Our team participates in CanSat Competition 2019 and has to develop a CanSat that will descent without a parachute, using only freely spining blades (autogyro descent). We used a simulation (provided to us by NTUA's Fluid Mechanics Department) that takes as input the blade's chord and twist at different values of the radius and computes the angular momentum at the equilibrium point as well as the terminal velocity of our CanSat.
In order to make our control system's job easier we want to design a blade that minimizes the angular momentum of the rotor so as to constrain the gyroscopic effects. To achieve that we did the following:
The blade is characterized by 2 functions associating chord with radius and twist with radius. We modeled the chord-radius and twist-radius functions as Bezier Curves with 4 and 3 control points respectively. Therefore, we are able to uniquely determine a blade using only 7 variables:
The angular velocity and terminal descent rate
computed from
the simulation are a function of our state vector
.
We want the minimum angular velocity and a terminal velocity in the range [11,14] m/s, so the cost function will look as following:
(Please note that all of the above functions have values in the real numbers.)
Where f is a function that has nearly zero values in the region of interest ( [11,14] ) and grows exponentially outside this region.For the function f we used:
In order to determine the desired blade, we have to find the state vector that minimizes our cost function
:
In real life, there are some manufacturing constraints for the blade so the accepted state vectors must lie in a set, let it be , that is the set of all blades with certain characteristics.
To minimize the function over
we will use the Projected
Gradient Descent Algorithm. Starting from a blade with state vector
we are using the following update law, to iterratively improve our blade:
Where is a positive constant.
The partial derivatives are calculated as:
Which can also be written as: