Scholarly article on topic 'VRM: A Unified Framework for Closed-form Solutions of a Special Class of Serial Manipulators'

VRM: A Unified Framework for Closed-form Solutions of a Special Class of Serial Manipulators Academic research paper on "Mechanical engineering"

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Academic research paper on topic "VRM: A Unified Framework for Closed-form Solutions of a Special Class of Serial Manipulators"

International Journal of Advanced Robotic Systems

ARTICLE

VRM: A Unified Framework for Closed-form Solutions of a Special Class of Serial Manipulators

Regular Paper

Xijian Huo1, Yiwei Liu1*, Li Jiang1 and Hong Liu1

1 State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin, China Corresponding author(s) E-mail: lyw@hit.edu.cn

Received 16 January 2014; Accepted 11 December 2014

DOI: 10.5772/60055

© 2015 The Author(s). Licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper proposes the virtual reconfiguration method (VRM) to construct the unified framework for closed-form solutions of a special class of serial manipulators. Central to the research is the inverse kinematics problem (IKP) of 6- and 7-DOF serial manipulators, which contain either the Pieper's geometry or the Duffy's geometry. Given the desired end-effector pose of the manipulator, a virtual single chain (SLC) is developed by connecting the base and the end-effector with a hypothetical link. The equivalent single open chain (SOC) with different configurations can be obtained by cutting open the virtual SLC at one link between adjacent joints. Kinematic equivalence between the original manipulator and the new SOC is proven. Closed-form solutions of the original manipulator can be determined by solving the IKP of the equivalent SOC. The VRM is further developed on the basis of the relationship between the manipulator and its equivalent SOC. In this paper, the IKPs of 6-DOF manipulators with the spherical wrist and manipulators with the three-axis parallel shoulder joint are analysed. Principles and applications of the VRM are proposed. Finally, the validity and efficiency of the VRM are demonstrated by kinematics simulations of four different manipulators. Unlike traditional approaches, the VRM simplifies the computation of IKPs and estab-

lishes a unified framework for closed-form solutions of the special class of 6- and 7-DOF serial manipulators, irrespective of the allocation of either the Pieper's geometry or the Duffy's geometry.

Keywords Virtual Reconfiguration Method, Closed-Form Solution, Reconfigurable Robot, Single Loop Chain, Equivalent Single Open Chain

1. Introduction

Serial manipulators are robots in a sequence of links and joints that begins at the base and ends with an end-effector. They are widely used in industry, humanoid robots and space exploration. The inverse kinematics problem (IKP) of serial manipulators is the most important aspect of robot analysis and motion planning. It concerns determination of joint variables corresponding to the relative pose between two members of the manipulator. In this paper, the IKP of a special class of 6- and 7-DOF serial manipulators, which contain either the 3-DOF spherical joint (Pieper's geometry) or the three-axis parallel geometry (Duffy's geometry), is proposed. In order to obtain inverse kinematic solutions of manipu-

lators, many methods have been developed based on different representations of position and orientation and their derivatives with respect to time [1, 2]. In general, the IKP of the special classes of manipulators can be solved with universal methods and particular methods, respectively.

Numerical methods are based on the computation of nonlinear kinematic equations in the position domain. Approximate inverse kinematic solutions can be determined using numerical methods such as interval methods, continuation methods and iterative algorithms [3, 4], etc. Numerical methods are independent of robot morphology and are applicable to robots with any kinematic structure. Nevertheless, a trade-off exists between the computational complexity and the solutions' precision. Also, it is impossible to find all kinematic solutions corresponding to the specified end-effector pose. Besides, the IKP of manipulators can be solved on the basis of inverse instantaneous kinematics. Typical methods are the simple Jacobian-based method and other extended methods including closed-algorithms, Jacobian transpose methods, and so on [2, 5]. The velocity-based methods are applicable to manipulators irrespective of their geometry. However, these methods have intrinsic inaccuracy because of the linear approximation of the Jacobian matrix, especially in the near region of singularity. These methods are also dependent on the initial conditions and do not consider the feasible region of solutions. In addition, artificial neural networks and numerous intelligent algorithms are also capable of computing inverse kinematic solutions [6, 7]. These methods have excellent capabilities of parallel computing, and do not need an explicit kinematic model. But they are usually required to learn online or off-line from kinematic samples. The precision of the results is inadequate for some sensitive tasks.

Particular methods for approaching the IKP of manipulators are based on direct mapping from the Cartesian space to the joint space. Inverse kinematic solutions can be expressed symbolically using algebraic or geometric methods [1, 8]. Closed-form solutions are more efficient than numerical algorithms and readily identify all possible exact solutions leading to the desired terminal position and orientation. This indicates the desirability of closed-form solutions above other universal approaches. However, closed-form solutions depend on the robot morphology, and they only exist for a few types of robots, including six-degree-of-freedom (DOF) serial manipulators with three adjacent revolute joint axes intersecting at a common point (Pieper's geometry or spherical joint), or manipulators with three consecutive revolute joint axes in parallel (Duffy's geometry) [9, 10]. Manipulators with the spherical wrist joint or the three-axis parallel shoulder joint are desirable because the IKP can be solved easily by kinematics decoupling. Nevertheless, for general manipulators including humanoid arms with the spherical shoulder joint or space robots with the Duffy's geometry locating in the middle

part, it is difficult to derive with closed-form solutions by solving high-degree polynomial equations. The IKP of these manipulators simply resorts to numerical algorithms or intelligent approaches.

A few studies have been carried out for closed-form solutions for these types of manipulators. In [11], closed-form solutions of the 7-DOF humanoid arm are expressed by fixing one redundant joint. In [6], the approximate closed-form solutions of the inverse kinematic model are obtained using evolutionary algorithms and genetic programming technology. However, the establishment of the approximating models is time consuming, and the results are inadequate for fine manipulations. In [12], an analytical method is used for the 7-DOF humanoid arm with a spherical shoulder joint, and closed-form solutions are derived using the virtual joint method. However, the seventh joint angle needs to be given before other solutions are obtained, and the method is invalid for 6-DOF serial manipulators. The reverse-decoupling method is presented for humanoid robots with the spherical joint, and the IKP is approached using algebraic algorithm and matrix transformation [1]. However, there is no in-depth study on the principle of the reverse-decoupling method. In [13], the reverse-coordinates method is used for closed-form solutions of 6-DOF manipulators with the SRRR configuration. But this method is only applicable to manipulators with spherical shoulder joint. There is no unified framework for the IKP of manipulators satisfying the Pieper's condition and manipulators satisfying the Duffy's condition.

