Relative geodesics in the special Euclidean group

Abstract

We introduce a measurement (the discrepancy) of the minimum energy needed to transform from a standard parametrized planar curve c 0 to an observed curve c 1 . To this end, we say that a curve of transformations in the special Euclidean group SE (2) is admissible if it maps the source curve to the target curve under the point-wise action of SE (2) on the plane. After endowing the group SE(2) with a leftinvariant metric, we define a relative geodesic in SE(2) to be a critical point of the energy functional associated to the metric, over all admissible curves. The discrepancy is then defined as the value of the energy of the minimizing relative geodesic. In the first part of the paper, we derive a scalar ODE which is a necessary condition for a curve in SE(2) to be a relative geodesic, and we discuss some of the properties of the discrepancy. In the second part of the paper, we consider discrete curves, and by means of a variational principle, we derive a system of discrete equations for the relative geodesics. We finish with several examples.

Optimal Control of Left-Invariant Multi-Agent Systems with Asymmetric Formation Constraints

“…In the following, we will use the Cayley map to provide a parametrization of a neighborhood of the identity in SE(2) by means of the Lie algebra se(2), but it is possible to replace the Cayley map by any other local diffeomorphism satisfying (58) from se(2) to SE(2), such as the exponential map. The Cayley map, however, has the advantage that it is efficiently computable, and its derivative is particularly easy to characterize as it have been shown in [6], [15], [11], [21], among others.…”

“…For the Lie group SE(2), the Cayley map and its derivatives were computed in [20] (see also [11]). For the discrete time equations describing necessary conditions for normal extrema in the optimal formation problem, we will need also the right-trivialized derivative of the Cayley map (see [14] for details), dCay : se(2) × se(2) → se * (2) given by a map (linear in its second argument) defined as dCay ω (η) := (T Cay(ω) · η)Cay(ω) −1 ;…”

Importance and Effectiveness of Representing the Shapes of Cosserat Rods and Framed Curves as Paths in the Special Euclidean Algebra

“…Evidently, there is a considerable simplification in the model, since only three scalar fields determine the shape of a rod constrained in accord with (8) and (9). Perhaps surprisingly, however, there is absolutely no simplification in the Lie algebra and group necessary to describe the system.…”

“…Hence, we can describe without any approximation certain curved configurations, as long as their shapes correspond to piecewise constant paths in se(3). Moreover, the internal constraints of unshearability (8) and inextensibility (9) discussed in Sect. 2.2 can be imposed exactly because they are compatible with piecewise constant values of the fields v 1 , v 2 , and v 3 .…”

“…Moreover, the role of the special Euclidean algebra is also essential in connection with the geometric mechanical concepts described, for instance, in the works by Simo, Marsden and Krishnaprasad [6], Simo, Posbergh and Marsden [7], Holm, Noakes and Vankerschaver [8], and Eldering and Vankerschaver [9] and with the G-strand equations discussed by Holm and Ivanov [10]. It should be noted, however, that these authors apply geometric concepts to study the dynamics of rods, whereas we focus on the description of shapes.…”