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hand linkages

linkages and drives

We will first make the distinction between linkages ( or mechanisms ) and drives.

  Shafts, cables, and chains typically transmit continuous power. Gearboxes will change low torque, high RPM motion into high torque, low RPM motion. These are all typical parts of a ( power ) drive. Linkages on the other hand, transmit forces over a limited range of motion in a back-and-forth fashion. We can call such linkages mechanisms in the strict mechanical sense of the word, even if they use cables.

  Viewed in this way, purists will call a cable winch in a harbour crane a mechanism. Most people would be more comfortable calling this a drive because of the "almost infinite" travel of the cable. In small haptics devices, the travel of the cable drive is limited between two end stops. These can rightly be called cable mecha­nisms. They act just like push-pull rods, or other finite-stroke mechanisms.

  These haptic drives by the way are routinely called "capstan drives". This is certainly wrong, because the end of the cable is fixed to the motor axis. This makes the motor axis a winch, not a capstan.

applications of hand linkages

Hand linkages are not so much a separate class of linkage. They are more of an application area for lightweight and low friction mechanisms, typically consisting of pushrods and bellcranks, torque tubes and cables.

  The video below shows an example which uses all of these components.

Goertz nuclear lab manipulator.

Hand force mechanisms are common in aircraft flight controls, but also in haptics ( including exoskeletons ), and they may be useful in humanoid robots. The crucial difference with power drives is not even the limited range of motion, but the fact that these are two-way mechanisms. The human user drives the movement, but equally important is the information transmitted back to the user from the other end of the chain, on the forces encoun­tered there in reaction to this movement.

  In master-slave systems, we would call these the forces "from the environment". In aircraft flight controls, these are the aerodynamic forces from the control surfaces, which give crucial information on the state of the aircraft to the pilot. Such crisp force information can only be obtained via a stiff, low mass, low friction mech­anism.

  The video shows an early application of a hand linkage, for a remote manipulator in a nuclear lab. The dexter­ity achieved is remarkable. A point of interest is that often in such mechanisms, the user initiates motions which are relatively gentle in the remote environment. We do not want to break glass in the lab. But the lab objects may well transmit back small, but crisp contact forces, signalling contact or even collisions in the remote environ­ment of delicate but hard tools and objects.

typical components

Hand linkages typically use one or more of the following components :

- cables running over pulleys.

- pushrods.

- torque tubes.

We will see that each have their merit. Cable solutions are lowest in weight, although this is not necessarily true of their moving mass since they also tend to have the largest travel of the moving parts, in particular the cable itself.

current motivation

The direct motivation for this section on hand linkages is the design of a haptic joystick for use in MRI. By "haptic" we mean a computer-controlled, force-sensitive so-called "manipulandum", capable of operating close to the bore of the MRI tunnel.

  Many machines of this type have been built over the years, but they do not seem to be very successful. Or maybe the world does not really need them, but we will leave that possibility aside for now.

  Most solutions focus on finding actuators and force sensors that are allowed in the MRI room. This leads to aluminium induction motors, active hydraulics, etc. Another solution, rarely seen, is to perform the actuation and measurement outside the MRI room, and transfer the movement to the MRI table using a simple, passive mecha­nism. The reason this road is rarely taken is probably the perception that it will be very hard transfer hand movement rigidly, lightly and with negligible friction over the required distance of almost 10 metres.

  Yet there are plenty of examples of this in other fields.

manual control of aircraft

Most aircraft do not have power steering. The elevator, ailerons and rudder are controlled mechanically via a control stick and foot pedals. The transmission is via cross-stranded steel cables, over pulleys of approximately 80 mm in diameter, ending in a lever at the control surface.

  A more expensive, higher-quality version uses lightweight push rods. Each individual push rod is less than 1 metre long to prevent buckling, and is supported by smooth-running levers of approximately 50 mm radius.

Changes in direction are negotiated by forked ( "butterfly" ) bellcranks. The places in the aircraft's structure where the pulleys and rockers exert a reaction force must be extremely rigid, to avoid losing energy from the linkage. They are usually located at the corners of a rigid spar and a rib or frame, so that they do not flex under varying control forces.

  Manual aircraft controls of are extremely rigid, light and free of play, because the pilot must be able to feel the aerodynamic forces on the control surfaces directly. When the pilot releases the stick, the control surface itself must move the stick to the momentary equilibrium position, or possibly oscillate around it.

application in MRI

The forces and distances from the control stick to the control surface are of the same order as those in an MRI chamber, and sometimes longer.

  Large commercial aircraft from the 1950s, including the first generation of jet transports, still had manual controls. It should therefore be possible to apply this principle to transfer the movements of a haptic drive from outside the MRI chamber to a manipulandum close to the MRI tunnel. The only difference with an aircraft is that no metal parts are allowed.

  An added complication is that the bed slides back and forth during use, but this problem is no larger than in an aircraft with folding wings.

requirements on the mechanism

The linkage system should be light and rigid, and as free as possible from play and friction. This note is an attempt to examine these properties and optimize them where possible.

a simple manipulandum

the spring loaded joystick

The simplest form of a haptic manipulandum is a joystick like the one shown in figure 1. The handle is driven by a force-sensitive motor, via a mechanical transmission. The transmission has stiffness, mass, friction and play. The stiffness and the mass together form a spring-mass system with at least one natural frequency.

  In practice, the handle also has its own mass, as does the human hand holding it, although the user is free to let go of the joystick. The controlled servo motor will also behave like a spring-mass system, although we can also block the output shaft.

Figure 1 :   Haptic joystick.

bandwidth

A common quality measure in servo systems is the bandwidth. Roughly speaking, this is the highest frequency that the system can reasonably transmit. It usually corresponds to the lowest frequency at which the system will go in resonance. At that frequency, the output of the system no longer follows the input, but overshoots it by far.

  The joystick in Figure 1 behaves like a simple spring-mass system. Initially, we model the mass of the entire linkage between the motor and the stick as a single mass ( "lumped mass" ) and the flexibility as a single spring. The ( undamped ) natural frequency of a spring-mass system is then given by the following equation :

Where :

  K     serial stiffness of all links.

  m     serial mass (concentrated at the "open" end of the linkage)

mass at the user's hand

Torques and moments of inertia do not mean much to most people. Force and mass are more familiar.

  With a joystick-like manipulum comes a lever length. A lever of one metre would be nice for calculations, but it is not a common size for a joystick, and therefore still not very intuitive. A common lever length for a joystick is :

This is the distance between the pivot point of the joystick and the "point of contact" of the hand gripping it.

  The mass of the human hand itself is approximately 0.5 [ kg ], and is located on a "lever length" of approx­imately 0.05 [ m ] around the wrist.

stiffness and mass

It is not trivial to set requirements for the stiffness and mass of a haptic manipulandum. Bandwidth appears to be not the only criterion. We will see that it is largely determined by material properties and the distance to be covered, rather than by geometry. Also, the same bandwidth can be achieved with large forces and large masses, or with small forces and small masses.

  As a first example let us take the Haptic Master, a relatively large admittance-controlled device. In free motion, this device has a stable "apparent mass" ( the perceived mass, equal to the virtual mass set in the admittance controller, plus the physical mass of the end effector attached outside the force sensor ) of approximately 3 kg :

With a blocked motor ( or with an infinitely rigid virtual wall or perturbation ), the perceived stiffness is approximately:

Expressing this in terms of the bandwidth that the virtual controller can impose on the end effector, we have :

Our second example is the Sensable Phantom, a much smaller, impedance-controlled device. This type of device has a much lower stiffness of approximately 2,000 [N/m]. It does not feel like a hard wall or tabletop, but rather like a rubber eraser. On the other hand, the perceived mass is also much lower : approximately 50 grams with a pen-shaped handle ( "stylus" ). The bandwidth of the Phantom as a perturbator is then :

This may sound great, but the forces are very small. You can feel them in your fingertips, but they wil not perturb your whole hand.

  The human hand weighs approximately 0.5 [kg]. With the Haptic Master, this is much less than the total virtual mass of 3 [kg], and you can use the HM to perturb the entire hand. This would be impossible with a Sensable Phantom, so perhaps it is better to take the mass of the human hand into account when determining the band­width.

reflected properties

reflecting stiffness and mass

For a given movement of the reference point, all parts of the linkage will move, each in their own proportion. This proportion can be referred to as the "gearing ratio" or the "mechanical advantage". The latter name is more appropriate to forces, which are inversely proportional to the displacements.

  Converting the stiffness and mass of a component to the reference point is traditionally called "reflecting". At the reference point, we will talk about the "reflected mass" and sometimes also of the "reflected stiffness".

"reflecting" components to the end point

We are looking for ways to combine the contributions of a collection of separate components into a single number for the end-to-end stiffness at the end point, and a single number for the mass at that location.

  The most intuitive approach for manual operation is the stiffness and mass expressed in a linear force and a linear displacement. It can also be expressed as a torque and a rotation, but for most people these are less familiar.

  The reference point in the mechanism can be chosen freely and does not even have to be physically present in the mechanism. But in a joystick, the grip point of the handle is the obvious choice.

