Model and data driving fused mechanical arm gravity compensation algorithm

Abstract

The invention provides a model and data driving fused mechanical arm gravity compensation algorithm, and aims to solve the problem of poor mechanical arm gravity compensation precision caused by model uncertainty. According to the algorithm, firstly, a statics equation is established, secondly, multiple data are trained through a Bayesian neural network (BNN), a predicted torque and a standard are generated, then, through a Bayesian fusion method, a model calculation result and a neural network fitting result are subjected to Bayesian fusion according to a standard deviation, and a fusion estimation torque is generated. In order to solve the problem that the generalization ability of a data driving model is insufficient in an unseen scene, the algorithm uses a physical model as a constraint condition to limit the range of fusion torque, high deviation caused by insufficient model generalization is reduced, and then stability is ensured. Compared with a traditional modeling method, the fusion strategy is combined with the self-adaptability of physical constraint and data driving, the influence of modeling errors on the control precision is effectively reduced, and the precision and robustness of gravity compensation are remarkably improved.

Description

Technical Field

[0001] The embodiments of the present invention relate to the technical field of robotic arm gravity compensation, and specifically to a robotic arm gravity compensation algorithm that integrates model and data driving. Background Art

[0002] The gravity term of the robotic arm, a nonlinear perturbation in the system, significantly increases labor intensity for wearable robotic arms and significantly restricts motion accuracy and stability for industrial robotic arms. Therefore, it is typically compensated for through feedforward control. However, traditional gravity compensation methods rely on modeling accuracy, constructing an analytical expression for the gravity torque by combining the weight of the rod, the center of gravity position, and kinematic and static analysis. However, in practical applications, uncertainties such as assembly errors, zero-point calibration errors, and external cables lead to deviations between the model and actual working conditions, making the system sensitive to model errors and uncertainties.

[0003] In recent years, data-driven approaches have become a research hotspot in the control field, demonstrating strong adaptability by building datasets and combining them with machine learning or data fitting techniques. However, these approaches are limited by the quality and quantity of training data, and their generalization capabilities to unseen scenarios are weak, making it difficult to ensure stability under complex working conditions. Accurately compensating for the gravitational torque of a robotic arm is a pressing and challenging technical issue. Summary of the Invention

[0004] In order to solve the practical problem of insufficient accuracy of gravity compensation of robotic arms, the present invention proposes a gravity compensation algorithm that integrates model and data driving, taking a wearable robotic arm as an example. This algorithm is based on the idea of ​​"co-optimization of theoretical constraints and data adaptation", making full use of the physical reliability and analytical advantages of the model, as well as the adaptability and robustness of the experimental numerical values ​​of the Bayesian neural network (BNN), effectively overcoming the limitations of traditional theoretical modeling deviations and insufficient generalization ability of data driving. By constraining the threshold range of the neural network fitting through the standard deviation of the model, it is ensured that the compensation value remains within a reasonable and stable range, thereby achieving simple, efficient, stable, reliable and high-precision gravity compensation control of the robotic arm.

[0005] The present invention discloses a robotic arm gravity compensation algorithm based on data-driven fusion, which specifically includes the following steps:

[0006] Step 1: Robotic arm system modeling: Based on the DH method, establish the robot arm base coordinate system Z0 and each joint axis coordinate system as follows Figure 2 As shown; further derive the kinematic model and calculate the homogeneous transformation matrix; derive the statics equation based on the virtual displacement principle, and use the statics equation to calculate the gravitational torque of the model in combination with the mass of the rod and the position of the center of mass.

[0007] Step 2: Training Data Collection and Bayesian Neural Network Construction: The actuator is set to Cyclic Synchronous Position (CSP) mode, and current and joint angle data are collected in real time while in static equilibrium at the target position. This data is mean filtered, normalized, and outliers removed to obtain a processed current and corresponding angle dataset. A BNN model is then used to train this preprocessed data offline to generate the fitted torque mean and standard deviation.

[0008] Step 3: Bayesian Fusion and Constrained Optimization: The fitted torque output by the BNN is Bayesian-fused with the model's calculated torque based on the standard deviation. Using the model torque as a physical constraint, the resulting torque is optimized to avoid significant errors caused by insufficient generalization of the neural network model.

