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1 Stiffness Analysis of 3-d.o.f. Overconstrained Translational Parallel Manipulators Anatoly Pashkevich, Damien Chablat, Philippe Wenger the VJM method is based on the expansion of the traditional rigid Abstract— The paper presents a new stiffness modelling model by adding the virtual joints (localized springs), which method for overconstrained parallel manipulators, which is describe the elastic deformations of the links. The VJM technique applied to 3-d.o.f. translational mechanisms. It is based on a is widely used at the pre-design stage. multidimensional lumped-parameter model that replaces the Next section introduces a general methodology to derive the link flexibility by localized 6-d.o.f. virtual springs. In contrast to kinematic and stiffness model. Section 3 describes the manipulator other works, the method includes a FEA-based link stiffness compliant elements and the link stiffness evaluation methods. evaluation and employs a new solution strategy of the Finally in section 4, we apply our method on two application kinetostatic equations, which allows computing the stiffness examples. matrix for the overconstrained architectures and for the singular manipulator postures. The advantages of the developed II. GENERAL METHODOLOGY technique are confirmed by application examples, which deal with comparative stiffness analysis of two translational parallel manipulators. A. Manipulator Architecture Let us consider a general 3 d.o.f. translational parallel I. INTRODUCTION / RELATED WORKS manipulator, which consists of a mobile platform connected to a fixed base by three identical kinematics chains (Fig. 1). Each chain R elative to serial manipulators, parallel manipulators are claimed to offer an improved stiffness-to-mass ratio and better accuracy. This feature makes them attractive for innovative includes an actuated joint “Ac” (prismatic or rotational) followed by a “Foot” and a “Leg” with a number of passive joints “Ps” inside. Certain geometrical conditions are assumed to be satisfied machine-tool structures for high speed machining [1, 2, 3]. When a with respect to the passive joints to eliminate the platform rotations parallel manipulator is used as a Parallel Kinematic Machine and to achieve stability of its translational motions. (PKM), stiffness becomes a very important issue in its design [4, 5, Base 6, 7]. This paper presents a general method to compute the stiffness Ac Ac Ac analysis of 3-dof overconstrained translational parallel F F F manipulators. Ps Ps Ps Generally, the stiffness analysis of parallel manipulators is based Ps L Ps Ps L Ps Ps L Ps Ps Ps Ps Ps Ps Ps on a kinetostatic modeling [8], which proposes a map of the Ps Ps Ps stiffness by taking into account the compliance of the actuated Mobile platform joints. However, this method is not appropriate for PKM whose Fig. 1. Schematic diagram of a general 3-d.o.f. translational parallel legs are subject to bending [9]. manipulator (Ac – actuated joint, Ps – passive joints, F – foot, L - Leg) Several methods exist for the computation of the stiffness Typical examples of such architectures are: matrix: the Finite Element Analysis (FEA) [10], the matrix (a) 3-PUU translational PKM (Fig 2a); where each leg consists structural analysis (SMA) [11], and the virtual joint method (VJM) of a rod ended by two U-joints (with parallel intermediate and that is often called the lumped modeling [8]. exterior axes), and active joint is driven by linear actuator [13]; The FEA is proved to be the most accurate and reliable, however (b) Delta parallel robot (Fig 2b) that is based on the 3-RRPaR it is usually applied at the final design stage because of the high architecture with parallelogram-type legs and rotational active computational expenses required for the repeated re-meshing of the joints [14]; complicated 3D structure over the whole workspace. The SMA also (c) Orthoglide parallel robot (Fig 2c) that implements the 3- incorporates the main ideas of the FEA, but operates with rather PRPaR architecture with parallelogram-type legs and translational large elements – 3D flexible beams describing the manipulator active joints [10]. structure. This leads obviously to the reduction of the Here R, P, U and Pa denote the revolute, prismatic, universal computational expenses, but does not provide clear physical and parallelogram joints, respectively. relations required for the parametric stiffness analysis. And finally, It should be noted that examples (b) and (c) illustrate