Geometric constructions of connections on the higher order principal prolongations of a principal bundle are considered. Moreover, the existing differences among connections on non-holonomic, semiholonomic and holonomic principal prolongations are discussed.
We recall several different definitions of semiholonomic jet prolongations of a fibered manifold and use them to derive some interesting properties of prolongation of a first order connection to a third order semiholonomic connection.
We construct a privileged system of coordinates with respect to the controlling distribution of a trident snake robot and, furthermore, we construct a nilpotent approximation with respect to the given filtration. Note that all constructions are local in the neighbourhood of a particular point. We compare the motions corresponding to the Lie bracket of the original controlling vector fields and their nilpotent approximation.
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