Multi-dimensional dynamic measurement uncertainty analysis for the six-axis force sensor used in docking system

Abstract

The tough space environment has a great influence on the reliability of the measurement results owing to the rapidly aging sensor. Also, the measurement uncertainty caused by the random factors seriously affects the accuracy of the sensor. Aiming at the multidimensional dynamic uncertainty problem of the six-axis force measurement, a six-axis force sensor for docking system is presented based on the piezoelectric force sensing element. The decoupling algorithm of the six-axis force sensor is proposed. Grey prediction model is employed to establish the dynamic uncertainty model in single channel. Residual test, correlation test and posterior error test are conducted to verify the reliability of the prediction model. The multidimensional dynamic uncertainty model of the six-axis force sensor is derived by combination of the decoupling algorithm and the dynamic uncertainty in single channel. The multidimensional dynamic uncertainty model of the six-axis force sensor can avoid the error signals for the force feedback control in the space environment, and the model can also be employed for the measurement of six-axis force in any other harsh conditions.

Keywords

Six-axis force sensor multi-dimensional dynamic measurement parametric uncertainty grey prediction model

Introduction

Peripheral type docking mechanism is usually used for spacecraft. And the docking mechanism may be damaged for large impact force. So, the six-axis force of the docking mechanism should be measured to release the impact force. Different types of six-axis force sensor have been developed in recent years. Yao et al. (2015, 2016) proposed a fully pre-stressed dual-layer parallel six-axis wrist force sensor, whose force isotropy index was analyzed. Yuan et al. (2015) developed a six-axis force sensor with using rectangular cross-sectional beams to measure forces and moments exerted on humanoid robot foot for controlling the robot. Sun et al. (2015) designed a six-axis force/moment sensor with through-hole beam for the space robot, and the nonlinearity, repeatability, stability, hysteresis, sensitivity and accuracy were analyzed. Wang et al. (2012) proposed a fully pre-stressed dual-layer six-axis force/moment sensor based on a modified Stewart platform architecture, and the measurement error in all the directions was discussed. Kang et al. (2014) presented the design optimization of a mechanically decoupled six-axis force/moment (F/T) sensor by minimization of cross coupling error. Mastinu et al. (2011) and Gobbi et al. (2011) presented a high precision six-axis load cell with a three spoke structure, the error analysis including linearity, sensitivity, accuracy and repeatability was conducted. Wang (2013) developed a novel six-axis force sensor combined parallel plate structures and one-block structure for wind tunnel model test, and the sensitivity design error, the relative measurement error and the interference error were given. Shi et al. (2012) proposed a new large range flexible joints 6-UPUR six-axis force sensor. The influences of system noise, calibration matrix errors, processing and installation errors, structural deformation on the platform, hydraulic loading system and data acquisition system errors were discussed. As above, researches on the six-axis force sensor were always focused on the structure and the deterministic error. The multi-dimensional dynamic uncertainty of the six-axis force sensor has been lacking investigation. However, the space environment is so bad that the reliability of the measurement results will be reduced. The measurement uncertainty caused by the random factors seriously affects the accuracy of the six-axis force sensor. Therefore, the multi-dimensional dynamic measurement uncertainty of the six-axis force sensor should be considered.

In this paper, a piezoelectric six-axis force sensor for the docking system is presented. Also, the decoupling algorithm of the six-axis force sensor is proposed. Grey system model is employed to predict the dynamic uncertainty of each channel. Residual test, correlation test and posterior error test are conducted to verify the reliability of the prediction model. Combining the decoupling algorithm with the dynamic uncertainty in single channel, the multidimensional dynamic uncertainty of six-axis force sensor is deduced.

Establishment of the measurement system

Peripheral docking mechanism is shown in Figure 1, and four three-axis force sensors are arranged in the docking ring. The six-axis force can be calculated according to the decoupling algorithm by the measurement values of the four three-axis force sensors.

Figure 1.

Structure of docking ring with force sensors. (a) Peripheral docking mechanism; (b) Local enlarged drawing of the docking ring.

The entire measurement system architecture is shown in Figure 2. The charge signals in 12 channels are amplified and converted to voltage signals by the charge amplifier. Also, the voltage signals are received by the data acquisition card.

Figure 2.

The entire measurement system architecture.

Multidimensional dynamic uncertainty of the six-axis force sensor

Piezoelectric type six-axis force sensor is a complex dynamic measurement system, and it is difficult to establish an accurate mathematical model of multidimensional dynamic measurement uncertainty. Therefore, the complex modeling process of the multidimensional dynamic measurement uncertainty can be completed by three steps:

The relationship between the six-axis force and the forces in 12 channels is presented by the decoupling algorithm.

The measurement uncertainty in single channel is deduced by grey prediction model.

Combining the decoupling algorithm of the six-axis force with the grey prediction model of single channel, the multidimensional dynamic measurement uncertainty can be evaluated.

The decoupling algorithm of the six-axis force sensor with spatial layout

The six-axis force sensor is composed of four three-axis force sensors, whose spatial layout are shown in Figure 3. O-XYZ is the coordinate system of the six-axis force sensor center. Also, four three-axis force sensors are evenly distributed around the center axis of the six-axis force sensor.

Figure 3.

Four points layout of the three-axis force sensor.

If the force vector $F = {[F_{x}, F_{y}, F_{z}, M_{x}, M_{y}, M_{z}]}^{T}$ is applied on point O, the model of the six-axis force sensor can be established in equation (1) based on the force balance condition (Sun, 1988) .

{\begin{matrix} F_{x} = \sum_{i = 1}^{4} F_{xi} \\ F_{y} = \sum_{i = 1}^{4} F_{yi} \\ F_{z} = \sum_{i = 1}^{4} F_{zi} \\ M_{x} = b (- F_{z 1} + F_{z 2} + F_{z 3} - F_{z 4}) \\ M_{y} = b (- F_{z 1} - F_{z 2} + F_{z 3} + F_{z 4}) \\ M_{z} = b (F_{x 1} - F_{x 2} - F_{x 3} + F_{x 4} + F_{y 1} + F_{y 2} - F_{y 3} - F_{y 4}) \end{matrix}

(1)

where, b is the projection distance between the center of the three-axis force sensor and the X or Y axis.

Dynamic measurement uncertainty model of single channel in the six-axis force sensor

All the data are acquired without load on the six-axis force sensor because the measurement value drift of the force sensing element is only considered, and other errors will be taken if the sensor is loaded. However, the drift value may be positive or negative. And GM(1,1) model employed to predict the measurement uncertainty is not suitable for the data less than zero. Then the constant 100 is added to each measurement value for the convenient calculation. Also, the constant 100 will be subtracted during the process of obtaining the uncertainty. Each channel is measured once every minute in 10 minutes. The 12 channels signal can be denoted as $F_{xi} (i = 1, 2, 3, 4)$ , $F_{yi} (i = 1, 2, 3, 4)$ and $F_{zi} (i = 1, 2, 3, 4)$ . The 10 groups measurement data in 12 channels are listed in Table 1.

Table 1.

10 Groups measurement data in 12 channels.

F_x1	100.000	99.999	99.999	99.999	100.001	99.998	99.998	99.999	99.999	100.000
F_x2	100.000	99.992	99.993	99.993	99.993	99.993	99.993	99.992	99.991	99.992
F_x3	99.999	100.000	100.000	100.000	100.002	100.003	100.004	100.005	100.006	100.007
F_x4	100.000	99.995	99.996	99.998	99.998	99.999	100.000	100.000	100.000	100.000
F_y1	100.003	100.003	100.004	100.004	100.004	100.005	100.006	100.007	100.006	100.007
F_y2	100.003	100.004	100.004	100.006	100.006	100.007	100.007	100.007	100.008	100.007
F_y3	100.003	100.004	100.005	100.007	100.007	100.007	100.007	100.008	100.008	100.007
F_y4	99.995	99.995	99.997	99.998	99.999	99.999	100.000	100.000	100.000	100.000
F_z1	100.002	100.014	100.029	100.035	100.045	100.054	100.061	100.067	100.072	100.074
F_z2	99.995	99.996	99.996	99.998	99.999	100.000	100.001	100.001	100.003	100.000
F_z3	100.012	100.013	100.014	100.014	100.014	100.015	100.015	100.017	100.017	100.018
F_z4	100.012	100.013	100.015	100.016	100.018	100.019	100.021	100.022	100.023	100.024

Establishment of grey prediction model GM(1,1)

(1) Generation of the accumulative sequence and verification of the applicability

Suppose the number of the measurement value is n before time t, and the data sequence is denoted as $X^{(0)}$ , which can be expressed as

X^{(0)} = (x^{(0)} (1), x^{(0)} (2), \dots, x^{(0)} (k), \dots, x^{(0)} (n))

(2)

Resample data sequence $X^{(0)}$ , and the data can be returned to the set with equal probability during the process of resampling. The resampling is repeated B times, then B groups of resampling samples are obtained as

X_{Bootstrap} = (X_{1}, X_{2}, \dots, X_{b}, \dots, X_{B})

(3)

where, $X_{b}$ is the bth resampling sample, and $X_{b}$ can be expressed as

X_{b} = {x_{b} (k)} (b = 1, 2, \dots, B)

(4)

where, $x_{b} (k)$ is the kth resampling data in $X_{b}$ .

