Force Measuring System with Zero-Compliance Suspension

Abstract

The principle and classification of force measuring system with zero-compliance suspension are described. A zero-compliance suspension consists of series-connected suspensions, one of which operates to cancel the variation of length of the other suspension. As a result, the total length of the connected suspensions is invariant even when force acts between the ends. In the force measuring system, one end is fixed to the base and force to be measured is applied to the other end (point of application). The force is estimated from the displacement of the connection point of the suspensions that is to be detected (detection point). This principle of measuring force is explained in a basic model. Measuring systems are classified according to the location of active element, the actuator of the active element, and the location of positive-stiffness element. Several examples of measurement system are described. Then, control methods of achieving zero-compliance are explained for an apparatus using double series magnetic suspension. Finally, measurement results by the developed apparatus are shown. It is confirmed experimentally that the displacement of the detection point is proportional to external force acting on the point of application and high-resolution measurement is possible by the developed system.

1. Introduction

Force is one of the most fundamental physical quantities. Accurate force measurement is required in a lot of engineering systems [1]. A variety of force measurement methods have been proposed and in practical use. They are classified in a number of different ways. One classification is according to the system structure: open-loop system or closed-loop system. The former corresponds to the deflection method [2, 3], while the latter corresponds to the null method [2, 3, 4].

Most of the developed devices are of the former type [5, 6, 7]. One of the most popular devices is load cell [1]. In devices of this type, the stiffness of the mechanical conversion part is made lower to achieve higher resolution. However, such low-stiffness mechanism has several drawbacks. One of them is the deflection of the point of application of force by the measurand (force). It may cause measurement error, because the distance between the force source and the point of application changes in measuring. Another is the limited bandwidth of measurement.

The first drawback can be overcome by the null method (second category). However, high-gain feedback is implemented to ensure against phase lag in measurement [8, 9, 10]. It often makes the control signal noisy, which may result in significant degradation of measurement resolution, because the force is estimated from the control signal.

Force measurement using zero-compliance mechanism has been proposed to overcome the above-mentioned problems [11]. The original concept of zero-compliance mechanism is just a series-connected two springs, one of which has positive stiffness and the other has negative stiffness, whose amplitude is equal to that of the former. In this condition, the total stiffness becomes infinite. It enables the point of application of force to keep the original position even when force acts on it like in the null method, while the connection point of the two springs (detection point) deflects in proportional to the applied force like in the deflection method.

In this chapter, the principle of measuring force is explained in a basic model. Then, measuring systems are classified according to the location of active element, the actuator of the active element, and the location of positive-stiffness element. Several examples of measurement system are briefly introduced. One of them is micromachined. It will suit microforce measurement because of its higher resolution. Then, control methods of achieving zero-compliance are explained in an apparatus using double series magnetic suspension. Finally, measurement results by the apparatus are shown to demonstrate the effectiveness of the proposed measuring system.

2. Principles of measurement

See Figure 1.

Figure 1.

Schematic drawings of measurement mechanism.

2.1 Zero-compliance mechanism

The principle of force measurement with zero-compliance mechanism is illustrated in Figure 1. Suspension I and Suspension II are connected in series at the point X. This point becomes the detection point. Force to be measured is applied to the other end of Suspension II (Y). This point is called the point of application. The stiffness of the series-connected suspensions becomes

$$k_c=\frac{k_1{k}_2}{k_1+k_2},$$E1

where $k_c$: stiffness of the series-connected suspensions, $k_i$: stiffness of each suspension $\left(i=1,2\right)$. This equation indicates that the resultant stiffness becomes lower than the stiffness of each suspension when both suspensions have positive stiffness $\left(k_i>0\right)$. Nevertheless, the situation dramatically changes if one of the suspensions is allowed to have negative stiffness. Especially, when

$$k_1=-k_2$$E2

is satisfied, the stiffness of the connected suspensions becomes infinite as

$$\left|k_c\right|=\infty .$$E3

It means that the point of application is invariant (no deflection), which is just a characteristic of measurement by the null method. Meanwhile, the detection point moves from the original position in proportion to the applied force as

$$x=\frac{f}{k_1}=-\frac{f}{k_2},$$E4

where x: displacement of detection point and f: force acting on the point of application (Figure 1b). Therefore, the force can be estimated from the displacement of the detection point, which characterizes measurement by the deflection method. Eq. (4) indicates that each suspension should have low stiffness for high resolution in measurement.

2.2 Classification of mechanisms

In explaining the principle of operation in the previous section, pure springs with positive and negative stiffness were assumed. However, to achieve the zero-compliance characteristic by a stable system, active control is necessary [12]. It indicates that at least one of the suspensions must be an active element equipped with an actuator. Hence, the measurement systems can be classified according to [13, 14]

• location of active element,

• actuator used in active element,

• location of positive (or negative) stiffness element.

