A method for calculating the joint motion parameters of a hyperbolic manipulator using a nonlinear mixed friction model

Abstract

A method for calculating the joint motion parameters of a quasi-hyperbolic manipulator using a nonlinear mixed friction model relates to the field of joint mechanics calculation in robotics. Although the traditional calculation model can describe the relationship between friction force and relative motion speed or contact surface positive pressure to a certain extent, it ignores the nonlinear characteristics of friction force, resulting in large errors in motion parameters and unable to meet the current design requirements for high-precision manipulators with the rapid development of industrial automation technology. Therefore, in order to improve the accuracy of the calculation of the joint motion parameters of the manipulator, the present invention proposes a method for calculating the joint motion parameters of a quasi-hyperbolic manipulator using a nonlinear mixed friction model. The friction model is based on the Gauss friction model and combines the Coulomb friction model, the viscous friction model and the Stribeck friction model to provide a more accurate pre-calculation method for the subsequent structural design of the manipulator, the manufacturing of rods and fuselage, and the trajectory control of the end effector.

Description

Technical Field

[0001] This research intersects multiple disciplines, including robotics, tribology, and dynamic modeling, specifically focusing on the precise calculation of joint parameters for high-precision, high-load-capacity industrial robotic arms. With the rapid development of industrial automation technology, robotic arms have found widespread application in a variety of fields, including manufacturing and logistics. The friction characteristics of robotic arm joints are key factors affecting their motion accuracy, dynamic performance, and service life. Developing a nonlinear mixed friction model that can accurately calculate robotic arm joint parameters is crucial for improving the overall performance of robotic arms. Background Art

[0002] Traditional friction models for robotic arm joints are primarily based on simple linear relationships, such as the Coulomb friction model and the viscous friction model. These models can, to a certain extent, describe the relationship between friction and relative velocity or contact surface pressure, but they ignore the nonlinear characteristics of friction, such as the variation in the friction coefficient and the conversion between static and kinetic friction. Consequently, these traditional models often exhibit significant errors when calculating robotic arm joint parameters, failing to meet the design requirements for high-precision robotic arms. In recent years, with the advancement of tribological research, the nonlinear characteristics of friction have been gradually recognized, leading to the development of a series of more complex friction models, such as the Stribeck friction model and the Gaussian friction model. The Stribeck friction model describes the complex relationship between friction and relative sliding velocity, including static friction, kinetic friction, and the transition between them. The Gaussian friction model, based on statistical principles, more accurately describes the impact of joint friction on the mechanism's driving force. However, these models still face challenges in practical applications, such as difficulty in determining parameters and high computational complexity.

[0003] Therefore, in order to overcome the limitations of traditional friction models and improve the calculation accuracy of robot arm joint parameters, the present invention proposes a method for calculating robot arm joint motion parameters using a nonlinear mixed friction model. This method can more comprehensively describe the friction characteristics of the robot arm joints and provide a more accurate pre-calculation means for the design, manufacture and control of the robot arm. Summary of the Invention

[0004] This algorithm proposes a method for calculating the motion parameters of robotic arm joints using a nonlinear mixed friction model. This model establishes a nonlinear mixed friction model for the robotic arm joints. Based on the Gaussian friction model, this model combines the Coulomb friction model, the viscous friction model, and the Stribeck friction model. This model can more accurately describe the friction characteristics of robotic arm joints and more precisely calculate the relevant parameters of robotic arm joint motion.

[0005] 1. Limit the range of motion angles and speeds of each joint, and determine the range of friction parameters for each joint based on the contact material and lubrication conditions of the joints. This ensures that the calculated range matches the actual range of motion of the robotic arm, avoiding overcalculation.

[0006] 2. Use the "Delete" function in SolidWorks to remove components that do not affect the calculation results, resulting in a simpler and more intuitive 3D model. The removed components are the motor, support structure, and shock absorber structure. The "Measure" function in SolidWorks can be used to obtain the basic parameters of each connecting rod, including rod mass, center of mass position, moment of inertia, and moment of inertia tensor.

[0007] 3. Model the robotic arm within the software, primarily using the Robotic toolkit within Matlab. This toolkit must first be installed and debugged repeatedly to ensure it is compatible with the computer's software environment and runs correctly. Execute the build command to generate an animation file of the robotic arm's motion. Play it back step by step to check for discontinuities and excessive jitter during the motion of each joint.

