Disscussion: “Kinematics of the Translational 3-URC Mechanism” [Di Gregorio, R., 2004, ASME J. Mech. Des., 126, pp. 1113–1117]

Kong, Xianwen; Gosselin, Clément M.

doi:10.1115/1.2205875

The author of (1) proposed a 3-URC translational parallel mechanism (TPM) and presented a comprehensive study on the kinematics of the 3-URC TPM. He concluded that “only one solution exists both for the direct and for the inverse position analyses.” However, we do not agree with his result on the inverse position analysis and his statement that Ref. 2 “presented a class of TPMs with linear input-output equations that contain some translational 3-URC mechanisms.”

In this discussion, we will show that the 3-URC TPM does not belong to the class of TPMs with linear input-output equations (2,3,4,5 6) by investigating the inverse position analysis of the 3-URC TPM.

Leg $i$ of a 3-URC TPM is shown in Fig. 1. In addition to the notations used in (1), $h_{i}$ is used to denote a unit vector directed from $A_{i}$ to $C_{i}$ ⁠. For leg $i$ ⁠, the following holds: $w_{1 i} ∙ v_{i} = 0$ ⁠, $w_{1 i} ∙ w_{2 i} = 0$ ⁠, and $w_{2 i} ∙ h_{i} = 0$ ⁠.

Figure 1

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URC leg

In the coordinate system

O - X Y Z

⁠, we have

A_{i} + h_{i} h_{i} - d_{i} w_{2 i} + s_{i} w_{1 i} = B_{i 0}

1

where

w_{2 i} = \cos θ_{1 i} v_{i} + \sin θ_{1 i} w_{1 i} \times v_{i}

w_{2 i} \times w_{1 i} = - \cos θ_{1 i} w_{1 i} \times v_{i} + \sin θ_{1 i} v_{i}

h_{i} = \cos θ_{2 i} w_{1 i} + \sin θ_{2 i} w_{2 i} \times w_{1 i}

B_{i 0} = P + R_{b p} {}^{p}{(B_{i 0} - P)}

The inverse position analysis of the 3-URC TPM can be performed by solving Eq. 1 for $θ_{1 i}$ ⁠, $θ_{2 i}$ ⁠, and $s_{i}$ in sequence.

Solution for θ1i

Taking the inner product of Eq. with

w_{2 i}

⁠, we obtain

w_{2 i} ∙ (A_{i} - B_{i 0}) - d_{i} = 0

2

i.e.,

(B_{i 0} - A_{i}) ∙ (w_{1 i} \times v_{i}) \sin θ_{1 i} + (B_{i 0} - A_{i}) ∙ v_{i} \cos θ_{1 i} + d_{i} = 0

3

Define an angle

α_{i}

by

4

\cos α_{i} = [(B_{i 0} - A_{i}) ∙ (w_{1 i} \times v_{i})] ∕ a_{i}

\sin α_{i} = [(B_{i 0} - A_{i}) ∙ v_{i}] ∕ a_{i}

where

a_{i} = {[(B_{i 0} - A_{i}) ∙ v_{i}]}^{2} + {[(B_{i 0} - A_{i}) ∙ (w_{1 i} \times v_{i})]}^{2}

⁠.

Equation can be rewritten as

\sin (θ_{1 i} + α_{i}) = - d_{i} ∕ a_{i}

5

Solving

\sin^{2} (θ_{1 i} + α_{i}) + \cos^{2} (θ_{1 i} + α_{i}) = 1

⁠, we obtain two solutions for

\cos (θ_{1 i} + α_{i})

as

\cos (θ_{1 i} + α_{i}) = \pm {[1 - \sin^{2} (θ_{1 i} + α_{i})]}^{1 ∕ 2}

6

Equations , show that there are two solutions for

(θ_{1 i} + α_{i})

⁠. For each

(θ_{1 i} + α_{i})

⁠, one solution for

θ_{1 i}

can be obtained as

7

\sin θ_{1 i} = \sin (θ_{1 i} + α_{i}) \cos α_{i} - \cos (θ_{1 i} + α_{i}) \sin α_{i}

\cos θ_{1 i} = \cos (θ_{1 i} + α_{i}) \cos α_{i} + \sin (θ_{1 i} + α_{i}) \sin α_{i}

From Eqs. 4,5,6,7, we learn that there are usually two solutions for $θ_{1 i}$ ⁠.

Solution for θ2i

For each

θ_{1 i}

obtained using Eq. ,

\sin θ_{2 i}

can be obtained by taking the inner product for Eq. with

w_{2 i} \times w_{1 i}

as

\sin θ_{2 i} = {\sin θ_{1 i} [(B_{i 0} - A_{i}) ∙ v_{i}] - \cos θ_{1 i} [(B_{i 0} - A_{i}) ∙ (w_{1 i} \times v_{i})]} ∕ h_{i}

Substituting Eq. into the above equation, we obtain

\sin θ_{2 i} = - a_{i} \cos (θ_{1 i} + α_{i}) ∕ h_{i}

8

Solving

\sin^{2} θ_{2 i} + \cos^{2} θ_{2 i} = 1

⁠, we obtain two solutions for

\cos θ_{2 i}

as

\cos θ_{2 i} = \pm {(1 - \sin^{2} θ_{2 i})}^{1 ∕ 2}

9

Equations 8,9 show that there exist two solutions for $θ_{2 i}$ for a given $θ_{1 i}$ ⁠.

Solution for si

Once

θ_{1 i}

and

θ_{2 i}

have been determined,

s_{i}

can be obtained by taking the inner product of Eq. with

w_{1 i}

as

s_{i} = [(B_{i 0} - A_{i}) ∙ w_{1 i}] - h_{i} \cos θ_{2 i}

10

Number of Solutions to the Inverse Displacement Analysis

The above analysis shows that for a given position of the moving platform, there are usually two solutions (Eqs. 4,5,6,7) for the input $θ_{1 i}$ for each leg $i$ and four sets of solutions (Eqs. 4,5,6,7,8,9,10)² for the joint variables in each leg $i$ ⁠. Thus, for a given position of the moving platform, there are usually eight $(= 2^{3})$ sets of solutions for the inputs $θ_{11}$ ⁠, $θ_{12}$ ⁠, and $θ_{13}$ and 64 $(= 4^{3})$ sets of solutions for all the joints variables in the 3-URC TPM.

In summary, it has been shown that for a given position of the moving platform, there are usually two solutions for each input and eight sets of solutions for all the inputs in the 3-URC TPM. Thus, we have proved that the 3-URC TPM does not belong to the class of TPMs with linear input-output equations (2,3,4,5,6). In fact, it belongs to the class of linear TPMs, whose forward displacement analysis can be performed by solving a set of linear equations, dealt with systematically in (5,6). The work reported in (5,6) is an extension of the work reported in (3 4).

From Eqs. 5,6,8,9,10 in this paper or Eq. (11b) in (1), it is learned that there are usually two solutions for $s_{i}$ for a given position of the moving platform.

1.

Di Gregorio

,

R.

, 2004, “

Kinematics of the Translational 3-URC Mechanism

,”

ASME J. Mech. Des.

1050-0472,

126

(

6

), pp.

1113

–

1117

.

Crossref

2.

Kong

,

X.

, and

Gosselin

,

C. M.

, 2002, “

A Class of 3-DOF Translational Parallel Manipulators With Linear Input-Output Equations

,”

Proceedings of the Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators

,

Québec, Canada

, October 3–4, pp.

25

–

32

.

3.

Gosselin

,

C. M.

, and

Kong

,

X.

, 2002, “

Cartesian Parallel Manipulators

,” International Patent No. WO 02/096605 A1.

4.

Gosselin

,

C. M.

, and

Kong

,

X.

, 2004, “

Cartesian Parallel Manipulators

,” U.S. Patent No. 6729202.

5.

Kong

,

X.

, and

Gosselin

,

C. M.

, 2002, “

Type Synthesis of Linear Translational Parallel Manipulators

,”

Advances in Robot Kinematics—Theory and Applications

,

J.

Lenarčič

and

F.

Thomas

, eds.,

Kluwer Academic

,

Dordrecht

, pp.

411

–

420

.

6.

Kong

,

X.

, and

Gosselin

,

C. M.

, 2004, “

Type Synthesis of 3-DOF Translational Parallel Manipulators Based on Screw Theory

,”

ASME J. Mech. Des.

1050-0472,

126

(

1

), pp.

83

–

92

.

Crossref

View Original Article

by American Society of Mechanical Engineers

Disscussion: “Kinematics of the Translational 3-URC Mechanism” [Di Gregorio, R., 2004, ASME J. Mech. Des., 126, pp. 1113–1117]

Solution for θ1i

Solution for θ2i

Solution for si

Number of Solutions to the Inverse Displacement Analysis

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Disscussion: “Kinematics of the Translational 3-URC Mechanism” [Di Gregorio, R., 2004, ASME J. Mech. Des., 126, pp. 1113–1117]

Solution for θ1i

Solution for θ2i

Solution for si

Number of Solutions to the Inverse Displacement Analysis

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