Coordinate transformation from Cartesian to Frenet Frame

Question

Suvo Ganguli 2021-3-9

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回答： Cameron Stabile 2023-1-26

Can anyone tell me how to convert from Cartesian to Frenet Frame for a vehicle driving on a curved road? In other words how do I convert from (x,y) -> (s,d) where is along the curve and d is perpendicular to the curve?

Is there a Matlab code someone can share?

For example if the vehicle is 1 m off centerline and driving along the arc of a circle at speed 1 m/s, then the coordinates at every 1 sec in Cartesian and Frenet frame are given by:

x = cos(theta) ... [with the appropriate scalings for radius and speed]

y = sin(theta) ... [with the appropriate scalings for radius and speed]

s = (0, 1, 2, ...)

d = (1, 1, 1, ...)

I have been looking into Matlab codes but the solutions I got are in the form of (T, N, B) - the tangent, normal and binormal. How do I convert them to distance along the centerline and perpendicular to the centerline?

Thanks.

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M I 2021-5-21

i recently found this: https://github.com/fjp/frenet/blob/master/matlab/Cart2FRT.m

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Answer 1

Cameron Stabile 2023-1-26

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Hi Suvo,

The referencePathFrenet feature from the Navigation Toolbox might be of use to you. The feature fits a piece-wise clothoid spline between a set of

or

waypoints, after which you can convert between Cartesian

and Frenet

space.

There are a number of tools at your disposal, which loosely fall into the following categories:

Projection XY point to Curve:

closestPoint - Find closest point on reference path to global point
closestProjections - Find orthogonal projections between path tangent vector and query point
closestPointsToSequence - Projects sequence of points onto path

Conversion between Cartesian

and Frenet

frenet2global - Convert Frenet states to global states
global2frenet - Convert global states to Frenet states

Evaluating Curve at Arclength (S)

interpolate - Interpolate reference path at provided arc lengths
position - Return xy-position at arclength
tangentAngle - Return tangent angle at arclength
curvature - Return curvature at arclength
changeInCurvature - Return change-in-curvature at arclength

There are also a number of examples that show how this can be applied to road-based planners, a good one to get you started could be Highway Trajectory Planning Using Frenet Reference Path.

Hope this helps,

Cameron