Method for modeling and stability analysis of tethered asteroid exploration system

Abstract

The application discloses a kind of tethered asteroid exploration system's dynamics modeling and stability analysis method, it is related to deep space exploration technical field.The method described in the application is proposed to a tethered probe operating in the gravitational field near irregular asteroid, including S1 based on multi-flexible body method to the tethered asteroid exploration system discretization, S2 the dynamics equation of multi-flexible body system is established using the second Lagrange equation of dissipative system, S3 stability analysis is carried out based on phase plane method and the like steps.The method described in the application can be used to analyze the dynamic response of tethered probe under initial conditions, accurately obtain the feasible region of tethered probe stable operation, and can accurately analyze the influence of tether mass, elasticity and tension on the stability of probe tethering, so as to provide theoretical support for the engineering implementation of tethered asteroid exploration system.

Description

Technical Field

[0001] This invention relates to the field of deep space exploration technology, specifically to a method for dynamic modeling and stability analysis of a tethered asteroid exploration system. Background Technology

[0002] Asteroid tethered exploration is a novel approach to asteroid observation. The probe flies alongside the asteroid and is connected via a flexible tether, similar to a tethered satellite. This method allows for long-term asteroid observation and offers advantages such as topological stability, favorable observation conditions, and no fuel consumption. It provides a highly feasible approach for asteroid surface sampling, probe-assisted landing, robotic landing, and attachment, making it a promising in-situ asteroid exploration method.

[0003] Tethered asteroid exploration systems have broad application prospects. Once connected to an asteroid, the tether allows for close-range observation and can also be used for orbital maneuvers or de-rotation of large targets like asteroids. Furthermore, tethered asteroid exploration systems can be used to assist landing. Due to the weak gravitational field of asteroids, asteroid probes are highly susceptible to bounce upon landing. A method such as first connecting the probe to the asteroid with a tether, then retrieving the tether, allows the probe to approach the asteroid and land, as seen with the Rosetta comet probe.

[0004] In a tethered system consisting of a probe and an asteroid, the gravitational pull of the asteroid on the probe cannot be ignored. Furthermore, asteroids generally possess a certain degree of spin, which significantly impacts the probe's motion and the dynamics of the tether. In studies of the dynamics and stability of tethered asteroid systems, the tether is often simplified to a massless rigid rod, neglecting the influence of the asteroid's gravitational field on the tether, thus affecting the dynamic response analysis and stability of the tethered system. Moreover, in actual probes tethering asteroids, the tether length can reach several kilometers, and its mass should not be neglected. Therefore, conducting multi-flexible body dynamics modeling and stability analysis of tethered asteroid probe systems is of great significance and value for asteroid approach detection and sampling missions. Summary of the Invention

[0005] The purpose of this invention is to provide a dynamic modeling and stability analysis method for a tethered probe operating in a gravitational field near an irregular asteroid. This method is used to analyze the dynamic response of the probe under initial conditions, accurately obtain the feasible region for stable operation of the probe, and precisely analyze the influence of tether mass, elasticity, and tension on the tethered stability of the probe, thus providing theoretical support for the engineering realization of a tethered asteroid exploration system.

[0006] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0007] A method for dynamic modeling and stability analysis of a tethered asteroid exploration system, comprising the following steps:

[0008] S1, Discretization of the tethered asteroid detection system based on the multi-flexible-body method;

[0009] The tethered asteroid detection system consists of an asteroid, a tethered asteroid detector, and a flexible tether.

[0010] Based on the system composition of the tethered asteroid exploration system, a coordinate system is established to describe the motion of the tethered asteroid probe and the flexible tether. The probe-tether system is discretized into a multi-flexible-body model, and the generalized coordinates used to describe the motion of the probe are analyzed.

