An autonomous relative navigation method for close-range space targets

Abstract

An embodiment of the present invention provides a method for autonomous relative navigation of close-range space targets, including: constructing a system model for autonomous relative navigation of the close-range space target based on sequence images, the system model including a system state equation and a measurement equation; performing observability analysis on the system model to determine the minimum number of observations of key surface points of the space target to ensure observability of the system model; optimizing the number and position of multiple surface key points based on the minimum number of observations of the space target to obtain optimal surface key points; optimizing the system model using the optimal surface key points to obtain an optimized system model; and estimating the motion state of the close-range space target using the optimized system model. Research on the minimum number of observations of feature points of non-cooperative space targets can reduce the computational burden on board the satellite.

Description

Technical Field

[0001] The present invention relates to the field of navigation, and in particular to an autonomous relative navigation method for close-range space targets. Background Art

[0002] Autonomous relative navigation of space targets has wide applications in areas such as space debris removal, on-orbit repair of faulty spacecraft, and deep space exploration. The targets of these missions are all non-cooperative targets with unknown dimensions and no cooperative identification. All require spacecraft to accurately estimate the relative position, attitude, and other related states of the non-cooperative targets at close range. Therefore, close-range relative navigation of non-cooperative targets is a key technology to ensure the implementation of these missions. Monocular cameras, as standard optical measurement sensors on spacecraft, offer high measurement accuracy, rich information, and are compact and energy-efficient. These cameras align with the current trend of spacecraft miniaturization and are an ideal source of measurement information for achieving autonomous relative navigation of close-range space targets.

[0003] When a spacecraft is very close to a non-cooperative target, the projection of the non-cooperative target onto the camera pixel plane is a two-dimensional image with rich texture. Image processing techniques can be used to extract rich keypoint information from the non-cooperative target's surface. However, due to the lack of prior information about the non-cooperative target and its unknown motion state, a single moment-instant measurement cannot accurately determine the target's relative motion state. Therefore, it is usually necessary to continuously observe the non-cooperative target in space for a period of time to obtain a sequence of images of the non-cooperative target. By analyzing and processing the information in these images, more usable information about the non-cooperative target can be obtained. Furthermore, using the image information of the non-cooperative target obtained by a single camera, keypoint extraction and matching can be used to obtain the two-dimensional pixel coordinates of keypoints on the non-cooperative target's surface. However, due to the lack of distance scale information, the obtained information is essentially line-of-sight measurement information of the keypoints on the non-cooperative target's surface, and the relative distance and depth information (distance scale information) of the target's feature points cannot be obtained. As a result, the system cannot fully and accurately estimate the target's state due to the limited measurement information obtained. Therefore, the sequential image autonomous navigation system is a typical under-observed system. At present, how to conduct observability analysis and optimize observation capabilities for under-observed systems such as autonomous navigation of sequential images has become a hot research issue.

[0004] Research on the observability of underobserved systems has been conducted. Woffinden and Geller characterized the conditions required for underobserved systems to meet observability requirements. Song Liang conducted an observability analysis of the position-related states of non-cooperative targets, assuming that their posture-related states were fully observable. However, current research on the observability of underobserved systems for non-cooperative targets has primarily focused on studying the unobservable states present in the system and the conditions for meeting observability. There is a lack of research on the minimum number of observations required for underobserved systems to meet observability requirements. This research, which addresses the boundary conditions for the observability of underobserved systems, can provide a theoretical basis for optimizing the observability of non-cooperative targets.

[0005] The optimization methods for the observation capability of under-observed systems, i.e., autonomous navigation systems of target sequence images, mainly include two methods: information mining and information optimization.

[0006] This information mining primarily involves acquiring additional range and depth information through methods such as orbital maneuvers and camera offsets. However, these methods are all based on the assumption that the target is relatively far from the spacecraft. In this case, the target can be approximated as a point mass, and only the relative distance to the target needs to be considered for state estimation. However, when the target is closer to the spacecraft, the available measurement information consists of rich surface features, requiring estimation of parameters related to the target's attitude.

[0007] The information optimization method primarily optimizes measurement data. Because target images captured by optical sensors are rich in information and feature-rich, limited onboard computing and storage resources necessitate optimization. This allows the system to minimize the computational and storage burdens onboard without sacrificing navigation accuracy. Information optimization is an effective way to enhance observation capabilities. However, current research on feature point optimization primarily focuses on planetary landing using vision-assisted inertial navigation. This method is unsuitable for non-cooperative targets with negligible gravitational pull, such as small celestial bodies and spacecraft. Furthermore, limited research is currently underway on key point optimization for these non-cooperative targets. Summary of the Invention

[0008] The embodiment of the present invention provides an autonomous relative navigation method for close-range space targets, which can solve the technical problem in the prior art that the surface key points of close-range space targets cannot be optimized when observing close-range space targets.

[0009] To achieve the above objectives, an embodiment of the present invention provides a method for autonomous relative navigation of close-range space targets, comprising:

[0010] For close-range space targets, a system model for autonomous relative navigation of space targets is constructed based on sequence images, wherein the system model includes system state equations and measurement equations;

[0011] Conduct observability analysis on the system model to determine the minimum number of observations of key points on the surface of the space target to make the system model observable;

[0012] Based on the minimum number of observations of the space target, the number and position of multiple surface key points are optimized to obtain the optimal surface key points, and the system model is optimized through the optimal surface key points to obtain the optimized system model;

[0013] The optimized system model is used to estimate the motion state of close-range space targets.

