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  • La page Présentation de COPRIN
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    Still a long way to go on the road for parallel mechanisms
    J-P. Merlet
    INRIA Sophia-Antipolis, France
     
    A keynote speech to be presented at the
    ASME 2002 DETC Conference, Montréal



    Introduction

    After spending almost 20 years in the laboratories for preliminary studies parallel robots are now used in real-life applications in domains such as fine positioning devices, motion generators, ultra-fast pick and place robot and will probably find their use in the field of machine-tools, medical application, haptic devices, entertainment...

    This interest come from the potentially interesting features of parallel mechanisms, the most noticeable being:

    which in a very large number of cases may overcome the drawbacks of the more complex kinematics and smaller workspace.

    But a fact is that these advantages are only potential and any real parallel robot will present in practice impressing performances only if all its components (either hardware or software) present a high level of performance. In this paper we will review some key issues in this field, without pretending to be exhaustive1.

    The various layers of a parallel robot system

    Like their serial counterpart parallel robots are constituted of various layers (figure 1).

    Figure 1: The various layers of a parallel robot
    \begin{figure}\begin{center} \ifx\cs\tempdima \newdimen \tempdima\fi \ifx\cs\tem... ...xternal} \end{picture}\setlength{\unitlength}{1cm}\par\end{center} \end{figure}

    The mechanism layer is the robot itself with first a theoretical model constituted of :

    But the practical realization of the robot will differ from this theoretical model and we will find the real robot with: The control layer is constituted of: The design and simulation layer is constituted of As an option we may have also a calibration layer whose purpose is to obtain a better match between the theoretical model of the geometry and its real geometry by using either the sensors of the robot or additional sensors.

    We may also distinguish between on line and off-line layers (mostly the design and simulation layers) although elements of one category may be used by elements of the other categories.

    At this level of description it may appear that there is no difference between a serial robot and a parallel robot. But we will see that some underlying problems at each layer are very specific for each category of robot.

    Mechanism

    Mechanical architectures

    More than 100 different mechanical architectures of parallel robots have already been proposed and it is probable that not all of them have been discovered. Unfortunately there are not so many proposed architecture that have only 4 or 5 d.o.f.2while many applications require such number of d.o.f. Hence a recent trend is to propose parallel robots with less than 6 d.o.f [12,14,28,40,57,68].

    This is clearly an interesting research area but many questions arise with this type of robots:

    Redundancy is also an interesting and open research area [57]. In the field of parallel robot for machine-tools redundancy has been used to increase the workspace of the robot (such as in the Eclipse parallel robot [65]) and to deal with singularities. Another form of redundancy is the concept of modular robots [33,34,71] in which additional actuators allow to adapt the geometry of the robot according to the task to be performed The main unsolved problem for redundant parallel robot is to determine how to use the redundancy for an optimal use of the robot.

    MEMS parallel robots is also an exciting research area. Indeed the motion principle of such mechanism can be used at any scale, from very large motion platform for driving or flight simulators to micro scale robots. Already parallel robots with size of a few millimeters have been built [3,6,44,60] while the concept of even smaller robots has been proposed [42]. The current technology for actuators and sensors does not allow yet for the development of robot in the millimeter (or lower) size but this will probably change in the near future. The change in size will have a large influence on the physics of such system (gravity will have a very low influence while atomic forces will become preponderant) and new types of analysis will be required.

    Joints, actuators and sensors

    Parallel robots require higher kinematic pairs with relatively large amplitude of motion and, in some cases, relatively high load. Current available joints (either ball-and-socket or U-joints) are not completely satisfactory from this view point although recent products like the INA or Hephaist joints have been developed especially for parallel robots [21]. Hence the development of higher kinematic pairs with 2 to 4 d.o.f. is a key issue. As for any mechanical joints these joints must have a low friction, no hysteresis and must have a very reduced backlash. But in addition these joints must be designed so that it is possible to add sensors to measure partly or totally the amplitude of the motion of the joints (which is important for the forward kinematics as mentioned in the next section). Note also that flexible joints is also an interesting field of research, especially for micro-robot [56].

    As for the actuators many robot are using linear actuators.In the field of machine tools some parallel robot such as the Urane SX of Renault Automation are using linear electric motor which exhibit impressing accelerations. But there is a lack of linear actuators and sensors for micro parallel robots [44].

    Parallel structures offer also the use of interesting alternate actuators such as:

    Kinematics

    Inverse kinematics

    Everybody will agree that inverse kinematics (IK) is one of the basic element of any robot controller. Fortunately it is known that inverse kinematics is usually straightforward for any parallel robot. More precisely in most cases

    The later point is a key difference with serial robot and allows for very fast IK provided that the controller hardware is appropriate. It may be thought that the development of a dedicated IC for the IK will be a major component of an effective parallel robot controller.

