Vectorial parameterizations of pose

Robotics and computer vision problems commonly require handling rigid-body motions comprising translation and rotation – together referred to as pose. In some situations, a vectorial parameterization of pose can be useful, where elements of a vector space are surjectively mapped to a matrix Lie group. For example, these vectorial representations can be employed for optimization as well as uncertainty representation on groups. The most common mapping is the matrix exponential, which maps elements of a Lie algebra onto the associated Lie group. However, this choice is not unique. It has been previously shown how to characterize all such vectorial parameterizations for SO(3), the group of rotations. Some results are also known for the group of poses, where it is possible to build a family of vectorial mappings that includes the matrix exponential as well as the Cayley transformation. We extend what is known for these pose mappings to the $4 \times 4$ representation common in robotics and also demonstrate three different examples of the proposed pose mappings: (i) pose interpolation, (ii) pose servoing control, and (iii) pose estimation in a pointcloud alignment problem. In the pointcloud alignment problem, our results lead to a new algorithm based on the Cayley transformation, which we call CayPer.

Matrix iteration algorithms for solving the generalized Lyapunov matrix equation

In this paper, we first recall some well-known results on the solvability of the generalized Lyapunov equation and rewrite this equation into the generalized Stein equation by using Cayley transformation. Then we introduce the matrix versions of biconjugate residual (BICR), biconjugate gradients stabilized (Bi-CGSTAB), and conjugate residual squared (CRS) algorithms. This study’s primary motivation is to avoid the increase of computational complexity by using the Kronecker product and vectorization operation. Finally, we offer several numerical examples to show the effectiveness of the derived algorithms.

On the Dimension of Certain Graded Lie Algebras Arising in Geometric Integration of Differential Equations

Many discretization methods for differential equations that evolve in Lie groups and homogeneous spaces advance the solution in the underlying Lie algebra. The main expense of computation is the calculation of commutators, a task that can be made significantly cheaper by the introduction of appropriate bases of function values and by the exploitation of redundancies inherent in a Lie-algebraic structure by means of graded spaces. In many Lie groups of practical interest a convenient alternative to the exponential map is a Cayley transformation, and the subject of this paper is the investigation of graded algebras that occur in this context. To this end we introduce a new concept, a hierarchical algebra, a Lie algebra equipped with a countable number of m-nary multilinear operations which display alternating symmetry and a ‘hierarchy condition’. We present explicit formulae for the dimension of graded subspaces of free hierarchical algebras and an algorithm for the construction of their basis. The paper is concluded by reviewing a number of applications of our results to numerical methods in a Lie-algebraic setting.

Time-dependent treatment of scattering: potential referenced and kinetic energy referenced modified Cayley approaches to atom–diatom collisions and time-scale separations

The time-dependent Lippmann–Schwinger equation describing atom–diatom collisions is expressed in terms of a general reference Hamiltonian, Hr, whose dynamics are easily solved in one representation, and a corresponding disturbance Hamiltonian, Hd, whose dynamics are easily solved in a different representation. The wavefunction at time t + τ t is then expressed in terms of its value at a previous time t by means of a simple quadrature approximation. The resulting expression for ψ(t + τ) has a form similar to that occurring in earlier numerical unitary solutions to the time-dependent Schrodinger equation via a Cayley transformation. The structure of the new equations is made explicit for (a) the choice where Hr is taken to be the kinetic energy and Hd is the potential energy and (b) the choice where Hr is taken to be the potential energy and Hd is the kinetic energy. In addition, we also deal with several alternatives for treating the binding potential of the diatom. Several alternatives for choosing representations are then explored for reducing the equations to a form amenable to computation. The short time structure of the equations is discussed in terms of a multiple time-scales analysis. Keywords: molecular collisions, multiple time scales, quantum dynamics.

Dissipative operators with finite dimensional damping

SynopsisLetG: ε(G)⊂ℋ → ℋ be a maximal dissipative operator with compact resolvent on a complex separable Hilbert space ℋ andT(t) be theCosemigroup generated byG. A spectral mapping theorem σ(T(t))\{0} = exp (tσ(G))/{0} together with a condition for 0 ε σ(T(t)) are proved if the set {xε ⅅ(G) | Re (Gx, x) = 0} has finite codimension in ε(G) and if some eigenvalue conditions forGare satisfied. Proofs are given in terms of the Cayley transformationT= (G+I)(G−I)−1ofG. The results are applied to the damped wave equationutt+ γutx+uxxxx+ ßuxx= 0, 0 ≦t< ∞ 0 <x< 1, β, γ ≧ 0, with boundary conditionsu(0,t) =ux(0,t) =uxx(1,t) =uxxx(1,t) = 0.