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Mathematical Structure of
Kinematic Model
Hitesh Mohapatra
https://www.linkedin.com/in/hiteshmohapatra/
Manipulator
The mechanical structure of a manipulator consists of rigid bodies (links)
connected by means of joints, is segmented into an arm that ensures mobility
and reachability, a wrist that confers orientation and an end-effectors that
performs the required task.
Links and Joints
Degrees of Freedom
The number of independent movements that an object can perform in 3-D space. A rigid
body free in space has six degrees of freedom –
 3 translations (T1, T2, T3), representing linear motions along three perpendicular axes,
specify the position of the body in space.
 3 rotations (R1, R2, R3), which represent angular motions about three axes, specify the
orientation of the body in space.
NOTE: THE DEGREE OF FREEDOM OF A KINEMATIC CHAIN IS EQUAL TO THE NUMBER OF
JOINTS IN THE CHAIN
Frames
Mapping
 Changing the description of a point in space from one frame to another
frame.
 There are three possibilities:
1. MAPPING BETWEEN ROTATED FRAME S
2. MAPPING BETWEEN TRANSLATED FRAMES
3. MAPPING BETWEEN ROTATED AND TRANSLATED FRAMES
Mapping between Rotated frames
Mapping between Translated
frames
O1P =O1O2 + O2P
=>1P = 2P+ 1D2
Mapping between Rotated and
Translated frames
Kinematic Model (Introduction)
Kinematics is the study of the robot’s movements with regard to a reference
system.
To program the tool motion and joint-link motions, a mathematical model of
the manipulator is required.
The relation between the joint-variables and the position and orientation of the
end-effector is the kinematic model.
Kinematic model is represented by the Homogeneous Transformation Matrix.
Joint-Link parameters
Link Parameters:
Link length & Link Twist
Joint Parameters:
Joint Distance & Joint Angle
Direct & Inverse Kinematics
Kinematic modelling is split into two problems:
Direct Kinematics: This model gives the positions and orientation of the
end-effector as a function of the joint-link parameters
Inverse Kinematics: For a given position and orientation of the end-
effector, it is required to find a set of joint-variables that would bring
the end-effector in the specified position and orientation.
Denavit-Hartenberg Notation
(assigning frames)
Kinematic Relationship b/w adjacent
links
Manipulator Transformation Matrix
The above equation is a
function of the joint-link
parameters.
This equation is called the
Kinematic model.

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Mathematical Structure of Kinematic Model

  • 1. Mathematical Structure of Kinematic Model Hitesh Mohapatra https://www.linkedin.com/in/hiteshmohapatra/
  • 2. Manipulator The mechanical structure of a manipulator consists of rigid bodies (links) connected by means of joints, is segmented into an arm that ensures mobility and reachability, a wrist that confers orientation and an end-effectors that performs the required task.
  • 4. Degrees of Freedom The number of independent movements that an object can perform in 3-D space. A rigid body free in space has six degrees of freedom –  3 translations (T1, T2, T3), representing linear motions along three perpendicular axes, specify the position of the body in space.  3 rotations (R1, R2, R3), which represent angular motions about three axes, specify the orientation of the body in space. NOTE: THE DEGREE OF FREEDOM OF A KINEMATIC CHAIN IS EQUAL TO THE NUMBER OF JOINTS IN THE CHAIN
  • 6. Mapping  Changing the description of a point in space from one frame to another frame.  There are three possibilities: 1. MAPPING BETWEEN ROTATED FRAME S 2. MAPPING BETWEEN TRANSLATED FRAMES 3. MAPPING BETWEEN ROTATED AND TRANSLATED FRAMES
  • 8. Mapping between Translated frames O1P =O1O2 + O2P =>1P = 2P+ 1D2
  • 9. Mapping between Rotated and Translated frames
  • 10. Kinematic Model (Introduction) Kinematics is the study of the robot’s movements with regard to a reference system. To program the tool motion and joint-link motions, a mathematical model of the manipulator is required. The relation between the joint-variables and the position and orientation of the end-effector is the kinematic model. Kinematic model is represented by the Homogeneous Transformation Matrix.
  • 11. Joint-Link parameters Link Parameters: Link length & Link Twist Joint Parameters: Joint Distance & Joint Angle
  • 12. Direct & Inverse Kinematics Kinematic modelling is split into two problems: Direct Kinematics: This model gives the positions and orientation of the end-effector as a function of the joint-link parameters Inverse Kinematics: For a given position and orientation of the end- effector, it is required to find a set of joint-variables that would bring the end-effector in the specified position and orientation.
  • 14. Kinematic Relationship b/w adjacent links
  • 15. Manipulator Transformation Matrix The above equation is a function of the joint-link parameters. This equation is called the Kinematic model.