Trajectory tracking control method for complex small celestial body's surface bouncing motion

Abstract

The present invention discloses a method for tracking and controlling the bouncing movement trajectory of a complex small celestial body, which belongs to the field of deep space exploration. The method is as follows: by introducing a pulse collision model to simulate the contact and collision process between a cube detector and the ground, the two contact states of sliding and sticking, as well as the velocity change of the contact point, are analyzed to establish the detector collision state differential equation; by introducing a stretching variable to simplify the numerical integration process, the collision state differential equation is solved to obtain the state parameters of the detector after the collision; by designing a sliding mode control law when the detector contacts the ground, the desired take-off state is tracked, and the velocity and angular velocity of the detector after the collision are compensated to obtain the corrected detector take-off state; the corrected state parameters are used as the initial state of the next bounce to realize the continuous bouncing movement of the detector on the surface of the complex small celestial body, thereby achieving the detector's accurate tracking of the nominal trajectory.

Description

Technical Field

[0001] The present invention relates to a method for achieving precise tracking of a nominal trajectory by controlling a detector during a collision to change the next take-off state during continuous bouncing movement on the surface of a complex small celestial body, and belongs to the field of deep space exploration. Background Art

[0002] Small-body exploration is becoming a hot topic in deep space exploration. Among these missions, exploration methods primarily include flybys, flybys, attachments, surface movements, and sample return. Mobile exploration on the surfaces of small bodies is a key approach. Unlike planetary surfaces, the gravity of small bodies is extremely low, and the surface environment is complex and dynamic. This makes it extremely difficult to navigate and control traditional wheeled probes. Currently, a surface mobility method known as bouncing has emerged. Bouncing offers advantages such as the ability to traverse obstacles and travel long distances in a short period of time. However, the probe's attitude is uncertain during descent, and collisions with small bodies result in energy loss, making long-distance movement and accurate tracking of the nominal trajectory impossible. Therefore, it is necessary to study the state changes of a probe during a collision. By applying control during the collision process to alter the next takeoff state and compensate for energy loss, the probe can accurately track its nominal trajectory while continuously bouncing on the complex surface of a small body. The cube-shaped detector will undergo complex changes in its posture during the bouncing movement, resulting in an uncertain state of the detector after contact with the ground; at the same time, the collision with the ground will cause energy loss, and it will not be able to reach the target area according to the designed nominal trajectory.

[0003] The existing method for tracking and controlling the trajectory of complex small celestial bodies bouncing on the surface does not consider the influence of the probe's shape and posture on the bouncing movement process, and treats the probe as a particle model; it only considers the control of the probe during flight and ignores the influence of the collision process, resulting in the cumulative increase of the probe trajectory tracking error; the traditional particle instantaneous collision model uses a simplified collision model and central collision process, and does not consider the influence of the complex contact surface environment on the collision process, and is unable to handle the collision problem of the cube probe.

[0004] Among the complex small celestial body surface bouncing motion trajectory tracking control methods that have been developed, the prior art [1] (see Bellerose J, Scheeres D J. Dynamics and Control for Surface Exploration of Small Bodies [C]. AIAA / AAS 2008 Astrodynamics Specialist Conference, Honolulu, Hawaii, August 18-21, 2008: AIAA 2008-6251.) designed a guidance algorithm based on a parabolic motion model to solve the problem of small celestial body surface bouncing motion trajectory tracking. However, this method is based on a simplified motion model, so the control accuracy is poor.

[0005] In the prior art [2] (see Liu Yanjie. Research on trajectory optimization and guidance methods for small celestial body attachment detection [D]. Beijing Institute of Technology, 2017), a second-order sliding surface is designed to track the first-order sliding surface, and the analytical expression of the guidance acceleration is derived using the second-order sliding surface. The obtained guidance acceleration is used to achieve the precise transfer of the detector in a single bounce. However, this method treats the detector as a simplified particle model and does not consider the impact of attitude changes and energy loss during the collision process on the motion process. In addition, using multiple pulses to correct the trajectory consumes a large amount of fuel and electricity. Summary of the Invention

[0006] In response to the above technical problems in the background technology, the main purpose of the present invention is to provide a method for tracking and controlling the bouncing movement trajectory of the surface of a complex small celestial body. By introducing a pulse collision model to simulate the contact and collision process of the cube detector with the ground, the two contact states of sliding and sticking and the velocity change of the contact point are analyzed, and the detector collision state differential equation is established; by introducing a stretching variable to simplify the numerical integration process, the collision state differential equation is solved to obtain the state parameters of the detector after the collision; by designing a sliding mode control law when the detector contacts the ground, the expected take-off state is tracked, and then the velocity and angular velocity of the detector after the collision are compensated to obtain the corrected detector take-off state; the corrected state parameters are used as the initial state of the next bounce to realize the continuous bouncing movement of the detector on the surface of a complex small celestial body, thereby realizing the accurate tracking of the nominal trajectory of the detector.

[0007] The present invention is achieved through the following technical solutions.

