A reliability calculation method for multi-robot collaborative system based on boundary estimation

Abstract

The present invention relates to a method for calculating the motion reliability of a multi-robot collaborative system. The purpose is to propose a method for calculating the reliability of multi-industrial robots based on boundary estimation, so as to solve the problem of motion reliability analysis of a multi-robot collaborative system and ensure that the robots can accurately, stably and reliably collaborate in actual use. The technical scheme is as follows: the present invention regards the joint clearance, the connecting rod length deviation and the connecting rod offset deviation as normally distributed variables, uses the probability distribution model of the above variables, establishes the function function of the robot through the forward kinematics of the robot, expands the function function in the first order at the mean point of the variable based on the Taylor series, and uses the mean and standard deviation of the variable to respectively calculate the mean and standard deviation of the robot function function, thereby obtaining the reliability index of the robot, and uses the integral to calculate the motion failure probability of the robot.

Description

Technical Field

[0001] The present invention relates to a method for calculating the motion reliability of a multi-robot cooperation system, and particularly to a method for calculating the motion reliability of a multi-robot series system based on boundary estimation, which is applicable to the multi-robot cooperation systems widely used in fields such as automobile production, pharmaceutical manufacturing, and parts assembly. Background Art

[0002] Robots have been widely used in industrial production operations such as handling, welding, spraying, loading and unloading, and palletizing due to their flexible operating space, fast beat response, and relatively low cost. With the rapid development of high-tech fields such as aerospace, the parts to be processed are becoming larger and more complex. Compared with a single-robot system, a multi-robot cooperation system has a larger operating space, higher reliability and flexibility, and greater load capacity, and thus can undertake more complex processing tasks, which has received extensive attention and research. Due to inevitable wear and force deformation during the design, processing, assembly, and use of a single robot, there are inevitable errors in the robot body, mainly including the joint clearance at the rotating joint and the deviation of the connecting rod size. The deviation of the connecting rod size reduces the motion accuracy of the robot end. The existence of the joint clearance not only affects the motion accuracy but also causes joint impact, exacerbates vibration, and even leads to the motion failure of the robot. The motion state of the multi-robot cooperation system will also be affected by the motion instability of a single robot. Therefore, it is of great significance to study the motion reliability of the multi-robot cooperation system under the influence of the above uncertainties. In current industrial production, the research on multi-robot cooperation systems mainly focuses on the development of control strategies. By establishing a complex coupled mechanical model of multi-robots and using control algorithms to monitor and regulate the state of multi-robots during collaborative motion, it is mainly based on dynamics. Since a single robot may have motion failure under the influence of connecting rod deviation and joint clearance, it will lead to the possibility of motion failure of the multi-robot collaborative system. For example, in collaborative handling and assembly tasks, the failure of any robot to meet the requirements will result in the failure of the overall task. Summary of the Invention

[0003] The object of the present invention is to overcome the deficiencies of the above background art and propose a method for calculating the reliability of multi-industrial robots based on boundary estimation to solve the problem of motion reliability analysis of multi-robot cooperation systems and ensure that robots can cooperate accurately, stably, and reliably in actual operation.

[0004] The technical solution provided by the present invention is as follows: A method for calculating the motion reliability of a multi-robot cooperation system based on boundary estimation is carried out according to the following steps:

[0005] Step 1: Establish a D-H link coordinate system that includes a base coordinate system and joint local coordinate systems according to the link arm structure of the robot;

[0006] As Figure 2 shown, first, take the bottom center of the robot as the origin to establish a base coordinate system O_XYZ, where the Z-axis is along the axial direction of the first rotary joint of the robot, the X-axis direction can be arbitrarily selected, and the direction of the Y-axis is determined by the X-axis and Z-axis according to the right-hand rule;

[0007] As Figure 3 shown, establish a local coordinate system O i-1 X i-1 Y i-1 Z i-1 at the rotary joint of the i-th link arm. The origin O i-1 of the local coordinate system is the rotation center of the i-th joint. The Z i-1 -axis is along the axial direction of the i-th joint, the X i -axis is the current joint axial direction, and the direction of the Y i -axis is determined by the X i -axis and Z i -axis according to the right-hand rule. Then, the position relationship between the local coordinates O i X i Y i Z i of the i-th link arm and the local coordinates of the (i - 1)-th link arm can be obtained through the following transformation: Rotate the local coordinate system O i- 1 X i-1 Y i-1 Z i-1 around the Z i-1 -axis by an angle θ i to make the X i-1 axis parallel to the X i axis, and translate along the Z i-1 axis by d i to obtain the coordinate system O′ i X′ i Y′ i Z′ i . Then, translate the coordinate system O′ i X′ i Y′ i Z′ i along the X′ i axis by a i and rotate around the X′ i axis by an angle α i to obtain the coordinate system O i X i Y i Z i .

[0008] The above transformation relationship is represented as a 4×4 homogeneous matrix Specifically, it refers to the following expression:

[0009]

[0010] where a i is the length of the i-th link arm, d i is the offset of the i-th link, α i is the twist angle of the i-th link, θ i is the joint rotation angle of the i-th link arm, and the robot is a serial or parallel robotic arm structure;

[0011] Step 2: Obtain the homogeneous matrix of the robot end position relative to the base coordinate system through the forward kinematics of the robot link arms The coordinate value of the theoretical position of the robot end relative to the base coordinate system O_XYZ is obtained as P(x, y, z), and the above homogeneous matrix Adopts the following expression:

[0012]

[0013] where n represents the number of degrees of freedom of the robot, R represents the attitude matrix of the robot end in the base coordinate system, P(x, y, z) represents the theoretical position coordinates of the robot end, and П represents continuous multiplication;

[0014] Step 3: Define the dimensional deviations of the link arm length a i and the link offset d i at the i-th link arm of the robot as random variables that follow a normal distribution. The means of the link arm length and the link offset are μ ai and μ di , respectively, and the standard deviations are σ ai and σ di ; at the same time, define the joint rotation angle θ i of the rotating joint including the joint clearance at the i-th link arm of the robot as a random variable that follows a normal distribution. The mean and standard deviation of the joint rotation angle are μ θi and σ θi ;

[0015] Step 4: Use the random number function normrnd in MATLAB to generate random numbers that follow a normal distribution to represent the link arm length, the link offset, and the joint rotation angle;

[0016] Step 5: Define the motion error ε of the robot end as the distance deviation between the spatial position of the robot in the ideal state and the actual position considering the influence of errors such as joint clearances. Through the random numerical values generated in Step 4, use the homogeneous matrix of the forward kinematics expression in Step 2 The coordinate values of the actual end position of the robotic arm during movement with respect to the base coordinate system are P'(x', y', z'), and the expected coordinate values of the robotic arm in the ideal state are P(x, y, z). Then, the following formula is used to calculate the end movement error ε of the robotic arm:

[0017]

[0018] Step 6: Combine the link lengths, link offsets, and joint angles of the robotic arm to form a vector X = [a 1 , a 2 , a 3 , d 1 , d 4 , d 6 , θ 1 , θ 2 , …, θ 6 . Set the positioning accuracy of the end of the robotic arm to δ. When the end movement error ε of the robotic arm exceeds the allowable positioning accuracy, the movement of the robotic arm fails. Thus, the functional function expression of the robotic arm is established as:

[0019] Z = g(X) = δ - ε

[0020] Step 7: Expand the robotic arm movement functional function g(X) at the mean point μ x of the vector X using the Taylor series to obtain the expression:

[0021]

[0022] Step 8: Calculate the mean μ z of the functional function Z according to the functional function expression in Step 7:

[0023] μ z = g(μ X )

[0024] Step 9: Calculate the standard deviation σ z of the functional function Z according to the functional function expression in Step 7:

[0025]

[0026] Step 10: Set the reliability index β of the robotic arm, and its expression is:

[0027] β = μ Z / σ Z

[0028] Step 11: According to the definition of motion reliability, use the integral function normcdf of MATLAB to calculate the motion failure probability P f of the robotic arm as:

[0029]

[0030] wherein represents the cumulative probability density function of the normal distribution;

[0031] Step 12: For a multi-robot cooperation system including m robots, regarding a single robot as a sub-component of the system, in the processes of collaborative handling, assembly, etc., forming a series system with multiple robots. According to the definition of the reliability of the series system, if any robot fails during the movement, the system fails. Define the failure event of the i-th robot as E i , and define the reliable event of the i-th robot as The motion failure probability of the multi-robot series system can be expressed as follows:

[0032] P(E) = P(E 1 ∪E 2 ……∪E m )

[0033] where ∪ represents the OR operation in set operations

[0034] If the robots are independent of each other, the motion failure probability of the series system composed of m robots can be obtained as:

[0035]

[0036] Step 13: In the actual collaborative motion of the series system composed of multiple robots, all robots contact the same object through the end effector, and the robots affect each other through the contact force generated by gripping the object, so there is a correlation. Set the correlation coefficient between the i-th robot and the j-th robot as ρ ij (1 ≤ i, j ≤ m). Since the motion failure of a certain robot causes the change of the contact force between the remaining robots and the handled object and the load borne by the end of the remaining robots increases, which increases the probability of the motion failure of the remaining robots, so ρ ij > 0;

[0037] Step 14: For the i-th robot and the j-th robot, set the probability of simultaneous motion failure as P(E i E j ), and the probability of the j-th robot's motion failure after the i-th robot's motion failure as P(E j |E i ). According to the conditional probability criterion, it can be known that

[0038] P(E i |E j ) = P(E i )P(E j |E i)

[0039] Since ρ ij > 0, so

[0040] P(E j |E i ) ≥ P(E j )

[0041] Therefore, it can be obtained that:

[0042] P(E i E j ) ≥ P(E i )P(E j )

[0043] Thus, the formula for calculating the motion failure probability of the series system composed of m robots described in the twelfth step can be converted into the following inequality:

[0044]

[0045] Where represents the upper bound (maximum value) of the motion failure probability of the series system composed of m robots, and max{P(E 1 ), P(E 2 ), …, P(E m )} represents the lower bound (minimum value) of the motion failure probability of the series system, and max represents the maximum value among all possible values.

[0046] The dimensional deviations in the third step include the link length deviation and the link offset deviation. The link dimensions mainly include the link length and the link offset in the D-H parameters of the robot. Therefore, the link dimensional deviation includes the link length deviation and the link offset deviation. Since there are inevitable errors in the machining, manufacturing, and assembly processes of the link arms, which are mainly manifested as differences between the link dimensions and the theoretical values, the error parameters in the third step include the joint clearance and the link dimensional deviation. The link dimensions mainly include the link length and the link offset in the D-H parameters of the robot. Therefore, the link dimensional deviation includes the link length deviation and the link offset deviation. The link length deviation and the link offset deviation of the robot are mainly caused by factors such as machining, assembly, and force deformation. Therefore, the link dimensional deviation is a random variable that follows a normal distribution.

[0047] In the third step (such as Figure 5As shown in the figure: The adjacent link arms of the robot are connected by a revolute pair composed of a rotating shaft and a bearing. Therefore, the joint clearance is manifested as the non - coincidence of the rotation centers of the rotating shaft and the bearing. During the operation of the robot, the rotating shaft randomly moves within the bearing, causing shutdown shocks and vibrations, resulting in an error between the actual rotation angle and the theoretical rotation angle of the joint. Therefore, the distance between the rotation centers of the rotating shaft and the bearing is used as the joint clearance. The rotation center of the rotating shaft falls within a circular area formed by the inner ring of the bearing. Due to the relative movement between the rotating shaft and the bearing being affected by various external factors such as contact forces, collision forces, and frictional forces, the landing position of the rotating shaft is random. And due to the influence of the self - gravity of the rotating shaft, the landing points are more concentrated at the bottom of the bearing. After multiple movement results, a normal distribution is formed, that is, the error of the joint clearance is a random variable that conforms to the normal distribution.

[0048] In the fourth step mentioned above: In the present invention, the MATLAB random number function normrnd is used to generate the link length parameters and link offset parameters that contain deviations and conform to the normal distribution, as well as the joint rotation angle parameters that contain joint clearances and conform to the normal distribution. In the method of the present invention, there are link size errors for all link arms of the robot, and there are joint clearances for all rotating joints.

[0049] In the fifth step, without considering the influence of joint clearances and link size deviations, the actual position and the theoretical position calculated by the forward kinematics equation at the end of the robot are completely coincident. However, due to the existence of joint clearances at the joints of the robot and certain errors in the link sizes, the actual end position of the robot does not coincide with the theoretical end position of the robot, and there is a certain error. Therefore, the distance by which the actual position deviates from the theoretical position is used as the motion error of the robot.

[0050] In the sixth step, the joint clearances and link size deviations of the robot are random variables that satisfy the normal distribution. Therefore, during the movement, there are errors in the rotation angles of each joint and the link size parameters of the robot. There is an error between the actual position and the theoretical position of the robot end obtained through the forward kinematics equation. Due to the randomness of the joint clearances and link size deviations, the motion error of the robot also has randomness, so it is regarded as a random variable.

[0051] In the seventh step, due to the random characteristics of the motion error at the end of the robot, the value of the performance function also has certain random characteristics, so the value of the performance function is regarded as a random variable. The calculation of the end error is based on the forward kinematics equation of the robot. The forward kinematics equation includes the link length dimensions, link offset dimensions, and joint rotation angles of the robot. Therefore, the performance function of the robot is a non - linear function of the link length dimensions, link offset dimensions, and joint rotation angles. The performance function is linearly expanded to the first order at the mean points of the link length dimensions, link offset dimensions, and joint rotation angles through the Taylor series to simplify the performance function.

[0052] In the eighth step, the mean value of the robot function function is calculated through the mean values of the link length, link offset, and joint rotation angle of the robot. Where E represents the expected function.

[0053] In the ninth step, the standard deviation of the robot function function is calculated through the standard deviations of the link length, link offset, and joint rotation angle of the robot. Where k represents the number of random variables (link length, link offset, and joint rotation angle), and X i represents the i-th variable, and represent the mean value and standard deviation of the i-th variable, respectively.

[0054] In the tenth step, through the mean value μ Z and standard deviation σ z of the function function, the robot reliability index β is calculated; β = μ Z / σ Z ;

[0055] In the eleventh step, according to the robot reliability index β, the probability density function of the normal variable is integrated over the interval (-∞, β] to obtain the motion failure probability of the robot.

[0056] In the eleventh step, represents the cumulative probability density function of the normal distribution and has the following expression:

[0057]

[0058] In the twelfth step (as shown in Figure 6 ), m robots cooperate to move an object through the end effector. All robots must ensure reliable motion to complete the movement of the object. When a certain robot has a motion failure, the object will slide due to unbalanced forces, and the handling task of the entire multi-robot cooperation system will fail, resulting in a motion failure of the system. Therefore, the m robots form a series system.

[0059] In the twelfth step, U represents the OR operation in set operations.

[0060] In the thirteenth step, according to the definition of the series system, the motion reliability of the multi-robot cooperation system including m robots is Because So the upper bound of the motion reliability of the multi-robot cooperation system is where (1 ≤ i ≤ m). Since there is a positive correlation ρ ij ≥ 0 between the motion failures of the i-th robot and the j-th robot, so P(E j |E i ) ≥ P(E j ), that is According to the conditional probability criterion: So Therefore, the lower bound of the motion reliability of the multi-robot cooperation system is The value range of the motion reliability of the multi-robot cooperation system is jointly composed of the upper bound and the lower bound:

[0061]

[0062] For the i-th robot, because So the value range of the motion failure probability of the multi-robot cooperation series system is: After sorting out, we get:

[0063] The motion failure probability of the series system in the fourteenth step is an interval, and its value range is determined by the upper bound and the lower bound respectively, not a specific value.

[0064] In the present invention, the joint clearance, the link length deviation, and the link offset deviation are regarded as variables subject to normal distribution. The probability distribution models of the above variables are fully utilized. The functional function of the robot is established through the forward kinematics of the robot. Based on the Taylor series, the functional function is expanded to the first order at the mean point of the variables. The mean and standard deviation of the robot's functional function are calculated by using the mean and standard deviation of the variables respectively, so as to obtain the reliability index of the robot, and the motion failure probability of the robot is calculated by using integration. The cooperative motion system composed of multiple robots is regarded as a series system. From the perspective of kinematics, the reliability of the multi-robot cooperation system is analyzed and calculated in the present invention, avoiding the complex dynamic modeling of the multi-robot cooperation system, fully utilizing the correlation existing in the cooperative motion of the robots, constructing an inequality of the motion failure probability of the multi-robot cooperation system based on the conditional probability criterion, and calculating the upper bound (maximum value) and the lower bound (minimum value) of the motion failure probability of the multi-robot cooperation system respectively, so as to obtain the interval range of the motion failure probability of the multi-robot cooperation system, avoiding the increased computational complexity caused by solving the exact solution.

[0065] The advantages and beneficial effects of the present invention are as follows:

[0066] 1. For the multi-robot cooperation system with joint clearance, link length deviation, and link offset deviation, the present invention solves the problem of motion reliability analysis of the multi-robot cooperation system, which is the technical basis for ensuring the accurate, stable, and reliable cooperative operation of the robot in actual applications.

[0067] 2. The method of the present invention establishes an inequality for the motion failure probability of a multi-robot cooperation system based on the motion failure probability of a single robot and the positive correlation between robots, calculates the upper bound (maximum value) and the lower bound (minimum value) of the system failure probability respectively, thereby constructing a value range interval to provide a basis for evaluating the motion reliability of the multi-robot cooperation system. Description of the Drawings

[0068] Figure 1 is the flow block diagram of the method of the present invention.

[0069] Figure 2 is a schematic diagram of a rotary joint type robot applicable in the present invention.

[0070] Figure 3 is a schematic diagram of the link coordinate system of the robot in the present invention.

[0071] Figure 4 is a schematic diagram of the motion error at the end of the robot in the present invention.

[0072] Figure 5 is a schematic diagram of the clearance of the rotary joint of the robot in the present invention.

[0073] Figure 6 is a schematic diagram of the motion of the multi-robot cooperation system in the present invention. Detailed Description of the Invention

[0074] Based on the motion failure probability of a single robot and the positive correlation between robots, the present invention establishes an inequality for the motion failure probability of a multi-robot cooperation system, calculates the upper bound (maximum value) and the lower bound (minimum value) of the system failure probability respectively, thereby constructing a value range interval to provide a basis for evaluating the motion reliability of the multi-robot cooperation system for a robot or mechanical system composed of rotary joints; through statistical analysis of the random variables composed of the joint clearance and the link dimension deviation, the probability distribution type is determined, the error of the robot end position is obtained through robot kinematics, the performance function of the robot is established and expanded by Taylor series, the mean value and standard deviation of the performance function are calculated respectively, and then the robot reliability index is calculated, and the robot motion failure probability is obtained by integration. Based on the motion state of a single robot, a series system reliability model for multi-robot cooperation is constructed, and using the correlation and conditional probability criterion in the process of robot cooperative motion, an inequality for the failure probability of the multi-robot cooperation system is established, and the upper bound (maximum value) and the lower bound (minimum value) of the motion failure probability of the multi-robot system are calculated respectively, thereby constructing the value range of the motion failure probability of the multi-robot cooperation system.

[0075] The following further describes the present invention in detail with reference to the drawings and specific embodiments.

[0076] The basic principle of the present invention mainly lies in that: the joint clearances, link length deviations, and link offset deviations of the robot are random variables that follow a normal distribution. The joint clearances will cause a random error with a normal distribution within a certain range in the joint rotation angles of the robot. The link size deviations will cause changes in the physical parameters of the robot. The joint clearances, link length deviations, and link offset deviations will cause the actual position of the robot end to deviate from its expected position through the kinematics of the robot. The distance between the actual position and the ideal position is the motion error of the robot end position, and this error also belongs to a random variable. According to the end positioning accuracy of the robot, a function function of the robot system is established, and a calculation formula for the robot reliability is established through the function function. The value of the function function also belongs to a random variable. The function function is expanded in a Taylor series at the mean points of the joint rotation angles, link lengths, and link offsets. The mean and standard deviation of the robot function function are calculated using the mean and standard deviation of the joint rotation angles, link lengths, and link offsets, and then the robot reliability index is calculated, and the robot motion failure probability is obtained through integration. The multi-robot cooperation system is regarded as a series system, a calculation function for the failure mode of the series system is established, and an inequality for the motion failure probability of the multi-robot cooperation system is constructed using the positive correlation between the robots. The upper boundary (maximum value) and lower boundary (minimum value) of the system motion failure probability are calculated respectively, and finally the value range of the motion failure probability of the multi-robot cooperation system is obtained to complete the motion reliability assessment.

[0077] Embodiment 1:

[0078] Appendix Figure 1 The block diagram in Figure 6 shows the specific flowchart of the novel robot reliability calculation method proposed by the present invention; as

[0079] The first step:

[0080] Establish the D-H link coordinate system of the robot. The corresponding physical parameters of the robot are as follows: the link length a 1 = 40 (mm), a 2 = 315 (mm), a 3 = 70 (mm), the link offset d 1 = 330 (mm), d 4 = 310 (mm), d 6= 70 (mm), the torsional angle α of the connecting rod 1 = -90°, α 3 = -90°, α 4 = 90°, α 5 = -90°. At the target point, the theoretical rotation angles of the 6 rotating joints of the robot R1 are respectively: The theoretical rotation angles of the 6 rotating joints of the robot R2 are respectively: The theoretical rotation angles of the 6 rotating joints of the robot R3 are respectively:

[0081] Step 2:

[0082] Based on the DH coordinate system of the robot and the joint rotation angle values, the spatial attitude and coordinate values of the end point of the robot can be solved by using forward kinematics. For the robot R1, it can be calculated that:

[0083]

[0084] Therefore, the theoretical position coordinates P of the end of the robot R1 1 (x, y, z) is P 1 (396.25, 44.80, 519.24).

[0085] Similarly, based on the parameters and angle values in the first step, for the robots R2 and R3, they can be obtained respectively:

[0086]

[0087] Step 3:

[0088] Define the joint clearance between the connecting rod arms themselves and between the connecting rod arms, the connecting rod length deviation, and the connecting rod offset deviation as random variables that satisfy a certain distribution type, and randomly generate random values that satisfy the distribution type within the range of the error parameters as the error parameters;

[0089] Specific example of joint clearance: At the connection of two joints as Figure 5 shown, the end of the connecting rod 3 of one joint is fixedly connected to the rotating shaft 1, the end of the connecting rod 3 of the other joint is fixedly connected to the bearing 2, the rotating shaft 1 is sleeved in the bearing 2. Due to the clearance between the outer ring of the rotating shaft 1 and the inner ring of the bearing 2, the rotation centers of the rotating shaft and the bearing do not coincide, and the distance between the rotation centers of the rotating shaft and the bearing is used as the center offset 4, so that there is a joint clearance ζ during the rotation of the joint.

[0090] The connecting rod dimension deviation belongs to the normal distribution, and their respective means and standard deviations are respectively:

[0091] μa1 = 40 (mm), σ a1 = 0.04 (mm), μ a2 = 315 (mm), σ a2 = 0.32 (mm),

[0092] μ a3 = 70 (mm), σ a3 = 0.70 (mm), μ d1 = 330 (mm), σ d1 = 0.33 (mm),

[0093] μ d4 = 310 (mm), σ d4 = 0.31 (mm), μ d6 = 70 (mm), σ d6 = 0.07 (mm),

[0094] The joint clearance ζ follows a normal distribution, and its mean and standard deviation are respectively Then the actual joint angle of the robot is:

[0095]

[0096] where i represents the i-th joint of the robot.

