Received 16 April 2014Received in revised form 1 January 2015Accepted 18 January 2015Available online 2 March 2015

A novel method for force analysis of the overconstrained lower mobility parallel mechanisms

Keywords:

y some scholars in theevolute joints such asn practical production,2-dofmechanism [11],

Mechanism and Machine Theory 88 (2015) 3148

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Mechanism and Machine Theory

j ourna l homepage: www.e lsev ie r .com/ locate /mechmtand the 3-dof translational PMs such as 3-PRRR mechanism [12], Delta parallel robot [13], and Orthoglide parallel robot [14], are alloverconstrained LMPMs. The notations of S, U, P, and R represent the spherical joint, universal joint, prismatic joint, and revolutejoint, respectively.PM [5], which is just the overconstrained LMPM. The large-scale six-dimensional force sensors developed bworld [6,7] also adopted the overconstrained PM n-SS (n 7). Many single-loop mechanisms with only rBennett linkage are all overconstrained mechanisms [8,9]. Actually, planar linkage mechanisms, widely used iaremostly overconstrained planar PMs. Besides, the spherical PMs such as 3-RRR3-dofmechanism [10] and5RThe overconstrained lower mobility parallel mechanism (LMPM) is the LMPM that includes common constraints or redundantconstraints [14]. As a special kind of parallel mechanism (PM), it has the characteristics of simple structure, high precision, andhigh rigidity, which has played an important role in many cases that require high load carrying capacity. For example, the body of anew generation of parallel machine tool XT 700 developed currently employed the 2-UPR + SPR three degrees of freedom (dofs)1. Introduction Corresponding author. Tel./fax: +86 335 807 4581.E-mail addresses: ydxu@ysu.edu.cn (Y. Xu), yszhao@

1 Tel./fax: +86 335 807 4581.

http://dx.doi.org/10.1016/j.mechmachtheory.2015.01.000094-114X/ 2015 Elsevier Ltd. All rights reserved.(LMPM) is proposed. Firstly, denitions of the constraint wrenches and overconstraint wrenchesare put forward. Then, overconstrained LMPMs are classied into two classes based on character-istic of the elastic deformations generated at the end of the supporting limbs, which are limbstiffness decoupled overconstrained LMPM and limb stiffness coupled overconstrained LMPM,respectively. Corresponding to the limb stiffness decoupled and coupled overconstrainedLMPMs, stiffness matrix of the supporting limb's overconstraint wrenches and constraintwrenches are introduced and dened respectively. Next, taking the link's spatial composite elasticdeformations into account, the stiffness matrix of the supporting limb's overconstraint wrenchesor constraint wrenches is established, and general analytical expression for the solution of eachsupporting limb's overconstraint wrenches or constraint wrenches is derived. Furthermore, theinternal relation between the constraint wrenches and the actual joint reactions is discussed.Finally, based on Adams simulation software, a method to set up the force simulation model ofthe overconstrained LMPM is put forward, and the simulation results and the theoretical calculationones are basically consistent, which effectively veries the correctness of the proposedmethod forforce analysis of the overconstrained LMPM.

2015 Elsevier Ltd. All rights reserved.Lower mobility parallel mechanismOverconstraintForce analysisStiffness matrixForce simulation modelA method for force analysis of the overconstrained lowermobility parallel mechanism

Yundou Xu1, Wenlan Liu 1, Jiantao Yao 1, Yongsheng ZhaoParallel Robot and Mechatronic System Laboratory of Hebei Province, Key Laboratory of Advanced Forging and Stamping Technology and Science of Ministry of NationalEducation Yanshan University, Qinhuangdao, Hebei 066004, PR China

a r t i c l e i n f o a b s t r a c t

Article history:ysu.edu.cn (Y. Zhao).

4

Force analysis of the LMPM is one of the important steps for mechanical design, simulation, and control. Not only the actuatedforces/torques but also constraint wrenches of the LMPM should be studied [1517], since constraint wrenches are also very vitalfor force analysis of the LMPMs. However, the statically indeterminate problem arises in force analysis of the overconstrainedLMPM, which increases the complexity and difculty of analysis. For this reason, up to now, investigations focusing on the forceanalysis of the overconstrained LMPM at home and abroad have been relatively less.

