ARTICLE IN PRESS

Robotics and Computer-Integrated Manufacturing 24 (2008) 402–414 www.elsevier.com/locate/rcim

An investigation on mobility and stiffness of a 3-DOF translational parallel manipulator via screw theory Qingsong Xu, Yangmin Li Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Av. Padre Toma´s Pereira S.J., Taipa, Macao SAR, PR China Received 21 September 2006; received in revised form 31 January 2007; accepted 8 February 2007

Abstract This paper analyzes the mobility and stiffness of a three-prismatic-revolute-cylindrical (3-PRC) translational parallel manipulator (TPM). Firstly, the original 3-PRC TPM is converted into a non-overconstrained manipulator since there exist some practical problems for the overconstrained mechanism. By resorting to the screw theory, it is demonstrated that the conversion brings no influences to the mobility and kinematics of the manipulator. Secondly, the stiffness matrix is derived intuitively via an alternative approach based upon screw theory with the consideration of actuations and constraints, and the compliances subject to both actuators and legs are taken into account to establish the stiffness model. Furthermore, the stiffness performance of the manipulator is evaluated by utilizing the extremum stiffness values over the usable workspace, and the influences of design parameters on stiffness properties are presented, which will be helpful for the architecture design of the TPM. r 2007 Elsevier Ltd. All rights reserved. Keywords: Parallel manipulators; Mobility; Stiffness; Analysis; Screw theory; Mechanism design

1. Introduction A parallel manipulator typically consists of a mobile platform that is connected to a fixed base by several limbs or legs in parallel. Generally, if carefully designed, parallel manipulators can provide several attractive advantages over their serial counterparts in terms of high stiffness, high accuracy, and low inertia [1], which enable them become challenging alternatives for wide applications such as in assembly lines, flight simulators, machine tools, ultraprecision instruments, medical devices, and so on. Recently, the progress on the development of parallel manipulators with less than six degree-of-freedom (DOF) has been accelerated because these limited-DOF parallel manipulators own several other advantages including the total cost reduction in manufacturing and operations in addition to the inherent merits of parallel mechanisms.

Corresponding author. Tel.: +853 3974464; fax: +853 838314.

E-mail address: [email protected] (Y. Li). 0736-5845/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.rcim.2007.02.022

As a result, limited-DOF parallel manipulators have been investigated and applied more and more extensively. Among the limited-DOF manipulators, those translational parallel manipulators (TPM) possessing three spatial pure translational DOF have drawn particular interests from numerous researchers, since they satisfy the requirements of many specific applications. Various TPM architectures have been proposed in the literatures [2,3], and the type syntheses of a 3-DOF TPM have been carried out by investigations based on the screw theory [4,5], group theoretic approach [6,7], and several other methods [8–11]. In our previous research [12], a 3-PRC (three-prismaticrevolute-cylindrical) TPM with three P joints intersecting at a common point was presented and designed so as to eliminate all the singularities from the workspace. It should be noted that this 3-PRC TPM possessed the same mechanism with the one that was proposed in [13] with three P joints parallel to one another, and the used PRC linkage was equivalent to the PRˇRˇP (letters Rˇ denote the revolute joints with parallel axes) translational parallel kinematic chain that was enumerated by the type synthesis in [5].

ARTICLE IN PRESS Q. Xu, Y. Li / Robotics and Computer-Integrated Manufacturing 24 (2008) 402–414

As an overconstrained mechanism, the 3-PRC TPM has a very simple structure. However, in practice, the problems of variable frictions in passive joints and large reaction moment have to be considered so as to assure the mobility of the mobile platform for a 3-PRC TPM. Otherwise, the mobile platform may not move or the manipulator cannot work if there are some kinematic errors. Unfortunately, due to the unavoidable errors arising from the manufacturing tolerance and imperfect assembly, there will always exist kinematic errors. Considering this point, the original 3-PRC TPM will be converted into a non-overconstrained TPM in this paper in order to solve the movability problem by resorting to the screw theory [14–17]. Additionally, since high stiffness is one of the advantages of parallel manipulators, it is necessary to investigate the stiffness issue of the 3-PRC TPM in more detail. Actually, the stiffness of a manipulator has direct impact on its position accuracy. Hence, in the early design stage, it is desired to perform the stiffness modeling and evaluation of a parallel manipulator for the precise manipulation purpose. The second objective of this paper is to accomplish the stiffness analysis of the 3-PRC TPM since there are no efforts made toward the stiffness characterization of this type of manipulator yet. In the remainder of this paper, after a brief review of the screw and reciprocal screw systems in Section 2, and a short description of the 3-PRC TPM in Section 3, the mobility of the manipulator is analyzed in Section 4, where the overconstrained conditions of the mechanism are eliminated without any influences on its mobility and kinematics. Then in Section 5, the stiffness model is identified with the consideration of compliances subject to both actuators and legs, and the overall stiffness matrix is established. Afterwards, the stiffness assessment is carried out in Section 6 along with the derivation for the influences of design parameters on stiffness characteristics. Finally, some concluding remarks are summarized in Section 7. 2. Overview of screw and reciprocal screw systems In screw theory, a unit (normalized) screw is defined by a pair of vectors: " # s $^ ¼ , (1) r  s þ hs where s is a unit vector directing along the screw axis, r denotes the position vector pointing from an arbitrary point on the screw axis to the origin of the reference frame, the vector r  s defines the moment of the screw axis with respect to the origin of the reference frame, and h represents the pitch of the screw. If the pitch equals to zero, the screw becomes:   s ^$ ¼ . (2) rs

While in case of infinite pitch, the screw reduces to:   0 $^ ¼ . s

403

(3)

A screw can be used to represent a twist or a wrench. With $F and $L respectively denoting the vectors of the first and last three components of a screw $, then $F and $L respectively represent the angular and linear velocities when $ refers to a twist, and the force and couple vectors when $ refers to a wrench. Two screws, namely, $r and $, are said to be reciprocal if they satisfy the following condition [15]: ~ r T $ ¼ 0, $r  $ ¼ ½D$

