# Wheel resistance to forward movement

According to wikipedia, rolling or wheel resistance to forward movement calculations can be simplified if the vehicle does not move fast, which is our case. Not having found the rolling resistance coefficient for a tire on grass, we take the one of sand, 0.3.

The rolling resistance is given by:

Rolling resistance

\[ F_{rr}=C_{rr}\cdot F_{g} \]

Gravitational force

\[ F_{g}=m \cdot g \]

where:

\( F_{rr} \) : rolling resistance force in N.

\( C_{rr} \) : rolling resistance coefficient.

\( m \) : mass of the vehicle (the mower for us) in Kg.

\( g \) : constant of gravitation, in \( m/s^{2} \)

With our specifications, \( F_{rr} \) is, for a flat vehicle, all wheels are identical and the weight is evenly distributed:

\[ F_{rr}=C_{rr}\cdot m\cdot g=20\cdot 9.81\cdot 0.3=58.8N\approx 60N \]

# Total resistance to forward movement

Taking into account the slope, the calculation is a little more complex: we must add the components due to gravitation. In addition, the rolling or wheel resistance to forward movement decreases with increasing gradient.

The total resistance is due to the force of gravity and rolling resistance, that is to say:

\[ \overrightarrow{F_{rtot}}=\overrightarrow{F_{rr}}+\overrightarrow{F_{rg}} \]

where:

\( \overrightarrow{F_{rr}} \) is the rolling resistance force.

\( \overrightarrow{F_{rg}} \) is the strength of resistance due to gravitation.

\( \overrightarrow{F_{rtot}} \) is the total resistance force: that which the motors must overcome.

From the diagram above, and the triangle of forces, we calculate:

Rolling resistance:

\[ F_{rr}=C_{rr}\cdot F_{p}= C_{rr}\cdot F_{g}\cdot cos(\alpha ) \]

Gravity resistance:

\[ F_{rg}=F_{g}\cdot sin(\alpha ) \]

Gravitational force:

\[ F_{g}=m\cdot g \]

by replacing the terms, you get:

\[ F_{rtot}=C_{rr}\cdot F_{g}\cdot cos(\alpha )+F_{g}\cdot sin{\alpha }=F_{g}\cdot (C_{rr}\cdot cos(\alpha )+sin(\alpha ))=m\cdot g\cdot (C_{rr}\cdot cos(\alpha )+sin(\alpha )) \]

Based on the general specifications, it is easy to calculate the total resistance force in the grass for a maximum gradient of 45°.

\[ F_{rtot}=20\cdot 9.81\cdot (0.3\cdot 0.7+0.7)=177.5N\approx 180N \]

Note: the values calculated above correspond to the maximum forces at which the robot could be confronted.