A method for controlling a tethered docking-based on-orbit servicing robot

Abstract

This invention discloses a tether-based approach docking control method for an on-orbit service robot. Focusing on the approach docking process, it studies the relative dynamics and control of the tether-based on-orbit service robot. A relative dynamic equation is established for the service robot to approach a customer satellite using tether tension. The influence of the customer satellite's orbital eccentricity, orbital altitude, and the robot's tether-retrieval rate on the system's relative motion is analyzed, along with the pitch motion characteristics under constant tether length. A stable tether-retrieval control law with a constant pitch angle constraint is proposed. By designing a tether length variation law, the robot can smoothly retrieve the tether to approach the customer satellite. Numerical simulation is used to verify the tether-retrieval control law. This control method can be used on elliptical orbits with non-zero eccentricity. By maintaining a constant pitch angle through tether-retrieval control, it is beneficial for changing the system's relative motion state and stabilizing the pitch motion.

Description

Technical Field

[0001] This invention relates to the field of space tethered satellite control technology, specifically to a tethered approach docking method for controlling the tethering of an on-orbit service robot. Background Technology

[0002] On-orbit servicing refers to a class of space missions conducted in space by humans, robots (or robotic satellites), or through human-robot collaboration, aimed at extending the lifespan of spacecraft and enhancing their mission capabilities. These missions primarily include space assembly, maintenance, and servicing. On-orbit servicing encompasses various space operations such as on-orbit monitoring, on-orbit refueling, on-orbit repair, and space debris removal. Many of these operations require the servicing spacecraft to approach and dock with the target spacecraft. However, conventional rendezvous and docking require complex cooperative / non-cooperative relative state measurement systems, and sometimes even specialized robotic arms to complete the capture and docking. This not only demands cooperation from the customer satellite but also significantly increases the quality and complexity of the servicing spacecraft.

[0003] A space tether system is a spacecraft system that connects two or more spacecraft together using flexible tethers. It offers several advantages, such as: 1) providing safety for operations through tethers; 2) assisting the spacecraft's guidance, navigation, and control systems; 3) enabling non-cooperative missions; and 4) good maneuverability. Therefore, servicing spacecraft can use space tethers to connect with customer satellites in the vicinity and then actively retrieve the tether to approach and dock with the customer satellite. This approach significantly reduces the complexity and cost of on-orbit servicing systems, making nanosatellites / CubeSats possible as servicing spacecraft. The relative motion control of the servicing spacecraft using space tethers is crucial for its approach to customer satellites and, to a certain extent, determines whether subsequent on-orbit servicing missions can continue. Therefore, it is necessary to carefully study the relative dynamics of spacecraft under tether tension and then design control laws that meet mission requirements to address this problem in existing technologies. Summary of the Invention

[0004] This invention aims to provide a rope-retrieval control method for on-orbit service robots based on rope-tethered close-in docking. For systems operating on elliptical orbits with non-zero eccentricity, rope retrieval control can maintain a constant pitch angle, which is beneficial for changing the relative motion state of the system and stabilizing the pitch motion, thereby solving the problems mentioned in the background art.

[0005] To achieve the above objectives, the present invention provides the following technical solution: a method for controlling the tethering of an on-orbit service robot based on tethered docking, wherein...

[0006] By utilizing the process of retracting the rope to achieve near docking, we study the relative dynamics and control problems of a tethered on-orbit service robot.

[0007] Establish the relative dynamic equations of an on-orbit service robot that utilizes a tethered system to approach a customer's satellite;

[0008] The relative dynamics equations of the tethered system are used to analyze the effects of the customer satellite's orbital eccentricity, orbital altitude, and robot tether retrieval rate on the system's relative motion, as well as the pitch motion characteristics with a constant tether length.

[0009] A stable rope retrieval control law with a fixed pitch angle constraint is proposed. By designing a rope length variation law, the robot can smoothly retrieve the rope and approach the customer satellite.

[0010] Numerical simulation was used to verify the rope-reeling control law.

[0011] Preferably, the service robot is transported to the vicinity of the customer satellite by an orbital transfer vehicle equipped with a high-precision target perception and relative state measurement system. The throwing system on the orbital transfer vehicle throws the other end of the tether to the customer satellite and achieves strong adhesion. The service robot and the customer satellite form a tethered system. The service robot actively retracts the tether to get closer to the customer satellite and finally docks.

