The above chapter have introduced the design idea of the novel six-axis wrist force sensor. In this chapter, some structure of the designed sensor will be analyzed with mechanics.
3.1. Comparison between H-Beam and Floating Beam
The H-beam is defined as an “H” shape beam, which is composed of two single beams connected in the middle. The shape of two single beams is not fixed. Here, for the convenience of calculation, the floating beam is a uniform beam with rectangular cross section, and the H-beam is composed of two single beams the same size as the floating beam.
The H-beam and the floating beam in X-axis are taken for mechanical analysis. The H-beam and floating beam have the same length
l, height
h, and thickness
t that also means the width of the beam. The upper and lower ends of the beams are restrained. The displacement
y and rotation angle
θ of point A are compared under the same force or torque. The mechanical analysis of H-beam and floating beam under different working conditions are as follows [
22]:
Under Fx, the floating beam has only an X-directional displacement , and the H-beam has only an X-directional displacement . E is elasticity modulus. The moment of inertia . It can be seen that the X-directional displacement of the floating beam is twice that of the H-beam.
Under Fy, the floating beam has only a Y-directional displacement , and the H-beam only has a Y-directional displacement . The cross-sectional area . It can be seen that the Y-directional displacement of floating beam is twice that of the H-beam.
Under Fz, the floating beam has only a Z-directional displacement , and the H-beam has only an Z-directional displacement . The moment of inertia . It can be seen that the Z-directional displacement of floating beam is twice that of the H-beam.
Under Mx, the floating beam has only a X-directional rotation angle , and the H-beam has only an X-directional rotation angle . The moment of inertia . It can be seen that the X-directional rotation angle of floating beam is twice that of H-beam.
Under My, the floating beam has only a Y-directional rotation angle , and the H-beam has only an Y-directional rotation angle . G is shear modulus. It can be seen that the torsional moment of inertia Ip1 of the floating beam is much smaller than the torsional moment of inertia I′p1 of the H-beam. So, the Y-directional rotation angle of the floating beam is much larger than that of the H-beam.
Under Mz, the floating beam has only a Z-directional rotation angle , and the H-beam has only an Z-directional rotation angle . It can be seen that the torsional moment of inertia Ip2 of the floating beam is much smaller than the torsional moment of inertia I′p2 of the H-beam. So, the Z-directional rotation angle of the floating beam is much larger than that of the H-beam.
From the above mechanical analysis, it can be seen that the stiffness of the H-beam is greater than that of the floating beam in all directions. Where the rotational stiffness of Y-axis and Z-axis is much larger than that of the floating beam, and the stiffness of the other four directions is twice that of the floating beam.
As the theoretical model is simplified and the calculation results are complex and inaccurate, the finite element simulation is necessary. We suppose that
l = 40 mm,
t = 1.5mm, and the two beams spacing of H-beam
l1 = 7 mm. The material is LY12 aluminum alloy. The upper and lower ends of H-beam and floating beam are fixed. Measuring
Fx = 50 N,
Fy = 50 N,
Fz = 50 N,
Mx = 2.5 Nm,
My = 2.5 Nm, and
Mz = 2.5 Nm respectively, the finite element simulation results are shown in
Table 1.
It can be seen from
Table 1 that the results of the finite element simulation and mechanical analysis are basically the same. So, the above mechanical analysis is correct. After the H-beam is used to replace the floating beam, its stiffness in all directions are improved, especially in the directions of around Y-axis and around Z-axis. However, this change hinders the measurement of force or torque by the main beam. In other word, it reduces the sensor sensitivity. So in the following part, we will introduce a method that punching holes in the main beam to improve the sensitivity and solve the questions about the displacement and rotation angle of H-beam.
3.2. Comparison between Single Beam and Parallel Beam
We define that the parallel beam is composed of two single beams that the two ends are connected, and the shape of two single beams are not fixed. Here, for the convenience of calculation, the single beam is a uniform beam with rectangular cross section, and the parallel beam is composed of two uniform single beams with a rectangular cross section.
It can be seen from
Table 1 that the floating beam mainly restricts the degrees of freedom of
y,
z, and
θx. From the perspective of the entire elastomer, when the central plate is subjected to the force of
Fz or the moment of
My, in order to facilitate analysis and simplify calculation, the main beam can be equivalent to a cantilever beam. The force analysis of single beam and parallel beam in the shape of cantilever beam is carried out under the action of
F force or
M moment. The moment diagrams are as follows
Figure 9a,b [
23].
In
Figure 9a,b
,
,
,
,
, where
K is the stiffness ratio between the parallel beam and the vertical beam,
l is the length of the beam [
23].
According to the , if the section of single beam is rectangle b1 × h1, and the section of parallel beam is two rectangles b2 × h2, then the moment of inertia , . If , then . When , , the strain of F measured with the parallel beam will be greater than that of the single beam, and . Therefore, an appropriate value of K should be selected, such as take K = 0.01. At this time, , , . From these data, it can be seen that the force of F is measured with the place near the fixed end and the moment of M is measured with the loading end. When , , the strain of F measured with the parallel beam will be greater than that with the single beam, and the strain of M measured with the parallel beam will be also greater than that with the single beam. Above all, if an appropriate K value can be obtained, the strain measured with parallel beam will be bigger than that with single beam.