Kinematic calibration of a 3-DoF rotational parallel manipulator using laser tracker

Robotics and Computer-Integrated Manufacturing 41 (2016) 78–91

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Robotics and Computer-Integrated Manufacturing journal homepage: www.elsevier.com/locate/rcim

Kinematic calibration of a 3-DoF rotational parallel manipulator using laser tracker Tao Sun, Yapu Zhai, Yimin Song n, Jiateng Zhang Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University, Tianjin 300072, China

art ic l e i nf o

a b s t r a c t

Article history: Received 3 April 2015 Received in revised form 11 December 2015 Accepted 23 February 2016

This paper proposes a laser tracker based kinematic calibration of a 3-degree-of-freedom (DoF) rotational parallel manipulator that would be applied in tracking and positioning fields. The process is implemented in this paper by four steps: 1) formulation of the geometric error model of this manipulator by means of screw theory considering all possible geometric source errors, which is followed by the verification of this error model employing SolidWorkss software. 2) sensitivity analysis of all geometric source errors based upon Monte Carlo method and remove some errors that have little influence on the pose accuracy of the moving platform in order to decrease the difficulty and complexity of the kinematic calibration. 3) error parameter identification and kinematic calibration experiment using laser tracker. 4) error compensation by amending controller model. Kinematic calibration experiment results of this 3-DoF rotational parallel manipulator show that three angular deviations are improved from 1.97°, 0.24° and 1.75° to 0.53°, 0.10° and 0.19° respectively within the prescribed workspace. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Rotational parallel manipulator Kinematic calibration Geometric error model Sensitivity analysis Calibration experiment

1. Introduction Three degree-of-freedom (DoF) rotational parallel manipulators using several open-loop limbs to link a moving platform and a fixed base have recently attracted great interest from both academia and industry [1,2]. The high stiffness, good accuracy and superior dynamic performance allow these parallel manipulators to be potentially utilized in tracking and positioning fields [3–5]. As with other parallel manipulators, the geometric accuracy is a crucial index and important performance of 3-DoF rotational parallel manipulators. It can be improved by kinematic calibration supposed that the component manufacturing and assembly accuracy can be effectively guaranteed by means of manufacturing and assembly processes [6,7]. This approach is implemented generally by four steps: 1) formulation of an error model considering all possible geometric source errors; 2) evaluation of the source errors affecting the pose accuracy of the end-effector by means of sensitivity analysis; 3) kinematic calibration employing built-in sensors or external measuring equipment; 4) error compensation by amending the controller model [8–11]. The geometric error model is the basis of error analysis and kinematic calibration of parallel manipulators, involving a mapping between the pose error of the moving platform and the geometric source errors [12–15]. If the source errors of the n

Corresponding author. Tel.: +86 22 87402015. E-mail address: [email protected] (Y. Song).

http://dx.doi.org/10.1016/j.rcim.2016.02.008 0736-5845/& 2016 Elsevier Ltd. All rights reserved.

geometry are very small compared to the nominal sizes, there are three main methods at hand available to formulate the geometric error model: vector loop method [16], matrix method [17] and screw method [18]. Comparing with the other two methods, the screw method based geometric error modeling would be carried out in a concise, distinct and clear-meaning manner [19,20]. In order to evaluate the effects of geometric source errors to the pose accuracy of the moving platform, the sensitivity analysis is carried out generally after formulating the geometric error model [21]. The Monte Carlo method [22] and sensitivity coefficient method are two common methods. Comparing with the sensitivity coefficient method, the Monte Carlo method based sensitivity analysis is beneficial to obtain more accurate terminal error bounds by sufficient sampling points within workspace and calculation numbers [23]. The kinematic calibration is an effective and low-cost way to improve the geometric accuracy of parallel manipulators comparing with another way that enhances the manufacturing and assembly accuracy of the components directly [9,24,25]. By formulating the error function between the theoretical values and measured values obtained from inbuilt sensors or external measuring equipment, the parameter identification can be carried out [26–28]. The identification results are then utilized to amend the controller model to improve the geometric accuracy of parallel manipulators [29,30]. Since the inbuilt sensors need to be considered in the design procedure of parallel manipulators, the external measuring equipment is more flexible and less constraints to measure the required information [31–34].

T. Sun et al. / Robotics and Computer-Integrated Manufacturing 41 (2016) 78–91

The aforementioned literature review shows that the existing kinematic calibration approach or flow of parallel manipulators is relative mature and general, however, the diverse parallel manipulators have different topology and characteristics that makes the kinematic calibration flow relate directly to these characteristics [35,36]. It is noted that the kinematic calibration of pure rotational parallel manipulator is a challenging issue. In this paper, the kinematic calibration of a 3-DoF rotational parallel manipulator using laser tracker is proposed. Having outlined the significance and general procedure of the kinematic calibration of parallel manipulators in Section 1, the paper is organized as follows. In Section 2, the inverse position analysis and motion mapping model of the 3-DoF rotational parallel manipulator are carried out. Considering all possible geometric source errors, the error model of the rotational parallel manipulator is formulated by means of screw theory, and then the model effectiveness is verified using software approach in Section 3. In Section 4, the sensitivity analysis is implemented employing Monte Carlo method. The kinematic calibration simulation of the rotational parallel manipulator using laser tracker is carried out after verifying the effectiveness of the parameter identification model in Section 5. In Section 6, the experiment is implemented to verify the validity of the proposed method. The conclusions are drawn in Section 7.

(a) Virtual prototype w s6,i

u

A3 U Joint

A1

O′

s5,i

2. Motion mapping model

s2,i

2.1. Inverse position analysis As shown in Fig. 1(a), the 3-PSU&S parallel manipulator [1] consists of a fixed base, a moving platform, a passive properlyconstrained S limb and three actuated non-constrained PSU limbs with an axis-symmetrical layout. Herein, P, S and U represent the prismatic, the spherical and the universal joint, respectively. As shown in Fig. 1(b), each PSU limb connects the fixed base at one end by a prismatic joint whose axis is perpendicular to the plane of the fixed base, and links the moving platform at the other end by a universal joint whose center is denoted by point Ai ( i = 1, 2, 3). The axes of the prismatic joints intersect with the fixed base plane at points Bi (i = 1, 2, 3). Without loss of generality, points Ai ( i = 1, 2, 3) and Bi ( i = 1, 2, 3) form two equilateral triangles whose circumcircle radii are represented by a and b, respectively. Let Ci ( i = 1, 2, 3, 4 ) be the centers of the S joints, the projections of C4 on the fixed base and the moving platform are points O and O′, respectively. The S limb links the fixed base at O and the moving platform at O′. In addition, h, h0, l and di respectively denote the lengths of C4 O , C4 O′, Ci Ai and Ci Bi ( i = 1, 2, 3). In order to describe the motions of the 3-PSU&S parallel manipulator, two coordinate frames are needed to be established. As shown in Fig. 1(b), a global coordinate frame O − xyz is established at point O with z-axis upwardly perpendicular to the fixed base, xaxis pointing from point O to point B1, and y-axis satisfying right hand rule. Similarly, a moving coordinate frame O′ − uvw is assigned to point O′ with u-axis pointing point A1, w-axis always vertical to the plane of the moving platform, and v-axis satisfying right hand rule. Since the 3-PSU&S parallel manipulator has three rotational DoFs, the Tilt-and-Torsion Angle method [37] is employed to describe the pose of its moving platform. Therefore, the orientation matrix of frame O′ − uvw with respect to frame O − xyz can be described as

