Kinematics analysis of an offset 3-UPU translational parallel robotic manipulator

Robotics and Autonomous Systems 42 (2003) 117–123

Kinematics analysis of an offset 3-UPU translational parallel robotic manipulator Ping Ji a,∗ , Hongtao Wu b a

b

Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Department of Mechanical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, People’s Republic of China Received 6 March 2002; received in revised form 3 September 2002 Communicated by F.C.A. Groen

Abstract The parallel robotic manipulator has attracted many researchers’ attention and it also has growing applications to different areas. This paper proposes a 3-UPU (universal–prismatic–universal) translational parallel robotic manipulator with an equal offset in its six universal joints, based on the zero offset 3-UPU parallel manipulator. The kinematics of the new manipulator is analyzed and its inverse and forward kinematics solutions are provided. The conclusion is that its forward kinematics has 16 solutions instead of two in the zero offset manipulator. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Robotic manipulator; Kinematics; Parallel manipulator

1. Introduction The parallel robotic manipulator has attracted many researchers’ attention [1–3] and it also has growing applications to robotics, machine tools, positioning systems, measurement devices, and so on. It has been proved that such a spatial close-loop manipulator has its great potential and advantage over the traditional serial manipulator. The parallel manipulator is able to provide a very complicated motion for the end effector in a flexible way with many degrees of freedom (DOF) by use of a combination of prismatic, revolute, universal or spherical joints. So, the family of parallel manipulators is very extensive. ∗

Corresponding author. E-mail address: [email protected] (P. Ji).

This paper studies an offset 3-UPU (universal– prismatic–universal) parallel manipulator, as shown in Fig. 1. A special 3-UPU parallel manipulator with zero offsets in the six universal joints, that is, e = 0, was proposed by Tsai in 1996 [4,5]. Tsai’s 3-UPU manipulator has three extensible limbs (legs) that connect the base and the mobile platform through universal joints, and it can provide a 3-DOF pure translational motion. Due to zero offsets in the universal joints, the kinematics of the manipulator is simple and straightforward [4]. Tsai and Stamper [5] also proposed a 3-RUU translational parallel manipulator with non-zero offset U joints and pointed out that the forward kinematics for this 3-RUU manipulator has sixteen solutions. The mobility analysis, that is, rotation singularities and translation singularities, of the zero offset 3-UPU manipulator was performed by

0921-8890/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 8 8 9 0 ( 0 2 ) 0 0 3 2 4 - X

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P. Ji, H. Wu / Robotics and Autonomous Systems 42 (2003) 117–123

Fig. 1. The 3-UPU parallel manipulator.

Di Gregorio and Parenti-Castelli [6]. Earlier, these two researchers also studied the general 3-RRPRR manipulator, and the mounting conditions for the pure translation of the manipulator [7]. However, the kinematics of the general 3-RRPRR manipulator has not been analyzed yet. The offset 3-UPU parallel manipulator studied in this paper, shown in Fig. 1, is an extension of Tsai’s 3-UPU manipulator since it has non-zero but the same offsets in the universal joints (e = 0). On the other hand, the offset 3-UPU manipulator is a special case of the general 3-RRPRR manipulator since all the offsets in the universal joints of the new manipulator are the same, and the two revolute axes in a universal joint are perpendicular each other. According to the mounting conditions for the pure translation [7], the offset 3-UPU manipulator still provides 3 DOF pure translational motion. However, the kinematics of the manipulator is complicated, and this paper will study this problem.

