3.1. Architecture of the Cylindrical Traction Force Sensor
Referring the force sensor in literature [
28], the cylindrical traction force sensor is designed, as shown in
Figure 5a. The cylindrical traction force sensor is consisting of a cylinder-shaped elastic structural body, a connecting fitting and a shell. The cylinder-shaped ESB shown in
Figure 5b is the core of the traction force sensor, and it has layer A (black area), layer B (red area) and layer C (blue area). Compared with the diaphragm type ESB [
29], cross beam type ESB [
30,
31,
32], parallel type ESB [
22,
33], etc., the cylinder-shaped ESB is hollow, and the free space inside can be used as the connection channel between the contact force sensor and the traction force sensor. The layer A consists of
A1,
A2,
A3, and
A4, and layer C is composed by
C1,
C2,
C3, and
C4 (
Figure 6)
. A1,
A2,
A3, and
A4 are uniformly distributed along the circumference, the
C1,
C2,
C3, and
C4 are also uniformly distributed along the circumference. In addition, the angle between
A1 and
C1 is 45 degrees, and the angle between the slots in layer A and the slots in layer C are 45 degrees or times of 45 degrees. The fixed end of cylinder-shaped ESB is fixed to the connecting fitting shown in
Figure 5c by screw fastening, and the contact force sensor is fixed to the central column of it by screw fastening. Then, the connecting fitting can be fixed to the end flange of a robot and provide rigid support for the ESB and the contact force sensor. The shell shown in
Figure 5d is secured to the free end of the ESB by screw fastening, and it can transfer the traction force exerted by human hands to the free end of ESB, as shown in
Figure 2.
3.2. Basic Force Measurement Principle of the Cylindrical Traction Force Sensor
The basic structure of the cylinder-shaped ESB can be illustrated by
Figure 6. Under the traction force, the ESB will produce bending deformation and shear deformation, which will lead to the occurrence of normal stress and shear stress in the ESB. The normal stress mainly exists in layer A and layer C, which is relatively small. Therefore, the traction force sensor uses shear stress to measure traction force.
The layer A of ESB, which is used to measure the force
FX along the X-axis and the force
FY along the Y-axis, consists of
A1,
A2,
A3, and
A4. When the force
FX is applied on the ESB, the
A2 and
A4 will produce shear stress. The strain values of the two points on the same diameter in the outside surface of
A2 and
A4 have the same signs, as shown in
Figure 7a. Besides, under the moment
MZ, the
A1,
A2,
A3, and
A4 will produce shear deformation. The strain values of the two points on the same diameter in the outside surface of
A2 and
A4, respectively, have apposite signs, and the strain values of the two points on the same diameter in the outside surface of
A1 and
A3 respectively have apposite signs, as shown in
Figure 7b. When the moment
MZ act on the cylinder-shaped ESB, the sum of strain values of the two points on the same diameter in the outside surface of
A2 and
A4 respectively is zero. Then, by measuring the sum of strain values of the points in the outside surface of
A2 and
A4, respectively and using this characteristic, the force
FX can obtain. Similar to
FX, the force
FY can be measured by measuring the sum of strain values of the points in the outside surface of
A1 and
A3, respectively.
The layer C of ESB used for the measurement of moment
MZ is composed by
C1,
C2,
C3, and
C4. Under the moment
MZ, the strain values of the two points on the same diameter in the outside surface of
C1 and
C3, respectively, own different signs, and the sign of strain values of the points in the outside surface of
C2 is opposite to that of the points on the same diameter in
C4, as shown in
Figure 8c. When the force
FX or the force
FY or the combination of both is acting on the ESB, the sign of strain values of the points located at the outside surface of
C1 is the same as that of the point on the same diameter in
C3, the same as the
C2 and
C4 (
Figure 8a,b). Then, under the force
FX or the force
FY or the combination of both, the difference of strain values of the two points on the same diameter in the outside surface of
C1 and
C3 or
C2 and
C4, respectively, is zero. However, when the
FX,
FY, and
MZ act on the cylinder-shaped ESB, the difference of strain values of the two points on the same diameter in the outside surface of
C1 and
C3, respectively, is not zero, same thing with
C2 and
C4. By using this property, the moment
MZ can be detected by measuring the difference of strain values of the points in the outside surface of
C1, and
C3 and the difference of strain values of the points in the outside surface of
C2 and
C4.
The layer B of the ESB is a ring connected to layer A and layer C respectively, and it can measure the force
FZ, the moment
MX and the moment
MY. Under the force
FZ, layer A bears axial pressure (
Figure 9a). When this axial pressure is transmitted to the layer B, there is the axial shear stress in layer B, as shown in
Figure 10.
Figure 10 illustrates the basic constitutional unit of the ESB, and the expansion diagram of ESB is shown in
Figure 11. The axial pressure induced by force
FZ causes shear deformation of
B1,
B2,
B3,
B4,
B5,
B6,
B7, and
B8 (
B1−B8), and then generate axial shear stress in the axial cross section of
B1−B8. Moreover, the sign of strain values of the points in the outside surface of
B1−B8 are the same. Besides, under the moment
MX,
A2 and
A4 are subjected to the axial pressure in opposite direction, respectively (
Figure 9b), which causes shear deformation in
B3,
B4,
B7, and
B8. The sign of strain values of the points in the outside surface of
B3 and
B4, respectively, is opposite to that of in
B7 and
B8. Then, by using this property, the force
FZ can be measured by measuring the sum of strain values of the points in the outside surface of
B1−B8, and the moment
MX can be measured by the difference between the strain values of the points in the outside surface of
B3 and
B7 and that of in
B4 and
B8. Similar to moment
MX, moment
MY can also be measured.
3.3. Mechanical Model of the Cylindrical Traction Force Sensor
To meet the design requirement of traction force sensor, the selection of ESB structural sizes should be carried out on the basis of theoretical analysis. Therefore, based on theory of mechanics, we analyze the mechanical properties of the ESB and establish the mechanical model of the ESB, which is of great significance for the determination of structural sizes of ESB and for the understanding of the mechanism of force perception and the mechanical properties of the ESB.
3.3.1. The Mechanics Analysis under the FX
When the traction force
FX exerts on ESB, the circular ring between layer A and the free end of the ESB will produce shear deformation along the force direction. According to the mechanics of materials, the direction of the shear stress of a point on the excircle of the circular ring coincides with the tangential direction of the excircle of it, and the angle between its direction vector and the direction of force
FX is an acute angle, as shown in
Figure 12. According to the calculation method of shear stress, the shear stress of point
e can be calculated by using the following equation.
where
is the static moment of
segment ring with regard to Z-axis,
is the moment of inertia with respect to Z-axis,
D is the diameter of the excircle of the ESB,
d is the diameter of the inner circle of the ESB,
is the acute angle between point
a and point
e about the Z-axis.
According to Equation (1), the shear stresses of point
a and point
c are zero, and the shear stresses of point
b and point
f are the largest. Then, the distribution of shear stress values of points in the outer surface of the circular ring is shown in
Figure 13.
Based on Equation (1) and
Figure 13, without considering stress concentration, the distribution of shear stress values of the points in the outside surface of layer A is shown in
Figure 14a. According to
Figure 14a, the shear stress in
A1 and
A3 is small, while that of
A2 and
A4 is large. Therefore, the shear stress of the points in
A2 and
A4 can be utilized to measure
FX. In addition, the largest shear stress value in
A2 and
A4 caused by force
FX is as follows.
Similar to layer A, the distribution of shear stress values of the points in the outside surface of layer C can be obtained, as shown in
Figure 14b. According to
Figure 14b, the values of shear stress of the points in
C1,
C2,
C3, and
C4 are not too large nor too small. Besides, the direction of shear stress at points in
C2 is the same as that of the points in
C4 and the direction of shear stress at points in
C1 is the same as that of the points in
C3 (
Figure 8a).
Unlike layer A and layer C, under the
FX, the layer B bears no shear stress. For layer B, the shear stress in
A2 and
A4 transmits to
B34 and
B78 that connects with layer A, which induces the occurrence of the normal stress in layer B, as shown in
Figure 15. This paper utilizes the shear stress in the ESB to measure the traction force. Therefore, the normal stress in layer B will not affect the measurement of
MX,
MY and
FZ.
3.3.2. The Mechanics Analysis under the FY
According to the basic structure of the ESB, under the
FY, the deformation of the ESB is similar to that of under the
FX. Similar to Equation (2), the following formula is important for the measurement of force
FY.
However, unlike under force FX, the points with the largest shear stress are in A1 and A3 and the points that owns zero shear stress value exist in A2 and A4. Hence, the indirect measurement of force FY can be achieved by using the shear stress values of the points in A1 and A3.
3.3.3. The Mechanics Analysis under the Force FZ
Under the force
FZ, the
A1,
A2,
A3, and
A4 bear axial pressure, the
C1,
C2,
C3, and
C4 also under axial pressure. Therefore, the shear stress of the points in the outside surface of layer A and layer C is zero. According to
Figure 10 and
Figure 11, under the
FZ, the cross sections along Z-axis of
B1−
B8 will bear shear force, and the shear stress of the points in
B1−B8 can be calculated using the following equation.
where
is the area of the cross section along Z-axis of layer B (
Figure 10),
is the wall thickness of layer B,
is the height of layer B.
Then, based on Equation (4), the force FZ can be measured by detecting the shear stress values of the points in B1−B8.
3.3.4. The Mechanics Analysis under the Moment MX
When the moment
MX acts on the cylinder-shaped ESB, the force/moment applied on
A1,
A2,
A3, and
A4 can be simplified as shown in
Figure 9b, which leads to the occurrence of normal stress in the
A1,
A2,
A3, and
A4. In addition, under the
MX,
C1,
C2,
C3, and
C4 will also produce normal stress, but no shear stress. When the
MX is positive,
A2 bears the largest tension and
A4 understands the largest pressure. However, the normal stress in
A1 and
A3 is close to zero, because the central axis of twist goes through
A1 and
A3. For the layer B, the tension applied on
A2 will transmit to
B34, and the tension in
B34 will cause shear stress in the outside surface of
B3 and
B4. Similarly, the outside surface of
B7 and
B8 will also produce shear stress. Because the normal stress in
A1 and
A3 is approximately zero, the tensions/pressures applied on
B3 and
B4 or
B7 and
B8 induced by moment
MX approximate to
. Then, the largest shear stress value of the points in the outside face of
B3,
B4,
B7 and
B8 can be figured out.
Although both FZ and MX cause shear stress in B3, B4, B7, and B8, the sign of shear stress incurred by MX in B3 and B4 is apposite to that of B7 and B8, the sign of shear stress caused by FZ in B3 and B4 is the same as that of B7 and B8. Then, the shear stress of the points in the outside surface of B3 and B4 minus the shear stress of the points in the outside surface of B7 and B8 is the shear stress caused by MX. On the contrary, the shear stress of the points in the outside surface of B3 and B4 add the shear stress of the points in the outside surface of B7 and B8 is the shear stress caused by FZ. Therefore, by using this property, FZ and MX can be measured, respectively.
In addition, the
FY applied on the ESB generates the moment around X-axis at layer B, as shown in
Figure 16. Therefore, the moment measured by using the shear stress in
B3,
B4,
B7, and
B8 is the superposition of the true moment
MX and the moment caused by
FY. However, the true moment
MX applied on the ESB is the moment value we need to measure. The force
FY is measurable by using the shear stress in
A1 and
A3, and the moment arm of the moment caused by
FY is available. Then, the moment caused by
FY can be calculated out, after which the true moment
MX is obtainable.
3.3.5. The Mechanics Analysis under the MY
Under the
MY, the deformation of ESB is similar to that of under the
MX. Therefore, similar to Equation (5), the following equation can be obtained.
Unlike under moment MX, under the MY, A1, and A3 bear the largest tension or pressure, and the normal stress in A2 and A4 is close to zero. Then, the points in the outside surface of B1, B2, B5, and B6 produce relatively large shear stress. Under the combined action of MY and FZ, both will cause shear stress in B1, B2, B5, and B6. The sign of shear stress incurred by MY in B1 and B2 is apposite to that of B5 and B6, the sign of shear stress caused by FZ in B1 and B2 is the same as that of B5 and B6. By using this property, the FZ and MX can be measured respectively. In addition, the FX applied on the shell will also produce moment around Y-axis. Therefore, the measurement of true moment MY applied on the ESB also needs to wipe off the moment around Y-axis caused by FX.
3.3.6. The Mechanics Analysis under the Moment MZ
When the moment
MZ act on the cylinder-shaped ESB, the
A1,
A2,
A3, and
A4 all produce shear stress and the value of shear stress can be calculated using the following equation.
where
is the area of the cross section of layer A perpendicular to Z-axis,
is the radius of the excircle of the ESB,
r is the ratio between the arc length of four grooves and the arc length of
A1,
A2,
A3 and
A4.
Under the
MZ, the direction of shear stress of the points in the outer surface of
A1 and
A2 is opposite to that of
A3 and
A4 respectively, as shown in
Figure 7b. However, under the
FX, the direction of shear stress of the points in
A2 and
A4 is the same (
Figure 7a). Similarly, under the
FY, the direction of shear stress of the points in
A1 and
A3 is the same. Therefore, the measurement of
FX or
FY that using the shear stress of the outside surface of
A2 and
A4 or
A1 and
A3 respectively will not be affected by
MZ.
For layer B, the shear stress in
A1,
A2,
A3, and
A4 transmits to
B12,
B34,
B56, and
B78 that connects with layer A. Then,
B1−
B8 under the normal stress, which does not affect the measurement of
MX,
MY and
FZ. The normal stress in
B1−
B8 causes stress in
B23,
B45,
B67, and
B81, which induces the shear stress in
C1,
C2,
C3, and
C4. The values of the shear stress in the outside surface of
C1,
C2,
C3, and
C4 are the same as that of layer A, which can be figured out by using Equation (7). Similarly, the direction of the shear stress of the points in the outside surface of
C1 is opposite to that of
C3, the direction of the shear stress of the points in the outside surface of
C2 is opposite to that of
C4 (
Figure 8c). Besides,
FX and
FY also affect the shear stress of the points in the outside surface of
C1,
C2,
C3, and
C4. However, according to
Figure 8a,b, under
FX and
FY, the direction of the shear stress of the points in the outside surface of
C1 is the same as that of
C3 and the shear stress of the points in the outside surface of
C2 is the same as that of
C4. Hence, the shear stress in the outside surface of
C1,
C2,
C3, and
C4 can be used to detect
MZ without being affected by
FX and
FY.
3.5. Measurement of the Traction Force
Given the true sensitivities of the traction force sensor, the traction force can be calculated according to the measured strain values. By combining Equations (2)−(8), the traction force can be figured out, as follows.
where
,
,
,
,
and
are the shear strain values caused by
FX,
FY,
FZ,
,
, and
MZ, respectively.
In order to measure the shear strains caused by external forces/moments, the strain gauges need to be pasted to the ESB in a manner of ±45° with the axis of the ESB and the strain gauges pasted in different regions are formed into six electric bridges. The output of an electric bridge is voltage, not strain value. Then, Equation (10) can be rewrote to exhibit the mapping relation between the voltage changes of electric bridges and the external forces.
where
is
;
is the diagonal matrix in Equation (10);
is the coefficient matrix of strain transfer of electric bridge, the elements in
is the strain values corresponding to unit voltage; the elements in
are the change values in the output voltage of the electric bridges;
is equal to
.
Owing to the moment
in
includes the moment caused by
FY and the moment
contains the moment induced by
FX, the amendment is necessary to get the real moment
MX and
MY applied on the traction force sensor. The following equation can eliminate the errors in
and
.
where
is the moment arm from the application point of the force
FX to the moment measuring point,
is the moment arm from the application point of the force
FY to the moment measuring point, and in ideal circumstances,
is equal to
.
3.6. The Realization of Traction Force Sensor
After the traction force sensor is machined, it is necessary to paste strain gauges for the shear stress measurement on the surface of the cylinder-shaped ESB. To measure the traction force, we pasted 48 miniature strain gauges on the outer surface of the cylinder-shaped ESB, and the distribution diagram of these strain gauges is shown in
Figure 17. The blue rectangles in
Figure 17 represent strain gauges, and the red squares in
Figure 17 represent the connecting terminals of strain gauges. In order to measure the shear stress of one point, two strain gauges are pasted on the same area at an angle of 90°, and the angle between the two strain gauges and the direction of shear stress is 45° and −45° respectively. Therefore. One strain gauge is used to detect tensile stress caused by shear stress, and the other is used to measure compression stress induced by shear stress. Moreover, the strain gauges pasted in
A1,
A2,
A3,
A4,
C1,
C2,
C3, and
C4 should be pasted in the area that bears the largest shear stress under the
FX,
FY, and
MZ, that is, the middle position of these areas. However, based on the analysis in
Section 3.3.3,
Section 3.3.4 and
Section 3.3.5, the strain gauges pasted in
B1−
B8 can be arranged as
Figure 17 shows. After the strain gauges were pasted, the cylinder-shaped ESB is shown in Figure 19a.
After the pasting of the strain gauges, the strain gauges pasted in different areas are connected to form six electric bridges. The four strain gauges pasted in A2 and A4 are connected to form the 1st electric bridge to measure the strain caused by the force FX. The strain gauges pasted in A1 and A3 are connected to form the 2nd electric bridge to measure the strain caused by the force FY. The strain gauges pasted in B12, B21, B32, B41, B52, B61, B72, and B81 are connected to form the third electric bridge to measure the strain induced by the force FZ. Similarly, the strain caused by the moment can be measured by the fourth electric bridge made up of strain gauges stuck in B31, B42, B71, and B82, the indirect measurement of the moment is obtainable by the fifth electric bridge made up of strain gauges pasted in B11, B22, B51, and B62, the strain caused by the moment MZ can be measured by the sixth electric bridge made up of strain gauges pasted in C1, C2, C3, and C4.
As presented in
Section 3.2 and
Section 3.3, this paper utilizes the sum or the difference of strain values of the points in the outside surface of the ESB to measure the traction force. The sum of strain values of the points in the outside surface of
A2 and
A4 respectively is used to represent the force
FX. Therefore, the connection mode of the four strain gauges pasted in
A2 and
A4 is shown in
Figure 18a.
and
represent the changes in the resistance values of the strain gauges caused by the force
FX and the moment
MZ respectively. In addition, the minus sign and plus sign of
and
indicate that the strain gauge is compressed and stretched respectively. According to the measurement principle of electric bridges, when
is zero, the output voltage is zero even if
is not equal to zero. However, the output voltage is not zero when
is equal to zero and
is not zero. Therefore, the 1st electric bridge can measure the force
FX. In order to measure the forces
FY and
FZ, the connection mode of the second and the third electric bridges is basically the same as that of the 1st electric bridge.
According to
Section 3.2 and
Section 3.3.4, the difference between the strain values of the points in the outside surface of
B3 and
B4 and that in
B7 and
B8 is used to represent the moment
MX. Therefore, the connection mode of the eight strain gauges pasted in
B31,
B42,
B71, and
B82 is shown in
Figure 18b.
and
represent the changes in the resistance values of the strain gauges caused by the force
FZ and the moment
MX respectively. According to the measurement principle of electric bridges, when
is zero, the output voltage is zero even if
is not equal to zero. However, the output voltage is not zero when
is equal to zero and
is not zero. Therefore, the fourth electric bridge can measure the moment
MX. In order to measure the moments
MY and
MZ, the connection mode of the fifth and the sixth electric bridges is basically the same as that of the fourth electric bridge.
