Training method of KAN model based on budgeted boundary function selection, end position prediction method of mechanical arm and mechanical arm

Abstract

This invention relates to the field of robotic arm control technology, and particularly to a training method for a KAN model based on budgeted edge function selection, a method for predicting the end effector position of a robotic arm, and a robotic arm itself. The training method includes: acquiring a training dataset and a reference model for the robotic arm joints; iteratively training the reference model using the training dataset to obtain a target model, which includes a family of target affine functions for each edge; after convergence of the iterative training, constructing a spectral complexity certificate for the reference model, determining the certificate weight for each edge of the reference model, and projecting each edge by least squares onto an elementary function menu to obtain the affine function menu and projection residual for each edge; and determining the family of target affine functions for each edge from the affine function menu based on the projection residual and a preset budget multiplier. Through this training method, each edge of the model is a readable closed-form expression, enhancing the interpretability of the model and improving the prediction efficiency of the robotic arm's end effector position.

Description

Technical Field

[0001] This invention relates to the field of robotic arm control technology, and in particular to a training method for a KAN model based on budgeted side function selection, a method for predicting the end-effector position of a robotic arm, and a robotic arm. Background Technology

[0002] End-effector position prediction (i.e., forward kinematics solution) for multi-joint robotic arms is a core component of robot motion planning, servo control, and state monitoring. In practical engineering deployments, the end-effector position prediction model (i.e., kinematics prediction model) typically needs to run on target platforms such as embedded robot controllers, edge computing units, or high-frequency monitoring loops. These target platforms impose strict limitations on the number of parameters, memory usage, and inference latency of the prediction model. Furthermore, robotic arm applications are widely involved in reliability-critical scenarios such as human-robot collaboration and industrial safety, requiring the deployed model to be auditable. This means that engineers can examine the model's internal expressions item by item, verify them at operational limits, and trace the prediction results back to the physical mechanism, rather than treating the model as a black box to be trusted. Currently, data-driven proxy models for forward kinematics of robotic arms mainly fall into three categories: (1) Multilayer Perceptron (MLP) surrogate model. MLP is composed of dense linear transformation and fixed node activation functions in concatenation, used to fit the mapping from the joint angle of a robotic arm to the end position. The disadvantages of the MLP model are: First, the hidden units and weight matrices do not have readable physical meaning, the model is completely black box, cannot be audited item by item, and it is difficult to meet the interpretability requirements of safety-critical scenarios. Second, the parameter efficiency per unit of accuracy is not high under small parameter budget, making it difficult to balance accuracy and resource constraints in embedded controllers.

[0003] (2) Gradient boosting tree (e.g., XGBoost) surrogate models. Gradient boosting tree models are a strong baseline for medium-sized tabular regression tasks. The disadvantages of this type of model are: First, the ensemble of hundreds of deep decision trees is also uninterpretable and cannot provide a closed-form expression. Second, the inference process of the model requires traversing each tree, resulting in high inference latency per sample, which is not conducive to high-frequency control loops. Third, tree models lack inductive bias for the inherent sine / cosine periodic combination structure in the forward kinematics of the robotic arm, which significantly limits accuracy in kinematic tasks dominated by trigonometric function composition.

[0004] Therefore, existing methods for predicting the end effector position of robotic arms suffer from problems such as lack of interpretability and low prediction efficiency. Summary of the Invention

[0005] This application provides a training method for a KAN model based on budgeted edge function selection, a method for predicting the end effector position of a robotic arm, and a robotic arm. It addresses the technical problems of uninterpretability and low prediction efficiency in existing robotic arm end effector position prediction methods. It enables the transformation of any trained B-spline KAN model into a compact network where each edge is a named closed elementary function without retraining from scratch, enhancing model interpretability. Furthermore, it differentiates the pricing of edge parameters through the model's spectral complexity certificate and selects the function family edge-by-edge under explicit global parameter budgeting. This achieves high accuracy, low latency, low parameter count, and edge-by-edge auditable robotic arm end effector position prediction under embedded resource constraints, thereby improving the model's prediction efficiency.

[0006] In a first aspect, embodiments of the present invention provide a training method for a KAN model based on budgeted side function selection, comprising: Obtain the training dataset and reference model for the robotic arm joints, wherein each training data point in the training dataset is a normalized sample pair. The Let the joint angle vector of the robotic arm be a sampled value. The standardized label for the position of the end effector of the robotic arm at the same sampling time is the reference model, which is the reference B-spline KAN model. The reference model is iteratively trained using the training dataset until the iteratively trained reference model meets the convergence condition. The reference model that meets the convergence condition is then determined as the target model. The target model includes a family of target affine functions for each edge. After the iterative training satisfies the convergence condition, the spectral complexity certificate of the reference model is constructed, the certificate weight of each edge of the reference model is determined, and the edge is least squared projected to the elementary function menu to obtain the affine function menu and projection residual of each edge. Based on the projection residual of each edge and the preset budget multiplier, the target affine function family of each edge is determined from the affine function menu of each edge.

[0007] Preferably, the step of constructing the spectral complexity certificate of the reference model and determining the certificate weight of each edge of the reference model includes: Construct the Lipschitz matrix for each layer based on the unary edge activation function of each edge in each layer of the reference model; The spectral complexity certificate is obtained based on the Lipschitz matrix of each layer; Based on the spectral complexity certificate, the certificate weight of each edge is obtained.

[0008] Preferably, the spectral complexity certificate is obtained based on the Lipschitz matrix of each layer, as shown in the following formula: ; in, The spectral complexity certificate is the product of the maximum singular values ​​of each layer, and β is a configuration representing the set of function family choices for all edges of the reference model. Represents the maximum singular value. For the reference model of the first The Lipschitz matrix of the layer, where L is the total number of layers in the reference model; For each of the aforementioned edges Based on the spectral complexity certificate, the certificate weight of each edge is obtained, as shown in the following formula: ; in, The first part of the reference model represents the first part. layer, For the first The node number on the input side of the layer. For the first The node number on the output side of the layer. For the edge Certificate weight, = For the edge The Lipschitz constant of the unary edge activation function in its domain (i.e., the element in the i-th row and j-th column of the Lipschitz matrix). For the edge The unary edge activation function on, for The largest left singular vector, for The largest right singular vector.

[0009] Preferably, the step of projecting each edge by least squares onto the elementary function menu to obtain the affine function menu and projection residual for each edge includes: Affine function menu for each of the aforementioned edges for: ; in, Let be the input scalar for this edge. for A family of functions, The elementary function menu includes multiple preset function families. a family of functions The basic function form, For input scaling parameters, To input translation parameters, For amplitude parameters, This is the output bias parameter.

[0010] On each of the aforementioned edges, based on each of the aforementioned function families, in the domain A predetermined number of grid points are selected, and the univariate edge activation function curve of the edge is fitted using least squares to obtain the projection residual of the edge. , For the edge Projection to the family of functions The least squares fitting residual in the sense of the second norm (i.e., the square integral norm).