2. Manipulator description Fig. 2 shows a schema of the ith limb (i = 1, 2, 3), which connects point Bi on the mobile platform and point Ai on the base by a passive universal (U) joint, an active prismatic (P) joint, and another passive universal (U) joint. So, it is called a 3-UPU parallel manipulator. W1i and W2i are the two rotating axes of

Fig. 2. The ith limb of the offset 3-UPU manipulator.

the bottom universal joint in the ith branch limb (i = 1, 2, 3), while W3i and W4i are the two rotating axes of the top universal joint of the limb, as shown in Fig. 2. Consequently, θ 1i , θ 2i , θ 3i and θ 4i are the rotation angles around its corresponding axis. As mentioned earlier, e is the offset in a universal joint. Besides, W1i is perpendicular to W2i and similarly, W3i is perpendicular to W4i . Due to e = 0, this manipulator is called the offset 3-UPU parallel manipulator. Fig. 3 shows the arrangement of three vertices Ai on the fixed base, which form a triangle A1 A2 A3 . Let the global frame Σ be OXYZ, attached to the base, with the origin O being the circumcenter of the triangle A1 A2 A3 . Here, the Z axis is chosen to be parallel to one of the universal axes (see Fig. 3), the X axis is vertical to the plane spanned by the triangle A1 A2 A3 while the Y axis is determined by the right-hand rule. Similarly, the mobile frame Σ P , that is, OP XP YP ZP , attached to the mobile platform, can be determined as shown in Fig. 4. Besides, r is the circumradius of

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119

3) of a point in a local frames Σ i and its coordinates {X, Y, Z} in the global frame Σ have the following relationship:      1 0 0    Xi   X    Yi =  0 cos βi sin βi  Y         Z Zi 0 −sin βi cos βi (i = 1, 2, 3).

Fig. 3. The base and the global frame.

triangle A1 A2 A3 in the base (shown in Figs. 2 and 3), while rP is the circumradius of triangle B1 B2 B3 with the circumcenter OP in the mobile platform (shown in Figs. 2 and 4). For the convenience of kinematics analysis, three local frames, that is, Σ 1 : OX1 Y1 Z1 , Σ 2 : OX2 Y2 Z2 , and Σ 3 : OX3 Y3 Z3 , are set up and attached to the base in order to represent the position and the first axis of a universal joint. Actually, frame Σ i (i = 1, 2, 3) is rotated along the X axis by angle β i from frame Σ. Besides, β1 = 0 since Σ 1 is already chosen to be parallel to Σ. Thus coordinates {Xi , Yi , Zi } (i = 1, 2,

That is,    1 0   X1    Y1 =  0 cos β1     Z1 0 −sin β1    X  = Y ,     Z

   X   sin β1  Y     Z cos β1 0

       1 0 0  X2   X    Y2 =  0 cos β2 sin β2  Y    Z   0 −sin β2 cos β2  Z  2     X   Y cos β2 + Z sin β2 = ,   −Y sin β + Z cos β   2 2

Fig. 4. The mobile platform and its frame.

(1)

(1a)

(1b)

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     1 0 0    X3   X    Y3 =  0 cos β3 sin β3  Y         Z Z3 0 −sin β3 cos β3   X     Y cos β3 + Z sin β3 = .     −Y sin β3 + Z sin β3

As a matter of fact, each branch limb is a 3-DOF serial manipulator with three independent joint variables θ 1i , θ 2i , and Li due to (2a) and (2b) as discussed by Di Gregorio and Parenti-Castelli [7]. (1c) 3. Inverse kinematics

Due to the special structure of the offset 3-UPU manipulator, the following two equations exist: θ1i = −θ4i

(i = 1, 2, 3),

(2a)

θ2i = −θ3i

(i = 1, 2, 3).

(2b)

The above results were also obtained by Tsai [4] and actually they can be derived from the mounting conditions for a general 3-RRPRR manipulator [7]. As a matter of fact, the manipulator is required to be assembled to satisfy W1i = W4i

(i = 1, 2, 3),

(3a)

W2i = W3i

(i = 1, 2, 3),

(3b)

so, the manipulator will provide a pure translational motion. Consequently, frame Σ P in the mobile platform is always parallel to the global frame Σ. Now from Fig. 2, the vector from Ai to Bi in a local frame Σ i (i = 1, 2, 3) is        cos θ1i cos θ2i    cos θ1i   {Ai Bi } = e sin θ1i + Li sin θ1i cos θ2i         0 sin θ2i     cos(−θ4i )   + e sin(−θ4i ) (i = 1, 2, 3). (4a)     0 Due to condition (2a), Eq. (4a) becomes        cos θ1i cos θ2i    cos θ1i   {Ai Bi } = 2e sin θ1i + Li sin θ1i cos θ2i         0 sin θ2i     (Li cos θ2i + 2e)cos θ1i   = (Li cos θ2i + 2e)sin θ1i (i = 1, 2, 3).     Li sin θ2i (4b)

Let the coordinates of OP be (X, Y, Z) in the global frame Σ. For the inverse kinematics problem, the origin point OP (X, Y, Z) of the mobile platform in the global frame Σ is given and the link lengths Li (i = 1, 2, 3) are unknown, and need to be found. Since each branch limb can be regarded as a 3-DOF serial manipulator with its joint variables {θ 1i , θ 2i , Li }, point OP in a local frame Σ i (i = 1, 2, 3) can be represented as          Xi   0   0  Yi = {Ai Bi } + 0 − 0 (i = 1, 2, 3),             Zi r rP (5) where {Xi , Yi , Zi } is the coordinates of OP in the local frame Σ i (i = 1, 2, 3), or       Xi     (Li cos θ2i + 2e) cos θ1i   Yi = (Li cos θ2i + 2e) sin θ1i (i = 1, 2, 3),         Zi Li sin θ2i +dr (6) where dr = r − rP is the difference of the two radii. For a given position vector {Xi , Yi , Zi } (i = 1, 2, 3), Eq. (6) has three unknowns θ 1i , θ 2i , and Li . By eliminating θ 1i and θ 2i , Eq. (6) becomes a quartic equation in terms of e, dr and Li : [Xi2 + Yi2 + (Zi − dr)2 − 4e2 − L2i ]2 −16e2 [L2i − (Zi − dr)2 ] = 0

(i = 1, 2, 3).

(7)

Actually, Eq. (7) is a quadratic polynomial in terms of L2i :  L2i = Xi2 + Yi2 + (Zi − dr)2 + 4e2 ± 4e Xi2 + Yi2 (i = 1, 2, 3),

(8a)

P. Ji, H. Wu / Robotics and Autonomous Systems 42 (2003) 117–123

or

121

can be obtained: 

 Li = Xi2 +Yi2 +(Zi − dr)2 + 4e2 ± 4e Xi2 + Yi2 (i = 1, 2, 3).

(8b)

The above equation implies each Li has two solutions   LBi = Xi2 +Yi2 +(Zi − dr)2 + 4e2 + 4e Xi2 + Yi2 (i = 1, 2, 3),

− 16e2 [L21 − (Z − dr)2 ] = 0,

 LSi = Xi2 +Yi2 +(Zi − dr)2 + 4e2 − 4e Xi2 + Yi2 (9b)

If e (the offset) is zero, the two solutions merge into one, and this is the solution to Tsai’s 3-UPU manipulator.

+ (−Y sin β2 + Z cos β2 − dr)2 − 4e2 − L22 ]2 − 16e2 [L22 − (−Y sin β2 + Z cos β2 − dr)2 ] = 0, (10b) [X2 + (Y cos β3 + Z sin β3 )2 + (−Y sin β3 + Z cos β3 − dr)2 − 4e2 − L23 ]2 − 16e2 [L23 −(−Y sin β3 + Z cos β3 − dr)2 ] = 0. (10c) Each of Eqs. (10a)–(10c) represents one quartic equation in terms of X, Y, and Z. In order to solve the above equation set, let P = X2 + Y 2 + Z 2 ,

4. Forward kinematics

(10a)

[X2 + (Y cos β2 + Z sin β2 )2

(9a)



(i = 1, 2, 3).

[X 2 + Y 2 + (Z − dr)2 − 4e2 − L21 ]2

(11)

so, Eqs. (10a)–(10c) can be rewritten as

When three link lengths Li (i = 1, 2, 3) are given, the forward kinematics is to find out OP {X, Y, Z}. From Eqs. (1) and (7), the following three equations

[P − 2 dr Z + dr 2 − 4e2 − L21 ]2 − 16e2 [L21 − (Z − dr)2 ] = 0,

Fig. 5. A butterfly (13).