[0097] Step 5:

[0098] By constructing the mathematical model of the deviation distribution of the joint clearance, link length, and link offset of the robot in Step 4, and using the random number function normrnd in MATLAB to generate random numbers that satisfy the normal distribution to represent the link arm length, link offset, and joint angle. Under the influence of the link dimension deviation and joint clearance, through the forward kinematics of the robot constructed in Step 1 and Step 2, the actual position coordinates of the robot's end are calculated as P'(x', y', z'), so the position error of the robot's end is:

[0099]

[0100] Step 6:

[0101] Given that the positioning accuracy of the robot's end is δ = 2.5 (mm), the performance function of the robot R 1 is established

[0102] Z = g(X) = δ - ε

[0103] Step 7:

[0104] Substitute the performance function at the variable X = [a 1 , a 2 , a 3 , d1 , d 4 , d 6 , θ 1 , θ 2 , …, θ 6 mean point

[0105] That is Perform Taylor series expansion at the point.

[0106] Eighth step:

[0107] Use the mean value of the random variable X Calculate the mean value of the performance function

[0108] Ninth step:

[0109] Use the standard deviation of the variable X That is Calculate the standard deviation of the performance function

[0110] Tenth step:

[0111] According to the calculation results of the seventh to ninth steps, obtain the reliability index of the robot R1 Similarly, the robot R can be calculated 2 The mean value and standard deviation of the performance function are respectively Then the robot R 2 The reliability index is The robot R 3 The mean value and standard deviation of the performance function are respectively So the robot R 3 The reliability index is

[0112] Eleventh step:

[0113] According to the reliability index β = [β 1 , β 2 , β 3 , complete the integral operation through the MATLAB integral function normcdf:

[0114]

[0115] Obtain the failure probability of the robot R 1 as P f1 = 1.120×10 -3 . Similarly, the failure probabilities of the robots R 2 and R 3 can be calculated as P f2= 0.8917×10 -3 and P f3 = 1.2959×10 -3 。

[0116] The twelfth step:

[0117] In multi-robot cooperative motion, set the correlation coefficient between robot R 1 and robot R 2 to be ρ 12 = 0.8, the correlation coefficient between robot R 1 and robot R 3 to be ρ 13 = 0.8, the correlation coefficient between robot R 2 and robot R 3 to be ρ 23 = 0.8.

[0118] The thirteenth step:

[0119] Calculate the lower bound (minimum value) of the motion failure probability of the multi-robot cooperation system as P(E)≥max{P f1 , P f2 , P f3}, that is, P(E)≥max{1.120×10 -3 , 0.8917×10 -3 , 1.2959×10 -3}, P(E)≥1.2959×10 -3 。

[0120] Calculate the upper bound (maximum value) of the motion failure probability of the multi-robot cooperation system as P(E)≤1 - (1 - P f1 )×(1 - P f2 )×(1 - P f3 ), that is, P(E)≤3.3042×10 -3 。

[0121] According to the boundary values of the motion failure probability of the multi-robot cooperation system, that is, the maximum value and the minimum value, construct the interval of the motion failure probability of the multi-robot cooperation system: P(E)∈[1.2959×10 -3 , 3.3042×10 -3 .

