Kinodynamic Planning

A geometric planner only checks positions. A real robot also has limits on speed and acceleration, so a fast robot cannot turn sharply without slowing down first, and it needs room to brake. Kinodynamic planning respects these limits, planning the path and the motion together.

Top speed Max sideways acceleration Max braking and acceleration

Speed limits from the path

On a curve of radius $\rho$ and curvature $\kappa = 1/\rho$, the sideways acceleration is $v^2 \kappa$, so the speed must stay below a curvature limit. The robot must also be able to brake in time to reach each slow section, which gives a backward pass along the path:

$$v(s) \le \sqrt{\frac{a_{\text{lat}}}{\kappa(s)}} \qquad v(s)^2 \le v(s + \Delta s)^2 + 2\,a_{\text{long}}\,\Delta s$$

A forward pass with the acceleration limit finishes the profile. This is the standard time-optimal path parameterization used for robot arms and vehicles.

What to notice

The robot that ignores dynamics charges into the corner at top speed, and with limited grip it swings wide off the path. The planned robot brakes before the corner, takes it slowly, and speeds up after. Raise the sideways limit, like better tires, and it can take the corner faster. Full kinodynamic planners, such as kinodynamic RRT, also change the path itself, for example to swing wide into a corner, rather than only the speed along a fixed path.