overconstrained mechanisms, where some kinematic constrains are redundant but do not affect the resulting degrees of freedom. A. Pashkevich is with the IRCCyN (UMR CNRS 6597), Nantes, France However, most of the past works deal with non-overconstrained and with the Department of Automatics and Production Systems, École des architectures, which motivates the subject of this paper [8]. Mines de Nantes, France (anatol.pashkevich@emn.fr); D. Chablat is with the IRCCyN (UMR CNRS 6597), Nantes, France B. Basic Assumptions (Damien.Chablat@irccyn-nantes.fr); P. Wenger is with the IRCCyN (UMR CNRS 6597), Nantes, France To evaluate the manipulator stiffness, let us apply a modification (Philippe.Wenger@irccyn-nantes.fr). 2 (a) 3-PUU translational PKM [13] (b) Delta parallel robot [14] (c) Orthoglide parallel robot [10] Fig. 2. Typical 3 d.o.f. translational parallel mechanisms of the virtual joint method (VJM), which is based on the lump (i) a rigid link from the manipulator leg the end-effector (part of modeling approach [8, 10]. According to this approach, the original the movable platform) described by the constant homogenous i rigid model should be extended by adding the virtual joints matrix transformation Ttool ; (localized springs), which describe elastic deformations of the The expression defining the end-effector location subject to links. Besides, virtual springs are included in the actuating joints to variations of all coordinates of a single kinematic chain may be take into account stiffness of the control loop. To overcome written as follows difficulties with parallelogram modeling, let us first replace the Ti Tbase Va (q0i 0i ) Tfoot Vs (1i , 6i ) i manipulator legs (see Fig. 3) by rigid links with configuration- (1) dependent stiffness. Vu1 (q1i , q2 ) Tleg Vs (7i , 12 ) Vu 2 (q3 , q4 ) Ttool i i i i i This transforms the general architecture into the extended 3- where matrix function Va (.) is either an elementary rotation or xUU case allowing treating all the considered manipulators in the translation, matrix functions Vu1 (.) and Vu 2 (.) are compositions of similar manner. Under such assumptions, each kinematic chain of two successive rotations, and the spring matrix Vs (.) is composed the manipulator can be described by a serial structure (Fig. 3), of six elementary transformations. In the rigid case, the virtual joint which includes sequentially: coordinates 0i , 12 are equal to zero, while the remaining ones i Base platform (rigid) End-effector i (both active q0 and passive q0i , q4i ) are obtained through the (rigid) inverse kinematics, ensuring that all three matrices Ti , i 1,2,3 Ac Rigid Foot U Rigid Leg U are equal to the prescribed one that characterizes the spatial 1-d.o.f. 6-d.o.f. 6-d.o.f. location of the moving platform (kinematic loop-closure spring spring spring equations). Particular expressions for all components of the product Fig. 3. Flexible model of a single kinematic chain (1) may be easily derived using standard techniques for the (a) a rigid link between the manipulator base and the ith homogenous transformation matrices. It should be noted that the actuating joint (part of the base platform) described by the constant kinematic model (1) includes 18 variables (1 for active joint, 4 for i homogenous transformation matrix Tbase ; passive joints, and 13 for virtual springs). However, some of the (b) a 1-d.o.f. actuating joint with supplementary virtual spring, virtual springs are redundant, since they are compensated by which is described by the homogenous matrix function corresponding passive joints with aligning axes or by combination Va (q0i 0i ) where q 0 is the actuated coordinate and 0 is the i i of passive joints. For computational convenience, nevertheless, it is virtual spring coordinate; not reasonable to detect and analytically eliminate redundant (c) a rigid “Foot” linking the actuating joint and the leg, which is variables at this step, because the developed below technique described by the constant homogenous transformation matrix Tfoot ; allows easy and efficient computational elimination. (d) a 6-d.o.f. virtual joint defining three translational and three C. Differential Kinematic Model rotational foot-springs, which are described by the homogenous matrix function Vs (1i , 6i ) , where {1i , 2i , 3i } and {4i , 5i , 6i } To evaluate the manipulator ability to respond to the external correspond to the elementary translations and rotations forces and torques, let us first derive the differential equation respectively; describing relations between the end-effector location and small (e) a 2-d.o.f. passive U-joint at the beginning of the leg allowing variations of the joint variables. For each ith kinematic chain, this two independent rotations with angles {q1i , q2} , which is described i equation can be generalized as follows i i by the homogenous matrix function Vu1 (q1 , q2 ) ; t i J i θi J iq qi , i 1,2,3 , (2) (f) a rigid “Leg” linking the foot to the movable platform, which is described by the constant homogenous matrix transformation where the vector δt i (δpxi , δp yi , δpzi , δ xi , δ yi , δ zi ) describes T Tleg ; the translation δpi (δpxi , δp yi , δpzi )T and the rotation (g) a 6-d.o.f. virtual joint defining three translational and three δi (δ xi , δ yi , δzi )T of the end-effector with respect to the rotational leg-springs, which are described by the homogenous Cartesian axes; vector θi (0i , 12 )T collects all virtual i matrix function Vs (7i , 12 ) , where {7i , 8i , 9i } and i joint coordinates, vector qi (q1i , q4i )T includes all passive {10 , 12 , 12} correspond to the elementary translations and i i i joint coordinates, symbol '' stands for the variation with respect rotations, respectively; to the rigid case values, and J , J q are the matrices of sizes 613 (h) a 2-d.o.f. passive U-joint at the end of the leg allowing two and 64 respectively. It should be noted that the derivative for the i i independent rotations with angles {q3 , q4} , which is described by actuated coordinate q 0i is not included in J q but it is represented i the homogenous matrix function Vu 2 (q3 , q4 ) ; i in the first column of J through variable 0i . The desired matrices J , J q , which are the only parameters of the differential model (2), may be computed from (1) analytically, using some 3 software support tools, such as Maple, MathCAD or Mathematica. should be stressed that, in contrast to other works, these matrices However, a straightforward differentiation usually yields very are assumed to be non-diagonal. This allows taking into account awkward expressions that are not convenient for further complicated coupling between rotational and translational computations. On the other hand, the fractionized structure of (1), deformations, while usual lump-based approach does consider this where all variables are separated, allows applying an efficient semi- phenomena [8]. For analytical convenience, expressions (6) may be analytical method. To present this technique, let us assume that for collected in a single matrix equation the particular virtual joint variable 0i the model (1) is rewritten as τ i K θ θi , i 1, 2,3 (7) Ti H V j ( ) H , 1 i 2 (3) ij j ij where τ ( , ) is the aggregated vector of the virtual i i 0 i 12 T where the first and the third multipliers are the constant joint reactions, and K θ diag ( Kact , K Foot , K Leg ) is the aggregated homogenous matrices, and the second multiplier is the elementary spring stiffness matrix of the size 1313. Similarly, one can define translation or rotation. Then the partial derivative of the the aggregated vector of the passive joint reactions homogenous matrix Ti for the variable ij at point ij 0 may be τiq ( qi1 , qi4 )T but all its components must be equal to zero: computed from a similar product where the internal term is τiq 0, i 1,2,3 (8) replaced by V j (.) that admits very simple analytical presentation. In particular, for the elementary translations and rotations about the To find the static equations corresponding to the end-effector X-axis, these derivatives are: motion t i , let us apply the principle of virtual work assuming that the joints are given small, arbitrary virtual displacements 0 0 0 1 0 0 0 0 (θi , qi ) in the equilibrium neighborhood. Then the virtual work VTranx 0 0 0 0 ; V 0 0 1 0 . (4) 0 0 0 0 Rot x 0 1 0 0 of the external force fi applied to the end-effector along the 0 0 0 0 0 0 0 0 corresponding displacement t i J iθi J iq qi is equal to the Furthermore, since the derivative of the homogenous matrix sum (fiT J i )θi (fiT J iq )qi . For the internal forces, the virtual work is τ iθ θi since the passive joints do not produce the T Ti H1 V j ( ij ) Hij may be presented as ij 2 force/torque reactions (the minus sign takes into account the 0 iz iy pix iz 0 ix piy adopted directions for the virtual spring forces/torques). Therefore, Ti , (5) because in the static equilibrium the total virtual work is equal to iy ix 0 piz zero for any virtual displacement, the equilibrium conditions may 0 0 0 0 be written as then the desired jth column of J can be extracted