positioning error compensation for industrial robots based

Research ArticlePositioning Error Compensation for Industrial Robots Based onStiffness Modelling

Yingjie Li Guanbin Gao and Fei Liu

Faculty of Mechanical and Electrical Engineering Kunming University of Science and Technology Kunming 650500 China

Correspondence should be addressed to Guanbin Gao gbgao163com

Received 10 August 2020 Revised 6 September 2020 Accepted 17 September 2020 Published 20 November 2020

Academic Editor Kailong Liu

Copyright copy 2020 Yingjie Li et al )is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Insufficient stiffness of industrial robots is a significant factor which affects its positioning accuracy To improve the positioningaccuracy a novel positioning error compensation method based on the stiffness modelling is proposed in this paper First thepositioning errors considering the end load and gravity of industrial robots due to stiffness are analyzed Based on the results ofanalysis it is found that the positioning errors can be described by two kinds of deformation errors at joints the axial deformationerror and the radial deformation error )en the axial deformation error is modelled by the differential relationship of kinematicsequations)emodel of radial deformation error is deduced through the recurrence method and rotation transformation betweenjoints Finally these two models are transformed into a Cartesian coordinate system and a positioning error compensationmethod based on these two models is presented Simulations based on the finite element analysis are implemented to verify thepositioning error compensation method )e results show that the suggested method can efficiently predict the positioning erroraccording to the gravity and loads so that the positioning accuracy of industrial robots can be improved with theproposed method

1 Introduction

With the rapid development of intelligent manufacturingindustrial robots (noted as robots) have been widely appliedin automobile manufacturing logistics systems mechanicalprocessing food packaging industries etc [1] According tothe report of the International Federation of Robotics (IFR)more than 72 of industrial robots are used in the low-precision occasions eg sorting palletizing handling spotwelding painting and assembly of simple parts [2] for thepositioning and the trajectory accuracy of industrial robotsis still relatively low [3] Kinematic calibration is the generalway to improve the absolute positioning and trajectoryaccuracy of industrial robots [4] However kinematic cali-bration can only compensate the static kinematic parametererrors of industrial robots while the dynamic factors egloads of the end effector (noted as EE) speed accelerationgravity and poses also influence the positioning accuracygreatly One of the most notable characteristics of industrialrobots is the open-chain structure that provides high

flexibility and large working space for industrial robots[5 6] However the open-chain structure also results in alow stiffness and an error accumulation amplification whichare the main reasons for the low accuracy of industrialrobots )erefore improvement of the stiffness error hasalways been an important research field for industrial robotsAt present there are mainly three methods to improve thestiffness ie structural strengthening method stiffnesscontrol method and working space compensation method

)e structural strengthening is an approach to improvethe absolute accuracy of robots which enhances the integralstiffness of robots by changing the material properties of therobotrsquos mechanical parts and transmission parts )is ap-proach requires the designers to consider the low posi-tioning accuracy caused by insufficient stiffness in thedesign Hence the designers need to estimate the robotrsquosstiffness in advance according to its application fieldsTyapin et al [7] proposed a method to model the physicalstiffness of the driver reducer and transmission parts of therobots respectively With the corresponding weight

HindawiComplexityVolume 2020 Article ID 8850751 13 pageshttpsdoiorg10115520208850751

coefficients obtained from the model the robotrsquos stiffnesscan be further integrated and estimated However thestiffness model is not precise enough according to its ex-perimental results Liu et al [8] presented a method ofmodelling the rotating joint of the robots based on theanalysis of the contact relationships of the robotrsquos jointsAlthough one can calculate the stiffness of each joint of therobots accurately with this method its calculation process iscomplicated and the consumption cost of calculation isconsiderable By using structural strengthening the stiffnessof the robots can be improved in the stage of manufactureHowever after manufacture the stiffness of robots deter-mined by the material properties and structure will not bechanged

)e stiffness control method aims to control the robots atsome certain poses in which the stiffness of robots is rela-tively high [9 10] Abele et al [11 12] presented an adaptivemachining method after measuring workpiece shape toovercome the problem of low positioning accuracy causedby the weak stiffness when machining )e results show thatthe machining accuracy of robots can be improved bychanging the robotrsquos poses to adjust the torque at the jointwhich can enhance the robotrsquos stiffness indirectly By takingthe stiffness ellipsoid as the index to assess stiffness Guoet al [13] improved the robotrsquos stiffness by optimizing itsworking poses with the maximum joint angle as the con-straints Combining with the redundancy Zanchettin et al[14] presented a method of optimizing the robotrsquos poses toenhance the stiffness when the robots perform drilling tasksFrom the above methods it can be found that the main ideaof the stiffness control method is to strengthen the EEoperation stiffness of the robots by changing or optimizingthe robotrsquos working poses [15] Nevertheless the precon-dition of optimizing the pose to improve the positioningaccuracy is that there is a contact force at the EE of robots)erefore when there are no contact forces eg paintingprealignment and measurement the stiffness controlmethod does not work

Based on the kinematic and dynamic relationship ofrobots the working space compensation methods can es-tablish a mathematical model that can describe the rela-tionship between the positioning error and end loads alongwith the gravity of the robot And it can be used to predictthe positioning error in the working space to improve thepositioning accuracy Salisbury proposed the traditionalstiffness model of robots based on the kinematic and statictheories [16] )e stiffness modelling for robots in theCartesian coordinates was studied by connecting the stiff-ness of each joint [17ndash19] Abele et al [11] provided astiffness model without calculating the inverse of Jacobianmatrix which can simplify the calculation of error com-pensation In [20] the stiffness matrix of joints was identifiedby measuring the EE displacement and rotation of robots)en the stiffness matrix in the Cartesian coordinates wasderived by the stiffness mapping model from the joint to EESun et al [21] proposed a method to calculate the EEtranslation stiffness for serial robots However according tothe experimental results this method is not accurate in termsof prediction of the positioning error Overall working space

compensation methods are simpler and more universal thanstructural strengthening methods and stiffness controlmethods in practice while the low accuracy limits its ap-plication )e main reason is that the current methods onlyconsider the rotary deformation around the axis of jointswhich does not contain the rotary deformation around theradial direction of joints

To improve the positioning accuracy a novel posi-tioning error compensation model based on the stiffnessfor industrial robots is proposed in this paper whichdescribes the relationship between the positioning errorand the EE load and gravity of robots First according tothe NewtonndashEuler method the driving torque for eachjoint is calculated )rough the deformation assumption ofjoints the torque is connected with the rotation defor-mation at joints Moreover the torque is also decomposedalong the axial and radial directions of joints which isconsistent with the axial deformation and radial defor-mation Positioning error models including the axial andradial deformation are established by means of kinematicsdifferential and recurrence methods respectively In ac-cordance with the small deformation assumption twokinds of positioning error models are linearized Finally acomplete error compensation model is derived accordingto the error models of axial and radial deformation whichcan effectively predict and compensate the positioningerror and improve the positioning accuracy for industrialrobots

)e major contributions of this paper include thefollowing

(1) A novel positioning error model is proposed for n-degree-of-freedom (DOF) industrial robots based onthe relationship between its kinematics and dy-namics parameters which can be applied to arbitrarymulti-DOF serial robots

(2) )e NewtonndashEuler method is introduced to calculatethe balance torque of joints which makes it con-venient to calculate the positioning error caused bythe EE loads and gravity of the robot

(3) )e radial deformation error at joints is modelledand included into the stiffness error model whichimproves the accuracy of prediction and compen-sation for positioning error compared to the tradi-tional methods

(4) )e positioning error model is linearized by intro-ducing proper assumptions which reduces thecomplexity of the proposed method and makes itconvenient to applications

)e rest of the paper is organized as follows in Section 2the displacement and deformation at joints of robots areanalyzed with the gravity and EE load In Section 3 six basicassumptions are provided and the positioning error ofrobots is derived In Section 4 the finite element analysis andthree-dimensional model of a general six-DOF industrialrobot are used to validate the effectiveness and correctness ofthe proposed method Finally conclusions for this paper aregiven in Section 5

2 Complexity

Notations )roughout the paper R denotes the set of realnumbers Rn is the Euclidean space of n-dimensional realvectors Rmtimesn is the space of m times n real matrix Intimesn denotesthe identity matrix inRntimesn 0 stands for the base coordinatesystem of the robot i is the coordinate system of link i ofthe robot iTj isin R4times4 is the homogeneous transform from ito j iRj isin R3times3 is the rotation transform from i to jiPj isin R3 represents the description of coordinate origin ofj in i iPCj isin R3 denotes the centroid position of link j ofrobots in i fi isin R3 stands for forces of link i-1 acting onlink i ni isin R3 stands for torque of link i-1 acting on link iAnalogously ifj isin R3 and inj isin R3 are descriptions of fj

and nj in i respectively _θi isin R euroθi isin R are the angularspeed and angular acceleration of link i relative to ii 1113954Zj isin R3 is a vector of z-axis of j in i iωj isin R3 andi _ωj isin R3 represent the angular speed and angular acceler-ation of link j relating to 0 in i i _vj isin R3 is defined as theacceleration of j in i i _vCj isin R3 denotes the acceleration ofcentroid of link j in i middot means the vector norm of avector in Rn mi+1 means the mass of link i+1 of industrialrobots To simplify expressions cos(θ) sin(θ) and1 minus sin(θ) are replaced by cθ sθ and vθ respectively(z(middot)z(x)) means the gradient operator to variable xS(middot) isin R3times3 is defined as

S(middot)

0 minusVz Vy

Vz 0 minusVx

minusVy Vx 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (1)

where Vx isin R Vy isin R and Vz isin R represent the threecomponents of a vector Further 1113957S(middot) isin R4times4 can be definedas

1113957S(middot) S(middot)3times3 0

0 01113890 1113891

4times4 (2)

)e sign function is defined as sign(middot)

2 Analysis of Deformation at Joints

A general industrial robot is shown in Figure 1 When forcesare applied to the robot and the linkrsquos gravity is consideredbalance forces and torques are generated at each joint andlink According to the theories of material mechanics if theobject is subjected to forces and torques its shape will bechanged eg tension compression shear torsion andbending [22] )e effect of applied forces at joints and linksof the robot is more complex than the results of theoreticalanalysis because the joint of robots consists of many ele-ments in reality eg motors drive shafts gears and re-ducers Hence the deformation of robots in reality is acombination of the above five deformations However sincethe deformation at joints is significantly larger than thedeformation at links the deformation at joints is mainlystudied in this paper

