to the weight that produces the controlling input to stabilize.

The processor 130 of the control system 101 includes a control input module 208 that seeks commands 107 to the force, torque, and on-board momentum exchange devices during the current iteration by optimizing the current cost function 209. Run various modules. The control input module optimizes the current cost function using the spacecraft's current model 201, subject to constraints 205 on spacecraft operation and 206 on current control inputs.

In one embodiment, the optimization of the cost function 209 in the control input module 208 is formulated as a quadratic programming (QP). The quadratic programming can be solved quickly and efficiently on resource-constrained hardware such as spacecraft with limited computing power. A linear-quadratic MPC (LQ-MPC) is used to take advantage of the quadratic programming.

For example, the control system also includes a current model module 201 for linearizing the nominal model 202 at the desired target location 165 on the target orbit and determining the disturbance force at the desired target location 165. In some embodiments, this linearization results from the LQ-MPC utilizing a linear prediction model. Module 201 finds the current model of the spacecraft throughout the MPC Horizon for the current time point. Module 201 can also receive the current state of spacecraft 106 to determine the state of the spacecraft relative to linearization.

In one embodiment, the control system obtains the current control input 104 from the commanded full force and total torque of the spacecraft and applies it to the individual thrusters, where the thrusters generate the desired force generated by the control input module 208. And a force torque map module 204 that reverses the control input constraint 206 is also provided to ensure that the torque 107 is fully applied to the spacecraft. The force torque map module 204 passes the commands calculated in the control input module 208 as the current control input 104 along with the individual thruster commands to the onboard momentum exchange device without modification.

The control system also includes a cost function module 207 that finds the current cost function 209. For example, this cost function module updates the previous cost function based on changes in the target motion of the spacecraft, eg, changes in the desired motion 103. This is because different motions may require different cost functions for the spacecraft physical quantities 105 to meet their desired objectives. The cost function module can also update the stability component 295 of the cost function if the desired trajectory requires updated weights based on that trajectory. Since the control steps are performed iteratively, the current model and current cost function become the previous model and previous cost function for subsequent iterations. For example, the previous model, the previous cost function, and the previous control input are sought as the current model, the current cost function, and the current control input in the previous iteration.

FIG. 5 shows a block diagram of the method performed by the module of control system 101. This method iteratively controls the movement of the spacecraft using the control inputs obtained using the spacecraft model based on the optimization of the cost function. This method finds the current state of the spacecraft as a result of control using the previous control inputs obtained for the previous iterations by optimizing the previous cost function using the spacecraft's previous model. (210). The current state of the spacecraft can be determined using hardware, software, or communication with the ground, such as GPS, relative distance measurements, stellar trackers, and horizon sensors.

In one embodiment, this method updates the model before finding the current control input (230). For example, the model update involves linearizing the spacecraft model 112 at the desired target location 165 in the target orbit over the future predicted horizon for the current time point. The model update also calculates the predicted disturbance force 230 over the same horizon at the target location and combines this with the dynamic prediction model to form the overall prediction. Finally, in some embodiments, model update 230 can also update the stability component 295 of the cost function to the correct value over the future prediction horizon for the current point in time.

This method then uses the current model and the current cost function to find the current control input that controls the spacecraft in the current iteration (240). For example, this method uses updated current cost functions and current spacecraft models so that the predicted future spacecraft state and inputs meet constraints on spacecraft operation and control inputs. And future input of forces, torque, and commands to the onboard momentum exchange device during a fixed time of the future, at least the length from the current time to the acquisition of new spacecraft state measurements. Find the sequence (240). The first part of this input sequence for a duration equal to the time required to obtain new measurements of the spacecraft's condition is converted from force and torque to individual thruster profiles (250), It is applied to the spacecraft as the current control input, along with commands to the onboard momentum exchange device (260). Based on the spacecraft's current state, the spacecraft's current model, and the spacecraft's current control inputs, the spacecraft's next state is determined and the controller receives a new state measurement. Wait (270).

