State space represenation of double integrator car model in the curvilinear domain.

The curved road is defined equation is given as the curvature k(s), where 's' is a distance travelled along the curve.

The road heading angle θ as well as coordinates may be calculated by integrating the curvature as follows (2):

The main advantage of curvilinear coordinates approach is the use of them in tracking the orientation of the vehicle based on the calculus of the vehicle forward

where is the yaw rate, are the lateral offest on the road strip while α is the vehicle relative heading to the road. In summary, the vehicle dynamics is describe by means of two control inputs plus eight state variables :

The equations of motion can be summarised into:

My doubt is how to represent the yaw rate, if it is not present in the set of states, does yaw rate need to be on of the control input ? I am referring to a paper titled as "Minimisation of Motion Sickness in Autonomous Vehicles". My main agenda is to write this into proper state space form.

Answer 1

[Aditya]

Hi Ishu,

I understand that want to figure out how to include yaw in the dynamics.

You cannot include yaw in the control input as it is not a parameter that you can directly control.

You can calculate the yaw from other observable parameters: the track orientation angle given by θ and the angle between the vehicle and the track given by ξ. The absolute yaw angle ψ = θ + ξ. Refer [1] for further information.

The other way could be to include yaw in your state space itself. [2]

[1]: Perantoni G, Limebeer DJ. “Optimal control for a formula one car with variable parameters.” Vehicle System Dynamics. 2014 May 4;52(5):653-78.

[2]: Lot R, Dal Bianco N. “Lap time optimisation of a racing go-kart. ” Vehicle System Dynamics. 2016 Feb 1;54(2):210-30.