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

3.1.1 Description of coordinate system

Example 1


  • Robot Reference Frames

    • World frame

    • Joint frame

    • Tool frame





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 ==>

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

  • Coordinate Transformation

    • Rotation only

How to relate the coordinate in these two frames?

  • Basic Rotation

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

    • Since

  • Basic Rotation Matrix

    • Rotation about x-axis with

  • Is it True?

    • Rotation about x axis with

Basic Rotation Matrices

  • Rotation about x-axis with

  • Rotation about y-axis with

  • Rotation about z-axis with

  • Basic Rotation Matrix

    • Obtain the coordinate of from the coordinate of

Dot products are commutative!

<== 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.

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

  • 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}

  • 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

  • Coordinate transformation from {B} to {A}

  • Homogeneous transformation matrix

Rotation matrix

Position vector


Homogeneous Transformation

  • Special cases

    1. Translation

    2. Rotation

Example 3

  • Rotation about the X-axis by

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 Representation

  • A frame in space (Geometric Interpretation)




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

Homogeneous Transformation

  • Translation


Composite Homogeneous Transformation Matrix

Transformation matrix for adjacent coordinate frames

Chain product of successive coordinate transformation matrices

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 Angles Representation ( , , )

    • Many different types

    • Description of Euler angle representations

Euler Angle I, Animated







v "







  • Euler Angle I

Euler Angle I

Resultant eulerian rotation matrix:

Euler Angle II, Animated











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



  • Matrix with Euler Angle II

Quiz: How to get this matrix ?


  • Description of Roll Pitch Yaw



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.


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.


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.


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.


•high transfer efficiency

•high positioning accuracy


•long service life

•good synchronization

Installation modes





3.2.7 Belt and chain drives

3.3 Oriented mechanisms

Linear oriented Guider

Configuration combination

3.4 Execute mechanisms

Mechanical Equipment

Electrical Equipment

Hydraulic Equipment

homework: page 115, problem 5,8

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