RESEARCH ARTICLE

Precise semi-analytical inverse kinematic solution for 7-DOF offset manipulator with arm angle optimization

  • Boyu MA ,
  • Zongwu XIE ,
  • Zainan JIANG ,
  • Hong LIU
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  • School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China; State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150001, China

Received date: 10 Oct 2020

Accepted date: 16 Jan 2021

Published date: 15 Sep 2021

Copyright

2021 The Author(s) 2021. This article is published with open access at link.springer.com and journal.hep.com.cn

Abstract

Seven-degree-of-freedom redundant manipulators with link offset have many advantages, including obvious geometric significance and suitability for configu-ration control. Their configuration is similar to that of the experimental module manipulator (EMM) in the Chinese Space Station Remote Manipulator System. However, finding the analytical solution of an EMM on the basis of arm angle parameterization is difficult. This study proposes a high-precision, semi-analytical inverse method for EMMs. Firstly, the analytical inverse kinematic solution is established based on joint angle parameterization. Secondly, the analytical inverse kinematic solution for a non-offset spherical–roll–spherical (SRS) redundant manipulator is derived based on arm angle parameterization. The approximate solution of the EMM is calculated in accordance with the relationship between the joint angles of the EMM and the SRS manipulator. Thirdly, the error is corrected using a numerical method through the analytical inverse solution based on joint angle parameterization. After selecting the stride and termination condition, the precise inverse solution is computed for the EMM based on arm angle parameterization. Lastly, case solutions confirm that this method has high precision, and the arm angle parameterization method is superior to the joint angle parameterization method in terms of parameter selection.

Cite this article

Boyu MA , Zongwu XIE , Zainan JIANG , Hong LIU . Precise semi-analytical inverse kinematic solution for 7-DOF offset manipulator with arm angle optimization[J]. Frontiers of Mechanical Engineering, 2021 , 16(3) : 435 -450 . DOI: 10.1007/s11465-021-0630-x

1 Introduction

Space robots play an irreplaceable role in human space exploration [14]. The Space Station Remote Manipulator System (SSRMS) [5], special purpose dexterous manipulator [6], and European Robotic Arm [7] applied to the International Space Station [8] are all 7-degree-of-freedom (7-DOF) redundant manipulators. Space robots also include the Chinese Space Station Remote Manipulator System (CSSRMS) [9], TianGong-2 manipulator [10], Robonaut 2 [11], Japanese Experiment Module Remote Manipulator System [12], and ETS-VII manipulator [13]. CSSRMS consists of a core module manipulator (CMM) and an experimental module manipulator (EMM) [14], which are also 7-DOF redundant manipulators with link offset. The EMM can not only be mounted at the end of the CMM to perform precise operations but also operate independently. They work together to maintain CSSRMS. The kinematics of these manipulators is particularly important when they are performing given tasks. The manipulator discussed in this study is the EMM.
Seven-DOF redundant manipulators have more joint DOFs than the dimension of its operating space. When they perform given tasks, additional constraints, namely, obstacle avoidance [15,16], singularity handling [17,18], joint torque optimization [19,20], and fault-tolerant control [2123], must be considered. However, the inverse kinematic solution of redundant manipulators is more difficult than that of non-redundant manipulators. The solution can be solved in the position domain [24] or velocity domain [25,26]. The precision of the position-domain solution is high, but multiple solutions need to be dealt with. The self-motion of manipulators can be controlled in the velocity domain by adopting the gradient projection method to optimize the given tasks [27,28]. However, the solution in the velocity domain produces several numerical errors. This study investigates the inverse kinematic solution of redundant manipulators in the position domain.
Seven-DOF redundant manipulators are classified into two types: Non-offset spherical–roll–spherical (SRS) and offset manipulators. For convenience, offset manipulators are referred to as EMMs. The shoulder and wrist joints of the SRS manipulator have no offset, and its shoulder and wrist are spherical joints. The EMM has offsets at the shoulder and wrist joints. Therefore, its inverse solution is relatively complex. A few extant studies have examined the inverse kinematics of EMMs.
Focusing on the SRS manipulator, Lee and Bejczy [29] proposed a method to solve the analytical inverse kinematic solution for redundant manipulators on the basis of joint angle parameterization, but the selection of joint parameters in this method is difficult. Zu et al. [30] proposed a quadratic calculation method based on the errors caused by the gradient projection method and the accuracy of the joint angle parameterization method. This method reaches the given trajectory and improves the precision of the end-effector. Kreutz-Delgado et al. [31] presented the concept of arm angle. The geometric significance of the arm angle parameterization method is more obvious than that of the joint angle parameterization method. The former is suitable for configuration control of redundant manipulators. However, this method has two algorithm singularities, namely, the reference plane and the arm plane are not unique. The elbows of the SRS manipulator and the EMM in this work have offsets; hence, the second singularity does not exist. Shimizu et al. [32] solved the inverse kinematics of the SRS manipulator via arm angle parameterization and reported that this method is suitable for joint limits. Xu et al. [33] proposed the dual arm angle to avoid the singularity that arises when the reference plane is not unique and derived the absolute reference attitude matrix of the elbow. Solving the inverse kinematics of the SRS manipulator on the basis of arm angle parameterization is more convenient. Zhou et al. [34] presented a method for the inverse kinematics of the SRS manipulator with joint and attitude limits. In accordance with the arm angle range corresponding to the restriction of each joint, they obtained the arm angle range of the manipulator and selected the best arm angle. Then, the inverse kinematics of the manipulator was solved in the position domain. Dereli and Koker [35] proposed a swarm optimization method called firefly algorithm to solve the inverse kinematics of redundant manipulators. Other methods can be found in Refs. [3638]. However, the previous method based on arm angle parameterization is only suitable for the SRS manipulator.
Focusing on the EMM, Crane et al. [39] proposed an inverse kinematic solution method based on spatial polygon projection. A 6-DOF subchain can be formed by parameterizing the value of joint angles θ 1, θ 2, or θ 7. By projecting it in space, eight solutions can be solved. This method is inconvenient because it involves the complex geometry of space. Yu et al. [40] presented a method using a virtual spherical wrist to replace the real offset wrist of a redundant manipulator with an offset wrist and assumed that the two wrists have the same joint angle. The attitude of the end-effector of the manipulator with a spherical wrist was calculated by using that of the manipulator with an offset wrist. Then, the inverse kinematics solution of the manipulator with a spherical wrist was solved. Simulations for a single target point and continuous trajectory verified this method. Abbasi et al. [41] used the arm angle rate as an additional task constraint and controlled SSRMS by adopting the augmented Jacobian method. Lu et al. [42] proposed a method in which the orientation matrices and arm angle do not need to change during the movement of a redundant manipulator with link offset. The method is based on damped least-squares, and the inverse kinematic solution is solved by iterative calculation in the velocity domain. This method was verified through a simulation in which the end-effector tracked a circle with a constant orientation and arm angle. Jin et al. [43] proposed an optimization algorithm for determining the optimal elbow orientation on the basis of particle swarm optimization and solved the inverse kinematics by using the relationship between elbow orientation and joint angles. Xu et al. [44] solved the inverse kinematic solution of the SRS manipulator through arm angle parameterization and generalized the results to the EMM (referred to as the SSRMS manipulator in their paper) in accordance with the relationship of joint angles between the EMM and the SRS manipulator. However, the inverse kinematic solution for the EMM was approximate based on arm angle parameterization. The arm angles of the eight solutions had uncontrollable errors.
The geometric significance of the arm angle parameteri-zation method is obvious, and it is suitable for configuration control and obstacle avoidance. The precise analytical solution cannot be solved based on arm angle parameteri-zation because of the particularity of the configuration for the EMM, but an approximate solution was solved in Ref. [44]. No previous study has presented a method to solve the precise solution for the EMM on the basis of arm angle parameterization. The current study proposes a high-precision, semi-analytical inverse solution method for the EMM. We define nominal arm angle ψ and actual arm angle ψ˜. The approximate solution based on arm angle parameterization is corrected by the analytical solution based on joint angle parameterization. Then, the precise solution is solved. The superiority of the arm angle parameterization method in parameter selection is verified using cases. The studied cases prove that this method has high precision.
This paper is organized in the following manner. In Section 2, we establish a model of the EMM and provide the analytical inverse kinematic solution based on joint angle parameterization. In Section 3, the approximate analytical inverse kinematic solution for the EMM is derived based on arm angle parameterization. In Section 4, the approximate solution is corrected by a numerical method, and the high precision of this method is verified by cases. In Section 5, the superiority of the arm angle parameterization method over the joint angle parameterization method in terms of parameter selection is analyzed.

