Kinematic analysis of the planar motion of vehicles when travelling along tractrix curves

A detailed analysis of the path traversed by the rear wheels of the bus is useful in highway design, and in the placement of curbs at intersections. In fact, when a bus turns a corner, the rear wheel could move into the path of a passing motorist or pedestrian by generating an hazard situation, as in the cases of traffic circles and changing lanes. Of course, the same problems regard long trucks, semi-trailer trucks and articulated vehicles, but also bicycles (tandem) and motorbikes. In particular, the path of the rear wheels of a bus or a long vehicle in general, is a tractrix or equi-tangential curve, which is different by that traced by the front wheels. The planar motion of the chassis of the vehicle can be represented by means of the centrodes, which are traced by the instant center of rotation on the ground (fixed centrode) and on the moving plane that is attached to the vehicle (moving centrode), respectively. Thus, the pure-rolling motion of the moving centrode that is represented by the axis of the rear wheel, on the fixed centrode (catenary) can reproduce the planar motion of the chassis. This paper deals with the formulation of a suitable algorithm for the kinematic analysis of the planar motion of vehicles that travel corners, traffic circles and changing lanes. The tractrix is considered as the involute of the fixed centrode (catenary), which takes also the rule of evolute. The inflection and the cuspidal or return circles, along with the Euler-Savary equation and the Aronhold Theorems, give other useful information on the planar motion of the chassis.