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 $$\alpha$$.

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.

Tags: