# Dynamic Characteristics Analysis of the Six-Axis Force/Torque Sensor

Authors: Liyue Fu; Aiguo Song
Journal: Journal of Sensors
 (2018)

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## Abstract

In this study, dynamic characteristics of a robot six-axis wrist force/torque (F/T) sensor with crossbeam elastomer are analyzed by two methods of model identification, a method for simultaneous identification of order and parameters of the model (SIM) and a method based on the differential evolution (DE) algorithm. Firstly, by establishing the simplified mechanical model and finite element (FE) model, respectively, natural frequency of the six-axis wrist F/T sensor is calculated. Secondly, dynamic calibration experiment is conducted. Lastly, two dynamic models of the sensor are identified by SIM and DE methods and the dynamic characteristics of the sensor, such as natural frequency and working band, are further analyzed. Comparing experimental values with the theoretical values, the results show that this sensor has a wide dynamic range with the first natural frequency at more than 1600 Hz, working bands (±5%) are more than 400 Hz, and the step response oscillation is intense. This study can provide a reference for the application of the six-axis F/T sensor in the field of dynamic measurement.


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## Body

## 1. Introduction

A six-axis force/torque (F/T) sensor can measure the tangential force terms alongX-, Y-, and normal force term along Z-axis (FX, FY, and FZ) as well as the moment terms about X-, Y-, and Z-axis (MX, MY, and MZ) simultaneously. Six-axis F/T sensor can be used in robotics, aerospace, automotive, industrial areas [1]. Dynamic force measurement is more and more used in various industries, which make the dynamic performance analysis of six-axis F/T sensors particularly important. In the field of micromachining, multiaxis F/T sensors are used to measure 3D machining forces which involve very high frequencies due to the high spindle speeds used during the process [2]. As research in vibration-based damage identification increases, multiaxis F/T sensors with high dynamic ranges are used in real-time structural health monitoring [3]. With the quick development of the robot field and the needs of special areas, such as precise assembly, contour tracing, and two-hand coordination [4], six-axis F/T sensor has become one of the most important sensors in the field of the intelligent robot. When F/T sensor is used to sense the collision between robot and environment, it is necessary to detect the size and direction of the dynamic collision force [5]. As a detecting element in the force reflection control system, it should respond quickly to the load, namely, having excellent dynamic characteristics [6]. These applications demonstrate the importance of the research on the dynamic characteristics of the F/T sensor. This study arises from the demand for dynamic measurements in various industries.The acquisition of the dynamic characteristics of the six-axis F/T sensor is based on the dynamic calibration experiment. The dynamic calibration of a six-axis F/T sensor is to obtain the relationship between the input and output of the sensor when performing varied six-axis forces. Essentially, it is a process to acquire dynamic characteristics’ indices such as natural frequency, time constant, and damping ratio. Huang verified that the frequency band has the most influence for the sensor and test system and therefore becomes the most important practical index for dynamic calibration [7]. Regular incentives are impulse, step, and sinusoidal forces, and the corresponding calibration methods are called impulse response, step response, and frequency response methods, respectively [8]. Previously, the research on dynamic characteristics is mainly based on the impulse response method. Xu and Li used a pendulum to exert an impact force along one axis of a multiaxis wrist F/T sensor [9] and developed a dynamic test to compare the dynamic characteristics between the wrist sensor and a JR3® multiaxis F/T sensor [10]. However, only a few channels were utilized, since it is difficult to calibrate the torque, which cannot fully reflect the six-axis wrist F/T sensor’s performance. Liu et al. developed a method-based correlation wavelet for measuring the dynamic performance index, but the cantilever was tested, and the robot’s multiaxis wrist force sensor was only simulated [11]. In this study, the negative step excitation is constructed by cutting the rope of hanging weight to obtain the step response to carry out the dynamic calibration experiment of the six-axis wrist F/T sensor.There has been limited literatures on dynamic performance analysis of the multiaxis F/T sensor, but most of them are not comprehensive enough. Song et al. studied dynamic performance of the 2-axis force sensor applied for the force feedback maglev control system by using an impulse stimulation method [12], yet dynamic indices were not attained. Xu and Zhu carried out the dynamic calibration experiment of the six-axis wrist F/T sensor, without building the necessary difference equation model or transfer function model [13]. Ballo et al. employed a six-axis F/T sensor for the frontal impact test on a Hybrid III 50th percentile dummy [14], but the dynamic analysis method was not given. Thus, the purpose of this paper is to analyze the dynamic characteristics of the six-axis F/T sensor comprehensively.The rest of this paper is organized as follows. Section2 describes the mechanical structure of the F/T sensor. The mechanical model is established, and the relationship between the natural frequency and weight of the elastomer is obtained in Section 3. In Section 4, the first six natural frequencies and vibration modes are acquired by modeling and modal analysis of the sensor in ANSYS software. In Section 5, dynamic calibration experiment is conducted and the dynamic models are established by model identification. The methods of model identification used in this section are a method for simultaneous identification of order and parameters of the model (SIM) and a method based on the differential evolution (DE) algorithm. Furthermore, the dynamic performance indices of the sensor are achieved. Finally, the concluding remarks follow in Section 6.


