Kinematics and inverse dynamics analysis for a general 3-PRS spatial parallel mechanism Yangmin Li and Qingsong Xu

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1 Robotia (2005) volume 23, pp Cambridge University Press doi: /s Printed in the United Kingdom Kinematis and inverse dynamis analysis for a general 3-PRS spatial parallel mehanism Yangmin Li and Qingsong Xu Dept. of Eletromehanial Engineering, Faulty of Siene and Tehnology, University of Maau, Av. Padre Tomás Pereira S. J., Taipa, Maao SAR (P. R. of China) ymli@uma.mo (Reeived in Final Form: June 15, 2004) SUMMARY In this paper, the kinematis and inverse dynamis of a novel kind of mehanism alled a general 3-PRS parallel mehanism is investigated. In the kinematis study, the inverse kinematis solution is derived in losed form, and the forward kinematis problem is resolved by the Newton iterative method seeking for an on-line solution to this issue. The inverse dynamis analysis is approahed with two methods: Lagrangian formulations and priniple of virtual work. After deriving the dynami model by a Lagrangian formulation approah, the simulation results of two introdued examples quantitatively and qualitatively verify the auray of the derived dynami equations. By introduing a simplifying hypothesis, a simplified dynami model is set up using priniple of virtual work, also a omputer simulation is performed on this redued model. The simulation results demonstrate that the simplified dynami model is reasonable under suh kind of assumptions through omparison with the preise model derived from the Lagrangian formulation. The inverse dynamis analysis provides a sound basis to develop ontrollers for ontrolling over a general 3-PRS parallel robot. KEYWORDS: Parallel mehanism; Kinematis; Inverse dynamis; Simulation. 1. INTRODUCTION Parallel robots have reeived inreasing attention due to their inherent advantages over onventional serial mehanism, suh as high rigidity, high load apaity, high veloity, and high preision. A definite advantage of parallel robots is the fat that, in most ases, atuators an be plaed on the truss, thus ahieving a limited weight for the moving parts, whih makes it possible for parallel robots to move at a high speed. The major drawbak of parallel robots is their limited range of motion. Considering the tradeoff between high operational preision and large working volume, parallel atuated mehanisms beome ideal devies for appliations that require high positional auray within a limited workspae. There are many works onerning 6-DOF parallel robots. 1,2 Although general 6-DOF parallel mehanisms, * Corresponding author suh as Gough-Stewart platforms, have attrated onsiderable researh interests in the appliation field of manufature, in a sense it is not neessary to use 6-DOF in most ases. In reent years, parallel mehanisms with less than 6-DOF have attrated the attention of many researhers. Many 3-DOF parallel manipulators have been designed; extensive researh work foused on the famous DELTA robot with three translational DOF, 3 Tsai s manipulator whih was similar to DELTA robot but not a version of DELTA, 4 and other 3-RPS parallel mehanisms whih were exploited as miromanipulators. 5 The stati balaning problem of the 3-RRS spatial parallel mehanisms was disussed by using ounter-weights and springs. 6 Anew type of 4-DOF parallel robots alled H4 was proposed. 7 Although several 3-PRS spatial parallel mehanisms were designed and analyzed, 8,9 and different methods in atuators arrangement in these mehanisms were used, they still ould be onsidered as the same type of mehanisms and be resolved using the same kinematis tehnique. In addition, although the parallel mehanisms with less DOF have been investigated extensively on a kinemati sope, investigations on their dynamis are relatively few. The dynami model of a DELTA parallel robot based on the virtual work priniple was established. 10 The dynamis of a 3-DOF in-parallel atuated manipulator was analyzed using the Lagrange approah. 11 A dynami analysis of a 3-PRS mehanism with the atuators perpendiular to the base platform was performed. 12 The inverse dynamis analysis for a novel 3- DOF spatial parallel manipulator using the Newton-Euler method was investigated. 13 In this paper, we propose a new general type of 3- PRS parallel mehanism by modifying the urrent existed struture of a 3-PRS parallel mehanism, whih will be more suitable for pratial appliations. The kinematis and dynamis analysis for this general 3-PRS parallel mehanism will beome more ompliated than the urrent existed one. 14 Based on our previous researh results, 15 we analyze kinematis with onstraint onditions and obtain a onstrained Jaobian matrix. Hene not only the inverse kinematis problem is resolved in losed form, but also the forward kinematis problem is resolved numerially. The lassial Newton iterative method is exploited for seeking an on-line solution to the forward kinemati problem. The key issue in the dynamis analysis is to establish an inverse dynamis model of the parallel manipulator, whih an solve

2 220 Parallel mehanism Fig. 1. A general 3-PRS parallel mehanism. the required atuator torques given a desired trajetory of the moving platform. The dynamis of the parallel robots, whih is a very important step to design a ontroller, tends to be very ompliated in ase of the atuating legs having losed-loop onnetions with eah other. The losed mehanial hains make the dynamis of parallel robots oupled and highly nonlinear. The rest of the paper is organized as follows: The mehanism is desribed and the kinematis analysis is presented in Setion 2. In Setion 3, the dynami model using Lagrange equation is established. Then, by introduing hypothesis of simplifiation, the dynami equations are derived through the priniple of virtual work. Two examples are hosen to illustrate the dynami simulation and verify the auray of the derived dynami models. And the simulation results also show the effiieny of the adopted simplifiation hypothesis in Setion 4. Some onlusions and future researh diretions are given in Setion KINEMATICS ANALYSIS 2.1. Desription of a general 3-PRS parallel mehanism The arhiteture of a general 3-PRS parallel mehanism is shown in Figure 1, whih is omposed of a moving platform, a fixed base, and three supporting limbs with idential geometrial struture. Eah limb onnets the fixed base to the moving platform by a prismati joint, a revolute joint and a spherial joint in series. A linear atuator atuates the prismati joint of eah limb. Thus, the base is attahed to the moving platform by three idential PRS linkages. Sine the joint DOF of eah limb is equal to five, eah limb provides one onstraint to the moving platform, hene a 3-PRS mehanism has three DOF, whih are two DOF of rotation about two perpendiular axes interseting at the moving platform enter and onstruting a horizontal plane, and one DOF of a vertial translated motion. As shown in Figure 2, a Cartesian referene oordinate frame O{x,y,z} is attahed at the entered point O of the fixed triangle base platform A 1 A 2 A 3. And another oordinate frame P {u,v,w} is attahed on the moving platform at point P whih is the entered point of triangle B 1 B 2 B 3. The x-axis is along OA 1 diretion, and the u-axis is along PB 1 diretion. α is defined as the atuator s arrangement angle. Fig. 2. Shemati representation of a general 3-PRS parallel mehanism. In this study, we assume that β = 120 and γ = 240 for brevity. For a general 3-PRS parallel mehanism, let d = [d 1 d 2 d 3 ] T be a set of the joint spae variables. The transformation from the moving platform to the fixed base an be desribed by a position vetor p = [p x p y p z ] T, and a rotation matrix A R B in [3 3]. Let u, v and w be three unit vetors defined along the u, v and w axis of the moving frame P {u,v,w} aordingly. Then the rotation matrix an be expressed in terms of the diretion osines of u, v and w as u x v x w x A R B = u y v y w y (1) u z v z w z The orientation of the moving platform an be desribed by three Euler angles φ, θ and ψ, whih are angles rotated about z, x, and y axis of the fixed referene frame in sequene, i.e. A R B = R y (θ)r x (ψ)r z (φ) θφ + sψsθsφ θsφ + sψsθφ ψsθ = ψsφ ψφ sψ sθφ + sψθsφ sθsφ + sψθφ ψθ where denotes osine and s denotes sine. A R B an also be desribed by Z-Y-Z Euler angles in terms of θ 1, θ 2,andθ 3. Let B b i be the vetor from P to B i, whih an be expressed in the moving frame as follows: B b 1 = [b 0 0] T B b 2 = [ b/2 3 b/2 0] T B b 3 = [ b/2 3 b/2 0] T The position vetor q i an be obtained by (2) (3) q i = p + A R B B b i (4)

3 Parallel mehanism 221 Substituting Equations (1) and (3) into Equation (4) yields p x + bu x q 1 = p y + bu y p z + bu z p x bu x /2 + 3 bv x /2 q 2 = p y bu y /2 + 3 bv y /2 p z bu z /2 + (5) 3 bv z /2 p x bu x /2 3 bv x /2 q 3 = p y bu y /2 3 bv y /2 p z bu z /2 3 bv z / Constraint onditions Considering the mehanial onstraints imposed by revolute joints, the spherial joints B i an only move in the planes defined by vetors d i and l i (i.e. the ith atuator and the ith leg). Therefore the following three equations hold q 1y = 0 (6) q 2y = 3 q 2x (7) q 3y = 3 q 3x (8) Substituting the omponents of q i from Equation (5) into above three equations, yields p y + bu x = 0 (9) v x = u y (10) p x = b(u x v y )/2 (11) Hene Equations (9) to (11) impose three onstraints on the motion of the moving platform Jaobian matrix analysis From Figure 2, vetor loops