# An Introduction to Kinematics and Applications

## Perfect Position and Repeatability through Kinematic Couplings

To begin with, the basic fulfillment of perfection in the Kinematic coupling depends on the very nature of spherical geometry. No other physical form has a single point, as its source of origin. The exact physical position of a perfect sphere at any moment in time can be defined as a single point in three-dimensional space. This point is obviously the center of all radii of the sphere’s outer surface. The physical position of any rigid body is determined by constraints placed on its basic six degrees of freedom. Every point on a rigid body will remain in exactly the same place no matter what position the body is placed in. A rigid body can be moved side to side, which is the “X” axis. It can be moved forward and back, which is the “Y” axis. It can be moved up and down, which is the “Z” axis. These three linear axii are mutually perpendicular. It can also be rotated around each of these axii. It can “roll” around the “X” axis. It can pitch around the “Y” axis and it can yaw around the “Z” axis.

A Kinematic coupling is a very deterministic mechanical system. By this we mean that you always know exactly how it will perform. In addition to its positional repeatability and its ability to automatically adjust to severe mechanical twisting or bending, it will accommodate dramatic temperature variations without loss of accuracy. These abilities are due to the perfect self-alignment of all six axii of all true Kinematic couplings.

Basically a Kinematic coupling consists of two mechanical devices or platforms, one fixed and one that is portable or removable. The two main features of a Kinematic coupling are its ability to be separated and then to be returned to exactly the same position and its positional stability. A Kinematic coupling restricts all six degrees of mechanical freedom without over constraining any one of them.

There is a good deal of confusion as to what constitutes a true Kinematic coupling. Although there are many variations we divided Kinematic couplings into two classes. The first and probably best known is the Kelvin Clamp which in its simplest form consists of three spheres on one platform and a conical cup, a Vee-block and a flat surface on the other platform. This system was first described by Lord Kelvin in the late eighteen hundreds. The second class consists of the three spheres on one platform and three Vee-blocks on the other platform. This lesser known arrangement was first promulgated by Maxwell in the same year that Kelvin published.

Each of these configurations has its merits and each has its limitations, but both of these arrangements are truly Kinematic couplings. If a large temperature gradient is involved you should seriously consider the three Vee-block configuration as its symmetrical restriction of two degrees of freedom, by each of the three couples will yield a much more stable location.

When we use the terms perfect position and perfect repeatability, we unleash another concept that must be addressed. Variations in these six degrees of freedom are rigid body errors only but in addition there is any number of other flexible or bending errors possible. We might think of the bodies or platforms of the Kinematic coupling system as being in equilibrium instead of being rigid because the Kinematic concept is very good at placing or replacing a body in this condition and besides no mechanical device is truly rigid.

Like any other field of engineering, Kinematic couplings have their conditions. The four conditions of Kinematic couplings are their required repeatability, i.e. accuracy, their required load carrying capacity, their required resonant frequency and then there are the financial limitations. We might add to this, design convenience but, at least part of this is associated with cost.

A condition closely associated with a systems repeatability, i.e. accuracy is what we will call the “slipperiness” of the Kinematic components. The slippery that I am referring to, is a coined term and may not coincide exactly with the dictionary definition. The material that the Kinematic components are made of has a considerable effect on their slipperiness. The two obvious ways are the coefficients of friction and the stiffness of the component parts; but the chemical compatibility of the two materials in contact is the hidden dilemma. We have changed our outlook on materials over the last two years. We are recommending hybrid components and exotic materials for more applications today. The high performance materials are carbides, nitrides and oxides.

A key problem with Kinematic couplings is their propensity to fret. The fretting is a chemical reaction on the surface of ferritic ( steel ) based materials that causes erosion when they are subjected to elastic stresses. What must be realized is that the contact interfaces are actually slipping over each other elastically. This is happening on a micro scale, but it is happening. Fretting will eventually adversely affect the performance of ferrite-based materials even those made of hardened Martensitic stainless steels, including the highest quality 440C alloy. Mixing cermets or ceramics with conventional steel alloys greatly reduces or totally eliminates this problem. By making the intricate geometric components for the Kinematic coupling out of easy to machine stainless steel and the simple spherical shaped components out of the exotic materials, high performance Kinematic coupling systems can be produced that are still very economical.

The surface texture which includes both waviness and microfinish, has at least as much effect on slipperiness as the materials used. The difference in slipperiness of commercially ground and precision lapped surfaces is literally orders of magnitude. You will have a corollary if you think of the surface texture as the tiny teeth of two very fine pitch gears that will mesh with each other to hinder and eventually stop all relative motion between the surfaces of the Kinematic components. The problem caused by rough surface texture is that the movement between the two components stops before the coupling reaches true Kinematic equilibrium. This has a macro effect on the repeatability because each time the two platforms are brought into contact, they are at a little different attitude so they stop in a different position.

The geometry of the components affects the slipperiness due to both the regularity of the geometric shape and to the fundamental nature of the shape. For the same two materials, the interface between a sphere and a cylinder, contrary to expectations based on the high Hertzian elastic deformations, will slip very well because there is less compliance. A sphere against a flat plane will tend to stick on first contact, then slip and then stick again and so on. When the two parts of a Kinematic coupling approach each for a rendezvous, they will be misaligned to some degree. After the first actual contact, they must slip over each other while seeking equilibrium.

Modern Kinematic coupling designs are getting more and more sophisticated. The demand for accuracy, i.e. repeatability, has gone from microns to nanometers. A few years ago the resonant frequency of an ordinary well-designed Kinematic platform was 800 hertz. With a little care we could achieve 1200 hertz and anything more was out of the question. During this period, three microns was considered good repeatability, i.e. accuracy.

Theoretically, a higher load on the couplings will raise the resonant frequency but increasing the load will dramatically reduce the accuracy, i.e. repeatability of the system and even worse there will be fretting that will get worse as the loads grow higher. By using hybrid materials, we are able to hold the fretting to a minimum while increasing the stiffness and thus raising the resonate frequency while still maintaining good repeatability. With careful design using hybrid materials and increasing the wrap around, the resonant frequency can reach 3000 hertz. Wrap around increases as the geometry of the two elements of the couple more nearly match.

As our customers needs keep driving the accuracy, i.e. repeatability requirements and the resonant frequency of Kinematic systems up, we have responded by using stiffer and stiffer materials with more and more wrap around and much higher preload. We are successfully producing Kinematic systems with repeatability measured in nanometers and resonant frequencies in the 6000-hertz range. The newest approach that yields outstanding results but at a premium cost actually uses diamond Kinematic components.

A Kinematic coupling system needs a certain minimum mechanical preload to bring the coupling’s two sets of elements into structural equilibrium. This preload consists of enough force to achieve elastic compliance between the surfaces of the two sets of Kinematic elements.

Dead weight, i.e. gravity, is the ideal loading force but for a variety of reasons it cannot always be used. In some cases the mass of the portable element of the Kinematic coupling system can’t be properly balanced, therefore it may be located at an angle to the gravitational force. It may even be up side down. Then there are the modern situations where the platform is in outer space and there is no gravity. One readily available solution for these anomalies is the application of a magnetic field or magnetic fields, between the two Kinematic platforms.

In the recent past, our designs for magnetically preloaded Kinematic couplings were based on the premise that the magnetic fields providing the preload had to be exactly in line with the axis of each couple. We had a magnet in the center of the three ball couple that attracted the mating steel sphere. We did the same thing with the Vee block and the two cylinder couples. For the flat to the sphere couple we precision lapped the flat face of the magnet and had the steel ball physically contact it. In these designs we were severely limited by the low stiffness or Young’s Modulus of the steel balls and the highest resonant frequency that we were able to achieve was in the 3000-Hertz range. When this level is acceptable, these designs perform well and are inexpensive.

More recently, we have found that the magnetic fields are more forgiving than we had imagined. If the pattern of the magnetic forces is well balanced, the entire magnetic preload can be outboard of the couples and still provide nanometric repeatability. What this outboard placement of the preload does is to allow us to use spheres that are much stiffer than steel with heavy preloads that raise the resonant frequency into the 6000-hertz range.

Rare earth magnets have so many advantages over other magnetic materials that they are usually the magnetic material of choice. Unlike Alnico ®, rare earth magnets have no aspect ratio so very thin sections can provide powerful magnetic fields. In terms of magnetic field strength, rare earth magnets made of neodymium boron iron are quite cost effective. The material is readily available in a broad array of sizes and shapes. We offer several standard sizes of naked cylindrical magnetic disks for off the shelf delivery. The magnetic field strength of these disks can be multiplied by simply stacking a series of self-adhering disks together.

With our extensive capabilities in Electrical Discharge Machining, diamond grinding and diamond lapping, we can economically configure this hard brittle material into standard and custom shapes.

## Shrouded Magnets

A substantial improvement over simple naked magnets is the shrouded design. In this off the shelf version, the rare earth magnet is surrounded by a high permeability metal cup or shroud. This high permeability metal shroud acts as a magnetic short circuit thus shielding the stray magnetic field emanating from the back of the magnet. The outside diameter of the magnet itself is separated from the cup by a nonmagnetic ring. With this design the outside ring of the shroud becomes a magnetic force path of its own thus multiplying the magnetic pull. The high permeability metal shroud or cup has a fine pitch thread machined on its outside diameter. The non-magnetic ring separating the magnet from the shroud has four notches cut in the top. These notches allow a nonmagnetic spanner wrench Part Number SW-1 to rotate the magnet. These design features provide an excellent way of mounting the magnets and it provides a fine-adjustment of the magnetic pull. After adjusting its position the magnet can be locked into position by setscrews, thread locking compound, a split or pinch clamping mechanism.

For less critical applications, the shrouded magnetic assembly can simply be glued into an oversize counter bore in the platform. The shrouded magnets are manufactured with either a North-seeking pole up or a South-seeking pole up. By placing a North-seeking magnet in one plate of the Kinematic platform and a South-seeking magnet in the other, very powerful magnetic forces can be applied in a limited area.

High permeability metal disks are available with the same dimensional parameters as those of the shrouded magnets. They are for use when the less powerful force of a single magnet is desired. This variety of magnetic components will allow you to hedge your bets when the desired magnetic force isn’t perfectly known. The high permeability cup and the disk have a black oxide coating to improve corrosion resistance. A clean room version is also offered with both the cup and the magnet electroless nickel-plated. We will custom manufacture any magnetic assemblies of any design required to satisfy our customers’ needs, including metric dimensions and metric threads.

## Aspect Ratio

What we would like to see in a Kinematic coupling is the component parts forming a nice equilateral triangle. Unfortunately, in the real world this isn’t always possible. We had to develop a high performance Kinematic coupling to hold a 34-inch long Coordinate Measuring Machine calibration artifact that was only 3 inches wide. We choose three of our standard Vee blocks and three truncated and threaded spheres as a simple cost-effective solution. The design looked good on paper but its repeatability was only in the range of thousandths of an inch and the artifact shifted position when the probing force of the measuring head was applied to the artifact. This was one of only a very few actual examples that I have observed of an under-constrained Kinematic coupling system.

When we built our spherical measuring machine about five years ago, we experimented to determine what configuration of components gave us the best repeatability. We were able to get the test sphere to repeat its single axis position, within plus or minus one nanometer by running it against diamond spheres that were positioned at an angle of 120 degrees. Based on this information, we redesigned this problem Kinematic coupling to use one set of 3 spherical segments that were widely separated to give them approximately a 120-degree contact with the mating sphere. To this, we added a sphere in contact with a standard Vee-block and a sphere against a flat. The artifact now repeated so exactly that we couldn't measure the error with the Coordinate Measuring Machine. This problem spawned what is now the Rose Bud line of Kinematic components shown in our Kinematic Catalog #105B.