## 2. Sensor Structure and Calibration Method

As shown in Figure1(a), the prototype of the six-axis wrist F/T sensor for the robot consists of four parts: pedestal, crossbeam elastomer, top cover, and calibration pillar. The sensor’s housing is designed as cylindrical structure to connect with the robotic arm and claws. Crossbeam elastomer is the core component of the sensor, and the performance of the sensor mainly relies on it. Figure 1(b) shows the structure of the crossbeam elastomer. It comprises of four crossbeams, four compliant beams, a central platform, and four rims, which are characterized by a compliant beam flexible link at the connection between a crossbeam and two rims. The force or torque is applied to the six-axis F/T sensor through the calibration pillar which transmits the force or torque to the crossbeam elastomer by connecting with the central platform. Figure 2(a) illustrates the loading setup of six-axis forces or torques. The calibration methods of various channels are depicted in Figures 2(b)–2(e).Figure 1
Prototype of the six-axis wrist F/T sensor for the robot. (a) Mechanical composition of the sensor. (b) Crossbeam elastomer.
(a)
(b)Figure 2
Calibration of the six-axis F/T sensor. (a) Calibration experiment setup. (b) Calibration inFz channel. (c) Calibration in Fx channel. (d) Calibration in My channel. (e) Calibration in Mz channel.
(a)
(b)
(c)
(d)
(e)The excitation force applied to the sensor through the calibration pillar caused deformation of the crossbeam elastomer. Since the compliant beam is very thin, it can be regarded as the flexible link when the applied force is perpendicular to the plane of it and it can be regarded as the ideal rigid body when the applied force is parallel to the plane. Therefore, the force applied to the sensor leads to the bending deformation of two crossbeams whose deformation degree is proportional to the applied force and tension or compression deformation of the other two crossbeams whose deformation degree can be neglected. The resistive strain gauges attached to the bending crossbeams can detect the amplitude of the corresponding force.


## 3. Theoretical Analysis

### 3.1. Simplified Statics Model

To get the equivalent stiffness value of the flexible link of the crossbeam elastomer, simplified statics model is established. Figure3(a) shows the schematic diagram of crossbeam elastomer, where F represents the force applied to the elastomer along Y direction through central platform. Since its width (or height) is much less than the length, the crossbeam can be regarded as a slender beam. When the central platform is subjected to a tensile force along Y direction, bending occurs on crossbeams AB and CD and compliant beams JK and MN. The compliant beams EF and GH are considered as the roller support of the crossbeams AB and CD. As the compliant beams are laminated elements, compliant beams JK and MN are idealized as flexible bodies and crossbeams PQ and RS are idealized as rigid bodies. Thus, the beam AD can be simplified as a simply supported beam with a fixed hinge at one end and a movable hinge at the other, whereas the beams JK and MN as cantilever beams. Figure 3(b) depicts the mechanical model of the whole crossbeam elastomer, where ΔXe and ΔXc are the displacement of midpoints of beams AD and JK, respectively, and FQ and FC are shear forces on beams AD and JK, respectively. The simplified statics model of the crossbeams AB and CD is shown as Figure 3(c).Figure 3
The crossbeam elastomer of the six-axis wrist F/T sensor. (a) The schematic diagram of the elastomer. (b) Mechanical model of crossbeam elastomer underFy. (c) The simplified statics model of the elastomer under Fy. (d) Deformation of the beam AD under force.
(a)
(b)
(c)
(d)According to the principle of the balanced force and moment, the supporting reaction forces of the supports areFA=FB=Fe/2, where Fe represents the force applied to the beam AD which is component of force F and Fe=2FQ. The central platform can be regarded as a lumped mass owing to smaller size relative to the crossbeam, that is, the BC segment in Figure 3(d) can be concentrated on a point O. The shear force and bending moment on an arbitrary section of the beam are as follows:
(1)FQ=Fe2,0≤x≤l2,−Fe2,l2<x≤l,Mx=Fex2,0≤x≤l2,Fel−x2,l2<x≤l.The deformation of the crossbeam can be expressed by the deflection and rotation angle of a point on this beam. As is shown in Figure3(d), the point m on this beam is shifted to the point m1 under the action of the force Fe, the distance between the two points along Y direction is deflection ωx, and the angle at which the cross section turns relative to the original position is rotation angle φx. Both the bending moment and the shear force can cause the deformation of the beam [15].
(2)ωx=vmx+vqx,φx=γmx+γqx,where vmx are γmx are deflection and rotation angle brought on by bending moment, respectively, and vqx and γqx are deflection and rotation angle brought on by shear force, respectively. Based on the differential equation of the deflection and equations (1) and (2), the deflection and rotation angle of a point on this beam are derived [16]:
(3)ωx=Fe2EIx36+C1x+Fex2KGA+C3,0≤x≤l2,Felx24EI−Fex312EI+C2x+C4+Fex2KGA,l2<x≤l,φx=Felx24EI+C1,0≤x≤l2,Fe2EIlx−x22+C2,l2<x≤l,where C1, C2, C3, and C4 are undetermined coefficients whose values can be derived in virtue of boundary and continuous conditions. The final solution of the deflection of the beam is computed as follows:
(4)ωx=Fel2x16EI−Fex312EI+Fex2KGA,0≤x≤l2,Fex312EI−Felx24EI+3Fel2x16EI−Fel348EI+Fel−x2KGA,l2<x≤l,so that the midspan deflection, namely, the displacement of the beam AD, is shown as equation (5). In the same way, the displacement of the beam JK is summarized as equation (6).
(5)ωl2=Fel348EI+Fel4KGA=ΔXe,(6)ΔXc=FGlc324EIc+2FGlcKGAc,where E and G are the elastic modulus and shear modulus of the material, respectively, G=E/1+μ, A and Ac are the sectional area of the beams AD and JK, respectively, and I and Ic are the moment of inertia of the beams AD and JK, respectively. Based on the moment of inertia of a rectangular section, I and Ic can be expressed as equations (7) and (8), respectively.
(7)I=bh312,(8)Ic=bh2312.The material of the crossbeam elastomer of the six-axis wrist F/T sensor is aluminium alloy 2024. Parameters of the crossbeam elastomer are shown in Table1, where b and h denote the width and thickness of crossbeam, respectively, h2 and l2 represent the height and length of compliant beam, respectively, and E and μ are the elastic modulus and Poisson’s ratio of aluminium alloy 2024 which is used to make of the crossbeam elastomer. Moreover, equation (9) can be obtained according to the geometric characteristic of the deformation.
(9)ΔXe=ΔXc.Table 1
Parameters of the cr⋅ossbeam elastomer.
Parameter
b (mm)
h (mm)
l (mm)
h2 (mm)
l2 (mm)
Density (kg/m3)
E (MPa)
μ
Value
3.8
3.8
64
1
32
2820
72,000
0.3By substituting equations (5)–(8) to equation (9), the formula is derived as follows:
(10)FQ=8Fc=0.4F.Subsequently, the equivalent stiffness coefficient is deducted as follows:(11)k=FΔXe=1l3/60EI+l/5KGA.Substituting the parameters in Table1 into equation (7), the equivalent stiffness coefficient can be calculated as
(12)k=1.824×105.


### 3.2. Simplified Dynamics Model

The six-axis wrist F/T sensor is a multiple input and multiple output (MIMO) system. When the sensor is constrained on a foundation surface, the joint between the foundation surface and the pedestal is a rigid connection, and thus the input-output characteristics in any direction can be simplified as a single degree-of-freedom (DOF) system. When the sensor is mounted on the wrist of the robot, the whole mechanism consists of three components: the pedestal of the sensor, the crossbeam elastomer, and the robotic hand. Every component has one translation/rotation DOF in any direction of the six-axis F/T sensor, so the mechanism is a 3-DOF system. As shown in Figure4, m, u, k, and c are the mass, displacement of the mass, equivalent stiffness, and damp, respectively, and subscripts h, t, and k are the hand of the robot, the crossbeam elastomer, and pedestal of the sensor, respectively.Figure 4
The simplified dynamics models of the six-axis wrist F/T sensor for the robot. (a) A single DOF system. (b) A 3-DOF system.
(a)
(b)Assuming thatkh, kt, and kk are all compressed and the displacements of the mass change according to sinusoidal law, the displacements of the masses and the vibration equation are expressed as equations (13) and (14). Then, equation (13) is substituted with equation (14), and equation (15) is obtained.
(13)uhutuk=φhφtφksinωt+θ,(14)mh000mt000mku¨hu¨tu¨k+ch−ck0−chch+ct00−cct+cku̇hu̇tu̇k+kh−kh0−khkh+kt−kt0−ktkt+kkuhutuk=0,(15)Zφhφtφksinωt+θ=0,where Z is mechanical impedance matrix and can be denoted as the following form:
(16)Z=−mhω2+jωch+kh−jωck−kh0−jωch−kh−mtω2+jωch+ct+kh+kt00−jωct−kt−mkω2+jωck+ct+kk+kt.The eigenvalue of the matrixZ is the natural frequency of the system. Considering that the joint between robotic hand and the elastomer and the joint between robotic arm and the pedestal are rigid connections, the 3-DOF system can be simplified to a single-DOF system as shown in Figure 4(a). Moreover, the mass is m=mh+mt, and thus, the natural frequency of the system is resolved in equation (17). Obviously, in case of considering the equivalent mass of the robotic hand, mounted resonance frequency of the six-axis wrist F/T sensor is diminished. In fact, the change of gripping workpiece mass, the mass, shape, and mass distribution of end effector will affect the dynamic characteristics of the sensor. Before testing and evaluating the dynamic properties of the wrist force sensor, it is necessary to determine the state of the sensor and the environment in which it is used. And the unified basis and meaningful results can be obtained under a uniform load.
(17)ω=kmh+mt.In view of the equivalent stiffness coefficient and the mass of the crossbeam elastomer and calibration pillar (4.9×10−2kg), the natural frequency of the six-axis wrist F/T sensor in Fy direction can be calculated to be 1642 Hz. Similarly, the natural frequencies of the sensor’s other channels can be solved as shown in Table 2.Table 2
The results of simplified dynamics analysis of the six-axis F/T sensor.
Channel
Natural frequency (Hz)
Fx
1642
Fy
1642
Fz
1973
Mx
3376
My
3376
Mz
4892