2 Analytical solution based on joint angle parameterization

2.1 Kinematics modeling of the EMM

The configuration of the EMM used in this study is (R–Y–P)–P–(P–Y–R). a i (i=0, 1,..., 8) is the link length of the EMM. The axes of joints 3, 4, and 5 are parallel. Shoulder offset a 1 and wrist offset a 7 are not zero. The D–H frame [45] is shown in Fig. 1(a). The initial configuration of the EMM is shown in Fig. 1(b). The D–H parameters are listed in Table 1. Σi(i= 0,1, ..., 7,EE ) (EE corresponds to end coordinate system ΣEE) is the coordinate system of this manipulator. The axes of base coordinate system Σ0 and end coordinate system ΣEE are parallel to that of the hand-eye-camera coordinate systems at the beginning and end of this manipulator, respectively.
Fig.1 Experimental module manipulator: (a) D–H frame and (b) initial configuration.

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Tab.1 D–H parameters of the experimental module manipulator
Link i ai 1( DH)/m αi1(DH)/(° ) di (DH)/m θi(DH)/( ° )
1 0 90 a0 0
2 0 90 a1 0
3 0 –90 a2 -90
4 a3 0 a4 0
5 a5 0 a6 90
6 0 90 a7 0
7 0 –90 a8 0
8 0 90 0 90

Note: a i1 (DH) is the length of link, which is defined as the distance from z i1 to zi measured along xi 1 positive direction; αi1(DH) is the torsion angle of link, which is defined as the angle from z i1 to zi measured about xi 1 positive direction; di (DH) is the offset distance of link, which is defined as the distance from xi 1 to xi measured along zi positive direction; and θi(DH) is the rotation angle of link, which is defined as the angle from xi 1 to xi measured about zi positive direction. Link 1 is relative to coordinate system Σ0, and link 8 is relative to coordinate system Σ7.

