Stewart platforms without computer

many ways are there to move H in such a way that P i belongs to S 1 , i = 0; : : : ; 5 ? We show in this paper that generically there are at most 40 solutions. This problem is the geometric version of a control problem for Stewart robots. A (generalized) Stewart platform is a solid with 6 points P 0 ; : : : ; P 5 on it attached through 6 legs to 6 xed points Q 0 ; : : : ; Q 5 in the space R 3. Assuming that the lengths of the legs can be varied arbitrarily (within the physical limits), the problem, rst considered by D. Stewart 8], is to control the position of the body. y In mathematical terms, we can identify the space of positions of the solid with the space SO(3) R 3 of rotations and translations of R 3. We have a map: The coordinates in the target play the role of control parameters, whereas a point in the source represents a position of the solid. The problem is to study the map. Note that both the source and target of have dimension 6, so we might expect that for a generic choice of the points P i and Q i the bers of are nite. See 7] for a more detailed discussion of Stewart platforms. The aim of this paper is to use the intersection theory in algebraic geometry to show that for a generic choice of the points P i and Q i the number of possible positions of the platform for 6 given lengths is at most 40. A rst version of this paper was inspired from results of D. Lazard 6], who considered the case of a planar platform, (i.e. the P i 's lying in a plane as well as the Q i 's) using formal calculus manipulations on a computer; he also found that the number of possible positions for given lengths of the legs is at most 40 and conjectured that this should hold in general, but could only prove that it is bounded by 320 in the general case. Our rst version in turn inspired to D. Lazard new ideas that enabled him to simplify enormously y In fact a similar device was designed already in 1947 by V.E. Gough; see 8].