Advances in Robotics, Mechatronics and Circuits
Linearity Error of Force Transducers arising
from Nonlinear Elasticity
Jang H. Yi
the material, making the degree of nonlinearity more or less
severe depending on the conditioning.
In this paper, the contributions of the nonlinear elastic
response of an aluminum alloy to the transducer linear error will
be numerically calculated and analyzed. The nonlinear
stress-strain behavior will be modeled using three regression
methods, and it will be shown that the nonlinear elasticity may
cause the transducer linearity error.
Abstract—The linearity error of strain gage-based force
transducers due to the nonlinear stress-strain behavior of an aluminum
alloy is numerically calculated and analyzed using three regression
models. It is found that a typical linearity error of 0.03% full scale can
be obtained when the maximum deviation from the linear stress-strain
behavior is in the range of 1%−2% of tensile strength depending on the
regression model used. The difference between the average stress in a
tension gage and the average stress in a compression gage does not
appear to significantly affect the linearity error.
II. LINEARITY ERROR
Keywords— linearity, load cells, strain gages, tensile modulus.
A
A bending beam load cell shown in Fig. 1 is designed to
produce an output signal Vo/Vs (mV/V) given by
Vo
R4
R2
(1)
.
=
−
Vs R3 + R4 R1 + R2
I. INTRODUCTION
strain gage-based force transducer such as a load cell
consists of an elastic element and strain gages installed on
the surface of the elastic element and converts a load-induced
deformation into an electrical signal proportional to the load.
Force transducers have been used in robots for measuring forces
exerted on joints or forces generated by actuators [1]. In
general, the electrical output of the transducer is a nonlinear
function of the input deformation, typically expressed in
microstrain units [2]. The maximum deviation of the transducer
output from the straight line between the transducer output with
no load applied (zero output) and the output at rated capacity
(full scale output) is referred to as the linearity error.
Commercial load cells are ranked into differing accuracy
classes. A specific accuracy class specifies a combined error
including temperature effects, linearity error and hysteresis,
typically less than 0.1% of full scale output [3]. Various
methods for compensating for these errors have been proposed
and analyzed [4]−[7].
It was experimentally determined that microstructural
changes induced from heat treatments can affect the hysteresis
performance of force transducers [8]. It was also found that the
linearity and hysteresis performance parameters of a overloaded
load cell can be essentially restored by subjecting the load cell
to conditioning similar to that which it received in initial
manufacture [9]. These findings suggest that deviations from
the previous state of crystalline structure resulting from a
particular type of conditioning may affect the elastic response of
where the gage resistors, R1 and R3 (nominal resistance 350 Ω),
are subjected to an average tensile stress of about 40 MPa and
R2 and R4 are subjected to an average compressive stress of 40
MPa when a 1 kgf load is applied as shown in the Wheatstone
bridge circuit in Fig. 2. For the load cell shown in Fig. 1,
Wheatstone bridge nonlinearity does not contribute to the
transducer linearity error [10]. Let E be the linearity error given
by
Vo / Vs full load
V
(2)
E = o half load −
.
Vs
2
The full scale output Vo / Vs full load is the output signal under full
load (1kgf). A tensile modulus of 73 GPa results in an average
strain of 550 µε (microstrain) under full load, which in turn
yields a change in resistance of K×0.00055×350 Ω = 0.38 Ω,
where K = 2 is the gage factor. Then from Eq. (1)
Vo
350.38 349.62
(3)
−
≈ 1.08 mV / V .
full load =
Vs
700
700
Ideally the half scale output should be exactly one half (0.54
mV/V) of the full scale output. However, commercial load cells
show deviations from this ideal linear behavior. It has been
observed in our lab that the linearity error of the load cell shown
in Fig. 1 is typically 0.03% full scale (0.0003 mV/V), or less.
This work was supported in part by the Korean Evaluation Institute of
Industrial Technology under the World Class 300 Project.
J. H. Yi is with CAS Corporation, Yangju-si, Gyeonggi-do, Republic of
Korea (phone: +82-10-3169-2010; fax: +82-31-836-0730; e-mail:
yijangho.cas@ gmail.com).
ISBN: 978-1-61804-242-2
243
Advances in Robotics, Mechatronics and Circuits
Fig. 1 Aluminum alloy load cell.
Fig. 3 Stress as a linear (solid) and nonlinear (dashed) function of
strain.
Fig. 2 Wheatstone bridge circuit.
