The Cartesian Coordinate System

C H A P T E R 7

Throughout the CNC process of CAD, CAM, and tool referencing, some type of coordinate system must be used to keep track of referencing. What takes place on the computer should match that of what occurs at the physical ma-chine. Most readers will recall this coordinate system from one or more basic mathematics courses taken in the past. It is the Cartesian coordinate system that is typically used in the types of applications discussed throughout this book. There are other coordinate systems widely used in multi-axis robotics applications (kinematics, polar) that are not addressed in this text. If you are a person who has difficulty with math, please do not fret! You will not need to understand or compute any algebra or trigonometric equations. In fact, all that is involved is some very basic geometry.

Figure 7-1 shows a two-dimensional (i.e., 2D) representation of the Carte-sian coordinate system. Particular points of interest are that the X and Y lines that are orthogonal (meaning, at 90 degrees to each other). Also note that where the lines of intersection cross is defined to have values of X = 0 and Y = 0.

Figure 7-2 shows the same 2D Cartesian coordinate system with a bit more information superimposed. Note that the intersection of the X and Y axes are the "zero" points for each axis, respectively. Hence, any values to the right of the Y axis will have a positive X value. Conversely, any values above the X axis will have a positive Y value. By definition, the Cartesian coordinate system is said to have four "Quadrants" (again, refer to Fig. 7-2). Arbitrary X and Y point values have been selected in each of the quadrants to show their corresponding X and Y numerical values. It is also customary to refer to the coordinate axes in a certain "order." That being X, then Y, and, finally, Z. There are actually several more possible axis designators, but we

FIGURE 7 - 1 Defining the coordinate system.

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FIGURE 7 - 2 Coordinate system with quadrants and sample points.

will restrict our discussions to three dimensions. By using the X, Y, Z pattern it is not always necessary to denote the axis by its letter. A common shorthand method is to just use the numerical values for each axis while still adhering to the same pattern. For example, one way to represent the coordinate point of X = 4, Y = 7, and Z = 2, with the shorthand notation, would be <4,7 ,2>. Please refer back to Fig. 7-2 for various examples of this notation. Also pay particular attention to the fact that at least one of the X or Y values in quadrants II, III, and IV have at least one negative location value, whereas all coordinate values in quadrant I are positive values.

The most common physical topology of CNC-based implementations can be generalized by using either the mill or lathe architecture. From a gener-alized perspective to aid in the understanding of the Cartesian coordinate system, the term mill can be referred to as a metal working mill, but also a table router, plasma cutting table, water jet table, laser table, or even an XYZ pick and place machine. The thing that is common to all of the aforemen-tioned systems is that they all are based upon three sets of orthogonally (i.e., right-angle) placed axes. The addition of the third axis is visualized as a line extending at right angles through the intersection of the XY plane and is at right angles to this plane. Positive Z values are determined from the right-hand rule. Using your right hand, extend your four fingers in the positive X direction. Then, curl your fingers into the positive Y direction. Your thumb will now be pointing in the positive Z direction of the coordinate system. There is more discussion on the Z axis later on.

Although it is altogether possible to define the X and Y zero-points of your CNC tabletop as being directly in the center of the quadrants as shown in

The Table or Mill Topology

FIGURE 7 - 3 Showing typical XY zero locations on a table.

T h e C a r t e s i a n C o o r d i n a t e S y s t e m 1 3 1

FIGURE 7 - 4 T h e t w o most common Z zero locations.

Figs. 7-1 and 7-2, it is universally accepted to adopt a "quadrant I" coordi-nate strategy. By defining the lower-left corner of the CNC work table as the X = 0 and Y = 0 being your reference point, this allows the CNC operator and programmer to work with positive values of both X and Y. (Please see Fig. 7-3.) (Important note: Although any point on the table can be defined as the zero reference point, we will adopt the "lower-left" corner as our X, Y zero reference point throughout this book unless mentioned otherwise.) Both rectangles shown in Fig. 7-3 are valid layouts as they both use the lower-left corner as a zero reference point.

Now that you are familiar with the two-dimensional coordinate system and the X, Y zero location as a reference point, all that remains in our simple 3D model is the addition of the Z axis. In practice, the zero reference point for the Z plane of motion is defined by the user (i.e., programmer and operator) at the time the machine code file is generated. Generally, there are two zero reference points used: The top of the material to be worked or the top of the table surface, usually referred to as the spoil board. It is important to note that the top of the spoil-board surface in actuality is the bottom of the material. Novice users typically have an easier time learning the concept of zeroing the Z axis (i.e., tip of tool cutter) at the top of the material to be machined. Hence, any Z axis values that are negative will be in the negative Z direction and will be cutting into the material. Conversely, any positive Z axis values will be above the material to be machined. (Please refer to Fig. 7-4 for a graph-ical representation of both Z zero points commonly used.) Just as knowing where your zero reference point is located in the X and Y planes, it is equally important to understand where your "Z zero" reference is located.

By using the table or mill topology, we can now extrapolate this same co-ordinate convention system to describe a rotary or lathe topology. Referring to Fig. 7-5, you will notice that a basic lathe configuration consists of two

Lathe/Rotary Topology

FIGURE 7 - 5 A top -down view of a lathe showing the axes.

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FIGURE 7-6 Lathe view showing possible X and Z zeroing locations.

orthogonal sets of axes (i.e., the X and Z). Unfortunately, there are not any typical and straightforward zeroing points used on the lathe as compared to the mill topology. The most generalized zero point along the Z axis is usu-ally the point where the tip of the cutting tool abuts the end of the stock being turned. With regard to the X zero point, the two most common zeroing points are located either at the center line of the rotating stock or at the perimeter of the rotating stock. (See Fig. 7-6 for detail on the zeroing points located on a lathe.) Note that the lathe chuck is holding the rotating stock. It is customary to denote the longer axis that is inline with the spinning material to be the Z axis. The shorter axis is referred to as the X and it is placed at a 90 degree angle to the Z.