In Figure 1 two types of deformation are shown whichinclude the rotary deformation around the axis of joints andthe linear deformation along the certain direction )e jointof the robot deflects angle Δφ around its axis and its linear

deformation is Δl under effects of gravity G and the endload F

According to the NewtonndashEuler method [23] the re-lationship between the motion and the driving force (ordriving torque) of industrial robots can be described throughthe following dynamic equations

i+1_ωi+1

i+1i R

i_ωi +

i+1i R

i_ωi times _θi+1

i+1 1113954Zi+1 + euroθi+1i+1 1113954Zi+1

i+1_vi+1

i+1i R

i_ωi times

iPi+1 +

i_ωi times

i_ωi times

iPi+1 +

i_vi1113960 1113961

i+1_vCi+1

i+1

_ωi+1 timesi+1

PCi+1+

i+1ωi+1 timesi+1ωi+1 times

i+1PCi+1

+i+1

_vi+1i+1Fi+1 mi+1

i+1_vCi+1

i+1Ni+1 Ci+1Ii+1i+1 _ωi+1 +

i+1ωi+1 timesCi+1Ii+1

i+1ωi+1

(3)

ifi

ii+1R

i+1fi+1 +

iFi (4)

ini

iNi +

ii+1R

i+1ni+1 +

iPCi

timesiFi +

iPi+1i times

ii+1R

i+1fi+1

(5)

where iFi isin R3 and iNi isin R3 are the inertia force and torqueof link i in i respectively Ci+1Ii+1 isin R3times3 denotes the inertialtensor matrix of link i+1 in Ci+11113864 1113865 that is the coordinatesystem of centroid of link i+1 When the robot stops at aposition some variables in equations (4) and (5) are con-stant eg _θi+1 0 euroθi+1 0 i+1ωi+1 (0 0 0)T andi+1 _ωi+1 (0 0 0)T(i+1) )us the above dynamic equationsare further simplified as

i+1_vi+1

i+1i R

i_vi

i+1_vCi+1

i+1

_vi+1

i+1Fi+1 mi+1

i+1_vCi+1

(6)

ifi

ii+1R

i+1fi+1 +

iFi (7)

ini

ii+1R

i+1ni+1 +

iPCi

ini times

iFi +

iPi+1 times

ii+1R

i+1fi+1 (8)

It should be noted that although the robot is at stationarystate the value of i _vi is not zero According to the weakprinciple of equivalence the gravity applied to the robot isequivalent to a case that the robot has an initial accelerationwhich is opposite to the gravity direction In this paper thegravity of the robot is considered and 0 _v0 is a three-di-mensional vector that is opposite to the gravity

From equations (7) and (8) conclusions can be obtainedthat even if the robot is at stationary state the balance forceand torque exist at the joints of the robot To display theforce and torque clearly the joint is taken out from the robotas a separate body in Figure 2 in which F and M representthe resistant force and torque respectively Fd andMd denotethe driving force and torque respectively Ma and Mr standfor axial and radial components of torqueM)us under theinfluence of force F and torqueM there will be two kinds ofdeformation at the joint ie the rotary deformation aroundthe torquersquos axis and the linear deformation along the forcedirection

Complexity 3

From the above analysis there are two types of defor-mation at the joint under influence of the gravity and endload Although the deformation at the joint may be verysmall in practice the joint as vital but weak parts of therobot will greatly affect the positioning accuracy of therobots It can be observed from (a) and (b) in Figure 1 thatthe rotary deformation and linear deformation (these twodeformations are usually appearing together) lead to thepositioning error of the robots Hence it is essential todescribe the positioning error caused by joint deformationwith a mathematical model and to more accurately predictthe positioning error Hence to resolve these problems ageneralized mathematical model for industrial robots isproposed to predict the positioning error in this paper

3 Positioning Error Modelling forIndustrial Robots

Some assumptions should be introduced before establishingthe positioning error model because they are the basis of theproposed method

Assumptions are as follows

(i) )e industrial robots only contain rotating jointsbut not moving joints

(ii) )e elastic deformation of the robotrsquos link is neg-ligible compared to its deformation at joints

(iii) )e effect caused by the rotary deformation at jointson the positioning error of the robots is much larger

than the effect caused by the linear deformation ofthe joint on the end positioning error

(iv) )e rotary angle Δφ isin R caused by the rotary de-formation at the joint is small enough so that thefollowing equations can be regarded as meaningfulwithin the allowable range of accuracy

sin(Δφ) Δφ cos(Δφ) 1 (9)

(v) )ere is a linear relationship between the rotarydeformation Δφ at the joint and the torque n isin R3

applied to the joint as shown belowΔφ Cn (10)

where C isin R is the flexibility coefficient of the joint)erefore the stiffness coefficient can be defined asK (1C)

(vi) )ere are two types of rotary deformation at each jointie the rotary deformation Δθ isin R around jointrsquos axisand the rotary deformation Δc isin R around the radialdirection of the joint According to assumption (iv)the following equations can be obtained

Δθ Ca na

Δc Cr nr

(11)

where Ca isin R denotes the axial stiffness coefficient of thejoint and Cr isin R denotes the radial stiffness coefficient of the

Md

Fd

M

x F

z

y

(a)

x

zMa

Mr

M

y

(b)

Figure 2 )e force and torque at the joint (a) )e torque and force at the joint (b) )e decomposition of torque M

Δφ

G

F

(a)

G

Δl

F

(b)

Figure 1 Two types of deformation at the joint (a) )e rotary deformation at the joint (b) )e linear deformation at the joint

4 Complexity

joint With assumption (v) we can define Ka (1Ca) andKr (1Cr) na isin R3 and nr isin R3 stand for axial torque andradial torque at the joint

Remark 1 In practice general robots consist of six revolutejoints [24 25] )us assumption (i) is appropriate forgeneral-purpose robots It is also shown that the deforma-tion of the joint due to the insufficient stiffness of driving andtransmission system accounts for 70 of the total defor-mations which are caused by the external load or gravity[26] Accordingly assumption (ii) is true in this paper Asshown in Figure 1 although the rotary deformation andlinear deformation at the joint may be tiny the effects causedby the rotary deformation on the positioning error aresignificant because of the magnifying effect of the link Forthis reason assumption (iii) is reasonable In accordancewith [27 28] the joint stiffness of industrial robot is 1NμmIn other words a force of 1000N is required to generatedeformation of 1mm However the maximum end load ofmost industrial robots is less than 1000N When the de-formation at the joint is less than 1mm it can be reckonedthat assumption (iv) is meaningful In the light of Hookersquoslaw of the material assumption (v) is feasible In the pre-vious discussions there is a resistant torque M at the joint)e vector of torque M can be further decomposed alongtwo directions ie the axial and radial direction of joint asshown in (b) of Figure 2 Hence it can be considered that therotary deformation consists of rotary deformation aroundjointrsquos axis and the radial direction of the joint Since thesetwo kinds of deformation are different in essence Ca and Cr

are required to describe the relationship between the rotarydeformation and torque applied to the joint )us as-sumption (vi) can also be valid

Now it is considered that the force Fe isin R3 is applied tothe EE of the robot and the gravity of the robot is alsoincludedWhen the robot is stationary the driving torque ini

of joint i can be obtained by equations (6) to (8) )is torquecan be decomposed as

ini

inai +

inri (12)

)us the rotary deformations around axial and radialdirection at joint i are

Δθi Caiinai

Δci Criinri

(13)

According to the two types of rotary deformations at thejoint in equation (13) the positioning error model of axialand the radial deformation will be established respectivelyin the next sections and the total positioning error modelwill be derived finally

31 Positioning Error Model of Axial Deformation around theJoint When the EE of the robot is at a point Pe0 in Cartesiancoordinate system and its coordinates are0Pe0 (0px 0py 0pz)T in 0 )e joint angle correspondingto 0Pe0 isΘ (θ1 θ2 θ3 θN)T Since 0Pe0 is a function ofjoint angle Θ the differential operation of 0Pe0 to Θ is asfollows

d(0px)

d(0py)

d(0pz)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z0px

zθ1

z0px

zθ2middot middot middot

z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

dθ1

dθ1

⋮

dθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(14)

where N is the number of joints According to assumption(iv) equation (14) can be written as

0Δxa

0Δya

0Δza

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Δθ1

Δθ1

⋮

ΔθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(15)

where 0Δxa 0Δya and 0Δza represent the three componentsof positioning error of the EE due to the rotary deformationaround jointrsquos axis in 0 Combined with equation (13)equation (15) can be further written as

0ΔPa

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1na1 0 middot middot middot 0

0 2na2 middot middot middot 0

⋮ ⋮ ⋱ ⋮

0 0 middot middot middot NnaN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ca1

Ca1

⋮

CaN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)

Complexity 5

Equation (16) preliminarily indicates the relationshipbetween the torque applied to the joint and the positioningerror However equation (16) still cannot describe this re-lationship sufficiently )e main reason is that the directionof Δθi (positive or negative) is not associated with thesubjected torque in Δθi Cai

inai )ereforeΔθi Cai

inai can be rewritten as (17) by introducing a signfunction as

Δθi Caiinaisign(minus

ini(z)) (17)

where ini(z) isin R stands for the component z of ini Inequation (17) a negative sign is added before ini(z) becausethe driving torque of joint and the subjected torque are a pairof balance torques Hence on the basis of equations (15) and(17) a complete positioning error model of axial defor-mation around the joint is given as follows

0ΔPa

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

times

1n1sign(minus1n1(z)) 0 middot middot middot 0

0 2na2sign(minus2n2(z)) middot middot middot 0

⋮ ⋮ ⋱ ⋮

0 0 middot middot middot NnaNsign(minusNnN(z))

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ca1

Ca1

⋮

CaN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

In particular equation (18) indicates that the positioningerror 0ΔPa of the robots due to the rotary deformationaround the joint is a function of the variables EE position0Pe0 (or joint angle Θ (θ1 θ2 θ3 θN)T) end load F(or1na1

2na2 NnaN) and the gravity of the robot Moreovereven though the end load F does not change the positioningerror 0ΔPa is also different when the robot is at variousconfigurations

32 Positioning Error Modelling of Radial Deformationaround the Joint

321 Rotation Transformation around Arbitrary AxisFor the integrity of the modelling process the concept ofrotation transformation around the arbitrary axis will beintroduced in this section It is assumed that the vector AK

(kx ky kz)T is an identity vector in A According to theright-hand rule the rotation transformation matrix of ro-tating θ around axis of AK is as follows

RK(θ)

kxkxvθ + cθ kxkyvθ minus kzsθ kxkzvθ + kysθ

kxkyvθ + kzsθ kykyvθ + cθ kykzvθ minus kxsθ

kxkzvθ minus kysθ kykzvθ + kxsθ kzkzvθ + cθ

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

cθ middot I3times3 + vθ middotA

KA

KT + sθ middotA

K

(19)