[0008] The present invention discloses a method for tracking and controlling the trajectory of complex small celestial body surface bounces. This method addresses the problem of tracking the continuous bounce trajectory of a probe. The dynamic equations for the probe after takeoff are established in both the fixed coordinate system of the small celestial body and the coordinate system of the small celestial body surface. By analyzing the initial takeoff process, the optimal takeoff angle, takeoff speed, and takeoff angular velocity are calculated. Based on the nominal movement trajectory after path planning, the displacement required for each bounce, as well as the magnitude and direction of the velocity required at takeoff, are obtained. Using the calculated takeoff angle, heading angle, takeoff speed, and takeoff angular velocity as the initial dynamic state parameters, the trajectory of the probe's first bounce is obtained. An impulse collision model is established to analyze the force conditions at the contact point where the probe collides with the ground, and the state equation for the entire probe is derived. The normal impulse obtained after the probe collision is substituted as the independent variable into the collision state equation. A scaling variable is defined to divide the collision process into a sliding process and a sticking process, and a switch is set to distinguish between the two processes. A sliding mode control law is designed to correct the velocity and angular velocity of the probe during the collision process to compensate for the energy loss caused by the collision. The corrected state parameters are used as the initial state of the next bounce to realize the continuous bouncing movement of the probe on the surface of a complex small celestial body, thereby achieving the precise tracking of the probe to the nominal trajectory.

[0009] The present invention discloses a method for tracking and controlling the trajectory of a complex small celestial body bouncing on its surface, comprising the following steps:

[0010] Step 1: Establish the detector's fixed coordinate system O on the small celestial body. B -X B Y B Z B The dynamic and kinematic equations for the probe's hopping motion in the surface coordinate system O-XYZ are derived. Analysis of the kinematic equations derives the expected takeoff velocity, heading angle, and takeoff angle. The displacement sequence derived from the nominal trajectory derives the velocity and acceleration sequences for the probe's continuous hopping motion on the surface of a small celestial body.

[0011] Regarding the single bounce motion problem of the probe, in the fixed coordinate system of the small celestial body, the dynamic equation of the probe after take-off is expressed as

[0012]

[0013] Among them, r B 、v B are the position and velocity vector of the detector respectively, ω is the spin angular velocity of the small celestial body, and g is the gravitational acceleration vector of the small celestial body.

[0014] In the surface coordinate system, the dynamic equation of the detector is:

[0015]

[0016] Among them, r and v are the position and velocity vectors of the detector respectively, and ρ is the position vector of the origin of the surface coordinate system relative to the center of the small celestial body. is the matrix for transforming from the body coordinate system to the surface coordinate system.

[0017] In the known nominal bounce trajectory, the collision point of each bounce of the detector is R i , the corresponding velocity vector is v hi , the angular velocity vector is ω hi The displacement sequence of the nominal trajectory of the detector bouncing movement is R = [R0, R1, R2…] T , the velocity sequence is v h =[v h0 ,v h1 ,v h2 …] T And the angular velocity sequence is ω h =[ω h0 ,ω h1 ,ω h2 …] T .

[0018] In the surface coordinate system, the starting point of the detector's movement is defined as the origin of the coordinate system. The direction opposite to the line connecting the landing point of the first bounce and the starting point is defined as the positive direction of the X axis. The positive direction of the Z axis is the opposite direction of the component of the local gravity vector perpendicular to the XY plane. The Y axis conforms to the right-hand rule. The kinematic equation of the detector in the surface coordinate system is:

[0019]

[0020] Among them, v hi is the speed required for each jump, β i is the angle between the detector's initial take-off velocity direction and the XY plane, which is the detector's take-off angle. It is the angle between the component of the detector's initial take-off velocity in the XY plane and the X-axis, that is, the detector's heading angle.

[0021] Define the distance the detector moves in a single jump as R i , then the expected take-off speed of the detector each time it bounces is:

[0022]

[0023] The expected take-off speed v i Take-off angle β i Taking the derivative and setting it equal to zero gives the take-off angle corresponding to the minimum take-off speed:

[0024]

[0025] Given the expected take-off velocity v of the detector hi , heading angle and take-off angle β i In the case of , the center-of-mass velocity vector of the detector is:

[0026]

[0027] Assume that the contact point when the probe collides with the ground is P. When the probe rotates around point P and jumps, the center of mass velocity is v = ω × r P , where ω is the angular velocity vector of the detector, r P is the position vector of the collision point in the detector body coordinate system. Assume that the side length of the cube detector is 2l, and the detector is set to jump with the edge when it jumps, then ω z =0, the formula for the detector’s angular velocity is:

[0028]

[0029] The velocity vector and angular velocity vector required by the detector for each bounce are calculated using equations (4), (6), and (7).

[0030] Step 2: Establish the attitude dynamics model and attitude kinematics model of the detector in the body coordinate system, define the error angular velocity and error quaternion of the detector, solve the rotation matrix of the body coordinate system relative to the target coordinate system, and obtain the error attitude kinematics model and error attitude dynamics model.

[0031] The attitude dynamics model of the cube detector is:

[0032]

[0033] Where ω is the attitude angular velocity of the probe, J is the moment of inertia matrix of the probe, u is the control torque, and d is the external interference torque.

[0034] Assume that the quaternion of the detector body coordinate system relative to the inertial coordinate system is Angular velocity is ω=[ω x ω y ω z ] T , then the attitude kinematics of the detector is:

[0035]

[0036] Among them, q0 is the quaternion's standard part, q v is the vector part of the quaternion, and

[0037] The expected quaternion of the detector is q f , the desired angular velocity is ωf , define the error quaternion as q e ,have:

[0038]

[0039] Then the rotation matrix from the target coordinate system to the body coordinate system is:

[0040]

[0041] The error angular velocity of the body coordinate system relative to the target coordinate system is defined as:

[0042] ω e =ω-C e ω f (12)

[0043] Combining equations (9), (10), and (12), the attitude kinematic equation of the detector based on the error quaternion is:

[0044]

[0045]

[0046] Substituting Equation (12) into Equation (8), the attitude dynamics model of the detector based on the error angular velocity is obtained as follows:

[0047]

[0048] Step 3: The collision between the cube detector and the ground is a non-central collision. A velocity change model of the collision point is established, and an inverse inertia matrix is ​​defined to represent the influence of the force exerted by the ground on the contact point of the detector on the velocity of the contact point. This matrix is ​​used to establish the collision model in the subsequent step 4 and improve the accuracy of the detector collision model state update.