of the ith leg an be written as p + b i = a i + d i d i0 + ll i0 (12) where l i0 is a unit vetor of the ith leg in the diretion of C i B i, d i0 represents a unit vetor of the ith linear atuator. The three unit vetors an be expressed as d 10 = [ α 0 sα] T d 20 = [α/2 3 α/2 sα] T (13) d 30 = [α/2 3 α/2 sα] T Vetors a i in the base platform an be expressed as a 1 = [a 0 0] T a 2 = [ a/2 3 a/2 0] T (14) a 3 = [ a/2 3 a/2 0] T Differentiating Equation (12) with respet to time yields υ p + ω p b i = υ i d i0 + lω i l i0 (15) where υ p is the three-dimensional linear veloity of the moving platform, ω p is the angular veloity of the moving platform, υ i represents the ith atuator linear veloity, and ω i is the 3-dimensional angular veloity of the ith leg. The passive variables ω i an be eliminated by dot-multiply both sides of Equation (15) with l i0, that is l i0 υ p + (b i l i0 ) ω p = υ i l i0 d i0 (16) where and represents the dot produt and ross produt between vetors respetively. Let Ẋ p = [ ] T υ p ω p and d = [υ 1 υ 2 υ 3 ] T be the vetor of moving platform veloities and vetor of atuator joint rates respetively. The following equation an be derived from Equation (16) where J x = l T 10 l T 20 J x Ẋ p = J q d (17) (b 1 l 10 ) T (b 2 l 20 ) T l30 T (b 3 l 30 ) T 3 6 l 10 d J q = 0 l 20 d l 30 d When the mehanism is away from singularity, we have d = J Ẋ p (18) where J = J 1 q J x is a matrix in [3 6]. Equation (18) represents the inverse veloity solution of a general 3-PRS parallel mehanism. The onstraint equations (9) (11) an be detailed as p y + bψsφ= 0 (19) θsφ + sψsθφ = ψsφ (20) p x = b (θφ + sψsθsφ ψφ) (21) 2 Equations (19) and (20) yield p y = bψsφ (22) ( ) sψsθ φ = tan 1 ψ + θ (23) Let p z, ψ and θ be speified independent variables. Substitution of these three values into Equations (21) (23) an alulate the onstrained variables p x, p y and φ. Let Ẋ = [υ pz ω px ω py ] T, then Ẋ p = G Ẋ (24)

4 222 Parallel mehanism where Therefore p x p x p x p z ψ θ p y p y p y p z ψ θ G = φ φ φ p z ψ θ Let ϕ i be measured from C i B i to the fixed base platform, then the three vetors q 1, q 2 and q 3 an also be expressed as: a d 1 α lϕ 1 q 1 = 0 d 1 sα lsϕ 1 (a d 2 α lϕ 2 )/2 q 2 = 3(a d2 α lϕ 2 )/2 (31) d 2 sα lsϕ 2 (a d 3 α lϕ 3 )/2 q 3 = 3(a d 3 α lϕ 3 )/2 d = J Ẋ (25) where J = JG is a Jaobian matrix in [3 3] whih inludes the effet of the mehanial onstraints on the mehanism. J is defined as the onstrained Jaobian matrix of a general 3-PRS parallel mehanism Inverse kinematis analysis The inverse kinematis problem resolves the atuated variables from a given position and orientation of the output platform. From Figure 2, we an obtain L i = d i d i0 + ll i0 (26) L i = q i a i (27) where q i is expressed by Equation (4). Sine p and A R B are known values, L i is also known. Equaiton (26) yields L i d i d i0 = ll i0 (28) Squaring both sides of Equation (28) and rearranging it yields d 2 i 2d i L i d i0 + L 2 i l2 = 0 (29) Solving Equation (29), the inverse kinematis solutions an be derived as d i = (L i d i0 ) ± (L i d i0 ) 2 L 2 i + l2 (30) We an see that there exist two solutions for eah atuator, there are eight possible solutions totally for a given platform position and orientation. In this paper, only the negative square root is seleted as a solution in ase of the three legs are inlined inward from top to bottom Forward kinematis analysis Given a set of atuated inputs, the position and orientation of the output platform is resolved by the forward kinematis. The forward kinemati problem an be resolved by nonlinear equations solving method. d 3 sα lsϕ 3 The geometri distane between two spherial joints B i and B j (i j) is equal to a onstant, that is B i B j = 3b (i j). We an obtain three equations as follows: That is [q i q i+1 ] T [q i q i+1 ] 3b 2 = 0 (32) e 1i ϕ i ϕ i+1 + e 2i sϕ i sϕ i+1 + e 3i ϕ i + e 4i sϕ i + e 5i ϕ i+1 + e 6i sϕ i+1 + e 7i = 0 (33) where e 1i = l 2, e 2i = 2l 2, e 3i = [(2d i + d i+1 )α 3a]l, e 4i = 2l(d i d i+1 )sα, e 5i = [(d i + 2d i+1 )α 3a]l, e 6i = 2l(d i d i+1 )sα, e 7i = (d i d i+1 ) 2 + 3d i d i+1 2 α 3a(d i + 2d i+1 )α + 3(a 2 b 2 ) + 2l 2, i = 1, 2, 3. Substituting the trigonometri identities sϕ i = 2t i and 1+ti 2 ϕ i = 1 t i 2 (t 1+ti 2 i = tan ϕ i ) into Equation (33) yields three fourthdegree polynomials in t 1, t 2, and t 3 2 : R 1i t 2 i t 2 i+1 +R 2it i t 2 i+1 +R 3it 2 i t i+1 +R 4i t 2 i +R 5i t 2 i+1 +R 6i t i t i+1 +R 7i t i +R 8i t i+1 +R 9i = 0 (34) where R 1i = e 1i e 3i e 5i + e 7i, R 2i = 2e 4i, R 3i = 2e 6i, R 4i = e 1i e 3i + e 5i + e 7i, R 5i = e 1i + e 3i e 5i + e 7i, R 6i = 4e 2i, R 7i = 2e 4i, R 8i = 2e 6i, R 9i = e 1i + e 3i + e 5i + e 7i, i=1, 2, 3. By applying Bezout s elimination method, the entire system equation an be redued to a 16th-degree polynomial in single variable. 14 Tsai et al. 9 solved the three angles ϕ 1, ϕ 2 and ϕ 3 for a 3-PRS parallel mehanism in ase of three atuator rails parallel to one another and perpendiular to the base platform using Bezout s elimination method and an optimization tehnique. The three angles ϕ 1, ϕ 2 and ϕ 3 for a general 3-PRS parallel mehanism an also be solved by the same methods. One ϕ 1, ϕ 2 and ϕ 3 are solved, the position vetor of the moving platform is obtained by p = 1 3 (q 1 + q 2 + q 3 ) (35)