[0011] Preferably, determining the target affine function family for each edge from the affine function menu for each edge, based on the projected residual of each edge and a preset budget multiplier, includes: Based on the certificate weight and projection residual of each edge, and the preset budget multiplier, the certificate weighting rule is obtained, and then the target affine function family of each edge is obtained according to the certificate weighting rule. The certificate weighting rule is as follows: ; in, a family of functions The effective degrees of freedom after removing redundancy from reparameterization. For the edge The family of target affine functions.

[0012] Preferably, determining the target affine function family for each edge from the affine function menu for each edge, based on the projected residual of each edge and a preset budget multiplier, includes: Based on the projection residual of each edge and the preset budget multiplier, a uniform rule is obtained, and then based on the uniform rule, the target affine function family of each edge is obtained. The uniformity rule is as follows: ; in, a family of functions The effective degrees of freedom after removing redundancy from reparameterization. For the edge The family of target affine functions.

[0013] Preferably, it further includes: obtaining an explicit PAC-Bayes generalization upper bound expression based on a finite set of configurations; the finite set of configurations The configuration consists of all candidate configurations generated by running the certificate weighting rule and the uniformity rule on each value of the preset budget multiplier grid, or may also be incorporated into a single function family configuration. The number of configurations in the finite configuration set. For configuration, the configuration represents the set consisting of the function family selections for all edges; The explicit PAC-Bayes generalization upper bound expression is: ; in, In order to be in The above is based on the certificate weighting rules from the finite set of configurations. The selected configuration, For a unified notation of the model, any configuration The corresponding KAN network model is denoted as (The basic function form whose subscript in the function expression is the family of functions) (Distinguish between the meanings of the subscripts) For selected configuration The corresponding edge-by-edge closed network model (i.e., the target model before fine-tuning), where n is the total number of sample pairs in the training dataset, and the first edge-by-edge closed network model in the training dataset. Each sample pair is denoted as , Used to construct the PAC-Bayes prior distribution Each sample pair is denoted as , Used to calculate empirical risk For the model Overall risk For the model exist Experience and risk For only Configuration of the above calculation The projection residual, For configuration The total effective degrees of freedom (i.e., parameter budget). Let be the normalization constant of the PAC-Bayes prior distribution. , The a priori temperature index, This is a confidence parameter.

[0014] Preferably, it also includes: obtaining the minimum projection residual and the median of the projection residual for each edge during the process of edge-by-edge least squares projection onto the elementary function menu; The median of the projected residual is compared with a preset first residual threshold and a second residual threshold, wherein the first residual threshold is less than the second residual threshold; When the median of the projection residuals is not greater than the first residual threshold, it is determined that the elementary function menu can fully cover the target mapping (the target mapping is the mapping from the joint angle vector to the position of the robotic arm end effector, as characterized by the training dataset), and the trained target model is deployed. When the median of the projected residual is not less than the second residual threshold, it is determined that the elementary function menu is insufficient, and the iterative training is re-executed after expanding the elementary function menu or using richer basis functions; When the median of the projected residuals is between the first residual threshold and the second residual threshold, it is determined that the elementary function menu is substantially sufficient, and the trained target model is deployed. Furthermore, when the minimum projected residual of any edge exceeds a preset edge residual threshold, the edge index is used to determine the appropriate method. Locate the corresponding joint pathway and expand the elementary function menu or increase the data sampling density for that joint pathway.

[0015] Based on the same inventive concept, in a second aspect, the present invention also provides a method for predicting the end-effector position of a robotic arm, comprising: Obtain the input vector and target model at the current moment, wherein the input vector is the joint angle vector of the robotic arm, and the target model is a model trained by the training method based on the budgeted side function selection of the KAN model as described in the first aspect; The target model is used to process the input vector to output the predicted position value of the robotic arm's end effector.

[0016] Based on the same inventive concept, in a third aspect, the present invention provides a robotic arm for performing the robotic arm end-effector position prediction method described in the second aspect.

[0017] One or more technical solutions in the embodiments of the present invention have at least the following technical effects or advantages: In this embodiment of the invention, a pre-trained reference B-spline KAN model is used as the reference model, and iterative training is performed on the reference model using a training dataset. After convergence of the iterative training, the activation function of each edge is projected onto a menu composed of elementary function families (i.e., elementary function menu) using least squares. A spectral complexity certificate based on the product of the maximum singular values ​​of the layer-by-layer Lipschitz matrices is constructed, and the effective degrees of freedom of each edge are priced differently based on the first-order sensitivity of each edge to the certificate (i.e., certificate weight). A unique function family is selected edge by edge under global parameter budget constraints. The reference model that satisfies the convergence condition is determined as the target model when the convergence condition is met. After fine-tuning, the target model is deployed on an embedded robotic arm controller for online prediction of the end effector position.

[0018] Therefore, the training / prediction method of this invention can transform any trained B-spline KAN model into a compact network where each edge is a named closed elementary function without retraining from scratch. That is, each edge of the model is a readable closed expression, auditable edge-by-edge and verifiable against kinematic physics, enhancing the model's interpretability. By using spectral complexity certificates to differentiate the parameters of each edge and selecting function families edge-by-edge under explicit global parameter budgeting, high accuracy, low latency, low parameter count, and edge-by-edge auditable robotic arm end-effector position prediction are simultaneously achieved under embedded resource constraints, improving the model's prediction efficiency. Furthermore, the training method of this invention can also provide menu sufficiency diagnostic indicators that can be computed before deployment. Attached Figure Description

[0019] Various other advantages and benefits will become apparent to those skilled in the art upon reading the following detailed description of preferred embodiments. The accompanying drawings are for illustrative purposes only and are not intended to limit the invention. Furthermore, the same reference figures denote the same parts throughout the drawings. In the drawings: Figure 1 A schematic diagram of the steps of the training method for the KAN model based on budgeted side function selection in an embodiment of the present invention is shown. Figure 2 A schematic diagram of the network structure of the reference model (reference B-spline KAN) before transformation is shown in an embodiment of the present invention; Figure 3 A schematic diagram of the network structure of the transformed target model (BudgetKAN) in an embodiment of the present invention is shown; Figure 4 A flowchart illustrating the steps of the robotic arm end-effector position prediction method in an embodiment of the present invention is shown. Detailed Implementation

[0020] Exemplary embodiments of the present disclosure will now be described in more detail with reference to the accompanying drawings. While exemplary embodiments of the present disclosure are shown in the drawings, it should be understood that the present disclosure may be implemented in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the disclosure to those skilled in the art.

[0021] Example 1 The first embodiment of the present invention provides a training method for a KAN model based on budgeted side function selection, such as... Figure 1 As shown, it includes: S101, Obtain the training dataset and reference model for the robotic arm joints, where each training data point in the training dataset is a normalized sample pair. , Let be the joint angle vector of the robotic arm at a certain sampling time. The standardized labels are for the position quantities of the end effector of the robotic arm at the same sampling time, and the reference model is the reference B-spline KAN model. S102, the reference model is iteratively trained using the training dataset until the iteratively trained reference model meets the convergence condition. The reference model that meets the convergence condition is then determined as the target model. The target model includes the target affine function family for each edge. S103. After the iterative training satisfies the convergence condition, construct the spectral complexity certificate of the reference model, determine the certificate weight of each edge of the reference model, and project the edge-by-edge least squares onto the elementary function menu to obtain the affine function menu and projection residual of each edge. S104. Based on the projection residual of each edge and the preset budget multiplier, determine the target affine function family for each edge from the affine function menu of each edge.