[0049] The point where the probe collides with the ground is P, and the velocity of the contact point is v P , the vector from the center of mass of the detector to the contact point is r OP , then the velocity of the contact point is:

[0050] v P =v+ω×r OP (16)

[0051] During the contact process, the detector will be subjected to the contact force F acting on the contact point. C and contact torque T C =r OP ×F C Taking the derivative of the velocity at the contact point, we get:

[0052]

[0053] in,

[0054]

[0055]

[0056]

[0057] Substituting equations (18), (19), (20) into equation (17), we can obtain:

[0058]

[0059] Define the inverse inertia matrix This matrix represents how the force applied to the contact point changes the velocity of the contact point and will be used to establish the contact model.

[0060] Step 4: Substitute the normal impulse as the independent variable into the probe dynamics equation constructed in Step 1, derive the relationship between the probe state variables and the normal impulse, and establish a pulse collision model for the probe on the surface of a complex asteroid. Divide the probe's collision state into a sliding state and a viscous state, derive the corresponding impulse state equations for each, and combine them with the velocity variation model at the collision point obtained in Step 3 to construct the probe's collision state equation. By introducing a stretching variable and decomposing the contact point velocity to simplify the numerical integration process, a differential equation for the probe's collision state on the surface of a complex asteroid is established. By solving the differential equation for the collision state, the probe's state parameters after the collision are obtained.

[0061] When the detector collides with the ground, the force F at the contact point C A pulse will be generated, that is, dP=F C t. The collision occurs in a very short time, and the position and posture of the detector will not change during the collision. During the collision, due to the normal contact force F n Strictly monotonically increasing, so the normal impulse it receives It is also monotonically increasing. Therefore, the normal impulse p is brought into the dynamic equation of the detector as an independent variable to obtain:

[0062]

[0063]

[0064] The contact point velocity change expression is:

[0065]

[0066] When the normal velocity v of the contact point Pz When ≥0, the collision process of the detector ends.

[0067] The movement of the probe after colliding with the ground may be subject to two conditions: sliding and sticking. When sliding occurs, the contact point will be subjected to the normal contact force impulse and the tangential friction force impulse. The tangential velocity of the contact point is defined to satisfy And v Px , v Py The angle Φ satisfies have to:

[0068] v Px =v PT cosΦand v Py =v PT sinΦ (25)

[0069] When v PT >0, the friction impulse satisfies dP f =μ·dP N , where μ is the coefficient of kinetic friction, then the derivative of the total impulse with respect to the normal impulse during sliding is:

[0070]

[0071] When v PT = 0, the friction impulse satisfies dP f =μ s ·dP N , where μ s is the static friction coefficient and μ s ≤μ, then Substituting into formula (24) we get:

[0072]

[0073] Right now,

[0074]

[0075] in,

[0076]

[0077]

[0078] At this time, the detector is in a viscous state, μ s and Φ s is a fixed value, there is,

[0079]

[0080] According to the above dynamic equation, the collision state of the detector is defined as:

[0081]

[0082] Among them, ε is used to represent the moment when the collision ends. When the detector is in the collision process, v PT will tend to zero, that is, v Px and v Py will tend to zero, and the direction of Φ will be impossible to determine, so we need to use v PT and Φ replace v Px and v Py .Right now,

[0083] X C =[v PT Φ v Pz P v ω ε] T (33)

[0084] and

[0085]

[0086]

[0087] Introduce the stretch variable τ, let

[0088]

[0089] Using the chain rule, for any variable a, the equation of motion with respect to dτ is expressed as:

[0090]

[0091] During the sliding of the detector, the PT When integrating, although v PT →0 can be achieved with a finite normal impulse, but it is difficult to converge numerically. Therefore, the integral of τ is used to replace the integral of p, and then τ→∞ replaces v PT →0. When the sticking state occurs, continue to integrate p. Define the switch variable η to control the switching between sliding and sticking states:

[0092]

[0093] Then the derivative expression of the collision state is:

[0094]

[0095] Where σ is the integral variable. When the detector is in a sliding state, σ = τ; when it is in a viscous state, σ = p.

[0096] Step 5: After a collision with a complex asteroid, the probe's state changes, and some energy is lost. Based on the post-collision state and the desired takeoff state, the probe's error state is derived. A sliding mode surface containing the error velocity and error angular velocity is designed. The stability of the probe's control system is analyzed using the Lyapunov function, and a sliding mode control law that meets the stability requirements of the probe's control system is derived. The sliding mode control law is designed to track the desired takeoff state, thereby compensating for the probe's velocity and angular velocity after the collision, resulting in a corrected takeoff state. This corrected state parameter serves as the initial state for the next hop, enabling the probe to continuously hop on the surface of a complex asteroid, thereby accurately tracking the nominal trajectory.

[0097] The error between the detector's velocity after the collision and the expected take-off velocity is v t =v h -v, the error between the angular velocity and the expected angular velocity is ω t =ω h -ω, select the sliding surface as:

[0098] s t =v t +k t ω t (40)

[0099] to s t The derivative is:

[0100]

[0101] Choose the reaching law as The Lyapunov function is but Satisfies uniform asymptotic stability.

[0102] The stability of the detector control system is analyzed by Lyapunov function, and the sliding mode control law that satisfies the stability of the detector control system is set as:

[0103]

[0104] The desired take-off state is tracked through the sliding mode control law to obtain the corrected take-off state of the detector; the corrected state parameters are used as the initial state of the next bounce, and the detector is continuously bounced on the surface of a complex small celestial body until it reaches the target area, thereby achieving accurate tracking of the nominal trajectory of the detector.

[0105] Beneficial effects:

[0106] 1. The existing methods for tracking and controlling the trajectory of bounce movement of small celestial body probes only consider the motion of the probe as a point mass, without considering the impact of the probe's attitude change on its motion state during a ground collision. The method for tracking and controlling the trajectory of bounce movement of complex small celestial bodies disclosed in the present invention considers the impact of the cubic probe's attitude on the bounce movement process, establishes a velocity change model of the collision point, defines an inverse inertia matrix to represent the impact of the force exerted by the ground on the probe's contact point on the velocity of the contact point, and establishes a collision dynamics model about the contact point. This method can process the update of the probe's state quantity during a non-central collision and reduce the error of the probe's state quantity after the collision.

[0107] 2. Existing methods for tracking and controlling the trajectory of bounce motion on the surface of small celestial body probes use simplified collision models and a central collision process, without considering the impact of complex contact surfaces on the probe's motion. The complex small celestial body surface bounce motion tracking and control method disclosed in this invention uses a pulse collision model to simulate the probe's collision process, analyzes the dynamic model under both sliding and sticking conditions, and introduces a stretching variable to simplify the numerical integration process, enabling the probe to simulate the complex planar non-central collision process, thereby improving the accuracy of the estimation of the probe's state parameters after the collision.

[0108] 3. Existing methods for tracking the trajectory of bouncing motion on the surface of small celestial bodies ignore the energy loss of the detector during the collision process, resulting in an increase in the cumulative trajectory error. The present invention discloses a method for tracking the trajectory of bouncing motion on the surface of a complex small celestial body. By designing a sliding mode control law that includes error velocity and error angular velocity, the flywheel is controlled to enable the detector to obtain compensation velocity and angular velocity after the detector collides, thereby compensating for the energy loss caused by the collision. The corrected state parameters are used as the initial state for the next bounce, thereby achieving continuous bouncing motion of the detector on the surface of a complex small celestial body, thereby achieving accurate tracking of the detector's bouncing motion trajectory on the surface of a complex small celestial body. BRIEF DESCRIPTION OF THE DRAWINGS

[0109] Figure 1 This is a flow chart of the method for tracking and controlling the trajectory of a complex small celestial body bouncing on the surface of the present invention;

[0110] Figure 2 This is a diagram showing the trajectory tracking results of the detector bouncing on the surface of a complex small celestial body in an example of the present invention;

[0111] Figure 3 The three-axis position change curve of the detector in the embodiment of the present invention when bouncing and moving on the surface of a complex small celestial body;

[0112] Figure 4 The curve of the velocity change of the center of mass of the three axes of the detector in the embodiment of the present invention when bouncing on the surface of a complex small celestial body;

[0113] Figure 5 The three-axis angular velocity variation curve of the detector in the embodiment of the present invention when bouncing on the surface of a complex small celestial body;

[0114] Figure 6 It is the velocity change curve of the center of mass of the probe in the embodiment of the present invention during the last collision process during the bouncing movement on the surface of a complex small celestial body;

[0115] Figure 7 It is the angular velocity variation curve of the detector in the embodiment of the present invention during the last collision process during the bouncing movement on the surface of a complex small celestial body; DETAILED DESCRIPTION

[0116] In order to better illustrate the purpose and advantages of the present invention, the invention is further described below with reference to the accompanying drawings and examples.

[0117] like Figure 1 As shown, the method for controlling the surface bounce trajectory of a weakly gravitational small celestial body disclosed in this embodiment is specifically implemented in the following steps:

[0118] Step 1: Establish the detector's fixed coordinate system O in the small celestial body. B -X B Y B Z B The dynamic and kinematic equations for the probe's hopping motion in the surface coordinate system O-XYZ are derived. Analysis of the kinematic equations derives the expected takeoff velocity, heading angle, and takeoff angle. The displacement sequence derived from the nominal trajectory derives the velocity and acceleration sequences for the probe's continuous hopping motion on the surface of a small celestial body.

[0119] Taking the small celestial body Eros as an example, the method disclosed in the present invention is simulated and verified under simulated surface terrain conditions. The physical parameters of the small celestial body are: three-dimensional size 34.4km×11.2km×11.2km, mass 6.69×10 15 kg, density 2.67×10 3 kg / m 3 , the angular velocity of the spin around its principal axis of inertia is 3.31×10 -4 rad / s, and the gravitational constant is 6.67×10 -11 Nm 2 / kg 2 .

[0120] Regarding the single bounce motion problem of the probe, in the fixed coordinate system of the small celestial body, the dynamic equation of the probe after take-off is expressed as

[0121]

[0122] Among them, r B、v B are the position and velocity vector of the detector respectively, ω is the spin angular velocity of the small celestial body, and g is the gravitational acceleration vector of the small celestial body.

[0123] In the surface coordinate system, the dynamic equation of the detector is:

[0124]

[0125] Among them, r and v are the position and velocity vectors of the detector respectively, and ρ is the position vector of the origin of the surface coordinate system relative to the center of the small celestial body. is the matrix for transforming from the body coordinate system to the surface coordinate system.

