  Tensegrities synergistically combine tensile (cable) and rigid (link)
elements to achieve structural integrity, making them lightweight, packable,
and impact resistant. Consequently, they have high potential for locomotion in
unstructured environments. This research presents geometric modeling of a
Tensegrity eXploratory Robot (TeXploR) comprised of two semi-circular, curved
links held together by 12 prestressed cables and actuated with an internal mass
shifting along each link. This design allows for efficient rolling with
stability (e.g., tip-over on an incline). However, the unique design poses
static and dynamic modeling challenges given the discontinuous nature of the
semi-circular, curved links, two changing points of contact with the surface
plane, and instantaneous movement of the masses along the links. The robot is
modeled using a geometric approach where the holonomic constraints confirm the
experimentally observed four-state hybrid system, proving TeXploR rolls along
one link while pivoting about the end of the other. It also identifies the
quasi-static state transition boundaries that enable a continuous change in the
robot states via internal mass shifting. This is the first time in literature a
non-spherical two-point contact system is kinematically and geometrically
modeled. Furthermore, the static solutions are closed-form and do not require
numerical exploration of the solution. The MATLAB simulations are
experimentally validated on a tetherless prototype with mean absolute error of
4.36{\deg}.
