Notes on Motion Planning

where g1(x), . . . , gr(x) span R for all x. Note that r may be greater than m. This is because the vector fields are a function of position, and there may not be a fixed set of m brackets that span R ∀x. For an example, see [1], page 238. The new inputs v are referred to as fictitious inputs, because the v’s corresponding to the gi, i > m do not correspond to real inputs. Further, we require that the “new” brackets appended to the original system to create the extended system be in a Philip Hall basis for the Lie algebra L(g). This requirement arises from the fact that later calculations will involve Philip Hall coordinates, which require that the basis elements of the Lie algebra be a Philip Hall basis. Now, if the fictitious inputs were real, then the solving the motion planning problem would be trivial. If we let γ(t) represent a desired trajectory, then we simply substitute γ(t) into Equation 1 to get