(4)

where ‘‘’’ represents the reciprocal product operator, and ~ which is used to interchange the first and last the matrix D, three components of a screw ($r ), is defined by:   0 I ~D  , (5) I 0 where 0 and I denote a zero matrix and an identity matrix in 3  3, respectively. The physical meaning of reciprocal screws is that the wrench $r produces no work along the twist of $. Concerning a n-DOF spatial serial kinematic chain with n 1-DOF joints ðnp6Þ, the joint screws (twists) associated with all the joints form an n-order twist system or n-system if the n joint screws are linearly independent. The instantaneous twists of the end-effector can be described as follows: $¼

n X

q_ i $^ i ,

(6)

i¼1

where q_ i is the intensity and $^ i is the unit screw associated with the ith joint. The reciprocal screw system of the twist system consists of 6-n linearly independent reciprocal screws (wrenches) and is called a (6-n)-order wrench system or (6-n)-system. In what follows, the relevant results of screw theory are utilized for both the mobility and stiffness investigations of a 3-DOF TPM. 3. Architecture description of the manipulator The schematic diagram of a 3-PRC TPM is shown in Fig. 1. It consists of a mobile platform, a fixed base, and three limbs with identical kinematic structure. Each limb connects the fixed base to the mobile platform by a P (prismatic) joint, a R (revolute) joint, and a C (cylindrical) joint in sequence, where the P joint is driven by a linear actuator assembled on the fixed base. Thus, the mobile platform is attached to the base by three identical PRC linkages. To facilitate the analysis, as shown in Figs. 1 and 2, we assign a fixed Cartesian frame Ofx; y; zg at the centered point O of the fixed base, and a moving Cartesian frame

ARTICLE IN PRESS Q. Xu, Y. Li / Robotics and Computer-Integrated Manufacturing 24 (2008) 402–414

404

A2

Base platform z

y

2

3

A1 

x

O C2

A3

P joint



R joint

C1 C3 D l20

w

C joint

v B1(B'1)

u

P

Mobile platform

B3(B'3)

Fig. 1. Schematic representation of a 3-PRC TPM.

z

y Ai

ai

d max d max pd i p , ð7Þ 2 2 smax smax  psi p ð8Þ 2 2 for i ¼ 1, 2, and 3. Furthermore, in order to achieve a symmetric workspace of the manipulator, both DA1 A2 A3 and DB1 B2 B3 are assigned to be equilateral triangles. The following mobility analysis shows that in order to keep the mobile platform from changing its orientation, it is sufficient for the joint axes within the same limb to satisfy some certain geometric conditions. That is, the R joint axis ðs2;i Þ and C joint axis (s3;i or s4;i ) within the ith limb are parallel to the same unit ! vector si0 , which is perpendicular to the leg direction C i Bi , for i ¼ 1, 2, and 3. 

l10

l 30 (B'2) B2

as the layout angle of actuators. In order to obtain a compact architecture, the value of a is designed within the ! range of ½0 ; 90 . Angle ji is defined from the x-axis to OAi ! in the fixed frame, and also from the u-axis to PB0i in the moving frame. Without loss of generality, let the x-axis ! ! point along OA1 , and the u-axis along PB01 . Then, we have j1 ¼ 0 . Additionally, let d max and smax denote the maximum stroke of linear actuators and C joints, respectively, i.e.,

Mi

O Ci

x Ni

s2,i

p

s 1,i

) (s 5,i

4. Mobility analysis and elimination of overconstrained conditions

Bi

l li0

k2,i k1,i

w

v Bi

P

ki

s3,i (s4,i)

B'i u

Fig. 2. Representation of vectors.

Pfu; v; wg on the triangle mobile platform at the centered point P, along with the z- and w-axes perpendicular to the platform, and the x- and y-axes parallel to the u- and v-axes, respectively. In addition, the ith limb C i Bi ði ¼ 1; 2; 3Þ with the length of l is connected to the mobile platform at Bi which is a point on the axis of the ith C joint. B0i denotes the point on the mobile platform that is coincident with the initial position of Bi , and the three points B0i , for i ¼ 1, 2, and 3, lie on a circle of radius b. The three rails M i N i intersect one another at point D and intersect the x2y plane at points A1 , A2 , and A3 that lie on a circle of radius a. The sliders of P joints C i are restricted to move along the rails between M i and N i . Moreover, the axis of P joint is perpendicular to the axes of R and C joints within the ith limb. Angle a is measured from the fixed base to rails M i N i and is defined

4.1. Mobility determination of a 3-PRC TPM The mobility determination, i.e., the DOF identification, is the first and foremost issue in designing a parallel manipulator. The general Gru¨bler–Kutzbach criterion is useful in mobility analysis for many parallel manipulators, however it is difficult to directly apply this criterion directly to mobility analysis of some kinds of limited-DOF parallel manipulators. For example, the number of DOF of a 3-PRC TPM given by the general Gru¨bler–Kutzbach criterion is F ¼ lðn  j  1Þ þ

j X

f i ¼ 6  ð8  9  1Þ þ 12 ¼ 0,

(9)

i¼1

where l represents the dimension of task space, n is the number of links, j is the number of joints, and f i denotes the DOF of joint i. The zero number of DOF of a 3-PRC TPM given by the general Gru¨bler–Kutzbach criterion reveals that the 3-PRC TPM is an overconstrained parallel manipulator. Another drawback of the general Gru¨bler–Kutzbach criterion is that it can only derive the number of DOF of some mechanisms but cannot obtain the properties of the DOF, i.e., whether they are translational or rotational DOF. On the contrary, we can effectively determine the mobility of a 3-PRC TPM by resorting to the screw