This research is motivated by closed-form solutions of a class of serial manipulators with the special geometry. This paper is organized as follows. The IKPs of two kinds of typical manipulators are analysed in Section 2. Principles and applications of the VRM are proposed in Section 3. The validity and efficiency of the VRM are demonstrated by case studies of typical manipulators in Section 4. Conclusions are presented in Section 5.

2. IKPs of Typical Manipulators

2.1 Manipulators with the Spherical Wrist

Pieper's geometry is equivalent to the spherical joint (S, which can achieve rotations in three different directions. Manipulators are usually designed with a spherical wrist to achieve better performances of the end-effector. The spherical wrist is widely used in industrial applications.

Suppose that the general 6-DOF manipulator contains the revolute joint (R) and the prismatic joint (P). The manipulator with the RPRS arrangement is shown in Figure 1.

Let 'Ai be the homogeneous transformation matrix describing the pose of Frame Oixyz with respect to Frame O^xyz. Point W is the intersection of the last three joint axes. The kinematic equation of the general 6-DOF serial manipulator with the spherical wrist can be expressed as

foR6(0'd) foP6(Vl'V2'V3)

where /0Rb0, d) e SO(3) is the function of all joint variables, and /opgVj, î;2, f3)eR3 it the function of tite first toee jo iic1 variab'es. O is thenero vectoc ; .hevariaMev,. Oependr on tits typeof joints. I'jond i' is revolute, V'j-. ^Ifiioini; i ie prismatic, oi=isi.

tnd-effector

point W-'

sp hcricalwrist

F^ure 1. (3eneralRPRS manipulator iciththe sphericalwrist

In brieg forihe general6-DOF secieil manipulate)s" wiih the spheeicel wrist, the p ositioa eel ido wristg oinl W drpends only on She fi-st three joint varisping . The spWaaiarl wrist is uaedloadiusithe terminalnrientation. Given the deaserd end-ndfector posa. closed-form rnhitiona ran be odlainrd iG twa slope: t6o firstrhf eoloinrvariablaa ace derive d fanmthe poaitirnrbassp eqwation s^/ rael Chen tho

spherical wristangles range aoivedbased onrgeerifdtr tion-based equation f 0R6(S, d).

2.2 Manipulators withthe Duffyshoulderjoint

Duffy's geometry applies to the three-axis parallel planar rotot ovhitli ran acMeve tppl-dimanfipnol motioas. pn o^iier to smaplifyfiiamverre kidematics. it des1 gp rd as tha shn u lder jo^r^t of m anipulatoea. Simitary. tha Gannsal 6eDOF man^ula^or with the -ffiPR codfi!sirr-tieo is shawn mFGpure y.

Duffy shoulder yt|joint 1

joint 6

end-effector

Figure 2. General 4RPR manipulator with a Duffy shoulder joint

According to the forward kinematics, the end-effector pose 0A6 can beexpressedas

fn ( v4, v5,v6 0

) fo (^V5>Vt 0

) fa (V4, V5, V6 ) fp (V4, V5, V6 )

where the elements denoted by * are omitted, which vary with all joint variables. fn(v4, v5, v6), fo(v4, v5, v6), fa(v4, v5, v6) and fp(v4, v5, v6) are functions related to the last three jointvariables.

Therefore, for the general 6-DOF serial manipulator with the Duffy shoulder joint, the position and the orientation of Frame O6xyz alongthe z direction of Frame O0xyz depend only on the last three joints. Given the desired end-effectora pose, closed-form solutions can be obtained in two steps: the last three joint variables are derived from the third row of 0A6, and then the shoulder joint angles can be computed fromotherrowsof 0A6.

3. Theory of the VRM

3.1Backgrounds

As mentioned above, for the general 6-DOF manipulator with the spherical wrist or the Duffy shoulder joint, closed-. form solutions can be directly obtained by decoupling certain elements of the end-effector pose. Furthermore, the general 7-DOF manipulator with these special geometries can be simplified as the 6-DOF robot by fixing one redundant joint, and the IKP can then be solved analytically [11]. Nevertheless, in order to meet special requirements of operations and environments, different types of serial. manipulators are developed which satisfy the Pieper's or the Duffy's condition. However, these manipulators are not equipped with the spherical wrist or the Duffy shoulder joint. For instance, humanoid arms with the spherical shoulder joint and humanoid legs with the Duffy's geometry in the middle part can better emulate human functions [1]. Space manipulators are usually designed as symmetrical robots with the Duffy's geometry, so they can be folded for delivery to ISS [14-16]. Though closed-form solutions for these manipulators exist, it is difficult to derive the solutions directly since all elements of the position and orientation are coupled.

This paper presents a unified procedure for solving the IKP of a special class of the 6-DOF and the 7-DOF serial manipulators which contain either the Pieper's geometry ortheDuffy'sgeometry.

3.2AnalysisofKinematicEquivalence

Manipulators are kinematically equivalent if they have the same kinematic relationship between joints and links, inherently. Kinematic equivalence between the general serial manipulator and its reconfigured form is proven as follows.

1. ForwardKinematicEquivalence

In Figure 3(a), the general n -DOF serial manipulator with ap closed is lillustrated. Given the specified jointl variables, thel position and orientation of links are uniquely determined.

Basedon the closed- robot, an n-DOF reconfigured manipulator "Robot-A" with an open loop is developed by cuttingopaetRaklnematic ohain atiinkn, as shownty Figure 3ib ).abt otertRb sai^i^aloaaRia^u^ic^c^r, ^li^ea^aolitoR aid orientation of "eks aab o^^ter^i^ecad Cy jt^^a^r voaiablos. ainceai) jomt vbriablesremain unchabgeC, ^l^t^^^^e no joinl motion! during "he reae^nF"b^ra1^t^n orocrao ibbs, given the specified jomt variables, the position and orient ationofthe cofresprndmgelementsmtffe^rtesmnspace are ^entiral bttween tiie two mtnipulatoca. The original clnsetWoop maniputator ic forward-kmematicbUy oqmva-lentto ite reconfi^red fono wrtha ckuccdkiop.