(12a)

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P. Ji, H. Wu / Robotics and Autonomous Systems 42 (2003) 117–123

[P − 2 dr Z cos β2 +2 dr Y sin β2 +dr 2 −4e2 − L22 ]2 − 16e

2

[L22

− (−Y sin β2 + Z cos β2 − dr) ] = 0, 2

(12b) [P − 2 dr Z cos β3 +2 dr Y sin β3 +dr 2 − 4e2 − L23 ]2 − 16e2 [L23 − (−Y sin β3 + Z cos β3 − dr)2 ] = 0.

β1 = 0, β2 = 2π/3, and β3 = 4π/3. If three limb lengths are: L1 = 1.486, L2 = 1.386, and L3 = 1.576, the forward kinematics can be found by use of the above results. In this case, Eqs. (12a)–(12c) are − 0.529984[2.2082 − (0.382 + Z)2 ] + (−2.19477 + P + 0.764Z)2 = 0,

(15a)

(12c) Now each of Eqs. (12a)–(12c) is a quadratic polynomial in three variables {P, Y, Z}. By using the traditional elimination method [8,9], the final equation is an eighth polynomial in one variable. Eq. (12a) represents an ellipse in the P–Z plane. Eqs. (12b) and (12c) are two ellipsoids in the space of (P, Y, Z). By eliminating Y from (12b) and (12c), the result is a fourth polynomial in the P–Z plane, and it represents a curve like a “butterfly”, a typical one is shown in Fig. 5. The fourth polynomial is in the form of

− 0.529984[1.921 − (0.382 + 0.866Y − 0.5Z)2 ]

a4 (Z)P 4 +a3 (Z)P 3 +a2 (Z)P 2 +a1 (Z)P +a0 (Z)=0.

By eliminating Y from Eqs. (15b) and (15c), the butterfly (13) becomes

(13) So, the final solution for P and Z is given by the intersection of the ellipse (12a) and the butterfly (13) and there are at most eight intersection points. By eliminating P from (12a) and (13), an eighth polynomial in Z is s8 Z + s7 Z + s6 Z + s5 Z + s4 Z + s3 Z 8

7

6

+ s2 Z + s1 Z + s0 = 0. 2

5

4

3

+ (−1.90757 + P + 0.661643Y − 0.382Z)2 = 0, (15b) − 0.529984[2.48378 − (0.382 − 0.866Y − 0.5Z)2 ] + (−2.47035 + P − 0.661643Y − 0.382Z)2 = 0. (15c)

81.7209 − 179.76P + 142.29P 2 − 48.125P 3 + 5.85045P 4 + 93.1875Z − 153.359PZ + 79.5426P 2 Z − 12.9979P 3 Z + 41.7691Z 2 − 44.9452PZ2 + 11.2523P 2 Z 2 + 9.1366Z 3 − 4.74884PZ3 + 0.865295Z 4 = 0.

(16)

(14)

So, there are at most eight real solutions in terms of P, Y, and Z. From Eq. (11), there are at most 16 real solutions in terms of X, Y, Z and they are symmetrical to the Y–Z plane, which means eight solutions are above the Y–Z plane and the others are below the plane. For each solution of (X, Y, Z), (Xi , Yi , Zi ) (i = 1, 2, 3) can be easily obtained from Eq. (1). Finally, once (Xi , Yi , Zi ) (i = 1, 2, 3) are known, parameters θ 1i and θ 2i (i = 1, 2, 3) can be found out from Eq. (6).

5. Numerical example A numerical example is presented here to verify the forward kinematics solution. The geometric parameters of the manipulator are: e = 0.182, dr = 0.38,

Finally, by eliminating P in Eqs. (15a) and (16), an eighth polynomial about Z is 0.194954 − 1.20028Z − 178.795Z 2 − 113.225Z 3 + 1123.83Z 4 + 327.219Z 5 − 2119.7Z 6 − 25.5739Z 7 + 943.637Z 8 = 0.

