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

doi:10.1007/s11465-021-0630-x

Front. Mech. Eng.

2021, Vol. 16

Issue (3) : 435-450 https://doi.org/10.1007/s11465-021-0630-x

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

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

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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.

Keywords 7-DOF redundant manipulator inverse kinematics semi-analytical arm angle link offset

Corresponding Author(s): Zainan JIANG

Just Accepted Date: 29 March 2021 Online First Date: 17 June 2021 Issue Date: 24 September 2021

Cite this article:

Boyu MA,Zongwu XIE,Zainan JIANG, et al. Precise semi-analytical inverse kinematic solution for 7-DOF offset manipulator with arm angle optimization[J]. Front. Mech. Eng., 2021, 16(3): 435-450.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-021-0630-x
https://academic.hep.com.cn/fme/EN/Y2021/V16/I3/435

Fig.1 Experimental module manipulator: (a) D–H frame and (b) initial configuration.

Link $i$	$a i − 1 (DH)$ /m	$α i − 1 (DH)$ /( $°$ )	$d i (DH)$ /m	$θ i (DH)$ /( $°$ )
1	0	90	$a 0$	0
2	0	90	$a 1$	0
3	0	–90	$a 2$	-90
4	$a 3$	0	$a 4$	0
5	$a 5$	0	$a 6$	90
6	0	90	$a 7$	0
7	0	–90	$a 8$	0
8	0	90	0	90

Tab.1 D–H parameters of the experimental module manipulator

Fig.2 Reference plane, arm plane, and arm angle of the experimental module manipulator.

Fig.3 Inverse kinematic solution diagram of the spherical–roll–spherical manipulator on the basis of arm angle parameterization.

Joint angle	EMM	SRS manipulator	Relationship
$θ 1$ , $θ 2$	$(a 0 + p x − a 8 n x) c 2 + [(a 8 n y − p y) c 1 + (p z − a 8 n z) s 1] s 2 + (a 2 + a 4 + a 6) = 0$	$(a 0 + p x − a 8 n x) c 2 ′ + [(a 8 n y − p y) c 1 ′ + (p z − a 8 n z) s 1 ′] s 2 ′ + (a 2 + a 4 + a 6) = 0$	$θ 1 = θ ′ 1$ , $θ 2 = θ ′ 2$
$θ 6$	$c 6 = − (n z s 1 − n y c 1) s 2 − n x c 2$	$c 6 ′ = − (n z s 1 ′ − n y c 1 ′) s 2 ′ − n x c 2 ′$	$θ 6 = θ ′ 6$
$θ 7$	${c 7 = (s z s 1 − s y c 1) s 2 + s x c 2 s 6, s 7 = − (a z s 1 − a y c 1) s 2 + a x c 2 s 6$	${c 7 ′ = (s z s 1 ′ − s y c 1 ′) s 2 ′ + s x c 2 ′ s 6 ′, s 7 ′ = − (a z s 1 ′ − a y c 1 ′) s 2 ′ + a x c 2 ′ s 6 ′$	$θ 7 = θ ′ 7$
$θ 4$	$c 4 = A 2 + B 2 − a 32 − a 52 2 a 3 a 5$	$c 4 ′ = A ′ 2 + B ′ 2 − a 32 − a 52 2 a 3 a 5$	The EMM $a 1 ≠ 0$ and $a 7 ≠ 0,$ and the SRS manipulator $a 1 = a 7 = 0$ . Therefore, $A ≠ A ′$ , $B ≠ B ′$ , $θ 4 ≠ θ ′ 4,$ $θ 3 ≠ θ ′ 3,$ and $θ 5 ≠ θ ′ 5 .$ $θ 345$ has no relationship with $a 1$ and $a 7$ ; hence, $θ 345 = θ 3 ′ 4 ′ 5 ′ .$
$θ 3$	${s 3 = A (a 3 + a 5 c 4) − B a 5 s 4 a 32 + a 52 + 2 a 3 a 5 c 4, c 3 = B (a 3 + a 5 c 4) + A a 5 s 4 a 32 + a 52 + 2 a 3 a 5 c 4$	${s 3 ′ = A ′ (a 3 + a 5 c 4 ′) − B ′ a 5 s 4 ′ a 32 + a 52 + 2 a 3 a 5 c 4 ′, c 3 ′ = B ′ (a 3 + a 5 c 4 ′) + A ′ a 5 s 4 ′ a 32 + a 52 + 2 a 3 a 5 c 4 ′$
$θ 5$	$θ 5 = (θ 3 + θ 4 + θ 5) − (θ 3 + θ 4) ? ? ? ? ? ? ? ? ? ? ? ? ? ? = arctan ? 2 (s 345, c 345) − (θ 3 + θ 4)$	$θ 5 ′ = (θ 3 ′ + θ 4 ′ + θ 5 ′) − (θ 3 ′ + θ 4 ′) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = a r c t a n 2 (s 3 ′ 4 ′ 5 ′, c 3 ′ 4 ′ 5 ′) − (θ 3 ′ + θ 4 ′)$
$θ 3 + θ 4 + θ 5$	${s 345 = − n z c 1 + n y s 1 s 6, c 345 = − [(s z c 1 + s y s 1) s 7 + (a z c 1 + a y s 1) c 7]$	${s 3 ′ 4 ′ 5 ′ = − n z c 1 ′ + n y s 1 ′ s 6 ′, c 3 ′ 4 ′ 5 ′ = − [(s z c 1 ′ + s y s 1 ′) s 7 ′ + (a z c 1 ′ + a y s 1 ′) c 7 ′]$

