I want to know how to make the mathematical model of Quadcopter? Is there any differential equation for quadcopter? I want to simulate quadcopter as a mathematical model so I want to know how can I make equations to convert the Speed of motors into the desired yaw, pitch and roll to achieve desired real world x,y,z coordinates. Please help if any body know any reference or direct code that simulates the quadcopter.

• – Amzoti Sep 11 '13 at 12:49

Yes, there are. See the fantastic paper by Lee, Leok, and McClamroch:

See equations (2)-(5).

Off hand there are some physics principles that can help.

Gravity, air resistance, as well a thrust of the propellars provide forces acting on the drone. For forces, there is usually a torque as well. Forces move the copter foward/backward, left/right,up/down. Torque provides pitch, yaw, and spin.

Forces have an x,y, and z component. So does torque.

Gravity points downward acting at the center of mass. The propellars provide a net force on the drone. They also act to provide torques about the center of mas.

By Newton's Second Law, the net force from thrust has to exceed the other forces acting on the quad copter like surface normal force and gravity.

Force is $$(a_x,a_y,a_z)$$ for say he front left copter.

Torque is $$\vec{\tau}=\vec{F}\times \vec{r}$$ where $$\vec{r}$$ is the displacement vector from the center of mass of the copter to center of mass of the propellar.

So if the front left and front right copters provide identical thrusts along the z-axis, there are equal and opposite components vying to spin the drone around the forward axis. they help contribute to a torque forcing the drone to spin along the side to side axis. At equilibrium this is balanced by the other two copters.

Propellars generate thrust. The faster the propeallars, the more thrust, but the relationship might not be linear. The force of thrust is usually in the direction of the axis of the copters.

If the rear copters spin faster than the forward copters there will be a torque tilting the copter forward. This will shift the thrust of the propellars somewhat forward providing acceleration forward. It's the torque that move the copter laterally, combined with other influences.