[0008] 4. Perform kinematic calculations. The main purpose of this calculation is to verify that the set working conditions are achievable. Kinematic calculations include two steps: forward kinematics calculations and inverse kinematics calculations. The calculation steps are as follows: Figure 1 As shown. During the calculation process, forward kinematics research is the problem of calculating the position and posture of the tool coordinate system relative to the fixed coordinate system. After the joint angle is given, the general expression of the matrix transformation relationship between two adjacent joint coordinate systems is obtained. The link transformation matrix is ​​multiplied to obtain a transformation matrix of coordinate system {N} relative to coordinate system {0}, which is a function of the joint variables. By obtaining the values ​​collected by the sensors at the joint positions, the position and posture of the robot end link in the Cartesian coordinate system can be represented by this matrix. Inverse kinematics research is to find the joint angle so that the tool coordinate system is in a specific position and posture in the fixed coordinate system. For robots with 6 revolute joints, the coordinate systems of the three rear joints intersect at the same point, and the Pieper solution is usually used for analysis.

[0009] 5. To calculate the friction torque, the process requires first determining the forces and torques on the joints. The manipulator joints in industrial robot systems are divided into two types: rotary joints and translational joints. The various combinations of the former can cover a wider range of working areas and are more widely used due to their simple structure and smoother movement. The rotational joints will generate three-directional constraint forces F x 、F y 、F z , the constraint moments T in three directions x 、Ty and T z After that, the calculation model of the nonlinear mixed friction model is constructed, which includes the three friction types as follows:

[0010] (1) Coulomb friction. Assume that the joint is only affected by the Coulomb friction torque after overcoming the maximum static friction torque. The Coulomb friction torque T generated by the joint constraint force and constraint torque is fc and the maximum static friction torque T fs It can be expressed as

[0011] T fc =μ cc N1R n +μ cc (N2+N3)R p

[0012] T fs =μ sc N1R n +μ sc (N2+N3)R p

[0013] μ cc and μ sc are the Coulomb friction coefficient and the static friction coefficient, R n is the friction arm, R p is the pin radius. The equivalent pressure N1 generated by the axial constraint is |F z |,|F z | indicates F z Take the absolute value; equivalent pressure under lateral constraint Normal pressure due to restraint moment R b The bending reaction arm.

[0014] (2) Viscous friction model. The viscous friction model is a nonlinear function. Within the angular velocity range, after overcoming the Coulomb friction torque, as the speed increases, a viscous friction resistance torque will appear between the objects in relative motion, and it is linearly proportional to the speed. After static-dynamic conversion, the change in friction torque in the low-speed zone is proportional to the joint angular velocity. is linearly proportional, and its mathematical model is T v represents the viscous friction torque, B represents the viscous friction factor, Represents the joint angular velocity.

[0015] (3) Stribeck friction. The Stribeck effect is reflected in the transition stage between low speed and high speed. After the robot joint overcomes the static friction, the friction force is discontinuous at the moment when the joint switches from static friction to sliding friction. In the low-speed area, the joint friction torque decreases with the increase of angular velocity; in the transition stage, the Stribeck friction force becomes dominant. In the low-speed area, the Stribeck friction torque value is used instead of the viscous friction torque value. The formula of the Stribeck friction model is:

[0016]

[0017] T s is the Stribeck friction torque, θ sc is an empirical constant related to the nonlinearity of the Stribeck curve. The Stribeck curve is determined in a laboratory setting using unidirectional friction testing of a pin against a rotating disk under submerged lubrication conditions. A converging gap model is created based on the contact geometry of the pin end faces.

[0018] This results in a calculation model for the nonlinear mixed friction model. The nonlinear joint mixed friction model can better reflect the friction phenomena in actual operation, especially the nonlinear behavior caused by large speed changes. The expression of the friction torque of the joint in the case of rotational motion is:

[0019]

[0020] Where, T out is the external torque, |T out | represents the absolute value of the external torque; sgn(T out ) is a symbolic function. The specific calculation principle is as follows: out When the value is greater than 0, sgn(T out ) output is 1, when T out When the value is less than 0, sgn(T out ) output is -1, T out When the value is equal to 0, sgn(T out ) output is 0. This model accurately calculates the kinematic and dynamic parameters of the robotic arm under different working conditions, such as speed, acceleration, torque, etc., providing basic data support for subsequent optimization design.

[0021] Combined with the actual situation of joint movement, this function is plotted and analyzed in detail. The function graph is as follows: Figure 2As shown in the figure. When the joint is about to start rotating, the friction force is dominated by the maximum static friction force, and the movement is hindered until the external force overcomes the maximum static friction force, resulting in relative sliding. At this point, the friction force is no longer dominated by the maximum static friction force, but gradually decreases to a smaller value. This is due to the change in the lubrication state between the two surfaces. First, due to the presence of viscosity in the lubricant, it cannot completely fill the gap. In the transition stage with low speed, the lubricant can only provide some lubrication at the boundary and is in a boundary lubrication state, which is accompanied by the Stribeck phenomenon. Then, as the speed increases, the lubricant is better filled between the two contact surfaces, and is in a partial fluid lubrication state. At this time, the friction force is dominated by viscous friction.

[0022] 6. According to the dynamic calculation of joint torque, the construction of the dynamic equation of the robot arm usually adopts the Newton-Euler method or the Lagrange method. The former divides the robot arm into multiple rigid bodies or links, and considers the force and torque relationship between each rigid body. It is more suitable for rigidly connected robot arm systems, so its generalized form is selected to calculate the driving torque. The Newton-Euler recursive dynamics algorithm has two steps in the solution process. The speed and acceleration of each link are calculated recursively from the robot base to the end of the joint; then the interaction force and torque between the links and the driving torque of the joint are calculated recursively from the end link of the robot to the base. The specific process and parameters used are as follows Figure 3 The graph compares the joint torque values ​​with and without considering the influence of friction. The graph uses time as the horizontal axis and joint torque value as the vertical axis. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] Figure 1 A flowchart of an embodiment of a method for calculating kinematics of a robotic arm according to the present invention;

[0024] Figure 2 This is the function graph of the nonlinear mixed friction model proposed in the present invention;

[0025] Figure 3 A flowchart of an embodiment of a method for calculating the dynamics of a robotic arm according to the present invention;

[0026] Figure 4 This is a schematic diagram of the calculation system for a certain type of robotic arm currently on sale in the market used in the calculation examples of the present invention;

[0027] Figure 5 is the joint velocity value of a certain commercially available robotic arm used in the examples of the present invention;

[0028] Figure 6 These are the joint torque results of a commercially available robotic arm used in the examples of this invention. DETAILED DESCRIPTION

[0029] Various exemplary embodiments of the present invention will now be described in detail with reference to the accompanying drawings. It should be noted that, unless otherwise specifically stated, the relative arrangements of components and steps, numerical expressions and numerical values ​​set forth in these embodiments do not limit the scope of the present invention.

[0030] Technologies, methods, and equipment known to ordinary technicians in the relevant art may not be discussed in detail, but where appropriate, the technologies, methods, and equipment should be considered part of the specification.

[0031] A method for calculating joint motion parameters of a hyperbolic manipulator using a nonlinear mixed friction model, comprising:

[0032] (1) By limiting the range of motion angles and speeds of each joint in the working condition, the range of friction parameters of each joint is determined according to the two conditions of the material and lubrication conditions of the joint; the friction parameters include the viscous friction coefficient B, the Stribeck curve parameter θ sc , Coulomb friction torque T fc , Maximum static friction torque value T fs ;

[0033] (2) Using SolidWorks 3D software, disassemble the rod structure and use the "delete" or "hide" function to remove the components that do not participate in the calculation from the calculation list to obtain a simplified shell 3D model. Use the built-in calculation function of the software to obtain the basic parameters of each connecting rod and draw a parametric system diagram; the basic parameters of the connecting rod include the connecting rod mass, the center of mass position of the connecting rod, the connecting rod moment of inertia, and the connecting rod inertia tensor value. The basic parameters are all expressed in matrix form, and the number of rows is the number of movable joints of the robotic arm;

[0034] (3) After installing the Robotic toolkit in Matlab software, the animation file of the robot arm motion process is gradually output through repeated debugging of the input running program;

[0035] (4) By constructing and executing the kinematic solution code, the motion parameters of each joint when the friction effect is ignored are obtained as the judgment conditions of the nonlinear mixed friction model; the motion parameters of each joint include the joint motion speed, the coordinate point of the end effector motion trajectory, and the joint driving torque.