    Forward kinematics

    The major kinematics problem is the forward kinematics (FK), which consists in finding the possible pose of the platform for given joint coordinates (the solutions are called the assembly modes of the robot for the given joint coordinates). The FK is a more complex problem than its dual IK counterpart for serial robot (F. Freudenstein mentioned that this was the Himalaya of modern kinematics). The need of the FK is a controversed question. It may be thought that FK is an academic question that may be useful only off-line for simulation purposes as a parallel robot will be position controlled using IK only. In my opinion pure position control is very difficult for parallel robot especially when there are constraints on both the trajectory and the velocity of the robot (for example when the robot is used as a machining tool). In that case velocity control, which imply solving the FK, is much more appropriate.

    FK is an area where a lot of progress has been made thanks to a collaborative work with mathematicians (which has benefited from this problem: solving the FK of a Gough platform is considered now as a classical bench in algebraic geometry). Although there are many mechanical architectures of parallel robots the FK problem for most of them may be reduced to solve the FK for a few key architectures. For example solving the FK for the Gough platform [25] allows to solve the FK of the Hexa [58] or the Hexaglide [29] although the mechanical architectures of these robots are quite different.

    It is now well known that the FK of the Gough platform may have up to 40 solutions [61,63] and that all these 40 solutions may be real [18]. Numerous works have provided a deep understanding of the problem [20,39,51], which in turn has led to efficient algorithms for determining all the solutions of the FK [30,64,70] using elimination, Gröebner basis or interval analysis. Although impressing progress have been made these algorithms are not yet real-time and furthermore it cannot be said that FK is a fully solved problem. Indeed the true FK problem is to determine the current pose of the platform being given the joint coordinates. The algorithms provide all the solutions and hence it is necessary to sort the solutions to determine the current pose. Hence the true unsolved FK problem is to complement the current algorithms with a sort algorithm that will reject solutions that cannot be reached from an initial assembly mode by a singularity and interference free trajectory (and it is unclear if this criteria will be sufficient to eliminate all but one solution).

    For real-time purpose many authors have proposed the use of the Newton-Raphson iterative scheme that assumes that an estimate of the solution is known. This scheme allows for possibly determining one solution of a non-linear square system of equations but there are many ways to model FK equations, not all of them being equivalent in term of quality of the result, computation time or size of the convergence domain [50]. Furthermore it is not so well known that the Newton scheme may converge toward a solution that is not the closest to the estimate, whatever close is the estimate to this desired solution. Interval analysis based methods are good alternate with a similar computation time than Newton scheme and guarantee on the results. These methods share with the Newton scheme the possibility of a distributed implementation and we believe that this opportunity must be used in a robot controller to speed up the FK which is essential for the control of the robot.

    Another interesting possibility is to have a number of sensor which is larger than the number of d.o.f. of the platform. The extra sensors may allow to determine the current pose of the platform (and may also be used for the calibration of the robot, see the corresponding section). But it is necessary to:

    1. determine the number and location of the extra sensor(s) so that a unique solution of the FK is found
    2. study the influence of the sensor errors on the FK
    3. carefully determine the speed-up that the extra sensors allow for the FK
    Although this field has been recently investigated [4,5,45,55,67] many problems are still unsolved, especially for point 2.

    Singularities

    There are various ways to introduce the concept of singularities but the most spectacular one is to consider the static behavior of the robot. Let ${\cal F}$ be the wrench applied on the platform of the robot and ${\bf\tau}$ the set of joint forces. These quantities are linearly related by

    \begin{displaymath} {\cal F}=J^{-T}(X) {\bf\tau} \end{displaymath}

    where $J^{-T}$ is the transpose of the inverse jacobian matrix of the robot that is pose dependent. Each component of the joint forces vector may thus be obtained as a ratio:

    \begin{displaymath} \tau_i = \frac{A}{\vert J^{-T}\vert} \end{displaymath}

    where $A$ is the minor associated to $\tau_i$. Hence, provided that $A$ is not 0, the joint force $\tau_i$ will go to infinity at any pose, called singular poses, where the determinant of $J^{-T}$ is 0, causing a breakdown of the robot (in fact the breakdown will occur well before reaching the singularity).

    Although the condition $\vert J^{-T}\vert=0$ seems to be a simple condition as the matrix $J^{-T}$ has an analytical form, the full calculation of this determinant leads to a complex expression with a large number of terms (especially if the robot has 6 d.o.f.) which is useless in practice.

    We have now a better understanding of singular configurations. They will occur for specific geometrical configurations of the robot that may be determined, whatever is the number of d.o.f of the robot, using line geometry [47]. We have now efficient algorithm that allows to determine if singular configurations exist either in the reachable workspace of the robot or in a specific workspace for the platform [49]. We may also test in near real-time the presence of singularity on any arbitrary trajectory [43].

    But this does not mean that all problems related to singular configurations are solved. For example:

    Workspace

    It is well known that a main drawback of parallel robot is their reduced workspace. Furthermore computing this workspace is not an easy task as, at the opposite of classical serial robot, the translational and orientation workspace are coupled. Classically a first approach to solve this problem is to fix the values of some d.o.f. until only 3 d.o.f. are free. This is usually done by fixing either the orientation of the platform or the location of its center. In the first case the geometrical approach that determine geometrically the possible motion of the center of the platform for each kinematic chains leads usually to the best result as it provides exact calculation with a compact storage and easy representation [24]. Orientation workspace is more difficult to deal with as there is no universal way to represent this workspace.

