Trajectory tracking control for front-steered ground vehicles

The ability to follow a desired trajectory is an important part of many autonomous vehicle navigation and hazard avoidance systems. An important requirement for trajectory tracking controllers is appropriate consideration of the vehicle dynamics, especially with regard to wheel slip. When wheel slip is small, the vehicle dynamics are greatly simplified. When wheel slip does occur, however, it can cause a loss of control, such as in the video below showing a lane change maneuver on snow and ice (please skip to 2 minutes 34 seconds if the video does not automatically do so).

One approach to dealing with the loss of control when wheel slip is large is the use of electronic yaw stability control systems. These systems operate by precisely controlling the brakes at individual wheels to minimize sideslip. Such a system is illustrated in the video above at 3 minutes, 55 seconds. These systems have been shown to reduce the risk of crashes, and fatal crashes in particular.

Electronic stability control systems are effective in reducing wheel slip, but they currently do not consider the effect that stability control has on altering the vehicle path and its ability to avoid collisions with hazards. Also, there is evidence that vehicles can be controlled precisely even with large amounts of wheel slip, as evidenced by the Ken Block Gymkhana video shown below (please skip to 2 minutes, 7 seconds if the video does not automatically do so).

Rather than minimizing wheel slip like a conventional stability controller, we suggest compensating for slip while following a desired trajectory, in a similar manner to expert rally drivers. We recently considered a controller that employs feedback control of tire friction forces to control the position of the front center of oscillation along a desired trajectory. This work was inspired by previous work by Ackermann and is similar to concurrent work being done at Stanford's Dynamic Design Lab.

The trajectory tracking controller is based on a planar half-car model that has one steerable wheel at the front and one wheel at the rear. It is also sometimes called a bicycle model, though it is only 2-dimensional and cannot tip over like a 3-dimensional bicycle. An illustration of this model is given below. Friction force Ff and Fr act at the front and rear wheels, and the speed at the center of gravity (c.g.) is V.

Illustration of half-car vehicle model (aka bicycle model)

The trajectory tracking controller controls the position of a point near the front wheels to follow a desired trajectory. The vehicle behavior is illustrated in the animation below for a sinusoidal trajectory with low acceleration. It can be seen that the front of the vehicle follows the desired trajectory, and the vehicle orientation oscillates a small amount.

Trajectory tracking: sinusoid, low acceleration from Steven Peters on Vimeo.

For reference trajectories with higher acceleration, the vehicle orientation can oscillate much more wildly. In the animation below, the vehicle follows a sinusoidal trajectory with higher acceleration. The high levels of acceleration cause the vehicle orientation to oscillate wildly, so that the required steering angles may exceed the vehicle steering limits.

Trajectory tracking: sinusoid, high acceleration from Steven Peters on Vimeo.

These results indicate that the acceleration and shape of the trajectory to be followed have an impact on the vehicle orientation, also known as yaw dynamics. If the yaw dynamics tend to oscillate wildly, the steering limits may prevent the vehicle from following the desired trajectory. In Steve Peters's PhD thesis, structure was identified in the yaw dynamics, and a Lyapunov stability analysis was used to identify stability conditions and estimates of the basin of attraction for steady turning. If the stability conditions are met, then the vehicle can turn along circular paths without oscillating wildly. One of the stability conditions is shown below, which relates the path acceleration, the friction coefficient, and a term related to the friction circle at the rear wheel. These concepts were first discussed in a conference paper presented at IROS 2011 and are detailed more fully in Steve Peters's PhD thesis and a paper submitted to Vehicle System Dynamics.

Stability condition for steady turning

This research applies directly to situations with large amounts of wheel slip. It addresses the oscillations and decreases in slip when following particular trajectories and can also be used to define controllers for drifting turns at large sideslip angles, as shown in the following three figures.

Stability condition for steady turning

Stability condition for steady turning

Stability condition for steady turning