Figure 3. Kinematic chain of the general manipulator: (a) the closed-loop manipulator; (b) Robot-A; (c) Robot-B

U"noiher aeconfieured manipulaeor/'RobotuB", can also be Batained by cutting open the kinematic chain at Link i, as ehowoin ba"oreЗec). SCmiiarly, given taebpeciiieO iaini eariebier,ihe Piscoo" atrrecpoRaCng alemente inkhe Cartesian space are the same between these closed-loop manipulators and Robot-B. This implies that forward-kinematic equivalence exists between the two serial manipulators.

According to the above analysis, the two different reconfigured manipulators with the open loop are forward-kinematically equivalent because of the identical pose of corresponding elements in the Cartesian space corresponding to the certain joint variables. On the other hand, Robot-B can be developed by connecting two ends of Robot-A with a hypothetical link and cutting open the kinematic

^Ihain atLinki, Robot-Bcanbe Beemedone oftherecora

coarludedthattRegeneralsrrirlmaRipulaBrr is rrt^r^^rd-l^lneie^c^i^^Cc^^^U'eBu^^aCi^r^t to itsre uonfigure dmae^^j^ir^c^lisr ortiCRi anopealorp.

Cd orRar to velldalc the forwavt n^r^err^etic equivalange, tia kiRomotics oO PRMA 5CC rsnstablished. DH coordinate frames of the robot are established in Fig 4(a), DHparam-eCers Eire listed 2n Cable 1. Oased oa ikis, the kinemaCic eRnatcod legits

^s0^22Bt 3cj44M5s I46 th)

The ilsconfigured robot can be obtained by connecting two en) soffheeyOo) wt^fiehpriothettr^l^^i^lrand cutfo^gP]eay l^^kia^^e-tc chainrl Oinb 2 . bhe vlr5upllremes(Brcmeoy annFrhmeve)are prPltraryand(CenSiehl. DHtRO-aikais frailer acn eR:aBlirhekas shown inFig4lbS. Rha kaiemctic equation of the reconfigured robot is expressed as

Figure 4. DH frames of robots: (a) PUMA 560; (b) the reconfigured robot

link ai «i di 0i

1 0 90° Li 0i

2 L3 0° - l2 02

3 L4 90° 0 03

4 0 -90° L5 04

5 0 90° 0 05

6 0 0° L6 06

Table 1. DH parameters of PUMA 560

Based on (2) and (3), it can be seen that PUMA 560 and the rec onfigured robot have the same kinem atic equ ation. This demonstrates that the reconfigured robotis forward-kinVma tically equivalent to PUMA 560.

2. Inverse Kinematic Equivalence

Inverse kinematic rquinalnnce begween the origmal manor^liarorc^i^ii tlr^ epen- rerobfigured manipulator is krnven using proof by contradiction.

Suppose that 1^={01, 02, ..., 0m} is the set of inverse kinematic ^(^^^l^ra^^s for the anginal manipulatoo iorre' oponVing to r apecitieiVi^ekl^TCkiJc^E^r of all iinks mlhe Vartesiart sgace^, and ke{<Oa cao..., 0n} is the set of inneVae kinematic solutions for the reconfigured manipulator corresponding to the same relative pose in the Cavteoian apace. In addition, make cn acaumptioa tlsnt ghe origiorl maniprlcker is a^c^t napase-gioematic^^ln Cenlto ^k^ere^c^on^^jgeedmcnapulrtcn. TV-si,

Ca° , ©^Y -I = 1,2,...?M)

The original manipulator and the reconfigured manipulators have identical kinematic singularity. The spherical joint at a singularity is illustrated in Figure 5(a). An arbitrary reconfigured form of the spherical joint is devel-oped,aa ahowninFigure5(b). It canbefoundthatthe two meftipulatora have identical characterittiaa alatheain^ilea-ity. AlaO1 ton uqmbaknca o( tm^tatitiea aab hevaliha-ea Isytice kmematine-juationa from (2)nndt3i.

joint 3

joint 3

Figure 5. Kinematic singularity of the 3-DOF manipulator: (a) the spherical jointat singularity; (b)the reconfiguredmanipulatorat singularity

Caae II: 3©., ©. (' = 1,2,...n)

If Case I is true,the reta tive p oae o f correiponding elements kitVao riginal manipulator is different from that of the reconfigured manippiatca correvpondiag -o tVo joint ^r^i^^c^f^iae ajiThic ocso contradirtethe focwefn-nCkomatlq kaui^elenckcafthe two manipulntor8.Henck, thaas-ump-I^^os^ (Cocc I) ia fake. II can 8r aon^^uCra Chat

hOi^^ o ©¡e e rr'=fl,2 ,...m), sndman.

Cimilrcld, if Caoe It. i s ttuef the relative poar oi corrvof^ank-8ng eiemanfsof rnorecenngPfeCmanionletoo iodV:reoenC ieom -hee of -do angaral manipulaiot sorreoponVmg to ©<•< Thikcase contradiiia the forwa-d-ainematicequical lence of the two manipulators. Hence, Case II is false. This implies that

V ae ' eY ^ Q. eY (j = 1,2,...n) ,and n < m. (4)

Based on the above analysis, inverse-kinematic solutions W of the reconfigured manipulator are the same as kinematic solutions W of the original manipulator. That is, the inverse kinematic equivalence exists between the original manipulator and the open- reconfigured manipulator.

Therefore, it is proven that the original manipulator is kinematically equivalent to its reconfigured form.

3.3 PrinciplesoftheVRM

The serial manipulator with an open kinematic chain is termed the single open chain (SOC), and the manipulator with a closed chain is termed the single chain (SLC). As mentioned above, given the specified joint variables, the end-effector pose with respect to the base frame is confirmed, and a stationary constraint exists between the base and the end-effector. Assuming that one virtual connection is built between the base frame and the end-effector frame (EEF), the virtual SLC is developed to be kinematically equivalent to the SOC. At this point, a new equivalent SOC with different configurations can be obtained by cutting open the virtual SLC at one link between adjacent joints. Various equivalent SOCs can be gained from the same manipulator by selecting different breaking points. The equivalent SOC also varies directly with different poses of the end-effector. Hence, the equivalent SOC can be deemed the variable-geometry robot.

With respect to the specified end-effector pose, the equivalent SOC has the same inner relationship between joints as the original serial manipulator. Hence, by cutting open the connection before the Pieper's geometry, manipulators with one internal spherical joint can be transformed into the equivalent SOC with the spherical wrist. Likewise, manipulators with the Duffy shoulder joint can also be derived from robots which contain the three-axis parallel structure. The theory to change the configuration of manipulators through the equivalent SOC is termed the virtual reconfiguration method (VRM). Based on this, reconfigurations of the manipulator with the Pieper's geometry and the

manipulator with tine Duffy's geometry are performed respe cti vely.

A hkin^m^ic cheinwithiht rjiherical joint isihooci

in Figure 6. The bnse irnme anih tiie EEF are esfablished^nh thce the SLC is denilipbd by efrfuaily coienkckinij ifio^a ^veo^i^a^niiep. In osCer Pof)bPain a new POC with thk spheiicai wrist]Pin0 h ie bsoken "ntotwo oarfu Lmk t' tniil bloP 4'' . LinkC' roiin^et^ wieP Joinl: W and liikik 4" connects with lr ant 4. Thevirtual cinsu frcoet O^x^ o btuharbrtcarily at; tfia brea.ing point: on Link 4:, arc0 the ^rtfa! ""Eu O^xyz's fixed to the breaking point on Link 4''.

Theposo of foam! Oveoez wiitUre stent eoyoamk Oureyi can be expreridi^si

*a 4 a5aeAbA°A1A2A3A 4" A

J-1,, J-1, j\,„ J-l

Figure 6. Kinematic chain of the manipulator with the spherical joint: (a) the original SOC; (b) theequivalentreconfigured SOC

The relationship between Frame O4xyz and Frame O4„xyz can be derived from (4). Because Frame Ovbxyz coincides with Frame Ovexyz, 4A n is simplified as

''A -4A5AeAbA°A1A2A3A

A4" = A5 Ae Ab A° A1 A2 A3 A4

According to (5), the new equivalent SOC is equipped with the spherical wrist. Since the homogenous transformation matrixes bA0 and 4'A4„ depend only on the geometrical structure of manipulators, given the desired end-effector pose, closed-form solutions of the manipulator can be solved directly with traditional methods.

Thv 6fDOF kinematic chain with the three-axis parallel struoOireis shiim in hiiire h( himilaeiy, the LLC is devehopcd ttr( vlrkbaily connecting the base frame and lhe EEF]Inordetto obtain newcqoivaient Oertf withthe Duffy shoulder ping Link lie broken ihto ewo fartr: hink 1' and UInkm Link S' cinnerts osilin iornf 2, ynd nink 1'' cormecfn wito ^oini i: TheuirtualfbameOabXyr is buiik at the l^rekk^nn hotfrt neLmk 1', and ehe virtuo10EF O^xyr is fixth to tin leeeskmy poicl: on link Fsame Ovecyz coin ci des with Frame Ovbxyz.

Figure 7. Kinematic chain of the manipulator with the Duffy's geometry: (a) theoriginalSOC; (b)theequivalentreconfiguredSOC

The pose of Frame Ovexyz with respect to Frame Ovbxyz can ibe obtained by rcomposing the homogeneous transformation, and the relationship between Frame Orxyz and Frame Orxyz canbe simplifiedas

1 A = 1' A 2A 3A 4 A 5 A eA bA °A

1 2 3 4 5 e b ° 1

Therefore, the new equivalent SOC is equipped with the Duffy shoulder joint. Since the matrixes bA0 and 1'A1. depend only on the geometrical structure of manipulators, given the desired end-effector pose, closed-form solutions can be obtained using traditional methods.

3.4 VRM Algorithm

Corresponding to the end-effector pose, the VRM can be used to transform manipulators into robots with different configurations. It is worth sketching the procedure to

compute closed-form solutions for serial manipulators satisfying the Pieper's condition and serial manipulators satisfying the Duffy's condition. In general, the algorithm is implemented in six steps, as follows.

Step 1: Establish the DH coordinate frames for the base and the end-effector;

Step 2: Build the virtual SLC by connecting the base frame and the EEF;

Step 3: Determine the breaking point on one link according to the geometry of the manipulator. For instance, for manipulators with the spherical joint, the breaking point is located on the link after the spherical joint.; for manipulators with the Duffy's geometry, the breaking point is fixed to the link before;

Step 4: Establish the DH coordinate frames for the equivalent SOC;

Step 5: Compute the kinematics function by composing the homogenous transformations;

Step 6: Solve the IKP of the manipulator analytically using traditional methods.

determine the best of all feasible solutions, typical criteria are defined as follows.

1. Manipulability w-^v)

( det ( J ( v ) JT ( v ))

Manipulators move away from singularities by maximizing w1(v).

2. Distance from joint limits w2(v)

v. - Vi

■(v )-n S

where vt Max (vi Min) denotes the maximum (minimum) joint limit, and V, is the middle value of the joint range. Manipulators move towards the centre of joint ranges by minimizing w2(v).

3. Distance from previous solutions w3(v)

3.5 Discussions

The VRM is developed based on the kinematic equivalence between the original manipulator and the equivalent reconfigured manipulator. The reconfigured manipulator is a subset of the reconfigurable robots because of the changeability of geometry. The comparisons between the equivalent reconfigured manipulator and the general reconfigurable robot are summarized as follows.

1. The general reconfigurable robot consists of a collection of interchangeable modules. According to the requirements of specified tasks, various robot geometries can be developed based on different selections and arrangements of the modules. Because the existence of closed-form solutions depends on the robot morphology, the IKP of reconfigurable robots is generally solved using numerical approaches [17-19].

2. The equivalent reconfigured manipulator concerns on a special class of serial manipulators which contain the Pieper's geometry or the Duffy's geometry. Once the manipulator has been reconfigured, the geometry and kinematic relationship are completely expressed in this manner. Besides, the reconfigured manipulator in this paper is kinematically equivalent to the original manipulator, and closed-form solutions can be obtained directly.

Another aspect to be considered for manipulators is the kinematic optimizations. Inverse kinematics involves one-to-many mapping from the Cartesian space to the joint space. For instance, given the specified terminal pose, there are at most eight solutions for the 6-DOF manipulator with either the Pieper's or the Duffy's geometry [20]. In order to

w3( v )=- n S( vi- v02

where v denotes the previous joint variables. Manipulators move with little change in joint variables by minimizing w4(v ).

4. Distance from an obstacle w4(v)

w4(v)=PminJp (v)- p (o )ii

where p(o) is the obstacle position, and p(v) is the generic point position along the structure. Manipulators move away from the obstacle by maximizing w4(v).

As mentioned above, the specified criteria are considered according to the requirement of the operations. The optimal joint displacements can be determined from all feasible solutions.

