Strojniški vestnik - Journal of Mechanical Engineering 63(2017)12, 715-724 © 2017 Journal of Mechanical Engineering. All rights reserved.
D0l:10.5545/sv-jme.2017.4529	Original Scientific Paper
Received for review: 2017-04-24 Received revised form: 2017-08-30 Accepted for publication: 2017-09-05
Rapid and Automatic Zero-Offset Calibration of a 2-DOF Parallel Robot Based on a New Measuring Mechanism
Jiangping Mei - Jiawei Zang - Yabin Ding*- Shenglong Xie - Xu Zhang
Ministry of Education, Tianjin University, Key Laboratory of Mechanism Theory and Equipment Design, China
This paper deals with the rapid and automatic zero-offset calibration of a 2-DOF parallel robot using distance measurements. The calibration system is introduced with emphasis on the design of a new measuring mechanism. A simplified error model of the robot is proposed after the sensitivity analyses of source errors, based on which a zero-offset identification model is developed using the truncated singular value decomposition (TSVD) method, and then it is modified with the manufacturing and assembly errors of the measuring mechanism (MAEMM). Furthermore, an optimization approach for selecting measurement positions is proposed by considering the condition number of the identification matrix. Finally, simulations and experiments are carried out to verify the effectiveness of the zero-offset calibration method. The results show that the identification model has good identifiability and robustness, and the position accuracy after calibration can be significantly improved.
Keywords: parallel robot, calibration, zero offset, measuring mechanism Highlights
•	A new measuring mechanism is designed for the zero-offset calibration of a 2-DOF parallel robot.
•	The zero-offset identification model is truncated by the TSVD method and modified with the MAEMM.
•	The measurement positions are optimized based on the condition number of the identification matrix.
•	Simulations and experiments are carried out to verify the effectiveness of the calibration method.
0 INTRODUCTION
Parallel robots have been widely used in many fields. This can be exemplified by the well-known Delta robot [1], including many applications of its modified versions, [2] to [4]. In recent years, the 2-DOF translational parallel robots has drawn ongoing interest from academia and industry due to their compact configurations and high stiffness, such as the very successful 4-PP [5] and 4-PP-E [6] simple decoupled XY parallel robots with enhanced stiffness, the Diamond [7] for high speed operation and the large-workspace 2-DOF parallel robot for solar tracking systems [8].
Position errors of parallel robots are mainly caused by their zero offsets, i.e. the errors between the nominal and actual initial positions of active links (see Fig. 1), provided that adequate fundamental geometric accuracy can be achieved at the manufacturing and assembly levels, [9] and [10]. The zero offsets may be caused by the control faults, collisions, or looseness of active joints at any time in practical applications. Therefore, to ensure the position accuracy, it is necessary to eliminate the zero offsets when they occur.
It is well recognized that the zero-offset calibration, one of the kinematic calibrations, is a practical and economical w y to reduce zero offsets, [11] and [12]. The zero-offset calibration pays more
attention to the calibration of the zero offsets than the geometric errors. Furthermore, a fine calibration of the zero offsets is the premise to ensure the calibration accuracy of the geometric errors [13]. In general, the calibration can be implemented by four sequential processes, i.e. error modelling, measurement, identification and compensation such that the zero offsets affecting the position accuracy can be suppressed [14].
a) Active link Actual initial position
Zero offset!
	r IRi"*- 1
Base	1 '
	j/tu ■ ■ ■ ■ ■
b)
Nominal initial position
^^Zero offset I Actual initial position Active link
/**•.... ........
Nominal initial position**.......■*■'
Fig. 1. Zero offsets of parallel robots; a) parallel robot with revolute joint; and b) parallel robot with prismatic joint
The methods of the zero-offset calibration can be classified into self/autonomous calibration [15] and external calibration [16]. Compared with the self/autonomous calibration that realizes the
*Corr. Author's Address: Tianjin University, Key Laboratory of Mechanism Theory and Equipment Des ign, No. 92 Weijin Road, Nankai District, Tianjin, China, dingyabin@126.com
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identification of the zero offsets through minimizing the discrepancies between the measured and computed values of joint space sensors, the external calibration finishes the same work using task space sensors. Furthermore, the external calibration can be classified into the coordinate-based approach and the distance-based approach, [17] to [20]. In comparison with the coordinate-based approach, the advantage of the distance-based approach lies in that it is invariant with the chosen reference frame. Hence, it has been widely applied for the calibrations. For the data acquisition during the measurement process, it is usually implemented using a large metrology device, e.g. a laser tracker or interferometer, which is costly and inconvenient to use. Meanwhile, to ensure the identifiability, the number of measurement positions usually tends to be overlarge, which reduces the measurement efficiency. Therefore, the problem of how to make the measurement process in a time and cost-effective manner needs to be further studied.
