This document provides an overview and summary of a term project report on approximation in 2D CNC motion. The report discusses the history and basic components of numerical control machines, including lathes, mills, and 3D printers. It describes how 2D motion is achieved through stepper motors and algorithms for linear and circular interpolation. Examples are provided and references are listed at the end.

1. Term Project Report
Industrial Building & Automation Systems
(Fall 2016-2017)
Approximation in 2D CNC Motion
Prepared by:
Subkhiddin Mukhidinov
Department of Mechanical Engineering
Piraeus University of Applied Sciences
Athens, January 2017

2. Table of Contents
 Introduction
 History
 Basic Components of NC Machines
 Computer Numerical Controlled Machines
 Lathe
 Mill
 3D Printer
 2D (x-y) Motion
 Stepper Motor
 Algorithm to Approximate
 Linear Interpolation
 Circular Interpolation
 Examples
 References

3. INTRODUCTION
Numerical control can be defined as a technique of controlling a machine
tool by the direct insertion of numerical data at some point of the system. The
functions that are controlled on the machine tool are displacement of the slide
members, spindle speeds, tool selection etc. At first, the numerical control was
used to produce geocentrically complex parts but later used for added efficiency
in medium batch production of turned and milled parts presently, Numerical
control is employed in all sectors of production.
Rapid development in the field ofelectronics suchas integrated circuit, large scale
integrated circuits and development of minicomputer lead to the development of
minicomputers based CNC systems. Further development and the electronic
“chip” revolution have ushered in the current generation “compactand powerful”
Microprocessor based CNC systems.
Development of computer numerically controlled (CNC) machines is an
outstanding contribution to the manufacturing industries. It has made possiblethe
automation of the machining process with flexibility to handle small to medium
batch of quantities in part production.
Initially, the CNC technology was applied on basic metal cutting machine like
lathes, milling machines, etc. Later, to increase the flexibility of the machines in
handling a variety of components and to finish them in a single setup on the same
machine, CNC machines capable of performing multiple operations were
developed. To start with, this concept was applied to develop a CNC machining
centre for machining prismatic components combining operations like milling,
drilling, boring and taping. Further, the concept of multi-operations was also
extended for machining cylindrical components, which led to the development of
turning centres.
Computer Numerical Control (CNC) is a specialized and versatile form of Soft
Automation and its applications cover many kinds, although it was initially
developed to control the motion and operation of machine tools.
Computer Numerical Control may be considered to be a means of operating a
machine through the use of discrete numerical values fed into the machine, where
the required 'input' technical information is stored on a kind of input media such
as floppy disk, hard disk, CD ROM, DVD, USB flash drive, or RAM card etc.
The machine follows a predetermined sequence of machining operations at the
predetermined speeds necessary to produce a workpiece of the right shape and

4. size and thus according to completely predictable results. A different productcan
be produced through reprogramming and a low-quantity production run of
different products is justified.
History
The development of numerical control owes much to the United States air
force, which recognized the need to develop more efficient manufacturing
methods for modern aircraft. Following World War II, the components used to
fabricate jet aircraft became more complex and required more machining. Most
of the machining involved milling operations, so the Air Force sponsored
a research project at Massachusetts Institute ofTechnology to develop a prototype
NC milling machine. This prototypewas produced by retrofitting a conventional
tracer mill with numerical control servomechanisms for the three axes of the
machine. In March 1952, the MIT Labs held the first demonstration of the NC
machine. The machine tool builders gradually began developing their
own projects to introduce commercial NC units. Also, certain industry users,
especially airframe builders, worked to devise numerical control machines to
satisfy their own particular production needs. The Air force continued its
encouragement of NC development by sponsoring additional research at MIT to
design a part programming language that could be used in controlling N.C.
machines.

5. Basic Components of NC System
An operational numerical controlsystem consists of the following three basic
components:
1. Program of instructions.
2. Controller unit, also called machine tool unit.
3. Machine toolor other controlled process.
The program of instructions serves as input to the controller unit, which in turn
commands the machine tool or other process to be controlled
Programof Instructions
The program of instructions is the detailed step by step set of instructions
which tell the machine what to do. It is coded in numerical or symbolic form on
some type of input medium that can be interpreted by the controller unit. The
most common one is the 1-inch-wide punched tape. Over the years, other forms
of input media have been used, including punched cards, magnetic tape, and
even 35mm motion picture film.
There are two other methods of input to the NC system which should be
mentioned. The first is by manual entry of instructional data to the controller
unit. This is time consuming and is rarely used
except as an auxiliary means of controlor when one or a very limited no. of
parts to be made. The second method of input is by means of a direct link with
the computer. This is called direct numerical
control, or DNC.