Equation (19) is also called Rodiguesrsquos formula but it isnot linearized form concerning θ In the light of assumption(iv) when the rotary angle Δθ is small enough equation (19)can be further simplified as

RK(Δθ)

1 0 0

0 1 0

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0 minuskz ky

kz 0 minuskx

minusky kx 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦Δθ I3times3 + S(

AK )Δθ

(20)

Equation (20) indicates that the rotation transforma-tion matrix can be handled via a linearized function withrespect to the variable Δθ after using assumption (iv)which is beneficial to the linearization of the positioningerror model

322 Modelling of Radial Deformation As mentionedabove it is assumed that the end point of the robot is still atpoint Pe0 With the influence of end loads and gravity eachjoint will have a slight rotary deformation Δci around theradial direction of ini )us the coordinate of EE positioncan be described by (21) with the rotary deformation of eachjoint

1Pe1 R1n1

Δc1( 11138571Pe0

2Pe2 R2n2

Δc2( 11138572Pe1

3Pe3 R3n3

Δc3( 11138573Pe2

⋮

iPei Rini

Δci( 1113857iPeiminus1

⋮

NPeN RNnN

ΔcN( 1113857N

PeNminus1

(21)

where N is the number of joint iPei stands for the EE po-sition with the rotary deformation Δci at joint i in i Itshould be noted that in (21) the coordinates of all points aredescribed by homogeneous coordinates and the rotationtransformation is also the homogeneous form in order to beconvenient for the following coordinate transformation)en the variation of each joint before and after rotarydeformation can be obtained based on equation (21) asfollows

6 Complexity

1ΔPr1 1Pe1 minus

1Pe0

2ΔPr2 2Pe2 minus

2Pe1

3ΔPr3 3Pe3 minus

3Pe2

⋮iΔPri

iPei minus

iPeiminus1

⋮NΔPrN

NPeN minus

NPeNminus1

(22)

where iΔPri denotes the variation for the EE position at jointi before and after rotary deformation in i Next the analysisabout iΔPri will be performed First 0ΔPri is obtained bytransforming iΔPri into 0

0ΔPri 0i T

iΔPri (23)

Combining with equations (20) (21) (22) and (23) thefollowing equation can be derived

0ΔPri 0i T1113957S(minus

i1113954nri)

iPeiminus1 middot Δci (24)

where i1113954nri represents an identity vector corresponding to inriIt can be found by observing equation (21) that iPeiminus1 isdeduced from the term of iPe0 in equation (24) )us iPeiminus1

can also be expanded to iPe0 According to (21) and (22) thefollowing equations are obtained

iPeiminus1

iiminus1T

iminus1Peiminus1

iminus1Peiminus1

iminus1ΔPriminus1 +iminus1

Peiminus2

iminus1ΔPriminus11113957S minus

iminus1nriminus1

iminus1Peiminus2 middot Δciminus1

(25)

Next the following equation is obtained according toequation (25)

iPeiminus1

iiminus1T

1113957S(minusiminus1

1113954nriminus1)iminus1

Peiminus2 middot Δciminus1 +iminus1

Peiminus21113960 1113961 (26)

Equation (26) gives a recursive relationship between EEposition iPeiminus1 before the rotary deformation at joint i andthe EE position iminus1Peiminus2 before the rotary deformation atjoint i-1 Based on (26) equation of iPeiminus1 including iPe0minus1 iswritten as follows

iPeiminus1 1113944

iminus1

k1

ikT1113957S(minus

k1113954nrk)

kPekminus1 middot Δck1113960 1113961 +

i1T

1Pe0 (27)

Although iPeiminus1 has been expanded to iPe0 the aboveequation still contains 2Pe1 3Pe2 iminus1Peiminus2 Substitutingequation (27) into equations (24) and (28) we have


i1113954nri) 1113944

iminus1

k1

ikT1113957S(minus

k1113954nnk)

kPekminus1 middot ΔckΔci1113960 1113961 +

i1T

1Pe0 middot Δci

⎧⎨

⎩

⎫⎬

⎭ (28)

In (28) since Δc satisfies assumption (iv) and ΔckΔci ishigh-order terms of Δc then ΔckΔci can be removed WhenΔckΔci 0 equation (28) will be further simplified as


i1113954nri)

iPe0 middot Δci (29)

where the last component of 0ΔPri is zero since the ho-mogeneous coordinates are employed ie0ΔPri (0Δxr

0Δyr0Δzr 0)T Accordingly a transforma-

tion for equation (29) should be performed to eliminate thelast component of 0ΔPri that is zero )e specific trans-forming process is as follows

0ΔPri 0i T1113957S minus

i1113954nri1113872 1113873

iPe0 middot Δci

0Δxr0Δyr0Δzr

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i R

0Piorg

0 11113890 1113891

S minusi1113954nri1113872 1113873 00 0

1113890 1113891iPe0(3times1) 11113960 1113961 middot Δci

0i R S minus

i1113954nri1113872 1113873

iPe0(3times1)01113960 1113961 middot Δci

0Δxr0Δyr0Δzr

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i RS minus

i1113954nri1113872 1113873

iPe01113960 11139613times1 middot Δci

0Pri(3times1)

0i RS minus

i1113954nri1113872 1113873


(30)

which shows that the EE position of robots iPe0 and thetorque of the joint can be calculated by the rotary defor-mation Δci and the positioning error 0Pri caused by therotary deformation the around radial direction of the jointcan also be calculated )en the total positioning error dueto the rotary deformation Δci is determined as

0Pr(3times1) 1113944

N

i1

0Pr(3times1) (31)

From equations (13) (30) and (31) the positioning errormodel after the rotary deformation Δci is derived as follows

0Pr(3times1) 1113944

N

i1[0i RS(minus

i1113954nri)

iPe0]3times1 middot Cri

inri (32)

when Yi inri[0i RS(i1113954nri)iPe0]3times1 equation (32) will be

written as

0Pr(3times1) Y1 Y2 middot middot middot YN1113858 1113859

Cr1

Cr2

⋮

CrN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)

Similar to equation (18) equation (33) illustrates thatwhen the force Fe is applied to the EE of the robot and its

Complexity 7

gravity is considered the radial torque at the joint will lead tothe positioning error 0ΔPr(3times1)

33 ErrorModelling including Axial and Radial DeformationAccording to the analysis in Sections 31 and 32 two po-sitioning error models that correspond to the axial and radialdeformation at the joint respectively have been acquiredSince the positioning errors 0ΔPa and 0ΔPr are in 0 theycan be composited to obtain a complete positioning errormodel which includes both the influence of axial and radialdeformation as shown in the following equation

0ΔP 0ΔPa +

0ΔPr (34)

Combining with equations (18) and (33) the completepositioning error model is derived as follows

0ΔP 0JPaΛa01113957JPrΛr1113960 11139613times2N

Ca

Cr

1113890 11138912Ntimes1

(35)

where

0JPa

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

01113957JPr col1 col2 middot middot middot coli middot middot middot colN1113858 1113859 (36)

with arbitrary column

coli [0i RS(minus

i1113954nri)

iPe0]3times1

Ca Ca1 Ca2 Ca3 CaN( 1113857T

Cr Cr1 Cr2 Cr3 CrN( 1113857T

Λa

1na1sign(minus1n1(z)) 0 middot middot middot 0


⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Λr

1nr1 0 middot middot middot 0

0 2nr1 middot middot middot 0

⋮ ⋮ ⋱ ⋮

0 0 middot middot middot NnrN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(37)

Equation (35) describes the effect of the end load and thegravity on the positioning error at an arbitrary position inthe working space of the robot and it can be rewritten in asimplified form as follows

0ΔP Φ(FΘ)C (38)

where Φ(FΘ) is a simplified expression of0JPaΛa

01113957JPrΛr1113960 11139613times2N and C means (Ca Cr)

T

Remark 2 Two conclusions can be obtained according to(35) (i) )e mapping relationship between the positioningerror of the robot and its loads including the end load andgravity is linear (ii) )e positioning error is affected by theload as well as the poses of robots It should be noted that Ca

and Cr stand for the overall stiffness of all components thatmake up the joint in (35) Hence they cannot be used tomeasure the stiffness of a specific part of joints Moreover Ca

and Cr describe the torsional stiffness and bending stiffness

of joints respectively )is is consistent with the practicalsituation in which the torsional stiffness is different from thebending stiffness Finally since Ca and Cr are the integratedstiffness they cannot be obtained directly by measurementHowever many mathematical methods can be used toidentify Ca and Cr eg least square method [29 30] geneticalgorithm [31] particle swarm optimization algorithm [32]Kalman filtering algorithm [33] etc

4 Simulations

)e proposed positioning error model will be verified bysimulations )e procedure of the verification consists of sixsteps as shown in Figure 3 It is noted that if dynamic pa-rameters of the robot are known step S3 can be omitted Inthis paper dynamic parameters are obtained by Computer-Aided Design (CAD) method so step S3 is represented by adotted box here

8 Complexity

41 Kinematics Modelling A 6-DOF general-purpose robotis used to verify the effectiveness and generality of theproposed method First the coordinate systems of the robotare established according to the D-H method as shown inFigure 4 To describe the EE position of the robot the originof 6 is set at its EE )en kinematics parameters of therobot are obtained as shown in Table 1 based on the co-ordinate systems in Figure 4

According to the kinematics parameters in Table 1 thekinematic model of the robot is established by equation (39)Moreover the homogeneous transformation matrix i

jT andthe rotation transformation matrix i

jR can be obtainedwhich will be used in the positioning error model

iminus1i T

cθi minussθi 0 aiminus1

sθicαiminus1 cθicαiminus1 minussαiminus1 minussαiminus1di

sθisαiminus1 cθisαiminus1 cαiminus1 cαiminus1di

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i T 1113945

i

k1

kminus1k T

ijT

i0T

0jT

(39)

42 Dynamics Parameters To acquire the driving torque ini

and dynamics parameters of the robot are established on thebasis of NewtonndashEuler method as shown in equations (3)(4) and (5) Generally the dynamics parameters are ob-tained by identification [34] or calculation from the designparameters )e dynamics parameters of the robot can beacquired by Computer-Aided Design (CAD) method basedon the kinematics model as shown in Table 2

43 Simulation and Analysis Combining with the abovekinematics model and dynamics parameters a simulationenvironment is constructed by using finite element method)e data set from finite element simulation is defined aspractical values which are used to identify the unknownparameters Ca and Cr )en the identified parameters areput into the positioning error model Finally the predicted

