Pedestrians may have a different desired speed on a stair than on a horizontal plan. Therefore, it is necessary to calculate a "smooth" transition in the desired speed, when pedestrians move on planes with a different inclination. In this way "jumpy" changes in the desired speed are avoided.

Definitions

Assume the following scenario, with two horizontal planes and a stair, where \(z_0<z_1\) and the inclination of the stair \(\alpha\).

Speed curve in the transition area between levels and stairs

The agent has a desired speed on the horizontal plane \(v^0_{\text{horizonal}}\) and a different desired speed on the stair \(v^0_{\text{stair}}\).

Given a stair connecting two hirozontal floors, we define the following functions:

\[f(z) = \frac{2}{1 + \exp\Big(-c\cdot \alpha (z-z_1)^2)\Big)} - 1,\]

and

\[g(z) = \frac{2}{1 + \exp\Big(-c\cdot \alpha ((z-z_0)^2)\Big)} - 1.\]

Increasing function $$f(z)$$ and decreasing function $$g(z)

Function of the desired speed

Taking the previously introduced quantities into consideration, we can define the desired speed od the agent with respect to its \(z-\)component as

\[v^0(z) = v^0_{\text{horizonal}}\cdot\Big(1 − f(z)\cdot g(z)\Big) + v^0_{\text{stair}}\cdot f(z)\cdot g(z),\]

\(c\) is a constant.

The following figure shows the changes of the desired speed with repsect to the inclination of the stair \(\alpha\). The steepter the inclination of the stair, the faster is the change of the desired speed.

Transition area of levels and stairs

Tags: jpscore model