2.2 Analytical solution of each joint angle

Define θ i (i=1, 2,..., 7) as the joint angle of the EMM. The homogeneous transformation matrix between the adjacent link coordinate systems of the manipulator is
T i i1=[ cosθisinθi0 ai1 sin θicos αi 1cos θicosαi1sinαi1 disinαi1 sinθisinαi1 cosθisinαi1 cosαi1 dicosαi1 00 01].
For convenience of expression, we define ci=cos θ i and si=sin θ i.
The forward kinematic equation of the manipulator is
TEE0= T00T10 T2 1T32 T4 3 T 5 4 T6 5T76 TEE7.
The homogeneous transformation matrix of end coordinate system ΣEE relative to base coordinate system Σ0 is
TEE0=[ nx s xa x px nysya y py nzsza z pz00 01],
where [ nxnyn z ]T, [s x sysz] T, and [ axaya z ]T are unit vectors represented by end coordinate system ΣEE in base coordinate system Σ0, and [p x pypz] T is the position of the origin of end coordinate system ΣEE in base coordinate system Σ0.
Given the value of θ1, the analytical inverse kinematic solutions based on joint angle parameterization are as follows.
θ2 has two solutions, namely,
θ2=arcsin (a 2+a4+ a6) a 2+ b2φ ,
θ2=πarcsin (a 2+a4+ a6) a 2+ b2φ ,
where a=a0 +pxa8nx, b=( a8n y py)c 1+(pz a8n z) s1, and φ=arctan2(a,b).
θ6 has two solutions, namely,
θ6=±arccos[ ( nz s1ny c1) s2 n xc 2] .
θ7 has only one solution, which is
θ7=arctan2( (az s1ay c1) s2 +axc 2 s6, (szs1s yc1)s 2+ sx c2s6) .
When s 6= 0, the axes of joints 5 and 7 are parallel, and the manipulator is in a singular configuration. Equation (7) cannot solve θ7. We have to solve it by s6=0 and c6=±1. Then, θ7 can take arbitrary values by solving it. Usually, θ7=0.
θ4 has two solutions, namely,
θ 4= ±arccos A2+B2 a 32 a522a3a5,
where
A=a7 [c 7 (( ayc1 azs1)c2+ ax s2)+s 7(( sy c1sz s1) c2+sxs2) ]+a8 [( ny c1nz s1) c2+nxs2]+(pzs 1pyc 1 )c 2 (p x+ a0)s2,
B= a1+a 7 [( ay s1+azc1)c7 +(sy s1+szc1)s7]+a8(nys1+n zc 1 )pys1p zc1 .
θ3 has only one solution, which is
θ3=arctan2[A (a 3+a5c4)Ba5s4,B( a3+ a5c 4 )+A a5s4].
θ5 has only one solution, namely,
c345=[( szc1+ sy s1) s7+( azc1+ ay s1) c7].
When s 6 0,
s 345= nzc1+ ny s1s6,
θ5=(θ3+ θ4+θ5 )(θ3+θ4 )=arctan 2( s345, c345)(θ3+ θ4),
where c345=cos( θ3+ θ4+θ5), s345=sin( θ3+ θ4+θ5), and θ345= θ3+θ4 +θ 5.
When s 6= 0, (θ 3+θ4+ θ5) can be solved in accordance with θ7 =0.
In summary, when θ1 or θ2 is given, we can solve each joint angle of the manipulator in accordance with T EE 0. Eight solutions are available.

3 Approximate analytical solution based on arm angle parameterization

The analytical solution cannot be solved based on arm angle parameterization because of the particularity of the configuration for the EMM. We can solve the analytical solution of the SRS manipulator with the same end position and attitude based on arm angle parameterization and generalize the results to the EMM in accordance with the relationship of joint angles between the EMM and the SRS manipulator. However, these results are approximate, and the errors are uncontrollable.

3.1 Definition of arm angle and arm plane

Shoulder point S is the intersection point of the rotation axes of joints 1 and 2, elbow point E is the origin of coordinate system Σ4, and wrist point W is the intersection point of the rotation axes of joints 6 and 7. In other words, S, E, and W are the origins of D–H coordinate systems Σ1, Σ4, and Σ6, respectively. e is the vector from S to E, and w is the vector from S to W. Vector V is a unit vector parallel to the rotation axis of joint 1, and it is defined as V 0=[ 1 0 0] T. Vector d is the projection of vector e on vector w. Vector p is perpendicular to w and passes through point E (defined as p= e d). Vector k is on the plane that contains vectors V and w and perpendicular to w.
The reference plane is the plane that contains vectors V and w. The arm plane is the plane that contains points S, E, and W. Arm angle ψ is the angle at which the reference plane rotates around vector w to coincide with the arm plane, as shown in Fig. 2.
Fig.2 Reference plane, arm plane, and arm angle of the experimental module manipulator.

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If the configuration of the manipulator is known, the arm angle at the current moment can be calculated. The calculation processes are as follows:
w= SW 0= w0= PEE0 [ a00 0 ]T REE0[ a8 00]T,
T e 0= T00 T10 T2 1T32 T4 3 T00 T10= [ ** *e x* ** e y* ** e z0 00 1 ],
e= S0E=e 0=[ exeye z ]T,
where P EE 0 and REE0 are the position vector and rotation matrix of end coordinate system ΣEE in base coordinate system Σ0, respectively, w 0 and e0 are vectors expressed by w and e in base coordinate system Σ0, respectively, * means that the calculation results are not listed.
The projection of vector e on vector w is
d=w^( w^T e), w^ = w w ,
where w ^ is the unit vector of w, denotes the norm of a vector.
The unit vector perpendicular to w on the arm plane is
p^= p p ,p= e d= (I3w^ w^T)e,
where p ^ is the unit vector of p, I3 is a 3×3 identity matrix.
The unit vector perpendicular to w on the reference plane is
k^=kk, k=w×(w× V) ,
where k ^ is the unit vector of k.
The expressions of arm angle ψ obtained from Fig. 2 are
{ cosψ =k^Tp^, sinψ=(w^× k^)T p^= w^ T (k^× p^).
Therefore, arm angle ψ can be calculated as follows:
ψ=arctan 2(sinψ,cosψ)=arctan2 ( w^T( k^ × p^), k^T p ^ ).

3.2 Analytical solution of the SRS manipulator based on arm angle parameterization

Shoulder offset a1 and wrist offset a7 of the SRS manipulator are zero. Define θ i (i=1, 2,..., 7) as the joint angle of the SRS manipulator. In the following derivation, the origin of coordinate system Σ4 is on the reference plane. When arm angle ψ=0, the upper right of each parameter is marked with 0 to distinguish it. Its inverse kinematic solution diagram is shown in Fig. 3 on the basis of arm angle parameterization.
Fig.3 Inverse kinematic solution diagram of the spherical–roll–spherical manipulator on the basis of arm angle parameterization.

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3.2.1 Analytical solution of θ 4

θ 4 has no relation to arm angle ψ. Two solutions of θ 4 can be solved in accordance with the geometric relationship. As shown in Fig. 3, points E and W are projected onto a plane perpendicular to the axis of joint 1, which passes through point S. The feet of perpendicular are E and W . β is the angle between SE and E W. Therefore, the geometric expressions are as follows:
{ SW2=SW2 WW2= w2 ( a2+a4 +a6)2 ,SW2=SE2+ EW22 SE EWcos β= a32+a5 22a3a5cosβ.
Two solutions of β can be solved as follows:
β=±arccosa3 2+a52+(a2+ a4+a6)2 w22 a3a 5.
Figure 3 shows that θ4+β=π, so θ 4 also has two solutions as follows:
θ 4=π±arccosa3 2+a52+(a2+ a4+a6)2 w22 a3a 5.

3.2.2 Analytical solution of R40

When arm angle ψ=0, the rotation matrix of coordinate system Σ4 relative to base coordinate system Σ0 is R4ψ= 0 0, and the rotation matrix that represents the rotation of arm angle ψ about vector w is Rψ 0. Rotation matrix R40 can be calculated as follows:
R 4 0= Rψ0 R4ψ=0 0.
Therefore, we need to calculate R 4ψ=0 0 and Rψ 0.
w can be solved from Eq. (13), and w 0 0= w. The expression of e 0 0 obtained from Fig. 3 is
0e 0=R(l,α)0 w 0 0w 0 e 0 0,
where l=V×w, e 0 02= a32+ (a 2+ a4)2, and R(l,α )=I3+[ u l×]sin α+[ u l× ]2(1 cos α). w 0 0 and e 0 0 are vectors expressed by w 0 and e0 when arm angle ψ=0, respectively. α is the angle from w 0 0 rotation to e 0 0. [ u l×] denotes the skew-symmetric matrix of vector l.
By using the cosine law for the triangle SE W, we obtain
a52+ a62= e 0 02+ w 0 02 2 e 0 0 w 0 0 cosα .
Referring to the configuration of the manipulator, we know that parameter α90°. Therefore, α has only one solution, that is,
α=arccos 0e 0 2+0 w 0 2 a5 2 a6 220 e 00 w 0.
These parameters are substituted into Eq. (25) to solve e 0 0.
In accordance with Fig. 3, the expressions can be obtained as follows:
{ a6z40 0+a 5 x 400= 0w 0 0e 0, (a2+ a4) z40 0+a 3 x 300= 0e 0,
where x40 0, z40 0, and y40 0 in the following derivation are unit vectors of three coordinate axes of coordinate system Σ4 relative to base coordinate system Σ0 when arm angle ψ=0, and x30 0 has the same meaning.
From the characteristic of the rotation matrix, we obtain
R34= R4 1 3= [ c4 s4 0 s4 c 4000 1 ] 1=[ x 3 4 y34 z3 4] ,
where x34, y34, and z34 are unit vectors of three coordinate axes of coordinate system Σ3 relative to coordinate system Σ4.
Therefore,
x30 0= R4ψ=0 0 x3 4=[ x 400 y40 0 z400][c 4 s4 0]= c4 x40 0 s4 y40 0 .
x30 0 is substituted into Eq. (28). Both sides of these equations are calculated by cross-multiplication. The new equation is simplified to
a3a6 s4x40 0+[ a3 a6c4 a 5( a2+a4)] 0y40 a3a5s4 z 400= w 0 0× e0 0.
In accordance with Eqs. (28) and (31), we obtain
{ x40 0=a3 a6 s4( w0 0× e0 0)a32 a5 s4 2( e0 0 w0 0)a 32 s4 2( a52+ a62)+[ a5 (a2+a4) a 3a 6c4]2 [a5( a2+a 4) a3a6c4] [a 6 e0 0+(a 2+a4) (e00 w0 0)]a 32 s4 2( a52+ a62)+[ a5 (a2+a4) a 3a 6c4]2,z40 0= w0 0 e0 0 a5 x40 0a6 , y40 0= z40 0× x 400= a3 c4 x40 0+( a2+a4) z 400 e 0 0 a3s4.
Therefore, the initial attitude matrix is
R 4ψ=0 0=[ x40 0 y400 z40 0] .
From the definition of R ψ0, we obtain
Rψ 0=I3+[ uw×]sinψ +[uw×] 2(1 cosψ),
where [uw ×] denotes the skew-symmetric matrix of vector w.
In summary, R 4 0 can be solved by Eq. (24).

3.2.3 Analytical solution of θ 1, θ 2, and θ 3

Joint angle θ4 of the SRS manipulator has been solved, so R 4 3 is a known quantity.
On the one hand, R 3 0 can be obtained as follows:
R 3 0= R4 0R413= Rψ 0R4ψ= 0 0 R4 1 3= [ as11 as12as13 as21 as22as23 as31 as32as33],
where asij(i ,j=1,2, 3) represent the element of the ith row and jth column of matrix R 3 0.
On the other hand, in accordance with forward kinematics,
R30= R00R10 R2 1R32=[ s2s 3 s2c3 c 2 s1c3 c1c 2 s3 s1s3 c1c 2 c3 c1s2 s 1 c2s3 c1c 3 c 1 s3+s 1 c2c3s1 s2].
Equations (35) and (36) are equal, and θ 1, θ 2, and θ 3 can be solved as follows:
θ 2=±arccos( a s13 ),
θ 1=arctan2(as33 s2,a s23s2 ) ,
θ 3=arctan2(as11 s2, as12s 2 ).

