[1] Donald et al. developed the first polynomial-time approximation schemes (PTAS) for the problem.
By providing a provably polynomial-time ε-approximation algorithm, they resolved a long-standing open problem in optimal control.
Their first paper considered time-optimal control ("fastest path") of a point mass under Newtonian dynamics, amidst polygonal (2D) or polyhedral (3D) obstacles, subject to state bounds on position, velocity, and acceleration.
Later they extended the technique to many other cases, for example, to 3D open-chain kinematic robots under full Lagrangian dynamics.
[2] [3] More recently, many practical heuristic algorithms based on stochastic optimization and iterative sampling were developed, by a wide range of authors, to address the kinodynamic planning problem.