[0126] In the known nominal bounce trajectory, the collision point of each bounce of the detector is R i , the corresponding velocity vector is v hi , the angular velocity vector is ω hi Assume that the displacement sequence of the detector's bouncing trajectory is R = [R0, R1, R2, R3, R4, R5] T , where R0 = [0,0,0] T , R1=[23,26,0] T , R2=[40,38,0] T , R3=[56,50,0] T , R4=[72,80,0] T , R5=[90,90,0] T The velocity sequence is v h =[v h0 ,v h1 ,v h2 …] T And the angular velocity sequence is ω h =[ω h0 ,ω h1 ,ω h2 …] T .

[0127] In the surface coordinate system, the starting point of the detector's movement is defined as the origin of the coordinate system. The direction opposite to the line connecting the landing point of the first bounce and the starting point is defined as the positive direction of the X axis. The positive direction of the Z axis is the opposite direction of the component of the local gravity vector perpendicular to the XY plane. The Y axis conforms to the right-hand rule. The kinematic equation of the detector in the surface coordinate system is:

[0128]

[0129] Among them, v hi is the speed required for each jump, β i is the angle between the detector's initial take-off velocity direction and the XY plane, which is the detector's take-off angle. It is the angle between the component of the detector's initial take-off velocity in the XY plane and the X-axis, that is, the detector's heading angle.

[0130] Define the distance the detector moves in a single jump as R i , then the expected take-off speed of the detector each time it bounces is:

[0131]

[0132] The expected take-off speed v i Take-off angle β i Taking the derivative and setting it equal to zero gives the take-off angle corresponding to the minimum take-off speed:

[0133]

[0134] Given the expected take-off velocity v of the detector hi , heading angle and take-off angle β i In the case of , the center-of-mass velocity vector of the detector is:

[0135]

[0136] The contact point when the probe collides with the ground is P. When the probe rotates around point P and jumps, the center of mass velocity is v = ω × r P , where ω is the angular velocity vector of the detector, r P is the position vector of the collision point in the detector body coordinate system. Assuming the side length of the cube detector is 0.4m, then l = 0.2m, and the mass of the detector m = 3kg. When the detector takes off, set the edge jump mode, then ω z =0, the formula for the detector’s angular velocity is:

[0137]

[0138] The desired take-off angle β of the detector can be obtained by equations (45), (47), and (48): i =45°, heading angle The velocity sequence is v h =[v h0 ,v h1 ,v h2 ,v h3 ,v h4 ] T , where v h0 =[0.0731,0.0828,0.1559] T , v h1 =[0.0706,0.0498,0.1204] T , vh2 =[0.0676,0.0507,0.1183] T , v h3 =[0.0528,0.0989,0.1517] T , v h4 =[0.0761,0.0423,0.1183] T ; The angular velocity sequence is ω h =[ω h0 ,ω h1 ,ω h2 ,ω h3 ,ω h4 ] T , where ω h0 =[-0.4138,0.3657,0] T ,ω h1 =[-0.2419,0.3529,0] T ,ω h2 =[-0.2535,0.3381,0] T ,ω h3 =[-0.4945,0.2638,0] T ,ω h0 =[-0.2113,0.3803,0] T .

[0139] Step 2: Establish the attitude dynamics model and attitude kinematics model of the detector in the body coordinate system, define the error angular velocity and error quaternion of the detector, solve the rotation matrix of the body coordinate system relative to the target coordinate system, and obtain the error attitude kinematics model and error attitude dynamics model.

[0140] The attitude dynamics model of the cube detector is:

[0141]

[0142] Where ω is the angular velocity of the detector, J = diag(0.005, 0.005, 0.005) kg·m 2 is the moment of inertia matrix of the detector, u is the control torque, and d is the external interference torque.

[0143] The quaternion of the detector body coordinate system relative to the inertial coordinate system is Angular velocity is ω=[ω x ω y ω z ] T , then the attitude kinematics of the detector is:

[0144]

[0145] Among them, q0 is the quaternion's standard part, q v is the vector part of the quaternion, and

[0146] The expected quaternion of the detector is q f =[1,0,0,0] T , the desired angular velocity is ω f =[0,0,0] T , define the error quaternion as q e ,have:

[0147]

[0148] Then the rotation matrix from the target coordinate system to the body coordinate system is:

[0149]

[0150] The error angular velocity of the body coordinate system relative to the target coordinate system is defined as:

[0151] ω e =ω-C e ω f (53)

[0152] Combining equations (50), (51), and (53), the attitude kinematic equation of the detector based on the error quaternion is:

[0153]

[0154]

[0155] Substituting Equation (53) into Equation (49), the attitude dynamics model of the detector based on the error angular velocity is obtained as follows:

[0156]

[0157] Step 3: The collision between the cube detector and the ground is a non-central collision. A velocity change model of the collision point is established, and an inverse inertia matrix is ​​defined to represent the influence of the force exerted by the ground on the contact point of the detector on the velocity of the contact point. This matrix is ​​used to establish the collision model in the subsequent step 4 and improve the accuracy of the detector collision model state update.

[0158] The point where the probe collides with the ground is P, and the velocity of the contact point is v P , the vector from the center of mass of the detector to the contact point is r OP , then the velocity of the contact point is:

[0159] v P =v+ω×r OP (57)

[0160] During the contact process, the detector will be subjected to the contact force F acting on the contact point. C and contact torque T C =r OP ×F C Taking the derivative of the velocity at the contact point, we get:

[0161]

[0162] in,

[0163]

[0164]

[0165]

[0166] Substituting equations (59), (60), (61) into equation (58), we can obtain:

[0167]

[0168] Define the inverse inertia matrix This matrix represents how the force applied to the contact point changes the velocity of the contact point and will be used to establish the contact model.