3.2.4 Analytical solution of θ 5, θ 6, and θ 7

On the one hand, EE R 4 can be obtained as follows:
EE R4= REE10R40= REE1 0 R ψ0R4ψ= 0 0= [ aw11 aw12aw13 aw21 aw22aw23 aw31 aw32aw33],
where awij(i ,j=1,2, 3) represent the element of the ith row and jth column of matrix EE R 4.
On the other hand, in accordance with forward kinematics,
EER4=( R5 4R65 R7 6 R EE 7) 1= [ s5s 6 c5s6 c6c 5 s7+s 5 c6c7s 5 s7c5 c6c 7 s6c7 c 5 c7s5 c6s 7 s 5 c7+c 5 c6s7s 6 s7].
Equations (40) and (41) are equal, and θ 5, θ 6, and θ 7 can be solved as follows:
θ 6=±arccos aw13,
θ 5=arctan2( aw11s6, aw12s6),
θ 7=arctan2( aw33s6, aw23s6).

3.3 Approximate analytical solution of the EMM based on arm angle parameterization

3.3.1 Relationship of joint angles between the EMM and the SRS manipulator

The difference between the EMM and the SRS manipulator is that offset a 1 and a 7 of the shoulder and wrist are not zero. In accordance with the solutions of the EMM based on joint angle parameterization in Section 2.2, we can obtain the relationship of joint angles between the EMM and the SRS manipulator, as shown in Table 2. With the same TEE0, joint angles θ1, θ2, θ6, and θ7 of the two types of manipulators are the same, but θ 3, θ4, and θ5 are different. θ3 +θ 4+ θ5 is also the same.
Tab.2 Relationship of joint angles between the experimental module manipulator (EMM) and the spherical–roll–spherical (SRS) manipulator
Joint angle EMM SRS manipulator Relationship
θ1, θ2 (a 0+pxa8 nx)c2 +[ (a 8 ny p y) c1+( pza 8 nz)s 1 ]s 2+( a2+a4 +a6)=0 (a 0+pxa8 nx) c2+ [( a8nypy )c 1+ (p z a8n z) s1]s 2+(a2+ a4+a6)=0 θ1= θ1, θ2= θ2
θ6 c6=(nzs 1nyc 1 )s 2 nx c2 c 6= (nzs1 ny c1) s2 nxc 2 θ6= θ6
θ7 { c 7= (s zs 1syc 1 )s 2 + sx c2s 6, s 7= (az s1ay c1) s2 + a xc2 s6 { c 7= (szs1 sy c1)s2 + s x c2s 6,s7=( azs1 a yc 1) s2 + ax c2 s6 θ7= θ7
θ4 c4=A2+B 2 a32 a522 a3a 5 c4 = A 2+ B 2 a32 a522 a3a 5 The EMM a 1 0 and a70, and the SRS manipulator
a1 =a7=0
. Therefore, AA, B B, θ4 θ4,θ3 θ 3, and θ5 θ 5.θ345 has no relationship with a1 and a7; hence, θ345= θ34 5.
θ3 { s 3=A(a 3+ a5c 4)Ba5 s4a3 2+a52+2a3a5c4,c3=B( a3+a5 c4)+A a5s 4 a32+a5 2+2a3a5c4 { s 3= A(a 3+a5c4) Ba5s4 a32+a 52+2a 3 a5c 4,c3= B(a3+a 5 c4)+ Aa5s4 a32+a 52+2a 3 a5c 4
θ5 θ5=(θ3+ θ4+θ5 )(θ3+θ4 )=arctan2(s345 , c 345)(θ3 +θ 4) θ5 =(θ3+θ4+θ5)( θ3 +θ 4) = ar ct an 2( s345, c34 5 )(θ3 +θ 4)
θ3+ θ4+θ5 { s 345= nzc 1+nys1 s6, c 345=[( szc1+ sy s1)s7 +(a zc 1+ays1) c7] { s345= nzc1 +nys 1s 6, c3 45=[( szc1 +sys 1)s 7 +(a zc1+a ys1)c 7]

Note: A= a7[c7((ayc1a zs1 )c2+ ax s2) +s 7( (syc 1szs 1)c2+ sx s2)]+a 8[(n yc1 n zs 1)c2+ nx s2] +(p zs1 p yc 1)c 2 (px+a0 )s 2, B= a 1+ a7[(ays 1+azc1)c7 +( sys1+ sz c1) s7]+ a8( ny s1+nzc1) py s1pz c1; A=a 8 [(nyc1 nz s1) c2+ nx s2]+(pzs1 py c1)c2 (px+ a0) s2, B=a 8(nys1 +nzc 1)pys1 pz c1.

3.3.2 Approximate analytical solution of each joint angle

In accordance with the analysis in Table 2 and the solutions of the SRS manipulator based on arm angle parameterization in Section 3.2, the analytical inverse kinematic solution of the EMM based on arm angle parameterization can be obtained as follows:
θ1=arctan2(as33 s2, as23s 2 ),
θ2=±arccos( a s13 ),
θ6=±arccos aw13,
θ7=arctan2( aw33s6,aw23s6),
θ4=±arccos A2+B 2a32 a522 a3a 5,
θ3=arctan2[A (a 3+a5c4)Ba5s4,B( a3+a5 c4)+ Aa5s4],
θ5=(θ 3+θ4+ θ5)(θ 3+θ4)=arctan2( s345, c345)(θ3 +θ 4).
The expressions of related parameters are shown in Table 2. The symbol of θ 4 must be consistent with that of θ 4.
Given the value of arm angle ψ, eight solutions of the EMM can be solved based on arm angle parameterization. θ3, θ4, and θ5 calculated by these formulas are different because the offsets of the two manipulators are different. Therefore, the actual arm angles corresponding to the eight solutions are not the given ψ. These errors are uncontrollable.
We define the given ψ as the nominal arm angle, and ψ˜ corresponding to the solution calculated by ψ is called the actual arm angle.