[0012] The service robot and customer satellite connected by tethers are abstracted into a space tether system. In the space tether system, the customer satellite and the service robot are connected by flexible tethers. The customer satellite and the service robot are orbiting the Earth in the same orbital plane in the Earth's gravitational field, and each is subjected to the tension of the tether.

[0013] Treating the satellite and robot as point masses, neglecting the mass of the rope, assuming the rope is inextensible and can only generate tension, not thrust, with the force directed only along the rope, and controlling the rope to remain taut during system motion, the distance between the satellite and robot is the rope length. Treating the Earth as a sphere, disregarding its oblateness, and considering only Earth's gravity as the external force, ignoring drag and other interference, in a near-focal coordinate system... The true near point angle is used in the middle. and Earth's center distance Determine the position of the customer satellite in the center-of-mass orbital coordinate system. Used rope length and pitch angle To determine the position of the service robot relative to the customer's satellite, based on the above requirements, the system's relative dynamic equations are as follows:

[0014] (1)

[0015] In the formula, the dots above the letters represent the derivative with respect to time. The gravitational constant of Earth, The tension in the rope, For equivalent quality, and ( The quality of customer satellites and service robots are respectively.

[0016] If we neglect the effect of the tension of the rope on the motion of the customer satellite in equation (1), then the customer satellite orbit is a standard Keplerian orbit, and the first and second equations of equation (1) can be further simplified:

[0017] (2)

[0018] In the formula, , and These are the semi-drillion, semi-major axis, and eccentricity of the customer satellite's orbit, respectively. ;

[0019] Considering the length of the rope It is much smaller than the geocentric distance of the customer's satellite orbit. Therefore, they have an approximate relationship: , Therefore, equations 3 and 4 can be simplified to:

[0020] (3)

[0021] In the motion control of the tethered system, we focus on the relative motion of the two spacecraft, transforming the equations into dimensionless equations. The transformed formulas are as follows:

[0022] ,

[0023] ,

[0024] In the formula This represents the maximum rope length, which is the initial or final rope length when the customer's satellite recovers or releases the service robot; therefore, it always has... This holds true, and the corresponding dimensionless equation is as follows:

[0025] (4)

[0026] In the formula, the apostrophe indicates the angle relative to the true anterior point. The derivative of .

[0027] Preferably, the orbital eccentricity of the customer spacecraft It has a direct impact on the form of the system equations, and will further affect the system's control performance.

[0028] exist When the customer satellite's orbit is circular, the system is autonomous and has an equilibrium point. Substituting into equation (4), we get:

[0029] (5)

[0030] exist When the customer satellite orbit is elliptical, the system is non-autonomous and there is no equilibrium point.

[0031] exist At that time, the pitch angle can be controlled to remain at Unchanged, will , , Substituting into equation (4) part 2, we obtain the rope length. Equation of change:

[0032] (6)

[0033] Preferably, when the rate of change of the pitch angle with respect to the true anomaly angle is the same, the higher the orbital altitude of the customer satellite, the smaller the rate of change of the pitch angle over time. It can be seen that for the same process of change Due to area rate constant, The rate of change of pitch angle is the sum of orbital altitude and Earth's radius. With height (or The value decreases as the orbital period increases. This reflects that when the system is running in high orbit, as the orbital period increases, the relative motion of the system tends to slow down, and the time required to reach the same motion state is longer;

[0034] When the relative motion state changes with the true anomaly angle for the same reason, the higher the orbital altitude of the customer satellite, the smaller the tether tension required. It can be seen that for the same input (Output the same relative motion state), tether tension With track height (or (Increases and decreases.)

[0035] Preferably, when the robot rapidly (e.g., a few meters per second) retracts the rope, the angular velocity of the orbital satellite is not higher than the order of magnitude of... , Compared to It can be ignored, so the second equation of equation (3) can be approximated as:

[0036] (7)

[0037] Multiply both sides of the equation by Furthermore, we can obtain:

[0038] (8)

[0039] Therefore Represented as:

[0040] (9)

[0041] In the formula, the subscript 0 indicates the corresponding initial value.

[0042] Preferably, when the customer satellite moves in a circular orbit, it makes... ,get Furthermore, we can conclude that:

[0043] (10)

[0044] The points can be obtained as follows:

[0045] (11)

[0046] When the rope length remains constant, the change in pitch angle depends on the initial state.

[0047] Preferably, the constraints considered in the design of the control law are that the rate of change of rope length and the rate of change of pitch angle are close to zero, so as to keep the rope length and pitch angle basically constant, expressed as:

[0048] (12)

[0049] In the formula, Indicates the designed terminal rope length, subscript The value at the end, and These are the upper and lower limits of the feasible pitch angle, respectively.

[0050] Preferably, in the stable rope retrieval control law with constant pitch angle constraint, denoted as... , As a set value, it is obtained from equation (6). Integrating, we get:

[0051] (13)

[0052] Let be the initial true anterior angle, corresponding to the start of rope winding. Further, we obtain:

[0053] (14)

[0054] Substituting equation (14) into the first equation of equation (4), we get:

[0055] (15)

[0056] This control law allows for rope retrieval control while maintaining a constant pitch angle; rope retrieval is... from Change to The process, according to the equation, requires solving... ,get The range of values ​​for is:

[0057] (16)

[0058] A rope can only generate tension, not thrust, therefore the control input... Must meet According to equation (15), we get:

[0059] (17)

[0060] Equation (17) needs to be applied to any Established, Let be the true anterior angle at the end, corresponding to the end of the rope pull, which can be obtained from equation (17):

[0061] (18)

[0062] For convenience, construct a bivariate function. :

[0063] (19)

[0064] To ensure that inequality (17) holds, The following conditions must be met:

[0065] (20)

[0066] From the equation As can be seen from this, a bivariate function Compared to The period is ,and The periods of the value ranges are the same.

[0067] Preferably, the numerical simulation verification of the rope-reeling control law uses the PD control law as a comparison. The PD control law is as follows:

[0068] (twenty one)

[0069] In the formula, , .

[0070] Compared with the prior art, the beneficial effects of the present invention are:

[0071] This control method is used on elliptical orbits with non-zero eccentricity. The pitch angle of the rope system can be kept constant by controlling the rope winding, which is beneficial for changing the relative motion state of the system and stabilizing the pitch motion. Attached Figure Description

[0072] Figure 1 This is a schematic diagram illustrating the approach and docking process of the tethered on-orbit service robot of the present invention.

[0073] Figure 2 This is a schematic diagram illustrating the rate of change of pitch angle at different track altitudes according to the present invention;

[0074] Figure 3 This is a schematic diagram illustrating the tension of the rope at different track heights according to the present invention;

[0075] Figure 4 This is a schematic diagram showing the rate of change of rope length and pitch angle during rapid rope winding in this invention;

[0076] Figure 5 This is a schematic diagram illustrating the rate of change of pitch angle and its approximate error during rapid rope retraction in this invention.

[0077] Figure 6 This is a schematic diagram of the phase trajectory of the pitch angle change in this invention;

[0078] Figure 7 Functions of this invention A schematic diagram of a three-dimensional curved surface;

[0079] Figure 8 For the purposes of this invention The following function Follow A diagram illustrating the changes;

[0080] Figure 9 For the purposes of this invention Values Schematic diagram of the change curve;

[0081] Figure 10 This is a schematic diagram of the rate of change curve of the rope length control law of the present invention. Detailed Implementation

[0082] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0083] Please see Figure 1-10 The present invention provides a technical solution: a rope-retracting control method for on-orbit service robots based on rope-tied close-in docking.

[0084] This study investigates the relative dynamics and control of a tethered on-orbit servicing robot by utilizing the tether retrieval process to achieve near docking. The relative dynamic equations of the on-orbit servicing robot relative to the customer satellite are established. Using these equations, the influence of the orbital eccentricity, orbital altitude, and robot tether retrieval rate on the relative motion of the servicing spacecraft are analyzed, along with the pitch motion characteristics under constant tether length. A stable tether retrieval control law with a constant pitch angle constraint is proposed to design tether length variations, ensuring stable relative motion of the space tethered system during tether retrieval. Numerical simulation is used to verify the tether retrieval control law.

[0085] The service robot is transported to the vicinity of the customer spacecraft by an orbital transfer vehicle equipped with a high-precision target perception and relative state measurement system. The throwing system on the orbital transfer vehicle throws the other end of the tether to the customer satellite and achieves strong adhesion. The service robot and the customer satellite form a tethered system. The service robot actively retracts the tether to get closer to the customer satellite and finally dock.

[0086] The service robot and customer satellite connected by a tether are abstracted into a space tether system. In the space tether system, the customer satellite and the service robot are connected by a flexible tether. The customer satellite and the service robot are orbiting the Earth in the same orbital plane within the Earth's gravitational field, and each is subjected to the tension of the tether.

[0087] Treating the customer satellite and service robot as point masses, neglecting the mass of the rope, assuming the rope is inextensible and can only generate tension, not thrust, with the force directed only along the rope, and keeping the rope taut during system motion, the distance between the satellite and robot is the rope length. Treating the Earth as a sphere, disregarding its oblateness, and considering only gravity as an external force, ignoring drag and other interference, in a near-focal coordinate system... The true near point angle is used in the middle. and Earth's center distance Determine the position of the customer satellite in the center-of-mass orbital coordinate system. Used rope length and pitch angle To determine the position of the service robot relative to the customer's satellite, based on the above requirements, the system dynamics equations are as follows:

[0088] (1)

[0089] In the formula, the dots above the letters represent the derivative with respect to time. The gravitational constant of Earth, The tension in the rope, For equivalent quality, and ( ( ) refers to the quality of customer satellites and service robots, respectively.

[0090] If we neglect the effect of the tension of the rope on the motion of the customer satellite in equation (1), then the customer satellite orbit is a standard Keplerian orbit, and the first and second equations of equation (1) can be further simplified:

[0091] (2)

[0092] In the formula, , and These are the semi-drillion, semi-major axis, and eccentricity of the customer satellite's orbit, respectively. ;

[0093] Considering the length of the rope It is much smaller than the geocentric distance of the customer's satellite orbit. Therefore, they have an approximate relationship: , Therefore, equations 3 and 4 of equation (1) can be simplified to:

[0094] (3)

[0095] In the motion control of the tethered system, we focus on the relative motion of the two spacecraft, transforming the equations into dimensionless equations. The transformed formulas are as follows:

[0096] ,

[0097] ,

[0098] In the formula This represents the maximum rope length, which is the initial or final rope length when the customer's satellite recovers or releases the service robot; therefore, it always has... This holds true, and the corresponding dimensionless equation is as follows:

[0099] (4)

[0100] In the formula, the apostrophe indicates the angle relative to the true anterior point. The derivative of the equation clearly indicates that the control input appears in the first equation for the change of rope length. There is no torque term in the second equation related to the pitch angle motion, therefore the relative motion control system is underactuated.

[0101] Customer satellite orbital eccentricity It has a direct impact on the system's configuration and will further affect the system's control performance.

[0102] exist When the customer satellite's orbit is circular, the system is autonomous and has an equilibrium point. When the customer satellite orbit is elliptical, the system is non-autonomous and there is no equilibrium point.

[0103] The difference in gravitational force experienced by the two spacecraft within the tethered system and the Earth's center distance from the customer satellite. and the initial relative position of the service robot Regarding the customer satellite's movement in a circular orbit, the system begins its motion from any true angle of apogee. All are the same; subsequent relative motion states are only the same as the initial relative states. The system equations are related to the initial true anomaly angle but not to the initial true anomaly angle, thus the system equations are autonomous. However, when the customer satellite is in an elliptical orbit, due to... The initial true anomaly angle of the system changes, and the subsequent relative motion state depends not only on the initial relative motion state but also on the initial true anomaly angle. Therefore, the system equations are non-autonomous.

[0104] when At that time, Substituting into equation (4), we get:

[0105] (5)

[0106] exist At this time, the pitch angle can be maintained at Unchanged, will , , Substituting into equation (4) part 2, we get:

[0107] (6)

[0108] When the change in rope length satisfies equation (6), the pitch angle of the state variable can be maintained. This equation remains unchanged and serves as the basis for subsequent control law design.

[0109] For the same rate of change of the pitch angle with respect to the true anomaly angle, the higher the orbital altitude of the customer satellite, the smaller the rate of change of the pitch angle over time. It can be seen that for the same process of change Due to the rate of change in area It is a constant. The rate of change of pitch angle is the sum of orbital altitude and Earth's radius. With height (or The decrease in relative motion as the orbital period increases reflects that when the system is running in high orbit, the relative motion process of the system tends to slow down as the orbital period becomes longer, and it takes longer to reach the same motion state.

[0110] When the relative motion state changes with the true anomaly angle for the same reason, the higher the orbital altitude of the customer satellite, the smaller the tether tension required. It can be seen that for the same input (Output the same relative motion state), tether tension With track height (or (increases and decreases)

[0111] The pitch rate and the amplitude of the tether tension both decrease as the track rises. Although the number of track rotations throughout the process is almost the same, the track period increases with altitude, which in turn makes the time required to complete the retrieval process longer.

[0112] The orbital angular velocity of Earth-orbiting satellites is no higher than the order of magnitude of When the rope is pulled in quickly (e.g., a few meters per second), Compared to It can be ignored, so the second equation of equation (3) can be approximated as:

[0113] (7)

[0114] Multiply both sides of the equation by Furthermore, we can obtain:

[0115] (8)

[0116] Therefore Represented as:

[0117] (9)

[0118] In the formula, the subscript 0 indicates the corresponding initial value.

[0119] From equation (9), we can see that during the rapid rope retrieval process, the pitch rate is... With the rate of change of rope length It is irrelevant, but relative to the length of the rope. It relates to the square of the number, when the rope is finished being pulled in. It could become very large, such as a rope decreasing from 100m to 1m. It can reach ,if Not satisfied, The value will be very large, causing the tether to quickly wrap around the customer satellite at the end. However, for the release process, such as releasing the tether from 1m to 100m, This will make This is the key difference between reeling in and releasing the rope. Rapidly reeling in the rope will cause a sharp change in the pitch angle, but rapidly releasing the rope will suppress the change in the pitch angle.

[0120] Therefore, in the design of the control law, it is necessary to take advantage of the completely controllable rope length and ensure that the rope is wound up slowly. and The values ​​are of a considerable magnitude, and this method indirectly controls the pitch angle within a reasonable range by controlling the rope winding rate.

[0121] The system is set to operate in low Earth orbit from 400km to 600km apogee. The service robot approaches the customer satellite at a speed of no more than 6m / s at a distance of 100m, and continues until it reaches a position of 2m. The entire process takes less than 35s.

[0122] First, consider the scenario where the customer satellite moves in a circular orbit. ,get get:

[0123] (10)

[0124] The points can be obtained as follows:

[0125] (11)

[0126] When the rope length remains constant, the change in pitch angle depends on the initial state. The law governing the change in pitch angle can be studied through phase diagram analysis. Let... ,exist Draw different on a plane value corresponding Follow The changes.

[0127] If the system runs on a circular track, with the rope length remaining constant, the direction corresponding to the local plumb bob... It is the stable equilibrium position for pitch motion, corresponding to the local horizontal direction. It is an unstable equilibrium position.

[0128] The control design constraints are: the rate of change of rope length and the rate of change of pitch angle are close to zero to keep the rope length and pitch angle essentially constant, that is:

[0129] (12)

[0130] In the formula, Indicates the designed terminal rope length, subscript The value at the end, and These are the upper and lower limits of the feasible pitch angle, respectively.

[0131] During the retrieval process, the expected pitch angle variation range depends on the attitude of the customer satellite and is not necessarily near 0. This varies depending on the customer satellite's body coordinate system. With orbital coordinate system When they do not overlap, the pitch angle needs to be controlled. Variations near non-zero values ​​mean that, for certain observation tasks, service robots may need to observe customer satellites in a specific direction, which also requires... Since it is a non-zero fixed value, the equation is valid for... The constraints are Instead , This is the maximum amplitude value.

[0132] remember , As a set value, it is obtained from equation (6). Integrating, we get:

[0133] (13)

[0134] This is the initial true advance angle, corresponding to the start of rope winding. Further, we obtain:

[0135] (14)

[0136] Substituting equation (14) into the first equation of equation (4), we get:

[0137] (15)

[0138] This control law allows for rope retrieval control while maintaining a constant pitch angle; rope retrieval is... from Change to The process, according to the equation, requires... Thus obtain The range of values ​​for is:

[0139] (16)

[0140] A rope can only generate tension, not thrust, therefore the control input... Must meet According to equation (15), we get:

[0141] (17)

[0142] Equation (17) for any Established, Let be the true anterior angle at the end, corresponding to the end of the rope pull, which can be obtained from equation (17):

[0143] (18)

[0144] For convenience, construct a bivariate function. :

[0145] (19)

[0146] To ensure that inequality (17) holds, The following conditions must be met:

[0147] (20)

[0148] Due to the equation As can be seen from this, a bivariate function Compared to The period is ,and The periods of the value ranges are the same.

[0149] Take different ,get The curve formed is composed of a bivariate function. As the continuity is known, with The increase of , the pitch angle that satisfies the inequality The range is smaller, which indicates that if the service robot maintains a pitch angle Under unchanged conditions, the cable is retrieved and the satellite approaches the customer's satellite. The feasible range depends on the customer's satellite orbital eccentricity. It decreases as it increases.

[0150] As a special case, explore For example, at this time It can be easily obtained from the equation: ,according to The definition obviously has This indicates that in maintaining At that time, the rope length can only vary within a limited range.

[0151] Numerical simulation was used to verify the rope take-up control law, with the PD control law used for comparison. The PD control law is as follows:

[0152] (twenty one)

[0153] In the formula, , .

[0154] Both the invented control law (15) and the PD control law can pull the service robot from a position 100m away to a position 1m away, and the speed at the end of the rope retraction is very close to 0. Both control laws meet the rope length control requirements. The pitch angle change and pitch rate caused by the control law (15) are very small. In contrast, the PD control law makes the pitch angle and pitch rate very large, causing the tether to wrap around the customer satellite quickly, which is not allowed in practice. The tension is always kept in line with the mechanical characteristics of the rope. The control law (15) generates a very small tension to complete the relative motion control, which is in line with the mechanical characteristics of the microgravity environment. The PD control law generates a large tension when the rope is retracted, which is caused by the large pitch rate.

[0155] This control method can maintain the system's pitch angle constant by winding the rope on an elliptical orbit with a non-zero eccentricity, which is beneficial for changing the system's relative motion state and stabilizing the pitch motion.

[0156] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A method for controlling the tethering of an on-orbit service robot based on tethered docking, characterized in that: By utilizing the process of retracting the rope to achieve near docking, we study the relative dynamics and control problems of a tethered on-orbit service robot. Establish the relative dynamic equations of an on-orbit service robot that utilizes a tethered system to approach a customer's satellite; The relative dynamics equations of the tethered system are used to analyze the effects of the customer satellite's orbital eccentricity, orbital altitude, and robot tether retrieval rate on the system's relative motion, as well as the pitch motion characteristics with a constant tether length. A stable rope retrieval control law with a fixed pitch angle constraint is proposed. By designing a rope length variation law, the robot can smoothly retrieve the rope and approach the customer satellite. Numerical simulation was used to verify the smooth rope take-up control law; The service robot is transported to the vicinity of the customer satellite by an orbital transfer vehicle equipped with a high-precision target perception and relative state measurement system. The throwing system on the orbital transfer vehicle throws the other end of the tether to the customer satellite and achieves strong adhesion. The service robot and the customer satellite form a tethered system. The service robot actively retracts the tether to get closer to the customer satellite and finally docks. The service robot and customer satellite connected by tethers are abstracted into a space tether system. In the space tether system, the customer satellite and the service robot are connected by flexible tethers. The customer satellite and the service robot are orbiting the Earth in the same orbital plane in the Earth's gravitational field, and each is subjected to the tension of the tether. Treating the satellite and robot as point masses, neglecting the mass of the rope, assuming the rope is inextensible and can only generate tension, not thrust, with the force directed only along the rope, and controlling the rope to remain taut during system motion, the distance between the satellite and robot is the rope length. Treating the Earth as a sphere, disregarding its oblateness, and considering only Earth's gravity as the external force, while ignoring drag, this is done in a near-focal coordinate system. The true near point angle is used in the middle. and Earth's center distance Determine the position of the customer satellite in the center-of-mass orbital coordinate system. Used rope length and pitch angle The system's relative dynamic equations are as follows to determine the position of the service robot relative to the customer's satellite: （1） In the formula, the dots above the letters represent the derivative with respect to time. The gravitational constant of Earth, The tension in the rope, For equivalent quality, and , These refer to the quality of customer satellites and service robots, respectively. If we neglect the effect of the tension of the rope on the motion of the customer satellite in equation (1), then the customer satellite orbit is a standard Keplerian orbit, and the first and second equations of equation (1) can be further simplified: （2） In the formula, , and These are the semi-drillion, semi-major axis, and eccentricity of the customer satellite's orbit, respectively. ; Considering the length of the rope It is much smaller than the geocentric distance of the customer's satellite orbit. Therefore, they have an approximate relationship: , Therefore, equations 3 and 4 of equation (1) can be simplified to: （3） In the motion control of the tethered system, the focus is on the relative motion of the two spacecraft. Equation (3) is transformed into a dimensionless equation, and the transformation formula is as follows: ， In the formula This represents the maximum rope length, which is the initial or final rope length when the customer's satellite recovers or releases the service robot; therefore, it always has... The equation (3) is true, and the dimensionless equation corresponding to it is as follows: （4） In the formula, the apostrophe indicates the angle relative to the true anterior point. The derivative of .