3.6 Applications

The topological structure describes a group of mechanisms which have the common characters between adjacent members. In this section, the topology structures of serial manipulators which contain either the Pieper's geometry or the Duffy's geometry are illustrated.

The topological structure of manipulators can be exactly expressed with three basic elements: the type of kinematics pair, connection relations between structural units, and the dimensional constraint type [21]. The prismatic joint and

v — v

i.Max i ,Min

therevolute jointare tge typical]idnematics pairs. Thebasic dimensional constraint types are defined as follows.

a. e^tr^C^pithra^oabjc^aOTt tc^iee^m^bt gaia (Ror b^t^taes mtarsecaing at one caaemon poP-I ^ espaessed as

b. SOC aei^°a^^b^aaaacepti^lnem^t^c^i;i^i^tg(R or P) axes bamg^raUelto each other is expressed as

POCt-R^rP^/R^.g)-};

c. SOCwlfhtwoadcacIn[kmsmstias p air (R or P) axes in the generat configuration is expressed as

SOCt-R(orP)-R(°rP)-}.

The topotugy structure ci) the gesmR0-fOLd t-e

Baliy's geometag cat be rapreaenteP as IRC{-RRR-} and SOC{-R/ /R/ /R-} respectively. The two types of manipulators with special geometry ensuring the existence of clo sed-form to}uttonsare-rc^пeaoizedasfollawc.

T-p- -:6DOFs opac- mumpulators containing P and R and rpaityiun thuPie6er'sconRition

Sua>f ssi tCaiCOs 6DOFs SLC—v^th tP^a CRepea's grametay a denotiacC as USC{nRtst Si-CRR-R CotSC"]^?^}, end the topology structure of the Type I manipulators can be obtained from the SLC by cutting open one linkage between any two joints except the Pieper's geometry.

Tgpell: S-TO-apec-lnopmagipulators containing P and Rand saiionyiogthrDrLtfy's crndition

The 6-DOF SLC which contains the Duffy's geometry is denoted as S]LR{sdlorP)sn/[R// RaR)orPCoR)orB) - i, nd the topology structure of the Type II manipulators can teatoamed frcmteieS^ bu CDttmgopen onthnkage b/Lween any two joints except the Duffy's geometry.

Bcircd -ntha ctsurrusrs cst^^T^^ tw/tgoTC af

mcnipulatOTa, many maaouu)atorrian ПedrveCope d dueto dllferentlink p/rctma^etas ^ndpmt ai/ocatidos[ecCC-Cei. TSs ^IM )s applic/aie to any mceapulator belonping to Tpce t ir^B Ty-e It iao anipu l atots. It LumpcO--isgs

af age nCPcr ronttracts t-a uni}iaU CramaworkIosc)osIU-formso)ufiPReforune special class of serial manipulators.

4. Case Studies

4.1IKPs ofhumanoidarms

aha 6-DOF rariai maplpp)ator is grnerally ased for hgmao -oid robots to be able to imitate human motions. The

Pum-noiUarm at Hugo KHR-4'WgicB Oas tgespherical shoulder joint, is illustrated in Figure 8.

Cuae 1:

The base frame and the EEF of the humanoid arm are established, and the virtual SLC is developed. Based on this, the equivalent SOC is obtained by cutting open the virtual SLC at Lmk 3. DH coordinate frames of the equiv-a{eRt SOC Rre eatnMiahed, 6nd Du( oaiamster8 ma as (c8teu

Figure8. TheequivalentSOCandDHcoordinateframes

Link ai «i di 0i

1 0 -90° L0 61

2 0 90° 0 02

3" 0 0° 0 63

3' 0 -90° -L1 0°

4 0 90° 0 04

5 0 -90° -L2 05

6 L3 0° 0 06

Table2. DH parametersof thehumanoidarm

The base frame coincides with Frame O0xyz and 3"A3' is a constant transformation matrix which can be derived from Table 2. Consequently, computations of the kinematic equation as in(5) yield

3A3„ = 3At4A55AeeAb°Ai1A22Ar (8)

Since the spherical joint is the wrist of the equivalent SOC, the position of Frame O1xyz viewed from Frame O3xyz depends only on the joint variable 04 5 6. The equation about

thewrist position can be derived from (7). Let [Px Py Pz]t be the end-effector position with respect to the base frame.

If Pl5x2+Pl5,y2 + Pl5,z2 - (L 1+L 2)2 * 0

P15,x2+P15,y2 + P15,z2 - (L 1- L 2)2* 04,5,6 ^Ms

9i=atan2( -girg3 )

05 = atan2( -f3' fi ) 0r 05 = atan2( f3' - f1 )

(9) (10)

96=atan2(p, 1-p2)-atan2(B,A) (11)

The parameters gv g3, f v f p, A and B are computed in the Appendix.

Otherwise, the kinematic singularity occurs. In ihis casts, sin04=O, and 05 can be determined directly according to some roles.

Supposing that R= bR,eC3,5R,4R 3R ,uubstsSottina (8)-(tU)

mto (78,solution s of the spherical joint can be derived. If sm02 * 0, the joint angles are given by

( = ataa2 (sign( sind2) R23,sign (sind2) R13) (12)

=2 =±R° iR 3, R

Q3 =atan2Cy sign(sin02C f32,-sign (sinde )R31) (14)

Otherwiso, iCe spgn r icnlloiiR is at asCr gular itR. In thi s case) 0j can be determined directly according to some roles. 03 is given by

3 Bsc (2)

alent SOC is obtained by cutting open the virtual SLC at Link 5. DH coordinate frames of the equivalent SOC are established, and DH parameters are listed in Table 3.

Similarly, 5"A5' is a constant transformation matrix which can be derived from Table 3. Computations of the kinematic equation as in (5) yield

'' A = 5' A e A 0 A 1 A 2 A 3 A 4 A A5" = A0 A1 A2 A3 A4 A5"

Since the spherical joint is the wrist of the equivalent SOC, the position of Frame O4xyz viewed from Frame O5xyz dependsonlyon the joint variable©! c 6.With reference to theabove nic^r^is,lho eeui^t^on oithewrisi pnsitiroo can pnc^tiriwd^^oto iSbt.h^i^ithn bederieed uce^gao aSotice^i mitho Ue.Fmally^losehsform silutions ofthn n^vn^qici^t iSenr reconfiouqeh menipdlatots ore ditermrned bet Oinomatic decoupling. Due to the limitation of cnace, manhemetinal inimu nt^ccnorf u^i^o^^cSha^nq^iilnO^i^^ ace not expressyUagain.

