Kinetic description of collision avoidance in pedestrian crowds by sidestepping

Figure 1.  Calculation of the time to collision.

Figure 2.  The state variables describing pedestrians moving with constant speed with $\alpha_d = 0$.

Figure 3.  Comparison between (5) (blue curves computed with $\gamma = 0.8$, $\tau = 1$) and (6) (red curves). Left: $\theta_j = \pi$; right: $\theta_j = -\frac{\pi}{4}$ (see the text for details).

Figure 4.  The set of pairs $(\rho,\,|\alpha_c|)\in [0,\,1]\times [0,\,\pi]$ (shaded area) for which Theorem 4.1 guarantees alignment in the case study of Section 4.2.

Figure 5.  Alignment to the desired direction $\alpha_d = 0$ for the case study of Section 4.2 with $\rho = \frac{1}{2}$, $\alpha_c = \frac{\pi}{5}$. Left: $t = 10^{-2}$, right: $t = 10$.

Figure 6.  The mapping $t\mapsto\bar{\theta}(t)$ computed numerically (blue solid curve) and estimated theoretically by (19) (red dashed curve). Left: $(\rho,\,\alpha_c) = \left(\frac{1}{2},\,\frac{\pi}{5}\right)$; right: $(\rho,\,\alpha_c) = \left(\frac{1}{3},\,\frac{3}{5}\pi\right)$.

Figure 7.  The mapping $t\mapsto{{\mathit{\bar{\theta }}}_{\rm{MC}}}(t)$ for: (left) $\rho = \frac{1}{2}$ and different choices of $\alpha_c\in (0,\,\pi]$; (centre) $\alpha_c = \frac{\pi}{5}$ and different choices of $\rho\in (0,\,1]$; (right) $\rho = 1$, $\alpha_c = \frac{2}{5}\pi$ and different values of $\bar{\theta}_0$ obtained by varying the parameter $\hat{\theta}$ in (22). Dashed curves indicate the cases in which alignment fails.

Figure 8.  Conditions (14) (blue) with $\bar{\theta}_0 = \frac{\pi}{2}$ and (31) (red) for two different choices of $a(\rho)$. Purple areas correspond to the pairs $(\rho,\,|\alpha_c|)$ satisfying both conditions. The values $\rho_1$, $\rho_2$ in the right graph are $\rho_{1,2} = \frac{21\mp\sqrt{105}}{42}$, i.e. $\rho_1\approx 0.256$, $\rho_2\approx 0.744$.

Figure 9.  Top left: snapshots in time of the solution to (27) with initial condition (20). Top right: the same solution plotted in the $\theta t$-plane. Bottom: snapshots in time of the solution to (27) with initial condition (22) for two different choices of $\hat{\theta}$, $\sigma^2$.

Figure 10.  The mapping $t\mapsto{{\mathit{\bar{\theta }}}_{\rm{FEM}}}(t)$ for: (left) $\rho = \frac{1}{2}$ and different choices of $\alpha_c\in (0,\,\pi]$; (right) $\alpha_c = \frac{\pi}{5}$ and different choices of $\rho\in (0,\,1]$. Alignment is always observed.

Figure 11.  Test 1 -Alignment in the spatially inhomogeneous case and time trend of ${{\mathit{\bar{\theta }}}_{\rm{MC}}}$ over the whole space domain $\Omega$.

Figure 12.  Test 2 -Lane formation in pedestrian counterflow. For pictorial purposes, in order to represent the two crowds in the same picture, the blue density is plotted on a negative scale. In white areas where the velocity is non-zero the two densities take nearly the same values.

Figure 13.  Test 3 -Crossing flows and stripe formation.