[0169] Step 4: Substitute the normal impulse as the independent variable into the probe dynamics equation constructed in Step 1, derive the relationship between the probe state variables and the normal impulse, and establish a pulse collision model for the probe on the surface of a complex asteroid. Divide the probe's collision state into a sliding state and a viscous state, derive the corresponding impulse state equations for each, and combine them with the velocity variation model at the collision point obtained in Step 3 to construct the probe's collision state equation. By introducing a stretching variable and decomposing the contact point velocity to simplify the numerical integration process, a differential equation for the probe's collision state on the surface of a complex asteroid is established. By solving the differential equation for the collision state, the probe's state parameters after the collision are obtained.

[0170] When the detector collides with the ground, the force F at the contact point C A pulse will be generated, that is, dP=F C t. The collision occurs in a very short time, and the position and posture of the detector will not change during the collision. During the collision, due to the normal contact force F n Strictly monotonically increasing, so the normal impulse it receives It is also monotonically increasing. Therefore, the normal impulse p is brought into the dynamic equation of the detector as an independent variable to obtain:

[0171]

[0172]

[0173] Then the contact point velocity change formula can be expressed as:

[0174]

[0175] When the normal velocity v of the contact point Pz When ≥0, the collision process of the detector ends.

[0176] The movement of the probe after colliding with the ground may be subject to two conditions: sliding and sticking. When sliding occurs, the contact point will be subjected to the normal contact force impulse and the tangential friction force impulse. The tangential velocity of the contact point is defined to satisfy And v Px , v Py The angle Φ satisfies have to:

[0177] v Px =v PT cosΦ and v Py =v PT sinΦ (66)

[0178] When v PT >0, the friction impulse satisfies dP f =μ·dP N , where μ = 0.2 is the coefficient of kinetic friction, then the derivative of the total impulse with respect to the normal impulse during sliding is:

[0179]

[0180] When v PT = 0, the friction impulse satisfies dP f =μ s ·dP N , where μ s is the static friction coefficient and μ s ≤μ, then Substituting into formula (65) we get:

[0181]

[0182] Right now,

[0183]

[0184] in,

[0185]

[0186]

[0187] At this time, the detector is in a viscous state, μ sand Φ s is a fixed value, there is,

[0188]

[0189] According to the above dynamic equation, the collision state of the detector is defined as:

[0190]

[0191] Among them, ε is used to represent the moment when the collision ends. When the detector is in the collision process, v PT will tend to zero, that is, v Px and v Py will tend to zero, and the direction of Φ will be impossible to determine, so we need to use v PT and Φ replace v Px and v Py .Right now,

[0192] X C =[v PT Φ v Pz P v ω ε] T (74)

[0193] and

[0194]

[0195]

[0196] Introduce the stretch variable τ, let

[0197]

[0198] Using the chain rule, for any variable a, the equation of motion with respect to dτ is expressed as:

[0199]

[0200] During the sliding of the detector, the PT When integrating, although v PT →0 can be achieved with a finite normal impulse, but it is difficult to converge numerically. Therefore, the integral of τ is used to replace the integral of p, and then τ→∞ replaces v PT →0. When the sticking state occurs, continue to integrate p. Define the switch variable η to control the switching between sliding and sticking states:

[0201]

[0202] Then the derivative expression of the collision state is:

[0203]

[0204] Where σ is the integral variable. When the detector is in a sliding state, σ = τ; when it is in a viscous state, σ = p.

[0205] Step 5: The state of the detector will change after a collision with the surface of a complex small celestial body, and some energy will be lost. The error state of the detector is obtained based on the state of the detector after the collision and the expected take-off state. A sliding surface containing error velocity and error angular velocity is designed. The stability of the detector control system is analyzed by the Lyapunov function, and a sliding mode control law that meets the stability requirements of the detector control system is derived. The expected take-off state is tracked by designing a sliding mode control law, and then the velocity and angular velocity of the detector after the collision are compensated to obtain the corrected take-off state of the detector; the corrected state parameters are used as the initial state of the next bounce to realize the continuous bounce movement of the detector on the surface of a complex small celestial body, thereby realizing the accurate tracking of the nominal trajectory of the detector. Let the error between the velocity of the detector after the collision and the expected take-off velocity be v t =v h -v, the error between the angular velocity and the expected angular velocity is ω t =ω h -ω, select the sliding surface as:

[0206] s t =v t +k t ω t (81)

[0207] to s t The derivative is:

[0208]

[0209] Choose the reaching law as The Lyapunov function is but Satisfies uniform asymptotic stability.

[0210] The stability of the detector control system is analyzed by Lyapunov function, and the sliding mode control law that satisfies the stability of the detector control system is set as:

[0211]

[0212] The desired take-off state is tracked through the sliding mode control law to obtain the corrected take-off state of the detector; the corrected state parameters are used as the initial state of the next bounce, and the detector is continuously bounced on the surface of a complex small celestial body until it reaches the target area, thereby achieving accurate tracking of the nominal trajectory of the detector.