    Another approach is to calculate an approximation either of the border or of the whole workspace using a numerical method [1,26,48]. Some of these approaches have the advantage to be able to deal also with limits on the motion of the passive joints and to allow for workspace verification (i.e. to check if a desired workspace is included in the workspace of the robot). They may also calculate various types of workspace (for example to determine all the possible locations of the center of the platform such that it is possible to have any orientation of the platform within some prescribed ranges for the orientation angles).

    In this field remains two unsolved problems:

    Motion planning

    Motion planning is a classical problem for serial robot. But in the case of parallel robots the problem is somewhat different: while for serial robot obstacle avoidance is the main reason for motion planning, its counterpart for parallel robot is the workspace. Possible problems are:

    1. verify if a given trajectory lie completely within the workspace of the robot
    2. determine if two poses may be reached by a singularity and interference free trajectory that lie completely within the workspace of the robot
    Problem 1 can be solved for almost any arbitrary time-function trajectory using interval analysis [43], while problem 2 has no known solution at this time.

    Calibration

    Calibration is a well known problem for serial robots and is now a well-treated problem. It may be thought that the calibration of parallel robots may rely on the methods developed for serial robot but unfortunately this is not exactly the case. Indeed there is a major difference between both robots: for serial robot small errors on the geometrical parameters induce large errors on the positioning of the end-effector while for parallel robots these errors will also be small. Simulation for calibration is essential: it allows to determine how much a calibration method is sensitive to noise in the measurements and to numerical errors. It allows for example to show that methods directly adapted from the calibration of serial robots may lead to results that are worse than the initial guess as soon as the simulated measurement noise is realistic $\ldots$.

    There are two types of calibration methods:

    The first method is difficult and tedious to use in practice but may give good results. The second method may be less accurate but is easy to use and has also the advantages that it can be fully automatized.

    An interesting theoretical problem is to determine what are the measurement configurations of the platform that will lead to the best calibration. Then there is also the problem to put calibration in use in a realistic, industrial environment.

    Dynamics

    Another advantage of parallel robots is that they can reach a high acceleration and velocity, due to their light mobile mass [13,29,59]. But control of such robots is a difficult task: although numerous works have reported methods for computing the dynamic model of a parallel robot they are all computer intensive (and involves also solving the FK problem). An important problem is to determine what should be the computation time of the calculation of the dynamic model so that its use in a control loop will really leads to an improvement of the performances of the robot. This is a very complex issue especially if it is considered that the control algorithm is not continuous. The second key issue in this field is to implement the control scheme. In my opinion the involved computation time implies the use of a distributed computation scheme: implementation considerations will hence have a large influence on the choice for the control algorithm and for the dynamic model.

    Synthesis

    It is well known that the performances that will be reached by a mechanism depends upon:

    This is especially true for closed-loop mechanisms that are highly sensitive to both factors. Hence to design a parallel mechanism so that its performances fit at best a list of requirements both aspects must be addressed:

    Synthesis of parallel robot is an open field (there is a very limited number of papers addressing this issue) and, in my opinion, one of the main issue for the development of parallel robots in practice. The use of parallel structures in the field of machine-tool has shown that designers which have a deep understanding of open-loop mechanisms but have a total lack of experience in closed-loop have focused only on the development of the basic mechanical components of their machine and have almost completely neglected the analysis part. Many such machines have thus suffered from elementary errors: a direct consequence was a reinforcement of a trend that claim that parallel structures is only an academic field that will never be put in practice. As for any human activity one single failure has more influence than numerous success.

    Topology synthesis

    This is a very complex problem for parallel mechanism at the opposite of open-loop mechanism for which the number of possible kinematic combinations is relatively reduced. Currently topological synthesis for parallel robots is restricted to find a mechanism with a given number of d.o.f without considering other performance criterion and is still mostly done intuitively. There is total lack of automated tool for topological synthesis and even no existing convention for naming a closed-loop mechanism. Although over 100 mechanical architectures of parallel mechanisms have already been proposed I feel that not all possible structures have been found

    An additional difficulty for closed-loop mechanisms is that topological synthesis cannot be considered independently from dimensional synthesis: it is usually not possible to compare a-priory the performances of two mechanical designs just by inspection of their topology at the opposite of open-loop mechanisms for which such qualitative comparison is sometime possible. For example the workspace volume of a Cartesian robot using 3 linear actuators of stroke $L$ is roughly $L^3$ while this volume for a 3R robot whose links has length $L$ is roughly $4\pi (2L)^3/3 \approx 34 L^3$: hence in general a 3R robot will have a much more larger workspace than a Cartesian robot, at least for a similar dimensioning.

    A first approach to topology synthesis is based on the Gruëbler mobility formula. Its use is quite simple but this formula does not take into account the geometry of the arrangement of the kinematic pairs and hence may lead to invalid results. Furthermore a Gruëbler based topological synthesis approach cannot benefit from the use of specific geometric arrangements that allow for specific motions.