[0022] In this embodiment, a pre-trained reference B-spline KAN model is used as the reference model, and iterative training is performed on the reference model using a training dataset. After convergence of the iterative training, the activation function of each edge is projected onto a menu composed of elementary function families (i.e., elementary function menu) using least squares. A spectral complexity certificate based on the product of the maximum singular values ​​of the layer-by-layer Lipschitz matrices is constructed, and the effective degrees of freedom of each edge are priced differently based on the first-order sensitivity of each edge to the certificate (i.e., certificate weight). A unique function family is selected edge by edge under global parameter budget constraints. This process continues until the reference model trained iteratively meets the convergence condition, at which point the reference model that meets the convergence condition is determined as the target model. After fine-tuning, the target model is deployed on an embedded robotic arm controller for online prediction of the end effector position.

[0023] Therefore, the training method in this embodiment can transform any trained B-spline KAN model into a compact network where each edge is a named closed elementary function without retraining from scratch. That is, each edge of the model is a readable closed expression, auditable edge-by-edge and verifiable against kinematic physics, enhancing the model's interpretability. By using spectral complexity certificates to differentiate the pricing of parameters on each edge and selecting function families edge-by-edge under explicit global parameter budgeting, high accuracy, low latency, low parameter count, and edge-by-edge auditable robotic arm end-effector position prediction are simultaneously achieved under embedded resource constraints, improving the model's prediction efficiency. Furthermore, the training method in this embodiment can also provide menu sufficiency diagnostic indicators that can be computed before deployment.

[0024] Below, in conjunction with Figure 1 This embodiment details the specific process of training the KAN model based on budgeted side function selection: First, step S101 is executed to obtain the training dataset and reference model for the robotic arm joints, wherein each training data point in the training dataset is a normalized sample pair. , Let be the joint angle vector of the robotic arm at a certain sampling time. The standardized labels are for the position quantities of the end effector of the robotic arm at the same sampling time, and the reference model is the reference B-spline KAN model.

[0025] Specifically, the process of obtaining the training dataset for the robotic arm joints involves collecting data through the robotic arm's joint encoders. Joint angle data for each joint (taking an eight-bar linkage robotic arm as an example). Simultaneously record (or measure by a high-precision motion capture / calibration device) the position of the corresponding robotic arm end effector. This positional quantity is the Euclidean distance from the end effector reference point to a preset fixed reference point, or the Cartesian coordinate component of the end effector in the base coordinate system. The joint angle data is standardized, and each joint angle data point is mapped to the network domain via an affine transformation. In the process, the joint angle vector is obtained. Based on the joint angle vectors and the position quantities of the robotic arm's end effector collected at the same sampling time, sample pairs are obtained. This process yields the training dataset. Sample pairs can also be divided into training, validation, and test datasets according to a preset ratio. For example, the dataset can be divided into training / validation / test sets at 60% / 20% / 20%, and the preset ratio can be set according to actual needs. The purpose of this step is to provide normalized sample pairs for reference model training and subsequent projection selection. .

[0026] A sample pair is denoted as . Let be the input vector of the model at the same sampling time, where the th is... Each component For the robotic arm at this sampling time, the first... The vector elements corresponding to the joint angle data of each joint, i.e., the joint rotation angles read from the joint encoder. Mapped to the network domain by affine transformation The dimensionless value after that. This refers to the standardized label for the position of the robotic arm's end effector at the same sampling time. Specifically, it is taken as the Euclidean distance from the end effector reference point to a pre-set fixed reference point, or it can be taken as the Cartesian coordinate component of the end effector in the base coordinate system. The total number of sample pairs is denoted as n, and the first... Each sample pair is denoted as , Used to construct the PAC-Bayes prior distribution. Each sample pair is denoted as , Used to calculate empirical risk.

[0027] The process of obtaining the reference model involves building a model with a depth (i.e., the total number of layers in the model) of... KAN network ,in For configuration, that is, the set of function families selected from all edges of the entire network (to avoid the input translation parameters of the edge functions). Obfuscation, configuration using Greek letters (Representation); Both the reference model and the target model can be uniformly written in the above composite form, with only the configuration of the two being different. The difference lies in the configuration: the reference model is configured as edge-wise B-splines, while the target model is configured as an edge-wise closed family of elementary functions selected according to the following steps. .in, Or it can be set according to actual needs. The specific value. This is the layer number. (Number) The layer mapping relationship is denoted as , Indicates will dimensional vector mapping to Dimensional vector. This refers to the input dimension of the model network, which is the number of joints in the robotic arm involved in the modeling. Taking an eight-link robotic arm as an example... According to the first Layer mapping relationship Apply a unary edge activation function to each input node. And sum them: (1); In the In the layer, This is the index of the node on the input side of this layer (i.e., the node in the previous layer). This refers to the index of the node on the output side of this layer (i.e., the node in the next layer); triplet An edge that is a unique identifier for the reference model, namely "the first" edge. Layer The input node points to the first Layer The edges connecting the "output nodes". (This appears throughout the example.) , (including unary edge activation functions) Lipschitz matrix elements Certificate weight Projection residuals Select a family of functions Singular vector components and (etc.) all adopt the same definition as above: The node number is always the one on the output side (i.e., the row number of the matrix). The node indices are always the input side (i.e., matrix columns), and their meaning remains unchanged throughout any step. When the layer number is not required, the edge... Abbreviated as .

[0028] Each edge in the reference model Pick Cubic B-splines for each interval are superimposed with learnable SiLU residual terms. This represents the number of spline intervals (i.e., nodes). As an example, the reference model employs a two-layer structure. Hide width From the verification set The (determination coefficient) is selected, the final layer output dimension is 1, corresponding to the predicted output of a single terminal position, and the Adam optimizer (learning rate) is used. Weight decay (Cosine annealing) Training until convergence. The purpose of this step is to obtain a high-precision reference fit for the kinematic mapping of the robotic arm, which serves as the "teacher" for subsequent edge-by-edge projection. This step can also be omitted, and an existing, deployed, or third-party trained B-spline KAN model can be directly used, which reflects the "retraining-free transformation" feature of this invention.

[0029] Specifically, the convergence condition can be any one or a combination of the following conditions: ① The number of training epochs reaches a preset maximum; ② The loss on the validation dataset does not improve within a preset number of consecutive epochs, i.e., the early stopping condition. For example, if the early stopping patience value is set to 50, the convergence condition is satisfied when the loss on the validation dataset does not decrease for 50 consecutive training epochs; ③ The change in training loss between adjacent training epochs is less than a preset change threshold. The preset change threshold can be set according to actual needs. As an example, the reference model uses the Adam optimizer (learning rate...). Weight decay The model is trained using a cosine annealing learning rate scheduler, with condition ② above as the convergence criterion. In practice, early stopping occurs between approximately 360 and 1000 training epochs, and the loss on the validation dataset plateaus within 5% of its final value between approximately 200 and 890 training epochs. Meeting the convergence criterion indicates successful model training: for each edge of the reference model that meets the convergence criterion, edge-by-edge function family selection is performed based on the certificate weights and projected residuals calculated all at once after training. The resulting target affine function family for each edge constitutes the target model.

[0030] After the iterative training meets the convergence condition, step S103 is executed to construct the spectral complexity certificate of the reference model, determine the certificate weight of each edge of the reference model, and project each edge by least squares onto the elementary function menu to obtain the affine function menu and projection residual of each edge.

[0031] Specifically, the process of constructing the spectral complexity certificate of the reference model and determining the certificate weight of each edge of the reference model is as follows: The first step is to construct the Lipschitz matrix for each layer based on the unary edge activation function of each edge in each layer of the reference model. Specifically, for each layer... Constructing the Lipschitz matrix layer by layer Lipschitz matrix The size is . The element (i.e., the element in the i-th row and j-th column of the matrix) is the Lipschitz constant of the unary edge activation function for that edge. Based on the Lipschitz constant of the unary edge activation function for each edge in each layer, the Lipschitz matrix of each layer is obtained. .

[0032] The second step is to obtain the spectral complexity certificate based on the Lipschitz matrix of each layer. Specifically, the network-level spectral complexity certificate (a computationally calculable and verifiable upper bound of model complexity) is defined as the product of the maximum singular values ​​of each layer: (2); in, Let β be the spectral complexity certificate, and β be the configuration, which represents the set of function family choices for all edges of the reference model. Represents the maximum singular value. For the reference model, the first The Lipschitz matrix of the layer, where L is the total number of layers in the reference model.

[0033] The third step is to obtain the certificate weight for each edge based on the spectral complexity certificate. Specifically, for each edge... The certificate weight is defined as the first-order sensitivity of the certificate logarithm to the Lipschitz term of that edge: (3); in, For the edge Certificate weight, = For the edge The Lipschitz constant of the unary edge activation function in its domain (i.e., the element in the i-th row and j-th column of the Lipschitz matrix). For the edge The unary edge activation function on, for The largest left singular vector, for The largest right singular vector. According to Perron–Frobenius theory, we can assume sequential nonnegativity, such that the weights of all certificates are equal. .

[0034] The purpose of determining the spectral complexity certificate and the certificate weight for each edge is to provide a quantitative impact of each edge on the overall network (i.e., the entire model) capacity, thus providing a basis for differentiated pricing. In practical applications, the impact of this step on joint angle prediction and robotic arms lies in the certificate weights. This paper presents a quantitative ranking of which joint angle paths have the greatest impact on the stability of the end-effector position prediction. In multi-joint robotic arms, joint angle deviations near the base are amplified step-by-step along the kinematic chain to the end-effector, and the edges corresponding to these joints often have a large certificate weight. This step explicitly expresses this fact in a computable numerical form, allowing subsequent steps to prioritize these edges with limited parameter budgets. This controls the amplification factor of joint encoder quantization error, transmission backlash, and measurement noise propagating from the model to the end-effector position prediction. The upper bound of this amplification factor is the certificate weight. The direct engineering benefit is that when the robotic arm is moving at high speed, experiencing joint vibration, or when the encoder accuracy decreases, the predicted end position is less likely to jump abruptly, thus ensuring the stability and safety of downstream trajectory planning and servo feedforward commands.

[0035] The process of projecting each edge by least squares onto the elementary function menu to obtain the affine function menu and projection residual for each edge involves setting the elementary function menu. The elementary functions menu includes several preset function families. For example, the elementary functions menu contains six function families. , lin is a family of linear functions, sin is a family of sine functions, cos is a family of cosine functions, tanh is a family of hyperbolic tangent functions, and gauss is a family of Gaussian functions. It is a family of absolute value functions. a family of functions The effective degrees of freedom after removing redundancy from reparameterization. for One of the function families. Or, a menu of elementary functions with multiple function families set according to actual needs. For example, the elementary function menu can add other bounded closed function families such as exponential, logarithmic, sigmoid, and ReLU smooth approximation, or be customized according to the specific robot arm configuration (such as retaining only the trigonometric function family).

[0036] Each edge, under affine reparameterization, is unified as follows: the affine function menu for each edge. for: (4).

[0037] in, Let be the input scalar of this edge, that is... Or the output of the previous layer node, for A family of functions, This is a menu for elementary functions. a family of functions The basic function form, For input scaling parameters, To input translation parameters, For amplitude parameters, This is for outputting bias parameters. It should also be noted that in this embodiment, subscripts are used. Specifically refers to the dimension of each layer, without subscript. Specifically refers to the output bias parameter of the side function.

[0038] On each edge, based on each family of functions, in the domain Take a preset number of grid points, such as within the defined domain. Take a preset number of grid points (e.g., 200), fit the univariate edge activation function curve of the edge using least squares, and obtain the projection residual of the edge. , For the edge Projection to the family of functions The least squares fitting residuals in the sense of the quadratic norm (i.e., the square integral norm). The preset number can be set according to actual needs.

[0039] The step of determining the affine function menu and projected residuals for each edge quantifies the approximation ability of each candidate family of closed functions for that edge. A byproduct of this step is the optimal (smallest) projected residuals for each edge. and the median of the projected residuals , , For all sides of the network The diagnostic index steps are constructed simultaneously using the median of the median. In practical applications, the impact of this step on joint angle prediction and the robotic arm is that the edge-by-edge projection residual directly corresponds to "the error introduced by replacing the spline edge with a named elementary function," and its magnitude determines the prediction deviation of the end position of the transformed model on the joint path. Since the projection residual can be calculated before deployment and without training any projection model, it is possible to determine whether the accuracy meets the positioning tolerance requirements of the robotic arm before writing the edge-by-edge parameter table into the robotic arm controller. If the projection residuals of some edges are abnormally large, they can be indexed by edge. It can directly locate specific joints and expand the elementary function menu or increase the data sampling density of the working range of that joint, so that the end-positioning error is not discovered until the machine is put into trial operation, thus avoiding repeated trial and error and downtime on the production line.

[0040] After executing step S103, step S104 is executed, whereby the target affine function family for each edge is determined from the affine function menu for each edge based on the projection residual of each edge and the preset budget multiplier.

