1. KINEMATIC CHAINS

2. Kinematic Chains
This lecture continues the discussion on a body that cannot be treated as a single particle but as a combination of a large number of particles. Many machines can be viewed as an assemblage of rigid bodies called kinematic chains. This lecture continues the discussion on the analysis of kinematic chains with focus on robots.
After this lecture, the student should be able to:
Appreciate the concept of kinematic pairs (joints) between rigid bodies
Define common (lower) kinematic pairs
Distinguish between open and closed kinematic chains
Appreciate the concept of forward and inverse kinematics and dynamics analysis
Express a finite motion in terms of the transformation matrix

3. General Rigid-Body Motion
General Motion =
Translation +
Rotation
The general motion is equivalent to that of the action of a screw, which can be described using 4 “screw” parameters:
Axis of rotation
The angle of rotation 
Axis of rotation passes through the point
The displacement component parallel to the direction of rotation

4. Kinematic Pair
A kinematic pair is the coupling or joint between 2 rigid bodies that constraints their relative motion.
The kinematic pair can be classified according to the contact between the jointed bodies:
Lower kinematic pairs: there is surface contact between the jointed bodies
Higher kinematic pairs: the contact is localized to lines, curves, or points

5. Lower Kinematic Pairs 
Revolute/pin joint s 
Cylindrical joint  x y
Planar joint 
Spherical joint  
Prismatic joint s

6. Kinematic Chain
A kinematic chain is a system of rigid bodies which are joined together by kinematic joints to permit the bodies to move relative to one another.
Kinematic chains can be classified as:
Open kinematic chain: There are bodies in the chain with only one associated kinematic joint
Closed kinematic chain: Each body in the chain has at least two associated kinematic joints
A mechanism is a closed kinematic chain with one of the bodies fixed (designated as the base)
In a structure, there can be no motion of the bodies relative to one another

7. Open and Closed Kinematic Chains
Base (fixed)
Open kinematic chain
Kinematic joint
Rigid bodies
Structure

8. Robots
Robots for manipulation of objects are generally designed as open kinematic chains
These robots typically contain either revolute or prismatic joints

9. X Notation Z-axis C B “O” Y-axis A X-axis
The position of point “C” expressed in frame {a} is denoted by
For example, is the position of point “C” relative to frame {b} expressed in terms of frame {a}

10. X Transformation matrix
Consider the 2 frames {a} and {b}. Notice that for a vector
Example: X A B C
X-axis
Y-axis
Z-axis
“O”

11. X Example: Transformation matrix
What is the transformation matrix for the case below? X A B C
X-axis
Y-axis
Z-axis
“O”

12. Example: Transformation matrixB C
X-axis
Y-axis
Z-axis
“O”

13. Transformation matrix and Position
If R is the identity matrix, then there is no rotation and the transformation matrix will represent the pure displacement of the origins of the frames of reference:

14. Summary
Many machines can be viewed as an assemblage of rigid bodies called kinematic chains. This lecture continues the the discussion on the analysis of kinematic chains with focus on robots.
The following were covered:
The concept of kinematic pairs (joints) between rigid bodies
Definition of common (lower) kinematic pairs
Open and closed kinematic chains
The concept of forward and inverse kinematics and dynamics analysis
Finite motion in terms of the transformation matrix