## 3.1. Simplified Statics Model

To get the equivalent stiffness value of the flexible link of the crossbeam elastomer, simplified statics model is established. Figure3(a) shows the schematic diagram of crossbeam elastomer, where F represents the force applied to the elastomer along Y direction through central platform. Since its width (or height) is much less than the length, the crossbeam can be regarded as a slender beam. When the central platform is subjected to a tensile force along Y direction, bending occurs on crossbeams AB and CD and compliant beams JK and MN. The compliant beams EF and GH are considered as the roller support of the crossbeams AB and CD. As the compliant beams are laminated elements, compliant beams JK and MN are idealized as flexible bodies and crossbeams PQ and RS are idealized as rigid bodies. Thus, the beam AD can be simplified as a simply supported beam with a fixed hinge at one end and a movable hinge at the other, whereas the beams JK and MN as cantilever beams. Figure 3(b) depicts the mechanical model of the whole crossbeam elastomer, where ΔXe and ΔXc are the displacement of midpoints of beams AD and JK, respectively, and FQ and FC are shear forces on beams AD and JK, respectively. The simplified statics model of the crossbeams AB and CD is shown as Figure 3(c).Figure 3
The crossbeam elastomer of the six-axis wrist F/T sensor. (a) The schematic diagram of the elastomer. (b) Mechanical model of crossbeam elastomer underFy. (c) The simplified statics model of the elastomer under Fy. (d) Deformation of the beam AD under force.
(a)
(b)
(c)
(d)According to the principle of the balanced force and moment, the supporting reaction forces of the supports areFA=FB=Fe/2, where Fe represents the force applied to the beam AD which is component of force F and Fe=2FQ. The central platform can be regarded as a lumped mass owing to smaller size relative to the crossbeam, that is, the BC segment in Figure 3(d) can be concentrated on a point O. The shear force and bending moment on an arbitrary section of the beam are as follows:
(1)FQ=Fe2,0≤x≤l2,−Fe2,l2<x≤l,Mx=Fex2,0≤x≤l2,Fel−x2,l2<x≤l.The deformation of the crossbeam can be expressed by the deflection and rotation angle of a point on this beam. As is shown in Figure3(d), the point m on this beam is shifted to the point m1 under the action of the force Fe, the distance between the two points along Y direction is deflection ωx, and the angle at which the cross section turns relative to the original position is rotation angle φx. Both the bending moment and the shear force can cause the deformation of the beam [15].
(2)ωx=vmx+vqx,φx=γmx+γqx,where vmx are γmx are deflection and rotation angle brought on by bending moment, respectively, and vqx and γqx are deflection and rotation angle brought on by shear force, respectively. Based on the differential equation of the deflection and equations (1) and (2), the deflection and rotation angle of a point on this beam are derived [16]:
(3)ωx=Fe2EIx36+C1x+Fex2KGA+C3,0≤x≤l2,Felx24EI−Fex312EI+C2x+C4+Fex2KGA,l2<x≤l,φx=Felx24EI+C1,0≤x≤l2,Fe2EIlx−x22+C2,l2<x≤l,where C1, C2, C3, and C4 are undetermined coefficients whose values can be derived in virtue of boundary and continuous conditions. The final solution of the deflection of the beam is computed as follows:
(4)ωx=Fel2x16EI−Fex312EI+Fex2KGA,0≤x≤l2,Fex312EI−Felx24EI+3Fel2x16EI−Fel348EI+Fel−x2KGA,l2<x≤l,so that the midspan deflection, namely, the displacement of the beam AD, is shown as equation (5). In the same way, the displacement of the beam JK is summarized as equation (6).
(5)ωl2=Fel348EI+Fel4KGA=ΔXe,(6)ΔXc=FGlc324EIc+2FGlcKGAc,where E and G are the elastic modulus and shear modulus of the material, respectively, G=E/1+μ, A and Ac are the sectional area of the beams AD and JK, respectively, and I and Ic are the moment of inertia of the beams AD and JK, respectively. Based on the moment of inertia of a rectangular section, I and Ic can be expressed as equations (7) and (8), respectively.
(7)I=bh312,(8)Ic=bh2312.The material of the crossbeam elastomer of the six-axis wrist F/T sensor is aluminium alloy 2024. Parameters of the crossbeam elastomer are shown in Table1, where b and h denote the width and thickness of crossbeam, respectively, h2 and l2 represent the height and length of compliant beam, respectively, and E and μ are the elastic modulus and Poisson’s ratio of aluminium alloy 2024 which is used to make of the crossbeam elastomer. Moreover, equation (9) can be obtained according to the geometric characteristic of the deformation.
(9)ΔXe=ΔXc.Table 1
Parameters of the cr⋅ossbeam elastomer.
Parameter
b (mm)
h (mm)
l (mm)
h2 (mm)
l2 (mm)
Density (kg/m3)
E (MPa)
μ
Value
3.8
3.8
64
1
32
2820
72,000
0.3By substituting equations (5)–(8) to equation (9), the formula is derived as follows:
(10)FQ=8Fc=0.4F.Subsequently, the equivalent stiffness coefficient is deducted as follows:(11)k=FΔXe=1l3/60EI+l/5KGA.Substituting the parameters in Table1 into equation (7), the equivalent stiffness coefficient can be calculated as
(12)k=1.824×105.