[0041] Specifically, based on the certificate weight and projected residual of each edge, as well as a preset budget multiplier, a certificate weighting rule is obtained. Then, based on this rule, the target affine function family for each edge is derived. The certificate weighting rule is as follows: (5), Formula (5) indicates that a unique family of functions is selected for each edge through a certificate weighting rule, that is, the objective of each edge is "projection residual + certificate weight × effective degree of freedom price", so that the parameter price is proportional to the first-order influence of the edge on the network-level spectral complexity certificate. The parameter price here refers to the parameter cost required to select a certain family of functions for the edge: in Formula (5) One item is the price, where the budget multiplier is... The benchmark unit price for each effective degree of freedom, and the certificate weight. The unit price coefficient of this edge, and the effective degrees of freedom. The number of degrees of freedom purchased; the more sensitive to spectral complexity certificates (i.e. The larger the edge (the higher the value of the edge), the higher the cost of increasing the same number of degrees of freedom, thus making the parameter cost of each edge more proportional to the cost of that edge. The first-order influence is proportional. Among them, a family of functions The effective degrees of freedom after removing redundancy from reparameterization. For the edge The target affine function family. Given budget multipliers. In log-interval grid The above is a verification set The selected parameter can also be determined directly based on a pre-defined upper limit, because the scale of the projected model is... A deterministic function.

[0042] As an example of comparison and alternatives, based on the projected residuals of each edge and a preset budget multiplier, a uniformity rule is obtained. Then, based on the uniformity rule, the target affine function family for each edge is obtained. The uniformity rule is: (6) This formula (6) represents the selection of a unique family of functions for each edge by uniformity rules.

[0043] The two rules differ only by a certificate weighting factor. This can be theoretically proven (parameter set inequalities): For each edge under the same multiplier, .

[0044] That is, on the highly sensitive edge ( On certificate-weighted rules, the family of functions that is more expensive than the uniform rule will not be chosen, especially on low-sensitivity edges. The system will not choose a cheaper family of functions, thus achieving a directional redistribution of the budget between edges.

[0045] Furthermore, when each layer satisfies the spectral gap condition (in The first-order certificate difference is computable in closed-form. For the edge The certificate weight (see equation (3), which is abbreviated here as edge weight) (Note the subscript) and The certificate weighting rule and the uniform rule are respectively the edges. The selected family of functions, and Function families , Effective degrees of freedom; and They are the edges Projection as a family of functions , The Lipschitz constant of the subsequent unary edge activation function; Represents the maximum singular value. Represents the second largest singular value. (where Frobenius norm is the upper bound of the Lipschitz matrix perturbation caused by projection), and each layer satisfies At that time, the complexity of certificate weighting rules A certificate of complexity not greater than that of a uniform rule ,Right now Any dependency The generalized capacity bound is correspondingly tightened. The above inequality applies to the same budget multiplier. The certificates of the two selection rules are compared below; in actual selection, each of the two rules selects its own budget multiplier from the verification set, and the certificate value at the selected point is not a quantity compared by the conclusion.

[0046] It should be noted that the family of target affine functions for each edge of the reference model that satisfies the convergence condition forms the target model.

[0047] The core of this invention is the step of determining the target affine function family for each edge from the affine function menu for each edge. Its function is to deterministically transform the reference spline network into an edge-by-edge closed elementary function network under explicit budgeting. In practical applications, the impact of this step on joint angle prediction and the robotic arm is that it directly transforms the hard constraint of "how large a model the embedded controller allows" into parameter selection for each joint path. On the one hand, since the number of parameters in the projected model is a budget multiplier... For deterministic functions, the upper limit of parameters can be determined first based on the storage and computing power limits of the robotic arm controller, and then the solution can be obtained by inverse problem-solving. This allows for a one-time acquisition of a model that meets the upper limit, eliminating the need for repeated retraining and trial and error, thus shortening the selection and integration cycle of robotic arm control software. Furthermore, under the same parameter upper limit, certificate-weighted pricing allows the budget to be allocated from edges with minimal impact on the end effector position (such as the approximately linear path for attitude fine-tuning joints, which can be achieved using only...). The linear function family) is transferred to the side with the greater influence (such as the periodic joint pathway that bears the main displacement), and preserved. The family of sine / cosine functions maintains the same end-position prediction accuracy as uniform pricing statistics in the positive kinematics task of the robotic arm (paper test: certificate weighted rule). For uniform regularity Both methods select their respective budget multipliers from the validation set, significantly improving the robustness of accuracy to the values ​​of the budget multipliers—on this task, the test accuracy fluctuation of the certificate-weighted rule across all 8 values ​​of the budget multiplier grid is less than [a certain value]. And uniform regularity in Time accuracy collapse is as high as Simultaneously, the total effective degrees of freedom of the projected model is a deterministic function of the budget multipliers, allowing direct selection based on preset parameter upper limits. Under the premise of satisfying the aforementioned spectral gap conditions (compared at the same budget multipliers), it also makes the Lipschitz amplification factor of the model smaller. The Lipschitz amplification factor refers to the maximum amplification factor of a unit perturbation at the input end propagating to the output end through the model; its upper bound at the network level is the spectral complexity certificate. The smaller this coefficient, the lower the impact of input disturbances such as joint angle measurement noise and encoder quantization error on the end-effector position prediction value is suppressed, and the more stable the model output is. Ultimately, this translates to higher end-effector positioning accuracy or a shorter control cycle on a controller with the same computing power, improving the model's prediction efficiency.

[0048] After obtaining the target model, fine-tuning and embedded deployment are required. Specifically, the selected edge-by-edge closed parameters... As initial values, the entire closed network (i.e., the target model) undergoes short-range fine-tuning. During fine-tuning, the same optimizer settings as during training are used to recover the small amount of accuracy loss introduced by projection. Subsequently, the target model is exported as a parameter table of edge-by-edge quadruples and deployed to the embedded robotic arm controller or edge detection unit. The edge-by-edge parameter table records the selected function family identifier for each edge of the target model. and its four parameters , , , During online prediction, closed-form scalar evaluation is performed on each side of the real-time acquired joint angle vectors. The algorithm then sums up layer by layer to output the predicted position of the end effector, which can be used for motion planning, servo feedforward, or state monitoring. Since each edge is evaluated only once using an elementary function, the computational cost and latency of inference are much lower than those of spline mesh evaluation and tree traversal. This step completes the closed loop from model transformation to engineering deployment.

[0049] Before obtaining and deploying the target model, the sufficiency and auditability of the menu need to be diagnosed. In the diagnostic indicator step, it is determined whether the elementary function menu is sufficient for the current robotic arm task: during the edge-by-edge least squares projection onto the elementary function menu, the minimum projection residual and the median of the projection residual for each edge are obtained.

[0050] The specific determination method is as follows: The median of the projected residuals... The residual threshold is compared with a preset first residual threshold and a second residual threshold. The first residual threshold is less than the second residual threshold. For example, the first residual threshold is set to 0.013 and the second residual threshold is set to 0.030. The thresholds can be calibrated according to the actual task. Not greater than the first residual threshold (e.g.) When it is determined that the elementary function menu can fully cover the target mapping (the target mapping is the input-output mapping from joint angle vectors to end effector position quantities characterized by the training dataset, which is also the forward kinematics mapping of the robotic arm, and is the object that the reference model and the target model want to approximate), the closed model and the projected baseline (the closed model is the target model obtained by the method of this embodiment, where each edge is a named closed elementary function; the projected baseline refers to the comparison method of only performing edge-by-edge least squares projection, without budgeting the function family selection, and assigning a fixed number of basis functions to each edge, such as the fixed assignment of each edge as described in the technical effect section) are compared. The accuracy of the P-KAN (percentage of basis functions) is basically the same, and the residuals are low, so the trained target model can be safely deployed; when Not less than the second residual threshold (e.g.) When the elementary function menu is insufficient, it is determined that the elementary function menu needs to be expanded or a richer set of basis functions should be used before re-executing projection and selection; that is, iterative training should be re-executed after using a richer set of basis functions. Between two thresholds (e.g.) When the value is approximately 0.015 to 0.025, the elementary function menu is considered sufficiently sufficient, and the trained target model is deployed. At this point, the method of this invention has an advantage in accuracy or parameter efficiency compared to the projective baseline. Furthermore, when the minimum projection residual of any of the edges exceeds a preset edge residual threshold, the edge index of that edge is used... The corresponding joint path is located, and the elementary function menu is expanded or the data sampling density is increased for that joint path. The preset edge residual threshold can be set according to actual needs. Specifically, the minimum projected residual for each edge... Check each edge individually: If individual edges... Relative to the overall network level (e.g., relative projection residual median) If the preset multiple is abnormally large, then the edge index of that edge will be used. Once the corresponding joint pathway is located, the elementary function menu can be expanded or the data sampling density of the working range of that joint can be increased accordingly. The above stratification is an empirical range obtained on a multi-dataset suite; the median of the projection residual and the minimum projection residual for each side can be calculated using only the trained reference model, without any fine-tuning of the projected model or test labels, thus constituting a pre-deployment criterion for "diagnosis first, deployment later".

[0051] Meanwhile, since each edge is a named closed expression, the learned edge functions (especially the frequencies and phases of the sine and cosine families) can be compared edge by edge with the known joint angle trigonometric function structure in the forward kinematics of the robotic arm, enabling white-box auditing. If the edge with the highest certificate weight is occupied by a single function family with a narrow coefficient distribution, this phenomenon itself is a failure warning signal that can be directly read from the fitted formula.

[0052] Furthermore, an explicit PAC-Bayes generalization upper bound can be computed based on a finite set of configurations. It consists of all candidate configurations generated by running the certificate weighting rule and the uniformity rule on each value of the budget multiplier subgrid (optionally incorporated into a single family of function configurations). This refers to the number of configurations. It should be noted that this is a finite set of configurations. With the Elementary Functions menu They are objects at different levels: The elements are the family of functions that can be used for a single edge. The elements represent a configuration for the entire network (i.e., a family of function choices for all edges). Assume the single-sample loss is bounded to... (For example, the squared error truncated on the standardized target), by Constructing the PAC-Bayes prior distribution from the projected residuals ( The configured enumeration index, and the node sequence number. , (Irrelevant), then not less than The probability holds the following generalized upper bound: (7); in, In order to be in The above is based on the certificate weighting rules from a finite set of configurations. The selected configuration, For a unified notation of the model, any configuration The corresponding KAN network model is denoted as (The basic function form whose subscript in the function expression is the family of functions) (Distinguish between the meanings of the subscripts) That is, the selected configuration The corresponding edge-by-edge closed network model (i.e., the target model before fine-tuning), where n is the total number of sample pairs in the training dataset, and the first edge-by-edge closed network model in the training dataset. Each sample pair is denoted as , Used to construct the PAC-Bayes prior distribution Each sample pair is denoted as , Used to calculate empirical risk For the model Overall risk For the model exist Experience and risk For only Configuration of the above calculation The projection residual, For configuration The total effective degrees of freedom (i.e., parameter budget). Let be the normalization constant of the PAC-Bayes prior distribution. , The a priori temperature index, This is a confidence parameter.

[0053] Enumerable number of configurations Satisfies: two selection rules (i.e., certificate weighting rule and uniformity rule) and 8 points (here, 8 is the budget multiplier subgrid). The number of possible values ​​is independent of the number of joints in an eight-bar linkage robot; each of the two rules operates on eight possible values. Budget multiplier grid combination If incorporated into a single function family configuration (a single function family configuration refers to a configuration that forces all edges in the entire network to use the same function family; there are 6 options for the elementary function menu containing 6 function families, such as using the linear function family for all edges, or the sine function family for all edges, etc.), then... )but Therefore, all terms on the right-hand side of equation (7) above can be directly calculated numerically. When the a priori temperature index At that time, the prior distribution degenerates into a finite set of loci. The uniform distribution on the upper bound degenerates into the classical form. An upper bound on the type. This provides a verifiable generalization guarantee for deploying the target model.

[0054] The model transformation relationship in this embodiment is as follows: Figure 2 and Figure 3 As shown. Figure 2 The network structure of the reference model (i.e., the reference B-spline KAN) before transformation is shown: taking an eight-bar robotic arm as an example, the input layer consists of 8 joint angles. to (Right now (The data is then collected in the hidden layer and converged to the output node.) (i.e., end position quantity), each edge contains The cubic B-spline coefficient grid of each interval is superimposed with SiLU residual terms. The number of parameters on the edges increases unbounded with the accuracy requirement, and the spline coefficients do not have a readable physical meaning. Figure 3 The network structure of the target model (i.e., BudgetKAN) after transformation by the training method of this embodiment is shown: the network topology is consistent with the reference model, but each edge is replaced with a unique named closed elementary function. , . Figure 3 Different line types are used to distinguish the selected function families for each edge (solid lines for tanh family, dashed lines for gauss family, dotted lines for sin / cos family, and dashed-dot lines for abs / lin family), and the line width represents the magnitude of the edge function. After conversion, each edge is described only by the function family identifier and four closed parameters, allowing for edge-by-edge printing, auditing, and verification against the trigonometric function physical structure of the robotic arm joint angles.

[0055] One or more technical solutions in the embodiments of the present invention have at least the following technical effects or advantages: (1) Improved prediction accuracy: On the positive kinematics task of an eight-bar linkage robotic arm (kin8nm, 8 input joint angles), the coefficient of determination of the test set of this invention reaches 100%. It significantly outperforms Gradient Boosting Tree (XGBoost). (difference) , (This represents the largest single-pair difference among all comparisons), and is also superior to parameter-matched MLP. ( ) and the reference B-spline KAN without budget constraints ( The mechanism involves certificate-weighted selection that precisely allocates the sine / cosine function family (accounting for 40% of the total edges) to the edges with the largest decrease in projected residuals. This matches the periodic dependency structure of the robotic arm's joint angles, representing a domain inductive bias that tree models cannot replicate. Projected into practical applications: Under the same set of joint angle inputs, the coefficient of determination is... Upgraded to This means that the variance in the terminal positions that the model failed to explain is due to Down to , decrease by approximately This is equivalent to reducing the root mean square error of the end position prediction to approximately [percentage missing] of the original value. In practical applications, when a robotic arm is tracking a trajectory, feedforward compensation based on the predicted end-effector position more closely approximates the actual end-effector position, reducing end-effector positioning deviation and overshoot. This improves the first-pass success rate for tasks sensitive to positioning tolerances, such as gripping, inserting, and grinding. In human-robot collaborative scenarios, more accurate end-effector position prediction means that the safety envelope can be set more compactly without increasing the risk of collision, thereby increasing the available workspace and permissible operating speed of the robotic arm.

[0056] (2) Improve parameter efficiency: Compared with edge-by-edge fixed allocation The projected baseline P-KAN with one basis function saves 7%–14% of effective edge degrees of freedom at the same or better accuracy level (approximately 15% saving in the positive kinematics task of the robotic arm, i.e., 984 pairs of 1152 effective degrees of freedom, main benchmark experimental caliber). Compared with the degenerate spline KAN with the same number of parameters, the test... The advantage ranged from 0.077 to 2536 units (four orders of magnitude on a logarithmic scale) across all eight datasets, with the advantage on the robotic arm's forward kinematics task being [missing value]. Projecting to practical applications: Reduced effective edge degrees of freedom. – This directly corresponds to a reduction in the size of the edge-by-edge parameter table written to the robotic arm controller and the corresponding reduction in resident memory, allowing for more joints (larger...) to be supported within the same MCU / DSP storage budget. (This allows for the creation of) or wider hidden layers to provide space. When a single industrial computer controls multiple robotic arms simultaneously, or when a set of kinematic models needs to be stored for different loads / conditions of the same robotic arm, more models can be accommodated without expanding the hardware.

[0057] (3) Improved inference efficiency: Closed-loop edge-by-edge evaluation reduces the amount of inference computation by approximately 71% compared to the reference spline network. The average single-sample inference latency is 0.148 milliseconds, which is faster than XGBoost's 0.245 milliseconds and 5–22 times faster than P-KAN, meeting the real-time requirements of embedded high-frequency control loops. Projected into practical applications: The above latency values ​​are measured values ​​from a horizontal comparison of the methods on the Apple M4 platform, reflecting the relative magnitude of the inference overhead of each method. Its engineering implications are: In a typical kHz (period) In a millisecond (ms) servo control loop, the time slice for end-effector position prediction is significantly smaller than a control cycle. Therefore, a kinematic proxy model can be directly embedded into the real-time loop for feedforward compensation, collision warning, and status monitoring without reducing the control frequency. Furthermore, since edge-by-edge closed-form evaluation is a branchless, fixed-length operation, its execution time remains largely unchanged with the input. In contrast, gradient boosting trees require traversing each tree, and the path length varies with the input. Therefore, this invention is more convenient for estimating the worst-case execution time of the control cycle and is better suited for hard real-time robotic arm control systems.

[0058] (4) Enhanced auditability: With all random seeds (15 / 15), the learned closed-loop edge functions recover the periodic physical structure of the robot arm's kinematics more effectively than the projected baseline (scored separately at the structure and function layers according to a predefined white-box scoring protocol), while incurring a retention set accuracy cost of only 0.006; the expression for each edge It can be printed directly, checked item by item, and verified against physical mechanisms. Applied to practical applications: it can be printed directly and compared with the first... Check the angular frequency parameters of all edge functions related to each joint. Does it match the actual rotation range of the joint? In the eight-link forward kinematics task, the true angular frequency of the joint angle in normalized coordinates is approximately... The distance between the fitted frequency and the true value in this invention is One octave band, while the projected baseline is One octave band. On a true periodic input, the amplitude ratio of the expression of this invention placed on a named periodic function (sine / cosine) reaches... While projected baselines are only This allows for assessment before installation whether the model has truly learned the periodic structure of joint rotation, rather than simply fitting data noise. If the robotic arm is modified, a joint's limit is changed, or a joint sensor drifts, the edge requiring correction can be directly located from the expression, without having to retrain the entire model as a black box. For scenarios such as human-machine collaboration, medical robots, and aerospace assembly where the model must be submitted for safety review, this is a prerequisite for practical application.

[0059] (5) Enhanced diagnosability: The edge-by-edge projection residuals can be calculated from the reference fit before any fine-tuning of the projection model. Empirically, tasks can be correctly stratified according to menu sufficiency (e.g., high residual tasks). (corresponding to the relatively disadvantageous scenarios of this method); while the measured median of the projected residuals for the robotic arm's positive kinematics task targeted by this application is... (Approximately the diameter of another experimental pipeline) This falls within the "sufficient menu" range, consistent with the performance of this method on this task, providing a selection basis for "diagnosis first, deployment later"; simultaneously, the upper bound of PAC-Bayes is determined by the number of configuration options. It can be directly calculated numerically, providing an explicit generalization guarantee. Projecting this to practical applications: before writing the model into the robotic arm controller, it can be based on... The current robotic arm task is categorized under "Sufficient Menu" ( The closed-loop model has roughly the same accuracy as the projected baseline and can be deployed directly; the "menu is basically sufficient" (meaning the closed-loop model has the same accuracy as the projected baseline and the "menu is basically sufficient"). The method of this invention has advantages) and "insufficient menu" The selection process is categorized into three types: those requiring expanded menus or richer basis functions, forming a "diagnosis first, deployment later" approach to avoid repeated trial and error and downtime adjustments on the production line. The PAC-Bayes upper bound can serve as a quantization generalization guarantee delivered with the model, used for the safety review and acceptance of robotic arm control software.

[0060] Example 2 Based on the same inventive concept, the second embodiment of the present invention also provides a method for predicting the end effector position of a robotic arm, such as... Figure 4 As shown, it includes: S201, Obtain the input vector and target model at the current moment. The input vector is the joint angle vector of the robotic arm, and the target model is the model trained by the training method of the KAN model based on budgeted side function selection in Example 1. S202 processes the input vector using the target model and outputs the predicted position of the robotic arm's end effector.

[0061] Specifically, after deploying the target model to application terminals such as robots, robotic arms, and edge units, the prediction method (i.e., the inference process) differs from the training method (i.e., the modeling process). Each control cycle in the online inference process executes the following steps: ① Read the current state of the robotic arm... The joint angle θ of each joint encoder. ② In the step of obtaining the training dataset based on the training method, the fixed affine coefficients map θ to the domain. , to obtain the input vector ; ③ Calculate the closed scalar value for each edge according to the edge-by-edge parameter table. ④ Forward propagation is completed by summing within layers and compositing between layers; ⑤ The output is inversely transformed using the standardized coefficients fixed in the training dataset acquisition step of the training method to obtain the predicted position value of the robotic arm's end effector. This online inference process is a branchless, iterative, fixed-length operation, and its computational load and latency are basically constant for any input, facilitating worst-case execution time analysis in the real-time control loop. Only when the robotic arm undergoes structural modification, load replacement, joint limit change, or long-term wear causes the prediction residual to exceed the set threshold, is it necessary to return to the offline training stage to re-execute the training method and reissue the edge-by-edge parameter table. Since the reference model of the training method can be directly connected to the existing spline KAN model, this re-conversion does not require retraining from scratch.