79

s3,i

h0 s 1,4 C4

v lw i

C1

a A2

S Joint s3, 4 s2,4

C3

P Joint

C2

s4,i

h

di z

s1,i x

B3 B1

O

b

B2 y

(b) Schematic diagram Fig. 1. 3-PSU&S rotational parallel manipulator (a) virtual prototype (b) schematic diagram.

R (ψ , θ , ϕ) = ⎡⎣ u v w ⎤⎦ ⎡ Cψ Cθ C (ϕ − ψ ) −Cψ CθS (ϕ − ψ ) Cψ Sθ ⎤⎥ ⎢ ⎥ ⎢ − Sψ S (ϕ − ψ ) − Sψ C (ϕ − ψ ) ⎥ ⎢ = ⎢ Sψ Cθ C (ϕ − ψ ) −Sψ CθS (ϕ − ψ ) Sψ Sθ ⎥ ⎥ ⎢ + Cψ C (ϕ − ψ ) ⎥ ⎢ + Cψ S (ϕ − ψ ) ⎢⎣ −Sθ C (ϕ − ψ ) SθS (ϕ − ψ ) Cθ ⎥⎦

(1)

where θ denotes the angle between the unit vector w and z-axis, ψ represents the angle between x-axis and the vector w′ which is the projection of w on x–y plane, ϕ is the angle rotating about z-axis. Besides, C and S represent Cosine and Sine, respectively. As shown in Fig. 1(b), the closed-loop vector equation of the 3PSU&S parallel manipulator can be formulated as

80

T. Sun et al. / Robotics and Computer-Integrated Manufacturing 41 (2016) 78–91

bi + di e + lwi = he + R (h0 w + ai0 ), i = 1, 2, 3

(2)

where e = ( 0 0 1)T and wi represents the unit vector along the direction of the link from Ci to Ai. bi is the position vector of point Bi in frame O − xyz and

(

T

b1 = ( b 0 0 ) , b 2 = − 1 b 2

3 b 2

T

0

) , b = (− 3

1 b 2

−

3 b 2

T

(

3 a 2

T

0

) , a = (− 30

1 a 2

−

)

3 a 2

)

Based upon Eq. (2), the inverse position problem can be solved

(3) ^ ^ $ wa,1, i $ t = $ wa,1, i

where

6

∑ δρa, j, i $^ ta, j, i, i = 1, 2, 3

(6)

j=1

l2 − (h0 R13 + aR11 − b)2 − (h0 R23 + aR21)2 ⎛ 3 ^ ^ ^ $ wc, i,4 $ t = $ wc, i,4 ⎜⎜ ∑ δρa, j,4 $ ta, j,4 + ⎝ j=1

2

⎛ 1 3 1 ⎞ l2 − ⎜ h0 R13 − aR11 + aR12 + b⎟ 2 2 2 ⎠ ⎝

3

⎞

j=1

⎠

∑ δρc, j,4 $^ tc, j,4 ⎟⎟, i = 1, 2, 3

⎛ wi ⎞ ^ ⎟, i = 1, 2, 3 $ wa,1, i = ⎜ ⎝ ai × wi ⎠ ⎛ s2,4 ⎞ ^ ⎛ n1,4 ⎞ ^ ⎛ s1,4 ⎞ ^ $ wc,1,4 = ⎜ ⎟, $ wc,2,4 = ⎜ ⎟, $ wc,3,4 = ⎜ ⎟ ⎝ h 0 × s2,4 ⎠ ⎝ h 0 × n1,4 ⎠ ⎝ h 0 × s1,4 ⎠

2

⎛ 1 3 1 ⎞ l2 − ⎜ h0 R13 − aR11 − aR12 + b⎟ 2 2 2 ⎠ ⎝

Eqs. (6) and (7) can be written in matrix form as

2 ⎛ 1 3 3 ⎞ − ⎜ h0 R23 − aR21 − aR22 + b⎟ 2 2 2 ⎠ ⎝

Jx $ t = Jρ δρ

(8)

Provided that matrix Jx is not singular, Eq. (8) can be rewritten

herein Rij denotes the ith row jth column element of the matrix R.

as

$ t = Jx−1 Jρ δρ 2.2. Motion mapping model

6

j=1

⎤ ⎡J ⎡J ⎤ ⎛ δρ ⎞ ρa xa ⎥, δρ = ⎜ a ⎟, Jρc = E3, Jρa = diag (s1,T i wi ), i Jx = ⎢ ⎥, Jρ = ⎢ Jρc ⎥⎦ ⎝ δρc ⎠ ⎣ Jxc ⎦ ⎣⎢ = 1, 2, 3 herein Jxa and Jxc are respectively represented as

(4) Jxa

$t =

3

3

j=1

j=1

∑ δρa, j,4 $^ ta, j,4 + ∑ δρc, j,4 $^ tc, j,4

(5)

^ where $ta, j, i and δρa, j, i are known as the jth unit twist of permissions ^ and its intensity in the ith limb. $tc, j,4 and δρc, j,4 are known as the jth unit twist of restrictions and its intensity in the passive limb. And

⎛ (ai − lwi ) × s2, i ⎞ ^ ⎛ (ai − lwi ) × s3, i ⎞ ⎛ s1, i ⎞ ^ ^ $ ta,1, i = ⎜ ⎟, $ ta,2, i = ⎜ ⎟, $ ta,3, i = ⎜ ⎟ s2, i s3, i ⎝ 0⎠ ⎝ ⎠ ⎝ ⎠

(9)

where

Mainly drawing on screw theory, the twist of the point O′ can be represented by a linear combination of the twist of 1-DoF joint in the limbs since all limbs share the same moving platform, i.e.,