6. Controller Unit
The second basic component of NC system is the controller unit. This
consists of electronics and hardware that read and interpret the program of
instructions and convert it to mechanical actions of the machine tool. The typical
elements of the controller unit include the tape reader, a data buffer, signal
output channels to the machine tool, and the sequence controls to coordinate the
overall operation of the foregoing elements.
The tape reader is an electrical-mechanical device for the winding and reading
the punched tape containing the program of instructions. The signal output
channels are connected to the servomotors and other controls in machine tools.
Most N.C. tools today are provided with positive feedback controls for this
purposeand are referred as closed loop systems. However there has been growth
in the open loop systems which do not make use of feedback signals to the
controller unit. The advocates of the open loop conceptclaim that the reliability
of the system is great enough that the feedback controls are not needed.
Machine Tool
The third basic component of an NC system is the machine tool or other
controlled process. It is part of the NC system which performs useful work. In
the most common example of an NC system, one designed to perform
machining operations. The machine tool consists of the worktable and spindle as
well as the motors and controls necessary to drive them. It also includes the
cutting tools, work fixtures and other auxiliary equipment needed in machining
operation.

7. Descriptionof NC Machines
Motion is controlled along multiple axes, normally at least two (X and
Y), and a toolspindle that moves in the Z (depth). The position of the tool is
driven by direct-drive stepper motor or servo motors in order to provide highly
accurate movements, or in older designs, motors through a series of step down
gears. Open-loop control works if the forces are kept small enough and speeds
are not too great. On commercial metalworking machines, closed loop controls
are standard and required to provide the accuracy, speed,
and repeatability demanded.
As the controller hardware evolved, the mills themselves also evolved.
One change has been to enclose the entire mechanism in a large box as a safety
measure, often with additional safety interlocks to ensure the operator is far
enough from the working piece for safe operation. Most new CNC systems built
today are 100% electronically controlled.
CNC-like systems are now used for any process that can be described as a
series of movements and operations. These include laser cutting, welding,
friction stir welding, ultrasonic welding, flame and plasma cutting, bending,
spinning, hole-punching, pinning, gluing, fabric cutting, sewing, tape and fibre
placement, routing, picking and placing, and sawing.
Mills
CNC milling is a specific form of computer numerical controlled (CNC)
machining. Milling itself is a machining process similar to both drilling and
cutting, and able to achieve many of the operations performed by cutting and
drilling machines. Like drilling, milling uses a rotating cylindrical cutting tool.
However, the cutter in a milling machine is able to move along multiple axes,
and can create a variety of shapes, slots and holes. In addition, the work-piece is
often moved across the milling tool in different directions, unlike the single axis
motion of a drill.
CNC milling devices are the most widely used type of CNC machine.
Typically, they are grouped by the number of axes on which they operate, which
are labelled with various letters. X and Y designate horizontal movement of the
work-piece (forward-and-back and side-to-side on a flat plane). Z represents
vertical, or up-and-down, movement, while W represents diagonal movement
across a vertical plane. Most machines offer from 3 to 5 axes, providing
performance along at least the X, Y and Z axes. Advanced machines, such as 5-

8. axis milling centres, require CAM programming for optimal performance due to
the incredibly complex geometries involved in the machining process.These
devices are extremely useful because they are able to produceshapes that would
be nearly impossible using manual tooling methods. Most CNC milling
machines also integrate a device for pumping cutting fluid to the cutting tool
during machining.
Lathes
A metal lathe or metalworking lathe is a large class of lathes designed for
precisely machining relatively hard materials. They were originally designed to
machine metals; however, with the advent of plastics and other materials, and
with their inherent versatility, they are used in a wide range of applications, and
a broad range of materials. In machining jargon, where the larger context is
already understood, they are usually simply called lathes, or else referred to by
more-specific subtype names (toolroom lathe, turret lathe, etc.). These
rigid machine tools remove material from a rotating workpiece via the
(typically linear) movements of various cutting tools, such as toolbits and drill
bits.