ARTICLE IN PRESS Q. Xu, Y. Li / Robotics and Computer-Integrated Manufacturing 24 (2008) 402–414

theory. For a limited-DOF parallel manipulator, the motion of each limb that can be treated as a twist system is guaranteed under some exerted structural constraints which are termed as a wrench system. The wrench system is a reciprocal screw system of the twist system for the limb. The mobility of the manipulator is then determined by the effect of linear combination of the wrench systems for all limbs. With x ¼ ½ox oy oz T and t ¼ ½ux uy uz T respectively denoting the vectors for the angular and linear velocities, the twist of the mobile platform can be defined as $p ¼ ½xT tT T . Considering that a C joint is equivalent to the combination of a P joint with a coaxial R joint, the connectivity of each limb for a 3-PRC TPM is equal to 4 since each limb consists of four 1-DOF joints. Hence, the instantaneous twist $p of the mobile platform can be expressed as a linear combination of the four instantaneous twists, i.e., $p ¼ d_ i $^ 1;i þ y_ 2;i $^ 2;i þ y_ 3;i $^ 3;i þ s_i $^ 4;i

(10)

for i ¼ 1, 2, 3, where y_ j;i is the intensity and $^ j;i denotes a unit screw associated with the jth joint of the ith limb with respect to the instantaneous reference frame P, and " $^ 1;i ¼ " $^ 2;i ¼ "

s1;i

,

ð11Þ #

ci  s2;i s3;i

bi  s3;i " # 0 ¼ , s4;i

,

ð12Þ

# ,

ð13Þ

ð14Þ

can be identified, where sj;i represents a unit vector along the jth joint axis of the ith limb, 0 denotes a 3  1 zero ! ! vector, bi ¼ PB , ci ¼ PC ¼ bi  lli0 , and s2;i ¼ s3;i ¼ s4;i ¼ si0 since the R and C joint axes are parallel to each other. The screws that are reciprocal to all the joint screws of one limb of a 3-PRC TPM form a 2-system. Hence, two reciprocal screws of the ith limb can be identified as two infinite-pitch wrench screws as follows: " $^ r;1;i ¼ " $^ r;2;i ¼

0 h1;i 0 h2;i

For simplicity, let h1;i lie in the u–v plane and h2;i be vertical to the u–v plane, respectively, i.e., h1;1 ¼ ½1 0 0T , pffiffi h1;2 ¼ ½12 23 0T , pffiffi h1;3 ¼ ½12  23 0T , h2;1 ¼ h2;2 ¼ h2;3 ¼ ½0 0 1T . It is observed that the six wrench screws are linearly dependent and form a screw system of order 3, namely a 3-order wrench system of the mobile platform. Since the directions of each R and C joint axes satisfy the conditions described earlier, i.e., they are invariable, the wrench system restricts three rotations of the mobile platform with respect to the x-, y-, and z-axes of the fixed frame at any instant. Thus leads to a TPM with three translational DOF along the x-, y-, and z-axes of the fixed frame. It should be noted that the mobility of a 3-PRC TPM can also be determined by adopting other methods, such as a recent theory of DOF for complex spatial mechanisms proposed in [18], or a group-theoretic approach recommended in [19], etc. 4.2. Elimination of overconstraints of a 3-PRC TPM

s2;i

$^ 3;i ¼ $^ 4;i

#

0

405

As an overconstrained mechanism, the problems of variable friction in passive joints and large reaction moment in the 3-PRC TPM have to be considered to assure the mobility of the mobile platform. Otherwise, the mobile platform may not move or the manipulator cannot work if there are some kinematic errors. In this paper, a R joint with its axis along the unit vector s5;i is added in each limb of the 3-PRC TPM to eliminate the overconstrained conditions. It will be demonstrated that the added R joints are all inactive and do not affect the mobility and kinematics of the manipulator at all. For the convenience of assembly, the R joint is added with the common axis of the P joint in the ith limb, i.e., s5;i ¼ s1;i and di ¼ ci , which allows the generation of a 3-CRC parallel manipulator indeed. The DOF number of the 3-CRC parallel manipulator can be given by the general Gru¨bler–Kutzbach criterion as F ¼ lðn  j  1Þ þ

j X

f i ¼ 6  ð8  9  1Þ þ 15 ¼ 3,

i¼1

#

(17) ,

ð15Þ

,

ð16Þ

#

where h1;i and h2;i are two different arbitrary vectors perpendicular to si0 of the ith limb. $^ r;1;i and $^ r;2;i denote two unit couples of constraints imposed by the joints of the ith limb, and are exerted on the mobile platform.

which indicates that it is not an overconstrained manipulator. In addition, the joint screw of the added R joint can be expressed as " # s5;i $^ 5;i ¼ , (18) di  s5;i where di is the position vector of the center of the added R joint with respect to the reference frame P.

ARTICLE IN PRESS Q. Xu, Y. Li / Robotics and Computer-Integrated Manufacturing 24 (2008) 402–414

406

Then, the instantaneous twist $p of the mobile platform can be expressed as a linear combination of the five instantaneous twists, i.e., $p ¼ d_ i $^ 1;i þ y_ 2;i $^ 2;i þ y_ 3;i $^ 3;i þ s_i $^ 4;i þ y_ 5;i $^ 5;i

(19)

for i ¼ 1, 2, and 3. The screws that are reciprocal to all the joint screws of a limb of the manipulator form a 1-system. Hence, one reciprocal screw of the ith limb can be identified as an infinite-pitch wrench screw as follows: " # 0 ^$c;i ¼ , (20) ki where ki is a unit vector defined by s2;i  s5;i . (21) ki  ks2;i  s5;i k It is observed that $^ c;i in Eq. (20) represents a unit couple of constraints imposed by the joints of the ith limb, and the couple is exerted on the mobile platform around the direction of ki . Taking the reciprocal product of both sides of Eq. (19) with $^ c;i , yields ~ $^ c;i T $p ¼ 0, ½D

(22)

which can be rewritten into the matrix form: Jc $p ¼ 0, where 2

kT1

6 T Jc ¼ 4 k2 kT3

(23)