from Ti (using J i fi τ i ; J iq fi 0 . T T the matrix elements T14 , T24 , T34 , T23 , T31 , T12 ). (9) The Jacobians J q can be computed in a similar manner, but the This gives additional expressions describing the force/torque derivatives are evaluated in the neighborhood of the “nominal” propagation from the joints to the end-effector. Hence, the values of the passive joint coordinates q ij nom corresponding to the complete kinetostatic model consists of five matrix equations (2), rigid case (these values are provided by the inverse kinematics). (7)…(9) where either fi or t i are treated as known, and the However, simple transformation qij qij nom qij and remaining variables are considered as unknowns. Obviously, since corresponding factoring of the function separate kinematic chains posses some degrees-of-freedom, this Vq j (qij ) Vq j (qij nom )Vq j ( qij ) allow applying the above approach. system cannot be uniquely solved for given fi . However, vice It is also worth mentioning that this technique may be used in versa, for given end-effector displacement t i , it is possible to analytical computations, allowing one to avoid bulky compute both the corresponding external force fi and the internal transformations produced by the straightforward differentiating. variables, θi , τ i , qi (i.e. virtual spring reactions and D. Kinetostatic and Stiffness Models displacements in passive joints, which may also provide useful For the manipulator kinetostatic model, which describes the information for the designer). Since matrix K θ is non-singular (it force-and-motion relation, it is necessary to introduce additional describes the stiffness of the virtual sprigs), the variable θi can be expressed via fi using equations τi K θ θi and J i fi τ i . T equations that define the virtual joint reactions to the 1 This yields substitution θi (K θ J i ) fi allowing reducing the T corresponding spring deformations. In accordance with the adopted stiffness model, three types of virtual springs are included in each kinetostatic model to system of two matrix equations 1 J i ) fi J iq q i t i ; J iq fi 0 T T kinematic chain: (J i K θ (10) 1-d.o.f. virtual spring describing the actuator compliance; 6-d.o.f. virtual spring describing compliance of the foot; with unknowns fi and qi . This system can be also rewritten in a 6-d.o.f. virtual spring describing compliance of the leg. matrix form Assuming that the spring deformations are small enough, the Siθ J iq fi t i required relations may be expressed by linear equations iT (11) J q 0 qi 0 i i i i 1 K act i0 ; K Foot ; K Leg , 1 7 where the sub-matrix Siθ J i K θ J i describes the spring T i i (6) i6 i 12 i i compliance relative to the end-effector, and the sub-matrix J q takes into account the passive joint influence on the end-effector where j is the generalized force for the jth virtual joint of the ith i motions. Therefore, for a separate kinematic chain, the desired kinematic chain, Kact is the actuator stiffness (scalar), and, K Foot , stiffness matrix K i defining the motion-to-force mapping K Leg are 66 stiffness matrices for the foot and leg respectively. It fi Ki ti , can be computed by direct inversion of relevant 4 1010 matrix in the left-hand side of (11) and extracting from it the of the chains may include several redundant springs that are totally 66 sub-matrix with indices corresponding to S iθ . It is also worth compensated by relevant passive joints. However, there is no need mentioning that computing S iθ requires 66 inversions only, since to eliminate these springs from the model manually, since they do 1 K 1 diag ( Kact , K 1 , K 1 ) . Solvability of system (11) in general Foot Leg not increase the matrix sizes in system (11). This allows including case, i.e. for any given J i and J iq , cannot be proved. Moreover, if in the model 6-d.o.f. virtual springs of general type, without any the matrix J iq is singular, the passive joint coordinates qi can not modifications. Another advantage of the proposed technique is that be found uniquely. From a physical point of view, it means that if it can be generalized easily. Within this paper, it is applied to the the kinematic chain is located in a singular posture, then certain stiffness modeling of 3-d.o.f. translational manipulators with displacements t i can be generated by infinite combinations of the actuators located between the base and the foot. However, it can be passive joints. But for the variable fi the corresponding solution is easily modified to cover other actuator locations, which