CN107443370A - 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 method based on quadravalence moments estimation.Establish the D H link rod coordinate systems of robot and build 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 preceding Fourth-order moment of the systemic-function function and computing function function that characterize robot motion's reliability, the probability density function of systemic-function function is solved using quadravalence moment estimation method, integration is carried out and obtains machine human reriability.The present invention takes full advantage of the probability distribution information of joint space and connecting rod dimensional discrepancy, therefore few sample is only needed to complete the Calculation of Reliability to robot system, it significantly shorten the time and improve efficiency, suitable for series connection and the industrial robot of other any rotary joint types.
Description
Technical field
The present invention relates to a kind of robot reliability's computational methods, more particularly, to a kind of based on quadravalence moments estimation
Industrial robot reliability calculation method, it is related to widely used series connection and parallel connection type in the fields such as industrial production, medicine manufacture
Robot.
Background technology
With the arrival in industrial 4.0 epoch, the processing of such as large complicated parts in many fields, assembling 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 the abrasion and stress deformation etc. unavoidably in design, parts machining, assembling and use due to robot 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 cause the end movement precision of robot to reduce, and the presence of joint space not only influences kinematic accuracy,
Can also joint be caused to impact, aggravation vibration even causes the motion failures of robot, therefore to robot in above-mentioned uncertainty
Under the influence of motion reliability consideration it is significant.In current industrial production, for entering to machine human reriability
The method that row calculates mainly includes single order analysis method for reliability, especially Monte Carlo Method of Stochastic, Monte Carlo
Method of Stochastic is even more to widely use, but because Monte Carlo simulation needs relatively to be defined based on very big sample value
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, because its computational accuracy is limited, it is used primarily in that some are simple in construction, it is non-thread
In the property relatively low mechanical system of degree, the requirement in the fields such as Aero-Space, robot movement-control system can not be met.
The content of the invention
To solve the problems, such as robot reliability's calculating, the purpose of the present invention is to propose to a kind of work based on quadravalence moments estimation
Industry robot reliability's computational methods, the robot formed for series, parallel and any rotary joint or mechanical system, lead to
The stochastic variable for crossing the composition to joint space and connecting rod dimensional discrepancy carries out statistical analysis, determines its probability distribution,
The sample of the error composition of robot end position is obtained by robot kinematics, establishes the function letter of robot system
Number, and by carrying out statistical analysis to terminal position error, the probability density function expression formula of power function is obtained, so as to quick
Accurately robot reliability is calculated.
The technical solution adopted by the present invention is to use following steps：
The first step：The D-H comprising basis coordinates system and local joint coordinate system is established according to the linkage arm arrangement of robot to connect
Bar coordinate system；
As shown in Figure 2, base coordinate system O_XYZ, wherein Z axis edge are established using the bottom centre of robot as origin first
The axial direction of first rotary joint of robot, X-direction are arbitrarily chosen, and the sensing of Y-axis is according to right-hand rule by X-axis and Z
Axle determines.
As shown in Figure 3, a local coordinate system O is established at i-th of pitman arm rotary jointi-1Xi-1Yi-1Zi-1, office
The origin O of portion's coordinate systemi-1For the pivot in the i-th joint, Zi-1Axial direction of the axle along the i-th joint, XiAxle is current joint
Axial direction, YiThe sensing of axle is according to right-hand rule by XiAxle and ZiAxle determines, then the local coordinate O of i-th of pitman armiXiYiZi
It can be obtained relative to the position relationship between the local coordinate of the i-th -1 pitman arm by such as down conversion：By local coordinate system Oi- 1Xi-1Yi-1Zi-1Around Zi-1Axle anglec of rotation θiSo that Xi-1With XiAxle is parallel, and along Zi-1Axle translates diObtain coordinate system O 'iX′iY′iZ′i, by coordinate system O 'iX′iY′iZ′iAlong X 'iTranslate aiAnd around X 'iAxle anglec of rotation αiObtain coordinate system OiXiYiZi。
Above-mentioned transformation relation is expressed as the homogeneous matrix of one 4 × 4Specifically 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, θiFor i-th
The joint rotation angle of individual pitman arm, described robot are mechanical arm configurations in series or in parallel.