[0062] Example 3 Based on the same inventive concept, the third embodiment of the present invention also provides a robotic arm, which is used to perform any of the steps of the robotic arm end-effector position prediction method described in the preceding embodiment two.

[0063] Those skilled in the art will understand that although preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments as well as all changes and modifications falling within the scope of the invention.

[0064] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, this invention also intends to include these modifications and variations.

Claims

1. A training method for a KAN model based on budgeted side function selection, characterized in that, include: Obtain a training dataset and a reference model for the robotic arm joints, wherein each training data point in the training dataset is a normalized sample pair. The Let the joint angle vector of the robotic arm be a sampled value. The standardized label for the position of the end effector of the robotic arm at the same sampling time is the reference model, which is the reference B-spline KAN model. The reference model is iteratively trained using the training dataset until the iteratively trained reference model meets the convergence condition. The reference model that meets the convergence condition is then determined as the target model. The target model includes a family of target affine functions for each edge. After the iterative training satisfies the convergence condition, a spectral complexity certificate for the reference model is constructed, the certificate weight of each edge of the reference model is determined, and each edge is least squared projected onto an elementary function menu to obtain the affine function menu and projection residual for each edge; wherein, constructing the spectral complexity certificate for the reference model and determining the certificate weight of each edge of the reference model includes: Construct the Lipschitz matrix for each layer based on the unary edge activation function of each edge in each layer of the reference model; The spectral complexity certificate is obtained based on the Lipschitz matrix of each layer; Based on the spectral complexity certificate, the certificate weight of each edge is obtained; Based on the projection residual of each edge and the preset budget multiplier, the target affine function family of each edge is determined from the affine function menu of each edge.

2. The training method for the KAN model based on budgeted side function selection as described in claim 1, characterized in that, The spectral complexity is obtained based on the Lipschitz matrix of each layer, as shown in the following formula: ； in, The spectral complexity certificate is the product of the maximum singular values ​​of each layer, and β is a configuration representing the set of function family choices for all edges of the reference model. Represents the maximum singular value. For the reference model of the first The Lipschitz matrix of the layer, where L is the total number of layers in the reference model; For each of the aforementioned edges Based on the spectral complexity certificate, the certificate weight of each edge is obtained, as shown in the following formula: ； in, The first part of the reference model represents the first part. layer, For the first The node number on the input side of the layer. For the first The node number on the output side of the layer. For the edge Certificate weight, = For the edge The Lipschitz constant of the unary edge activation function over its domain is the element in the i-th row and j-th column of the Lipschitz matrix. For the edge The unary edge activation function on, for The largest left singular vector, for The largest right singular vector.

3. The training method for the KAN model based on budgeted side function selection as described in claim 2, characterized in that, The edge-by-edge least squares projection onto the elementary function menu yields the affine function menu and projection residual for each edge, including: Affine function menu for each of the aforementioned edges for: ; in, Let be the input scalar for this edge. for A family of functions, The elementary function menu includes multiple preset function families. a family of functions The basic function form, For input scaling parameters, To input translation parameters, For amplitude parameters, For output bias parameters; On each of the aforementioned edges, based on each of the aforementioned family of functions, in the domain A predetermined number of grid points are selected, and the univariate edge activation function curve of the edge is fitted using least squares to obtain the projection residual of the edge. , For the edge Projection to the family of functions The least squares fitting residual in the sense of the second norm.

4. The training method for the KAN model based on budgeted side function selection as described in claim 3, characterized in that, The step of determining the target affine function family for each edge from the affine function menu based on the projected residual of each edge and a preset budget multiplier includes: Based on the certificate weight and projection residual of each edge, and the preset budget multiplier, the certificate weighting rule is obtained, and then the target affine function family of each edge is obtained according to the certificate weighting rule. The certificate weighting rule is as follows: ; in, a family of functions The effective degrees of freedom after removing redundancy from reparameterization. For the edge The family of target affine functions.

5. The training method for the KAN model based on budgeted side function selection as described in claim 4, characterized in that, The step of determining the target affine function family for each edge from the affine function menu based on the projected residual of each edge and a preset budget multiplier includes: Based on the projection residual of each edge and the preset budget multiplier, a uniform rule is obtained, and then based on the uniform rule, the target affine function family of each edge is obtained. The uniformity rule is as follows: ; in, a family of functions The effective degrees of freedom after removing redundancy from reparameterization. For the edge The family of target affine functions.

6. The training method for the KAN model based on budgeted side function selection as described in claim 5, characterized in that, Also includes: Based on a finite set of configurations, an explicit PAC-Bayes generalization upper bound expression is obtained; the finite set of configurations The configuration consists of all candidate configurations generated by running the certificate weighting rule and the uniformity rule on each value of the preset budget multiplier grid, or may also be incorporated into a single function family configuration. The number of configurations in the finite configuration set, For configuration, the configuration represents the set of function family choices for all edges; The explicit PAC-Bayes generalization upper bound expression is: ； in, In order to be in The above is based on the certificate weighting rules from the finite configuration set. The selected configuration, For a unified notation of the model, any configuration The corresponding KAN network model is denoted as , For selected configuration The corresponding edge-wise closed network model, where n is the total number of sample pairs in the training dataset, and the first edge-wise closed network model in the training dataset... Each sample pair is denoted as , Used to construct the PAC-Bayes prior distribution Each sample pair is denoted as , Used to calculate empirical risk For the model Overall risk For the model exist Experience and risk For only Configuration for calculation The projection residual, For configuration Total effective degrees of freedom Let be the normalization constant of the PAC-Bayes prior distribution. , The a priori temperature index, This is a confidence parameter.

7. The training method for the KAN model based on budgeted side function selection as described in claim 1, characterized in that, Also includes: During the process of edge-by-edge least squares projection onto the elementary function menu, the minimum projection residual and the median of the projection residual for each edge are obtained. The median of the projected residual is compared with a preset first residual threshold and a second residual threshold, wherein the first residual threshold is less than the second residual threshold; When the median of the projection residual is not greater than the first residual threshold, it is determined that the elementary function menu can fully cover the target mapping, and the trained target model is deployed. When the median of the projected residual is not less than the second residual threshold, it is determined that the elementary function menu is insufficient, and the iterative training is re-executed after expanding the elementary function menu or using richer basis functions; When the median of the projected residuals is between the first residual threshold and the second residual threshold, it is determined that the elementary function menu is substantially sufficient, and the trained target model is deployed. Furthermore, when the minimum projected residual of any edge exceeds a preset edge residual threshold, the edge index is used to determine the appropriate method. Locate the corresponding joint pathway and expand the elementary function menu or increase the data sampling density for that joint pathway.

8. A method for predicting the end-effector position of a robotic arm, characterized in that, include: Obtain the input vector and target model at the current moment, wherein the input vector is the joint angle vector of the robotic arm, and the target model is a model trained by the training method based on the budgeted side function selection KAN model as described in any one of claims 1-7; The target model is used to process the input vector to output the predicted position value of the robotic arm's end effector.

9. A robotic arm, characterized in that, Used to perform the robotic arm end-effector position prediction method as described in claim 8.

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