Dimensional synthesis of a 3-DOF parallel manipulator based on

SCIENCE CHINA Technological Sciences •

January 2010

ARTICLES •

Vol. 53 No.1: 168−174

doi: 10.1007/s11431-009-0375-y

Dimensional synthesis of a 3-DOF parallel manipulator based on dimensionally homogeneous Jacobian matrix SUN Tao1, SONG YiMin1†, LI YongGang2 & LIU LinShan1 2

1 School of Mechanical Engineering, Tianjin University, Tianjin 300072, China; School of Mechanical Engineering, Tianjin University of Technology and Education, Tianjin 300222, China

Received May 18, 2009; accepted August 19, 2009

A study of dimensional synthesis of a 3-DOF parallel manipulator with coupling of translation and rotation is carried out. The architecture of the manipulator is composed of a moving platform attached to a fixed base through three identical PRS (prismatic-revolute-spherical) serial limbs, whose unique topology leads to the physical unit inconsistency of the conventional Jacobian matrix and the emergence of the parasitic motion. Then this paper introduces a kinetostatic performance index of the manipulator based on the condition of a dimensionally homogeneous Jacobian matrix, later, the workspace of the aforementioned manipulator is searched and the influence of the crucial design variables on the workspace is analyzed. Finally, a dimensional synthesis method of the manipulator is proposed, which may be regarded as a nonlinear programming problem with subject to the parasitic motion and other several engineering constraints. PKMs (parallel kinematic machines), dimensional synthesis, dimensionally homogeneous Jacobian, 3-PRS manipulator, workspace Citation:

Sun T, Song Y M, Li Y G, et al. Dimensional synthesis of a 3-DOF parallel manipulator based on dimensionally homogeneous Jacobian matrix. Sci China Tech Sci, 2010, 53: 168−174, doi: 10.1007/s11431-009-0375-y

1 Introduction Development of the limited-DOF (degree of freedom) PKMs (parallel kinematic machines) has been a research hotspot in the PKMs due to the merits in terms of simple structure, lower cost and easy control etc. comparing with 6-DOF PKMs. The appearance and application of the limited-DOF PKMs with coupling of translation and rotation provide an option for a bottleneck problem of the manufacture or assembly for large components in the aircraft and automobile industry. For instance, a well-designed limited-DOF PKM with coupling of translation and rotation may be integrated into a numerical control manufacturing cell as a plug-and-play module, which can be driven along an extra long track or other auxiliary apparatus in large *Corresponding author (email: [email protected]) © Science China Press and Springer-Verlag Berlin Heidelberg 2010

aircraft and automobile components machining or assembly. As an outstanding representation of the limited-DOF PKMs with coupling of translation and rotation, the 3-PRS (prismatic-revolute-spherical) manipulator has been applied to many aspects because of compact architecture, excellent kinematic and dynamic performance, for example, the famous Sprint Z3 Head made by the DS Technologie Company [1, 2] in Germany, as well as a telescope application proposed originally in ref. [3], and so on. The underlying architecture of the 3-PRS manipulator is an axis-symmetric parallel mechanism with three identical PRS limbs, herein the prismatic joints are considered to be actuated. Some theoretical research results of this manipulator have been published in the past decades. The kinematic/dynamic analysis, including inverse and forward kinematics, dynamic model and so on, of the 3-PRS manipulator was investigated elaborately by Tsai et al. [4], Li et al. [5] and Zhang et al. [6]. Refs. [7–9] made research on tech.scichina.com