[0036] (5) By constructing and executing the calculation model of the nonlinear mixed friction model and running it, the motion parameters of each joint are obtained when the influence of friction on the running results is considered. By constructing and executing the dynamic solution code, the torque values ​​of each joint are obtained when the influence of friction is ignored. The model includes four judgment intervals. When the speed falls in different intervals, the code will run different calculation formulas. The overall expression of the model is: Among them, the robot arm joint speed External torque Tout , viscous friction coefficient B, Coulomb friction torque T fc , maximum static friction torque T fs , an empirical constant θ related to the nonlinearity of the Stribeck curve sc The calculated output parameter is the friction torque of the joint of the robot arm

[0037] (6) In the same time period, the joint torque values ​​with and without friction are compared, and a comparison chart is drawn by controlling the variables to show the change trend.

[0038] It is obtained using Matlab built-in calculation commands. When solving the dynamics forward, the input of the corresponding command is the trajectory point and the output is the joint torque.

[0039] The joint driving torque It is a matrix. The number of matrix rows is the number of movable joints of the robot arm or the number of joints studied. The number of matrix columns is the data set capacity related to the calculation step size. Within the specified working time, if the calculation step size is smaller, the number of matrix columns is larger. If the calculation step size is larger, the number of matrix columns is smaller.

[0040] The various joint motion parameters when considering the influence of friction on the operation results include the joint motion speed under the influence of friction, the end effector motion trajectory coordinate points under the influence of friction, the joint friction torque

[0041] The joint friction torque It is a matrix. The number of matrix rows is the number of movable joints of the robot arm or the number of joints studied. The number of matrix columns is the data set capacity related to the calculation step size. Within the specified working time, if the calculation step size is smaller, the number of matrix columns is larger. If the calculation step size is larger, the number of matrix columns is smaller.

[0042] 1. Using a commercially available robotic arm as an example, the specific implementation of the present invention is described. This robotic arm has six rotating joints and is commonly used in welding scenarios. The motion conditions of robotic arms in factories are complex and varied, requiring full assurance of the reliability and safety of the robotic arm structure and constraints and restrictions on the joint operating speed and angle range. The range of motion of the first three joints of this robotic arm is (-170°, 170°), (-160°, 90°), and (-90°, 175°), while the angular range of the rear three joints is plus or minus 180°.

[0043] The parameters of the robot arm links are as follows.

[0044]

[0045]

[0046] The first three joints of the robotic arm use rotary vector reducers, and the last three joints use hypoid gear reducers. The models of the hypoid joints are HY1-4, HY1-5, and HY1-6 respectively.

[0047] The viscous friction coefficient of the first three joints is 2.0 Nms / rad, the empirical constant related to the nonlinearity of the Stribeck curve is 0.05, the Coulomb friction torque is 10 N·m, and the maximum static friction torque is 20 N·m; the viscous friction coefficient of the last three joints is 0.5 Nms / rad, the empirical constant related to the nonlinearity of the Stribeck curve is 0.03, the Coulomb friction torque is 1.25 N·m, and the maximum static friction torque is 3.6 N·m.

[0048] 2. Using SolidWorks, disassemble the motor, reducer, and support structure in the robotic arm's 3D model and evaluate and measure the six connecting rods to obtain the basic parameters of each connecting rod, as shown in the table below.

[0049]

[0050] Determine and draw the system construction results of the robot arm as follows Figure 4 As shown, the kinematics of the robot arm are calculated based on this.

[0051] 3. Consider a 2-second operating condition of the robotic arm. Assume both the upper and lower arms are extended to a horizontal position. The joints at both ends of the lower arm and elbow rotate. Because their rotation axes intersect perpendicularly, an external torque is generated on the components. Run the Build command to output an animation of the robotic arm's operation, checking step by step to ensure smooth joint operation under this condition.

[0052] 4. Perform kinematic calculations on the robot arm. The main purpose of kinematic calculations is to ensure that the set working conditions are achievable. The output joint motion speed is the basic judgment condition for the subsequent calculation of joint friction torque. Draw the motion of each joint as shown in the following figure. Figure 5 As shown in the figure, this image further verifies that the joint operates smoothly under this working condition. Joint 1 does not move, so the curve is a straight line. Joint 2 has the same speed as joints 3, 4, 5, and 6, but in opposite directions, so the curves increase and decrease in opposite directions, but the absolute value remains the same.

