CN113885316B - Stiffness modeling and identification method for a seven-degree-of-freedom collaborative robot - Google Patents

Abstract

The present invention relates to the field of collaborative robots, in particular to a stiffness modeling and identification method for a seven-degree-of-freedom collaborative robot, which includes the following steps: Step 1: Carry out kinematic modeling of the robot, and define robot joint parameters; Stiffness modeling; Step 3: Select the inverse condition number

As an observation index of the optimal method, solve the inverse condition number of each joint

The individual effects of , and according to the inverse condition number of each joint pair

Obtain a well-recognized region in the joint space; Step 4: Calculate the joint stiffness. The present invention researches the selection of poses for the stiffness identification of a seven-degree-of-freedom robot, and uses the inverse condition number as an observation index to determine a good identification area with high flexibility of the robot, improves the identification accuracy of the stiffness model, and can effectively perform the optimal configuration choose.

Description

A stiffness modeling and identification method for a seven-degree-of-freedom collaborative robot

Technical Field

The invention relates to the field of collaborative robots, and in particular to a stiffness modeling and identification method for a seven-degree-of-freedom collaborative robot.

Background Art

As China's industrialization process continues to advance, the market demand for industrial robots is also changing. The need for collaboration between robots and humans has given rise to the development of collaborative robots.

Collaborative robots refer to robots that can interact directly with people in a designated collaborative area. They are characterized by human-machine integration, safety and ease of use, sensitivity, precision, and flexibility. They not only meet the flexible manufacturing needs of small and medium batches, multiple varieties, and user customization in the industrial field, but also have potential application prospects in the fields of social services and rehabilitation medicine to cope with aging. They have become an important direction for leading the development of future robots. The design concept of lightweight and high load-to-weight ratio of collaborative robots requires integrated robot joints, compact structure, and lightweight arm rods, which introduces a large number of flexible factors into collaborative robots. The influence of structural parts and support parts such as connecting rods on the stiffness of the entire robot cannot be ignored, which brings difficulties to the improvement of the stiffness of the entire machine, thereby affecting the dynamic performance and accuracy of the robot. The virtual joint method is a common method for robot stiffness modeling, but the modeling workload is large. For collaborative robots, how to reduce the modeling workload and ensure the accuracy of modeling is an urgent problem to be solved.

Summary of the invention

The purpose of the present invention is to provide a stiffness modeling and identification method for a seven-degree-of-freedom collaborative robot. The stiffness identification posture selection of the seven-degree-of-freedom robot is studied, and the inverse condition number is used as an observation index to determine the good identification area with higher flexibility of the robot, thereby improving the recognition accuracy of the stiffness model and effectively selecting the optimal configuration.

The objective of the present invention is achieved through the following technical solutions:

A seven-degree-of-freedom collaborative robot stiffness modeling and identification method comprises the following steps:

Step 1: Kinematic modeling of the robot and definition of robot joint parameters;

Step 2: Model the stiffness of the robot;

作为最优方法的观察性指标，求解各关节对逆条件数

的个体影响，并根据各关节对逆条件数

的影响获得关节空间内的良好识别区域；Step 3: Choose the inverse condition number

As an observation indicator of the optimal method, solve the inverse condition number of each joint

The individual influence of each joint on the inverse condition number

The influence of obtaining a good identification area in the joint space;

Step 4: Calculate joint stiffness.

In step 1, each two adjacent connecting rods are described by improved DH parameters as:

是第i-1连杆的齐次变换矩阵，表示第i连杆坐标系到第i-1连杆坐标系的变换；in

is the homogeneous transformation matrix of the i-1th link, which represents the transformation from the i-th link coordinate system to the i-1th link coordinate system;

The kinematic equation of the seven-degree-of-freedom robot is derived as:

In step 2, the stiffness model K _X of the redundant robot is simplified to:

K _X =(J) ^T K _Θ J (6);

In the above formula (6), J represents the Jacobian matrix of the robot, and K _Θ represents the joint stiffness matrix.

