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The problem of rigid-body attitude tracking in the presence of exogenous disturbances is addressed. Attitude is parameterized using the rotation matrix, an element of the Special Orthogonal Group SO(3), as it provides a singularity-free and unambiguous attitude description. The closed-loop stability and robustness properties of a PD-type state-feedback control law, proposed in literature for attitude tracking using rotation matrices, are investigated using the nonlinear H∞ control framework. Starting from a dissipation inequality, sufficient conditions are derived which ensure that the closed-loop energy gain from bounded, finite-energy exogenous disturbances to a specified error signal respects a given upper bound. Then, the sufficient conditions are reformulated using the state and input matrices for the translational double integrator, and recast as linear matrix inequalities (LMIs). Lastly, the reformulated LMIs are used to synthesize controller gains for the proportional and derivative state-feedback terms in the original SO(3) control law. The controller synthesis problem for a microsatellite is considered as a case study. The controller gains are obtained using the proposed LMI-based procedure, and the tracking and disturbance rejection capabilities of the SO(3) controller are illustrated.

In this article, we find the subalgebra of the orthogonal Lie algebra corresponding to the DSER elementary orthogonal group on a quadratic space with a hyperbolic summand.

This paper is devoted to some selected topics of the theory of special orthogonal group SO(3). First, we discuss the trace of orthogonal matrices and its relation to the characteristic polynomial; on this basis, the partition of SO(3) formed by conjugation classes is described by trace mapping. Second, we show that every special orthogonal matrix can be expressed as the product of three elementary special orthogonal matrices. Explicit formulas for the decomposition as needed for applications in differential geometry and physics as symmetry transformations are given.

The complete classification of the finite simple groups that are \\((2,3)\\)-generated is a problem which is still open only for orthogonal groups. Here, we construct \\((2, 3)\\)-generators for the finite odd-dimensional orthogonal groups \\(\\Omega _{2k+1}(q)\\), \\(k\\geq 4\\). As a byproduct, we also obtain \\((2,3)\\)-generators for \\(\\Omega _{4k}^+(q)\\) with \\(k\\geq 3\\) and q odd, and for \\(\\Omega _{4k+2}^\\pm (q)\\) with \\(k\\geq 4\\) and \\(q\\equiv \\pm 1~ \\mathrm {(mod~ 4)}\\).

The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems, dynamic low-rank decompositions and model reduction. With this mostly expository work, we aim to provide a collection of the essential facts and formulae on the geometry of the Grassmann manifold in a fashion that is fit for tackling the aforementioned problems with matrix-based algorithms. Moreover, we expose the Grassmann geometry both from the approach of representing subspaces with orthogonal projectors and when viewed as a quotient space of the orthogonal group, where subspaces are identified as equivalence classes of (orthogonal) bases. This bridges the associated research tracks and allows for an easy transition between these two approaches. Original contributions include a modified algorithm for computing the Riemannian logarithm map on the Grassmannian that is advantageous numerically but also allows for a more elementary, yet more complete description of the cut locus and the conjugate points. We also derive a formula for parallel transport along geodesics in the orthogonal projector perspective, formulae for the derivative of the exponential map, as well as a formula for Jacobi fields vanishing at one point.

Let Γ n be an n × n Haar-invariant orthogonal matrix. Let Z n be the p × q upper-left submatrix of Γ n , where p = p n and q = q n are two positive integers. Let G n be a p × q matrix whose pq entries are independent standard normals. In this paper we consider the distance between √nZ n and G n in terms of the total variation distance, the Kullback-Leibler distance, the Hellinger distance, and the Euclidean distance. We prove that each of the first three distances goes to zero as long as pq/n goes to zero, and not so if (p, q) sits on the curve pq = σn, where σ is a constant. However, it is different for the Euclidean distance, which goes to zero provided pq²/n goes to zero, and not so if (p, q) sits on the curve pq² = σn. A previous work by Jiang (2006) shows that the total variation distance goes to zero if both p/ √n and q/ √n go to zero, and it is not true provided p = c √n and q = d √n with c and d being constants. One of the above results confirms a conjecture that the total variation distance goes to zero as long as pq/n → 0 and the distance does not go to zero if pq = σn for some constant σ.

In this work the matrix exponential function is solved analytically for the special orthogonal groups SO(n) up to n=9. The number of occurring k-th matrix powers gets limited to 0≤k≤n−1 by exploiting the Cayley–Hamilton relation. The corresponding expansion coefficients can be expressed as cosine and sine functions of a vector-norm V and the roots of a polynomial equation that depends on a few specific invariants. Besides the well-known case of SO(3), a quadratic equation needs to be solved for n=4,5, a cubic equation for n=6,7, and a quartic equation for n=8,9. As an interesting subgroup of SO(7), the exceptional Lie group G[sub.2] of dimension 14 is constructed via the matrix exponential function through a remarkably simple constraint on an invariant, ξ=1. The traces of the SO(n)-matrices arising from the exponential function are sums of cosines of several angles. This feature confirms that the employed method is equivalent to exponentiation after diagonalization, but avoids complex eigenvalues and eigenvectors and operates only with real-valued quantities.

Descriptions of various subsets of \\(\\mathbb{SO}(3)\\) are encountered frequently in robotics, for example, in the context of specifying the orientation workspaces of manipulators. Often, the Cartesian concept of a cuboid is extended into the domain of Euler angles, notwithstanding the fact that the physical implications of this practice are not documented. Motivated by this lacuna in the existing literature, this article focuses on studying sets of rotations described by such cuboids by mapping them to the space of Rodrigues parameters, where a physically meaningful measure of distance from the origin is available and the spherical geometry is intrinsically pertinent. It is established that the planar faces of the said cuboid transform into hyperboloids of one sheet and hence, the cuboid itself maps into a solid of complicated non-convex shape. To quantify the extents of these solids, the largest spheres contained within them are computed analytically. It is expected that this study would help in the process of design and path planning of spatial robots, especially those of parallel architecture, due to a better and quantitative understanding of their orientation workspaces.

A computation shows that there are 7777 (up to scalar shifts) possible pairs of integer coefficient polynomials of degree five having roots of unity as their roots and satisfying the conditions of Beukers and Heckman, so that the Zariski closures of the associated monodromy groups are either finite or the orthogonal groups of non-degenerate and non-positive quadratic forms. Following the criterion of Beukers and Heckman, it is easy to check that only 44 of these pairs correspond to finite monodromy groups, and only 1717 pairs correspond to monodromy groups, for which the Zariski closure has real rank one. There are 5656 pairs remaining, for which the Zariski closures of the associated monodromy groups have real rank two. It follows from Venkataramana that 1111 of these 5656 pairs correspond to arithmetic monodromy groups, and the arithmeticity of 22 other cases follows from Singh. In this article, we show that 2323 of the remaining 4343 rank two cases correspond to arithmetic groups.