sensors Article
Economical High–Low Temperature and Heading Rotation Test Method for the Evaluation and Optimization of the Temperature Control System for High-Precision Platform Inertial Navigation Systems Qiang Yang , Rong Zhang * and Haixia Li * Department of Precision Instrument, Tsinghua University, Beijing 100084, China;
[email protected] * Correspondence:
[email protected] (R.Z.);
[email protected] (H.L.); Tel.: +86-10-6277-1335 (R.Z.); +86-10-6279-2116 (H.L.) Received: 22 October 2018; Accepted: 13 November 2018; Published: 15 November 2018
Abstract: Inertial navigation systems (INSs) use the temperature control system to ensure the stability of the temperature of the inertial sensors for improving the navigation accuracy of the INSs. That is, the temperature control accuracy affects the performance of the INSs. Thus, the performance of temperature control systems must be evaluated before their application. However, nearly all high-precision INSs are large and heavy and require long-term testing under many different experimental conditions. As a result, conducting an outdoor navigation experiment, which involves high–low temperature and heading rotation tests, is time consuming, laborious, and costly for researchers. To address this issue, an economical high–low temperature and heading rotation test method for high-precision platform INSs is proposed, and an evaluation system based on this method is developed to evaluate the performance of the temperature control systems for high-precision platform INSs indoors. The evaluation system uses an acrylic chamber, exhaust fans, temperature sensors, and an air conditioner to simulate the environment temperature change. The outer gimbals of the platform INSs are utilized to simulate the heading rotation. The temperature control system of a high-precision platform INS is evaluated using the proposed evaluation method. The temperature difference of the gyros is obtained in the high–low temperature test, and the temperature fluctuation of the temperature control system is observed in the rotation test. These tests verify the effectiveness of the proposed evaluation method. Then, the corresponding optimization method for the temperature control system of this high-precision platform INS is put forward on the basis of the test results of the evaluation system. Experimental results show that the maximum temperature differences of the two gyros between high- and low-temperature tests are decreased from 1.51 ◦ C to 0.50 ◦ C, and the maximum temperature fluctuation value of the temperature control system is decreased from 0.81 ◦ C to 0.27 ◦ C after the proposed evaluation and optimization processes. Therefore, the proposed methods are cost effective and useful for evaluating and optimization of the temperature control system for INSs. Keywords: inertial navigation system; temperature control system; evaluation method; optimization method; gimbal; platform; high–low temperature test; heading rotation
1. Introduction Inertial navigation systems (INSs) can provide position, velocity, and attitude independently [1]. Thus, they are used extensively in land [2,3], marine [4,5], and aerospace navigation [6]. A direct and effective way to improve the navigation accuracy of an INS is to increase the accuracy of its inertial Sensors 2018, 18, 3967; doi:10.3390/s18113967
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sensors. Unfortunately, the fluctuation in environment temperature [7] and the increase in temperature in running the navigation system lead to zero bias [8–10], scale factor errors [9,11] of the sensitive element, and deterioration of relevant electric circuits. These problems cause inertial sensor errors. To compensate for the inertial errors caused by the fluctuation in the environment temperature [12], two kinds of methods have been proposed as follows. One is the passive method [10,13], which can include thermal compensation, thermal shunting of elements, and special structural design. Yang et al. [13] used tri-axial Micro-Electro-Mechanical System (MEMS) gyroscope outputs in different temperatures to calculate the error parameters of different temperature points for error compensation of gyroscope caused by temperature fluctuation. The other is the active method, which involves designing a temperature control system [14–20] to keep the temperature of the inertial sensors within a certain range while maintaining a high accuracy value. The former method requires a full understanding of all temperature effects. However, many temperature effects cannot be predicted effectively or modeled precisely [21]. Thus, the latter method, which entails adding a temperature control system for the INS, is often adopted. With regard to current studies, Giordano et al. [18] used cooling fans to control the temperature ultrasound obstacle detection system. Zhu et al. [19] utilized heaters to stabilize the temperature of the gyroscope. Cao et al. [20] used the thermoelectric coolers, fan and heater elements and temperature sensors to adjust the temperature of the sensor box. Li et al. [22] adopted fans, heater elements and fuzzy control algorithm to control the temperature of the INS. Xia et al. [23] adopted thermoelectric coolers and fuzzy-PID controllers to develop a temperature control system for silicon micro-gyroscope. Recently, Mironova et al. [24–26] used the thermoelectric cooler, fan and Lyapunov based control strategy for workpiece clamping system. The Lyapunov based control strategy guarantee the control of the nonlinear thermal system thermal systems and it has a potential use in INSs. Section 2 briefly introduces the temperature control system of the high-precision INS. The control accuracy of the temperature control system affects that of the inertial sensors. The performance of the temperature control system should be evaluated before its application. However, a car test, a sea trial, or a space experiment is costly for researchers. Thus, a low-cost indoor evaluation system is urgently needed. On the one hand, the change in ambient temperature of INSs also changes the temperature fields outside the temperature control system. On the other hand, the heading rotation of the vehicle leads to relative position change of the gimbals, which changes the temperature fields inside the temperature control system. Both factors cause temperature fluctuation in the temperature control system. Therefore, the changes in heading rotation [27] and environment temperature are the main