Excessive Transverse Coordinates for Orbital Stabilization of (Underactuated) Mechanical Systems

On Orbital Stabilization as an alternative to Reference Tracking Control

“…The task is as before to converge to the periodic orbit (4). Towards this end, we compared the following three controllers: the u P D -controller (3) with (k p , k d ) = (4, 4); the controller (8) taken as u I = −ω 2 q − sgn + (q)I with I given by (5); and the u yz -controller (12) with (k y , k z ) = (4, 4). 3 Figure 1 shows the obtained phase portraits and control inputs from numerically simulating the system (13) when starting at (q 0 ,q 0 ) = (0, −3aω/2) with a = π/2, ω = π and with white noise added to the measurements.…”

“…Recently, however, some methods have appeared in the literature which try to make it a more viable option for common robotics tasks. These include immersion and invariance-based approaches [3] and transverse linearization-based approaches [4], [5]. We will in this paper focus on the latter type of approach.…”

“…In this regard, orbital stabilization of periodic motions of a fully actuated robot manipulator was considered in [4] by stabilizing an excessive set of so-called transverse coordinates. This approach has recently been further extended in [5], [6] to underactuated systems, with the additional relaxation that the so-called projection operators -mappings recovering the current "position" along the nominal motion -only have to be state-dependent and not strictly configuration-based as in [4]. Moreover, new insights were there provided in regards to a useful tool for orbitally stabilizing feedback design, as well as for general analysis of orbital (in-)stability in the large: the transverse linearization.…”

“…In this paper, we consider the approach and analysis tools outlined [5], [6] in relation to the particle system (13). The simplicity of this system allows for constructive-and illustrative examples, both highlighting some advantageous and disadvantageous of orbitally stabilizing controllers in comparison to basic (PD+feedforward-) reference tracking controllers.…”

“…The simplicity of this system allows for constructive-and illustrative examples, both highlighting some advantageous and disadvantageous of orbitally stabilizing controllers in comparison to basic (PD+feedforward-) reference tracking controllers. Furthermore, it facilitates the demonstration of how analysis-and construction of new types orbitally stabilizing feedback controllers can be achieved using the transverse linearization-based tools provided in [5], [6].…”