A spatiotemporal identification method of human upper limb extremity stiffness based on block least squares and Gaussian process fitting

Abstract

This invention discloses a spatiotemporal identification method for the stiffness of the extremities of the human upper limb based on block least squares and Gaussian process fitting. The method comprises the following steps: normalizing and gridding the workspace; performing recursive least squares on each subspace based on motion and force data collected during human-computer interaction, and iteratively updating the stiffness vectors within the subspace; and fitting the stiffness using an online Gaussian process with noise. The method enables real-time stiffness identification and continuous identification within the workspace. It can identify the stiffness of the extremities of the human upper limb in one, two, or three dimensions, depending on the application, with high accuracy.

Description

Technical Field

[0001] The present invention belongs to the field of robotics technology and relates to a human upper limb stiffness measurement technology, and in particular to a spatiotemporal identification method of human upper limb end stiffness based on block least squares and Gaussian process fitting. Background Art

[0002] Robotic arm robots are currently widely used in production and life, such as providing daily services and medical care. Considering that robots usually share the workspace with users and conduct human-machine interaction, the robots need to be able to identify the user's upper limb status (especially the stiffness of the upper limb end) in real time and adjust the control strategy to ensure the naturalness and safety of human-machine interaction. Taking the rehabilitation treatment of hemiplegic stroke patients as an example, the patient usually grasps the handle of the tool end of the robotic arm, and the robot provides force through the tool end to assist the patient in task-oriented rehabilitation training. During the treatment process, the robot needs to identify the patient's motor ability to adjust the appropriate level of assistance. Therefore, in rehabilitation treatment, the identification of the stiffness of the patient's upper limb end is particularly important.

[0003] Current research generally uses motion data and force data from the user's human-computer interaction process to identify the stiffness of the upper limb end. These data can be obtained through encoders and force sensors installed on the robotic arm. The stiffness of the human upper limb is generally divided into static stiffness and dynamic stiffness. The perturbation method is usually used to identify static stiffness, that is, the user holds the handle at the end of the robot and remains still. The robot then applies a small perturbation to the end of the upper limb and collects the upper limb interaction force caused by the perturbation. The stiffness parameters are obtained by solving the stiffness model of the upper limb end. For the identification of dynamic stiffness, the user's motion data and force data are usually collected while the user is performing the task, and the user's upper limb stiffness is then identified in real time.

[0004] For real-time considerations, it is often necessary to identify the dynamic stiffness of the user's upper limb end in scenarios such as human-machine collaboration and rehabilitation therapy. Traditional solutions can only identify the change of stiffness over time, but cannot reflect its change in the workspace. In addition, the stiffness usually identified is a scalar value k or a diagonal matrix. It cannot well characterize the stiffness of the upper limb. Summary of the Invention

[0005] In response to the existing upper limb extremity stiffness identification problem, the present invention provides a spatiotemporal identification method for the human upper limb extremity stiffness based on block least squares and Gaussian process fitting. This method can take into account the changes in upper limb extremity stiffness over time and space, realize multi-dimensional upper limb extremity stiffness identification, and achieve high identification accuracy. The method first normalizes and grids the workspace of the user's upper limb, and collects the motion data and force data of the user during the operation task using the robotic arm. The recursive least squares method is used to identify the stiffness of the user's upper limb in each block subspace. Finally, the Gaussian process is used to fit the upper limb stiffness. Based on the fitted Gaussian process regression model, the stiffness at any position in the workspace can be estimated, realizing the spatiotemporal stiffness identification of the human upper limb extremity stiffness.

[0006] In order to achieve the above object, the present invention provides the following technical solutions:

[0007] A method for upper limb stiffness identification based on block least squares and Gaussian process regression includes the following steps:

[0008] Step 1: Normalize and grid the workspace to normalize the user's upper limb workspace Ω to the standard space The normalized workspace is gridded and evenly divided into L subspaces, denoted as Initialize a set of covariance matrices for each subspace and the stiffness vector Used to perform recursive least squares calculations;

[0009] Step 2: Perform recursive least squares on each subspace, and record the sample value of motion data and force data collected at time n during the human-computer interaction as (x n ,Δx n ,f n ), where first the position value x in the sample data is n Normalize to get Then according to the normalized position value Determine whether the sample belongs to the jth subspace Finally, according to the recursive least squares algorithm, the sample (Δx n ,f n ) as new input, the subspace The covariance matrix P within j and the stiffness vector k j Perform iterative update; where Δx n and f n are the upper limb end position deviation and human-machine interaction force at time n respectively;

