inverse kinematics of a 7r 6-dof robot with nonspherical...

Research ArticleInverse Kinematics of a 7R 6-DOF Robot with NonsphericalWrist Based on Transformation into the 6R Robot

XuhaoWang Dawei Zhang and Chen Zhao

Key Laboratory of MechanismTheory and Equipment Design of Ministry of Education School of Mechanical EngineeringTianjin University Tianjin 300072 China

Correspondence should be addressed to Chen Zhao zhaochentjueducn

Received 23 December 2016 Revised 13 April 2017 Accepted 26 April 2017 Published 18 May 2017

Academic Editor Oscar Reinoso

Copyright copy 2017 Xuhao Wang et al This 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

The 7R 6-DOF robots with hollow nonspherical wrist have been provenmore suitable for spray painting applications However theinverse kinematics of this kind of robot is still imperfect due to the coupling between position and orientation of the end-effector(EE) In this paper a new and efficient algorithm for the inverse kinematics of a 7R 6-DOF robot is proposedThe proposed inversekinematics algorithm is a two-stepmethodThe geometry of the 7R 6-DOF robot is analyzed A comparison between the 7R 6-DOFrobot and the well-known equivalent 6R robot is made Based on this comparison a rational transformation between these twokinds of robots is constructed Then the general inverse kinematics algorithm of the equivalent 6R robot is applied to calculate theapproximate solutions of the 7R 6-DOF robot in the first step The Damped Least-Squares (DLS) method is employed to derivethe exact solutions in the second step The accuracy and efficiency of the algorithm are tested on a 7R 6-DOF painting robot Theresults show that the proposed algorithm is more advantageous in the case without an approximate solution such as the initialpoint of a continuous trajectory

1 Introduction

In robotics inverse kinematics is one of the most traditionalresearch areas It is necessary for robot design trajectoryplanning and dynamic analysis of robots There are mainlytwo types of inverse kinematic techniques namely analyticalmethods and numerical methods Analytical solutions existonly for some special geometric structure that is threeadjacent axes intersect at one point or parallel to each other[1]The robot with a spherical wrist is a good example As theposition and orientation of the end-effector (EE) are deter-mined respectively by the first three joints and the last threejoints which is convenient to control and teaching sphericalwrists are widely used in industrial robots In this paper theserial 6R manipulator with a spherical wrist is called equiva-lent 6R robot Many efficient inverse kinematics methods [2ndash4] have been presented for the equivalent 6R robot Howeverin some industrial applications such as spray painting the 7R6-DOF robot with hollow nonspherical wrist has been provedto be advantageous due to wider range of motion of thewrist As shown in Figure 1 the wrist of 7R 6-DOF robot has

4 revolute joints the second and third of which are coupledwith the relation 1205796 = minus1205795 It means that an extra revolutejoint is added to enlarge the range ofmotion of the wrist Andin order to avoid introduction of redundancy a constraintis introduced on the second and third revolute joints of thewrist resulting in a couple joint Figure 2 is the configurationof the 7R 6-DOF robot

Because of the nonspherical wrist the analytical solutionof the 7R 6-DOF robot is nonexistent In order to getinverse kinematics of the robots with nonspherical wrist Tsaiand Morgan [5] proposed a higher dimensional approachwith eight second-degree equations As an improvementRaghavan and Roth [6] used dialytic elimination to derivea 16 degree polynomial Manocha and Canny [7] proposedsymbolic preprocessing and matrix computations to convertthe inverse kinematics to an eigenvalue problem In recentyears [8ndash12] have studied the inverse kinematics of general6R robots However the problem is that these methods canonly be applied to 6R robots In some references themethodsbased on heuristic search techniques such as neural networksolution [13 14] genetic algorithms [15 16] and simulated

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 2074137 12 pageshttpsdoiorg10115520172074137

2 Mathematical Problems in Engineering

Joint 1

Joint 7

Joint 6 Joint 5

Joint 4

Joint 3

Joint 2

The wrist

Figure 1 Prototype of a 7R 6-DOF robot

X3

Y3

Z3

Z4

d5X2

X4

Y2

Z1

Z2

d4

a2

X7

X6

X0

X5

X1

Y6

Y7

Y5

Y0Y1

Y4

Z0

Z7

Z6

Z5

d6

d7

Figure 2 Configuration of the 7R 6-DOF robot

annealing [17] are developed for the solution of inversekinematics problem These methods convert the kinematicsproblem into an equivalentminimization problem and gener-ally suffer from time-consuming and low accuracy For serialrobots the numerical iterative techniques such as Newton-Raphson approach [18] the steepest descent approach [19]and the Damped Least-Squares (DLS) approach [20 21]are often applied The drawbacks of a numerical iterativealgorithm are slow iterations and sensitivity to the initialvalue and normally just one solution instead of all solutionscan be derived

For the 7R 6-DOF robot Wu et al [22] proposed a two-step method the approximate analytical solutions are firstlyderived through solving the 7R robot with spherical wristby introducing a virtual wrist center and the Levenberg-Marquardt (LM) method is used to calculate the exactsolutions This is an interesting approach but a complexpolynomial system needs to be solved in the first step whichis time-consuming In this paper a new and efficient two-step method is presented As the major improvement arational transformation between the 7R 6-DOF robot and

X3

Y3

Z3

X2

X4

Y2

Z1

Z2

X6

X0

X5

X1

Y6

Y5

Y0

Y1

Y4

Z0

Z4

Z6

Z5

deq6

deq4

aeq2

Figure 3 Configuration of the equivalent 6R robot

the well-known equivalent 6R robot is constructed Thegeneral inverse kinematics algorithm of the equivalent 6Rrobot is used to calculate the approximate solutions of the7R 6-DOF robot in the first step Then a general iterativealgorithm that is the DLS method is employed to get theexact solutions The approximate solutions derived from thefirst step can offer good initial value to the DLS method andmake it computationally efficient To verify the accuracy andefficiency of this method three simulations are implemented

The paper is organized in the following manner InSection 2 the inverse kinematics of the equivalent 6R robotis briefly reviewed Section 3 describes the efficient transfor-mation between the 7R 6-DOF robot and the equivalent 6Rrobot where a rational assumptionwill be given In Section 4the DLS method is reviewed and the new inverse kinematicsalgorithm is presented In Section 5 three simulations areimplemented to illustrate the accuracy and efficiency of theproposed method The results are presented and discussedSection 6 is the conclusion

2 Inverse Kinematics Algorithm ofthe Well-Known Equivalent 6R Robot

As stated previously the equivalent 6R robot is a well-knowntypical structure of serial robots The inverse kinematicsmethods have been proposed by [2ndash4] Before deriving theinverse kinematics of the 7R 6-DOF robot firstly we brieflydescribe that of the equivalent 6R robot For the equivalent6R robot the configuration is shown in Figure 3 For serialrobots DenavitndashHartenberg (DH) parameters are widelyused to describe the position and orientation of the EE Thetransformation matrix relating the joint 119894 to joint 119894 minus 1 couldbe given by

119894minus1119894T = [[[[[

[

c119894 minuss119894 0 119886119894minus1s119894c120572119894minus1 c119894c120572119894minus1 minuss120572119894minus1 minus119889119894s120572119894minus1s119894s120572119894minus1 c119894s120572119894minus1 c120572119894minus1 119889119894c120572119894minus10 0 0 1

]]]]]] (1)

where s119894 = sin 120579119894 c119894 = cos 120579119894 c120572119894 = cos120572119894 and s120572119894 = sin120572119894

Mathematical Problems in Engineering 3

Then the forward kinematics of the manipulator could beformulated by01T (1205791) 12T (1205792) 23T (1205793) 34T (1205794) 45T (1205795) 56T (1205796) = Tend (2)

where Tend is the configuration of the EE with respect to thebase frame For the inverse kinematics Tend is known anddescribed by (3) where n o and a are three unit orientationvectors and p is the position vector of Tend

Tend = [n o a p0 0 0 1] =

[[[[[[

119899119909 119900119909 119886119909 119901119909119899119910 119900119910 119886119910 119901119910119899119911 119900119911 119886119911 1199011199110 0 0 1

]]]]]] (3)

As a result the inverse kinematics problem is to calculatethe joint angles 120579119894 through the matrix (2) To derive thesolutions (2) is firstly rearranged as

12T (1205792) 23T (1205793) 34T (1205794) 45T (1205795) 56T (1205796)= 01T (1205791)minus1 Tend (4)