## Cryogenic and High Temperature Applications

For both elevated and depressed temperature applications there is one main design consideration that must be addressed. This is the large dimensional changes that can take place due to thermal expansion or contraction of the structures. Even if the two structural parts of the mechanism are made from identical materials there is bound to be mechanical changes due to variations in the time of thermionic equilibration, caused by differences in the cross sectional areas. This will cause an oscillation between the elements of the Kinematic couples not just an expansion or contraction. In order to minimize the effect of these movements on the position of the payload on the Kinematic coupling, a symmetrical design should be chosen. Placing three sets of identical components in an equilateral pattern will yield the most dimensionally stable payload possible. Three spheres and three Vee-blocks or one of its derivatives is the obvious choice. The conical cup, vee and flat will equilibrate off axis, as the temperature changes, due to the uneven constraints placed on each one of the three spheres. The choices for materials to construct the Kinematic components in high and low temperature applications are surprisingly similar. In the range from plus 600-900 degrees F. down to 200 degrees F. below zero common materials will function with limitations. For very low loads with no shock or vibration, tungsten carbide balls and 440c stainless steel components will function satisfactorily. As the loads go up or shock and vibration are factors, nickel based, participation hardened (strengthened) materials, must be used for the complex components, while tungsten carbide balls will still function. The interfaces between the platforms and the individual Kinematic components are limited to purely mechanical clamps to hold them in place, as no organic glues will stand up under either of the temperature extremes.

## Dynamic Kinematic Couplings

Many if not most Kinematic couplings are not used to restrict all six degrees of freedom. Our definition of a Kinematic coupling here is of a fully self-aligning highly deterministic system without any over constraint.

## The Pivot

Simple couples that restrict all three linear motions but allow full rotation in pitch, yaw and roll provide the basis for the pivot. A sphere in a conical cup is the simplest design. Generating a very high quality cone verges on the impossible, but there are at least three viable alternatives to the cone that all work quite well.

## The Flat Prismatic Couple

One approach is to use three flat surfaces in an internal prismatic configuration forming a trihedral cup. This is often referred to as the Kelvin Clamp as Lord Kelvin first described it in his book. The flat prismatic design has the highest load carrying capacity in a given size envelope but it is also the most expensive to produce. Refer to “Flat Prismatic Components” in our Kinematic Catalog #105B.

## The Inclined Cylindrical Couple

Another approach is to use three very high quality inclined cylinders spaced at 120-degree intervals, in which to rotate the sphere. This approach was used very successfully in a 1940’s device for evaluating the surface quality of precision balls. The inclined cylinder design is in the mid-range of both cost and load carrying capacity. This design does require some fairly sophisticated machining to mount the cylinders.

## The Ball Couple

The three ball Kinematic coupling is probably the best all around approach for building a pivot. Precision balls are of a very high quality and they are inexpensive. The three ball couple has the lowest load rating for a given footprint, but it is the most accurate and it is inexpensive. When the load carrying capacity goes up, larger and larger diameter balls must be used. This soon causes the three-ball coupling to take on gigantic proportions. By cutting small segments out of a larger diameter sphere, the excellent qualities of the three-ball coupling and high load carrying capacity of the larger radius are combined in a compact envelope. This form of device is listed in our Kinematic Catalog #105B under the “Rose Bud”. Our indexing master sphere, which is used for the ultimate calibration of roundness measuring machines, is a typical application of the three ball coupling used as a pivot: see it in the Authenticator section

## The Kinematic Hinge

An excellent ultra precise hinge can be generated by adding a Vee and a second sphere to any one of the previously described pivot designs. Combining a Vee and one form or another of the conical cup will restrict all of the “X”-“Y”- Z” linear motions as well as the pitch and yaw rotational motions, which leaves only the roll motion. The Vee may consist of two flat inclined planes as in the conventional Vee-block or it may be fabricated from two parallel cylinders. Although less frequently used, the design may be reversed somewhat by using a single cylinder and two spheres. (See cylinders, Vee-blocks, the quarter round and flat prismatic components in our Catalog #105B).

## The Tilt

A tilt is a complete Kinematic coupling based on the combination of a hinge and a stop. The stop is provided by a sphere on a flat plane. There is an actuator that lifts the rear end of the portable platform separating the sphere and the flat plane. When the activating force is removed, the Kinematic platform returns to its original position exactly. The activation can be accomplished thru a jackscrew, a motor driven cam, a pneumatic or hydraulic driven piston, a hand-operated lever or any number of other designs. Tilts are used to provide clearance for inserting testing devices into inaccessible areas, for positioning read-write heads on a memory disc, for inserting the location tooth in precision angular dividing devices and similar high precision functions.

## Planar Kinematics

A tripod formed by three spheres that rests on a flat plane can mechanically describe X-Y-Z parameters of that plane with excellence. Spherical segments of large radius are used for the feet. These feet are almost always made of tungsten carbide, spherical segments, to hold down the wear.