In step 2, the stiffness model K _X of the redundant robot is simplified as follows:

The stiffness matrix is used to define the stiffness performance of the robot at the end point, specifically:

F = K _X Δt (3);

In the above formula (3), F is the external force and torque vector applied to the robot endpoint, and Δt represents the elastic deformation of the robot endpoint in Cartesian space;

The stiffness model _KX of the redundant robot is expressed as:

K _X =(J) ^T (K _Θ -K _C ) ^J (4);

是关节位移矢量；In the above equations (4) and (5), J represents the Jacobian matrix of the robot, K _C is the complementary stiffness matrix of the stiffness model, K _Θ represents the joint stiffness matrix,

is the joint displacement vector;

For redundant manipulators, the Jacobian matrix J is not invertible, and it is assumed that the Jacobian matrix J of the robot does not change with the end load, that is, the influence of the complementary stiffness matrix K _C on the stiffness model can be ignored, and the stiffness model is simplified as:

K _X =(J) ^T K _θ J (6).

In step 3, the inverse condition number is defined as follows:

In the above formula (7), κ _F represents the condition number of the Jacobian matrix based on the Frobenius norm, and tr(g) represents the trace of the matrix;

The Jacobian matrix J of the robot is normalized by the characteristic length L. The relationship between the dimensionally uniform Jacobian matrix J _N and the unnormalized Jacobian matrix J is expressed as:

In step 3, the characteristic length L of the robot is derived as follows:

a _max = max{a _i },

d _max =max{d _i }(i=1, 2,...7),

M＝max{a _max ,d _max }，

Where a _i is the length of the robot link, and d _i is the offset of the link;

来寻找特征长度L，把所有设计变量放到新的设计向量中，即：By minimizing the condition number of the Jacobian matrix J _N

To find the characteristic length L, put all the design variables into the new design vector, that is:

Solving for the vector value x can be transformed into solving an optimization problem:

的个体影响及获得良好识别区域的过程如下：In step 3, solve the inverse condition number of each joint

The individual impact and the process of obtaining a good recognition area are as follows:

的数学模型表示为：Will

The mathematical model is expressed as:

表示当x_i不变，变化其他m-1个影响因素所得的条件数平均值，即：The individual effect of influencing factor _xi

It means the average value of the condition number when _xi remains unchanged and changes other m-1 influencing factors, that is:

Discretize each influencing factor x _i into s _i small intervals within the allowable variation range [a _i ,b _i ], and use the equally divided points as discrete nodes (s _i +1), so:

In the above formula (13), p _j is the discrete point number, x _j (p _j ) represents the value of the jth influencing factor at the p _j discrete point;

The influencing factors _xi and the discrete points of each influencing factor _xi are used to construct an orthogonal array (OA) as shown in the following equation (14):

Among them, Λ _j,i is the serial number of the discrete point of the i-th influencing factor in the j-th experimental unit;

The influence of individual utility f( _xi ) is solved by the following formula (16):

Where x _m (Λ _j,m ) represents the value of the influence factor x _m at the Λ _j,m discrete point;

数产生重大影响，根据各影响因素x_i对逆条件数

的影响获取关节空间中的良好识别区域。Perform an ANOVA to determine whether the influencing factor _xi has an effect on the inverse condition number.

The inverse condition number is significantly affected by each influencing factor x _i.

The influence of is used to obtain well-identified regions in the joint space.

数产生重大影响，归因于以下假设检验：In step 3, we perform variance analysis to determine whether the influencing factor _xi has an effect on the adverse condition.

The significant impact is attributed to the following hypothesis testing:

The total average value μ( _xi ) of the influencing factors _xi is derived as:

The sum of squares of the errors of the influencing factors _xi , _SE ( _xi ), is obtained as follows:

The sum of squares of influencing factors _xi , S _A ( _xi ), yields:

Therefore, the F-distribution test statistic for the test statistic _Ti is:

没有重大影响的因素，并简化逆条件数

数学模型，同时通过比较T_i得出各影响因素x_i对逆条件数

的影响顺序，T_i越大，第i个因素对逆条件数

的影响越大。Ignoring the inverse condition number by ANOVA

There are no significant factors, and the inverse condition number is simplified

Mathematical model, and by comparing _Ti, we can get the inverse condition number of each influencing factor _xi

The order of influence, the larger the _Ti , the greater the i-th factor's influence on the inverse condition number

The greater the impact.