factors of the temperature fluctuation in the temperature control system. At present, researchers in related fields propose evaluation methods for different INSs. Cao et al. [20] conducted a stability experiment inside a room without considering different environment temperature points. Li et al. [22] performed a temperature test by using a lamp to change the internal temperature of the INS. Xia et al. [23] tested the performance of the temperature control system in a temperature chamber that can change the external temperature of the inertial sensors. Levinson et al. [16] used a SMV temperature chamber to change the external temperature of a two-axis indexed Laser Gyro MK49 system. In summary, temperature tests have been adopted widely to test the performance of temperature control systems. However, the effect of the heading rotation is often neglected in the temperature tests. Temperature chambers to support the heading rotation and temperature change for the gyroscopes of low-precision, small-sized, and lightweight MEMS INSs are available [13]. Nevertheless, no evaluation system can support the high–low temperature and heading rotation of the high-precision, large-sized, and heavy platform INSs. The evaluation system should contain a rotation functional unit (e.g., a turntable to rotate the INS [28–31]) and a temperature control functional unit (e.g., a high–low temperature chamber) to change the temperature of the environment. The weight of high-performance INSs often reaches 100–300 kg {e.g., MK39 (weight = 317.5 kg) [32], MK49 (weight = 317.5 kg) [33], AN/WSN-5L (weight = 172.7 kg) [34], NCS-1 (weight = 77.3 kg), and ESGN (weight = 405 kg) [35]}. To date, no indoor equipment can support high–low temperature and rotation test for the aforementioned systems.
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study proposes an economical indoor evaluation method to evaluate the temperature control system for high-precision platform INSs withindoor rotation gimbals for solving above-mentioned issue.control This study proposes an economical evaluation method tothe evaluate the temperature On for the high-precision basis of the testplatform results ofINSs the evaluation system, thefor corresponding optimization method system with rotation gimbals solving the above-mentioned issue. is proposed for the temperature control system of a high-precision platform INS. The temperature On the basis of the test results of the evaluation system, the corresponding optimization method is control system in temperature this study is control developed by Li al. [22] in accordance with special requirements proposed for the system of et a high-precision platform INS.the The temperature control of our previously developed high-precision INS. The with current involves system in this study is developed by Li et al. platform [22] in accordance the study specialmainly requirements of the our evaluation and optimization of the temperature control system of this INS. previously developed high-precision platform INS. The current study mainly involves the evaluation remainder paper is organized as follows. The principle and structure of typical and The optimization of of thethe temperature control system of this INS. temperature control systems of high-precision INSs and the control system ofofour INS The remainder of the paper is organized as follows. temperature The principle and structure typical are introduced in Section 2. The economical indoor evaluation method that contains high–low temperature control systems of high-precision INSs and the temperature control system of our temperature test method (Section 3.1) and heading rotation test method method that (Section 3.2)high–low for the INS are introduced in Section 2. The economical indoor evaluation contains temperature control systems of high-precision platform INSs is introduced in Sections 3. The temperature test method (Section 3.1) and heading rotation test method (Section 3.2) for the temperature evaluation results of the temperature control system of a high-precision platform INS are provided control systems of high-precision platform INSs is introduced in Section 3. The evaluation results in 4. The corresponding optimization method for the temperature control systeminisSection given in4. ofSection the temperature control system of a high-precision platform INS are provided Section 5. The test results after optimization for the temperature control system are described The corresponding optimization method for the temperature control system is given in Sectionin5. Section 6. results Finally,after the conclusions and future research directions discussed in Sectionin 7. Section 6. The test optimization for the temperature controlare system are described Finally, the conclusions and future research directions are discussed in Section 7. 2. Temperature Control Systems of INSs 2. Temperature Control Systems of INSs 2.1. Typical Temperature Control System of INSs 2.1. Typical Temperature Control System of INSs The temperature control system is often utilized to ensure that the temperature of the inertial The control often utilized to ensure thattemperature the temperature of the for inertial sensors is temperature close to constant andsystem is less is affected by the environment fluctuation the sensors is close to constant and is less affected by the environment temperature fluctuation for the INSs. INSs. That is, the temperature control system keeps the temperature of the inertial sensors within a That is,range the temperature control system the value. temperature of the inertial sensors within a certain certain while maintaining a high keeps accuracy A typical temperature control system uses range while a high accuracy value. A with typical system heaters heaters (fans)maintaining to heat (cool) the INS in accordance thetemperature temperaturecontrol measured by uses the sensors (fans) to1).heat (cool) the INS in accordance with the temperature measured by the sensors (Figure 1). (Figure
Host computer display
Drive Circuits
DSP/SCM
Drive Circuits
Temperature sensors +/Valve
Heaters Fans
Inertial sensors
Temperature control system of INS
Temperature sensors External envionment
Figure1.1.Platform Platformofofthe thetemperature temperaturecontrol controlsystem systemhardware. hardware. Figure
