Main Content

referencePathFrenet

Smooth reference path fit to waypoints

Since R2020b

expand all in page

Description

The referencePathFrenet object fits a smooth, piecewise, continuous curve to a set of waypoints given as [x y] or [x y theta]. After fitting, points along the curve, the path points are expressed as [x y theta kappa dkappa s], where:

  • x y and theta— SE(2) state expressed in global coordinates, with x and y in meters and theta in radians

  • kappa — Curvature, or inverse of the radius, in m-1

  • dkappa — Derivative of curvature with respect to arc length in m-2

  • s — Arc length, or distance along path from path origin, in meters

Depiction of global Frenet coordinates

Using this object, convert trajectories between global and Frenet coordinate systems, interpolate states along the path based on arc length, and query the closest point on a path from a global state.

The object expresses Frenet states as a vector of form [S dS ddS L dL ddL], where S is the arc length and L is the perpendicular deviation from the direction of the reference path. Derivatives of S are relative to time. Derivatives of L are relative to the arc length, S.

Depiction of Frenet states relative to a reference path

Creation

Syntax

refPathObj = referencePathFrenet(waypoints)
refPathObj = referencePathFrenet(___,Name,Value)

Description

refPathObj = referencePathFrenet(waypoints) fits a piecewise, continuous set of curves to the specified waypoints. The waypoints argument sets the Waypoints property.

example

refPathObj = referencePathFrenet(___,Name,Value) sets properties using one or more name-value pair arguments in addition to the input arguments in the previous syntax. You can specify the DiscretizationDistance and MaxNumWaypoints properties as name-value arguments.

Properties

expand all

Presampled points along the path, specified as a P-by-2 matrix with rows of form [x y] or P-by-3 matrix with rows of form [x y theta]. Specify x and y in meters and theta in radians. P is the number of presampled points, and must be greater than or equal to two.

Data Types: single | double

This property is read-only.

Total arclength along the path in meters.

Data Types: single | double

This property is read-only.

Clothoid parameters at start of segments, specified as an N-by-6 matrix where N is the total number of waypoints. Each row contains the parameters of the clothoidal segment to connect to the next consecutive waypoint and is in the form [x y theta kappa dkappa s] where:

  • x y and theta — SE(2) state expressed in global coordinates, with x and y in meters and theta in radians

  • kappa — Curvature, or inverse of the radius, in m-1

  • dkappa — Derivative of curvature with respect to arc length in m-2

  • s — Arc length, or distance along path from path origin, in meters

Data Types: single | double

Arc length between interpolated path points, specified as a positive scalar in meters. The object uses interpolated path points to accelerate performance of the transformation functions frenet2global and global2frenet. A smaller discretization distance can improve accuracy at the expense of memory and computational efficiency.

Example: refPath = referencePathFrenet(waypoints,'DiscretizationDistance',0.4)

Data Types: single | double

Maximum number of waypoints allowed in the path, specified as either Inf for resizeable path or a positive integer to enforce a static limit.

If MaxNumWaypoints is set to Inf, to generate code, DynamicMemoryAllocation must be set to 'on'.

Example: refPath = referencePathFrenet(waypoints,'MaxNumWaypoints',10)

Data Types: single | double

Object Functions

closestPointFind closest point on reference path to global point
closestPointsToSequenceProjects sequence of points onto path
closestProjectionsFind orthogonal projections between path tangent vector and query point
curvatureReturn curvature at arclength
changeInCurvatureReturn change-in-curvature at arclength
frenet2globalConvert Frenet states to global states
global2frenetConvert global states to Frenet states
interpolateInterpolate reference path at provided arc lengths
positionReturn xy-position at arclength
tangentAngleReturn tangent angle at arclength
showDisplay reference path in figure
copyCopy reference path

Examples

collapse all

Open Live Script

Generate alternative trajectories for a reference path using Frenet coordinates. Specify different initial and terminal states for your trajectories. Tune your states based on the generated trajectories.

Generate a reference path from a set of waypoints. Create a trajectoryGeneratorFrenet object from the reference path.

waypoints = [0 0; ...
	50 20; ...
	100 0; ...
	150 10];
refPath = referencePathFrenet(waypoints);
connector = trajectoryGeneratorFrenet(refPath);

Generate a five-second trajectory between the path origin and a point 30 m down the path as Frenet states.

initState = [0 0 0 0 0 0];  % [S ds ddS L dL ddL]
termState = [30 0 0 0 0 0]; % [S ds ddS L dL ddL]
[~,trajGlobal] = connect(connector,initState,termState,5);

Display the trajectory in global coordinates.

show(refPath);
hold on
axis equal
plot(trajGlobal.Trajectory(:,1),trajGlobal.Trajectory(:,2),'b')
legend(["Waypoints","Reference Path","Trajectory to 30m"])

Figure contains an axes object. The axes object contains 3 objects of type line. One or more of the lines displays its values using only markers These objects represent Waypoints, Reference Path, Trajectory to 30m.

Create a matrix of terminal states with lateral deviations between –3 m and 3 m. Generate trajectories that cover the same arc length in 10 seconds, but deviate laterally from the reference path. Display the new alternative paths.

termStateDeviated = termState + ([-3:3]' * [0 0 0 1 0 0]);
[~,trajGlobal] = connect(connector,initState,termStateDeviated,10);

clf
show(refPath);
hold on
axis equal
for i = 1:length(trajGlobal)
    plot(trajGlobal(i).Trajectory(:,1),trajGlobal(i).Trajectory(:,2),'g')
end
legend(["Waypoints","Reference Path","Alternative Trajectories"])
hold off

Figure contains an axes object. The axes object contains 9 objects of type line. One or more of the lines displays its values using only markers These objects represent Waypoints, Reference Path, Alternative Trajectories.

Specify a new terminal state to generate a new trajectory. This trajectory is not desirable because it requires reverse motion to achieve a longitudinal velocity of 10 m/s.

newTermState = [5 10 0 5 0 0];
[~,newTrajGlobal] = connect(connector,initState,newTermState,3);

clf
show(refPath);
hold on
axis equal
plot(newTrajGlobal.Trajectory(:,1),newTrajGlobal.Trajectory(:,2),'b');
legend(["Waypoint","Reference Path","New Trajectory"])
hold off

Figure contains an axes object. The axes object contains 3 objects of type line. One or more of the lines displays its values using only markers These objects represent Waypoint, Reference Path, New Trajectory.

Relax the restriction on the longitudinal state by specifying an arc length of NaN. Generate and display the trajectory again. The new position shows a good alternative trajectory that deviates off the reference path.

relaxedTermState = [NaN 10 0 5 0 0];
[~,trajGlobalRelaxed] = connect(connector,initState,relaxedTermState,3);

clf
show(refPath);
hold on
axis equal
plot(trajGlobalRelaxed.Trajectory(:,1),trajGlobalRelaxed.Trajectory(:,2),'g');
hold off

Figure contains an axes object. The axes object contains 3 objects of type line. One or more of the lines displays its values using only markers

More About

expand all

References

[1] Werling, Moritz, Julius Ziegler, Sören Kammel, and Sebastian Thrun. "Optimal Trajectory Generation for Dynamic Street Scenarios in a Frenet Frame." 2010 IEEE International Conference on Robotics and Automation. 2010, pp. 987–993.

[2] Bertolazzi, Enrico, and Marco Frego. “Fast and Accurate Clothoid Fitting.” Mathematical Methods in the Applied Sciences 38, no. 5 (March 30, 2015): 881–97. https://doi.org/10.1002/mma.3114.

Extended Capabilities

Version History

Introduced in R2020b

See Also

Objects

Functions

Topics