    Alternative approaches are:

    In my opinion this area should be expanded and a standard way of describing parallel structure is needed (especially for an automated analysis of their performances as presented in the next section). Note also that an important point for topology synthesis has already been mentioned in the Mechanical architectures section: a structure may be based on special geometrical arrangements of the links leading to some specific properties for the mechanism but in practice the geometry may not exactly fulfill the theoretical constraints. It is hence necessary to examine carefully what will be the effect of the manufacturing errors on the motion of the mechanism.

    Dimensional synthesis

    Parallel mechanisms are highly sensitive to dimensioning: a classical example is that by changing the radius of the platform of Stewart-Gough platform by 10% we may change the minimal stiffness of the robot over its workspace by 700%.

    I have already discussed existing dimensional synthesis method [46] but, in my opinion, none of them are appropriate for parallel robots which have usually a large number of design parameters. Furthermore these methods lead to a unique solution: in the case of parallel robots we believe that there will be usually not a single solution to a design problem and furthermore that providing only one design solution is not realistic. Indeed:

    Therefore a design methodology should provide not only one single solution but, if possible, all the possible design solutions, or, at least, an approximation of the set of all design solutions.

    Performance analysis

    Whatever is the design methodology it will be necessary to have a performance analysis module. Being given a mechanism of known topology and dimensions the aim of a performance analysis module is to determine what are the performances of the mechanism. In the synthesis domain such module is used mostly to compare different design solutions, while for simulation purposes the objective will be to determine the performances of the robot.

    Performance analysis is difficult for parallel robot. Indeed most interesting performances index are related to the determination of the optimum of a function over a given set. For example the accuracy index consists in determining the worst case positioning error $\Delta {\bf X}$ of the platform being given the sensor accuracy $\Delta \rho$, over the workspace of the robot. Both quantities are related by

    \begin{displaymath} \Delta \rho = J^{-1}(X) \Delta {\bf X} \end{displaymath}

    where $J^{-1}$ is the inverse jacobian matrix of the robot, which is pose dependent. Hence determining the accuracy index is equivalent to solving a constrained optimization problem. In this case the problem is quite difficult as an analytical form of $J^{-1}$ is usually known, while $J$ (which will allow to obtain an analytical form of the criteria to be optimized) has a complex form. It must be noted that the exact calculation of the accuracy index is still an open problem and that it is the case for most performance index of parallel robots.

    A key point for performance analysis for synthesis is that the result must be guaranteed as it will be used to compare different design solutions. Hence the usual method of discretizing the workspace and computing the accuracy index at a limited number of poses within the workspace is not a valid approach. But guaranteeing the result does not mean that the index should be computed exactly, even in the computer science signification of this term (i.e. by relying on the accuracy of the computer). Indeed as the result will be used for comparison purposes it has to be calculated only up to an accuracy that allows for a right choice between different solutions. For example if is possible to compute for two robots that their accuracy index lie in the ranges $[a_1, b_1]$ for the robot 1 and $[a_2, b_2]$ for robot 2 with $b_1 < a_2$, then we may conclude that the robot 1 is more accurate than the robot 2, even if the width $b_1-a_1$ of the range (i.e. the accuracy with which we have computed the accuracy index of robot 1) is quite large. In my opinion any performance analysis module should take advantage of this property to speed-up the calculation of the performance index as any design methodology will use extensively the performance analysis module.

    Dimensioning methodology

    As mentioned in the previous section a design methodology should allow to determine not one single solution but a set of possible solutions and ideally all the design solutions.

    Mathematically speaking let ${\cal P}$ be the set of $n$ design parameters and let us introduce the parameters space as a $n$ dimensional space in which each dimension corresponds to one of the $n$ design parameters. In the parameters space a point represents an unique robot geometry and the purpose of the dimensioning methodology should be to determine the regions of the parameters space such that if a point belong to a region, then the corresponding robot fulfill the requirements.

    Clearly determining these regions is not an easy task but a possible approach is to determine them incrementally: for each requirement $i$ the region ${\cal R}_i$ corresponding to the robots that satisfy the requirement are computed and the design region will be obtained as the intersection of all the ${\cal R}_i$. Alternatively as soon as one of the ${\cal R}_i$ has been calculated it can be used as starting point for the determination of a region ${\cal R}_{ij}$ where both the requirements $i$ and $j$ are satisfied and so forth. Such approach has been proven to be effective for the workspace requirement [48,49].

    In my opinion the development of a generic optimal design and simulation software for parallel robot is one of the most exciting tasks in this field. Such software should be able to deal with any mechanical architecture and requirements: clearly this will represent a huge development both at the theoretical and software level that justify a collaborative work of academics (from may different fields), companies that develop parallel robots and end-users. This is why the Computational Kinematics Technical Committee of IFToMM has launched the Parallel Kinematic Initiative (PKI)3 for encouraging collaborations in this field.