S1 Establish kinematic modelof the robots with the D-H method

S2 Establish dynamic modelof the robots with

the NewtonndashEular method

S3 Perform dynamicparameters identification

S6 Evaluate the proposedmodel

S5 Acquire data used tovariation by finite element

simulation

S4 Build end positioningerror model with the proposed

method in this paper

ijT i

jR iPj

iPCimi

Figure 3 )e flowchart of verification for the proposed model

Table 1 Kinematics parameters of the robot

i αiminus1 (rad) aiminus1 (m) Diminus1 (m) θiminus1 (rad)1 0 0 0504 θ12 π2 0170 0101 θ2 + π23 0 070 minus0120 θ34 π2 0140 0760 θ45 minusπ2 0 0 θ5 + π6 minusπ2 0 0170 θ6

Z6

X6Z5X5

X4 Z4

Z3

X3

X2X1

Z1

Z2

Z0X0

Figure 4 Coordinate systems of the robot

Table 2 )e mass and centroid position of each link

i (mikg) irxi (m) iryi (m) irzi (m)

1 31138 0126 minus0076 minus01852 52132 0390 0 minus00253 48813 0124 minus0076 00054 35088 0 0 minus01905 6675 0 0183 06 1893 0 0 minus0014

Complexity 9

values according to the proposed method are compared withthe results of finite element simulation

To estimate unknown parameters Ca and Cr a group ofjoint angles are chosen arbitrarily in the working space of therobot ie Θ (44∘ minus45∘ 20∘ 45∘ minus30∘ 80∘)T It should benoted that Θ needs to be converted into a radian systemwhen calculating A group of end loads F are used as shownin Table 3 In the light of Θ and F the regression matrixΦ(FΘ) is calculated )e positioning error 0ΔP can beacquired from the finite element model as shown in Figure 5)ere are 10 groups of data in Table 3

Since (35) is linear with respect to parameters Ca and Crthe least square method is used to estimate the unknownparameters Ca and Cr )e results of parameter estimation areas shown in Table 4 It can be found that the values of 1113954Ca1 1113954Ca6and 1113954Cr6 are zero Nevertheless this does not mean that the realstiffness coefficient at joints 1 and 6 is zero but means that theirchanges have no effects on the positioning error In Table 4except the case where estimated parameter C is zero it can bealso found that some identified values are negative Accordingto Δφ Cn in assumption (v) when flexibility coefficient Cis positive it indicates that the direction of rotary deformationΔφ and joint torque n are the same And when C is negative it

indicates that the direction of Δφ is opposite to the direction ofn Meanwhile it also shows that the stiffness parametersCa andCr do not possess practical physical significance but merelymathematical meaning in the proposed model

Tomeasure the accuracy of the identified parameters theindex of relative error is introduced A linear model can beexpressed as Ax b and the relative error about the esti-mated value 1113954x can be defined as follows

er b minus A1113954x

b1113888 1113889 times 100 (40)

According to (40) it can be found that lim1113954x⟶xer 0 In

other words the value of |er| describes a degree of closeness

Nodex y zValue

3893108e + 03 115e + 03 745mm374854356e ndash 02mm

UX (mm)142688081e ndash 01

130360141e ndash 01

118032187e ndash 01

105704241e ndash 01

933762938e ndash 02

810483471e ndash 02

687204003e ndash 02

563924611e ndash 02

440645143e ndash 02

317365676e ndash 02

194086209e ndash 02

708067603e ndash 03

ndash524726976e ndash 03

Figure 5 )e finite element model

Table 3 Settings of the end load for parameters identification

FN 0Δx (mm) 0Δy (mm) 0Δz (mm)(0 0 minus500)T 56939times10minus2 50193times10minus2 minus37822times10minus1

(0 0 minus450)T 55263times10minus2 48607times10minus2 minus36290times10minus1

(0 0 minus330)T 51240times10minus2 44801times 10minus2 minus32616times10minus1



(0 0 minus88)T 43131times 10minus2 37124times10minus2 minus25206times10minus1


(0 0 minus30)T 41187times10minus2 35285times10minus2 minus23431times 10minus1


(0 0 0)T 40184times10minus2 34333times10minus2 minus22512times10minus1

Table 4 )e results of parameters identification

Parameters Value (radNbullm) Parameters Value (radNbullm)1113954Ca1 0 1113954Cr1 99312times10minus5

1113954Ca2 39922times10minus5 1113954Cr2 minus67172times10minus5

1113954Ca3 minus50087times10minus5 1113954Cr3 minus24600times10minus4

1113954Ca4 25951times 10minus5 1113954Cr4 13356times10minus3

1113954Ca5 39436times10minus4 1113954Cr5 10381times 10minus4

1113954Ca6 0 1113954Cr6 0

10 Complexity

between the estimated value 1113954x and the true value x Inaddition the value of |er| can also be used to measure thedegree of closeness between the predicted value A1113954x and thetrue value b According to (40) the relative error with respectto the identified parameters 1113954Ca and 1113954Cr is obtained as shownin Figure 6

It can be seen from Figure 6 that the relative errorbetween the theoretical values Φ(FΘ)1113954C and the measuredvalues 0ΔP is very small in the three directions and it isbetween minus00025 and 00025

20 groups of different end loads are selected randomly toverify the effectiveness of the presented method as shown in

0 50 100 150 200 250 300 350 400 450 500End load (N)

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

25

3

Rela

tive e

rror

()

times10ndash3 e relative error for identified parameters

x-directiony-directionz-direction

Figure 6 )e accuracy of identified parameters

2 4 6 8 10 12 14 16 18 20ndash4

ndash3

ndash2

ndash1

0

1

2

3 times10ndash3 e relative error between the predicted value and the measured value

Rela

tive e

rror

()

Number of points


Figure 7 )e accuracy for predicted value 0ΔP

Complexity 11

the first column of Table 5 )en the three components ofthe positioning error corresponding to each load can beacquired with finite element simulations as shown in the lastthree columns of Table 5 )e predicted values Φ(FΘ)1113954C ofthe positioning error can be calculated based on these endloads and identified parameters 1113954Ca and 1113954Cr )e relative errorbetween the predicted values Φ(FΘ)1113954C and the measuredvalues 0ΔP is calculated by (40) )e results are as shown inFigure 7

From Figure 7 it can be found that the relative errorsbetween the predicted values Φ(FΘ)1113954C and the measuredvalues 0ΔP are very small in all the three directions )ey areall in the range of [minus0004 0003] Compared with therelative error shown in Figure 6 the relative error in Figure 7is larger )e main reason is that the former group of data isinvolved in the parameter identification but the latter is onlyused to predict the positioning error of the robot With apredicted accuracy of 99996 the accuracy of the model isquite high in predicting and compensating positioning error0ΔP In practice the predicted accuracy may reduce when thereal data used to identify the parameters contains the noise

5 Conclusion

)e main factors that affect the positioning accuracy ofrobots were analyzed considering the end loads and gravityBased on the results of the analysis it is found that thepositioning error can be described by two parameters iethe axial deformation and the radial deformation at the jointA prediction and compensation model of positioning errorwas proposed based on the two kinds of deformations )epositioning error can be calculated according to the loadsand gravity though the model for n-DOF industrial robotsFinite element simulation was used to verify the proposedmodel )e results of simulation showed that the proposed

positioning error model can predict positioning errorsFuture work will focus on the verification of the presentedmodel by means of experiments and applying it to predictthe positioning error under different loads to improve thepositioning accuracy of industrial robots

Data Availability

Data were curated by the authors and are available uponrequest from the corresponding author

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (Grant no 51865020)

References

[1] Q Wu Y J Liu and C S Wu ldquoAn overview of current sit-uations of robot industry developmentrdquo in Proceedings of the4th Annual International Conference on Wireless Communi-cation and Sensor Network WCSN) Wuhan China 2018

[2] J L ZhangW H Liao Y BuW Tian and J H Hu ldquoStiffnessproperties analysis and enhancement in robotic drilling ap-plicationrdquo International Journal of Advanced ManufacturingTechnology vol 106 no 11-12 pp 5539ndash5558 2020

[3] S Y Chen T Zhang and M Shao ldquoA 6-DOF articulatedrobot stiffness researchrdquo in Proceedings of the Proceedings ofthe 12th World Congress on Intelligent Control and Auto-mation WCICA) New York NY USA 2016

[4] G Gao G Sun J Na Y Guo and X Wu ldquoStructural pa-rameter identification for 6 DOF industrial robotsrdquo

Table 5 )e data for model verification

FN)e measured values )e predicted values

0Δx (mm) 0Δy (mm) 0Δz (mm) 0Δx (mm) 0Δy (mm) 0Δz (mm)(0 0 minus7880)T 42823times10minus2 36833times10minus2 minus24925times10minus1 42824times10minus2 36832times10minus2 minus24925times10minus1

(0 0 minus48530)T 56446times10minus2 49727times10minus2 minus37371times 10minus1 56445times10minus2 49727times10minus2 minus37372times10minus1

(0 0 minus47858)T 56221times 10minus2 49513times10minus2 minus37166times10minus1 56220times10minus2 49514times10minus2 minus37165times10minus1

(0 0 minus24269)T 48315times10minus2 42031times 10minus2 minus29943times10minus1 48316times10minus2 42033times10minus2 minus29942times10minus1

(0 0 minus40014)T 53589times10minus2 47025times10minus2 minus34763times10minus1 53592times10minus2 47026times10minus2 minus34764times10minus1





(0 0 minus49774)T 56260times10minus2 42823times10minus2 minus37201times 10minus1 56261times 10minus2 42822times10minus2 minus37202times10minus1

(0 0 minus32787)T 51170times10minus2 49550times10minus2 minus32551times 10minus1 51169times10minus2 49551times 10minus2 minus32552times10minus1




(0 0 minus33936)T 51554times10minus2 49145times10minus2 minus32903times10minus1 51555times10minus2 49144times10minus2 minus32901times 10minus1





(0 0 minus8559)T 43050times10minus2 37048times10minus2 minus25133times10minus1 43051times 10minus2 37046times10minus2 minus25132times10minus1

12 Complexity

Mechanical Systems and Signal Processing vol 113 pp 145ndash155 2018

[5] C Chen F Peng R Yan et al ldquoStiffness performance indexbased posture and feed orientation optimization in roboticmilling processrdquo Robotics and Computer-IntegratedManufacturing vol 55 pp 29ndash40 2019

[6] S Wang J Na and Y Xing ldquoAdaptive optimal parameterestimation and control of servo mechanisms theory andexperimentsrdquo IEEE Transactions on Industrial Electronicsvol 68 no 1 pp 598ndash608 2020

[7] I Tyapin G Hovland and T Brogardh ldquoMethod for esti-mating combined controller joint and link stiffnesses of anindustrial robotrdquo in Proceedings of the 12th IEEE InternationalSymposium on Robotic and Sensors Environments ROSE)New York NY USA 2014

[8] Z F Liu J J Xu Q Cheng Y S Zhao and Y H PeildquoRotation-joint stiffness modeling for industrial robots con-sidering contactsrdquo Advances in Mechanical Engineeringvol 10 no 8 p 13 2018

[9] A Ajoudani N G Tsagarakis and A Bicchi ldquoChoosing posesfor force and stiffness controlrdquo IEEE Transactions on Roboticsvol 33 no 6 pp 1483ndash1490 2017

[10] M Koehler A M Okamura and C Duriez ldquoStiffness controlof deformable robots using finite element modelingrdquo IEEERobotics and Automation Letters vol 4 no 2 pp 469ndash4762019

[11] E Abele M Weigold and S Rothenbucher ldquoModeling andidentification of an industrial robot for machining applica-tionsrdquo CIRP Annals vol 56 no 1 pp 387ndash390 2007

[12] G-C Vosniakos and E Matsas ldquoImproving feasibility ofrobotic milling through robot placement optimisationrdquo Ro-botics and Computer-Integrated Manufacturing vol 26 no 5pp 517ndash525 2010

[13] Y Guo H Dong and Y Ke ldquoStiffness-oriented postureoptimization in robotic machining applicationsrdquo Robotics andComputer-Integrated Manufacturing vol 35 pp 69ndash76 2015

[14] A M Zanchettin P Rocco A Robertsson and R JohanssonldquoExploiting task redundancy in industrial manipulatorsduring drilling operationsrdquo in Proceedings of the 2011 IEEEInternational Conference on Robotics and Automation ICRA)Shanghai China 2011

[15] S Wang and J Na ldquoParameter estimation and adaptivecontrol for servo mechanisms with friction compensationrdquoIEEE Transactions on Industrial Informatics vol 16 no 11pp 6816ndash6825 2020

[16] G Alici and B Shirinzadeh ldquoEnhanced stiffness modelingidentification and characterization for robot manipulatorsrdquoIEEE Transactions on Robotics vol 21 no 4 pp 544ndash5642005

[17] F Demeester and H Van Brussel ldquoExperimental compliancebreakdown of industrial robotsrdquo Journal of Mechanical De-sign vol 116 no 4 pp 1065ndash1072 1994

[18] G Hovland E Berglund and O Sordalen ldquoIdentification ofjoint elas-ticity of industrial robotsrdquo in Proceedings of the 6thInternational Symposium on Experimental Robotics pp 455ndash464 Sydney Australia 1999

[19] S Wang M Huang and K Wang ldquoAn indirect measurementmethod for joint stiffness of flexible manipulatorsrdquo MachineTool amp Hydraulics no 8 pp 154ndash177 2004

[20] C Dumas S Caro S Garnier and B Furet ldquoJoint stiffnessidentification of six-revolute industrial serial robotsrdquo Roboticsand Computer-Integrated Manufacturing vol 27 no 4pp 881ndash888 2011

[21] J B Sun W M Zhang Z H Liu and M Assoc ldquoComptranslation stiffness calculation for serial robotsrdquo in Pro-ceedings of the 4th International Conference on Robotics andArtificial Intelligence pp 87ndash91 ICRAI) New York NY USA2018

[22] R Denzer F J Barth and P Steinmann ldquoStudies in elasticfracture mechanics based on the material force methodrdquoInternational Journal for Numerical Methods in Engineeringvol 58 no 12 pp 1817ndash1835 2003

[23] T Xu J Fan Y Chen et al ldquoDynamic identification of theKUKA LBR iiwa robot with retrieval of physical parametersusing global optimizationrdquo IEEE Access vol 8 pp 108018ndash108031 2020

[24] S Panda D Mishra and B B Biswal ldquoAn approach fordesign optimization of 3R manipulator using AdaptiveCuckoo Search algorithmrdquo Mechanics Based Design ofStructures and Machines vol 48 no 6 pp 773ndash798 2020

[25] R Wang A W Wu X Chen and J Wang ldquoA point anddistance constraint based 6R robot calibration methodthrough machine visionrdquo Robotics and Computer-IntegratedManufacturing vol 65 p 7 2020

[26] Y Zhang C Liu and P Liu 6R Industrial Robot StiffnessAnalysis Machinery Design amp Manufacture no 2 pp 257ndash260 2015

[27] M Cordes and W Hintze ldquoOffline simulation of path de-viation due to joint compliance and hysteresis for robotmachiningrdquo International Journal of Advanced Manufactur-ing Technology vol 90 no 1-4 pp 1075ndash1083 2017

[28] N R Slavkovic D S Milutinovic and M M Glavonjic ldquoAmethod for off-line compensation of cutting force-inducederrors in robotic machining by tool path modificationrdquo In-ternational Journal of Advanced Manufacturing Technologyvol 70 no 9-12 pp 2083ndash2096 2014

[29] Z Bingul and O Karahan ldquoDynamic identification of StaubliRX-60 robot using PSO and LS methodsrdquo Expert Systems withApplications vol 38 no 4 pp 4136ndash4149 2011

[30] Q Chen X Yu M Sun C Wu and Z Fu ldquoAdaptive re-petitive learning control of PMSM servo systems withbounded nonparametric uncertainties theory and experi-mentsrdquo IEEE Transactions on Industrial Electronics p 1 2020

[31] J W Ma Y Liu S F Zang and L Wang ldquoRobot pathplanning based on genetic algorithm fused with continuousbezier optimizationrdquo Computational Intelligence and Neu-roscience vol 2020 Article ID 9813040 10 pages 2020

[32] G B Gao F Liu H J San X Wu and W Wang ldquoHybridoptimal kinematic parameter identification for an industrialrobot based on BPNN-PSOrdquo Complexity vol 11 2018

[33] I Ullah X Su X W Zhang and D Choi ldquoSimultaneouslocalization andmapping based on kalman filter and extendedkalman filterrdquo Wireless Communications amp Mobile Com-puting vol 2020 p 12 Article ID 2138643 2020

[34] S Wang L Tao Q Chen J Na and X Ren ldquoUSDE-basedsliding mode control for servo mechanisms with unknownsystem dynamicsrdquo IEEEASME Transactions onMechatronicsvol 25 no 2 pp 1056ndash1066 2020

Complexity 13

coefficients obtained from the model the robotrsquos stiffnesscan be further integrated and estimated However thestiffness model is not precise enough according to its ex-perimental results Liu et al [8] presented a method ofmodelling the rotating joint of the robots based on theanalysis of the contact relationships of the robotrsquos jointsAlthough one can calculate the stiffness of each joint of therobots accurately with this method its calculation process iscomplicated and the consumption cost of calculation isconsiderable By using structural strengthening the stiffnessof the robots can be improved in the stage of manufactureHowever after manufacture the stiffness of robots deter-mined by the material properties and structure will not bechanged

)e stiffness control method aims to control the robots atsome certain poses in which the stiffness of robots is rela-tively high [9 10] Abele et al [11 12] presented an adaptivemachining method after measuring workpiece shape toovercome the problem of low positioning accuracy causedby the weak stiffness when machining )e results show thatthe machining accuracy of robots can be improved bychanging the robotrsquos poses to adjust the torque at the jointwhich can enhance the robotrsquos stiffness indirectly By takingthe stiffness ellipsoid as the index to assess stiffness Guoet al [13] improved the robotrsquos stiffness by optimizing itsworking poses with the maximum joint angle as the con-straints Combining with the redundancy Zanchettin et al[14] presented a method of optimizing the robotrsquos poses toenhance the stiffness when the robots perform drilling tasksFrom the above methods it can be found that the main ideaof the stiffness control method is to strengthen the EEoperation stiffness of the robots by changing or optimizingthe robotrsquos working poses [15] Nevertheless the precon-dition of optimizing the pose to improve the positioningaccuracy is that there is a contact force at the EE of robots)erefore when there are no contact forces eg paintingprealignment and measurement the stiffness controlmethod does not work

Based on the kinematic and dynamic relationship ofrobots the working space compensation methods can es-tablish a mathematical model that can describe the rela-tionship between the positioning error and end loads alongwith the gravity of the robot And it can be used to predictthe positioning error in the working space to improve thepositioning accuracy Salisbury proposed the traditionalstiffness model of robots based on the kinematic and statictheories [16] )e stiffness modelling for robots in theCartesian coordinates was studied by connecting the stiff-ness of each joint [17ndash19] Abele et al [11] provided astiffness model without calculating the inverse of Jacobianmatrix which can simplify the calculation of error com-pensation In [20] the stiffness matrix of joints was identifiedby measuring the EE displacement and rotation of robots)en the stiffness matrix in the Cartesian coordinates wasderived by the stiffness mapping model from the joint to EESun et al [21] proposed a method to calculate the EEtranslation stiffness for serial robots However according tothe experimental results this method is not accurate in termsof prediction of the positioning error Overall working space

compensation methods are simpler and more universal thanstructural strengthening methods and stiffness controlmethods in practice while the low accuracy limits its ap-plication )e main reason is that the current methods onlyconsider the rotary deformation around the axis of jointswhich does not contain the rotary deformation around theradial direction of joints

To improve the positioning accuracy a novel posi-tioning error compensation model based on the stiffnessfor industrial robots is proposed in this paper whichdescribes the relationship between the positioning errorand the EE load and gravity of robots First according tothe NewtonndashEuler method the driving torque for eachjoint is calculated )rough the deformation assumption ofjoints the torque is connected with the rotation defor-mation at joints Moreover the torque is also decomposedalong the axial and radial directions of joints which isconsistent with the axial deformation and radial defor-mation Positioning error models including the axial andradial deformation are established by means of kinematicsdifferential and recurrence methods respectively In ac-cordance with the small deformation assumption twokinds of positioning error models are linearized Finally acomplete error compensation model is derived accordingto the error models of axial and radial deformation whichcan effectively predict and compensate the positioningerror and improve the positioning accuracy for industrialrobots

)e major contributions of this paper include thefollowing

(1) A novel positioning error model is proposed for n-degree-of-freedom (DOF) industrial robots based onthe relationship between its kinematics and dy-namics parameters which can be applied to arbitrarymulti-DOF serial robots

(2) )e NewtonndashEuler method is introduced to calculatethe balance torque of joints which makes it con-venient to calculate the positioning error caused bythe EE loads and gravity of the robot

(3) )e radial deformation error at joints is modelledand included into the stiffness error model whichimproves the accuracy of prediction and compen-sation for positioning error compared to the tradi-tional methods

(4) )e positioning error model is linearized by intro-ducing proper assumptions which reduces thecomplexity of the proposed method and makes itconvenient to applications