may be unique (since the matrix J i is obviously non-singular if at least included in the foot or in the leg. A further generalization is related one 6 d.o.f. spring is included in a serial kinematic chain). On the to a number of kinematic chains and their similarity. They are also other hand, the singularity may produce an infinite number of not crucial assumptions and influence on the Jacobian computing stiffness matrices for the same spatial location of the end-effector only. But after the Jacobians are determined, the stiffness matrices and for different values qi provided by the inverse kinematics. A for separate chains may be computed in the same manner and then special technique to tackle this case, based on the singular value aggregated. decomposition, has been also developed. After the stiffness matrices K i for all kinematic chains are computed, the stiffness of III. PARAMETERS OF THE COMPLIANT ELEMENTS the entire manipulator can be found by simple addition The adopted stiffness model of each kinematic chain includes K m 31 K i . This follows from the superposition principle, i three compliant components, which are described by one 1-d.o.f. because the total external force corresponding to the end-effector spring and two 6-d.o.f. springs corresponding to the actuator, and displacement t (the same for all kinematic chains) can be to the foot/leg links (see Fig. 3). Let us describe particular expressed as f 31 fi where fi Ki t . It should be stressed i techniques for their evaluation. that the resulting matrix K i is not invertible, since some motions of the end-effector do not produce the virtual spring reactions A. Actuator Compliance (because of passive joints influence). However, for the entire The actuator compliance, described by the scalar parameter manipulator, the stiffness matrix K m is s positive definite and 1 kact Kact , depends on both the servomechanism mechanics and invertible for all non-singular (for the rigid model) postures. the control algorithms. Since most of modern actuators implement E. Comparison with Other Results the digital PID control, the main contribution to kact is done by the mechanical transmissions. The latter are usually located outside the The main advantage of the proposed methodology is its feedback-control loop and consist of screws, gears, shafts, belts, applicability to overconstrained mechanisms. To describe it in etc., whose flexibility is comparable with the flexibility of the more details, let us briefly review an alternative technique [8]. The manipulator links. Because of the complicated mechanical structure latter is originated from the same principal equations but the of the servomechanisms, the parameter kact is usually evaluated solution strategy includes straightforward elimination of the from static load experiments, by applying the linear regression to passive joint variables qi using the differential kinematic the experimental data. equations (2) only. Obviously, the feasibility of this step depends on the solvability of the equivalent matrix system B. Link Compliance I J 1 t J1 θ1 Following a general methodology, the compliance of a q1 q I J 2 J 2 θ 2 (12) manipulator link (foots and legs) is described by 66 symmetrical 3 q 2 J 3 θ3 q I J q q 1 positive definite matrices K leg , K 1 corresponding to 6-d.o.f. 3 foot springs with relevant coupling between translational and rotational where t and qi are treated as unknowns. In the non- deformations. This distinguishes our approach from other lumped constrained case (for the 3-PUU architecture, for instance) the modeling techniques, where the coupling is neglected and only a matrix in the left-hand side of (14) is square, of size 1818, so it subset of deformations is taken into account (presented by a set of can be inverted usually. However, for overconstrained 1-d.o.f. springs). The simplest way to obtain these matrices is to manipulators, this matrix is non-square, so the system cannot be approximate the link by a beam element for which the non-zero solved uniquely. For example, for manipulators with the elements of the compliance matrix may be expressed analytically: parallelogram-type legs (Orthoglide, Delta, etc.) the matrix size is L L3 L3 L L 1815. So, in [10] three additional (virtual) passive joints were k11 ; k22 ; k33 ; k44 ; k55 ; EA 3EI z 3EI y GJ EI y introduced to solve the problem. But, obviously, such a (13) modification changes the manipulator architecture and its stiffness L L2 L2 k66 ; k35 ; k26 matrix, doubting validity of the corresponding