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 $% $ 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:

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 $z-$component as

$c$ 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.

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