)e rest of the paper is organized as follows in Section 2the displacement and deformation at joints of robots areanalyzed with the gravity and EE load In Section 3 six basicassumptions are provided and the positioning error ofrobots is derived In Section 4 the finite element analysis andthree-dimensional model of a general six-DOF industrialrobot are used to validate the effectiveness and correctness ofthe proposed method Finally conclusions for this paper aregiven in Section 5

2 Complexity



S(middot)

0 minusVz Vy

Vz 0 minusVx

minusVy Vx 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (1)



0 01113890 1113891

4times4 (2)







i+1_ωi+1

i+1i R

i_ωi +

i+1i R

i_ωi times _θi+1

i+1 1113954Zi+1 + euroθi+1i+1 1113954Zi+1

i+1_vi+1

i+1i R

i_ωi times

iPi+1 +

i_ωi times

i_ωi times

iPi+1 +

i_vi1113960 1113961

i+1_vCi+1

i+1

_ωi+1 timesi+1

PCi+1+


i+1PCi+1

+i+1

_vi+1i+1Fi+1 mi+1

i+1_vCi+1

i+1Ni+1 Ci+1Ii+1i+1 _ωi+1 +


i+1ωi+1

(3)

ifi

ii+1R

i+1fi+1 +

iFi (4)

ini

iNi +

ii+1R

i+1ni+1 +

iPCi

timesiFi +

iPi+1i times

ii+1R

i+1fi+1

(5)


i+1_vi+1

i+1i R

i_vi

i+1_vCi+1

i+1

_vi+1

i+1Fi+1 mi+1

i+1_vCi+1

(6)

ifi

ii+1R

i+1fi+1 +

iFi (7)

ini

ii+1R

i+1ni+1 +

iPCi

ini times

iFi +

iPi+1 times

ii+1R

i+1fi+1 (8)



Complexity 3















Δθ Ca na

Δc Cr nr

(11)


Md

Fd

M

x F

z

y

(a)

x

zMa

Mr

M

y

(b)


Δφ

G

F

(a)

G

Δl

F

(b)


4 Complexity






ini

inai +

inri (12)


Δθi Caiinai

Δci Criinri

(13)



d(0px)

d(0py)

d(0pz)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

dθ1

dθ1

⋮

dθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(14)


0Δxa

0Δya

0Δza

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Δθ1

Δθ1

⋮

ΔθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(15)


0ΔPa

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ca1

Ca1

⋮

CaN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)

Complexity 5





ini(z)) (17)


0ΔPa

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

times



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ca1

Ca1

⋮

CaN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)






RK(θ)




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


KA

KT + sθ middotA

K

(19)


RK(Δθ)

1 0 0

0 1 0

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0 minuskz ky

kz 0 minuskx

minusky kx 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦Δθ I3times3 + S(

AK )Δθ

(20)



1Pe1 R1n1

Δc1( 11138571Pe0

2Pe2 R2n2

Δc2( 11138572Pe1

3Pe3 R3n3

Δc3( 11138573Pe2

⋮

iPei Rini


⋮

NPeN RNnN

ΔcN( 1113857N

PeNminus1

(21)


6 Complexity

1ΔPr1 1Pe1 minus

1Pe0

2ΔPr2 2Pe2 minus

2Pe1

3ΔPr3 3Pe3 minus

3Pe2

⋮iΔPri

iPei minus

iPeiminus1

⋮NΔPrN

NPeN minus

NPeNminus1

(22)


0ΔPri 0i T

iΔPri (23)



i1113954nri)




iPeiminus1

iiminus1T

iminus1Peiminus1

iminus1Peiminus1


Peiminus2


iminus1nriminus1


(25)


iPeiminus1

iiminus1T




Peiminus21113960 1113961 (26)


iPeiminus1 1113944

iminus1

k1

ikT1113957S(minus

k1113954nrk)


i1T

1Pe0 (27)



i1113954nri) 1113944

iminus1

k1

ikT1113957S(minus

k1113954nnk)


i1T

1Pe0 middot Δci

⎧⎨

⎩

⎫⎬

⎭ (28)



i1113954nri)






i1113954nri1113872 1113873

iPe0 middot Δci

0Δxr0Δyr0Δzr

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i R

0Piorg

0 11113890 1113891

S minusi1113954nri1113872 1113873 00 0


0i R S minus

i1113954nri1113872 1113873


0Δxr0Δyr0Δzr

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i RS minus

i1113954nri1113872 1113873


0Pri(3times1)

0i RS minus

i1113954nri1113872 1113873


(30)


0Pr(3times1) 1113944

N

i1

0Pr(3times1) (31)


0Pr(3times1) 1113944

N

i1[0i RS(minus

i1113954nri)


inri (32)


written as


Cr1

Cr2

⋮

CrN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)


Complexity 7



0ΔP 0ΔPa +

0ΔPr (34)



Ca

Cr

1113890 11138912Ntimes1

(35)

where

0JPa

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



coli [0i RS(minus

i1113954nri)

iPe0]3times1



Λa



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Λr



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(37)


0ΔP Φ(FΘ)C (38)



T





4 Simulations


8 Complexity





iminus1i T




0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i T 1113945

i

k1

kminus1k T

ijT

i0T

0jT

(39)










simulation



ijT i

jR iPj

iPCimi




Z6

X6Z5X5

X4 Z4

Z3

X3

X2X1

Z1

Z2

Z0X0





Complexity 9







b1113888 1113889 times 100 (40)



Nodex y zValue

3893108e + 03 115e + 03 745mm374854356e ndash 02mm

UX (mm)142688081e ndash 01

130360141e ndash 01

118032187e ndash 01

105704241e ndash 01

933762938e ndash 02

810483471e ndash 02

687204003e ndash 02

563924611e ndash 02

440645143e ndash 02

317365676e ndash 02

194086209e ndash 02

708067603e ndash 03




















1113954Ca6 0 1113954Cr6 0

10 Complexity




0 50 100 150 200 250 300 350 400 450 500End load (N)

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

25

3

Rela

tive e

rror

()




2 4 6 8 10 12 14 16 18 20ndash4

ndash3

ndash2

ndash1

0

1

2


Rela

tive e

rror

()

Number of points



Complexity 11



5 Conclusion



Data Availability




Acknowledgments


References



























12 Complexity
































Complexity 13



S(middot)

0 minusVz Vy

Vz 0 minusVx

minusVy Vx 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (1)



0 01113890 1113891

4times4 (2)







i+1_ωi+1

i+1i R

i_ωi +

i+1i R

i_ωi times _θi+1

i+1 1113954Zi+1 + euroθi+1i+1 1113954Zi+1

i+1_vi+1

i+1i R

i_ωi times

iPi+1 +

i_ωi times

i_ωi times

iPi+1 +

i_vi1113960 1113961

i+1_vCi+1

i+1

_ωi+1 timesi+1

PCi+1+


i+1PCi+1

+i+1

_vi+1i+1Fi+1 mi+1

i+1_vCi+1

i+1Ni+1 Ci+1Ii+1i+1 _ωi+1 +


i+1ωi+1

(3)

ifi

ii+1R

i+1fi+1 +

iFi (4)

ini

iNi +

ii+1R

i+1ni+1 +

iPCi

timesiFi +

iPi+1i times

ii+1R

i+1fi+1

(5)


i+1_vi+1

i+1i R

i_vi

i+1_vCi+1

i+1

_vi+1

i+1Fi+1 mi+1

i+1_vCi+1

(6)

ifi

ii+1R

i+1fi+1 +

iFi (7)

ini

ii+1R

i+1ni+1 +

iPCi

ini times

iFi +

iPi+1 times

ii+1R

i+1fi+1 (8)



Complexity 3















Δθ Ca na

Δc Cr nr

(11)


Md

Fd

M

x F

z

y

(a)

x

zMa

Mr

M

y

(b)


Δφ

G

F

(a)

G

Δl

F

(b)


4 Complexity






ini

inai +

inri (12)


Δθi Caiinai

Δci Criinri

(13)



d(0px)

d(0py)

d(0pz)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

dθ1

dθ1

⋮

dθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(14)


0Δxa

0Δya

0Δza

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Δθ1

Δθ1

⋮

ΔθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(15)


0ΔPa

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ca1

Ca1

⋮

CaN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)

Complexity 5





ini(z)) (17)


0ΔPa

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

times



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ca1

Ca1

⋮

CaN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)






RK(θ)




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


KA

KT + sθ middotA

K

(19)


RK(Δθ)

1 0 0

0 1 0

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0 minuskz ky

kz 0 minuskx

minusky kx 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦Δθ I3times3 + S(

AK )Δθ

(20)



1Pe1 R1n1

Δc1( 11138571Pe0

2Pe2 R2n2

Δc2( 11138572Pe1

3Pe3 R3n3

Δc3( 11138573Pe2

⋮

iPei Rini


⋮

NPeN RNnN

ΔcN( 1113857N

PeNminus1

(21)


6 Complexity

1ΔPr1 1Pe1 minus

1Pe0

2ΔPr2 2Pe2 minus

2Pe1

3ΔPr3 3Pe3 minus

3Pe2

⋮iΔPri

iPei minus

iPeiminus1

⋮NΔPrN

NPeN minus

NPeNminus1

(22)


0ΔPri 0i T

iΔPri (23)



i1113954nri)




iPeiminus1

iiminus1T

iminus1Peiminus1

iminus1Peiminus1


Peiminus2


iminus1nriminus1


(25)


iPeiminus1

iiminus1T




Peiminus21113960 1113961 (26)


iPeiminus1 1113944

iminus1

k1

ikT1113957S(minus

k1113954nrk)


i1T

1Pe0 (27)



i1113954nri) 1113944

iminus1

k1

ikT1113957S(minus

k1113954nnk)


i1T

1Pe0 middot Δci

⎧⎨

⎩

⎫⎬

⎭ (28)



i1113954nri)






i1113954nri1113872 1113873

iPe0 middot Δci

0Δxr0Δyr0Δzr

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i R

0Piorg

0 11113890 1113891

S minusi1113954nri1113872 1113873 00 0


0i R S minus

i1113954nri1113872 1113873


0Δxr0Δyr0Δzr

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i RS minus

i1113954nri1113872 1113873


0Pri(3times1)

0i RS minus

i1113954nri1113872 1113873


(30)


0Pr(3times1) 1113944

N

i1

0Pr(3times1) (31)


0Pr(3times1) 1113944

N

i1[0i RS(minus

i1113954nri)


inri (32)


written as


Cr1

Cr2

⋮

CrN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)


Complexity 7



0ΔP 0ΔPa +

0ΔPr (34)



Ca

Cr

1113890 11138912Ntimes1

(35)

where

0JPa

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



coli [0i RS(minus

i1113954nri)

iPe0]3times1



Λa



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Λr



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(37)


