New math breakthrough lets cars park like humans using polar coordinates

A team led by Velimir Todorovski, Kwang Hak Kim, Alessandro Astolfi, and Miroslav Krstic has published a paper on arXiv (2607.26442) that solves a long-standing control problem for car-like vehicles at parking speeds. The kinematic bicycle model, standard for low-speed maneuvers, has resisted smooth static feedback stabilization in Cartesian coordinates due to Brockett's condition—a topological obstruction that prevents continuous feedback from achieving stability. The researchers bypass this by converting the system into polar coordinates with additional range-normalized coordinates that encode human-like parking geometry. This transformation reveals a strict-feedback form, enabling a nonconventional backstepping design that yields smooth feedback laws achieving global exponential stabilization. The resulting trajectories resemble those performed by human drivers, using only feedback control.

The work sits at the intersection of control theory, robotics, and optimization, with potential applications in autonomous vehicle parking systems. Unlike existing methods that often use complex path planning or discontinuous control, this approach uses smooth feedback alone, simplifying implementation. The key innovation lies in the coordinate transformation, which exploits the natural geometry of parking maneuvers. By proving global exponential stability, the authors guarantee that the vehicle will converge to the target pose from any initial condition. This could lead to more robust and natural-feeling parking assist features in cars, reducing the need for multi-step planning algorithms. The paper is available for review and has been submitted to a conference or journal.