CN107443370B - A kind of industrial robot reliability calculation method based on quadravalence moments estimation - Google Patents
A kind of industrial robot reliability calculation method based on quadravalence moments estimation Download PDFInfo
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Abstract
The invention discloses a kind of industrial robot reliability calculation methods based on quadravalence moments estimation.It establishes the D-H link rod coordinate system of robot and constructs forward kinematics equation, the joint space of robot and connecting rod dimensional discrepancy are considered as stochastic variable, calculate kinematic error of the distance of true location point and desired locations point as robot, the kinematic error of robot is considered as stochastic variable and calculates preceding Fourth-order moment, establish the system function function of characterization robot motion's reliability and the preceding Fourth-order moment of computing function function, the probability density function of system function function is solved using quadravalence moment estimation method, is carried out integral and is obtained machine human reriability.The present invention takes full advantage of the probability distribution information of joint space and connecting rod dimensional discrepancy, therefore only need few sample that can complete the Calculation of Reliability to robot system, it significantly shortens the time and improves efficiency, suitable for connecting and the industrial robot of other any rotary joint types.
Description
Technical field
The present invention relates to a kind of robot reliability's calculation method, more particularly, to a kind of based on quadravalence moments estimation
Industrial robot reliability calculation method is related to widely used series connection and parallel connection type in the fields such as industrial production, medical manufacture
Robot.
Background technique
With the arrival in industrial 4.0 epoch, the processing of such as large complicated components in many fields, assembly and carrying, nothing
The welding of chucking appliance system, spraying and engraving etc., the requirement to the kinematic accuracy of robot, stability and reliability is also increasingly
Height, but due to robot inevitably abrasion and stress deformation etc. in design, parts machining, assembly and use, so that machine
The ontological existence of device people inevitable error, main joint space and connecting rod size including at rotary joint it is inclined
Difference, the deviation of connecting rod size reduce the end movement precision of robot, and the presence of joint space not only influences kinematic accuracy,
It also will cause joint impact, aggravation vibration even causes the motion failures of robot, therefore to robot in above-mentioned uncertainty
Under the influence of the reliability consideration of movement be of great significance.In current industrial production, be used to machine human reriability into
The method that row calculates mainly includes single order analysis method for reliability, Monte Carlo Method of Stochastic, especially Monte Carlo
Method of Stochastic be even more be widely used, but due to Monte Carlo simulation need based on very big sample value could be compared with subject to
The true reliability for estimating system, it is not only with high costs in practice, and also efficiency is relatively low, single order reliability side
Although method can reduce the sample size of simulation, since its computational accuracy is limited, it is simple to be used primarily in some structures, non-thread
In the property lower mechanical system of degree, it is unable to satisfy the requirement in the fields such as aerospace, robot movement-control system.
Summary of the invention
To solve the problems, such as that robot reliability calculates, the purpose of the present invention is to propose to a kind of works based on quadravalence moments estimation
Industry robot reliability's calculation method, the robot formed for series, parallel and any rotary joint or mechanical system, lead to
It crosses and statistical analysis is carried out to the stochastic variable of the composition of joint space and connecting rod dimensional discrepancy, determine its probability distribution,
The sample that the error composition of robot end position is obtained by robot kinematics, establishes the function letter of robot system
Number, and by for statistical analysis to terminal position error, the probability density function expression formula of power function is obtained, thus quickly
Accurately robot reliability is calculated.
The technical solution adopted by the present invention is that using following steps:
Connect step 1: establishing the D-H comprising basis coordinates system and local joint coordinate system according to the linkage arm arrangement of robot
Bar coordinate system；
As shown in Fig. 2, base coordinate system O_XYZ is established using the bottom centre of robot as origin first, wherein Z axis edge
The axial direction of first rotary joint of robot, X-direction are arbitrarily chosen, and the direction of Y-axis is according to right-hand rule by X-axis and Z
Axis determines.