[0011] S2, The dynamic equations of a multi-flexible body system are established based on the second kind of Lagrange equations for dissipative systems;

[0012] The motion of the detector is described using the exterior angle β as a generalized coordinate, and the tether length l is used as the coordinate. i (i = 1, 2, ..., n) is used as a generalized coordinate to describe the motion of the tether node i, where i = 1, 2, ..., n represents the tether node number and n is the total number of tether nodes;

[0013] The dynamic equations for the motion of the tethered asteroid probe are as follows:

[0014]

[0015] In the formula, m k Let l represent the mass of node k in the tether. i Let i represent the length of the i-th segment of the rope. i This represents the rate of change of the length of the i-th segment of the tether, with σ2 and σ3 as intermediate variables, expressed as σ2 = R. lx ωcos(λ+β)+R ly ωsin(λ+β) and Where β represents the exterior angle, Represents the rate of change of the exterior angle. R represents the angular acceleration at the exterior angle, ω represents the asteroid's rotational angular velocity, λ represents the latitude of the anchor point, and R lx and R ly These represent the distances from the asteroid's center of mass to the anchor point in the x and y directions, respectively. and These represent the components of the gravitational acceleration of node k caused by the asteroid in the x and y directions, respectively.

[0016] The dynamic equation of the node in the flexible tether is:

[0017]

[0018] In the formula, K represents the equivalent stiffness of the tether, and c dj Let l represent the equivalent damping coefficient of the j-th segment of the tether. j0 Let σj represent the initial length of the j-th tether segment, and σ1 be an intermediate variable, expressed as σ1 = R. lx ωsin(λ+β)-R ly ωcos(λ+β), The derivative represents the rate of change of the length of the i-th segment of the tether;

[0019] The dynamic equations of the probe's motion and the dynamic equations of the nodes in the tether together constitute the multi-flexible body dynamic equations of the tethered asteroid exploration system.

[0020] S3, Stability analysis of the system based on the phase plane method;

[0021] Analysis of the exterior angle β and the rate of change of the exterior angle based on the phase plane diagram. The value of affects the stability of the tethered system. The stability region of the detector is the condition that the exterior angle and the rate of change of the exterior angle should satisfy under the initial conditions.

[0022] First, the number of segments and material of the flexible tether are selected to obtain the equivalent stiffness K of the tether, the total number of nodes n of the tether, and the mass of each node of the tether. These parameters are then substituted into the multi-body dynamics equations of the tethered asteroid exploration system. Next, the order of the multi-body dynamics equations of the tethered asteroid exploration system is reduced to obtain a reduced set of first-order differential equations. Finally, given the exterior angle β and the rate of change of the exterior angle... The range of values ​​for is defined by the exterior angle β and the rate of change of the exterior angle. The values ​​within the range of the given values ​​are used as initial values, and simulation calculations are performed based on the reduced first-order differential equations to obtain the dynamic response curves of the exterior angle and the rate of change of the exterior angle under a certain initial value. The dynamic response curves are then plotted on the xy plane to obtain the phase trajectory. Finally, the system stability under a certain initial value is analyzed based on the phase trajectory to obtain the feasible region that allows the detector to remain stable.

[0023] Furthermore, the detector's three-dimensional coordinates in the centroid-fixed coordinate system are determined by the anchor point's longitude. The five independent coordinates of the anchor point latitude λ, the in-plane angle α, the out-plane angle β, and the tether length l are completely determined.

[0024] Furthermore, in step S1, the flexible tether is discretized into n segments and forms n nodes, and the mass of each segment of the tether is evenly distributed on each node. The connection model between the nodes adopts a massless spring and damper model, and each node is subjected to elastic force and damping force.

[0025] Furthermore, in step S3, it is assumed that the tether is inextensible, and the rate of change of length of each segment of the tether and the derivative of the rate of change of length of each segment of the tether are both zero.

[0026] Furthermore, in step S3, the fourth-order Runge-Kutta method is used to perform simulation calculations on the reduced-order first-order differential equation system.

[0027] Compared with the prior art, the beneficial effects achieved by the present invention are as follows:

[0028] 1) The dynamic modeling and stability analysis method of the tethered asteroid exploration system proposed in this invention can accurately analyze the influence of factors such as tether mass, length, elasticity and tension on the tether stability of the probe, including the influence of these factors on the dynamic characteristics of the probe and the position of the gravitational equilibrium point.

[0029] 2) This invention is of great significance for applications such as tethered asteroid approach and sampling missions. Compared with the traditional assumption of a massless rigid rod, the tether is regarded as an elastic cable with mass and damping, which can only withstand tension. Under this assumption, the multi-flexible body dynamics model of the probe system can accurately reflect the dynamic characteristics of the system and accurately obtain the feasible domain of stable motion of the probe, laying a theoretical foundation for the engineering implementation of tethered asteroid exploration missions. Attached Figure Description

[0030] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the structures shown in these drawings without creative effort.

[0031] Figure 1 This is a flowchart of the dynamic modeling and stability analysis method described in a specific embodiment of the present invention.

[0032] Figure 2 This is a schematic diagram of the composition and coordinate system of a tethered asteroid exploration system.

[0033] Figure 3 This is a schematic diagram of the multi-flexible body dynamics model of the tethered asteroid exploration system.

[0034] Figure 4 This is a diagram showing the distribution of the gravitational equilibrium points of the Eros asteroid.

[0035] Figure 5 It is an exterior angle phase plane diagram under the assumption of a mass and a damped elastic rope;

[0036] Figure 6 It is an out-of-plane phase plane diagram under the assumption of a massless rigid rod.

[0037] Legend: 1. Asteroid; 2. Probe; 3. Tether; 4. Anchor point;

[0038] L1 – First gravitational equilibrium point; L2 – Second gravitational equilibrium point; L3 – Third gravitational equilibrium point; L4 – Fourth gravitational equilibrium point. Detailed Implementation

[0039] 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 a part of the embodiments of the present invention, and not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0040] For clarity, the physical meanings of the symbols used in this specification are shown in Table 1 below.

[0041] Table 1. Symbols and their meanings

[0042]

[0043] The coordinate system used in this invention specification is as follows:

[0044] O a -X a Y a Z a Coordinate system – a centroid-fixed coordinate system, origin O a Located at the asteroid's center of mass, coordinate axis O a Z a Along the principal axis of the asteroid's maximum inertia. The equatorial plane is perpendicular to coordinate axis O. a Z a , passing through the center of mass O a The plane. Coordinate axis O a X a Located within the equatorial plane of the asteroid, and along the direction of the asteroid's principal axis of minimum inertia, coordinate axis O a Y a Together with the other two axes, this forms a right-handed coordinate system. The fixed coordinate system moves with the asteroid's spin. The equatorial plane is the X-axis. a O a Y a Face, at this time Z a =0.

[0045] O l -X l Y l Z l Coordinate system—anchor point coordinate system, fixed to coordinate system O by the asteroid's center of mass. a -X a Ya Z a Around O a Z a Rotate the axis by latitude λ, then around O a X' a Shaft (O after rotation) a X a (Axis) Rotation Longitude get.

[0046] O b -X b Y b Z b Coordinate system—the tethered coordinate system, derived from the anchor point coordinate system O. l -X l Y l Z l Around O l X l Rotate the axis of the plane by an interior angle α, and then around O l Z' l Shaft (O after rotation) l Z l Axis, i.e., O l Z b (Axis) Rotational out-of-plane angle β.

[0047] For tethered asteroid exploration systems, this invention studies the dynamics and stability of tethered probes near irregular asteroids, and proposes a multi-flexible-body dynamics modeling and stability analysis method for tethered asteroid exploration systems. The steps include:

[0048] S1, Discretization of the tethered asteroid detection system based on the multi-flexible-body method;

[0049] Based on the system composition of the tethered asteroid probe system, a coordinate system describing the motion of the probe and the tether is established. The probe-tether system is considered a multi-flexible-body system, and the multi-flexibility method is used to discretize the elastic tether, analyzing the generalized coordinates used to describe the probe's motion. Multi-flexible systems are typically composed of multiple flexible (and possibly rigid) objects; the multi-flexible-body method connects the flexible objects in a multi-flexible system through kinematic constraints and force elements (such as springs and dampers), and then uses methods such as Newtonian mechanics or analytical mechanics to establish the system's dynamic equations.

