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                            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                              δi  (δ xi , δ yi , δzi )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 613
    (h) a 2-d.o.f. passive U-joint at the end of the leg allowing two                               and 64 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 1313. 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 iθi  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 66 stiffness matrices for the foot and leg respectively. It                    fi  Ki  ti , can be computed by direct inversion of relevant
                                                                                                                                                       4




1010 matrix in the left-hand side of (11) and extracting from it the           of the chains may include several redundant springs that are totally
66 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 66 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   31 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   31 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 66 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 1818, 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
1815. 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. Gosselin, “Kinematics and Singularity Analysis of
                                                                                                                a Novel Type of 3-CRR 3-DOF Translational Parallel Manipulator,”
can be extended to other parallel architectures composed of several
                                                                                                                The International Journal of Robotics Research, vol. 21(9), pp. 791–
kinematic chains with rotational/prismatic joints and limb- or                                                  798, September 2002.
parallelogram-based links. So, future work will focus on the                                          [10]      F. Majou, C. Gosselin, P. Wenger and D. Chablat. “Parametric
stiffness modeling of more complicated parallel mechanism with                                                  stiffness analysis of the Orthoglide,” Mechanism and Machine
another actuator location (such as the Verne machine [16]) and also                                             Theory, vol. 42(3), pp. 296–311, March 2007.
on the experimental verification of the stiffness models for the                                      [11]      B.C. Bouzgarrou, J.C. Fauroux, G. Gogu and Y. Heerah, “Rigidity
                                                                                                                analysis of T3R1 parallel robot uncoupled kinematics,” In: Proc. of
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                                                                                                                March 2004.
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                                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 [Nmm/rad]                    ktran [N/mm]    krot [Nmm/rad] ktran [N/mm]               krot [Nmm/rad]

                   3-PUU manipulator                    2.7810-4                    20.910-7                   10.910-4         24.110-7              71.310-4            25.810-7

               3-PRPaR manipulator                      2.7810-4                    1.9410-7                   9.8610-4         2.0610-7              21.210-4            2.6510-7