## Linear Motion or Sliding Kinematics

Linear motion Kinematic coupling systems are commonplace in industry. To start with, sliding Kinematics is not a bearing system. They are low speed, low load devices but they are able to define an axis of linear motion to almost any level of accuracy. The basic premise is that five of the six degrees of freedom are constrained, leaving one linear axis free to translate. A variety of designs can be used, each with its advantages and each having limitations. Very high quality spheres are readily available at low cost so they are the most frequent choice as one set of the elements in sliding or linear Kinematic systems. These linear designs use the Stator elements as linear ways that will define the travel along one axis just as accurately as the geometric quality of the Kinematic elements.

## The Geometric Way

A geometric way consists of a long Vee-block and a parallel flat plane. The moving platform has three spheres mounted to the bottom of the moving platform in a triangular pattern. Two of the spheres ride in the groove of the Vee-block and one rests on the Flat Plane. The five contact points each restrict one degree of freedom leaving only one linear motion unimpeded.

The alternative geometric way uses a positive or male Vee, rail and a parallel flat plane. Machining a high quality male vee can be simplified by producing, rectangular rail with a square cross section that has two surfaces ground or ground and lapped and then mounting it in a machined vee groove. Four spherical contacts are required on the vee rail and one on the flat plane to provide the necessary constraints of the five degrees of freedom.

## The Cylindrical Way

Linear ways that are comprised of cylinders have some unique features. To start out with, if we reverse the design and use four spheres or four inclined cylinders as the stator and a large diameter long cylinder as the mobile element, we get a simple, very accurate device with one linear and one rotary degree of freedom. It can translate along the linear axis of the cylinder and it can roll around that same axis. By adding a parallel flat rail and another sphere we constrain the roll and only the linear translation remains. Another version of the cylindrical rail scenario, that yields excellent results, uses two long parallel cylinders. Four spheres or four cylinders placed at right angles ride on the one cylindrical rail and a high quality cylinder rides along the top of the other long cylindrical rail at right angles to the travel.

## Many Variations

The limitations of linear kinematics are only restricted by the ingenuity of the designer. About thirty-five years ago, we designed and built a measuring machine to evaluate the mechanical accuracy of the first star gazing, inertial guidance platform. We had to measure the attitude of the main mirror to a reference datum with a certainty of one microinch (25.4 nanometers). With a little thought, we realized that a variation of the geometric way would work. By running three spherical points of contact against one flat reference datum we restricted the first three degrees of freedom. We then constrained the other two degrees of freedom by running the flat right angle side of the moving platform against two more spherical contacts. We drove the platform with an ordinary laboratory micrometer that was isolated from the platform by a floating sphere in a version of the Jones Wobble Pin.

## Vertical or Off Axis Kinematic Couplings

How do you design a Kinematic coupling system when gravity is not forcing the components into contact? There seems to be three fundamental approaches to solving this dilemma.

## Brute Force

The first is to simply use Brute Force. This technique may sound crude for such a sophisticated science, but it works, and sometimes it is the only practical solution. The fundamental of this approach is to overcome the upsetting force of gravity by providing a counter force and a slight preload to hold the two Kinematic platforms in reliable contact. The force required to perform this task is most often provided by permanent magnets but springs or a cable and counter weight can yield acceptable results. Electro magnets can be used when the heat they generate is tolerable. Electro magnets work well when you want to turn off the counter force. With the counter force off, the portable platform is free to be removed. By reversing the polarity of the excitation voltage while using the combination of a set of the Electro Magnets and a set of Permanent Magnets you can actually eject the platform.

When using the Brute Force method the mass of the portable platform and that of the payload on board should be held to a very minimum. This will reduce the counter force required, to adequately seat the three Kinematic spheres in their mating components. One application where the Brute Force concept shines is when the entire Kinematic coupling is up side down, with the portable platform actually hanging.

The Brute Force design is very forgiving so that any combination of Kinematic components that restricts all six degrees of freedom will work well.

## Stand It Up

The second approach for vertical or off axis systems is The Stand Up Platform. In this design, you literally stand the platform up on end in two widely separated sets of Kinematic components and then pull the third set of components into contact, using some form of preloading force. You basically form a Kinematic hinge with the two sets of components on the bottom and then swing the third set into contact with a preload. The best design is to stand the platform up on a sphere in a conical cup or one of it’s derivatives on one end and a sphere in a Vee-block or one of its derivatives on the other and to swing the third sphere into contact with a flat component. This design can support very heavy payloads in vertical or angular attitudes.

## Hang It Up

The third and truly beautiful approach is to literally hang the portable Platform on the base. In this design approach, there is an overhanging lip at the top of the portable platform. A pair of widely separated Kinematic components is situated under this lip and the mating set is located on top of the base platform. Like the stand up platform, you use a sphere in a conical cup or one of its derivatives and a sphere in a Vee-block or one of its derivatives to form the hinge, but this time they are up on top of the platforms. The real beauty of this design is that the mass of the over hanging payload will automatically swing the platform and the third sphere into solid contact with the third flat component of the Kinematic coupling. This design has good load carrying capacity.

## Fail Safe Kinematic Couplings

How can you mechanically secure a Kinematic Coupling System to eliminate any possibility that it will separate? You need the excellent accuracy of the Kinematic design but extreme financial loss or even the loss of Human life itself, make it mandatory that the system be fail safe.

A similar problem occurs when the Kinematic Coupling System is subject to severe vibration or negative acceleration. The same set of circumstances occur again when the Kinematic platform moves from a plane parallel with the earth’s surface to one that is perpendicular to it, or even upside down. These conditions require that we break the Kinematic coupling credo of never over constraining the system. The challenge here, is how do we safely secure the platforms with the least possible detrimental effect?

The first requirement is that the mechanical force holding the platforms together be applied as exactly on axis as possible. The most reliable way to do this is to use a threaded shaft that passes concentrically thru the centers of the individual couples. The second requirement is that the amount of force that is applied be as repeatable as possible. The third is to provide redundancy in the form of a clamp to clamp the clamp.

Self-alignment between the axis of the clamping device and the axis of each Kinematic couple is a must. A special spherical tilt with a free-floating base is the heart of the fail-safe tie down system. All six degrees of freedom for the tie down system must be constrained. To do this with a minimum of over constraint even the most minute variations in alignment must be compensated for.

Duplicating the clamping force, time after time, is required to achieve exact repeatability of all the elastic deformations of the component parts of the Kinematic coupling. The simplest method of applying almost exactly the same force to each couple and repeating it time after time is to use clocking nuts. The torque required to advance the nut will go up abruptly when it makes contact with the top of the spherical tilt. The clocking or indicating point is noted and from this point on any advance of the nut will resolve as elastic deformation of the Kinematic coupling system. If the rotational advance of all three of the clocking nuts are held equal, the force on the three couples will be uniform and by returning to this point each time it will repeat very well. The final step is to install a jam nut on top of the clocking nut to clamp the clamp.

A quick analysis of the fail-safe tie down’s design will account for each of the six degrees of freedom. The free floating lower half of the spherical tilt will compensate for any X or Y-axis misalignment. Any movement along Z-axis and any tendency to yaw are eliminated when the clocking nut is secured. The roll and pitch errors are nullified by the self-adjustment of the spherical tilt. The small error sources that are left are mainly elastic due to pent up torque, caused by the tightening of the clocking nut on the threaded shaft.

## Really Large to Gigantic Kinematic Couplings

The one factor limiting the load carrying capacity of a Kinematic coupling is the contact stresses between the spherical diameter and the mating components. How do you solve a design problem that seems to be impossible? Up to certain load limits, conventional designs can still be used, but with component parts made from high stiffness, high strength materials and very large spherical diameters. As loads continue to go up, the size of the component parts take on gigantic proportions of their own. To cope with this situation, we simply make the contact surfaces as separate inserts instead of creating the complete geometrical shape. Instead of fourteen-inch diameter, complete balls, that weigh in at 880 pounds each, we substitute two-inch diameter two-inch thick inserts with a seven-inch radius on one face. To improve the situation even more, the inserts can be made from a cermet that has a young’s modulus that is three times that of steel. See our technical data sheet: Split Kinematics without Abbé offset in the Kinematic section.

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