5 Parallel mehanism 223 Table I. Parameters of a general 3-PRS parallel mehanism. Parameter Value a 400 mm b 200 mm l 550 mm m p 1.0 kg m s 0.2 kg m l 0.2 kg g 0.98 m/s 2 Components of rotation matrix A R B an be found by u = q 1 p b (36) v = q 2 q 3 3b (37) w = u v (38) Then Euler angles ψ, θ and φ an be solved easily. In this paper, the forward kinemati problem is resolved numerially by the lassial Newton iterative method. For a ertain poses of the general 3-PRS parallel mehanism, a system with three equations an be written as f (p z,ψ,θ) = d(p z,ψ,θ) d given = 0 (39) where d(p z,ψ,θ) is the joint spae oordinate vetor generated from the inverse kinemati solution, and d given is the known joint spae oordinate vetor whih an be measured. Let X n denote a given set (p n z,ψn,θ n ), then the following equation holds aording to the Newton iterative method Applying f (Xk ) X ( f (X X k+1 = X k k ) 1 ) f (X k ) (40) X = J (X k ), Equation (40) an be written as X k+1 = X k (J (X k )) 1 (d(x k ) d given ) (41) Starting with an initial estimated value X 0, the iterative proess will end one the maximum of the absolute value of d(x k ) d given is less than a speified tolerane. Sine there are many solutions, it is neessary to start with an initial guess losed to the atual pose of the moving platform. Suh an initial guess an be hosen by either the known desired pose or the pose of a previous point on the trajetory at a short time interval in the past. A omputer program is developed to implement the Newton iterative method. For the kinemati parameters of a general 3-PRS parallel mehanism shown in Table I, let the moving platform be in the Cartesian pose [p z ψ θ] T = [ ] T (42) Table II. Results of the forward kinematis analysis using Newton iterative method. p z (mm) ψ (rad) θ (rad) Then the onstrained variables an be derived by Equations (21) (23). [ px p y φ ] T = [ ] T (43) where the unit of linear displaement is mm and the unit of angular displaement is rad. The joint spae oordinate vetor an be derived from the inverse kinematis solution [d 1 d 2 d 3 ] T = [ ] T (44) where all units of the joint lengths are mm. We assume that d given is given by Equation (44), whih may be measured from joint spae, and we need to find an approximate numerial solution for X. Applying the numerial tehnique presented above with an initial guess of the following equation X 0 = ( 468, 0.35, 0.25) (45) and a speified tolerane 1e-6 as the stopping riterion, the onvergene is reahed after four iterations as shown in Table II. 3. INVERSE DYNAMICS ANALYSIS USING LAGRANGIAN FORMULATION In this setion, we fous on the dynami modeling of a general 3-PRS parallel mehanism using the Lagrange method. For the inverse dynamis problem, the time history of a desired trajetory is given and the problem is to determine the fores of atuators and/or torques required to produe that motion. The first type of Lagrange s equations is appliable to mehanial systems with either holonomi or nonholonomi onstraints. The onstraint equations and their first and seond derivatives must be involved into the equations of motion to produe a number of equations in aordane with the number of unknowns. In this regard, Lagrange s equations of the first type are more suitable for modeling the dynamis of parallel manipulators, whih are ompliated in ase of the existene of multiple losed-loop hains. 14 Theoretially, the dynami analysis an be aomplished by using just three generalized oordinates sine a general 3-PRS parallel mehanism has 3 DOF. However, this will lead to a umbersome expression for the Lagrange funtion due to the omplex kinematis of the mehanism. Instead, the first type of Lagrange s equations introdue three redundant oordinates ϕ 1, ϕ 2 and ϕ 3. Thus we have ϕ 1, ϕ 2, ϕ 3, d 1,