^ $ t = ∑ δρa, j, i $ ta, j, i , i = 1, 2, 3

(7)

where

2 ⎛ 1 3 3 ⎞ − ⎜ h0 R23 − aR21 + aR22 − b⎟ 2 2 2 ⎠ ⎝

M3 =

⎛ s1,4 ⎞ ⎛ n1,4 ⎞ ^ ⎛ s2,4 ⎞ ^ ^ ⎟ ⎟, $ tc,3,4 = ⎜ ⎟, $ tc,2,4 = ⎜ $ tc,1,4 = ⎜ ⎝ 0 ⎠ ⎝ 0 ⎠ ⎝ 0 ⎠ herein as shown in Fig. 1(b), sj, i ( i = 1, 2, 3, j = 1, 2, ⋯ , 6) is a unit vector along the jth 1-DoF joint of the ith limb, and the joint axes are arranged such that s1, i//s2, i , s3, i ⊥s2, i , s3, i ⊥s4, i , s4, i//s5, i , ai = Rai0, n1,4 = s1,4 × s2,4, h0 = − h0 Rw Taking the generalized inner product on the both sides of Eqs. ^ ^ (4) and (5) with $ wa,1, i and $ wc, i,4 respectively yields

⎧ d1 = h + h0 R33 + R31a − M1 ⎪ ⎪ ⎪ d2 = h + h0 R33 − 1 aR31 + 3 aR32 − M2 ⎨ 2 2 ⎪ 1 3 ⎪ d = h + h0 R33 − aR31 − aR32 − M3 ⎪ ⎩ 3 2 2

M2 =

⎛ h 0 × s2,4 ⎞ ^ ⎛ h 0 × s3,4 ⎞ ^ ⎛ h 0 × s1,4 ⎞ ^ $ ta,1,4 = ⎜ ⎟, $ ta,2,4 = ⎜ ⎟, $ ta,3,4 = ⎜ ⎟ ⎝ s2,4 ⎠ ⎝ s3,4 ⎠ ⎝ s1,4 ⎠

T

0

as

M1 =

⎛ ai × s6, i ⎞ ⎟, i = 1, 2, 3 =⎜ ⎝ s6, i ⎠

T

0

ai0 represents the initial vector of point Ai in frame O′ − uvw , and

a10 = ( a 0 0) , a20 = − 1 a 2

⎛ (ai − lwi ) × s4, i ⎞ ^ ⎛ ai × s5, i ⎞ ^ ^ ⎟, $ ta,6, i $ ta,4, i = ⎜ ⎟, $ ta,5, i = ⎜ s4, i ⎝ s5, i ⎠ ⎝ ⎠

⎡ T ⎡ w T ( a × w )T ⎤ 1 1 ⎢ s2,4 ⎥ ⎢ 1 T T T = ⎢ w2 ( a2 × w2 ) ⎥, Jxc = ⎢ n1,4 ⎢ ⎥ ⎢ ⎢ T ⎢⎣ w3T ( a3 × w 3 )T ⎥⎦ ⎣ s1,4

( h 0 × s2,4 )T ⎤⎥ ( h 0 × n1,4 )T ⎥⎥ ( h 0 × s1,4 )T ⎥⎦

where E3 is the third-order identity matrix.

3. Geometric error model and its verification 3.1. Geometric error model On the basis of Fig. 1(b), some reference coordinate frames are established to describe the geometric errors of the 3-PSU&S parallel manipulator as shown in Fig. 2. A coordinate frame R0, i for the

T. Sun et al. / Robotics and Computer-Integrated Manufacturing 41 (2016) 78–91

z′ z 7,i ( z 4,4 ) x7,i

z 6,i P5 , i ( P6 ,i ) z5,i z2,i

y′

wi

x′

y7,i

z1,4

P2 ,4 ( P3, 4 )

Trans (u, κ )is the homogeneous transformation matrix for a translation along the “u” axis with a distance κ , (Rot (u, η)) is the homogeneous transformation matrix for a rotation about the “u” axis by an angle η . In addition, θa, j, i ( i = 1, 2, 3; j = 1, 2, ⋯ , 6) represents a distance (an angle) along the jth joint axis in the ith PSU limb. Then, it is noted that θa,1, i = di . According to Ref. [15], the pose error twist can be expressed by either of the two ways (i.e., via the PUS limb or the S limb). When considering the PUS limb, the pose error twist is described as

y4,4 x4,4

O′ z3,4 z 2,4

6

$t =

z3,i

81

∑ Δθa, j, i $^ ta, j, i

+ $ G, i , i = 1, 2, 3 (11)

j=1

P2,i ( P3,i P4,i ) z 4,i

6

$ G, i =

∑ AdgOj −′1,i j − 1P j, i j − 1Δj, i

+ Adg O ′ 6P 7, i6Δ7, i 6, i

j=1

z1,i x( x0,i )

z ( z0,i , z 0,4 )

P1,4

P1,i

where Δθa, j, i denotes the error of θa, j, i , Δθa,1, i = Δdi . And Adg j − 1 is the

x0,4 O

j, i

6 × 6 adjoint transformation matrix of frame Rj, i with respect to

y0,4

y ( y 0,i ) Fig. 2. The geometric error frames of 3-PSU&S parallel manipulator.

ith PSU limb (i¼1, 2, 3) is defined by rotating frame O − xyz about z-axis by 2π(i−1) /3. A coordinate frame O′ − x′y′z′ is fixed to point O′, and its axes are parallel to those of frame O − xyz . zj, i axis denote the 1-DoF joint axis of PSU limb and S limb, and the xj, i axis is perpendicular to both zj, i axis and zj + 1, i axis. P1, i is the intersection of z1, i axis and the plane of the fixed base, while Pj, i represents the intersection of zj, i axis and xj − 1, i axis. Similarly, several body fixed frames Rj, i locating at Pj, i are defined. A body fixed frame R7, i is assigned to point O′ with x7, i axis pointing to point P6,1 and z7, i axis always vertical to the plane of the moving platform. For the S limb, a frame R0,4 is defined by making frame O − xyz rotate about z-axis by π . A body fixed frame R 4,4 is established at the point O′ with the z 4,4 axis collinear with the z7, i axis and the x 4,4 axis pointing to the midpoint of line P6,2 P6,3. The relationships between the frames of the PSU limb are given by the transformations

⎧ O ′T = Trans ( − r ) ⎪ O ⎪ O T = Rot (z, β ) i ⎪ 0, i ⎪ 0T = Trans (x, b) 1, i ⎪ ⎪1 ⎛ π ⎞ ⎪ T 2, i = Trans ( z, di ) Rot ⎜⎝ z, 2 + θa,2, i ⎟⎠ ⎪ ⎛ π⎞ ⎛ ⎞ π ⎪2 + θa,3, i ⎟ ⎪ T 3, i = Rot ⎜⎝ x, ⎟⎠ Rot ⎜⎝ z, − ⎠ 2 2 ⎨ ⎪ ⎛ ⎞ π ⎪ 3T 4, i = Rot ⎜ x, ⎟ Rot ( z, θa,4, i ) ⎝ 2⎠ ⎪ ⎪ 4 T = Trans (x, l) Rot (z, θ ) a,5, i ⎪ 5, i ⎪5 ⎛ π⎞ ⎛ π ⎞ + θa,6, i ⎟ ⎪ T 6, i = Rot ⎜⎝ x, ⎟⎠ Rot ⎜⎝ z, ⎠ 2 2 ⎪ ⎪6 ⎛ π⎞ ⎛ π⎞ ⎪ T 7, i = Trans (z, − a) Rot ⎜⎝ x, ⎟⎠ Rot ⎜⎝ z, ⎟⎠ ⎩ 2 2 where