As shown in Fig. 3, a local coordinate system O is established at i-th of link arm rotary jointi-1Xi-1Yi-1Zi-1, office
The origin O of portion's coordinate systemi-1For the rotation center in the i-th joint, Zi-1Axial direction of the axis along the i-th joint, XiAxis is current joint
Axial direction, YiThe direction of axis is according to right-hand rule by XiAxis and ZiAxis is determining, then the local coordinate O of i-th of link armiXiYiZi
It can be obtained by such as down conversion relative to the positional relationship between the local coordinate of (i-1)-th link arm: by local coordinate system Oi- 1Xi-1Yi-1Zi-1Around Zi-1Axis rotates angle, θiSo that Xi-1With XiAxis is parallel, and along Zi-1Axis translates diObtain coordinate system O 'iX′iY′iZ′i, by coordinate system O 'iX′iY′iZ′iAlong X 'iTranslate aiAnd around X 'iAxis rotates angle [alpha]iObtain coordinate system OiXiYiZi。
Above-mentioned transformation relation is expressed as one 4 × 4 homogeneous matrixSpecifically refer to following expression:
Wherein, aiFor i-th of connecting rod arm lengths, diFor the inclined square of i-th of connecting rod, αiFor i-th of connecting rod torsion angle, θiIt is i-th
The joint rotation angle of a link arm, the robot are mechanical arm configurations in series or in parallel.
Step 2: obtaining robot end position relative to the neat of basis coordinates system by the positive kinematics of robot links arm
SubmatrixIt is P (x, y, z) that the theoretical position of robot end, which is obtained, relative to the coordinate value of basis coordinates system O_XYZ, above-mentioned
Homogeneous matrixUsing following expression:
Wherein, the number of degrees of freedom, of n expression robot, attitude matrix of the R expression robot end in base coordinate system, P (x,
Y, z) indicate that robot end's position coordinates, Π indicate even to multiply symbol；
Step 3: by the connecting rod arm lengths a at i-th of link arm of robotiWith connecting rod offset distance diDimensional discrepancy be defined as
Meet the stochastic variable of normal distribution, the mean value of connecting rod arm lengths and connecting rod offset distance is respectively μaiAnd μdi, standard deviation is respectively σai
And σdi；Simultaneously by the joint rotation angle θ of the rotary joint at i-th of link arm of robot comprising joint spaceiIt is defined as meeting equal
The stochastic variable of even distribution, coboundary of the joint rotation angle in the case where being uniformly distributed are θi+Δi, lower boundary θi-Δi；
Step 4: generating the random number expression connecting rod for meeting normal distribution using the random number functions normrnd of MATLAB
Arm lengths and connecting rod offset distance, meeting equally distributed random number using random number functions unifrnd generation indicates joint rotation angle；
Step 5: considering that robot moves back and forth, in jth time moves back and forth, due to link arm length variation, even
The influence of bar offset distance deviation and joint space, robot actual terminal position meeting deviation theory position, as shown in Fig. 3, partially
From distance definition be robot end's kinematic error ε, by the 4th step generate random number, utilize second step positive kinematics
The homogeneous matrix of expression formulaThe calculating robot's coordinate value of actual end position relative to base coordinate system in jth time movement
For P ' (x ', y ', z), then it is calculated using the following equation and obtains robot end's kinematic error εj:
Step 6: the positioning accuracy for setting robot end is less than positioning as Δ r with the end movement error ε of robot
The probability of precision Δ r is considered as stochastic variable, machine as robot motion's reliability, by actual robot end movement error ε
People is every once to be moved back and forth, then there is an end movement error amount, is repeated the 4th step~the 5th step and is carried out n times, obtains altogether
N number of robot end's kinematic error value is obtained, all robot end's kinematic error values are formed into a sample data；
Step 7: analyzing the sample data obtained in the 6th step, the mean value of calculating robot's end movement error ε
με, calculation formula are as follows:
Wherein, εjIndicate robot end's kinematic error in jth time movement；
Step 8: analyzing the sample data obtained in the 6th step, preceding the four of calculating robot's end movement error ε
Rank central momentCalculation formula are as follows:
Wherein, k indicates order, value k=0,1,2,3,4；