[0050] The tethered asteroid exploration system consists of three parts: the asteroid, the tethered asteroid probe (referred to as the probe), and the flexible tether (referred to as the tether). The asteroid revolves in an elliptical orbit around the Sun and spins about its principal axis of maximum inertia. The probe is connected to the asteroid via the tether and follows the asteroid's spin, achieving hovering and companionship. One end of the tether is anchored to the asteroid's surface at a point called the anchor point, and the other end is connected to the probe. The tether can be taut or slack.

[0051] To describe the positions of the detector and the tether, three coordinate systems are established: the center-of-mass fixed coordinate system O a X a Y a Z a Anchor point coordinate system and tethered coordinate system

[0052] The detector-tether system is discretized based on the multi-flexible-body method, with the tether divided into n segments, each of length l. i (i = 1, 2, ..., n). The mass of the tethering rope is uniformly distributed at each node, and the mass of each node is represented by m. k (k = 1, 2, ..., n) represents the sequence. When k = n, m n =m a The detector mass is represented by a massless spring and damper model. The nodes are subjected to elastic force (i.e., tension) and damping force.

[0053] For the three-dimensional coordinates of the detector in the centroid-fixed coordinate system, the longitude of the anchor point is... The five independent coordinates of the anchor point latitude λ, the in-plane angle α, the out-plane angle β, and the tether length l are completely determined.

[0054] S2, Establish the dynamic equations of the multi-flexible body system;

[0055] Considering the mass, elasticity, and damping of the tether, the Lagrangian function of the detector-tether system is written, and the governing equations of the multi-flexible body system are obtained based on the second kind of Lagrangian equation for dissipative systems.

[0056] The second type of Lagrange equations based on dissipative systems are used to describe non-conservative systems involving viscous or frictional dissipation. These dissipative systems are typically influenced by factors such as velocity and surface properties. In the Lagrange equations, dissipative drag is treated as a generalized drag, which depends on velocity, position, or time. The Rayleigh dissipation function is a convenient method for incorporating linear velocity-dependent dissipative forces into Lagrangian and Hamiltonian mechanics. For details, see Douglas Cline's "VARIATIONAL PRINCIPLES in CLASSICAL MECHANICS".

[0057] The expression for the second kind of Lagrange equation based on dissipative systems is:

[0058]

[0059] In the formula, L = TU is the Lagrangian function, which can be constructed using kinetic and potential energy; q j For generalized coordinates; Q j For generalized forces, the virtual work method can be used to solve for them; Ψ is the Rayleigh dissipation function.

[0060] The probe is located within the asteroid's gravitational field and is also subjected to the tension of the tether, with damping within the tether also considered. Damping force is a dissipative force, so it is solved using the second kind of Lagrange equations for dissipative systems.

[0061] Choose the generalized coordinates as β and l1, l2, ..., l n Since the tether has mass, its kinetic energy needs to be considered. The nodes m on the tether... k (k = 1, 2, ..., n-1) and detector node m k The coordinates of (k=n) can be represented as

[0062]

[0063] In the formula, R l R is the distance from the asteroid's center of mass to the anchor point. lx R ly Its weight.

[0064] The speed of each node is

[0065]

[0066] In the formula, ω a =(0,0,ω) T Let be the asteroid's spin angular velocity, in the coordinate system O fixed at its center of mass. a X a Y a Z a The middle represents; R l =(R lx ,R ly ,R lz ) T In the centroid-fixed coordinate system O a X a Y a Z a The middle represents the coordinate system O fixed by the center of mass. a X a Y a Z a To the tethered coordinate system The coordinate transformation matrix is ​​A ba;l i =(0,l i ,0) T ;ω b For the tethered coordinate system angular velocity.

[0067] The total kinetic energy of the tether and the probe is

[0068]

[0069] In the formula,

[0070] σ1=R lx ωsin(λ+β)-R ly ωcos(λ+β)

[0071] σ2=R lx ωcos(λ+β)+R ly ωsin(λ+β)

[0072]

[0073] Since the tension in the tethered rope is a force, the potential energy of the system can be written as:

[0074]

[0075] In the formula, U gk U is the gravitational potential energy per unit mass at node k. ek Let be the elastic potential energy of the k-th segment of the rope.

[0076] The elastic potential energy of the rope exists only when the rope is under tension. The elastic potential energy U of the k-th segment of the rope is... ek It can be represented as

[0077]

[0078] In the formula, l k0 This represents the nominal length of the k-th segment of the rope, i.e., the length when the tension is 0. For equivalent stiffness, if the tether is divided into n equal segments, then the equivalent stiffness of each segment is the same. E is the elastic modulus of the tether, and A is the cross-sectional area of ​​the tether.

[0079] Similarly, the damping force only exists when the rope is under tension, and the Rayleigh dissipation function Ψ can be expressed as:

[0080]

[0081] In the formula, c dk =2m k ζω nk ζ is the equivalent damping coefficient of the k-th tether segment; ζ is the damping ratio. Let be the natural frequency of the k-th segment of the rope.

[0082] The generalized coordinate β describes the motion of the detector. The dynamic equation of the detector's motion can be obtained from the Lagrange equations:

[0083]

[0084] Generalized coordinates l j (j=1,2,…,n) describes the motion of the tethered nodes. The dynamic equations of the nodes in the tether, obtained from the Lagrange equations, are as follows:

[0085]

[0086] The multibody dynamics equations of the tethered asteroid exploration system established by the Lagrange method are Equations (9) and (10).

[0087] S3, Stability analysis of the system based on the phase plane method.

[0088] When calculating the stability region, it is assumed that the tether is inextensible, and in equations (9) and (10) and If both are 0, then only the exterior angle β is an unknown in the control equations of the tethered detection system. The exterior angle β and its rate of change are analyzed based on the phase plane diagram. The influence of the value of on the stability of the tethered system.

[0089] The phase plane method is a method for analyzing the stability of first-order / second-order and nonlinear systems. The phase plane is a plane composed of system variables and their derivatives used to describe the system state. The phase trajectory is the trajectory drawn on the phase plane as the system variables and their derivatives change over time. The stability of the system can be determined based on the phase trajectory. For details, see Paul Dawkins' "Differential Equations".

[0090] Ideally, when the tethered detector is in a hovering state, the exterior angle β and the rate of change of the exterior angle... Both are 0, meaning the probe rotates with the asteroid's spin and remains stationary relative to the asteroid. However, in reality, under initial conditions, the exterior angle β and the rate of change of the exterior angle are... There are certain deviations in all cases, namely, the probe is not directly above the anchor point, and its velocity relative to the asteroid is not zero. In this case, the probe will either oscillate stably or fall onto the asteroid's surface under the combined influence of the asteroid's gravity and the tension of the tether. The probe's stability region is defined by the conditions that the exterior angle and the rate of change of the exterior angle must satisfy under the initial conditions.

[0091] Specifically, firstly, the number of segments and material of the flexible tether are selected to obtain the equivalent stiffness K of the tether, the total number of nodes n of the tether, and the mass of each node of the tether. These parameters are then substituted as known values ​​into the multi-body dynamics equation of the tethered asteroid exploration system. Next, the multi-body dynamics equation of the tethered asteroid exploration system is reduced in order to obtain a reduced set of first-order differential equations. That is, the n+1 ordinary differential equations in the dynamics equations represented by equations (9) and (10) are reduced to a set of 2n+2 first-order ordinary differential equations. Finally, the exterior angle β and the rate of change of the exterior angle are given. The range of values ​​for is defined by the exterior angle β and the rate of change of the exterior angle. The values ​​within the range of the given values ​​are used as initial values, and simulation calculations are performed based on the reduced first-order differential equations to obtain the dynamic response of the exterior angle and the rate of change of the exterior angle under a certain initial value. These responses are then plotted on the xy plane to obtain the phase trajectory. Finally, the system stability under a certain initial value is analyzed based on the phase trajectory to obtain the feasible region that allows the detector to remain stable.

[0092] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings.

[0093] Figure 1 A flowchart of the present invention is shown.