III. NONLINEAR ELASTICITY
Most metals are known to exhibit linear elasticity in the
elastic region as illustrated in Fig. 3. In this work, it is assumed
that the aluminum alloy machined to make the load cell in Fig. 1
has a nonlinear stress-strain property as illustrated in Fig. 3
(dashed curve), and only the contributions of this material
nonlinearity to the transducer linear error will be examined
although other factors such as hyperelastic properties of strain
gage backing materials used in the industry may also contribute
to the linearity error.
To characterize the nonlinear stress-strain behavior of the
aluminum alloy, three regression models are considered as
shown in Fig. 4. A third degree polynomial model is shown as
solid curve; a third order Ogden model conventionally used to
describe the nonlinear stress-strain behavior of rubbers is shown
as dotted curve [11]; and an inverted sinusoidal curve is shown
as dashed curve. The maximum deviation from the linear
stress-strain behavior is set to approximately 1% of the tensile
strength (320 MPa) of the aluminum alloy for the three models.
Fig. 4 Nonlinear stress-strain models: polynomial (solid), Ogden
(dotted), and sinusoid (dashed).
IV. FINITE ELEMENT ANALYSIS
Commercially available finite element analysis software was
used to compute the stress distribution over the gage areas. The
3D load cell model consists of nine parts: the body, the four
gage areas (the dimension of each gage area is 3 mm × 1.5 mm),
and the four transitional “wells” that surround the gage areas
(except the surfaces) and properly connect the elements in two
adjacent meshes as shown in Fig. 5. The element size of the
body mesh was set to 1 mm. The gage areas are composed of
much finer elements of 0.1 mm to provide smoother stress
distributions.
Fig. 5 Meshes generated with different element sizes.
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244
Advances in Robotics, Mechatronics and Circuits
V. SIMULATION RESULTS
sinusoidal model results in the largest error due to its steepest
initial slope (see Fig. 4), whereas the Ogden model results in the
smallest error due to its least severe degree of nonlinearity up to
2,000 microstrain (see Fig. 4).
Table I shows how the ratio of the average stress in R2 to the
average stress in R1 gets closer to unity as d gets larger when the
sinusoidal model is employed (fourth column). The linearity
error in the last column, however, is not significantly affected by
the change in d. Thus, the difference between the average stress
in the tension gage and the average stress in the compression
gage does not appear to play a significant role in determining the
linearity error as far as the contributions of the nonlinear
elasticity are concerned.
The distribution of stress in the horizontal direction (positive
going from R1 to R2 in Fig. 1) under full load (1 kgf) is shown in
Fig. 6. Salient stress concentrations are shown over the gage
areas. The maximum stress (46.9 MPa) was produced in R1 and
another tension gage R4 exhibited 4% less peak stress (45.01
MPa); the minimum stress (−46.9 MPa) was produced in R3 and
the peak stress over R2 was −45.01 MPa.
Fig. 6 Stress distribution.
Fig. 7 Load cell with R1 and R2 separated by 42 mm.
Fig. 9 Linearity error with varying stress-strain nonlinearity.
B. Degree of Stress-Strain Nonlinearity
As mentioned above, the maximum deviation from the linear
stress-strain behavior was set to about 3.2 MPa (1% of tensile
strength) for the results presented thus far. Fig. 9 shows how the
linearity error of the load cell in Fig. 1 changes as the degree of
stress-strain nonlinearity varies. Assuming that the linearity
error should not exceed 0.03% full scale, the degree of
stress-strain nonlinearity should be less than 1% of tensile
strength if the actual stress-strain behavior of the aluminum
alloy follows the sinusoidal model, or less than 2% of tensile
strength if it follows the polynomial model.
Fig. 8 Linearity error when d = 12 mm and d = 42 mm.
Table I. Effects of the distance between R1 and R2 (sinusoidal
stress-strain model).
d=12
d=42
avg.
stress
in R1
(MPa)
39.336
39.334
avg.
stress
in R2
(MPa)
−36.465
−38.021
avg.
stress
ratio
|R2/R1|
0.927
0.967
Vo/Vs
(half
load)
(mV/V)
0.5359
0.5469
Vo/Vs
(full
load)
(mV/V)
1.0711
1.0932
Linearity
error
(% full
scale)
0.0325
0.0310
ACKNOWLEDGMENT
The author would like to thank J. W. Choi and J. H. Park for
their invaluable support.
A. Distance between R1 and R2
Let d be the distance between R1 and R2 (d = 12 mm in Fig. 1).
Consider another load cell with d = 42 mm shown in Fig. 7. This
load cell is designed to produce comparable stresses over the
gage areas. The linearity error obtained from the
aforementioned regression models is shown in Fig. 8. The
ISBN: 978-1-61804-242-2
245
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