According to the GM(1,1) model, the accumulated sequence of resampling data sequence ${X_{b}}^{(0)}$ can be expressed as

{X_{b}}^{(1)} = ({x^{(1)}}_{b} (1), {x^{(1)}}_{b} (2), \dots, {x^{(1)}}_{b} (k) \dots, {x^{(1)}}_{b} (n))

(5)

where, ${x^{(1)}}_{b} (k) = \sum_{i = 1}^{k} {x^{(0)}}_{b} (i)$

If ${X_{b}}^{(1)}$ meets the quasi-exponential law and quasi-smooth condition, ${X_{b}}^{(1)}$ will meet the qualifications to establish the grey model GM(1,1).

For the six-axis force sensor, the resampling data series of the measurement value of $F_{x 1}$ is used as the original data sequence ${X_{b}}^{(0)}$ . Then the data sequence ${X_{b}}^{(1)}$ can be obtained according to equation (5), and the two data sequence can be shown in Table 2.

Table 2.

Data sequence of ${X_{b}}^{(0)}$ and ${X_{b}}^{(1)}$ .

${X_{b}}^{(0)}$	99.998	100.00	99.999	100.00	100.000	99.999	99.999	99.999	99.999	99.998
${X_{b}}^{(1)}$	99.998	199.998	299.997	399.997	499.997	599.996	699.995	799.994	899.993	999.991

Test formulas for quasi-smooth property and quasi-exponential law are used to test the fitness of $x_{b}^{1} (k)$ for grey system model. According to the test formula for quasi-smooth property, the smooth ratio $ρ (k)$ of $x_{b}^{1} (k)$ can be expressed as Liu et al. (2013)

ρ (k) = \frac{x_{b}^{0} (k)}{x_{b}^{1} (k - 1)}, k = 2, 3, \dots, n

(6)

Substitute $x_{b}^{0} (k)$ and $x_{b}^{1} (k - 1)$ into equation (6), each $ρ (k)$ is less than 0.5.

Test formula of quasi-exponential law is as follows

σ (k) = \frac{x_{b}^{1} (k)}{x_{b}^{1} (k - 1)}, k = 2, 3, \dots, n

(7)

Substitute $x_{b}^{1} (k)$ and $x_{b}^{1} (k - 1)$ into equation (7), $σ (k)$ is less than 2.

Then $ρ (k)$ and $σ (k)$ meet the requirement. ${X_{b}}^{(1)}$ can be used to establish the grey prediction model GM(1,1).

In order to smooth the data sequence, consecutive neighbor sequences of $Z_{b}^{(1)}$ are obtained as follows

Z_{b}^{(1)} = (z_{b}^{(1)} (2), z_{b}^{(1)} (3), \dots, z_{b}^{(1)} (k) \dots, z_{b}^{(1)} (n)) k = 2, 3, \dots, n

(8)

where, $z_{b}^{(1)} (k) = 0.5 x_{b}^{(1)} (k) + 0.5 x_{b}^{(1)} (k - 1)$

The differential equation of GM (1,1) grey prediction model is as follows

\frac{d x^{(1)}}{dt} + a x^{(1)} = u

(9)

Equation (9) is discretized, and then the discretized grey prediction model can be expressed as

{\hat{x}}_{b}^{(0)} (k + 1) + {az}_{b}^{(1)} (k + 1) = u

(10)

Set $B = {[\begin{matrix} - z_{b}^{(1)} (2) & - z_{b}^{(1)} (3) & \dots & - z_{b}^{(1)} (10) \\ 1 & 1 & \dots & 1 \end{matrix}]}^{T}$ , $Y = {[x_{b}^{(0)} (2), x_{b}^{(0)} (3), \dots, x_{b}^{(0)} (k), \dots, x_{b}^{(0)} (n)]}^{T}$ , then equation (10) can be translated as follows