## 3.2. Simplified Dynamics Model

The six-axis wrist F/T sensor is a multiple input and multiple output (MIMO) system. When the sensor is constrained on a foundation surface, the joint between the foundation surface and the pedestal is a rigid connection, and thus the input-output characteristics in any direction can be simplified as a single degree-of-freedom (DOF) system. When the sensor is mounted on the wrist of the robot, the whole mechanism consists of three components: the pedestal of the sensor, the crossbeam elastomer, and the robotic hand. Every component has one translation/rotation DOF in any direction of the six-axis F/T sensor, so the mechanism is a 3-DOF system. As shown in Figure4, m, u, k, and c are the mass, displacement of the mass, equivalent stiffness, and damp, respectively, and subscripts h, t, and k are the hand of the robot, the crossbeam elastomer, and pedestal of the sensor, respectively.Figure 4
The simplified dynamics models of the six-axis wrist F/T sensor for the robot. (a) A single DOF system. (b) A 3-DOF system.
(a)
(b)Assuming thatkh, kt, and kk are all compressed and the displacements of the mass change according to sinusoidal law, the displacements of the masses and the vibration equation are expressed as equations (13) and (14). Then, equation (13) is substituted with equation (14), and equation (15) is obtained.
(13)uhutuk=φhφtφksinωt+θ,(14)mh000mt000mku¨hu¨tu¨k+ch−ck0−chch+ct00−cct+cku̇hu̇tu̇k+kh−kh0−khkh+kt−kt0−ktkt+kkuhutuk=0,(15)Zφhφtφksinωt+θ=0,where Z is mechanical impedance matrix and can be denoted as the following form:
(16)Z=−mhω2+jωch+kh−jωck−kh0−jωch−kh−mtω2+jωch+ct+kh+kt00−jωct−kt−mkω2+jωck+ct+kk+kt.The eigenvalue of the matrixZ is the natural frequency of the system. Considering that the joint between robotic hand and the elastomer and the joint between robotic arm and the pedestal are rigid connections, the 3-DOF system can be simplified to a single-DOF system as shown in Figure 4(a). Moreover, the mass is m=mh+mt, and thus, the natural frequency of the system is resolved in equation (17). Obviously, in case of considering the equivalent mass of the robotic hand, mounted resonance frequency of the six-axis wrist F/T sensor is diminished. In fact, the change of gripping workpiece mass, the mass, shape, and mass distribution of end effector will affect the dynamic characteristics of the sensor. Before testing and evaluating the dynamic properties of the wrist force sensor, it is necessary to determine the state of the sensor and the environment in which it is used. And the unified basis and meaningful results can be obtained under a uniform load.
(17)ω=kmh+mt.In view of the equivalent stiffness coefficient and the mass of the crossbeam elastomer and calibration pillar (4.9×10−2kg), the natural frequency of the six-axis wrist F/T sensor in Fy direction can be calculated to be 1642 Hz. Similarly, the natural frequencies of the sensor’s other channels can be solved as shown in Table 2.Table 2
The results of simplified dynamics analysis of the six-axis F/T sensor.
Channel
Natural frequency (Hz)
Fx
1642
Fy
1642
Fz
1973
Mx
3376
My
3376
Mz
4892


## 4. Modal Analysis

Vibration modal is one of the intrinsic characteristics of elastic structures. The modal analysis of the elastomer is carried out in ANSYS software, and the inherent vibration properties of the six-axis wrist F/T sensor are studied, including the undamped natural frequency, damped ratio, and vibration modal. The modal extraction method is set to block lanczos, and the number of extraction is 6. The results are shown in Table3, and six vibration modes are shown in Figures 5(a)–5(f).Table 3
The results of modal analysis of the six-axis F/T sensor.
Set
Natural frequency (Hz)
Vibration mode
1
1660
Translation alongX-axis
2
1661
Translation alongY-axis
3
2025
Translation alongZ-axis
4
3577
Rotation aroundX-axis
5
3578
Rotation aroundY-axis
6
5094
Rotation aroundZ-axisFigure 5
The first six-order vibration mode of the six-axis wrist F/T sensor for the robot. (a) The first-order mode. (b) The second-order mode. (c) The third-order mode. (d) The fourth-order mode. (e) The fifth-order mode. (f) The sixth-order mode.
(a)
(b)
(c)
(d)
(e)
(f)


## 5. Dynamic Calibration Experiment and Model Identification

### 5.1. Dynamic Calibration Experiment

The step response method is used to calibrate the six-axis F/T sensor. The negative step excitation is constructed by cutting the rope of hanging weight to obtain the step response to carry out the dynamic calibration experiment of the six-axis wrist F/T sensor. One end of the rope is hung on the calibration pillar, and the other end suspended the weight by a pulley. The rope is cut off at a given time, thus a negative step excitation is applied to the six-axis F/T sensor. The cotton rope, fishing line, and ordinary plastic rope are, respectively, used to suspend weights during calibration experiment. Figure6 shows the three ropes. The test results show that the fishing line has the best effect among the three ropes; hence, the fishing line with diameter of 0.4 mm is used in the dynamic calibration experiment. The step responses of the sensor have been acquired via a high-speed Smacq® USB-4620 data acquisition card and transmitted to the host computer. The sampling frequency of this data acquisition card can be changed by software and set to 100 kHz in the dynamic calibration experiment. Figure 2(a) presents the experimental facilities.Figure 6
The three ropes used in dynamic calibration experiment.The method of constructing step excitation by cutting the rope of hanging weight during dynamic calibration experiment is simple, but the step rise time is long. Force excitation can be generated in other ways, such as dynamic pressure generator based on a double-acting pneumatic actuator [17].


