Page 109 - Textos de Matemática Vol. 43

STATE EQUIVALENCE TO THE SECOND-ORDER CHAINED FORM 97
In contrast to systems with first-order nonholonomic constraints, systems with second-order nonholonomic constraints include the drift-term (that is, a vector field describing the evolution not subject to controls, i.e., u = 0) which complicates significantly their analysis, thus making it more challenging. It has been shown that several nonholonomic systems can be transformed into the second-order chained form (SCF) via feedback and coordinate transformations (see e.g., [1, 6, 23]). This form has the advantage of considerably simplifying the dynamical equations of the system, being much more suitable to deal with than the equations of the original system.
In this paper, we are interested in knowing which nonlinear systems can be trans- formed, under a state-transformation, into the (SCF)0, that is, in finding geometric conditions allowing to characterise this class of systems. This problem has already been discussed in [21], where a geometric characterisation of the (SCF)0 was given by the following formulation: The system ⌃ is locally state equivalent, at z0 2 R6, to a system of the form (SCF)0, at the origin of R6, if and only if ⌃ satisfies in a neighborhood of z0 conditions of the following type:
(SCF1) Mechanical conditions;
(SCF2) Absence of dissipative-type forces and uncontrolled forces conditions; (SCF3) Symmetric product and Lie product conditions.
At this point we confine ourselves to distinguish between the type of conditions involved; they are discussed in detail in Section 2.1 of the present paper (see also Theorem 4 of [21]). We observe that this characterisation involves conditions with a clear geometric meaning: conditions (SCF1) are responsable for the mechanical structure of the system; conditions (SCF2) imply the absence of dissipative and gy- roscopic type terms and the absence of uncontrolled forces; and conditions (SCF3), related to the symmetric product and the Lie product of the involved vector fields, dictate the particular form of the (SCF)0. That result gives thus a characterisation of the (SCF)0 in terms of geometric conditions whose role is well understood, and, to the best of our knowledge, it is the first geometric characterisation of the (SCF)0 available in the literature. The used approach is based in geometric tools developed by the authors in [22] and [20].
In the present paper, which is an extended version of [21], we revisit the geo- metric interpretation of the (SCF)0 as described in that paper, but also we give an alternative geometric characterisation involving conditions which are easier to check. This yields a new approach which constitutes the novelty of the present pa- per. The aim of the paper is also to compare both sets of conditions and to discuss their geometric interpretations.
The rest of the paper is organized as follows. In Section 2 we present our main results, which are two di↵erent characterisations of the second-order chained form (SCF). In Subsection 2.1 we revisit the main result of [21], and in Subsection 2.2 we give a new result based on a di↵erent approach. Section 3 provides some background on previous works related directly to the present work. Namely, we introduce the