The measurement of hydraulic cylinder displacement has been addressed from different fields. The detection principle of magnetic grating is able to realize the high integration and accuracy. In this paper, a signal response quality evaluation algorithm for devising and optimizing a high-accuracy displacement measuring system is proposed. On the basic of signal response quality evaluation method, structure variables are optimized to enhance the working performance. By defining the parameters, an optimum structure cylinder prototype is made and tested to provide better estimates. Experimental results on working characteristic are presented to verify the effectiveness of the optimized structure. The efficiency of the proposed signal response quality evaluation function is therefore demonstrated through the working performance.
In the past decades, the development of electro-hydraulic actuators (EHA) allows flight-control systems to balance requirements of high power, lightweight, safety, fast response, and continuity of service [
Studies emerged have paved a way for research on cylinder displacement sensing methods of EHA. In the industrial practice, two kinds of position sensing devices are used: one is linear variable inductor displacement transducer (LVDT), and the other is magnetostrictive displacement sensor [
Previously, Murakami and Kitsunai applied the idea of equipping the hydraulic cylinder with a stroke sensor based on a magnetoresistor sensor to detect positions [
Structure of magnetic-grating-like stroke-sensing cylinder.
The permanent magnet generates magnetic fields while the sensor detects the magnetic field intensity and thereby measures the displacement. The piston rod is made of ferromagnetic materials with the repeating grooves on the body. The intensity variation is sampled as a function of the relative position between the permanent magnet and the piston rod by positioning the receiving magnetic force across a range of lateral offsets. When the piston rod is moving, the permanent magnet forms constant magnetic scales on the piston rod in combination with the repeating grooves. The magnetic induction line is thus periodically modulated. Supposing the piston rod is moving with a fixed speed, the induction line along with the sensor is simulated, as shown in Figure
Periodic change of magnetic field intensity during the piston rod moving.
At the position of 0 mm
At the position of 0.4 mm
At the position of 0.8 mm
The sensor can effectively detect the magnetic field depending on the magnetic scale, in which the displacement is measured in a magnetic-grating sensing way. The Hall sensor is employed to sense signals on the cylinder’s surface with magnetic scale [
Curve of Hall sensor response by simulation.
The problem of magnetic-grating-like stroke-sensing cylinder employment arises in precision measurement, since current laboratory research is still not able to address this issue. Specifically, with the added sliding mode control, Yang et al. did not give the emphasis on accuracy improvement. Most of the previous research aimed at setting up a preliminary system for sensing the stroke of the cylinder. To the best of our knowledge, specific structure design and optimization have not yet been established and no quantitative information is given.
In this paper, we will consider the reason for the low accuracy of magnetic-grating-like hydraulic cylinder integrated displacement sensor. A set of structural optimization method is proposed for addressing the design of the sensor. The research contributions of this paper can be summarized as follows:
Response quantification criteria setup: the study develops a function which combined the signal subdivision principle and sensor output used for evaluating the measurement accuracy. Structural parameter optimization: define the structural variables by referring to the response quantification function and compare the influence of each component on optimization results. Performance testing design: to prove the technical efficacy of proposed optimization framework, the prototype is produced and experimental platform is designed. Through the characteristic testing, the measurement accuracy of magnetic-grating-like stroke-sensing cylinder can be obtained.
This paper will introduce signal subdivision-related knowledge and present specific response quality assessment method in Section
The fundamental principle of stroke measurement is the evaluation of the relative displacement. When the piston rod moves with the distance of one slot, one complete period of harmonic signal will therefore be generated. The signal, which is sensed by the Hall sensor, reflects the variation of magnet field intensity. In this way, the measurement accuracy means the length of one-slot range. According to previous study, the responses of the grating sensor are sine and cosine signals [
Lot of research has been conducted to address signal subdivision for performance criteria. Tangent-cotangent subdivision technique is a state-of-the-art method for grating signal segmentation according to Moiré Fringe electric subdivision technique [
For simplicity, the expression is written in terms of angle, rather than radian.
In this way, the displacement can be calculated with the angle
Tangent/cotangent variation for angle increments.
In line with Figure
During the parameter modification process, the variable
We refer readers to [
In order to verify the correctness of the adopted response quality evaluation method, we compute the error of predefined samples with its numerical change in combination with sin/cosine signal. In this test, white noise is added into the sinusoidal signal and cosine signal (Figure
(a) Sine/cosine signal with white noise. (b) White noise error from quality evaluation function before modification. (c) White noise error from quality evaluation function after modification.
For the purpose of structure optimization, the algorithm is implemented as the following steps:
Take one working period of a specific sensor as the research sample.
On the basic of signal subdivision principle, set the sensing response as harmonic signal.