(17)

The above equation results in eight solutions in terms of P, Y, Z, and the final 16 solutions are listed in Table 1. If e = 0, that is, the zero offset 3-UPU manipulator, the solutions will be P = 2.35025, X = ±1.472887, Y = −0.42529 and Z = 0.00506981, that is, only two solutions. Compared the two solutions in the zero offset 3-UPU manipulator, obviously, the offset 3-UPU parallel manipulator can reach a larger space due to the non-zero offset (e = 0).

P. Ji, H. Wu / Robotics and Autonomous Systems 42 (2003) 117–123 Table 1 The final solutions Solution

P

X

Y

Z

1 2 3 4 5 6 7 8

2.48980 1.47616 1.17091 3.21158 2.40438 1.54796 1.57198 2.37061

±0.407328 ±0.202637 ±0.991043 ±1.739793 ±0.707189 ±0.540142 ±0.846806 ±0.988666

−1.38315 −1.10073 −0.432851 −0.428745 −0.289322 −0.288197 0.714177 0.935370

0.640921 −0.472749 −0.0372116 0.0296588 −1.34928 1.08312 −0.58724 0.719883

6. Conclusion This paper proposed an offset 3-UPU parallel manipulator based on Tsai’s zero offset 3-UPU manipulator. This new manipulator still provides a 3-DOF pure translational motion, however, the kinematics is more complicated. In terms of forward kinematics, this manipulator has 16 solutions instead of two solutions in Tsai’s manipulator. The disadvantage of this manipulator lies in its singularities. The manipulator suffers a rotation singularity. When the rotation singularity occurs, the mobile platform may rotate instantaneously or finitely. The singularity problem of the manipulator needs a further study.

Acknowledgements The authors would like to thank The Hong Kong Polytechnic University for the support of this project under project No. G-T056. References [1] K. Bürüncük, Y. Tokad, On the kinematic of a 3-DOF Stewart platform, Journal of Robotic Systems 16 (1999) 105–118. [2] C. Innocenti, Analytical-form direct kinematics for the second scheme of a 5–5 general-geometry fully parallel manipulator, Journal of Robotic Systems 12 (1995) 661–676.

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[3] C.M. Gosselin, L. Parreault, C. Vaillancourt, Simulation and computer-aided kinematic design of three-degree-of-freedom spherical parallel manipulators, Journal of Robotic Systems 12 (1995) 857–869. [4] L.W. Tsai, Kinematics of a three-DOF platform with three extensible limbs, in: Recent Advances in Robot Kinematics, Kluwer Academic Publishers, Dordrecht, 1996, pp. 401–410. [5] L.W. Tsai, R. Stamper, A parallel manipulator with only translational degrees of freedom, ASME 96-DETC-MECH1152, USA, 1996. [6] R. Di Gregorio, V. Parenti-Castelli, Mobility analysis of the 3-UPU parallel manipulator assembled for a pure translational motion, in: Proceedings of the 1999 IEEE/ ASME International Conference on Advanced Intelligent Mechatronics, Atlanta, USA, September 19–23, 1999, pp. 520–525. [7] R. Di Gregorio, V. Parenti-Castelli, A translational 3-DOF parallel manipulator, in: Advances in Robot Kinematics: Analysis and Control, Kluwer Academic Publishers, Dordrecht, 1998, pp. 49–59. [8] G. Salmon, Higher Algebra, 5th ed., 1885, Chelsea, New York, 1964. [9] B. Roth, Computational advances in robot kinematics, in: J. Lenarcic, B. Ravani (Eds.), Advances in Robot Kinematics and Computational Geometry, Kluwer Academic Publishers, Dordrecht, 1994, pp. 7–16.

Ping Ji is an Associate Professor in the Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University. His research areas include robotics, CAD/CAM and production management.

Hongtao Wu is a Professor in the Department of Mechanical Engineering, Nanjing University of Aeronautics and Astronautics, China. His research interests are parallel mechanisms, rigid body systems.