Second step：Robot end position is obtained 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 basis coordinates system O_XYZ coordinate value, above-mentioned
Homogeneous matrixUsing following expression：
Wherein, the number of degrees of freedom, of n expressions robot, attitude matrix of the R expression robot ends in base coordinate system, P (x,
Y, z) robot end's position coordinates is represented, П represents even to multiply symbol；
3rd step：By the connecting rod arm lengths a at i-th of pitman arm of robotiWith connecting rod offset distance diDimensional discrepancy be defined as
Meet the stochastic variable of normal distribution, the average of connecting rod arm lengths and connecting rod offset distance is respectively μaiAnd μdi, standard deviation is respectively σai
And σdi；The joint rotation angle θ of the rotary joint of joint space will be included at i-th of pitman arm of robot simultaneouslyiIt is defined as meeting
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；
4th step：Meet that the random number of normal distribution represents connecting rod using MATLAB random number functions normrnd generations
Arm lengths and connecting rod offset distance, meet that equally distributed random number represents joint rotation angle using random number functions unifrnd generations；
5th step：Consider that robot moves back and forth, in jth time moves back and forth, due to pitman arm length variation, even
Bar offset distance deviation and the influence of joint space, the actual terminal position meeting deviation theory position of robot, as shown in Figure 3, partially
From distance definition be robot end kinematic error ε, by the 4th step generate random number, utilize second step positive kinematics
The homogeneous matrix of expression formulaCalculating robot in jth time motion actual end position relative to base coordinate system coordinate value
For P ' (x ', y ', z '), then calculated using below equation and obtain robot end's kinematic error εj：
6th step：The positioning precision of robot end is set as Δ r, positioning is less than with the end movement error ε of robot
Actual robot end movement error ε is considered as stochastic variable, machine by precision Δ r probability as robot motion's reliability
People is often once moved back and forth, then in the presence of an end movement error amount, repeats the step of the 4th step~the 5th and carry out n times, obtain 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；
7th step：The sample data obtained in 6th step is analyzed, the average of calculating robot's end movement error ε
με, calculation formula is：
Wherein, εjRepresent robot end's kinematic error in jth time motion；
8th step：The sample data obtained in 6th step is analyzed, preceding the four of calculating robot's end movement error ε
Rank central momentCalculation formula is：
Wherein, k represents exponent number, 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；
9th step：According to the definition of robot motion's reliability, the power function g (ε) of robot system, expression formula are established
For：
Z=g (ε)=Δ r- ε
Wherein, Z represents the value of power function, and Δ r represents the positioning precision of robot end, and ε represents robot end
Kinematic error；
Tenth step：According to robot end's kinematic error ε mean μεAnd its preceding fourth central square WithThe power function g (ε) for the robot system established by the 9th step, the function of calculating robot's system
Function g (ε) mean μZ, the formula of calculating is：
μZ=Δ r- με
The preceding fourth central square of calculating robot's systemic-function function againCalculation formula is：
Wherein, k represents specific exponent number, 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；
11st step：It is the canonical statistics Y that average is 0, variance is 1 by power function value transform, transformation for mula
For：
Calculate canonical statistics Y preceding fourth central squareCalculation formula is：
Wherein, k represents specific exponent number, value k=0,1,2,3,4；
12nd step：The comentropy for defining stochastic variable Y is H [f (y)], and comentropy expression is：
H [f (y)]=- ∫Yf(y)ln[f(y)]dy
Wherein, f (y) represents stochastic variable Y probability density function；
Using canonical statistics Y preceding fourth central square as constraints, Lagrange is constructed according to principle of maximum entropy
Function, Lagrangian expression formula are：
Wherein, λkFor Lagrange coefficient, λ corresponding to kth rankk=(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 is：
13rd step：According to previous step derivation process, the probability density letter for obtaining stochastic variable Y is calculated using below equation
Number f (y), expression formula are specially：
14th step：Due to machine human reriabilityRefer to robot connecting rod dimensional discrepancy, connecting rod offset distance deviation with
And under the influence of joint space, the probability of the kinematic error ε of robot end within the scope of given positioning precision Δ r, i.e.,
In certain motion process, the end movement error ε of robot<Δ r probability, so after power function, robot fortune
Dynamic reliability is converted further into：
Wherein, Pr { Z≤0 } represents the probability for meeting the condition of Z≤0, and Pr { Z≤0 } is equivalent to
According to stochastic variable Y probability density function f (y), machine human reriability is calculatedCalculating process is expressed
Formula is specially：
Wherein, σzRepresent the variance of robot system power function, μzRepresent 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 parameters
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 pitman arm, be mainly shown as between connecting rod size and theoretical value and deposit in processing and manufacturing, assembling process
In difference, therefore the error parameter includes joint space and connecting rod dimensional discrepancy, and connecting rod size mainly includes robot DH and joined
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 be primarily due to process, assemble and the factor such as stress deformation causes, because
This length of connecting rod deviation is to meet the stochastic variable of normal distribution.