    Controller

    The developments proposed in the previous sections will lead to an effective system only if the robot controller allows for dealing with the specificities of parallel robots. Unfortunately the current trend, especially in the field of machine tool, is to try to adapt existing hardware for the purpose of controlling parallel robots. If this trend may be justified when starting a project with parallel robots it will drastically penalize the performance of the system on the long term. If we take as example the machine-tool field we may analyze the errors on the fabricated parts that are due to each element of the system: the CAD system that is used to define the parts and which is generating motions for the platform, the controller that monitor the execution of these motions and finally the platform itself. Using current technology it can be shown that the CAD system is responsible of approximatively 20% of the errors, the platform (if optimally designed) less than 10% while the controller induces 70% of the errors. Hence research should focus on the the CAD system (but existing methods may already improve this part) and mostly on the controller. The hardware of the controller should support:

    Conclusion

    In this paper we have tried to outline some open problems in the field of parallel robots (without pretending to be exhaustive). Some of these problems are long term while other are key issue for the short term possibilities of using parallel robots in practice. In the last 20 years we have gained a better understanding of the behavior of these complex closed-loop mechanisms but there are still many unsolved and exciting problems. If we compare this 20 years to the 200 years that has been necessary to reach the current level of achievement for serial mechanisms we may conclude that there is still a long way to go on the road for parallel mechanisms.

    Bibliography

    1
    Adkins F.A. and Haug E.J.
    Operational envelope of a spatial Stewart platform.
    ASME J. of Mechanical Design, 119(2):330-332, 6January February March April May June July August September October November December Summer 1997.

    2
    Albus J., Bostelman R., and Dagalakis N.
    The NIST ROBOCRANE.
    J. of Robotic Systems, 10(5):709-724, 7January February March April May June July August September October November December Summer 1993.

    3
    Arai T., Stoughton R., and Jaya Y.M.
    Micro hand module using parallel link mechanism.
    In Japan/USA Symp. on Flexible Automation, pages 163-168, San Francisco, 7January, February, March, April, May, June, July, August, September, October, November, December, Summer 13-15, 1993.

    4
    Baron L. and Angeles J.
    The kinematic decoupling of parallel manipulators using joint-sensor data.
    IEEE Trans. on Robotics and Automation, 16(6):644-651, 12January February March April May June July August September October November December Summer 2000.

    5
    Bonev J., I.A.and Ryu.
    A new method for solving the direct kinematics of general 6-6 Stewart platforms using three linear extra sensors.
    Mechanism and Machine Theory, 35(3):423-436, 3January February March April May June July August September October November December Summer 2000.

    6
    Breguet J-M., Pernette E., and Clavel R.
    Stick and splip actuators and parallel architectures dedicated to microrobotics.
    In Microrobotics: componants and applications, SPIE Photonic East, pages 13-24, Boston, 11January February March April May June July August September October November December Summer 1996.

    7
    Chakarov D. and Parushev P.
    Synthesis of parallel manipulator with linear drive modules.
    Mechanism and Machine Theory, 29(7):917-932, 10January February March April May June July August September October November December Summer 1994.

    8
    Chen I-M. and others .
    Self-calibration of three-legged modular reconfigurable parallel robots based on measurement residues.
    In F.C. Park C.C. Iurascu, editor, Computational Kinematics, pages 117-132. 5January, February, March, April, May, June, July, August, September, October, November, December, Summer 20-22, 2001.

    9
    Chirikjian G.S.
    Group theoretical synthesis of binary manipulators.
    In 11th RoManSy, pages 107-114, Udine, 7January, February, March, April, May, June, July, August, September, October, November, December, Summer 1-4, 1996.

    10
    Chirikjian G.S.
    A binary paradigm for robotic manipulators.
    In IEEE Int. Conf. on Robotics and Automation, pages 3063-3070, San Diego, 5January, February, March, April, May, June, July, August, September, October, November, December, Summer 8-13, 1994.

    11
    Clavel R.
    DELTA, a fast robot with parallel geometry.
    In 18th Int. Symp. on Industrial Robot, pages 91-100, Lausanne, 4January, February, March, April, May, June, July, August, September, October, November, December, Summer 26-28, 1988.

    12
    Clavel R. and others .
    A new parallel kinematics able to machine 5 sides of a cube-shaped object: Hita STT.
    In 1st Int. Colloquium, Collaborative Research Centre 562, pages 107-118, Braunschweig, 5January, February, March, April, May, June, July, August, September, October, November, December, Summer 29-30, 2002.

    13
    Codourey A.
    Control algorithm and controller for the direct drive Delta robot.
    In 3rd IFAC/IFIP/IMACS Symp. Syroco'91, Vienne, 9January, February, March, April, May, June, July, August, September, October, November, December, Summer 16-18, 1991.

    14
    Company O. and Pierrot F.
    A new 3T-1R parallel robot.
    In ICAR, Tokyo, 11January February March April May June July August September October November December Summer 1999.

