Implementation of path tracking with a linear/non-linear bicycle model. We use the PID and standley controllers to control the longitudinal and lateral movements, respectively. We use the key idea of ref.[1], while replacing the vehicle dynamics in Carla simulator with linear/non-linear bicycle models.

We use the kinematic and dynamic bicycle models as mentioned in ref.6.

Fig.1 Kinematic Bicycle Model

• Linear Bicycle Model.

• Non-linear bicycle model.

The control inputs are [throttle, steering].

• PID controller.

Given the current speed v(t) we minimize the error term e = v_desired − v_current using a PID controller for the throttle value. The range for the throttle values is [-1, 1]. The formula is

Where KP, KI and KD are proportional, integral and derivative parameters, respectively.

• Stanley Controller.

For lateral control, we adapt the standley control(To learn more about the Stanley Control, check out ref.5). There are two error metrics: the distance to centerline d(t) and the relative angle ψ(t). The control law to calculate the steering angle δ_{SC}(t) at the current vehicle speed v(t) is given by

where k is a gain parameter.

We test vehicle models with PID and standley controllers.

Fig.2 Speed tracking of linear vehicle model

Fig.3 Path tracking of linear vehicle model

The testing results on non-linear bicycle models.

Fig.4 Speed tracking of non-linear vehicle model

Fig.5 Path tracking of non-linear vehicle model

Here we enlarge the throttle by 5 times for better visualization.

• For better tracking accuracy, we use the linearly interpolation between waypoints. While We can also use better methods like spline interpolation, for example. [see ref.2]
• To better improvement, we can use seperate longitudinal and lateral bicycle model. In our non-linear bicycle model, we simply consider Fx as the driving force, while this is not how it is done in a real vehicle (engine -> torque converter -> transmission -> wheel). For better performance, please go to ref.3 and ref.4.
• Refine the performance.
• For better tracking performance, we can also try control methods, like MPC. [see ref.7]

1. Self Driving Cars Longitudinal and Lateral Control Design

2. Path tracking simulation with Stanley steering control and PID speed control.

3. Model predictive control for autonomous driving of a truck

4. Longitudinal Vehicle Model Implementation

5. Snider, J. M., "Automatic Steering Methods for Autonomous Automobile Path Tracking", Robotics Institute, Carnegie Mellon University, Pittsburg (February 2009).

6. [Kong, Jason, et al. "Kinematic and dynamic vehicle models for autonomous driving control design." 2015 IEEE Intelligent Vehicles Symposium (IV). IEEE, 2015.] (https://borrelli.me.berkeley.edu/pdfpub/IV_KinematicMPC_jason.pdf)

7. model_predictive_speed_and_steer_control