[0014] The above technical solution has the following beneficial effects: Under the conditions of limited onboard resources, for the observation capability of the autonomous relative optical navigation system for sequence images of close-range space targets, it is demonstrated that at least four consecutive sequence images are required to fully observe the target's motion state. Research is conducted on the minimum number of observations of surface key points of non-cooperative space targets, which can reduce the computational burden on board. Quantitative analysis of system observability is used to optimize the sequence of surface key points, and combined with a simulated annealing algorithm, the optimal estimation of the motion state of close-range space targets is achieved. It is assumed that multiple surface key points are continuously visible. However, in reality, due to the continuous and large-scale motion of the target, surface key points may be obscured from view. Further research is needed to investigate high-precision autonomous navigation methods when some surface key points cannot be continuously observed. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0016] Figure 1 This is a flow chart of a method for autonomous relative navigation of close-range space targets according to an embodiment of the present invention;

[0017] Figure 2 This is a schematic diagram of autonomous navigation of a sequence of images of a close-range space target according to an embodiment of the present invention;

[0018] Figure 3 1 is a schematic diagram of autonomous relative optical navigation observation based on camera bias according to an embodiment of the present invention;

[0019] Figure 4 This is a surface key point optimization method based on a simulated annealing algorithm according to an embodiment of the present invention;

[0020] Figure 5is the change of the rank of the observability matrix with time when the number of surface key points is different in the embodiment of the present invention;

[0021] Figure 6 is the variation trend of GDOP with the number of surface key points in the embodiment of the present invention;

[0022] Figure 7 is the variation trend of the observability of the surface key point system and each state with the sampling time in the embodiment of the present invention;

[0023] Figure 8 This is a comparison of the state estimation results of the five preferred surface key points of an embodiment of the present invention and 30 randomly selected groups of feature points. DETAILED DESCRIPTION

[0024] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0025] The present invention proposes a method for optimizing the observability of an autonomous relative navigation system for sequence images of near-space targets. System observability refers to the ability of the system to determine its state based on observed quantities within a limited timeframe. To address the insufficient observability of this system, the observability of the sequence image autonomous relative navigation system is analyzed under limited onboard computing and storage resources, and the minimum number of observations required to meet observability is derived. The method also theoretically analyzes the minimum number of surface key point observations required for the autonomous relative navigation system to converge on its observable state. Based on this, a method for optimizing surface key points for non-cooperative targets is proposed, using system observability as the objective function. This method ultimately achieves high-precision estimation of non-cooperative target autonomous relative optical navigation. Finally, simulation experiments verify the correctness of the theoretical observability analysis, as well as the feasibility and efficiency of the system observability optimization method. This method effectively improves the system's observation capability and autonomous relative navigation accuracy while reducing the onboard computing burden.

[0026] like Figure 1 As shown, in combination with an embodiment of the present invention, a method for autonomous relative navigation of a close-range space target is provided, comprising:

[0027] S101: For a close-range space target, construct a system model for autonomous relative navigation of the space target based on sequence images, wherein the system model includes a system state equation and a measurement equation;

[0028] S102: Perform observability analysis on the system model to determine the minimum number of observations of key points on the surface of the space target so that the system model is observable;

[0029] S103: Based on the minimum number of observations of the space target, the number and position of the plurality of surface key points are optimized to obtain the optimal surface key points, and the system model is optimized using the optimal surface key points to obtain the optimized system model;

[0030] S104: Estimate the motion state of the close-range space target using the optimized system model.

[0031] Preferably, the space target refers to a non-cooperative target, and the close distance refers to a distance between the space target and the spacecraft observing the space target being much smaller than the orbital radius of the spacecraft;

[0032] Constructing a system model for autonomous relative navigation of a space target based on sequence images means using a spacecraft camera carried by a spacecraft to measure sequence images of a space target over a period of time, and constructing a system model for autonomous relative navigation of the space target based on the surface key points of the space target in the acquired sequence images. The system model is used to estimate the motion state of the space target relative to the spacecraft.

[0033] Preferably, first, for close-range space targets, a system model for optical autonomous relative navigation of space targets is constructed based on sequence images.

[0034] This section will establish the system state equation and measurement equation of the optical autonomous relative navigation system for space targets, providing a theoretical basis for the following system observability analysis and feature point (i.e., surface key point) optimization.

[0035] 1.1 The system state equation for constructing the space target autonomous relative navigation system is as follows:

[0036] The monocular pose estimation problem of space targets based on sequential images refers to the spacecraft visually observing the target through a single optical camera carried by it, and estimating the target's relative position, relative velocity, relative rotation quaternion, and angular velocity to inertia ratio (i.e., state parameters) based on the sequential image information.

[0037] When performing monocular pose estimation on a space target, the following coordinate systems are defined: spacecraft system {B}, spacecraft orbit system {O}, spacecraft camera system {C}, target system {T}, and inertial system {I}. The mutual transformation relationship between these coordinate systems is defined as follows: the rotation matrix of the spacecraft camera system {C} relative to the spacecraft system {B} is There is also a translation vector d; the rotation matrix of the spacecraft system {B} relative to the spacecraft orbital system {O} is The rotation matrix of the spacecraft camera system {C} relative to the spacecraft orbit system {O} is The rotation matrix of the target system {T} relative to the spacecraft orbital system {O} is

[0038] At this time, the state quantity is expressed as: are respectively represented as the position r of the center of mass of the space target relative to the spacecraft camera system {C} Τ =[x,y,z], speed The absolute attitude (target attitude) of the target system {T} relative to the inertial system {I} is usually expressed as quaternion q Τ =[q1,q2,q3,q0], absolute angular velocity ω Τ =[ω1,ω2,ω3], inertia ratio and the space target surface n p The coordinates of the surface key points relative to the target system {T} The overall expression is ρ m Τ =[x m ,y m ,z m ],m=1,2,...,n p , ρ m is the coordinate of the mth surface key point in the target system {T} (using rectangular coordinate system). The system state equation is constructed as follows:

[0039] Assume that the spacecraft and the space target are in two-body motion, and that the distance between the space target and the spacecraft is much smaller than the orbital radius of the spacecraft. Without considering the perturbation force, the position and velocity of the space target relative to the spacecraft camera system {C} satisfy the relative motion state equation (CW equation):

[0040]

[0041] in, is the average angular velocity of the spacecraft, μ is the gravitational constant of the Earth, r c represents the spacecraft orbit radius, Indicates that it corresponds to The linear acceleration, Indicates that it corresponds to The linear acceleration, Indicates that it corresponds to Linear acceleration.

[0042] According to the attitude kinematics equation, the motion equation of the absolute attitude quaternion q of the space target relative to the inertial system {I} is expressed as:

[0043]

[0044] Where q = [q1q2q3q0] Τ =[q v q0] Τ , is the estimated value of angular velocity ω, ω x ,ω y ,ω z are the angular velocities of the target along the three axes in the inertial system {I}.

[0045] After linearizing equation (1.2), we get:

[0046]

[0047] in, represents the differential of the absolute attitude quaternion estimation error δq, δq v is the vector part of δq, δω is the angular velocity error, Represents ω x The estimated value of Represents ω y The estimated value of Represents ω z estimated value.

