This talk will outline a comprehensive set of modeling, analysis and design techniques for a class of mechanical systems. We concern ourselves with simple mechanical control systems, that is, systems whose Lagrangian is kinetic energy minus potential energy. Example devices include robotic manipulators, aerospace and underwater vehicles, and mechanisms that locomote exploiting nonholonomic constraints. Borrowing techniques from nonlinear control and geometric mechanics, we propose a coordinateinvariant control theory for this class of systems. First, we take a Riemannian geometric approach to modeling systems dened on smooth manifolds, subject to nonholonomic constraints, external forces and control forces. We also model mechanical systems on groups and symmetries. Second, we analyze some control-theoretic properties of this class of systems, including controllability, averaged response to oscillatory controls, and kinematic reductions. Finally, we exploit the modeling and analysis results to tackle control design problems. Starting from controllability and kinematic reduction assumptions we propose some algorithms for generating and tracking trajectories.

/pdf/geometric-control-of-mechanical-systems-59lvmkw96d.pdf

Geometric control of mechanical systems

I. On the interior geometry of metric spaces.- 1. Preliminaries.- 2. The Hopf-Rinow Theorem.- 3. Spaces with curvature bounded from above.- 4. The Hadamard-Cartan Theorem.- 5. Hadamard spaces.- II. The boundary at infinity.- 1. Closure of X via Busemann functions.- 2. Closure of X via rays.- 3. Classification of isometries.- 4. The cone at infinity and the Tits metric.- III. Weak hyperbolicity.- 1. The duality condition.- 2. Geodesic flows on Hadamard spaces.- 3. The flat half plane condition.- 4. Harmonic functions and random walks on ?.- IV. Rank rigidity.- 1. Preliminaries on geodesic flows.- 2. Jacobi fields and curvature.- 3. Busemann functions and horospheres.- 4. Rank, regular vectors and flats.- 5. An invariant set at infinity.- 6. Proof of the rank rigidity.- Appendix. Ergodicity of geodesic flows.- 1. Introductory remarks.- Measure and ergodic theory preliminaries.- Absolutely continuous foliations.- Anosov flows and the Ho continuity of invariant distributions.- Proof of absolute continuity and ergodicity.

Lectures on Spaces of Nonpositive Curvature

If M is a differentiable ra-dimensional manifold and V a linear connection for M, then the 2 rc-dimensional manifold TM, which is the total space of the tangent bündle of M, admits an almost complex structure /, naturally determined by V *). (I learned of this almost complex structure, which occurs e. g. in the theory of partial differential equations on Riemannian manifolds, frorn Professor W. Ambrose. I wish to thank him very much for the stimulating conversations which I have had with him on that topic.) We shall give here a computation of the Eckmann-Frölicher torsion tensor for this almost complex structure /, which implies the following result: / is integrable if and only if the linear connection has vanishing torsion and curvature). An appendix is devoted to some questions on the geometry of the tangent bündle TM which arise in connection with the construction of / and which can be answered easily by methods similar to those which we have used in order to compute the EckmannFrölicher torsion tensor of /. We list here only two of these results: If g is a Riemann metric for M and V its Levi-Civita connection, then TM admits a canonical hermitian metric hg with respect to the almost complex structure / on TM, which is determined (see above) by VWe prove, (confer Appendix (iii)) that hg is kählerian. If V is any linear connection for M, then the distribution of the \"horizontal subspaces\" on TM is invariant under the action of the multiplicative group R* of non vanishing real numbers on TM. We prove (confer Appendix (iv)) that if oppositely an n-dimensional distribution on TM is given, which is invariant under the action of the group JR* on TM and which contains no nonzero vertical\" vector, then this distribution

On the Geometry of the Tangent Bundle.

We construct supersymmetric field theories on Riemannian three-manifolds $ \mathcal{M} $
 , focusing on $ \mathcal{N} $
 = 2 theories with a U(1)R symmetry. Our approach is based on the rigid limit of new minimal supergravity in three dimensions, which couples to the flat-space supermultiplet containing the R-current and the energy-momentum tensor. The field theory on $ \mathcal{M} $
 possesses a single supercharge if and only if $ \mathcal{M} $
 admits an almost contact metric structure that satisfies a certain integrability condition. This may lead to global restrictions on $ \mathcal{M} $
 , even though we can always construct one supercharge on any given patch. We also analyze the conditions for the presence of additional supercharges. In particular, two supercharges of opposite R-charge exist on every Seifert manifold. We present general supersymmetric Lagrangians on $ \mathcal{M} $
 and discuss their flat-space limit, which can be analyzed using the R-current supermultiplet. As an application, we show how the flat-space two-point function of the energy-momentum tensor in $ \mathcal{N} $
 = 2 superconformal theories can be calculated using localization on a squashed sphere.

/pdf/supersymmetric-field-theories-on-three-manifolds-1o2i1gaopi.pdf

Supersymmetric Field Theories on Three-Manifolds

Relativistic transport theory

H.Poincare used the tangent sphere-bundles of ovaloids in three dimensional Euclidean space, i.e. the phase spaces of the ovaloids, to prove the existence of certain closed geodesies on the ovaloids. He introduced a Riemannian metric on the tangent sphere-bundles and considered the geodesic flow on it. As the metric of tangent bundles of Riemannian manifolds seems to be important, we would like to study differential geometry of tangent bundles of Riemannian manifolds by introducing on it natural Riemannian metrics. In this papar we shall do it by restricting ourselves only to the tangent bundles T{M).

https://projecteuclid.org/download/pdf_1/euclid.tmj/1178244668

On the differential geometry of tangent bundles of riemannian manifolds ii

then M is said to be a differentiable manifold with almost complex structure. (Tensor fields of the form given above may exist only for some manifolds with even dimension.) We shall call φ the fundamental collineation of the almost complex structure. The set of differentiable manifolds with almost complex structure is wider than the set of complex manifolds. Every differentiable manifold with almost complex structure φ admits a poistive definite Riemannian metric g such that

/pdf/on-differentiable-manifolds-with-certain-structures-which-262e3d52vw.pdf

On differentiable manifolds with certain structures which are closely related to almost contact structure i

/pdf/on-differentiable-manifolds-with-contact-metric-structures-3m002wpnve.pdf

On differentiable manifolds with contact metric structures

/pdf/on-the-tangent-sphere-bundle-of-a-2-sphere-4y6ar395zv.pdf

On the tangent sphere bundle of a 2-sphere

/pdf/on-complete-flat-surfaces-in-hyperbolic-3-space-1hjx5nu6g4.pdf

Shigeo Sasaki

Papers

On the differential geometry of tangent bundles of riemannian manifolds ii

On differentiable manifolds with certain structures which are closely related to almost contact structure i

On differentiable manifolds with contact metric structures

On the tangent sphere bundle of a 2-sphere

On complete flat surfaces in hyperbolic $3$-space