Claims

1. A method for calculating the motion reliability of a multi-robot collaborative system based on boundary estimation, comprising the following steps: Step 1: Establish a DH link coordinate system including a base coordinate system and a joint local coordinate system according to the robot's link arm structure; Step 2: Obtain the homogeneous matrix of the robot end position relative to the base coordinate system through the positive kinematics of the robot link arm The coordinates of the theoretical position of the robot end relative to the base coordinate system O_XYZ are P(x, y, z). The above homogeneous matrix Use the following expression: Where n represents the number of degrees of freedom of the robot, R represents the posture matrix of the robot end in the base coordinate system, P(x, y, z) represents the position coordinates of the robot end, and Π represents continuous multiplication; Step 3: Set the length of the link arm at the i-th link arm of the robot to a i and connecting rod offset d i The dimensional deviation of is defined as a random variable that satisfies the normal distribution, and the means of the connecting rod arm length and the connecting rod offset are μ ai and μ di , and the standard deviations are σ ai and σ di ; At the same time, the joint angle θ of the revolute joint containing the joint gap at the i-th link arm of the robot i It is defined as a random variable that satisfies the normal distribution, and the mean and standard deviation of the joint angle are μ θi and σ θi ; Step 4: Use MATLAB's random number function normrnd to generate random numbers that satisfy the normal distribution to represent the link arm length, link offset, and joint angle; Step 5: Use the random values ​​generated in step 4 to obtain the homogeneous matrix of the positive kinematic expression in step 2 The coordinate value of the actual end position of the robot during motion relative to the base coordinate system is calculated as P'(x', y', z'), and the expected coordinate value of the robot in an ideal state is P(x, y, z). The following formula is then used to calculate the robot end motion error ε: Step 6: The robot's connecting rod length, connecting rod offset and joint angle are combined into a vector X = [a1, a2, a3, d1, d4, d6, θ1, θ2, ..., θ6], and the positioning accuracy of the robot end is set to δ. When the robot's end motion error ε exceeds the allowable positioning accuracy, the robot motion fails. The function expression of the robot is thus established as: Z = g(X) = δ - ε; Step 7: Transform the robot motion function g(X) at the mean point μ of vector X x , using Taylor series to perform first-order expansion, we can get the expression: Step 8: Calculate the mean μ of the performance function Z according to the performance function expression in step 7 z : m z =g(μx); Step 9: Calculate the standard deviation σ of the performance function Z according to the performance function expression in step 7 z : Step 10: Set the reliability index β of the robot, which is expressed as: β=μ Z / s Z ; Step 11: According to the definition of motion reliability, calculate the robot's motion failure probability P through MATLAB's integral function normcdf f for: in Represents the cumulative probability density function of the normal distribution; Step 12: For a multi-robot collaborative system consisting of m robots, a single robot is regarded as a sub-component of the system. In the process of collaborative handling and assembly, multiple robots are combined into a series system. According to the definition of series system reliability, if any robot fails during the movement, the system fails. The failure event of the i-th robot is defined as E i , define the reliable event of the ith robot as The motion failure probability of a multi-robot serial system can be expressed as follows: P(E)=P(E1UE2.......UE m ); Where ∪ represents the OR operation in set operation; If the robots are independent of each other, the failure probability of the series system consisting of m robots can be obtained as: Step 13: In the actual coordinated motion of a series system composed of multiple robots, all robots contact the same object through the end fixture. The contact force generated by the robots through the clamping of the object affects each other, so there is a correlation. The correlation coefficient between the i-th robot and the j-th robot is set to ρi j (1≤i, j≤m); due to the failure of the motion of a certain robot, the contact force between all the remaining robots and the transported objects changes and the load on the ends of all the remaining robots increases, which increases the probability of motion failure of the remaining robots, so ρ ij >0; Step 14: For the i-th robot and the j-th robot, define the probability of simultaneous motion failure as P(E i E j ), the probability of the jth robot motion failure after the i-th robot motion failure is P(E j |E i ), according to the conditional probability criterion, we know P(E i AND j )=P(E i )P(E j |And i ) Because ij >0, so P(E j |And i )≥P(E j ) So we can get: P(E i AND j )≥P(E i )P(E j ); Therefore, the calculation formula for the motion failure probability of the series system composed of m robots described in step 12 can be converted into the following inequality: in represents the upper limit of the motion failure probability of the series system composed of m robots, that is, the maximum value, max{P(E1), P(E2), ..., P(E m )} represents the lower boundary of the motion failure probability of the series system, i.e., the minimum value, and max represents the maximum value among all possible values.

2. The method for calculating motion reliability of a multi-robot collaborative system based on boundary estimation according to claim 1, characterized in that: The dimensional deviation in the third step includes connecting rod length deviation and connecting rod offset deviation.

3. The method for calculating motion reliability of a multi-robot collaborative system based on boundary estimation according to claim 1, characterized in that: In the eleventh step, The cumulative probability density function of the normal distribution has the following expression:

4. The method for calculating motion reliability of a multi-robot collaborative system based on boundary estimation according to claim 1, characterized in that: The probability of motion failure of the series system in the fourteenth step is an interval, and the value range is determined by the upper boundary and the lower boundary respectively, and is not a specific value.

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