    15
    Danescu G., Jacquet P., and Dahan M.
    A method for the design of parallel structures.
    In 2nd Japan-France Congress on Mechatronics, pages 671-674, Takamatsu, 11January, February, March, April, May, June, July, August, September, October, November, December, Summer 1-3, 1994.

    16
    Daney D.
    Self calibration of Gough platform using leg mobility constraints.
    In 10th World Congress on the Theory of Machines and Mechanisms, pages 104-109, Oulu, 6January, February, March, April, May, June, July, August, September, October, November, December, Summer 20-24, 1999.

    17
    Daney D. and Emiris I.Z.
    Variable elimination for reliable parallel robot calibration.
    In F.C. Park C.C. Iurascu, editor, Computational Kinematics, pages 133-144. 5January, February, March, April, May, June, July, August, September, October, November, December, Summer 20-22, 2001.

    18
    Dietmaier P.
    The Stewart-Gough platform of general geometry can have 40 real postures.
    In ARK, pages 7-16, Strobl, 6January February March April May June July August September October November December Summer 29- 7January February March April May June July August September October November December Summer 4, 1998.

    19
    Ebert-Uphoff I. and Chirikjian G.S.
    Inverse kinematics of discretely actuated hyper-redundant manipulators using workspace densities.
    In IEEE Int. Conf. on Robotics and Automation, pages 139-145, Minneapolis, 4January, February, March, April, May, June, July, August, September, October, November, December, Summer 24-26, 1996.

    20
    Faugère J.C. and Lazard D.
    The combinatorial classes of parallel manipulators.
    Mechanism and Machine Theory, 30(6):765-776, 8January February March April May June July August September October November December Summer 1995.

    21
    Franke H.J., Otremba R., and Janicke T.
    Methodical development of optimized passive joints.
    In 1st Int. Colloquium, Collaborative Research Centre 562, pages 119-130, Braunschweig, 5January, February, March, April, May, June, July, August, September, October, November, December, Summer 29-30, 2002.

    22
    Geng Z. and Haynes L.S.
    An effective kinematics calibration method for Stewart platform.
    In ISRAM, pages 87-92, Hawaï, 8January, February, March, April, May, June, July, August, September, October, November, December, Summer 15-17, 1994.

    23
    Glazunov V.A. and others .
    Classification principles and analysis methods for parallel-structure spatial mechanisms.
    J. of Machinery Manufacture and Reliability, (1):41-49, 1990.

    24
    Gosselin C.
    Determination of the workspace of 6-dof parallel manipulators.
    ASME J. of Mechanical Design, 112(3):331-336, 9January February March April May June July August September October November December Summer 1990.

    25
    Gough V.E.
    Contribution to discussion of papers on research in automobile stability, control and tyre performance, 1956-1957.
    Proc. Auto Div. Inst. Mech. Eng.

    26
    Haugh E.J., Adkins F.A., and Luh C.M.
    Operational envelopes for working bodies of mechanisms and manipulators.
    ASME J. of Mechanical Design, 120(1):84-91, 3January February March April May June July August September October November December Summer 1998.

    27
    Hervé J.M.
    Group mathematics and parallel link mechanisms.
    In 9th World Congress on the Theory of Machines and Mechanisms, pages 2079-2082, Milan, 8January February March April May June July August September October November December Summer 30- 9January February March April May June July August September October November December Summer 2, 1995.

    28
    Hesselbach J., Plitea N., Frindt M., and Kusiek A.
    A new parallel mechanism to use for cutting convex glass panels.
    In ARK, pages 165-174, Strobl, 6January February March April May June July August September October November December Summer 29- 7January February March April May June July August September October November December Summer 4, 1998.

    29
    Honegger M., Codourey A., and Burdet E.
    Adaptive control of the Hexaglide, a 6 dof parallel manipulator.
    In IEEE Int. Conf. on Robotics and Automation, pages 543-548, Albuquerque, 4January, February, March, April, May, June, July, August, September, October, November, December, Summer 21-28, 1997.

    30
    Husty M.L.
    An algorithm for solving the direct kinematic of Stewart-Gough-type platforms.
    Mechanism and Machine Theory, 31(4):365-380, 1996.

    31
    Husty M.L. and Karger A.
    Architecture singular parallel manipulators and their self-motions.
    In ARK, pages 355-364, Piran, 6January, February, March, April, May, June, July, August, September, October, November, December, Summer 25-29, 2000.

    32
    Innocenti C.
    Algorithms for kinematic calibration of fully-parallel manipulators.
    In J-P. Merlet B. Ravani, editor, Computational Kinematics, pages 241-250. Kluwer, 1995.

    33
    Ji P. and Wu H.T.
    A fast solution to identity placement parameters for modular platform manipulators.
    J. of Robotic Systems, 17(5):251-253, 2000.

    34
    Ji Z. and Song P.
    Design of a reconfigurable platform manipulator.
    J. of Robotic Systems, 15(6):341-346, 1998.

    35
    Karger A. and Husty M.
    Classification of all self-motion of the original Stewart-Gough platform.
    Computer-aided design, 30(3):205-215, 1998.

