We provide two sections from the book (in preparation) Intelligent and Autonomous Road Vehicles, by Ozguner, Acarman and Redmill. 2.3.2. Steering control using point mass model: Open loop commands We consider the point mass model on the plane so as to present motion in the lateral and in the longitudinal directions, x = Vs cosθ y = Vs sin Θ Θ = r (6) where x represents position of the point mass vehicle model in the longitudinal direction, y represents position of the point mass vehicle model in the lateral direction, and Θ, [rad] respresents angle between the longitudinal and the lateral directions of the model or steering wheel angle and r denotes the commanded steering wheel turn rate [rad/sec]. Note that this model assumes the speed to be constant. In fact, if the angle is zero, traveling along the x direction only, we will end up with a first order differential equation in x, contrary to (2.1).
Figure 2.27: The time responses of lane change maneuvering. Position in the longitudinal direction versus position in the lateral direction is plotted. We consider fixed speed [V s = constant]. We simulate lane change maneuvering in Fig.2.27. Wheel steering angle is applied at t=5 sec., and point mass vehicle model s motion is plotted in Fig.2.28. Changing lane is accomplished when the position of the vehicle model is shifted from the center of left lane to the center of the right lane. Lane width is chosen to be 3.25 meters therefore left lane center is placed to be at 1.625 meters and the center of the right lane is placed to be at 4.875 meters.
Figure 2.28: The time responses of positions in the longitudinal and in the lateral direction. Figure 2.29: The time responses of steering wheel angle and steering wheel angle rate. Wheel steering angle Θ, and wheel steering angular rate is plotted r, in Fig.2.29 to accomplish lane change maneuvering.
Figure 2.30: The time responses of double lane change maneuvering. Position in the longitudinal direction versus position in the lateral direction is plotted. The time responses of positions in the longitudinal and in the lateral direction are added. Double lane change maneuvering is accomplished by a right lane change operation followed by a left lane change operation, i.e., shifting the center of gravity of the vehicle model from the center of the left lane to the center of the right lane and vice versa, see Fig.2.30. Wheel steering angle, Θ, and wheel steering angular rate is plotted r, in Fig.2.31 to accomplish double lane change maneuvering task.
Figure 2.31: The time responses of steering wheel angle and steering wheel angle rate. 2.3.3. Steering control using point mass model: Closed loop commands We shall now consider steering with a closed loop command, coming from sensors detecting our status with respect to the roadway. Figure 2.32: Tracking the road curvature. It is assumed that the car is able to measure the lane deviation at some look-ahead distance away. We shall assume that this deviation is measured in terms of an angle?.
(We shall discuss the technology needed to make such a measurement in a later chapter. At this point, let us assume such a measurement is possible, it is instantaneous, and it is noise-free.) Consider the basic equations for the point mass with fixed speed again: x = Vs cosθ y = Vs sin Θ Θ = r We propose a steering wheel command r = K? Simulation to test such a lane tracking algorithm will rely on making a representation of the curved road. Such a representation can be done in terms of a data file (a point by point representation of the curve) or a polynomial or other closed for expression. We shall now move on to a very specific curve, namely a ninety degree corner. Turning a corner is an important maneuvering task. Reference establishment is shown in Fig. 2.33. A 2-step reference trajectory is proposed so as to accomplish the corner turn. Figure 2.33: Turning corner and reference establishment. The point mass vehicle, coming to the crossroad from the left side, is going to turn right. Reference establishment for turning right perpendicularly is constituted in two steps:
1) Lane keeping: coming onto the corner while maintaining the reference position in the lateral direction, denoted by y 0 and increasing the reference position in the longitudinal direction, denoted by x 0 in Fig 2.33, reference is established to approach to the corner: d x0 2 d y0 = 2 2) Approaching to the center of the corner, just turning the corner, a new reference is established satisfying a fixed reference position in the longitudinal direction, denoted by x 0 =-d/2, and decreasing reference position in the lateral direction, denoted by y 0 =-d/2 to the final reference value y f d x0 = 2 d y0 = y f 2 Figure 2.34: The time responses of turning corner maneuvering. Positions in the longitudinal direction versus lateral direction is plotted when speed is constant and variable.
