Kinematic Couplings and the Parameters that Affect Their Design

What are the parameters that influence the design of a kinematic clamp, from the rudimentary to the ultimate?

The Payload

To begin with, the payload is the sole reason for the existence of the kinematic clamp. The kinematic system is built to support without stress, and to locate the payload, and then to precisely relocate it, time after time. We must know the size, the weight and the shape of the payload that is to be carried by the kinematic clamp.

The size of the payload will determine the physical dimensions of the kinematic platforms. The weight of the payload will determine the minimum dimensions of the kinematic elements. That is the diameter of the spheres, and the size of the mating components, that will safely carry that load. It is rather obvious that the defining element will be the sphere, in systems where they are used.

The shape of the payload will influence several factors of the design. The simplest is how the payload will be attached to the kinematic platform. A tall payload may require additional preload of the kinematic clamp to prevent the platform from toppling over.

The Aspect Ratio

A long, narrow payload will require the use of entirely different kinematic elements, than those for a nice equilateral shaped payload. If the aspect ratio is greater than three to one, you will need to divide the positions of the kinematic elements.

You will need to divide the power of the kinematic elements to promote rotation on the roll axis and dramatically restrict linear movement along that long axis. This is done by always using the Kelvin kinematic design, consisting of a trihedral clamp (cone-like device) , a vee and a flat on one platform and three spheres on the other, instead of using the more popular Maxwell design consisting of three vees and three spheres.

A similar approach is used when the mass of the payload is off center with the majority of the weight over to one side. Here again, we use the Kelvin system, with the more powerful trihedral (cone-like) element centered under the point of maximum load.

Six Degrees of Freedom

The six degrees of freedom, or movement, for any mechanical body, correspond to each of the six axii of that body. There are the three linear degrees of freedoms that are at right angles to each other. There is the back and forth, usually designated “X”, the in and out, usually designated “Y”, and the up and down, usually designated “Z”. Then there is a rotational degree of freedom around each of these axii. There is the roll around the back and forth “X” axis. There is the pitch around the in and out “Y” axis and there is the yaw around the up and down “Z” axis.

To be technically correct the level of required repeatability or accuracy should be specified separately for each individual axis or degree of freedom. This can have a profound effect on the choice of design, the placement of component parts and ultimately on the cost. The tolerances for the linear variations are usually spelled out in micrometers or micro-inches and the rotary variations are in arc seconds or micro-radians.

Magnetic Properties

Ferrite, which is basically unalloyed steel, nickel and cobalt are the three main materials (outside the rare earth’s) that manifest strong magnetic properties. In a generally conceptual manner, there are two magnetic properties that may affect the choice of materials used in kinematic clamping systems. The first is the magnetic permeability of the elements, which is manifested as the strong attraction by another magnetic field. The higher the permeability, the stronger the magnetic attraction will be.

Kinematic components with a high magnetic permeability are essential, when axial i.e. on axis, magnetic preloading is used. In these situations, the powerful rare earth magnets are placed directly in line with the ball, in the gap between the vees, directly in the center of the trihedral clamp, and surrounding the flat. The second magnetic property is retentivity. This is the tendency of a material that has been exposed to a magnetic field to permanently retain some of that magnetism. This is the tendency for a material to become a permanent magnet on its own account.

Materials with either of these magnetic properties can be disastrous when used in the vicinity of many delicate scientific instruments. When the problem of magnetic sensitivity occurs, various ceramics or metallic titanium are the most frequent material choices to cope with this situation.

Vacuum Compatibility

Kinematic pallets used in vacuum environments are coming into wide use in the electronic circuit processing industry and in outer space. In some vacuum applications, magnetic transparency is a key requirement. This is particularly true in E-beam applications. The use of ceramics solves most of the problems here.

The problem of attaching the exotic kinematic components to these platforms has been solved through the use of high strength, vacuum compatible epoxy glue. The life cycle of the rubbing surfaces of the kinematic components has been extended almost indefinitely through the use of vacuum compatible lubricants.

Longevity

Through the careful choice of materials with nanometer quality surface texture and the use of carefully selected, oxygen resistant lubricants, the life expectancy of kinematic systems have been extended almost indefinitely, even when the system is pushed to its load-carrying limits.

Horizontal – Vertical – Or?

Is the kinematic platform held horizontal or is it located at an angle to gravity? The question here is really how will the clamp be preloaded? Dead weight, which is simply the force of gravity, if it can be applied uniformly, is the very best preload. Dead weight can only be applied in certain angular positions.

When the kinematic clamps are used in off axis positions, or in space applications where there is no gravity, mechanical or magnetic preloads are required. Mechanical preload can be a positive clamping force or it can be a spring loading. Simple, compact and very powerful magnetic preload can be provided through the use of rare earth permanent magnets. See the paragraph on “tunable magnetic preload”.

Temperature Variability

There are two schools of thought about choosing the basic kinematic design, to best cope with any temperature changes that occur during operation. One approach is to use the Kelvin system because there is no physical change, in the position of that one spot, on the payload, that is positioned directly over the trihedral (cone-like) Kelvin clamp. This one very powerful kinematic element restricts three degrees of mechanical freedom all by itself.

