Overview
This module determines the desired heading and altitude of the aircraft and manages the waypoints (also called “path nodes”) of the plane’s flight path. The module takes in GPS coordinates and altitude measurements and calculates the heading and altitude the plane needs to stay on course. Additionally, the module takes instructions from the state machine to modify the waypoints in the plane’s flight path.
How does Waypoint/Path Management Work?
Note that a lot of this information comes from the PicPilot docs, which was last updated on January 6, 2016.
Introduction
Path management is the process of dynamically determining the optimal course to navigate a set of pre-defined waypoints. It can be broken down into three sections: straight path following, orbit following, and the blending of straight path and orbit following.
Straight Path Following
Straight path following is basically following a line (shocking).
The principle is simple. Given two points, and the location of your vehicle, determine what heading it should follow. If your plane is on the desired path, the heading of the plane will be the same direction as the line connecting the two points. However, things get interesting when the plane is slightly off the path. In this case, the plane should face slightly towards the path to reduce its cross-track error (the distance between the plane and the line connecting the two waypoints). We don’t make the plane follow a path perpendicular to the line since it may result in the plane crossing the line, resulting in the same issue all over again. Now, if the plane is an infinite distance from the line, then the plane should head at a heading perpendicular to the line, in order to regain distance. This is the premise of the straight line path following algorithm. As a result, a heading vector field would look as such:
As the plane moves further away from the line, the angle between the plane's desired path and the line approaches 90º.
The equations that govern this behaviour are not complicated. There are only a few things required to make this calculation. Firstly, you must know the heading of the line. If you have the XY coordinates of the line endpoints, you can easily determine that through simple trigonometry. If you only have the GPS coordinates, you should use the Haversine Formula in order to get the XY coordinates.
Once you have the coordinates, subtracting them will give you the direction of travel (in an XY plane). Furthermore, by applying the arctan function on the direction of travel, one will determine the line’s heading. In addition, the value should be between –pi and +pi. Thus, any 2pi corrections that need to be made can be done at this point.
The cross-track error is calculated as follows (can be derived with trigonometry):
cross_track_error = cos(courseAngle) * (positionY - targetWaypointY) - sin(courseAngle) * (positionX - targetWaypointX)
The following diagram provides a good visualization of the variables in this equation:
The cross track error is then useful to determine the desired heading of the aircraft. Once again, using the arctan function is suitable to do so:
desired_heading = 90 - rad2deg(courseAngle - MAX_PATH_APPROACH_ANGLE * 2/PI * atan(k_gain[PATH] * pathError))
Note, as the atan term increases, the heading approaches the (courseAngle -__MAX_PATH_APPROACH_ANGLE).
Note, that the courseAngle is calculated with respect to the x-axis, on the x-y plane. In other words, a path going from West to East would have a courseAngle of 0 degrees. This would be 90 degrees in terms of a true heading. Therefore, to get the true heading, you must subtract the courseAngle from 90 degrees.
Orbit Following
Orbit following is less straight-forward as straight path following, but it basically involves following a curve of a certain radius. An orbit is depicted by a radius, a center location, and the direction of travel (clockwise or anti-clockwise).
In order to maintain a certain radius, the Euclidean distance needs to be calculated between the center of the orbit and the plane itself. The goal of this function is to maintain this Euclidean distance constant. The Euclidean distance can be calculated as such:
float orbitDistance = sqrt(pow(position[0] - center[0],2) + pow(position[1] - center[1],2));
position[0] → x coordinate of plane’s current position
position[1] → y coordinate of plane’s current position
center[0] → x coordinate of orbit center
center[1] → y coordinate of orbit center
This value is then used to determine the equivalent of cross_track _error, but for an orbit. This is done very easily. The term d (Euclidean distance) subtracted by the ρ (desired radius) provides the relative error, which must be minimized.
orbit_cross_track_error = 90 - rad2deg(courseAngle + direction \* (PI/2 + atan(k\_gain[ORBIT] \* (orbitDistance - radius)/radius)))
This equation is actually very similar to the equation governing straight line path following. The arctan function forces the heading to converge onto the orbit. The direction of travel (λ) which can be either 1 or -1, reverses the effect of the heading perturbations. This is then added onto the course angle as a perturbation. Once again, a gain value needs to be tuned to determine the rate of convergence.
The course angle can be determined easily based on the location of the curve. For instance, if the vehicle is in the first quadrant of the circle/orbit, the heading will range between 270° and 0°, assuming a counter-clockwise rotation. This course angle can be calculated using this equation:
float courseAngle = atan2(position[1] - center[1], position[0] - center[0]);
Blending Straight Path and Orbit Following
The orbit following and straight path following algorithms are used in combination in order to assemble a path. Both algorithms alternate in usage. Every corner uses the orbit following algorithm. Every straight line uses the straight path following algorithm.
In order to put the two together, you must draw in orbits between each set of points. The additional restriction is that they must be tangent to both lines, as depicted in this diagram:
Between waypoints we use straight line following. When switching between straight line paths, we use orbit path following for a smooth transition. Notice that the two straight line paths are tangential to the circle defining the orbit.
Figuring out where the tangent will touch requires some basic trigonometry. [WILL TOUCH ON LATER]. The coordinates at which this occurs can be calculated using:
NextX/Y/Z refer to the coordinates of the point W_(I+1) and targetX/Y/Z refer to the coordinates of the point W_i.
he turning angle can be calculated via the following equation:
float turningAngle = acos(-deg2rad(waypointDirection[0] * nextWaypointDirection[0] + waypointDirection[1] * nextWaypointDirection[1] + waypointDirection[2] * nextWaypointDirection[2]));
Index 0 is the x-coordinate
Index 1 is the y-coordinate
Index 2 is the z-coordinate
This is simply the dot product of the (W_i - W_(i-1)) vector and the (W_(i+1) - W_i) vector. Given the dot product formula, you can use the arccos function to determine the angle between the two lines.
At the points where the straight line paths touch the orbit path, there is a checkpoint. Imagine a giant plane perpendicular to the path. As soon as the plane crosses this boundary, the next step is executed. For instance, if the vehicle is travelling straight along a path, then passes the plane, it will initiate a turn (orbiting algorithm). Once it passes the next plane, it will initiate the straight line path following algorithm once again.
In order to detect if a vehicle passes the boundary, the dot product of two vectors must be taken. If the value is positive, it is an indicator that the vehicle has crossed the boundary.
The dot product is:
Both vectors have X, Y, and Z components. Likewise, equations that depict the half plane are stated above.
Note that in the code, all "direction vectors" such as W_i – W_(i-1) are normalized.
For every pair of "checkpoints" the path index is incremented once they are passed. The index is used to identify the data in a linked list through the wireless communications.
Implementation
Managing Path Data
Inputs and Outputs
Inputs
Current GPS Coordinates (longitude and latitude)
Current Altitude
Current Heading
Outputs
Desired Heading
Desired Altitude
Distance to next waypoint
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