[0009] Step 4: Real-time compensation: The fusion torque generated in step 3 is converted into a control current value and output to the joint motor controller in real time through the Ethercat control bus. The motor torque is dynamically adjusted to achieve online real-time gravity compensation control of the exoskeleton robotic arm.

[0010] Effective gain:

[0011] (1) This algorithm integrates the theoretical model with the adaptive prediction capability through Bayesian fusion, which can effectively reduce the impact of interference items such as assembly error, zero point calibration deviation and external cables on modeling accuracy, and achieve high-precision and robust gravity compensation control.

[0012] (2) The algorithm combines the theoretical constraints of the statics model with the uncertainty estimation of BNN, and uses a constrained optimization method to limit the fusion torque to a reasonable range of the physical model, effectively reducing the generalization error of the data-driven model in unseen scenarios, thereby ensuring the stability and reliability of gravity compensation in any state.

[0013] (3) This algorithm is streamlined and effectively reduces computational complexity through kinematic modeling, static analysis based on the virtual displacement principle, and a lightweight BNN training architecture. Compared with traditional methods based on model parameter identification, this algorithm is more efficient, convenient, safe, and reliable. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] To more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for the embodiments or the description of the prior art. Obviously, the drawings described below are merely exemplary, and those skilled in the art can, without inventive effort, derive other implementation drawings based on the provided drawings.

[0015] Figure 1 A flow chart of a manipulator gravity compensation algorithm based on model and data-driven fusion provided by the present invention;

[0016] Figure 2 A schematic diagram of the joint axes of an exoskeleton robotic arm according to an embodiment of the present invention;

[0017] Figure 3 This is a curve showing the change in the BNN training and validation loss function according to an embodiment of the present invention.

[0018] Figure 4 This is a schematic diagram of the prediction results of the BNN model for test data according to one embodiment of the present invention;

[0019] Figure 5 A schematic diagram of the mean current value and the standard deviation of the calculated value of a data set according to an embodiment of the present invention;

[0020] Figure 6 A comparison curve of the BNN fitting current and the model calculated current according to an embodiment of the present invention;

[0021] Figure 7 This is a comparison curve between the Bayesian fusion current and the model-calculated current according to an embodiment of the present invention. DETAILED DESCRIPTION

[0022] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0023] like Figures 1 to 7 As shown, the present invention discloses a new model-driven and data-driven fusion robot arm gravity compensation algorithm. The present invention uses a wearable upper limb exoskeleton robot arm as an example to illustrate the specific implementation of the algorithm in detail, including the following steps:

[0024] Step 1: For the robotic arm system, a mathematical model of the robotic arm is established, including forward kinematics and statics equations based on the principle of virtual displacement.

[0025] Step 1.1 First, establish the joint coordinates of the robot arm, unify the coordinates of the robot arm to the base coordinate system Z0, and establish the kinematic equation of the robot arm based on the coordinate transformation relationship:

[0026]

[0027] in, The homogeneous transformation matrix that the robot arm needs to go through from the i-1th link to the i-th link is expressed as:

[0028]

[0029] θ j Indicates the jth joint of the robot arm. The routine of the present invention needs to compensate for the gravity torque of the exoskeleton robot arm shoulder joint and elbow joint, totaling 4 degrees of freedom, so j = 4. In addition, since the exoskeleton robot arm shoulder joint of this routine is connected by a planar six-bar linkage Figure 2 The motor rotation axis Z shown in 21 The directional movement is equivalent to moving to the shoulder joint movement axis Z2, and the passive link gravity introduced is represented by m2, m3, m4, m5, and m6 in the figure respectively, and the mass of the remaining active joint rods is represented by m1, m7, and m8.

[0030] Step 1.2: Split the rods of the exoskeleton robot arm 3D model into independent related rods according to the above joint coordinate system, measure the mass and center of mass position of each rod of the model relative to the link's own coordinate system, and obtain accurate measurement results using Solidworks software.

[0031] Step 1.3: For the exoskeleton multi-link (rigid body) system, establish the statics equations based on the virtual displacement principle. The virtual displacement principle states that the sum of the virtual work done by all active forces in any virtual displacement is equal to zero.