[0048] For a freely tumbling space target, according to its angular velocity dynamics equation, the equation for the target's rotational angular velocity ω relative to the inertial system {I} is expressed as:

[0049]

[0050] in, Inertia ratio estimated value of;

[0051] Linearize equation (1.4) as follows:

[0052]

[0053] Where δω is the angular velocity error, represents the inertia ratio error, and Respectively expressed as:

[0054]

[0055] in, express The estimated value of express The estimated value of express estimated value.

[0056] Since the space target is a rigid body, its principal inertia and inertia ratio are both constant, so the inertia ratio of the space target is Expressed as:

[0057]

[0058] in, are the principal inertias of the target system {T} respectively.

[0059] In addition, under the target system {T}, the coordinates of the key points on the surface of the spatial target are expressed as:

[0060] In summary, the system state equation of the space target autonomous relative navigation system is expressed as:

[0061]

[0062] Among them, δx k represents the state estimation error variable at time k, ε k It means that the k moment satisfies the zero mean and is a normally distributed process noise with a variance of Q.

[0063]

[0064] in, Represented as a 0 matrix.

[0065] The system state equation of formula (1.8) is a discrete-time system state equation. By differentially solving formula (1.8), the discrete-time system state transition equation of δx from the i-th sampling moment to the j-th sampling moment can be obtained as:

[0066]

[0067] In formula (1.9), δx represents the state estimation error variable;

[0068]

[0069] Among them, R represents the rotation matrix corresponding to the absolute attitude quaternion. The relationship between the rotation matrix and the quaternion is as follows:

[0070]

[0071] In order to distinguish the integral representation, the outermost integral is represented as The innermost integral is expressed as The intermediate layer integral is expressed as t i represents the i-th moment, t j represents the jth moment, τ=tj -t i Indicates that from the t i Time to t j The time interval of the moment, M in Mdt represents the The N in Ndt represents the

[0072] 1.2 Measurement equation

[0073] For close-range space targets, a system model for autonomous relative navigation of space targets is constructed based on sequence images, which also includes:

[0074] Construct the measurement equation of the space target autonomous relative navigation system, including:

[0075] Autonomous navigation of sequence images of close-range space targets means: when the distance between the space target and the spacecraft is relatively close, it only relies on the optical sensor (spacecraft camera) carried by the spacecraft to measure the sequence images of the space target over a period of time, obtains the measurement information of the key points of the space target surface in the image through image processing and other means, and estimates the position, speed, absolute attitude (quaternion), angular velocity, inertia ratio and other motion states of the space target relative to the spacecraft by combining the orbit and attitude dynamics of the space target. Figure 2 shown.

[0076] Since a single optical camera cannot obtain distance information, the optical autonomous navigation system of the space target is not fully observable. However, the appropriate camera bias can provide relative pseudo-range information so that all states can be observed. Therefore, the measurement equation based on the camera bias is given below. The measurement diagram of the key points of the surface of the camera and the space target is as follows Figure 3 .

[0077] At this time, the unit vector of the mth surface key point in the spacecraft camera system {C}, that is, the observation equation of the system is expressed as:

[0078]

[0079] Where v represents the measurement noise, which has a mean of zero and a variance of Normal distribution; Expressed as:

[0080]

[0081] Among them, r m C is the position vector of the mth surface key point of the space target in the spacecraft camera system {C}, r Cis the position vector of the center of mass of the space target relative to the spacecraft camera system {C}, r is the position vector of the center of mass of the space target in the spacecraft system {B}, and d is the offset vector of the camera center of mass relative to the spacecraft system {B}.

[0082] q r The rotation quaternion of the target system {T} relative to the spacecraft camera system {C} can be expressed as: Among them, q is the rotation quaternion of the space target in the inertial system {I}, q s is the rotation quaternion of the spacecraft camera system {C} relative to the inertial system {I} (which can be obtained from the camera installation matrix and is generally considered a known quantity).

[0083] Represents the relative position estimation value, δr represents the relative position estimation error, Indicates q r The estimated value of δq represents the absolute attitude quaternion estimation error, represents ρ m The estimated value of δρ m represents the position estimation error of the mth surface key point, S(ρ m ) is expressed as:

[0084] Calculate the partial derivatives of each component in equation (1.19) corresponding to the measurement equation:

[0085]

[0086] make Combined with (1.20), we have:

[0087]

[0088] The Jacobian matrix of the linearized measurement equation (1.18) is:

[0089]

[0090] Observability Analysis

[0091] Conduct observability analysis on the system model to determine the minimum number of observations of key points on the surface of the space target to make the system model observable, including:

[0092] To address the issue of onboard computing resources, it is necessary to minimize the number of consecutive observations of surface key points. As shown in Section 1.1, the system state equation (1.8) for the space target's autonomous relative navigation system also estimates the three-dimensional coordinates of the space target's surface key points within the target's own system as state variables. If the number of observations is too small, the state of the space target's autonomous relative navigation system will not converge, making it unobservable. Therefore, before optimizing surface key points, an observability analysis of the space target's autonomous relative navigation system is necessary.

[0093] Therefore, this section analyzes the rank of the observability matrix of the autonomous relative navigation system of space targets and theoretically solves the minimum number of observations of key points on the surface to ensure that the autonomous relative navigation system meets the observability.

[0094] The Jacobian matrix (1.22) after linearization of the system state transfer equation (1.9) and the measurement equation (1.18) of the aforementioned space target autonomous relative navigation system is used to observe τ from the i-th sampling moment. N After that, the observable matrix of the discrete-time space target autonomous relative navigation system can be expressed as:

[0095]

[0096] Among them, Φ (i+τ,i) =Φ (i+1,i) Φ (i+2,i+1) …Φ (i+τ,i+τ-1) , τ N Indicates the τth N observations, i represents the initial sampling time. When the rank is full, the space target autonomous relative navigation system can be observed. From the Jacobian matrix (1.22) after the linearization of the measurement equation, we know that H k The dimension of the matrix is ​​3n p ×(15+3n p ), it can be seen that when the space target autonomous relative navigation system based on camera bias observes three non-collinear unknown surface key points, the states of all space targets can be observed, so The rank of the p .

