Motion planning for mobile robots and autonomous vehicles is a key challenge in modern robotics. The main difficulty lies in accommodating non-trivial kinematic constraints, such as trajectory smoothness and minimum turning radius limitations. One of the most effective approaches to this problem is planning in a discretized state space, which utilizes a precomputed set of kinematically feasible trajectories known as motion primitives. The quality and generation speed of these primitives directly determine the efficiency of the entire planning process. This paper proposes a novel method for generating motion primitives for systems with bicycle-type kinematics. The method is based on representing the trajectory as a curve with a polynomial curvature function. Unlike existing approaches that use polynomial coefficients as parameters, we introduce a new reparameterization based on curvature values at key points along the trajectory. This reparameterization has a more intuitive physical meaning, which improves the convergence and stability of the numerical solution-finding method. The task of generating a primitive connecting two given states (position, orientation, curvature) is reduced to solving a system of nonlinear equations using Newton's method. The conducted experiments demonstrate the advantage of the proposed reparameterization in terms of speed and robustness compared to the baseline approach.