的影响确定优化关节，并绘制关节轮廓图，分别计算全局区域、良好识别区域和其他区域的逆条件数，良好识别区域的平均逆条件数、最大逆条件数和最小逆条件数均大于全局区域和其他区域的逆条件数。In step 3, the inverse condition number is calculated based on each influencing factor x _i

The influence of is used to determine the optimized joints, draw the joint contour diagram, and calculate the inverse condition numbers of the global area, well-recognized area and other areas respectively. The average inverse condition number, maximum inverse condition number and minimum inverse condition number of the well-recognized area are all greater than those of the global area and other areas.

In step 4, the joint stiffness value is calculated by the least squares method.

The advantages and positive effects of the present invention are:

1. The present invention studies the selection of stiffness identification postures for a seven-degree-of-freedom robot, takes the inverse condition number as the observation index to determine a good identification area with high flexibility of the robot, selects multiple groups of measured postures in this area for stiffness identification, reduces the error sensitivity of the least squares solution, and improves the recognition accuracy of the stiffness model.

2. The present invention adopts an influencing factor separation method (IFSM) based on orthogonal design experiment (ODE) to separate the individual influence of joints on the inverse condition number, and finds out the change law of each joint according to the inverse condition number. Through the variance analysis (ANOVA) of the experimental results, the joints that have a significant impact on the index can be found, thereby obtaining a good recognition area of the robot in the joint space and reducing the modeling workload.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG1 is a flow chart of a design method of the present invention;

FIG2 is a schematic diagram of a seven-degree-of-freedom redundant robot according to the present invention;

FIG3 is a schematic diagram of a virtual joint of a seven-degree-of-freedom redundant robot in FIG2 ;

FIG4 is a diagram showing the relationship between the inverse condition number and influencing factors according to an embodiment of the present invention;

FIG. 5 is a joint contour diagram showing a good recognition area determined in an embodiment of the present invention.

DETAILED DESCRIPTION

The present invention will be further described below in conjunction with the accompanying drawings.

As shown in Figures 1 to 5, the present invention includes the following steps:

Step 1: Perform kinematic modeling of the robot and define the robot joint parameters.

As shown in Figures 2 and 3, the seven-degree-of-freedom redundant robot can be seen as consisting of eight links and seven joints. The DHm parametric modeling method can describe the motion characteristics of the links through four parameters: link angle α _i-1 , link length a _i-1 , link offset d _i , and joint angle θ _i. Among them, α _i-1 and a _i-1 describe the motion characteristics of link i-1 itself, and d _i and θ _i describe the connection relationship between link i-1 and link i.

The general formula of the connecting rod transformation of the coordinate system O _{i -xi y i} z _i relative to the coordinate system O _i-1 -xi _-1 y _i-1 zi _-1 is:

In the above formula (1), cθ _i =cosθ _i , sθ _i =sinθ _i , cα _i-1 =cosα _i-1 , sα _i-1 =sinα _i-1 .

Every two adjacent links can be described by the improved DH parameters (i.e., formula (1)). The position and posture of the robot end can be obtained by multiplying the homogeneous transformation matrix of the links. The kinematic equation of the seven-degree-of-freedom robot can be derived as follows:

分别表示姿态矩阵和位置矩阵。In the above formula (2),

Represent the attitude matrix and position matrix respectively.

θ_i表示第i个关节的位移角。The robot joint displacement vector can be obtained from the robot kinematic equation of formula (2):

_θi represents the displacement angle of the i-th joint.

Step 2: Use the virtual joint method (VJM) to model the robot's stiffness, specifically:

(2.1) The stiffness performance of the robot at the end point is defined by the stiffness matrix, specifically:

F = K _X Δt (3);

In the above formula (3), F is the external force and torque vector applied to the robot endpoint, and Δt represents the elastic deformation of the robot endpoint in Cartesian space.

(2.2) The stiffness model K _X of the redundant robot is expressed as:

K _X =(J)T(K _Θ -K _C )J (4);

是关节位移矢量，由步骤一的机器人运动学方程获得。In the above equations (4) and (5), J represents the Jacobian matrix of the robot, K _C is the complementary stiffness matrix of the stiffness model, K _Θ represents the joint stiffness matrix,

is the joint displacement vector, obtained from the robot kinematic equation in step 1.

