Submitted to ICAR 2003   D. Chablat Ph. Wenger F. Majou 1/6  The Optimal Design of Three Degree-of-Freedom Parallel Mechanisms for Machining Applications  Damien Chablat - Philippe Wenger – Félix Majou Institut de Recherche en Communications et Cybernétique de Nantes 1 1, rue de la Noë, 44321 Nantes, France Damien.Chablat@irccyn.ec-nantes.fr 1   IRCCyN: UMR CNRS 6596, École Centrale de Nantes, Université de Nantes, École des Mines de Nantes  Abstract  The subject of this paper is the optimal design of a parallel mechanism   intended   for   three-axis   machining applications.   Parallel   mechanisms   are   interesting alternative designs in this context but most of them are designed for three- or six-axis machining applications. In the last case, the position and the orientation of the tool are coupled and the shape of the workspace is complex. The   aim   of   this   paper   is   to   use   a   simple   parallel mechanism   with   two-degree-of-freedom   (dof)   for translational motions and to add one leg to have one-dof rotational   motion.   The   kinematics   and   singular configurations   are studied   as well   as an   optimization method.   The   three-degree-of-freedom   mechanisms analyzed   in   this   paper   can   be   extended   to   four-axis machines by adding a fourth axis in series with the first two.  Key Words : Parallel Machine Tool, Isotropic Design, and Singularity.  1   Introduction  Parallel   kinematic   machines   (PKM)   are   commonly claimed   to   offer   several   advantages   over   their   serial counterparts, like high structural rigidity, high dynamic capacities   and   high   accuracy   [1].   Thus,   PKM   are interesting alternative designs for high-speed machining applications. The first industrial application of PKMs was the Gough platform, designed in 1957 to test tyres [2]. PKMs have then been used for many years in flight simulators and robotic applications [3] because of their low moving mass and high dynamic performances [1]. This is why parallel kinematic   machine   tools   attract   the   interest   of   most researchers   and   companies.   Since   the   first   prototype presented   in   1994   during   the   IMTS   in   Chicago   by Gidding&Lewis (the VARIAX), many other prototypes have appeared. To design a parallel mechanism, two important problems must be solved. The first one is the identification of singular configurations, which can be located inside the workspace. For a six-dof parallel mechanism, like the Gough-Stewart   platform,   the   location   of   the   singular configurations is very difficult to characterize and can change under small variations in the design parameters [3]. The second problem is the non-homogeneity of the performance   indices   (condition   number,   stiffness...) throughout the workspace [1]. To the authors' knowledge, only one parallel mechanism is isotropic throughout the workspace   [4]   but   the   legs   are   subject   to   bending. Moreover, this concept is limited to three-dof mechanisms and   cannot   be   extended   to   four   or   five-dof   parallel mechanisms. Numerous   papers   deal   with   the   design   of   parallel mechanisms [4,5]. However, there is a lack of four- or five-dof   parallel   mechanisms,   which   are   especially required for machining applications [6]. To decrease the cost of industrialization of new PKM and to reduce the problems of design, a modular strategy can be applied. The translational and rotational motions can be divided into two separated parts to produce a mechanism where the direct kinematic problem is decoupled. This simplification   yields   also   some   simplifications   in   the definition of the singular configurations. The organization of this paper is as follows. Next section presents design problems of parallel mechanisms. The kinematic   description   and   singularity   analysis   of   the parallel mechanism used, are reported in sections 3.1 and 3.2. Sections 3.3 and 3.4 are devoted to design and the optimization.
Submitted to ICAR 2003   D. Chablat Ph. Wenger F. Majou 2/6  2   About parallel kinematic machines 2.1   General remarks  In a PKM, the tool is connected to the base through several kinematic chains or legs that are   mounted in parallel. The legs are generally made of telescopic struts with fixed foot points (Figure 1a), or fixed length struts with moveable foot points (Figure 1b).  P   P Figure 1a: A bipod PKM   Figure 1b: A biglide PKM For machining applications, the second architecture is more appropriate because the masses in motion are lower. The linear joints can be actuated by means of linear motors or by conventional rotary motors with ball screws. A classification of the legs suitable to produce motions for parallel kinematic machines is provided by [6] with their degrees of freedom and constraints. The connection of identical or different kinematic legs permits the authors to define   two-,   three-,   four-   and   five-dof   parallel mechanisms. However, it is not possible to remove one leg from a four-dof to produce a three-dof mechanism because no modular approach is used.  2.2   Singularities  The singular configurations (also called singularities) of a PKM   may   appear   inside   the   workspace   or   at   its boundaries. There are two types of singularities [7]. A configuration where a finite tool velocity requires infinite joint   rates   is   called   a   serial   singularity.   These configurations   are   located   at   the   boundary   of   the workspace. A configuration where the tool cannot resist any effort and in turn, becomes uncontrollable is called a parallel singularity. Parallel singularities are particularly undesirable because they induce the following problems (i) a high increase in forces in joints and links, that may damage the structure, and (ii) a decrease of the mechanism stiffness that can lead to uncontrolled motions of the tool though actuated joints are locked. Figures   2a and   2b show the singularities for the biglide mechanism of Fig.   1b. In Fig. 2a, we have a serial singularity. The velocity amplification factor along the vertical direction is null and the force amplification factor is infinite. Figure   2b shows a parallel singularity. The velocity amplification factor is infinite along the vertical direction and the force amplification factor is close to zero. Note that a high velocity amplification factor is not necessarily desirable   because   the   actuator   encoder   resolution   is amplified and thus the accuracy is lower.  P  P Figure 2a: A serial singularity Figure 2b: A parallel singularity The determination of the singular configurations for two- dof mechanisms is very simple; conversely, for a six-dof mechanism   like   the   Gough-Stewart   platform,   a mechanism with six-dof, the problem is very difficult [3]. With a modular architecture, when the position and the orientation of the mobile platform are decoupled, the determination of the singularities is easier.  2.3   Kinetostatic performance of parallel mechanism  Various performance indices have been devised to assess the   kinetostatic   performances   of   serial   and   parallel mechanisms. The literature on performance indices   is extremely rich to fit in the limits of this paper (service angle, dexterous workspace and manipulability...) [9]. The main problem of these performance indices is that they do not take into account the location of the tool frame. However, the Jacobian determinant depends on this location [9] and this location depends on the tool used. To the authors' knowledge there is no parallel mechanism, suitable   for   machining,   for   which   the   kinetostatic performance   indices   are   constant   throughout   the workspace (like the condition number or the stiffness...). For a serial three-axis machine tool, a motion of an actuated joint yields the same motion of the tool (the transmission factors are equal to one). For a parallel machine,   these   motions   are   generally   not   equivalent. When the mechanism is close to a parallel singularity, a small joint rate can generate a large velocity of the tool. This means that the positioning accuracy of the tool is lower in some directions for some configurations close to parallel singularities because the encoder resolution is