[0032] ∑δW F =0

[0033] So suppose any connecting rod M i The generalized active force and virtual displacement on the i and δ ri , then the projection expression of the rectangular coordinate system is:

[0034]

[0035] If the exoskeleton manipulator is regarded as a mass system with ideal constraints, without loss of generality, then in this state, the sum of the virtual work done by the output torque of each active joint of the exoskeleton manipulator, the gravity of each link of the exoskeleton manipulator and other constraint forces in any virtual displacement is equal to 0. Select the angle θ of each active joint j As a generalized coordinate, assume that a small virtual displacement δθ is generated at each active joint angle j , then the joint active output torque τ j And the gravity of each connecting rod G i The sum of virtual work generated δ W for:

[0036]

[0037] Establish the coordinate system of each link of the exoskeleton manipulator, and the coordinate equation of the center of gravity of each link is:

[0038]

[0039] in: 0 x ci is the center of mass position of link i relative to the base coordinate system, i x ci is the center of mass position of link i relative to the link coordinate system, 0 T i is the homogeneous transformation matrix of the connecting rod coordinate system relative to the base coordinate system. The coordinates of the center of mass of each member in the base coordinate system Z0 are obtained as follows:

[0040]

[0041] According to the virtual displacement principle in the rectangular coordinate system, gravity only produces virtual work in the vertical direction. Therefore, the virtual displacement principle is applied to each active joint. For joint j (j = 1, 2, 3, 4), the virtual displacement formula of joint j is expressed as shown below:

[0042]

[0043] Finally, the weight and center of mass parameter values ​​of each rod of the robotic arm are brought into the final calculation to obtain the torque τ of the four active joints of the robotic arm. m，j and joint angle θ j The relationship between them.

[0044] Step 2: Gravity compensation data collection and training, and establishment of a Bayesian neural network learning framework.

[0045] Step 2.1 Initialize the algorithm parameters. According to the kinematic equation of the robot arm established in step 1, set the initial zero position of each joint of the robot arm to θ 0，j , further define the shoulder and elbow joint range of motion space as: S j ={90°,80°,90°,120°}, divide the target position of the robot arm into units of 10° according to the requirements j , a total of 11200 sets of current values ​​I under static balance at the target position can be collected exp The joint angle θ with the robot arm exp The dataset is used for neural network training; it is worth noting that the dataset dimension can be set according to the specific situation of the robot arm's degrees of freedom. Although too little data is easy to collect, it will result in a decrease in training accuracy.

[0046] Step 2.2 Set the robot arm joint driver mode to periodic position mode (CSP), and set the zero point of each joint θ 0，j At the beginning, control each joint of the robot arm to move to the target point P in sequence j After the robot reaches the target position, it delays for 3 seconds to ensure that the robot is in a static balance state. The master station is controlled by Ethercat to read the joint angle θ of 1000 loop groups in real time. expAnd the current feedback value I exp The data set is further processed by mean filtering, normalization and outlier removal to finally obtain the filtered joint current value I exp,j and the corresponding joint angle θ exp,j value.

[0047] Step 2.3 Use BNN to train the preprocessed multiple sets of joint motor torque data to generate the predicted current value I nn and its uncertainty variance σ nn , characterized in that the network structure of the Bayesian neural network includes an input layer, a hidden layer and an output layer;

[0048] Step 2.3.1 The network input layer receives the pre-processed feature data. For the exoskeleton manipulator of the present invention, the input includes the motor current I of each joint. exp,j and angle θ exp,j (j=1,2,3,4) is the joint index), that is, each degree of freedom corresponds to 2 features, so the basic input dimension is N in =2D=8. Perform Z-score normalization on the input data, the formula is Among them, μ and σ are the mean and standard deviation of the training set features, respectively. The standard eliminates the dimension effect and accelerates the convergence of model training.

[0049] Step 2.3.2 Training goal setting: BNN is trained using the Evidence Lower Bound (ELBO) loss function, the optimizer is Adam, the learning rate is 0.001, the training is 100 epochs, and the batch size is 128. The training goal is to maximize the Evidence Lower Bound ELBO function:

[0050]

[0051] in: Expressed as data likelihood expectation, it measures the model's ability to fit the data; KL[q φ (w)||p(w)] is the KL divergence between the variational distribution and the prior distribution, which serves as a regularization term to prevent overfitting. In actual implementation, ELBO is expressed as:

[0052]

[0053] where w (i) ~q φ (w) is the weight sampled from the variational distribution, M is the number of samples, and β is the scaling factor of the KL divergence, which is used to balance fitting and regularization.

