Differential Geometry And Mechanical Systems Modeling Analysis And Nonlinear Control

Nonlinear control of mechanical systems is a very challenging discipline liesat the intersection among many disciplines. It is closely related with control theory,geometric mechanical and numerical analysis. Experts work on this disciplineusually background in mathematics, mechanic, control theory, aerospace, roboticsand so on. They also usually focus on different aspects. We mainly introduce howdifferential geometry (include Lie theory) is applied in nonlinear control systemsfrom modeling, analysis, and design.Different from the classic Newton and Lagrange method for the modeling ofmechanical systems, here differential manifold is used in the modeling and finallywe get a compact geometry model. The traditional model of kinematic equationsand dynamic equations are always illustrate by Euler angle and unit quaternionwhich is complicate, here we use a class of lie group like SO(3) and SE(3) todescribe the geometric structure of a body which makes the kinematic anddynamic equations looks much more tidier. So we can see the great potential of theuse of differential geometry.Here we make a correspondence between the under-actuated direction ofspacecraft and the point on a unit sphere and use the notion of geodesic indesigning feedback control of an under-actuated spacecraft and prove theexponential convergence by Lyapunov method. We also give a feedback controllaw which can be used in a class of systems and here theory of lie group is applied.We illustrate all the control law by numerical experiment.Disappointedly, we can not use the theory we described above in all controlsystems. In addition, the method we used in numerical analysis is4-order R-Kscheme which have dissipation, for the long time simulation the result is nottrustable. Are there any other schemes that could hold the topologic structure ofsystems? We should consider this problem between numerical analysis andgeometric control.