This paper presents a motion planner tailored for particular requirements for robotic car navigation. We leverage B-spline curve properties to include vehicle's constraint requirements, thus lowering the search dimensionality. An algorithm, which combines competent exploratory nature of the randomized search methods with vector-valued parameterization steering, is developed here. Vehicle's limitations, along with obstacle's constraints, are satisfied without being hindered by numerical integration and control space discretization of traditional randomized kinodynamic planners. We rely on newly developed theoretical underpinnings to overcome performance issues in rapidly exploring random tree (RRT) solutions. Rigorous simulations and analysis demonstrate that this new approach outperforms recently proposed planners by using an efficient bidirectional RRT-based search, by maintaining continuous state and control spaces, and generating C<sup>2</sup> continuous paths, which are realistic inputs suited for mobile robotic applications and passenger vehicles.