Review of the exponential and Cayley map on SE(3) as relevant for Lie group integration of the generalized Poisson equation and flexible multibody systems

The exponential and Cayley maps on SE(3) are the prevailing coordinate maps used in Lie group integration schemes for rigid body and flexible body systems. Such geometric integrators are the Munthe–Kaas and generalized- α schemes, which involve the differential and its directional derivative of the respective coordinate map. Relevant closed form expressions, which were reported over the last two decades, are scattered in the literature, and some are reported without proof. This paper provides a reference summarizing all relevant closed-form relations along with the relevant proofs, including the right-trivialized differential of the exponential and Cayley map and their directional derivatives (resembling the Hessian). The latter gives rise to an implicit generalized- α scheme for rigid/flexible multibody systems in terms of the Cayley map with improved computational efficiency.

Cayley Map for Symplectic Groups

Despite of their importance, the symplectic groups are not so popular like orthogonal ones as they deserve. The only explanation of this fact seems to be that their algebras can not be described so simply. While in the case of the orthogonal groups they are just the anti-symmetric matrices, those of the symplectic ones should be split in four blocks that have to be specified separately. It turns out however that in some sense they can be presented by the even dimensional symmetric matrices. Here, we present such a scheme and illustrate it in the lowest possible dimension via the Cayley map. Besides, it is proved that by means of the exponential map all such matrices generate genuine symplectic matrices.

The Cayley Isomorphism Property for Cayley Maps

The Cayley Isomorphism property for combinatorial objects was introduced by L. Babai in 1977. Since then it has been intensively studied for binary relational structures: graphs, digraphs, colored graphs etc. In this paper we study this property for oriented Cayley maps. A Cayley map is a Cayley graph provided by a cyclic rotation of its connection set. If the underlying graph is connected, then the map is an embedding of a Cayley graph into an oriented surface with the same cyclic rotation around every vertex.Two Cayley maps are called Cayley isomorphic if there exists a map isomorphism between them which is a group isomorphism too. We say that a finite group $H$ is a CIM-group if any two Cayley maps over $H$ are isomorphic if and only if they are Cayley isomorphic.The paper contains two main results regarding CIM-groups. The first one provides necessary conditons for being a CIM-group. It shows that a CIM-group should be one of the following$$\mathbb{Z}_m\times\mathbb{Z}_2^r, \\mathbb{Z}_m\times\mathbb{Z}_{4},\\mathbb{Z}_m\times\mathbb{Z}_{8}, \ \mathbb{Z}_m\times Q_8, \\mathbb{Z}_m\rtimes\mathbb{Z}_{2^e}, e=1,2,3,$$ where $m$ is an odd square-free number and $r$ a non-negative integer. Our second main result shows that the groups $\mathbb{Z}_m\times\mathbb{Z}_2^r$, $\mathbb{Z}_m\times\mathbb{Z}_{4}$, $\mathbb{Z}_m\times Q_8$ contained in the above list are indeed CIM-groups.

Rational Interpolation of Car Motions

This work introduces a general approach to the interpolation of the rigid-body motions of cars by rational motions. A key feature of the approach is that the motions produced automatically satisfy the kinematic constraints imposed by the car wheels, that is, cars cannot instantaneously translate sideways. This is achieved by using a Cayley map to project a polynomial curve in the Lie algebra se(2) to SE(2) the group of rigid displacements in the plane. The differential constraint on se(2), which expresses the kinematic constraint on the car, is easily solved for one coordinate if the other two are given, in this case as polynomial functions. In this way, families of motions obeying the constraint can be found. Several families are found here and examples of their use are shown. It is shown how rest-to-rest motions can be generated in this way and also how these motions can be joined so that the motion is continuous and differentiable across the join. A final section discusses the optimization of these motions. For some cost functions, the optimal motions are known but can be rather impractical to use. By optimizing over a family of motions which satisfy the boundary conditions for the motion, it is shown that rational motions can be found simply and are close to the overall optimal motion.

Unoriented Cayley maps

A Cayley map is an embedding of a Cayley graph into an orientable surface and it has been studied intensively for last decades [1, 8, 10, 11, 15, 16, 17, 18, etc]. In this paper we consider an embedding of a Cayley graph into an orientable or nonorientable surface. We call it a generalized Cayley map. We describe the automorphism group of a generalized Cayley map and determine when a generalized Cayley map can be regular. The Petrie dual of a generalized Cayley map is also studied. Finally, the first infinite family of graphs which can be underlying graphs of nonorientable regular maps is presented.

Geometric Approaches to Relativistic Kinematics

A comparison is made, between three methods for visualizing the Minkowski space, in view of their application to relativistic kinematics.The velocity manifold method, in which a vector is represented by a point on the upper sheet of the unit-radius hyperboloid, presents two advantages: there is no distortion of angles, and the method is independent of the choice of any particular frame or observer. It is, however, limited to the representation of positive timelike vectors.The Cayley map method allows the representation of spacelike as well as timelike vectors, but it has two drawbacks: it distorts the angles, and it depends on the choice of a reference system.In the third method, the use of a unified space–time formalism permits one to visualize any kind of vector. This method generalizes the velocity manifold method, and presents the same two advantages: it provides us with a conformal representation, which is independent of any particular observer.We also present a rather concise comparison between two covariant methods of calculation, considered from the point of view of relativistic kinematics. One method is the familiar tensor calculus; the other is the spinor calculus, based on the representation of four-vectors by matrices.