model. Besides, the EI z 2EI y 2EI z developed technique allows computing the stiffness matrix even for the singular manipulator postures and does not incorporate the Here L is the link length, A is its cross-section area, Iy, Iz, and J are least-square pseudo-inversions applied by other authors. This is the quadratic and polar moments of inertia of the cross-section, and achieved by applying another solution strategy, which considers E and G are the Young’s and Coulomb’s modules respectively. simultaneously the kinematic and static-equilibrium equations for However, for certain link geometries, the accuracy of a single-beam each kinematic chain separately. Some hidden conveniences are approximation can be insufficient. In this case the link can be included in the modeling stage. In particular, the kinematic models approximated by a serial chain of the beams, whose compliance is 5 evaluated by applying the same method (i.e. considering the B. Stiffness of Parallelogram Based Manipulator kinematic chain with 6-d.o.f. virtual springs, but without passive Before evaluation the compliance of the entire manipulator, let joints). This leads to the resulting compliance matrix us derive the stiffness matrix of the parallelogram. Using the K 1 Jb Kb1 JT , where J b and Kb1 incorporate the Jacobian and Link b adopted notations, the parallelogram equivalent model may be the compliance matrices for all virtual springs. written as C. FEA-based evaluation of stiffness TPlg R y (q2 ) Tx ( L) R y (q2 ) Vs (7 , 12 ) (14) For complex link geometries, the most reliable results can be where, compared to the above case, the third passive joint is obtained from the FEA modeling. To apply this approach, the CAD eliminated (it is implicitly assumed that q3 q2 ). On the other model of each link should be extended by introducing an auxiliary hand, the original parallelogram may be split into two serial 3D object, a “pseudo-rigid” body, which is used as a reference for kinematic chains (the “upper” and “lower” ones) the compliance evaluation. Besides, the link origin must be fixed relative to the global coordinate system. Then, sequentially and Tup Tz (d /2) R y (q q1up ) Tx ( L) (15) separately applying forces Fx , Fy , Fz and torques M x , M y , M z to Vs (1up , 6up ) R y (q q2 ) Tz (d /2) up the reference object, it is possible to evaluate corresponding linear and angular displacements, which allow computing the stiffness Tdn Tz (d /2) R y (q q1dn ) Tx ( L) matrix columns. The main difficulty here is to obtain accurate (16) displacement values by using proper FEA-discretization (“mesh Vs (1dn , 6dn ) R y (q q2 ) Tz (d /2) dn size”). Besides, to increase accuracy, the displacements must be where L, d are the parallelogram geometrical parameters, evaluated using redundant data set describing the reference body q1 , q2 , i {up, dn} are the variations of the passive joint i i motion. For this reason, it is worth applying a dedicated SVD- coordinates and the sub/superscripts “up” and “dn” correspond to based algorithm. As follows from our study, the single-beam the upper and lower chain respectively. Hence, the parallelogram approximation of the Orthoglide foot gives accuracy of about 50%, compliance matrix may be also derived using the proposed and the four-beam approximation improves it up to 30% only. technique that yields an analytical expression While the FEA-based method is the most accurate one, it is also the most time consuming. However, in contrast to the straightforward K11 0 0 0 0 0 0 K 22 0 0 0 K 26 FEA-modeling of the entire manipulator, which requires re- 0 0 0 0 0 0 computing for each manipulator posture, the proposed technique 2 d 2Cq K 22 d 2 S2 q K 22 0 0 0 K 44 0 (17) involves a single evaluation of link stiffness. K Plg 2 4 2 2 8 d Cq K11 0 0 0 0 4 0 IV. APPLICATION EXAMPLES 2 d S2 q K 22 d 2 Sq K 22 2 0 K 26 0 0 K 66 To demonstrate efficiency of the proposed methodology, let us 8 4 apply it to the comparative stiffness analysis of two where Cq cos(q); Sq sin(q) . Using this model and applying the 3-d.o.f. translational mechanism, which employ Orthoglide proposed technique, we computed the compliance matrices for architecture. CAD models of these mechanisms are presented in three typical manipulator postures (see table Table 1). As follows Fig. 4. from the comparison with the U-joint case, the parallelograms A. Stiffness of U-Joint Based Manipulator allow increasing the rotational stiffness roughly in 10 times. This justifies application of this architecture in the Orthoglide prototype First, let