Submitted to ICAR 2003   D. Chablat Ph. Wenger F. Majou 3/6 amplified. In addition, a high velocity amplification factor in one direction is equivalent to a loss of stiffness in this direction. The manipulability ellipsoid of the Jacobian matrix of robotic manipulators was defined two decades ago [8]. The   JJ -1   eigenvalues square roots,   γ 1   and   γ 2 , are the lengths of the semi-axes of the ellipse that define the two velocity amplification factors between the actuated joints velocities and the velocity vector   t &   ( λ 1   = 1/ γ 1   and  λ 2   =   1/ γ 2 ).   For   parallel   mechanisms   with   only   pure translation or pure rotation motions, the variations of these factors inside the Cartesian workspace can be limited by the following constraints min   max i   i   i  λ   λ   λ ≤   ≤  Unfortunately, this concept is quite difficult to apply when the   tool   frame   can   produce   both   rotational   and translational motions. In this case, indeed the Jacobian matrix is not homogeneous [9]. A first way to solve this problem is its normalization by computing its characteristic length [9-10]. The second approach is to limit the values of all terms of the Jacobian matrix to avoid singular configuration and to associate these values to a physical measurement (See section 3.4).  3   Kinematics of mechanisms studied 3.1   Kinematics   of   a   parallel   mechanism   for translational motions  The aim of this section is to define the kinematics and the singular   configuration   of   a   two-dof   translational mechanism (Figure 3), which can be extended to three- axis machines by adding a third axis in series with the first two. The output body is connected to the linear joints through a set of two parallelograms of equal lengths  i   i L   A B =   , so that it can move only in translation. The two legs are   PPa   identical chains, where   P   and   Pa  stand for Prismatic and Parallelogram joints, respectively. This mechanism can be optimized to have a workspace whose shape is close to a square workspace and the velocity amplification factors are bounded [11]. The joint variables   1 ρ   and   2 ρ   are associated with the two prismatic joints. The output variables are the Cartesian coordinates of the tool center point   T y x P   ] [ =   . To control the orientation of the reference frame attached to   P , two parallelograms   can   be   used,   which   also   increase   the rigidity of the structure, Figure 3.  i   e 1  A 1  B 1  P  e 2  j  θ   1 θ   2  i  x  y z  Figure 3: Parallel mechanism with two-dof To produce the third translational motion, it is possible to place orthogonally a third prismatic joint. The velocity   p &   of point   P   can be expressed in two different   ways.   By   traversing   the   closed   loop ) (   2 2 1 1   P B A P B A   −   in two possible directions, we obtain ) (   1 1 1 1   a b i a p   − × θ + =   & & &   (1a) ) (   2 2 2 2   a b i a p   − × θ + =   & & &   (1b) where   1 a   ,   1 b   ,   2 a   and   2 b   represent the position vectors of the points   1 A   ,   1 B   ,   2 A   and   2 B   , respectively. Moreover, the velocities   1 a &   and   2 a &   of   1 A   and   2 A   are given by 1 1 1   ρ =   & &   e a   and   2 2 2   ρ =   & &   e a   , respectively. For an isotropic configuration to exist where the velocity amplification factors are equal to one, we must have 0 .   2 1   = e e   [11] (Figure 5). We would like to eliminate the two passive joint rates   1 θ &  and   2 θ &   from Eqs. (1a-b), which we do upon dot-multiply the former by   T  ) (   1 1   a b   −   and the latter by   T  ) (   2 2   a b   −   , thus obtaining 1 1 1 1 1 1   ) ( ) (   ρ − = −   & &   e a b p a b   T T   (2a) 2 2 2 2 2 2   ) ( ) (   ρ − = −   & &   e a b p a b   T T   (2b) Equations (2a-b) can be cast in vector   form, namely  ρ & &   B p A   =   , with   A   and   B   denoted, respectively, as the parallel and serial Jacobian matrices,  ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − ≡   T T  ) ( ) ( 2 2 1 1  a b a b A   ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − ≡  2 2 2 1 1 1 ) ( 0 0 ) (  e a b e a b B   T T  where   ρ &   is defined as the vector of actuated joint rates and   p &   is the velocity of point   P ,   i.e. ,  [   ] 1   2  T  ρ   ρ = &   &   & ρ   and   [   ] T x   y = p &   &   &  When   A   and   B   are not singular, we obtain the relations,  ρ & &   J p   =   with   B A J   1 − =  Parallel singularities occur whenever the lines   1   1 A B   and 2   2 A B   are colinear,   i.e.   when   1   2   , k  θ   θ   π −   =   for   k   = 1,2,.... Serial   singularities   occur   whenever   1   1   1 ⊥   − e   b   a   or 2   2   2 ⊥   − e   b   a   . To avoid these two singularities, the range limits are defined in using suitable bounds on the velocity factor amplification (See section 3.3).
Submitted to ICAR 2003   D. Chablat Ph. Wenger F. Majou 4/6  3.2   Kinematics of a spatial parallel mechanism with one-dof of rotation  The aim of this section is to define the kinematics of a simple mechanism with two-dof of translation and one- dof   of   rotation.   To   be   modular,   the   direct   kinematic problem   must   be   decoupled   between   position   and orientation equations. A decoupled version of the Gough- Stewart Platform exists but it is very difficult to build because three spherical joints must coincide [12]. Thus, it cannot be used to perform milling applications. The main idea of the proposed architecture is to attach a new body with the tool frame to the mobile platform of the two-dof mechanism defined in the previous section. The new joint admits one or two-dofs according to the prescribed tasks. To add one-dof on the mechanism defined in section 3, we introduce one revolute joint between the previous mobile platform and the tool frame. Only one leg is necessary to hold   the   tool   frame   in   position.   Figure 4   shows   the mechanism obtained with two translational dofs and one rotational dof.  i   e 1  A 1  B 1  A 3  B 3  e 2  j  A   2  B 2  k e 3  θ   1 θ   2  i  β θ   3 α  P x y z  Figure 4: Parallel mechanism with two-dof of translation and one-dof of rotation The architecture of the leg added is   PUU   where   P   and   U  stand for Prismatic and Universal joints, respectively [6]. The new prismatic joint is located orthogonaly to the first two prismatic joints. This location can be easily justified because on this configuration,   i.e.   when   1   1   2   2 −   ⊥   − b   a   b   a  and   3   3   3 −   ⊥   − b   a   b   p   , the third leg is far away from serial and parallel singularities. Let   ρ &   be referred to as the vector of actuated joint rates and   p &   as the velocity vector of point   P , 1   2   2 [   ] T  ρ   ρ   ρ = &   &   &   & ρ   and   [   ] T x   y = p &   &   &  Due to the architecture of the two-dof mechanism and the location of   P , its velocity on the z-axis is equal to zero.   p &  can be written in three different ways by traversing the three chains   P B A   i i   , 1   1   1   1 (   )  θ =   +   ×   − p   a   i   b   a & &   &   (3a) 2   2   2   2 (   )  θ =   +   ×   − p   a   i   b   a & &   &   (3b) 3   3   3   3   3 (   )   (   )   (   )  θ   α β =   +   +   ×   −   +   ×   − p   a   j   k   b   a   j   p   b &   & &   &   &   (3c) where   i a   and   i b   are the position vectors of the points   i A  and   i B   for   1, 2,3 i   =   ,   respectively.   Moreover,   the velocities   1 a &   ,   2 a &   and   3 a &   of   1 A   ,   2 A   and   3 A   are given by 1 1 1   ρ =   & &   e a   ,   2 2 2   ρ =   & &   e a   and   3   3   3  ρ = a   e &   &   , respectively. We want to eliminate the passive joint rates   i θ &   and  α &  from   Eqs. (3a-c),   which   we   do   upon   dot-multiplying Eqs. (3a-c) by   i i   a b   −   , 1   1   1   1   1   1 (   )   (   ) T   T  ρ −   =   − b   a   p   b   a   e &   &   (4a) 2   2   2   2   2   2 (   )   (   ) T   T  ρ −   =   − b   a   p   b   a   e &   &   (4b) 3   3   3   3   3   3 3   3   3 (   )   (   ) (   )   (   )  T   T T  ρ β −   =   − +   −   ×   −  b   a   p   b   a   e b   a   j   p   b  &   & &   (4c) Equations (4a-c) can be cast in vector form, namely,  = t   J   & ρ   with   B A J   1 − =   and   T y x   ] [   β =   & & & t  where   A   and   B   are   the   parallel   and   serial   Jacobian matrices, respectively, 1   1 2   2 3   3   3   3   3 (   )   0 (   )   0 (   )   (   )   (   )  T T T   T  ⎡   ⎤ − ⎢   ⎥ ≡   − ⎢   ⎥ ⎢   ⎥ −   −   −   ×   − ⎣   ⎦  b   a A   b   a b   a   b   a   j   p   b  ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − − ≡  3 3 3 2 2 2 1 1 1 ) ( 0 0 0 ) ( 0 0 0 ) (  e a b e a b e a b B  T T T  There are two new singularities when one leg is added. The   first   one   is   a   parallel   singularity   when 3   3   3 (   )   (   )   0 T  −   ×   −   = b   a   j   p   b   ,   i.e. , when the lines   ) (   3 3   B A   and ) (   3   P B   are   colinear,   and   the   second   one   is   a   serial singularity   when   3   3   3 (   )   0 T  −   = b   a   e   ,   i.e. ,   3   3   3 −   ⊥ a   b   e   . However, these singular configurations are simple and can be avoided by proper limits on the actuated joints.  3.3   Optimization   of   the   useful   workspace   for translational motions  Two types of workspaces can be defined, (i) the Cartesian workspace is the manipulator’s workspace defined in the Cartesian space, and (ii) the useful workspace is defined as a subset of the Cartesian workspace. Workspace and size are prescribed where some performance indices are prescribed. For   parallel   mechanism,   the   useful   workspace   shape should be similar to the one of classical serial machine
Submitted to ICAR 2003   D. Chablat Ph. Wenger F. Majou 5/6 tools, which is parallelepipedic if the machine has three translational degrees of freedom for instance. So, a square useful workspace is prescribed here where the velocity amplification factors remain under the prescribed values. Two square useful workspaces can be used, (i) he first one has horizontal and vertical sides (Figure 5a) and (ii) the second   one   has   oblique   sides   but   its   size   is   higher (Figure 5b).  P B 1  A 1  e 1   e 2  B 2 Useful workspace  P Cartesian workspace  B 1  A   1  e 1   e 2  B 2 Close to singularity locus (a)   (b) Figure 5: Cartesian workspace and isotropic configuration To find the best useful workspace (center locus and size), we can shift the useful workspace along   x -axis   (   ) x Δ   and  y -axis   (   ) y Δ   (Figure 6)   and   the   velocity   amplification factors are computed for each configuration. This method was developed in [13].  P 1   P 2  P 4   P 3  P 1  P 2  P 4  P 3 (a)   (b)  Δ u   Δ u  Δ x  Δ y   x y   y x  Figure 6: Looking for the best useful workspace (center locus and size) In each case, velocity amplification factor extrema are located along the sides   i   j P P   : they start from 1 at point S, then they vary until they reach prescribed boundaries (1/ 3   3 i  λ ≤   ≤   ).   Then   computation   (which   analytical expressions   λ i )   has been obtained with Maple along the four sides of the square and for the same length of the leg equal to one. For the first mechanism, solution (a), the size of the optimal surface is equal to 0,89 m 2   and for the second mechanism, solution (b), the size is equal to 0,62 m 2   . The result obtained for the solution (b) is smaller than for the solution (a) but is more appropriate for the extension three-axis mechanism of Figure 4. In effect, we want the axis of rotation to be parallel to one of the side of the useful   workspace. In the next section, the lengths of the third leg will optimize to achieve this square useful workspace without singularity.  3.4   Optimization   of   the   third-axis   for   rotational motions  The aim of this section is to define the two lengths of the leg,   1   3 L   B P =   and   2   3   3 L   A B =   such that it is possible to achieve the maximum range variation of third axis  β without meeting a singular configuration throughout the square workspace with the size defined in the previous section. The fist step of this optimization is to find the location of the prismatic joint. When   P   is on the center of the square workspace, we chose to place the third leg furthest   away   from   singular   configuration,   i.e.   when 3   3 − b   a   and   3 e   are   colinear   and   3   3   3 −   ⊥   − b   a   p   b  (Figure 7a).  B 3   A 3  P   e 3  ρ 3 (a)  B 3  A 3  P   e 3  ρ 3 (b)   B   3  A 3  P  e 3  ρ   3 (d)  B 3  A 3  P  e 3  ρ 3 (c)  β  L 1  γ  σ  Figure 7: Optimal, serial and parallel configuration of the third leg As it is defined in Section 3.2, the third leg is in a singular configuration   whenever   3   3   3 (   )   (   )   0 T  −   ×   −   = b   a   j   p   b  (Figure 7b-c)   or   3   3   3 (   )   0 T  −   = b   a   e   (Figure 7d).   In   the optimization function, we set: 3   3 3 3   3 (   )   0, 2  T  −   > −  b   a   e b   a   (4a) 3   3   3 3   3   3 (   )   (   )   0, 2  T  −   − ×   > −   −  b   a   p   b j b   a   p   b   (4b) with arcsin(0, 2)   11,53 =   °  This   means   that   3   3 A B P  γ   =   ∠   is   in   [   ] 11,5 168,5   and 3   3   3 A B  σ   =   ∠   e   is   in   [   ] 78,5 78,5 −   .   The   result   of   this optimization as a function of   1 L   and   2 L   is depicted in Fig. 8.