0T

3

0T 7 5 T 0

(24) 36

is called the Jacobian of constraints [20]. Each row in Jc denotes a unit wrench of constraints imposed by the joints of a limb, the combination of which constrains the mobile platform to a 3-DOF motion. Hence, if ki (i ¼ 1, 2, and 3) are linearly independent, the unique solution to Eq. (23) is x ¼ 0,

(26)

In view of the condition that s2;i ¼ s3;i as5 , the following two equations can be derived from Eq. (26): y_ 2;i þ y_ 3;i ¼ 0, y_ 5;i ¼ 0.

non-overconstrained parallel manipulator as indicated by Eq. (17). Additionally, Eq. (27) reveals that the intermediate two R joints always rotate in the inverse directions at the same angular velocity. It follows that the two R joints rotate in the same motion range, which provide some guidelines for the selection of mechanical joints to construct a 3-PRC or a 3-CRC TPM prototype. The CAD model of a 3-CRC TPM is shown in Fig. 3, where the P joint within the first C joint is driven by a lead screw linear actuator, that is easy to implement in practice. It should be noted that the overconstraints of a 3-PRC TPM can be eliminated by other approaches as well. For example, one R joint can be added between the original P and R joints to construct a 3-PUC parallel manipulator, which is also a non-overconstrained manipulator. And it can be shown easily via screw theory that the 3-PUC parallel manipulator has the same mobility and kinematics with the original 3-PRC TPM.

(25)

which exhibits that the 3-CRC manipulator has three translational DOF. Substituting Eq. (25) into Eq. (19), results in y_ 2;i s2;i þ y_ 3;i s3;i þ y_ 5;i s5;i ¼ 0.

Fig. 3. A 3-CRC TPM.

ð27Þ ð28Þ

Eq. (28) reveals that the added R joints in a 3-CRC TPM do not rotate around the C joints at all and do not influence the mobility and kinematics of the original 3-PRC TPM, but they are introduced to eliminate the internal constraints in a 3-PRC TPM and lead to a

5. Stiffness model identification Concerning a rigid body elastically suspended by elastic devices, if only small displacements from its unpreloaded equilibrium position are considered, the overall spatial force–deflection relation of the mechanism is linear and described by a 6  6 symmetric positive semidefinite matrix [21], i.e., the stiffness matrix. Generally, the stiffness characteristics of a parallel manipulator can be described by the 6  6 stiffness matrix, which relates the vector of compliant deformations of the end-effector to an external static wrench applied on the manipulator [22]. By taking into account the flexibilities of every compliant elements, the stiffness model of 6-DOF parallel manipulators with 6 legs can be established straightforwardly via a proper investigation [23,24]. While for those limited-DOF parallel

ARTICLE IN PRESS Q. Xu, Y. Li / Robotics and Computer-Integrated Manufacturing 24 (2008) 402–414

manipulators, it is not easy to derive their overall stiffness matrices since the rank of conventional Jacobian matrix for a limited-DOF parallel manipulator is less than 6. There are also various examples illustrating how to perform the stiffness modeling in the literatures. For instance, the stiffness of a tripod-based parallel manipulator is modeled in [25] by decomposing the whole machine structure into two separated substructures, and formulating the stiffness model of each substructure by means of virtual work principle. A stiffness model of the 3-DOF CaPaMan parallel manipulator is established in [26] by taking into account the kinematic and static features of the three legs in view of the motions of every joint and link. In addition, the stiffness modeling of four tripod-based parallel manipulators is carried out in [27] with the consideration of actuator flexibility, and the leg bending and axial deformations, where the stiffness matrix is derived using the force and infinitesimal motion relationships in each serial limb, while the processes for the derivation of Jacobian matrices are not given in detail. An observation of the commonly used stiffness modeling of limited-DOF parallel manipulators reveals that it is not obvious what is the best way since these existing approaches are not intuitive enough. Recently, the screw theoretic approach is adopted in [28] to derive the stiffness model of a limited-DOF parallel manipulator with the determination of spring constants of six connecting springs, which is then applied for the stiffness modeling of the Tricept parallel manipulator with variable-length legs. In this paper, we will derive the stiffness model of a 3-PRC TPM based upon the screw theory as well. However, it will be illustrated that the stiffness model of a limited-DOF parallel manipulator with fixed-length legs is different from that with variable-length legs, since there are nine springs instead of only six ones connecting the mobile platform to the fixed base.

5.1. Derivation of the Jacobian of actuations The overall Jacobian of a 3-CRC TPM can be generated by the combination of the Jacobian of actuations with the Jacobian of constraints, where the former one has been derived as expressed in Eq. (24), and the latter one can be determined as follows. Let the P joint in each limb be locked, then the reciprocal screws of each limb form a 2-system which includes one screw $^ r;1;i identified earlier. The other basis screw being reciprocal to all the passive joint screws of the ith limb can be identified as a zero-pitch screw along the direction of the ith leg, i.e., " # li0 ^$a;i ¼ , (29) bi  li0 ! where li0 is a unit vector along C i Bi .

407

Taking the reciprocal product of both sides of Eq. (19) with $^ a;i , results in ½D~ $^ a;i T $p ¼ d_i lTi0 s1;i ,

(30)

which can be rewritten into the following matrix form: Ja $p ¼ q_ ,

(31)

where q_ ¼ ½d_1 d_2 d_3 T denotes the actuated joint rates and 3 2 ðb1  l10 ÞT lT10 6 lT s lT10 s1;1 7 7 6 10 1;1 7 6 T 6 ðb2  l20 ÞT l20 7 7 6 Ja ¼ 6 T (32) 7 lT20 s1;2 7 6 l20 s1;2 7 6 6 ðb  l ÞT lT30 7 30 5 4 3 T T l30 s1;3 l30 s1;3 36

is called the Jacobian of actuations, which transforms the angular and linear velocities of the mobile platform to the input joint rates. Here, it is assumed that all singularities have been eliminated from the workspace by the mechanism design rules proposed in [12]. 5.2. Stiffness modeling Referring to Eqs. (20) and (29), it can be observed that the wrench system, which is the infinite-pitch reciprocal screw system of constraints, exerts three constraint couples to the mobile platform with the direction of ki , and the wrench system which is the zero-pitch reciprocal screw system of actuations imposes three constraint forces to the mobile platform with the directions just along the legs. This means that each leg is subject to a force and a couple. With the assumption that the rigidities of the passive R and C joints and the mobile platform are infinite, the compliances subject to actuators and legs can be derived as follows. 5.2.1. Compliance subject to actuators Let us consider the compliance of actuators in their actuation directions. Referring to Fig. 3, for the lead screw actuation system, the torque ti of the ith actuator can be expressed as ti ¼ K a;i Dai ,