Figure 2.35: The time responses of the wheel angle and rate for turning corner maneuvering. The time response of the speed is plotted for the cases when speed is constant and variable. In Fig.2.34, turning maneuvering of a vehicle model is plotted. In this plot, the initial reference for the lateral position is chosen as the center of the lane y 0 =-1.625 meters. And x 0 is increasing until coming to the corner. To realize the turn maneuver, a second step of reference generation is established as x 0 =-1.625 meters meaning a perpendicular turn at the center and maintain its position in the longitudinal or x direction and leaving the corner by tending its position in the lateral or y direction to its final value denoted by y f, a negative value higher than y 0. In Fig. 2.34 and Fig.2.35, the turn maneuver with fixed speeds is accomplished. Since the vehicle motion is subjected to constant drive force, displacement in the lateral direction is maintained large around the corner. Speed can be reduced while approaching the corner and resumed to its fixed value after turning the corner. The time responses are plotted for steering wheel angle and variable speed to accomplish turning task. 2.3.4. The need for command sequencing It should be clear that some way for generating a command sequence is needed, as indicated by the corner turning example, or earlier when doing lane change. A lot can be accomplished by judicious use of reference signals while in lane-tracking mode. For example a lane shift operation can be accomplished by inserting an additional bias signal, while the lane sensor simply switches to acquiring data from the next lane. But even then,
a higher level authority has to initiate the insertion of the bias signal at the appropriate lane change time. And in the corner-turning example, some sensor should be aware of the approach of the intersection, so as to initiate the reference-changing operation. Command sequencing, as illustrated above, can be accomplished in one of two approaches, and possibly a combination of both. These are: Defining a finite set of states and transitioning through them. Each state leads to a set of feedback gains and/or reference signals. The model of the system is a combination of the vehicle dynamics jointly with a state machine, leading to a hybrid system. Defining a functional hierarchy that under certain conditions again leads to a different set of feedback gains and reference signals. The hybrid system model is probably hidden, but still exists. 3.1.3. Application to obstacle avoidance In this section, we consider obstacle avoidance with the following assumptions. 1. The road is a multiple-lane, structured road, with detectable lane markers; 2. The car that we are driving always runs on the right lane if there is no obstacle ahead. 3. When an obstacle is detected ahead, the obstacle avoidance task is accomplished by a series of actions of changing to the left lane, passing the obstacle, and changing back to the right lane. 4. During the obstacle avoidance process, no other vehicles or obstacles will appear in the scenario. For example, the left lane is always empty so a left lane change is always risk-free. Here we call an object obstacle based on its speed on the road. If an object has a speed less than certain threshold, v min, our car will regard it as an obstacle and try to avoid it, otherwise, the car will just follow it. A complete obstacle avoidance scenario can be described as follows. y x Car 1 Obstacle
As Figure 3.4. illustrates, the whole obstacle avoidance procedure is divided into five stages as follows: 1: In this stage, we assume that there is no object ahead (within distance d 0 ). Thus, Car1 runs along the right lane of a two-lane road at speed v1. Whenever an object is found within distance d 0, it enters stage 2. 2: In this stage, Car1 checks the speed of the object to see whether it is an obstacle or not. At the same time, Car1still keeps running on the right lane, but it may slow down little by little as it is approaching the object. When the distance between Car1 and the object decreases down to d 1, it will either enter stage if the detected object is considered as an obstacle, or just follow the object ahead, which will leads Car1 into another stage that is not shown in this scenario. : In this stage, Car1 turns left and changes to the left lane if it considers the object ahead as an obstacle. When the left lane changing is finished, Car1 enters stage 4. 4: In this stage, Car1 runs on the left lane until it has totally passed the obstacle. Then Car1 enters stage. : In this stage, Car1 turns right and changes back to the right lane. After that, Car1 switches back to stage 1. Based on the assumptions and analysis above, we can design the obstacle avoidance system as follows: Γ Finite State Machine Sensors Interface Ψ Φ D Signal Processing
Obviously, the system will be a hybrid system, as shown in Fig. 3.5. The continuous time system represents the acceleration system of Car1, which can switches among several dynamic models according to the state that Car1 is at. Those models include standard (normal) running, following, approaching (object observed), left lane changing, passing, and right lane changing. The state switching is controlled by a finite state machine. The interface provides the communication between the finite state machine and the continuous time system, which consists of two functions, ψ and φ. ψ is used to translate the state information to switching signals and send it to the acceleration system. While, φ is in charge of generating the events for the finite state machine according to the outputs of the system (y) and the data from the sensor system (D) based on some thresholds. Γ is the external control event, which can reset the state of the finite state machine. These control events are: Table 1: External Control Events Γ C1 C2 C3 Meaning Go to standard running state To make a left lane change To make a right lane change Several sensor systems are used in the system: 1) A vision system based on cameras in the front to detect the lane markers, which is used to get the longitude position of the car on the road. 2) An obstacle detection system based on radar in the front to detect the objects ahead, which is used to get the distances and velocities of the objects ahead. 3) A side-looking radar system to on each side of Car1, which is used to check if Car1 has passed the obstacle. The outline of the algorithm is represented by a finite state machine in the following figure. State normal, obstacle observed, right lane change, passing and left lane change are corresponding to the five stages outlined in Fig. 2.1, respectively. The follow state is for the case when there is a low speed vehicle ahead, but it is not
considered as an obstacle. In this case, Car1 is assumed to be controlled by its ACC system and follow the slow car ahead. Table 2 explains the events in the finite state machine, as well as the interface conditions to generate the events. Table 3 lists the parameters and variables used in the system. E3 Follow E1, C1 E1 E3 E2 Normal E2 Object observed E1 E9 C3 C2 E4, C2 Right lane change Left lane change E10 E5, C3 Passing E7 E8 E6 Figure 3.6: Finite state machine diagram
Table 2: Events Events E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 Interface conditions to generate the events (φ) d >= d 0 or v 2 >= v max d 1 <= d < d 0 and v 2 < v max d < d 1 and v min <= v 2 < v max d < d 1 and v 2 < v min Pass = true Pass = false LLCF= true LLCF = false RLCF = true RLCF = false Table 3: Variables and Parameters Constant Parameters Meaning d 0 d 1 v min v max Variables D v 2 v 1 The distance that an object ahead starts to be observed. The distance that the object avoidance procedure starts. The minimum lateral velocity of Car1 for following. The maximum velocity of Car1. Meaning The distance between Car1 and the object ahead. The velocity (lateral only) of the object ahead. The lateral velocity of Car1
Pass LLCF RLCF Passing process indicator Left lane change finished tag Right lane change finished tag As mentioned before, the interface part includes tow functions, ψ and φ. The detail thresholding functions of φ have been listed in Table 2. The function of ψ is shown in the following table. Table 4: Interfacing function y S: State of the finite state machine ψ(s): Model switches 1: Normal 1: Standard running 2: Object observed 2: Approaching 3: Follow 3: Following 4: Left lane change 4: Left lane changing 5: Pass 5: Passing 6: Left lane change 6: Left lane changing