[0010] Step 3: L subspaces Central location and the stiffness vector k j As a training dataset in K D =[k 1 k 2 …k L ] T ; Note that the noise matrix has a mean of 0 and a variance of Gaussian distribution but I is the identity matrix; then the stiffness vector follows a Gaussian distribution where μ is the mean stiffness, Is the kernel function; when given a normalized data input to be predicted It is expected to estimate the user's upper limb end stiffness k at this position * , according to the Bayesian principle, the posterior mean and covariance of the stiffness to be predicted are:

[0011]

[0012] Where, and are the prior means of the points to be predicted and the training data set respectively;

[0013] The posterior mean μ of the stiffness * Considered as the point to be predicted The stiffness k at * The estimated value of k * =μ * , thereby realizing the identification of stiffness at any position in the workspace.

[0014] Furthermore, in step 1, the workspace is Ω = {x}, which is the set of all reachable positions of the upper limb end. For stiffness identification in one-dimensional, two-dimensional, and three-dimensional cases, x = (x), x = (x, y), and x = (x, y, z), respectively, where x, y, and z represent the positions of the upper limb end in the Cartesian coordinate system. The workspace Ω is normalized using the following formula to obtain the normalized workspace:

[0015]

[0016] Among them, max(x), min(x), max(y), min(y), max(z), and min(z) represent the boundaries of the workspace on different coordinate axes respectively.

[0017] Furthermore, in step 1, the covariance matrix δ is a small positive number, I is the identity matrix; the stiffness vector O is a zero matrix, which means that it is assumed that the stiffness of the user's upper limb end is zero at the beginning.

[0018] Furthermore, in step 2, the recursive least squares algorithm uses the following formula:

[0019]

[0020] Where λ∈(0,1] is the forgetting factor to suppress data saturation, P n =(Δx' T Δx') -1 is the covariance matrix. For two-dimensional stiffness identification, Δx' n ∈{[Δx n,x 0Δx n,y ],[0Δx n,y Δx n,x ]}, f' n ∈{f n,x ,f n,y}; For three-dimensional stiffness identification,

[0021] Δx' n ∈{[Δx n,x 00Δx n,y Δx n,z 0],[0Δx n,y 0Δx n,x 0Δx n,z ],[00Δx n,z 0Δx n,x Δx n,y ]}, f' n ∈{f n,x ,f n,y ,f n,z}.

[0022] Furthermore, the formula used in the recursive least squares algorithm is obtained through the following process:

[0023] First, construct the upper limb end stiffness equation in multidimensional space:

[0024] f=K(x d -x)=KΔx (1)

[0025] Where f∈R m×1 Represents the human-computer interaction force, Δx∈R m×1 is the desired position x of the upper limb end d The difference between the actual position x, K∈R m×m is the stiffness matrix of the upper limb end, and m is the dimension of the stiffness to be identified;

[0026] Consider the least squares calculation of the two-dimensional upper limb end stiffness, assuming the two-dimensional stiffness matrix Is a symmetric matrix, that is, k xy =k yx , rewrite the stiffness matrix into vector form k = [k xx k yy k xy ] T , where k xx and k yy are the stiffness values ​​of the upper limb in the x-axis and y-axis directions, k xy and k yx is the stiffness value of the upper limb coupled on the x-axis and y-axis; when there are n sets of sampled interaction force data {f i =[f i,x ,f i,y ] T |i=1,2,…,n} and position difference data {Δx i =[Δx i,x ,Δx i,y ] T |i=1,2,…,n}

[0027] When the linear regression model F = ΔXk is constructed, where F = [f 1,x … f n,x f 1,y … f n,y ] T and The subscript i in the symbol represents the i-th group of sampled data, and the subscripts x and y represent the components of the data in the x- and y-axis directions, respectively;

[0028] Get the least squares solution:

[0029] K=(ΔX T ΔX) -1 ΔX T F (2)

[0030] Considering the least squares calculation of the three-dimensional upper limb end stiffness, first the stiffness matrix Rewritten as vector form k=[k xx k yy k zz k xy k xz k yz ] T , then build a linear regression model F = ΔXk, where F = [f 1,x …f n,x f 1,y …f n,y f 1,z …f n,z ] T ,

[0031] Its least squares solution is also formula (2). According to formula (2), the recursive least squares formula (3) of the stiffness matrix is ​​obtained.