According to the structure of the traction force sensor shown in
Figure 5, the cylinder-shaped ESB, connecting fitting and shell were assembled into a traction force sensor by screw fastening, as shown in
Figure 19b. The central column in the connecting fitting attaches the contact force sensor to the end of the traction force sensor and form a tandem force sensor. The output signal of the traction force sensor is voltage, and we developed a 12-channel signal acquisition instrument to realize the signal acquisition (
Figure 19c). The 6-channel in the signal acquisition instrument is used for the signal acquisition of the traction force sensor, and the other 6-channel is used for the information acquisition of the contact force sensor.
3.7. Calibration Experiment of Cylindrical Traction Force Sensor
Equations (11) and (12) exhibit that the traction force can be detected by measuring variation values of voltages of six electric bridges. To obtain the real matrix
in Equation (11) and the moment arm in Equation (12), calibration experiment is necessary. In the calibration experiment of traction force sensor, we use a 6-DOF industrial robot to finish the calibration experiment, as shown in
Figure 20. In the calibration experiment, the robot remains stationary during the calibration process to provide a rigid support for the sensor, and forces and torques are applied to the sensor by mounting weights on the loading structure. Moreover, the attitude of the traction force sensor can be changed by adjusting the posture of the robot so that the forces/moments in different directions can be applied to the sensor. After applying forces/torques to the sensor, the self-developed signal acquisition instrument collects the output voltages of the sensor.
In the calibration experiment, small force/moment ranges are adopted because humans like to guide robot with small forces/torques. During the calibration process,
FX and
FY adopt interval load of
,
FZ adopts interval load of
,
MX and
MY adopt interval load of
and
MZ adopts interval load of
. The moments applied to the sensor were achieved by mounting weights on the loading structure. Therefore, when moments were applied to the sensor, the weights will also exert forces on the sensor. After each loading, the output values of electric bridge were recorded. Each calibration experiment was repeated three times to ensure the availability and repeatability of the experimental data. Under the external force, the changes of output voltage value are shown in
Figure 21, and CH1, CH2, CH3, CH4, CH5, and CH6 represent the output voltage values of the first, second, third, fourth, fifth, and sixth electric bridge, respectively. Under the
FX and
MZ, CH1, CH5, and CH6 have significant output, and this certifies that
FX will induce the occur of moment around Y-axis; under the force
FY and moment
MZ, CH2, CH4 and CH6 have significant output, and this certifies that
FY will induce the occur of moment around X-axis. In addition,
Figure 21 shows CH1 is mainly sensitive to
FX, CH2 is mainly sensitive to
FY, CH3 is mainly sensitive to
FZ, CH4 is mainly sensitive to
MX, CH5 is mainly sensitive to
MY and CH6 is mainly sensitive to
MZ. All of this certifies the theoretical analysis in
Section 3.2.
After the calibration experiment, the least square method was used to calculate the calibration matrix
, as follows.
Plug the calibration matrix into Equation (11) and using Equation (12), the calculated forces/torques can be obtained, which are presented in
Figure 22. Then, the interference errors of the cylindrical traction force sensor are shown in
Table 2, which shows that most of the errors are not larger than 1.0%, and the measurement ranges are
,
,
,
,
,
, respectively.
Non-linear errors (NLES), hysteresis errors (HES) and repeatability errors (RES) are important indexes to show the static performance of a sensor. Five of the six NLES of the cylindrical traction force sensor are not larger than 0.70%, five of the six HES are not larger than 0.85% and four of the six RES are not larger than 0.80%, as
Table 3 shows. To demonstrate the measurement error visually of the sensor, several load and measurement experiments of forces/moments were conducted, and
Table 4 compares the calculated values with the actual values. The measurement errors in
Table 4 verified that the cylindrical traction force sensor can detect the external forces/torques applied to it, and the measurement errors are small.