The identification is the kernel process of calibration, and it is usually implemented using the least square (LS) method [21]. However, if the zero offsets are identified together with too many geometric errors, it may le d to a sharp increase in the condition number of the identification matrix and thereby cause the nonlinear ill-conditioning problem for identification model. To solve this problem, the ridge estimation (RE) method and the truncated singular value decomposition (TSVD) method have been widely adopted [22] and [23]. Some studies have indicated that the TSVD has better identification accuracy than the LS does, and it is easier to implement than the RE is. Though the nonlinear ill-conditioning problem can be solved to some extent by the RE or TSVD, the problem of how to further improve the identification accuracy of the zero offsets needs to be thoroughly investigated.
This paper deals with the rapid and automatic zero offset calibration of a 2-DOF parallel robot [24]. We focus on: 1) the design of a new measuring mechanism to make the measurement process in a time and cost-effective manner; 2) the development of a simplified error model containing the zero offsets of the robot; 3) the development of an identification method to solve the nonlinea ill-conditioning problem and improve the identifiability; 4) the selection of optimal measurement posit ons to further improve the identifiability and the measurement efficiency. Simulations and experiment are also carried out to validate the proposed calibration method.
1 SYSTEM DESCRIPTION
As shown in Fig. 2, the 2-DOF parallel robot is revolute jointed. Driven by two active proximal links, the robot can provide its moving platform with a 2-DOF translational moving capability.
Fig. 3 shows the new measuring mechanism which mainly consists of two revolute joints, two guide rods, a shipper rod and a linear scale. The two guide rods and the linear scale are arranged in parallel and fixed on two connecting plates. The shipper rod and the reading head of the linear scale are fixed on a slider which is vertically conneeted tothe trev gui de rods by linear bearings. The revolute joints 3 and 4 are fixed on the upper connecting pltte and leg end of the shipper rod, respectively, based on which the m eaturing mechanism can be connected to the base aed themovingplatformof the robot.
By letting ttie moving platform undergo several measurement positions, the distance changes between the revolute joints 3 an e 4 can be automatically obtained by the reading head and then transferred into the zero-offset calibration model in the robot controller. Thus, the zero offsets can be rapidly calibrated.
1	- Revolute joint 1
2	- Rcveluic joini 2
3	- Pasajvc proximal link
4	- Active proximal link
5	- Rotation shaft 1	7
6	- Rotation shaft 2 g
7	- Distal links
8	- Measuring mechanism
9	- Moving platform 9
Fig. 2. 3D model of the 2-DOF parallel robot
Linear scale , Lower connecting plate^ Revolutejoint 4 Reading head
Revclate jciet 3
Upper connecting plate Slider Shipper rod Guide rods Fig.3. 3D model of the measuring mechanism
2 KINEMATIC ANALYSES
The 2-DOF parallel robot can be simplified as shown in Fig. 4.Inthe O-xy coordinate system,the nominal position vector, r = (x y)T, of the reference point P can be written as:
r = ei + LOi + h wi, x =1,2	(1)
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where Lh l„ u, and w, are the nominal lengths and nominal unit orientation vectors of the proximal and distal links, respectively; e, is the nominal position vector of A{, and
u - (cosO, sine,)7,	sinç>,)T
e — (e e )T,
7 y ix iy ' 1
(2)
where 0, and (p, are the nominal rotation angles of the proximal and distal links, respectively.
Taking 2-norm on the two sides of Eq. (1), the solution of theinverse positional aialysiscan tiien be expressed as:
-C -JC2 - D2 + E2
= 2 arctan —-i-¡_,	(3)
D - E
i 1
where C, - -2L (y - eiy ), Ü = -2L (x - eix ) E.=(x-e. )2+(v-e. f+L-I2..
1 V ix /	iv /	i 1
Hence, w, and the position vector from Ol to 02, denoted by /. can be calculated as follows:
W = r - e. - L, À = r - c - d,	(4)
e
where c is the position vector from O to Ox; d is the position vector from O2 to P.
1 1 \ r

[0l
b
Fig. 4. Kinematic model of the 2-DOF parallel robot (Note: Ax (A2) is the nominal rotation centre of the revolute joint 1 (2); B1 (B2) is thenominal rotationcentreoftherotationshaftl (2); O1 (O2) is the nominal rotation centre of the revolute joint 3 (4); P is a reference point at the centre of the moving platform; Wt is the workspace; H is the distance between O and the upper boundary of the workspace; h is the height of the workspace; b is the width of the workspace)
To develop the forward positional model, rewrite Eq. (1) as:
rT r - 2(e, + Liui )T r + (e, + L u, )T(ei + Ltut ) = . (5)
Subtract the two equations in Eq. (5) with each other yields:
x = -
My + S F '
(6)
where F = 2[(e2 + L2u2)T -(e1 + L1u1)T]a1, M = 2[(e2 + L2U2)T - (ei + LUi)T ]a2,
5 = ||ei + Liui\f -||e2 + L^f -(/i2 -122), ai = (1 0)T, a2 = (0 1)T.
Substitute Eq. (6) into Eq. (5), then the quadratic equation ofy can be written as:
Ny2 + Q + Rt = 0,
where
R. = —— + 2—(ej + Lu)Ta, +\e, + Luf -12
1 ,-,2	j—t v 1 11' 1	1 1 1	1 '
F2 f
(7)
N=1+^, g=M
F2 1 f 2
T M
+ 2(et + Lu) (— ai - a2). F
According to the assembly mode of the robot, the y coordinate of P can be expressed as:
-Q -VQ2 - 4NRi
y = ■
2 N
(8)
Hence, substitute Eq. (8) into Eq. (6), then the x coordinate of P can be determined.
3 ERROR MODELLING AND SENSITIVITY ANALYSES
The first-order approximation of Eq. (1) can be formulated by:
Ar = Ae; + AL. ui + Li Au; + A/; +1 i Awt, (9)
where Ar = (Ax Aj)T is the position error vector of the reference point P; Ae^ = (Aeix Aey)T is the position error vector of A; AL, Al, Am, and Awj are the length errors and orientation error vectors of the proximal and distal links. Furthermore, the first-order approximation of u , can be written as:
A«,. = A0,.(-sin0, cose,)T = Qu,A0„ Q =
0 -1 1 0
,(10)
where A0,- is the zero offset of the robot.
Then, taking the dot product with AwT on the both sides of Eq. (9) (note that w ^ Aw,) yields:
w] Ar=w] Aet+ALwI U + L wT QuAdi+Al,. (11)
For a parallel robot, its error model is usually expressed in matrix form, such that the relationship between the position error and the source errors can be directly revealed by an error transfer matrix.
Rapid and Automatic Zero-Offset Calibration of a 2-DOF Parallel Robot Based on a New Measuring Mechanism
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According to Eq. (11), the error model of the robot can be expressed as
Ar = J 'Aq',
where J denotes the error transfer matrix, and
'Aq'
(12)
J '=w w2r
J' = (wTQu J2 = (WlQU2
J' 0 0 J'
Aq ' =
Aq2
"lx
W' U' 1),
2 y
W2 U2 1),
A?i = (LA 1 Aelx Aely AL1 ^ ,
Aq2 = (L24 2 Ae!x Aeiy AL2 Al2)T.
Since the robot has symmetrical geometry, the sensitivity analyses of the source errors can be studied by analysing the variation of Ap0 (Ap0 is the absolute distance error of P0, and P0 is the home position at which 61 = 0° and 02 = 180°) versus the source errors within the 1st limb.
Given L1 = L2 = L, l1 = l2 = l, e1x = -e2x = ex and e1y = e2y = ey, the nominal geometric parameters of the robot are listed in Table 1, and the results of the sensitivity analyses are presented in Fig. 5. It can be seen that the position accuracy is more sensitive to the zero offset than the geometric errors. Hence assume that the adequate fundamental geometric accuracy of the robot can be achieved, Eq. (12) can be simplified as follows:
Ar = J Aq,	(13)
where
j=[w w2r
wlQu 0 0 W2TQu2
Aq =
L1A01
l2aq2
Table 1. Nominal geometric parameters [mm]
ex L	l	H	b	h Cx	Cy dx	dy
80 0 375	825	632	480	150 0	79 0	51
a)
1= B
£
b)
<
0 0.2 0.4 0.6 0.8 A6>i [°]
1.2 1
0.8
0.6 0.4 0.2 0
0 0.2 0.4 0.6 0.8 1 Geometric error [mm]
Fig. 5. Sensitivity analyses; a) variation of Ap0 vs. A^0 and b) variations of Ap0 vs. geometric errors
4 ZERO OFFSET IDENTIFICATION
The zero-offset identification model is developed based on two adjacent measurement positions, Pk and Pk+1 (1 < k < K-1, and K is the total number of measurement positions). As shown in Fig. 6, considering the position errors of O1 and O2, then OjO2 and Oj'O^ of Pk, denoted by lk and pk, respectively, can be expressed as:
Xk =XkXt = rk - c - d,
Pk =Pt Pt = <- c'- d
(14)
(15)
where Xk and Xk are the length and unit orientation vector of ik; pk and pk are the length and unit orientation vector of pk; rk and r' are the nominal and actual position vectors of Pk; c' is the position vector from O to Oj'; d' is the position vector from O2 to Pk . Then, taking the first-order approximation of Eq. (14) yields:
AX,,Xk + X,AXk = Ar,, - Ar„
(16)
where Mk and AXh are the length error and orientation error vector of ik; hrk it the position error vhctoh of Pk; ArM = (&Mx ArMx)T is the MAEOmi; and wecan obtain:
ArM = Ac + Ad, Ac = c'- c, Ad = d'- d, (17)
AXk=pk-Xk,	(18)
where Ac' and Ad' are the position error vectors of O1 and O2, respectively.
Oi
d
PLl Pk+l	Pk Pk
Fig. 6. Error model of the measuring mechanism Note: O' (O2) is the actual rotation centre of the revolute joint 3 (4); Pk (Pk+1) and Pk (Pk'+1) are the kth ((k+ 1)th) nominal and actual measurement positions, respectively
Taking dot product with Xk on the both sides of Eq. (16) (note that Xk 1 AAk) yields:
AXk =Xk (Ar -ArM).
(19)
Then substituting Eq. (12) and Eq. (18) into Eq. (19), we can obtain:
1
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Strojniski vestnik - Journal of Mechanical Engineering 63(2017)12, 715-724
Pk =lk(J'Aq' -AM),
where J' is the error transfer matrix J of Pk. Rewriting Eq. (20) as:
Pk -h = gkAp>
^T
where gk = Xk [ J'k -1], I =
(20)
(21)
1 0 0 1
Ap' =
Aq' Aru
We can also get Eq. (22) according to Pk+1:
Pk-h= gk+\Ap',	(22)
where gk+i = A*+i [Jk+i -1 ]; pk+1 and pw are the length and unit orientation vector of pk+1; pk+1 is O[O'2 of Pk+1; Xk+1 and X k+i are the length and unit orientation vector of ik+1; ik+1 is OjO2 of Pk+1; and J'+1 is the error transfer matrix J of Pk+1.
Subtracting Eq. (22) with Eq. (21) leads to:
(Pk+l-Pk) - (K+i-K) = (gL - gk W. (23)
Hence, the matrix form of the identification model can be expressed as:
AX = G 'Ap\
(24)
where
	' (P	-P)	- (4 -	4i) ^		' g-g "
AA =	(Pk+1	-Pk )	- (4k+1	-4k )	, G =	gk+1 - gk
	, (PK -	Pk-i)	- (4k "	~4K -1)		_ gK - gK-1 _
It is easy to prove that rank (G') = 12 if K > 13 provided that Xi, X2, ■■■, Xk-1 and Xk are not colinear, then hp' can be identified using the LS method:
Ap' = [(G')t G']-1(G')t M
(25)
The singular value decomposition method is often used to study the identifiability, by which the identification matrix G' can be rewritten as:
G ' = U 'S '(V ')T
(26)
where U and V are (£-1)x(£-1) and 12x12 matrixes, respectively, and each of them is composed of a set of standard orthogonal bases; S' is a diagonal matrix composed of the singular values of G'. Hence, Eq. (25) can be rewritten as:
=	— v'„
t=\

(27)
where u't is the standard orthogonal basis of U; v't is the standard orthogonal basis of V; °'t is the singular value of G, and a\ >a'2 >■■■> a'12 > 0 .