[0032] Furthermore, in step 3, the kernel function S uses a Gaussian kernel function, that is, Where σ f and l are the hyperparameters of the Gaussian kernel, and the optimal solution of the hyperparameters is obtained by maximum likelihood estimation.

[0033] The beneficial effects of the present invention are:

[0034] 1. This method normalizes and grids the user's upper limb workspace and uses recursive least squares to identify the upper limb extremity stiffness within each subspace. This method, through real-time iteration, allows for the identification of upper limb extremity stiffness over time. A Gaussian process algorithm is then used to fit the stiffness within each subspace, enabling the identification of upper limb extremity stiffness over space. This method achieves high identification accuracy.

[0035] 2. The present invention uses a robotic arm with a force sensor for human-computer interaction and records motion data and force data during the task, thereby identifying the stiffness of the upper limb extremities of the human body in real time. It can realize one-dimensional, two-dimensional or three-dimensional stiffness identification of the upper limb extremities of the human body according to actual applications.

[0036] 3. The method of the present invention can be used to measure the stiffness of the human upper limb extremities in scenarios such as human-machine collaboration and rehabilitation treatment. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 This is a schematic diagram of the spatiotemporal identification method of human upper limb extremity stiffness based on block least squares and Gaussian process fitting proposed by the present invention;

[0038] Figure 2 is a schematic diagram of the motion trajectory and workspace segmentation of human-computer interaction in an example of the present invention;

[0039] Figure 3 This is a simulation result diagram of the stiffness identification of the user's upper limb end in the example of the present invention. Four sub-graphs are used to show the four element values ​​k in the stiffness matrix K. xx , k xy , k yx , k yy Changes within the workspace. The large white circle in the figure represents the desired motion trajectory, and the small white circle marks the stiffness identification value at that position. DETAILED DESCRIPTION

[0040] The technical solutions provided by the present invention will be described in detail below with reference to specific embodiments. It should be understood that the following specific embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention.

[0041] The spatiotemporal identification method of human upper limb end stiffness based on block least squares and Gaussian process fitting proposed in this invention has the following principles and steps: Figure 1 As shown in Figure 2, the identification of the two-dimensional upper limb end stiffness matrix during robot-assisted rehabilitation training is used as an example. Figure 2 A schematic diagram shows a user performing a circular motion with the assistance of a robotic arm. The method of the present invention collects motion data and force data during human-computer interaction to achieve spatiotemporal identification of the stiffness of the user's upper limb extremities. Specifically, the method of the present invention includes the following steps:

[0042] Step 1: Normalize and grid the workspace. First, denote the workspace of the user's upper limb as Ω = {x}, which is the set of all reachable positions of the upper limb end. For stiffness identification in one-dimensional, two-dimensional, and three-dimensional cases, denote x = (x), x = (x, y), and x = (x, y, z), respectively, where x, y, z represent the position of the upper limb end in the Cartesian coordinate system. In order to facilitate subsequent calculations, the workspace of the user's upper limb Ω = {x|(x, y)} is normalized to the standard space The normalization function is as follows:

[0043]

[0044] Among them, max(x), min(x), max(y), and min(y) represent the boundaries of the workspace on different coordinate axes respectively.

[0045] The normalized workspace is grid-divided into L subspaces, which are recorded as like Figure 2 As shown in the figure, in this example, the normalized working space is divided into 10×10 subspaces, and each subspace is numbered from left to right and from top to bottom, that is, L=100. Then, a set of covariance matrices is initialized for each subspace. and the stiffness vector Used to perform recursive least squares calculations. δ is a small positive number, I is the identity matrix, O is a zero matrix, which means that the stiffness of the user's upper limb end is assumed to be zero at the beginning to ensure that the robot can provide sufficient auxiliary force at the beginning. The estimated value of the stiffness is subsequently continuously adjusted through the recursive least squares algorithm.