In 3rd step：As shown in Figure 4, between the adjacent pitman arm of robot it is made up of rotating shaft and bearing
Revolute is formed by connecting, therefore joint space shows as rotating shaft and the pivot of bearing is misaligned, in robot running
In, rotating shaft causes shock and vibration of shutting down in the internal random play of bearing so that the actual rotational angle in joint and theoretical corner it
Between error be present, therefore using the distance between pivot of rotating shaft and bearing as joint space, the pivot of rotating shaft falls
In the border circular areas that bearing inner race is formed, 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：Generate 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 of normal distribution are closed, is generated using random function unifrnd comprising joint space
And meet equally distributed joint rotation angle parameter, in the inventive method, all pitman arms of robot have connecting rod size
Joint space be present 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 superposed, but due to the joint of robot exist joint space with
And connecting rod size has certain error, so that robot actual end position and robot theory terminal position do not weigh
Close, certain error be present, therefore the kinematic error of robot is used as using the distance of physical location deviation theory position.
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, be present error in each joint rotation angle and connecting rod dimensional parameters of robot, pass through
Error be present between robot end's physical location and theoretical position that forward kinematics equation obtains, due to joint space and connecting rod
The randomness of dimensional discrepancy, the kinematic error of robot also has randomness, so being regarded as stochastic variable.
In 7th step, because the kinematic error of robot end has stochastic behaviour, hence in so that power function value
Also there is certain stochastic behaviour, so power function value is considered as into 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 for characterizing robot end respectively
Count first moment about the origin (average) 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, the average using robot end's kinematic error enters one
Step obtains first moment about the origin (average) 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 canonical statistics Y mean μY, 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 canonical statistics Y probability distribution, therefore is
Reliability is calculated, it is necessary to obtain canonical statistics Y probability density estimation expression formula, canonical statistics Y's is equal
Value and preceding fourth central square can obtain, and using the principle of maximum entropy of probability theory, construct Lagrangian, subsequently enter
The Solve problems of canonical statistics Y probability density estimation are converted into meeting that the optimization of certain constraints is asked by one step
Topic.
In 13rd step, obtained by solving the optimization problem met under constraints in Lagrangian not
Parameter is known, so as to obtain the complete canonical statistics Y with parameter probability density estimation expression formula, by general
The integration of rate density function model, realize that the motion credibility of robot calculates.
Present invention utilizes joint space and length of connecting rod deviation and the probability Distribution Model 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, so as to eliminate due to by the error brought of nonlinear function linearisation, meanwhile, because joint space,
The randomness of length of connecting rod deviation and connecting rod offset distance deviation causes robot end's site error also to have certain randomness, this
Robot end's kinematic error is considered as stochastic variable by invention, and its average and preceding fourth central square have been asked for using statistical analysis,
The statistical nature of stochastic variable is taken full advantage of, avoids the loss of the statistical information included to stochastic variable to the full extent,
The ASSOCIATE STATISTICS feature of power function has further been calculated using the statistical nature of terminal position error, and it is former based on maximum entropy
Reason has asked for the probability density function expression formula of power function, finally using integration, obtains machine human reriability.
The present invention advantage and beneficial effect be it is following some：
1st, solves machine for the robot containing joint space, length of connecting rod deviation and connecting rod offset distance deviation, the present invention
Human reliability analysis problem, it is to ensure robot energy among practice as Robot Design, processing, the premise assembled
It is enough accurate, stably, the technical foundation of reliability service.
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, according to machine by end movement error
Device people positioning precision and reliability definition, the linear function function of robot system is established, is avoided directly to robot
Forward kinematics equation carries out Taylor series expansion, avoids and carries out nonlinear function to linearize caused error.
2nd, the inventive method, compared to now widely used Monte Carlo Analogue Method in fail-safe analysis, ensureing
As a result on 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.
3rd, the inventive method, in fail-safe analysis, compared to traditional single order fail-safe analysis method, can utilize random
The more statistical informations of variable.