The second approach is to choose the Maxwell system because any physical changes that occur due to temperature variations will cause a uniform expansion or contraction in all directions so the net change to the position of the payload will be nearly zero. What it boils down to, is that if you are only interested in one single spot on the payload you use the Kelvin system.   For all other situations, where there is a Delta-“T” that will affect the payload, you will use the symmetrical Maxwell system of three identical vees and three spheres.

Monolithic Structure

It is only in very special circumstances that it is either practical or economical to machine the kinematic details into the platform itself. This is based on the unsuitability of the platform material itself and the difficulty or even impossibility of generating the excellent geometric and surface quality, required for satisfactory kinematic performance.

Material for the Platforms

The factors that influence the choice of platform materials include the cost, the machinability, the stiffness, the thermal coefficient of expansion; the rate of thermal equilibration, the corrosive resistance, the magnetic properties and the materials ready availability. In some rare situations, the mass of the platform material may be critical. To begin with the choice of platform materials can be influenced by all of these factors in one way or another.

Mild Steel
Mild steel is stiff, (30,000,000 p.s.i. - Y.M.) relatively easy to machine, readily available, inexpensive, has a reasonably low coefficient of thermal expansion of 6.4 micro-inches per inch, per degree of Fahrenheit, but this material is quite heavy and has a very high magnetic permeability. It is very rust prone, but this can be dramatically improved by a thin electroplated coat of nickel.
Aluminum
Aluminum alloys are very popular. They are light in weight and not too expensive. They are easy to machine. They have a very fast rate of thermal equilibration. They are readily available and they are nonmagnetic, but on the other hand, they have a high coefficient of thermal expansion of 12.6 micro-inches per inch, per degree Fahrenheit, and they are only one-third as stiff as steel (10,000,000 p.s.i. – Y.M.). Aluminum is very prone to corrosion in certain environments, but the application of a thin coat of electro-less nickel will dramatically improve this. For clean room environments, aluminum tends to slough off turbidity. This is also controlled by the electro-less nickel coating.
Brass
Brass has most of the same properties as aluminum, but is very heavy and much more expensive. Brass is quite corrosive resistant in many environments.
Aluminum Oxide Ceramic
Aluminum oxide ceramic has some extremely attractive properties but it is very expensive and requires a long lead-time to acquire the material. It is lightweight, very stiff (45,000,000 p.s.i. - Y.M.), non-magnetic, and an excellent electrical insulator. It has a very low coefficient of thermal expansion. It can only be machined using diamond tools. It has a very slow rate of thermal equilibration. It is one of the most corrosive resistant of all materials.
Silicon Carbide Ceramic
Silicon Carbide has some properties that are even superior to the Aluminum Oxide Ceramic and it is available in an Electrical Discharge Machinable grade. It is expensive and requires a long lead time for acquisition.
Stainless Steel
Three hundred series, (18-8) stainless steel has one outstanding property, it is highly corrosive resistant. On the other hand, it is expensive, it is heavy, it is not easy to machine, it is only two-thirds as stiff as mild steel (20,000,000 p.s.i. – Y.M.) and its rate of thermal equilibration is terrible. In the annealed condition it is nearly nonmagnetic.
Titanium
Titanium is quite lightweight, approaching aluminum, but it is as strong as steel. It is readily available in plate form. It is very nonmagnetic. It is very corrosive resistant in many environments. It is only slightly more difficult to machine than steel. It is among the most expensive of commercially available materials.

Understanding the Load Carrying Capacity of a Kinematic Coupling

To start out with a kinematic clamp that restricts a single degree of freedom consists of “point” contact between the two mechanical elements. It is not kinematic if it consists of line contact and it certainly isn’t if it consists of any area of contact. It is obvious, but the case should be made that the point contact, leads to extremely high hurting elastic deformation.

Here is the dilemma, as soon as the two elements of any kinematic couple come into physical contact there will be Hertzian elastic deformation, creating at least a small area of contact and the system will no longer be truly kinematic.

Thinking in these purist terms it is easier to understand the importance of having an extremely high quality surface texture and the need to use the stiffest materials possible. You want the two misaligned surfaces to physically slide over each other, to achieve mechanical equilibrium. They should not stick and elastically comply or deform to finally come to rest in a highly stressed, metastable condition. This statement is not compatible with the frequent recommendation that the components should have a low Young’s Modulus and a high yield strength. I can only say that these writers don’t live in the real world where the number one function of a kinematic system is accurate repeatability.

Mathematical algorithms based on Hertzian theory will calculate the point at which permanent deformation of the kinematic elements will occur. You should apply the proportional limit not the ultimate yield for this calculation. We consider a 40% safety factor as a necessary minimum to insulate for shock and vibration. We strongly recommend the well published mathematical approach advocated by Dr. Alex Slocum of M.I.T. These calculations will coincide with real world experience.

Crossed Cylinders

Crossed cylinders or a cylinder between two spheres can provide excellent kinematic clamps, but the usual configuration is a sphere against another geometry, to provide the required points of contact.