From the last column of both sides of (4) we obtain

1198881119901119909 + 1199041119901119910 = 119889411990423 + 11988621198882 (5)minus1199041119901119909 + 1198881119901119910 = 0 (6)

119901119911 = minus119889411988823 + 11988621199042 (7)

From (6) two values of 1205791are obtained as

1205791 = atan 2 (119901119910 119901119909) 12057910158401 =

1205791 minus 119901119894 (1205791 gt 0) 1205791 + 119901119894 (1205791 le 0)

(8)

Calculating the sum of the squares of (5) (6) and (7) onboth sides we obtain

2119886211988941199043 = 1199012119909 + 1199012119910 + 1199012119911 minus 11988924 minus 11988622 (9)

The joint angle 1205793 is obtained as

1205793 = 2 atan((1 plusmn radic1 minus 1198962)119896 ) (10)

where 119896 = (119901119909 + 119901119910 + 119901119911 minus 1198894 minus 1198862)211988621198894In order to derive 1205792 rearrange (2) as

34T (1205794) 45T (1205795) 56T (1205796)= 23T (1205793)minus1 12T (1205792)minus1 01T (1205791)minus1 Tend (11)

Then 1205792 can be obtained by equating (1 4) and (2 4)matrix elements of each sides in (11)

1205792 = atan 2 (11990111991111988621198883 + 1198962 (11988621199043 + 1198894) 1198961 + 119896211988621198883)minus 1205793 (12)

where 1198961 = minus119901z1198894 minus 11990111991111988621199043 and 1198962 = 1199011199091198881 + 1199011199101199041

Equate (1 3) and (3 3) matrix elements of each sides in(11) we obtain

1205794 = atan 2 (1198861199091199041 minus 1198861199101198881 11988611991111990423 + 119886119910119888231199041 + 119886119909119888231198881) (13)

Similarly 1205795 and 1205796 can be obtained Following thecalculation above 8 solutions in total could be obtained

3 Transformation between the 7R 6-DOFRobot and Equivalent 6R Robot

In order to apply above algorithm to the inverse kinematicsof the 7R 6-DOF robot firstly the geometry of the 7R 6-DOFrobot is analyzedThe transformation between the 7R 6-DOFrobot and the equivalent 6R robot is constructed based on acomparison

As shown in Figure 1 the 7R 6-DOF robot consists of 7revolute joints Configuration of the first three joints that is119877 perp 119877119877 is widely used in industrial robots The last fourjoints construct the nonspherical wrist From the comparisonbetween the 7R 6-DOF robot and the equivalent 6R robotwhich is shown in Figure 3 configurations of the first threejoints are the same and the wrists are different In order toconstruct the transformation between the two kinds of robotthe first three joint angles are firstly set to be equal describedby (14) Then the transformation between the wrists can bederived separately Geometricalmodels of thewrists are givenin Figures 4 and 5 respectively The center of joint 3 is takenas the origin of the base frame For simplicity the notation4R wrist and equivalent 3R wrist are used in the followingtext As for the 4R wrist because the couple joint is the maindifference the kinematic analysis of the couple joint is firstlyexecuted

1205791 = 1205791eq1205792 = 1205792eq1205793 = 1205793eq

(14)

Figure 4 shows that the couple joint consists of twocoupled revolute joints As a result of rotating the couplejoint over an angle 1205795 axis 4 and axis 7 revolve around axis5 and axis 6 respectively And axis 4 and axis 7 intersect at achanging point 1198751015840119888 Ignoring the change of position a virtualaxis which is perpendicular to axis 4 and axis 7 could beconstructed at the point1198751015840119888 As a result a virtual equivalent 3Rwrist can be obtained as shown in Figure 5 Taking the changeof position into account an additional relation is given by(15) which could be derived in triangle O4O5O6 and triangleO41198751015840119888O6

119889eq4 = 1198894 + 1198895 cos (120573)cos (120579eq5 2)

119889eq6 = 1198897 + 1198895 cos (120573)cos (120579eq5 2)

(15)


Axis 5 Axis 6

Axis 7Axis 4

Virtual axis

Couple joint

120573120573

1205795

1205794

1205797

1205796

Z4

Z7

Z5

Z3

Y3

Z6

Y7

Y5

Y6Y4

03

07

06

05

04

Pc2

Pc1

Pc

P998400c

Figure 4 Configuration of the 4R wrist

Z5

Z6Z3Z4

Y6Y3Y4

Y5120579eq4 120579

eq6

120579eq5

0605 0403

Figure 5 Configuration of the equivalent 3R wrist

Based on DH parameters configuration of the EE withrespect to the base frame is given by

Tend4r = 34T 45T 56T 67TTend3r = 34Teq 4

5Teq 56T

eq (16)

The equivalent 3R wrist must generate the same configu-ration of EE as the 4R wrist meaning the following equation

Tend4r = Tend3r (17)

By equating the matrix elements of each side in (17) thetransformation between the twowrists could be obtained Butit is not a simple relation Given this situation the orientationof the EE with respect to the base frame is considered solelywhich is efficient for the 3-DOF wrist [23] Then (17) issimplified as

119877 (119911 1205794) 119877 (119909 1205724) 119877 (119911 1205795) 119877 (119909 1205725) 119877 (119911 1205796) 119877 (119909 1205726)sdot 119877 (119911 1205797) = 119877 (119911 120579eq4 ) 119877 (119909 120572eq4 ) 119877 (119911 120579eq5 )sdot 119877 (119909 120572eq5 ) 119877 (119911 120579eq6 )

(18)

Because the couple joint is the main difference we firstlykeep joint 4 and joint 7 at the initial position that is

1205794 = 1205797 = 0 By equating the matrix elements of each sidein (18) we obtain

c1205726s1205725s5 minus s1205726 (c5s5 minus c1205725c5s5) = c4eqs5eq (19)

minus s1205726 (c1205724s25 + c1205724c1205725c25 minus s1205724s1205725c5)

minus c1205726 (c1205725s1205724 + c1205724s1205725c5) = s4eqs5eq (20)

c1205726 (c1205724c1205725 minus s1205724s1205725c5)minus s1205726 (c1205725s1205724c25 + c1205724s1205725c5 + s1205724s

25) = c5eq (21)

Then 1205795eq is derived by substituting c5eq = (1minus1199092)(1+1199092)into (21) where 119909 = tan(1205795eq2)

1205795eq = 2 atan(plusmnradic1 minus 1198601 + 119860) (22)

where 119860 = c1205726(c1205724c1205725 minus s1205724s1205725c5) minus s1205726(c1205725s1205724c25 + c1205724s1205725c5 +s1205724s25)When s5eq = 0 we can obtain the value of 1205794eq with (19)

and (20)

1205794eq = atan 2 (119861 119862) (23)

where

119861 = sign (s5eq) (minuss1205726 (c1205724s25 + c1205724c1205725c25 minus s1205724s1205725c5)

minus c1205726 (c1205725s1205724 + c1205724s1205725c5)) 119862 = sign (s5eq) (c1205726s1205725s5 minus s1205726 (c5s5 minus c1205725c5s5))

(24)

According to the property of the wrist 1205796eq is equal tominus1205794eq that is 1205796eq = minus1205794eq Hence taking the displacementof joint 4 and joint 7 into account the value of 1205794eq and 1205796eqcan be obtained

1205794eq = atan 2 (119861 119862) + 12057941205796eq = minusatan 2 (119861 119862) + 1205797 (25)


Axis 5 Axis 6

Axis 7Axis 4

Virtual axis

04

05

06120573120573

d

12057961205795

Pc2

Pc1

Pc

P998400c

Figure 6 Transformation error described in the 4R wrist

As for the inverse transformation substitute c5 = (1 minus119910)(1 + 119910) into (21) Where 119910 = (tan(12057952))2 we can obtain

(119882 minus 119880 + 119881) 1199102 + 2 (119881 minus119882)119910 +119882 + 119880 + 119881 = 0 (26)

where119882 = s1205724s1205726 minus s1205726c1205725s1205724 119880 = minuss1205725(c1205726s1205724 + c1205724s1205726) and119881 = c1205724c1205725c1205726 minus s1205724s1205726 minus c5eqFrom (26) two values of 119910 can be obtained However one

of the values is negative which results in 1205795 with imaginarypart Then note 119910+ as the nonnegative solution of (26) 1205795 isobtained as

