Applications of vector Calculus in Robotics Engineering.

1. APPLICATION OF VECTOR CALCULUS IN
ROBOTICS ENGINEERING
PRESENTED BY:-
NAVEEN A
SAISARAN K
ROOPESH P
PAAVAI ENGINEERING COLLEGE, (AUTONOMOUS) NAMAKKAL.

2. INTRODUCTION:
 Vector Calculus was developed by J. Willard
Gibbs.
 He was born on February 11 1839 United States.
 He studied at the Yale University.
 He is the founder of vector calculus.
 He died on April 28 1903.

3. VECTOR CALCULUS:
 Vector Calculus is a branch of mathematics concerned with differentiation
and integration of vector fields. Vector has both magnitude and direction.
The term “VECTOR CALCULUS” is sometimes used as a synonym for the
broader subject of multivariable calculus, which includes vector calculus as
well as partial differentiation and multiple integration.
 In vector calculus, various differential operators defined on scalar
or vector fields are studied, which are typically expressed in terms
of the del operator.
= (∂⁄∂x, ∂⁄∂y, ∂⁄∂z)

4. Do you know how the
VECTOR CALCULUS is
applied in ROBOTICS?

5. Movement Imitation - Example
vector calculus in robotics for the research path integration in robot
Navigation and path optimization technique using vector calculus.
Future technology in vector graphics is a humanoid robot.
VECTOR CALCULUS IN ROBOTICS

6. Position of tip in (x,y) coordinates

7. APPLICATION OF VECTOR CALCULUS IN
ROBOTICS ENGINEERING
ROBOTS AS MECHANISM:
Multiple type robot have 3 Dimensional
(open loop, chain mechanisms)
Fig. 2.1 A one-degree-of-freedom
closed-loop four-bar mechanism
Fig. 2.2 (a) Closed-loop versus (b) open-
loop mechanism

8. Representation of a Point in
ROBOT Mechanism:
A point P in Robot mechanism:
3 coordinates relative to a reference frame.
It has both magnitude and direction.
^^^
kcjbiaP zyx 
Representation of a point in space

9. Representation of a vector in space
A Vector P in robot mechanism:
3 coordinates of its tail and of its head
^^^__
kcjbiaP zyx 













w
z
y
x
P
__
Representation of a Vector in
ROBOT Mechanism:

10. Notations:
 The Angular Velocity Vector :
Angular velocity vector of a body ( is for a point) As a frame
represents the orientation of the body = rotational velocity of the frame.
Rotational velocity of frame {B} relative to frame {A}:
- it’s direction represents the instantaneous axis of rotation of {B}
relative to {A}
- it’s magnitude represents the speed of rotation.
the angular velocity of {B} relative to {A} expressed in {C}
In the universe frame {U}
10
  A
Cv
A
B 
 
C
A
B 
U
C C 

11. Linear and rotational velocities of rigid bodies
2. Rotational velocity: (Rotation only)
Two frames {A} and {B} have the same origin for all the time only the relative
orientation is changing in time 
is fixed in {B} 
What is the velocity of point Q in {A}
If is changing in {B}
11
A
B
B
Q 0B
QV 
 
A A A
Q B
A A B
B B
V Q
R Q
  
  
B
Q
?A
QV
   
 
A
A A A B B
Q B B Q
A A B A B
B B B Q
V R Q V
R Q R V
   
   

12. Velocity propagation from link to link
i
i
1
1
i
iv

i
iv
1
1
i
i

Starting from the base the velocity of any link
(i+1) equal to the previous link (i) + the relative
Velocity between (i+1) and (i)
• Angular velocity propagation
   
1 1
1
1 1 1 1 1 1
i i i
i i i
i i i i
i i i i i iz R z
 
 
 

     
  
   
1
1
0
0
1
i
iz

 
 
 
  
 
     
1
1 1 1 1
1 1 1 1
1 1 1 10 0 1
i i i i
i i i i i
Ti i i i i i
i i i i i i i i
R z
R z R
  
    

   
   
   
  
      

13. Jacobians
• In robotics, the Jacobian relates the Cartesian velocities with
joint velocities
≡ vector of joint angles
≡ vector of joint velocities
• General Case:
13
Instantaneous, as  is changing  J( ) is changing
11 12 1
21 22 20
1 2
( )
n
n
m m mn
J J J
J J J
J
J J J
 
 
  
 
 
 

14. Fig. 2.17 The hand frame of the robot relative to the reference frame.
Forward Kinematics Analysis:
• Calculating the position and orientation of the hand of the robot.
If all robot joint variables are known, one can calculate where the robot is
at any instant.
.
ROBOT NAVIGATION AND OPTIMIZATION
TECHNIQUE

15.  Roll, Pitch, Yaw (RPY)
angles
Roll: Rotation of about -axis (z-axis of the moving frame)
Pitch: Rotation of about -axis (y-axis of the moving frame)
Yaw: Rotation of about -axis (x-axis of the moving frame)
FORWARD AND INVERSE MOVEMENT OF ROBOT

16. CONCLUSION:
We have presented a novel neural framework for robot
navigation in a cluttered environment.
“A MATHEMATICIAN IS A BLIND MAN IN A
DARK ROOM LOOKING FOR A BLACK CAT
WHICH ISN’T THERE.”
-CHARLES DARWIN

17. Thank
you