Particle-based model for skiing traffic

We develop and investigate a particle-based model for ski slope traffic. Skiers are modeled as particles with a mass that are exposed to social and physical forces, which define the riding behavior of skiers during their descents on ski slopes. We also report position and speed data of 21 skiers recorded with GPS-equipped cell phones on two ski slopes. A comparison of these data with the trajectories resulting from computer simulations of our model shows a good correspondence. A study of the relationship among the density, speed, and flow of skiers reveals that congestion does not occur even with arrival rates of skiers exceeding the maximum ski lift capacity. In a sensitivity analysis, we identify the kinetic friction coefficient of skis on snow, the skier mass, the range of repelling social forces, and the arrival rate of skiers as the crucial parameters influencing the simulation results. Our model allows for the prediction of speed zones and skier densities on ski slopes, which is important in the prevention of skiing accidents.

Figure 1

Skidded (a) and carved (b) turns from the frontal perspective. (a) During skidded turns, the angle $\chi$ between the skier's effective force ${\mathbf{F}}_{\mathrm{EFF}}$ and the lateral axis of the skis is greater than 90${}^{\circ }$. Therefore, the skis also slip away to the side. (b) During carved turns, the effective force ${\mathbf{F}}_{\mathrm{EFF}}$ is perpendicular to the lateral axis of the skis ($\chi ={90}^{\circ }$). The direction of motion is exclusively parallel to the skis.

Figure 2

Profile of a modern carving ski. The sidecut radius $R_{\mathrm{sc}}$ is an an intrinsic property of carving skis. It corresponds to the size of the turn the skis will make when set on edge. The sidecuts of modern carving skis are designed for a turn radius ranging from 7 to 15 m [35].

Figure 3

Should the angle between the desired direction ${\mathbf{e}}_{\mathrm{social}}$ and the current direction of motion ${\mathbf{e}}_{\dot{\mathbf{r}}}$ exceed the threshold angle $\delta$, the skier adjusts his or her direction by performing a corresponding turn from ${\mathbf{e}}_{\dot{\mathbf{r}}}$ to ${\mathbf{e}}_{\mathrm{social}}$. Otherwise, the skier maintains the current direction of motion. In the case illustrated here, the angle between ${\mathbf{e}}_{\mathrm{social}}$ and ${\mathbf{e}}_{\dot{\mathbf{r}}}$ is greater than $\delta$, and the skier will turn left.

Figure 4

Illustration of social forces. The slope is shown in top view. ${\mathbf{F}}_D$ attracts skier $a$ toward the next waypoint. The forces ${\mathbf{F}}_L$ and ${\mathbf{F}}_R$ repel $a$ from the left and right edge of the slope, respectively. ${\mathbf{F}}_A$ keeps $a$ away from other skiers, and ${\mathbf{F}}_O$ repels $a$ from obstacles.

Figure 5

The downhill force ${\mathbf{F}}_S$ parallel to the fall line can be decomposed into a lateral force ${\mathbf{F}}_{\mathrm{lat}}$ and a downhill force ${\mathbf{F}}_P$ accelerating the skier along ${\mathbf{e}}_{\dot{\mathbf{r}}}$.

Figure 6

Effective force during the descent on a straight line (a) and a carved turn (b). (a) Skier descending in a straight line. The effective force can be determined as ${\mathbf{F}}_{\mathrm{eff}}={\mathbf{F}}_{\mathrm{lat}}-{\mathbf{F}}_N$. (b) Skier performing the second half of a turn. The effective force can be determined as ${\mathbf{F}}_{\mathrm{EFF}}={\mathbf{F}}_{\mathrm{lat}}-{\mathbf{F}}_N-{\mathbf{F}}_C$.

Figure 7

Summer pictures of the two ski slopes at the ski resort Engelberg. (a) The slope Graustock is narrow at the beginning and gets wider. It flattens toward the end. (b) The slope Jochstock starts narrow and widens later on. Skiers also have to pass a counter slope.

Figure 8

Average speed measured during the experiment. The speed at each point of a slope was determined as the mean of all measured values observed within a distance of 10 m. The slopes are visualized in top view. As we might expect, skiers need some time to reach their traveling speed. Moreover, flattening (a) and counter slopes (b) slow them down. (a) Average speed of 35 descents on Graustock. (b) Average speed of 50 descents on Jochstock.

Figure 9

Speed distribution measured during the experiment (min, $p_{25}$, $p_{50}$, $p_{75}$, max). Apparently, skiers descend more slowly and sometimes even stop on Graustock. This might be due to the fact that this slope is classified as being more difficult than Jochstock (see Table ).

Figure 10

The speed for each point on the two slopes was determined as the average speed of all 600 simulated skiers observed within a distance of 10 m. The simulations confirm the observations in the experiment: skiers need some time to accelerate and are slowed down by flat areas (a) and counter slopes (b). Average speed of 600 skiers on (a) Graustock and (b) Jochstock.

Figure 11

Signed relative error of the simulated with respect to the measured average speed. There are some regions on Graustock with significant speed differences (a), whereas there is a good overall correspondence for the slope Jochstock (b). Relative speed error for (a) Graustock and (b) Jochstock.

Figure 12

Speed distribution in experiment and simulation (min, $p_{25}$, $p_{50}$, $p_{75}$, max). Apparently, the simulated speed is higher and less dispersed than the measured one on both slopes. The skiers in the simulation have a minimal initial speed of 5 km/h. Speed distribution on (a) Graustock and (b) Jochstock.

Figure 13

Maps of skier densities occurring on the two slopes. Skiers prefer riding in the center of the slopes. The density increases when the slope narrows and when skiers slow down due to flat areas (a) or counter slopes (b). Skier density on (a) Graustock and (b) Jochstock.

Figure 14

Recorded (a, b) and simulated (c, d) trajectories of 10 skiers on the two slopes. Apparently, the skiers pass the edges of the slopes from time to time during the experiment (a). Moreover, they perform fewer turns and prefer more direct descents in reality. Recorded trajectories on (a) Graustock and (b) Jochstock. Simulated trajectories on (c) Graustock and (d) Jochstock.

Figure 15

To obtain a spatial distribution of the average number of turns per skier, we split the slopes into sections 100 m in length. Apparently, the skiers perform more turns when the density increases due to speed loss in flat areas and narrow sections. In the experiment, the skiers perform more turns on Graustock than on Jochstock, most likely to better control their speed on the expert slope. Recorded turns on (a) Graustock and (b) Jochstock. Simulated turns on (c) Graustock and (d) Jochstock. Recorded and simulated turns on (e) Graustock and (f) Jochstock.

Figure 16

To obtain a spatial distribution of the mean of $\kappa$, we split the slopes into sections 100 m in length. In the experiment, the skiers deviate more from the fall line on Graustock than on Jochstock, most likely to decrease the acceleration on the expert slope. (a) Deviation $\kappa$ on (a) Graustock and (b) Jochstock.

Figure 17

Fundamental diagrams of traffic for three sections on Jochstock. Apparently, an increase in the skier density forces skiers to slow down. However, skiers do not stop and congestion does not even occur at an arrival rate exceeding the capacity of the corresponding ski lift. (a) Speed depending on the skier density. (b) Skier flow depending on the skier density.

Figure 18

Effect of increasing the range $R_A$ of the athlete repulsion on Jochstock. When skiers start slowing down at the counter slope, other snow-sport athletes approaching from behind are repelled earlier and try to keep away by riding toward the edges (b, d). Skier density for (a) athlete repulsion with a short range ($R_A=2$) and (b) athlete repulsion with a long range ($R_A=100$). Trajectories of 10 consecutive skiers for (c) athlete repulsion with a short range ($R_A=2$) and (d) athlete repulsion with a long range ($R_A=100$).

Figure 19

Influence of (a) $m$ and (b) $\mu$ on the speed distribution (min, $p_{25}$, $p_{50}$, $p_{75}$, max) on Jochstock. Skiers have a minimal initial speed of 5 km/h. The simulated speed (a) increases with increasing $m$ and (b) decreases with increasing $\mu$.

Figure 20

Median of the unsigned relative speed error depending on the kinetic friction coefficient $\mu$. The median of the relative error reaches a minimum at $\mu =0.125$ for Graustock and at $\mu =0.1$ for Jochstock. Apparently, the participants in our experiment slow down more on Graustock. This might be due to the fact that the terrain of this slope is classified as more difficult than Jochstock. Another reason could be different snow conditions on the two slopes.