Path following of a class of underactuated mechanical systems via immersion and invariance‐based orbital stabilization

Abstract

This article aims to provide a new problem formulation of path following for mechanical systems without time parameterization nor guidance laws, namely, we express the control objective as an orbital stabilization problem. It is shown that it is possible to adapt the immersion and invariance technique to design static state-feedback controllers that solve this problem. In particular, we select the target dynamics adopting the recently introduced Mexican sombrero energy assignment method. To demonstrate the effectiveness of the proposed method we apply it to control underactuated marine surface vessels. K E Y W O R D S energy shaping, immersion and invariance, orbital stabilization, path following 1 INTRODUCTION Tracking a geometric path in mechanical systems is a task often encountered in various application fields, for example, robotics, aerospace, and autonomous vehicles. There are two approaches to formulate this problem, depending on whether the predefined path is parameterized by time or not, and they are known as trajectory tracking or path following, respectively. The former approach largely dominates the control literature and it has been extensively studied. Unfortunately, for nonminimum phase systems, which are common in these applications, the achievable performance has a fundamental limitation, first identified in Reference 1. It is possible to circumvent this limitation removing the time

Orbital stabilization of underactuated mechanical systems without Euler-Lagrange structure after of a collocated pre-feedback

“…. Now, let us come back to the target dynamics (29). Since there is no equilibrium for the level set {ξ ∈ R 2 | H(ξ) = c} with c > H(ξ * 1 , 0), all the trajectories of ( 29), excluding the isolated equilbirum, are non-trivial periodic solutions.…”

“…Oscillating behaviour of dynamical systems is ubiquitous in biology, physics and engineering [3], [21], [27]. For the latter, we are concerned with the constructive perspective to generate stable oscillators for closed loop-known as the orbital stabilization problem [10]which widely appears in many engineering areas, including bipedal robots [19], exoskeletons [11], electrical motors [23], AC power converters [13], path following of closed orbits [24], [29], combustion oscillations, etc. Despite the fact that these control problems may be formulated as trajectory tracking, it brings the following merits to consider them as orbital stabilization:…”

Orbital synchronization of homogeneous mechanical systems with one degree of underactuation

“…As an example, robot manipulators in the automotive industry are employed to realize painting, welding, and assembling tasks in a synchronized and repetitive way. In the literature, one can find works contributing to impose periodic motion (seen as orbital behavior in the system's phase portrait) in underactuated mechanical systems (this is, those with less control inputs than degrees of freedom) 8‐12 . These systems have attracted a lot of interest due to their structures.…”

“…In the literature, one can find works contributing to impose periodic motion (seen as orbital behavior in the system's phase portrait) in underactuated mechanical systems (this is, those with less control inputs than degrees of freedom). [8][9][10][11][12] These systems have attracted a lot of interest due to their structures. It is true that they face challenges in their maneuverability and capacity to efficiently perform a given task in the same or similar manner as their fully actuated counterparts.…”

Space‐and‐time‐synchronized simultaneous fully‐actuated vehicle tracking/formation using cascaded prescribed‐time control

“…In practice, positioning and tracking problems of ground, marine vehicles and unmanned aerial vehicles (UAVs) are of special interest to many important applications which involve the trajectory/path following of autonomous vehicles, 1‐6 formation/containment control of multiple autonomous vehicles 7‐14 and so on. It is noted that there exist two important scales in positioning and tracking: space and time.…”