    36
    Kawamura S. and others .
    High-speed manipulation by using parallel wire-driven robots.
    Robotica, 18(1):13-21, 2000.

    37
    Khalil W. and Besnard S.
    Self calibration of Stewart-Gough parallel robot without extra sensors.
    IEEE Trans. on Robotics and Automation, 15(6):1116-1121, 12January February March April May June July August September October November December Summer 1999.

    38
    Kyatkin A.B. and Chirikjian G.S.
    Synthesis of binary manipulators using the Fourier transform on the Euclidean group.
    ASME J. of Mechanical Design, 121(1):9-14, 3January February March April May June July August September October November December Summer 1999.

    39
    Lazard D.
    Stewart platform and Gröbner basis.
    In ARK, pages 136-142, Ferrare, 9January, February, March, April, May, June, July, August, September, October, November, December, Summer 7-9, 1992.

    40
    Lenarcic J., Stanisic M.M., and Parenti-Castelli V.
    A 4-dof parallel mechanism simulating the movement of the human sternum-clavicle-scapula complex.
    In ARK, pages 325-332, Piran, 6January, February, March, April, May, June, July, August, September, October, November, December, Summer 25-29, 2000.

    41
    Masory O., Wang J., and Zhuang H.
    On the accuracy of a Stewart platform-part II: Kinematic calibration and compensation.
    In IEEE Int. Conf. on Robotics and Automation, pages 725-731, Atlanta, 5January, February, March, April, May, June, July, August, September, October, November, December, Summer 2-6, 1993.

    42
    Merkle R.C.
    A new family of six degree of freedom positional devices.
    http://nano.xerox.com/nanotech/6dof.html, 1994.

    43
    Merlet J-P.
    A parser for the interval evaluation of analytical functions and its applications to engineering problems.
    J. Symbolic Computation, 31:475-486, 2001.

    44
    Merlet J-P.
    Micro parallel robot mips for medical applications.
    In IEEE Int. Conf. on Emerging Technologies and Factory Automation, Antibes, 10January, February, March, April, May, June, July, August, September, October, November, December, Summer 15-18, 2001.

    45
    Merlet J-P.
    Closed-form resolution of the direct kinematics of parallel manipulators using extra sensors data.
    In IEEE Int. Conf. on Robotics and Automation, pages 200-204, Atlanta, 5January, February, March, April, May, June, July, August, September, October, November, December, Summer 2-7, 1993.

    46
    Merlet J-P.
    The need for a systematic methodology for the evaluation and optimal design of parallel manipulators.
    In 3rd Chemnitzer Parallelkinematik Seminar, pages 49-62, Chemnitz, 4January, February, March, April, May, June, July, August, September, October, November, December, Summer 23-25, 2002.

    47
    Merlet J-P.
    Singular configurations of parallel manipulators and Grassmann geometry.
    Int. J. of Robotics Research, 8(5):45-56, 10January February March April May June July August September October November December Summer 1989.

    48
    Merlet J-P.
    Determination of 6d workspaces of Gough-type parallel manipulator and comparison between different geometries.
    Int. J. of Robotics Research, 18(9):902-916, 10January February March April May June July August September October November December Summer 1999.

    49
    Merlet J-P. and Daney D.
    A formal-numerical approach to determine the presence of singularity within the workspace of a parallel robot.
    In F.C. Park C.C. Iurascu, editor, Computational Kinematics, pages 167-176. Seoul, 5January, February, March, April, May, June, July, August, September, October, November, December, Summer 20-22, 2001.

    50
    Merlet J-P., Perng M-W., and Daney D.
    Optimal trajectory planning of a 5-axis machine tool based on a 6-axis parallel manipulator.
    In ARK, pages 315-322, Piran, 6January, February, March, April, May, June, July, August, September, October, November, December, Summer 25-29, 2000.

    51
    Mourrain B.
    The 40 generic positions of a parallel robot.
    In Bronstein M., editor, ISSAC'93, ACM press, pages 173-182, Kiev (Ukraine), 7January February March April May June July August September October November December Summer 1993.

    52
    Mukherjee S. and Murlidhar S.
    Massively parallel binary manipulators.
    J. of Mechanical Design, 123(1):68-73, 2001 7January, February, March, April, May, June, July, August, September, October, November, December, Summer , .

    53
    Nahvi A. and Hollerbach J.M.
    The noise amplification index for optimal pose selection in robot calibration.
    In IEEE Int. Conf. on Robotics and Automation, pages 647-654, Minneapolis, 4January, February, March, April, May, June, July, August, September, October, November, December, Summer 24-26, 1996.

    54
    Parenti-Castelli V. and Di Gregorio R.
    Influence of manufacturing errors on the kinematic performance of the 3-UPU parallel mechanism.
    In 2nd Chemnitzer Parallelkinematik Seminar, pages 85-99, Chemnitz, 4January, February, March, April, May, June, July, August, September, October, November, December, Summer 12-13, 2000.