Formula used to calculate commands to thrusters In one embodiment of the invention, the spacecraft model 112 has six 2-axis electric thrusters 150 and three wires mounted on a rigid bus in an orthogonal mass-balanced configuration. Required for nadir-pointing spacecraft in geosynchronous orbit (GEO) equipped with a symmetric reaction wheel 151. The bus fixed coordinate system 174 is specified for the spacecraft, and the inertial coordinate system 171 is specified for determining the attitude of the spacecraft. The equation of motion of the spacecraft is given by the following equation.<maths num="1"><img file="JP6444293B2_D0001.tif" /></maths>Where r R³Is the spacecraft's position vector with respect to the center of the earth, F R³Is the vector of the external force applied by the thruster, a_p R³Is a vector of perturbation acceleration 114, m is the mass of the spacecraft, μ is the gravitational constant of the earth, and J R.^3×3Is the moment of inertia of the spacecraft bus and reaction wheel array, J_α R^3×3Is the moment of inertia of the reaction wheel array, ω R³Is the angular velocity of the bus coordinate system with respect to the inertial coordinate system, ν R³Is the angular velocity of the reaction wheel array, τ R³Is the torque applied by the thruster, ω^×Is the cross product matrix of ω, and R R^3×3Is a rotation dyadic that transforms the inertial coordinate system 171 into the bus coordinate system 174, which is decomposed in the bus coordinate system 174.

In another embodiment, equation (1) is used in place of an equation controlling a spacecraft that is in another orbit and has a momentum exchange device other than a reaction wheel.

In one embodiment, model (1) is linearized to form the current prediction model 201. For a small maneuver around a nominal circular orbit, the linearized equation approximates the relative motion of the spacecraft as:<maths num="2"><img file="JP6444293B2_D0002.tif" /></maths>Where δx, δy, and δz are components of the spacecraft's position vector with respect to the nominal location 165, F._x, F_y, F_zIs a component of the thrust vector, a_{p, x}, A_{p, y}, A_{p, z}Is a component of the perturbation acceleration vector,<maths num="3"><img file="JP6444293B2_D0003.tif" /></maths>Is the average motion of the nominal orbit.

In one embodiment, the attitude error rotation matrix<maths num="4"><img file="JP6444293B2_D0004.tif" /></maths>Using a set of 3-2-1 Euler angles (ψ, θ, φ)<maths num="5"><img file="JP6444293B2_D0005.tif" /></maths>It is parameterized as. Where R_dIs the desired attitude flight trajectory, C₁, C₂, And C₃Are the basic rotations of ψ, θ, and φ around the x-axis, y-axis, and z-axis, respectively. The linearization of postural mechanics and postural kinematics for the equilibrium y-axis (principal axis) spin having an angular velocity corresponding to the average motion n of the orbit is as follows.<maths num="6"><img file="JP6444293B2_D0006.tif" /></maths>Where δω₁, Δω₂, Δω₃, Δφ, δθ, δψ, and δν₁, Δν₂, Δν₃Is the relative angular velocity component of the spacecraft, the relative Euler angles of the spacecraft, and the relative angular velocity component of the reaction wheel array. That is, they are physical quantities that represent the error from the desired spacecraft angular velocity component, the desired Euler angles, and the desired reaction wheel array angular velocity component.

In the embodiment where the spacecraft is on GEO, the main perturbation accelerations are due to the gravity of the sun and moon, the solar radiation pressure, and the anisotropic geopotential, the non-spherical gravitational field of the earth. To do. These perturbations per unit mass, that is, the analytical formulas for disturbance acceleration, are given by the following formulas, respectively.<maths num="7"><img file="JP6444293B2_D0007.tif" /></maths>here,<maths num="8"><img file="JP6444293B2_D0008.tif" /></maths>Indicates a coordinate-independent (unresolved) vector, μ_sunAnd μ_moonIs the gravitational constant of the sun and moon, C_srpIs the solar radiation pressure constant, S is the area of the surface facing the sun, c_reflIs the surface reflectance and ρ_EIs the equatorial radius of the earth,<maths num="9"><img file="JP6444293B2_D0009.tif" /></maths>Is the z-axis unit vector of the inertial coordinate system centered on the earth, J₂Is the dominant coefficient in the geopotential perturbation model being considered, where additional higher-order terms are ignored. The sum of the individual disturbance accelerations in (4) gives the total disturbance accelerations considered in (1).

In some embodiments, a state space model is given by the following equation.<maths num="10"><img file="JP6444293B2_D0010.tif" /></maths>here,<maths num="11"><img file="JP6444293B2_D0011.tif" /></maths>Is.