[0046] Step 2: Iteratively update the stiffness vector in the subspace based on the motion data and force data collected during the human-computer interaction process to achieve real-time identification of stiffness. Assume that a set of sample data (x n ,Δx n ,f n ), where x n After normalization, (Δx n ,f n )=([0.02,0.05] T ,[5,14.6] T ), then the sampling position belongs to the subspace And we can get two sets of data for recursive least squares (Δx',f') = ([0.02, 0, 0.05], 5) and (Δx',f') = ([0, 0.05, 0.02], 14.6). Substituting these two sets of data into the recursive least squares formula (3), the subspace can be updated. The covariance matrix P within 23 and the stiffness vector k 23 .

[0047] Specifically, the detailed description of recursive least squares is as follows:

[0048] First, construct the upper limb end stiffness equation in multidimensional space:

[0049] f=K(x d -x)=KΔx (1)

[0050] Where f∈R m×1 Represents the human-computer interaction force, Δx∈R m×1 is the desired position x of the upper limb end d The difference between the actual position x, K∈R m×m is the stiffness matrix of the upper limb end, and m is the dimension of the stiffness to be identified.

[0051] One-dimensional upper limb extremity stiffness is a scalar quantity, making calculation relatively simple. However, for two- or three-dimensional upper limb extremity stiffness identification, a simplified approach is to assume that stiffness is decoupled in different dimensions. Based on this assumption, the resulting stiffness matrix is ​​a diagonal matrix. While assuming that upper limb extremity stiffness is decoupled in different dimensions can simplify the calculation process, some stiffness information is lost. In fact, relevant research indicates that the human upper limb extremity stiffness matrix is ​​a symmetric matrix. Therefore, the present invention conducts subsequent calculations based on the assumption that the stiffness matrix K is a symmetric matrix.

[0052] Consider the least squares calculation of the two-dimensional upper limb end stiffness, and record the two-dimensional stiffness matrix as where k xx and kyy are the stiffness values ​​of the upper limb in the x-axis and y-axis directions, k xy and k yx is the stiffness value of the upper limb coupled on the x-axis and y-axis, because the stiffness matrix K is a symmetric matrix, that is, k xy =k yx In order to facilitate subsequent calculations, the stiffness matrix is ​​rewritten as a vector form k = [k xx k yy k xy ] T . Then when there are n groups of sampled interaction force data {f i =[f i,x ,f i,y ] T |i=1,2,…,n} and position difference data {Δx i =[Δx i,x ,Δx i,y ] T |i=1,2,…,n}, a linear regression model F=ΔXk can be constructed, where F=[f 1,x f n,x f 1,y …f n,y ] T and The subscript i in the symbol represents the i-th group of sampled data, and the subscripts x and y represent the components of the data in the x- and y-axis directions, respectively.

[0053] We can further obtain the least squares solution:

[0054] K=(ΔX T ΔX) -1 ΔX T F (2)

[0055] Considering the least squares calculation of the three-dimensional upper limb end stiffness, the processing flow is similar to that of the two-dimensional one. First, the stiffness matrix Rewritten as vector form k=[k xx k yy k zz k xy k xz k yz ] T , then build a linear regression model F = ΔXk, where F = [f 1,x …f n,x f 1,y …f n,y f 1,z …f n,z ] T , Its least squares solution is also equation (2).

[0056] In order to realize the real-time least square solution of stiffness matrix in the process of human-computer interaction, the recursive least square formula of stiffness matrix can be obtained according to formula (2):

[0057]

[0058] Where λ∈(0,1] is the forgetting factor to suppress data saturation, P n =(Δx' T Δx') -1 Is the covariance matrix, used as an intermediate variable for auxiliary iterative calculation. For two-dimensional stiffness identification, Δx' n ∈{[Δx n,x 0Δx n,y ],[0Δx n,y Δx n,x ]}, f' n ∈{f n,x ,f n,y For three-dimensional stiffness identification,

[0059] Δx' n ∈{[Δx n,x 00Δx n,y Δx n,z 0],[0Δx n,y 0Δx n,x 0Δx n,z ],[00Δx n,z 0Δx n,x Δx n,y ]}, f' n ∈{f n,x ,f n,y ,f n,z}.