us derive the stiffness model for the simplified Orthoglide design [15]. mechanics, where the legs are comprised of equivalent limbs with U-joints at the ends. Accordingly, to retain major compliance properties, the limb geometry corresponds to the parallelogram bars V. CONCLUSIONS with doubled cross-section area. Let us assume that the world The paper proposes a new systematic method for computing the coordinate system is located at the end-effector reference point stiffness matrix of overconstrained parallel manipulators. It is based corresponding to the isotropic manipulator posture (when the legs on multidimensional lumped model of the flexible links, whose are mutually perpendicular and parallel to relevant actuator axes). parameters are evaluated via the FEA modeling and describe both For this assumption, the geometrical models of separate kinematic the translational/rotational compliances and the coupling between chains can be described by the expression (1) Because for the rigid them. In contrast to previous works, the method employs a new manipulator the end-effector moves with only translational solution strategy of the kinetostatic equations, which considers motions, the nominal values of the passive joint coordinates are simultaneously the kinematic and static relations for each separate subject to the specific constraints q3 q2 ; q4 q1 , which are kinematic chain and then aggregates the partial solutions in a total implicitly incorporated in the direct/inverse kinematics [10]. one. This allows computing the stiffness matrices for However, the flexible model allows variations for all passive joints. overconstrained mechanisms for any given manipulator posture, Using the link stiffness parameters obtained by the FEA-modeling including singular configurations and their neighborhood. Another and applying the proposed methodology, we computed the advantage is computational simplicity that requires low- compliance matrices for three typical manipulator postures, the dimensional matrix inversion compared to other techniques. principal components of which are presented in Table 1. Below, Besides, the method does not require manual elimination of the they are compared with the compliance of the parallelogram-based redundant spring corresponding to the passive joints, since this manipulator. operation is inherently included in the numerical algorithm. The efficiency of the proposed method was demonstrated through 6 application examples, which deal with comparative stiffness on Fundamental Issues and Future Research Directions for Parallel analysis of two parallel manipulators of the Orthoglide family (with Mechanisms and Manipulators, Quebec, Canada, October 2002. [7] A. Paskhevich, P. Wenger and D. Chablat, “Kinematic and stiffness U-joint based and parallelogram based links). Relevant simulation analysis of the Orthoglide, a PKM with simple, regular workspace and results have confirmed essential advantages of the parallelogram homogeneous performances,” In: IEEE International Conference On based architecture and validated adopted design of the Orthoglide Robotics And Automation, Rome, Italy, April 2007 prototype. Another contribution is the analytical stiffness model of [8] C.M. Gosselin, “Stiffness mapping for parallel manipulators,” IEEE the parallelogram, which was derived using the same methodology. Transactions on Robotics and Automation, vol. 6, pp. 377–382, 1990. While applied to the 3-d.o.f. translational mechanisms, the method [9] X. Kong and C. M. 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Mechanism and Machine Science, IFToMM, Besançon, June 2007. [6] T. Brogardh, “PKM Research - Important Issues, as seen from a Product Development Perspective at ABB Robotics,” In: Workshop A3 A3 A3 B3 i1 B3 B3 A2 i1 A1 j1 A2 j1 A2 C3 B2 B1 A1 C3 C3 B2 y x B1 A1 z C2 y x B1 C1 C2 y x z C1 C2 P P z C1 P (A) U-JOINT BASED ARCHITECTURE (B) PARALLELOGRAM BASED ARCHITECTURE (C) WORKSPACE AND CRITICAL POINTS Q1 AND Q2 FIG. 4. KINEMATICS OF TWO 3-DOF TRANSLATIONAL MECHANISMS EMPLOYING THE ORTHOGLIDE ARCHITECTURE TABLE I: TRANSLATIONAL AND ROTATIONAL STIFFNESS OF THE 3-PUU AND 3-PRPAR MANIPULATORS MANIPULATOR Point Q0 Point Q1 Point Q2 ARCHITECTURE x, y, z 0.00 mm x, y, z 73.65 mm x, y, z 126.35 mm ktran [N/mm] krot [Nmm/rad] ktran [N/mm] krot [Nmm/rad] ktran [N/mm] krot [Nmm/rad] 3-PUU manipulator 2.7810-4 20.910-7 10.910-4 24.110-7 71.310-4 25.810-7 3-PRPaR manipulator 2.7810-4 1.9410-7 9.8610-4 2.0610-7 21.210-4 2.6510-7

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