1205795 = 2 atan(plusmnradic119910+) (27)

Take the effect of the couple joint into account we canobtain

1205794 = 1205794eq minus atan 2 (119861 119862) 1205797 = 1205796eq + atan 2 (119861 119862) (28)

Because of the changing point 1198751015840119888 the inverse kinematicsalgorithm in Section 2 cannot be used to the equivalent 6Rrobot obtained from the transformation Given this situationan assumption is made that the point 1198751015840119888 is fixed It meansthat the points 1198751015840119888 1198751198881 and 1198751198882 coincide all the time as shownin Figure 4 With this assumption the inverse kinematicsalgorithm in Section 2 can be used to calculate approximatesolutions of the 7R 6-DOF robot

In order to have a further understanding the error of thistransformation with an assumption is investigated as followsAs shown in Figure 6 the intersection of axis 4 and axis 7 ispoint 1198751015840119888 The position of point 1198751015840119888 is changing while rotatingthe couple joint of the 4R wrist which is the source of theerror So the variation of point1198751015840119888 that is the distance betweenpoint1198751015840119888 and1198751198881 is calculated in triangleO4O5 119875119888 and triangleO41198751015840119888119875119888 as

119889 = 1198895 cos (120573) ( 1cos (120579eq5 2) minus 1) (29)

70

80

90

100

110

120

130

140

150

160

30 31 32 33 34 35 3629120573 (degree)

dmax

(mm

)

Figure 7 Tendency of variation on 119889max versus 120573 with 1198895 = 90mm

It is obvious that

119889max = 1198895 cos (120573) ( 1cos (120579eq5max2) minus 1) (30)

where 119889max is the maximum distance between point 1198751015840119888 and1198751198881 120579eq5max is the maximum joint angle of the equivalent 6Rrobot and 120579eq5max = 4120573 when 120573 le 45∘ For practical robots30∘ le 120573 le 35∘ As shown in Figure 7 it is the tendency ofvariation on 119889max versus 120573 with 1198895 = 90mm It is obviousthat the smaller1198895 and120573 are the smaller the distance betweenpoint 1198751015840119888 and 1198751198881 will be As a practical example the wrist ofABB IRB 5400 painting robot uses 120573 = 35∘ and parameter1198895 is small enough to ensure the assumption above Theeffect of the transformation error on the inverse kinematicssolution accuracy is discussed in Section 51 with a numericalsimulation

4 The DLS Method and the New InverseKinematics Algorithm

Because of the absence of the analytical solutions a numericaliterative algorithm is necessary and helpful for the inversekinematics of the 7R 6-DOF robot As a stable numericalalgorithm the Damped Least-Squares (DLS) method iswidely used for inverse kinematics of serial robots [20] Asthemain advantage the DLSmethodwith a varying dampingfactor could deal with the kinematic singularities of the robotproviding user-defined accuracy capabilities So in this paperthe DLS method is used and the main idea is described asfollows

41 The DLS Method Firstly the error between the desiredand actual EE configuration is investigated With the desiredconfigurationT119889 and the actual configurationT119886 the error ofconfiguration could be described in the tool (EE) coordinate


frame as (31)The orientation of the EE is described accordingto the 119885-119884-119883 Euler angles

e =[[[[[[[[[[[

119890119909119890119910119890119911119890120601119890120579119890120595

]]]]]]]]]]]

=

[[[[[[[[[[[[[[[

n119886 (p119889 minus p119886)o119886 (p119889 minus p119886)a119886 (p119889 minus p119886)(a119886 sdot o119889 minus a119889 sdot o119886)2(n119886 sdot a119889 minus n119889 sdot a119886)2(o119886 sdot n119889 minus o119889 sdot n119886)2

]]]]]]]]]]]]]]]

(31)

where [119890119909 119890119910 119890119911]119879 is the position error [119890120601 119890120579 119890120595]119879 is theorientation error of EE p119889 and p119886 are the position vectorof T119889 and T119886 respectively n119889 o119889 a119889 and n119886 o119886 a119886 are theorientation vector of T119889 and T119886 respectively

According to the differential theory we can obtaine = 119869Δq (32)

where e denotes the configuration error of EEΔq denotes theincrement of joint angle and 119869 is the Jacobian matrix

As for the general configuration where the Jacobianmatrix is nonsingular Δq could be calculated by (33) accord-ing to the least-squares principle

Δq = (119869119879119869)minus1 119869119879e (33)

When the Jacobian matrix is singular or at the neighbor-hood the damping factor 120582 is introduced to make a tradingoff accuracy against feasibility of the joint angle required andΔq could be described as

Δq = (119869119879119869 + 1205822I)minus1 119869119879e (34)

It is obvious that a suitable value for 120582 is essential Anda small value of 120582 gives accurate solution but low robustnessto the kinematic singularities Given this problem a methodto determine the value of 120582 is given based on the conditionnumbers of 119869 as follows

1205822 = (1 minus 120576119896)

2 12058221 119896 gt 1205760 119896 le 120576 (35)

where 119896 is the condition numbers of 119869 120576 is the critical valueof 119896 1205821 is the maximum value of 120582 and usually is set based ontrial and error

Then the iterative formula of joint angle will beq = q + Δq (36)

The exact inverse kinematics solution will be obtainedwith the rational termination condition

119901err le 120576119901119903err le 120576119903 (37)

where 119901err = radic1198902119909 + 1198902119910 + 1198902119911 and 119903err = radic1198902120601 + 1198902120579 + 1198902120595 are theposition error and orientation error of EE respectively 120576119901 and120576119903 are the critical value and usually are set based on trial anderror

42 New Inverse Kinematics Algorithm The new presentedalgorithm for the inverse kinematics of the 7R 6-DOF robotis shown in Algorithm 1 At line (1) DH parameters of thecorresponding equivalent 6R robot are calculated with thetransformation method shown in Section 3 At line (2) jointangle 120579eq of equivalent 6R robot is derived with the inversekinematics algorithm shown in Section 2 At line (3) theapproximate solutions of the 7R 6-DOF robot are obtainedwith the inverse transformation method shown in Section 3At lines (4)ndash(7) the exact inverse kinematic solutions of the7R 6-DOF robot can be derived with the DLS method

The new proposed algorithm is a two-step methodwhich incorporates the inverse kinematics algorithm of theequivalent 6R robot and the DLS method The approximatesolutions derived from the first step can offer good initialvalue to the DLS method This is the main advantage of theproposed algorithmover theDLSmethodMoreover theDLSmethod can derive just a single solution while the proposedalgorithm can derive all 8 solutionsThis is another advantageof the proposed algorithm over the DLS method It is notablethat [22] also presents a two-step method which is noted asWursquos method There are several similarities between the newproposed algorithm and Wursquos method (1) the approximatesolutions are derived by analytical method (2) the exactsolutions are derived by a numerical iterative technique withthe approximate solutions as the initial values And the maindifference is the way to derive the approximate solutions inthe first step Because a complex polynomial system needsto be solved the method used in the first step of Wursquosmethod (ie directly deriving the approximate solutions bysolving the 7R robot with spherical wrist) is time-consumingInstead the inverse kinematics algorithm of the equivalent 6Rrobot (which has been well studied and is available in mostcommercial robot controllers) is utilized in the first step of theproposed algorithm This improvement makes the proposedmethod more computationally efficient than Wursquos methodMoreover it is easier to find the correct approximate solutionfrom all possible ones by prescribing configuration indicators[24] for the equivalent 6R robot This is another advantage ofthe proposed method over Wursquos method

Algorithm 1 (new inverse kinematics algorithm)

(1) DH parameters of the equivalent 6R robot lArr DHparameters of the 7R 6-DOF robot

(2) 120579eq = IKeq(Tend)(3) 120579in = ITRANS(120579eq)(4) T119886 lArr DK(120579in)(5) TendT119886 rArr e

(6) 120579in = 120579in + Δ120579 = 120579in + (119869119879119869 + 1205822119868)minus1119869119879e(7) if 119901err le 120576119901 and 119903err le 120576119903 then 120579final = 120579in stop else go

to (4)Additionally the multiple solutions problem is discussed

It is well known that a general 6R 6-DOF robot can have atmost 16 inverse kinematic solutions [7] However with theproposed algorithm only 8 real number solutions can be


minus5

0

5

10

15

20

25

30

35

40

y

minus80 minus40 0 40 80 120minus1201205795eq (degree)