Load Carring Capacity

In rather general terms, the load carrying capacity of two given materials will depend on the spherical diameter and the differential rate of curvature between the two elements. You can think of it as the wrap around. A sphere against a sphere has less wrap around and will have much lower load carrying capacity than a sphere against a cylinder, of the same radius. In the same way the same sphere against a flat will have an even higher load carrying capacity. Far less often used is a sphere in contact with an internal cylinder or other symmetrical geometry, which will have even higher load carrying capacity.

As an example of this, you will occasionally see this design used in the so-called Cathedral or Gothic Arch, which consists of two symmetrical internal curves that act as a vee block. This design will accurately constrain two degrees of freedom.

Canoe Sphere

As the need for exact constraint has multiplied, in the pursuit of nanotolerances, the need for accurate location of very large loads, using kinematic devices, has become a reality. In an attempt to cope with these much higher loads some nonkinematic arrangements, such as cylinders in vee blocks and spheres in cones have been experimented with, but the simple solution to the dilemma of high loads is to stay with the fully kinematic technique, by simply using small segments of very large spheres, against the plane flat surfaces of vee blocks. When using this fundamentally kinematic design there are no loose pieces that must be tethered.

This is not a patented technique, so there is no usurious price premium to be paid for its use. An off the shelf device marketed as the “Canoe Sphere” with matching vee blocks is available. The well-tested design, of this device, utilizes two small segments of a huge 40 inch (one meter) spherical diameter. The load carrying capacity for each “Canoe Sphere” and Vee Block combination is 5,000 pounds (2268 kg) and the repeatability is excellent.

Humidity

It is common practice to use ceramic filled epoxy glue, such as an EG-3000 to rigidly fix the kinematic elements to the platforms. A note of caution here: epoxy glue is hygroscopic, so glued assemblies used in a high humidity environment over long periods of time, can be subject to physical movement. Under really wet conditions, the bond strength can be affected. The best practice is to use as thin a glue line as possible and to remove all overflow of the glue with isopropyl alcohol while it is still liquid. This arrangement simply reduces the conduit for moisture to a very minimum.

Vibration and Resonate Frequency

If you pluck the string of a guitar it will vibrate, at it’s resonate frequency, and you will hear it. If you tap a kinematic platform with a small hammer it will also ring at its natural or resonate frequency. The kinematic platform is not a good radiator of sound and it is highly damped by the preload so that all you will hear is the tap of the hammer. By attaching a vibrometer or by using a laser vibrometer, these brief bursts of energy can be documented. When excited, the kinematic platform will ring and so will the payload held on it!

If the kinematic system is subjected to shock or vibration, this excitation energy will cause a sympathetic vibration of the whole kinematic system and may adversely affect the function of the payload. The highest possible resonate frequency is normally desirable. If a given burst of mechanical energy impinges on a kinematic system, the amplitude of the platform’s displacement will be inversely proportional to it’s resonate frequency. The higher the frequency, the lower the amplitude of the displacement will be.

A plain vanilla kinematic system, with moderate preload, might have a resonate frequency of about 800 hertz. A system with a few whistles and bells can get up to 1200 hertz. Experimental systems with silicone carbide elements and carbon composite platforms have achieved 1800 hertz.

Heavy preload equals heavy damping. Stiff component elements reduce the rubbery interface between the elements. By building the systems with relatively large diameter steel balls and mating hardware constructed from extremely stiff (112 million p.s.i. - Y.M.) tungsten carbide, it is possible to coaxially preload each individual couple with a powerful rare earth magnet. This technique has been able to achieve resonate frequencies in the 3000 hertz realm.

Tunable Magnetic Preload

With the insight gained from coaxial preloading and a more advanced understanding of the “tunable” magnetic preloads, that can be placed on the bisectors of the lines between the kinematic elements, instead of coaxially, new highs in the resonate frequencies can be reached. By building all of the kinematic elements out of the extremely stiff tungsten carbide, with large radii and with very strong preload, new highs of 6000 hertz have been achieved. Stiff material with large radii and heavy preload yield previously unheard of resonate frequencies and its all basic physics.

Tungsten carbide is a cermet, not a ceramic. The individual particles are, in a sense, glued together by a metallic binder. This provides an inherently tough material compared with the rather friable nature of ceramic. Tungsten carbide has a higher Young’s Modulus of Elasticity in compression than it does in tension, so the normally reported 98 million p.s.i. in tension ends up 112 million p.s.i. in compression.

The “tunable” magnetic preload is achieved by simply sheathing the powerful rare earth magnets in threaded cups of high permeability steel. By matching a north seeking magnet on one platform and a south-seeking magnet on the other, very high preloads can be achieved, with component parts of modest size. The magnetic pull varies inversely with the square of the distance between the magnets, so that large changes in the pull can be achieved with small adjustments in the separation of the magnets. Using this technique the kinematic system can actually be tuned to achieve peak performance. The high permeability steel cup, surrounding the powerful rare earth magnet, acts as a magnetic shield to limit stray magnetic fields.

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