Traditional single order fail-safe analysis method with only first moment about the origin (average) and the second-order moment around mean (side of variable
Difference) statistical information, do not meet can there is larger error in the case of normal distribution strictly in variable.
The inventive method not only make use of the average and variance of stochastic variable, also further make use of three ranks of stochastic variable
(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 both needed for sample sizes it is less, the inventive method can obtain more accurate meter
Calculate result.
Brief description of the drawings
Fig. 1 is the FB(flow block) of the inventive method；
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.
Embodiment
Below in conjunction with the accompanying drawings and specific embodiment is described in further details to the present invention
The general principle of the present 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 joint space can cause the joint of robot to turn
Angle has one, and equally distributed random error, connecting rod dimensional discrepancy can cause the physical parameter of robot to send out within the specific limits
Changing, joint space, length of connecting rod deviation and connecting rod offset distance deviation can 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 motion of robot end position misses
Difference, the error fall within stochastic variable, according to the end positioning precision 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 falls within stochastic variable, using by
The sample of robot end's kinematic error composition, carries out statistical analysis, obtains representing the average of the stochastic variable of kinematic error
And preceding fourth central square, the average of power function is further asked for by power function using its average and preceding fourth central square
With preceding fourth central square, using the preceding quadravalence moment of the orign of power function as constraints, the principle of maximum entropy based on probability theory will
The problem has been converted further into meeting the Optimization Solution problem under constraints, establishes Lagrange's equation, solves its glug
Bright day coefficient, so as to obtain the probability density function of power function, by integration, calculate machine human reriability.
Embodiments of the invention：
Block diagram in accompanying drawing 1 is the particular flow sheet of novel robot reliability calculation method proposed by the present invention, such as attached
Shown in figure, as shown in Figure 2, the present invention artificially analyzes object, including pitman arm 1 with a 3DOF tandem type industrial machine
With rotary joint 2, for containing joint space and, the robot reliability of length of connecting rod deviation and connecting rod offset distance deviation calculate, knot
Close embodiment to illustrate, the angle being directed to represents that the dimensions length and the unit of distance being related to are to spend (°)
Millimeter (mm), idiographic flow of the invention mainly include the following steps that：
The first step：
The D-H link rod coordinate systems of robot are established, the corresponding physical parameter of robot includes as follows：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：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).
Second step：
It is full by joint space, length of connecting rod deviation and connecting rod offset distance deviation definition of the pitman arm itself between pitman arm
The stochastic variable of the certain distribution pattern of foot, randomly generates 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 explanation：Two joints junction of specific implementation is as shown in figure 5, the connecting rod in a joint
3 ends are fixedly connected with rotating shaft 1, and the end of connecting rod 3 in another joint is fixedly connected with bearing 2, and rotating shaft 1 is sleeved in bearing 2,
Because the outer ring of rotating shaft 1 and the inner ring of bearing 2 have gap, therefore the pivot for causing rotating shaft and bearing is misaligned, and to turn
The distance between pivot of axle and bearing is used as center offset distance 4 so that joint is present in rotary course between a joint
Gap Δ.
Connecting rod dimensional discrepancy belongs to normal distribution, and respective average and standard deviation are respectively：
μa2=400 (mm), σa2=0.35 (mm), μa3=600 (mm),
σa3=0.30 (mm), μd1=500 (mm), σd1=0.35 (mm)
Joint space Δ meets to be uniformly distributed, and scope is [- 0.1 °, 0.1 °], then the actual joint rotation angle of robot is：
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 site error is：
3rd step：
Repeat step 2 500 times, obtain by 500 random samples being made up of 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 moment
Second-order moment around mean (variance)Third central momentIn quadravalence
Heart square
4th step：
The positioning precision of given robot end is Δ r=2 (mm), thus establishes the power function Z=g (ε) of robot
=Δ r- ε, power function Z average and preceding fourth central square are calculated using stochastic variable ε average and preceding fourth central square
Respectively：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 precision, so machine human reriability can be with
Further it is expressed as：
5th step：
It is 0 that power function Z is converted into average, and variance is 1 canonical statistics Y, and transformational relation can be expressed as：
Can be in the hope of Y preceding fourth central square, corresponding relation according to Z preceding fourth central square：
Z related data, which is substituted into, which can try to achieve Y preceding fourth central square respectively, is： Because
Stochastic variable Y is canonical statistics, i.e., average is 0, variance 1, so its preceding fourth central square is equal to preceding quadravalence moment of the orign.