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the workspace analysis of the 3-PRS manipulator. Later, the parasitic motion of this manipulator of coupling of translation and rotation was introduced by Carretero [10], and the optimal selection of architectural-parameter values was carried out by Quasi-Newton algorithms with subject to the average and min-max parasitic motion respectively. Further researches demonstrate that the Jacobian matrix of 3-PRS manipulator is composed of non-homogeneous physical units, which may lack in physical significance. Therefore, refs. [11–13] proposed the calculation of a dimensionally homogeneous Jacobian matrix based on the distinct three EE points of a moving platform. It is noteworthy that the dimensional synthesis is one of the most important issues in the field of the PKMs and is increasingly attracting the interests of many researchers [14–16]. Ref. [17] determines the design parameter bounds and space of the limited-DOF PKMs by means of PFNM (parameter-finiteness normalization method), furthermore, the optimal dimensional results can be calculated by means of the design parameter space. Liu and Bonev [18] implemented the dimensional synthesis of two articulated tool heads with PKMs using design parameter space and performance atlas. However, their researches did not involve the formulation of kinematic performance index based on this dimensionally homogeneous Jacobian matrix and the optimal dimensional design considering the parasitic motion constraint. This paper formulates the kinematic performance index based on the dimensionally homogeneous Jacobian matrix and carries out the dimensional synthesis of the 3-PRS manipulator with subject to the parasitic motion and other engineering constraints. This paper is organized as follows: In Section 2 the description of the 3-PRS manipulator is briefly introduced. The dimensionally homogeneous Jacobian matrix is developed in Section 3 so as to acquire a suitable index for appraising its kinematic performance. To guarantee the demanded posture capability, the influence of main design variables on the workspace is analyzed in Section 4, and finally the dimensional variables of the manipulator are optimized by solving a constrained nonlinear programming problem.

2 Description of 3-PRS manipulator The schematic diagram of the 3-PRS manipulator is shown in Figure 1. As can be seen, this symmetric parallel manipulator with 3-DOF is composed of a fixed base, a moving platform and three identical PRS limbs. Herein, R and S represent the revolute joint and the spherical joint, respectively. The underlined P denotes the prismatic joint driven by a servomotor. Generally, the slideways of the sliders are all perpendicular to the fixed base. As shown in Figure 1, Ai, Ri (i=1–3) denote the centers of the spherical joint and the revolute joint, respectively. Without loss of generality, the fixed base and the moving

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Figure 1 Schematic diagram of 3-PRS manipulator.

platform are both equilateral triangles whose circumradius are nominated as rb and ra. A global coordinate system O-xyz and a moving coordinate system P-uvw are established, respectively, with O and P being located at centers of the equilateral triangles. Let x be parallel to OB1 and u be parallel to PA1, then z and w are normal to ΔB1B2B3 and ΔA1A2A3, respectively. Herein, both y and v satisfy the right-hand rule. Additionally, a body-fixed coordinate system Ri-uiviwi, in which vi takes along the axis of the revolute joint, and wi is coincident with RiAi.

3

Dimensionally homogeneous Jacobian matrix

The orientation of the moving platform of the 3-PRS manipulator may be described by α and β angles, which denote the swing angles of the x-axis and y-axis, respectively. Then, the dimensionally homogeneous Jacobian matrix can be formulated in this section. As shown in Figure 1, the closed-loop vector equation may be obtained as follows: p + ai = bi + qi z + lwi (i = 1 − 3) ,

(1)

where p, ai and bi represent position vectors of the point P, Ai and Bi in O-xyz; qi is the translational distance of the active prismatic joint with respect to the fixed base; l is the length of the link; z and wi denote the unit vectors of the axis z and RiAi, respectively. Taking the derivative of eq. (1) with respect to time yields vP + ωP × ai = qi z + ωi × lwi ,

(2)

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where vP and ωP are the linear and angular velocity of the moving platform, ωi denotes the angular velocity of the link, and qi is the linear rate of the slider. Taking dot product of eq. (2) with wi leads to wiT (v P + ωP × ai ) = qi wiz ,

Cq = Dp ,

(4)

where

⎡ w D=⎢ 1 ⎣a1 × w1

q = [ q1

q2

w2 z

q3 ] , T

w3 z ),

p = ⎡⎣v

T P

T

ω ⎤⎦ . T P

Therefore, the velocity mapping between the operational space and the joint space can be expressed by q = Jp ,

(5)

where J = C −1 D . Obviously, the elements in the former three columns of J are all dimensionless while the units of the other elements of J are distance/angle. In other words, the units of all elements in the Jacobian matrix J are not consistent. In order to evaluate the kinematic performance of the 3-PRS manipulator, the Three End-Effector Points Method [11, 12] may be utilized for developing a dimensionally homogeneous Jacobian matrix. Herein, Ai are selected as key points to depict the position of the moving platform, which can be written as p ′ = [a

T 1

where ai = [ Aix

Aiy

a

T 2

T T 3

a ] ,

(6)

Aiz ]T .