The TSVD method can be used to improve the identifiability of hp' by simply truncating the summation in Eq. (27) at an upper limit t < 12 before the small singular values start to dominate. However, if the zero offsets are identified together with too many geometric errors, the upper limit of t will be too large and then the TSVD may be performed like the LS that cannot overcome the nonlinear ill-conditioning problem of Eq. (25). Since it has been proved in Section 3 that the position accuracy is more sensitive to the zero offset than the geometric errors, the nonlinear ill-conditioning problem can be directly solved to some extent by neglecting the identification of the geometric errors according to the TSVD method, i.e. by substituting Eq. (13) and Eq. (18) into Eq. (19), then the identification model can be degenerated into the following form:
AX = GAp, G =
gi — gi gk+i - gk
gK — gK-1
Ap = (Aq Aru )T
gk =X J — I ] ,(28)
gk+i =X T+i[Jk+i — I]
where Jk and Jk+1 are the error transfer matrix J of Pk and Pk+1, respectively.
It can also be proved that rank (G) = 4 if K > 5 provided that Xi, X2, ■■■, Xk-1 and Xk are not colinear. Hence, hp can be calculated by
. ^(u, )T AX AP = L -vr-
t=1
(29)
where ut, vt and at are the standard orthogonal bases and singular value derived from the SVD format of the identification matrix G.
Since we neglect the identification of the geometric errors, the accuracy of hO, solved by Eq. (29) may be slightly decreased even though the nonlinear ill-conditioning problem can be solved. To improve the accuracy, the following aspects are considered: (1) the source errors should be identified multiple times; (2) the measuring mechanism is used as a metrology device and its measurement accuracy can be improved with the decrease of hrM; (3) hrM is independent of the source errors of the robot, and the smaller hrM the better the identifiability of hO,. Based on these considerations, the identification model can be modified as follows.
For the 1st identification, we use Eq. (29) to identify hp; for the jth (j > 2) identification, by modifying Eq. (20) with the former identification result of hrM as shown in Eq. (30), then the jth identification model can be redeveloped:
Rapid and Automatic Zero-Offset Calibration of a 2-DOF Parallel Robot Based on a New Measuring Mechanism
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Pk "I \ftJ>\ |=
( j) \\= ( fk '))T
\\fk j )\\
(Aq(j) "ArM) ), C30)
where 2 (j = fj + Aj, and f(> = \ .
5 OPTIMAL MEASU REMENT POSITIONS
AND ERROR COMPENSATION STRATEGY
The identification of Ap re quires the moving platform to undergo K > 5 measurement positions; meanwhile, these positions should converge to the boundaries of the workspace where the highest signal-to-noise ratio be may achieved. In addition, the moving platform should experience all the controllable degrees of freedom. Al shown in Fig. 7, the most straightforward way is to choose n evenly spaced positions on each of the upper and lower boundaries of the work space.
Since the identifiability can be imp rov e d with the decrease of the condition number of the identification matrix, to make the measurements in a time-effective manner, this selection problem of the optimal measurement positions can be solved by minimizing n, kj and Km subject to a given threshold e0 defined as the relative difference beftveen kj and Km vs. n, i.e.
min [n,ic1 ,k— s.t. s=——<sa, n>2,
(3^
where Kj denotes the condition number of the first identification; and Km denotes the mean condition number of the remaining identifications.
Pi-
-Pi—>-Po-r -•-•-

*~Po+1
--> P
* n
Pn + 1	^	P2 n-1^"P2 n
Fig. 7. Optimal measurement positions
After AO, of the jth and (/+1)th identifications are obtained, the position error of the robot can be reduced in an iterative manner by compensating the kinematic model in the robot controller with the identification results of AO, until the compensation accuracy ^ defined as follows converges within a given threshold

(j+D _ Aq(j))2 + (A0<j+1> - A02(j>)2 ] / 2 Thenthe compensation value of A6t is:
AO. = A0;(1) + A0;<2) + ••• + A0ii + ••• + where m is the compensationnumber. 720
AO
(m)
6 SIMULATION ANALYSES
In this section, simulations are carried out to investigate the accuracy and robustness of the zero-offset calibration method in depth.
