Fig 1.
A photograph of the rear-steered bicycle that Richard Klein and others deemed unridable.
Fig 2.
A: When the steer chair wraps around the sprockets in a simple loop, a leftward turn of the handlebar causes the rear-steered bicycle to execute a rightward turn. B: When the steer chain is in a figure eight configuration, the bicycle turns in the same direction that the handlebar is turned.
Fig 3.
(A) Modeling the lean dynamics of a bicycle as an inverted pendulum. (B) To balance the pendulum, one must accelerate the base in the direction of the lean. (C) A Carvallo-Whipple-type bicycle model.
Fig 4.
Mechanical equivalent of the simplified model of the front-steered bicycle.
The model consists of an inverted pendulum attached to a carriage simplifying the steer kinematics and its interaction with the lean dynamics.
Fig 5.
Rear-steered version of the simplified mechanical model of the bicycle.
Fig 6.
Steer geometry of the front-steered carriage.
Fig 7.
Time dependent steer input δ and its time derivative used in the numerical experiment.
The plot is based on values of A = π/6 = 30° and .
Fig 8.
Paths traced out by the two wheels (points u and s) of the carriage as the steer maneuver is executed.
(A) Front-steered carriage. (B) Rear-steered carriage. The green dashed curve in (B) shows the path taken by point c, aligned with the center of mass of the pendulum. Plots assumed a speed of v = 1.1m/s, and wheel base β = 1.09m. The plot of point c in panel B corresponds to a center of mass location α = 0.61β, the same as Klein’s RSB1. At the final points of the paths shown, two thick gray line segments indicate the locations and orientations of the wheels at that instant.
Fig 9.
Wheel accelerations of points u and s for the front-steered carriage as it executes the simple steer maneuver.
(A) Centripetal acceleration vectors on point u of the rear unsteered wheel. (B) Acceleration vectors on point s of the front wheel are enhanced due to steering action. (C) Two sources of acceleration that enhance each other. Parameters are the same as those listed in Fig 8.
Fig 10.
Wheel accelerations of points u and s for the rear-steered carriage as it executes the simple steer maneuver.
(A) As the carriage executes a left turn, acceleration vectors of point s of the rear-steered wheel. (B) Acceleration vectors of point u of the front unsteered wheel. (C) Two sources of acceleration that oppose each other on the rear-steered wheel. (D) Acceleration vectors on point c aligned with pendulum center of mass. Parameters are the same as those listed in Fig 8.
Fig 11.
Foliation of phase portraits for the bicycle model’s drift dynamics (19).
Fig 12.
(A) For the drift vector field . (B) For the control vector field
when u is a positive constant.
Fig 13.
Loss of controllability, from a state space perspective, for the front-steered bike model.
Integral curves projected onto a δ = const plane for the drift vector field and for the control vector field for the front-steered bike at the speed where the controllability is lost. Integral curves for the drift vector field are solid and shown in blue. Integral curves for the control vector field are dashed and shown in green.
Fig 14.
Loss of controllability and crossover for the rear-steered bike model.
(A) Integral curves projected onto a δ = const plane for the drift vector field and for the control vector field for the rear-steered bike at the speed where the controllability is lost. Integral curves for the drift vector field are solid and shown in blue. Integral curves for the control vector field are dashed and shown in green. (B) Same curves, but for a speed slightly faster than the crossover speed. (C) Same curves, but for a speed slightly slower than the crossover speed.
Fig 15.
Plot of balance authority, bu as a function of bike speed, v.
The solid curve is bu given by (24) for the rear-steered bike, while dashed curve is the corresponding measure for a typical front steer bike. Parameters β = 1.09m and α/β = 0.61 were measured from Klein’s RSB1, while parameters m = 94kg, h = 0.861m, g = 9.81N/kg, Ic1 = 80.8kg m2 were obtained from [20]. bu for the front steer bike is determined by changing the sign of the second term in (24), and using a value of α/β = 0.34.
Fig 16.
The “Easy Rear-Steered Bicycle” (EZRSB).
Built by the author’s students. The wheel base is β = 0.62m; and center of mass location α = 0.2β, leading to a crossover speed of approximately vcr = 0.4m/s.
Fig 17.
Balance authority for the “Easy Rear-Steered Bicycle” (EZRSB).
Comparisons are provided to the same measure on a typical front-steered bicycle (orange dashed), and to Klein’s “unridable” RSB1 (blue dashed).