Figure 8 The solutions of (26) (ie the values of 119910)Table 1 The D-H parameters of the 7R 6-DOF robot

119894-th 119886119894minus1 (mm) 120572119894minus1 (deg) 119889119894 (mm) 120579119894 (deg)1 0 0 0 602 0 90 0 minus303 1000 0 0 604 0 90 900 minus305 0 minus35 80 606 0 70 80 minus607 0 minus35 100 30

obtained for the 7R 6-DOF robot This can be explained by(26) which is used in the inverse transformationmethod thatis Algorithm 1 line (3) Given a solution of the equivalent6R robot two values of 119910 can be obtained However oneof the values is negative which results in 1205795 with imaginarypart Because there are 8 solutions for the equivalent 6Rrobot 8 values of 1205795 with imaginary part can be obtainedwhich are meaningless for practical application Finally only8 real number solutions are left In order to have a furtherunderstanding the solutions of (26) (ie the values of 119910) areanalyzedThe values of119910 are influenced by 1205795eq and are shownin Figure 8 It is obvious that there is only one value of119910whichis nonnegative when 1205795eq changes in its working space thatis (minus140∘ 140∘) Therefore in some degree we can draw theconclusion that there are at most 8 meaningful real numberinverse kinematic solutions when the 7R 6-DOF robot worksin its working space

5 Simulation and Discussion

In order to illustrate the accuracy and efficiency of thealgorithm proposed in this paper three simulations areimplemented on a practical 7R 6-DOF painting robot asshown in Figure 1 The DH parameters of the robot areshown in Table 1 The first simulation aims to validate theaccuracy of the algorithm by calculating 8 solutions of theinverse kinematics with a given configuration of EE In thesecond simulation the proposed algorithm is compared withthe DLS method and Wursquos method The accuracy of the

Table 2 The D-H parameters of the equivalent 6R robot

119894-th 119886119894minus1 (mm) 120572119894minus1 (deg) 119889119894 (mm)1 0 0 02 0 90 03 1000 0 04 0 90 965532165 0 minus90 06 0 90 16553216

algorithm is also assessed in the whole usable workspace ofthe robotThe third simulation aims to illustrate the efficiencyof the algorithm when it is tested in an offline programmingoperation

51 Simulation 1 Take q = [1205791 1205792 1205793 1205794 1205795 1205797] = [60∘ minus30∘60∘ minus30∘ 60∘ 30∘] as the target joint angle Through theforward kinematic equations the configuration of EE isdescribed asTend

= [[[[[[

minus0012894 090259766 043029197 75440005minus0120727 minus04285850 089539928 133344420992602 minus00404021 011449425 minus1326919

0 0 0 1

]]]]]] (38)

According to the new proposed algorithm the DHparameters of the corresponding equivalent 6R robot arederived with the method shown in Algorithm 1 line (1) asshown in Table 2 The 8 solutions of the equivalent 6R robotare calculatedwith equations inAlgorithm 1 line (2)Then theapproximate solutions of the 7R 6-DOF robot are obtainedwith the inverse transformation method in Algorithm 1 line(3) as listed in Table 3 At the end Table 4 lists the finalinverse kinematic solutions of the 7R 6-DOF robot And thecorresponding postures for the 8 inverse kinematic solutionsare shown in Figure 9 The algorithm is performed on adesktop computer platform (Pentium i7 28GHz 8GB RAMMATLAB software program)

It is obvious that 8 solutions are obtained for a given con-figuration of EE The first solution in Table 4 agrees with thegiven target joint angle and the deviation is less than 00001∘which illustrate the accuracy of the proposed algorithm It isworthwhile to make a comparison between the approximatesolutions in Table 3 and the exact solutions in Table 4 Asstated in Section 3 the deviations result from the error of thetransformation that is the distance between points 1198751015840119888 and1198751198881 described by (34) From the comparison the approximatesolutions are very close to the exact ones with the deviationless than 5∘Therefore the approximate solutions as the initialvalues could guarantee the DLS method to convergent to theexact solutions rapidly This is the advantage of the proposedalgorithm over the general DLS method of which the initialvalue is randomly selectedAdditionally as shown in Figure 9the corresponding postures of the robot are similar to thatof the equivalent 6R robot [24] which makes it easy toselect an optimal solution according to the joint working


Table 3 The approximate inverse kinematic solutions of the 7R 6-DOF robot

119894 1205791 (deg) 1205792 (deg) 1205793 (deg) 1205794 (deg) 1205795 (deg) 1205797 (deg)1 600406 minus322712 640328 minus288849 573570 2905152 600406 minus322712 640328 1993868 minus573570 minus19922023 600406 minus567935 1149672 minus209030 340025 2223994 600406 minus567935 1149672 1872190 minus340025 minus18588215 minus1199594 minus1232065 640328 72190 minus340025 17411796 minus1199594 minus1232065 640328 1590970 340025 2223997 minus1199594 minus1477288 1149672 193868 minus573570 16077988 minus1199594 minus1477288 1149672 1511151 573570 290515

Table 4 The inverse kinematic solutions of the 7R 6-DOF robot


Figure 9 Postures for the 8 inverse kinematic solutions of the 7R6-DOF robot

range restrictions which is a main advantage of the proposedalgorithm over Wursquos method

52 Simulation 2 In order to further illustrate the perfor-mance of the proposed algorithm it is compared with theDLS method and Wursquos method in this section In the firstcomparison the target joint angle in Simulation 1 is usedThe proposed algorithm and DLS method are utilized tosolve this inverse kinematics problem Here the terminationconditions of the two methods are set to be the same (ie119901err le 001mm and 119903err le 001∘) The correspondingresults are shown in Table 5 From Table 5 the proposedalgorithm takes about 64ms on an average It is notable thatthree different initial values are chosen for the DLS methodWhen the initial value is chosen as 120579in = [60∘ minus25∘ 60∘minus30∘ 60∘ minus60∘ 30∘] the computation time of DLS method

Table 5 The computation time of the proposed algorithm and DLSmethod

Method The initial value 120579in(deg)

Computation time(ms)

The proposedmethod 64

DLS method

[60 minus25 60 minus30 6060 30] 60

[0 0 0 0 0 0 0] 182[minus60 minus25 60 minus30

60 60 30] Failure

is 60ms which is slightly smaller than that of the proposedmethod However when the initial value is changed to 120579in =[0∘ 0∘ 0∘ 0∘ 0∘ 0∘ 0∘] the computation time will greatlyincrease to 182ms The DLS method even fails to convergentto the correct solution when the initial value is chosen as120579in = [minus60∘ minus25∘ 60∘ minus30∘ 60∘ minus60∘ 30∘] It means that theDLS method is too sensitive to the initial value which is ran-domly selected in practical usage So in some degree we drawthe conclusion that the proposed algorithm ismore stable andefficient than the DLS method Moreover the DLS methodcan derive only one solution while the proposed algorithmcan derive all 8 solutions as shown in Simulation 1 This isanother advantage of the proposed algorithm over the DLSmethod

In the second comparison the accuracy and efficiency ofthe algorithm are assessed in the whole usable workspace ofthe robot For the 7R 6-DOF painting robot under studied


Table 6 The usable workspace of the 7R 6-DOF robot

119894-th 1205791 (deg) 1205792 (deg) 1205793 (deg) 1205794 (deg) 1205795 (deg) 1205797 (deg)Range of joint angle [minus120 120) [minus30 135] [minus80 80] (minus360 360) (minus170 170) (minus360 360)

times10minus2

0 1000600 800200 400Simulation number

0

02

04

06

08

1

per

r(m

m)

(a)

0 200 400 600 800Simulation number

1000

times10minus2

0

02

04

06

08

1

per

r(m

m)

(b)

Figure 10 Position errors of the inverse kinematic solutions (a) the proposed method (b) modified Wursquos method

0

1times10minus4

r err

(deg

ree)

08

06

04

02


(a)

times10minus4

r err

(deg

ree)

0

02

04

06

08

1


(b)

