Guidance and Navigation
Scenario: The balloon has popped, and the glider has started its glide. Our goal is to glide home. But how?
Algorithm #1
Use GPS only. GPS provides a
good estimate of the glider’s position and earth fixed velocity. So why not just twiddle with the rudder
until the GPS ground track is pointed home. At first this seems like a simple and
elegant solution, and it might work if the wind speed is slow enough. But this algorithm gets confused in
strong winds. For example, what
happens when the wind speed is greater than the gliding speed? In this case, it is not possible to make
the ground track point to the goal.
In fact, we would need more information to determine if we are even
headed in the right direction.
Algorithm #2
Use the glider’s current position and the goal’s position to determine the bearing to the goal. Twiddle the rudder until a digital compass indicates that the nose of the glider is pointed toward the goal. This algorithm is simple. It does the right thing, even in the presence of strong headwinds. A crosswind will blow the glider off course, and it will follow a curved path (i.e. a pursuit curve) toward its goal.
Algorithm #3
(A work in progress.) Let’s assume we could estimate the wind speed and direction. With a good wind velocity estimate, we could determine the best heading to fly. Then, like algorithm #2, we point our nose the best heading using a digital compass.
To find the best direction to fly, we make use of the following observation: Flying toward a fixed target in the presence of wind is equivalent to flying toward a moving target with no wind, where the target is moving with the same speed, but opposite direction as the wind.
bearing angle to goal (geometric angle)
heading angle of wind
heading angle to goal (angle to point nose)
Wind
speed
Flight
speed
By drawing the triangle whose edges consist of the initial goal vector, the goal movement vector, and the flight vector, and solving using the law of sines, we get:
This algorithm has two main benefits over algorithm #2. First, for a given time interval, this algorithm will make more progress toward the goal if there is a cross wind. The second benefit is related to the fact that the 2-D digital compass we intend to use gives poor results when it is not horizontal. Since this algorithm requires less turning, and thus less banking, it might get better compass readings. The question is: Do these advantages outweigh the increase in complexity?
TODO:
Perform a quantitative analysis of the benefit of algorithm 3 over algorithm 2 assuming perfect knowledge. How much do we save?
Is there a (simple) way to do continuous correction to the heading and wind speed estimates using the more reliable GPS velocity data?
Discuss the method of estimating Vw, i.e. calculate a steady-state gliding airspeed based on altitude, and subtract from groundspeed.
Discuss the peculiarities of the lousy 2-D digital compass. Is it possible to make this compass work better using magnitude info?