Kinematics is a branch of classical mechanics that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the mass of each or the forces that caused the motion. 

1. KINEMATIC RELATIONSHIP
OF A OBJECT IN SPACE
Hitesh Mohapatra
https://www.linkedin.com/in/hiteshmohapatra/

2. KINEMATICS
Kinematics is a branch of classical mechanics that describes the motion of points,
bodies (objects), and systems of bodies (groups of objects) without considering the mass
of each or the forces that caused the motion.
Kinematics, as a field of study, is often referred to as the "geometry of motion" and is
occasionally seen as a branch of mathematics.

3.  KINEMATICS – the analytical study of the geometry of
motion of a mechanism:
• with respect to a fixed reference co-ordinate system,
• without regard to the forces or moments that cause the motion.
In order to control the motions and position of the robot we use close chain
and open chain manipulators.

4. OPEN CHAIN CLOSED CHAIN

6. 1.Length/Distance: One dimensional geometry deals with length and distance.
2.Width/Area: Two dimensional geometries are expressed as flat planes which have
length and width but no depth. A shadow is an example of a two dimensional
appearance. 2d shapes are typically measured in square units, such as cm2m2 or others
like acres.
3.Depth/Volume: Three dimensional geometries add the dimension of depth or height
so that they describe objects with volume. Volume should not be confused with weight as
two objects can be the same volume but one can be much heavier than the other. A
gallon of mercury is much heavier than a gallon of milk. 3d measures include cubic
units cm3cm3 , pints, quarts, tablespoons, and liters.

7. KINEMATIC RELATIONSHIP IN 1D

10. KINEMATIC RELATIONSHIP IN 2D

15. 3D
Kinematics
 Consists of two parts
 3D rotation
 3D translation
 The same as 2D
 3D rotation is more complicated
than 2D rotation (restricted to z-axis)
 Next, we will discuss the treatment
for spatial (3D) rotation

16. 3D ROTATION REPRESENTATIONS
 Euler Angle
 Axis-angle
 3X3 rotation matrix

17. EULER ANGLES
 The Euler angles are three angles introduced by Leonhard Euler to
describe the orientation of a rigid body with respect to a
fixed coordinate systems.

18. AXIS-ANGLE REPRESENTATION
 Rot(n,q)
 n: rotation axis (global)
 q: rotation angle (rad. or deg.)
 follow right-handed rule
 Rot(n,q)=Rot (-n,-q)
 Problem with null rotation: rot(n,0), any n
 Perform rotation
 Rodrigues formula
 Interpolation/Composition: poor
 Rot(n2,q2)Rot(n1,q1) =?= Rot(n3,q3)
We create matrix
R for rotation

19. ROTATION MATRIX
 Meaning of three columns
 Perform rotation: linear algebra
 Composition: trivial
 orthogonalization might be required
due to floating point errors
 Interpolation: ?
 
Ax
uAxuAxuAx
uxuxuxx
uuuaA ij














ˆˆˆ
ˆˆˆ
ˆˆˆ
332211
332211
321
 xRRxRRxRx
xRx
12122
1

