The goal of this project is to safely navigate a car around a virtual highway with other traffic that is driving +-10 MPH of the 50 MPH speed limit using a path planning algorithm. The autonomous car should drive passengers with comfort meaning with limited maximum acceleration and jerk. In this project, I decided to follow the approach described in here: Optimal Trajectory Generation For Dynamic Street Scenarious in A Frenet Frame
A Path-Planner generates a set of trajectories with different target speed, duration, and target driving lanes. Then, a cost function is calculated for every trajectory and the trajectory with the smallest cost is passed to a vehicle controller for execution. The Path-Planner contains stack machine with two states Lane-Keeping and Lane-Changing used to control the process of trajectory generation.
Motion Planning is implemented in the FRENET´ FRAME. All transformations from FRENET´ FRAME coordinates to Cartesian coordinates are supported by special class Waypoints, which loads waypoint coordinates as [x,y,s,dx,dy] values from special file, supplied to the project, and fits two bsplines x(s) and y(s) - the x and y coordinates as functions of longitudinal s-parameter.
Transformation from FRENET coordinates (s,d) to Cartesian coordinates is implemented in two steps: evaluation of splines Fx(s) and Fy(s) at the s-coordinate and adding shift on the d-value along normal to the splines:
[x y]^t = [Fx(s) Fy(s)]^t + d * [-Fy'(s) Fx'(s)]^t (1)
During the first run, the position of the vehicle in Frenet space is initialized by the simulator as a point (x, y) in Cartesian space, which is then transformed to a point (s, d) Frenet space in two steps. During the first step, the s-coordinate is computed using gradient descent optimization solver:
Jxy(s) = sqrt( (x – Fx(s))^2 + (y – Fy(s))^2 )
After that, during the second step, the d-coordinate is computed using formula (1) and the point (x, y), supplied by simulator. This Inverse transformation is added because the simulator doesn't provide the vehicle with the accurate position in Frenet space.
Velocity Keeping Trajectory is the common state where the car spends most of the time, trying to follow the leading car as close as possible with keeping speed limit and safety distance to it. The path for this state is generated as jerk minimizing trajectories using quintic polynomials Optimal Trajectory Generation For Dynamic Street Scenarious in A Frenet Frame:
s(t) = a0+ a1*t + a2*t^2 + a3*t^3 +a4*t^4 s’(t) = a1 + 2*a2*t + 3*a3*t^2 + 4*a4*t^3 (2) s’’(t) = 2*a2 + 6*a3*t +12*a4*t^2
The boundary conditions for the current state at the star t=0 of the trajectory are the longitudinal position s0, velocity s0_dot and acceleration s0_dot_dot. Using the start condition, the following three parameters can be found as:
s0 = a0 s0_dot = a1 (3) s0_dot_dot = 2*a2
The boundary conditions at the end t=T of the trajectory are the longitudinal velocity s1_dot and acceleration s1_dot_dot, which is commonly zero. Using (2) and (3), the other a3 and a4 parameters can be calculated using the following linear equation system:
s1_dot =s0_dot + s0_dot_dot * T + 3 * a3 * T^2 + 4*a4*T^3 s1_dot_dot = s0_dot_dot + 6 * a3 * T + 12 * a4 * T^2
Or using matrix form:
| s1_dot – s0_dot – s0_dot_dot * T | | 3T^2 4T^3 | | a3 | | | = | | x | | | s1_dot_dot – s0_dot_dot | | 6T 12T^2 | | a4 |
The boundary conditions at the start and end states for lateral component are: d0=d1=d, d0_dot=d1_dot=0 and d0_dot_dot=d1_dot_dot=0
This type of trajectory is generated as a combination of quartic polynomial trajectory, for the longitudinal component s, using Velocity Keeping Trajectories, with a quintic polynomial for the lateral component d, using the following linear equation system:
| d1 – d0 – d0_dot – 0.05 * d0_dot_dot * T^2 | | T^3 T^4 T^5 | | a3 | | d1_dot – d0_dot – d0_dot_dot * T | = | 3T^2 4T^3 5T^4 | x | a4 | | d1_dot_dot – d0_dot_dot | | 6T 12T^2 20T^3 | | a5 |
Every 20 ms the SDC vehicle receives sensor fusion data from the simulator as an array of [id, x, y, vx, vy, s, d] elements for every tracked vehicle, where id is unique ID of vehicle, (x,y) is position and (vx, vy) is vector speed of tracked vehicle, (s, d) is position in Frenet coordinates. Unfortunately, (s, d) in general is not accurate enough to be used directly in SDC vehicle. The data from sensor fusion are passed to the SensorFusion class which is used to predict the state (or Constant Velocity trajectory) of every tracked vehicle [s, d, vs] in every 20ms. The predicted state [s, d, vs] is the position (s, d) in Frenet coordinates and vs is longitudinal velocity. The position in Frenet space is recalculated using Waypoints class with data from sensor fusion: (x, y), s-coordinate and vector of speed (vx, vy) (see Coordinate System for Motion Planning).