βi = (i − 1)

2π , 3

(12)

frame Rj − 1, i .

j−1

⎡ ⎢E ⎢ 3 ⎢ ⎢ ⎣0

⎡ j−1 ⎢ r j, i ⎣

×

⎤⎤ ⎥⎥ ⎦⎥ j−1 r j, i ⎥, ⎥ ⎦

is the position vector of E3 the origin of frame Rj, i . Moreover, for any one PSU limb, all the possible sources of errors are listed as follow[15]:

P j, i =

⎧0 T 0 0 0 0 ⎪ Δ1, i = ( δx1, i δy1, i 0 δα1, i δβ1, i 0) ⎪1 T 1 1 1 ⎪ Δ2, i = ( δx2, i 0 0 δα2, i δβ2, i 0) ⎪ ⎪ 2 Δ3, i = (2δx3, i 0 0 2δα3, i 0 0)T ⎪ 3 T 3 3 3 ⎪ ⎪ Δ4, i = ( δx 4, i 0 δz 4, i δα4, i 0 0) ⎨ ⎪ 4 Δ5, i = ( 4δx5, i 0 4δz5, i 4δα5, i 0 0)T ⎪ ⎪ 5 Δ6, i = (5δx6, i 0 5δz6, i 5δα6, i 0 0)T ⎪ ⎪ 6P 6 Δ = − Ad 6 7P 7 Δ g7, i 6 6 ⎪ 7, i 7, i ⎪7 T 7 7 7 7 7 ⎪ ⎩ Δ6, i = ( δx6, i δy6, i δz6, i 0 δβ6, i δγ6, i )

(13)

herein δ xj þ 1,i, δ yj þ 1,i and δ zj þ 1,i are the translational errors for the center of frame Rj,i with respect to frame Rj  1,i, while j δαj þ 1,i, jδβj þ 1,i and jδγj þ 1,i denote the corresponding rotating errors. In practice, 7Δ6, i replaces 6Δ7, i in this paper since it is easier to evaluate the geometric errors of R6, i with respect to R7, i . Similar to the PSU limb, relating transformation matrices in the S limb are calculated as j

j

j

⎧O ⎪ T 0,4 = Rot (z, π) ⎪0 ⎛ π ⎞ + θa,1,4 ⎟ ⎪ T 1,4 = Trans (z, h) Rot ⎜ z, ⎝ 2 ⎠ ⎪ ⎪1 ⎪ T = Rot ⎜⎛ x, π ⎟⎞ Rot ⎜⎛ z, π + θa,2,4 ⎟⎞ ⎨ 2,4 ⎝ 2⎠ ⎝ 2 ⎠ ⎪ ⎛ π⎞ ⎛ π ⎞ ⎪2 ⎜ ⎟ ⎜ ⎟ ⎪ T 3,4 = Rot ⎝ x, 2 ⎠ Rot ⎝ z, 2 + θa,3,4 ⎠ ⎪ ⎛ π⎞ ⎪3 T 4,4 = Trans (y , − h0 ) Rot ⎜ x, ⎟ ⎪ ⎝ 2⎠ ⎩

(14)

where θa, j,4 is the angle about the jth joint axis in the S limb. When considering the S limb, the pose error twist is formulated as

(10)

Trans ( − r )(r = OO′) is the homogeneous transformation matrix of frame R with respect to frame R′,

3

$t =

∑ Δθa, j,4 $^ ta, j,4 j=1

+ $ G,4 (15)

82

T. Sun et al. / Robotics and Computer-Integrated Manufacturing 41 (2016) 78–91 T T T T ⎤T Δd a = ( Δd1 Δd2 Δd3) , Eae = diag (Eae, i ), ϵae = ⎡⎣ ϵae ,1 ϵae,2 ϵae,3⎦

3

∑ AdgOj −′1,4 j − 1P j,4 j − 1Δ j,4 + AdgO3,4′ 3P 4,4 3Δ4,4

$ G,4 =

(16)

j=1

εae, i =

( δx 0

0

δy1, i

2, i

0

δα1, i

0

δβ1, i

2

3

δx3, i

δx 4, i

3

δz 4, i Δli

5

δx6, i

7

δx6, i

7

δy6, i

7

δz6, i

T

)

where Δθa, j,4 is the error of θa, j,4 . And Adg j − 1 is the 6 × 6 adjoint j,4

transformation matrix of frame Rj,4 with respect to frame Rj − 1,4 . j−1

P j,4 =

⎡ ⎢E ⎢ 3 ⎢ ⎢ ⎣0

⎡ j−1 ⎢ r j,4 ⎣

×

⎤⎤ ⎥⎥ ⎦⎥ j−1 r j,4 ⎥, ⎥ ⎦

0

is the position vector of the origin of

δx2, i = 0δx1, i + 1δx2, i

E3 frame Rj,4 . For the S limb, all the possible sources of errors are listed as follow [15]:

⎧0 ⎪ Δ1,4 ⎪ 1Δ ⎪ 2,4 ⎨ ⎪ 2 Δ3,4 ⎪ ⎪ 3 Δ4,4 ⎩

= (0 0 Δh 0 0 0)T = (1δx2,4 0 0 1δα2,4 0 0)T = (2δx3,4 0

2

= (0 Δh0 0 j

δz3,4

δα3,4 0 0)

3

δα4,4

j

T

2

3

Eae, i

T

δβ4,4 0)

(17)

j

herein δxj + 1,4 , δyj + 1,4 and δzj + 1,4 are the translational errors for the center of frame Rj,4 with respect to frame Rj − 1,4 , while

j

δαj + 1,4 ,

j

j

δβj + 1, 4 and δγj + 1,4 are the corresponding rotating errors. Taking inner products on both sides of Eqs. (11) and (15) with ^ ^ $ wa,1, i and $ wc, i,4 respectively, the geometric error model of the 3PSU&S parallel manipulator can be expressed as

⎛ 6 ⎞ ^ ^ ^ $ wa,1, i $ t = $ wa,1, i ⎜⎜ ∑ Δθa, j, i $ ta, j, i + $ G, i ⎟⎟, i = 1, 2, 3 ⎝ j=1 ⎠

