Note: GPU acceleration required to meet realtime demands.
"Motion Planning for Autonomous Driving with Conformal Spatiotemporal Lattice" McNaughton et al.
"Runtime-Bounded Tunable Motion Planning for Autonomous Driving" Gu et al.
Solve the Boundary Value Problem connecting $\mathbf{X_s} = [x_0, y_0, \theta_0, \kappa_0]^\intercal$ to $\mathbf{X_e} = [x_e, y_e, \theta_e, \kappa_e]^\intercal$ while respecting vehicle kinematics.
"Reactive Nonholonomic Trajectory Generation via Parametric Optimal Control" Kelly et al.
Given a control sequence parameterized by $\begin{bmatrix} c_0...c_{n-1} & d_0...d_{n-1} & T_f \end{bmatrix}^\intercal$
and an initial state $\mathbf{X_{s}}$, the vehicle state can be found by numerically integrating the kinematic equations:
$ \mathbf{X}(t) = \mathbf{X_s} + \int_0^t{\mathbf{f}(\mathbf{x}(t'), \mathbf{u}(t'), t') dt'} $