Feedback control of a redundant wheeled snake mechanism using transverse functions SO(4)

The Transverse Function (TF) approach is applied to the control of a nonholonomic three-segments/snake-like wheeled mechanism, similar to the planar low-dimensional version of Hirose's Active Cord Mechanism (ACM) previously studied by the authors with the same control approach, but with two additional internal degrees of freedom (d.o.f.) whose actuation yields more flexible and efficient control solutions. From a theoretical point of view, these complementary d.o.f. modify the Control Lie Algebra of the system so that only first-order Lie brackets of the control vector fields are needed to satisfy the Lie Algebra Rank Condition (LARC). The fact that four independent (angular velocity) control inputs are used also implies for this system the existence of Transverse Functions (TF) defined on the six-dimensional special orthogonal group(4). Several examples of mechanisms whose control involve TF defined on(3) have been pointed out in the past. Beyond the specific control problem addressed here, a motivation for the present study is to illustrate for the first time how functions defined on the larger set(4) can be determined and used for the control of a physical system. This study is complemented with recalls concerning the parametrization of(4) by pairs of isoclinic quaternions and with the derivation of complementary differential calculus relations associated with this parametrization.