[0097] In the following, we increase the number of consecutive observations of key points on the surface to find The minimum number of observations required for the rank condition to hold, that is, the boundary condition for the observability of the autonomous relative navigation system for space targets.

[0098] For the observability matrix (1.23) of the space target autonomous relative navigation system, the τth N (τ N >1) block rows (dividing the matrix into blocks, a fast row of the block matrix), can be expressed as:

[0099]

[0100] in,

[0101] Theorem 1: For the unknown n of simultaneously observing space targets p An optical autonomous relative navigation system based on camera bias for each surface key point can make the state of the space target autonomous relative navigation system observable by observing the same set of surface key points at least four times in a row.

[0102] The process of proving Theorem 1 is as follows: When the number of consecutive observations is τ N =2, Expressed as:

[0103]

[0104] In formula (1.26), the superscripts (1) and (2) represent the first observation and the second observation.

[0105] The row-column transformation of equation (1.26) is expressed as:

[0106]

[0107] From formula (1.24), we can see that O (i+1,i) A total of 15+3n p Therefore, if we want to make the rank condition hold by increasing the number of observations, the boundary conditions are:

[0108] γ=rank(O (i+1,i) )=15+3n p (1.27)

[0109] That is, the matrix O (i+1,i) The rank should be equal to 15+3n p .

[0110] for According to formula (1.20):

[0111]

[0112] U m Performing eigendecomposition, we can get U m There are two unequal eigenvalues, namely U m The rank of is 2. Therefore, from formula (1.26), when the number of observations τ N =2, the observability matrix O (i+1,i) After the row and column transformation, there are 6+3n non-zero columns p Column, non-zero rows have 4n pAccording to the properties of the matrix rank, when n p When >6, the observability matrix O of formula (1.26) (i+1,i) The rank of γ = 6 + 3n p <15+3n p ; When n p <6, the observability matrix O of formula (1.26) (i+1,i) The rank is γ = 4n p <15+3n p , so the rank condition is not met when the number of observations is 2.

[0113] When the number of consecutive observations is τ = 3, Expressed as:

[0114]

[0115] The row-column transformation of equation (1.30) is expressed as:

[0116]

[0117] in,

[0118] Now we can see from (1.30) that after transforming the equation (1.30) into rows and columns, the number of non-zero columns is 12+3n. p Column, non-zero rows have 6n p OK. When n p >6 o'clock, O (i+1,i) The rank of γ = 12 + 3n p <15+3n p , when n p <4 o'clock, O (i+1,i) The rank is γ = 6n p <15+3n p , so when τ N =3, the rank condition is not met.

[0119] Similarly, we can derive τ N =4, after the row-column transformation, the number of non-zero columns is 15+3n p Column, non-zero rows have 8n p OK, when n p >3 o'clock, O (i+1,i) The rank is γ = 15 + 3n p , when n p <3 o'clock, O (i+1,i) The rank is γ = 8n p ≥15+3n p , so when τ N =4, the rank condition is satisfied.

[0120] The above theorem shows that when observing the unknown n p When there are at least four consecutive observations of the surface key points, the space target autonomous relative navigation system can be fully observable. This theorem can provide a theoretical basis for the surface key point selection process.

[0121] 3. Optimal Navigation Based on Sequence Feature Optimization

[0122] Preferably, based on the minimum number of observations of the space target, the number and position of multiple surface key points are optimized to obtain the optimal surface key points, and the system model is optimized by the optimal surface key points to obtain the optimized system model, which specifically includes:

[0123] Assume that during the pose estimation process of a space target optical autonomous relative navigation system, multiple continuously observable surface key points can be obtained through image processing. Given limited computational resources, to reduce the dimensionality of the state variables and computational complexity in the system state estimation, it is necessary to minimize the number of observed surface key points. Furthermore, given a fixed number of surface key points, the configuration of the selected surface key points will also affect the state estimation accuracy. Since observability reflects the state estimation performance of a space target autonomous relative navigation system, higher observability indicates higher state estimation accuracy.

[0124] Therefore, this section first analyzes the relationship between the number of surface key points and the observability of the space target autonomous relative navigation system, and reasonably selects the number of observed surface key points. Then, under the premise of a certain number of surface key points, the observability of the space target autonomous relative navigation system is used as an indicator to select appropriate surface key points by minimizing the pose estimation covariance.

[0125] According to the observability analysis in Section 2, the navigation pose requires multiple observations to be calculated. Four consecutive observations of a set of surface keypoints are sufficient to meet the observability requirements of the space target's autonomous relative navigation system. Because the pose of a space target changes continuously over time, the optimal surface keypoint at a given moment may not necessarily be optimal across multiple observations. Therefore, a relationship between the measurement residuals and state residuals from these four observations is established to optimize surface keypoint selection.

[0126] Combining equations (1.9) and (1.23), there is a linear relationship between the measurement residuals and the state residuals of four consecutive observations:

[0127]

[0128] Where O represents the observability matrix of 4 observations, see formula (1.22).

[0129] Using the least squares estimation for equation (1.32), we can get:

[0130] δx i =(O Τ O) -1 O Τ δz (1.32)

[0131] By the definition of covariance, we can get δx i The covariance matrix of is expressed as:

[0132] E(δx i (δx i ) Τ )=(O Τ O) -1 O Τ E(δz(δz) Τ )O(O Τ O) -1 (1.33)

[0133] According to formula (1.18), the measurement noise is independent and identically distributed, and the distribution of the measurement noise satisfies zero mean and variance Therefore, Equation (1.33) can be expressed as:

[0134]

[0135] It can be seen from formula (1.35) that the measurement noise variance is expressed by (O Τ O) -1 Projected to the navigation state estimation variance, the measurement noise variance When constant, the navigation error covariance is given by (O Τ O) -1 Therefore, (O Τ O) -1 It can be used to evaluate the observability of the autonomous relative navigation system of space targets. The smaller the eigenvalue, the higher the estimation accuracy of the corresponding eigenvector in the same direction. Therefore, the geometric dilution of precision GDOP of the autonomous relative navigation system of space targets is defined as:

[0136] traceP=tr((O Τ O) -1 )(1.35)

[0137] traceP is represented as a matrix ( Τ O) -1 The smaller the traceP is, the smaller the navigation error covariance is, the higher the navigation accuracy is, and the higher the observability of the optical autonomous relative navigation system is. Therefore, the minimum traceP value is used as the performance indicator for optimizing surface key points. By optimizing surface key points, the observability of the optical autonomous relative navigation system is improved and the navigation accuracy is optimized.