Divide the period into even parts which correspond to angle data and compute the theoretical tangent/cotangent value based on (
Take (
Modify the sensing response with (
Intelligent algorithms are employed for strategy facilitating and cost reducing [
A multiparameter optimization problem using genetic algorithm is formulated by defining the system figure and the objective function of response quality, respectively. The modal parameters (i.e., natural modes) of the real structure are numerically obtained using its finite element model, from which the reduced numerical modes correspond to the target structure. The original related parameters are discussed in Section Initialization: consider a population of individuals, and each individual is described by one chromosome with Fitness calculation: the optimum configuration is sought by minimizing the value of signal response quality. The fitness function of each individual in current population can be calculated according to ( Selection: the selection mechanism allows a small portion of chromosomes from the population for further evaluation [ Crossover: offspring is generated by two randomly selected parents exchanging genetic information with each other. The crossover step aims at leading to minor differences between parents and children, strengthening the exploitative power of reproduction [ Mutation: mutation provides an opportunity to search new areas of the solution space. By randomly altering the alleles of genes, the GAs can effectively avoid trap situations and maintain sufficient variance in the population [ Replace the initial population with new individuals of best chromosome. An outcome value closer to the target value via repetitive iterative process can be generated. Consequently, after a large number of iterations, the best chromosome in the population is translated as the selected solution.
A sensor’s performance characteristics are governed by its specific architecture. The parameters with large uncertainty in the design process can affect the measurement accuracy significantly. Aiming at investigating the structure design for increasing working performance, optimization techniques are used to determine the design parameters. According to paragraph II, the key of measuring is the response of the sensor output. That is, in other words, given a parameterized representation of a sensor, we are capable of predicting its measurement accuracy.
To assist the optimization process, there are seven variables involved in this system (Table
Nomenclature.
Symbol | Definition |
---|---|
Diameter of permanent magnet | |
Height of permanent magnet | |
Distance from bottom of permanent magnet to centre of sensor | |
Distance from surface of piston rod to centre of sensor | |
Slot shape of piston rod | |
Diameter of piston rod | |
Groove width of piston rod |
Parameter of magnetic-grating-like stroke sensing for optimization.
We notice that the diameter of piston rod in a suspended state, which avoids the direct contact between the magnet and the Hall sensor [
The conceptual framework for optimizing a candidate design is shown in Figure
Conceptual diagram of magnetic-grating-like stroke-sensing cylinder for evolving structure model.
To validate every analytical result further, other parameters are assumed fixed during the optimizing process. The sensor exists in the position between the permanent magnet and the piston rod, and the distance is named as
Signal response of
It can be seen that the signal intensity is affected by the distance between the Hall sensor and the piston rod. As seen here, a smaller distance has a stronger sensor response while the value over 0.7 mm changes only slightly the strength of sensor response. However,
Effect of different
The variation of permanent magnet shape has a direct effect on the magnetic induction, which is involved with the diameter
Signal response of different values of
The signal response quality is found to be improved and stabilized as diameter increases to over 10 mm as shown in Figure
Effect of different
For the height of permanent magnet
Signal response of
Figure
Effect of different
The mechanical position of permanent magnet plays an important role in changing the sensing signal as well. The physical distance between the permanent magnet and the piston rod
Signal response of
As seen here, the decreasing distance causes the increasing of signal response. However, the computing results of parameter values illustrate that the characteristic of piston rod can be clearly described at all values. In the practical design, different structures correspond to different locations of permanent magnet. The objective function curve of signal response quality is shown in Figure
Effect of different
The shape of piston rod slot is optimized by setting up the coordinate system of
Geometric illustration of the slot.
In the relationship,
A GA is employed to search the optimum solution of
GA optimization procedure.
The process starts with randomly generated population. Each chromosome represents a possible set of parameters associated with a fitness value. The total population of each generation is evaluated based on slot geometry where GA carries out a fitness-based selection. In our GA, the size of the population is chosen to be 40 on the basic of modeling and modification. Processes are subject to recombination to form a successor population. The single-point crossover is used, and the probability of mutation is set as 0.17%. During recombination, chromosomes with the higher fitness values are more likely to be selected from the population, which passed on to the successor population. This is an iterative process, through which various generations are evolved until some stopping criterion is specified, and an optimal solution is reached based on an increase. The GA search process for the quality evaluation function is shown in Figure
Evolution process of best (smallest) quality evaluation function.
Accordingly, a piston rod with optimized variables is obtained. Configuration uses the following parameter values:
Photograph of piston rod sample and its signal response.
The experiments are conducted to verify the working performance of magnetic-grating-like stroke-sensing cylinder. The hardware used to perform the experiments is shown in Figure
Schematic layout of test setup.
Stroke tests contain short-term test and long-term test. Both are basically designed for testing the error between theoretical stroke and experimental stroke. The working distance of short-term test is one grating, which is one thread lead of the piston rod. The output of Hall sensor is detected every 5°, and the subdivision points is 96. Figure
Output of theoretical and actual stroke.
