Risc Colloquium

The basic kinematic characteristics of a mechanism are encoded in its configuration space (c-space) - the solution variety $V$ of certain geometric constraints. Such fundamental geometric property is the finite (local) mobility of a given mechanism. However, there is yet no universal and conclusive treatment of that very basic matter in the theory of mechanisms. The mobility problem boils down to the determination of the (local) dimension of $V$. The actual finite motion at a given configuration is determined by a local approximation of the c-space. The lowest-order local approximation is the tangent cone $C_{q}V$, and the local mobility is the dimension of $C_{q}V$.
The c-space can be defined as an algebraic or analytic variety, and consequently different mathematical concepts can be used for its analysis. These are discussed in this presentation.
As prerequisite the notion of differential, local, and global mobility is recalled, and thereupon trivial, overconstrained, underconstrained, and kinematotropic mechanisms are introduced.
Theoretical kinematics traditionally the constraints are defined by analytic mappings, and $V$ is an analytic variety. The intrinsic structure of the Lie group $SE(3)$ gives rise to closed-form expressions for the tangent cone. It is shown that this approach allows for determination of the local mobility of any mechanism, but the necessary approximation order is yet unknown.
Most mechanisms comprise algebraic joints, and thus allow for an alternative algebraic description, so that $V$ is an algebraic variety. The tangent cone can be in principle computed with tools from computational algebraic geometry. They do, however, fail to provide the real dimension for underconstrained mechanisms. This is discussed for several examples. Furthermore the computational complexity makes application to complex mechanism impossible.
Kinematotropic mechanism are remarkable since they can change between motion modes (subvarieties) with different mobility. The transition is commonly accompanied by a non-smooth motion, which is undesirable. The Wunderlich mechanism, on the other hand, is an example where this transition is smooth. The general geometric conditions that ensure smooth transitions are yet unknown. Preliminary results will be presented using intersection theory.