  The control stick will often take the form of a joystick with a pivot point. The point of contact between the hand and the stick is then important when converting to a linear movement. As a first example, we take the effect ( the "reflection" ) of the mass m_C and the spring K_CD to the end point A of the joystick in figure 1.

reflected displacements

In the initial position of the joystick, x_A = 0, and the additional displacement under a pulling force is Δx_A = 0. We will call the displacement of point C, x_C, and the rotation of the joystick θ_B.

  With a displacement x_A of the end effector, rigid levers give the following transfer "gearings" :

And so:

or conversely:

As a sign convention we give all displacements the same sign when the mechanism moves in one direction.

reflected forces

Now apply a force F_A to the end effector of figure 1. The bending moment in the stick at point B is then:

This makes the force F_C at the end of the spring ( or pushrod ) CD equal to :

The two "gearing ratios" in series therefore yield:

or conversely:

The force transfer ratio is seen to be inversely proportional to the displacement transfer ratio. This is consistent with conservation of energy : force times distance remains equal at the input and output, so that no energy is stored in the mechanism.

reflected stiffness

From a static point of view, a force in A also causes a small displacement of A. This is because there is compliance in a number of elements. The simplest way to see this is in the extension or compression of the spring ( or pushrod ) CD, under a force F_C at its end point. Let us call the local stiffness ( force per relative length change ) K_CD. Then we have:

Combining (16) and (15), the influence in A of the deflection of the spring or push rod CD equals :

By the definition of stiffness, the following applies to the "reflected" stiffness of the pushrod CD at A:

The stiffness reflected from CD to A is therefore proportional to the square of the lever ratio between them.

If we pull on a spring element by a long lever, it feels less stiff for two reasons. First, we exert a greater force on the spring element. And second, our end point moves more for a given deflection of the spring element.

reflected mass

For the reflected mass, we look at the accelerations. For a mass in C, the following applies:

If we apply (****) to the forces, then the following applies:

If we apply (****) for the displacements, then the following applies:

Here we see that the role of m_C in A is taken by:

The mass of CD "reflected" from C to A is therefore proportional to the square of the lever ratio.

  If we try to accelerate a mass on a long lever with a short lever, it will feel heavier for two reasons. First, we are trying to accelerate the distant mass more, on a longer lever. Second, the "mass force" also acts on the longer lever, so that the force at our end becomes correspondingly larger.

reflected bandwidth

The stiffness and the mass of a component somewhere in the chain both reflect in the same way at the end point. If a component, such as a push rod, has a certain ratio between stiffness and mass, then nothing will change in the end bandwidth if we attach that component to a longer or shorter lever.

  This makes perfect sense if we look at figure 1. As long as point A is just a mathematical point with no mass of its own, it will not affect the natural frequency of mass m_C oscillating on spring K_CD. If the user moves point A back and forth at a given rate and with a given amplitude, then the forces at A will be smaller with a longer lever AB. The lever will move quadratically “lighter”. This applies to both the spring force and the mass.

  But when we release the lever, it will oscillate at the same rate as before.

rotating components

A mechanism may also contain rotating parts, like flywheels or torque tubes. The equations of motion of such parts use torques and moments of inertia. The displacements are in radians, the stiffness in Nm/rad, and the inertia in kg.m². However, these can then be converted unambiguously into linear forces and masses.

  As an example, consider the mass m _c in figure 1 as a rotating element, a single-sided, rotating "flywheel" with a radius of L_BC and a mass of m_C. This gives the object a moment of inertia ( around B ) of :

The angular rotation of this element is linked to x_C via:

We can interpret the spring force in C as a torque around B:

We can consider the stiffness as a torsional stiffness between the torque T_C and the rotation dθ_C :

With (24) and (25) the stiffness is therefore:

In this view, the natural frequency of the rotating element on the "torque spring" follows from the ratio between the torsional stiffness and the mass moment of inertia, but this ratio is unchanged from the linear view:

Figure 2 shows a ‘real’ torque spring, with a ring as an example of an object with a moment of inertia.

  And (28) applies to the angular momentum of any arbitrary object. A ring clearly has the same radius of inertia R for its entire mass, but for a solid disc for example, the following applies:

In that case, the so-called "radius of inertia" is :

Figure 2 :   Torque spring on ring.

pushrods

We will now discuss a number of individual structural elements. The first, and probably most imprtant of these is the pushrod.

stiffness, strength and mass of pushrods

The tensile strength, or the compression strength of a push rod until it buckles, is given by:

The mass is given by :

The stiffness is found from:

so

the achievable bandwidth of a single pushrod

From the equations above, a pushrod becomes linearly heavier and less stiff as it gets longer. The ratio between stiffness and mass therefore goes down with L² :

The bandwidth which the pushrod can transmit decreases with the square root of this ratio. In other words, the bandwidth decreases linearly with L :

For aluminium, the material properties are:

The maximum bandwidth of an isolated push rod is therefore:

The number itself does not mean much, but it gives us an order of magnitude.

numerical example for a pushrod

Let us say that in figure 1, the values of m and K are the mass and stiffness of a single pushrod. Consider a round aluminium pushrod with a cross-section of 20 x 1 mm and a length of 1 metre. The frontal area is to a good approximation, with R ′  = (½ D − ½ t ) :

It can also be calculated exactly with πR² − π (R − t )², or equally exactly with R ′ = ( R - ½ t ), but for a first idea we can simply stick to R = ½ D. The stiffness is:

and the mass is:

This confirms the bandwidth predicted in (39) for an aluminium push rod of length L = 1 :

This stiffness is a factor of 200 higher than the typical haptic stiffness requirement of 20,000 [ N / m ].

  The stiffness will change by a factor of 10  for the distance of L = 10 m that we have to cover in the MRI room. The stiffness will fall to K = 40 e3 N/m, and the mass will increase to 1.65 kg in that case. We will see what hap­pens to the stiffness when we vary the lever lengths. First, we look at the strength.

pre-buckling strength of pushrods

The tensile strength of a push rod, and its compressive strength before buckling, is determined by the allowable stress σ in the material:

Hand linkages can often use simple building grade aluminium, partly because of its availability and affordability, but also because buckling is likely to be more decisive for strength than pure tension or compression, and because the stiffness is the same for all grades of aluminium. For a typical grade like Alu 6060, the allowable tensile stress before deformation is :

For the round tube pushrod tube 20 x 1 mm from our example, this gives:

This equals 720 kg_f, which is a huge overkill for a manually operated device.

buckling strength of a pushrod

A pushrod under pressure can buckle. The classic buckling equation is Euler's :

For pinned ends without clamping, the buckling length L is equal to the actual length.

  The moment of inertia a single direction ( here referred to as xx ) is exactly half the polar moment of inertia J. See further down for more information on the polar moment of inertia of a circular cross-section. For thin-walled round tubes, the following applies:

and therefore

In our example of an aluminium push rod measuring 20 x 1 mm, the following applies:

For a pushrod with a typical length of 1 metre, this equals 214 kg_f. This is still a lot, but considerably less than the allowable tensile force. Typically, buckling determines the strength of a pushrod.

torque tubes

The next important structural element after the pushrod, is the torque tube.

  We will see that torque tubes, like the appearance of torque in any structure, are less desirable than push-pull tubes. But they can give a convenient, and sometimes unavoidable solution over short distances.

torque tube stiffness, strength, mass and moment of inertia

The torque tube is an alternative to the pushrod. Its disadvantage is that a torque tube is less rigid than a push rod, as we will see below. However, a torsion tube is sometimes a practical solution for a short section of a linkage, or as a connection between two pulleys.

  The properties of a torque tube are determined by its frontal material surface area A, its so-called moment of resistance W, and the polar moment of inertia J of the cross-section.

Definitions can be looked up in any text on strength of materials. For a thin-walled circular tube, good approximations are :

These approximations can be improved slightly by replacing R by ( R − ½ t ), or of course by using the exact formulas.

The relationship between A and J is also clear without calculation, because J is the frontal surface area multiplied by the square of the radius at which this surface area is located.

  In a thin-walled tube, all the material is located on the radius R, and so the following applies:

The shear stress in the material follows from:

The torsional strength at a given allowable shear stress will be :

The mass of the tube is:

The torsional strength per weight equals (56) divided by (57) :

Contrary to the pushrod, where the shape of the cross-section is irrelevant to the strength, a thin-walled torsion tube of a given strength and length becomes more weight efficient as the diameter increases. For equal weight, i.e. for equal R . t, the strength will increase if R increases and t decreases in equal measure.

  The torsional stiffness follows from :

This, incidentally is a special case of the more general Bredt Batho formula for the stiffness of a torque tube of an arbitrary cross-sectional shape:

  In this equation A is not the frontal surface area of the material, but instead the hollow frontal area enclosed by the tube, which is then normally referred to as a "torque box". This is the way in which thin-walled aircraft wings and fuselages get their torsional rigidity.
SAY SOMETHING ABOUT STAR SHAPES VERSUS CIRCLES, R^3

  The length s is the circumference of the hollow tube, measured along the outer wall. Actually, if the value of G and / or t varies along the circumference, we need to take the integral of ∫ 1 / (G . t ) . ds along the wall, and use the result ( upside down ) instead of ( G . t ) / s in (61). For a circular tube, (61) immediately results in (60).

