On the Development of a Path Tracking Controller by combining Optimal Preview Control and Pursuit Control Methods

In the development of advanced rider assistance systems, a self-driving bicycle or motorcycle can be very helpful. Such a system is not only able to perform more reproducible measurements than a human but can also be used for safety-critical maneuvers. An important aspect when developing a self-driving single-track vehicle is the development of a controller that can stabilize the vehicle and can follow a desired roll angle, steer angle or yaw rate. When such a controller has been implemented, the next step is to develop a path tracking controller so that the vehicle can not only be operated remotely but can also follow a predefined path (important for reproducible measurements). A simplified cascaded control loop is shown in Fig. 1. For that control loop it is assumed, that the bicycle travels at a constant speed and that the bicycle states are measurable or that an observer exists, which estimates the bicycle state vector from the measured signals. When using a cascaded controller, the inner loop should be at least 3-4 times faster than the outer loop. Otherwise, performance problems and instabilities of the whole system can occur. As a result, with a slow lateral-dynamics controller, the path-tracking controller must be even slower. In (Gabriel et al., 2022) a lateral dynamics controller was presented, which is able to stabilize the bicycle and track a desired yaw rate of the bicycle. That controller has a time constant of 1.5s, therefore the outer control loop must have a time constant of at least 4.5s. Various path-tracking controllers can be found in literature. They can be divided into two groups: On the one hand, there are geometric path tracking controllers such as the "Pure Pursuit Controller" and the "Staneley Controller", which completely neglect the dynamical behavior of the underlying system. Hence, it is important that the cascaded-control loop rule is followed, otherwise the whole system can become unstable (Heredia et al., 2007). On the other hand, there are linear and nonlinear model-based controllers that take the system dynamics into account and usually combine the path-tracking controller and the lateral-dynamics controller in a single controller. We have several requirements on the path-tracking controller, which none of the existing approaches can fulfill: - The controller should ensure good tracking behavior despite the relatively slow steering dynamics of the lateral-dynamics controller. - Tight corners should not be cut. In particular, curves with a minimum radius associated with a bicycle roll angle of 15-20°, should be possible. - The controller has to use little computing power so that it can be implemented on a microcontroller. The problem with the geometric controllers is, that they would require huge look-ahead distances to be stable. That would lead to cutting corners. The nonlinear model-based approaches cannot be used because the algorithms usually require a lot of computing power. The linear model-based approaches such as in (Sharp et al., 2001) are very promising but cannot be used due to the linear path model, which does not allow directional changes of >90° within the preview distance. To fulfill all our requirements, methods of the model-based controllers and the geometric controllers must be combined: Instead of using the Linear Quadratic Integral controller which was proposed in (Gabriel et al., 2022), an Optimal Preview controller (similar to the one in (Sharp et al., 2001)) is used as lateral-dynamics controller. This controller not only receives a reference yaw rate for the current time step but also a reference yaw rate trajectory for a specific preview time. By using a controller with preview, the inner loop becomes faster since it can react to setpoint changes in advance. The reference yaw rate trajectory must be generated by the path-tracking controller in the outer loop. Therefore, the path-tracking controller must first find an intermediate path which guides the bicycle from the current position back to a target point on the reference path. A reference yaw rate trajectory can be generated from that intermediate path. When the intermediate path is calculated it is important to ensure that the resulting reference yaw rate trajectory is realizable (no jumps in the reference yaw rate) and that the target point is reached with the correct conditions for the following path. Therefore, not only the current position and the target position, but also the current yaw angle and yaw rate as well as the target yaw angle and yaw rate are taken into account when calculating the intermediate path. To calculate an intermediate path with the required boundary conditions quintic polynomials can be used. The developed path-tracking controller was tested in simulations and experiments. Very good results were achieved, the requirements are fully satisfied, and the bicycle was able to follow specified paths with minimum radii smaller than.