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cart2frenet

Convert Cartesian states to Frenet states

Since R2019b

Description

example

cart2frenet(planner,cartesianStates) converts a six-element vector of cartesianStates [x, y, theta, kappa, speed, acceleration] to a six-element vector of Frenet states [s, ds/dt, d2s/dt2, l, dl/ds, d2l/ds2], where s is arc length from the first point in reference path, and l is normal distance from the closest point at s on the reference path.

Examples

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This example shows how to plan an optimal trajectory using a trajectoryOptimalFrenet object.

Create and Assign Map to State Validator

Create a state validator object for collision checking.

stateValidator = validatorOccupancyMap;

Create an obstacle grid map.

grid = zeros(50,100);
grid(24:26,48:53) = 1;

Create a binaryOccupancyMap with the grid map.

map = binaryOccupancyMap(grid);

Assign the map and the state bounds to the state validator.

stateValidator.Map = map;
stateValidator.StateSpace.StateBounds(1:2,:) = [map.XWorldLimits; map.YWorldLimits];

Plan and Visualize Trajectory

Create a reference path for the planner to follow.

refPath = [0,25;100,25];

Initialize the planner object with the reference path, and the state validator.

planner = trajectoryOptimalFrenet(refPath,stateValidator);

Assign longitudinal terminal state, lateral deviation, and maximum acceleration values.

planner.TerminalStates.Longitudinal = 100;
planner.TerminalStates.Lateral = -10:5:10;
planner.FeasibilityParameters.MaxAcceleration = 10;

Specify the deviation offset value close to the left lateral terminal state to prioritize left lane changes.

planner.DeviationOffset = 5;

Trajectory Planning

Initial cartesian state of vehicle.

initCartState = [0 25 pi/9 0 0 0];

Convert cartesian state of vehicle to Frenet state.

initFrenetState = cart2frenet(planner,initCartState);

Plan a trajectory from initial Frenet state.

plan(planner,initFrenetState);

Trajectory Visualization

Visualize the map and the trajectories.

show(map)
hold on
show(planner,'Trajectory','all')

This example shows how to partition the longitudinal terminal states in optimal trajectory planning using a trajectoryOptimalFrenet object.

Create and Assign Map to State Validator

Create a state validator object for collision checking.

stateValidator = validatorOccupancyMap; 

Create an obstacle grid map.

grid = zeros(50,100);
grid(25:27,28:33) = 1;
grid(16:18,37:42) = 1;
grid(29:31,72:77) = 1;

Create a binaryOccupancyMap with the grid map.

map = binaryOccupancyMap(grid);

Assign the map and the state bounds to the state validator.

stateValidator.Map = map;
stateValidator.StateSpace.StateBounds(1:2,:) = [map.XWorldLimits; map.YWorldLimits];

Plan and Visualize Trajectory

Create a reference path for the planner to follow.

refPath = [0,25;30,30;75,20;100,25];

Initialize the planner object with the reference path, and the state validator.

planner = trajectoryOptimalFrenet(refPath,stateValidator);

Assign longitudinal terminal state, lateral deviation, and maximum acceleration values.

planner.TerminalStates.Longitudinal = 100;
planner.TerminalStates.Lateral = -5:5:5;
planner.FeasibilityParameters.MaxAcceleration = 10;

Assign the number of partitions for the longitudinal terminal state.

planner.NumSegments = 3;

Trajectory Planning

Initial Frenet state of vehicle.

initFrenetState = zeros(1,6);

Plan a trajectory from initial Frenet state.

plan(planner,initFrenetState);

Trajectory Visualization

Visualize the map and the trajectories.

show(map)
hold on
show(planner,'Trajectory','all')
hold on

Generate Lane Boundaries

Calculate end of reference path as Frenet state.

refPathEnd = cart2frenet(planner,[planner.Waypoints(end,:) 0 0 0 0]);

Calculate lane offsets on both sides of the lateral terminal states with half lane width value.

laneOffsets = unique([planner.TerminalStates.Lateral+2.5 planner.TerminalStates.Lateral-2.5]);

Calculate positions of lanes in Cartesian state.

numLaneOffsets = numel(laneOffsets);
xRefPathEnd = ceil(refPathEnd(1));
laneXY = zeros((numLaneOffsets*xRefPathEnd)+numLaneOffsets,2);
xIndex = 0;

for laneID = 1:numLaneOffsets
    for x = 1:xRefPathEnd
        laneCart = frenet2cart(planner,[x 0 0 laneOffsets(laneID) 0 0]);
        xIndex = xIndex + 1;
        laneXY(xIndex,:) = laneCart(1:2);
    end
    xIndex = xIndex + 1;
    laneXY(xIndex,:) = NaN(1,2);
end

Plot lane boundaries.

plot(laneXY(:,1),laneXY(:,2),'LineWidth',0.5,'Color',[0.5 0.5 0.5],'DisplayName','Lane Boundaries','LineStyle','--')

Input Arguments

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Optimal trajectory planner in Frenet space, specified as a trajectoryOptimalFrenet object.

Vector of Cartesian states, specified as a 1-by-6 vector [x, y, theta, kappa, speed, acceleration].

  • x and y specify the position in meters.

  • theta specifies the orientation angle in radians.

  • kappa specifies the curvature in m-1.

  • speed specifies the velocity in m/s.

  • acceleration specifies the acceleration in m/s2.

Example: [110 110 pi/4 0 0 0]

Data Types: double

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced in R2019b