Mechanics of Quadcopter

Review of Kinematics, Rotation, Dynamics and Motor Mixing

1. Vijay Kumar Professor Mechanical Engineering Chitkara University Mechanics of Quadcopter

2. V. K. Jadon, Prof., Mechanical Engineering, Chitkara University Motor Mixing Dynamics of Drone The Flow …. Rotation Kinematics

3. Vectors A vector has magnitude and direction.         Both vectors represents same quantity       Commutative Law             Associative Law               Addition Subtraction           V K Jadon, Professor, Mechanical Engineering, Chitkara University

4. Vector Components         A component of a vector is the projection of the vector on an axis.                 Shifting a vector without changing its direction does not change its components. Using right angle triangle     Vector is given by and .   Vector is given by and .     Vector can be transformed into magnitude-angle notation     For 3D case, we need a magnitude and two angles or three components to represent a vector V K Jadon, Professor, Mechanical Engineering, Chitkara University

5. Unit Vector A unit vector has unit magnitude are the unit vectors along x, y and z axes respectively. These are used to express any vector.                                             V K Jadon, Professor, Mechanical Engineering, Chitkara University

6.             Vector in Space   In matrix form     Modified form to include scale factor Where       If scale up   If scale down   means components are infinite. It represents only the direction of the vector.   V K Jadon, Professor, Mechanical Engineering, Chitkara University

7. Multiplication of Vectors Scalar Product is regarded as product of magnitude of one vector and the scalar component of the second vector along the direction of first vector.         .           .         What is the angle between and   V K Jadon, Professor, Mechanical Engineering, Chitkara University

8. Multiplication of Vectors Vector Product of two vectors produces another vector whose magnitude is and acts along perpendicular to the plane that contains the two vectors.                         A vector lies in xy plane, has magnitude of 18 units and points in a direction 250 from the positive direction of x, Also, vector b has magnitude of 12 units and points along the positive direction of z. what is the vector product. V K Jadon, Professor, Mechanical Engineering, Chitkara University

9. Linear Velocity             What is the direction of velocity of particle when   What is the direction of velocity of particle when   What is the velocity of particle when   What is the direction of velocity of particle when             V K Jadon, Professor, Mechanical Engineering, Chitkara University

10.         rotation about x-axis                         rotation about y-axis Angular Displacement           rotation about y-axis rotation about x-axis                   Direction is given by Right Hand Rule   V K Jadon, Professor, Mechanical Engineering, Chitkara University Commutative Law not valid

11. Velocity and Acceleration                     Where , constitute a right hand coordinate system       If is constant                   If changes with time       Magnitude does not change   V K Jadon, Professor, Mechanical Engineering, Chitkara University

12. Vector in Rotated Frame                                   V K Jadon, Professor, Mechanical Engineering, Chitkara University

13. Rigid body Rotation                                   Position vector of point in a rigid body   Position vector of p oint in a rigid body after rotation of rigid body about origin   Overlapping two positions of rigid body before and after rotation.                                 Rotated becomes   V K Jadon, Professor, Mechanical Engineering, Chitkara University

14. Rigid body Rotation                                     In matrix form                       Projection of   Projection of     Projection of   Our aim is to find projections of rotated vectors on reference axes system in terms of the projections of original vector on reference axis. To find , in terms of ,   V K Jadon, Professor, Mechanical Engineering, Chitkara University

15.   Body Fixed Frame in Reference Frame                         V K Jadon, Professor, Mechanical Engineering, Chitkara University

16.                   Pure Rotation about an axes To find , in terms of and       Projection of   Projection of     Projection of   projection of along reference axis       projection of along reference axis   projection of along body fixed axis   projection of along body fixed axis   V K Jadon, Professor, Mechanical Engineering, Chitkara University

17.             Pure Rotation about an axes                       …( i )                 …(ii)           Vector in reference frame is obtained if we multiply vector in body frame (rotated fame) by rotation matrix.       Projection of unit vector of   Projection of unit vector of   On unit vector of   On unit vector of           V K Jadon, Professor, Mechanical Engineering, Chitkara University

18.   Pure Rotation about an axes (3D)   Vector in reference frame is obtained if we multiply vector in body frame (rotated fame) by rotation matrix.     Projection of unit vector of   Projection of unit vector of   On unit vector of   On unit vector of                   On unit vector of     Projection of unit vector of       V K Jadon, Professor, Mechanical Engineering, Chitkara University