(33)

where K a;i denotes the torsional stiffness of the actuator and Dai represents the angular deformation. Through the torque transmission, the force acting on the ith nut and its linear displacement can be respectively expressed by [29] fi ¼

2ti , d p tanðl þ cÞ

Dti ¼ LDai ,

ð34aÞ ð34bÞ

where d p and L respectively denote the pitch diameter and lead of the lead screw, and l ¼ tan1 ðL=ðpd p ÞÞ and c ¼ tan1 ðmc = cosðat ÞÞ respectively denote the lead angle and friction angle, with at and mc representing the radial angle

ARTICLE IN PRESS Q. Xu, Y. Li / Robotics and Computer-Integrated Manufacturing 24 (2008) 402–414

408

of thread and coefficient of friction between screw and the nut, respectively. Combining Eq. (33) with Eq. (34) allows the derivation of Dti ¼ C i f i ,

(35)

where Ci ¼

Ld p tanðl þ cÞ 2K a;i

(36)

denotes the compliance of the ith linear driving device. The compliance of the ith actuator due to the constraint force in the ith leg’s direction can be derived as C s;i ¼

lTi0 s1;i C i .

(37)

In view of Eqs. (36) and (37), one can get that C s;i ¼

Ld p tanðl þ cÞlTi0 s1;i . 2K a;i

(38)

5.2.2. Compliance subject to legs The ith leg C i Bi bears a constraint force ðF ci Þ along the direction of li0 and a constraint couple (M ci ) around the direction of ki exerted at the position of point Bi , where the force causes a linear elastic deformation ðDl i Þ of the leg along its axis direction. As illustrated in Fig. 2, the constraint couple can be decomposed into two couples with the directions of k1;i and k2;i , which are along and perpendicular to li0 , respectively. Therefore, the ith constraint couple causes a torsional deformation Dyi of the ith leg around the k1;i direction and a bending deformation Dbi about the direction of k2;i . Let C l;i , C y;i , and C b;i be the longitudinal, torsional, and bending compliances of the ith leg, respectively. Then, the elastic deformations of the ith leg caused by the constraint force and couple can be expressed as Dl i ¼

C l;i F ci

l c F , ¼ EA i

l kk1;i kM ci , GI p l Dbi ¼ C b;i M ci kk2;i k ¼ kk2;i kM ci , EI k

Dyi ¼ C y;i M ci kk1;i k ¼

l , EA l ¼ kk1;i k, GI p l ¼ kk2;i k. EI k

C y;i C b;i

1 K a;i ¼ C 1 a;i ¼ ðC s;i þ C l;i Þ ,

K c;1;i ¼ K c;2;i ¼

C 1 c;1;i C 1 c;2;i

ð45aÞ

1

ð45bÞ

1

ð45cÞ

¼ ðC y;i Þ , ¼ ðC b;i Þ

for i ¼ 1, 2, and 3. Consequently, the stiffness model of a 3-CRC TPM can be established by considering that the mobile platform is connected to the fixed base by three compliant limbs, each of which consists of one linear spring and two rotational springs as shown in Fig. 4.

5.3. Derivation of the overall Jacobian matrix Since the constraint couple of each limb is equivalently decomposed into two separate couples for the convenience of compliant analysis, Eq. (20) can be divided into $^ c;i ¼ $^ c;1;i þ $^ c;2;i , where " # 0 $^ c;1;i ¼ , ð46Þ k1;i " # 0 . ð47Þ $^ c;2;i ¼ k2;i

ð39Þ

Taking the reciprocal product of both sides of Eq. (19) with $^ c;k;i , leads to

ð40Þ

½D~ $^ c;k;i T $p ¼ 0

(48) A2

ð41Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where k1;i þ k2;i ¼ ki , kk1;i k ¼ lTi0 ki , kk2;i k ¼ 1  kk1;i k2 , l and A denote the length and cross-section area of each leg, E and G are the Young’s modulus and shear modulus of elasticity, I p represents the polar moment of inertia, and I k is the moment inertia around an axis parallel to k2;i , respectively. Then, from Eqs. (39)–(41), one can obtain C l;i ¼

5.2.3. Stiffness model From the above discussions, it is observed that the constraint couple of the ith limb is equivalent to two decomposed couples along the directions of k1;i and k2;i , respectively. In view of the inverse relationship between stiffness and compliance, the stiffness of actuations and constraints can be respectively obtained as

z

y

O

A1 

x

C2 A3

C1



Kc,2,1

C3

Ka,1

ð42Þ

Kc,1,1 B2

ð43Þ ð44Þ

B3

P

B1

Fig. 4. Stiffness model of a 3-CRC TPM.

