3 Mechanical systems 3.1 pose description and transformation 3.1.1 Description of coordinate system. Example 1. Preliminary. Robot Reference Frames World frame Joint frame Tool frame. T. P. W. R. O, O’. Coordinate Transformation Reference coordinate frame OXYZ
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3 Mechanical systems
3.1 pose description and transformation
3.1.1 Description of coordinate system
T
P
W
R
O, O’
Point represented in OXYZ:
Point represented in O’uvw:
Two frames coincide ==>
Mutually perpendicular
Unit vectors
Properties: Dot Product
Let and be arbitrary vectors in and be the angle from to , then
Properties of orthonormal coordinate frame
How to relate the coordinate in these two frames?
Dot products are commutative!
<== 3X3 identity matrix
Rotation of {B} with respect to {A}
Translation of the origin of {B} with respect to origin of {A}
1. Translation only
2. Rotation only
Rotation matrix
Position vector
Scaling
1. Translation
2. Rotation
(z’)
(y’)
(X’)
Principal axis n w.r.t. the reference coordinate system
?
Composite Homogeneous Transformation Matrix
Transformation matrix for adjacent coordinate frames
Chain product of successive coordinate transformation matrices
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
w'=
z
w'"=
w"
f
v'"
v "
v'
y
u'"
u'
=u"
x
Resultant eulerian rotation matrix:
w'=
z
w"
w"'=
v"'
v'
=v"
y
u"'
u"
Note the opposite (clockwise) sense of the third rotation, f.
u'
x
Quiz: How to get this matrix ?
Z
Y
X
Quiz: How to get rotation matrix ?
3.2 Transmission mechanisms
Mechanisms are motion converters in that they transform motion from one form to some required form.
Machine is a system that transmits or modifies the action of a force or torque to do useful work. (involving transmitting motion and force or torque)
Kinematics is the term used for the study of motion without regard to forces.
3.2.1 kinematic chains
Link is a part of a mechanism which has motion relative to some other part.
Joint is the point of attachment to other link which allows the links attached to have relative movements among each other.
Kinematic chain is a sequence of joints and links.
3.2.2 The four-bar chain
3.2.3 The slider-crank mechanism
The slider-crank mechanism is an extremely cost-effective means of converting rotary to linear motion.
The crank portion is the wheel that rotates about its center and has a rod of fixed lenght mounted to a point on its circumference; the other end of the connecting rod is attached to a linear stage which is constrained to move in only one dimension on a relatively frictionless surface.
At both its location the connecting rod is free to rotate thus the angle formed with the horizontal will change as a function of the disk’s position.
As the disk travels from 0 to 180° in the counterclockwise direction, the linear stage moves a distance equal to 2r: if the disk continues to travle from 180° back to 0° - still in counterclockwise direction, the load will move in the opposite direction over exactly the same linear distance.
3.2.4 cams
Cams – design criteria
Law of motion for the follower: position, velocity, acceleration and jerk
Machining problems: undercut
Pressure angle
3.2.5 Gear trains
Gear Train Ratio
The train ratio of a gear train is the ratio of the angular velocities of input and output members in the gear train. The train ratio here includes two factors, the magnitude and the relative rotating direction of the two members.
Advantages
Gear mechanisms are widely used in variety fields. They can be used to transmit motion and power between two any spatial rotating shafts and there are many advantages for them, such as, big range power, high transmission efficiency, exact transmission ratio, long life and reliable performance etc . .
Harmonic Drive
Harmonic Drive is a mechanical device for transmission of motion and power, which is based upon a unique principle of controlled elastic deformations of some thin-walled elements.
Components
The three basic components of a Harmonic Drive are: wave generator, flexspline and circular spline. Any one of the three components may be fixed, while one of the remaining two may be the driver, and the other the driven, to effect speed increase or speed decrease and at fixed speed ratio. Or, two of the three components may be the driver while the third the driven so as to effect differential transmission.
Characteristc
High reduction ratio with wide range
High precision
Small backlash
large torque capacity
High efficiency
Small size and light weight
Smooth running
Low noise
3.2.6 Ball screws
Ball screw assembly is consisted of screw, nut and ball. The function is transfer the rotary motion into linear motion or transfer the linear motion into rotary motion.
We have a great variety of ball screw at high performance, cost effective, and extensive for machine tool, production machinery, precision instrument, promote the CNC development of machine tools thoroughly. Our products have the great feature of long operating life, high accuracy to C3 and C5, high transfer efficiency and good synchronization, fast delivery on many models.
Characteristics
•high transfer efficiency
•high positioning accuracy
•reversibility
•long service life
•good synchronization
Installation modes
F—0
F—F
F—F
J—J
3.2.7 Belt and chain drives
Linear oriented Guider
Configuration combination
3.4 Execute mechanisms
Mechanical Equipment
Electrical Equipment
Hydraulic Equipment
homework: page 115, problem 5,8