IEEE Robotics & Automation Magazine - March 2017 - 71
moving components and therefore reduces friction; the placement of the motors and motor couplings allows for easy access
to, removal of, and installation of other motors of different
sizes; and shorter cable routing reduces the chance of the
transmission decabling. Independent axis control also means
that failure in the cable transmission (e.g., the cable snapping
or coming loose) in one axis does not affect (decable or loosen) any other axis.
Although motor and gearbox combinations are commercially much more common, cable drive transmission is the
standard for haptic devices, because it provides a nearfrictionless transmission and has no backlash, which nearly
no gearbox can achieve. (We note that harmonic drives are a
unique zero-backlash gearbox but still dissipate energy via the
wave gear from friction loss.) A high-tensile-strength cable is
necessary to maintain the stiffness of the transmission and
reflect high stiffnesses. At the same time, an ultraflexible cable
is advantageous, as it reduces the forces required to bend and
unbend the cable as the capstan rolls. Uncoated stainless steel
cables with a high count of individual steel fibers are used (we
use a 0.54-mm-diameter, 16-kg-rated stainless steel rope with
fibers in a 7 # 7 configuration).
As the cable wraps around, the grip of the cable on the
capstan increases exponentially (according to Fgrip = e ni ,
where n is the coefficient of friction between the steel cable
and the aluminum capstan), and therefore even a few turns
will immediately prevent the cable from slipping. We note
that dissimilar metals provide a higher coefficient of friction, so we attain high grip forces with aluminum and steel.
In practice, five turns is more than enough to prevent any
slipping between the capstan and the cable. This principle is
also how the final link's cable transmission (using the cable
loop and turnbuckle) works without slipping.
Regarding device compliance, increasing stiffness (i.e.,
reducing compliance) in the device's structure is done by
increasing the second moment of inertia of each link (e.g.,
making links wider so they do not twist), improving joint
stiffness (e.g., by increasing shaft diameters and increasing
the distance between the shaft bearings that hold the shafts
straight), and using a stiff material. Because plywood is a
layered composite, it is in fact quite stiff, is unlikely to split,
and yet is still reasonably light. It is also soft enough for
self-tapping holes and very minor misalignments that all
contribute to making the device more accessible and forgiving to build, without sacrificing substantial haptic fidelity.
Electrical System
The electrical system has two purposes: to drive the motors
and to measure their angular position. The torque of the
motor used is proportional to the current that is driven
through it, not the voltage it is supplied. Therefore, a current
or torque controller (in our case Maxon ESCON 50/5) is
connected between a generic power supply and the motor.
It is worth mentioning that the components used
(motors, amplifiers, encoders, and acquisition card) are of
professional laboratory quality and should not be confused
with hobbyist counterparts. While efforts to replace them
with lower-cost alternatives are most welcome, one has to be
careful to preserve the precision needed. For example, the
delay has to be less than 1 ms, and the resolution and quality
of the digital/analog converter sufficient. However, this also
brings to the surface the potentials of this starter kit, as it
allows users to explore what their haptic tolerance is for
lower-cost alternatives.
Mathematical Description and Analysis
The haptic device is displayed in a virtual environment as a
point (avatar) in the virtual environment; its location is
determined via a forward kinematics representation. The
forward kinematics is defined as f(i), where i = {i a, i b, i c} is
a vector of the joint angles. In this case, the manipulandum is
in the form of a classic RRR configuration manipulator: three
moving links that are serially linked through revolute (R)
joints (Figure 7). However, the motor for the end link is driven from the rotating base A, with the angle i c being defined
with respect to the spinning axis at z = nt z at A. This in fact
makes the equations of motion simpler, as can be seen in the
following forward kinematics model for the device:
N
cos i a (L b sin i b + L c sin i c)
vrP = f (i) = > sin i a (L b sin i b + L c sin i c) H,
L b cos i b - L c cos i c
(1)
where L is the length of each body's center of rotation to
the next. The partial derivatives of the forward kinematics
gives the device's Jacobian matrix J:
J=;
2f 2f 2f
E
2i a 2i b 2i c
(2)
- sin i a (L b sin i b + L c sin i c)
= > cos i a (L b sin i b + L c sin i c) f
0
L b cos i a cos i b L c cos i a cos i c)
L b sin i a cos i b L c sin i a cos i c) H,
- L b sin i b
L c sin i c
(3)
.
where v = Ji and v is the velocity of P. To give a force F at
the manipulandum, the body torque x is computed as
x
= J < F.
(4)
To see this derivation, we consider the ideal case where the
power delivered by the motors is transferred. completely to the
end effector (conservation of energy). Let i be the rotational
velocities of the system; then
.<
i x
.<
i x
.<
i
march 2017
= v< F
.
= ( J i) < F
.<
= i J< F
<
x = J F.
*
(5)
IEEE ROBOTICS & AUTOMATION MAGAZINE
*
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