Trajectories of motions

In our society, robots are playing an increasingly important role and there are many challenging open problems in robotics of a both theoretical and practical nature. Our goal is to study robots that perform complicated motions with two degrees of freedom, by decomposing these motions into revolutions along axes. We are interested in the curves or surfaces that are traced by a point on the robot as it performs its motion. Our point of view is the one of kinematics and thus we do not take speed or forces into account. In kinematics, a robot is considered as a linkage which is a collection of rigid bodies called links whose relative displacements are constrained by joints. Watt`s linkage is a type of mechanical linkage invented by James Watt around 1785 for his steam engine as it converts a rotational motion to an approximate straight-line motion. It was thought to be impossible to construct linkages with revolute joints that trace out a straight line segment exactly. However, such linkages were constructed by Sarrus, Peaucellier and Lipkin eighty years later. Scientists must have been astonished when Kempe described in 1876 a method that constructs for any given planar algebraic curve a linkage that traces out a portion of this curve. A recent version of Kempe`s universality theorem states that any surface in space is the trajectory of a linkage with only revolute joints. There are also methods for constructing a linkage for a given trajectory. However, examples suggest that the constructed linkages are more complicated than necessary. The horizon of this project is to construct a linkage with a minimal number of revolute joints that has any given curve or surface in space as trajectory. Towards this goal, we consider trajectories of linkages that are defined by bar-joint-frameworks of mobility at most two and we assume that these frameworks can be made rigid by adding only one bar. Thus, somewhat paradoxically, our goal is closely related to a problem in combinatorial rigidity theory: bounding the number of realizations of minimally rigid graphs. Another related objective is to characterize the geometry of linkages that are of mobility at most two. For example, we want to determine whether there exists a cyclic linkage of mobility two with five revolute joints and one prismatic joint such that no two axes are parallel and no three revolute axes are concurrent. We combine algebro geometric methods and combinatorial methods from both kinematics and rigidity theory.

In order to gain insight into motions of robots, the first objective of this project was to study trajectories of vertices of graphs that move in the plane. We call such moving graphs "calligraphs" and their trajectories are called "coupler curves". Their origins can be traced back to at least 1785, when James Watt used a calligraph to convert a rotational motion to an approximate straight-line motion for his steam engine. The project funded, in addition to the project leader, three external postdocs, and they are coauthors of three subsequent publications on coupler curves of calligraphs. The first publication uniquely assigns to each calligraph a vector consisting of three integers. This vector characterizes invariants of its associated planar coupler curve and links these invariants to the number of realizations of so called minimally rigid graphs. Such rigid graphs are well-known within the rigidity community and have applications in natural science and engineering. The second publication generalizes these results to calligraphs that move in the sphere instead of the plane and the third publication investigates the number of components of planar coupler curves. The first publication closely follows the initially proposed methods, although the proofs turned out to be much more challenging than expected. We did not foresee the use of the moduli space of stable rational curves with marked points for calligraphs in the sphere. Such moduli spaces are of recent interest in algebraic geometry. The horizon of this project is to find a linkage with a minimal number of links and revolute joints that has a given compact surface as trajectory. This turned out to be an unrealistic goal within the assigned four years, but we made considerable progress by the following readjustment of the remaining project objectives. The most basic linkages, namely linkages with at most two revolute joints, have either spheres or tori as trajectories. Spheres and tori of revolution are examples of "celestial surfaces", namely surfaces containing at least two circles through each point. Such embedded surfaces are of interest in kinematics, architecture and computer vision. We investigated combinatorial and topological aspects of celestial surfaces in six publications. In particular, in cooperation with the University of Innsbruck, we classified celestial surfaces as an application of an improved factorization method for bivariate quaternionic polynomials. Such polynomials play a central role in kinematics. The remaining publications funded by this project investigate curves on surfaces that behave like lines in the plane. In cooperation with the Johannes Keppler Unversity, we investigated automorphism of such surfaces and our results are of interest in geometric modeling. Another unforeseen outcome of this project are theorems on webs of curves on surfaces and we plan their further investigation.