1. KINEMATIC CHAINS
&
ROBOTS
(I)

2. Kinematic Chains and Robots (I)
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. General Rigid-Body Motion
Chasles Theorem:
A general finite rigid-body motion is equivalent to
A pure translation (sliding) along, by an amount us, and
A pure rotation about the axis of rotation, by an amount 
The general rigid-body motion can be characterized by the screw parameters

5. General Rigid-Body Motion
Given R, we can solve for the eigenvector of R corresponding to =1 to get
i.e. solve
where I is the 33 identity matrix
If given
find
Get  from

6. General Rigid-Body Motion
The previous deals with rotation. In general, if we are given the location of a point C at 2 time instance, i.e. given and
We can denote the motion of the point C due to both rotation and translation as
The displacement component of the motion parallel to is
Finally:

7. 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

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

9. 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

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

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

12. A Simple Planar Robot
Link (2)
Link (3): Gripper
Link (1)
Link (0): Base
This simple robot will be used throughout to illustrate simple concepts

13. Forward Kinematics Analysis
Consider the following motion: Y3 X3 ?
2=90° Y3 X0 Y0 X3
Given the dimensions of the linkages and the individual relative motion between links, how to find the position, velocity, acceleration of the gripper? (Forward kinematics analysis problem)

14. Inverse Kinematics Analysis
Again consider the following motion: Y3 X3
2=?° Y3 X0 Y0 X3
Given the dimensions of the linkages and the desired motion of the gripper, how to find the individual relative motion between links? (Inverse kinematics analysis problem)

15. Dynamics Analysis
Again consider the following motion: Y3 X3
2=?° and force required Y3 X0 Y0 X3
Given the inertia and dimensions of the linkages and the desired motion of the gripper, how to find the individual relative motion between links and the actuator forces to achieve this motion? (Dynamic analysis problem)

16. Notation
Consider the following motion. We will associate basis vectors with frame {b} and the (X, Y, Z) axis with frame {a}
X-axis
Y-axis
Z-axis
“O” X A B C
X-axis
Y-axis
Z-axis
“O” X A B C
We will denote the rotation matrix “R” that brings frame {a} to frame {b} as

17. Notation
Example
X-axis
Y-axis
Z-axis
“O” X A B C

18. 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}

19. 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”

20. Transformation matrix
So far, in the example we used the origins of the two frames are at the same point. What if the origin of frame {b} is at a distance defined by the vector
In this case, the point “B” is given by:
We can simplify the above equation to:
where
is called the Transformation Matrix of frame {b} w.r.t. frame {a}
is called the augmented vector

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

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

23. 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:

24. 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