# Absolute Positioning Kinematically

The accurate true positional location of a device and the accurate repeatability in the location of a device are two very different tasks. The simplest kinematic coupling will provide excellent repeatability in the location of the relationship between the two platforms of the Kinematic assembly in all six degrees of freedom. The accuracy of this relocation will be measured in Microinches or sub-micrometers.

One of the characteristic properties of all true kinematic couplings is that they are very forgiving. Some might say that all true kinematic couplings are by nature self aligning or self centering. When true positioning of a payload is not critical, this is an excellent quality to have. In situations where the true X-Y and Z positions of the payload is critical or more often, when the true X-Y and Z position of a large number of similar parts such as pallets is the task, the standard kinematic approach just will not do the job. Part of this difficulty is directly attributable to the excellent adaptability or self alignment of kinematic couplings. The two sections of a kinematic coupling will automatically find their mechanical equilibrium, so that true X-Y and Z positioning is made very difficult.

It is theoretically possible to achieve accurate true positioning of a subject in X-Y and Z, using any of the basic kinematic designs. This would be done by literally machining all of geometries of the component parts of the kinematic system and their exact location within micro inches (nanometers). The cost of doing this machining would be horrendous and the actual performance of the end product would be problematic. This statement is based the numerous efforts that we have made with real world projects. This broad approach just will not work.

There are six degrees of freedom in the relative movement of any mechanical body in relation to another body. There are two contact surfaces for each degree of freedom, so now we are talking about twelve surfaces and each one of these must be in true position in “X” in “Y” and in “Z”, so now we are talking 36 coordinate positions, held within micro inches (sub micrometers)! Only by physically constraining each and every one of the individual degrees of freedom to a specific Cartesian location can true positioning of an entire mechanism be achieved.

With this fundamental simplification of the problem, we can look at some of the methods to precisely locate each one of the six individual motions to a specific cartesion position. One of the important facts that must be addressed early on is that the cost of achieving any given task is always, and I do mean always, a key or even the key issue for choosing any design.

If we are going to succeed in accurately positioning the relationship between the two platforms in all six degrees of freedom and do it economically, we are going to have to be very clever in choosing a design that can actually be manufactured. To start out with, we will need to use some basic principals of physics, but we will use them in unique ways. Two basic conditions that we will use to our advantage are symmetry and uniformity. By choosing a modified Maxwell kinematic system, consisting of three vees and three spheres, instead the diverse components involved with the Kelvin kinematic system, we end up with perfect symmetry and we reduce the number of geometric forms to spheres, cylinders, and flats. As a philosophical principal, we will limit our designs, to the use of basic geometries that can be produced and inspected to near perfection. These near perfect parts will have to be produced, in quantity, economically.

A flat surface is one of those basic or fundamental geometries. A perfect flat or plane surface has an infinite radius of curvature. A flat surface of extreme accuracy can be generated with relative ease and it can be inspected absolutely, by comparing it with the very well known curvature of the earth’s surface with the help of reversal technology, or by using a plano-Intiferometer of known accuracy.

Technically a sphere has the same geometry as a flat plane but with a much smaller radius of curvature. A sphere can be mathematically generated, by basically simple machines that have been constructed to near mechanical perfection. This means a machine with a near perfect axis of rotation without any harmonic patterns in the drive with near zero vibration which is operating in a clean room under constant temperature conditions.

A sphere is the only geometry which has its entire surface area, equidistant from a single point in three dimensional space. The geometry of a sphere can be inspected absolutely. This is done by comparative axis of rotation evaluation using reversal technology. The sphere to be evaluated is rotated concentrically on the axis of rotation of a high quality spindle bearing. The path of the surface of the sphere is recorded, and the sphere is indexed 180° (this pattern of indexing is only one of a number of different arrangements that can be used) (See Donaldson-See Whitehead). The sphere is measured again. Any error of the sphere will move with the movement of the sphere. Any error of the spindle will remain in the same location. The absolute measurement of sphericity is somewhat more complex than this but this is a good general outline.

