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.

**Note:**This modelling of the desired speed in the transition area of planes and stairs is not validated, since experimental data are missing.

## Definitions

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

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

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

**Note:**The value of

*c*should be chosen so that the function grows fast (but smooth) from 0 to 1. However, in force-based models the speed is adapted exponentially from zero to the desired speed. Therefore, the parameter tau must be taken into consideration.