[0213] like Figure 2 and Figure 3 As shown in the figure, the blue trajectory is the nominal trajectory of the cube probe bouncing on the surface of a complex small celestial body, and the red trajectory is the actual tracking trajectory of the cube probe bouncing on the surface of a complex small celestial body. The landing point coordinates of the nominal trajectory are (90,90,0.2)m, and the landing point coordinates of the actual trajectory are (89.5,89.3,0.2)m. The landing error is within the allowable error range, and the actual trajectory tracking accuracy of the cube probe is relatively high. Figure 4 As shown, the cube detector collides with the ground during each bounce, and the center of mass velocity after the collision is updated and compensated under the control of the sliding mode control law designed by the present invention, achieving the speed and direction required for the next jump. Figure 5 As shown in FIG, the angular velocity of the cube detector after colliding with the ground reaches the magnitude and direction required for the next jump under the control of the sliding mode control law designed by the present invention. Figure 6 and Figure 7 As shown, after the probe collides with the ground, the sliding mode control law designed in this invention quickly controls the probe's center of mass velocity and angular velocity to the desired velocity and angular velocity, thus updating the probe's takeoff state. The simulation results above demonstrate that the method described in this invention enables the probe to track the trajectory of complex small celestial bodies bouncing on their surfaces, despite the energy loss and attitude changes caused by collisions, with excellent control effectiveness and trajectory tracking accuracy.

[0214] The above specific description further illustrates the purpose, technical solutions and beneficial effects of the invention in detail. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for tracking and controlling the bouncing trajectory of a complex small celestial body, characterized by: The following steps are included: Step 1: Establish the detector's fixed coordinate system O on the small celestial body. B -X B Y B Z B The dynamic equations and kinematic equations of the bouncing movement in the surface coordinate system O-XYZ are obtained; the expected take-off velocity, heading angle and take-off angle of the probe when taking off are obtained by analyzing the kinematic equations; the velocity sequence and acceleration sequence of the probe's continuous bouncing movement on the surface of the small celestial body are derived based on the displacement sequence obtained from the nominal trajectory; Step 2: Establish the attitude dynamics model and attitude kinematics model of the detector in the body coordinate system, define the error angular velocity and error quaternion of the detector, solve the rotation matrix of the body coordinate system relative to the target coordinate system, and obtain the error attitude kinematics model and error attitude dynamics model; Step 3: The collision between the cube detector and the ground is a non-central collision. A velocity change model of the collision point is established. An inverse inertia matrix is ​​defined to represent the effect of the force exerted by the ground on the contact point of the detector on the velocity of the contact point. This matrix is ​​used to establish the collision model in the subsequent step 4 and improve the accuracy of the detector collision model state update. Step 4: Substitute the normal impulse as the independent variable into the detector dynamics equation constructed in step 1, derive the relationship between the detector state quantity and the normal impulse, and establish a pulse collision model of the detector on the surface of a complex small celestial body; divide the collision state of the detector into a sliding state and a viscous state, derive the corresponding impulse state equations respectively, and combine them with the velocity change model of the collision point obtained in step 3 to construct the collision state equation of the detector; by introducing stretching variables and decomposing the contact point velocity to simplify the numerical integration process, establish the collision state differential equation of the detector on the surface of a complex small celestial body, and obtain the state parameters of the detector after the collision by solving the collision state differential equation; Step 5: After the probe collides with the surface of a complex small celestial body, its state will change and some energy will be lost. Based on the probe's state after the collision and the expected take-off state, the probe's error state is derived. A sliding mode surface containing the error velocity and error angular velocity is designed. The stability of the probe control system is analyzed using the Lyapunov function, and a sliding mode control law that meets the stability requirements of the probe control system is derived. The expected take-off state is tracked by the designed sliding mode control law, and the velocity and angular velocity of the probe after the collision are compensated to obtain the corrected probe take-off state. The corrected state parameters are used as the initial state of the next bounce to realize the continuous bouncing movement of the probe on the surface of a complex small celestial body, thereby achieving the precise tracking of the probe to the nominal trajectory.

2. The method for tracking and controlling the trajectory of a complex small celestial body bouncing on the surface of claim 1, wherein: Step 1 is implemented as follows: Regarding the single bounce motion problem of the probe, in the fixed coordinate system of the small celestial body, the dynamic equation of the probe after take-off is expressed as Among them, r B 、v B are the position and velocity vectors of the detector, ω is the spin angular velocity of the small celestial body, and g is the gravitational acceleration vector of the small celestial body; In the surface coordinate system, the dynamic equation of the detector is: Among them, r and v are the position and velocity vectors of the detector respectively, and ρ is the position vector of the origin of the surface coordinate system relative to the center of the small celestial body. is the matrix for transforming from the body coordinate system to the surface coordinate system; In the known nominal bounce trajectory, the collision point of each bounce of the detector is R i , the corresponding velocity vector is v hi , the angular velocity vector is ω hi ; The displacement sequence of the nominal trajectory of the detector's bouncing movement is R = [R0, R1, R2…] T , the velocity sequence is v h =[v h0 ,v h1 ,v h2 …] T And the angular velocity sequence is ω h =[ω h0 ,ω h1 ,ω h2 …] T ; In the surface coordinate system, the starting point of the detector is defined as the origin of the coordinate system. The positive direction of the X-axis is defined as the direction opposite to the line connecting the landing point of the first bounce and the starting point. The positive direction of the Z-axis is the opposite direction of the component of the local gravity vector perpendicular to the XY plane. The Y-axis conforms to the right-hand rule. The kinematic equation of the detector in the surface coordinate system is: Among them, v hi is the speed required for each jump, β i is the angle between the detector's initial take-off velocity direction and the XY plane, which is the detector's take-off angle. The angle between the component of the detector's initial take-off velocity in the XY plane and the X-axis, that is, the detector's heading angle; Define the distance the detector moves in a single jump as R i , then the expected take-off speed of the detector each time it bounces is: The expected take-off speed v i Take-off angle β i Taking the derivative and setting it equal to zero gives the take-off angle corresponding to the minimum take-off speed: Given the expected take-off velocity v of the detector hi , heading angle and take-off angle β i In the case of , the center-of-mass velocity vector of the detector is: The contact point when the probe collides with the ground is P. When the probe rotates around point P and jumps, the center of mass velocity is v = ω × r P , where ω is the angular velocity vector of the detector, r P is the position vector of the collision point in the detector body coordinate system; assuming the side length of the cube detector is 2l, and the detector is set to jump with the edge when it jumps, then ω z =0, the formula for the detector’s angular velocity is: The velocity vector and angular velocity vector required by the detector for each bounce are calculated using equations (4), (6), and (7).

3. The method for tracking and controlling the trajectory of a complex small celestial body bouncing on the surface of claim 2, wherein: Step 2 is implemented as follows: The attitude dynamics model of the cube detector is: Where ω is the angular velocity of the detector, J is the moment of inertia matrix of the detector, u is the control torque, and d is the external interference torque; The quaternion of the detector body coordinate system relative to the inertial coordinate system is Angular velocity is ω=[ω x ω y ω z ] T , then the attitude kinematics of the detector is: Among them, q0 is the quaternion's standard part, q v is the vector part of the quaternion, and The expected quaternion of the detector is q f , the desired angular velocity is ω f , define the error quaternion as q e ,have: Then the rotation matrix from the target coordinate system to the body coordinate system is: The error angular velocity of the body coordinate system relative to the target coordinate system is defined as: oh e =ω-C e oh f (12) Combining equations (9), (10), and (12), the attitude kinematic equation of the detector based on the error quaternion is: Substituting Equation (12) into Equation (8), the attitude dynamics model of the detector based on the error angular velocity is obtained as follows: 。 4. The method for tracking and controlling the trajectory of a complex small celestial body bouncing on the surface of claim 3, wherein: Step 3 is implemented as follows: The point where the probe collides with the ground is P, and the velocity of the contact point is v P , the vector from the center of mass of the detector to the contact point is r OP , then the velocity of the contact point is: v P =v+ω×r OP (16) During the contact process, the detector will be subjected to the contact force F acting on the contact point. C and contact torque T C =r OP ×F C ; Taking the derivative of the velocity at the contact point, we get: in, Substituting equations (18), (19), (20) into equation (17), we can obtain: Define the inverse inertia matrix This matrix represents how the force applied to the contact point changes the velocity of the contact point and will be used to establish the contact model.

5. The method for tracking and controlling the trajectory of a complex small celestial body bouncing on the surface of claim 4, wherein: Step 4 is implemented as follows: When the detector collides with the ground, the force F at the contact point C A pulse will be generated, that is, dP=F C t; The collision occurs in a very short time, and the position and posture of the detector will not change during the collision; During the collision, the normal contact force F n Strictly monotonically increasing, the normal impulse it receives It also increases monotonically; therefore, the normal impulse p is brought into the dynamic equation of the detector as an independent variable to obtain: The contact point velocity change expression is: When the normal velocity v of the contact point Pz When ≥0, the collision process of the detector ends; The movement of the detector after colliding with the ground may be sliding and sticking. When sliding occurs, the contact point will be subjected to the normal contact force impulse and the tangential friction force impulse. The tangential velocity of the contact point is defined to satisfy And v Px , v Py The angle Φ satisfies have to: in Px =in PT cosΦ and v Py =in PT sinΦ (25) When v PT >0, the friction impulse satisfies dP f =μ·dP N , where μ is the coefficient of kinetic friction, then the derivative of the total impulse with respect to the normal impulse during sliding is: When v PT = 0, the friction impulse satisfies dP f =μ s ·dP N , where μ s is the static friction coefficient and μ s ≤μ, then Substituting into formula (24) we get: Right now, in, At this time, the detector is in a viscous state, μ s and Φ s is a fixed value, there is, According to the above dynamic equation, the collision state of the detector is defined as: Among them, ε is used to represent the moment when the collision ends; when the detector is in the collision process, v PT will tend to zero, that is, v Px and v Py will tend to zero, and the direction of Φ will be impossible to determine. PT and Φ replace v Px and v Py ;Right now, X C =[v PT F v Pz P v ω e] T (33) and Introduce the stretch variable τ, let Using the chain rule, for any variable a, the equation of motion with respect to dτ is expressed as: During the sliding of the detector, the PT When integrating, v PT →0 can be achieved with a finite normal impulse, but it is difficult to converge numerically; the integral of τ is used to replace the integral of p, and then τ→∞ replaces v PT →0; when the sticking state occurs, continue to integrate p; define the switch variable η to control the switching between the sliding and sticking states: Then the derivative expression of the collision state is: Where σ is the integral variable. When the detector is in a sliding state, σ = τ; when it is in a viscous state, σ = p.

6. The method for tracking and controlling the trajectory of a complex small celestial body bouncing on the surface of claim 5, wherein: The implementation method of step 5 is: The error between the detector's velocity after the collision and the expected take-off velocity is v t =v h -v, the error between the angular velocity and the expected angular velocity is ω t =ω h -ω, select the sliding surface as: s t =v t +k t ω t (40) to s t The derivative is: Choose the reaching law as The Lyapunov function is but Satisfy uniform asymptotic stability; The stability of the detector control system is analyzed by Lyapunov function, and the sliding mode control law that satisfies the stability of the detector control system is set as: The desired take-off state is tracked through the sliding mode control law to obtain the corrected take-off state of the detector; the corrected state parameters are used as the initial state of the next bounce, and the detector is continuously bounced on the surface of a complex small celestial body until it reaches the target area, thereby achieving accurate tracking of the nominal trajectory of the detector.

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