PD-Like Regulation of Mechanical Systems With Prescribed Bounds of Exponential Stability: The Point-to-Point Case

Abstract

This letter discusses an extension of the famous PD regulator implementing point to point motions with prescribed exponential rates of convergence. This is achieved by deriving a novel global exponential stability result, dealing with mechanical systems evolving on uni-dimensional invariant manifolds of the configuration space. The construction of closed loop controllers enforcing the existence of such manifolds is then discussed. Explicit upper and lower bounds of convergence are provided, and connected to the gains of the closed loop controller. Simulations are carried out, assessing the effectiveness of the controller and the tightness of the exponential bounds.

Actuating Eigenmanifolds of Conservative Mechanical Systems via Bounded or Impulsive Control Actions

“…Therefore, if M is strict, we always have the freedom of imposing a one dimensional feedback action τ * , by making it tangent to the manifold. Note that Lemma 1 is coherent with [23,Theorem 1]. We use this result to derive a manifold preserving controller that injects or removes energy from the system.…”

“…2(a). In our previous work [23] we have proven that this can happen only if kinetic energy is constant along the trajectories of M (x)ẍ + C(x,ẋ)ẋ = 0 contained in the projection of M.…”

“…We consider the system depicted in Fig. 3 with potential energy V (X(x m )) = 1 2 kx 2 m , with k > 0, x m = 0.53x 1 + 0.66x 2 + 0.53x 3 , and perform simulations on the strict mode discussed in [23], using (12) to achieve a desired level of energy. The starting configuration is on the mode, close to the equilibrium.…”

Using Nonlinear Normal Modes for Execution of Efficient Cyclic Motions in Articulated Soft Robots

“…The state evolves under the control action (6). When the system is in a neighborhood of equilibrium configuration x m = 0 (i.e.…”

“…In a series of recent works [3,1,7,6], we have proposed a framework for exciting efficient oscillatory motions in robots immersed in a potential field. The long term aim of this theory is to equip generic soft robots (of both articulated and continuum kind) to execute periodic tasks efficiently and effectively.…”