[0060] Step 3, Gaussian process fitting, taking into account each subspace The stiffness vector identified internally is {k j |j=1,2,...L} is discrete, so an online Gaussian process with noise is used to fit the stiffness to achieve continuous identification of the stiffness in the workspace. Central location and the stiffness vector k j As a training dataset Perform Gaussian process fitting, where K D =[k 1 k 2 …k L ] T In this example, the normalized working space is divided into 10×10 subspaces, so K D=[k 1 k 2 …k 100 ] T ∈R 100×3 . Note that the noise matrix has a mean of 0 and a variance of Gaussian distribution but Then the stiffness vector follows a Gaussian distribution where μ is the mean stiffness, is the kernel function. In this example, the kernel function S uses the Gaussian kernel function, that is, Where σ f and l are the hyperparameters of the Gaussian kernel, and the optimal solution of the hyperparameters can be obtained by maximum likelihood estimation. It is expected to estimate the user's upper limb end stiffness k at this position * , according to the Bayesian principle, the posterior mean and covariance of the stiffness to be predicted can be obtained as:

[0061]

[0062] Where, and are the prior means of the points to be predicted and the training data set, respectively.

[0063] Then the posterior mean of stiffness μ can be * Considered as the point to be predicted The stiffness k at * The estimated value of k * =μ * , which makes it possible to identify the stiffness at any position in the workspace.

[0064] In this example, when using Gaussian process to estimate the stiffness of the upper limb end, the prior mean of the predicted point in formula (4) is and the prior mean of the training dataset are zero vector and zero matrix respectively. That is, considering the actual situation of robot-assisted rehabilitation training, it is assumed a priori that the user's stiffness is zero to ensure that the robot can provide sufficient assistance at the beginning of the task.

[0065] When identifying the upper limb end stiffness, first give a normalized position point to be predicted in the workspace According to the training set D and Gaussian process formula (4), the stiffness estimation vector of the position point can be obtained This vector can be used directly for robot-assisted training calculations or rewritten as a matrix It is used for subsequent calculations to realize the stiffness identification of the upper limb end at any position in the workspace.

[0066] Matlab was used to simulate the robot assisting the user in drawing a circle. The radius of the circle was normalized to 0.8, and the stiffness matrix of the user's upper limb end in the first quadrant was simulated to be In the remaining quadrants The stiffness identification results after block least squares and Gaussian process fitting using the method proposed in this patent are as follows: Figure 3 As shown, the four sub-graphs show the four element values ​​k in the stiffness matrix K. xx , k xy , k yx , k yy The figure shows the changes in the workspace, and the stiffness identification values ​​at certain locations are marked with white circles. It can be seen from the figure that this method can effectively identify the changes in the stiffness of the upper limb end with the workspace, and the identification accuracy is high.

[0067] The technical means disclosed in the solutions of the present invention are not limited to those disclosed in the above-mentioned embodiments, but also include technical solutions composed of any combination of the above-mentioned technical features. It should be noted that those skilled in the art may make various improvements and modifications without departing from the principles of the present invention, and such improvements and modifications are also considered to be within the scope of protection of the present invention.

Claims

1. A method for upper limb stiffness identification based on block least squares and Gaussian process regression, characterized in that: The steps include: Step 1: Normalize and grid the workspace to normalize the user's upper limb workspace Ω to the standard space The normalized workspace is gridded and evenly divided into L subspaces, denoted as Initialize a set of covariance matrices for each subspace and the stiffness vector Used to perform recursive least squares calculations; Step 2: Perform recursive least squares on each subspace, and record the sample value of motion data and force data collected at time n during the human-computer interaction as (x n ,Δx n ,f n ), where first the position value x in the sample data is n Normalize to get x n , and then according to the normalized position value Determine whether the sample belongs to the jth subspace Finally, according to the recursive least squares algorithm, the sample (Δx n ,f n ) as new input, the subspace The covariance matrix P within j and the stiffness vector k j Perform iterative update; where Δx n and f n are the upper limb end position deviation and human-machine interaction force at time n respectively; Step 3: L subspaces Central location and the stiffness vector k j As a training dataset in Note that the noise matrix has a mean of 0 and a variance of Gaussian distribution but I is the identity matrix; then the stiffness vector follows a Gaussian distribution where μ is the mean stiffness, Is the kernel function; when given a normalized data input to be predicted It is expected to estimate the user's upper limb end stiffness k at this position * , according to the Bayesian principle, the posterior mean and covariance of the stiffness to be predicted are: Where, and are the prior means of the points to be predicted and the training data set respectively; The posterior mean μ of the stiffness * Considered as the point to be predicted The stiffness k at * The estimated value of k * =μ * , thereby realizing the identification of stiffness at any position in the workspace.