[0054] Assume the output follows a Gaussian distribution: The prior distribution is the standard normal distribution. The KL divergence term can be calculated analytically:

[0055]

[0056] Step 2.3.3 Stochastic Variational Inference Training: The training process uses stochastic variational inference (SVI). The specific steps are as follows: Initialize the variational parameter φ and repeat the following steps until convergence: randomly extract batches w from the training data; extract the θ from the variational distribution q by re-parameterizing the θ. φ (w) Sampling weight w = μ φ +σ φ ∈, is standard Gaussian noise; calculate the unbiased estimate of the ELBO; and use the Adam optimizer to update the variational parameter φ.

[0057] Step 2.3.4 The output layer consists of two parallel fully connected layers: the first output layer generates the predicted mean μ, and the second output layer generates the logarithmic standard deviation log sigma The logarithmic standard deviation is converted to standard deviation through exponential transformation: σ=exp(log sigma ) Finally, the model outputs the predicted current value μ=I nn and its uncertainty standard deviation σ nn , the output dimension is 1. The parameterization ensures that the standard deviation is always positive and the gradient is kept stable during training.

[0058] Training results Figure 3 As shown in the figure, we use 20% of the data as the validation set and 80% of the data as the training set. Finally, we can see that the training loss and the validation loss function Loss drop sharply in the first 20 iterations and successfully converge to a smaller value after 100 iterations, indicating that the training is successful. Figure 4 As shown in the figure, the X-axis represents the actual current value, the Y-axis represents the fitted current mean, each blue point represents the actual value of a test sample and the predicted mean, the red dot represents the ideal prediction line, and the scatter points close to the red dotted line indicate that the predicted value is close to the actual value and the model prediction is accurate.

[0059] Figure 5 The figure shows the mean current for each joint across all data sets, along with its standard deviation compared to the model-calculated value. Because the driver rated current for joints 1 and 2 was set to 7A, and for joints 3 and 4 to 5A, the figure does not show differences per unit current. However, the average current for the elbow joint (Joint 4) is significantly lower than that of the first three joints. Furthermore, the joint standard deviation increases as the joint approaches the base Z0, presumably due to cumulative factors such as assembly quality and measurement error.

[0060] 100 joint angles within the joint motion range are randomly selected as the algorithm verification set. The Bayesian neural network fitting current value and the model calculation current value are as follows: Figure 6 As shown in the figure, it can be seen that the overall trend of the fitted current values ​​of each joint is consistent with the calculated current values, and the floating deviation is within a reasonable range, which further verifies the rationality and effectiveness of the BNN fitting results.

[0061] Step 3 is based on the fusion of model and data driven Bayesian. nn Standard deviation σ nn The torque τ calculated by the model m and standard deviation σ m Perform Bayesian fusion to obtain the final fusion torque τ f .

[0062] Step 3.1 The normalization of torque and current is the basis for establishing the relationship between motor current and torque. t (Unit: Nm / A, indicating the torque generated per ampere of current), the following relationship between motor torque and control current is established: τ nn，j =K t，j I nn，j .

[0063] Where: τ nn，j Indicates the torque value of joint j based on the model (unit: Nm), K t，j represents the electrical torque constant of joint j (unit: Nm / A), I nn，j Represents the current value of joint j obtained in step 2 after BNN fitting.

[0064] Step 3.2 The current value I after processing according to step 2 exp,j and the corresponding angle θ exp,j As the true value, calculate the torque τ derived from the model for each joint j m,j The standard deviation σ m,j First, calculate the residual of the model torque: e j =τ exp,j -τ m,j . Then, calculate the standard deviation: Where: b represents the number of samples, e j,i represents the residual of the i-th sample, represents the mean of the residuals.