6 224 Parallel mehanism d 2 and d 3 as the generalized oordinates. To derive the dynami equations of a general 3-PRS parallel mehanism, the kineti and potential energies for all omponents of the mehanism must be expressed in terms of the hosen generalized oordinates and their derivatives Constraint equations The onstraint equations are obtained from the fat that the distane between two adjaent spherial joints equals to 3 b,i.e. Ɣ i = B i B i+1 2 3b 2 = [q i q i+1 ] T [q i q i+1 ] 3b 2 = 0, (i = 1, 2 and 3) (46) 3.2. Kineti energy expression The kineti energy of the moving platform an be expressed as K p = K Tp + K Rp (47) where K Tp and K Rp represents translational and rotational kineti energy respetively. Let m p represents the mass of the moving platform, then we have K Tp = 1 2 m p v 2 p (48) where v p = ṗ = [ṗ x ṗ y ṗ z ] T is the linear veloity of the moving platform. Let S = ṘR 1, where R denotes the rotation matrix A R B, then 0 ω pz ω py S = ω pz 0 ω px (49) ω py ω px 0 From Equation (49) we an get the angular veloity of the moving platform ω p = [ω px ω py ω pz ] T (50) Thus the rotational kineti energy an be expressed as K Rp = 1 2 ωt p I p ω p (51) where I p represents the inertial matrix of the moving platform with respet to the fixed referene frame, and it an be expressed as I p = RI p RT (52) where I p is the inertial matrix of the moving platform with respet to the moving oordinate P {u, v, w}, and an be derived by b I p = m b p (53) 0 0 b 2 4 The kineti energy of three links is K l = 3 (K Tli + K Rli ), (i = 1, 2, and 3) (54) i=1 where K Tli and K Rli represents the translational and rotational kineti energy of the ith link, respetively. Let ω li represents the angular veloity of the ith link, then K Rli = 1 2 ωt li I li ω li, (i = 1, 2, and 3) (55) where I li = 1 12 m ll 2 denotes the rotational inertia of the ith link about its mass enter, here m l is the mass of eah link. And the three angular veloities an be expressed as ω l1 = ϕ 1 [0 1 0] T ω l2 = ϕ 2 [ 3/2 1/2 0] T (56) ω l3 = ϕ 3 [ 3/2 1/2 0] T The translational kineti energy of the ith link an be expressed as K Tli = 1 2 m l υli 2, (i = 1, 2, and 3) (57) where υ li = d i + l 2 (ω li l i0 ) is the linear veloity of the mass enter of the ith link. The kineti energy of the three sliders is K s = 3 i=1 ( ) 1 2 m s d 2 i (58) where m s denotes the mass of eah slider. Therefore, the total kineti energy of all omponents is given by 3.3. Potential energy expression The total potential energy is derived by P = m p gp z + K = K p + K l + K s (59) + 3 ( m l gd i sα) i=1 3 ( m s g(d i sα + lsϕ i )) (60) i=1 where g is the gravity aeleration Lagrangian formulation The Lagrange funtion is defined as L = K P (61)

7 Parallel mehanism 225 Therefore the Lagrangian equations of motion an be derived by the six generalized oordinates and their derivatives d dt ( L where q j ) L q j = Q j + k i=1 { ϕj j = 1, 2, 3 q j = d j 3 j = 4, 5, 6 { Tj j = 1, 2, 3 Q j = F j 3 j = 4, 5, 6 λ i Ɣ i q j, (i = 1, 2,...,6) (62) in whih T j (j = 1, 2, 3) denotes the fritional torque of the ith link and F j (j = 4, 5, 6) represents the atuating fore along the diretion of the ith slider. In the following disussion, we assume the fritional torque is zero. For j = 1, 2, and 3, we have Ɣ 1 Ɣ 2 Ɣ 3 λ 1 + λ 2 + λ 3 = d ( ) L L (63) ϕ j ϕ j ϕ j dt ϕ j ϕ j parameters of the general 3-PRS parallel mehanism is given in Table I. Case 1: In the first example, we let the moving platform move straight along z-axis at a onstant speed of 5 mm/s, that is p x = 0 p y = 0 p z = z0 5t θ 1 = 0 θ 2 = 0 θ 3 = 0 where z 0 denotes the initial position of the enter of the moving platform in z-axis diretion, and t is the time variable. We assume that z 0 = 460 mm, whih is within the workspae. One simulation result is shown in Figure 3 in ase of For j = 4, 5, and 6, we have F j 3 = d ( L dt d j 3 ) L d j 3 λ 1 Ɣ 1 d j 3 λ 2 Ɣ 2 d j 3 λ 3 Ɣ 3 d j 3 (64) Equations (63) form a set of three linear equations in three unknowns from whih the three Lagrange multipliers an be determined. One the Lagrange multipliers are found, the atuator fores an be solved from the seond group of Equations (64) Dynami simulations Although it is more pratial to assume the mehanism has two orientation DOF in addition to a third DOF in the z-axis diretion, it is possible that the Cartesian oordinates of the entered point of the moving platform an be ontrolled at a sarifie of orientation DOF, whih an be illustrated by the following two ases. The rotation matrix an be expressed ompatly with three Euler angles in terms of Z-Y-Z Euler angles θ 1, θ 2 and θ 3. The three onstraint Equations (9) (11) yield p x = b 2 (1 θ 2) (2θ 1 ) (65) p y = b 2 (1 θ 2) s(2θ 1 ) (66) θ 3 = θ 1 (67) whih denote the relations between the onstraint and unonstraint variables. The following dynami simulations are implemented using Mathematia software and animations of results are performed with Matlab software. The arhiteture and dynami Fig. 3. Simulation results of ase 1 when α = 30. (a) Time history of joint displaement, veloity, aeleration and fore; (b) Joint fore versus joint displaement.

