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 and the inclination of the stair .

The agent has a desired speed on the horizontal plane and a *different* desired speed on the stair .

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

and

## 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 component as

is a constant.

The following figure shows the changes of the desired speed with repsect to the inclination of the stair . 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.