Tab.2 Relationship of joint angles between the experimental module manipulator (EMM) and the spherical–roll–spherical (SRS) manipulator

Inverse solution	$θ 1$ /(° )	$θ 2$ /(° )	$θ 3$ /(° )	$θ 4$ /(° )	$θ 5$ /(° )	$θ 6$ /(° )	$θ 7$ /(° )	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

Tab.3 Eight approximate solutions calculated by nominal arm angle

ψ = 135 °

and their actual arm angle

ψ ˜

in Case 1

Fig.4 Flowchart for the algorithm for correcting the arm angle errors.

Inverse solution	$θ 1$ /(° )	$θ 2$ /(° )	$θ 3$ /(° )	$θ 4$ /(° )	$θ 5$ /(° )	$θ 6$ /(° )	$θ 7$ /(° )	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

Tab.4 Eight precise solutions corrected to nominal arm angle

ψ = 135 °

and their actual arm angle

ψ ˜

in Case 1

Fig.5 Experimental module manipulator configuration of the first precise solution in Case 1.

Fig.6 Curves of actual arm angles

ψ ˜

correspond to

θ 1

in Case 1: (a)

θ 2 +

θ 4 −

θ 6 +

; (b)

θ 2 +

θ 4 −

θ 6 −

; (c)

θ 2 −

θ 4 −

θ 6 +

; (d)

θ 2 −

θ 4 −

θ 6 −

; (e)

θ 2 +

θ 4 +

θ 6 +

; (f)

θ 2 +

θ 4 +

θ 6 −

; (g)

θ 2 −

θ 4 +

θ 6 +

; and (h)

θ 2 −

θ 4 +

θ 6 −

Inverse solution	$θ 1$ /(° )	$θ 2$ /(° )	$θ 3$ /(° )	$θ 4$ /(° )	$θ 5$ /(° )	$θ 6$ /(° )	$θ 7$ /(° )	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.5 Eight approximate solutions calculated by nominal arm angle

ψ = 135 °

and their actual arm angle

ψ ˜

in Case 2

Inverse solution	$θ 1$ /(° )	$θ 2$ /(° )	$θ 3$ /(° )	$θ 4$ /(° )	$θ 5$ /(° )	$θ 6$ /(° )	$θ 7$ /(° )	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

Tab.6 Eight precise solutions corrected to nominal arm angle

ψ = 135 °

and their actual arm angle

ψ ˜

in Case 2

Fig.7 Experimental module manipulator of the first precise solution in Case 2.

Fig.8 Curves of actual arm angles

ψ ˜

correspond to

θ 1

in Case 2: (a)

θ 2 +

θ 4 −

θ 6 +

; (b)

θ 2 +

θ 4 −

θ 6 −

; (c)

θ 2 −

θ 4 −

θ 6 +

; (d)

θ 2 −

θ 4 −

θ 6 −

; (e)

θ 2 −

θ 4 +

θ 6 +

; (f)

θ 2 −

θ 4 +

θ 6 −

; (g)

θ 2 +

θ 4 +

θ 6 +

; and (h)

θ 2 +

θ 4 +

θ 6 −

Combination of joint angles	Type of interval	Interval of actual arm angle $ψ ˜$ /(° )
$θ 2 +$ , $θ 4 −$ , $θ 6 +$	Non-solution	$(16.1116, ? ? 26.2073) ∪ (168.8087, ? 172.7647)$
$θ 2 −$ , $θ 4 −$ , $θ 6 +$	Repetition-solution	$[16.1116, ? 26.2073] ∪ [168.8087, ? 172.7647]$
$θ 2 +$ , $θ 4 −$ , $θ 6 −$	Non-solution	$(31.4828, ? 36.0285) ∪ (151.4337, ? 162.3077)$
$θ 2 −$ , $θ 4 −$ , $θ 6 −$	Repetition-solution	$[31.4828, ? 36.0285] ∪ [151.4337, ? 162.3077]$
$θ 2 −$ , $θ 4 +$ , $θ 6 +$	Non-solution	$(− 166.8049, ? − 160.3144) ∪ (− 29.4358, ? − 17.4838)$
$θ 2 +$ , $θ 4 +$ , $θ 6 +$	Repetition-solution	$[− 166.8049, ? − 160.3144] ∪ [− 29.4358, ? − 17.4838]$
$θ 2 −$ , $θ 4 +$ , $θ 6 −$	Non-solution	$(− 167.1600, ? − 156.2214) ∪ (− 31.5175, ? − 25.6658)$
$θ 2 +$ , $θ 4 +$ , $θ 6 −$	Repetition-solution	$[− 167.1600, ? − 156.2214] ∪ [− 31.5175, ? − 25.6658]$