8 226 Parallel mehanism Table III. The atuated work and total energy net inrements at various atuator layout angles. α (deg.) W a (J ) P (J ) K (J ) U (J ) δw a (%) e e e e e e e e α = 30. From Figure 3, we an see that the three joint displaements, veloities, aelerations, and joint fores (the negative signs mean that the atuator fores are along the diretion of d i0 ) are idential sine the symmetri struture of the mehanism. From viewpoint of energy onservation, the work done by three atuators W a equals to the inrements of total energy of the system U that an be expressed by the kineti and potential energy K and P,i.e. U = V 2 V 1 = (K 2 + P 2 ) (K 1 + P 1 ) = (P 2 P 1 ) + (K 2 K 1 ) = P + K (68) From Figure 3(b) we an see that the relation between joint fore and displaement is approximately linear, therefore, the approximated value of the work done by the three atuators an be expressed as W a 1 2 (F max + F min )(d max d min ) (69) At different atuator layout angles, the orresponding W a and U are listed in Table III. From Table III, we an see that there is a little deviation between W a and U, whih is desribed by δw a = [(W a U)/ U] 100% The deviation partly omes from the fat that the relationship between joint fore and displaement is not linear atually. But we notie that when α = 90, W a equals to U exatly, the simulation result is shown in Figure 4 when α = 90.We an see that the three atuator fores are onstants in terms of F m during the moving platform moves at a onstant veloity, so the work done by three atuators exatly equals to W a = F m (d max d min ) (70) Therefore, the deviation between W a and U is zero. This simulation results verify partially the auray of the derived dynami equations quantitatively. Case 2: In the seond example, we let the moving platform trak a helial path with the radius of r and the pith h. The equation Fig. 4. Simulation results of ase 1 when α = 90. (a) Time history of joint displaement, veloity, aeleration and fore; (b) Joint fore versus joint displaement. of the helial path with respet to the fixed frame is given by x = r sin(ηt) y = r os(ηt) z = z0 + h (71) H t

9 Parallel mehanism 227 Fig. 5. Simulation results of ase 2: time history of joint displaement, veloity, aeleration and fore. Fig. 6. The range of atuator displaements versus atuator layout angles. Table IV. The inrements of total energy at different atuator layout angles. α (deg.) U (J ) where T is the time required to travel one pith, η is the angular frequeny, and t is the time variable. We assume that z0 = 460 mm, r = 5 mm, h = 10 mm, η = π rad/s, and 5 T = 10 s. Substituting equation (71) into Equations (65) and (66) yields Hene we an obtain 2θ 1 + ηt = nπ, n = 0, ±1,... os (θ 2 ) = 1 2r b θ 1 = 1 2 ηt + π 2 ( θ 2 = os 1 1 2r ) b θ 3 = θ 1 The simulation results in ase of the atuator layout angle α = 30 are shown in Figure 5. As the atuator layout angle is hanged, the inrement of total energy U(the work done by atuators) is shown in Table IV. The ranges of atuator displaements and joint fores versus atuator layout angles are shown in Figure 6 and 7, respetively. Fig. 7. The range of atuator fores versus atuator layout angles. When traking the same helial path, we an see from Table IV that the work done by atuators at various atuators layout angles is almost equivalent. From Figure 6 we an see that the atuators an move within the minimum motion range when the atuator layout angle is around 60, but it needs the maximum values of atuated fore at the same time, whih an be observed from Figure 7. This an be explained from the viewpoint of work that is a produt of fore and the orresponding displaement, that is, while doing the idential work, the smaller displaement requires the larger fore. The simulation results verify the auray of the derived dynami equations qualitatively. 4. DYNAMICS MODELING WITH PRINCIPLE OF VIRTUAL WORK 4.1. Simplifying hypothesis For a general 3-PRS parallel mehanism, the omplexity of the dynami model partly omes from the three moving legs.