Figure 11 Orientation errors of the inverse kinematic solutions (a) the proposed method (b) modified Wursquos method

the usable workspace is shown in Table 6 And 1000 groupsof target joint angles are randomly selected in the usableworkspace For comparisonWursquos method [22] is also utilizedto solve the inverse kinematics Because the main differencebetween the proposed algorithm andWursquos method is the firststep that is the method to derive the approximate solutionsthe second step of Wursquos method is changed from LMmethodto DLS method for simplicity And the termination condi-tions of the two methods are set to be the same (ie 119901err le001mm and 119903err le 001∘)The corresponding solution errorsand the computation times are depicted in Figures 10 11 and12 respectively For simplicity the average and maximum

computation times of the two methods are also summarizedin Table 7 From Figures 10 and 11 the inverse kinematicsolutions for all 1000 configurations are successfully derivedwith the two methods and the solution errors are rathersmall that is 119901err le 001mm and 119903err le 00001∘ Howeverthe computation time of the proposed method is less thanthat of Wursquos method as shown in Figure 12 and Table 7 Inother words the proposedmethod ismore efficient thanWursquosmethod This is because directly solving the 7R robot withspherical wrist inWursquos method is more time-consuming thansolving the equivalent 6R robot in the new proposedmethodMoreover it is easier to find the correct approximate solution


The proposed methodModified Wursquos method


0

002

004

006

008

01

012

014

016

018

02

Com

puta

tion

time (

s)

Figure 12 Computation time of the proposedmethod andmodifiedWursquos method

Table 7 The computation time of the proposed algorithm andmodified Wursquos method

Method Average computationtime 119879average (ms)

Maximumcomputation time119879maximum (ms)

The proposedmethod 603 128

Modified Wursquosmethod 906 1699

from all possible ones by prescribing configuration indicators[24] for the equivalent 6R robot

Additionally it is notable that the computation time ofthe algorithm is influenced by the termination conditionsthat is the prescribed solution error 119901err and 119903err In otherwords in order to improve the accuracy of the solutionthe computation time will increase From the perspective ofapplication the termination conditions used in this simula-tion are rational

53 Simulation 3 Thenewproposed inverse kinematics algo-rithm is implemented in an offline programming operation tospray paintingThe graphic simulator of the 7R 6-DOF robotand the task is shown in Figure 13 The trajectory of EE is aclassical raster trajectory in the plane 119909-119900-119910 on the surface ofthe workpiece The orientation of EE is constant with the 119909-axes parallel to that of the base frame and the 119911- and 119910-axesopposite to that of the base frame respectively The durationof the motion is determined as 20 s For trajectory trackingfirstly discrete points are sampled from the trajectory Theinverse kinematic solution for each sample point is derivedLater on a further interpolation in joint space is implementedto make the trajectory smooth enough And the calculated

Workpiece Trajectory

Spray gun

Z120782

Zt

Y120782

X120782

Yt

Xt

Ot

ZsYs

XsOs

O

Figure 13 Graphic simulator of the 7R 6-DOF robot and thetrajectory

joint positions velocities and acceleration of the 7R 6-DOFrobot are given in Figure 14

It is obvious that the offline programming operation iscompleted with the new proposed algorithm The algorithmtakes 15ms on an average on a desktop computer platform(Pentium i7 28GHz 8GB RAM MATLAB software pro-gram) which illustrates the efficiency By comparing simula-tion 3 and simulation 2 the computing time in simulation 3 isless than 25 percent of that in simulation 2The reason is thatwe use the solution of the last step as the initial value of thenext step when the robot is following a continuous trajectoryIn other words the presented algorithm is only used for theinitial point of the continuous trajectory So the presentedalgorithm may be not so great advantageous than generaliterative method during a continuous trajectory However asfor the initial point of the continuous trajectory that is thepoint without an approximate solution the new proposedalgorithm in this paper is significant because with an initialsolution derived in the first step the proposed algorithmis less time-consuming Additionally 8 solutions will beobtained for a given EE configuration According to the jointworking range restrictions and so on the optimal solutioncould be selected

6 Conclusion

A new and efficient two-step algorithm for the inversekinematics of a 7R 6-DOF robot is proposed and studied inthis paper In the first step a rational transformation betweenthe 7R 6-DOF robot and the well-known equivalent 6R robotis constructedThen8 approximate solutions of the 7R6-DOFrobot are derived with the general inverse kinematics algo-rithm of the equivalent 6R robot In the second step the DLSmethod is employed to derive the exact solutions

Compared with other methods the proposed algorithmis systematic The accuracy and efficiency has been evaluatedwith three simulations In the first simulation 8 solutionsare obtained for a given EE configuration of the 7R 6-DOF


5 1510 200Time (s)

Joint 1Joint 2Joint 3


minus1

minus05

0

05

1

15

2

Join

t pos

ition

(rad

)

(a)

minus02

minus015

minus01

minus005

0

005

01

015

Join

t velo

city

(rad

s)

50 10 15 20Time (s)



(b)

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

Join

t acc

eler

atio

n (r

ads

2)

5 10 15 200Time (s)



(c)

Figure 14 Simulation results of the 7R 6-DOF robot (a) joint position (b) joint velocity (c) joint acceleration

robot The error is less than 00001 degree which illustratesthe accuracy of this proposed algorithm In order to furtherillustrate the performance of the proposed algorithm it iscompared with the DLS method and Wursquos method in thesecond simulation The accuracy of the algorithm is alsoassessed in the whole usable workspace of the robot In thethird simulation the algorithm is implemented in an offlineprogramming operation The results show that the proposedalgorithm is efficient for offline programming and it is moreadvantageous in the case without an approximate solutionsuch as the initial point of the continuous trajectory

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This work was supported by Tianjin Science and TechnologyCommittee [Grant no 15ZXZNGX00200]

References

[1] D L Pieper The Kinematics of Manipulators under ComputerControl Stanford University Stanford Calif USA 1968

[2] J M SeligGeometric Fundamentals of Robotics Monographs inComputer Science Series Springer New York NY USA 2005

[3] J Huang X Wang D Liu and Y Cui ldquoA new method forsolving inverse kinematics of an industrial robotrdquo inProceedingsof International Conference on Computer Science and ElectronicsEngineering ICCSEE rsquo12 pp 53ndash56 China March 2012


[4] H Liu Y Zhang and S Zhu ldquoNovel inverse kinematicapproaches for robot manipulators with Pieper-Criterion basedgeometryrdquo International Journal of Control Automation andSystems vol 13 no 5 pp 1242ndash1250 2015

[5] L W Tsai and A P Morgan ldquoSolving the kinematics of themost general six- and five-degree-of-freedom manipulators bycontinuation methodsrdquo Transactions of the ASMEmdashJournal ofmechanisms Transmissions and Automation in Design vol 107no 2 pp 189ndash200 1985

[6] M Raghavan and B Roth ldquoInverse kinematics of the general 6Rmanipulator and related linkagesrdquo Transactions of the ASMEmdashJournal of Mechanical Design vol 115 no 3 pp 502ndash508 1993

[7] D Manocha and J F Canny ldquoEfficient inverse kinematics forgeneral 6R manipulatorsrdquo IEEE Transactions on Robotics andAutomation vol 10 no 5 pp 648ndash657 1994

[8] M L Husty M Pfurner and H-P Schrocker ldquoA new andefficient algorithm for the inverse kinematics of a general serial6RmanipulatorrdquoMechanism andMachineTheory vol 42 no 1pp 66ndash81 2007

[9] S Liu S Zhu and X Wang ldquoReal-time and high-accurateinverse kinematics algorithm for general 6R robots based onmatrix decompositionrdquo Jixie Chinese Journal of MechanicalEngineering vol 44 no 11 pp 304ndash309 2008

[10] S G Qiao Q Z Liao SMWei andH J Su ldquoInverse kinematicanalysis of the general 6R serial manipulators based on doublequaternionsrdquoMechanism andMachineTheory vol 45 no 2 pp193ndash199 2010

[11] Y Wei S Jian S He and Z Wang ldquoGeneral approach forinverse kinematics of nR robotsrdquo Mechanism and MachineTheory vol 75 no 1 pp 97ndash106 2014

[12] S Kucuk andZ Bingul ldquoInverse kinematics solutions for indus-trial robot manipulators with offset wristsrdquoApplied Mathemati-cal Modelling Simulation and Computation for Engineering andEnvironmental Systems vol 38 no 7-8 pp 1983ndash1999 2014

[13] R Koker T Cakar and Y Sari ldquoA neural-network committeemachine approach to the inverse kinematics problem solutionof robotic manipulatorsrdquo Engineering with Computers vol 30no 4 pp 641ndash649 2014

[14] M Asadi-Eydivand M M Ebadzadeh M Solati-Hashjin CDarlot and N A Abu Osman ldquoCerebellum-inspired neuralnetwork solution of the inverse kinematics problemrdquo BiologicalCybernetics vol 109 no 6 pp 561ndash574 2015