As to the first criterion of classification (location of active element), three ways are possible:

1. Suspension I,

2. Suspension II,

3. Suspension I and Suspension II.

Simply to achieve zero-compliance, (I) or (II) is sufficient. It is advantageous in reducing component count and cost. The third may be adopted when a more flexible system is required, for example, both stiffness are adjustable.

As to the second criterion of classification (actuator), the followings are candidates:

1. (M) Electromagnet,

2. (V) Voice coil motor,

3. (E) Electrostatic actuator,

4. (P) Piezoelectric actuator.

It is to be mentioned that any other linear actuator is applicable.

As to the third criterion of classification (location of positive-stiffness element),

1. (U) Suspension I,

2. (L) Suspension II.

Figure 2 shows how the mechanism operates in each case. In the upper location (U), the detection point moves in the same direction as the applied force. In the lower location (L), it moves in the direction opposite to the applied force.

Figure 2.

Operation to applied force.

3. Examples of mechanism and measuring apparatus

A simple calculation indicates 24 combinations for the realization of zero-compliance mechanism. The selection should be done according to target force (gravitational, magnetic, electrostatic, atomic, etc.), the magnitude/order of force (mega to pico-Newton), the size of instrument (macro- to micro-scale), and environment (clean, chemical, vacuum, operation temperature, etc.). The optimization is still under investigation.

Among the possible combinations, the first developed apparatus was a combination of (I)-(M)-(U) [15], whose system configuration and picture are shown in Figures 3 and 4. It was based on double series magnetic suspension. In this suspension system, two floators (objects to be suspended) are controlled with a single electromagnet. The force of the electromagnet directly acts on the first floator equipped with a permanent magnet at the bottom. It corresponds to the detection point. The attractive force of the permanent magnet acts on the second floator made of ferromagnetic material. It corresponds to the point of application. The force increases as the gap between the floators decreases. Thus, it acts as a spring with negative stiffness. When a downward force acts on the point of application, the second floator is indirectly controlled to keep the original position by the electromagnet. Simultaneously, the detection point deflects downward for the gap between the two floators to decrease, and the increased magnetic force cancels the force applied to the point of application to achieve zero-compliance state.

Figure 3.

Measurement system using double series magnetic suspension.

Figure 4.

Picture of fabricated apparatus using double series magnetic suspension.

The second developed apparatus was a combination of (I)-(V)-(L) [16], whose system configuration is shown in Figure 5. In this configuration, Suspension II has positive stiffness and suspends the point of application from the detection point. The motion of the detection point is controlled by a VCM. Figure 6 shows a picture of the fabricated apparatus. It uses a one-piece parallel spring mechanism where a VCM is installed with as an actuator as shown in Figure 7. Such structure is effective for downsizing the measurement mechanism. The spring mechanism has sixteen circular hinges. It operates like a series-connected springs. The VCM is located between the top and the middle of the frame to make the stiffness of the upper spring negative. The point of application is attached to the bottom of the frame. The motions of the point of application and the detection point are restricted solely to the vertical translation by the spring mechanism.

Figure 5.

System using positive spring and voice coil motor.

Figure 6.

Picture of fabricated apparatus with one-piece parallel spring mechanism and voice coil motor.

Figure 7.

Picture of parallel spring mechanism with VCM.

One of the more effective ways of downsizing the measurement mechanism is the application of MEMS technology in which electrostatic actuator is mostly used. An apparatus based on MEMS technology is a combination of (I)-(E)-(U), whose configuration is shown in Figure 8. A photo and a schematic drawing of the measuring device are shown in Figure 9 [17]. Electrostatic actuator is used in Suspension I (active element), whose electrode is shown in Figure 10. The displacement sensors are capacitive and have same electrodes. Suspension II (positive-stiffness element) is also fabricated by MEMS technology as shown in Figure 11. To restrict the motion of the point of application to a single translational motion, the point of application is also suspended by this spring. In addition, an electrostatic actuator is also implemented to the point of application to give load to the point of application for testing the performances. Fine force measurement performances have been improved dramatically [17]. It demonstrates that the applicability of the proposed force measuring system will be widened by micromachining.

Figure 8.

System using positive spring and electrostatic actuator.

Figure 9.

Measurement device fabricated by MEMS technology. (a) Picture. (b) Schematic drawing.

Figure 10.

Enlarged picture of comb-shaped electrode.

Figure 11.

Enlarged picture of leaf spring.