For redundant manipulators, the Jacobian matrix J is not invertible, so the Moore-Penrose inverse is used, and it is assumed that the Jacobian matrix J of the robot does not change with the end load, that is, the influence of the complementary stiffness matrix K _C on the stiffness model can be ignored, and the stiffness model can be simplified as follows:

K _X =(J) ^T K _Θ J (6);

Step 3: Select observable indicators, solve the influence of each joint on the observable indicators, and solve the better identification area in the joint space, specifically:

作为最优方法的可观察性指标，具体为：(3.1) Select the inverse condition number

The observability indicators used as the best approach are:

(3.1.1) The inverse condition number of the Jacobian matrix based on the Frobenius norm is defined as follows:

越接近1，机器人配置的灵活性就越好，当

时，机器人的配置称为各向同性，并且此配置的灵活性最高，而

越小，机器人的配置就越接近奇点。In the above formula (7), κ _F represents the condition number of the Jacobian matrix based on the Frobenius norm, tr(g) represents the trace of the matrix, and the inverse condition number is

The closer it is to 1, the better the flexibility of the robot configuration.

When , the robot configuration is called isotropic and has the highest flexibility, while

The smaller it is, the closer the robot's configuration is to the singularity.

对刚度识别的结果高度敏感，因此将其用作选择最佳机器人配置的标准，并且应尽可能接近1。Since the inverse condition number

The result of the stiffness identification is highly sensitive, so it is used as a criterion for selecting the best robot configuration and should be as close to 1 as possible.

(3.1.2) The Jacobian matrix J of the robot is normalized by the characteristic length L. The Jacobian matrix with uniform dimensions is represented by J _N , and its relationship with the unnormalized Jacobian matrix J can be expressed as:

In the above formula (8), L represents the characteristic length of the robot obtained.

The derivation of L is as follows:

a _max = max{a _i },

d _max =max{d _i }(i=1, 2,...7),

M＝max{a _max ,d _max }，

Where _ai is the robot link length and d _i is the link offset.

来寻找特征长度L，把所有设计变量放到新的设计向量中，即：Therefore, the Jacobian matrix J _N can be minimized by

To find the characteristic length L, put all the design variables into the new design vector, that is:

Solving for the vector value x can be transformed into solving an optimization problem:

的显著影响，具体为：(3.2) Obtain the individual effect of each joint and the inverse condition number of the joint through IFSM

The significant impacts are:

的个体影响，并根据逆条件数

找出每个关节的变化规律，通过实验结果的方差分析(ANOVA)，可以找出对指标有显着影响的关节，从而获得关节空间内机器人的良好识别区域。(3.2.1) The influence factor separation method (IFSM) based on orthogonal design experiment (ODE) is used to separate the joint effect on the inverse condition number.

The individual influence of

By finding out the changing pattern of each joint, through the analysis of variance (ANOVA) of the experimental results, the joints that have a significant impact on the indicators can be found, so as to obtain a good recognition area of the robot in the joint space.

的数学模型表示为：Will

The mathematical model is expressed as:

表示当x_i不变，变化其他m-1个影响因素所得的条件数平均值，即：The individual effect of joint influencing factor _xi

It means the average value of the condition number when _xi remains unchanged and changes other m-1 influencing factors, that is:

Obviously, it is very difficult to find the multiple integral for the above formula, and it is almost impossible to do so in practice. However, when analyzing the robot condition number model, we usually only need to understand the changing trend of the condition number κ _F with the displacement angle θ _i of each joint of the robot, rather than the precise changing curve. Therefore, the complex mathematical derivation of multiple integrals can be transformed into a series summation problem, that is, the influencing factors _xi of each joint are discretized into _si small intervals within their allowable range of variation [a _i , _bi ], and the equally divided points are used as discrete nodes (s _i +1).

then:

In the above formula (13), _pj is the discrete point number, and _xj ( _pj ) represents the value of the jth influencing factor at the _pj discrete point.

(3.2.2) A method of separating influencing factors is used to obtain the influence of each factor with less computational effort. This method can quickly separate the influence of each factor x _i from the inverse condition number mathematical model of the above formula (11).

Assuming that the number of discrete points of each influencing factor _xi is the same, that is, s _i = s, according to the number of influencing factors _xi and their discrete points, the influencing factors _xi and the discrete points of each influencing factor _xi are used to construct the orthogonal array (OA) L _c (s ^m ) shown in the following equation (14). Each column of the orthogonal array (OA) represents an influencing factor _xi , and each row represents an experimental unit under different discrete point combinations of factors, and the total number of experiments is c = s ² .

