State space represenation of double integrator car model in the curvilinear domain.
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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):
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1269475/image.png)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1269440/image.png)
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 ![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1269445/image.png)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1269445/image.png)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1269450/image.png)
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 :
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1269455/image.png)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1269460/image.png)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1269465/image.png)
The equations of motion can be summarised into:
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1269470/image.png)
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.
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Accepted Answer
Aditya
on 23 Jan 2023
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.
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