    55
    Parenti-Castelli V. and Di Gregorio R.
    A new algorithm based on two extra sensors for real-time computation of the actual configuration of the generalized Stewart-Gough manipulator.
    ASME J. of Mechanical Design, 122:294-298, 9January February March April May June July August September October November December Summer 2000.

    56
    Pernette E. and others .
    Design of parallel robots in microrobotics.
    Robotica, 15(4):417-420, 7January February March April May June July August September October November December Summer - 8January February March April May June July August September October November December Summer , 1997.

    57
    Pierrot F.
    Parallel mechanisms and redundancy.
    In 1st Int. Colloquium, Collaborative Research Centre 562, pages 261-277, Braunschweig, 5January, February, March, April, May, June, July, August, September, October, November, December, Summer 29-30, 2002.

    58
    Pierrot F., Dauchez P., and Fournier A.
    Hexa: a fast six-dof fully parallel robot.
    In ICAR, pages 1159-1163, Pise, 6January, February, March, April, May, June, July, August, September, October, November, December, Summer 19-22, 1991.

    59
    Pierrot F., Dauchez P., and Fournier A.
    Fast parallel robots.
    Journal of Robotic Systems, 8(6):829-840, 12January February March April May June July August September October November December Summer 1991.

    60
    Portman V.T., Sandler B-Z, and Zahavi E.
    Rigid 6x6 parallel platform for precision 3D micromanipulation: theory and design application.
    IEEE Trans. on Robotics and Automation, 16(6):6290643, 12January February March April May June July August September October November December Summer 2000.

    61
    Raghavan M.
    The Stewart platform of general geometry has 40 configurations.
    In ASME Design and Automation Conf., volume 32-2, pages 397-402, Chicago, 9January, February, March, April, May, June, July, August, September, October, November, December, Summer 22-25, 1991.

    62
    Rao A.C.
    Topological characteristics of linkage mechanisms with particular reference to platform type robots.
    Mechanism and Machine Theory, 30(1):33-42, 1January February March April May June July August September October November December Summer 1995.

    63
    Ronga F. and Vust T.
    Stewart platforms without computer?, 1992.
    Preprint.

    64
    Rouillier F.
    Real roots counting for some robotics problems.
    In J-P. Merlet B. Ravani, editor, Computational Kinematics, pages 73-82. Kluwer, 1995.

    65
    Ryu R.J. and others .
    Eclipse: an overactuated parallel mechanism for rapid machining.
    In 12th RoManSy, pages 79-86, Paris, 7January, February, March, April, May, June, July, August, September, October, November, December, Summer 6-9, 1998.

    66
    Schmid H.A.
    Spreadbands drive parallel robots.
    Industrial robot, 28(4):320-327, 2001.

    67
    Tancredi L. and Teillaud M.
    Application de la géométrie synthétique au problème de modélisation géométrique directe des robots parallèles.
    Mechanism and Machine Theory, 34(2):255-269, 2January February March April May June July August September October November December Summer 1999.

    68
    Tönshoff K., Grendel H., and Kaak R.
    A hybrid manipulator for laser machining.
    In First European-American Forum on Parallel Kinematic Machines, Milan, 8January February March April May June July August September October November December Summer 31- 9January February March April May June July August September October November December Summer 1, 1998.

    69
    Verhoeven R. and Miller M.
    Estimating the controllable workspace of tendon-based Stewart platforms.
    In ARK, pages 277-284, Piran, 6January, February, March, April, May, June, July, August, September, October, November, December, Summer 25-29, 2000.

    70
    Wampler C.W.
    Forward displacement analysis of general six-in-parallel SPS (Stewart) platform manipulators using soma coordinates.
    Mechanism and Machine Theory, 31(3):331-337, 4January February March April May June July August September October November December Summer 1996.

    71
    Yang G., Chen I-M., and Yeo S.H.
    Design consideration and kinematic modeling for modular reconfigurable parallel robots.
    In 10th World Congress on the Theory of Machines and Mechanisms, pages 1079-1084, Oulu, 6January, February, March, April, May, June, July, August, September, October, November, December, Summer 20-24, 1999.

    72
    Yang P-H. and Waldron J.K.
    Coordination of parallel arrays of binary actuators.
    In 13th RoManSy, pages 43-50, Zakopane, 7January, February, March, April, May, June, July, August, September, October, November, December, Summer 3-6, 2000.

    73
    Zhuang H., Yan J., and Masory O.
    Calibration of Stewart platforms and other parallel manipulators by minimizing inverse kinematic residuals.
    J. of Robotic Systems, 15(7):395-405, 1998.

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    Footnotes

    ... exhaustive1
    The references in this paper are not exhaustive: further references and information on parallel robot may be found at
    http://www-sop.inria.fr/coprin/equipe/merlet/merlet_eng.html or
    http://www/parallelmic.org
    ... d.o.f.2
    It can be shown that parallel robots with as many identical kinematic chains as d.o.f. cannot have 4 or 5 d.o.f. except if special kinematic chains are used
    ... (PKI)3
    http://www-sop.inria.fr/EJCK/PKI/PKI.html
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    2002-09-25