This section addresses the relationship between the road plane and its projective transform, the image plane, and how this affects the determination of the vehicle speed, and steering control.
Fig. 5. Road tracking control
First the vision based algorithm is used to determine the potential road region boundaries and thus find possible road parameters in each window from bottom to horizon. As shown in Fig. 5, while at point p, the vehicle driver approximately predicts the next point on the road to reach at reasonably enough distance. Here as the image plane is split into multiple overlapping windows of pixel size l, we can say that the next points to reach are a and b (Fig. 5). A human driver would change the steering angle to match the new heading for the vehicle at point a, χa. In this “predictive control” algorithm, the controller approximately predicts the heading at the next point for the vehicle to reach and adjusts the speed and the steering to reach that point with the required heading.
The road curve parameters are determined from frame to frame considering the x- shift and the shift in α from the bottom-most window to the uppermost window. If L0i and L0j define the possible boundaries of the road region in the bottommost window, say (w0), the current vanishing point (x0, y0) is the intersection of L0i and L0j. The change in the x co-ordinate of the vanishing point is a function of road curvature in the image plane. Considering that the road profile in the image plane follows a clothoid trajectory, the following Eq(4) holds as explained in [2].
(4)
As it is assumed that l - the height of each window - is small enough to approximate that part with a straight line, the discrete form of the clothoid is considered here. Assuming the camera is installed at the center of the vehicle with a pan angle of zero with respect to vehicle heading, the current heading, χ0 is considered to be zero. The same reason applies for projecting the current vehicle position, p, from the center of the scene, a. The steering change required at point p depends on the difference between the current vehicle heading (χP) and the required vehicle heading (χa).
(5) Eq (5) estimates the curvature from point a to point b as shown in Fig. 5. The same equation can be used to predict the approximate window size l. As l should be sufficiently small to be able to approximate the boundaries as small discrete tangent line segments, Eq(6) and (7) are used as constraints in addition to determine l. The following constraints are used to track the road boundaries from the bottommost window, w0, to the topmost window, w3:
(6)
(7)
SAE Int. J. Commer. Veh. | Volume 3 | Issue 1
Eq(6) means the y-shift in the consecutive vanishing points from frame to frame should be negligible as we assumed that the road is flat. Where, Δx′ is the horizontal distance between two consecutive vanishing points. Δx′ is thus a function of change of heading angles (Eq(11)). The vehicle might not always be at the center of the lane and the control algorithm should be designed in a way to be able to track the lane center irrespective of the initial position of the vehicle. If the vehicle is tracking the center of the lane correctly then θ = α.
Therefore, the required steering in the image plane, θ, can be calculated using the lane detection output (x′, α) by Eq(8).
(Refer Fig. 2 and 4.)
(8) Where, 2b, 2a : Width and height of the image plane in pixels, respectively. To control the vehicle smoothly, the steering angle needs to be refreshed based on an optimal rate of change within a given distance of the look-ahead, which is l/2 in this case. (See Fig. 5.)
From Eqs(2) and (1) we have,
(9)
(10) Let the actual heading of the vehicle in the ground plane - with respect to Z-axis - be χg, which corresponds to the vehicle heading with respect to the y-axis in the image plane - χi. Z′ is the ground plane distance from camera location to the closest ground point visible in the camera scene (Fig. 1): we have,
(11)
(12)
Therefore left turns in ground plane are preserved as left turns in image plane and the similar is true for right turns. This makes steering control in the image plane possible without the need to transform to ground co-ordinates.
Let ẏ be the vehicle speed in the image plane (pixels/second) and Ż be the speed in the ground plane(m/s). Therefore differentiating Eq(1) and (9),
(13)
(14)
Now, be the rate of change of steering angle in image plane. be the steering i.e. rate of change of vehicle heading. Therefore, the following estimates are used:
(15)
(16) Where, θP and ωp are the steering angle and rate of change of steering angle to be applied at p to track the lane correctly.
Let, the change in heading required from point p to point a be Δχp.
(17)
(18)
(19) Where the speed in the image plane from point p to a is given by
SAE Int. J. Commer. Veh. | Volume 3 | Issue 1
(20) As per Eq(13),
(21) Therefore, from Eq(14),
(22) Thus, from Eq(22), the ground plane speed, Ż, and the rate of change of steering angle in image plane, ωi, are dependent on each other. This is intuitive as we can say that when the road is bending or for higher vehicle speeds, the steering needs to be updated faster. Thus, once the safe speed Ż for the vehicle at a given curvature is calculated as per vehicle dynamics, the corresponding ωi can be calculated recursively as the vehicle moves from current point to the predicted point using Eq(13), (18), and (22). Eq(22) will hold valid for Ż, as long as Δχ ≠ Δχ0.
(23) According to Eq(23), ωi = 0 i.e. no change in vehicle heading is predicted, which is true for straight roads (curvature zero).
Eq(22) thus implies infinite vehicle speed (Ż = ∞) on a straight road. Thus infinite speed is translated to a specific maximum vehicle speed, Żmax. Żmax can be determined by various factors such as speed limit specified on the road, maximum speed of the vehicle as per its dynamic constraints, frame rate, etc.
As the relations between image plane and ground plane velocities are derived in terms of image plane parameters, the vehicle control parameters can be computed from the image plane and a transformation into the ground plane is not required. Therefore this method is called “predictive control in the image plane”. Moreover, the Eq(22) shows that the vehicle controls are co-dependent and can be calculated concurrently to make sure the correct vehicle response for tracking the desired lane. This illustrates the potential of PCIP for autonomous vehicle control.
IV. STABILITY ANALYSIS AND RESULTS
The vehicle stability plays a big role in vehicle control. As we are dealing with image plane co-ordinates, the image plane control parameters are to be incorporated in the vehicle control system to achieve a desired ground plane response i.e.
path following. This section presents a case study to illustrate the stability control possible with PCIP.
Fig. 6. Simple PID controller block diagram A simple PID controller is employed for the vehicle steering to study the stability in lane following. Kp, KI, and KD are generally determined experimentally to achieve a desired response. This paper presents some initial analysis on vehicle stability. For the rate of steering angle change required, ω is calculated using Eq(22) and Eq(18). e(t) is the error in the desired and current heading of the vehicle which will be used to correct the heading and thus determine the steering control.
Therefore,
(24) ωp can be calculated using the current vehicle speed and Eq(22) for a particular value of l on a given curvature. Thus considering only PD control (KI = 0), the values of required KP and KD, can be calculated using Eq(24). This will ensure that the vehicle response will match the speed and steering angle response required to track the road correctly. Once the PD gain values are determined approximately, Eq(24) will determine the correct ω for a required heading χa, which is determined by the image based lane detection. Further, Eq(22 can be used to determine the correct vehicle speed at the calculated ω. The gain values can be changed dynamically in case both the desired ω and Ż are known in any situations.
This can be determined from vehicle dynamics. In other words, if the vehicle is moving slower, the steering angle will be changed slowly by choosing appropriate PD controller gain values.
SAE Int. J. Commer. Veh. | Volume 3 | Issue 1