Tab.7 Changes in actual arm angle

ψ ˜

in the interval

[− π, ? π]

$a i$ $(i = 0, ? ? 1, ? ..., ? 8)$	Link length of the EMM
$a s i j$ $(i, ? j = 1, ? 2, ? 3)$	Element of the ith row and jth column of matrix $R 3 0$
$a w i j$ $(i, ? j = 1, ? 2, ? 3)$	Element of the ith row and jth column of matrix $EE R 4$
$d$	Vector which is the projection of vector $e$ on vector $w$
$e$	Vector from shoulder point S to elbow point E
$e 0$	Vector expressed by $e$ in base coordinate system $Σ 0$
$e 0 0$	Vector expressed by $e 0$ when arm angle $ψ = 0$
$I 3$	$3 × 3$ identity matrix
$k$	Vector which is on the plane that contains vectors $V$ and $w$ and perpendicular to $w$
$k^$	Unit vector of $k$
$l$	Multiplication cross vector of $V$ and $w$ , defined as $l = V × w$
$p$	Vector which is perpendicular to $w$ and passes through point E, defined as $p = e − d$
$p^$	Unit vector of $p$
$P E E 0$	Position vector of end coordinate system $Σ EE$ in base coordinate system $Σ 0$
$R i j$ $(i, ? j = 0, ? 1, ? ..., ? 7, ? EE)$	Rotation matrix of coordinate system $Σ i$ relative to coordinate system $Σ j$ (EE corresponds to end coordinate system $Σ EE$ )
$R 4 ψ = 0 0$	Rotation matrix of coordinate system $Σ 4$ relative to base coordinate system $Σ 0$ when $ψ = 0$
$R ψ 0$	Rotation matrix which represents the rotation of arm angle $ψ$ about vector $w$
$T i j$ $(i, ? j = 0, ? 1, ? ..., ? 7, ? EE)$	Homogeneous transformation matrix of coordinate system $Σ i$ relative to coordinate system $Σ j$
$[u l ×]$	Skew-symmetric matrix of vector $l$
$[u w ×]$	Skew-symmetric matrix of vector $w$
$V$	Unit vector parallel to the rotation axis of joint 1
$V 0$	Vector expressed by $V$ in base coordinate system $Σ 0$ , defined as $V 0 = [− 1 00] T$
$w$	Vector from shoulder point S to wrist point W
$w^$	Unit vector of $w$
$w 0$	Vector expressed by $w$ in base coordinate system $Σ 0$
$w 0 0$	Vector expressed by $w 0$ when arm angle $ψ = 0$
$x 3 4$ , $y 3 4$ , and $z 3 4$	Unit vectors of three coordinate axes of coordinate system $Σ 3$ relative to coordinate system $Σ 4$
$x 40 0$ , $y 40 0$ , and $z 40 0$	Unit vectors of three coordinate axes of coordinate system $Σ 4$ relative to base coordinate system $Σ 0$ when arm angle $ψ = 0$ , and $x 30 0$ has the same meaning
$α$	Angle from $w 0 0$ rotation to $e 0 0$
$β$	Angle between $S E ⊥$ and $E ⊥ W ⊥$
$ψ$	Nominal arm angle of the EMM
$ψ ˜$	Actual arm angle of the EMM
$θ i$ $(i = 1, ? 2, ? ..., ? 7)$	Joint angle of the EMM
$θ ′ i$ $(i = 1, ? 2, ? ..., ? 7)$	Joint angle of the SRS manipulator
$θ 2 −$ and $θ 2 +$	$θ 2$ calculated by Eqs. (4) and (5)
$θ 4 −$ and $θ 4 +$	Formulas of the negative and positive values in accordance with $θ 4$ calculated by Eq. (8)
$θ 6 −$ and $θ 6 +$	Formulas of the negative and positive values in accordance with $θ 6$ calculated by Eq. (6)
$Δ θ$	Stride of $θ 1$
$ε$	Iteration termination condition
$Σ i$ $(i = 0, ? 1, ? ..., ? 7, ? EE)$	Coordinate system of the manipulator

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