2. The upper limb stiffness identification method based on block least squares and Gaussian process regression according to claim 1 is characterized in that: In step 1, the workspace is Ω = {x}, which is the set of all reachable positions of the upper limb end. For stiffness identification in one-dimensional, two-dimensional, and three-dimensional cases, x = (x), x = (x, y), and x = (x, y, z), respectively, where x, y, and z represent the positions of the upper limb end in the Cartesian coordinate system. The workspace Ω is normalized using the following formula to obtain the normalized workspace: Among them, max(x), min(x), max(y), min(y), max(z), and min(z) represent the boundaries of the workspace on different coordinate axes respectively.

3. The upper limb stiffness identification method based on block least squares and Gaussian process regression according to claim 1 is characterized in that: In step 1, the covariance matrix P0 j =δI, δ is a small positive number, I is the identity matrix; Stiffness vector O is a zero matrix, which means that it is assumed that the stiffness of the user's upper limb end is zero at the beginning.

4. The upper limb stiffness identification method based on block least squares and Gaussian process regression according to claim 1 is characterized in that: In step 2, the recursive least squares algorithm uses the following formula: Where λ∈(0,1] is the forgetting factor to suppress data saturation, P n =(Δx' T Δx') -1 is the covariance matrix. For two-dimensional stiffness identification, Δx' n ∈{[Δx n,x 0Δx n,y ],[0Δx n,y Δx n,x ]}, f' n ∈{f n,x ,f n,y }; For three-dimensional stiffness identification, Δx' n ∈{[Δx n,x 00Δx n,y Δx n,z 0],[0Δx n,y 0Δx n,x 0Δx n,z ],[00Δx n,z 0Δx n,x Δx n,y ]},f' n ∈{f n,x ,f n,y ,f n,z }。 5. The upper limb stiffness identification method based on block least squares and Gaussian process regression according to claim 4 is characterized in that: The formula used in the recursive least squares algorithm is obtained through the following process: First, construct the upper limb end stiffness equation in multidimensional space: f=K(x d -x)=KΔx (1) Where f∈R m×1 Represents the human-computer interaction force, Δx∈R m×1 is the desired position x of the upper limb end d The difference between the actual position x, K∈R m×m is the stiffness matrix of the upper limb end, and m is the dimension of the stiffness to be identified; Consider the least squares calculation of the two-dimensional upper limb end stiffness, assuming the two-dimensional stiffness matrix Is a symmetric matrix, that is, k xy =k yx , rewrite the stiffness matrix into vector form k = [k xx k yy k xy ] T , where k xx and k yy are the stiffness values ​​of the upper limb in the x-axis and y-axis directions, k xy and k yx is the stiffness value of the upper limb coupled on the x-axis and y-axis; when there are n sets of sampled interaction force data {f i =[f i,x ,f i,y ] T |i=1,2,…,n} and position difference data {Δx i =[Δx i,x ,Δx i,y ] T |i=1,2,…,n}, construct a linear regression model F=ΔXk, where F=[f 1,x …f n,x f 1,y …f n,y ] T and The subscript i in the symbol represents the i-th group of sampled data, and the subscripts x and y represent the components of the data in the x- and y-axis directions, respectively; Get the least squares solution: K=(ΔX T (ΔX) -1 ΔX T F (2) Considering the least squares calculation of the three-dimensional upper limb end stiffness, first the stiffness matrix Rewritten as vector form k=[k xx k yy k zz k xy k xz k yz ] T , then build a linear regression model F = ΔXk, where F = [f 1,x …f n,x f 1,y …f n,y f 1,z …f n,z ] T , Its least squares solution is also formula (2). According to formula (2), the recursive least squares formula (3) of the stiffness matrix is ​​obtained.

6. The upper limb stiffness identification method based on block least squares and Gaussian process regression according to claim 1 is characterized in that: In step 3, the kernel function S uses the Gaussian kernel function, that is, Where σ f and l are the hyperparameters of the Gaussian kernel, and the optimal solution of the hyperparameters is obtained by maximum likelihood estimation.

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