### 5.2. Model Identification Based on the SIM Method

Based on the acquired input and output data, the dynamic models are established by model identification. The methods of model identification used in this paper are the SIM method and DE method. The SIM method is a method for simultaneous identification of order and parameters of the model. This method can directly calculate the minimum value of the index function corresponding to all the models of interest, and then the order is determined. Compared with the least square method, this approach has some important features, such as less calculation, simpler program, and more flexible applications. Meanwhile, the problem of numerical “morbid” caused by ill-conditioned regular equations in the least square method is avoided.Let the differential equation of the six-axis F/T sensor inFx channel is described as follows:
(18)Bd−1uk−Ad−1yk=εk,where uk and yk are the input and output observations of the sensor, respectively, d−1 is a backward shift operator, εk is the fitting error, Ad−1=1+a1d−1+a2d−2+⋯+and−n, and Bd−1=b0+b1d−1+b2d−2+⋯+bnd−n.The negative step excitation constructed and step response acquired in experiment are shown in Figure7. The relationship curve between the index function, described as the sum of squared residuals, and model order is depicted in Figure 8. It can be seen that the changes after J3 is very slow, and thus the model order is 3. Then, the model parameter vector can be expressed as equation (19) by the SIM method model identification. The difference equation and transfer function of the sensor are shown in equations (20) and (21), respectively.
(19)θn=0.00830.58410.0047−0.2608−0.0011−1.30760.0037T,(20)yn−1.3076yn−1−0.2608yn−2+0.5841yn−3=0.0037xn−0.0011xn−1+0.0047xn−2+0.0083xn−3,(21)Gs=8.2e−4s3+4430s2−5.36e8s+8.532e13s3+8.643e5s2+5.494e9s+8.587e13.Figure 7
The input and output signals of the sensor inFx channel in dynamic calibration experiment.Figure 8
The relationship curve between the index function and model order.The negative step excitation constructed is used to get the model’s response. Figure9(a) shows that the calculated values of the model fit well with the experimental response curve, and this model can be used. Bode diagram of this model is depicted in Figure 9(b).Figure 9
The model response and bode diagram of the sensor. (a) The model response in comparison with test response. (b) Bode diagram of the model.
(a)
(b)From the bode diagram and transfer function, we can see that the natural frequency of the model is 1720 Hz and the working bands of amplitude error within±10% and within ±5% are 560 Hz and 414 Hz, respectively. Compared with the aforementioned theoretical analysis and modal analysis, this model identified by the SIM method is correct.So the dynamic models from other channels can be also identified by the SIM method. The dynamic performance indices of the sensor can be obtained as shown in Table4.Table 4
The dynamic performance indices of the model identified by the SIM method.
Index
Fx
My
Mz
Fz
Natural frequency (Hz)
1720
3530
5060
2020
Working band (±10%) (Hz)
560
997
1622
652
Working band (±5%) (Hz)
414
693
1294
475


### 5.3. Model Identification Based on the DE Algorithm

Differential evolution (DE) algorithm is a kind of the optimization algorithm based on swarm intelligence theory, which is often used to solve the problem of optimization search in complex real space. The special memory ability of the DE algorithm enables it to dynamically track the current search situation to adjust its search strategy. It has strong global convergence ability and robustness and does not need the characteristics’ information of the problem. The parameter identification of the complex system can be realized.The basic idea of the DE algorithm is that starting from a random initial population, a new individual is generated by summing weighted vector difference between any two individuals in the population and the third individual within the certain rules. By comparing to a predetermined individual, if the fitness of the new individual is better, the new individual will replace the old one in the next generation; otherwise the old remains. Through continuous iterative computation, we retain good individuals, eliminate inferior individuals, and guide the search process to approach the optimal solution.Assumed to be a three-order model, the discrete transfer function is expressed as equation (22) and parameters to be identified can be written as follows: θ^=b3a3b2a2b1a1. The ranges of these parameters are −5,5. In the DE algorithm, the mutation factor is 0.8, the cross factor is 0.7, the size of samples is 100, and the maximum of iterations is 500. After 60 steps of iteration, the best sample can be obtained as equation (23). Figures 10(a) and 10(b) depict the fitness-iteration relational curve and the model response, respectively.
(22)Hz=b3z−3+b2z−2+b1z−1a3z−3+a2z−2+a1z−1+1,(23)θ^=0.0063−0.8597−0.012812.7020.00798−2.841.Figure 10
The dynamic model identification of the six-axis wrist F/T sensor inFx channel. (a) The fitness-iteration relational curve. (b) The response of the model obtained by DE algorithm identification.
(a)
(b)In the light of the transfer function identified by the DE algorithm, the bode diagram can be depicted as shown in Figure11.Figure 11
The bode diagram of the model identified by the DE algorithm in the sensor’sFx channel.From Figure11 and equation (23), it can be obtained that the natural frequency is 1680 Hz and the working bands of amplitude error within ±10% and within ±5% are 530 Hz and 395 Hz, respectively. These results agree with the model identified by the SIM method.In the same way, the model identification results ofMy, Mz, and Fz channels are shown as Figure 12. The transfer functions of the three channels are identified as equations (24)–(26). The dynamic performance indices are shown as Table 5.
(24)Hz=−0.01755z−1+0.04019z−2−0.02001z−31−2.797z−1+2.615z−2−0.816z−3,(25)Hz=−0.00606z−1+0.009565z−2−0.003314z−31−1.009z−1−0.9739z−2+0.9831z−3,(26)Hz=−0.0007735z−1−0.002627z−2+0.005051z−31−1.053z−1−0.8864z−2+0.941z−3.Figure 12
The model’s identification results of other channels. (a)My channel. (b) Mz channel. (c) Fz channel.
(a)
(b)
(c)Table 5
The dynamic performance indices of the model identified by the DE algorithm.
Index
Fx
My
Mz
Fz
Natural frequency (Hz)
1680
3460
4910
1986
Working band (±10%) (Hz)
530
980
1610
648
Working band (±5%) (Hz)
405
678
1285
466Combined with the aforementioned models, the dynamic performance indices are integrated into Table6. It can be seen in the table that the analysis results of these models are consistent and the first natural frequency is more than 1600 Hz.Table 6
The natural frequencies derived from various models.
Model
Fx
My
Mz
Fz
Simplified dynamics model
1642 Hz
3376 Hz
4892 Hz
1973 Hz
Finite element model
1660 Hz
3578 Hz
5094 Hz
2025 Hz
The model identified by the SIM method
1720 Hz
3530 Hz
5060 Hz
2020 Hz
The model identified by the DE algorithm
1680 Hz
3460 Hz
4910 Hz
1986 Hz
Maximum difference among the above models
4.5%
5.6%
4.0%
2.6%


## 5.1. Dynamic Calibration Experiment

The step response method is used to calibrate the six-axis F/T sensor. The negative step excitation is constructed by cutting the rope of hanging weight to obtain the step response to carry out the dynamic calibration experiment of the six-axis wrist F/T sensor. One end of the rope is hung on the calibration pillar, and the other end suspended the weight by a pulley. The rope is cut off at a given time, thus a negative step excitation is applied to the six-axis F/T sensor. The cotton rope, fishing line, and ordinary plastic rope are, respectively, used to suspend weights during calibration experiment. Figure6 shows the three ropes. The test results show that the fishing line has the best effect among the three ropes; hence, the fishing line with diameter of 0.4 mm is used in the dynamic calibration experiment. The step responses of the sensor have been acquired via a high-speed Smacq® USB-4620 data acquisition card and transmitted to the host computer. The sampling frequency of this data acquisition card can be changed by software and set to 100 kHz in the dynamic calibration experiment. Figure 2(a) presents the experimental facilities.Figure 6
The three ropes used in dynamic calibration experiment.The method of constructing step excitation by cutting the rope of hanging weight during dynamic calibration experiment is simple, but the step rise time is long. Force excitation can be generated in other ways, such as dynamic pressure generator based on a double-acting pneumatic actuator [17].


## 5.2. Model Identification Based on the SIM Method

Based on the acquired input and output data, the dynamic models are established by model identification. The methods of model identification used in this paper are the SIM method and DE method. The SIM method is a method for simultaneous identification of order and parameters of the model. This method can directly calculate the minimum value of the index function corresponding to all the models of interest, and then the order is determined. Compared with the least square method, this approach has some important features, such as less calculation, simpler program, and more flexible applications. Meanwhile, the problem of numerical “morbid” caused by ill-conditioned regular equations in the least square method is avoided.Let the differential equation of the six-axis F/T sensor inFx channel is described as follows:
(18)Bd−1uk−Ad−1yk=εk,where uk and yk are the input and output observations of the sensor, respectively, d−1 is a backward shift operator, εk is the fitting error, Ad−1=1+a1d−1+a2d−2+⋯+and−n, and Bd−1=b0+b1d−1+b2d−2+⋯+bnd−n.The negative step excitation constructed and step response acquired in experiment are shown in Figure7. The relationship curve between the index function, described as the sum of squared residuals, and model order is depicted in Figure 8. It can be seen that the changes after J3 is very slow, and thus the model order is 3. Then, the model parameter vector can be expressed as equation (19) by the SIM method model identification. The difference equation and transfer function of the sensor are shown in equations (20) and (21), respectively.
(19)θn=0.00830.58410.0047−0.2608−0.0011−1.30760.0037T,(20)yn−1.3076yn−1−0.2608yn−2+0.5841yn−3=0.0037xn−0.0011xn−1+0.0047xn−2+0.0083xn−3,(21)Gs=8.2e−4s3+4430s2−5.36e8s+8.532e13s3+8.643e5s2+5.494e9s+8.587e13.Figure 7
The input and output signals of the sensor inFx channel in dynamic calibration experiment.Figure 8
The relationship curve between the index function and model order.The negative step excitation constructed is used to get the model’s response. Figure9(a) shows that the calculated values of the model fit well with the experimental response curve, and this model can be used. Bode diagram of this model is depicted in Figure 9(b).Figure 9
The model response and bode diagram of the sensor. (a) The model response in comparison with test response. (b) Bode diagram of the model.
(a)
(b)From the bode diagram and transfer function, we can see that the natural frequency of the model is 1720 Hz and the working bands of amplitude error within±10% and within ±5% are 560 Hz and 414 Hz, respectively. Compared with the aforementioned theoretical analysis and modal analysis, this model identified by the SIM method is correct.So the dynamic models from other channels can be also identified by the SIM method. The dynamic performance indices of the sensor can be obtained as shown in Table4.Table 4
The dynamic performance indices of the model identified by the SIM method.
Index
Fx
My
Mz
Fz
Natural frequency (Hz)
1720
3530
5060
2020
Working band (±10%) (Hz)
560
997
1622
652
Working band (±5%) (Hz)
414
693
1294
475