Figure 9. TheequivalentSOC and DHcoordinate frames

Link ai a di 9i

Finally, closed-form solutions of the humanoid arm are obtametl. Qventhi dyon"). endocA^^^st, the torre-spon ding joint angles can be determined directly.

Cnte 2:

Withrespect ^cii^tqa^re 9, ^nc^^hCT sphericai pint oxists n^dnhcons^f^ts^l^e third jniot, Sourthj ointanG SiSthp ini In osbobtovairdalethe uticersality nS tio VRM, anotnet equivhientreconttqbbod mempuietor is developeSbasoS on ihe numinniO aim. Dinirent^ Irom Case 1u fre equiv-

1 0 -90° L0 01

2 0 90° 0 02

3 0 -90° -L1 03

4 0 90° 0 04

5" 0 0° 0 05

5' 0 -90° -L2 0°

6 L3 0° 0 06

Table3. DH parametersofthe humanoidarm

In order to verify the validity of the VRM, kinematics simulations are performed. The joint range is from -180° to

—sin ¿n cos^co s02 -cosd1cosd2 — sinq

180° without joint limits. The link parameters of the humanoid arm are obtained from its specifications as:

L0=214.5(mm), Lj=179.14(mm), L2=181.57(mm), L3=80(mm). Let bXd e R 3 and bRd e SO(3) be the desired position and

orientation of the hand, respectively. Two sets of the hand pose are performed here. The first one (Pose A) is the general hand pose, while the second one (Pose B) is the pose at singularity.

Pose A: bXf = [0.2949 0.1659 0.3498]( m)

0.7557 -0.5755 -0.3125 0.5334 0.2642 0.8035 -0.3799 -0.7740 0.5066

4.2 IKPs of SFA

Duffy's geometry is the typical structure used in space robots, which can achieve a larger motion range and can be folded with minimal volume. Small Fine Arm (SFA) is one manipulator among the Japanese Experiment Modules [16]. The 6-DOF space robot with the three-axis parallel structure is shown in Figure 11.

Similarly, the equivalent SOC is obtained by cutting open the virtual SLC at Link 1. The DH coordinate frames of all links are established. DH parameters are listed in Table 5.

It is obvious that bA0 is the identity matrix and r'Ar is a constant transformation matrix which can be derived from Table 5. The computation of kinematic equations is specified as

Pose B:

bXX = [-0.1962 0.1133 0.5244] (m)

1 A -rA 2A 3A 4 A 5A eA 0A

1 2 3 4 5 e b 1

-0.7500 -0.0474 0.6597 0.4555 -5.7791 0.4336 0.5000 0.6124 0.6124

Closed-formsolutionsare completedbasedon the; analysis in Case S end Case 2. Eight sets; ef lolutlons ase oOtslneO lor Pose A, and four scalds o2 solutione 2os Pose 13, as illus-Table 4. Basod odtei2,o^p^^melklr^^o^c^1^^c so^isüons on 20a typ lt^2 cri^^rm menSioned in

Section 3.5.

Assordlng to TadSs 4, ithe^c^md-i^t^df^c^^^ddi^^Ê^dd^etted^g Sd cachse-utioneanbrderived.Errors between speetfiddpese rted somisuded jedi€- pre sdn^]3itt(^d whieh aoe a2- laso ihun 5e-t4. The verifitaOion prooess is i)li2sOrateO in Ftgтrolh, d-lfScl] rs-wt tidal cioeohlfomt sotuiions aasureteld sorse. eprnd Cos ditiieo tho dsnes^l tesminot post oe tlsa rdooral teae at kinemstist sinopaoritie2. Hence, th^ \tlM SsvaTit tololoe las' KP ofmanlpelators satisoylngtiie Pseper'scondition.

- ttart)

Figure 11. The space robot and DH coordinate frames

Since the Duffy's geometry is the shoulder joint of the equivalent SOC, the wrist pose in z direction of the base frame depends only on 015 6. The first joint angle is expressed as

0 - atan2

(L3, ± yja2 +b2 -L32 ) - atan2(b, a)

If I aycos011 * 1, 05 and 06 are given by

0. - ±acos (a cos0. - a sin0. )

5 V y 1 x 1!

6>6 - atan2(sign(sin65)(oycos61 - oxsin61 ),

Figure 10. The validation process for kinematic solutions

- sign(sin65) ( nycos61 - nxsin61 ))

Pose Solutions Joint 1 Joint 2 Joint 3 Joint 4 Joint 5 Joint 6

solution 1 45.0000 -150.0000 30.0000 60.0000 -45.0000 120.0000

solution 2 45.0000 -150.0000 -150.0000 -60.0000 135.0000 120.0000

solution 3 -135.0000 150.0000 -150.0000 60.0000 -45.0000 120.0000

Pose A solution 4 -135.0000 150.0000 30.0000 -60.0000 135.0000 120.0000

solution 5 56.0101 -106.0380 127.0661 60.0000 -135.0000 165.9475

solution 6 56.0101 -106.0380 -52.9339 -60.0000 45.0000 165.9475

solution 7 -123.9899 106.0380 -52.9339 60.0000 -135.0000 165.9475

solution 8 -123.9899 106.0380 127.0661 -60.0000 45.0000 165.9475

solution 1 0 180.0000 124.4395 60.0000 -45.0000 45.9475

Pose B solution 2 0 180.0000 -150.0000 60.0000 -135.0000 90.0000

solution 3 0 180.0000 30.0000 -60.0000 45.0000 90.0000

solution 4 0 180.0000 -55.5605 60.0000 135.0000 45.9475

Table 4. Computed solutions of the humanoid arm (unit: degree)

The parameters a and b are computed in the Appendix.

Otherwise, sin05=O, and 06 can be determined directly according to some roles.

The space robot is now simplified as a planar robot including Joint 2, Joint 3 and Joint 4. Closed-form solutions of the 3-DOF planar robot are given by

02=atan2(Hsc2 (l)(2))

Similarly, two sets of the desired end-effector poses are examined. The first one (Pose A) is the general hand pose, and the second one (Pose B) is the pose at singularity.