ARTICLE IN PRESS Q. Xu, Y. Li / Robotics and Computer-Integrated Manufacturing 24 (2008) 402–414

for k ¼ 1 and 2, and i ¼ 1, 2, and 3, which can be rewritten into the matrix form: Jcr $p ¼ 061 , where 2

(49)

409

matrices are ð55aÞ va ¼ diag½K a;1 ; K a;2 ; K a;3 , vc ¼ diag½K c;1;1 ; K c;2;1 ; K c;1;2 ; K c;2;2 ; K c;1;3 ; K c;2;3 . ð55bÞ

kT1;1

6 T 6k 6 2;1 6 T 6 k1;2 6 Jcr ¼ 6 T 6 k2;2 6 6 T 6 k1;3 4 kT2;3

0

3 T

T

7 0T 7 7 7 0T 7 7 7 0T 7 7 7 0T 7 5 0T

(50)

Moreover, let Dx ¼ ½Dx Dy Dz and Dh ¼ ½Dyx Dyy Dyz T be the infinitesimal displacements of translation and rotation of the mobile platform with respect to three axes of the reference frame. Then, applying the principle of virtual work by neglecting the gravitational effect, allows the generation of wT DX ¼ sTa Dqa þ sTc Dqc , T

66

is the extended Jacobian of constraints. Then, complementing Eq. (31) with Eq. (49) allows the generation of q_ o ¼ J $p ,

(51)

(56)

T T

where DX ¼ ½Dx Dh  denotes the mobile platform’s twist deformation in the axis coordinate. Substituting Eq. (54) into Eq. (56), results in wT DX ¼ DqTa va Dqa þ DqTc vc Dqc .

In addition, substituting Eq. (53) into Eq. (56), leads to ~ ~ sTa ½Ja DDX þ sTc ½Jcr DDX ¼ sTa Dqa þ sTc Dqc ,

where q_ o ¼ ½d_ 1 d_ 2 d_3 0 0 0 0 0 0T , and " # Ja J¼ Jcr

(52)

96

is called the overall Jacobian of a 3-CRC TPM, which includes the effects of both actuations and constraints.

(57)

(58)

which holds for any values of sa and sc . Consequently, the virtual displacements of actuations and constraints can be derived as ~ Dqa ¼ ½Ja DDX, ~ Dq ¼ ½Jcr DDX.

ð59aÞ ð59bÞ

c

Then, by substituting Eq. (59) into Eq. (57), one can obtain

5.4. Stiffness matrix determination In view of Plu¨cker’s conventions, a twist is expressed in the axis coordinate and a wrench is described in the ray coordinate. While in above discussions, the twist $p ¼ ½xT tT T is expressed in the Plu¨cker ray coordinate. The ~ p ¼ ½tT xT T . conventional twist can be derived as T ¼ D$ Assuming that there are no preload effects on the manipulator, the stiffness model of the TPM is established based on the following discussions. Let f ¼ ½f x f y f z T and m ¼ ½mx my mz T respectively denote an external force and torque exerted on the mobile platform. The wrench can be expressed as w ¼ ½f T mT T in the Plu¨cker ray coordinate. Additionally, let sa ¼ ½ta;1 ta;2 ta;3 T and sc ¼ ½tc;1;1 tc;2;1 tc;1;2 tc;2;2 tc;1;3 tc;2;3 T represent the reaction forces/torques of actuations and constraints, respectively. In the absence of gravity, the external wrench is balanced by the reaction forces/torques exerted by the actuations and constraints, i.e., ~ T sa þ ½Jcr D ~ T sc , w ¼ ½Ja D

(53)

where the reaction forces/torques can be expressed as sa ¼ va Dqa ,

ð54aÞ

sc ¼ vc Dqc

ð54bÞ

with Dqa and Dqc denoting the displacements of actuations and constraints, respectively. And the diagonal

~ T v ½Ja DDX ~ wT DX ¼ DXT ½Ja D a ~ T v ½Jcr DDX. ~ þ DXT ½Jcr D c

ð60Þ

Since Eq. (60) holds for any arbitrary virtual displacements DX, we conclude that w ¼ KDX,

(61)

where ~ T v ½Ja D ~ þ ½Jcr D ~ T v ½Jcr D ~ K ¼ ½Ja D a c

(62)

is defined as the 6  6 stiffness matrix of a 3-CRC TPM including the effects of actuations and constraints, which can be further written into the form: ~ ~ T v½JD, K ¼ ½JD

(63)

where the 9  9 diagonal matrix v ¼ diag½va vc . An observation of the units of matrix Ja in Eq. (32) reveals that the last three columns are dimensionless while the first three ones are related to the unit of length which is introduced by the position vector bi . It is necessary to homogenize the units of the Jacobian matrices so as to generate a stiffness matrix and performance index invariant of the length unit adopted. The dimensionally homogeneous Jacobian of actuations can be achieved by Jah ¼ Ja D

(64)

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410

with   1 1 1 D ¼ diag 1; 1; 1; ; ; , b b b

(65)

where the mobile platform radius b is chosen as the characteristic length [30] to homogenize the dimension of the Jacobian matrix. In consequence, the overall Jacobian in Eq. (52) becomes " # Jah J¼ , (66) Jcr 96

which is homogeneous in terms of units. 6. Stiffness evaluation of the TPM For a given design of a parallel manipulator, the stiffness varies with the variation of the manipulator configurations within its workspace as well as the direction of the applied wrenches. Once the stiffness model is obtained, it is desired to predict its stiffness characteristics over the workspace in order to assess whether the design is satisfied with the stiffness requirements or even further to perform an optimal design with the stiffness considered especially in the design stage. As far as the approaches for the stiffness evaluation are concerned, several different performance indices have been proposed and utilized in the literatures. A simple way to predict the stiffness is to use stiffness factors, i.e., the terms of the stiffness matrix [22,25]. Besides, the stiffness can be evaluated using the eigenvalue of stiffness matrix which is experienced in the direction of the corresponding eigenvector [22,31]. It has been shown that the stiffness is bounded by the minimum and maximum eigenvalues of the stiffness matrix [32]. Based on this concept, the stiffness has been predicted by the minimum, maximum, and average eigenvalues, even magnitude of the ratio of the maximum and minimum eigenvalues of the stiffness matrix [32]. Additionally, the determinant of stiffness matrix, that is the product of its eigenvalues, has been adopted to assess the stiffness of parallel manipulators [26,33]. And the trace of the compliance matrix (the inverse of the stiffness matrix), which is the sum of its eigenvalues, has been used to measure the kinetostatic property of tripod-based parallel manipulators [27]. Furthermore, similar to the condition number of Jacobian matrix, the condition number of the stiffness matrix has been introduced, based on which, a global stiffness index defined as the inverse of the condition number of stiffness matrix integrated over the reachable workspace and divided by the workspace volume is presented to assess the stiffness of a 3-DOF spherical parallel manipulator [34] and the hexaslide machine tools [24]. Among these usually used stiffness performance indices, the stiffness factors are preferred to be applied to evaluate the stiffness matrix only with a diagonal form. Because for a stiffness matrix with the generic form, the off-diagonal