Simultaneously with second-order moment around meanAs the variance of robot end's kinematic error, i.e.,Wherein σεTable
Show the standard deviation of robot end's kinematic error；
Step 9: establishing the power function g (ε) of robot system, expression formula according to the definition of robot motion's reliability
Are as follows:
Z=g (ε)=Δ r- ε
Wherein, Z indicates the value of power function, and Δ r indicates the positioning accuracy of robot end, and ε indicates robot end
Kinematic error；
Step 10: according to the mean μ of robot end's kinematic error εεAnd its preceding fourth central square WithPass through the power function g (ε) for the robot system that the 9th step is established, the function of calculating robot's system
The mean μ of energy function g (ε)z, the formula of calculating are as follows:
μz=Δ r- με
The preceding fourth central square of calculating robot's system function function againCalculation formula are as follows:
Wherein, k indicates specific order, value k=0,1,2,3,4；
The zeroth order central moment of the power function of robot systemSingle order central momentTwo
Rank central momentThird central momentFourth central square
Simultaneously with second-order moment around meanAs the variance of robot system power function, i.e.,Wherein σZTable
Show the standard deviation of robot system power function；
Step 11: being the canonical statistics Y that mean value is 0, variance is 1, transformation for mula by power function value transform
Are as follows:
Calculate the preceding fourth central square of canonical statistics YCalculation formula are as follows:
Wherein, k indicates specific order, value k=0,1,2,3,4；
Step 12: the comentropy for defining stochastic variable Y is H [f (y)], comentropy expression are as follows:
H [f (y)]=- ∫Yf(y)ln[f(y)]dy
Wherein, f (y) indicates the probability density function of stochastic variable Y；
Using the preceding fourth central square of canonical statistics Y as constraint condition, Lagrange is constructed according to principle of maximum entropy
Function, Lagrangian expression formula are as follows:
Wherein, λkFor the corresponding Lagrange coefficient of kth rank, λk=(λ0, λ1..., λ4)；
Partial derivative is asked to probability density function f (y) using LagrangianL, so that partial derivative is 0 at extreme point,
Derivation process expression are as follows:
Step 13: being calculated using the following equation the probability density letter for obtaining stochastic variable Y according to previous step derivation process
Number f (y), expression formula specifically:
Step 14: due to machine human reriabilityRefer to robot connecting rod dimensional discrepancy, connecting rod offset distance deviation with
And under the influence of joint space, probability of the kinematic error ε of robot end within the scope of given positioning accuracy Δ r, i.e.,
In certain motion process, the probability of the end movement error ε < Δ r of robot, so after power function, robot fortune
Dynamic reliability is converted further into:
Wherein, Pr { Z≤0 } indicates that the probability for meeting the condition of Z≤0, Pr { Z≤0 } are equivalent to
According to the probability density function f (y) of stochastic variable Y, machine human reriability is calculatedCalculating process expression
Formula specifically:
Wherein, σzIndicate the variance of robot system power function, μzIndicate that the power function g's (ε) of robot system is equal
Value.
In the second step, including joint space and connecting rod dimensional discrepancy, connecting rod size mainly include robot DH parameter
In length of connecting rod and connecting rod offset distance, therefore connecting rod dimensional discrepancy includes length of connecting rod deviation and connecting rod offset distance deviation.Due to
Unavoidably there is error in processing and manufacturing, assembling process in link arm, be mainly shown as between connecting rod size and theoretical value and deposit
In difference, therefore the error parameter includes joint space and connecting rod dimensional discrepancy, and connecting rod size mainly includes robot DH ginseng
Length of connecting rod and connecting rod offset distance in number, therefore connecting rod dimensional discrepancy includes length of connecting rod deviation and connecting rod offset distance deviation, machine
The length of connecting rod deviation and connecting rod offset distance deviation of device people is primarily due to the factors such as processing, assembly and stress deformation and causes, because
This length of connecting rod deviation is to meet the stochastic variable of normal distribution.