Lu et al. [18] and Li et al. [19] solved the actuated forces/torques of the 4-dof overconstrained RRPU + 2-UPU PM and the 5Rspherical PM, respectively, but the constraint wrenches were not taken into account, so it did not involve the statically inde-terminate problem. Pashkevich et al. [20] analyzed stiffness of Delta PM and Orthoglide PM based on a multidimensionallumped model of the exible links. Static analysis of the overconstrained PMs that each supporting leg only supplies a con-straint force along the axial direction of the leg (UPS leg) was carried out in [6,7,21,22]. Song et al. [23] identied the solvabilityof every unknown joint forces/torques in static analysis of the overconstrained mechanisms by using mobility equation.Wojtyra [24] used techniques of Jacobian matrix analysis to detect constraints and joints for which reactions can be uniquelydetermined despite the existence of redundant constraints. Frczek et al. [25] studied motion uniqueness of overconstrainedmechanisms with joint friction. Wojtyra et al. [26] also discussed which parts of the overconstrained mechanism should bemodeled as exible bodies to obtain the unique reaction solution. Zahariev et al. [27] proposed an approach to dynamicsimulation of over-constrained multibody systems for which the kinematic constraints are substituted by elastic forces inthe exible links. Huang et al. [28] solved the actuated forces/torques and joint reactions of the 4-R(CRR) PM withoverconstraint coaxial forces based on reciprocal screw theory, but how to get stiffness of the supporting limbs consideringexibility of the links was not studied.

Except the above literatures, there are very few investigations on force analysis of overconstrained LMPMs. Thus, in thiswork, a novel method for force analysis of overconstrained PMs will be proposed. This method is mainly based on the constraintwrenches applied to the moving platform by the supporting limbs, and then combining the theory of link's spatial compositeelastic deformations, takes the mapping relation between the constraint wrenches and the deformations of the supportinglimbs into account.

This paper is organized as follows. Following the Introduction, constraintwrenches of the overconstrained LMPMare analyzed andthe actuated forces/torques solution is obtained in Section 2. The magnitudes of the overconstraint wrenches are solved in Section 3.The joint reactions are obtained in Section 4. And then, in Section 5, numerical example and verication are illustrated. Finally,conclusions are drawn in Section 6.

32 Y. Xu et al. / Mechanism and Machine Theory 88 (2015) 3148Fig. 1. A general overconstrained LMPM.

2. Con

of dofsis also

For ththree

numbearlywrencnot umeet

33Y. Xu et al. / Mechanism and Machine Theory 88 (2015) 3148wrenches $ r,t are the maximum linear independence group of the constraint wrenches of the overconstrained LMPM, and its rankis equal to (6 n).

Note that the mass of all the supporting limbs and friction of all joints are neglected in this work, which would beconsidered further in our future work. Then considering force balance of the moving platform, the following equation canbe obtained

$F $er;1 $er;2 $er;d $r;1 $r;l $a;1 $a;2 $a;nh i

GFf f 1

where GfF is the 6 6 square matrix mapping f into external wrench $ F, which can be expressed as

GFf $^er;1 $^er;2 $^

er;d $^r;1 $^r;l $^a;1 $^a;2 $^a;n

h i2

inwhich $^ stands for the unit screw of $. f is the vector composed of themagnitudes of the constraint and actuationwrenches, which isgiven by

f f e1 f e2 f el f 1 f l w1 w2 wn T 3er of overconstraint wrenches is (m l), denoted by $ r,v (v = l + 1, l + 2,, m) and its maximum number of lin-independent is d = 6 n l. Thus, the overconstraint wrenches can be equivalently transformed into p constrainthes that are linearly independent, denoted by $ r,ue (u = 1, 2, , d). The equivalent constraint wrenches may benique, but it would not affect the results obtained nally, an arbitrary set of equivalent constraint wrenches thatthe requirements can be chosen. The combination of the equivalent constraint wrenches $ r,ue and the non-overconstraintAssumed that Ni constraint wrenches and one actuation wrench are exerted to the moving platform by supporting limb i, for i=1 ~M1, which represent supporting limbs of type i,Nj actuationwrenches are exerted to themoving platform by supporting limb j, forj=(M1+1) ~ (M1+M2), which represent supporting limbs of type ii, and one actuationwrench is exerted to themovingplatformbysupporting limb k, for k = (M1 + M2 + 1) ~ (M1 + M2 + M3), which represent supporting limbs of type iii. Then the followingequation can be obtained

M1 M3 nXM1i1

Ni XM1M2

jM11Nj m:

8>:

The unit twist of the actuated jointwithin the supporting limb i (or limb k) is denoted by $^i (or $^k), and the corresponding actuationwrench is denoted by $ a,i (or $ a,k), and the constraint wrenches exerted by supporting limb i (or limb j) are denoted by $ r,i1, $ r,i2,,$r;iNi (or $ r,j1, $ r,j2,, $r; jN j), respectively, just as shown in Fig. 1. For without loss of generality, not just as shown in Fig. 1, supportinglimb i (or limb k) can contain more than one actuated joint, namely, can exert more than one actuation wrench to the moving plat-form. It is noted that all the actuation and constraint wrenches studied in this paper are only restricted to pure forces or pure couples.

A reference coordinate frame O-xyz is attached at point O on the moving platform, and the wrenches described below are allexpressed in the reference coordinate frame O-xyz.

Firstly, the constraint wrenches that are independent from all the other single constraint wrench and constraint wrenches are de-ned as the non-overconstraint wrenches, and the remaining, i.e., the constraint wrenches linearly dependent are called as theoverconstraint wrenches. Judgment of the non-overconstraint wrenches and overconstraint wrenches can refer to linear algebraand Grossmann geometry knowledge [32,33]. For example, for the 4-UPU LMPM analyzed in [34], each UPU supporting limb suppliesa constraint couple perpendicular to the axes of the U joint. The four constraint couples are linearly dependent, whose maximumnumber of linearly independent is two, and they restrain two rotational dofs of themovingplatform. There are no constraintwrenchesindependent from the other single constraint wrench and combination one. Thus, the four constraint couples are the overconstraintwrenches, and there do not exist non-overconstraint wrenches.

Assume that the number of non-overconstraint wrenches of the LMPM is l, denoted by $ r,t (t = 1, 2,, l). Then thee limbs of type iii, do not exert constraint wrenches to the moving platform, only contain actuated joints. Let the number oftypes of supporting limbs beM1,M2, andM3, respectively.The supporting limbs are classied into three types, which are type i, type ii and type iii, respectively. For the limbs of type i, notonly exert constraint wrenches to the moving platform, but also contain actuated joints, that is, exert actuation wrenches to themoving platform. For the limbs of type ii, only exert constraint wrenches to the moving platform, do not contain actuated joints.nstrained LMPM and is less than six. Assume actuation redundancy [29], that is, the number of actuators is greater than thatof themechanism, is not considered here, then the number of the actuationwrenches [30,31] imposed on themoving platformn.In this section, a general model of the overconstrained LMPM is built up, as shown in Fig. 1. Assume that the number of the con-straint wrenches imposed on the moving platform is m, then m N 6 n must be satised, where n represents the mobility of theovercostraint wrenches analysis and actuated forces/torques solution of the overconstrained LMPM

in which fue is themagnitude of $ r,ue , ft is themagnitude of $ r,t, andwq (q=1, 2,, n) is themagnitude of $ a,i or $ a,k. $ F is the externalwrench applied at point O on the moving platform including the gravity and inertial force of the moving platform, which can beexpressed by

$F Fx Fy Fz Mx My Mz T

: 4

In this case, the problem becomes statically determinate. Left multiplying both sides of Eq. (1) by [GfF]1 yields

f GFfh i1

$ F : 5

From the above equation, it can be seen that the magnitudes the non-overconstraint wrenches and actuation wrenches can besolved directly without considering the structural stiffness of the mechanism.

Then the actuated forces/torques can be yielded as [3537]

w 6

where is the vector of the actuated forces/torques of the actuated joints, i.e., 1 2 n T,w is the vector of the magni-tudes of the actuation wrenches $ a,i or $ a,j, i.e.,w w1 w2 wn T, and is the diagonal n nmatrix, which is given as

$^1$^a;1 0

0 $^n$^a;n

2664

3775 7

where represents the reciprocal product.

34 Y. Xu et al. / Mechanism and Machine Theory 88 (2015) 3148Fig. 2. 2-UPR + SR two-rotational overconstrained PM.

Sinsuppo

Whjoints

Heelastictuatio

35Y. Xu et al. / Mechanism and Machine Theory 88 (2015) 3148overconstraint wrenches belong to the stiffness decoupled limbs, such overconstrained LMPM is called as limb stiffness decoupledoverconstrained LMPM, such as 2-UPR + SR two-rotational overconstrained PM.

The second class of supporting limbs' feature is that its end would generate elastic deformations along the axes of theoverconstraint wrenches under the action of the actuation wrench or non-overconstraint wrenches, this class of supportinglimbs are called as stiffness coupled limbs. When there are one or more supporting limbs that contain overconstraintwrenches of the overconstrained LMPM belong to the stiffness coupled limbs, such overconstrained LMPM is called as limbstiffness coupled overconstrained LMPM, such as 3-PRRR orthogonal 3-dof translational PM and 5R spherical 2-dof rotationalPM.

In the following, the concepts of the limb stiffness decoupled overconstrained LMPM and the coupled one will be explained inmore detail combining the specic examples.

3.1.1. Limb stiffness decoupled overconstrained LMPM2UPR + SR two-rotational PMThe 2-UPR + SR two-rotational overconstrained PM, as shown in Fig. 2, is composed of a moving platform, a xed base, two UPR

supporting limbs, and a SR supporting limb. The limb UPR connects the moving platform to the base by an R joint on the movingplatform at Ri (i= 1, 2), an actuated P joint along the limb, and a U joint on the base Ui in sequence. The axis of the U joint withinthe U1P1R1 limb connected to the base is denoted by r1, and that close to the moving platform is denoted by r2. The axis of the Ujoint within the U2P2R2 limb connected to the base is denoted by r3, and that close to the moving platform is denoted by r4. Andthe SR limb connects the moving platform to the base by an R joint on the moving platform at R3, and an S joint on the base at S insequence.

For the UPR limb, the axis of the R joint is parallel to the axis of the U joint close to the moving platform, the axis of theactuated P joint is perpendicular to the axis of the R joint, the axes of the R joints of two UPR limbs are parallel to each other,and r1 and r3 are coincident. The axis of the R joint of the SR limb is perpendicular to the axis of the R joint of the UPR limb.The three points, S, U1, and U2, on the base constitute an isosceles triangle with SU1 = SU2, the length of the bottom side U1U2is equal to 2a, and the length of the height SA is equal to h1. Another three points, R1, R2, and R3, on the moving platform alsoconstitute an isosceles triangle with R3R2 = R3R1, the length of the bottom side R1R2 is equal to 2b, the length of the heightR3B is equal to h2. When the mechanism is in the initial conguration, the points R1, R2, U1, and U2 are located on a sameplane, and the plane is perpendicular to the planes SU1U2 and R1R2R3, the vector AB is also perpendicular to the planes SU1U2and R1R2R3, whose length is equal to l0, just as shown in Fig. 2.

For the purpose of analysis, a xed coordinate frame A: xyz is attached at point S on the base, with x-axis pointing along vec-tor SA and y-axis parallel to vector U1U2. A moving coordinate frame B: uvw is attached at point R3 on the moving platform,with u-axis pointing along vector R3B, v-axis parallel to vector R1R2, and w-axis perpendicular to the plane R1R2R3, just asshown in Fig. 2.

Here, zyx Euler angles , , and are used to describe the orientation of the moving coordinate frame Bwith respect to the xedcoordinate frame A, which can be expressed as

ABR ; ; Rot z; Rot y; Rot x;

cc csssc csc sssc sss cc ssccss cs cc

24

35: 8

Assume that theposition vector of the coordinate frame B's originR3 in the coordinate frameA is x0 y0 z0 T, and then the lengthof the two actuated UPR limbs can be derived as

l1 bss0 2 a bcy0 2 bcs 2

ql2

bss 2 abcy0 2 bcs 2

q8