6.1 Simulation Parameters
The given source errors are listed in Table 2. This is because: 1) the investigation of the identification accuracy requires the given values of AO, to cover a certain range; 2) the different attainable geometric accuracies of the robot should be considered; 3) the MAEMM can be roughlymeasured,and AhMx £md ArMy are about 1 mm and 0.5 mm, respectively.
Given ArjJj = (1 0.5)T (2 < j < m) , it can be seen from Fig. 8 that, for each simulation group, k1 and Km both monotonically increase with the increase of n. Meanwhile, Km is less than Kj corresponding to the same n, meaning that the identifiability can be slightly improved by using Eq. (30). Furthermore, given e0 = 0.01, it can also be seen that the minimum n is 3, whic h leads to K = 2n = 6 optimal mvasurement positions.
Based on the optimal positions, (Xk - A.+) can be obtained by the inverse positional analysis, and (pk - Pk+\) can be dnrivmb from a^e forwnrd nosilional model containing the source errors and considering the measurement errors. Then AO, can be calibrated using the proposed method.
n = 3	
K1 = 475.34	
■ Km = 472.54	
	K1 "
	
	
3 5 7 9 11 13 15 17 19 21 n
Fig. 8. Variations of kj and Km vs. n
In the calibration, the measurement errors are mainly caused by the linear scale and servo motor, which can be reasonably set as follows.
Since the maximum measurement error of the linear scale is ± (3 + /0/1000)x10-3 mm (l0 is the measuring range of the linear scale and l0 = 350 mm); meanwhile, the output of the reading head can be reset
(32)
after each measurement, the measurement error of the linear scale corresponding to Pk, denoted by mk, can be set as the Gaussian distributed error with mean 0
(33)	and variance m2, and m can be calculated by:
* = 1 f 3 + A 3 ^ 1000,
44	Mei, J. - Zang, J. - Ding. Y.- Xie, S. - Zhang, X.
x10-
(34)
Strojniski vestnik - Journal of Mechanical Engineering 63(2017)12, 715-724
Table 2. Set values of the source errors
Group	Zero offset [°]					Geometric error [mm]					MAEMM [mm]
	A01	A02	Aelx	A^iv	Ae2x	Ae2v	AL1	AL2	Al1	Al2	ArMx ArMv
1	2	-1									
2	-1	0.5	0.03	-0.02	-0.02	0.01	0.01	0.02	-0.03	-0.02	
3	0.5	0.25									1 0.5
4	2	-1									
5	-1	0.5	0.003	-0.002	-0.002	0.001	0.001	0.002	-0.003	-0.002	
0.5
0.25
6
Table 3. Simulation results of A0, and Ap0
Group	Zero offset	Compensation value [°]	s [°]	Apo (before) [mm]	Apo (after) [mm]
1	A01	2.033	0.033	10.933	0.402
	A02	-0.957	0.043		
2	A01	-1.049	0.049	5.482	0.423
	A02	0.463	0.037		
3	A01	0.536	0.036	3.796	0.434
	A02	0.297	0.047		
4	A01	2.024	0.024	10.896	0.228
	A02	-0.978	0.022		
5	A01	-1.025	0.025	5.5195	0.256
	A02	0.474	0.026		
6 -	A01	0.476	0.024	3.798	0.239
	A02	0.225	0.025		
Considering that the number of pulses per revolution of the servo motor is 1*104 and the maximum number of pulse error sper revolution is 4, the motion error of the servo motor corresponding to Pk, denoted by 4, can also be set as the Gaussian distributed error with mean 0 and variance £2, and £ can be calculated by:
r 4 ^
4-r *360°,	(35)
vr 104 )

where n = 20 is the reduction ratio of the reducer.
6.2 Simulation Results and Discussion
Given = 0.1°, the compensation value of A0, the absolute difference between the set and compensation values of A0, denoted by 8, and Ap0 before and after calibration are listed in Table 3. It can be seen that 8 is around 0.040° in the first three groups and 0.025° in the last three groups. This indicates that the identification accuracy is invariant with the set values of A0, and that it can be slightly improved with the decrease of the geometric errors. Furthermore, Ap0 can be significantly reduced after calibration, and the
maximum Ap0 after calibration is 0.434 mm in the first three groups and 0.256 mm in the last three groups.
As shown in Table 4, for each group, the maximum absolute distance error, denoted by Apmax, of the six optimal measurement positions can be reduced to a certain value after calibration. Since these positions are along the boundaries of the workspace where the position errors usually tend to be much larger than those of the internal positions, we can infer that the position accuracy throughout the workspace of the robot can be well improved after the calibration.