To find optimal trajectories, I used the same approach described in Optimal Trajectory Generation For Dynamic Street Scenarios in A Frenet Frame. At every time step multitude of trajectories is generated by varying the target velocity s_dot_dot and the time duration T of trajectories. For lane changing trajectories, the target d-coordinate is variated as a center of target lane in Frenet coordinates. Then, all generated candidate-trajectories are evaluates using the cost function Jt (see below) and the optimal trajectory is selected as a trajectory with minimal cost function value. The cost function is:
Jt = Kv * Jv + Ka * Ja + Kj * Jj + Ks * Js
where: Jv is terms to penalize the deviation from target velocity, calculated as integral of the following function, normalized by horizon of prediction T:
Jv(t) = (sqrt( s_dot(t)^2 + d_dot(t)^2 ) – v_max)^2
Ja is terms to penalize the maximum acceleration, calculated as integral of the following function, normalized by horizon of prediction T:
Ja(t) = s_dot_dot(t)^2 + d_dot_dot(t)^2
Jj is terms to penalize the maximum jerk, calculated as integral of the following function, normalized by horizon of prediction T:
Jj(t) = s_dot_dot_dot(t)^2 + d_dot_dot_dot(t)^2
The safety distance Js of the trajectory in relation to the predicted trajectory of the i-th tracked other vehicle is calculated as integral of the following function, normalized by horizon of prediction T:
Js,i(t) = 0 if d >= d_min Js,i(t) = (0.95*exp(-a*d) + 0.05*exp(-a*d_min) – exp(-a*ds) ) / (exp(-a*d_min) – exp(-a*ds)) if d < d_min
where: a-parameter is used to tuning penalty of cost function. d-paramener is distance to tracked vehicle, calculated as:
d = s(t) – si(t)
ds-parameter is safety distance, calculated as:
ds(v) = v * ts + d_min
ts = 1.2 sec
The state machine is implemented as object-oriented class PathPlanner, which supports two states: Lane Keeping and Lane Changing. The transaction between two states is implemented using a queue, which contains the active state at the end of it. The switching between two states is initiated by one of these two states Lane Keeping or Lane Changing.
The Lane Keeping state is the default state of the car, implemented as a class LaneKeeping, and is used to generate velocity keeping trajectories that allow the car to keep the lane close to the speed limit with keeping a safe distance to the leading cars. In addition, this state generates candidate trajectories that allows the car to change the lane. When the car follows to a slowly leading vehicle, the cost of this trajectory is increased due to Jv increasing. As soon as a trajectory with a different target lane has a lower total cost, the lane keeping state is constructed and returned to PathPlanner. The PathPlanner adds this new state at the end of the queue to use this state for optimal trajectory generation during next iteration. During evaluation of all candidate trajectories, the predicted trajectories of closest leading cars at the same lane are used for estimation and keeping safety distance cost Js to the nearest leading car.
The Lane Changing state is implemented as a class LaneChanging, and is responsible for executing a safe lane change maneuver. When this state is started to execute, the car has already started to move into the direction of target lane. The lane changing state starts to generate target trajectories with variation of target velocity and duration T along s-direction. For the lateral d-direction, a quintic polynomial trajectory is generated with the target d-coordinate in the center of the target lane and with a fixed lane change duration. During evaluation of all candidate trajectories, the predicted trajectories of all nearby cars at the same lane are used for estimation and keeping a safety distance cost Js to surrounding cars. When the lateral d-distance is close to the target d-value, which is the center of lane in Frenet coordinates, the changing lane state is deactivated by returning the nullptr into PathPlanner. The Path Planner detects the nullptr value and removes the Lane Changing state from the queue of active states. As a result, the Lane Keeping will be used during next iteration.
The implemented path planner is able to navigate safely through highway traffic with speed close to target speed limit. The car is able to keep lane with speed limit if the current lane is empty or execute safety maneuvers in order to overpass slow cars. Bellow, you can see some examples of car behavior:
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