In the third step: between the adjacent link arm of robot be made up of shaft and bearing as shown in Fig. 4
Revolute is formed by connecting, therefore joint space shows as shaft and the rotation center of bearing is not overlapped, in robot operational process
In, shaft causes shutdown shock and vibration in the internal random play of bearing so that the actual rotational angle in joint and theoretical corner it
Between there are errors, therefore using the distance between rotation center of shaft and bearing as joint space, the rotation center of shaft is fallen
In the border circular areas that bearing inner race is constituted, drop point site is random and is equiprobable, the i.e. mistake of joint space
Difference is to meet equally distributed stochastic variable.
In 4th step: generating comprising deviation and accord with using MATLAB random number functions normrnd in the present invention
The length of connecting rod parameter and connecting rod offset distance parameter for closing normal distribution, are generated using random function unifrnd comprising joint space
And meet equally distributed joint rotation angle parameter, in the method for the present invention, all link arms of robot have connecting rod size
There is joint space in error, all rotary joints.
In 5th step, under the influence of not considering joint space and connecting rod dimensional discrepancy, robot end is by just learning
Kinematical equation calculate physical location and theoretical position be completely coincident, but due to the joint of robot there are joint space with
And there are certain errors for connecting rod size, so that robot actual end position and robot theory terminal position do not weigh
It closes, there is a certain error, therefore using the distance of physical location deviation theory position as the kinematic error of robot.
In 6th step, the joint space of robot and connecting rod dimensional discrepancy are to meet to be uniformly distributed and normal state point respectively
The stochastic variable of cloth, therefore during exercise, there are errors for each joint rotation angle and connecting rod dimensional parameters of robot, pass through
There are errors between robot end's physical location that forward kinematics equation obtains and theoretical position, due to joint space and connecting rod
The kinematic error of the randomness of dimensional discrepancy, robot also has randomness, so being regarded as stochastic variable.
In 7th step, since the kinematic error of robot end has stochastic behaviour, so that power function value
Also there is certain stochastic behaviour, so power function value is considered as stochastic variable.Therefore by statistical method, can to by
The sample data of the kinematic error composition of robot end is analyzed, and obtains the kinematic error system of characterization robot end respectively
Count first moment about the origin (mean value) μ of characteristicε。
In 8th step, the zeroth order central moment of robot end's kinematic error is calculatedSingle order central moment
Second-order moment around mean (variance)Third central momentAnd fourth central square
In tenth step, then by power function expression formula, using the mean value of robot end's kinematic error into one
Step obtains first moment about the origin (mean value) μ of power function valueZ, zeroth order central momentSingle order central momentSecond-order moment around mean
(variance)And third central momentWith fourth central square
In 11st step, in order to simplify calculating process, robot power function Z is converted into canonical statistics Y,
And calculate the mean μ of canonical statistics YY, zeroth order central momentSingle order central momentSecond-order moment around mean (variance)And third central momentWith fourth central square
In 12nd step, the motion credibility of robot depends on the probability distribution of canonical statistics Y, therefore is
Calculating reliability, needs to obtain the probability density estimation expression formula of canonical statistics Y, canonical statistics Y's is equal
Value and preceding fourth central square are available, using the principle of maximum entropy of probability theory, construct Lagrangian, it is subsequent into
One step asks the optimization that the Solve problems of the probability density estimation of canonical statistics Y are converted into meeting certain constraint condition
Topic.
In 13rd step, obtained in Lagrangian not by solving the optimization problem met under constraint condition
Parameter is known, so that the probability density estimation expression formula of the completely canonical statistics Y with parameter is obtained, by general
The integral of rate density function model realizes that the motion credibility of robot calculates.
Present invention utilizes joint space and length of connecting rod deviation and the probability Distribution Models of connecting rod offset distance deviation, pass through machine
The positive kinematics of device people establish the function linear function of robot system, avoid the terminal position movement function to robot
Taylor series expansion, thus eliminate due to by nonlinear function linearize bring error, meanwhile, because joint space,
The randomness of length of connecting rod deviation and connecting rod offset distance deviation, which makes robot end's location error also, has certain randomness, this
Robot end's kinematic error is considered as stochastic variable by invention, has sought its mean value and preceding fourth central square using statistical analysis,
The statistical nature of stochastic variable is taken full advantage of, avoids the loss for the statistical information for including to stochastic variable to the full extent,
The ASSOCIATE STATISTICS feature of power function is further calculated out using the statistical nature of terminal position error, and former based on maximum entropy
Reason has sought the probability density function expression formula of power function, finally using integral, obtains machine human reriability.
Advantage and beneficial effect of the invention is following several points:
1, for the robot containing joint space, length of connecting rod deviation and connecting rod offset distance deviation, the present invention solves machine
Human reliability analysis problem is to guarantee robot energy among practice as Robot Design, the premise of processing, assembly
Enough accurate, stable, reliability service technical foundation.
The present invention is primarily based on joint space, length of connecting rod deviation and connecting rod offset distance deviation among calculating process
The Probability Analysis statistical nature of robot end's kinematic error, is considered as stochastic variable for end movement error, according to machine
Device people positioning accuracy and reliability definition establish the linear function function of robot system, avoid directly to robot
Forward kinematics equation carries out Taylor series expansion, avoids and carries out nonlinear function to linearize caused error.
2, the method for the present invention is guaranteeing compared to now widely used Monte Carlo Analogue Method in fail-safe analysis
As a result under the premise of accuracy, the demand to sample size can be greatly lowered, it is only necessary to which few sample size just obtains
Higher computational accuracy.
3, the method for the present invention, can be using at random compared to traditional single order fail-safe analysis method in fail-safe analysis
The more statistical informations of variable.
Traditional single order fail-safe analysis method is (square with only the first moment about the origin (mean value) and second-order moment around mean of variable
Difference) statistical information, can have large error when variable does not strictly meet normal distribution.
The mean value and variance of stochastic variable is not only utilized in the method for the present invention, and three ranks of stochastic variable are also further utilized
(degree of bias information of reflection stochastic variable probability density function) and fourth central square (the kurtosis letter of reflection stochastic variable probability function
Breath) statistical information, although sample size needed for the two is less, the method for the present invention can obtain more accurate meter
Calculate result.
Detailed description of the invention
Fig. 1 is the flow diagram of the method for the present invention；
Fig. 2 is the rotary joint humanoid robot schematic diagram being applicable in the present invention；
Fig. 3 is the link rod coordinate system schematic diagram of robot in the present invention；
Fig. 4 is robot end's kinematic error schematic diagram in the present invention；
Fig. 5 is robot rotary articulation gap schematic diagram in the present invention.
Specific embodiment
With reference to the accompanying drawing and specific embodiment is described in further details the present invention
Basic principle of the invention essentially consists in: the joint space of robot is to meet equally distributed stochastic variable, even
Pole length deviation and connecting rod offset distance deviation are to meet the stochastic variable of normal distribution, and the joint that joint space will cause robot turns
There are one, equally distributed random error, connecting rod dimensional discrepancy can make the physical parameter of robot send out in a certain range at angle
Changing, joint space, length of connecting rod deviation and connecting rod offset distance deviation will cause robot end by the kinematics of robot
Its desired location is deviateed in actual position, and the distance between physical location and ideal position are that the movement of robot end position misses
Difference, the error also belong to stochastic variable, according to the end positioning accuracy of robot, establish the power function of robot system,
The calculation formula of robot reliability is established by power function, the value of power function also belongs to stochastic variable, using by
The sample of robot end's kinematic error composition, carries out statistical analysis, obtains the mean value for indicating the stochastic variable of kinematic error
And preceding fourth central square, the mean value of power function is further sought by power function using its mean value and preceding fourth central square
With preceding fourth central square, using the preceding quadravalence moment of the orign of power function as constraint condition, the principle of maximum entropy based on probability theory will
The problem has been converted further into the Optimization Solution problem met under constraint condition, establishes Lagrange's equation, solves its glug
Bright day coefficient, by integral, calculates machine human reriability to obtain the probability density function of power function.