[0065] Step 3.3: The output of the Bayesian neural network is τ nn and σ nn τ calculated with the model m and σ m Perform Bayesian fusion to generate fusion torque estimate τ f The fusion process uses a weighted average method, and the weights are determined inversely by the standard deviation of each model: where ω nn and ωm Represent the Bayesian neural network and the weight of the model, satisfying ω nn +ω m = 1. Estimated fusion torque: τ f =ω nn ·τ nn +ω m ·τ m Standard deviation after fusion:

[0066] Step 3.4: τ calculated using the model m , σ m As a physical constraint, it ensures that the deviation between the fusion torque value and the model is maintained within the acceptable threshold. f The value range of τ is: f ∈[τ m -k·σ m ,τ m +k·σ m ], where k is an adjustable parameter, indicating the strictness of the constraint. The present invention selects k as 1. If the fusion torque τ f If it exceeds the range, it is constrained to the boundary value:

[0067] τ f =max(τ m -k·σ m ,min(τ f ,τ m +k·σ m ))

[0068] We compare the fused current value with the calculated current value again according to the constraint boundary conditions. If the error between the fused current value and the calculated current value is within the calculation model standard deviation σ m If the error between the two is greater than the standard deviation σ of the calculation model, then the value is directly taken as the final output compensation current. m , in order to ensure the safety of the experiment, we select τ m +k·σ m As the final output compensation current.

[0069] like Figure 7 As shown in the figure, the blue curve represents the fusion current of each joint, and the red dotted line represents the calculated current value. The difference between the two is represented by the red fill. It can be seen that the calculated current is more conservative in the current output than the fusion current value. This method can avoid the overfitting of the Bayesian neural network due to unseen scenarios and ensure that the output current for gravity compensation is within a safe and reasonable range.

[0070] Step 4 deploys the results of the robot arm gravity compensation algorithm fused with the model and data drive to the exoskeleton robot arm for real-time gravity compensation control. According to step 3, the algorithm will return the fused limited torque value τ f , and use this value for real-time control of the robot arm gravity compensation. The specific steps are as follows:

[0071] Step 4.1 Control the master station through Ethercat to obtain the current joint angle information θ of the robot arm in real time j .

[0072] Step 4.2 Input the joint angle information into the static model based on the virtual displacement principle and the BNN model respectively, and further obtain the fused joint output torque τ according to steps 1-3 f,j .

[0073] Step 4.3 Based on the torque constant K of different motors t，j Each joint motor outputs the joint torque τ f，j Converted to real-time control current signal I f，j , and finally sent to each joint driver through the Ethercat control master station, thereby realizing real-time online gravity compensation control of the robotic arm.

[0074] Although the present invention has been described in detail above using general descriptions and specific embodiments, it will be apparent to those skilled in the art that modifications and improvements may be made thereto. Therefore, such modifications and improvements, without departing from the spirit of the present invention, are intended to be within the scope of protection claimed herein.

[0075] appendix:

[0076] 1. The model finally calculates the simplified expression of joint torque:

[0077] τ1=sin(θ1)(2.3807+15.76045sin(θ3+θ4))-cos(θ1)·[0.5922+4.9509sin(θ2)+cos(θ2)(0.48671+15.76045cos(θ3+θ4)-4.98265sin(θ3+θ4))]

[0078] τ2=sin(θ1)[sin(θ2)(0.48671+15.76045cos(θ3)-4.98265sin(θ3))-5.4434cos(θ2)]

[0079] τ3=cos(θ1)(15.76045sin(θ3+θ4))+sin(θ1)cos(θ2)·(-15.76045cos(θ3+θ4)+4.98265sin(θ3+θ4))

[0080] τ4=sin(θ1)cos(θ2)(-15.76045cos(θ3+θ4)+4.98265sin(θ3+θ4))

[0081] 2. Table of mass center parameters of each link of the exoskeleton robot arm:

[0082] Table 1 Center of mass parameters of each link of the exoskeleton manipulator

[0083]