In the above formula (15), Λ _j,i is the serial number of the discrete point of the i-th influencing factor in the j-th experimental unit.

The influence of individual utility f( _xi ) is solved by the following formula (16):

Where x _m (Λ _j,m ) represents the value of the influence factor x _m at the Λ _j,mth discrete point.

数产生重大影响，并且可以将此问题归因于以下假设检验：(3.2.3) Perform variance analysis to determine whether the joint influencing factor _xi has an adverse effect on the adverse condition.

This issue can be attributed to the following hypothesis testing:

The total average value μ( _xi ) of the influencing factors _xi can be derived as:

The sum of squares of the errors of the influencing factors _xi , _SE ( _xi ), can be obtained as follows:

The sum of squares of influencing factors _xi , S _A ( _xi ), can be obtained as follows:

Therefore, the F-distribution test statistic for the test statistic _Ti is:

没有显着影响，相反当T_i>F_λ(s-1,s²-s)，表示要拒绝H₀的确定性为99％时，即影响因素x_i对逆条件数有重大影响。因此，通过方差分析(ANOVA)可以忽略对逆条件数

没有重大影响的因素，并简化逆条件数

数学模型，同时通过比较T_i，可以得出各因素x_i对逆条件数

从初级到次级的影响顺序，T_i越大，第i个因素对逆条件数

的影响越大。Let λ be the significance level, usually 0.01. If the test statistic T _i ≤F _λ (s-1,s ² -s), it means that there is 99% certainty that H ₀ can be accepted, that is, the influence factor _xi on the inverse condition number

There is no significant effect. On the contrary, when T _i >F _λ (s-1,s ² -s), it means that the certainty of rejecting H ₀ is 99%, that is, the influencing factor x _{i has a significant effect on the inverse condition number. Therefore, the influencing factor x i} on the inverse condition number can be ignored by analysis of variance (ANOVA).

There are no significant factors, and the inverse condition number is simplified

Mathematical model, by comparing _Ti , we can get the inverse condition number of each factor _xi

From primary to secondary influence order, the larger _Ti is, the greater the i-th factor is on the inverse condition number.

The greater the impact.

可以在分析过程中将其设置为任何值，本实施例中将关节1位移设为θ₁＝0°，将最后一个关节7位移设为θ₇＝0°，其他五个关节是影响因素x_i。For the seven-DOF collaborative robot shown in Figures 2 and 3, the displacement of the first and last joints does not affect the inverse condition number of the robot.

It can be set to any value during the analysis process. In this embodiment, the displacement of joint 1 is set to θ ₁ =0°, the displacement of the last joint 7 is set to θ ₇ =0°, and the other five joints are influencing factors x _i .

In this embodiment, the working range of each robot joint is evenly divided into 60 intervals, so the orthogonal array (OA) of the above formula (14) is composed of 5 factors, each factor contains 61 levels.

上的效果曲线，从图4可以看出，每个关节对逆条件数

的影响程度是不相等的，与关节2和关节3相比，关节4、关节5和关节6对逆条件数

的影响更大。Figure 4 shows the inverse condition number of the above five joints

From Figure 4, we can see that each joint has an inverse condition number

The degree of influence on the inverse condition number is not equal. Compared with joints 2 and 3, joints 4, 5 and 6 have a greater impact on the inverse condition number.

The impact is greater.

的影响最大，其次是关节6，而关节3对逆条件数

的影响最小，并且根据上述步骤(3.2.3)方差分析，有99％的把握可以判断关节3对逆条件数

没有明显影响。According to the above steps (3.2.3), the test statistic _Ti of each influencing factor _xi is calculated. Among the five joints, the test statistic _Ti of joint 4 is the largest, indicating that the inverse condition number

The largest effect is on joint 6, followed by joint 3, while joint 3 has a significant effect on the inverse condition number.

The effect of joint 3 on the inverse condition number is minimal, and according to the variance analysis in the above step (3.2.3), there is 99% confidence that joint 3 has a significant effect on the inverse condition number.

No noticeable impact.

的测量位姿，以实现刚度识别的良好收敛。(3.3) Obtain several well-identified regions in the joint space (regions with high robot flexibility) and select a set of regions with large inverse condition numbers in these regions.