10 228 Parallel mehanism We an simplify the dynami problem by negleting their rotational inertias aording to the following simplifying hypotheses: The rotational inertias of legs are negleted. The masses of legs are optimally divided into two portions and plaed at their two extremities, i.e. 1/2 at its upper extremity (slider) and 1/2 at its lower extremity (moving platform). Then, the equivalent mass of the slider and the moving platform an be written as ˆm s = m s m l, ˆm p = m p m l Dynami modeling From the kinematis analysis, we know that the onstrained Jaobian matrix of a general 3-PRS parallel mehanism is J, whih is a 3 3 matrix and desribes the relation between the atuators veloity and the derivatives of dependent variables of the moving platform. From Equation (25) we an generate δx = J 1 δd (72) Let f = [f 1 f 2 f 3 ] T be the atuator fore vetor, and δd = [δd 1 δd 2 δd 3 ] T be the orresponding virtual displaement vetor. Let F a = [F x F y F z N x N y N z ] T be the external fores atuated on the moving platform. Sine a general 3-PRS parallel mehanism has three DOF, the moving platform an only bears the fores F = [F z N x N y ] T, whih are a subset of F a, while the other omponents of F a are supported by the passive joint bearings. Assume that δx = [δp z δθ x δθ y ] T be the orresponding virtual displaement vetors. Then by adopting the priniple of virtual work, the following equation an be derived f T δd + G T s δd + F T δx + G T p δx f s T δd f p T δx = 0 (73) where G s = [ ˆm s g sin(α) ˆm s g sin(α) ˆm s g sin(α)] T represents the gravity fore of the sliders, G p = [ ˆm p g 00] T is the gravity fore of the moving platform, f s = [ ˆm s d 1 ˆm s d 2 ˆm s d 3 ] T is the inertial fore of the sliders. The inertial fore vetor of the moving platform is expressed as f p = ˆm p p z 0 0 Î xy [ θ x θ y ] where Î xy denotes the top-left 2 2 sub-matrix of inertial matrix Î p of the moving platform, whih an be expressed by replaing m p in Equation (52) with ˆm p. Substituting Equation (72) into (73) yields ( f T + G T s + F T J 1 + G T p J 1 fs T fp T J 1 ) δd = 0 (74) Sine this equation holds for any virtual displaement δd,we an get f T + G T s + F T J 1 + G T p J 1 f T s f T p J 1 = 0 (75) Taking the transpose of Equation (75) and rearranging it yields f = f s + ( J 1 ) T fp G s ( J 1 ) T Gp ( J 1 ) T F (76) Substituting the inertial fores into Equation (76) yields f = M s d + ( J 1 ) T Mp Ẍ G s ( J 1 ) T Gp ( J 1 ) T F (77) where ˆm s 0 0 [ ] ˆmp 0 M s = 0 ˆm s 0, M p =. 0 Iˆ 0 0 ˆm xy s From Equation (25), we an get Ẋ = (J ) 1 d (78) Differentiate Equation (78) with respet to time yields Ẍ = (J ) 1 d + ( J ) 1 d (79) Substituting Equation (79) into (77), and assuming that there are no external fores, whih leads to f = [ M s + ( J 1 + [( J 1 ) T ( Mp J 1 ) T ( ) Mp J 1 Ẍ] d )] d G s ( J 1 ) T Gp (80) Equation (80) represents the inverse dynamis equation of a general 3-PRS parallel mehanism generated by the priniple of virtual work, whih is more simplified than the one derived through the Lagrange formulation approah, and is more suitable for real-time ontrol Dynami simulation Let the moving platform trak a helial path desribed by Equations (71), and alulate the atuator fores through the derived dynamis model. We ompare the atuator fores generated by the two dynamis models derived through the two different approahes mentioned above, and the deviation an be alulated by δf = [(f pvw f Lag )/f Lag ] 100% (81) where f Lag and f pvw denote the atuator fores generated by either Lagrange equation or priniple of virtual work. From the simulation results shown in Figure 8, we an see that there is a deviation between the two dynamis models sine we introdue a simplifying hypothesis in the seond approah, but the deviation is not very large espeially when α is larger than 20 where δf is within the range of ± 5%. And this result demonstrates that the introdued simplifying hypothesis is reasonable.