The embodiment of the present invention:
Block diagram in attached drawing 1 is the specific flow chart of novel robot reliability calculation method proposed by the present invention, such as attached
Shown in figure, as shown in Fig. 2, the present invention artificially analyzes object with a 3DOF tandem type industrial machine, including link arm and
Rotary joint is calculated for containing the robot reliability of joint space and length of connecting rod deviation and connecting rod offset distance deviation, in conjunction with reality
It applies example to be illustrated, the angle being directed to is indicated with degree (°), and the unit of the dimensions length and distance that are related to is millimeter
(mm), detailed process of the invention mainly comprises the steps that
Step 1:
The D-H link rod coordinate system of robot is established, the corresponding physical parameter of robot includes the following: length of connecting rod a2=
400 (mm), a3=600 (mm), connecting rod offset distance d1=500 (mm), at target point, the theory of 3 rotary joints of robot turns
Angle is respectively as follows: θ1=100 °, θ2=-60 °, θ3=50 °, then the theoretical position coordinate of robot end is P (x, y, z), wherein x
=-137.59 (mm), y=780.36 (mm), z=-206.43 (mm).
Step 2:
It is full by joint space, length of connecting rod deviation and connecting rod offset distance deviation definition of the link arm itself between link arm
The stochastic variable of the certain distribution pattern of foot, is randomly generated the random number conduct for meeting distribution pattern in the range of error parameter
Error parameter, as shown in Figure 4；
Joint space concrete example illustrates: two joints junction of specific implementation as shown in figure 5, a joint connecting rod
3 ends are fixedly connected with shaft 1, and 3 end of connecting rod in another joint is fixedly connected with bearing 2, and shaft 1 is sleeved in bearing 2,
Since 1 outer ring of shaft and 2 inner ring of bearing are there are gap, the rotation center for causing shaft and bearing is not overlapped, and to turn
Offset distance 4 centered on the distance between rotation center of axis and bearing, so that there are between a joint in rotary course in joint
Gap Δ.
Connecting rod dimensional discrepancy belongs to normal distribution, and respective mean value and standard deviation are respectively as follows:
μa2=400 (mm), σa2=0.35 (mm), μa3=600 (mm),
σa3=0.30 (mm), μd1=500 (mm), σd1=0.35 (mm)
Δ satisfaction in joint space is uniformly distributed, and range is [- 0.1 °, 0.1 °], then the practical joint rotation angle of robot are as follows:
θ′i=θi+ Δ (i=1,2,3)
Under the influence of connecting rod dimensional discrepancy and joint space, robot is in jth time moves back and forth, end physical location
Coordinate is Pj′(xj', yj', zj'), then robot end's location error are as follows:
Step 3:
It repeats step 2 500 times, obtains the random sample being made of by 500 end movement error, carry out statistics credit
Analysis, difference can be in the hope of its mean με=1.284 (mm), zeroth order central momentSingle order central momentSecond-order moment around mean (variance)Third central momentFourth central square
Step 4:
The positioning accuracy of given robot end is Δ r=2 (mm), thus establishes the power function Z=g (ε) of robot
=Δ r- ε, using the mean value and preceding fourth central square of stochastic variable ε calculate power function Z mean value and preceding fourth central square
It is respectively as follows: μZ=0.716 (mm), According to the definition of robot reliability: in joint space and connecting rod
Under the influence of dimensional discrepancy, the terminal position error of robot is less than the probability of positioning accuracy, so machine human reriability can be with
Further it is expressed as:
Step 5:
It is 0 that power function Z, which is converted into mean value, and the canonical statistics Y that variance is 1, transformational relation can be expressed as:
It can be in the hope of the preceding fourth central square of Y, corresponding relationship according to the preceding fourth central square of Z are as follows:
The related data of Z is substituted into the preceding fourth central square that can acquire Y respectively are as follows:
Because stochastic variable Y is canonical statistics, i.e., mean value is 0, variance 1, so its preceding fourth central square is equal to preceding quadravalence original
Point square.