(7)

According to the mobility analysis of the 3-PRS manipulator, only three elements in eq. (6) are independent. Without loss of generality, Aiz are chosen to be independent ones. Rewriting eq. (7) as J ′Qp ′′ = q.

p ′ = [a1T

a2T

a3T ]T ,

J′ = ⎡ ∂q1 ⎢ ⎢ ∂A1x ⎢ ∂q ⎢ 2 ⎢ ∂A1x ⎢ ⎢ ∂q3 ⎢ ⎣ ∂A1x

∂q1 ∂A1 y

∂q1 ∂A1z

∂q1 ∂A2 x

∂q1 ∂A2 y

∂q1 ∂A2 z

∂q1 ∂A3 x

∂q1 ∂A3 y

∂q2 ∂A1 y

∂q2 ∂A1z

∂ q2 ∂A2 x

∂ q2 ∂A2 y

∂ q2 ∂A2 z

∂q2 ∂A3 x

∂ q2 ∂A3 y

∂q3 ∂A1 y

∂q3 ∂A1z

∂q3 ∂A2 x

∂q3 ∂A2 y

∂q3 ∂A2 z

∂q3 ∂A3 x

∂q3 ∂A3 y

⎡ ∂A1x ⎢ ⎢ ∂A1z ⎢ ∂A Q = ⎢ 1x ⎢ ∂A2 z ⎢ ⎢ ∂A1x ⎢⎣ ∂A3 z

4

A2 z ∂A1 y ∂A1z ∂A1 y ∂A2 z ∂A1 y ∂A3 z

(8)

By means of the projection method [11], all the other elements in eq. (6) may be computed. Thereby, the dimensionally homogeneous Jacobian matrix of the 3-PRS manipulator is formulated

∂q1 ⎤ ⎥ ∂A3 z ⎥ ∂q2 ⎥⎥ , ∂A3 z ⎥ ⎥ ∂q3 ⎥ ∂A3 z ⎥⎦

A3 z ]T , ∂A2 x ∂A1z

∂A2 y

0

∂A2 x ∂A2 z

∂A2 y

0

∂A2 x ∂A3 z

∂A2 y

1

∂A1z ∂A2 z ∂A3 z

∂A3 x ∂A1z

∂A3 y

1

∂A3 x ∂A2 z

∂A3 y

0

∂A3 x ∂A3 z

∂A3 y

0

∂A1z ∂A2 z ∂A3 z

T

⎤ 0⎥ ⎥ ⎥ 0⎥ . ⎥ ⎥ 1⎥ ⎥⎦

Workspace analysis

To meet the requirements of posture capability, the workspace of the 3-PRS manipulator should be searched first. In addition, it is of great importance to analyze the influence of design variables on the workspace. 4.1

Workspace search

In order to search the workspace, the following boundary conditions are taken into consideration. (1) The stroke of the P joint: qmin ≤ qi ≤ qmax ,

Taking the derivative of eq. (6) with respect to time and rewriting eq. (5), we have J ′p ′ = q.

(9)

Herein, p ′, J ′, p ′′ and Q in eqs. (7)–(9) can be calculated as

p ′′ = [ A1z

T

w3 ⎤ , a3 × w3 ⎥⎦

w2 a2 × w2

J ′′ = J ′Q.

(3)

where wiz represents the component of the unit vector wi along the axis z. Rewriting eq. (3) in matrix form yields

C = diag( w1z


(10)

where qmin and qmax denote the minimal and maximal strokes of the P joint. (2) The rotating angles of the R and S joints:

⎧⎪ θsi ≤ θsmax , ⎨ ⎪⎩ θ ri ≤ θ rmax ,

(11)

where θsi and θri denote the rotating angle of the S joint and that of R joint, and θsmax and θrmax are their maximal absolute values. (3) The interference of links: d′≥ d,

(12)

where d' is the minimal distance between links without in-

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terference, while d denotes the sectional diameter of the link. Given the boundary values tabulated in Table 1, the workspace of the 3-PRS manipulator, shown in Figure 2, may be achieved with the boundary search method. 4.2

Influence of crucial design variables on workspace

Let rb and l be normalized by ra such that

λ1 =

rb l , λ2 = . ra ra

(13)

Figures 3 and 4 demonstrate the variations of the workspace versus α and β, respectively. It can be seen from Figure 3 that, with the increment of λ1, the position of the workspace declines along the z-axis and the section area of the workspace diminishes simultaneously. Hence, a smaller λ1 is helpful to enhance the posture capability of the 3-PRS manipulator. As shown in Figure 4, the position of the workspace ascends along the z-axis with the increment of λ2. However, the section area of the workspace keeps still. In other words, λ2 does not affect the workspace shape or the posture capability of the 3-PRS manipulator. Table 1

Boundary values of constraints

Variable ra rb l qmin

Value 200 mm 300 mm 400 mm −300 mm

Variable qmax

θsmax θrmax d'

Value 300 mm 30° 45° 50 mm

Figure 2

5

Workspace of 3-PRS manipulator.

Dimensional synthesis

In this section, the dimensional synthesis of the 3-PRS manipulator is carried out so as to achieve the optimal global kinematic performance subjected to the parasitic motion and some related engineering constraints. 5.1

Performance indices

It has been well accepted that the condition number κ of the Jacobian matrix is one of the most suitable indices for evaluating the kinematic performance of PKMs. As κ is configuration-varying, a global condition index κ will be selected [19, 20]:

Figure 3 Variations of the workspace versus α and β.

Figure 4 Variations of the workspace versus α and β.

κ=

∫ V κ dV V

,

(14)

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where κ = σmin/σmax, σmax and σmin are the maximal and minimal singularity of J″, and V represents the workspace volume. Additionally, since κ cannot yet reflect the fluctuation of κ in the global workspace, another performance index is introduced [21]:

∫V (κ − κ ) dV 2

κ =

V

,

(15)

where κ can be considered as the standard deviation of κ with respect to its mean value κ in the workspace. 5.2

Constraints

5.2.1 Constraints of λ1 For a predefined workspace, it is important to synthesize an appropriate PKM module with a smaller mechanism volume. Therefore, the influence of design variables on the kinematic performance of the manipulator should be investigated. Without loss of generality, let λ 1∈[1,2] and λ∈[2,3]. The variations of κ and κ versus λ1 and λ2 are shown in Figures 5 and 6, respectively. As can be seen, both κ and κ increase monotonically with the increment of λ1, while decrease with the increment of λ2. Meanwhile, the influence of λ2 on the performance indices is not obvious once a smaller λ1 is selected. From the viewpoint of engineering, the choice of λ1 will affect the mechanical structure of the module, thus, a suitable constraint condition of λ1 might be [22]

λ1 ≥ λ1min ,

(16)

where λ1min is the minimum allowable value of λ1 in the mechanical engineering.

Figure 6 Variations of κ versus λ1 and λ2.

5.2.2 Constraints of parasitic motion As pointed out by Carretero [10], there exist parasitic motions in the 3-PRS manipulator due to its unique architecture. Based upon the inverse kinematic analysis, the parasitic motions of the moving platform can be formulated as ⎧ r ⎪ X = a (cos β cos γ + sin α sin β sin γ − cos α cos γ ), 2 ⎪ ⎪ (17) = − Y ra (sin α sin β cos γ + cos α sin γ ), ⎨ ⎪ ⎪γ = − tan −1 ⎛ sin α sin β ⎞ , ⎜ ⎟ ⎪⎩ ⎝ cos α + cos β ⎠

where X and Y denote the parasitic translations along the axes x and y, and γ is the parasitic rotation about the z-axis. Obviously, the parasitic translations are proportional to the circumradius of the moving platform. However, if the symmetry axis of payload is collinear with the z-axis, the parasitic rotation γ is of no consequence. Therefore, only the parasitic translations will be considered here. To highlight the importance of X and Y, the circumradius of the moving platform and its swing angles are supposed to be ra = 400 mm and α, β∈[−40°,40°] in the numerical simulation. The variations of the parasitic translations versus α and β are shown in Figure 7. It can be noted that the parasitic translations cannot be neglected with the increment of the swing angles of the moving platform. Therefore, an appropriate evaluation index of the parasitic translations can be expressed by

δ = X 2 +Y2.

