Deriving a Differential Equation for Acceleration
One important part of the model is determining how the vehicle responds to a given motor or brake torque. To express this relationship mathematically, a free body diagram of the vehicle is drawn as shown below:
From the above free body diagram, two equations of motion can be written, where m is the total mass of the vehicle, j is the total moment of inertia of the 4 wheels, Cv is the coefficient of a first order term corresponding to viscous damping and Cc is the coefficient of a constant term corresponding to coulombic friction.
Rearranging the above equations for Froad and expressing acceleration and angular motion in terms of linear velocity yields:
Setting the above two equations equal to each other and isolating for linear acceleration results in:
Note: rearranging the above equation to have mass and moment of inertia on the left hand side results in something of the form:
Force = f(velocity)
How to Determine Vehicle Parameters
Let us consider an experiment where a car is moving with a velocity v. At time t = 0, the brake torque and motor toque are both set to zero (no braking and no acceleration). Due to the first 4 losses in the above above sum the car will slowly come to a stop.
Let us assume that we log the speed of the car every 100ms as the car comes to a stop. By taking the derivative of this v vs t relationship using finite differences we get acceleration vs time. Since we know the velocity at each time stamp and have now computed acceleration at each time stamp, we can multiply the acceleration by mass and moment of inertia to graph the relationship between net force on the car and velocity:
Fitting a second order polynomial to this data we get a relationship of the the form: