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\title{Acceleration Analysis of 3DOF Parallel Manipulators}
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             \author[1]{Ajit Pal  Singh}

             \author[2]{Mr. Hassen  Nigatu}

             \author[3]{Solomon  Seid}

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\date{\small \em Received: 8 December 2011 Accepted: 4 January 2012 Published: 15 January 2012}

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\begin{abstract}
        


This paper presents a new approach to the velocity and acceleration analyses 3DOF parallel manipulators. Building on the definition of the ?acceleration motor?, the forward and inverse velocity and acceleration equations are formulated such that the relevant analysis can be integrated under a unified framework that is based on the generalized Jacobian. A new Hessian matrix of serial kinematic chains (or limbs) is developed in an explicit and compact form using Lie brackets. This idea is then extended to cover parallel manipulators by considering the loop closure constraints. A 3- PRS parallel manipulator with coupled translational and rotational motion capabilities is analyzed to illustrate the generality and effectiveness of this approach.

\end{abstract}


\keywords{Acceleration analysis, Kinematics, Parallel manipulators.}

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\let\tabcellsep& 	 	 		 \par
ower mobility parallel manipulators having fewer than six degrees of freedom (DOF) continue to draw interest from both industry and academia because they regularly offer improved balance between speed, accuracy, rigidity and reconfigurability when compared with conventional machine tools and industrial robots having serial architectures.\par
Velocity, accuracy, stiffness and rigid body dynamic behaviours are important performance factors that"s should be considered in the design of lower mobility parallel manipulators. Particularly in circumstances where high speed is the priority, rigid body dynamics become a major concern for the dynamic manipulability evaluation, motor sizing and controller design, all of which involve acceleration analysis as the prerequisite.\par
Although general, systematic approaches are available for velocity analysis of lower mobility parallel manipulators using either kinematic influence coefficient methods or screw theory based methods  {\ref (Huang et al., 2000;}\hyperref[b15]{Joshi \& Tsai, 2002;} {\ref Jhu et al., 2007)} it is by no means an easy task to use these approaches for acceleration analysis owing to the nonlinearity arising from the second order partial derivatives.\par
A number of approaches have been proposed for acceleration analysis of either serial or parallel manipulators. The most straightforward method is to take time derivatives of a set of velocity constraint equations. This, however, involves a tedious and laborious process as shown by many case-by-case studies \hyperref[b34]{(Tsai, 2000;}\hyperref[b16]{Khalil \& Guegan, 2004;}\hyperref[b18]{Li et al., 2005;}\hyperref[b3]{Callegari et al., 2006)}. Therefore, a recursive matrix method was proposed in order to reduce computational burdens \hyperref[b31]{(Staicu \& Zhang, 2008;}\hyperref[b30]{Staicu, 2009)}. Having a goal of achieving a general and compact form of the Hessian matrix, the kinematic influence coefficient method was proposed for dealing with the acceleration analysis of serial manipulators \hyperref[b33]{(Thomas \& Twsar, 1982)}. This idea was then extended to cover full and lower mobility parallel manipulators \hyperref[b11]{(Huang, 1985a;}\hyperref[b12]{Huang, 1985b;}\hyperref[b38]{Zhu, 2005;}\hyperref[b13]{Huang, 2006;}\hyperref[b36]{Zhu et al., 2007)}. Along this track, kinematic analysis of a number of parallel manipulators with different architectures has been carried out \hyperref[b5]{(Fang \& Huang, 1997;}\hyperref[b20]{Lu, 2006;}. Despite the plausibility and merits of the kinematic influence coefficient method, only an implicit form of the Hessian matrix can be achieved because of the unavoidable partial derivative implementations. Recently, an approach for acceleration analysis was proposed that introduced an auxiliary Hessian matrix derived from the differentiation of the auxiliary Jacobian of a class of parallel manipulators containing a passive properly constrained limb \hyperref[b21]{(Lu \& Hu, 2007a;}\hyperref[b22]{Lu \& Hu, 2007b;}. Its suitability for other types of parallel manipulators, however, remains an issue to be investigated. Screw theory based approaches \hyperref[b14]{(Hunt, 1978;}\hyperref[b26]{Mohamed \& Duffy, 1985;}\hyperref[b17]{Kumar, 1992;}\hyperref[b19]{Ling \& Huang, 1995;}\hyperref[b1]{Bonev et al., 2003;}\hyperref[b6]{Fang \& Tsai, 2003;}\hyperref[b40]{Zoppi, 2006)} could potentially be the most powerful method for acceleration analysis. In order to overcome the difficulty of expressing the twist derivatives in a screw form, a novel term named the "accelerator" \hyperref[b32]{(Sugimoto, 1990)} or "acceleration motor" \hyperref[b2]{(Brand, 1947)} was proposed and employed for the acceleration analysis of serial and parallel kinematic chains \hyperref[b28]{(Rico \& Duffy, 1996;}\hyperref[b29]{Rico \& Duffy, 2000)}. However, the terms associated with the second derivatives in the acceleration equations can only be written in a lengthy form of Lie brackets rather than in a compact form of the Hessian matrix. 