A cylinder has a some what more complex geometry than a sphere, but none the less it is a basic geometry. We can write a simple mathematical formula to describe it. Where a perfect sphere is a single point radius a perfect cylinder is generated by the rotation of a parallel line radius around a perfect axis of rotation. The radius must be constant and the line must be straight and parallel to the axis of rotation. Both of these properties can be measured absolutely, that is, they are traceable to nature. The cylindricity of a cylinder can be evaluated using the same axis of rotation and reversal technology method used for a sphere. A line around the bottom of the cylinder is aligned concentric to the axis of rotation of the spindle. The mounting plate holding the cylinder is then wobbled to bring the top of the cylinder concentric. A sample of measurements is then made up and down the cylinder.

A round, cylindrical part can be generated by smothering an ordinary quality cylindrical part with a generally cylindrical female lapping tool of good geometry, while the cylinder is rotated. The very round generally cylindrical form generated by this “ring lapping,” can then be rolled between two flat parallel plates, thus producing microinch perfect cylindrical parts. All of the parts in each batch of cylinders that are rotated together between the flat parallel plates will all be the same exact cylindrical diameter.

An important extension to the list of basic geometric shapes that can be produced to micro inch accuracy with relative easy is the truncated ball. This is another case where basic principles of physics are working with you, instead of against you. We can make almost perfect balls with relative easy. A perfect ball has one point in three dimensional space that is equidistant from all points on the surface so any flat machined on a ball, will automatically be exactly tangent to a parallel plane contacting the surface of the ball. The significance of this principal is that this is how a truncated ball will actually function in a kinematic coupling.

This makes a truncated ball one of those basic or natural forms that can be manufactured to near perfection and that can be produced economically. We have already discussed measuring the geometry of the ball absolutely. The absolute diameter of the ball can be accurately determined with considerable difficulty, but common diameters of a group of precision balls can be measured absolutely with relative ease, and a common ball diameter is all that we need.

Now, the only remaining challenge is to measure the height of the top of the ball from the truncated flat. Even this is made easy because we only need to confirm the consistency of this dimension for all of the balls used on a given program (any one set of pallets).

## The Tooling Ball Approach

One of the successful approaches to achieving exact positioning of the three spheres of a kinematic coupling in near perfect X-Y and Z locations is to use the “Tooling Ball” approach. We start out with an ultra-precise ball that has been drilled and threaded by Electrical Discharge Machining ( EDM ). This threaded ball is screwed into a close fitting spherical cup that is wetted with ceramic filled epoxy glue. This spherical cup is located on one end of the tooling ball blank. After screwing the ball into the cup, all excess epoxy glue is wiped away using a piece of absorbent material wetted in Isopropyl Alcohol.

After properly curing the glue, the cylindrical shank of the tooling ball is ground and lapped to an exact diameter. This cylinder is ground perfectly concentric to the point center of the ball. The adjacent flat annular shoulder is ground and lapped square to the cylinder and to an exact dimension, in relation to the point center of the sphere. This precise grinding and lapping is no small task, but it can be done. This task is achieved by using basic principles of physics and ultra precise grinding machines.

The next step is to jig bore or jig grind a three hole pattern that will depict the ideal X and Y locations of the tooling balls as perfectly as possible in one of the two Kinematic platforms. The substrata that these three holes are generated in should be lapped as flat and parallel as possible.

The top surface should be flat to provide as perfect a “Z” axis reference as possible. The rear surface should be as parallel as possible in order to provide a reference datum to locate this pallet on the Jig Borer or grinder table and to provide a vertical reference datum for locating the payload on the rear surface of the platform. The ends of these datum cylinders can incorporate extensive machining including minor diameters flats and threads to facilitate their use in accurately locating the payload. This approach will assure squareness of the finished cylindrical holes in relation to the top surface.