[15] F Chapelle and P Bidaud ldquoClosed form solutions for inversekinematics approximation of general 6RmanipulatorsrdquoMecha-nism and Machine Theory vol 39 no 3 pp 323ndash338 2004

[16] P Kalra P B Mahapatra and D K Aggarwal ldquoAn evolutionaryapproach for solving the multimodal inverse kinematics prob-lem of industrial robotsrdquo Mechanism and Machine Theory vol41 no 10 pp 1213ndash1229 2006

[17] R Koker ldquoA neuro-simulated annealing approach to the inversekinematics solution of redundant robotic manipulatorsrdquo Engi-neering with Computers vol 29 no 4 pp 507ndash515 2013

[18] O Khatib ldquoA unified approach for motion and force control ofrobot manipulators the operational space formulationrdquo IEEEJournal of Robotics and Automation vol 3 no 1 pp 43ndash53 1987

[19] W Wolovich and H Elliott ldquoA computational technique forinverse kinematicsrdquo in Proceedings of the 23rd IEEE Conferenceon Decision and Control IEEE Las Vegas Nev USA December1984

[20] S Chiaverini B Siciliano and O Egeland ldquoReview of thedamped least-squares inverse kinematics with experiments on

an industrial robot manipulatorrdquo IEEE Transactions on ControlSystems Technology vol 2 no 2 pp 123ndash134 1994

[21] W Xu J Zhang B Liang and B Li ldquoSingularity analysis andavoidance for robot manipulators with nonspherical wristsrdquoIEEE Transactions on Industrial Electronics vol 63 no 1 pp277ndash290 2016

[22] L Wu X Yang D Miao Y Xie and K Chen ldquoInversekinematics of a class of 7R 6-DOF robots with non-sphericalwristrdquo in Proceedings of 10th IEEE International Conference onMechatronics and Automation IEEE ICMA rsquo13 pp 69ndash74 JapanAugust 2013

[23] H Bruyninckx H Thielemans and J De Schutter ldquoEfficientkinematics of a spherical 4R wrist by means of an equivalent3R wristrdquo Mechanism and Machine Theory vol 33 no 6 pp649ndash659 1998

[24] C S G Lee and M Ziegler ldquoGeometric approach in solvinginverse kinematics of puma robotsrdquo IEEE Transactions onAerospace and Electronic Systems vol 20 no 6 pp 695ndash7061984

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Joint 1

Joint 7

Joint 6 Joint 5

Joint 4

Joint 3

Joint 2

The wrist

Figure 1 Prototype of a 7R 6-DOF robot

X3

Y3

Z3

Z4

d5X2

X4

Y2

Z1

Z2

d4

a2

X7

X6

X0

X5

X1

Y6

Y7

Y5

Y0Y1

Y4

Z0

Z7

Z6

Z5

d6

d7

Figure 2 Configuration of the 7R 6-DOF robot

annealing [17] are developed for the solution of inversekinematics problem These methods convert the kinematicsproblem into an equivalentminimization problem and gener-ally suffer from time-consuming and low accuracy For serialrobots the numerical iterative techniques such as Newton-Raphson approach [18] the steepest descent approach [19]and the Damped Least-Squares (DLS) approach [20 21]are often applied The drawbacks of a numerical iterativealgorithm are slow iterations and sensitivity to the initialvalue and normally just one solution instead of all solutionscan be derived

For the 7R 6-DOF robot Wu et al [22] proposed a two-step method the approximate analytical solutions are firstlyderived through solving the 7R robot with spherical wristby introducing a virtual wrist center and the Levenberg-Marquardt (LM) method is used to calculate the exactsolutions This is an interesting approach but a complexpolynomial system needs to be solved in the first step whichis time-consuming In this paper a new and efficient two-step method is presented As the major improvement arational transformation between the 7R 6-DOF robot and

X3

Y3

Z3

X2

X4

Y2

Z1

Z2

X6

X0

X5

X1

Y6

Y5

Y0

Y1

Y4

Z0

Z4

Z6

Z5

deq6

deq4

aeq2

Figure 3 Configuration of the equivalent 6R robot

the well-known equivalent 6R robot is constructed Thegeneral inverse kinematics algorithm of the equivalent 6Rrobot is used to calculate the approximate solutions of the7R 6-DOF robot in the first step Then a general iterativealgorithm that is the DLS method is employed to get theexact solutions The approximate solutions derived from thefirst step can offer good initial value to the DLS method andmake it computationally efficient To verify the accuracy andefficiency of this method three simulations are implemented

The paper is organized in the following manner InSection 2 the inverse kinematics of the equivalent 6R robotis briefly reviewed Section 3 describes the efficient transfor-mation between the 7R 6-DOF robot and the equivalent 6Rrobot where a rational assumptionwill be given In Section 4the DLS method is reviewed and the new inverse kinematicsalgorithm is presented In Section 5 three simulations areimplemented to illustrate the accuracy and efficiency of theproposed method The results are presented and discussedSection 6 is the conclusion

2 Inverse Kinematics Algorithm ofthe Well-Known Equivalent 6R Robot

As stated previously the equivalent 6R robot is a well-knowntypical structure of serial robots The inverse kinematicsmethods have been proposed by [2ndash4] Before deriving theinverse kinematics of the 7R 6-DOF robot firstly we brieflydescribe that of the equivalent 6R robot For the equivalent6R robot the configuration is shown in Figure 3 For serialrobots DenavitndashHartenberg (DH) parameters are widelyused to describe the position and orientation of the EE Thetransformation matrix relating the joint 119894 to joint 119894 minus 1 couldbe given by

119894minus1119894T = [[[[[

[

c119894 minuss119894 0 119886119894minus1s119894c120572119894minus1 c119894c120572119894minus1 minuss120572119894minus1 minus119889119894s120572119894minus1s119894s120572119894minus1 c119894s120572119894minus1 c120572119894minus1 119889119894c120572119894minus10 0 0 1

]]]]]] (1)

where s119894 = sin 120579119894 c119894 = cos 120579119894 c120572119894 = cos120572119894 and s120572119894 = sin120572119894




Tend = [n o a p0 0 0 1] =

[[[[[[

119899119909 119900119909 119886119909 119901119909119899119910 119900119910 119886119910 119901119910119899119911 119900119911 119886119911 1199011199110 0 0 1

]]]]]] (3)




1198881119901119909 + 1199041119901119910 = 119889411990423 + 11988621198882 (5)minus1199041119901119909 + 1198881119901119910 = 0 (6)

119901119911 = minus119889411988823 + 11988621199042 (7)


1205791 = atan 2 (119901119910 119901119909) 12057910158401 =

1205791 minus 119901119894 (1205791 gt 0) 1205791 + 119901119894 (1205791 le 0)

(8)


2119886211988941199043 = 1199012119909 + 1199012119910 + 1199012119911 minus 11988924 minus 11988622 (9)






1205792 = atan 2 (11990111991111988621198883 + 1198962 (11988621199043 + 1198894) 1198961 + 119896211988621198883)minus 1205793 (12)



1205794 = atan 2 (1198861199091199041 minus 1198861199101198881 11988611991111990423 + 119886119910119888231199041 + 119886119909119888231198881) (13)





1205791 = 1205791eq1205792 = 1205792eq1205793 = 1205793eq

(14)


119889eq4 = 1198894 + 1198895 cos (120573)cos (120579eq5 2)

119889eq6 = 1198897 + 1198895 cos (120573)cos (120579eq5 2)

(15)


Axis 5 Axis 6

Axis 7Axis 4

Virtual axis

Couple joint

120573120573

1205795

1205794

1205797

1205796

Z4

Z7

Z5

Z3

Y3

Z6

Y7

Y5

Y6Y4

03

07

06

05

04

Pc2

Pc1

Pc

P998400c


Z5

Z6Z3Z4

Y6Y3Y4

Y5120579eq4 120579

eq6

120579eq5

0605 0403




5Teq 56T

eq (16)





(18)







25) = c5eq (21)




and (20)

1205794eq = atan 2 (119861 119862) (23)

where



(24)




Axis 5 Axis 6

Axis 7Axis 4

Virtual axis

04

05

06120573120573

d

12057961205795

Pc2

Pc1

Pc

P998400c



(119882 minus 119880 + 119881) 1199102 + 2 (119881 minus119882)119910 +119882 + 119880 + 119881 = 0 (26)








