Please explain the maths behind calculating lateral deviation

Please explain me the maths behind the lateral deviation used in many exmples.

1. Curvature unit is m-1 and if we multiply by logitudinal velocity (m/s), we get s-1 which is rate of change / frequency. How are we subrating Yaw rate by it?
2. And how are we getting e1 from u3*u2 + u1?

[Andres Adam]

Hi Raghava, I cannot immediately recall the many examples that you are referring to. We also need a frame of reference and more context for this to make sense. Can you share the source of that image?

Answer 1

[Sam Chak]

Hi @Raghava Santhan Mysore Pavan

The lateral deviation of a car is typically covered in the field of Nonlinear Vehicle Dynamics (within the discipline of Automotive Engineering). However, this phenomenon can also be explained using the principles of mathematical physics, which are commonly studied at the A-level (UK) or Advanced Placement (US) educational levels.

Let us examine Figure 5, where e represents the lateral deviation, and denotes the heading angle. If the car is traveling in a straight line, the rate of change of the lateral deviation is the velocity perpendicular to the direction of motion (i.e., the x-axis of the car's body frame), which is known as the "lateral velocity" and is denoted by (as it is relative to the y-axis of the car's body frame).

Now, if the car is traveling along a curved path, the heading angle will be non-zero. In physics, we have learned that the arc length 's' (not to be confused with the length of the curvature path) of a circle is equal to the product of the heading angle and the radius of the circle r.

The rate of change of the arc length is given by the equation

where is denoted by (as the radial distance is aligned with the x-axis of the car's body frame). Therefore, by combining these equations, the total rate of change of the lateral deviation is the sum of and :

Yao, Q.; Tian, Y. A Model Predictive Controller with Longitudinal Speed Compensation for Autonomous Vehicle Path Tracking. Appl. Sci. 2019, 9, 4739. https://doi.org/10.3390/app9224739

[Raghava Santhan Mysore Pavan]

Thank you for the answer.

[Sam Chak]

You're welcome, @Raghava Santhan Mysore Pavan. If you find the explanation helpful, please consider clicking 'Accept' ✔ on the answer and voting 👍 for it. Your support is greatly appreciated!