This paper explores locally-optimal, efficient locomotion of a two-link planar robot balancing on a single, unactuated wheel. Because this model is essentially an acrobot mounted on a passive wheel, we name this model the acrollbot. By actuating an internal degree of freedom, the model can indirectly produce ground reaction forces yielding net accelerations and decelerations, to achieve locomotion. As with bipedal robot locomotion, this toy system is particularly challenging to control due to the need to balance continuously while controlling forward locomotion speed. However, unlike typical legged or rolling locomotion solutions, it is not immediately obvious how best to exploit actuation, internal reconfigurations, and motions to produce and control forward velocity along the ground, providing a useful benchmarking system for exploring optimization techniques. We use a direct collocation optimization framework to study this toy system, both to achieve a range of feasible locomotion solutions for nonintuitive dynamic robot models, and to investigate optimization of physical robot parameterizations, in the sense of improving locomotion efficiency. The framework and example presented throughout are designed with an aim toward bridging the gap between non-intuitive, data-driven optimization and model-based methods for design and control of underactuated and dynamically-stable locomotion.