⎞ ⎛ 3 ^ ^ ^ $ wc, i,4 $ t = $ wc, i,4 ⎜⎜ ∑ Δθa, j,4 $ ta, j,4 + $ G,4 ⎟⎟, i = 1, 2, 3 ⎠ ⎝ j=1

⎛ s2,4 ⎞ ^ ⎛ n1,4 ⎞ ^ ⎛ s1,4 ⎞ ^ $ wc,1,4 = ⎜ ⎟, $ wc,2,4 = ⎜ ⎟, $ wc,3,4 = ⎜ ⎟ ⎝ h 0 × s2,4 ⎠ ⎝ h 0 × n1,4 ⎠ ⎝ h 0 × s1,4 ⎠

Ece,1,4

⎛ ⎞T 0 ⎛ ⎞T 0 ⎜ ⎟ ⎜ ⎟ 1 0 ⎜ ⎟ ⎜ ⎟ ⎜ − Sθa,2,4 ⎟ ⎜ ⎟ 0 ⎟ , , Ece,2,4 = ⎜ =⎜ 0 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ Cθa,3,4 Sθa,2,4 ⎟ ⎜ − Sθa,3,4 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0 ⎝ h0 Cθa,2,4 ⎠

Ece,3,4

⎛ Δh ⎞ ⎛ ⎞T 1 ⎜1 ⎟ ⎜ ⎟ ⎜ δx2,4 ⎟ 0 ⎜ ⎟ ⎜2 ⎟ ⎜ ⎟ Cθa,2,4 ⎜ δx3,4 ⎟ ⎟ , εce = ⎜ 2 =⎜ ⎟ 0 ⎜ ⎟ ⎜ δz3,4 ⎟ ⎜ − Cθa,2,4 Cθa,3,4 ⎟ ⎜ Δh0 ⎟ ⎜ ⎟ ⎜⎜ ⎟⎟ ⎝ h0 Sθa,2,4 ⎠ 3 ⎝ δα4,4 ⎠

Without considering the home error temporarily, the error model can be rewritten by combining Eqs. (20) and (21) as

In this process of unfolding Eqs. (18) and (19), some elements of the error coefficient matrix are 0. The physical meaning of 0 is that the corresponding geometric errors have no effect on the pose, so they can be omitted. The error model can be rewritten as

Jxa $ ta = Jρa Δd a + Eae ϵae

(20)

Jxc $ tc = Ece εce

(21)

Jρa = diag (s1,T i wi ) = diag ( − Cθa,3, i Cθa,4, i ), i = 1, 2, 3

a,5, i

⎤T T T T Ece = ⎡⎣ Ece ,1,4 Ece,2,4 Ece,3,4 ⎦

(19)

⎛ wi ⎞ ⎟, i = 1, 2, 3 =⎜ ⎝ ai × wi ⎠

where

a,6, i

(18)

^ where $ wa,1, i is defined as the unit wrench of permission in the ith ^ PSU limb, $ wc, i,4 denotes the ith unit wrench of restriction in the S limb, and

^ $ wa,1, i

⎧ − Sθa,2, i Sθa,3, i Cθa,4, i + Cθa,2, i Sθa,4, i ⎫ ⎪ ⎪ Cθa,2, i Sθa,3, i Cθa,4, i + Sθa,2, i Sθa,4, i ⎪ ⎪ ⎪ ⎪ d θ θ θ θ θ C S C S S − ( + ) i a i a i a i a i a i ,2, ,3, ,4, ,2, ,4, ⎪ ⎪ ⎪ di ( − Sθa,2, i Sθa,3, i Cθa,4, i + Cθa,2, i Sθa,4, i )⎪ ⎪ ⎪ Sθa,3, i Cθa,4, i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Cθa,4, i ⎬ , i = 1, 2, 3 =⎨ ⎪ ⎪ Sθa,4, i ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ Cθa,5, i ⎪ ⎪ − Sθa,5, i ⎪ ⎪ ⎪ ⎪ − Sθa,6, i Cθa,5, i ⎪ ⎪ ⎪ ⎪ − Cθ Cθ ⎩ ⎭

Jx $ t = Ee εe

(22)

$ t = J x+ Ee εe

(23)

where

Jx+

is the inverse matrix of Jx , and

⎡ Eae 0 ⎤ T T T T T⎤ Ee = ⎢ ⎥, εe = ⎡⎣ εae ,1 εae,2 εae,3 εce ⎦ ⎣ 0 Ece ⎦ As the three PSU limbs connect the moving platform to the fixed base with an axis-symmetrical layout, it can be concluded that the characteristics of the geometric errors of the three PSU limbs are the same. Therefore, besides the 6 geometric errors of

T. Sun et al. / Robotics and Computer-Integrated Manufacturing 41 (2016) 78–91

83

Fig. 3. The verification flow of geometric error model.

Table 1 The given tolerances of geometric source errors. Error no.

Error

Value

Table 2 Typical poses within prescribed workspace.

Error no.

Error

Value

1

0

0.700

10

7

0.500

2

0

 0.500

11

7

0.400

δx2, i /mm δy1, i /mm

δx6, i /mm

δy6, i /mm

3

0

0.100

12

7

0.600

4

0

 0.100

13

Δh /mm

1.000

δα1, i /(°)

δβ1, i /(°)

δz6, i /mm

2

0.400

14

1

3

 0.300

15

2

0.400

7

3

 0.600

16

2

 0.700

8 9

0.800 0.500

17 18

Δh0 /mm

5

 0.800 0.100

δx 4, i /mm δz 4, i /mm

Δli /mm δx6, i /mm

δx2,4 /mm δx3,4 /mm δz3,4 /mm

3

δα4,4 /(°)

Pose II

Pose3 III

 35° 15° 60°

 45° 20° 120°

 55° 20° 240°

 0.600

5 6

δx3, i /mm

ψ /(°) θ /(°) ϕ /(°)

Pose I

the S limb, only the 12 geometric errors of one PSU limb are considered in Section 3.2 (Simulation and Verification of Geometric Error Model) and Section 4 (Sensitivity Analysis) for the sake of simplicity.

3.2. Simulation and verification of geometric error model In this section, the proposed geometric error model is verified using SolidWorkss software, which is implemented by the flow shown in Fig. 3. In order to verify the geometric error model, the values of the geometric source errors are given in Table 1. Without loss of generality, three typical poses within the prescribed workspace (ψ ∈ [0, 360∘], θ ∈ [0, 30∘], ϕ ∈ [ − 60∘ , − 15∘ ]) are selected to verify the error model. The general procedure for the simulation and verification of error model is shown in Fig.3, which is

Fig. 4. The position vector of measured point coordinate in SolidWorkss software.