Y = B \cdot {[a, u]}^{T}

(11)

According to the principle of least residual square sum, parameters a and u can be obtained as follows

[a, u]^{T} = {[B^{T} B]}^{- 1} B^{T} Y = [\begin{matrix} \hat{a} \\ \hat{b} \end{matrix}] = [\begin{matrix} 1.8333 E - 6 \\ 100.0002 \end{matrix}]

(12)

Substitute a and u into equation (10), the discretized grey prediction model is as follows

{\hat{x}}_{b}^{(1)} (k + 1) = (- 5.4546455 E 7) e^{(- 1.8333 E - 6) k} + (5.4546555 E 6)

(13)

According to equation (13), data sequence ${\hat{X}}_{b}^{(1)}$ can be obtained as ${\hat{X}}_{b}^{(1)}$ = [ 99.9980, 199.9980, 299.9977, 399.9973, 499.9967, 599.9959, 699.9950, 799.9938,899.9925, 999.9910].

Then predicted value of ${\hat{X}}_{b}^{(0)}$ can be obtained by the accumulative reduction equation as follows

\begin{array}{l} {\hat{x}}_{b}^{(0)} (k + 1) = (- 5.4546455 E + 7) \\ [e^{(- 1.8333 E - 6) K} - e^{(- 1.8333 E - 6) (k - 1)}] \end{array}

(14)

According to equation (14), the predicted value of $X_{b}^{(0)}$ is obtained as ${\hat{X}}_{b}^{(0)} =$ [99.9980, 100.000 99.9998 99.9996 99.9994 99.9992 99.9990 99.9989 99.9987 99.9985].

Test of the prediction accuracy

The precision test should be conducted before the grey prediction model is used to predict the data sequence. The most common test methods for prediction precision are the residual difference test, correlation test and posterior-variance-test.

(1) Residual test

Relative residual and average relative residual of the prediction model can be expressed as

e_{b}^{(0)} (k) % = \frac{x_{b}^{(0)} (k) - {\hat{x}}_{b}^{(0)} (k)}{x_{b}^{(0)} (k)} \times 100 %

(15)

\bar{Δ} = \frac{1}{n} \sum_{k = 1}^{n} | \frac{x_{b}^{(0)} (k) - {\hat{x}}_{b}^{(0)} (k)}{x_{b}^{(0)} (k)} |

(16)

where, $x_{b}^{(0)} (k)$ is the original data, ${\hat{x}}_{b}^{(0)} (k)$ is the prediction data.

Substitute the data sequence $X_{b}^{(0)}$ and ${\hat{X}}_{b}^{(0)}$ into equations (15) and (16), the residual and relative residual can be obtained in Table 3.

Table 3.

Residual test.

	Resampling sequence $X_{b}^{(0)}$	Prediction sequence ${\hat{X}}_{b}^{(0)}$	Residual $e_{b}^{(0)} (k)$	Relative residuals $e_{b}^{(0)} (k) %$
1	99.9980	99.9980	0.0000	0.0000
2	100.0000	100.0000	0.0000	0.0000
3	99.9990	99.9998	−0.0008	0.0008
4	100.0000	99.9996	0.0004	0.0004
5	100.0000	99.9994	0.0006	0.0006
6	99.9990	99.9992	−0.0002	0.0002
7	99.9990	99.9990	0.0000	0.0000
8	99.9990	99.9989	−0.0001	0.0001
9	99.9990	99.9987	0.0003	0.0003
10	99.9980	99.9985	−0.0005	0.0005

Table 3 shows that the maximum of relative residual is 0.0008%, and the average relative residual is 0.00029%.

(2) Test of correlation

Grey relational analysis is the similarity comparison of the predicted data sequence curve and the original data sequence curve. The correlation between the corresponding sequences is large if the two curves are close.