terms couple the forces/torques applied in the corresponding directions, the individual stiffness factors cannot totally reflect the stiffness property in any directions. Concerning the determinant or trace of the stiffness matrix, it cannot distinguish the situations in which the manipulator has a very low stiffness in one direction while a very high stiffness in another one, that leads to a high value of the determinant or trace although the low stiffness prohibits the use of the manipulator for some situations such as machine tool applications. In consequence, neither the determinant nor the trace of the stiffness matrix is a good choice for the stiffness evaluation of parallel manipulators. And the same problem arises for the average of the eigenvalues of the stiffness matrix. Additionally, although the condition number of stiffness matrix can indicate the ill-conditioning of the stiffness matrix, it does not provide enough information of the stiffness values. For a precise manipulation, the minimum stiffness over the workspace should be larger than a specified value in order to ensure the accuracy of the manipulation everywhere in the workspace. Hence, the minimum and maximum values of stiffness and their variances appear to be the most direct and reasonable indices for the stiffness evaluation, which are adopted as stiffness performance indices in this paper. The architectural parameters of a 3-CRC TPM are shown in Table 1. By adopting a numerical searching method described in [35], the reachable workspace of the manipulator is generated as illustrated in Fig. 5. In addition, the physical parameters of the designed TPM are elaborated in Table 2. According to Eqs. (38) and (42)–(44), the compliance parameters for the TPM can be calculated. Let the home position of the mobile platform be in the case of mid-stroke of linear actuators, i.e, d i ¼ 0 ði ¼ 1; 2; 3Þ, in which the stiffness matrix is calculated as follows: 2

0:3653 6 0 6 6 6 0 K0 ¼ 6 6 0 6 6 4 0:4871 0

0 0:3653

0 0

0 0:4871

0:4871 0

0 0 0

0

1:2990

0

0

0:4871

0

0:7277

0

0

0

0

0

0:7277

0

0

0

0

0

0:1132

3 7 7 7 7 7  105 , 7 7 7 5

(67) where the units of terms are N/m for {K 011 , K 022 , K 033 }, N/rad for {K 015 , K 024 }, Nm/m for {K 051 , K 042 }, and Nm/rad for {K 044 , K 055 , K 066 }. Table 1 Architectural parameters of a 3-CRC TPM Parameter

Value (m)

Parameter

Value (degree)

a b l d max smax

0.6 0.3 0.5 0.4 0.2

a j1 j2 j3

45 0 120 240

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0.1

411

Home position height

z (m)

0.2 0.3 0.4 0.5 0.6 0.1 0.05 y(

m)

0 -0.05 -0.1

-0.1

-0.05

0.05

0

0.1

)

x (m

Fig. 5. Reachable workspace of a 3-CRC TPM.

Table 2 Physical parameters of a 3-CRC TPM Parameter

Value

Parameter

Value

E G A Ip Iy

2:03  1011 N=m2 7:85  1010 N=m2 4:52  104 m2 3:26  108 m4 1:63  108 m4

K a;i dp L at mc

1:45  106 Nm=rad 31:75  103 m 6:35  103 m 14:5 0.20

In order to ensure the accuracy of the TPM manipulation anywhere within the workspace, the minimum stiffness over the workspace should be larger than a specified value. It follows that the minimum stiffness is the most important index for the 3-CRC TPM. In this subsection, both the minimum and maximum eigenvalues obtained through the conventional eigenvalue decomposition of the stiffness matrix are used as stiffness indices to have a global view of the stiffness values over the workspace. The distributions for the minimum ðK min Þ and maximum ðK max Þ stiffness in the planes of z ¼ 0:30 m, z ¼ z0 ¼ 0:40 m (home position height) and z ¼ 0:50 m are illustrated in Fig. 6. The first observation from Fig. 6 is that, similar to the reachable workspace, the distribution of stiffness in a x2y plane is 120 symmetrical about the axial directions of three actuated P joints. Additionally, in a particular height plane, the lowest value of minimum stiffness occurs around the boundary of the workspace, so does the highest value of the maximum stiffness, which arise from the reason that the manipulator approaches singular when it comes near the workspace boundary. Moreover, both the lowest and the highest stiffness values occur on the lowest height plane in the workspace. Since around the boundary of the reachable workspace, the TPM takes on a bad stiffness property, it is reasonable to restrict the TPM to operate in a subworkspace located within the reachable workspace. According to the

manipulator tasks and performances, there are several ways to define this subworkspace. Here, for the main purpose of examining the influences of the kinematic parameters on stiffness values, the subworkspace is assigned as a cylinder shape usable workspace with the radius of 0.08 m and height of 0.16 m, whose geometry center lies at the home position point of the mobile platform. In view of the stiffness distributions illustrated in Fig. 6, it can be deduced that both the lowest value of minimum stiffness and the highest value of maximum stiffness arise on the boundary of the lowest plane of the usable workspace, i.e., a circle of radius 0.08 m at the height of ðz0  0:08Þ m. This deduction will greatly simplify the procedure for the stiffness assessment. Following the aforementioned process, the analytical expression of stiffness value for the TPM can be easily obtained via a symbolic calculation tool such as Mathematica or Matlab, etc. However, it is a time-consuming work to derive the analytical solution, and the final symbolic expression is too long. For example, an attempt to derive the analytical solution of stiffness values via Matlab running on a personal computer (Intel Pentium 4 CPU 3.00 GHz, 512 MB RAM) with Microsoft Windows XP operating system exhibits that it needs more than 10 h to obtain solely the analytical solutions for the stiffness values. On the other hand, the numerical assessment of stiffness is quite straightforward and simple especially due to the aforementioned deduction. For instance, by equally sampling one circle ð360 Þ into 60 segments, and the range of actuators layout angles ð90 Þ into 90 segments, the calculation of the maximum and minimum stiffness values (Fig. 7(d)) requires only less than 1 min. The high computational efficiency of numerical method is very obvious. Hence, in order to have a quick view of the TPM stiffness, the numerical evaluation approach is adopted in this paper. Along with the variation of architectural parameters of the 3-CRC TPM, the tendency of variation on the