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1) Select one geometric source error according to Table 1. The given error is designed to be larger than possible error. 2) Choose one typical pose from Table 2. Three arbitrary poses are determined from the prescribed workspace. 3) Simulate the position and orientation errors with selected geometric source error and given pose. Firstly, add the geometric error to the virtual prototype with SolidWorkss software according to the assembling relation between components. Secondly, establish fixed and moving frames as defined in Section 2.1. The coordinates of verification points are obtained as

shown in Fig. 4. Then, calculate RM , RT , am and at . Finally, formulate ‖δrM ‖ and ‖δαM ‖. 4) Calculate the position and orientation errors with selected geometric source error and given pose. Firstly, establish the error model by Matlab programming. Secondly, introduce the same geometric source error and pose as step 3). Finally, run the program and achieve ‖δrC ‖ and ‖δαC ‖. 5) Compare the simulation from step 3) and calculation from step 4). 6) Go back to step 2). Select another pose and repeat step 3) and step 4).

Fig. 5. Comparison of two error values obtained from SolidWorkss software and geometric error model.

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Table 3 Prescribed workspace and dimensional parameters of the 3-PSU&S manipulator. Prescribed workspace (Unit: °)

dimensional parameters (Unit: mm) a

b

l

h

h0

ψ

θ

ϕ

160

216

280

567

75

[0, 360]

[0, 30]

[-60,-15]

Table 4 Design tolerances of the components. Error no.

Error

Tolerance

Error no.

Error

Tolerance

1

0

0.200

10

7

0.100

2

0

0.400

11

7

0.039

3

0

0.050

12

7

0.200

4

0

0.050

13

Δh /mm

0.100

5

2

0.022

14

1

0.022

6

3

0.022

15

2

0.022

7

3

0.100

16

2

0.100

8 9

Δli /mm

0.200 0.022

17 18

Δh0 /mm

5

3

0.100 0.020

δx2, i /mm δy1, i /mm δα1, i /(°)

δβ1, i /(°)

δx3, i /mm δx 4, i /mm δz 4, i /mm

δx6, i /mm

δx6, i /mm

δy6, i /mm

δz6, i /mm

δx2,4 /mm δx3,4 /mm δz3,4 /mm δα4,4 /(°)

Fig. 7. The pose error of the moving platform measured by laser tracker.

4. Sensitivity analysis 7) Go back to step 1). Choose another geometric source error and repeat Step 2) to Step 6). Noted that RM and RT represent the measured and the theoretical orientation matrix of frame O′ − uvw with respect to frame O − xyz , am and at represent the measured and the theoretical position vector of measured point in frame O − xyz , ‖δrM ‖ and ‖δrC ‖ represent the position error obtained by means of software and geometric error model, ‖δαM ‖ and ‖δαC ‖ represent the orientation error obtained by means of software and geometric error model. As shown in Fig. 5, the deviations of two error values obtained from SolidWorkss software and geometric error model at Pose I, II and III are around 10  3. Considering the issue that the nonlinear terms are neglected in the geometric error model, the geometric error model of the 3-PUS&S rotational parallel manipulator is regarded as valid.

As stated above, the effects of geometric source errors to the pose accuracy of the moving platform are demanded to be evaluated to find the errors having high sensitivity. Bearing it in mind, a probability model of geometric source errors is formulated based upon Eq. (23) as

$ t = Jc εe

(24)

where

Jc = J x+ Ee, Jc = ⎡⎣ Jc1 Jc2 Jc 3 Jc 4

T Jc5 Jc6 ⎤⎦

From Eq. (24), the sensitivity coefficient of the position and orientation volume errors [22] can be expressed as 3

κrm =

6

∑ Jci2, m , κ αm = ∑ Jci2, m , m = (1, 2, 3, ... , 18) i=1

i=4

(25)

where m denotes the eighteen types of the geometric source

Fig. 6. Sensitivity analysis of geometric source errors with respect to (a) μr and (b) μα .

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Fig. 8. Measured poses in the prescribed workspace.

δ r = κrm εem , δα = κ αm εem

(26)

where δr and δα denote the position and the orientation volume errors with respect to the moving platform, respectively. According to Ref. [22], when components are produced with large quantity, the component tolerance subjects to the zero-mean normal distribution if there is no special requirement. Thus, the zero-mean and normal distribution characteristic are assumed for geometric source error before utilizing Monte Carlo method. By taking into consideration the effects of component tolerances, the simulation results will be more realistic. According to the 3s manufacturing criteria, the standard deviation is one in six of the tolerance. It is noted that the output errors of the 3-PSU&S parallel manipulator are configuration-varying, and their maximum within the prescribed workspace can be selected as the global sensitivity evaluation index. Assuming that we have N times calculation in total by means of Monte Carlo Method, and the maximum value of output error can be obtained as

⎧ δ max rm = max (δ rm ) ⎨ ⎩ δ max αm = max (δαm )

(27)

Based upon Eq. (27), the maximum value of the mth type error can be obtained within the prescribed workspace when it acts on the terminal pose alone.

⎧ μ rm = max (δ max rm, n) ⎨ , n = 1~N ⎩ μαm = max (δ max αm, n )

Fig. 9. The simulation process of kinematic calibration.

errors. Therefore, the probability model of the geometric source errors can be rewritten as

(28)

where δmax rm, n and δmax αm, n denote the maximum values of the position and the orientation volume errors in the nth time calculation within the global workspace respectively when the mth type error acts on the terminal pose alone. In consequence, μrm and μαm are defined as the global sensitivity indices of the geometric source errors with respect to the output error of the 3-PSU&S parallel manipulator. By utilizing the sensitivity indices obtained in Eq. (28), the sensitivity analysis is carried out for the 3-PSU&S parallel manipulator whose prescribed workspace and nominal dimensional parameters are given in Table 3. The design tolerances of the components are shown in Table 4. It is noted that 5δx6, i , 2δx3, i , 3δx 4, i , 3δz 4, i , 7δx6, i , 7δy6, i (i¼ 1,2,3), 1

δx2,4 , 2δx3,4 , 2δz3,4 and 3δα4,4 represent the error of the adjacent axes of the S and U joints. As can be seen from Fig. 6, these errors have little influence on the pose accuracy of the moving platform and can be neglected when the calibration experiment is carried out.

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87

Table 5 Simulation results. Error no.

1

2

3

4

5

6

7

8

9

10

11

Error

Δd1 (mm)

0

0

0

0

Δl1 (mm)

5

Δd2 (mm)

0

0

0

Given value Without noise With noise

0.8 0.793 0.763

0.4 0.399 0.381

 0.3  0.304  0.268

0.05 0.048 0.048

0.01 0.009 0.009

1 0.994 0.988

 0.2  0.200  0.186

0.7 0.702 0.672

0.5 0.504 0.543

 0.2  0.203  0.226

0.1 0.106 0.106

Error No.