Some of the things that are working for you in this design are that the absolute size of the holes is not super critical. The location of the tooling balls is determined by the center lines of these holes and the fact that the holes will be an interference fit on the cylindrical shank of the tooling balls. The absolute size of these holes can vary a couple of ten thousandths of an inch with no ill effects.

When we are constructing Kinematic couplings that must depict true X-Y and Z positions within microinches, we simply cannot press fit cylinders into the holes accurately enough. Many years ago we conducted a study for the U.S. Air Force on how to accurately press fit a cylinder into a hole. It turns out that it simply can’t be done at least not to microinches.

During this program, we precision lapped cylindrical pins true within a couple of micro inches (0.05 micrometers). Then we honed and lapped the inside diameters of the holes almost perfect. When we pressed the lapped pins in the lapped holes the cylindrical axis of the pins moved all over the place.

What did finally work, was to heat up the body of the part and to deep freeze the pins before assembling them. The final refinement was to coat the pin with a thin layer of anhydrous lanolin. What occurres, is that as the two cylindrical interfaces returned to room temperature, a liquid film bearing develops, which self centers the cylindrical pin in the hole, perfectly. Over a twenty four hour period of time, the extreme pressure developed between the I.D. and the O.D. of the cylinders squeezes out all but a molecular layer of the lanolin from between the two cylindrical surfaces and the assembly became rigidly fixed.

## The Gear Plate Approach

An alternative to the tooling ball method for true positioning the three spherical contacts, of the Maxwell Kinematic system, in X-Y and Z, is to use the “gear plate” system. A gear plate consists of a metal plate with one surface lapped precisely flat. The thickness of this plate must be tall enough to come above the centerline of the truncated ball that it is constraining. A three hole pattern is precisely Jig Bored or Jig Ground through the gear plate. The flat lapped reference surface of the gear plate is placed against a flat plate that has been wobbled into be perpendicular to the jig boring or grinding spindle. As an alternative, a sub-plate mounted on the Jig Boring or Grinder table can be machined in place and that surface can be used for location.

The finished gear plate with the precise three hole pattern should be clamped to the flat lapped top surface of the pallet and pinned in three places. The balls usually used in the gear plat method are truncated and threaded balls as previously discussed. The holes in the Gear Plate should be machined the same diameter as the balls, or up to 0.0002 inches (.08mm) undersize.

## Flat Lapped Surface

In both the “tooling ball” and the “gear plate” techniques very flat lapped surfaces are involved. Tricks of the trade for economically lapping the flat and parallel surfaces dictate that these surfaces should have a limited area. This is done by using a series of balanced bosses on each side of the platform or by using two concentric annular rings, one on each side of the platform.

## The Three Vees

Following our basic principals, we will use two ultra precise cylinders to form each one of the three vees of the Maxwell system that are required to complete the coupling. The precise spacing and true positioning required between the two cylinders and the system is provided by two very precise right angle cylinders. The two horizontal cylinders are clamped in contact with the two vertical cylinders and the very flat surface of the pallet. Two 45 degree bar clamps, mounted in the centers, on the outer sides of the two horizontal cylinders are used to precisely clamp the two cylinders in place. This very simple and very inexpensive clamping device is extremely forgiving and not at all sensitive to the clamping force.

The top surface of the pallet used to locate the cylinders on should be lapped precisely flat, and the opposite surface should be lapped flat and parallel to this top surface. The “X” and “Y” locations of the vertical cylinders will very accurately position the axii of the Vee Blocks, formed by the pairs of cylinders. The precise location of these pairs of vertical cylinders is determined by sets of jig bored or jig ground holes in the pallet.

The length of some or all of the vertical cylinders can be made long enough to protrude above the back surface of the pallet to provide reference datum for locating the payload on the rear surface of the platform. The ends of these datum cylinders can incorporate extensive machining including minor diameters flats and threads to facilitate their use in accurately locating the payload. These pairs of cylinders are shrunk fit in the pallet just like the tooling balls, previously discussed.

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