[0138] The optimization problem of surface key points is described as: given n continuous visible surface key points, select the predefined nk(n k <n) optimal subset of surface key points. This problem is solved by introducing the indicator variable α m ∈{0,1}(m=1,...,n) constitutes an integer programming problem, the mth indicator variable corresponds to the mth surface key point, and each indicator variable indicates whether the corresponding surface key point is selected into the subset. The integer programming problem is expressed as:

[0139]

[0140] For this integer programming problem, when there are many surface key points, directly solving the planning problem will result in a high time cost and extremely low solution speed. Therefore, the simulated annealing algorithm is used to solve the optimal surface key points.

[0141] The specific steps are as follows Figure 4 As shown, the simulated annealing algorithm is used to solve Equation (1.37) to obtain the optimal surface key points, corresponding to Figure 4 The expression is as follows: First, initialize and define the number of optimal surface key points as n k , the temperature attenuation coefficient is β, optional n k The surface key points are taken as the initial solution, and the initial temperature T0 is given. The initial surface key points are taken as the optimal surface key points, and (O Τ O) -1 The trace λ0.

[0142] Set the current temperature of the simulated annealing algorithm to T, the number of iterations for each temperature T to N, the termination error of the algorithm to ε, and the number of iterations for the temperature T to drop to L. If the change in GDOP is less than the threshold ε after the temperature drops L times, the algorithm terminates.

[0143] At the beginning of each iteration, a group of random surface key points are selected as new surface key points in the neighborhood of the initial surface key point, and the corresponding traceP value λ is calculated. If λ < λ0, the new surface key point is accepted as the optimal surface key point. Otherwise, the probability of accepting the new solution is calculated according to the Metropolis criterion as p = exp(-(λ-λ0) / T). The acceptance probability decreases as the temperature T decreases. If p∈[0,1], the new surface key point is accepted as the optimal surface key point. After the judgment is completed, the next iteration begins. When the number of iterations reaches the optimal number of iterations for each temperature T, the termination condition is checked. If the termination condition is met, the program stops and the current surface key point is the optimal surface key point. Otherwise, the temperature T is updated according to the annealing strategy. k+1 =βT k , where 0<β<1 represents the temperature attenuation coefficient and k represents the number of iterations.

[0144] 4. Simulation Verification

[0145] To verify the feature point selection method for optical autonomous relative navigation of space targets, a space target optical autonomous relative navigation scenario was simulated using MATLAB. First, feature points (i.e., surface key points) were randomly generated on the 1m×1m target surface in the target body coordinate system. These random feature points were then projected onto the image plane using the camera's intrinsic and extrinsic parameters to obtain measurement information. The camera configuration used and the initial pose parameters of the simulated space target are shown in the table below.

[0146]

[0147]

[0148] 4.1 Simulation of the minimum number of consecutive observations of feature points

[0149] The observability analysis shows that to determine the target's posture from the image alone, at least three non-collinear unit vector observations are required. Therefore, in order to verify the correctness of the observability analysis of the system in Chapter 3, the following simulations are conducted to observe the number of feature points n. p =3,4,...,10, the rank of the observability matrix changes with the number of observations, such as Figure 5 shown.

[0150] Depend on Figure 5 It can be seen that the rank of the observability matrix gradually increases with the number of observations, indicating that the system's observable states gradually increase. Because the autonomous relative optical navigation system needs to estimate the position of the surface feature points of the space target within the target's own system, it is necessary to continuously observe the same set of feature points multiple times to increase the system's observable states. By the fourth observation, the observability matrix satisfies the system's observability rank condition for any number of feature points. This concludes that the system is fully observable after at least four consecutive observations. This simulation result is consistent with the conclusions drawn in Chapter 3, and the simulation experiment verifies the accuracy of the aforementioned theorem.

[0151] 4.2 Simulation of the observability of the number of feature points

[0152] The relationship between the number of feature points observed through numerical simulation and the observability of the system is as follows: Figure 6 shown.

[0153] Depend on Figure 6It can be seen that when observing multiple feature points on the surface of a space target, the system's GDOP gradually decreases with the increase in the number of observed feature points, and when the number of feature points increases from 3 to 4, the GDOP decreases the most, indicating that the navigation error covariance drops significantly and the system observability increases significantly; after that, with the increase in the number of observed feature points, the system observability will not increase significantly.

[0154] Since the Cramer-Rao lower bound represents the lower bound of the error covariance matrix, the trace of the CR lower bound can be used to measure the size of the system observability. Therefore, the traces of the CR lower bound of the corresponding system observability, relative position, relative velocity, absolute quaternion, angular velocity, inertia ratio and target surface feature point coordinates are simulated when observing different numbers of feature points. Figure 7 shown.

[0155] As can be seen from the figure above, when the number of observed feature points increases to 4, the error covariance decreases significantly. Further increasing the number of feature points does not significantly reduce the error covariance, but the computational complexity increases exponentially. Therefore, to achieve high state estimation accuracy, at least 4 feature points need to be observed. To balance computational complexity and state estimation accuracy, it is not necessary to observe too many feature points.

[0156] 4.3 Simulation of Feature Point Optimization

[0157] From the simulation part of the number of feature points above, it can be seen that in order to balance the feature points, at least 4 feature points need to be observed. Therefore, this section uses the simulated annealing algorithm mentioned in Chapter 4 to perform feature point optimization. The figure below shows the results of the relative position, relative velocity, absolute quaternion, absolute angular velocity, absolute inertia ratio and the error of the coordinates of the four feature points on the target surface obtained by using the optimized feature points as observation information and randomly selecting 30 groups of feature points as observation information, and performing state estimation through the IEKF (Iterative Extended Kalman Filter) algorithm. Among them, the red solid line represents the result of the state estimation of the optimized feature points, and the blue dotted line represents the state estimation result of 30 groups of random feature points, as shown below. Figure 8 shown.

[0158] From the results, it can be seen that the state estimation of the feature points selected by optimization has higher accuracy than that of the randomly selected feature points, which verifies the feasibility of the proposed method.

[0159] The beneficial technical effects achieved by the embodiments of the present invention are as follows:

[0160] Under the limited onboard resource conditions, autonomous relative optical navigation of close-range space target image sequences, with limited observation capabilities, demonstrates that at least four consecutive image sequences are required to fully observe the target's motion state. Research is conducted on the minimum number of observations of feature points of non-cooperative space targets, which can reduce the onboard computational burden. Quantitative analysis of system observability is used to optimize the selection of key points on the sequence's surface, and combined with a simulated annealing algorithm, the optimal estimation of the close-range space target's motion state is achieved. It is assumed that multiple surface key points are continuously visible. However, in reality, due to the target's continuous and large-scale motion, these key points may be obscured from view. Further research is needed to investigate high-precision autonomous navigation methods when some surface key points cannot be continuously observed.

[0161] It should be understood that the specific order or hierarchy of steps in the disclosed processes is an example of an exemplary method. Based on design preferences, it should be understood that the specific order or hierarchy of steps in the process can be rearranged without departing from the scope of the present disclosure. The accompanying method claims present elements of the various steps in an exemplary order and are not intended to be limited to the specific order or hierarchy described.

[0162] In the foregoing detailed description, various features are grouped together in a single embodiment to simplify the disclosure. This method of disclosure should not be interpreted as reflecting an intention that embodiments of the claimed subject matter require more features than are expressly recited in each claim. On the contrary, as reflected in the appended claims, the invention comprises less than all the features of any individual disclosed embodiment. The appended claims are hereby expressly incorporated into the detailed description, with each claim standing on its own as a separate preferred embodiment of the invention.

[0163] The above description of the disclosed embodiments is intended to enable any person skilled in the art to implement or use the present invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be applied to other embodiments without departing from the spirit and scope of the present disclosure. Therefore, the present disclosure is not limited to the embodiments presented herein but is intended to be accorded the widest scope consistent with the principles and novel features disclosed herein.

[0164] The foregoing description includes examples of one or more embodiments. Of course, it is not possible to describe all possible combinations of components or methods for the purposes of describing the above embodiments, but one of ordinary skill in the art will recognize that the various embodiments may be further combined and arranged. Therefore, the embodiments described herein are intended to encompass all such changes, modifications and variations that fall within the scope of the appended claims. Furthermore, to the extent the term "comprising" is used in the specification or claims, the term is intended to be encompassed in a manner similar to the term "including," as explained in terms of "including," used as a transitional word in the claims. Furthermore, any use of the term "or" in the specification of the claims is intended to mean a "non-exclusive or."

[0165] The specific implementation methods described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific implementation method 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 autonomous relative navigation of close-range space targets, characterized in that: include: For close-range space targets, a system model for autonomous relative navigation of space targets is constructed based on sequence images, wherein the system model includes system state equations and measurement equations; Conduct observability analysis on the system model to determine the minimum number of observations of key points on the surface of the space target to make the system model observable; Based on the minimum number of observations of the space target, the number and position of multiple surface key points are optimized to obtain the optimal surface key points, and the system model is optimized through the optimal surface key points to obtain the optimized system model; The optimized system model is used to estimate the motion state of close-range space targets.

2. The method for autonomous relative navigation of close-range space targets according to claim 1, characterized in that: A space target refers to a non-cooperative target, and a close range refers to a space target whose distance from a spacecraft observing the space target is much smaller than the orbital radius of the spacecraft; Constructing a system model for autonomous relative navigation of a space target based on sequence images means using a spacecraft camera carried by a spacecraft to measure sequence images of a space target over a period of time, and constructing a system model for autonomous relative navigation of the space target based on the surface key points of the space target in the acquired sequence images. The system model is used to estimate the motion state of the space target relative to the spacecraft.

3. The method for autonomous relative navigation of close-range space targets according to claim 1, characterized in that: For close-range space targets, a system model for autonomous relative navigation of space targets is constructed based on sequence images, including: Construct the system state equation of the space target autonomous relative navigation system, including: A single optical camera is carried on the spacecraft to visually observe the space target at a close distance. In the process of monocular pose estimation of the space target, the following coordinate systems are defined: spacecraft system {B}, spacecraft orbit system {O}, spacecraft camera system {C}, target system {T}, inertial system {I}; the mutual transformation relationship between the coordinate systems is defined as: the rotation matrix of the spacecraft camera system {C} relative to the spacecraft system {B} is There is also a translation vector d; the rotation matrix of the spacecraft system {B} relative to the spacecraft orbital system {O} is The rotation matrix of the spacecraft camera system {C} relative to the spacecraft orbit system {O} is The rotation matrix of the target system {T} relative to the spacecraft orbital system {O} is The state quantity of the space target is expressed as: The state quantities are the position r of the center of mass of the space target relative to the spacecraft camera system {C} Τ =[x,y,z], speed The absolute attitude quaternion of the target system {T} relative to the inertial system {I} is expressed as q Τ =[q1,q2,q3,q0], absolute angular velocity ω Τ =[ω1,ω2,ω3], inertia ratio and the space target surface n p The coordinates of the surface key points relative to the target system {T} The overall expression is ρ m Τ =[x m ,y m ,z m ],m=1,2,...,n p ,ρ m is the coordinate of the mth surface key point in the target system {T}; Assuming that the motion of the spacecraft and the space target is a two-body motion, without considering the perturbation force, the position and velocity of the space target relative to the spacecraft camera system {C} satisfy the relative motion state equation: in, represents the average angular velocity of the spacecraft, μ represents the Earth's gravitational constant, r c represents the spacecraft orbit radius, Indicates that it corresponds to The linear acceleration, Indicates that it corresponds to The linear acceleration, Indicates that it corresponds to Linear acceleration; The motion equation of the absolute attitude quaternion q of the target relative to the inertial system {I} is expressed as: Where q = [q1 q2 q3 q0] Τ =[q v q0] Τ , is the estimated value of angular velocity ω, ω x ,ω y ,ω z are the angular velocities of the target along the three axes in the inertial system {I}; Linearize equation (1.2) and we get: in, represents the differential of the absolute attitude quaternion estimation error δq, δq v is the vector part of δq, δω is the angular velocity error, Represents ω x The estimated value of Represents ω y The estimated value of Represents ω z estimated value of; For a freely tumbling space target, the equation for the angular velocity ω of the space target relative to the inertial system {I} is expressed as: in, Inertia ratio estimated value of; Linearize equation (1.4) as follows: Where δω is the angular velocity error, represents the inertia ratio error, and Respectively expressed as: in, express The estimated value of express The estimated value of express estimated value of; Since the space target is a rigid body, its principal inertia and inertia ratio are both constant, so the inertia ratio of the space target is Expressed as: in, are the principal inertias of the target system {T}; In the target system {T}, the coordinates of the key points on the surface of the space target are expressed as: The system state equation of the space target autonomous relative navigation system is expressed as: Among them, δx k represents the state estimation error variable at time k, ε k It means that k moment satisfies zero mean and is a normally distributed process noise with variance Q; in, Represented as a 0 matrix.