The measured pose is used to achieve good convergence of stiffness identification.

由于关节空间是对称的，因此仅将关节空间的一半用于分析和计算，每次计算使用两百万个采样点。经过计算验证，良好识别区域的平均逆条件数、最大逆条件数和最小逆条件数均大于全局区域和其他区域的逆条件数，证明基于正交设计实验(ODE)的影响因子分离方法(IFSM)的有效性。所述蒙特卡罗方法为本领域公知技术。Joints 4, 5, and 6 are optimized, and the working space of joint 5 is divided into 10 equal intervals. The joint contour diagram at the interval points of joint 5 is drawn, as shown in Figure 5. The inverse condition numbers of the global area, the well-recognized area, and other areas are calculated by the Monte Carlo method.

Since the joint space is symmetrical, only half of the joint space is used for analysis and calculation, and two million sampling points are used for each calculation. After calculation verification, the average inverse condition number, the maximum inverse condition number and the minimum inverse condition number of the well-identified area are all greater than the inverse condition number of the global area and other areas, proving the effectiveness of the influencing factor separation method (IFSM) based on orthogonal design experiment (ODE). The Monte Carlo method is a well-known technology in the art.

Among several well-identified regions in the acquired joint space, a set of measured poses with large inverse condition numbers in these regions are selected to achieve good convergence of stiffness identification.

Step 4: According to the stiffness model in step 2 and the good identification area of stiffness identification selected in step 3, the joint stiffness value is calculated by the least squares method, specifically:

According to the good identification area of stiffness identification selected in step 3 (the area with high flexibility of the robot), multiple groups of measured poses are selected in this area for stiffness identification and substituted into the stiffness model in step 2, and the residual sum of squares is minimized using the least squares method, as shown in the following formula:

In the above equations (22) and (23), _Fi is the i-th component of the force vector F applied to the robot endpoint, and c is the joint stiffness vector.

Each experiment obtained a 6×1 force vector, a 6×1 elastic displacement vector, and a 6×6 A matrix. Therefore, the size of the matrix changes from 6×7 to 6nq×7, and the value of c c _min minimizes the Euclidean norm Δ of the approximation error based on the generalized inverse of A.

c _min =(A ^T A) ^-1 A ^T Δt (24);

The above least squares calculation is a well-known technique in the art.

In this embodiment, the calculation and drawing of each step can be implemented by commercial software such as Matlab.

Claims (8)

1. A seven-degree-of-freedom collaborative robot stiffness modeling and identification method, characterized in that it includes the following steps: Step 1: Kinematic modeling of the robot and definition of robot joint parameters; Step 2: Model the stiffness of the robot; Step 3: Choose the inverse condition number

As an observation indicator of the optimal method, solve the inverse condition number of each joint

The individual influence of each joint on the inverse condition number

The influence of obtaining a good identification area in the joint space; Solve the inverse condition number of each joint

The individual impact and the process of obtaining a good recognition area are as follows: Will

The mathematical model is expressed as:

The individual effect of influencing factor _xi

It means the average value of the condition number when _xi remains unchanged and changes other m-1 influencing factors, that is:

Discretize each influencing factor x _i into s _i small intervals within the allowable variation range [a _i ,b _i ], and use the equally divided points as discrete nodes (s _i +1), so:

In the above formula (13), p _j is the discrete point number, x _j (p _j ) represents the value of the jth influencing factor at the p _j discrete point; The influencing factors _xi and the discrete points of each influencing factor _xi are used to construct an orthogonal array (OA) as shown in the following equation (14):

Among them, Λ _j,i is the serial number of the discrete point of the i-th influencing factor in the j-th experimental unit; The influence of individual utility f( _xi ) is solved by the following formula (16):

Where x _m (Λ _j,m ) represents the value of the influence factor x _m at the Λ _j,m discrete point; Perform an ANOVA to determine whether the influencing factor _xi has an effect on the inverse condition number.

The inverse condition number is significantly affected by each influencing factor x _i.