The resampling data sequence is denoted as

X_{b}^{(0)} = [x_{b}^{(0)} (1), x_{b}^{(0)} (2), \dots, x_{b}^{(0)} (k), \dots, x_{b}^{(0)} (n)] (k = 1, 2, \dots, n)

(17)

And the prediction data sequence is denoted as

{\hat{X}}_{b}^{(0)} = [{\hat{x}}_{b}^{(0)} (1), {\hat{x}}_{b}^{(0)} (2), \dots, {\hat{x}}_{b}^{(0)} (k), \dots, {\hat{x}}_{b}^{(0)} (n)] (k = 1, 2, \dots, n)

(18)

$X_{b}^{(0)}$ is set as the system characteristic sequence, and ${\hat{X}}_{b}^{(0)}$ is set as the related factors sequence. Then the grey correlation coefficient can be expressed as follows

γ_{k} = \frac{min_{k} | {\hat{x}}_{b}^{(0)} (k) - x_{b}^{(0)} (k) | + ρ max_{k} | {\hat{x}}_{b}^{(0)} (k) - x_{b}^{(0)} (k) |}{| {\hat{x}}_{b}^{(0)} (k) - x_{b}^{(0)} (k) | + ρ max_{k} | {\hat{x}}_{b}^{(0)} (k) - x_{b}^{(0)} (k) |}

(19)

where, $ρ$ is the resolution coefficient.

Then the grey correlation degree is as follows

γ = \frac{1}{n} \sum_{k = 1}^{n} γ_{k}

(20)

The greater the correlation degree, the higher the prediction accuracy.

In equation (19), the parameter $ρ$ is set as 0.5, and the grey correlation coefficient of the prediction data sequence and the original data sequence can be obtained in Table 4.

Table 4.

Grey correlation coefficient.

	1	2	3	4	5	6	7	8	9	10
$γ_{k}$	1.000	1.000	1.000	1.000	0.8333	1.000	1.000	1.000	1.0000	0.8333

According to equation (20), the grey correlation degree $γ$ is 0.9667, which is more than 0.6. Therefore, the grey prediction model can be used to predict the measurement data.

(3) Test of posterior error

Test of posterior error includes standard deviation ratio C and probability of small deviation p, which can be expressed as

C = \frac{s_{2}}{s_{1}}

(21)

p = P {| e (k) - \bar{e} | < 0.6745 s_{1}}

(22)

where, $s_{1}$ is the standard deviation of original data sequence $x_{b}^{(0)} (k)$ . $s_{2}$ is the standard deviation of residual data sequence $e_{b}^{(0)} (k)$ .

Then the standard deviation ratio C of the two data sequences can be obtained as follows

C = \frac{s_{2}}{s_{1}} = 0.560

(23)

Values $| e (k) - \bar{e} |$ and $0.6745 s_{1}$ in equation (22) can be calculated in Table 5.

Table 5.

Value of $| e (k) - \bar{e} |$ and $0.6745 s_{1}$ .

$\| e (k) - \bar{e} \|$	0.00003	0.00003	0.00071	0.00043	0.00063	0.00017	0.00003	0.00007	0.00033	0.00041
$0.6745 s_{1}$	0.00047	0.00047	0.00047	0.00047	0.00047	0.00047	0.00047	0.00047	0.00047	0.00047

According to Table 5, all the value of $| e (k) - \bar{e} |$ is less than $0.6745 s_{1}$ . The probability of small deviation p is 1. The standard deviation ratio $C = 0.56 < 0.65$ and the probability of small deviation $p = 1 > 0.95$ . The two indices are divided into four grades, which are shown in Table 6. Then the prediction accuracy is level three according to Table 6.

Table 6.

The accuracy of the model.

Accuracy level of model	p	c
Level one	$0.95 \leq p$	$c \leq 0.35$
Level two	$0.8 \leq p < 0.95$	$0.35 < c \leq 0.5$
Level three	$0.7 < p \leq 0.80$	$0.5 < c \leq 0.65$
Level four (unqualified)	$p < 0.70$	$0.65 < c$

The above precision tests show that the grey prediction model ${\hat{x}}_{b}^{(0)} (k + 1)$ meets the qualification of predicting the data sequence $x_{b}^{(0)} (k)$ for its high prediction accuracy.

Dynamic drift prediction model of single channel

${\hat{x}}_{b}^{(0)} (k + 1) = (- 5.4546455 E + 7) [e^{(- 1.8333 E - 6) K} - e^{(- 1.8333 E - 6) (K - 1)}]$ is the prediction model of the resampling data sequence for $F_{x 1}$ . The constant 100 is subtracted from ${\hat{x}}_{b}^{(0)} (k + 1)$ , and then the dynamic drift of the measurement value can be simplified as

{\hat{y}}_{b}^{(0)} (k + 1) = 100.0002 e^{(- 1.8333 E - 6) k} - 100

(24)

The uncertainty prediction sequence can be obtained from the measurement value prediction sequence as follows

{\hat{X}}_{t} = {{\hat{y}}_{b} (k + 1)} (b = 1, 2, \dots, B)

(25)

${\hat{X}}_{t}$ is sorted as the ascending order, and then the probability density function of the sample can be obtained. If the significant level is set as $α$ , the confidence level can be expressed as

P = (1 - α) \times 100 %

(26)

Under the confidence level P, the estimation interval can be denoted as

[x_{L}, x_{U}] = [x_{α / 2}, x_{1 - α / 2}]

(27)

where, $x_{α / 2}$ is the value of corresponding frequency $α / 2$ , $x_{1 - α / 2}$ is the value of corresponding frequency $1 - α / 2$ , $x_{L}$ is the lower boundary of the estimation interval, $x_{U}$ is the upper boundary of the estimation interval.

Then the extended uncertainty of the measurement value can be expressed as

U = x_{U} - x_{L}

(28)

Comparison of the grey prediction model with the bootstrap sample method

To verify the feasibility of the grey prediction model for the dynamic uncertainty, the traditional bootstrap sample method is used to obtain the uncertainty in single channel. Then the result is compared with the grey bootstrap model.

The measurement value sequence in any channel can be expressed as

X = (x (1), x (2), \dots, x (k), \dots, x (n))

(29)

where, $x (k)$ is the kth data, and the number of the measurement value is n.

Resampling is conducted with putting back in equal probability, and the number of measurement values is n in each resampling sample. And B groups bootstrap samples are obtained after resampling in B times, and then the bth bootstrap sample can be expressed as Efron (1979)

X_{b} = (x_{b} (1), x_{b} (2), \dots, x_{b} (k), \dots, x_{b} (n))

(30)

The mean value of the bootstrap sample can be calculated as

X_{b} = \frac{1}{n} \sum_{k = 1}^{n} x_{b} (k) (b = 1, 2, \dots, B)

(31)

Then a bootstrap sample with B groups of samples can be obtained as

X_{Bootstrap} = (X_{1}, X_{2}, \dots, X_{b}, \dots, X_{B})

(32)

The following process of calculating the uncertainty is the same as equations (26) to (28). Then the uncertainties with different confidence levels can be obtained in Table 7.

Table 7.

Uncertainties of 12 channels with bootstrap sample method.

Evaluation parameters	Confidence level p%
	90	92	94	96	98	100
UF_x1	0.0008	0.0008	0.0008	0.0009	0.0010	0.0016
UF_x2	0.0024	0.0025	0.0027	0.0028	0.0034	0.0054
UF_x3	0.0028	0.0030	0.0031	0.0035	0.0039	0.0057
UF_x4	0.0018	0.0019	0.0020	0.0022	0.0026	0.0036
UF_y1	0.0014	0.0016	0.0017	0.0019	0.0021	0.0033
UF_y2	0.0016	0.0017	0.0019	0.0021	0.0023	0.0036
UF_y3	0.0017	0.0017	0.0019	0.0020	0.0024	0.0035
UF_y4	0.0019	0.0021	0.0023	0.0024	0.0028	0.0043
UF_z1	0.0025	0.0026	0.0028	0.0030	0.0034	0.0052
UF_z2	0.0026	0.0028	0.0029	0.0031	0.0036	0.0055
UF_z3	0.0020	0.0020	0.0022	0.0023	0.0026	0.0043
UF_z4	0.0042	0.0044	0.0048	0.0052	0.0058	0.0088

To compare with the results of bootstrap sample method, B group data can be obtained from the grey prediction model at time k, which is expressed as

{\hat{X}}_{t} = {{\hat{y}}_{b}^{(0)} (k)} (b = 1, 2, \dots, B)

(33)

According to equations (26) to (28), the uncertainty of the 12 channels will be calculated. And then the uncertainties from the two different methods are compared. Figure 4 shows the relationship of confidence interval and uncertainty with bootstrap sample and grey prediction model method. Results show that the uncertainties obtained from the two methods are closed. However, the greatest advantage of the grey prediction method is the ability of prediction, which is not provided by the bootstrap method.

Figure 4.

Comparison of the bootstrap and GBM method.

Dynamic measurement uncertainty model of single channel

According to the comprehensive analysis of the measurement system error sources, the dynamic error model of the measurement system is obtained as Xie (2003)

Y (t) = (μ + b + δ (t)) + α + β + e

(34)

where, $μ$ is the standard value, b is the system deviation of the measurement system, $δ (t)$ is the variation of system deviation with time, $α$ is the error for inaccurate standard, $β$ is the error for measurement condition, e is the error for the dynamic error for repetitive test.

According to the synthesis principle of measurement uncertainty, the dynamic uncertainty of the measurement system is expressed as

u_{c} (t) = \sqrt{u_{δ}^{2} (t) + u_{α}^{2} + u_{β}^{2} + u_{e}^{2}}

(35)

where, $u_{δ} (t)$ , $u_{α}$ , $u_{β}$ and $u_{e}$ are the uncertainty component from $δ (t)$ , $α$ , $β$ and e.

The six-axis force is composed of signals in 12 channels in this measurement system. So, the combined uncertainty of the six-axis force sensor can be obtained by combination of the uncertainty in each channel. The uncertainty in the ith $(i = 1, 2, \dots, 12)$ channel can be expressed as

u_{ci} (t) = \sqrt{u_{δ i}^{2} (t) + u_{α i}^{2} + u_{β i}^{2} + u_{ei}^{2}}

(36)

$u_{β}$ is the component of the uncertainty for measurement condition. For the harsh space environment, the measurement condition changes dynamically. Then $u_{β}$ need not be discussed separately because the component $u_{β}$ is contained in $u_{δ i} (t)$ . Then the dynamic uncertainty evaluation model of the six-axis force sensor single channel can be obtained as

u_{ci} (t) = \sqrt{u_{δ i}^{2} (t) + u_{α}^{2} + u_{e}^{2}}

(37)

$u_{δ} (t) = [u_{δ F_{x 1}} (t), u_{δ F_{x 2}} (t), u_{δ F_{x 3}} (t), u_{δ F_{x 4}} (t), u_{δ F_{y 1}} (t), u_{δ F_{y 2}} (t), u_{δ F_{y 3}} (t), u_{δ F_{y 4}} (t), u_{δ F_{z 1}} (t), u_{δ F_{z 2}} (t), u_{δ F_{z 3}} (t), u_{δ F_{z 4}} (t)]$ is the uncertainty component in 12 channels. The prediction measurement value resampling sequence ${\hat{X}}_{t} = {{\hat{y}}_{b} (k + 1)} (b = 1, 2, \dots, B)$ is used to calculate the uncertainty according to equations (26) to (28). The prediction uncertainties in the 12 channels can be obtained in Table 8.

Table 8.

The prediction uncertainty of the 12 channels.

UF_x1	UF_x2	UF_x3	UF_x4	UF_y1	UF_y2	UF_y3	UF_y4	UF_z1	UF_z2	UF_z3	UF_z4
0.0010	0.0029	0.0034	0.0024	0.002	0.002	0.0019	0.0022	0.0028	0.0029	0.0025	0.0049

$u_{α}$ is the uncertainty of calibration errors, which is composed of the calibration error and standard force sensor error. The uncertainty of calibration machine is 1.0%, and the confidence coefficient is k=2. Then the uncertainty component $u_{α}$ is 0.5%.

Resolution of the piezoelectric six-axis force sensor is $e = 0.01$ , and the error is uniform distribution. Then uncertainty component of resolution is $u_{e} = 0.01 / 2 \sqrt{3}$ .

Substitute $u_{δ i} (t)$ , $u_{α}$ and $u_{e}$ into equation (37), the combined uncertainty in each channel can be obtained.

Combination of the multidimensional dynamic uncertainty

A major problem of the measurement uncertainty is propagation of the uncertainty. The multidimensional uncertainty propagation type is more complex than one dimension. The multidimensional uncertainty can be expressed as

V (Y) = J_{X_{n}} V_{X_{n}} {J_{X_{n}}}^{T}

(38)

where, $J_{X_{n}}$ is the Jacobian matrix of the input vector X and output vector Y . $V_{X_{n}}$ is the uncertainty covariance matrix of the input vector X .