4. The method for autonomous relative navigation of close-range space targets according to claim 3, characterized in that: The system state equations for constructing the space target autonomous relative navigation system also include: The system state equation of formula (1.8) is the discrete-time system state equation. Differentiate and solve formula (1.8) to obtain the discrete-time system state transition equation of δx from the i-th sampling moment to the j-th sampling moment, which is expressed as: In formula (1.9), δx represents the state estimation error variable; Among them, R represents the rotation matrix corresponding to the absolute attitude quaternion. The relationship between the rotation matrix and the quaternion is as follows: In order to distinguish the integral representation, the outermost integral is represented as The innermost integral is expressed as The intermediate layer integral is expressed as t i represents the i-th moment, t j represents the jth moment, τ=t j -t i Indicates that from the t i Time to t j The time interval of the moment, M in Mdt represents the The N in Ndt represents the 5. The method for autonomous relative navigation of close-range space targets according to claim 4, characterized in that: For close-range space targets, a system model for autonomous relative navigation of space targets is constructed based on sequence images, which also includes: Construct the measurement equations of the space target autonomous relative navigation system, including: Since a single optical camera cannot obtain the distance information of the space target, the autonomous navigation system of the space target is not fully observable. Based on the appropriate camera bias, relative pseudo-range information can be provided so that all states of the space target can be observed. Based on the camera bias, for the unit vector of the mth surface key point in the spacecraft camera system {C}, the observation equation of the autonomous relative navigation system of the space target is constructed as follows: Where v represents the measurement noise, which has a mean of zero and a variance of Normal distribution; Expressed as: Among them, r m C is the position vector of the mth surface key point of the space target in the spacecraft camera system {C}, r C is the position vector of the center of mass of the space target relative to the spacecraft camera system {C}, r is the position vector of the center of mass of the space target in the spacecraft system {B}, and d is the offset vector of the camera center of mass relative to the spacecraft system {B}; q r The rotation quaternion of the target system {T} relative to the spacecraft camera system {C} is expressed as: Among them, q is the rotation quaternion of the space target in the inertial system {I}, q s is the rotation quaternion of the spacecraft camera system {C} relative to the inertial system {I}; Represents the relative position estimation value, δr represents the relative position estimation error, Indicates q r The estimated value of , δq represents the absolute attitude quaternion estimation error, Represents ρ m The estimated value of δρ m represents the position estimation error of the mth surface key point, S(ρ m ) is expressed as: Find the partial derivatives of each component in equation (1.19): make Combined with (1.20), we have: The Jacobian matrix after linearizing equation (1.18) is:

6. The method for autonomous relative navigation of close-range space targets according to claim 5, characterized in that: Conduct observability analysis on the system model to determine the minimum number of observations of key points on the surface of the space target to make the system model observable, including: In the system state equation (1.8) of the space target autonomous relative navigation system, the three-dimensional coordinates of the key points on the surface of the space target in the target system are also estimated as state variables; By analyzing the rank of the observability matrix of the autonomous relative navigation system of space targets, the minimum number of observations of the key points on the surface of the space targets is theoretically solved to ensure that the autonomous relative navigation system meets the observability. The Jacobian matrix (1.22) after linearization of the system state transfer equation (1.9) and the measurement equation (1.18) of the space target autonomous relative navigation system is used to observe τ from the i-th sampling moment. N After that, the observable matrix of the discrete-time space target autonomous relative navigation system can be expressed as: in, τ N Indicates the τth N Second view, i represents the initial sampling time; when When the rank is full, the space target autonomous relative navigation system is observable; from the Jacobian matrix (1.22) after the linearization of the measurement equation, it can be seen that H k The dimension of the matrix is ​​3n p ×(15+3n p ), when the space target autonomous relative navigation system based on camera bias observes three non-collinear unknown surface key points, the states of all space targets can be observed. Therefore, The rank should be at most 15+3n p .

7. The method for autonomous relative navigation of close-range space targets according to claim 6, characterized in that: Conduct observability analysis on the system model to determine the minimum number of observations of key points on the surface of the space target to make the system model observable, including: By increasing the number of consecutive observations of the key points on the surface of the air target, we can find The minimum number of observations for which the rank condition is satisfied is used as the boundary condition for the observability of the autonomous relative navigation system for space targets. For the τth (τ>1) block row of the observability matrix (1.24) of the space target autonomous relative navigation system, it can be expressed as: in, Set Theorem 1, Theorem 1: For the unknown n of simultaneously observing space targets p The state of the space target autonomous relative navigation system based on camera bias can be observed by observing the same set of surface key points at least four times in a row.

