Gradient navigation model for pedestrian dynamics

We present a microscopic ordinary differential equation (ODE)-based model for pedestrian dynamics: the gradient navigation model. The model uses a superposition of gradients of distance functions to directly change the direction of the velocity vector. The velocity is then integrated to obtain the location. The approach differs fundamentally from force-based models needing only three equations to derive the ODE system, as opposed to four in, e.g., the social force model. Also, as a result, pedestrians are no longer subject to inertia. Several other advantages ensue: Model-induced oscillations are avoided completely since no actual forces are present. The derivatives in the equations of motion are smooth and therefore allow the use of fast and accurate high-order numerical integrators. At the same time, the existence and uniqueness of the solution to the ODE system follow almost directly from the smoothness properties. In addition, we introduce a method to calibrate parameters by theoretical arguments based on empirically validated assumptions rather than by numerical tests. These parameters, combined with the accurate integration, yield simulation results with no collisions of pedestrians. Several empirically observed system phenomena emerge without the need to recalibrate the parameter set for each scenario: obstacle avoidance, lane formation, stop-and-go waves, and congestion at bottlenecks. The density evolution in the latter is shown to be quantitatively close to controlled experiments. Likewise, we observe a dependence of the crowd velocity on the local density that compares well with benchmark fundamental diagrams.