3.3.3 Case analysis

The position and attitude of the end-effector are given as follows:
TEE0=[ 0.73890.3815 0.55541.4348 0.33000.9236 0.19531.96980.5875 0.03900.80831.03020 00 1 ].
Given nominal arm angle ψ=135°, eight solutions of the EMM can be solved in accordance with Section 3.3.2, and actual arm angle ψ˜ corresponding to each solution can be calculated in accordance with Section 3.1, as shown in Table 3.
Tab.3 Eight approximate solutions calculated by nominal arm angle ψ=135°and their actual arm angle ψ˜ in Case 1
Inverse solution θ1/(° ) θ2/(° ) θ3/(° ) θ4/(° ) θ5/(° ) θ6/(° ) θ7/(° ) Actual arm angle ψ˜/(° )
1 –75.0144 77.9015 –35.0088 –67.7947 –106.9448 108.4550 78.4236 129.7819
2 –75.0144 77.9015 –32.9623 –93.9476 97.1616 –108.4550 –101.5764 130.1939
3 104.9856 –77.9015 145.4975 –44.7304 –130.5154 108.4550 78.4236 142.4985
4 104.9856 –77.9015 157.5737 –83.3832 76.0612 –108.4550 –101.5764 142.5272
5 65.0858 129.3636 –170.4417 76.7806 –83.3659 87.0906 –58.9323 126.6470
6 65.0858 129.3636 –160.6816 39.4032 124.2514 –87.0906 121.0677 125.3823
7 –114.9142 –129.3636 12.8840 96.8902 –106.8012 87.0906 –58.9323 140.8476
8 –114.9142 –129.3636 10.8205 75.3257 96.8267 –87.0906 121.0677 140.3597
The actual arm angles calculated by eight solutions have errors. The maximum actual arm angle error is 9.6177°, and its relative error is up to 7.12%. This error cannot be ignored when precise control is required. Therefore, for the approximate inverse solution of the EMM based on arm angle parameterization, we need to correct actual arm angle ψ˜ in reference to nominal arm angle ψ. Then, we can calculate eight precise inverse solutions.

4 Precise semi-analytical solution based on arm angle parameterization

4.1 Solution process

We propose a numerical method to correct the solution errors of the EMM based on arm angle parameterization in Section 3 so that actual arm angle ψ˜ is infinitely close to nominal arm angle ψ. Eight precise solutions can be solved by this method. The algorithm flowchart is shown in Fig. 4.
Fig.4 Flowchart for the algorithm for correcting the arm angle errors.

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The analytical inverse kinematic solution is solved based on joint angle parameterization in Section 2.2. Given an arbitrary value of θ1, we can calculate the values of θ2, θ3, …, θ7. Thus, θ2, θ3, …, θ7 are all functions of θ1.
θj=f(θ 1),j=2, 3, ..., 7,
where f( ) is a nonlinear mapping function of each joint angle θj(j=2, 3, ..., 7) corresponding to θ1 in Section 2.2.
When the configuration of the manipulator is known, we can calculate actual arm angle ψ˜ as
ψ˜ =g(θ1 , θ2 , θ3 , θ4 , θ5 , θ6 , θ7 ),
where g( ) is a nonlinear mapping function with a known configuration for the manipulator.
When we correct the arm angle error of each solution, we retain the value of θ1 for this solution. In accordance with the actual precision requirement, we select stride Δθ of θ1. When θ1± Δθ, joint angles θ2, θ3, …, θ7 and actual arm angle ψ˜ can be calculated. The formulas for θ2, θ4, and θ6 depend on the symbols of the corresponding angles calculated by Eqs. (4) or (5), (8), and (6), respectively. θ3, θ5, and θ7 are calculated by Eqs. (9), (12), and (7), respectively. The change trend of actual arm angle ψ˜ is the trend that is close to or far from nominal arm angle ψ because actual arm angle ψ˜ changes monotonously in a 2π period of θ1 (analysis in Section 4.2). By comparing the magnitude of |ψ˜ψ| in two cases, the change trend of actual arm angle ψ˜ close to ψ is selected. Through stride Δθ of θ1, actual arm angle ψ˜ is updated continuously. The value of termination condition ε is selected reasonably. When | ψ˜ψ|<ε, the calculation stops. In other words,
lim Δ θ0|ψ ˜ ψ| =0.
Then, actual arm angle ψ˜ and θ1, θ2, …, θ7 are recorded at this moment. θ1, θ2, …, θ7 are the precise joint angles after the arm angle error correction. The errors of the eight solutions are corrected. Then, we obtain eight precise inverse kinematic solutions based on arm angle parameterization.
A summary of the approach is presented as follows:
Step 1: Given nominal arm angle ψ, select stride Δθ and termination condition ε in accordance with the actual situation. Select joint angle θ1 of each solution.
Step 2: Calculate actual arm angle ψ˜ when θ1± Δθ. Select the trend where ψ˜ is close to ψ.
Step 3: In accordance with this trend, calculate joint angles θ2, θ3, …, θ7 and ψ˜ through θ1 and Δθ.
Step 4: Change the value of θ1 through Δθ and go to Step 3. Exit until ε reaches its precision.
Step 5:Record θ1, θ2, …, θ7, and ψ˜.

4.2 Case correction and analysis

Given stride Δθ=0.001° and termination condition ε=0.0006°, eight approximate inverse kinematic solutions obtained by solving the case in Section 3.3.3 are corrected, as shown in Table 4. The manipulator configuration of the first precise solution is shown in Fig. 5.
Tab.4 Eight precise solutions corrected to nominal arm angle ψ=135°and their actual arm angle ψ˜ in Case 1
Inverse solution θ1/(° ) θ2/(° ) θ3/(° ) θ4/(° ) θ5/(° ) θ6/(° ) θ7/(° ) Actual arm angle ψ˜/(° )
1 –79.6594 80.0057 –31.7645 –68.5655 –107.4127 112.4957 81.6077 134.9996
2 –79.2564 79.8118 –29.8156 –94.8141 96.7095 –112.1390 –98.6737 135.0003
3 111.1286 –75.5521 143.0353 –46.8091 –128.5043 103.3635 74.3511 135.0000
4 111.4466 –75.4437 154.0291 –84.2827 77.8483 –103.1072 –105.8565 134.9999
5 57.8508 124.0487 –170.7387 75.5782 –86.7634 81.7248 –53.1516 135.0002
6 56.9938 123.3800 –162.8315 40.3714 119.9489 –81.0841 127.5950 134.9995
7 –109.5662 –132.8739 14.2388 97.0154 –104.6965 90.9200 –62.6529 134.9997
8 –109.8352 –132.7062 12.9465 74.4857 98.9454 –90.7312 117.5244 135.0001
Fig.5 Experimental module manipulator configuration of the first precise solution in Case 1.