12

13

14

15

16

17

18

19

20

21

22

Error

0

Δl2 (mm)

5

Δd3 (mm)

0

0

0

0

Δl3 (mm)

5

Δh (mm)

Given value Without noise With noise

0.01 0.011 0.011

1 1.001 1.033

 0.3 0.298  0.277

0.5 0.496 0.481

0.1 0.097 0.084

 0.2  0.205 -0.232

0.06 0.057 0.057

0.02 0.020 0.021

1 0.996 0.971

0.4  0.401  0.418

0.8 0.792 0.815

Error No.

23

24

25

26

Error

Δh0 (mm)

Δlcx (mm)

Δlcy (mm)

Δlcz (mm)

Given value Without noise With noise

0.8 0.799 0.779

0.5 0.502 0.472

0.6 0.592 0.567

0.8 0.807 0.821

δβ1,2 (°)

δx2,1 (mm)

δy1,1 (mm)

δz6,2 (mm)

δα1,1 (°)

δβ1,1 (°)

δx2,3 (mm)

δy1,3 (mm)

δz6,1 (mm)

δα1,3 (°)

δβ1,3 (°)

δx2,2 (mm)

δy1,2 (mm)

δz6,3 (mm)

Fig. 10. The deviation between given and identification values with and without noise.

Fig. 11. Physical prototype of 3-PSU&S parallel manipulator.

Fig. 12. Kinematic calibration experiment.

δα1,2 (°)

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Table 6 The given and measured pose parameters before and after calibration. Point no.

1 2 3 4 5 6 7 8 9 10 11 12

Given pose parameters (Unit: °)

Measured pose parameters before calibration (Unit: °)

Measured pose parameters after calibration (Unit: °)

ψ

θ

ϕ

ψ

θ

ϕ

ψ

θ

ϕ

15 45 75 105 135 165 195 225 255 285 315 345

10 10 10 10 10 10 20 20 20 20 20 20

 20  20  20  20  20  20  40  40  40  40  40  40

13.6384 43.5886 74.1526 104.5617 134.7275 164.7687 194.7310 224.0512 253.2430 283.0251 313.3350 343.7245

9.9499 9.9508 9.7976 9.9231 10.0254 10.0298 20.1434 20.2351 20.2101 20.0798 20.0134 19.9503

 21.7305  21.7321  21.6543  21.6486  21.7123  21.6685  41.6734  41.6654  41.7310  41.7208  41.6972  41.7481

14.6546 45.5032 75.5387 105.2826 134.9189 164.8065 194.5512 225.1487 255.0634 284.5434 315.1305 345.5374

9.9881 10.0456 10.0764 10.0843 10.0947 10.1034 20.0832 20.0612 19.9734 19.8987 19.8949 19.9384

 20.1853  20.1534  20.1143  20.1104  20.1115  20.0789  40.0945  40.1184  40.1134  40.1254  40.1798  40.1928

That is to say, the remaining 20 geometric errors need to be identified in the kinematic calibration.

As shown in Fig. 7, when the laser tracker is initialized, the coordinate frame Om − xm ym zm is established automatically. For the reference point P within the prescribed workspace, two closedloop vector equations can be formulated as

(29)

rc = rO ′ + Rl c = R t rm + r M

where rO ′, rc and rM represent the position vectors of points O′, P and Om in frame O − xyz , respectively; lc and rm are the position vectors of P in frames O′ − uvw and Om − xm ym zm ; Rt represents the orientation matrix of frame Om − xm ym zm with respect to frame O − xyz . Taking first-order perturbation on both sides of Eq. (29) yields

δrO ′ + δα × ( Rl c ) + R Δl c = R t Δrm + Δr M

(30)

where δrO ′ and δα represent the position and the orientation error, respectively, Δrm = ⌢ rm − r¯m , ⌢ rm and r¯m denote the theoretical and the measured position vector of point P in frame Om − xm ym zm , Δlc is the position error vector of point P in frame O′ − uvw , ΔrM represents the position error vector of point Om in frame O − xyz . Substituting Eq. (23) into Eq. (30) yields

JN ΔpN = R t Δrm + Δr M

(31)

where T JN = ⎡⎣ Jr − ⎡⎣ (Rl c ) × ⎤⎦ Jα R ⎤⎦, ΔpN = ⎡⎣ ϵT Δl cT ⎤⎦, ϵ = ( ϵ1 ϵ2 ϵ3 ϵ4 )

)

ϵ1 = Δd1 0δx2,1 0δy1,1 0δα1,1 0δβ1,1 Δl1 5δz6,1 , 2

(

ϵ3 = Δd3

0

0

0

δα1,2

0

δβ1,2 Δl2

5

)

0

0

0

δα1,3

0

δβ1,3 Δl3

5

)

δx2,2

δx2,3

δy1,2

δy1,3

)

(32)

R t ⎡⎣ Δθm × ⎤⎦ R m0 = [δα × ] R

5.1. Error parameter identification principle

ϵ2

(

where Δθm is the orientation error vector in the measuring process; Rm0 is the measured orientation matrix in theory. Noticing that Rt Rm0 = R , we can simplify Eq. (32) to

5. Kinematic calibration

( = ( Δd

R t E3 + ⎡⎣ Δθm × ⎤⎦ R m0 = ( E3 + [δα × ] ) R

δz6,2

δz6,3 , ϵ4 = ( Δh Δh0 )

herein ε T represents the error vector of the manipulator; Jq is the sub-matrix of Jc which corresponds to the columns of ε T ; Jr is the sub-matrix consisting of the first three rows of Jq and Jα is the submatrix which consists of the last three rows of Jq . By considering the orientation errors, the orientation parameter identification model is formulated as

(33)

′ is In the calibration experiments, if the orientation matrix R m obtained by measurement, then Eq. (33) can be rewritten as

′ − R m0 ) = [δα × ] R Rt ( R m

(34)

When the reference point P is located at the measured point l (l ) is the position error vector in frame (l = 1, 2, ⋅⋅⋅, L), Δrm Om − xm ym zm . The parameter identification model can be rewritten by combining Eqs. (31) and (34) as

Δr N = HN ΔpN

(35)

where

⎡ (1) ⎤ ⎡ J (1) ⎤ R t Δrm ⎥ ⎢ ⎢ N ⎥ ⎥ ⎢ ⎥ ⎢ (1) (1) ′ − R m0 ⎥ ⎢ Rt R m ⎢ MN(1) ⎥ ⎥ ⎢ ⎥ ⎢ (2) ⎥ ⎢ R t Δrm ⎢ J N(2) ⎥ ⎥ ⎢ ⎥ ⎢ (L) ⎡ ⎤ (L) 0 Δr N = ⎢ (2) (2) ⎥, HN = ⎢ (2) ⎥, MN = ⎣ [δα × ] R 3 × 3⎦ ′ R R R − M t m m0 ⎥ ⎢ ⎢ N ⎥ ⎥ ⎢ ⎢ ⋮ ⎥ ⋮ ⎥ ⎢ ⎢ (L) ⎥ ( ) L ⎢ R t Δrm ⎥ ⎢ JN ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ (L) ⎥ ⎢⎣ MN(L) ⎥⎦ ′ (L) − R m ⎢⎣ R t R m 0 ⎥ ⎦

