241031852 Kinematics of Robots

IntroductionKinematics of Robot Manipulator
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Outline
Review
Rotation Matrix, composite rotation matrix
Homogeneous Matrix
Direct kinematics
Denavit-Hartenberg Representation
Review
By general agreement a robot is:
A programmable machine that imitates the actions or appearance of an intelligent creature–usually a human.
To qualify as a robot, a machine must be able to:
1) Sensing and perception: get information from its surroundings
2) Carry out different tasks: Locomotion or manipulation, do something physical–such as move or manipulate objects
3) Re-programmable: can do different things
4) Function autonomously and/or interact with human beings
Why use robots?
Perform 4A tasks in 4D environments
The City College of New York
Manipulators
Joints: revolute or prismatic
Drive: electric or hydraulic
End-effector (tool) mounted on a flange or plate secured to the wrist joint of robot
Manipulators
Articulated: RRR
Hand coordinate:
tool mounting plate
Manipulators
Continuous path control
Spray painting, Arc welding, Gluing
Manipulators
Redundant, (7-n) => reaching around obstacles, avoiding undesirable configuration
Degree of Freedom (DOF)
Which one is more important?
Precision (validity) is defined as how accurately a specified point can be reached. It is a function of the resolution of the actuators, as well as its feedback devices. Most industrial robots can have precision of 0.001 inch or better.
Repeatability (variability) is how accurately the same position can be reached if the motion is repeated many times. The robot may not reach the same point every time, but will be within a certain radius from the desired point. The radius of a circle that is formed by this repeated motion is called repeatability. It is much more important than precision. If a robot is not precise, it will generally show a consistent error, which can be predicted and thus corrected through programming. However, if the error is random, it cannot be predicted and thus cannot be eliminated. Repeatability defines the extent of this random error. Manufactures must specify repeatability in conjunction with the number of tests, the applied payload during the tests, and the orientation of the arm. Have repeatability in the 0.001 inch range.
Depending on their configuration and the size of their links and wrist joints, robots can reach a collection of points called a workspace.
Payload is the weight a robot can carry and still remain within its other specification.
What is Kinematics
What is Kinematics
Example 1
Preliminary
WF is a universal coordinate frame.
JF is used to specify movements of each individual joint of the robot.
HF specifies movements of the robot’s hand relative to a frame attached to the hand
Preliminary
Two frames coincide ==>
Preliminary
Properties: Dot Product
Let and be arbitrary vectors in and be the angle from to , then
Preliminary
Preliminary
Basic Rotation
, , and represent the projections of onto OX, OY, OZ axes, respectively
Since
Preliminary
Preliminary
Basic Rotation Matrices
Preliminary
<== 3X3 identity matrix
Example 2
A point is attached to a rotating frame, the frame rotates 60 degree about the OZ axis of the reference frame. Find the coordinates of the point relative to the reference frame after the rotation.
Example 3
A point is the coordinate w.r.t. the reference coordinate system, find the corresponding point w.r.t. the rotated OU-V-W coordinate system if it has been rotated 60 degree about OZ axis.
Composite Rotation Matrix
rules:
if rotating coordinate O-U-V-W is rotating about principal axis of OXYZ frame, then Pre-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix
if rotating coordinate OUVW is rotating about its own principal axes, then post-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix
Example 4
Post-multiply if rotate about the OUVW axes
Pre-multiply if rotate about the OXYZ axes
Coordinate Transformations
position vector of P in {B} is transformed to position vector of P in {A}
description of {B} as seen from an observer in {A}
Rotation of {B} with respect to {A}
Translation of the origin of {B} with respect to origin of {A}
Coordinate Transformations
2. Rotation only
Homogeneous Representation
Homogeneous transformation matrix
Homogeneous Transformation
Special cases
1. Translation
2. Rotation
Example 5
O, O’
Example 6
Example 7
Find the homogeneous transformation matrix (T) for the following operations:
Homogeneous Representation
Principal axis n w.r.t. the reference coordinate system
(X’)
(y’)
(z’)
Chain product of successive coordinate transformation matrices
?
Example 8
For the figure shown below, find the 4x4 homogeneous transformation matrices and for i=1, 2, 3, 4, 5
Can you find the answer by observation based on the geometric interpretation of homogeneous transformation matrix?
a
b
c
d
e
Orientation Representation
Rotation matrix representation needs 9 elements to completely describe the orientation of a rotating rigid body.
Any easy way?
Euler Angles Representation
Euler Angle I Euler Angle II Roll-Pitch-Yaw
Sequence about OZ axis about OZ axis about OX axis
of about OU axis about OV axis about OY axis
Rotations about OW axis about OW axis about OZ axis
Euler Angle I, Animated
Euler Angle I
Euler Angle II, Animated
Note the opposite (clockwise) sense of the third rotation, f.
Quiz: How to get this matrix ?
X
Y
Z
Thank you!
Due: Sept. 16, 2008, before class
Next class: kinematics II

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241031852 Kinematics of Robots