Two Twistor actuators can be pre-twisted to form the jointed device
shown in Figure 2, where the Twistor-pair forms the flextural joint and the
bi-directional pneumatic actuator. Controllably varying the fluid pressure of
each Twistor rotates the member as indicated in Figure 2.
Two such crossed Twistor-pairs can then provide a fully-active flexural spherical joint, with open-loop proportional control. Such a gimbal-drive is shown in Figure 3.
This paper derives the characteristics of a Twistor-pair by first measuring
both volumes and torques under isobaric conditions at various angles and then
obtaining the consistent enthalpy function by integration of the
data.
An approach based directly upon the First Law of Thermodynamics
recognizes the near-reversible conversion of fluid available energy into
mechanical energy. Such methods have proved useful for Tugger and Twistor
design [1] and [16] but, can also provide rational characteristics for
computer-control.
This approach begins with the First Law statement:
dE = P× dV – T× da (1)
where Energy E = E (V,a)
This represents a constraint between Pressure, P; Volume, V; Torque, T; and Twist Angle, a. However, here we are interested in the situation where P and a are proper Input variables with V and T being the corresponding Output variables [6], [8], [9], [10], [11], and [12]. To create these conditions, we must perform a Legendre transformation upon Energy to form a complementary energy or generalized Enthalpy, H [13], [14], and [15].
The requisite Legendre transformation may be performed as follows.
Given E(V,a), the function H(P,a) is found directly from the thermodynamic relation:
H(P,a) = E – P× V
Differentiation yields, dH = -V× dP – T× da
From this last relation we then obtain the two Outputs:
Volume, V(P,a) = - ¶ H(P,a) / ¶ P
Torque, T(P,a) = - ¶ H(P,a) / ¶ a
These results suggest that the active transduction torque may be estimated
by direct integration of the volume changes with
pressure.
Volume data was collected on a single Twistor actuator. A Twistor in
the untwisted position is said to be at zero degrees. The data collected
resulted in volumetric measurements taken at 50, 60, 70, and 80 degrees of
Twist Angle; and at varied pressures of 35, 50, 60, and 70 psi. Measurements
were also made at 80 psi but, at this point the actuator began to balloon
excessively. This was anticipated in the design, where increasing the shell
wall thickness would support additional pressure but would detract from
low-pressure response. Table 1. shows the measured volumes.
| P / a | 50 | 60 | 70 | 80 |
| 35 | .0458 | .0409 | .0323 | .0256 |
| 50 | .0622 | .0561 | .0464 | .0354 |
| 60 | .072 | .0629 | .0519 | .0452 |
| 70 | .0805 | .0720 | .0622 | .0543 |
| 80 | .1239 | .1086 | .0867 | .0763 |
Table 1. Column 1 represents Pressure settings in pounds per square inch, Row 1 represents Twist Angle settings in degrees, and the interior points represent the measured Volumes in cubic inches.
This volume data was then fit to an equation that maintains the Maxwell Relations and the differential form of the First Law equation 1, [17], and [18].
This fitting process for volume employed Microsoft Excel. The resulting linear regression in English units is:
V(P,a) = 0.041093 + 0.001244× P – 0.031424× a - 0.000319× a× P
where Volume, V, is in Cubic-Inches, Pressure, P, is in Pounds per Square Inch, and Twist Angle, a, is in Radians.
Figure 4. shows the regression fit of the volume data as a function of Twist Angle at the four experimental pressures. V(P,a) is within one standard deviation from the experimental data results in Table 1.
From the volume data, V(P,a), alone, it is
only possible to obtain a partial enthalpy function through integration
in the form
Hv(P, a) = ò V(P,a)× dP.
Using the above fit to V(P,a), the integration yields:HV(P,a) = 0.041093× P + 0.001244× P2/2 – 0.031424× a× P - 0.000319× a× P2/2,
so now by differentiation with respect to a there results the active ( or transduced) torque:
TV(P,a) = - ¶ HV / ¶ a = 0.031424× P + 0.000319× P2/2.
However, this volumetric partial enthalpy is unable to provide the
pressure-dependent Coulomb torque component, which can be determined from
experimental measurements, as described in the next
section.
The previous results for the active torque, TV, indicate
that a single Twistor behaves like a vane-motor whose cross-section is
sensibly pressure dependent. However, if this resultant characteristic is to
be applied to the case of the Twistor-pair of Figure 2 or to the
Spherical-Joint of Figure 3, one must also take into account the net
Coulombian flexural torque variation of both opposed Twistors. Generally this
passive elastic torque will also be found to be pressure dependent.
In principle, if both Volume, V, and Torque, T, were independently varied and measured, they could be considered as Inputs, with Pressure, P, and Twist Angle, a, as respective Outputs. The corresponding potential would then be a generalized Free Energy, F(V, T). In the same way, if P and T were taken as Inputs and V and a as Outputs the appropriate potential is the Free Enthalpy, G(P, T).
However, upon pressurizing one actuator of a Twistor-pair, the torque was in fact directly measured by a torque-wrench for the very same pressures and angles used for the above volume data. In consequence, one may consider the Measured Torque, T, to be composed of two separate components: the Active Torque, TV, and a pressure-dependent Coulomb Torque, TC, in the fashion:
T(P, a) = TV(P, a) + TC(P, a).
Table 2 lists the measured data for T(P, a) while Table 3 lists the results for TC if TV is taken as that determined above from the volume measurements.
| P/a |
50 |
60 |
70 |
80 |
|
70 |
8.5 |
20 |
42.6 |
52 |
|
60 |
-21 |
-8 |
7 |
22 |
|
50 |
-31.4 |
-18.6 |
-5.4 |
9.4 |
|
35 |
-48 |
-36 |
-20.6 |
-4.6 |
|
0 |
-63.2 |
-50 |
-33.4 |
-19 |
Table 2. Column 1 represents Pressure settings in pounds per
square inch, Row 1 represents Twist Angle settings in degrees, and the
interior points represent the measured Torques in
Inch-Ounces.
| P/a |
50 |
60 |
70 |
80 |
|
70 |
-39.2 |
-27.7 |
-5.1 |
4.3 |
|
60 |
-60.35 |
-47.35 |
-32.35 |
-17.35 |
|
50 |
-62.92 |
-50.12 |
-36.92 |
-22.12 |
|
35 |
-68.72 |
-56.72 |
-41.32 |
-25.32 |
|
0 |
-63.2 |
-50 |
-33.4 |
-19 |
Table 3. Column 1 represents Pressure settings in pounds per square inch, Row 1 represents Twist Angle settings in degrees, and the interior points represent the Coulomb Torque in Inch-Ounces.
Inspection of Table 3 suggests the following model equation for TC(P, a):
TC(P, a) = a + b× P + c× P3 + d× a
As before, the regression parameters a, b, c, d, and e were determined using Microsoft Excel. The resulting model equation for T(P, a) then becomes:
T(P, a) = -135.385 + 0.050784× P + 0.005104× P2/2 + 0.000155× P3 + 83.36536× a.
Where Torque, T, is in Inch-Ounces, Pressure, P, is in Pounds per Square Inch, and Twist Angle, a, is in Radians.
Figure 5., shows the regression fit for the Torque as a function of Twist Angle at the five experimental pressures.
Of course, 4 pneumatic pressure regulators can directly manipulate the
Twistor-Gimbal Drive of Figure 3. However, sophisticated open-loop control of
both axes results upon supplying the 4 pressures from 4 electro-pneumatic
valves driven from either a solid-state PLC unit or through an interface from
a notebook computer. Figure 6., indicates such an arrangement where the PLC on
the left is moving the "eyeball" on the right through a simple tracking task.
The logic schedule makes use of the model equation above for torque as a
function of pressure and angle.
This analytical approach is very effective for estimating active torque
when only volume data is available for Twistors. It has been shown that the
volume equation in the form above satisfies the Maxwell relations and that the
intermediate evaluation of the (partial) Enthalpy serves as the means toward
calculating active Torque.
Although it is clear from the above test results that this particular Twistor has only a limited torque capability, nevertheless, simple scaling laws extrapolate volumes and active torques both to larger and smaller sizes. However, particularly for smaller-size Twistors, the Coulomb torques are very design dependent so that accurate total torque characteristics require prototype calibration. However, present experience with several sizes and designs indicates that the form of the model for T(P, a) remains nearly invariant, with only the parameters changing with design and size. For this reason the architecture of control programs for Twistor-pairs and Gimbal-drives can be held fixed.
A special thanks goes to Mario DiMarco of DM3 Electric Car Racing for
use of his experimental facility and for suggesting that we use air over water
to facilitate measuring the actuator chamber volumes and to Professor Rudy
Wojtecki of Kent State University, Trumbull Campus for his assistance in the
pneumatic computer contol of the Gimbal-drive.
Figure 3- Twistor Gimbal-Drive with Laser mounted
Figure 6. Mechatronic "eyeball" manipulated by a Twistor-Gimbal Drive.