Submitted to ICAR 2003   D. Chablat Ph. Wenger F. Majou 6/6 1.0   1.5   2.0   2.5   3.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0  L 1 L 2  L 2 =1,8L 1 Figure 8: Range of variation in degrees of  β   as a function of   1 L   and   2 L With a suitable chose of lengths, it is easy to obtain a range variation of  β   higher than 180°. So, this value can be reduced if we increase the constraint defined in Eqs. (4). However, when   2   1 1,8 L   L =   , the range of variation is optimal.  4   Conclusions  In this paper, a parallel mechanism with two degrees of position and one degree of rotation is studied. All the actuated joints are fixed prismatic joints, which can be actuated by means of linear motors or by conventional rotary motors with ball screws. Only three types of joints are used,   i.e. , prismatic, revolute and universal joints. All the singularities are characterized easily because position and orientation are decoupled for the direct kinematic problem and can be avoided by proper design. The lengths of the legs as well as their positions is optimized, to take into account the velocity amplification factors for the translational   motions   and   to   avoid   the   singular configuration for the rotational motions.  5   Acknowledgments  This   research   was   partially   supported   by   the   CNRS (Project ROBEA “Machine à Architecture compleXe”).  6   References  [1] Treib, T. and Zirn, O., “Similarity laws of serial and parallel manipulators   for   machine   tools,”   Proc.   Int.   Seminar   on Improving Machine Tool Performances, pp. 125-131, Vol. 1, 1998. [2]   Gough,   V.E.,   “Contribution   to   discussion   of   papers   on research in automobile stability, control an tyre performance,” Proc. Auto Div. Inst. Eng., 1956-1957. [3] Merlet, J-P, “The parallel robot,” Kluwer Academic Publ., The Netherland, 2000. [4] Kong,   K.   and   Gosselin,   C.,   “Generation   of   parallel manipulators with three translational degrees of freedom based on   screw   theory,”   Proc.   of   CCToMM   Symposium   on Mechanisms,   Machines   and   Mechantronics,   Saint-Hubert, Montreal, 2001. [5] Hervé, J.M. and Sparacino, F., 1991, “Structural Synthesis of Parallel Robots Generating Spatial Translation,” Proc. of IEEE 5th Int. Conf. on Adv. Robotics, Vol. 1, pp. 808-813, 1991. [6] Gao, F., Li, W., Zhao, X., Jin Z. and Zhao. H., “ New kinematic   structures   for   2-,   3-,   4-,   and   5-DOF   parallel manipulator   designs,”   Journal   of   Mechanism   and   Machine Theory, Vol. 37/11, pp. 1395-1411, 2002. [7] Gosselin, C. and Angeles, J., “Singularity analysis of closed- loop   kinematic   chains,”   IEEE   Transaction   on   Robotic   and Automation, Vol. 6, No. 3, June 1990. [8] Salisbury, J-K. and Craig, J-J., “Articulated Hands: Force Control and Kinematic Issues,” The Int. J. Robotics Res., Vol. 1, No. 1, pp. 4-17, 1982. [9] Angeles, J.,   Fundamentals of Robotic Mechanical Systems , 2 nd   Edition, Springer-Verlag, New York, 2002. [10] Chablat,   D.   and   Angeles,   J.,   “On   the   Kinetostatic Optimization of Revolute-Coupled Planar Manipulators,” J. of Mechanism and Machine Theory, vol. 37,(4).pp. 351-374, 2002. [11] Chablat,   D.,   Wenger,   Ph.   and   Angeles,   J., “Conception   Isotropique   d'une   morphologie   parallèle: Application à l'usinage,” 3 rd   International Conference On Integrated   Design   and   Manufacturing   in   Mechanical Engineering, Montreal, Canada, May, 2000. [12] Khalil, W. and Murareci, D., “Kinematic Analysis and Singular   configurations   of   a   class   of   parallel   robots,” Mathematics and Computer in simulation, pp. 377-390, 1996. [13] Majou F., Wenger Ph. et Chablat D., “A Novel method for the   design   of   2-DOF   Parallel   mechanisms   for   machining applications,”   8th   Int.   Symposium   on   Advances   in   Robot Kinematics, Kluwer Academic Publishers, ,2002.