(

)

(

)

(

)

Provided that HNT HN is not singular, the following equation can be obtained

ΔpN = (HNT HN )−1HNT Δr N

(36) HNT HN

It is noticed that the condition number of has a significant impact on the identification results. To obtain a smooth solution to the ill-posed problem [38], the standard Tikhonov regularization method [39] is employed in this paper. Hence, Eq. (36) can be rewritten as

ΔpN = (HNT HN + αI )−1HNT Δr N

(37)

where α is the regularization parameter to be determined by the GCV method and the Genetic Algorithms. Once ⌢ rm and Rt are obtained by utilizing the laser tracker, the geometric errors of each limb can be identified by Eq. (37).

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Table 7 Maximum orientation deviations before and after calibration. Orientation deviations

Before calibration

After calibration

Δψ /(˚) Δθ /(˚) Δϕ /(˚)

1.9749 0.2351 1.7481

0.5374 0.1034 0.1928

controllable degrees of freedom. Therefore, the methods of uniform in the prescribed workspace should be used. Moreover, it is believed that measuring enough poses is beneficial to increase the identification robustness [10]. However, the measured poses can not be as many as possible in consideration of the calibration efficiency. A compromise must be made between the identification robustness and the calibration efficiency. By trial and error, it is found that the number of identification equations should be at least twice more than that of parameters to be identified. Therefore, a total of 54 poses within the prescribed workspace are selected to be measured, which is shown in Fig. 8. 5.3. Example To verify the robustness of the identification algorithm, the Gaussian noise whose mean is zero and standard deviation is 0.002 is added to the theoretical value of the measured poses. With the help of SolidWorkss software, a virtual prototype is built which contains all geometric source errors. As shown in Fig. 9, the general procedure of the kinematic calibration simulation is: 1) determine a set of geometric error values and introduce them to the virtual prototype with the similar method in Section 3.2 2) drive the moving platform to the measured pose and measure the coordinates of the three points shown in Fig. 4 3) repeat with all the 54 measured poses 4) apply all the measuring coordinates to the parameter identification model 5) calculate α by GCV approach and obtain a set of error values 6) compare the geometric errors by identification model and the given errors After the simulation process shown in Fig. 9, the comparison of the given geometric errors, errors without considering noise and errors with considering noise are listed in Table 5, whose deviations are shown in Fig. 10. It is concluded from Table 4 and Fig. 10 that the deviation between geometric errors obtained by identification model and the given values are within 5%, which verifies the effectiveness of identification model considering noise or not.

6. Kinematic calibration experiment

Fig. 13. Error curves of 12 test poses in the direction ψ θ and ϕ (a) Orientation deviation before and after in the direction ψ (b) Orientation deviation before and after calibration in the direction θ (c) Orientation deviation before and after calibration in the direction ϕ .

5.2. Measured point planning In order to ensure the identifiability of the geometric source errors, the track of measured poses should go through all

To validate the kinematic calibration process of the 3-PUS&S rotational parallel manipulator, the kinematic calibration experiment is carried out in this section. As shown in Fig. 11, a physical prototype of the 3-PUS&S rotational parallel manipulator is established. It is noted that the pose of the moving platform is measured by means of the Leica-AT901LR Laser Tracker. The pose measuring procedure can be summarized as follows: (1) Determine w by measuring the coordinates of arbitrarily three non-collinear points on the moving platform plane; (2) Determine u by measuring the coordinates of the datum hole centers shown in Fig. 11 on the moving platform;

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T. Sun et al. / Robotics and Computer-Integrated Manufacturing 41 (2016) 78–91

(3) Determine v by right-hand rule and achieve the rotation matrix R; (4) Calculate the rotating angles θ , ψ and ϕ of the moving platform by the following Eq. (38)

⎧ θ = arccos ( R ) 33 ⎪ ⎪ ⎛ R23 ⎞ ⎪ Ψ = arctan ⎜ ⎟ ⎨ ⎝ R13 ⎠ ⎪ ⎛ R32 ⎞ ⎪ ⎪ ϕ = Ψ − arctan ⎜⎝ R ⎟⎠ ⎩ 31

(38)

The calibration experiment process shown in Fig. 12 is similar with the simulation process, that is to say, 54 measured poses shown in Fig. 8 are also used in the experiment to calibrate the 3-PUS&S rotational parallel manipulator. After performing the kinematic calibration, the identification parameters rather than the original parameters will be embedded into the control model. In order to evaluate the kinematic calibration experiment, the poses of another 12 test poses before and after kinematic calibration are obtained in Table 6. The corresponding curves are shown in Fig. 13. As shown in Table 7, the maximum orientation deviation before and after kinematic calibration of the 3-PSU&S parallel rotational manipulator is given. It is concluded from Table 7 that the pose accuracy of the 3PSU&S parallel manipulator is improved by at least 53.4%. Considering the manufacturing and assembly errors of the components, the pose accuracy after kinematic calibration is acceptable.

7. Conclusions In this paper, the kinematic calibration method of a 3-DoF rotational parallel manipulator (3-PUS&S parallel manipulator) using laser tracker is proposed. The conclusions are drawn as follows: 1) Considering all geometric source errors, the geometric error model of the 3-PSU&S parallel manipulator is formulated by means of screw theory in a concise and distinct manner. The validity of the error model is proved by utilizing the software method. 2) The sensitivity analysis of geometric errors is implemented by means of Monte Carlo method to evaluate the effects of geometric errors to the pose accuracy of the moving platform. In order to increase the identification robustness, the errors having high sensitivity are considered while the errors that have less influence on the pose accuracy of the moving platform are neglected when kinematic calibration is carried out. 3) By establishing the virtual prototype that has the geometric errors, the simulation of kinematic calibration is more close to the actual case. The simulation results of kinematic calibration verify the effectiveness of the parameter identification model. 4) The experimental results show that the proposed calibration method is valid and effective, and can improve the pose accuracy of the 3-PSU&S parallel manipulator.

Acknowledgments This research work was supported by the National Natural Science Foundation of China (NSFC) under Grant nos. 51475321 and 51205278, Tianjin Research Program of Application Foundation and Advanced Technology under Grant no. 15JCZDJC38900.

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