Undergraduate Seminar

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.
|𝑟𝑖
− 𝑟𝑗
| = 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 (Cont’d)
• 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 (Cont’d)
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:
𝜉 𝑇
= 𝑥 𝑦 𝑧 T
• The linear velocity of the quadcopter is:
𝜉 𝑇
= 𝑥 𝑦 𝑧 𝑇
• The angular position of the quadcopter is defined by the inertial
frame with three Euler angles ℵ.
ℵ 𝑇
= 𝜑 𝜃 ∅ T

8. Quadcopters: Kinematics
• The position of the quadcopter in space is thus:
𝑞 𝑇
= 𝑥 𝑦 𝑧 𝜑 𝜃 ∅ T
𝑞 𝑇
= 𝜉 ℵ T
• The time derivative of the angular position is:
ℵ 𝑇
= 𝜑 𝜃 ∅
𝑇

9. Quadcopters: Kinematics
• The quadcopter rotates hence, the angular velocity is:
− sin 𝜃 0 1
cos 𝜃 sin ∅ cos ∅ 0
cos 𝜃 cos ∅ − sin ∅ 0
𝜑
𝜃
∅
which is equal to:
∅ − 𝜑 sin 𝜃
𝜑 cos 𝜃 sin ∅ + 𝜃 cos ∅
𝜑 cos 𝜃 cos ∅ − 𝜃 sin ∅

10. Quadcopters: Dynamics
• The Lagrangian is by:
𝐿 𝑞, 𝑞 = 𝑇 − 𝑈
the quadcopter rotates as well as translates,
𝐿 𝑞, 𝑞 = 𝑇𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛𝑎𝑙 + 𝑇𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 − 𝑈
𝑇𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛𝑎𝑙 =
1
2
𝑚𝒗 𝟐
=
1
2
𝑚 𝜉 𝑇
𝜉

11. Quadcopters: Dynamics
𝑇𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛𝑎𝑙 =
1
2
𝑚( 𝑥2
+ 𝑦2
+ 𝑧2
) ; 𝑇𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 =
1
2
Ω 𝑇
𝐼Ω
𝐼 =
𝐼 𝑥𝑥 0 0
0 𝐼 𝑦𝑦 0
0 0 𝐼𝑧𝑧
1
2
𝐼 𝑥𝑥 ∅2
− 2∅ 𝜑 sin 𝜃 + 𝜑2
𝑠𝑖𝑛2
𝜃 + 𝐼 𝑦𝑦( 𝜑2
𝑐𝑜𝑠2
𝜃𝑠𝑖𝑛2
∅ + 2 𝜑 cos 𝜃 sin ∅ 𝜃 cos ∅ + 𝜃2
𝑐𝑜𝑠2
∅ + 𝐼𝑧𝑧( 𝜑2
𝑐𝑜𝑠2
𝜃𝑐𝑜𝑠2
∅ −
2 𝜑 cos 𝜃 cos ∅ 𝜃sin ∅ + 𝜃2
𝑠𝑖𝑛2
∅))

12. Quadcopters: Dynamics
• The Lagrangian thus is ;
𝐿 =
1
2
𝑚 𝑥2
+ 𝑦2
+ 𝑧2
+
1
2
𝐼 𝑥𝑥 ∅2
− 2∅ 𝜑 sin 𝜃 + 𝜑2
𝑠𝑖𝑛2
𝜃 + 𝐼 𝑦𝑦( 𝜑2
𝑐𝑜𝑠2
𝜃𝑠𝑖𝑛2
∅ + 2 𝜑 cos 𝜃 sin ∅ 𝜃 cos ∅ + 𝜃2
𝑐𝑜𝑠2
∅ + 𝐼𝑧𝑧( 𝜑2
𝑐𝑜𝑠2
𝜃𝑐𝑜𝑠2
∅ −
2 𝜑 cos 𝜃 cos ∅ 𝜃sin ∅ + 𝜃2
𝑠𝑖𝑛2
∅)) - 𝑚𝑔𝑧
The Lagrangian equation :
𝑑
𝑑𝑡
𝜕𝐿
𝜕 𝑞
−
𝜕𝐿
𝜕𝑞
=
𝐹𝜉
𝜏

13. Quadcopters: Dynamics
𝐹𝜉 =
𝐹𝑥
𝐹𝑦
𝐹𝑧
=
𝜌 sin ∅ sin 𝜑 + cos ∅ cos 𝜑 sin 𝜃
𝜌 cos ∅ sin 𝜃 sin 𝜑 − cos 𝜑 sin ∅
𝜌 cos 𝜃 cos ∅ − 𝑚𝑔
𝜏 =
𝜏 𝜑
𝜏 𝜃
𝜏∅
For a symmetrical body with unit inertia, 𝐼 𝑥𝑥 = 𝐼 𝑦𝑦 = 𝐼𝑧𝑧 = 1 and a rotation at
𝜃 = 360°,
𝐿 =
1
2
𝑚 𝑥2 + 𝑦2 + 𝑧2 + ∅2 + 𝜑2 + 𝜃2 − 𝑚𝑔𝑧

14. Quadcopters: The equations of motion
𝑚 𝑥 = 𝜌 sin ∅ sin 𝜑 + cos ∅ cos 𝜑 sin 𝜃
𝑚 𝑦 = 𝜌 cos ∅ sin 𝜃 sin 𝜑 − cos 𝜑 sin ∅
𝑚 𝑧 = 𝜌 cos 𝜃 cos ∅ − 𝑚𝑔
𝜑 = 𝜏 𝜑
𝜃 = 𝜏 𝜃
∅ = 𝜏∅

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). Lagrange’s Equations. (pdf version). Massachusetts Institute of
Technology.