Orbital stabilization of a class of underactuated mechanical systems

Abstract

Typically, the aim of a controller is to stabilize an equilibrium or a reference trajectory. For underactuated mechanical systems this objective might not be achievable because of the occurance of limit cycles due to discontinuous effects like actuator backlash or Coulomb friction. Therefore, an alternative approach is to explicitly impose a stable periodic orbit on the system by a suited controller. This way at least amplitude, frequency and settling time can be influenced directly.

Forced nonlinear oscillation of underactuated systems: the balance board example

“…We consider a linear controllable feedback system,ẋ = Ax + bu, with state x(t) ∈ R n and scalar input u(t) ∈ R. Clearly, this model might be the (jacobian oder exact) linearization of a nonlinear one. Further, we assume the application of a subordinate linear static state feedback [2] such that A has the special eigenvalue configuration spec(A) = {jω, −jω, λ 3 . .…”

“…As a consequence any periodic orbit corresponds to a level set of W (y) = α = const. For the construction of the feedback law that stabilizes the limit cycle with the desired amplitude α * we use the method of a Control-Lyapunov function (clf) [6], which is more systematic then the heuristic approach used in [2,7]. The clf-candidate incorporates the quadratic form W :…”

“…For reasonable physical parameters the balance board has an one-dimensional manifold of unstable equilibrium points. The system is motivated by a sports device, also known as "indoboard" and might be interpreted as an inverted version of the ball and beam benchmark system [1,2], although it differs in several features. In case of the balance board as sports device, limit cycles almost always occur and are much easier to stabilize than equilibriums by the operating human.…”