Fig 1.
Kinematic variables of the bicycle model plus the rider-controlled torques.
(A) Side view. In green, the bicycle rear frame, characterized by its lean angle ϕ1 over the roll axis (green arrow). In red, the bicycle front frame, characterized by its angle δ over the steering axis (red arrow). In blue, the rider’s upper body, characterized by its lean angle ϕ2 over the roll axis (blue arrow). In black, (1) the steering torque Tδ and the lean torque , which are both applied by the rider, and (2) the steering axis angle λ, which is set equal to 90 degrees for the purposes of the present paper (see text). (B) Rear view. In green, the bicycle rear frame (plus lower body) lean angle ϕ1. In blue, the rider’s upper body lean angle ϕ2. The symbol ⨂ denotes the CoG of the upper body (in blue), the lower body (in green), and the combined CoG (in black).
Fig 2.
Bicycle model without the known factors that affect bicycle self-stability.
Compared to Fig 1, this model has ice skates instead of wheels and a vertical steering axis.
Fig 3.
The relevant kinematic variables of the SPD in both an inertial (yellow origin) and a rider/bicycle-centered (purple origin) reference frame.
The inertial reference frame has an arbitrary origin, and the rider/bicycle-centered reference frame has its origin at the orthogonal projection of the combined CoG on the LoS. These reference frames have parallel coordinate axes. In green and blue, I depict the lean angles of the lower and the upper body (ϕ1 and ϕ2), and in red, I depict the yaw angle ψ of the LoS. The horizontal plane (road surface) is colored light yellow.
Fig 4.
Schematic representation of the simulation of the combined system in discrete time.
In red, green, blue and black, I show the variables that generated in, respectively, the mechanical, the sensory input, the computational, and the motor output system.
Fig 5.
Sensorimotor control of a mechanical system (in red) by input from a computational system (in blue).
The mechanical system is governed by the nonlinear differential equations , and the computational system produces an optimal control action u. The motor output system (in black) adds noise m to u and feeds this into the mechanical system. The sensory input system (in green) maps the state variables x to sensory variables, adds noise s and feeds the resulting sensory input y into the computational system. The computational system calculates an optimal internal state estimate
by integrating a linear differential equation (characterized by the matrices A, B, C, and the Kalman gain K) that takes the sensory feedback y as input. The optimal control action u is obtained from
and the linear quadratic regulator (LQR) gain -M.
Fig 6.
Simulation results for the model at its optimal parameter values.
(A, B) Percentage of trials with skidding, separately for the SDP (in A) and BDP (in B) simulations. (C, D) RMS combined CoG lean angle and maximum curvature, averaged over the successful trials in the SDP (in C) and BDP (in D) simulations.
Fig 7.
Simulation results for the SDP model with learned noise covariance matrices at 11 logarithmically spaced fractions of the actual noise covariance matrices.
(A, B) Percentage of trials in which skidding occurred, separately for trials in which the learned motor noise Σ (in A) and the learned sensor noise Ψ (in B) was manipulated. (C, D) RMS combined CoG lean angle and maximum curvature, averaged over the successful trials in which the learned motor noise (in C) and the learned sensor noise (in D) was manipulated.
Fig 8.
Simulation results for the BDP model with learned noise covariance matrices at 11 logarithmically spaced fractions of the actual noise covariance matrices.
See the caption of Fig 7.
Fig 9.
Simulation results as a function of 11 linearly spaced fractions of the actual speed for which the system matrix A was calculated.
Across all simulations, the actual speed was kept constant at v = 4.3 m/sec. (A, B) Percentage of trials with skidding, separately for the SDP (in A) and BDP (in B) simulations. (C, D) RMS combined CoG lean angle and maximum curvature, averaged over the successful trials in the SDP (in C) and BDP (in D) simulations. Values are omitted for speed fractions at which no successful trials were obtained.