[0094] Figure 2 This is a schematic diagram of the components and coordinate system of a tethered asteroid probe system. The asteroid's spin angular velocity is constant, and its mass is much greater than the probe's mass, so its motion is unaffected by the probe. The probe is located within the asteroid's sphere of influence and is only affected by the asteroid's gravity, ignoring the influence of other large celestial bodies. The anchor point is located within the equatorial plane, and the probe only moves within the equatorial plane, with no velocity component perpendicular to the equatorial plane.

[0095] Figure 3 This is a multi-flexible-body model of a tethered asteroid probe system. The probe is considered a point mass, and the tether is discretized into n segments with mass concentrated at the nodes. The nodes are connected by massless nonlinear springs and dampers, representing the elastic and damping characteristics of the tether, respectively. The nonlinear springs can only withstand tension, not compression.

[0096] This embodiment uses a probe tethered near the asteroid 433Eros as the subject of numerical simulation. The probe's mass is set to 100 kg. The asteroid 433Eros is elongated in shape, with dimensions of approximately 33 × 13 × 13 km and an average diameter of approximately 16.8 km. (See...) Figure 4 By utilizing the relationship between the gravitational acceleration near the asteroid and its rotational velocity, the location of the gravitational field equilibrium point near the asteroid can be determined. The asteroid 433Eros has five equilibrium points, one of which is located inside the asteroid and is therefore disregarded. The gravitational field equilibrium point is located in the center-of-mass fixed coordinate system O. a Xa Y a Z a The coordinates are shown in Table 1 and Figure 4 The anchor point for the tether should ideally be located at a high elevation on the asteroid. Figure 4 The surface of the asteroid corresponding to the second equilibrium point L2 in the gravitational field was selected as the anchor point. The distance between L2 and the surface of the asteroid is 3.52 km.

[0097] Table 1. Coordinates of the gravitational equilibrium point of the asteroid Eros

[0098]

[0099] The tether is made of nylon PA6 material, and its physical and chemical parameters are shown in Table 2. Based on its tensile strength and tether diameter, the tether can withstand a tensile force of up to 47N without failure.

[0100] Table 2 Physical and chemical parameters of the tethering rope

[0101]

[0102] To establish a multi-flexible-body model of the tethered detector system, the tether was divided into three segments, resulting in a set of governing equations for the detector-tether motion. The set of equations initially consisted of four second-order ordinary differential equations, which were reduced to eight first-order ordinary differential equations.

[0103] Simulations were conducted with tether lengths of l = 3.6 km, 5 km, and 10 km to analyze the stable region of the detector's outer surface angle, as shown in Table 3. As the tether length increases, the initial range of stable oscillations of the tethered system expands.

[0104] Table 3 Stability range for different rope lengths

[0105]

[0106] When the rope length l = 5 km, the detector height is higher than the gravitational field equilibrium point. This is achieved when the initial exterior angle is within ±57° and the rate of change of the exterior angle is -2 × 10⁻⁶. -4 rad / s ~ 3×10 -4 When the speed is within rad / s (equivalent to a detector velocity error of less than 1 m / s), the detector can remain stable. See Figure 5 Compared to the assumption of a massless rigid rod on the tether, the initial value range for stable oscillation of the tethered system is significantly reduced, see... Figure 6 .

[0107] The above description is merely a preferred embodiment of the present invention. The scope of protection of the present invention is not limited to the above embodiments. All technical solutions falling within the scope of the present invention's concept are within the scope of protection of the present invention. It should be noted that for those skilled in the art, any improvements and modifications made without departing from the principles of the present invention should also be considered within the scope of protection of the present invention.

Claims

1. A method for dynamic modeling and stability analysis of a tethered asteroid exploration system, characterized in that, The method includes the following steps: S1, Discretization of the tethered asteroid detection system based on the multi-flexible-body method; The tethered asteroid detection system consists of an asteroid, a tethered asteroid detector, and a flexible tether. Based on the system composition of the tethered asteroid exploration system, a coordinate system is established to describe the motion of the tethered asteroid probe and the flexible tether. The probe-tether system is discretized into a multi-flexible-body model, and the generalized coordinates used to describe the motion of the probe are analyzed. S2, The dynamic equations of a multi-flexible body system are established based on the second kind of Lagrange equations for dissipative systems; The motion of the detector is described using the exterior angle β as a generalized coordinate, and the tether length l is used as the coordinate. i As a generalized coordinate to describe the motion of the tether node i, where i = 1, 2, ..., n represents the tether node number and n is the total number of tether nodes; The dynamic equations for the motion of the tethered asteroid probe are as follows: In the formula, m k Let l represent the mass of node k in the tether. i This represents the length of the i-th segment of the rope. This represents the rate of change of the length of the i-th segment of the tether, with σ2 and σ3 as intermediate variables, expressed as σ2 = R. lx ωcos(λ+β)+R ly ωsin(λ+β) and Where β represents the exterior angle, Represents the rate of change of the exterior angle. R represents the angular acceleration at the exterior angle, ω represents the asteroid's rotational angular velocity, λ represents the latitude of the anchor point, and R lx and R ly These represent the distances from the asteroid's center of mass to the anchor point in the x and y directions, respectively. and These represent the components of the gravitational acceleration of the asteroid acting on node k in the x and y directions, respectively. The dynamic equations of the nodes in a flexible tether are: In the formula, K represents the equivalent stiffness of the tether, and c dj Let l represent the equivalent damping coefficient of the j-th segment of the tether. j0 Let σj represent the initial length of the j-th tether segment, and σ1 be an intermediate variable, expressed as σ1 = R. lx ωsin(λ+β)-R ly ωcos(λ+β), The derivative represents the rate of change of the length of the i-th segment of the tether; The dynamic equations of the probe's motion and the dynamic equations of the nodes in the tether together constitute the multi-flexible body dynamic equations of the tethered asteroid exploration system. S3, Stability analysis of the system based on the phase plane method; Analysis of the exterior angle β and the rate of change of the exterior angle based on the phase plane diagram. The value of affects the stability of the tethered system. The stability region of the detector is the condition that the exterior angle and the rate of change of the exterior angle should satisfy under the initial conditions. First, the number of segments and material of the flexible tether are selected to obtain the equivalent stiffness K of the tether, the total number of nodes n of the tether, and the mass of each node of the tether. These parameters are then substituted into the multi-body dynamics equations of the tethered asteroid exploration system. Next, the order of the multi-body dynamics equations of the tethered asteroid exploration system is reduced to obtain a reduced set of first-order differential equations. Finally, given the exterior angle β and the rate of change of the exterior angle... The range of values ​​for is defined by the exterior angle β and the rate of change of the exterior angle. The values ​​within the range of the given values ​​are used as initial values, and simulation calculations are performed based on the reduced first-order differential equations to obtain the dynamic response curves of the exterior angle and the rate of change of the exterior angle under a certain initial value. The dynamic response curves are then plotted on the xy plane to obtain the phase trajectory. Finally, the system stability under a certain initial value is analyzed based on the phase trajectory to obtain the feasible region that allows the detector to remain stable.

2. The method for dynamic modeling and stability analysis of a tethered asteroid exploration system according to claim 1, characterized in that, The three-dimensional coordinates of the detector in the centroid-fixed coordinate system are determined by the longitude of the anchor point. The anchor point latitude λ, in-plane angle α, out-plane angle β, and tether length l are determined by five independent coordinates.

3. The method for dynamic modeling and stability analysis of a tethered asteroid exploration system according to claim 1, characterized in that, In step S1, the flexible tether is discretized into n segments and forms n nodes, and the mass of each tether segment is evenly distributed on each node. The connection model between nodes adopts a massless spring and damper model, and each node is subjected to elastic force and damping force.

4. The method for dynamic modeling and stability analysis of a tethered asteroid exploration system according to claim 1, characterized in that, In step S3, when performing system stability analysis based on the phase plane method, the configuration is such that the tether is inextensible, and the rate of change of length of each tether segment and the derivative of the rate of change of length of each tether segment are both zero.

5. The method for dynamic modeling and stability analysis of a tethered asteroid exploration system according to any one of claims 1 to 4, characterized in that, In step S3, the fourth-order Runge-Kutta method is used to perform simulation calculations on the reduced-order first-order differential equation system.

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