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The actual arm angle ψ˜ of the eight solutions are all infinitely close to nominal arm angle ψ=135°. The maximum error is reduced from 9.6177° to 0.0005°. The minimum error is 0, which satisfies the actual precision requirement. The eight solutions are the precise inverse kinematic solutions of the EMM based on arm angle parameterization.
For analysis convenience, θ2 calculated by Eqs. (4) and (5) are denoted as θ2 and θ2+, respectively. When θ4 and θ6 are calculated by Eqs. (8) and (6), the formulas of the positive values are denoted as θ4 + and θ6+, and the formulas of the negative values are denoted as θ4 and θ6. In Table 4, the curves for the actual arm angles ψ˜ of the eight solutions corresponding to θ1 [π, π] are shown in Fig. 6. Regardless of which group of formulas is used for calculation, actual arm angle ψ˜ changes monotonously in the interval [ π, π]. When we provide the value of nominal arm angle ψ, θ1 has only one solution. Therefore, only eight inverse kinematic solutions exist for the EMM based on arm angle parameterization. The method proposed in this section is effective.
Fig.6 Curves of actual arm angles ψ˜ correspond to θ1 in Case 1: (a) θ2+, θ4, θ6+; (b) θ2+, θ4 , θ6; (c) θ2, θ4, θ6+; (d) θ2, θ4, θ6; (e) θ2+, θ4+, θ6 +; (f) θ2+, θ4+, θ6 ; (g) θ2, θ4+, θ6+; and (h) θ2, θ4+, θ6 .

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5 Superiority of the arm angle parameterization method in parameter selection

In Section 2, we solve the inverse kinematics of the EMM based on joint angle parameterization. If we provide the homogeneous transformation matrix of the end-effector in the reachable operating space, several given θ1 [ π, π] cannot solve the other six joint angles. With θ2 as an example, when θ 2 is calculated by Eqs. (4) and (5), the condition must be satisfied as follows:
1 (a2 +a4+ a6) a2+ b21.
If θ 1 does not satisfy this condition, the manipulator will be unsolvable. Therefore, the joint angle parameterization method has limitations on the given joint angle.
However, the arm angle parameterization method does not have these limitations on the given arm angle. In the case analysis in Section 3.3.3, given the homogeneous transformation matrix of the end-effector by Eq. (52), arbitrary θ1 [ π, π] can solve the other six joint angles. As shown in Fig. 6, arm angle ψ˜ and joint angle θ1 exhibit one-to-one correspondence in the interval [π, π].
When several given θ1 [ π, π] cannot solve the other six joint angles, we give the position and attitude of the end-effector as follows:
TEE0=[ 0.5187 0.41840.74560.87860.6013 0.44150.66601.70190.60780.79380.02261.17830 00 1 ].
When the EMM has solutions, the range of θ1 can be solved by Eq. (56) as follows:
θ1 [180°,137.1457 °] [79.2755°, 40.8543°]
[100.7245°, 180°].
Given nominal arm angle ψ=135°, eight approximate solutions can be solved in accordance with Section 3.3.2, as shown in Table 5. The actual arm angles calculated by the eight solutions have errors. The maximum actual arm angle error is 12.2831°, and its relative error is up to 9.10%. Given stride Δθ=0.001° and termination condition ε=0.002°, the correction results in accordance with Section 4.1 are shown in Table 6. The maximum error of actual arm angle ψ˜ is reduced from 12.2831° to 0.0020°. The minimum error is 0.0004°, which already satisfies the actual precision requirement. The manipulator configuration of the first precise solution is shown in Fig. 7.
Tab.5 Eight approximate solutions calculated by nominal arm angle ψ=135°and their actual arm angle ψ˜ in Case 2
Inverse solution θ1/(° ) θ2/(° ) θ3/(° ) θ4/(° ) θ5/(° ) θ6/(° ) θ7/(° ) Actual arm angle ψ˜/(° )
1 –74.5529 41.1411 59.0447 –147.0799 39.7875 96.3458 –25.6682 135.8676
2 –74.5529 41.1411 33.3418 –123.9961 –137.5933 –96.3458 154.3318 122.7169
3 105.4471 –41.1411 –127.4815 –121.8101 21.0440 96.3458 –25.6682 146.4159
4 105.4471 –41.1411 –151.1755 –97.6415 –159.4307 –96.3458 154.3318 136.0138
5 25.7588 113.7663 103.5857 97.3836 139.3401 121.8922 98.1554 138.8542
6 25.7588 113.7663 85.7979 125.7015 –51.1899 –121.8922 –81.8446 131.6935
7 –154.2412 –113.7663 –99.9945 121.7516 138.5523 121.8922 98.1554 139.3173
8 –154.2412 –113.7663 –124.9401 145.1176 –39.8680 –121.8922 –81.8446 125.0305
Tab.6 Eight precise solutions corrected to nominal arm angle ψ=135°and their actual arm angle ψ˜ in Case 2
Inverse solution θ1/(° ) θ2/(° ) θ3/(° ) θ4/(° ) θ5/(° ) θ6/(° ) θ7/(° ) Actual arm angle ψ˜/(° )
1 –74.2549 40.5634 58.1106 –147.0877 40.4691 96.8771 –25.9693 135.0011
2 -77.9949 51.0466 43.4482 –125.4135 –143.7159 –87.7922 159.8987 134.9980
3 113.0101 –31.4347 –137.9357 –123.3949 26.0090 105.8662 –30.5921 135.0012
4 105.9051 –40.2642 –151.8906 –97.6617 –159.0967 –97.1529 153.8738 135.0007
5 23.3078 116.7033 100.6682 97.9066 140.0811 118.3647 96.9436 135.0007
6 27.5818 111.2742 88.3354 125.4589 –52.2753 –124.7980 –80.9160 134.9985
7 –156.4982 –116.4861 –103.1285 121.4000 140.5163 118.6301 97.0378 135.0014
8 –150.2092 –107.8211 –117.2833 146.4267 –46.1905 –128.7216 –79.7563 135.0004
Fig.7 Experimental module manipulator of the first precise solution in Case 2.

