The　parametrization　of　a　rigid-body　rotation　is　a　classical　subject　in　rigid-body　dynamics.　Euler　angles,　the　rotation　matrix　and　quaternions　are　the　most　common　representations.　However,　Euler　angles　are　known　to　be　prone　to　singularities,　besides　not　being　frame-invariant.　The　full　3　x　3　rotation　matrix　conveys　all　the　motion　information,　but　poses　the　problem　of　an　excessive　number　of　parameters,　nine,　to　represent　a　transformation　that　entails　only　three　independent　parameters.　Quaternions　are　singularity-free,　and　thus,　ideal　to　study　rigid-body　kinematics.　However,　quaternions,　comprising　four　components,　are　subject　to　one　scalar　constraint,　which　has　to　be　included　in　the　mathematical　model　of　rigid-body　dynamics.　The　outcome　is　that　the　use　of　quaternions　imposes　one　algebraic　constraint,　even　in　the　case　of　mechanically　unconstrained　systems.　An　alternative　parametrization　is　proposed　here,　that　(a)　comprises　only　three　independent　parameters;　(b)　is　fairly　robust　to　representation　singularities;　and　(c)　satisfies　the　quaternion　scalar　constraint　intrinsically.　To　illustrate　the　concept,　a　simple,　yet　nontrivial　case　study　is　included.　This　is　a　mechanical　system　composed　of　a　rigid,　toroidal　wheel　rolling　without　slipping　or　skidding　on　a　horizontal　surface.　The　simulation　algorithm　based　on　the　proposed　parametrization　and　fundamentally　on　quaternions,　together　with　the　invariant　relations　between　the　quaternion　rate　of　change　and　the　angular　velocity,　is　capable　of　reproducing　the　falling　of　the　wheel　under　deterministic　initial　conditions　and　a　random　disturbance　acting　on　the　tilting　axis.　Finally,　a　comparative　study　is　included,　on　the　numerical　conditioning　of　the　parametrization　proposed　here　and　that　based　on　Euler　angles.　Ours　shows　as　broader　well-conditional　region　than　Euler　angles　offer.　Moreover,　the　two　parametrizations　exhibit　an　outstanding　complementarity:　where　the　conditioning　of　one　degenerates,　the　other　comes　to　rescue.