Claims

1. A robotic arm gravity compensation algorithm based on model and data-driven fusion, characterized in that: The following steps are involved: Step 1: For the robotic arm system, establish a mathematical model of the robotic arm, including forward kinematics and statics equations. The specific steps include: Step 1.1 Based on the Denavit-Hartenberg (DH) method, the robot base coordinate system Z0 and the coordinate systems of each joint axis are established, and the homogeneous transformation matrix and forward kinematic model of each joint of the robot are calculated by the DH method; Step 1.2 Based on the robotic arm CAD model, obtain the mass m of each joint link i and the center of mass position δ relative to its own joint coordinate system j ; Step 1.3 Based on the statics equation of the manipulator established by the virtual displacement principle, the angles θ of each joint of the manipulator are obtained. j and output torque τ m,j The corresponding relationship between them; Step 2: Gravity compensation data collection and training, establishing a Bayesian neural network learning framework. The specific steps include: Step 2.1 Initialize the algorithm parameters. According to the kinematic equation of the robot arm established in step 1, set the initial zero position S of each joint of the robot arm. j The range of motion of each joint is further divided, and multiple groups of target positions are divided within the set range. Step 2.2 Set the robot arm joint driver to the periodic position mode (CSP), and collect multiple sets of target position data current values ​​I in real time through the joint driver in the static balance state of the target position. exp During the acquisition, the collected data is processed by mean filtering, normalization and outlier elimination to obtain the joint current I exp,j , and record the joint angle θ corresponding to the filtered current value exp,j ; Step 2.3 Use Bayesian neural network (BNN) to train the preprocessed multiple sets of joint motor torque data to generate the predicted current value I nn and its uncertainty variance σ nn , characterized in that the network structure of the Bayesian neural network includes an input layer, a hidden layer and an output layer; Step 2.3.1 Input features include pre-processed joint motor current data I exp,j , joint angle θ exp,j ,Standardizing the input improves stability, and the input dimension can be determined according to the robot's degrees of freedom and the number of data features; In step 2.3.2, the hidden layer is designed as a two-layer fully connected network with 64 neurons in each layer, using the ReLU activation function. The network is trained using variational inference (SVI) and the ELBO loss function. The Adam optimizer is used with a learning rate of 0.001 for 500 epochs and a batch size of 128. Step 2.3.3 The output layer generates the predicted mean I through two fully connected layers. nn And logarithmic standard deviation log(σ), the output dimension is 1, and through σ=exp(log sigma ) is converted to standard deviation σ nn ; Step 3: BNN output I nn and standard deviation σ nn The torque τ calculated by the model m and standard deviation σ m Perform Bayesian fusion to obtain the final fusion torque τ f , the specific steps include: Step 3.1 Normalize the torque and current values ​​by the motor torque constant K t (Unit: Nm / A, the torque generated per ampere of current), establish the conversion relationship between motor current and torque, τ nn =K t I nn ; Step 3.2: The joint current value I exp,j and the corresponding angle θ exp,j As the true value, calculate the standard deviation σ of each joint j based on the model derivation m,j ; Step 3.3: The output of the Bayesian neural network is τ nn and σ nn τ calculated with the model m and σ m Perform Bayesian fusion to generate fusion torque estimate τ f ,The fusion process adopts the weighted average method, and the weights are determined inversely by the standard deviation of each model; Step 3.4: τ calculated using the model m ,σ m As a physical constraint, the fusion torque τ is limited f The value range of . Step 4: Real-time compensation: The fused torque τ f As the gravity compensation value, the motor torque constant K t The value is converted into a control current and output to the motor driver of the robot joint in real time through the Ethercat control bus for real-time online gravity compensation control.

2. The robot arm gravity compensation algorithm based on model and data driven fusion as claimed in claim 1 is characterized in that The homogeneous transformation matrix and forward kinematics model of the manipulator in step 1 are: in, Represents the homogeneous transformation matrix that the robot arm needs to go through from the i-1th link to the i-th link, θ j represents the j-th joint of the robotic arm; The static equation of the robotic arm based on the virtual displacement principle is expressed as: Among them G xj , G yj , G zj The table shows the generalized force components of the gravity of member j in the x, y, and z directions, δx j ,δx j ,δx j The table shows the coordinate position of the center of mass relative to the connecting rod, τ j and δθ j They represent the joint torque and the angular virtual displacement corresponding to joint j respectively.

3. The robot arm gravity compensation algorithm based on model and data driven fusion as claimed in claim 1 is characterized in that The step 3.3 converts the τ output by the Bayesian neural network into nn and σ nn τ calculated with the model m and σ m Perform the Bayesian fusion as shown below to obtain the following fused joint torque value τ f and standard deviation; Further use the model to calculate τ m ,σ m As a physical constraint, the fusion torque τ is limited f The threshold range is: t f ∈[τ m -k·s m ,t m +k·s m ]。

Images