Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 563Systems With Nonholonomic ConstraintsLet us now add to the data a distribution D defining nonholonomic constraints.One of the interesting things about this affine connection approachis that we can easily integrate into our framework systems with nonholonomicconstraints. As a simple example, consider a rolling disk (see Figure4.8), with two inputs: (1) a ‘rolling’ torque, F 1 = dθ and (2) a ‘spinning’torque, F 2 = dφ. It can be analyzed as a nonholonomic system (see [Lewis(1999); Lewis (2000a)]).Fig. 4.8Rolling disk problem (see text for explanation).The control equations for a simple mechanical control system with constraintsare:∇ ˙γ(t) ˙γ(t) = λ(t) + u a (t) Y a (γ(t)) [− grad V (γ(t))] , ˙γ(t) ∈ D γ(t) ,where λ(t) ∈ Dγ(t) ⊥ are Lagrangian multipliers.Examples1. Recall that for the simple robotic leg (Figure 4.5) above, Y 1 wasinternal torque and Y 2 was extension force. Now, in the following threecases:(i) both inputs active – this system is LCA and LCC (satisfies sufficientcondition);(ii) Y 1 only, it is LCA but not LCC; and(iii) Y 2 only, it is not LCA.In theses three cases, C hor is generated by the following linearly independentvector–fields:(i) both inputs: {Y 1 , Y 2 , [Y 1 , Y 2 ]};(ii) Y 1 only: {Y 1 , 〈Y 1 : Y 1 〉 , 〈Y 1 : 〈Y 1 : Y 1 〉〉}; and(iii) Y 2 only: 〈Y 2 〉.