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412

x 105 2600

1.3 Kmax

Kmin

2400 2200 2000 1800 0.1 0.05 0 y( m -0.05 -0.1 )

1.2 1.1 1 0.1 0.05 0 y( -0.05 -0.1 m)

0.1 0 0.05 -0.05 -0.1 ) x (m

0 -0.1-0.05

0.1 0.05 )

x (m

x 105 1.45 1.4

2000

Kmax

Kmin

2500

1500 1000

1.35 1.3 1.25

0.1 0.05 0 y (m -0.05 -0.1 )

0.1 0.05 0 y( m) -0.05 -0.1

0.1 0 0.05 -0.1-0.05 m) x(

0.1 0 0.05 -0.05 -0.1 ) x (m

2000

1.8

1500

1.78 Kmax

Kmin

x 105

1000

1.76 1.74 1.72

500 0.1 0.05 0 y( -0.05 -0.1 m )

0.1 0 0.05 -0.05 -0.1 ) x (m

0.1 0.05 0 y (m -0.05 -0.1 )

0.1 0 0.05 -0.05 ) -0.1 x (m

Fig. 6. The distributions for minimum and maximum stiffness in the planes of (a) and (b) z ¼ 0:30 m, (c) and (d) z ¼ 0:40 m, and (e) and (f) z ¼ 0:50 m, respectively.

minimum and maximum stiffness over the usable workspace is depicted in Figs. 7(a)–(d), from which it is clear to see the impacts of design parameters (a, b, l, and a) on the stiffness property of the manipulator. It can be observed from Figs. 7(a) and (b) that the increase of the fixed platform size has almost the same effect as the decrease of the mobile platform size on the stiffness properties as anticipated. In view of Fig. 7(c), one can see that as the leg length increases, the minimum stiffness values decrease monotonously, while the maximum stiffness does not vary uniformly which attains the lowest value at l ¼ 0:335 m. Additionally, as the actuators layout angle increases, the maximum stiffness decreases and reaches at a lowest value around 60 , while the variation of the minimum stiffness is not monotonously as elaborated in Fig. 7(d). It appears that the minimum stiffness arises to the highest value around 28 of actuators layout angle. It is noticeable that the lowest and highest values of the extremum stiffness can be generated alternatively by differentiating analytical formulas of stiffness values with respect to the design parameters once the analytical-form solutions are derived.

However, it is quite a time-consuming work, which is not preferred in this paper. Besides, these figures can also be used to design the manipulator in order to assure that the lowest stiffness is larger than a predefined value. For example, referring to Fig. 7(c), if the minimum stiffness is designed as K min 41000 N=m, then the length of the legs should be designed as lo0:56 m accordingly. 7. Conclusions In this paper, the mobility determination and stiffness analysis for a 3-DOF TPM are carried out based upon the screw theory. A 3-CRC TPM is constructed from a previously presented overconstrained 3-PRC TPM by adding a revolute joint in each limb. It has been demonstrated that the conversion does not bring any impact on the mobility and kinematics of original manipulator since the added joints are idle indeed, but it can eliminate the overconstrained conditions of a 3-PRC TPM. The stiffness matrix is derived based on an overall

ARTICLE IN PRESS Q. Xu, Y. Li / Robotics and Computer-Integrated Manufacturing 24 (2008) 402–414 x 105

3000

4

2000

3

1000

2

4000

x 105

0 0.10.150.20.250.30.350.40.450.5 b (m) x 106 6

1500

2.5

Kmin Kmax

1400

4000 1.5

2000 0.56

0 1 0.30.350.40.450.50.550.60.650.7 l (m)

Kmin

2 Kmax

Kmin

6000

Kmin Kmax

2000 2

0 1 0.40.450.50.550.60.650.70.750.8 a (m) 8000

4

Kmax

5 Kmin

Kmin

6 Kmin Kmax

5

1300

4

1200

3

1100

2

1000

1

Kmax

Kmin Kmax

4000

x 105 6

Kmax

5000

413

0

900 0 10 20 30 40 50 60 70 80 90 α (deg.)

Fig. 7. Stiffness indices versus design parameters of (a) the fixed platform size, (b) the mobile platform size, (c) the length of legs, and (d) actuators layout angle.

Jacobian via the theory of reciprocal screws in view of the effects of actuations and constraints. And the rigidities in both actuators and legs are taken into account to establish the stiffness model of the manipulator. Through a survey of the commonly used stiffness performance criteria, the minimum and maximum eigenvalues of the stiffness matrix over a cylinder usable workspace are adopted to evaluate the stiffness of the TPM. Furthermore, the variation tendencies of stiffness within the workspace are illustrated and the impacts of variation for design parameters on the stiffness characteristics are presented. The main contribution of this paper is the conversion of an overconstrained 3-PRC TPM into a non-overconstrained 3-CRC TPM without affecting the mobility and kinematic properties, and the stiffness modeling of the 3-DOF TPM via an alternative approach, i.e., the screw theoretic approach, along with the quantitative evaluation of the TPM stiffness. This paper provides a basis for the architectural design of a 3-CRC TPM with stiffness properties taken into account. Moreover, the methodology presented here can be extended for the stiffness analysis of other types of limited-DOF parallel manipulators (e.g., with architectures of 3-PUU, 3-PRS, 3-RPS, etc.) as well. Acknowledgments The authors appreciate the fund support from the research committee of University of Macau under Grant No. RG068/05-06S/07R/LYM/FST and Macao

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