119889 = 1198895 cos (120573) ( 1cos (120579eq5 2) minus 1) (29)

70

80

90

100

110

120

130

140

150

160

30 31 32 33 34 35 3629120573 (degree)

dmax

(mm

)


It is obvious that








e =[[[[[[[[[[[

119890119909119890119910119890119911119890120601119890120579119890120595

]]]]]]]]]]]

=

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

(31)





Δq = (119869119879119869)minus1 119869119879e (33)


Δq = (119869119879119869 + 1205822I)minus1 119869119879e (34)


1205822 = (1 minus 120576119896)

2 12058221 119896 gt 1205760 119896 le 120576 (35)




119901err le 120576119901119903err le 120576119903 (37)











minus5

0

5

10

15

20

25

30

35

40

y











= [[[[[[


0 0 0 1

]]]]]] (38)















DLS method

[60 minus25 60 minus30 6060 30] 60


60 60 30] Failure






times10minus2


0

02

04

06

08

1

per

r(m

m)

(a)


1000

times10minus2

0

02

04

06

08

1

per

r(m

m)

(b)


0

1times10minus4

r err

(deg

ree)

08

06

04

02


(a)

times10minus4

r err

(deg

ree)

0

02

04

06

08

1


(b)







0

002

004

006

008

01

012

014

016

018

02

Com

puta

tion

time (

s)











Spray gun

Z120782

Zt

Y120782

X120782

Yt

Xt

Ot

ZsYs

XsOs

O




6 Conclusion




5 1510 200Time (s)



minus1

minus05

0

05

1

15

2

Join

t pos

ition

(rad

)

(a)

minus02

minus015

minus01

minus005

0

005

01

015

Join

t velo

city

(rad

s)

50 10 15 20Time (s)



(b)

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

Join

t acc

eler

atio

n (r

ads

2)

5 10 15 200Time (s)



(c)





Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







Tend = [n o a p0 0 0 1] =

[[[[[[

119899119909 119900119909 119886119909 119901119909119899119910 119900119910 119886119910 119901119910119899119911 119900119911 119886119911 1199011199110 0 0 1

]]]]]] (3)




1198881119901119909 + 1199041119901119910 = 119889411990423 + 11988621198882 (5)minus1199041119901119909 + 1198881119901119910 = 0 (6)

119901119911 = minus119889411988823 + 11988621199042 (7)


1205791 = atan 2 (119901119910 119901119909) 12057910158401 =

1205791 minus 119901119894 (1205791 gt 0) 1205791 + 119901119894 (1205791 le 0)

(8)


2119886211988941199043 = 1199012119909 + 1199012119910 + 1199012119911 minus 11988924 minus 11988622 (9)






1205792 = atan 2 (11990111991111988621198883 + 1198962 (11988621199043 + 1198894) 1198961 + 119896211988621198883)minus 1205793 (12)



1205794 = atan 2 (1198861199091199041 minus 1198861199101198881 11988611991111990423 + 119886119910119888231199041 + 119886119909119888231198881) (13)





1205791 = 1205791eq1205792 = 1205792eq1205793 = 1205793eq

(14)


119889eq4 = 1198894 + 1198895 cos (120573)cos (120579eq5 2)

119889eq6 = 1198897 + 1198895 cos (120573)cos (120579eq5 2)

(15)


Axis 5 Axis 6

Axis 7Axis 4

Virtual axis

Couple joint

120573120573

1205795

1205794

1205797

1205796

Z4

Z7

Z5

Z3

Y3

Z6

Y7

Y5

Y6Y4

03

07

06

05

04

Pc2

Pc1

Pc

P998400c


Z5

Z6Z3Z4

Y6Y3Y4

Y5120579eq4 120579

eq6

120579eq5

0605 0403




5Teq 56T

eq (16)





(18)







25) = c5eq (21)




and (20)

1205794eq = atan 2 (119861 119862) (23)

where



(24)




Axis 5 Axis 6

Axis 7Axis 4

Virtual axis

04

05

06120573120573

d

12057961205795

Pc2

Pc1

Pc

P998400c



(119882 minus 119880 + 119881) 1199102 + 2 (119881 minus119882)119910 +119882 + 119880 + 119881 = 0 (26)








119889 = 1198895 cos (120573) ( 1cos (120579eq5 2) minus 1) (29)

70

80

90

100

110

120

130

140

150

160

30 31 32 33 34 35 3629120573 (degree)

dmax

(mm

)


It is obvious that








e =[[[[[[[[[[[

119890119909119890119910119890119911119890120601119890120579119890120595

]]]]]]]]]]]

=

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

(31)





Δq = (119869119879119869)minus1 119869119879e (33)


Δq = (119869119879119869 + 1205822I)minus1 119869119879e (34)


1205822 = (1 minus 120576119896)

2 12058221 119896 gt 1205760 119896 le 120576 (35)




119901err le 120576119901119903err le 120576119903 (37)











minus5

0

5

10

15

20

25

30

35

40

y











= [[[[[[


0 0 0 1

]]]]]] (38)















DLS method

[60 minus25 60 minus30 6060 30] 60


60 60 30] Failure






times10minus2


0

02

04

06

08

1

per

r(m

m)

(a)


1000

times10minus2

0

02

04

06

08

1

per

r(m

m)

(b)


0

1times10minus4

r err

(deg

ree)

08

06

04

02


(a)

times10minus4

r err

(deg

ree)

0

02

04

06

08

1


(b)







0

002

004

006

008

01

012

014

016

018

02

Com

puta

tion

time (

s)











Spray gun

Z120782

Zt

Y120782

X120782

Yt

Xt

Ot

ZsYs

XsOs

O




6 Conclusion




5 1510 200Time (s)



minus1

minus05

0

05

1

15

2

Join

t pos

ition

(rad

)

(a)

minus02

minus015

minus01

minus005

0

005

01

015

Join

t velo

city

(rad

s)

50 10 15 20Time (s)



(b)

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

Join

t acc

eler

atio

n (r

ads

2)

5 10 15 200Time (s)



(c)





Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Axis 5 Axis 6

Axis 7Axis 4

Virtual axis

Couple joint

120573120573

1205795

1205794

1205797

1205796

Z4

Z7

Z5

Z3

Y3

Z6

Y7

Y5

Y6Y4

03

07

06

05

04

Pc2

Pc1

Pc

P998400c


Z5

Z6Z3Z4

Y6Y3Y4

Y5120579eq4 120579

eq6

120579eq5

0605 0403




5Teq 56T

eq (16)





(18)







25) = c5eq (21)




and (20)

1205794eq = atan 2 (119861 119862) (23)

where



(24)




Axis 5 Axis 6

Axis 7Axis 4

Virtual axis

04

05

06120573120573

d

12057961205795

Pc2

Pc1

Pc

P998400c



(119882 minus 119880 + 119881) 1199102 + 2 (119881 minus119882)119910 +119882 + 119880 + 119881 = 0 (26)








119889 = 1198895 cos (120573) ( 1cos (120579eq5 2) minus 1) (29)

70

80

90

100

110

120

130

140

150

160

30 31 32 33 34 35 3629120573 (degree)

dmax

(mm

)


It is obvious that








e =[[[[[[[[[[[

119890119909119890119910119890119911119890120601119890120579119890120595

]]]]]]]]]]]

=

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

(31)





Δq = (119869119879119869)minus1 119869119879e (33)


Δq = (119869119879119869 + 1205822I)minus1 119869119879e (34)


1205822 = (1 minus 120576119896)

2 12058221 119896 gt 1205760 119896 le 120576 (35)




119901err le 120576119901119903err le 120576119903 (37)











minus5

0

5

10

15

20

25

30

35

40

y











= [[[[[[


0 0 0 1

]]]]]] (38)















DLS method

[60 minus25 60 minus30 6060 30] 60


60 60 30] Failure






times10minus2


0

02

04

06

08

1

per

r(m

m)

(a)


1000

times10minus2

0

02

04

06

08

1

per

r(m

m)

(b)


0

1times10minus4

r err

(deg

ree)

08

06

04

02


(a)

times10minus4

r err

(deg

ree)

0

02

04

06

08

1


(b)







0

002

004

006

008

01

012

014

016

018

02

Com

puta

tion

time (

s)











Spray gun

Z120782

Zt

Y120782

X120782

Yt

Xt

Ot

ZsYs

XsOs

O




6 Conclusion




5 1510 200Time (s)



minus1

minus05

0

05

1

15

2

Join

t pos

ition

(rad

)

