Detailed explanation of commonly used control methods for smart car planning and control

Control is the strategy that drives the vehicle forward. The goal of control is to use feasible control quantities to minimize the deviation from the target trajectory, maximize passenger comfort, etc.

As shown in the figure above, the modules associated with the input of the control module include planning module, positioning module and vehicle information, etc. The positioning module provides vehicle location information, the planning module provides target trajectory information, and vehicle information includes gear, speed, acceleration, etc. The control outputs are steering, acceleration and braking quantities.

The control module is mainly divided into horizontal control and vertical control. According to the different coupling forms, it can be divided into two methods: independent and integrated.

1 Control method

1.1 Decoupling control

So-called Decoupled control means to control horizontal and vertical control methods independently.

1.2 Coupling control

Coupling control takes into account the coupling problems that exist in horizontal and vertical control. A typical example is that a car cannot corner at high speed because when the longitudinal speed is too high, the lateral angular speed needs to be limited, otherwise the centripetal force cannot satisfy the centripetal acceleration.

The typical representative method of horizontal and vertical integration is linear time-varying model predictive control. This method adds horizontal and vertical integration on the basis of model predictive control. constraint. Such as maximum centripetal acceleration constraints, etc.

1.3 Lateral control

##As shown above, lateral control can be divided into geometric methods and kinematic model-based methods. methods and dynamic model-based methods.

1.3.1 Feedforward control

The so-called feedforward control is based on tracking point information. Control the amount to make appropriate compensation in advance. A typical example is to use the curvature information in the tracking sequence points to compensate for the rotation angle.

1.3.2 Chained Form

The chained system linearizes the nonlinear system into multiple layers, and decomposes the system layer by layer. In turn, the system can be slowed down, similar to the filtering system [3].

System model in frenet coordinates:

## After the reverse integral budget, the control rate can be obtained:

##1.3.3 Lyapunov

Based on Lyapunov The design of Nove stability method can be applied to kinematic and dynamic models. The basic idea is to first establish a kinematic or dynamic model, propose a tracking method based on the model, and then establish a Lyapunov function to prove the asymptotic stability of the closed-loop system through Lyapunov stability [4].

• Kinematic model

As shown in the figure above, the current point of the car is P, tracking the target point For Pr. is the pose difference between the current position and the target point, and are the reference velocity and angular velocity respectively. Design Lyapunov function:

Tracking rate design:

Finally, by limiting the constraint design parameters, the asymptotic stability of the tracking rate is proved, that is, when → ∞, → 0.

• Kinematic model

First establish the kinetic model:

Among them:

order

The error is:

Design cost function:

Design control rate:

Finally proves asymptotic stability.

1.3.4 Pure Pursuit

Pure tracking is a geometric path tracking controller. This controller uses the geometric relationship between the vehicle motion and the reference path to track the controller of the reference path. This control method uses the center of the vehicle's rear axle as a reference point.

According to the above picture, you can push out the front wheel turning command:

where R is the turning radius, L is the vehicle wheelbase, e is the lateral error between the vehicle's current attitude and the target waypoint, is the forward-looking distance and .

According to the experimental data in the figure above, as the forward-looking distance increases, the tracking jitter becomes smaller and smaller. Shorter front-sight distance provides more accurate tracking, while longer front-sight distance provides smoother tracking. Another characteristic of the PurePursuit is that excessive front sight distance can cause "cutting corners" when tracking turns. The Pure Pursuit is a hard trade-off between stability and tracking performance.

1.3.5 Stanley

Different from the pure chasing pure tracking method where the rear axis is the reference point, the Stanley controller uses the front axis as the reference point. It takes into account both heading and lateral errors. The Stanley controller not only considers heading errors but also lateral errors.

According to the above picture, you can push out the front wheel turning command:

According to the experimental data in the above figure, as k increases, the tracking performance will also improve. Stanley does not have enough stability like Pure Pursuit when the vehicle speed increases.

1.3.6 LQR

The method based on the vehicle kinematic model ignores the dynamic characteristics of the vehicle, so when the vehicle speed is too fast or the curvature When the change rate is too large, the algorithm cannot meet the vehicle's stability control requirements. For control methods based on vehicle dynamics models, the primary task is to model vehicle dynamics. Since the accurate two-degree-of-freedom dynamic model is nonlinear, in order to facilitate real-time tracking control calculations, it is usually necessary to make some simplified approximations based on the accurate two-degree-of-freedom dynamic model to obtain a linear two-degree-of-freedom dynamic model.

• Vehicle two-degree-of-freedom dynamic model:

• LQR:

The Linear Quadratic Regulator (LQR) is a model-based controller that uses the state of the vehicle to minimize the error. LQR theory is the earliest and most mature state space design method in modern control theory. LQR can obtain the optimal control law of state linear feedback and is easy to form closed-loop optimal control.

LQR optimal design means that the designed state feedback controller K should minimize the quadratic objective function J, and K is uniquely determined by the weight matrices Q and R, so The choice of Q and R is particularly important. The following formula is the LQR cost function:

According to the vehicle dynamics model and the LQR cost function, the algebraic Licati equation can be derived:

Finally, the feedback matrix is ​​calculated by iterating the Ricati equation, and then the optimal control amount is obtained based on the feedback matrix.

1.3.7 MPC

MPC (Model Prediction Control) is a method dedicated to extending the span of time, even to infinite time. The optimal control problem is decomposed into several optimization control problems with shorter time spans or limited time spans, and the optimal solution is still pursued to a certain extent.

MPC consists of the following three elements:

1. Prediction model: The prediction model can predict changes in system status very well in a short period of time;
2. Online rolling optimization: The results obtained by the prediction model There is still a deviation from the actual situation, so rolling optimization is used to find the local optimal solution at each moment. Usually, an objective (loss) function is designed and converted into a quadratic programming problem to find the optimal solution;
3. Feedback Correction: Re-predict and optimize based on the new state at the next point in time.

• Prediction model:

The prediction model can be derived based on the vehicle dynamics model in LQR.

• Scroll optimization:

MPC cost function:

The corresponding control instructions can be obtained by optimizing the solution based on the prediction model, vehicle lateral constraints, and cost function.

1.3.8 Comparison of horizontal control algorithms

1.4 Vertical direction

As shown in the figure above, vertical control generally adopts the cascade pid control method.

2 Detailed design

##The design of the controller is as shown in the figure above, where Controller As the base class, LonController, LonController and MPCController inherit this base class. LonController has derived subclasses such as LQRController, LyapunovController and StanleyController.

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