Closed trajectories and global controllability in the plane

In the plane, results are given about the structure of closedtrajectories which may occur as simple closed curves or generalclosed curves with self-intersections. Necessary conditionsfor the global controllability of nonlinear systems that arein the so-called linear-analytic form = f(x)+ug(x), where x R² and u 1, are given. It is provedthat, if there exists a closed trajectory of the system, then either contains a point where f and g are linearly dependent,or encloses some zeros of f + ug for all u [-1,1]. Then thisresult is used to prove that, if the linear-analytic systemis controllable and the vector field g is never zero in W R²,then W contains some zeros of f + ug for some u [-1,1]. A topologicalapproach is taken. Remarks are made about the size of the regionwhere a closed trajectory can lie, and about the shape of theclosed trajectories. Further implications are discussed.     

Linearized control systems and small-time reachable sets

Using linear approximations of nonlinear systems has long beena practice to design control laws. In this paper, an analysisis given involving linear approximation of the nonlinear controlsystem and small-time reachable sets in R². A useful concept,the swing–out, which is a measure of nonlinearity, isdefined. This is used to examine the relationship between thesmall-time reachable sets of the nonlinear control system andits linear approximation. Behaviour of the nonlinear systemunder a control law is examined within this context. More factsare given about the swing-out for some special cases.     

Cubic Splines on Curved Spaces

We consider a second-order problem in the calculus of variations,with an application to robotics in mind. The analysis is carriedout on a general Riemannian manifold M and then specializedto the case where M is the Lie group SO(3) of rotations in R³.For SO(3), the Euler-Lagrange equations reduce to interestingnonlinear systems of ordinary differential equations in R³.     

Discrete Signals and Continuous Dynamics

We show that, even for a control system with good constructibilityproperties, continuous signals or observers with memory maybe required in order to estimate continuously-changing states.     

Trajectory Generation and Obstacle Avoidance

We use methods from algebraic topology to study the problemof cartesian trajectory generation in roboties. We find thatcartesian trajectory generation may be impossible, or at bestirregular, or that singularities appear, unless the joint spaceand workspace of the robot can be matched sufficiently well.     

Feedback and Accessibility

We show that, for most control systems, smooth feedback lawsare insufficient to produce good accessibility properties, evenin a local sense.