Table 4. Apmax before and after calibration
Group	1	2	3	4	5	6
Apmax						
(before)	12.906	6.418	4.672	12.868	6.455	4.663
[mm]						
APmax (after) [mm]	0.493	0.531	0.533	0.279	0.314	0.293
Table 5 shows the absolute differences between the set values and identification results of ArMx and ArMy, denoted by 8Mx and 8My, respectively. It can be seen that, similar to the identification results of A0, the identification accuracies of ArMx and ArMy are
scarcely affected by the set values of A0, while they can be slightly improved with the decrease of the geometric errors of the robot.
Table 5. Absolute differences between the set values and identification results of ArMx and ArMy
Group	1	2	3	4	5	6
Smx [mm]	0.069	0.074	0.071	0.044	0.051	0.047
Smv [mm]	0.040	0.045	0.042	0.022	0.028	0.024
As shown in Table 6, of each group is about 472.50, which is almost the same as Km = 472.54 and less than k1 = 475.34 as shown in Fig. 8, further verifying that the identifiability of the identification
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a) 3.5 2.5
d)
0.5 -0.5 -1.5
3.5 2.5 1.51 0.5 -0.5 -1.5
► After
max ô=0.047°.
1.5Œ u=0.065°
-Q-A01 -v- A62 B- a
u
max u=0.091°
îbbo □□□□□□□□□□□□
max ô=0.048°^
b)
m tu
Q
1 3 5 7 9 11 13 15 17 19
m
► After
max ô=0.030°
A01
A62

-Q-
eeeeœeeeeeeeeeo £
. u=0.067° max u =0.066°
BBBBBQDDD^BBE] ^max ô =0.027°

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e)
00
tu
Q
1 3 5 7 9 11 13 15 17 19
m
1.5 1
0.5 0
-0.5 -1 -1.5
1.5 1
0.5 0
-0.5 -1 -1.5
-e- A01
t+After	I
max ô=0.047°^_ A621
c)
U=0.085° max u=0.088° max ô=0.057°
¿©éeeeee^
5 7 9 11 13 15 17 19
m
\+>After
max ô=0.036°
A6>! A6,
-O- a
j u=0.010° max u=0 096° -max ô=0.035°
M tu
Q
1 3 5 7
9 11 13 15 17 19
m
1.25 1
0.75 0.5[ 0.25 0
-0.25
1.25 1
0.75 0.55 0.25 0
-0.25
After
max ô=0.049 max ô = 0.048
>e
9 11 13 15 17 19
m
9 11 13 15 17 19
m
Fig. 9. Robustnessanalyses; a) group 1; b) group 2; c) group 3; d)group 4; e) group 5; and f) group 6
model modified using Eq. (30) can be slightly improved.
bhtwuen 0° to y.l°. Thtse obserqations iudicate tha9 tiie identification model has good robustneso.
Table 6. Mean condition number
Group
1
2
3
4
5
Table 7. Experimental results of A6, and Ap0
Group
Zero-offset
Compensation value [°]
AO,
1.929
0.071
AO,
-0.924
0.076
AO,
-0.935
0.065
AO,
0.572
0.072
AOi
0.569
0.069
AO
0.175
0.075
6
472.47 472.49 472.49 472.50 472.51 472.50
Ap0 Ap0 (before) (after) [mm] [mm]
12.062 0.732
6.964 0.605
4.945 0.711
Further research is performed to evaluate the robustness of the identification model. The variations of the compensation value of and the defined compensation accuracy p. versus m of each group are presented in Fig. 9. For each group, it can be observed that the compensation values of A0j and Ap2 both fluctuate slightly, but they can converge Po different values with the increase of m. Furthermore;, 37 is less than 0.060° in the first three groups and 0.040° in the last three groups, and p of each group after its value reduces to less than p0 = 0.1° for the first time is
7 EXPERIMENTAL VERIFICATION
Experiments are carried out on the 2-DOF parallel robot with the repeatability of ±0.05 mm over its workspace to verify the effectiveness of the zero-offset calibration method.