8. The method for autonomous relative navigation of close-range space targets according to claim 7, characterized in that: Conduct observability analysis on the system model to determine the minimum number of observations of key points on the surface of the space target to make the system model observable, including: The process of proving Theorem 1 is as follows: When the number of consecutive observations is τ N =2, Expressed as: In formula (1.26), superscripts (1) and (2) represent the first observation and the second observation; Transform the equation (1.26) into rows and columns to obtain: From formula (1.25), we can see that O (i+1,i) A total of 15+3n p Therefore, if we want to make the rank condition hold by increasing the number of consecutive observations of the key points of the surface of the space target, the boundary condition of the observability of the autonomous relative navigation system of the space target can be expressed as: γ=rank(O (i+1,i) )=15+3n p (0.27) Formula (1.28) represents the matrix O (i+1,i) The rank is equal to 15+3n p ; for According to formula (1.20): U m Performing eigendecomposition, we can get U m There are two unequal eigenvalues, indicating that U m The rank is 2; from formula (1.27), when the number of consecutive observations N = 2, the observability matrix O (i+1,i) After the row and column transformation, there are 6+3n non-zero columns p Column, non-zero rows have 4n p rows; According to the properties of the matrix rank, when n p When >6, the observability matrix O of formula (1.26) (i+1,i) The rank of γ = 6 + 3n p <15+3n p ; When n p <6, the observability matrix O of formula (1.26) (i+1,i) The rank is γ = 4n p <15+3n p , so the rank condition is not met when the number of observations is 2; When the number of consecutive observations is τ N =3, Expressed as: Transform the equation (1.30) into rows and columns to obtain: in, From (1.31), we can see that after transforming the equation (1.30) into rows and columns, the number of non-zero columns is 12+3n. p Column, non-zero rows have 6n p Line; when n p >6 o'clock, O (i+1,i) The rank of γ = 12 + 3n p <15+3n p , when n p <4 o'clock, O (i+1,i) The rank is γ = 6n p <15+3n p , so when τ N =3 does not meet the rank condition; When the number of consecutive observations is τ N =4, It means that after the row and column transformation, there are 15+3n non-zero columns. p Column, non-zero rows have 8n p OK, when n p >3 o'clock, O (i+1,i) The rank is γ = 15 + 3n p , when n p <3 o'clock, O (i+1,i) The rank is γ = 8n p ≥15+3n p , so when τ N =4 when the rank condition is satisfied; By proving that Theorem 1 holds, the autonomous relative navigation system of space targets based on camera bias can simultaneously observe the unknown n p When observing the surface key points of a space target, the same group of surface key points shall be observed at least 4 times in succession, so that the autonomous relative navigation system of the space target can be fully observed.

9. The method for autonomous relative navigation of close-range space targets according to claim 8, characterized in that: Based on the minimum number of observations of the space target, the number and position of multiple surface key points are optimized to obtain the optimal surface key points. The system model is optimized through the optimal surface key points to obtain the optimized system model, including: By analyzing the relationship between the number of surface key points and the observability of the space target autonomous relative navigation system, surface key points are selected and observed. Under the premise of determining the number of surface key points, the observability of the space target autonomous relative navigation system is used as an indicator to select appropriate surface key points by minimizing the pose estimation covariance. After observing a surface key point four times in a row, the observability of the space target autonomous relative navigation system can be satisfied. Since the position and posture of the space target change continuously over time, the optimal surface key point at a certain moment may not be the optimal when observed continuously at multiple moments. Therefore, the relationship between the measurement residual and the state residual of four consecutive observations is established to optimize the surface key point. Combining equations (1.9) and (1.23), there is the following linear relationship between the measurement residuals and the state residuals of four consecutive observations: Where O represents the observability matrix of 4 observations, see formula (1.22); Using the least squares estimation for equation (1.32), we can get: δx i =(O Τ O) -1OΤ δz (0.32) δx i The covariance matrix of is expressed as: E(δx i (δx i ) Τ )=(The Τ THE) -1 THE Τ E(δz(δz) Τ )O(O Τ THE) -1 (0.33) According to formula (1.18), the measurement noise is independent and identically distributed, and the distribution of the measurement noise satisfies zero mean and variance Therefore, Equation (1.34) can be expressed as: From formula (1.35), we can see that the measurement noise variance is given by (O Τ O) -1 Projected to the navigation state estimation variance, the measurement noise variance When a certain value is given, the navigation error covariance is given by (O Τ O) -1 Decision; (O Τ O) -1 Evaluate the observability of the autonomous relative navigation system of space targets; the smaller the eigenvalue, the higher the estimation accuracy of the corresponding eigenvector in the same direction; the geometric dilution of precision GDOP of the autonomous relative navigation system of space targets is expressed as: traceP=tr((O Τ O) -1 ) (0.35) Where traceP is represented by a matrix (O Τ O) -1 The smaller the traceP is, the smaller the navigation error covariance is, the higher the navigation accuracy is, and the higher the observability of the optical autonomous relative navigation system is. The minimum traceP value is used as the performance index of the preferred surface key point, and the preferred surface key point is described as: for a given n continuous visible surface key points, a predefined n k (n k <n) optimal subset of surface key points, by introducing the indicator variable α m ∈{0,1}(m=1,...,n) constitutes an integer programming problem, the mth indicator variable corresponds to the mth surface key point, and each indicator variable indicates whether the corresponding surface key point is selected into the subset; the integer programming problem is expressed as: The simulated annealing algorithm is used to solve equation (1.37) to obtain the optimal surface key points. The system model is optimized through the optimal surface key points to obtain the optimized system model.

10. The method for autonomous relative navigation of close-range space targets according to claim 9, characterized in that: The simulated annealing algorithm is used to solve Equation (1.37) to obtain the optimal surface key points, including: By initialization, the number of optimal surface key points is defined as n k , the temperature attenuation coefficient is β, optional n k The surface key points are taken as the initial surface key points, and the initial temperature T0 is given. The initial surface key points are taken as the optimal surface key points to calculate ( Τ O)- 1 The trace λ0; Set the current temperature of the simulated annealing algorithm to T, the number of iterations for each temperature T to N, the termination error of the algorithm to ε, and the number of iterations for the temperature T to decrease to L. If the change in GDOP is less than ε after the temperature decreases L times, the algorithm terminates. At the beginning of each iteration, a group of random surface key points are selected as new surface key points in the neighborhood of the initial surface key point, and the corresponding traceP value λ is calculated. If λ < λ0, the new surface key point is taken as the optimal surface key point. Otherwise, the probability of accepting the new surface key point is calculated according to the Metropolis criterion as p = exp(-(λ-λ0) / T). The probability decreases with the decrease of temperature T. If p∈[0,1], the new surface key point is accepted as the optimal surface key point. After the judgment is completed, the next iteration begins. When the number of iterations reaches the optimal number of iterations for each temperature T, the termination condition is checked. If the termination condition is met, the algorithm stops and the current surface key point is used as the optimal surface key point; otherwise, the temperature T is updated according to the annealing strategy. k+1 =βT k , where 0<β<1 represents the temperature attenuation coefficient and k represents the number of iterations.

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