The unique features of the zero-compliance-based measuring system are advantageous in measuring fine force that is a function of the distance from the source of force, for example, magnetic or atomic force. The critical problem to industrial use is that the complexity and component count of system inevitably increase in comparison to conventional measuring system, mainly because an active element is necessary.

4. Control and measurement results

4.1 Mathematical model

Control methods of achieving zero-compliance are discussed in this section. The measurement system shown in Figure 4 is targeted. The physical model is shown in Figure 12. The equations of motion are given by [13, 15]

Figure 12.

Physical model of measurement system.

$$m_d\ddot{x}\left(t\right)=-F_e+F_p+m_d\mathit{g}$$E5

$$m_p\ddot{y}\left(t\right)=-F_m+m_p\mathit{g}+f\left(t\right)$$E6

where $x$: displacement of the detection point, $y$: displacement of the point of application, $m_d$: mass of detection point, $m_p$: mass of the point of application, $F_e$: attractive force of the electromagnet $F_m$: attractive force of the permanent magnet, and $f$: force acting on the point of force. In the actual apparatus, the first floator is suspended by leaf spring to restrict the motion to a single-degree-of-freedom translational motion [15]. However, this effect is neglected here for simplification. It is to be noted that this effect has no influence on accuracy in static force measurement theoretically. In the neighborhood of the equilibrium states, the forces $F_e$ and $F_m$ are approximately given by

$$F_e=\left(m_d+m_p\right)\mathit{g}-k_{i}i+k_{s}x$$E7

$$F_m=m_p\mathit{g}+k_m\left(x-y\right)$$E8

where $i$: control current, $k_i$: coefficient of linearized characteristic equation of electromagnet (ratio of force to current), $k_s$: coefficient of linearized characteristic equation of the electromagnet (ratio of force to gap), $k_m$: coefficient of linearized characteristic equation of the permanent magnet (ratio of force to gap). The last coefficient $k_m$ is equal to the absolute value of the negative stiffness of Suspension II. The substitution of Eq. (7) and Eq. (8) into Eq. (5) and Eq. (6) leads to the linearized equations of motion:

$$m_d\ddot{x}\left(t\right)=k_{s}x+k_m\left(x-y\right)-k_{i}i,$$E9

$$m_p\ddot{y}\left(t\right)=-k_m\left(x-y\right)+f\left(t\right),$$E10

Laplace transforming Eq. (9) and Eq. (10) with null initial values leads to

$$m_d{s}^{2}X\left(s\right)=-\left(k_s+k_m\right)X\left(s\right)-k_{m}Y\left(s\right)-k_{i}I\left(s\right),$$E11

$$m_p{s}^{2}Y\left(s\right)=k_m\left(X\left(s\right)-Y\left(s\right)\right)+F\left(s\right),$$E12

where capitals are Laplace-transformed variables.

4.2 Controller

To achieve zero-compliance, there are several applicable control methods, such as

1. Integral feedback of the displacement of the point of application;

2. Displacement cancellation control;

3. Stiffness control includes proportional control and zero-power control [13].

We focus on the method (i). When the control method (i) with state feedback is applied, the control input is represented by

$$I\left(s\right)=\left(p_d+p_{v}s\right)X\left(s\right)+\left(q_d+q_{v}s+\frac{q_I}{s}\right)Y\left(s\right),$$E13

where $p_d$ and $p_v$: proportional and derivative feedback gains related to detection point, $q_d$, $q_v$, and $q_I$: proportional, derivative, and integral feedback gains related to point of application.

4.3 Analysis

Substituting Eq. (13) into Eqs. (11) and (12) and rearranging them gives

$$X\left(s\right)=-\frac{k_i{q}_v{s}^2+\left(k_m+k_i{q}_d\right)s+k_i{q}_I}{{\widehat{t}}_c\left(s\right)}F\left(s\right),$$E14

$$Y\left(s\right)=\frac{m_d{s}^2+k_i{p}_{v}s+k_i{p}_d-\left(k_m+k_s\right)}{{\widehat{t}}_c\left(s\right)}F\left(s\right),$$E15

where

$$\begin{array}{c}{\widehat{t}}_c\left(s\right)=m_d{m}_p{s}^5+m_p{k}_i{p}_v{s}^4+\left[m_p\left\{k_i{p}_d-\left(k_m+k_s\right)\right\}-m_d{k}_m\right]s^3\\ -k_m{k}_i\left(p_v+q_v\right)s^2-k_m\left\{k_i\left(p_d+q_d\right)-k_s\right\}s-k_m{k}_i{q}_I\end{array}$$E16

Assuming that the force acting on the point of force is constant as

$$F\left(s\right)=\frac{f_0}{s},$$E17

and the closed-loop is stable, the steady-state displacements are given by

$$x\left(\infty \right)=\underset{t\to \infty }{lim}x\left(t\right)=\underset{s\to 0}{lim}\mathit{sX}\left(s\right)=\frac{f_0}{k_m}$$E18

$$y\left(\infty \right)=\underset{t\to \infty }{lim}y\left(t\right)=\underset{s\to 0}{lim}\mathit{sY}\left(s\right)=0$$E19

It is proved by Eq. (19) that the point of application is invariant in position (see Figure 2(U)). It is shown by Eq. (18) that the detection point moves in the same direction of the applied force; the displacement is larger as the coefficient $k_m$ is smaller. From Eqs. (18) and (19), we get

$$y\left(\infty \right)-x\left(\infty \right)=-\frac{f_0}{k_m}.$$E20

It indicates that the distance between the detection point and the point of application decreases when the applied force is positive (downward) as if the stiffness between them is negative.

4.4 Measurement results

In the following figures, downward (positive) displacement is plotted to the downward direction for intuitive understanding. Figure 13 shows the control current, the displacement of the point of application, and the displacement of the detection point when static force is added to the point of application [15]. The static force was produced by adding weights to the point of application one by one. One weight corresponds to 24.5 [mN]. After 10 weights were added to the floator, weights were taken off one by one. This result demonstrates that the point of application is almost invariant, while the detection point moves proportionally to the applied force. It is also found that there are slight differences between increasing and decreasing. It was caused by magnetic hysteresis. Meanwhile, the force can be estimated from the control input (current) like by the conventional null method. Comparing it to the displacement of the detection point indicates that the linearity is worse.

Figure 13.

Measurement results; added removed weight corresponds to 24.5 mN.

To study the resolution of the measurement system, smaller weights were added to the point of application. Figures 14–16 show the measurement results when one weight corresponds to 2.45 [mN] (Figure 12), 0.245 [mN] (Figure 13), and 0.0245 [mN] (Figure 14). Figure 14 shows that both the control current and the displacement of the detection point vary proportionally according to the applied force; the linearity is again better in the latter. As shown by Figure 15, the displacement of the detection point is still proportional to the applied force (b), while such linearity is lost in the control current (a). This is due to noise including in the control signal. In Figures 13–15, the displacement of the detection point is proportional to the applied force. In contrast, such clear linear relation is not found in Figure 16. These results indicate that the resolution is 0.245-mN order in the developed apparatus. The mass of the floator with no weight is 69.2 [g]. It indicates that a change of 0.036% (the ratio of 0.025 to 69.2 [g]) in force can be detected by the proposed method.

Figure 14.

Measurement results; added weight corresponds to 2.45 mN.

Figure 15.

Measurement results; added weight corresponds to 0.245 mN.

Figure 16.

Measurement results; added weight corresponds to 0.0245 mN.

5. Conclusions

The principle of force measurement using zero-compliance suspension was explained in a basic model. The primitive zero-compliance is a series-connected two suspensions; one of them has negative stiffness whose absolute value is equal to that of the other suspension with positive stiffness. The connection point becomes the detection point in measurement systems. Practically, at least one of the suspensions must be an active element equipped with an actuator. The measurement systems were classified from three viewpoints related to such suspension. Three examples of measurement system with a possible configuration were presented. Then, the control methods of achieving zero-compliance were explained in a mathematical model of the first example that used double series magnetic suspension to achieve the zero-compliance characteristic. The measurement results were presented which demonstrate the high resolution of the developed system.

The measurement systems have a unique characteristic that the force is estimated from the deflection of the detection point like by a deflection method, while the position of the point of application of force is maintained at the original position (with no applied force) like by a null method. This characteristic is expected to lead to an innovation in the field of force measurement.

This chapter has focused on a single-axis force measurement. Several multi-axis force measurement systems have been also developed [18, 19]. In addition, a measurement device with a cantilever for detecting force has been developed [20]. This device targets scanning microscopes with a novel rigorous force detection mechanism with multi-degree-of-freedom-of-motion zero-compliance suspension [20]. The third example shown in Section 3 also targets such microscopes. The next step will be the development of a multi-degree-of-freedom-of-motion zero-compliance suspension based on the MEMS technology.

Acknowledgments

This work was financially supported in part by JSPS KAKENHI Grant Numbers: 25630073, 17H03188 and 21K18162.

Thanks

The authors thank Mr. Sekine, Mr. Takabayashi, and Mr. Monjiyama for their help in designing and fabricating the apparatuses shown in Section 3 and conducting experimental works shown in Section 4.