Similarly, the long-term test takes 100 continuous grating as the working distance. The maximum error of long-term is
Test error of forward and backward stroke.
The magnetic flux density is addressed relying on highly stabilized system output. In the power supply driving mode, the sensor is of different position of the stroke. The pass/fail of stability is based on the output of the sensor. The voltage is kept at 24.0 V for 1 hour by use of highly stabilized power supply, and the displacement is measured every 2 hours. The corresponding displacements
Sensor output of stability test 1
Time (h) | |
---|---|
0 | 62.539 |
2 | 62.526 |
4 | 62.539 |
6 | 62.526 |
8 | 62.526 |
12 | 62.539 |
14 | 62.539 |
16 | 62.526 |
18 | 62.539 |
20 | 62.526 |
22 | 62.539 |
24 | 62.539 |
Sensor output of stability test 2
Time (h) | |
---|---|
0 | 107.565 |
2 | 107.552 |
4 | 107.565 |
6 | 107.578 |
8 | 107.578 |
12 | 107.552 |
14 | 107.565 |
16 | 107.552 |
18 | 107.578 |
20 | 107.565 |
22 | 107.565 |
24 | 107.578 |
Arithmetic average difference can be calculated during relevant periods. The value of magnetic flux density stability error
The reproducibility test is designed to identify the repetitive feature of sensor output within the same stroke. The calibration is conducted three times while the piston rod repeats in a defined path. The recorded values, for each measurement point, are shown in Table
Sensor output of reproducibility test 1.
0.000 | 0.000 | 0.000 |
2.995 | 2.995 | 2.982 |
6.042 | 6.029 | 6.029 |
9.010 | 8.984 | 9.010 |
12.005 | 12.005 | 12.031 |
15.039 | 15.026 | 15.039 |
17.969 | 17.982 | 17.995 |
21.016 | 21.029 | 21.029 |
24.010 | 23.997 | 23.997 |
27.005 | 26.992 | 27.005 |
30.000 | 30.000 | 30.013 |
32.982 | 32.956 | 32.956 |
36.003 | 36.003 | 36.029 |
39.010 | 38.984 | 38.997 |
41.979 | 41.966 | 41.979 |
45.013 | 45.013 | 45.026 |
47.969 | 47.969 | 47.982 |
51.003 | 51.003 | 51.029 |
53.997 | 53.984 | 54.010 |
56.992 | 56.979 | 56.992 |
60.039 | 60.039 | 60.039 |
62.982 | 62.969 | 62.982 |
66.003 | 66.003 | 66.029 |
69.023 | 69.010 | 69.023 |
The absolute static error
The error determined by machining deviation is
In this research, the dynamic test deals with the working accuracy of different measuring modes. The setup for evaluating the dynamic error is as follows: a low-speed (0.1 m/s) and a high-speed (0.3 m/s) measurement stroke are carried out three times with the sampling rate of 100 Ksps.
Dynamic error is evaluated in accordance with the definition in [
The standard uncertainty of the positioning deviations obtained by a series of
Taking a coverage factor
We take the maximum value of the positioning repeatability at any position
The difference between the algebraic maximum and minimum of the positioning deviations at any position
Now, the coverage factor
The standard deviation of measured values is
In the working displacement of 0–90 mm, recordings corresponding to speed and position variation are shown in Figure
(a) Forward-stroke output in low speed. (b) Backward-stroke output in low speed. (c) Forward-stroke output in high speed. (d) Backward-stroke output in high speed.
The position deviation of proposed sensing stroke is within the range of
Positioning accuracy of dynamic testing.
As shown in Figure
In this paper, the devising and deploying of magnetic-grating-like stroke-sensing cylinder are specifically studied, according to its working mechanism and distinctive structure. Motivated by the significance of developing a high-accuracy displacement sensing device, capable of operating in different working distances and optimizing its performance, we set out to define the signal response quality of working parameters. The signal response quality evaluation algorithm is constructed to characterize sensor response in developing a stroke-sensing cylinder, which leads to the evaluation of structure variables mathematically. The optimal displacement sensor combines high response and state-of-the-art signal segmentation method.
The prototype is processed and tested. The proposed testing workbench, together with the magnetic-grating-like stroke-sensing cylinder, is set up in laboratory. Based on the static and dynamic testing, experimental outputs are recorded and demonstrate that it has a stable, repeatable performance and high measurement accuracy for a wide range of stroke. The testing results characterize the magnetic-grating-like stroke-sensing cylinder, which can replace current measurement devices and fulfill the demand of high-accuracy displacement detection.
The authors declare that they have no conflicts of interest.
This research is a general project supported by Shanxi Province Science Foundation for Youths (2015021123) and Shanxi Provincial Key Laboratory Open Fund (XJZZ201605).