0ΔP Φ(FΘ)C (38)



T





4 Simulations


8 Complexity





iminus1i T




0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i T 1113945

i

k1

kminus1k T

ijT

i0T

0jT

(39)










simulation



ijT i

jR iPj

iPCimi




Z6

X6Z5X5

X4 Z4

Z3

X3

X2X1

Z1

Z2

Z0X0





Complexity 9







b1113888 1113889 times 100 (40)



Nodex y zValue

3893108e + 03 115e + 03 745mm374854356e ndash 02mm

UX (mm)142688081e ndash 01

130360141e ndash 01

118032187e ndash 01

105704241e ndash 01

933762938e ndash 02

810483471e ndash 02

687204003e ndash 02

563924611e ndash 02

440645143e ndash 02

317365676e ndash 02

194086209e ndash 02

708067603e ndash 03




















1113954Ca6 0 1113954Cr6 0

10 Complexity




0 50 100 150 200 250 300 350 400 450 500End load (N)

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

25

3

Rela

tive e

rror

()




2 4 6 8 10 12 14 16 18 20ndash4

ndash3

ndash2

ndash1

0

1

2


Rela

tive e

rror

()

Number of points



Complexity 11



5 Conclusion



Data Availability




Acknowledgments


References



























12 Complexity
































Complexity 13















Δθ Ca na

Δc Cr nr

(11)


Md

Fd

M

x F

z

y

(a)

x

zMa

Mr

M

y

(b)


Δφ

G

F

(a)

G

Δl

F

(b)


4 Complexity






ini

inai +

inri (12)


Δθi Caiinai

Δci Criinri

(13)



d(0px)

d(0py)

d(0pz)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

dθ1

dθ1

⋮

dθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(14)


0Δxa

0Δya

0Δza

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Δθ1

Δθ1

⋮

ΔθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(15)


0ΔPa

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ca1

Ca1

⋮

CaN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)

Complexity 5





ini(z)) (17)


0ΔPa

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

times



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ca1

Ca1

⋮

CaN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)






RK(θ)




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


KA

KT + sθ middotA

K

(19)


RK(Δθ)

1 0 0

0 1 0

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0 minuskz ky

kz 0 minuskx

minusky kx 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦Δθ I3times3 + S(

AK )Δθ

(20)



1Pe1 R1n1

Δc1( 11138571Pe0

2Pe2 R2n2

Δc2( 11138572Pe1

3Pe3 R3n3

Δc3( 11138573Pe2

⋮

iPei Rini


⋮

NPeN RNnN

ΔcN( 1113857N

PeNminus1

(21)


6 Complexity

1ΔPr1 1Pe1 minus

1Pe0

2ΔPr2 2Pe2 minus

2Pe1

3ΔPr3 3Pe3 minus

3Pe2

⋮iΔPri

iPei minus

iPeiminus1

⋮NΔPrN

NPeN minus

NPeNminus1

(22)


0ΔPri 0i T

iΔPri (23)



i1113954nri)




iPeiminus1

iiminus1T

iminus1Peiminus1

iminus1Peiminus1


Peiminus2


iminus1nriminus1


(25)


iPeiminus1

iiminus1T




Peiminus21113960 1113961 (26)


iPeiminus1 1113944

iminus1

k1

ikT1113957S(minus

k1113954nrk)


i1T

1Pe0 (27)



i1113954nri) 1113944

iminus1

k1

ikT1113957S(minus

k1113954nnk)


i1T

1Pe0 middot Δci

⎧⎨

⎩

⎫⎬

⎭ (28)



i1113954nri)






i1113954nri1113872 1113873

iPe0 middot Δci

0Δxr0Δyr0Δzr

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i R

0Piorg

0 11113890 1113891

S minusi1113954nri1113872 1113873 00 0


0i R S minus

i1113954nri1113872 1113873


0Δxr0Δyr0Δzr

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i RS minus

i1113954nri1113872 1113873


0Pri(3times1)

0i RS minus

i1113954nri1113872 1113873


(30)


0Pr(3times1) 1113944

N

i1

0Pr(3times1) (31)


0Pr(3times1) 1113944

N

i1[0i RS(minus

i1113954nri)


inri (32)


written as


Cr1

Cr2

⋮

CrN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)


Complexity 7



0ΔP 0ΔPa +

0ΔPr (34)



Ca

Cr

1113890 11138912Ntimes1

(35)

where

0JPa

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



coli [0i RS(minus

i1113954nri)

iPe0]3times1



Λa



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Λr



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(37)


0ΔP Φ(FΘ)C (38)



T





4 Simulations


8 Complexity





iminus1i T




0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i T 1113945

i

k1

kminus1k T

ijT

i0T

0jT

(39)










simulation



ijT i

jR iPj

iPCimi




Z6

X6Z5X5

X4 Z4

Z3

X3

X2X1

Z1

Z2

Z0X0





Complexity 9







b1113888 1113889 times 100 (40)



Nodex y zValue

3893108e + 03 115e + 03 745mm374854356e ndash 02mm

UX (mm)142688081e ndash 01

130360141e ndash 01

118032187e ndash 01

105704241e ndash 01

933762938e ndash 02

810483471e ndash 02

687204003e ndash 02

563924611e ndash 02

440645143e ndash 02

317365676e ndash 02

194086209e ndash 02

708067603e ndash 03




















1113954Ca6 0 1113954Cr6 0

10 Complexity




0 50 100 150 200 250 300 350 400 450 500End load (N)

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

25

3

Rela

tive e

rror

()




2 4 6 8 10 12 14 16 18 20ndash4

ndash3

ndash2

ndash1

0

1

2


Rela

tive e

rror

()

Number of points



Complexity 11



5 Conclusion



Data Availability




Acknowledgments


References



























12 Complexity
































Complexity 13






ini

inai +

inri (12)


Δθi Caiinai

Δci Criinri

(13)



d(0px)

d(0py)

d(0pz)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

dθ1

dθ1

⋮

dθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(14)


0Δxa

0Δya

0Δza

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Δθ1

Δθ1

⋮

ΔθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(15)


0ΔPa

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ca1

Ca1

⋮

CaN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)

Complexity 5





ini(z)) (17)


0ΔPa

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

times



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ca1

Ca1

⋮

CaN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)






RK(θ)




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


KA

KT + sθ middotA

K

(19)


RK(Δθ)

1 0 0

0 1 0

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0 minuskz ky

kz 0 minuskx

minusky kx 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦Δθ I3times3 + S(

AK )Δθ

(20)



1Pe1 R1n1

Δc1( 11138571Pe0

2Pe2 R2n2

Δc2( 11138572Pe1

3Pe3 R3n3

Δc3( 11138573Pe2

⋮

iPei Rini


⋮

NPeN RNnN

ΔcN( 1113857N

PeNminus1

(21)


6 Complexity

1ΔPr1 1Pe1 minus

1Pe0

2ΔPr2 2Pe2 minus

2Pe1

3ΔPr3 3Pe3 minus

3Pe2

⋮iΔPri

iPei minus

iPeiminus1

⋮NΔPrN

NPeN minus

NPeNminus1

(22)


0ΔPri 0i T

iΔPri (23)



i1113954nri)




iPeiminus1

iiminus1T

iminus1Peiminus1

iminus1Peiminus1


Peiminus2


iminus1nriminus1


(25)


iPeiminus1

iiminus1T




Peiminus21113960 1113961 (26)


iPeiminus1 1113944

iminus1

k1

ikT1113957S(minus

k1113954nrk)


i1T

1Pe0 (27)



i1113954nri) 1113944

iminus1

k1

ikT1113957S(minus

k1113954nnk)


i1T

1Pe0 middot Δci

⎧⎨

⎩

⎫⎬

⎭ (28)



i1113954nri)






i1113954nri1113872 1113873

iPe0 middot Δci

0Δxr0Δyr0Δzr

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i R

0Piorg

0 11113890 1113891

S minusi1113954nri1113872 1113873 00 0


0i R S minus

i1113954nri1113872 1113873


0Δxr0Δyr0Δzr

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i RS minus

i1113954nri1113872 1113873


0Pri(3times1)

0i RS minus

i1113954nri1113872 1113873


(30)


0Pr(3times1) 1113944

N

i1

0Pr(3times1) (31)


0Pr(3times1) 1113944

N

i1[0i RS(minus

i1113954nri)


inri (32)


written as


Cr1

Cr2

⋮

CrN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)


Complexity 7



0ΔP 0ΔPa +

0ΔPr (34)



Ca

Cr

1113890 11138912Ntimes1

(35)

where

0JPa

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



coli [0i RS(minus

i1113954nri)

iPe0]3times1



Λa



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Λr



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(37)


0ΔP Φ(FΘ)C (38)



T





4 Simulations


8 Complexity





iminus1i T




0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i T 1113945

i

k1

kminus1k T

ijT

i0T

0jT

(39)










simulation



ijT i

jR iPj

iPCimi




Z6

X6Z5X5

X4 Z4

Z3

X3

X2X1

Z1

Z2

Z0X0





Complexity 9







b1113888 1113889 times 100 (40)



Nodex y zValue

3893108e + 03 115e + 03 745mm374854356e ndash 02mm

UX (mm)142688081e ndash 01

130360141e ndash 01

118032187e ndash 01

105704241e ndash 01

933762938e ndash 02

810483471e ndash 02

687204003e ndash 02

563924611e ndash 02

440645143e ndash 02

317365676e ndash 02

194086209e ndash 02

708067603e ndash 03




















1113954Ca6 0 1113954Cr6 0

10 Complexity




0 50 100 150 200 250 300 350 400 450 500End load (N)

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

25

3

Rela

tive e

rror

()




2 4 6 8 10 12 14 16 18 20ndash4

ndash3

ndash2

ndash1

0

1

2


Rela

tive e

rror

()

Number of points



Complexity 11



5 Conclusion



Data Availability




Acknowledgments


References



























12 Complexity
































Complexity 13





ini(z)) (17)


0ΔPa

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

times



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ca1

Ca1

⋮

CaN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)






RK(θ)




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


KA

KT + sθ middotA

K

(19)


RK(Δθ)

1 0 0

0 1 0

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0 minuskz ky

kz 0 minuskx

minusky kx 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦Δθ I3times3 + S(

AK )Δθ

(20)



1Pe1 R1n1

Δc1( 11138571Pe0

2Pe2 R2n2

Δc2( 11138572Pe1

3Pe3 R3n3

Δc3( 11138573Pe2

⋮

iPei Rini


⋮

NPeN RNnN

ΔcN( 1113857N

PeNminus1

(21)