9. The design of lathes can vary greatly depending on the intended
application; however, basic features are common to most types. These machines
consist of (at the least) a headstock, bed, carriage, and tailstock. Better machines
are solidly constructed with broad bearing surfaces (slide-ways) for stability,
and manufactured with great precision. This helps ensure the components
manufactured on the machines can meet the required tolerances and
repeatability.
3D Printers
3D Printing is an additive manufacturing process that creates a physical
object from a digital design. There are different 3D printing technologies and
materials you can print with, but all are based on the same principle: a digital
model is turned into a solid three-dimensional physical object by adding material
layer by layer.
Every 3D print starts as a digital 3D design file – like a blueprint – for a
physical object. Trying to print without a design file is like trying to print a
document ona sheet of paperwithout a text file. This design file is sliced into thin
layers which is then sent to the 3D printer. From here on the printing process
varies by technology, starting from desktop printers that melt a plastic material
and lay it down onto a print platform to large industrial machines that use a laser

10. to selectively melt metal powderat high temperatures. The printing can take hours
to complete depending on the size, and the printed objects are often post-
processed to reach the desired finish.
Available materials also vary by printer type, ranging from plastics to rubber,
sandstone, metals and alloys - with more and more materials appearing on the
market every year.
Although 3D printing is commonly thought of as a new ‘futuristic’ concept, it has
actually been around for more than 30 years. Chuck Hull invented the first 3D
printing process called ‘stereo lithography’ in 1983. In a patent, he defined stereo
lithography as ‘a method and apparatus for making solid objects by successively
“printing” thin layers of the ultraviolet curable material one on top of the other’.
This patent only focuses on ‘printing’ with a light curable liquid, but after Hull
founded the company ‘3D Systems’, he soon realized his technique was not
limited to only liquids, expanding the definition to ‘any material capable of
solidification or capable of altering its physical state’. With this, he built the
foundation of what we now know today as additive manufacturing (AM) – or 3D
printing.

11. X-Y Motions
A microcomputer can be used to controlthe motion of numerical control
machines. This article describes a straightforward method for approximating
diagonal lines and circular motion on an XY plane.
Many numerical controlmachines are powered by stepping motors. When
a pulse is sent to a stepping motor, the stepping motor alters its position by a
unit step. Two motors can be used to controlthe XY movements of an arm or
tool over a working plane.
If the pulses are generated by a device which can remember or generate a
specified train of pulses, repetitive operations such as grinding, painting, or
cutting can be performed hundreds of times with virtually no variation. A
microcomputer is an obvious choice to generate and remember the pulses.
Since stepper motors can move only in discrete steps, we must
approximate the actual curve by a series of small XY motions. Many algorithms
rely upon parametric functions such as sine and cosine to perform the necessary
calculations. Parametric functions, however, typically require a high degree of
numeric precision. Calculating sine and cosine values with a microcomputer can
be too time-consuming to be useful in a real-time application.
The following two algorithms require no parametric functions. This
makes them ideally suited to the computation and memory capacities Kenneth
and Melvin Goldberg 3913 Pine Street, Apt. F Philadelphia, Pennsylvania 19104
of microcomputers. Since these algorithms do not require a large amount of
complex mathematical calculation, they are fast enough to be used in real-time
applications. The program shown in listing 1 is written in C for fast execution,
portability, and ease of modification. The program, as shown, does not actually
control any stepping motors; rather, it provides a screen display consisting of +1,
-1, and 0. In an actual control situation, these values would be transferred to
stepper motors or a graphic display.

12. Stepper Motor
A stepper motor or step motor or stepping motor is a brushless DC electric
motorthat divides a full rotation into anumber ofequal steps. Themotor's position
can then be commanded to move and hold at one of these steps without any
feedback sensor (an open-loop controller), as long as the motor is carefully sized
to the application in respect to torque and speed.
Stepper motors effectively have multiple "toothed" electromagnets
arranged around a central gear-shaped piece of iron. The electromagnets are
energized by an external driver circuit or a micro controller. To make the motor
shaft turn, first, one electromagnet is given power, which magnetically attracts the
gear's teeth. When the gear's teeth are aligned to the first electromagnet, they are
slightly offset from the next electromagnet. This means that when the next
electromagnet is turned on and the first is turned off, the gear rotates slightly to
align with the next one. Fromthere the process is repeated. Each of thoserotations
is called a "step", with an integer number of steps making a full rotation. In that
way, the motor can be turned by a precise angle.
Types
There are three main types of stepper motors:
1. Permanent magnet stepper
2. Hybrid synchronous stepper
3. Variable reluctance stepper
Permanent magnet motors use a permanent magnet (PM) in the rotor and operate
on the attraction orrepulsion between the rotorPM and the stator electromagnets.
Variable reluctance (VR) motors have a plain iron rotor and operate based on the
principle that minimum reluctance occurs with minimum gap, hence the rotor
points are attracted toward the stator magnet poles.