11 Parallel mehanism 229 general 3-PRS parallel mehanism. Moreover the modeling methods and valid verifiation approahes presented in this paper an also be applied to other parallel mehanisms with less DOF. Aknowledgments The authors appreiate the fund support from the researh ommittee of University of Maau under grant no.: RG004/03-04S/LYM/FST and RG024/03-04S/LYM/FST. Fig. 8. Fore deviation of two dynamis models. 5. CONCLUSIONS In this paper, the kinematis and inverse dynamis analysis for a general 3-PRS spatial parallel mehanism has been presented. Closed form solutions for the inverse kinematis problem are derived. The forward kinematis problem of a general 3-PRS parallel mehanism is resolved by using Newton iterative numerial method. An illustrated numerial example shows that an aeptable solution an be reahed after a few iterations when using the Newton method, providing that an initial guess is losed enough to the atual solution. The kinematis analysis has laid a good foundation for the dynamis analysis. The dynamis model of a general 3-PRS parallel mehanism is established through two approahes: Lagrange equation and priniple of virtual work. By expressing the kineti and potential energy of all the omponents of a general 3-PRS parallel mehanism in terms of the hosen generalized oordinates, we derive the dynami model using the first type of Lagrange s equation. The simulation results of two ases of study quantitatively from viewpoint of onservation of energy and qualitatively from viewpoint of doing work show the validity of the derived dynamis model. By introduing a simplifying hypothesis, the dynamis model of a general 3-PRS parallel mehanism is derived by means of the priniple of virtual work, whih is more simplified than the one derived by Lagrangian equation. The moving platform is ommanded to trak a helial path, and the atuator fores are alulated with the two dynamis models. By omparing the atuator fores alulated through the two dynamis models, we an see that the effiieny of the adopted hypothesis an simplify the dynami model greatly, and the derived dynamis model an be more suitable for real time ontrol. The study presented here provides a sound basis for the future researh work on the kinematis and dynamis ontrol of a Referenes 1. J.-P. Merlet, Diret Kinematis of Parallel Manipulators, IEEE Trans. on Robotis and Automation 9(6), (1993). 2. K. Harib and K. Srinivasan, Kinemati and Dynami Analysis of Stewart Platform-Based Mahine Tool Sturtures, Robotia 21, Part 1, (1989). 3. R. Clavel, DELTA, a Fast Robot with Parallel Geometry, Pro. of 18th International Symposium on Industrial Robot, Lausanne (1988) pp L.W,Tsai,G.C.WalshandR.E.Stamper, Kinematisofa Novel Three DOF Translational Platform, Pro. of the IEEE Conf. on Robotis and Automation (1996) Vol. 4, pp K.-M. Lee and S. Arjunan, A Three-Degrees-of-Freedom Miromotion in-parallel Atuated Manipulator, IEEE Trans. on Robotis and Automation 7(5), (1991). 6. J. Wang and C. M. Gosselin, Stati Balaning of Spatial Three-Degree-of-Freedom Parallel Mehanism, Mehanism and Mehine Theory 34(3), (1999). 7. F. Pierrot and O. Company, H4: a New Family of 4-DOF Parallel Robots, Pro. of the IEEE/ASME Conf. on Advaned Intelligent Mehatronis (1999) pp J. A. Carretero, R. P. Podhorodeski, M. A. Nahon and C. M. Gosselin, Kinemati Analysis and Optimization of a New Three Degree-of-Freedom Spatial Parallel Manipulator, ASME J. Mehanial Design 122(1), (2000). 9. M.-S. Tsai, T.-N. Shiau, Y.-J. Tsai and T.-H. Chang, Diret Kinemati Analysis of a 3-PRS Parallel Mehanism, Mehanism and Mahine Theory 38(1), (2003). 10. A. Codourey, Dynami Modeling and Mass Matrix Evaluation of the DELTA Parallel Robot for Axes Deoupling Control, Pro. of the IEEE/RSJ Conf. on Intelligent Robots and Systems (1996) Vol. 3, pp K. M. Lee and D. K. Shah, Dynami Analysis of a Three- Degree-of-Freedom in-parallel Atuated Manipulator, IEEE J. Robotis and Automaton 4(3), (1988). 12. M. S. Tsai and W. H. Yuan, Dynami Analysis of a 3 PRS Parallel Mehanism, Pro. of 12th National Symposium on Automation, Taiwan (2001) No. 4303C-1, pp Y.-W. Li, J.-S. Wang, L.-P. Wang and X.-J. Liu, Inverse Dynamis and Simulation of a 3-DOF Spatial Parallel Manipulator, Pro. of the IEEE Conf. on Robotis and Automation (2003) pp L. W. Tsai, Robot Analysis: The Mehanis of Serial and Parallel Manipulator (John Wiley & Sons, New York, USA, 1999). 15. Y. Li and Q. Xu, Kinematis and Stiffness Analysis for a General 3-PRS Spatial Parallel Mehanism, 15th CISM- IFToMM Symposium on Robot Design, Dynamis and Control, Montreal, Canada, (June 14 18, 2004) Romo4 15.