  The torsional stiffness per weight is:

In contrast to the pushrod, a torque tube of given stiffness therefore becomes lighter with the square of the diameter.

  TBW.

  The rotational mass moment of inertia is:

The ratio between the torsional stiffness and the moment of inertia of a torsion tube is:

The bandwidth of an isolated torque tube is the square root of the stiffness divided by the inertia, here (64) divided by (63) :

The shear modulus G is approximately 0.4 times the linear stiffness or Young's modulus E. With the E of aluminium from (37) we have :

The square root of 0.4 is approximately 0.63. Relative to the pushrod (39), the bandwidth of the torque tube becomes:

  Again, the number in itself does not mean much but it does provide a kind of upper limit. That limit is somewhat lower than for the pushrod, in a material properties ratio of √( G / E ).

  In both cases, the bandwidth follows from a gearing ( in this case shear ) where the stiffness and the moving mass depend equally on the radius on which the material rotates. In the case of a push rod, this radius is the length of the lever to which the pushrod is attached. In the case of the torque tube, the radius is that of the tube itself.

torsional rigidity of a solid shaft

For comparison we look at a solid torque shaft. In a solid shaft, not all mass is located on the outer radius, Instead it is distributed over all radii. The polar moment of inertia is the integral of the square of the radial distances. This works out to half the value for a thin walled tube of the same mass.

Figure 3 :   Polar moment of inertia of tube and shaft.

torque tube numerical example

A torque tube should preferably have a larger diameter than a pushrod, but let us take as an example the same tube which we used before as a pushrod. This was a round alu tube of 20 x 1 mm with a length of 1 metre. We have :

Reflected by (27) to a point A by lever of 0.05 m, this is :

The same factor applies to the mass in A :

As a check, this gives for the bandwidth :

  As predicted in (67), the bandwidth for the torque tube is less than that for the pushrod by a factor of √(G/E).This is a factor of around 0.6, which does not really change the order of magnitude for the bandwidth.

  However, the stiffness and the moving mass are on a completely different scale than in the pushrod. For the same physical tube and lever, the reflected moving mass of the torque tube is a factor of 25 lower :

  There is a shortcut to this ratio as well. The material, and therefore the stiffness and mass, of the pushrod rotates on the lever, at a radius of 0.05 [m]. The material of the torque tube however rotates on the tube's wall radius of 0.01 [m]. So, there is a difference in radius by a factor of 5, and the radius squared gets a factor of 25 :

  For the stiffness, there is an additional material factor of G / W ≅ 0.4. This makes for a total ratio of 1 / 62.5, which confirms the result of our earlier long detour :

torque tubes swinging on eccentric bearings

A torque tube does not need to be supported in its centerline. It can also make a pivoting, swinging motion on bearings located parallel to the centerline, outside the tube.

  Eccentricity does not affect torsional stiffness, because a torque has no point of application ( consider the case of an ultracentrifuge if this concept is unfamiliar to you ). However, the swinging motion does play a role in the moving mass.

  We already had the polar mass moment of inertia (54) of the tube itself. If the tube turns on a bearing located on an arm of length e away the centerline, then Steiner's displacement theorem adds another term of A . e² to the polar moment of inertia around the bearing :

An external bearing is always outside the radius R. If the eccentricity is a certain percentage of R, let's say e ≅ 1.4 R, then this yields:

In conclusion, the eccentricity of the bearing makes no difference to the torsional stiffness, but it does increase the mass moment of inertia of the tube by a factor of approximately three. Whether this is a problem depends on the application.

levers

lever stiffness

Intuitively, we would measure the stiffness of a lever by the lateral movement of its tip under a side force. But for concatenating a chain of stiffnesses later, it is easier to measure the lever stiffness as the rotation of the root of the lever under a torque, while blocking the sideways displacement of the tip. See the sketch :

Figure 4 :   Lever bending.

The ratio between the rotation of the base and the displacement of the tip is:

The angle dθ is not the same as the rotation of the end point of the lever when the base is clamped.

  We are really talking about the lateral tip displacement dx, with the correct displacement equation as found in texts on strength of materials.

lever mass

The moment of inertia of a lever ( and of the joystick itself ) follows from a few standard formulas, including Steiner's. For a straight stick attached to a pivot point, the following approximation applies:

The lever mass moment of inertia is reflected to the output position using the standard equations of (****).

Whether a more accurate estimate ( and optimization ) is worthwhile depends on the details.