For the six-axis force sensor, the output vector is $Y = [F_{x}, F_{y}, F_{z}, M_{x}, M_{y}, M_{z}]$ . Also, the input vector is the 12 channels signals $X = [F_{x_{1}}, F_{x_{2}}, F_{x_{3}}, F_{x_{4}}; F_{y_{1}}, F_{y_{2}}, F_{y_{3}}, F_{y_{4}}; F_{z_{1}}, F_{z_{2}}, F_{z_{3}}, F_{z_{4}}]$ . Partial derivative is conducted with respect to X , and then the Jacobian matrix $J_{X_{n}}$ can be obtained as

J_{X} = [\begin{matrix} \partial y_{1} / \partial x_{1} & \partial y_{1} / \partial x_{2} & \dots \dots & \partial y_{1} / \partial x_{12} \\ \partial y_{2} / \partial x_{1} & \partial y_{2} / \partial x_{2} & \dots \dots & \partial y_{2} / \partial x_{12} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ ⋮ & ⋮ & ⋮ & ⋮ \\ \partial y_{6} \partial x_{1} & \partial y_{6} / \partial x_{2} & \dots \dots & \partial y_{6} / \partial x_{12} \end{matrix}]

(39)

The relationship between Y and X has been given in equation (1), then equation (39) can be calculated as follows

J_{X} = [\begin{matrix} 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - b & b & b & - b \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - b & - b & b & b \\ b & - b & - b & b & b & b & - b & - b & 0 & 0 & 0 & 0 \end{matrix}]

(40)

Then the uncertainty covariance matrix of 12 channels can be expressed as

V (X) = [\begin{matrix} u^{2} (x_{1}) & u (x_{1}, x_{2}) & \dots \dots & u (x_{1}, x_{12}) \\ u (x_{1}, x_{2}) & u^{2} (x_{2}) & \dots \dots & u (x_{2}, x_{12}) \\ \begin{matrix} ⋮ \\ ⋮ \end{matrix} & \begin{matrix} ⋮ \\ ⋮ \end{matrix} & \begin{matrix} ⋮ \\ ⋮ \end{matrix} & \begin{matrix} ⋮ \\ ⋮ \end{matrix} \\ u (x_{1}, x_{12}) & u (x_{2}, x_{12}) & \dots \dots & u^{2} (x_{12}) \end{matrix}]

(41)

The measurement signals in 12 channels are independent of each other, so the covariance between them is 0. Then the uncertainty of the covariance matrix can be simplified as

V (X_{12}) = [\begin{matrix} u^{2} (x_{1}) & 0 & \dots \dots & 0 \\ 0 & u^{2} (x_{2}) & \dots \dots & 0 \\ \begin{matrix} ⋮ \\ ⋮ \end{matrix} & \begin{matrix} ⋮ \\ ⋮ \end{matrix} & \begin{matrix} ⋮ \\ ⋮ \end{matrix} & \begin{matrix} ⋮ \\ ⋮ \end{matrix} \\ 0 & 0 & \dots \dots & u^{2} (x_{12}) \end{matrix}]

(42)

Substitute equation (39) and equation (42) into equation (38), the multidimensional measurement uncertainty of the piezoelectric six-axis force sensor can be calculated as follows

V (Y) = [\begin{matrix} 0.0253 & 0 & 0 & 0 & 0 & - 0.0004 \\ 0 & 0.0245 & 0 & 0 & 0 & - 0.000015 \\ 0 & 0 & 0.0267 & - 0.0005 & 0.0004 & 0 \\ 0 & 0 & - 0.0005 & 0.0043 & - 0.0002 & 0 \\ 0 & 0 & 0.0004 & - 0.0002 & 0.0043 & 0 \\ - 0.0004 & - 0.000015 & 0 & 0 & 0 & 0.0080 \end{matrix}]

(43)

The diagonal elements mean the auto-uncertainty of force and moment, and the non-diagonal elements mean the cross-uncertainty of the force and moment.

Conclusions

A piezoelectric six-axis force sensor for docking system is proposed. The relationship between the six-axis force and the forces in 12 channels is presented by the decoupling algorithm. Also, a single channel dynamic uncertainty prediction model based on the grey theory is established. The reliability of the prediction model is verified by the residual test, correlation test and posterior error test. Comparison between traditional bootstrap samples method and the grey prediction model is conducted, which verifies the feasibility of the grey prediction model. Finally, multidimensional dynamic uncertainty of six-axis force sensor is obtained by the combination of the decoupling algorithm and the dynamic uncertainty components in each channel. This will provide an important reference for the future development of the six-axis force sensor used under harsh conditions.

Footnotes

Declaration of conflicting interest

The authors declare that there is no conflict of interest.

Funding

This work was financially supported in part by National Natural Science Foundation of China (No.51105050) and Scientific research project of Liaoning education department (JDL2016027).

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