6th step：
Based on principle of maximum entropy in probability theory, using stochastic variable Y preceding quadravalence moment of the orign as constraints, and define
Stochastic variable Y comentropy is：
H [f (y)]=- ∫Yf(y)ln[f(y)]dy
Wherein, f (y) is stochastic variable Y probability density function,
Establishing Lagrange's equation is：
Stochastic variable Y PDF estimation is obtained according to Optimization Solution process：
Wherein, λ=(λ0, λ1..., λ4) it is Lagrange coefficient.
7th step：
According to the 5th step, stochastic variable Y preceding fourth central square has been tried to achieve, the stochastic variable Y established using the 6th step
Probability density function expression formula, construction includes the equation group of Lagrange coefficient：
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) be respectively：λ0=0.0437, λ1=0.0617, λ2=0.2935, λ3=-0.1538, λ4=1.0131, so as to
To obtain stochastic variable Y probability density function f (y) expression formula.
8th step：
According to the 4th step, the 5th step, machine human reriability can be converted into
The probability density function of stochastic variable Y according to required by the 7th step, robot reliability can pass through following methods
Try to achieve：
Under the influence of joint space and connecting rod dimensional discrepancy is considered, end positioning precision, the terminal position of robot are given
The probability that error is less than positioning precision is that i.e. machine human reriability is
Embodiment using the more authoritative Monte Carlo Analogue Method of tradition carry out it is same implement obtained reliability as
0.91236。
As can be seen here, the inventive method can calculate exactly obtains industrial machine human reriability,
And in being embodied, traditional Monte Carlo Analogue Method make use of 100,000 sample values, the calculating time is
The sample value that the 1182.63s present invention utilizes is 500, and the calculating 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.
The present invention not only has the advantages that efficiency high, computational accuracy are good as can be seen here, and greatly reduces amount of calculation,
Reduce and calculate the time, and there is notable technique effect in robot reliability analyzes.
Claims (1)
1. a kind of industrial robot reliability calculation method based on quadravalence moments estimation, it is characterised in that comprise the following steps：
The first step：The D-H connecting rods comprising basis coordinates system and local joint coordinate system are established according to the linkage arm arrangement of robot to sit
Mark system；
Second step：Homogeneous square of the robot end position relative to basis coordinates system is obtained 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 basis coordinates system O_XYZ coordinate value, above-mentioned homogeneous
MatrixUsing following expression：
<mrow>
<msubsup>
<mi>A</mi>
<mi>n</mi>
<mn>0</mn>
</msubsup>
<mo>=</mo>
<msubsup>
<mi>&Pi;</mi>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mi>n</mi>
</msubsup>
<msubsup>
<mi>A</mi>
<mi>i</mi>
<mrow>
<mi>i</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mi>R</mi>
</mtd>
<mtd>
<mi>P</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
Wherein, n represents the number of degrees of freedom, of robot, and R represents attitude matrix of the robot end in base coordinate system, P (x, y, z)
Robot end's position coordinates is represented, П represents even to multiply symbol；
3rd step：By the connecting rod arm lengths a at i-th of pitman arm of robotiWith connecting rod offset distance diDimensional discrepancy be defined as meeting
The average 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；The joint rotation angle θ of the rotary joint of joint space will be included at i-th of pitman arm of robot simultaneouslyiIt is defined as meeting uniformly
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；
4th step：Meet that the random number of normal distribution represents pitman arm length using MATLAB random number functions normrnd generations
Degree and connecting rod offset distance, meet that equally distributed random number represents joint rotation angle using random number functions unifrnd generations；
5th step：The random number generated by the 4th step, utilizes the homogeneous matrix of second step positive kinematics expression formulaCalculate
Robot actual end position in jth time motion 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：
<mrow>
<msub>
<mi>&epsiv;</mi>
<mi>j</mi>
</msub>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<msup>
<msub>
<mi>x</mi>
<mi>j</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mo>-</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mrow>
<mo>(</mo>
<msup>
<msub>
<mi>y</mi>
<mi>j</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mo>-</mo>
<mi>y</mi>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mrow>
<mo>(</mo>
<msup>
<msub>
<mi>z</mi>
<mi>j</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mo>-</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
6th step：The positioning precision of robot end is set as Δ r, the step of the 4th step~the 5th is repeated and carries out n times, obtain altogether N number of
Robot end's kinematic error value, all robot end's kinematic error values are formed into a sample data；
7th step：The sample data obtained in 6th step is analyzed, the mean μ of calculating robot's end movement error εε, meter
Calculating formula is：
<mrow>
<msub>
<mi>&mu;</mi>
<mi>&epsiv;</mi>
</msub>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>N</mi>
</mfrac>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mi>N</mi>
</munderover>
<msub>
<mi>&epsiv;</mi>
<mi>i</mi>
</msub>
</mrow>
Wherein, εjRepresent robot end's kinematic error in jth time motion；
8th step：The sample data obtained in 6th step is analyzed, in the preceding quadravalence of calculating robot's end movement error ε
Heart squareCalculation formula is：
Wherein, k represents exponent number, 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；
9th step：According to the definition of robot motion's reliability, the power function g (ε) of robot system is established, expression formula is：
Z=g (ε)=Δ r- ε
Wherein, Z represents the value of power function, and Δ r represents the positioning precision of robot end, and ε represents robot end's motion
Error；
Tenth step：According to robot end's kinematic error ε mean μεAnd its preceding fourth central square
WithThe power function g (ε), the power function g (ε) of calculating robot's system for the robot system established by the 9th step
Mean μZ, the formula of calculating is：
μZ=Δ r- με
The preceding fourth central square of calculating robot's systemic-function function againCalculation formula is：
Wherein, k represents specific exponent number, 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 systemic-function function；
11st step：It is the canonical statistics Y that average is 0, variance is 1 by power function value transform, transformation for mula is：
<mrow>
<mi>Y</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>Z</mi>
<mo>-</mo>
<msub>
<mi>&mu;</mi>
<mi>Z</mi>
</msub>
</mrow>
<msub>
<mi>&sigma;</mi>
<mi>Z</mi>
</msub>
</mfrac>
</mrow>
Calculate canonical statistics Y preceding fourth central squareCalculation formula is：
Wherein, k represents specific exponent number, value k=0,1,2,3,4；
12nd step：The comentropy for defining stochastic variable Y is H [f (y)], and comentropy expression is：
H [f (y)]=- ∫Yf(y)ln[f(y)]dy
Wherein, f (y) represents stochastic variable Y probability density function；
Using canonical statistics Y preceding fourth central square as constraints, Lagrangian is constructed according to principle of maximum entropy,
Lagrangian expression formula is：
Wherein, λkFor Lagrange coefficient, λ corresponding to kth rankk=(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 is：
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<mi>L</mi>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>y</mi>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</mrow>
13rd step：According to previous step derivation process, the probability density function f for obtaining stochastic variable Y is calculated using below equation
(y), expression formula is specially：
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>y</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>exp</mi>
<mrow>
<mo>(</mo>
<mo>-</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mn>4</mn>
</munderover>
<msub>
<mi>&lambda;</mi>
<mi>k</mi>
</msub>
<msup>
<mi>y</mi>
<mi>k</mi>
</msup>
<mo>)</mo>
</mrow>
</mrow>
2
14th step：According to stochastic variable Y probability density function f (y), machine human reriability is calculatedCalculating process
Expression formula is specially：
<mrow>
<msubsup>
<mi>P</mi>
<mi>r</mi>
<mi>s</mi>
</msubsup>
<mo>=</mo>
<mn>1</mn>
<mo>-</mo>
<msubsup>
<mo>&Integral;</mo>
<mrow>
<mo>-</mo>
<mi>&infin;</mi>
</mrow>
<mrow>
<mo>-</mo>
<msub>
<mi>&mu;</mi>
<mi>z</mi>
</msub>
<mo>/</mo>
<msub>
<mi>&sigma;</mi>
<mi>z</mi>
</msub>
</mrow>
</msubsup>
<mi>exp</mi>
<mrow>
<mo>(</mo>
<mo>-</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mn>4</mn>
</munderover>
<msub>
<mi>&lambda;</mi>
<mi>k</mi>
</msub>
<msup>
<mi>y</mi>
<mi>k</mi>
</msup>
<mo>)</mo>
</mrow>
<mi>d</mi>
<mi>y</mi>
</mrow>
Wherein, σzRepresent the variance of robot system power function, μzRepresent the power function g (ε) of robot system average.
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