(18)

For some engineering applications, the constraint equation of parasitic motion could be expressed by

δ ≤ δ max , Figure 5 Variations of κ versus λ1 and λ2.

where δmax is the maximal allowable value of δ.

(19)

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the maximum values of θri and θsi are 10.11° and 49.33°, respectively. Because θri is relatively small, only the constraint equation of θsi is considered:

θsi

max

≤ θ0 ,

(20)

where θ0 is the maximal allowable value of θsi. 5.3

Optimal design

By integrating eqs. (14) and (15), the global kinematic performance of the 3-PRS manipulator may be defined as f = κ 2 + ( wη κ )2 , Figure 7 Variations of parasitic motion versus swing angles of the moving platform.

5.2.3 Constraints of rotation angles To obtain the constraint conditions of the rotating angles, some related parameter values are set as λ1=1, λ2=2, and α, β∈[−40°, 40°]. Figures 8 and 9 show the variations of θri and θsi versus α and β. As can be seen from the figures,

Figure 8 Variations of θri versus α and β.

(21)

where wη is a weighted coefficient for emphasizing the importance of κ . For a given task, once parameters λ1min, θ 0 and δ max are pre-determined by the designer, the dimensional synthesis of the 3-PRS manipulator can be carried out as a constrained nonlinear programming problem: ⎧min f ( x ) = f (λ1 , λ2 ), x∈R ⎪ ⎪⎪ ⎧λ ≥ λ , 1 1min ⎨ ⎪⎪ ⎪s.t. ⎨ θsi max ≤ θ 0 , ⎪ ⎪ ⎪⎩ ⎩⎪δ ≤ δ max .

(22)

Given |αmax|=|βmax|=40°, θ0∈[45°, 55°] and wη=5. Figure 10 demonstrates the variations of f and |θsi |max versus λ1 and λ2. It can be seen that both f and |θsi |max increase monotonically with the increment of λ1, while decrease monotonically with the increment of λ2. In other words, a bigger λ2 should be selected when large swing angles and small manipulator volume are required, while a smaller λ2 might be chosen to meet the demands of stiffness and accuracy of the module. In addition, Figure 10 also shows that a smaller λ1 is helpful to improve operational performance of the manipulator. However, the examination of eq. (16) leads to the optimal condition, i.e. λ*1 =λ1max=1. Later, given the maximal parasitic translation constraint, the optimized λ*1 and λ*2 can be obtained by solving eq. (22). To validate the proposed method, a PKM module with the 3-PRS topology is synthesized. The following variables are pre-determined as ra=260 mm, λ1∈[1, 2], λ2∈[2, 3], θ 0=45°, δmax=15 mm, α , β ∈ [−40D , 40D ] , d=50 mm, h= 420 mm (height of workspace). By utilizing the dimensional synthesis method aforementioned, the other dimensional parameters are computed as follows:

Figure 9 Variations of θsi versus α and β.

qmax=765 mm, rb=260 mm, l=1014 mm.

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This work was supported by the National Natural Science Foundation of China (Grant Nos. 50535010, 50675151), National High-Tech Research and Development Program of China (Grant No. 2007AA042001), and Ph.D. Programs Foundation of Ministry of Education of China (Grant No. 20060056018). 1 2

3

4 5 6 7 8 9 10 11 12 Figure 10 Variations of |θsi |max and f versus λ1 and λ2.

6

Conclusion

13 14 15

This paper investigates into the dimensional synthesis of the 3-PRS manipulator with subject to the parasitic motion and several engineering constraints, and the kinematic variables are optimized by solving a nonlinear programming problem. The evaluation index of kinematic performance of the manipulator is selected as the global condition number of the dimensionally homogeneous Jacobian matrix. The constraints considered in the research include the predefined workspace, the manipulator volume, the parasitic motion of the moving platform, the rotating angles of the joints, etc. The influences of crucial design variables on the kinematic performance of the manipulator are carefully analyzed. Later, the simulation results prove the validity of the proposed dimensional synthesis method, which may be further utilized for visual prototype design of some similar PKMs with coupling of translation and rotation so as to meet the requirements of larger component manufacture or other special applications.

16 17 18 19 20 21 22

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