3 Mechanical systems
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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

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3 mechanical systems 3 1 pose description and transformation

3 Mechanical systems

3.1 pose description and transformation

3.1.1 Description of coordinate system


Example 1

Example 1


Preliminary

Preliminary

  • Robot Reference Frames

    • World frame

    • Joint frame

    • Tool frame

T

P

W

R


3 mechanical systems 3 1 pose description and transformation

O, O’

  • Coordinate Transformation

    • Reference coordinate frame OXYZ

    • Body-attached frame O’uvw

Point represented in OXYZ:

Point represented in O’uvw:

Two frames coincide ==>


3 mechanical systems 3 1 pose description and transformation

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


3 mechanical systems 3 1 pose description and transformation

  • Coordinate Transformation

    • Rotation only

How to relate the coordinate in these two frames?


3 mechanical systems 3 1 pose description and transformation

  • Basic Rotation

    • , , and represent the projections of onto OX, OY, OZ axes, respectively

    • Since


3 mechanical systems 3 1 pose description and transformation

  • Basic Rotation Matrix

    • Rotation about x-axis with


3 mechanical systems 3 1 pose description and transformation

  • Is it True?

    • Rotation about x axis with


Basic rotation matrices

Basic Rotation Matrices

  • Rotation about x-axis with

  • Rotation about y-axis with

  • Rotation about z-axis with


3 mechanical systems 3 1 pose description and transformation

  • Basic Rotation Matrix

    • Obtain the coordinate of from the coordinate of

Dot products are commutative!

<== 3X3 identity matrix


Example 2

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.


Composite rotation matrix

Composite Rotation Matrix

  • A sequence of finite rotations

    • matrix multiplications do not commute

    • 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


Coordinate transformations

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}


3 mechanical systems 3 1 pose description and transformation

  • Two Special Cases

    1. Translation only

    • Axes of {B} and {A} are parallel

      2. Rotation only

    • Origins of {B} and {A} are coincident


Homogeneous representation

Homogeneous Representation

  • Coordinate transformation from {B} to {A}

  • Homogeneous transformation matrix

Rotation matrix

Position vector

Scaling


Homogeneous transformation

Homogeneous Transformation

  • Special cases

    1. Translation

    2. Rotation


Example 3

Example 3

  • Rotation about the X-axis by


Homogeneous transformation1

Homogeneous Transformation

  • Composite Homogeneous Transformation Matrix

  • Rules:

    • Transformation (rotation/translation) w.r.t (X,Y,Z) (OLD FRAME), using pre-multiplication

    • Transformation (rotation/translation) w.r.t (U,V,W) (NEW FRAME), using post-multiplication


Homogeneous representation1

Homogeneous Representation

  • A frame in space (Geometric Interpretation)

(z’)

(y’)

(X’)

Principal axis n w.r.t. the reference coordinate system


Homogeneous transformation2

Homogeneous Transformation

  • Translation


3 mechanical systems 3 1 pose description and transformation

?

Composite Homogeneous Transformation Matrix

Transformation matrix for adjacent coordinate frames

Chain product of successive coordinate transformation matrices


Orientation representation

Orientation Representation

  • Rotation matrix representation needs 9 elements to completely describe the orientation of a rotating rigid body.

  • Any easy way?

Euler Angles Representation


3 mechanical systems 3 1 pose description and transformation

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 Angles Representation ( , , )

    • Many different types

    • Description of Euler angle representations


Euler angle i animated

Euler Angle I, Animated

w'=

z

w'"=

w"

f

v'"

v "

v'

y

u'"

u'

=u"

x


3 mechanical systems 3 1 pose description and transformation

  • Euler Angle I


Euler angle i

Euler Angle I

Resultant eulerian rotation matrix:


Euler angle ii animated

Euler Angle II, Animated

w'=

z

w"

w"'=

v"'

v'

=v"

y

u"'

u"

Note the opposite (clockwise) sense of the third rotation, f.

u'

x


3 mechanical systems 3 1 pose description and transformation

  • Matrix with Euler Angle II

Quiz: How to get this matrix ?


3 mechanical systems 3 1 pose description and transformation

Z

  • Description of Roll Pitch Yaw

Y

X

Quiz: How to get rotation matrix ?


3 mechanical systems 3 1 pose description and transformation

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 mechanical systems 3 1 pose description and transformation

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 mechanical systems 3 1 pose description and transformation

3.2.2 The four-bar chain


3 mechanical systems 3 1 pose description and transformation

3.2.3 The slider-crank mechanism


3 mechanical systems 3 1 pose description and transformation

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.


3 mechanical systems 3 1 pose description and transformation

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 mechanical systems 3 1 pose description and transformation

3.2.4 cams


3 mechanical systems 3 1 pose description and transformation

Cams – design criteria

Law of motion for the follower: position, velocity, acceleration and jerk

Machining problems: undercut

Pressure angle


3 mechanical systems 3 1 pose description and transformation

3.2.5 Gear trains


3 mechanical systems 3 1 pose description and transformation

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.


3 mechanical systems 3 1 pose description and transformation

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 . .


3 mechanical systems 3 1 pose description and transformation

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.


3 mechanical systems 3 1 pose description and transformation

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.


3 mechanical systems 3 1 pose description and transformation

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 mechanical systems 3 1 pose description and transformation

3.2.6 Ball screws


3 mechanical systems 3 1 pose description and transformation

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.


3 mechanical systems 3 1 pose description and transformation

Characteristics

•high transfer efficiency

•high positioning accuracy

•reversibility

•long service life

•good synchronization


3 mechanical systems 3 1 pose description and transformation

Installation modes

F—0

F—F

F—F

J—J


3 mechanical systems 3 1 pose description and transformation

3.2.7 Belt and chain drives


3 3 oriented mechanisms

3.3 Oriented mechanisms


3 mechanical systems 3 1 pose description and transformation

Linear oriented Guider


3 mechanical systems 3 1 pose description and transformation

Configuration combination


3 mechanical systems 3 1 pose description and transformation

3.4 Execute mechanisms

Mechanical Equipment

Electrical Equipment

Hydraulic Equipment

homework: page 115, problem 5,8


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