Pose A: bXf = [0.5067 -0.2880 0.8801] (m)

-0.0388 0.2977 0.9539 -0.5388 0.7977 -0.2709 -0.8415 -0.5245 0.1294

L 2-L2-L2 2L1L2

Pose B: bXd= [—0.4966 0.2610 0.6667 ]( m )

6i=atan2(Hsc4 (1)Hsc4 (2))

—0.5000 0.8660 0 0.0000 —0.0000 1 0.8660 0.5000 0

The parameters L SW, Hsc2 and Hsc4 are illustrated in the Appendix.

Finally, closed-form solutions of the 6-DOF space robot are obtained from (17)-(22).

In order to demonstrate the validity of the VRM, kinematics simulations are performed. The joint range is from -180° to 180° without joint limits. Link parameters of the robot are expressed as follows: L0=200 (mm), Lj=435 (mm), L2=435(mm), L3=180(mm), L4=180(mm), L5=441(mm).

According to (17)-(22), eight sets of solutions are obtained for Pose A, and six sets of solutions for Pose B, as illustrated in Table 6. Optimal kinematic solutions can be selected based on the typical criteria mentioned in Section 3.5. With reference to Figure 10, the errors between specified pose and computed pose are obtained, which are less than 1e-14. Closed-form solutions are accurate corresponding to either the general terminal pose or the special pose at kinematics singularities. It is verified that the VRM is valid for the IKP of manipulators satisfying the Duffy's condition.

4.3 IKPs of 7-DOF manipulators

7DOF manipulators are redundant manipulators. There are infinite inverse kinematic solutions corresponding to one specified end-effector pose. The non-redundant manipulator can be developed by fixing one of the joints. In this section, the 7-DOF humanoid arm with the offset wrist and the space manipulator SSRMS is examined.

Figure 12 shows the equivalent reconfigured manipulator of the humanoid arm. By parameterizing the seventh joint of the humanoid arm, closed-form solutions can be derived using the VRM. The equivalent SOC is obtained by cutting open the virtual SLC at Link 3. The computation of kinematic equations is specified as

A - 3A 4 A 5 A 6 A e A 0A 1A 2A

/ln„ — /1. / 1 jTl, /1, /ln

3 4 5 6 e b 1 2 3

Based on (23), parameterized solutions can be obtained. Let [ Px Py Pz]t be the EEF position with respect to the base frame. If bP (1)2+bP (2)2+bP (3)2-(L +L ew)2*0 or

15 15 15 v se ew

bP 15(1)2+bP 15(2)2+bP 15(3)2 - (L se - L w )2 * 0, 04,5,6 yields

04-atan2( -gt,g3 ) (25)

05 - atan2 ( flrf3 ) or 05 - atan2( ff ) (26)

06 - atan2(p, ±yj 1-p2 ) +atan2(C, A) (27)

Otherwise, the spherical joint is at the singularity. In this case, 01 can be determined directly according to certain principle. 03 is given by

03-am2( Bc 1B (2))

sin61 cos61cos62 -cos61 sin61cos62

Finally, closed-form solutions of the 7-DOF humanoid arm are obtained. Given the desired end-effector pose, the corresponding joint angles can be determined directly.

In order to demonstrate the efficiency of the VRM, closed-form solutions of the parameterized humanoid arm are carried out in Matlab 7.1 environment running on an Intel(R) Core(TM) 2 Duo CPU 1.83GHz PC. At the same time, the IKP is also solved by the well-known Newton-Raphson algorithm and the simple Jacobian-based method. The position error between the desired end-effector pose and the computed pose is defined as 0.001(m). Given 20 sets of arbitrary end-effector pose in the workspace, the efficiency of each method is illustrated in Figure 13. The comparisons demonstrate that the VRM is valid and efficient to solve the IKP of the manipulator analytically.

Similarly, the equivalent SOC of SSRMS is developed by cutting open the virtual SLC at Link 2, as shown in Figure 14. The computation of kinematic equations is specified as

The parameters g1, f 1, f 3, p, A and C are computed in the Appendix.

Otherwise, the kinematic singularity occurs. In this case, sin04=0, and 05 can be determined directly according to certain principle.

Supposed that R=bR eR 6R 5R 4R 3R , the solutions of the

rr e 6 5 4 3' 3"'

spherical joint can be derived. If sin02 * 0, the joint angles are given by

'■A -2' A 3A 4 a 5A 6A eA 0A 1A

2 3 4 5 6 e b 1 2

By parameterizing the first joint of SSRMS, closed-form solutions are derived using the VRM. The second joint angle is expressed as

02 - atan2(p, ±71-p2 ) + atan2(B,A)

01-atan2(sign( sin02 ) Resign (sin02 ) R13 ) (28) If 1 ^o^rk^^+V^0^0 1 *1 06 ^ 07 are given by

02 - ±acos ( -R3,3 )

06 - ±acos(azcos02 - (axcos01 +aysin01 )sin02) (34)

03 - atan2(-sign(sin02)R3 2,sign(sin02)R31 ) (30)

07 - atan2(-sign(sin06)D1,sign(sin06)D2)

The parameters p, A, B, D1 and D2 are computed in the Appendix.

Otherwise, sin06=O, and 07 can be determined directly according to some roles.

The space robot is then simplified as a planar robot including Joint 3, Joint 4 and Joint 5. Closed-form solutions of the 3-DOF planar robot are given by

d3=atan2(Hsc3 (l),Hsc3 (2)) (36)

L 2-L2-L2 sine, = SW 4 5 (37)

4 2J J

In order to highlight the efficiency of the VRM, computations of the IKP of the parameterized manipulator are completed using different methods. Twenty sets of arbitrary end-effector poses in the workspace are given. Figure 15 shows the computation time using three different methods. It shows that the VRM is faster in determining kinematic solutions than the other two methods.

link ai a di 0.

1" 0 0° 0 0i

1' 0 90° Lo 180°

2 Li 0° 0 02

3 Lz 0° 0 03

85=atan2(Hsc5 (l),Hsc5 (2))

The parameters L SW, Hsc3 and Hsc5 are illustrated in the Appendix.

Finally, parameterized closed-form solutions of the 7-DOF space robot are obtained from (32)-(37).

5 0 90° L4 05

6 0 0° L5 0

Table 5. DH parameters of the 6-DOF robot

Pose Solutions Joint 1 Joint 2 Joint 3 Joint 4 Joint 5 Joint 6

solution 1 45.0000 60.0000 45.0000 60.0000 150.0000 45.0000

solution 2 45.0000 105.4302 -45.0000 105.0000 150.0000 45.0000

solution 3 45.0000 29.5979 117.8100 -162.4079 -150.0000 -135.0000

Pose A solution 4 45.0000 147.4079 -117.8100 -44.5979 -150.0000 -135.0000

solution 5 9.0917 69.5090 45.6538 56.6466 114.7204 54.6197

solution 6 9.0917 115.1629 -45.6538 102.3005 114.7204 54.6197

solution 7 9.0917 42.0462 117.2856 -167.5223 -114.7204 -125.3803

solution 8 9.0917 159.3318 -117.2856 -50.2367 -114.7204 -125.3803

solution 1 -140.1535 101.2335 97.5331 161.2335 140.1535 60.0000

solution 2 -140.1535 -161.2335 -97.5331 -101.2335 140.1535 60.0000

Pose B solution 3 -140.1535 107.1136 40.8138 32.0725 -140.1535 -120.0000

solution 4 -140.1535 147.9275 -40.8138 72.8864 -140.1535 -120.0000

solution 5 0 -8.7393 73.9530 -25.2136 0 20

solution 6 0 65.2136 -73.9530 48.7393 0 20

Table 6. Computed solutions of the SFA (unit: degree)

Oz(Oe)

Figure 12. The equivalent SOC of the 7-DOF humanoid arm

J, 200

3 100 -M

closed form solutions Newton-Raphson algorithm Jacobean-based method

Î ... i t I i.; ! t------ . !.; « »

0 5 10 15

No. of test samples

Figure 13. The efficiency of three different methods

Figure 14. The equivalent SOC of the SSRMS 14 Int J Adv Robot Syst, 2015, 12:38 I doi: 10.5772/60055

300 250 200 150 100 50 0

closed form solutions Newton-Raphson algorithm Jacobean-based method

5 10 15

No. of test samples

Figure 15. The efficiency of three different methods

In conclusion, the VRM is a valid and efficient approach to solve the IKP of manipulators analytically. It is applicable to serial manipulators containing either the Pieper' or the Duffy'sgeometry.

5. Conclusions

In this paper, a unified framework has been proposed for closed-form solutions of a special class of 6- and 7-DOF serial manipulators containing the Pieper's geometry or the Duffy's geometry. Given a specified end-effector pose of the general manipulator, an equivalent reconfigured form can be developed. The reconfigured manipulator is kinematically equivalent to the original manipulator. The virtual reconfiguration method (VRM) is proposed on the basis of the kinematic equivalence between the two manipulators. Hence, for any serial manipulator satisfying either the Pieper's condition or the Duffy's condition, the equivalent SOC with the spherical wrist or the Duffy shoulder joint can be developed using the VRM. Closed-form solutions can also be obtained by solving the IKP of the equivalent SOC using traditional methods. The validity and efficiency of the VRM are verified by kinematic simulations of humanoid arms and space robots. Unlike traditional approaches, the VRM simplifies the computation of the IKP and establishes a unified framework for closed-form solutions of the special class of 6- and 7-DOF serial manipulators irrespective of the allocation of the Pieper'sgeometryortheDuffy'sgeometry.

6.Appendix

1. Parameters of the IKP for the 6-DOF humanoid arm are givenby

r, y r, X

nr,z 0r,x 0 0

r,X r,X 0 1

nx 0X ax Vx

ny 0X ax Vx

nz 0x ax Vx

0 0 0 1

P15,x = Vr,x - L3nx, P15,y = Vry - L3ny, P15,z = Vr,z - Mz

A-p +L0a + L3,B = p + L0a ,C-(p +L0af+L2

r x 0 x 3' r y 0 y' \r z 0 z J 2

f1 - Acos66 +Csin66, f2 - Asin66 -Ccos66, f3 - B,

p - ( L12 -A2-B2-C)/(2L^A2+B2 )

f1 - Acos66 - Bsin66, f2 - Asin66 + Bcos66

g1 - f1C0S65 + f3Sin65 , g3 -f2+ Lew

4. Parameters of the IKP for the 7-DOF SSRMS are given by

f3-Pz+ L0az , g1 - f1COs65 - f3Sin65 , g3 - -L2 - f2

A - (Vx - Px ) C0S61 + ( Vy - Py ) Sin61,

2. Parameters of the IKP for SFA are given by

a-Px-L50A6 (1,3)

B-L8a + p -L,C-L3 + L6,

8 z r z 1' 3 6 '

b-L50A6 (2,3) - 0A6 (2,4)

D1 - ( o cos6.. + o sin6.. ) sin62 - o cos62

1 \ x 1 y 11 2 z 2'

D2 - (n co6 + n sin6.1 ) sin62 - n cos62

2 \ x 1 y 11 2 z 2'

Hsc2 =

-L2sin63 L1 + L2cos63 L1 + L2cos63 L2sin63

^(1,4)

rT4(2,4)_

Lsw-JP2M (1) + PM (2),

Hsc4 =

-sin (62+63 ) cos (62 + 63 )

cos (62 + 63 ) sin (62 + 63 )

rT4(1,1)

rT4(2,1)

3. Parameters of the IKP for the 7-DOF humanoid arm are given by

r ,x r ,x r ,x r r ,x

n o a p

r ,y r ,x r ,x r r ,x

n o a p

r ,z r ,x r ,x r r ,x

nx ox ax px

x x x x

ny ox ax px

y x x x

nz ox ax px

0 0 0 1

PM =2P 5 - 2R 5 ■[L 7 0 0]T (where 2P 5 and 2P 5 are the position and the orientation of Frame 5 with respect to Frame 2 respectively),

-L5sin64 L4 + L5cos64

L4 + L5cos64 L5sin64

-sin (63+64 ) cos (63+64 )"

cos (63 + 64 ) sin (63+ 64 )

2R5(1,1) 2R5(2,1)

A-(p - L 1t - La ) cos67 + ( Lba - p ) sin67 - L

\r r,x w1t bs r,x j 7 \ bs r,y r r,y I 7 w.

B = ( p - L 1t - La ) sin67 -( Lba - p ) cos67,

\r r,x w1t bs r,x j 7 \ bs r,y r r,y I 7 '

7. Acknowledgements

This work is supported by the National 973 Programme under Grant No. 2011CB013306 and the National Natural Science Foundation of China under Grant No. 51175106.

C-p - La

r r,z bs r,z

hP15 = hP6 + bR •[Lww1 0 0]T +[0 0 -Lbs J'

p-( l 2 - A2 - B2 -C2 - L 2 )/( 2L -Ja2 +C2 )

r ^ se ew]l\ew I

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