\section[{L}]{L}\par
Introduction achieve an explicit and compact form of the Hessian matrix. Having outlined in Section I the significance of acceleration analysis and its existing problems, the paper is organized as follows. Sections II and III systematically develop the formulations of forward/inverse velocity and acceleration models of serial and parallel kinematic chains, leading to new expressions of the Hessian matrices in a general and compact form. A practical illustration is presented in Section IV before conclusions are drawn in Section V.\par
Based upon the authors" previous work \hyperref[b9]{(Huang et al., 2011)}, this section briefly addresses the velocity analysis of an f-DOF parallel manipulator using the "generalized Jacobian" and adds extensions necessary to its use in acceleration analysis. Without loss generality, assume that the manipulator is composed of l ( 1 f l f ? ? ? ) limbs connecting the platform with the base, each essentially containing i n ( 1, 2, , il ? ? ) 1-DOF joints with at most one of them actuated. Thus, two families of parallel manipulators can be taken into account. The first family covers fully parallel manipulators having f constrained active limbs (  6   i   n ? for all limbs). The second family contains those having f unconstrained active limbs (i.e. 6\par
i n ? for each of these f limbs) plus one properly constrained passive limb designated by 1 lf ?? . Any other parallel manipulators not belonging to these two families can be dealt with in a manner similar to that used below.\par
It has been shown \hyperref[b9]{(Huang et al., 2011}) that entire set of the variational twists of the platform spans a 6-dimensional vector space T , known as the twist space. As the dual space of T , the entire set of wrenches exerted on the platform spans a 6dimensional vector space W, named the wrench space. For an f-DOF manipulator, T can be decomposed into an f dimensional subspace, a ? TT , and a 6-f dimensional subspace, c ? TT , known respectively as the twist subspaces of permissions and restrictions. Correspondingly, W can also be decomposed into two subspaces, a ? WW and c ? WW , known as the wrench subspaces of actuations and constraints. It has been proved that the following commutative relationships hold:\par
Direct sum:ac ?? T T T , ac ?? W W W (1a) Orthogonality: ac ? ? WT, ca ? ? WT\textbf{(1b)}\par
Duality:* aa ? WT , * cc ? WT (1c) a) Velocity analysis of a limb Let , , , ?a ta j i a i ? \$ T ( 1,2, , a i j n ? ? ), , , , ?a wa k i a i ? \$ W ( 1,2, , a i k n ? ? ), , , , ?c tc j i c i ? \$ T ( 1,2, ,6 c i j n ? ? ?\par
) and , , ,?c wc k i c i ? \$ W ( 1,2, ,6 c i k n ? ? ?\par
) be the basis elements of four vector subspaces associated with the ith limb. Note that the commutative relationships given in Eq. (1a, b, c) also hold for each limb since all limbs share the same platform. The variational twist of the platform can then be represented by a linear combination of the basis elements of , ai T and, ci T ,,\textbf{6 , , , , , , , , 1 1 ?î}i a a c c a c t ta tc ta i tc i n n a j i ta j i c j i tc j i j j ii ?? ?? ? ? ? ? ? ? ? ? ? ? ? ?? J ? \$ \$ \$ \$ \$ \$ \$ , 1, 2, , il ? ?\textbf{(2)}\par
where,? ? T TT t ? ? =r ? \$ ,, J J J i a i c i ?? ? ?? ,, J J J i a i c i ?? ? ?? , ,\textbf{1, , , ??i a}i ta i ta n i ?? ? ?? J ? \$\$ , ,\textbf{1, ,6 ,}\par
??J i c i tc i tc n i ? ?? ? ?? \$\$ ? ? ? T TT ,, i a i c i ? ? ? ? ? ? ? ? ? T , ,\textbf{1, ,2, , , i a}i a i a i a n i ? ?? ?? ?? ? ? ? ? ? T , ,\textbf{1, ,2, ,6 ,}i c i c i c i c n i ? ?? ?? ?? ? ? ? ?\par
?r and ?? denote, respectively, the linear variation of the reference point and the angular variation of the platform. ? ? ? ? T T T T T T , , , i a i c i i a i ? ? ? ? ? ? ? ? ? ? ? ?? 0 ,, a i a i ? ? ? ? ? , ci ? ? 0 ? ? ? T , ,\textbf{1, ,1, , , ( A )}\par
ear 2012 Y© 2012 Global Journals Inc. (US) ? ? ? ? T T T T T T t t ? ? ? ? ? r ? v ? \$ \$ II. 
\section[{Velocity Analysis}]{Velocity Analysis}\par
Drawing primarily on the generalized Jacobian but also on the strengths of the kinematic influence coefficient method and the concept of accelerator, we propose a new approach for acceleration analysis of lower mobility parallel manipulators. Its goal is to where, ? i are individual joint velocities. Thus \hyperref[b9]{(Huang et al., 2011)} t i i ? J ? ? \$ , 1, 2, , il ? ?\textbf{(3)}\par
where, Using the commutative relationships given in Eq. ( \hyperref[formula_0]{1}) and taking inner products on both sides of Eq. ( \hyperref[formula_4]{2}) with 
\section[{? ?}]{? ?},, ?a wa k i \$ ( 1, 2, , a i k n ? ? ) and ,, ?c wc k i \$ ( 1,2, ,6 c i k n ? ? ?\par
) leads, after the same replacements, to1 i i t ? ? ? J ? \$ , 1, 2, , il ? ? (4) , 1 , L ai i L ci ? ?? ? ?? ?? ?? J J J TT ,1, ,\textbf{1, ,1, , TT , , , , , , ???î i}i wa i wa i ta i L ai wa n i wa n i ta n i ?? ?? ? ?? ?? ?? ?? J ? \$ \$ \$ \$ \$ \$ TT ,1\textbf{,1 ,1 , TT ,6 , ,6 , ,6 , ???î i}i wc wc tc L ci wc n i wc n i tc n i ? ? ? ?? ?? ? ?? ?? ?? ?? J ? \$ \$ \$ \$ \$ \$\par
Here the superscript L simply identifies that the matrix applies to a single limb. Thus, we have,, L a i a i t ? ? J ? \$ , 1, 2, , il ? ?\textbf{(5)}\par
b) Velocity analysis of a parallel manipulator Building upon the work in Section 2.1, the velocity modeling of a parallel manipulator can be carried out with little extra effort. Let ,, ?k wa g k \$ be the unit wrench of actuations associated with the one actuated joint, numbered k g , in the kth ( 1, 2, ,k f ? ? ) limb and ,, ?c wc k i \$ be the th c k ( 1,2, ,6 ck kn ?? ? ) unit wrench of constraints in the ith ( 1, 2, , i l ? ? ) limb.\par
Again, using the commutative relationships given in Eq. ( \hyperref[formula_0]{1}), taking inner products on both sides of Eq. ( \hyperref[formula_4]{2}) with ,, ?k wa g k \$ and ,, ?c wc k i \$ , respectively, and making replacements as at Eq. ( \hyperref[formula_12]{3}) results int ? Jq ? \$\textbf{(6)}\par
where,a c ?? ?? ?? J J J = 1 1 1 2 2 2 T T , ,\textbf{1 , ,1 , ,1 T T , ,2 , ,2 , ,2 T T , , , , , , ????f f f}\par
wa g wa g ta g wa g wa g ta g a wa g f wa g f ta g f? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? J ? \$ \$ \$ \$ \$ \$ \$ \$ \$ ,1 ,2 , c c c cl ?? ?? ?? ? ?? ?? ?? ?? J J J J ? TT ,1, ,\textbf{1, ,1, TT ,2, ,2, ,2, , T T ,6 , ,6 , ,6 ,}????î i i wc i wc i tc i wc i wc i tc i ci wc n i wc n i tc n i ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? J ? \$ \$ \$ \$ \$ \$ \$ \$ \$ ? ? T T T a qq ?? =0 ? ? 12 T , ,1 , ,2 , , f a a g a g a g f q q q ? ? ? ? ? ? q\par
In order to distinguish the joint rates of actuated joints from those of the passive joints, we use ,, k a g k q ? to represent the rate of actuated joint numbered k g in the kth limb. For convenience, ,, k a g k q ? will be simplified ask q ? in what follows. J is an ? ? ? ? 1 66 l i i fn ? ? ? ? ? matrix known as the generalized Jacobian of parallel manipulators with 6 f ? DOF. Since ? ? ? ? 1 66 l i i fn ? ? ? ? ?\par
, the left pseudo-inverse of J exists. Using superscript P to identify explicitly platform terms, this leads toP t a a ? ?? J q J q ?? \$ , ? ? 1 TT PP ac ? ? ?? ?? ?? J J J J J J\textbf{(7)}\par
Substituting Eq. ( \hyperref[formula_24]{7}) into Eq. (  {\ref 4}), i ? ? can then be expressed in terms of q ?1 LP i i i ?? ?? ? J J q J q ? ?? , 1, 2, , il ? ? (8)\par
where,1 LP ii ?? ? J J J is a ? ? ? ? 1 66 l i i fn ? ? ? ? ? matrix.\par
Furthermore, the linear map between a q ? and where,, ai ? ? is , , , L P LP a i a i a a a i a ?? ? J J q J q ? ?? , 1, 2, , il ? ? (9)\par
,,LP L P a i a i a ? J J J is a i nf ? matrix.\par
Following the scheme in Section II, acceleration analysis will first be carried out on an -DOF i n limb, with the results then being extended to cover an f-DOF parallel manipulator. 
\section[{a) Acceleration analysis of a limb}]{a) Acceleration analysis of a limb}\par
Taking the variation of Eq. ( \hyperref[formula_4]{2}) and expressing the derivatives of screws in the form of Lie brackets as given in \hyperref[b8]{(Gallardo et al., 2003)}, yields2 iii ? ?? AJ ? \$ , 1, 2, , il ? ? (\textbf{10})\par
where,? ? ? ? T TT 22 ? ? ? ? ?? ? ? ? ?? ?? Ar ? r ? ? ? ? ? T TT 2 2 2 ,, i a i c i ? ? ? ?? ? ?? ?? ? ? ? ? ? T 2 2 2 2 , ,\textbf{1, ,2, , ,}i a i a i a i a n i ? ? ? ? ? ? ? ? ? ? ? ? T 2 2 2 2 , ,\textbf{1, ,2, ,6 ,}i c i c i c i c n i ? ? ? ? ? ? ? ? ? ? ? ,1,\textbf{,1, ,2, ,2, ,6 , ,6 , ,2, ,2, ,3, ,3, ,6 , ,6 , ,5 , ,5 , ,6 , ,6 , ???î i}i i i i i i i a i ta i a i ta i c n i tc n i a i ta i a i ta i c n i tc n i c n i tc n i c n i tc n i ?? ?? ?? ?? ?? ?? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? \$ \$ \$ \$ \$ \$ \$ \$ \$ 2 ? r , 2 ? ? , 2 ,, a a j i ? ?\par
, and \$ denotes the Lie product \hyperref[b7]{(Gallardo, 2006)}.\par
From the properties of the Lie product, i \$ can also be written as, , , i a i ac i c i ? ? ? \$ \$ \$ \$\textbf{(11)}\par
where,, ,\textbf{1, ,2, ,1, ,2, ,1, , , ,1, , , ,2, ,3, ,2, , , ,2, , , ,2, , , , 1, , , , 1, , , ????î i}i i i i i i i a i a i a i ta i ta i a i a n i ta i ta n i a i a i ta i ta n i a i a n i ta i ta n i a n i a n i ta n i ta n i ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? \$ \$ \$ \$ \$ \$ \$ \$ \$ \$\$ , ,\textbf{1, ,1, ,1, ,1, ,1, ,6 , ,1, ,6 , ,2, ,1, ,2, ,1, ,2, ,6 , ,2, ,6 , , , ,1, , , ,1,}????î i i i i i 
\section[{ac i a i c i ta i tc i a i c n i ta i tc n i a i c i ta i tc i a i c n i ta i tc n i a n i c i ta n i tc i ?? ?? ?? ?? ?? ?? ?? ?? ?? ??}]{ac i a i c i ta i tc i a i c n i ta i tc n i a i c i ta i tc i a i c n i ta i tc n i a n i c i ta n i tc i ?? ?? ?? ?? ?? ?? ?? ?? ?? ??}? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ , ,\textbf{,6 , , , ,6 ,}\par
?îi i i a n i c n i ta n i tc n i ?? ?? ? ? ? ? ?? ? ? ? \$ \$ , ,\textbf{1, ,2, ,1, ,2, ,1, ,6 , ,1, ,6 , ,2, ,3, ,2, ,3, ,2, ,6 , ,2, ,6 , ,5 , ,6 , ,5 , ,6 ???????î}i i i i i i c i c i c i tc i tc i c i c n i tc i tc n i c i c i tc i tc i c i c n i tc i tc n i c n i c n i tc n i tc ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ , i ni ? ? ? ? ?\par
Then, Eq. ( \hyperref[formula_29]{10}) can be rewritten as2T i i i i i ? ? ? ?? AJ ? ? H ? , 1, 2, , il ? ?\textbf{(12)}\par
,, ,a i ac i i ci ?? ? ?? ?? HH H H 0 ,1, ,\textbf{2, ,1, ,3, ,1, , , ,2, ,3, ,2, , , , , 1, , , ????????i i}\par
ii ta i ta i ta i ta i ta i ta n i ta i ta i ta i ta n i aita n i ta n i ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ?? ?? ? ?? ?? ?? ?? ?? ?? ?? H ? ? ? ? ? ? ? ? ? 0 \$ \$ \$ \$ \$ \$ 0 0 \$ \$ \$ \$ 0 0 0 \$ \$ 0 0 0 0 ,1, ,\textbf{1, ,1, ,6 , , , , ,1, , , ,6 , ?????i}i i i ta i tc i ta i tc n i ac i ta n i tc i ta n i tc n i ? ? ?? ? ? ? ? ? ? ? ? ?? ?? ? ?? ? ? ? ? ?? ? ? ? ? ?? H ? ? ? ? ? \$ \$ \$ \$ \$ \$ \$ \$ ,1, ,\textbf{2, ,1, ,3, ,1, ,6 , ,2, ,3, ,2, ,6 , , ,5 , ,6 , ????????i}i ii tc i tc i tc i tc i tc i tc n i tc i tc i tc i tc n i ci tc n i tc n i ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? ? ? ? ? ? ? H ? ? ? ? ? ? ? ? ? 0 \$ \$ \$ \$ \$ \$ 0 0 \$ \$ \$ \$ 0 0 0 \$ \$ 0 0 0 0 666 i ?? ? H ?\par
is known as the Hessian matrix of the ith limb. It is a three-dimensional matrix having six layers, each containing a 66 ? upper triangular matrix as shown in Fig.  {\ref 1}, where ? ?i K ?? ( 1, 2, , 6 i K ? ? ) denotes the th i K element of the Lie bracket ? ? ?? . The constituent parts of i H , 6 , ii nn ai ?? ? H ? , ? ? 66 , ii nn ac i ? ? ? ? H ? and ? ? ? ? 6 6 6 , ii nn ci ? ? ? ? ? H ?\par
, are also three-dimensional matrices having six layers.\par