The influence of obtains the well-identified regions in the joint space; Perform an ANOVA to determine whether the influencing factor _xi has an adverse effect on the condition

The significant impact is attributed to the following hypothesis testing: H ₀ :f( _xi (1))=f( _xi (2))=…=f( _xi (s)) H ₁ : f(x _i (1)), f(x _i (2)),…, f(x _i (s))Not all equal (17); The total average value μ( _xi ) of the influencing factors _xi is derived as:

The sum of squares of the errors of the influencing factors _xi , _SE ( _xi ), is obtained as follows:

The sum of squares of influencing factors _xi , S _A ( _xi ), yields:

Therefore, the F-distribution test statistic for the test statistic _Ti is:

Ignoring the inverse condition number by ANOVA

There are no significant factors, and the inverse condition number is simplified

Mathematical model, and by comparing _Ti, we can get the inverse condition number of each influencing factor _xi

The order of influence, the larger the _Ti , the greater the i-th factor's influence on the inverse condition number

The greater the impact, Step 4: Calculate joint stiffness. 2. The method for stiffness modeling and identification of a seven-degree-of-freedom collaborative robot according to claim 1, characterized in that: In step 1, each two adjacent connecting rods are described by improved DH parameters as:

in

is the homogeneous transformation matrix of the i-1th link, which represents the transformation from the i-th link coordinate system to the i-1th link coordinate system; The kinematic equation of the seven-degree-of-freedom robot is derived as:

3. The stiffness modeling and identification method of a seven-degree-of-freedom collaborative robot according to claim 1, characterized in that: In step 2, the stiffness model K _X of the redundant robot is simplified to: K _X =(J) ^T K _Θ J (6); In the above formula (6), J represents the Jacobian matrix of the robot, and K _Q represents the joint stiffness matrix. 4. The method for stiffness modeling and identification of a seven-degree-of-freedom collaborative robot according to claim 3, characterized in that: In step 2, the stiffness model K _X of the redundant robot is simplified as follows: The stiffness matrix is used to define the stiffness performance of the robot at the end point, specifically: F = K _X Δt (3); In the above formula (3), F is the external force and torque vector applied to the robot endpoint, and Δt represents the elastic deformation of the robot endpoint in Cartesian space; The stiffness model _KX of the redundant robot is expressed as: K _X =(J) ^T (K _Θ -K _C )J (4);

In the above equations (4) and (5), J represents the Jacobian matrix of the robot, K _C is the complementary stiffness matrix of the stiffness model, K _Q represents the joint stiffness matrix,

is the joint displacement vector; For redundant manipulators, the Jacobian matrix J is not invertible, and it is assumed that the Jacobian matrix J of the robot does not change with the end load, that is, the influence of the complementary stiffness matrix K _C on the stiffness model can be ignored, and the stiffness model is simplified as: K _X =(J) ^T K _θ J (6). 5. The method for stiffness modeling and identification of a seven-degree-of-freedom collaborative robot according to claim 1, characterized in that: In step 3, the inverse condition number is defined as follows:

In the above formula (7), k _F represents the condition number of the Jacobian matrix based on the Frobenius norm, and tr(g) represents the trace of the matrix; The Jacobian matrix J of the robot is normalized by the characteristic length L. The relationship between the dimensionally uniform Jacobian matrix J _N and the unnormalized Jacobian matrix J is expressed as:

6. The method for stiffness modeling and identification of a seven-degree-of-freedom collaborative robot according to claim 5, characterized in that: In step 3, the characteristic length L of the robot is derived as follows: a _max = max{a _i }, d _max =max{d _i }(i=1, 2,...7), M＝max{a _max ,d _max }，

Where a _i is the length of the robot link, and d _i is the offset of the link; By minimizing the condition number of the Jacobian matrix J _N

To find the characteristic length L, put all the design variables into the new design vector, that is:

Solving for the vector value x can be transformed into solving an optimization problem:

7. The method for stiffness modeling and identification of a seven-degree-of-freedom collaborative robot according to claim 1 is characterized in that: in step 3, the inverse condition number is determined according to each influencing factor x _i

The influence of is used to determine the optimized joints, draw the joint contour diagram, and calculate the inverse condition numbers of the global area, well-recognized area and other areas respectively. The average inverse condition number, maximum inverse condition number and minimum inverse condition number of the well-recognized area are all greater than those of the global area and other areas. 8. The stiffness modeling and identification method of a seven-degree-of-freedom collaborative robot according to claim 1 is characterized in that: in step 4, the joint stiffness value is calculated by the least squares method.

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