Step 6:
Based on principle of maximum entropy in probability theory, using the preceding quadravalence moment of the orign of stochastic variable Y as constraint condition, and define
The comentropy of stochastic variable Y are as follows:
H [f (y)]=- ∫Yf(y)ln[f(y)]dy
Wherein, f (y) is the probability density function of stochastic variable Y,
Establish Lagrange's equation are as follows:
The PDF estimation of stochastic variable Y is obtained according to Optimization Solution process:
Wherein, λ=(λ0, λ1..., λ4) it is Lagrange coefficient.
Step 7:
According to the 5th step, the preceding fourth central square of stochastic variable Y has been acquired, the stochastic variable Y established using the 6th step
Probability density function expression formula, construction include Lagrange coefficient equation group:
It can solve to obtain Lagrange coefficient λ=(λ using equation group0, λ1..., λ4), and then determine stochastic variable Y
Probability density function expression formula fY(y), the data in abovementioned steps are substituted into, solution obtains Lagrange coefficient λ=(λ0,
λ1..., λ4) it is respectively as follows: λ0=0.0437, λ1=0.0617, λ2=0.2935, λ3=-0.1538, λ4=1.0131, so as to
To obtain the expression formula of the probability density function f (y) of stochastic variable Y.
Step 8:
According to the 4th step, the 5th step, machine human reriability be can be converted
According to the probability density function of stochastic variable Y required by the 7th step, robot reliability can pass through following methods
It acquires:
Under the influence of considering joint space and connecting rod dimensional discrepancy, end positioning accuracy, the terminal position of robot are given
The probability that error is less than positioning accuracy is that i.e. machine human reriability is
Embodiment is equally implemented reliability obtained with the more authoritative Monte Carlo Analogue Method of tradition
0.91236。
It can be seen that the method for the present invention, which can be calculated accurately, obtains industrial machine human reriability,
And in being embodied, traditional Monte Carlo Analogue Method is utilized 100,000 sample values, calculates the time and is
The sample value that the 1182.63s present invention utilizes is 500, and calculating the time is 12.64s compared to traditional Monte Carlo Analogue Method
It is greatly reduced, nearly 100 times can be improved in efficiency, and computational accuracy is high.
It can be seen that the present invention not only has many advantages, such as that high-efficient, computational accuracy is good, but also calculation amount is greatly reduced,
Reduce and calculate the time, and there is significant technical effect in robot reliability's analysis.
Claims (1)
1. a kind of industrial robot reliability calculation method based on quadravalence moments estimation, it is characterised in that the following steps are included:
It is sat step 1: establishing the D-H connecting rod comprising basis coordinates system and local joint coordinate system according to the linkage arm arrangement of robot
Mark system；
Step 2: obtaining homogeneous square of the robot end position relative to basis coordinates system by the positive kinematics of robot links arm
Battle arrayIt is P (x, y, z) that the theoretical position of robot end, which is obtained, relative to the coordinate value of basis coordinates system O_XYZ, above-mentioned homogeneous
MatrixUsing following expression:
Wherein, n indicates that the number of degrees of freedom, of robot, R indicate attitude matrix of the robot end in base coordinate system, P (x, y, z)
Indicate that robot end's position coordinates, П indicate even to multiply symbol；
Step 3: by the connecting rod arm lengths a at i-th of link arm of robotiWith connecting rod offset distance diDimensional discrepancy be defined as meeting
The mean value of the stochastic variable of normal distribution, connecting rod arm lengths and connecting rod offset distance is respectively μaiAnd μdi, standard deviation is respectively σaiWith
σdi；Simultaneously by the joint rotation angle θ of the rotary joint at i-th of link arm of robot comprising joint spaceiIt is defined as meeting uniform
The stochastic variable of distribution, coboundary of the joint rotation angle in the case where being uniformly distributed are θi+Δi, lower boundary θi-Δi；
Step 4: indicating that link arm is long using the random number that the random number functions normrnd generation of MATLAB meets normal distribution
Degree and connecting rod offset distance, meeting equally distributed random number using random number functions unifrnd generation indicates joint rotation angle；
Step 5: the random number generated by the 4th step, utilizes the homogeneous matrix of second step positive kinematics expression formulaIt calculates
Robot actual end position in jth time movement relative to the coordinate value of base coordinate system is P ' (x ', y ', z '), then use with
Lower formula, which calculates, obtains robot end's kinematic error εj:
Step 6: the positioning accuracy for setting robot end as Δ r, repeats the 4th step~the 5th step and carries out n times, obtain altogether N number of
All robot end's kinematic error values are formed a sample data by robot end's kinematic error value；
Step 7: analyzing the sample data obtained in the 6th step, the mean μ of calculating robot's end movement error εε, meter
Calculate formula are as follows:
Wherein, εjIndicate robot end's kinematic error in jth time movement；
Step 8: analyzing the sample data obtained in the 6th step, in the preceding quadravalence of calculating robot's end movement error ε
Heart squareCalculation formula are as follows:
Wherein, k indicates order, value k=0,1,2,3,4；
Simultaneously with second-order moment around meanAs the variance of robot end's kinematic error, i.e.,Wherein σεExpression machine
The standard deviation of device people's end movement error；
Step 9: establishing the power function g (ε) of robot system, expression formula according to the definition of robot motion's reliability are as follows:
Z=g (ε)=Δ r- ε
Wherein, Z indicates the value of power function, and Δ r indicates the positioning accuracy of robot end, and ε indicates robot end's movement
Error；
Step 10: according to the mean μ of robot end's kinematic error εεAnd its preceding fourth central square
WithPass through the power function g (ε), the power function g (ε) of calculating robot's system of the robot system that the 9th step is established
Mean μZ, the formula of calculating are as follows:
μZ=Δ r- με
The preceding fourth central square of calculating robot's system function function againCalculation formula are as follows:
Wherein, k indicates specific order, value k=0,1,2,3,4；
Simultaneously with second-order moment around meanAs the variance of robot system power function, i.e.,Wherein σZExpression machine
The standard deviation of device people's system function function；
Step 11: being the canonical statistics Y that mean value is 0, variance is 1, transformation for mula by power function value transform are as follows:
Calculate the preceding fourth central square of canonical statistics YCalculation formula are as follows:
Wherein, k indicates specific order, value k=0,1,2,3,4；
Step 12: the comentropy for defining stochastic variable Y is H [f (y)], comentropy expression are as follows:
H [f (y)]=- ∫Yf(y)ln[f(y)]dy
Wherein, f (y) indicates the probability density function of stochastic variable Y；
Using the preceding fourth central square of canonical statistics Y as constraint condition, Lagrangian is constructed according to principle of maximum entropy,
Lagrangian expression formula are as follows:
Wherein, λkFor the corresponding Lagrange coefficient of kth rank, λk=(λ0, λ1..., λ4)；
Partial derivative is asked to probability density function f (y) using LagrangianL, so that partial derivative is 0 at extreme point, derivation
Process expression are as follows:
Step 13: being calculated using the following equation the probability density function f for obtaining stochastic variable Y according to previous step derivation process
(y), expression formula specifically:
Step 14: calculating machine human reriability according to the probability density function f (y) of stochastic variable YCalculating process
Expression formula specifically:
Wherein, σzIndicate the variance of robot system power function, μzIndicate the mean value of the power function g (ε) of robot system.
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