Benchmark Results on the Linearized Equations of Motion of an Uncontrolled Bicycle

In this paper we present the linearized equations of motion for a bicycle as a benchmark. The results obtained by pencil-and-paper and two programs are compared. The bicycle model we consider here consists of four rigid bodies, viz. a rear frame, a front frame being the front fork and handlebar assembly, a rear wheel and a front wheel, which are connected by revolute joints. The contact between the knife-edge wheels and the flat level surface is modelled by holonomic constraints in the normal direction and by non-holonomic constraints in the longitudinal and lateral direction. The rider is rigidly attached to the rear frame with hands free from the handlebar. This system has three degrees of freedom, the roll, the steer, and the forward speed. For the benchmark we consider the linearized equations for small perturbations of the upright steady forward motion. The entries of the matrices of these equations form the basis for comparison. Three different kinds of methods to obtain the results are compared: pencil-and-paper, the numeric multibody dynamics program SPACAR, and the symbolic software system AutoSim. Because the results of the three methods are the same within the machine round-off error, we assume that the results are correct and can be used as a bicycle dynamics benchmark.