Measuring Mechanism
Fig. 10. Expariment set-ups; a) zero offset adjustment set-up; b) calibration set-up; and c) verification set-up
As shawn in Fig. 10a, in order to compare the experiments with the simulations, a digital level with the maximum observed deviation of 0.1° is usedto adjust the two active proximal links to the horizontal position before each experiment, and then
722
Mei, J. - Zang, J. - Ding. Y.- Xie, S. - Zhang, X.
K
ID
À
1
2
3
Strojniski vestnik - Journal of Mechanical Engineering 63(2017)12, 715-724
A61 and A62 can be roughly regarded as 0°. After that, Af: and Ad2 are set as listed in the first three groups of Table 2, respectively, by driving the two active proximal links to the corresponding positions. Havingbuilt the calibration set-up as shown in Fig. 10b, the experiments of the zero-offset calibration can be implemented, and the position errors before and after cahtoation are measured with e LEICA AT901 laser tracf er with the maximum observed deviation of 0.016 mm as shown in F id 10t.
Table 8. Identification results of ArMx and ArMy
Group
1
2
ArMx [mm]
0.842
0.835
0.852
ArMv [mm]
0.592
0.576
0.583
Likewise, givene0 = 0.1°, the experimental results are listed in Tables 7 and 8, from which we can determine that, similar to the simulation results, the identification accuracy is invariant with the set values of A6h and the Ap0 of each group can be significantly reduced after calibration. We can also determine that the maximum 3 and Ap0 after calibration are 0.076° and 0.732 mm, respectively, which are slightly larger than 0.049° and 0.434 mm of the simulations. Since it has been proved via the simulation analyses that the identifiability will decrease with the increase of the geometric errors, the slight decrease of the identification accuracy in the experiments is due to the fact that the actual geometric errors of the robot are larger than those given in the simulations.
As shown in Fig. 11, in order to carry out deeper investigation on the position accuracy, the absolute distance error, denoted by Ap, before and after calibration of K = 42 evenly spaced measurement positions along the boundaries of the workspace are measured by the laser tracker, and the results are presented in Fig. 12. Furthermore, the maximum position error Apmax of these positions are listed in Teble 9. Itcanbe seen that iiiemaximum p^om^^i^n error along the workspace of each group can also be significantly reduced to less than 0.85 mm after the calibration.
Pf Pc	Pi
P42
I
P
39
P38 P37	P3E ^	P23 P22
Fig. 11. The 42 measurement positions
^ 16 § 12
0
a) 0 3 6 9 e2 15 18 21 24 27 30 33 36 39 42
K
12
a a
8
o 4 4
Before
After
-B-
b) 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42
K
6
4
d 2
<
0
c) 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42
K
Fig. 12. Ap before and after calibration; a) group 1; b) group 2; and c) group 3
Table 9. Apmax before and after calibration
Group
1
APmax (before ) [mm]
16.901
8.326
5.448
APmax (after ) [mm]
0.847
0.623
0.825
8 CONCLUSIONS
To realize the rapid and automatic zero offset calibration of a 2-DOF parallel robot, a measuring mechanism is designed, and based on which a zero-offset calibration method is proposed in this paper. Compared with large measurement devices, the measuring mechanism is more convenient to use and it can make the measurements in a time and cost-effective manner. By using the TSVD method, the nonlinear ill-conditioning problem of the identification model can be solved. The identification model modified with the former identification result oftheMAEMMcan helptoimprove the identifiability of the zero offsets. The optimization approach for selecting measurement positions is able to maximize the measurement efficiency and further improve the identifiability. The simulation and experimental results of the calibration show that the identification model has good identifiability and robustness, and the position error after calibration can be significantly reduced. The proposed measuring mechanism and zero offset calibration method are also useful for the kinematic calibration of other planar or spatial parallel robots. It should be noted that since the spatial parallel
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Rapid and Automatic Zero-Offset Calibration of a 2-DOF Parallel Robot Based on a New Measuring Mechanism
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Strojniski vestnik - Journal of Mechanical Engineering 63(2017)12, 715-724
robots usually have cylindrical workspaces, if the measuring mechanism is used for the calibration of these parallel robots, its two revolute joints should be replaced by universal or spherical joints, so that the measurement positions can be more reasonably selected in those cylindrical workspaces.
9 ACKNOWLEDGEMENTS
This work is supported by the National Natural Science Foundation of China (Grant No. 51475320 and 51420105007), and the Key Technologies R & D Program of Tianjin (Grant No. 15ZXZNGX00220).
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