(a)

minus02

minus015

minus01

minus005

0

005

01

015

Join

t velo

city

(rad

s)

50 10 15 20Time (s)



(b)

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

Join

t acc

eler

atio

n (r

ads

2)

5 10 15 200Time (s)



(c)





Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Axis 5 Axis 6

Axis 7Axis 4

Virtual axis

04

05

06120573120573

d

12057961205795

Pc2

Pc1

Pc

P998400c



(119882 minus 119880 + 119881) 1199102 + 2 (119881 minus119882)119910 +119882 + 119880 + 119881 = 0 (26)








119889 = 1198895 cos (120573) ( 1cos (120579eq5 2) minus 1) (29)

70

80

90

100

110

120

130

140

150

160

30 31 32 33 34 35 3629120573 (degree)

dmax

(mm

)


It is obvious that








e =[[[[[[[[[[[

119890119909119890119910119890119911119890120601119890120579119890120595

]]]]]]]]]]]

=

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

(31)





Δq = (119869119879119869)minus1 119869119879e (33)


Δq = (119869119879119869 + 1205822I)minus1 119869119879e (34)


1205822 = (1 minus 120576119896)

2 12058221 119896 gt 1205760 119896 le 120576 (35)




119901err le 120576119901119903err le 120576119903 (37)











minus5

0

5

10

15

20

25

30

35

40

y











= [[[[[[


0 0 0 1

]]]]]] (38)















DLS method

[60 minus25 60 minus30 6060 30] 60


60 60 30] Failure






times10minus2


0

02

04

06

08

1

per

r(m

m)

(a)


1000

times10minus2

0

02

04

06

08

1

per

r(m

m)

(b)


0

1times10minus4

r err

(deg

ree)

08

06

04

02


(a)

times10minus4

r err

(deg

ree)

0

02

04

06

08

1


(b)







0

002

004

006

008

01

012

014

016

018

02

Com

puta

tion

time (

s)











Spray gun

Z120782

Zt

Y120782

X120782

Yt

Xt

Ot

ZsYs

XsOs

O




6 Conclusion




5 1510 200Time (s)



minus1

minus05

0

05

1

15

2

Join

t pos

ition

(rad

)

(a)

minus02

minus015

minus01

minus005

0

005

01

015

Join

t velo

city

(rad

s)

50 10 15 20Time (s)



(b)

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

Join

t acc

eler

atio

n (r

ads

2)

5 10 15 200Time (s)



(c)





Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






e =[[[[[[[[[[[

119890119909119890119910119890119911119890120601119890120579119890120595

]]]]]]]]]]]

=

[[[[[[[[[[[[[[[


]]]]]]]]]]]]]]]

(31)





Δq = (119869119879119869)minus1 119869119879e (33)


Δq = (119869119879119869 + 1205822I)minus1 119869119879e (34)


1205822 = (1 minus 120576119896)

2 12058221 119896 gt 1205760 119896 le 120576 (35)




119901err le 120576119901119903err le 120576119903 (37)











minus5

0

5

10

15

20

25

30

35

40

y











= [[[[[[


0 0 0 1

]]]]]] (38)















DLS method

[60 minus25 60 minus30 6060 30] 60


60 60 30] Failure






times10minus2


0

02

04

06

08

1

per

r(m

m)

(a)


1000

times10minus2

0

02

04

06

08

1

per

r(m

m)

(b)


0

1times10minus4

r err

(deg

ree)

08

06

04

02


(a)

times10minus4

r err

(deg

ree)

0

02

04

06

08

1


(b)







0

002

004

006

008

01

012

014

016

018

02

Com

puta

tion

time (

s)











Spray gun

Z120782

Zt

Y120782

X120782

Yt

Xt

Ot

ZsYs

XsOs

O




6 Conclusion




5 1510 200Time (s)



minus1

minus05

0

05

1

15

2

Join

t pos

ition

(rad

)

(a)

minus02

minus015

minus01

minus005

0

005

01

015

Join

t velo

city

(rad

s)

50 10 15 20Time (s)



(b)

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

Join

t acc

eler

atio

n (r

ads

2)

5 10 15 200Time (s)



(c)





Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





minus5

0

5

10

15

20

25

30

35

40

y











= [[[[[[


0 0 0 1

]]]]]] (38)















DLS method

[60 minus25 60 minus30 6060 30] 60


60 60 30] Failure






times10minus2


0

02

04

06

08

1

per

r(m

m)

(a)


1000

times10minus2

0

02

04

06

08

1

per

r(m

m)

(b)


0

1times10minus4

r err

(deg

ree)

08

06

04

02


(a)

times10minus4

r err

(deg

ree)

0

02

04

06

08

1


(b)







0

002

004

006

008

01

012

014

016

018

02

Com

puta

tion

time (

s)











Spray gun

Z120782

Zt

Y120782

X120782

Yt

Xt

Ot

ZsYs

XsOs

O




6 Conclusion




5 1510 200Time (s)



minus1

minus05

0

05

1

15

2

Join

t pos

ition

(rad

)

(a)

minus02

minus015

minus01

minus005

0

005

01

015

Join

t velo

city

(rad

s)

50 10 15 20Time (s)



(b)

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

Join

t acc

eler

atio

n (r

ads

2)

5 10 15 200Time (s)



(c)





Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of
















DLS method

[60 minus25 60 minus30 6060 30] 60


60 60 30] Failure






times10minus2


0

02

04

06

08

1

per

r(m

m)

(a)


1000

times10minus2

0

02

04

06

08

1

per

r(m

m)

(b)


0

1times10minus4

r err

(deg

ree)

08

06

04

02


(a)

times10minus4

r err

(deg

ree)

0

02

04

06

08

1


(b)







0

002

004

006

008

01

012

014

016

018

02

Com

puta

tion

time (

s)











Spray gun

Z120782

Zt

Y120782

X120782

Yt

Xt

Ot

ZsYs

XsOs

O




6 Conclusion




5 1510 200Time (s)



minus1

minus05

0

05

1

15

2

Join

t pos

ition

(rad

)

(a)

minus02

minus015

minus01

minus005

0

005

01

015

Join

t velo

city

(rad

s)

50 10 15 20Time (s)



(b)

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

Join

t acc

eler

atio

n (r

ads

2)

5 10 15 200Time (s)



(c)





Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







times10minus2


0

02

04

06

08

1

per

r(m

m)

(a)


1000

times10minus2

0

02

04

06

08

1

per

r(m

m)

(b)


0

1times10minus4

r err

(deg

ree)

08

06

04

02


(a)

times10minus4

r err

(deg

ree)

0

02

04

06

08

1


(b)







0

002

004

006

008

01

012

014

016

018

02

Com

puta

tion

time (

s)











Spray gun

Z120782

Zt

Y120782

X120782

Yt

Xt

Ot

ZsYs

XsOs

O




6 Conclusion




5 1510 200Time (s)



minus1

minus05

0

05

1

15

2

Join

t pos

ition

(rad

)

(a)

minus02

minus015

minus01

minus005

0

005

01

015

Join

t velo

city

(rad

s)

50 10 15 20Time (s)



(b)

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

Join

t acc

eler

atio

n (r

ads

2)

5 10 15 200Time (s)



(c)





Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







0

002

004

006

008

01

012

014

016

018

02

Com

puta

tion

time (

s)











Spray gun

Z120782

Zt

Y120782

X120782

Yt

Xt

Ot

ZsYs

XsOs

O




6 Conclusion




5 1510 200Time (s)



minus1

minus05

0

05

1

15

2

Join

t pos

ition

(rad

)

(a)

minus02

minus015

minus01

minus005

0

005

01

015

Join

t velo

city

(rad

s)

50 10 15 20Time (s)



(b)

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

Join

t acc

eler

atio

n (r

ads

2)

5 10 15 200Time (s)



(c)





Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





5 1510 200Time (s)



minus1

minus05

0

05

1

15

2

Join

t pos

ition

(rad

)

(a)

minus02

minus015

minus01

minus005

0

005

01

015

Join

t velo

city

(rad

s)

50 10 15 20Time (s)



(b)

minus05

minus04

minus03

minus02

minus01

0

01

02

03

04

Join

t acc

eler

atio

n (r

ads

2)

5 10 15 200Time (s)



(c)





Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




inverse kinematics of a 7r 6-dof robot with nonspherical...

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