13. System
A stepper motor system consists of three basic elements, often combined
with some type of user interface (host computer, PLC or dumb terminal):
Indexers
The indexer (or controller) is a microprocessor capable of generating step
pulses and direction signals for the driver. In addition, the indexer is typically
required to perform many other sophisticated command functions.
Drivers
The driver (or amplifier) converts the indexer command signals into the power
necessary to energize the motor windings. There are numerous types of drivers,
with different voltage and current ratings and construction technology. Not all
drivers are suitable to run all motors, so when designing a motion control system,
the driver selection process is critical.
Stepper motor
The steppermotor is an electromagnetic device that converts digital pulses
into mechanical shaft rotation. Advantages of step motors are low cost, high
reliability, high torque at low speeds and a simple, rugged construction that
operates in almost any environment. The main disadvantages in using a stepper
motor is the resonance effect often exhibited at low speeds and decreasing torque
with increasing speed.

14. Linear Interpolation
Approximating diagonal lines with unit steps in two dimensions can be
accomplished with the following algorithm. 1. Define the starting position (Xl,
Y1) and ending position (X3, Y3). Define the feed rate (f). Feed rate is the speed
at which the tool being controlled moves. The Electronic Industries Association
(EIA) recommends the following notation for linear interpolation:
G01 (X1, Y1, X3, Y3) f
where
Xl, Y1 is starting position
X3, Y3 is ending position
f is feed rate
2. Initialize variables. Setthe current relative position (X2, Y2) ofthe toolto (0,0).
This effectively sets the current tool position to the starting point. Set the step
count number to zero.
3. Calculate the direction in which to move the tool. When following a straight
line from one point to another, all X motion is in the same direction, as is all Y
motion. The direction is determined by the sign of the difference (DX and DY)
between the ending and the starting positions. DX=X3-Yl. DY=Y3-Yl.
4. Calculate the difference between the absolute values of DX and DY. This
determines FXY, a variable which is used to control the movements along the X
and Y axes. FXY=DX-DY.
5. Generate output pulses to move the tool until the endpoint is reached. This is
the heart of the program. The proportionate stream of XY pulses is generated by
manipulating variable FXY. Each time a step is taken in the X direction, the
absolute value of DY is subtracted from FXY. When FXY becomes negative, a
step is taken in the Y direction, and the absolute value of DX is added to FXY.
The sign ofFXY determines the appropriate step needed to approximate a straight
line.
6. A delay loop controls the feed rate. This loop may include extra delay for the
initial steps. "Ramping up" the feed rate in this manner is useful in real-world
situations where the inertia of a machine may have a significant effect on the

15. system. Table 1 shows the output generated when a starting point of (0,0), an
ending point of (3, -7), and a feed rate of 300 are given to the program shown in
listing 1.
Circular Interpolation
A conceptually similar nonparametric algorithm can provide the necessary
XY steps for approximating a circular path. The equation for a circle is:
FXY = X2 + Y2 – R2
FXY = positive when (X, Y) is outside circle
0 when (X, Y) is on circumference
negative when (X, Y) is inside circle
DX = 2X
DY = 2Y
The variable FXY determines the direction in which the tool is moved at
each point on the circle. The motion is always perpendicular to the instantaneous
circular radius. The tangent to a circle is always perpendicular to the radius. The
X and Y components of the radius are defined by the partial derivatives of FXY.
We propose to step the machine tool around the circle by comparing the
current tool position to the ideal radius. We perform this comparison by tracking
the value of FXY. We know that the tool has crossed the circumference and must

16. be corrected when the sign of FXY changes. The appropriate correction (±X, ±Y)
depends on the quadrant in which the tool is located.
This algorithm's simplicity lies in the fact that the only information required
to determine the properoutputis the sign ofFXY, its derivatives, and the direction
of rotation. (0= clockwise, 1= counter clockwise). If we denote positive by 1 and
negative by 0, then we can organize the 16 possible combinations of values as
shown in table 2. Table 3 shows the output generated for a typical circular
approximation.
Figure 1 demonstrates the raggedness found in part of an enlarged path of
a typical circle. The raggedness is greatly decreased in large diameter circles.
With a radius of 1000 steps, a circle will appear smooth to the naked eye.

17. REFERENCES
1. Introduction to Numerical ControlMachining
http://www.mfg.mtu.edu/cyberman/machtool/auto/nc/intro.html
2. Numerical Control
https://en.wikipedia.org/wiki/Numerical_control#History
3. 3D Printers
https://www.3dhubs.com/what-is-3d-printing
4. XY Interpolation Algorithms – UC Berkeley
http://goldberg.berkeley.edu/pubs/XY-Interpolation-Algorithms
5. Stepper Motors
https://en.wikipedia.org/wiki/Stepper_motor#Types