Since it is used as a prediction model in the MPC method, (5) is discretized using a sampling period of ΔT seconds, which gives the following equation.<maths num="12"><img file="JP6444293B2_D0012.tif" /></maths>Where x_kIs the time step k Z⁺Is the state in, u_kIs the time step k Z⁺Is the control vector in<maths num="13"><img file="JP6444293B2_D0013.tif" /></maths>Is the continuous time system realization value (A) in (5)_c, B_c) Is a discretized matrix obtained based on.

Estimating the disturbance acting on the spacecraft In some embodiments, model (8) is enhanced using the prediction model 203 of disturbance acceleration (4) to obtain the following equation:<maths num="14"><img file="JP6444293B2_D0014.tif" /></maths>Where a_{p, k}Is the total disturbance acceleration predicted at time step k based on the transmission at the desired position 165, O_{H / E}Is a from the inertial coordinate system 171_{p, k}Is a rotation matrix that transforms the components of to the components of the same acceleration in the desired reference frame of reference 170.

The desired position 165 for predicting disturbance acceleration is used in (9) due to the non-linearity of the analytical equation in (4).

FIG. 6 shows the spacecraft 102 displaced from its desired position 165 at time step k = 0. Since the desired positions 601, 602, and 603 on the nominal orbit 160 at time steps k = 1, k = 2, ..., k = N are known in advance, a_{p, k}Can be predicted from the disturbance forces 604, 605, and 606 based on the analytical equation (4) at the time steps k = 1, k = 2, ..., k = N. Since the spacecraft position is constrained within the tight window 166, the difference between the disturbance acceleration at the desired positions 601, 602, and 603 and the disturbance acceleration at the true satellite position 907 is unknown, but ignored. can do. Therefore, some embodiments seek disturbance forces as if the spacecraft were in the target position for the entire period of retreat horizon.

Constraints on Thruster Input In some embodiments, the constraint 205 on spacecraft operation is at least partially imposed by δy and δz corresponding to the orbital stationkeeping window 166, using the following relationship:<maths num="15"><img file="JP6444293B2_D0015.tif" /></maths>Where λ_{1, max}Is the maximum allowable longitude error, λ_{2, max}Is the maximum allowable latitude error.

In some embodiments of the invention, the spacecraft is equipped with six biaxial thrusters. T = [T₁ T₂ T₃ T₄ T₅ T₆] Is defined. Where T_iIs the force exerted by each biaxial thruster. Constraints on the size of individual thrusters, ie<maths num="16"><img file="JP6444293B2_D0016.tif" /></maths>Is related to the constraints on the force F and torque τ of the control input via the force torque map 204. The combination of (11) and (12) provides a constraint on the control input 206 for force and torque that effectively combines (2) with (3). That is, the thruster generates both a force for trajectory control and a torque for attitude control.<maths num="17"><img file="JP6444293B2_D0017.tif" /></maths>

FIG. 7 shows an example of a region 703 of feasible force and torque values for a pair of thrusters. Here, the maximum force 701 that a thruster pair can generate trades off with the maximum torque 702 that this pair can produce. Constraint (13) allows the control system 101 to simultaneously achieve the dual purpose of keeping the spacecraft in box 166 and unloading the stored excess momentum. Guarantee to generate a feasible command 104 to.

In some embodiments, the relative Euler angles (δφ, δθ, δψ) are small in the following equations to maintain the orientation of the spacecraft even while unloading the stored excess momentum. It is constrained to be within the permissible range.<maths num="18"><img file="JP6444293B2_D0018.tif" /></maths>

In some embodiments, the current cost function 209 has a variety of purposes, eg, the purpose of quantifying displacement from a nominal orbital position.₁, Purpose of quantifying Euler angles error and penalizing spacecraft angular velocity components J₂Purpose of penalizing the generation of force and torque using thrusters J₃, And the purpose of penalizing reaction wheel momentum J₄Consists of costs associated with ,. In some embodiments, these costs J₁~ J₄Is given by the following equation.<maths num="19"><img file="JP6444293B2_D0019.tif" /></maths>

Each purpose J₁~ J₄Is the weight w_iMultiplied by the total cost function J in the following equation_totCombined with.<maths num="20"><img file="JP6444293B2_D0020.tif" /></maths>

Weights assigned to each purpose w_iDefines its relative importance. The greater the weight assigned to a given purpose, the higher the priority of that purpose when the cost function is optimized.

Based on (6) and (7), J_totCan be described as the following equation for the formalized state space.<maths num="21"><img file="JP6444293B2_D0021.tif" /></maths>Here, Q and R are the weights w assigned to each purpose._iIs a symmetric positive-definite weighting matrix that encodes the

Cost Function Stability Objectives In some embodiments, if the desired orbit is not circular, for example elliptical or otherwise non-circular and periodic, a model of spacecraft motion around that orbit 201 Can be linear and time invariant. In such an embodiment, the stability component 295 of the cost function 209 is determined based on the following Periodic Difference Riccati Equation (PDRE).<maths num="22"><img file="JP6444293B2_D0022.tif" /></maths>Where A_k, B_kIs the matrix of model 201 at time step k, P_k, Q_k, And R_kIs a symmetric positive-definite weighting matrix. Matrix Q_kAnd R_kIs considered to be the same as the weighting matrix in (19).

For embodiments where the linearization is time invariant, such as motion around a nominal circular orbit, GEO, the stability component 295 is the solution of the following discrete algebra (positive: Algebraic) Riccati equation (DARE): It is calculated based on.<maths num="23"><img file="JP6444293B2_D0023.tif" /></maths>Here, A and B are matrices of the model in (8), and P, Q, and R are symmetric positive-definite weighting matrices. Similar to the above, the matrices Q and R are considered to be the same as the weighted matrix in (19).

Control Input Computation In some embodiments, the control input module 208 takes the form of a finite horizon numerical optimization problem, as in the following equation.<maths num="24"><img file="JP6444293B2_D0024.tif" /></maths>This equation uses the current cost function 209 and the current linearized spacecraft model 201, which uses (9) to predict the evolution of the state across the horizon, and (10), (13), and (14). Formed from the spacecraft constraint 206 used. Where P_N, Q_k, R_kIs the matrix given in (15), D is the matrix that implements the forces and torques available at the same time, as in (13), and x (t) is the state at the current time step. .. Problem (17) is solved using a numeric solver, which is an input sequence U = [u that minimizes the current cost function subject to problem constraints.₁... u_N]^TTo find out.

First input in this input sequence u₁Is considered to be the output 107 of the input calculation 208. Input u₁Is passed to the force torque map module 204, which realizes that the force torque map module can be achieved simultaneously by the spacecraft thrusters due to the input that inverts (12) and thus satisfies (13). Configure command 104 to the thruster by getting the possible values. At the next time step t + 1, the model and cost function are updated, the state is updated, and the numerical optimization problem is solved again.

If the orbit is such that the spacecraft model 201 is time invariant, then A₁= A₂= ... = A and B₁= B₂= ... = B, P in (17)_N, Q_k, And R_kIs given by the matrices P, Q, and R in (16). P to the cost function of (17)_NAlternatively, the inclusion of P ensures local stability of the target position, such as near the origin where the constraint has no effect, and in the absence of disturbance prediction, the solution in (17) is a periodic LQR controller or LQR controller. Is equivalent to any of the solutions.

The embodiments described above of the present invention can be carried out in any of a number of methods. For example, these embodiments can be implemented using hardware, software, or a combination thereof. When implemented in software, software code can be run on any suitable processor or collection of processors, whether located on a single computer or distributed among multiple computers. .. Such processors can be implemented as integrated circuits with one or more processors in the integrated circuit components. However, the processor can be implemented using any suitable type of circuit section.

Furthermore, it should be understood that the computer can be embodied in any of a plurality of forms such as rack mount computers, desktop computers, laptop computers, minicomputers, or tablet computers. Such computers can be interconnected in any suitable form by one or more networks, including enterprise networks or local area networks such as the Internet or wide area networks. Such networks can be based on any suitable technology, can operate according to any suitable protocol, and can include wireless networks, wired networks, or fiber optic networks.

The various methods or processes outlined herein can also be encoded as software that can be run on one or more processors using any one of the various operating systems or platforms. .. In addition, such software can be written using any of a number of suitable programming languages and / or programming tools or scripting tools.

Moreover, the embodiment of the present invention can also be embodied as a method. An example of this method is provided. The steps performed as part of this method can be ordered in any suitable way. Thus, it is possible to construct embodiments in which the actions are performed in a different order than those illustrated, and these embodiments show some actions as sequential actions in the illustrated embodiments. Can also include running at the same time.