2. The method for controlling the tethering of an on-orbit service robot based on tethered docking as described in claim 1, characterized in that: The customer satellite orbital eccentricity It has a direct impact on the form of the system equations, and will further affect the system's control performance. exist When the customer satellite's orbit is circular, the system is autonomous and has an equilibrium point. Substituting into equation (4), we get: （5） exist When the customer satellite orbit is elliptical, the system is non-autonomous and there is no equilibrium point. exist At that time, the pitch angle can be controlled to remain at Unchanged, will , , Substituting into equation (4) part 2, we obtain the rope length. Equation of change: （6）。 3. The method for controlling the tethering of an on-orbit service robot based on tethered docking as described in claim 1, characterized in that: When the rate of change of the pitch angle with respect to the true anomaly angle is the same, the higher the orbital altitude of the customer satellite, the smaller the rate of change of the pitch angle over time. It can be seen that for the same process of change Due to area rate constant, The rate of change of pitch angle is the sum of orbital altitude and Earth's radius. With height or The decrease due to the increase of the orbital period reflects that when the system is running in high orbit, as the orbital period becomes longer, the relative motion process of the system tends to slow down, and the time required to reach the same motion state is longer. When the relative motion state changes with the true anomaly angle for the same reason, the higher the orbital altitude of the customer satellite, the smaller the tether tension required. It can be seen that for the same input tether tension Depending on the orbital height or Increases and decreases.

4. The method for controlling the tethering of an on-orbit service robot based on tethered docking as described in claim 1, characterized in that: When the robot rapidly retracts the rope, the angular velocity of the orbital satellite is no higher than [a certain value]. , Compared to It can be ignored, so the second equation of equation (3) can be approximated as: （7） Multiply both sides of the equation by Furthermore, we can obtain: （8） Therefore Represented as: （9） In the formula, the subscript 0 indicates the corresponding initial value.

5. The method for controlling the tethering of an on-orbit service robot based on tethered docking as described in claim 1, characterized in that: When the customer satellite moves in a circular orbit, ,get Furthermore, we can conclude that: （10） The points can be obtained as follows: （11） In the formula, This represents the pitch angle at the initial moment. When the rope length remains constant, the change in the pitch angle is related to the initial state.

6. The method for controlling the tethering of an on-orbit service robot based on tethered docking as described in claim 1, characterized in that: The constraints considered in the design of the smooth rope take-up control law are that the rate of change of rope length and the rate of change of pitch angle are close to zero, so as to keep the rope length and pitch angle basically constant, expressed as: （12） In the formula, Indicates the designed terminal rope length, subscript The value at the end. and These are the upper and lower limits of the feasible pitch angle, respectively.

7. The method for controlling the tethering of an on-orbit service robot based on tethered docking as described in claim 1, characterized in that: In the stable rope retrieval control law with a constant pitch angle constraint, let , As a set value, it is obtained from equation (6). Integrating, we get: （13） Assuming the initial true anterior angle, corresponding to the start of rope winding, we further obtain: （14） Substituting equation (14) into the first equation of equation (4), we get: （15） This smooth rope retrieval control law can achieve rope retrieval control while maintaining a constant pitch angle, thus enabling rope retrieval... from Change to The process, according to the equation, requires solving... ,get The range of values ​​for is: （16） A rope can only generate tension, not thrust, therefore the control input... Must meet According to equation (15), we get: （17） Equation (17) needs to be applied to any Established, Let be the true anterior angle at the end, corresponding to the end of the rope pull, which can be obtained from equation (17): （18） For convenience, construct a bivariate function. : （19） To ensure that inequality (17) holds, The following conditions must be met: （20） From the equation As can be seen from this, a bivariate function Compared to The period is ,and The periods of the value ranges are the same.

8. The method for controlling the tethering of an on-orbit service robot based on tethered docking as described in claim 1, characterized in that: The numerical simulation verifies the smooth rope take-up control law, using the PD control law as a comparison. The PD control law is as follows: （21） In the formula, , .

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