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The curves for the actual arm angles ψ˜ of the eight solutions corresponding to θ1 [ π, π] are shown in Fig. 8. For this case, the change interval of θ1 is not continuous. For the eight groups of formulas, the changes in actual arm angle ψ˜ in the interval [π, π] are shown in Table 7.
Fig.8 Curves of actual arm angles ψ˜ correspond to θ1 in Case 2: (a) θ2+, θ4, θ6+; (b) θ2+, θ4 , θ6; (c) θ2, θ4, θ6+; (d) θ2, θ4, θ6; (e) θ2, θ4+, θ6+; (f) θ2, θ4+, θ6 ; (g) θ2+, θ4+, θ6 +; and (h) θ2+, θ4+, θ6 .

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Tab.7 Changes in actual arm angle ψ˜ in the interval [π, π ]
Combination of joint angles Type of interval Interval of actual arm angle ψ˜/(° )
θ2+, θ4, θ6+ Non-solution (16.1116,26.2073 )(168.8087,172.7647)
θ2, θ4, θ6+ Repetition-solution [16.1116 ,26.2073][ 168.8087,172.7647]
θ2+, θ4, θ6 Non-solution (31.4828,36.0285)(151.4337,162.3077)
θ2, θ4, θ6 Repetition-solution [31.4828 ,36.0285][ 151.4337,162.3077]
θ2, θ4+, θ6+ Non-solution (166.8049,160.3144) (29.4358 ,17.4838)
θ2+, θ4+, θ6 + Repetition-solution [166.8049,160.3144][29.4358, 17.4838]
θ2, θ4+, θ6 Non-solution (167.1600,156.2214) (31.5175 ,25.6658)
θ2+, θ4+, θ6 Repetition-solution [167.1600,156.2214][31.5175, 25.6658]
Therefore, for ψ˜[ π, π], the arm angles ψ˜ solved by four groups of formulas have non-solution intervals and those solved by the four other groups of formulas have repetition-solution intervals.
When formulas θ4 and θ6+ are used for calculation, θ2+ corresponds to the non-solution interval of ψ˜, and θ2 corresponds to the repetition-solution interval of ψ˜. The ranges of the two intervals are the same. Therefore, when ψ˜ takes an arbitrary value in the interval [π, π], although the change intervals of ψ˜ are not continuous, two solutions can be solved after the two situations are combined. Similarly, we combine θ2+ and θ2 corresponding to θ4 and θ6, θ4+ and θ6 +, and θ4+ and θ6. When ψ˜ takes an arbitrary value in the interval [π, π], two solutions can also be solved for each situation. For the eight situations, we can solve the eight solutions of the EMM for arbitrary ψ˜[ π, π].
In summary, when we provide the arbitrary position and attitude of the end-effector in the reachable operating space, several given θ1 [π, π] cannot solve the EMM based on joint angle parameterization. However, arbitrary ψ˜[ π, π] can solve the EMM based on arm angle parameterization. That is, the arm angle parameterization method has great superiority over the joint angle parameterization method in parameter selection.
The approximate solutions are solved for the EMM based on arm angle parameterization in Section 3.3.3. The numerical correction method uses θ 1 of the approximate solutions as the initial calculation value. By selecting the appropriate stride Δθ, we can solve the precise solutions quickly. If the initial value θ1 of calculation is selected arbitrarily, for the situation where arbitrary θ1 [π, π] can solve the other six joint angles, the amount of calculation will increase greatly although the precise solutions can be calculated by the correction method. The efficiency will be decreased greatly. For the situation where several given θ1 [π, π] cannot solve the other six joint angles, the change intervals of θ1 are not continuous. If the calculation reaches the end of an interval, it will be stopped, and the program will report errors. Therefore, the precise solutions cannot be calculated in this situation.
When stride Δθ is too small, although the method has high precision, the amount of calculation increases greatly. In general, the higher the resolution we select, the lower the efficiency and the more optimal the solution we obtain. In accordance with the actual situation, we select stride Δθ that not only satisfies the precision of calculation but also makes efficiency the highest.

6 Conclusions

This study proposes a high-precision, semi-analytical inverse method for 7-DOF redundant manipulators with link offset on the basis of arm angle parameterization. Previous studies could not solve precise inverse solutions by using arm angle parameterization because of the particularity of the configuration for the EMM, and the solutions that generalize the analytical results of the SRS manipulator to the EMM are approximate. In addition, the errors are uncontrollable. We propose a semi-analytical method and define the nominal and actual arm angles. The approximate solutions based on arm angle parameterization are corrected by the analytical solution based on joint angle parameterization. After selecting the stride and termination condition in accordance with the actual situation, the precise inverse solutions are calculated based on arm angle parameterization. The high precision of this method is verified by cases, and the minimum error approaches zero. The arm angle parameterization method has great superiority over the joint angle parameterization method in parameter selection.

Acknowledgements

This work was supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 51521003) and the Major Research Plan of National Natural Science Foundation of China (Grant No. 91848202).

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