In acceleration analysis where only ideal motions of the platform are considered, replacements can be made in Eq. ( \hyperref[formula_42]{12}) such that: The bracket Thus,? ? ? ? T T T T T T , , , i a i c i i a i ? ? ? ? ? ? ? ? ? ? ? ?? 0 ? ? ? ? ? ? T T T T 2 2 2 T , , , i a i c i i a i ? ? ? ?? ? ? ? ?? ?? ? ? ? ? ? ?? ?? 0 ? ? ? ? ? ? T T T TT T 22 ? ? ? ? ?? ?? ? ? ? ? ? ? ? ?? ??T i i i i i ?? AJ ? ? H ? ?? ? ? , 1, 2, , il ? ?\textbf{(13)}\par
. limb with v? and ? ? being the linear acceleration of the reference point and the angular acceleration of the platform \hyperref[b8]{(Gallardo, 2003)}. The th i K element in A has the expression T ,,IIIi i i K i K i i i K i ?? AJ ? ? H ? ?? ? ? , 1, 2, , il ? ? (15) ( A ) ear 2012 Y © 2012 Global Journals Inc. (US)\par
Fig.  {\ref 1} : Hessian matrix, H i , of the i th limb.\par
,1, ,?0 0 0 0 0 0 0 i i i i i i i i i i ta i ta i ta i ta i ta i tc n i K K K ta i ta i ta i tc n i K K tc n i tc n i K ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ non-zero element zero element 1 2 3 4 5 6 1 2 3 4 5 6 1 i K 6 The K i th layer ( 1, 2, ,6 i K ? ? )\textbf{2, ,1, ,3, ,1, ,6 , ,2, ,3, ,2, ,6 , ,5 , ,6 , ?????0 ???0 0}\par
where, , i iK J is theth i K row of i J , while , i iK H is the th i K layer of i H .\par
In addition, the inverse acceleration equation of the ith limb can easily be obtained, by recalling Eq. (  {\ref 4}), as? ? 1 T T 1 i i t i i i t ? ? ? ?? ? J A J H J ?? \$\$ , 1, 2, , il ? ? (\textbf{16})\par
Acceleration analysis of a parallel manipulator Following the same approach as at Eq. ( \hyperref[formula_3]{6}), the acceleration equation of a parallel manipulator is obtained by taking inner products on both sides of Eq. ( \hyperref[formula_49]{13}) with ,, ?k wa g k \$ and ,, ?c wc k i \$ , respectively, and noting the relationship given in Eq. (  {\ref 8}), to giveT ?? JA q q Hq ?? ? ?\textbf{(17)}\par
where,? ? T T a qq ?? ?? =0 ? ? 12 T , ,1 , ,2 , , f a a g a g a g f q q q ? ?? ?? ?? ?? ? q Here, ,, k a g k q ??\par
is the acceleration of the actuated joint numbered k g in the k th limb ( 1, 2, ,k f ? ? ). For convenience,\par
,, k a g k q ?? will be simplified as k q ?? in what follows.NNN ?? ? H ? ( ? ? 1 6 l i i N f n ? ? ? ? ?\par
) is known as the Hessian matrix of an f-DOF parallel manipulator. It is a three dimensional matrix composed off N N a ?? ? H ? and ? ? 1 6 l i i n N N c ? ? ? ? ? ? H ?\par
. The expression for the th a K acceleration equation of a parallel manipulator:? ? T ? ?? A J q q Hq ?? ? ?\textbf{(19)}\par
Furthermore, substituting Eqs. (  {\ref 8}) and ( \hyperref[formula_61]{19}) into Eq. ( \hyperref[formula_54]{16}) expresses the joint acceleration in the ith limb in terms of the velocity and acceleration of the actuated joints:? ? ? ? T T 1 T LP LP LP i i i i i i ? ? ? ? ? J q q Hq J q J H J q ?? ?? ? ? ? ? , (\textbf{20})1, 2, , il ? ? ( th c K ) layer of a H ( c H\par
) is given in Fig. \hyperref[fig_1]{2}. Eq. ( \hyperref[formula_56]{17}) readily yields the inverse acceleration equation of a parallel manipulator,TT q JA J HJ tt ?? \$\$ ??\textbf{(18)}\par
Moreover, multiplying both sides of Eq. ( \hyperref[formula_56]{17}) by the left pseudo-inverse of J gives the forward The above analyses formulate the forward/inverse velocity and acceleration equations of lower mobility parallel manipulators in a consistent manner under the umbrella of the generalized Jacobian. The velocity and acceleration of each 
\section[{DOF 1?}]{DOF 1?}\par
actuated joint of the manipulator can be evaluated using Eqs. (  {\ref 8}) and \hyperref[b20]{(20)}. Note, also, that the velocity and acceleration analyses given for a limb are also valid for serial manipulators with 6 f ? DOF.\par
Detailed execution of velocity and acceleration analyses for a 3-PRS parallel manipulator serves to i. 
