Quadcopters as Rigid Bodies
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The document discusses rigid bodies and quadcopters. It defines a rigid body as a system of particles where the distance between particles remains constant. A quadcopter is described as an aerial vehicle with four independently controlled rotors that can be modeled as a rigid body. The document covers quadcopter kinematics and dynamics using the Lagrangian formalism to derive the equations of motion. It concludes that rigid bodies do not deform and a quadcopter's motion can be described using Lagrangian mechanics.
Quadcopters as Rigid Bodies
- 1. A Seminar on Rigid Bodies: Quadcopters by: Ahiante Stephen Oriasotie, 1487/2013
- 2. Rigid Bodies A rigid body is a system of particles in which the distance between any two particles is constant. | | = ＄ It is a physical system of particles that does not deform; assuming that elasticity and breakage are the limits. No real body is absolutely rigid but there exist cases where a body can be regarded as rigid. Rigid bodies can either translate or rotate or exhibit both.
- 3. Rigid Bodies (Contd) Like any other physical system, forces act on rigid bodies. The forces acting on a rigid body could either be external or internal. External forces act outside the rigid body and cause motion, Internal forces act within the system and hold together the particles forming the system. A rigid body has six degrees of freedom: three translational and three rotational coordinates.
- 4. Quadcopters Quadcopters are also known as quadrotors or quadrotor helicopters. They are helicopters with four equally spaced, independently controlled rotors. A helicopter is an aerial vehicle. A quadcopter can be regarded as a rigid body.
- 5. Quadcopters (Contd) Quadcopters are used as a typical design for unmanned aerial vehicles due to its simple structure.
- 6. Quadcopters: Frames of Reference
- 7. Quadcopters: Kinematics The linear position of the quadcopter is given by: = T The linear velocity of the quadcopter is: = The angular position of the quadcopter is defined by the inertial frame with three Euler angles . = T
- 8. Quadcopters: Kinematics The position of the quadcopter in space is thus: = T = T The time derivative of the angular position is: =
- 9. Quadcopters: Kinematics The quadcopter rotates hence, the angular velocity is: sin 0 1 cos sin cos 0 cos cos sin 0 which is equal to: sin cos sin + cos cos cos sin
- 10. Quadcopters: Dynamics The Lagrangian is by: , = the quadcopter rotates as well as translates, , = ＄＄ + ＄＄ ＄＄ = 1 2 = 1 2
- 11. Quadcopters: Dynamics ＄＄ = 1 2 ( 2 + 2 + 2 ) ; ＄＄ = 1 2 錫 = ヰ 0 0 0 0 0 0 腫ю 1 2 ヰ 2 2 sin + 2 2 + ( 2 2 2 + 2 cos sin cos + 2 2 + 腫ю( 2 2 2 2 cos cos sin + 2 2 ))
- 12. Quadcopters: Dynamics The Lagrangian thus is ; = 1 2 2 + 2 + 2 + 1 2 ヰ 2 2 sin + 2 2 + ( 2 2 2 + 2 cos sin cos + 2 2 + 腫ю( 2 2 2 2 cos cos sin + 2 2 )) - The Lagrangian equation : = 告
- 13. Quadcopters: Dynamics 告 = 告 告 告 = sin sin + cos cos sin cos sin sin cos sin cos cos = For a symmetrical body with unit inertia, ヰ = = 腫ю = 1 and a rotation at = 360属, = 1 2 2 + 2 + 2 + 2 + 2 + 2
- 14. Quadcopters: The equations of motion = sin sin + cos cos sin = cos sin sin cos sin = cos cos = = =
- 15. Conclusion Rigid bodies do not deform; neglecting breakage and elasticity A quadcopter is an aerial vehicle with four rotors independently controlled Quadcopters can be considered a rigid bodies As a rigid body, the motion of a quadcopter can be described using the Lagrangian formalism
- 16. References Bostrom, A. (2012). Rigid Body Dynamics. (pdf version). Retrieved from: http://www.am.chalmers.se/~paja/RBD/Handouts/Compendium.pdf Kilby, T & Kilby, B. (2016). Make: getting started with drones. (pdf version). Retrieved from: http://bookzz.org/book/2610493/fe7cce Teppo, L. (2011). Modelling and control of quadcopter. (M.Sc. Thesis). (pdf version). School of Science. Independent research project in applied mathematics How, Deyst. (2003). Lagranges Equations. (pdf version). Massachusetts Institute of Technology.