## 5.3. Model Identification Based on the DE Algorithm

Differential evolution (DE) algorithm is a kind of the optimization algorithm based on swarm intelligence theory, which is often used to solve the problem of optimization search in complex real space. The special memory ability of the DE algorithm enables it to dynamically track the current search situation to adjust its search strategy. It has strong global convergence ability and robustness and does not need the characteristics’ information of the problem. The parameter identification of the complex system can be realized.The basic idea of the DE algorithm is that starting from a random initial population, a new individual is generated by summing weighted vector difference between any two individuals in the population and the third individual within the certain rules. By comparing to a predetermined individual, if the fitness of the new individual is better, the new individual will replace the old one in the next generation; otherwise the old remains. Through continuous iterative computation, we retain good individuals, eliminate inferior individuals, and guide the search process to approach the optimal solution.Assumed to be a three-order model, the discrete transfer function is expressed as equation (22) and parameters to be identified can be written as follows: θ^=b3a3b2a2b1a1. The ranges of these parameters are −5,5. In the DE algorithm, the mutation factor is 0.8, the cross factor is 0.7, the size of samples is 100, and the maximum of iterations is 500. After 60 steps of iteration, the best sample can be obtained as equation (23). Figures 10(a) and 10(b) depict the fitness-iteration relational curve and the model response, respectively.
(22)Hz=b3z−3+b2z−2+b1z−1a3z−3+a2z−2+a1z−1+1,(23)θ^=0.0063−0.8597−0.012812.7020.00798−2.841.Figure 10
The dynamic model identification of the six-axis wrist F/T sensor inFx channel. (a) The fitness-iteration relational curve. (b) The response of the model obtained by DE algorithm identification.
(a)
(b)In the light of the transfer function identified by the DE algorithm, the bode diagram can be depicted as shown in Figure11.Figure 11
The bode diagram of the model identified by the DE algorithm in the sensor’sFx channel.From Figure11 and equation (23), it can be obtained that the natural frequency is 1680 Hz and the working bands of amplitude error within ±10% and within ±5% are 530 Hz and 395 Hz, respectively. These results agree with the model identified by the SIM method.In the same way, the model identification results ofMy, Mz, and Fz channels are shown as Figure 12. The transfer functions of the three channels are identified as equations (24)–(26). The dynamic performance indices are shown as Table 5.
(24)Hz=−0.01755z−1+0.04019z−2−0.02001z−31−2.797z−1+2.615z−2−0.816z−3,(25)Hz=−0.00606z−1+0.009565z−2−0.003314z−31−1.009z−1−0.9739z−2+0.9831z−3,(26)Hz=−0.0007735z−1−0.002627z−2+0.005051z−31−1.053z−1−0.8864z−2+0.941z−3.Figure 12
The model’s identification results of other channels. (a)My channel. (b) Mz channel. (c) Fz channel.
(a)
(b)
(c)Table 5
The dynamic performance indices of the model identified by the DE algorithm.
Index
Fx
My
Mz
Fz
Natural frequency (Hz)
1680
3460
4910
1986
Working band (±10%) (Hz)
530
980
1610
648
Working band (±5%) (Hz)
405
678
1285
466Combined with the aforementioned models, the dynamic performance indices are integrated into Table6. It can be seen in the table that the analysis results of these models are consistent and the first natural frequency is more than 1600 Hz.Table 6
The natural frequencies derived from various models.
Model
Fx
My
Mz
Fz
Simplified dynamics model
1642 Hz
3376 Hz
4892 Hz
1973 Hz
Finite element model
1660 Hz
3578 Hz
5094 Hz
2025 Hz
The model identified by the SIM method
1720 Hz
3530 Hz
5060 Hz
2020 Hz
The model identified by the DE algorithm
1680 Hz
3460 Hz
4910 Hz
1986 Hz
Maximum difference among the above models
4.5%
5.6%
4.0%
2.6%


## 6. Conclusions

In this study, a systematic theoretical analysis, modal analysis, dynamic calibration experiment, and model identification are carried out for testing and verifying the six-axis wrist F/T sensor. The SIM method and DE algorithm are used to identify the dynamic models. From Tables2–6, the maximum difference among the aforementioned four models within 5.6% and the dynamic performance indices obtained by the identified models are consistent with the results from the theoretical analysis and thus showing the identified models are reasonable. The results from the dynamic analysis for the six-axis wrist F/T sensor provide a theoretical basis for the application of the sensor in the dynamic working environment.The results show that this sensor has a wide dynamic range with the first resonant frequency at more than 1600 Hz, working bands (±5%) are more than 400 Hz, and the step response oscillation is intense. To widen its working band and reduce the overshoot, it is necessary to add a compensation link to the sensor which will be the future work. In this dynamic calibration experiment, each channel is individually excited and only this channel’s responses are acquired simultaneously, so the coupling is weak and there are no decoupling in this experiment. However, the coupling is serious in actual dynamic applications, and dynamic decoupling of the six-axis F/T sensor will also be the future work.



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*Source: 6216979-2018-12-23.xml*