The aim of this seminar was to provide an account -of the state of the art in Computational Kinematics. We understand here under this term that branch of kinematics research involving intensive computations not only of the numerical type, but also of a symbolic nature.

Research in kinematics over the last decade has been intensive in the computational aspects of kinematics problems. In fact. this work has been prompted by the need to answer fundamental questions such as the number of solutions, whether real or complex, that a given problem can admit. Problems of this kind occur frequently in the analysis and synthesis of kinematic chains, when finite displacements are considered. The associated models, that are derived from kinematic relations known as closure equations, lead to systems of nonlinear algebraic equations in the variables or parameters sought. What we mean by algebraic equations here is equations whereby the unknowns are numbers, as opposed to differential equations, where the unknowns are functions. The algebraic equations at hand can take on the form of multivariate polynomials or may involve trigonometric functions of unknown angles.

Because of the nonlinear nature of the underlying kinematic models, purely numerical methods tum out to be too restrictive, for they involve iterative procedures whose convergence cannot, in general, be guaranteed. Additionally, when these methods converge, they do so to only isolated solutions, and the question as to the number of solutions to expect still remains. These drawbacks have been overcome with the development of continuation techniques that are meant to produce all solutions to a given problem. While continuation technqiues have provided solutions to a number of problems. they are still subjected to the uncertainties of iterative techniques. Hence, alternative approaches have been sought, that rely on modern software and hardware for symbolic computations. Commercial software of this kind is now very reliable and widespread; it has naturally found its way into kinematics research. ln fact, current research in kinematics involves symbolic manipulations that were impossible to even imagine as recently as fifteen years ago, when the first symbolic manipulation packages started coming out of the computer science laboratories. Among other techniques, resultant methods based on dyalitic elimination were discussed critically at the seminar, while Gröbner bases were proposed as a powerful elimination technique .

The seminar covered trends and progress attained in Computational Kinematics in a broad class of problems. The workshop topics were divided into seven parts, namely:

• kinematics algorithms, whereby general kinematics problems are discussed in light of their solution algorithms;
• redundant manipulators, which is self-descriptive;
• kinematic and dynamic control, in which the link between kinematics and the disciplines of dynamics and control is established;
• parallel manipulators, self-descriptive as well;
• motion planning, touching upon computational geometry;
• kinematics of mechanisms, in which the main issue is the presence of closed kinematic chains;
• a special section of miscellaneous contributions that were scheduled after the original program was mailed to the participants.

All contributions to the seminar, except those of the special section were published under the title "Computational Kinematics" in volume 28 of the series "Solid Mechanics and its applications" by Kluwer Academic Publishers, Dordrecht.

IBFI Schloß Dagstuhl is herewith given due acknowledgement for its financial and logistical support and encouragement. This support made it possible to bring together specialists of various disciplines working in the area. Among the participants, who met for one week at Schloß Dagstuhl, we count engineers, computer scientists and biomechanicists, all of whom share a common interest, namely, Computational Kinematics. Prof. Dr. Reinhard Wilhelm, Scientific Director of IBFI, and his staff are especially acknowledged for their support and their outstanding hospitality that contributed significantly to the success of the workshop.

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