19. Rotation Matrix of 3D Frames             =               Rotation about   1 1 =               Rotation about   =                     =                           Rotation about   , represent the components of vector   V K Jadon, Professor, Mechanical Engineering, Chitkara University

20.                             Rotation about   Rotation about       Initial Position of frame Final Position of frame       V K Jadon, Professor, Mechanical Engineering, Chitkara University Pure Rotation about an axes (3D)

21. A point is marked to a rigid body. The body is rotated by about of the reference frame. Find the coordinate of point reference axis.                         V K Jadon, Professor, Mechanical Engineering, Chitkara University

22.                         p                     )       Alternate Solution-Derived by one of Student Jaskaran Singh V K Jadon, Professor, Mechanical Engineering, Chitkara University

23.             p   p     p   p   V K Jadon, Professor, Mechanical Engineering, Chitkara University

24. Fundamental of Fluids Laminar flow Fluid flows in layers which does not cross each other. Turbulent flow The path traced by fluid particles crosses each other due to high velocity and low viscosity Compressible Flow Density changes during the flow Incompressible Flow Density remains constant Steady flow Flow parameters such as pressure, velocity etc. does not change w.r.t. time Unsteady flow Flow parameters change w.r.t. time Continuity equation Bernoulli’s Equation Mass flow rate is constant at every cross section.     compressible flow   incompressible flow     static pressure head   dynamic pressure head   datum head   V K Jadon, Professor, Mechanical Engineering, Chitkara University

25.     Dynamic pressure difference is responsible for Drag. Static pressure difference is responsible for Lift. More flow velocity is required at the top region of aerofoil compared to bottom to reach at a particular point. Due to this, the dynamic pressure increases at top region and static pressure decreases. Due to this low static pressure, the upward force (lift) is created. 1 2 Thrust           V K Jadon, Professor, Mechanical Engineering, Chitkara University

26. Quadcopter Dynamics 1 2 3 4                               V K Jadon, Professor, Mechanical Engineering, Chitkara University

27. Quadcopter Dynamics                 One for each degree of freedom Translation along x-axis   Rotation about x-axis   Translation along y-axis   Translation along x-axis   Rotation about y-axis   Rotation about z-axis   are known as Euler Angles   are called as Roll, Pitch, and Yaw Angles   Forward and backward Left and Right Up and Down , , , thrust force at rotors 1, 2, 3, 4 respectively   , , , moment reaction at rotors 1, 2, 3, 4 respectively   Six equations to describe the motion of Quadcopter Six degrees of freedom       V K Jadon, Professor, Mechanical Engineering, Chitkara University

28.                 Force and Moment Resultant force on the quadrotor + + +   Resultant moment on the quadrotor           V K Jadon, Professor, Mechanical Engineering, Chitkara University

29.                 Upward motion + + +   Downward motion + + +   Hovering motion + + +                                   V K Jadon, Professor, Mechanical Engineering, Chitkara University

30.                 Left/Right Translation: Rolling + + +                         + + +         Rolling           V K Jadon, Professor, Mechanical Engineering, Chitkara University

31.                 Forward/Backward Translation : Pitching + + +                   + + +         Pitching           V K Jadon, Professor, Mechanical Engineering, Chitkara University

32.                 Yawing     + + +         Yawing                         V K Jadon, Professor, Mechanical Engineering, Chitkara University

33.                                                 Motor Commands V K Jadon, Professor, Mechanical Engineering, Chitkara University

34. Sensors System State Controller Reference State         Controller compute the motor command to achieve the desired state Control Block Diagram   V K Jadon, Professor, Mechanical Engineering, Chitkara University

35. Quadcopter Dynamics                                       Resultant force on the quadrotor + + +   Resultant moment on the quadrotor       V K Jadon, Professor, Mechanical Engineering, Chitkara University

36.                               Angular momentum of body B with respect to A. It is a 3D vector   Angular velocity of body B with respect to A. It is a 3D vector Inertia tensor with CG of the body as center Quadcopter Dynamics V K Jadon, Professor, Mechanical Engineering, Chitkara University    

37. Let are the body fixed frame in the direction of principal axis of the body with CG as origin. The angular velocity of B w.r.t. A         Using the expression     As are in the direction of principal axis = = = = = =0         Quadcopter Dynamics   both are 3x1 matrix.   To multiply these two quantities, we will use skew symmetric matrix of .   V K Jadon, Professor, Mechanical Engineering, Chitkara University

38. Thanks V K Jadon, Professor, Mechanical Engineering, Chitkara University