6 Complexity

1ΔPr1 1Pe1 minus

1Pe0

2ΔPr2 2Pe2 minus

2Pe1

3ΔPr3 3Pe3 minus

3Pe2

⋮iΔPri

iPei minus

iPeiminus1

⋮NΔPrN

NPeN minus

NPeNminus1

(22)


0ΔPri 0i T

iΔPri (23)



i1113954nri)




iPeiminus1

iiminus1T

iminus1Peiminus1

iminus1Peiminus1


Peiminus2


iminus1nriminus1


(25)


iPeiminus1

iiminus1T




Peiminus21113960 1113961 (26)


iPeiminus1 1113944

iminus1

k1

ikT1113957S(minus

k1113954nrk)


i1T

1Pe0 (27)



i1113954nri) 1113944

iminus1

k1

ikT1113957S(minus

k1113954nnk)


i1T

1Pe0 middot Δci

⎧⎨

⎩

⎫⎬

⎭ (28)



i1113954nri)






i1113954nri1113872 1113873

iPe0 middot Δci

0Δxr0Δyr0Δzr

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i R

0Piorg

0 11113890 1113891

S minusi1113954nri1113872 1113873 00 0


0i R S minus

i1113954nri1113872 1113873


0Δxr0Δyr0Δzr

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i RS minus

i1113954nri1113872 1113873


0Pri(3times1)

0i RS minus

i1113954nri1113872 1113873


(30)


0Pr(3times1) 1113944

N

i1

0Pr(3times1) (31)


0Pr(3times1) 1113944

N

i1[0i RS(minus

i1113954nri)


inri (32)


written as


Cr1

Cr2

⋮

CrN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)


Complexity 7



0ΔP 0ΔPa +

0ΔPr (34)



Ca

Cr

1113890 11138912Ntimes1

(35)

where

0JPa

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



coli [0i RS(minus

i1113954nri)

iPe0]3times1



Λa



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Λr



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(37)


0ΔP Φ(FΘ)C (38)



T





4 Simulations


8 Complexity





iminus1i T




0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i T 1113945

i

k1

kminus1k T

ijT

i0T

0jT

(39)










simulation



ijT i

jR iPj

iPCimi




Z6

X6Z5X5

X4 Z4

Z3

X3

X2X1

Z1

Z2

Z0X0





Complexity 9







b1113888 1113889 times 100 (40)



Nodex y zValue

3893108e + 03 115e + 03 745mm374854356e ndash 02mm

UX (mm)142688081e ndash 01

130360141e ndash 01

118032187e ndash 01

105704241e ndash 01

933762938e ndash 02

810483471e ndash 02

687204003e ndash 02

563924611e ndash 02

440645143e ndash 02

317365676e ndash 02

194086209e ndash 02

708067603e ndash 03




















1113954Ca6 0 1113954Cr6 0

10 Complexity




0 50 100 150 200 250 300 350 400 450 500End load (N)

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

25

3

Rela

tive e

rror

()




2 4 6 8 10 12 14 16 18 20ndash4

ndash3

ndash2

ndash1

0

1

2


Rela

tive e

rror

()

Number of points



Complexity 11



5 Conclusion



Data Availability




Acknowledgments


References



























12 Complexity
































Complexity 13

1ΔPr1 1Pe1 minus

1Pe0

2ΔPr2 2Pe2 minus

2Pe1

3ΔPr3 3Pe3 minus

3Pe2

⋮iΔPri

iPei minus

iPeiminus1

⋮NΔPrN

NPeN minus

NPeNminus1

(22)


0ΔPri 0i T

iΔPri (23)



i1113954nri)




iPeiminus1

iiminus1T

iminus1Peiminus1

iminus1Peiminus1


Peiminus2


iminus1nriminus1


(25)


iPeiminus1

iiminus1T




Peiminus21113960 1113961 (26)


iPeiminus1 1113944

iminus1

k1

ikT1113957S(minus

k1113954nrk)


i1T

1Pe0 (27)



i1113954nri) 1113944

iminus1

k1

ikT1113957S(minus

k1113954nnk)


i1T

1Pe0 middot Δci

⎧⎨

⎩

⎫⎬

⎭ (28)



i1113954nri)






i1113954nri1113872 1113873

iPe0 middot Δci

0Δxr0Δyr0Δzr

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i R

0Piorg

0 11113890 1113891

S minusi1113954nri1113872 1113873 00 0


0i R S minus

i1113954nri1113872 1113873


0Δxr0Δyr0Δzr

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i RS minus

i1113954nri1113872 1113873


0Pri(3times1)

0i RS minus

i1113954nri1113872 1113873


(30)


0Pr(3times1) 1113944

N

i1

0Pr(3times1) (31)


0Pr(3times1) 1113944

N

i1[0i RS(minus

i1113954nri)


inri (32)


written as


Cr1

Cr2

⋮

CrN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)


Complexity 7



0ΔP 0ΔPa +

0ΔPr (34)



Ca

Cr

1113890 11138912Ntimes1

(35)

where

0JPa

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



coli [0i RS(minus

i1113954nri)

iPe0]3times1



Λa



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Λr



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(37)


0ΔP Φ(FΘ)C (38)



T





4 Simulations


8 Complexity





iminus1i T




0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i T 1113945

i

k1

kminus1k T

ijT

i0T

0jT

(39)










simulation



ijT i

jR iPj

iPCimi




Z6

X6Z5X5

X4 Z4

Z3

X3

X2X1

Z1

Z2

Z0X0





Complexity 9







b1113888 1113889 times 100 (40)



Nodex y zValue

3893108e + 03 115e + 03 745mm374854356e ndash 02mm

UX (mm)142688081e ndash 01

130360141e ndash 01

118032187e ndash 01

105704241e ndash 01

933762938e ndash 02

810483471e ndash 02

687204003e ndash 02

563924611e ndash 02

440645143e ndash 02

317365676e ndash 02

194086209e ndash 02

708067603e ndash 03




















1113954Ca6 0 1113954Cr6 0

10 Complexity




0 50 100 150 200 250 300 350 400 450 500End load (N)

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

25

3

Rela

tive e

rror

()




2 4 6 8 10 12 14 16 18 20ndash4

ndash3

ndash2

ndash1

0

1

2


Rela

tive e

rror

()

Number of points



Complexity 11



5 Conclusion



Data Availability




Acknowledgments


References



























12 Complexity
































Complexity 13



0ΔP 0ΔPa +

0ΔPr (34)



Ca

Cr

1113890 11138912Ntimes1

(35)

where

0JPa

z0px

zθ1

z0px


z0px

zθN

z0py

zθ1

z0py


z0py

zθN

z0pz

zθ1

z0pz


z0pz

zθN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



coli [0i RS(minus

i1113954nri)

iPe0]3times1



Λa



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Λr



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(37)


0ΔP Φ(FΘ)C (38)



T





4 Simulations


8 Complexity





iminus1i T




0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i T 1113945

i

k1

kminus1k T

ijT

i0T

0jT

(39)










simulation



ijT i

jR iPj

iPCimi




Z6

X6Z5X5

X4 Z4

Z3

X3

X2X1

Z1

Z2

Z0X0





Complexity 9







b1113888 1113889 times 100 (40)



Nodex y zValue

3893108e + 03 115e + 03 745mm374854356e ndash 02mm

UX (mm)142688081e ndash 01

130360141e ndash 01

118032187e ndash 01

105704241e ndash 01

933762938e ndash 02

810483471e ndash 02

687204003e ndash 02

563924611e ndash 02

440645143e ndash 02

317365676e ndash 02

194086209e ndash 02

708067603e ndash 03




















1113954Ca6 0 1113954Cr6 0

10 Complexity




0 50 100 150 200 250 300 350 400 450 500End load (N)

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

25

3

Rela

tive e

rror

()




2 4 6 8 10 12 14 16 18 20ndash4

ndash3

ndash2

ndash1

0

1

2


Rela

tive e

rror

()

Number of points



Complexity 11



5 Conclusion



Data Availability




Acknowledgments


References



























12 Complexity
































Complexity 13





iminus1i T




0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0i T 1113945

i

k1

kminus1k T

ijT

i0T

0jT

(39)










simulation



ijT i

jR iPj

iPCimi




Z6

X6Z5X5

X4 Z4

Z3

X3

X2X1

Z1

Z2

Z0X0





Complexity 9







b1113888 1113889 times 100 (40)



Nodex y zValue

3893108e + 03 115e + 03 745mm374854356e ndash 02mm

UX (mm)142688081e ndash 01

130360141e ndash 01

118032187e ndash 01

105704241e ndash 01

933762938e ndash 02

810483471e ndash 02

687204003e ndash 02

563924611e ndash 02

440645143e ndash 02

317365676e ndash 02

194086209e ndash 02

708067603e ndash 03




















1113954Ca6 0 1113954Cr6 0

10 Complexity




0 50 100 150 200 250 300 350 400 450 500End load (N)

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

25

3

Rela

tive e

rror

()




2 4 6 8 10 12 14 16 18 20ndash4

ndash3

ndash2

ndash1

0

1

2


Rela

tive e

rror

()

Number of points



Complexity 11



5 Conclusion



Data Availability




Acknowledgments


References



























12 Complexity
































Complexity 13







b1113888 1113889 times 100 (40)



Nodex y zValue

3893108e + 03 115e + 03 745mm374854356e ndash 02mm

UX (mm)142688081e ndash 01

130360141e ndash 01

118032187e ndash 01

105704241e ndash 01

933762938e ndash 02

810483471e ndash 02

687204003e ndash 02

563924611e ndash 02

440645143e ndash 02

317365676e ndash 02

194086209e ndash 02

708067603e ndash 03




















1113954Ca6 0 1113954Cr6 0

10 Complexity




0 50 100 150 200 250 300 350 400 450 500End load (N)

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

25

3

Rela

tive e

rror

()




2 4 6 8 10 12 14 16 18 20ndash4

ndash3

ndash2

ndash1

0

1

2


Rela

tive e

rror

()

Number of points



Complexity 11



5 Conclusion



Data Availability




Acknowledgments


References



























12 Complexity
































Complexity 13




0 50 100 150 200 250 300 350 400 450 500End load (N)

ndash25

ndash2

ndash15

ndash1

ndash05

0

05

1

15

2

25

3

Rela

tive e

rror

()




2 4 6 8 10 12 14 16 18 20ndash4

ndash3

ndash2

ndash1

0

1

2


Rela

tive e

rror

()

Number of points



Complexity 11



5 Conclusion



Data Availability




Acknowledgments


References



























12 Complexity
































Complexity 13



5 Conclusion



Data Availability




Acknowledgments


References



























12 Complexity
































Complexity 13
































Complexity 13

positioning error compensation for industrial robots based

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