[0053] 5. Calculate the torque values ​​of each joint using dynamic methods, ignoring the influence of friction. The inertia moment value is obtained by extrapolation, and the driving moment value is obtained by interpolation. This yields the first set of output data.

[0054] 6. Calculate the friction torque value of each joint in this working condition. The calculation formula is the nonlinear mixed friction model mentioned in the invention content. The judgment condition of this model is the joint speed. Get the second set of output data from Figure 6 The results show that, compared to traditional calculation methods, the required driving torque for the robotic arm is greater when friction is considered, with the average increment for each joint being 23.48%. This clearly demonstrates the significant impact that friction at the joints has on the design of the robotic arm's joint motors and reducers. The nonlinear mixed friction model explored in this article is accurate and significant, providing more precise data for subsequent research on lightweight design, robotic arm trajectory control, and lifespan calculation.

[0055] The nonlinear mixed friction model of this invention can more accurately describe the friction characteristics of the robot arm's joints, thereby more precisely calculating the relevant parameters of the robot arm's motion. This model provides a more precise calculation method for the design, manufacture, and control of the robot arm, helping to improve the robot's motion accuracy and stability.

Claims

1. A method for calculating the joint motion parameters of a hyperbolic manipulator using a nonlinear mixed friction model, characterized in that: include: (1) By limiting the range of motion angles and speeds of each joint in the working condition, the range of friction parameters of each joint is determined according to the two conditions of the material and lubrication conditions of the joint; the friction parameters include the viscous friction coefficient B, the Stribeck curve parameter θ sc , Coulomb friction torque T fc , Maximum static friction torque value T fs ; (2) Using SolidWorks 3D software, disassemble the rod structure and use the "delete" or "hide" function to remove the components that do not participate in the calculation from the calculation list to obtain a simplified 3D shell model. Use the built-in calculation function of the software to obtain the basic parameters of each connecting rod and draw a parametric system diagram; The basic parameters of the connecting rod include the connecting rod mass, the connecting rod center of mass position, the connecting rod moment of inertia, and the connecting rod inertia tensor value. The basic parameters are all expressed in matrix form, and the number of rows is the number of movable joints of the robot arm; (3) After installing the Robotic toolkit in Matlab software, the animation file of the robot arm motion process is gradually output through repeated debugging of the input running program; (4) By constructing and executing the kinematic solution code, the motion parameters of each joint when the friction effect is ignored are obtained as the judgment conditions of the nonlinear mixed friction model; the motion parameters of each joint include the joint motion speed, the coordinate point of the end effector motion trajectory, and the joint driving torque T θ ; (5) By constructing and executing the calculation model of the nonlinear mixed friction model and running it, the motion parameters of each joint are obtained when the influence of friction on the running results is considered. By constructing and executing the dynamic solution code, the torque values ​​of each joint are obtained when the influence of friction is ignored. The model includes four judgment intervals. When the speed falls in different intervals, the code will run different calculation formulas. The overall expression of the model is: Among them, the robot arm joint speed External torque T out , viscous friction coefficient B, Coulomb friction torque T fc , maximum static friction torque T fs , an empirical constant θ related to the nonlinearity of the Stribeck curve sc The calculated output parameter is the friction torque of the joint of the robot arm (6) In the same time period, the joint torque values ​​with and without friction are compared, and a comparison chart is drawn by controlling the variables to show the change trend.

2. The method according to claim 1, wherein: It is obtained using Matlab built-in calculation commands. When solving the dynamics forward, the input of the corresponding command is the trajectory point and the output is the joint torque.

3. The method according to claim 1, wherein: The joint driving torque It is a matrix. The number of matrix rows is the number of movable joints of the robot arm or the number of joints studied. The number of matrix columns is the data set capacity related to the calculation step size. Within the specified working time, if the calculation step size is smaller, the number of matrix columns is larger. If the calculation step size is larger, the number of matrix columns is smaller.

4. The method according to claim 1, wherein: The various joint motion parameters when considering the influence of friction on the operation results include the joint motion speed under the influence of friction, the end effector motion trajectory coordinate points under the influence of friction, the joint friction torque 5. The method according to claim 1, wherein: The joint friction torque It is a matrix. The number of matrix rows is the number of movable joints of the robot arm or the number of joints studied. The number of matrix columns is the data set capacity related to the calculation step size. Within the specified working time, if the calculation step size is smaller, the number of matrix columns is larger. If the calculation step size is larger, the number of matrix columns is smaller.

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