A shape-centric, continuum-body approach to dynamical modeling and control of rigid-link, articulated snake-like robots is presented. Full 3D gait shapes are decomposed into planar, curve-based components. A planar time-varying curve, defined with respect to an additional average body curve and rigidly-attached body frame, describes the gait shape projection within the locomotion plane. This is paired with a coordinated body-ground contact pattern, defined to encode vertical lift of segments of the robot body off the ground plane. Generalized dynamics governing robot group motion are then derived from the planar shape. A modeling procedure guides specialization of the derived dynamics, to specific gaits. To establish the generalizability of this shape-centric modeling framework, models of several gaits are specialized: traveling wave rectilinear, sidewinding, lateral undulation and a turn-in-place gait. They are modeled dynamically, resulting in integral closed-form, second order systems. Surveying the averaged steady-behavior motion of a gait, over the gait's parameter space, leads to a control-to-action map. This is a reduced-order, control-parametrized motion model, and supports uniformity-of-structure across gaits. Locomotion trajectory planning and feedback tracking strategies are constructed around the control-to-action paradigm, and purposed to intelligently accomplish locomotion objectives within obstacle-strewn, planar environments.