Solution Found!
It has been shown (Pounds, 2011) that an unloaded UAV
Chapter 6, Problem 64(choose chapter or problem)
It has been shown (Pounds, 2011) that an unloaded UAV helicopter is closed-loop stable and will have a characteristic equation given by
\(\begin{aligned} & s^3+\left(\frac{m g h}{I}\left(q_2+k k_d\right)+q_1 g\right) s^2 \\ & +k \frac{m g h}{I} s+\frac{m g h}{I}\left(k k_i+q_1\right)=0 \end{aligned}\)
where m is the mass of the helicopter, g is the gravitational constant, I is the rotational inertia of the helicopter, h is the height of the rotor plane above the center of gravity, \(q_1\) and \(q_2\) are stabilizer flapping parameters, \(k, k_i\), and \(k_d\) are controller parameters; all constants >0. The UAV is supposed to pick up a payload; when this occurs, the mass, height, and inertia change to \(m^{\prime}, h^{\prime}\), and \(I^{\prime}\), respectively, all still >0. Show that the helicopter will remain stable as long as
\(\frac{m^{\prime} g h^{\prime}}{I^{\prime}}>\frac{q_1+k k_i-q_1 g k}{k\left(q_2+k k_d\right)}\)
Questions & Answers
QUESTION:
It has been shown (Pounds, 2011) that an unloaded UAV helicopter is closed-loop stable and will have a characteristic equation given by
\(\begin{aligned} & s^3+\left(\frac{m g h}{I}\left(q_2+k k_d\right)+q_1 g\right) s^2 \\ & +k \frac{m g h}{I} s+\frac{m g h}{I}\left(k k_i+q_1\right)=0 \end{aligned}\)
where m is the mass of the helicopter, g is the gravitational constant, I is the rotational inertia of the helicopter, h is the height of the rotor plane above the center of gravity, \(q_1\) and \(q_2\) are stabilizer flapping parameters, \(k, k_i\), and \(k_d\) are controller parameters; all constants >0. The UAV is supposed to pick up a payload; when this occurs, the mass, height, and inertia change to \(m^{\prime}, h^{\prime}\), and \(I^{\prime}\), respectively, all still >0. Show that the helicopter will remain stable as long as
\(\frac{m^{\prime} g h^{\prime}}{I^{\prime}}>\frac{q_1+k k_i-q_1 g k}{k\left(q_2+k k_d\right)}\)
ANSWER:Step 1 of 3
An unloaded UAV helicopter is closed loop stable and have characteristic equation,
Where,
we need to prove that helicopter will remain stable as long as,