\section[{IV. An xample E}]{IV. An xample E}\par
illustrate the generality and effectiveness of the proposed approach. Fig. \hyperref[fig_0]{3} shows a schematic diagram of a 3-PRS parallel manipulator which is used as a 3-axis module named the Sprint Z3 \hyperref[b35]{(Wahl, 2002)} as part of a 5-axis high-speed machining cell for extra large components. The manipulator consists of a base, a platform, and three identical limbs, each connecting the base with the platform in sequence by an actuated prismatic joint, a revolute joint, and a spherical joint.  f ? ? 1 6 l i i f n ? ? ? ? c H a H ? ? T T T , ,\textbf{,1, ,2, , , ,1, ,3, , , ,1, ,6 ,}T T T , ,\textbf{,2, ,3, , , ,2, ,6 , T , , , , , ????????0 ?????0 0}??0 0 0 c c c i c c i c c 
\section[{wc k i ta i ta i wc k i ta i ta i wc k i ta i tc n i LP wc k i ta i ta i wc k i ta i tc n i i wc k i tc k i wc k}]{wc k i ta i ta i wc k i ta i ta i wc k i ta i tc n i LP wc k i ta i ta i wc k i ta i tc n i i wc k i tc k i wc k}? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? J ? ? ? ? ? ? ? ? \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ \$\$ \$ T , ,\textbf{5 , ,6 ,}\par
?0 0 0 0c i i LP i i tc n i tc n i ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? J ? \$ \$\par
The K a th layer of H a (  Fig. \hyperref[fig_2]{4 (a, b, and c}) shows the CAD model of the selected example 3-PRS parallel kinematic machine, which helps to visualize and internalize the physical mechanism.1, 2, , a K f ? ? ; a kK ? ) The K c th layer of H c 1 1, 2, ,6 c k n ? ? ? ( 1 1 1,2, ,6 , ,\textbf{6}l c i i K n n ? ? ? ? ? ? ? ;; 1, 2, , i l ? ? ) ? ? 1 6 l i i f n ? ? ? ? ? ? 1 6 l i i n ? ? ? ? ? T T T , ,\textbf{,1, ,2, , , ,1, ,3, , , ,1, ,6 ,}T T T , ,\textbf{,2, ,3, , , ,2, ,6 , T , , , , , ????????0 ?????0 0}??0 0 0 J k k k i k k i k k wa g k ta k ta k wa g k ta k ta k wa g k ta k tc n k LP wa g k ta k ta k wa g k ta k tc n k i wa g k ta g k wa g ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ \$ ? ? ? ? ? ? ? ? T , ,\textbf{5 , ,6 , ?0 0 0 0}J k i i LP k k tc n k tc n k ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? \$ \$ ? 4-c 4-b 
\section[{a) Inverse kinematics}]{a) Inverse kinematics}\par
A reference frame R is attached to the base and a body fixed frame 0 R to the platform, with O and O? located at the centers of the equilateral triangles  
\section[{B B B ? and 1 2 3}]{B B B ? and 1 2 3}\par
A A A ? , as shown. The z and 0 z axes are normal to the planes of those triangles, the x axis is parallel to 21 BB and the 0 x axis is parallel to 21 AA. Also, an instantaneous reference frame R? is set with its origin at point O? and its three orthogonal axes remaining always parallel to those of R . Consequently, the orientation matrix of 0 R with respect to R can be obtained using three Euler angles, ? , ? and ? in terms of precession, nutation, and body rotation according to the z-x-z convention? ? ? ? ? ? Rot , Rot , Rot , C C S C S C S S C C S S S C C C S S S C C C C S S S S C C z x z ? ?? ? ? ? ? ?? ?? ? ? ? ? ? ?? ?? ?? R ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?\textbf{(21)}\par
where, "S" and "C" represent "sin" and "cos", respectively. Then, the position vector,? ? T x y z ? r ,\par
of O? can be expressed as The constraint imposed by the revolute joint restricts the translational motion of i A along the axis of the revolute joint in limb i. This leads to three additional constraint equations, 1, 3 3, r b s s a i i i i i ql ? ? ? ? , 1, 2,3 i ?\textbf{(22) where, 1,}s i i i i q B P ? ? ? T 1, 0 0 1 s i ? 3 3, s i i i l P A ? ? ? ? ? T T cos sin 0 b i ix iy iz i i b b b b ?? ?? ? ? T 0 a Ra i i ix iy iz a a a ?? ? ? T 0 cos sin 0 i i i a ? a ?? ? ? 11 2 1 63 i i ?? ? ? ? ? , 1\textbf{, 2,3 i}? ? T 2, 0 r a s ii ?? , 1, 2,3 i ?\textbf{(23) where, ? ? T 2}\par
? ?1 S2 1 C 2 xa ?? ? ? ?\textbf{(25)}\par
? ?1 C2 1 C 2 ya ?? ? ? ?\textbf{(26)}\par
Thus the three desired motions, ? , ? , and z , can be considered as three independent Cartesian variables, leaving the translations along the x and y axes, and rotation about the z? axis (angle? ) as the constrained variables. Given a set of ? , ? , and z , the inverse position problem is determined by, ? ? T 3 3, 1, 
\section[{r a b s s}]{r a b s s}i i i i i ql ? ? ? ? , 1, 2,3 i ? (27) ? ? T 3,\textbf{3 , 3 , 3 ,}s i x i y i z i s s s ? , ? ? 3, 3 1 x i ix ix s x a b l ? ? ? ? ? 3, 3 1 y i iy iy s y a b l ? ? ? ,\textbf{22 3 , 3 , 3 ,}1 z i x i y i s s s ? ? ? b) Velocity analysis\par
Given the bases for the four vector subspaces of its ith limb \hyperref[b9]{(Huang et al., 2011)}, the generalized Jacobians of the ith limb and the generalized Jacobian of the manipulator can be formulated as follows.\par
For the i th limb ( 1, 2,3 i ? ): , i a i ci ?? ? ?? J J J (28) ? ? , ,\textbf{1, ,2, ,3, ,4, ,5, 1, 3 3, 2, 3, 4, 5, 2, 3, 4, 5,}i i i i i i i i i i i i i l ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? \$ \$ \$ \$ \$ 0 ? ? 3 3, 1, ,\textbf{,1, 1,}\par
? a s n J ni i i c i tc i i l ?? ?? ?? ?? ?? ?? 
\section[{\$}]{\$}\par
For the parallel manipulator:J J J a c ?? ?? ?? =\textbf{(29)}? ? ? ? ? ? ? ? ? ? ? ? ?? ? ?? ?? ?? ?? ?? ?? ? ??\par
where, Substituting Eq. ( \hyperref[formula_91]{29}) into Eqs. \hyperref[b5]{(6)} and Eq. ( \hyperref[formula_24]{7}) generates the inverse and forward velocity equations of the manipulatorqJ t ? \$ ? (\textbf{30})1 Jq t ? ? \$ ?\textbf{(31)}\par
where,? ? T T T a qq ?? =0 and ? ? T 1 2 3\par
q= a q q q ? ? ? ? . 
\section[{c) Acceleration analysis}]{c) Acceleration analysis}\par
The Hessian matrix, H , of the manipulator can be found by substituting the expressions just found for ? ? T , , T 3 1, 3, J M J H ss c LP LP i c i i cK ii l ? , 1, 2,3 c Ki ??\textbf{(33)}\par
where, 11 J J J LP ii ?? ? 1, , 2, M M M ai ai ai ?? ? ?? ?? 0 0 ? ? ? ? T T T 3, 1,\textbf{3, 1, 4, 3, 1, 5, 1,}i i i i i i i i ai i i i i ll ?? ?? ?? ? ?? ?? ? ? ? ? T 3 4, 1, 2, T\textbf{3 5, 3, 3, 1, 0 0 sn}M s s s n ii ai i i i i l l ?? ?? ? ?? ? ?? ?? ?? ?? ?? 1, , M M ci ci ?? ? ?? ?? 0 00 T T T 3, 1, 4, 1, 5, 1, 1, TT 3 3 5, 3, 3 3, 1, 1 0 s n s n s n M s s s n i i i i i i ci i i i i l l l ?? ? ? ? ? ? ?? ? ? ? ?? ?? Here, , H a aK ( , H c cK ) represents the th a K ( th c K ) layer of a H ( c H ).\par
Then, substituting Eqs. (  {\ref 32}) and (33) into Eqs. ( \hyperref[formula_65]{18}) and ( \hyperref[formula_61]{19}), the inverse and forward acceleration equations of the manipulator areTT q JA J HJ tt ?? \$\$ ?? (34) ? ? 1T A J q q Hq ? ?? ?? ? ?\textbf{(35)}\par
where,? ? T T a qq ?? ?? =0 and ? ? T 1 2 3\par
q a q q q ? ?? ?? ?? ?? . 
\section[{d) Coordinate transformation for numerical simulation}]{d) Coordinate transformation for numerical simulation}\par
Numerical simulations for the inverse velocity and acceleration, require the explicit relationships of the velocity twist and accelerator to the first and second derivatives of three independent coordinates, ? , ? , and z because they are used for path planning.\par
Taking the time derivative of Eqs. ( \hyperref[formula_82]{25}) and ( \hyperref[formula_83]{26})gives v r J g vc ?? ?? (36) ? ? ? ? C2 1 C 0.5 S2 S 0 S2 1 C 0.5 C2 S 0 0 0 1 v a ? ? a ? ? a ? ? a ? ? ?? ? ? ? ?? ? ? ? ?? ?? ?? J , g c ? ? z ?? ?? ? ?? ?? ?? ? ? ? ?\par
Then, taking the time derivative of Eq. (  {\ref 36}) results inT v J g g H g v c c v c ?? ? ?? ? ?\textbf{(37)}\par
where,333 H v ?? ? ? is a three dimensional matrix with , H vi ( 1, 2,3 i ? ) being its ith layer; g c ? ? z ?? ?? ? ?? ?? ?? ?? ?? ?? ?? , ? ? ,1 2 S2 1 C C2 S 0 C2 S 0.5 S2 C 0 0 0 0 H v a ? ? a ? ? a ? ? a ? ? ?? ?? ?? ? ? ? ?? ?? ?? ? ? ,2 2 C2 1 C S2 S 0 S2 S 0.5 C2 C 0 0 0 0 v a ? ? a ? ? a ? ? a ? ? ?? ? ?? ?? ?? ?? ?? H ,3 3 3 v ? ? H 0\par
The angular velocity vector of the platform,? ? T x y z ? ? ? ? ?\par
, can be derived by recalling, e.g. ? ? T 1 2 3 a q q q ? q ? ? ? ? and ? ? T 1 2 3 q a q q q ? ?? ?? ?? ?? can be evaluated using Eqs.\par
(36)-(  {\ref 40}), \hyperref[b5]{(6)}, and \hyperref[b18]{(18)}. Consider now a specific system having the geometry:\par
250 mm a ? , 312 5 mm . b ? , and 3 540 mm l ? Also, assume as an example that the path and motion rules of the platform are: When the platform of the manipulator moves according to the preceding rules, the velocity/acceleration of the actuated joints, the linear velocity/acceleration of the reference point O? , and the angular velocity/acceleration of the platform versus time can be evaluated using the proposed approach. These results, shown in Fig. \hyperref[fig_2]{4}, have been verified by a CAD model of the manipulator. There was no discernable difference between the results obtained using this approach and the CAD software. This paper presents a general and systematic approach for the forward and inverse velocity and acceleration analysis of lower mobility parallel manipulators using screw theory. With this approach, the process of acceleration modeling of serial and parallel kinematic chains can be integrated into the unified framework of the generalized Jacobian. It results in a new Hessian matrix being developed in a general and compact form. This allows rigid body dynamic modeling of lower mobility manipulators to be integrated into a single mathematical framework.\begin{figure}[htbp]
\noindent\textbf{3}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Fig. 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2}\includegraphics[]{image-3.png}
\caption{\label{fig_1}Fig. 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4}\includegraphics[]{image-4.png}
\caption{\label{fig_2}Fig. 4 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-5.png}
\caption{\label{fig_3}}\end{figure}
    \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.26241610738255033\textwidth}P{0.17684563758389263\textwidth}P{0.22248322147651006\textwidth}P{0.01711409395973154\textwidth}P{0.07986577181208054\textwidth}P{0.028523489932885907\textwidth}P{0.06275167785234899\textwidth}}
or\tabcellsep \tabcellsep \tabcellsep \\
, a i a i , ?? AJ? ??\tabcellsep \multicolumn{3}{l}{T , a i a i a i , , ? H ? ? ? ,}\tabcellsep 1, 2, , il ? ?\tabcellsep (14)\\
\multicolumn{2}{l}{where,}\tabcellsep \multicolumn{3}{l}{? ?? ? T T ? ? ?? ?? Av ? v ? ? ??}\tabcellsep T\tabcellsep becomes the\\
\multicolumn{2}{l}{accelerator of an}\tabcellsep i n ?\tabcellsep \multicolumn{2}{l}{DOF}\end{longtable} \par
 
\caption{\label{tab_5}}\end{figure}
 		 		\backmatter  			  				\begin{bibitemlist}{1}
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\bibitem[Mohamed and Duffy ()]{b26}\label{b26} 	 		‘A direct determination of the instantaneous kinematics of fully parallel robot manipulators’.  		 			M G Mohamed 		,  		 			J Duffy 		.  	 	 		\textit{Journal of Mechanisms, Transmissions, and Automation in Design}  		1985. 107  (2)  p. .  	 
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\end{bibitemlist}
 			 		 	 
\end{document}