Most high-precision INSs adopt the method of incorporating a closed chamber with Most high-precision INSs adopt the method of incorporating a closed chamber with an an approximately constant inner temperature. The temperature control system can adjust the internal approximately constant inner temperature. The temperature control system can adjust the internal temperature of the INSs automatically using heaters and fans depending on the difference of the temperature of the INSs automatically using heaters and fans depending on the difference of the current and target temperature. The heater is used to heat the internal temperature of the INS when current and target temperature. The heater is used to heat the internal temperature of the INS when the temperature is low or the temperature decreases quickly. The fan is used to cool the internal the temperature is low or the temperature decreases quickly. The fan is used to cool the internal temperature when the temperature is high or the temperature increases rapidly. Then, the current temperature when the temperature is high or the temperature increases rapidly. Then, the current temperature can be adjusted to the target temperature using the closed-loop temperature control temperature can be adjusted to the target temperature using the closed-loop temperature control
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method as as[24]. [24].Using Usingthe thetemperature temperature control system, internal temperature of INS the can INSbe can be method control system, thethe internal temperature of the kept kept within a certain to improve the accuracy of the inertial The temperature control within a certain range range to improve the accuracy of the inertial sensors.sensors. The temperature control system system used here is developed by[22]. Li etThe al. [22]. The principle detailed principle of the temperature control can used here is developed by Li et al. detailed of the temperature control can be found bereference found in [22]. reference [22]. The evaluation and optimization of the temperature control system are in The evaluation and optimization of the temperature control system are conducted conducted by us. by us. 2.2. 2.2. Temperature TemperatureControl ControlSystem SystemofofOur OurINS INS The The structure structure and and working working mechanism mechanism of of the the temperature temperature control control system system of of our our high-precision high-precision platform platformINS INS are are shown shown in in Figure Figure 2. 2. The The initial initial model model of the temperature temperature control control system system is is developed developed by etal. al.[22] [22]ininaccordance accordance with special requirements of high-precision this high-precision platform INS. by Li et with thethe special requirements of this platform INS. Since Since this study mainly focuses on the design of a generally applicable evaluation system and structure this study mainly focuses on the design of a generally applicable evaluation system and structure optimization control algorithm. In In thethe future, in order to optimization of ofINSs, INSs,we westill stilluse usethe thealgorithm algorithminin[22] [22]forfor control algorithm. future, in order further improve thethe control accuracy of temperature control system, we can [36],[36], fuzzy-PID to further improve control accuracy of temperature control system, we use can the usePID the PID fuzzyand basedbased control strategies to compare the current control strategies of ourofINSs. PID Lyapunov and Lyapunov control strategies to compare the current control strategies our INSs. Temperature Sensor Input Wind
Heat source
Fan
球罩
Outer Gimbal
Output
temperature control control system system of of our our high-precision high-precisionplatform platformINS. INS. Figure 2. Structure of temperature
Figure providesthe thephysical physicalstructure structure schematic sealed temperature control system. Figure 2 provides schematic of of thethe sealed temperature control system. The The measuring temperature sensors used temperaturecontrol controlsystem systemare are the the TMP275 digital measuring temperature sensors used forfor thethe temperature digital thermometers, thermometers, which are installed installed on the the inner inner walls walls of of the the chamber chamber with with aa temperature temperature resolution resolution ◦ of C. The of 0.0625 0.0625 °C. The inlet inlet air air temperature temperature of of the the inner inner channel channel is is measured measured by by the the four four symmetrical symmetrical distributions temperature sensors and the average value is selected the observed temperature distributionsofofthethe temperature sensors and the average value isasselected as the observed of the system. The closed structure can keep physical isolation of the measuring instrument with the temperature of the system. The closed structure can keep physical isolation of the measuring external environment. Electricenvironment. heaters are installed at the top the apparatus to provide energy for instrument with the external Electric heaters areofinstalled at the top of the apparatus to heating the inner chamber, and their heat power is adjusted by the pulse width modulation (PWM) provide energy for heating the inner chamber, and their heat power is adjusted by the pulse width control method steps. The method airflow between and environment chamber is driven by the modulation (PWM) control steps. Theambient airflow environment between ambient and chamber convection fans. convection intensity the airflow channel is determined by the is driven by the The convection fans.heat Thetransfer convection heat of transfer intensity of the airflow channel is fan speed. In normal working conditions, the system will adjust the temperature of the chamber determined by the fan speed. In normal working conditions, the system will adjust the temperature automatically controlling the and the fansheaters depending on the current and target temperatures. of the chamberby automatically by heaters controlling and fans depending on the current and target
temperatures. 3. Evaluation System of Temperature Control System 3. Evaluation System of Temperature Control System 3.1. High–Low Temperature Test Method To designTemperature a low-costTest high–low 3.1. High–Low Method temperature test system, the structure and equipment are designed as follows. The overall structure of the high–low temperature test chamber is made of To design a low-cost high–low temperature test system, the structure and equipment are acrylic plates consisting of a closed structure with an internal size of 150 cm × 120 cm × 90 cm designed as follows. The overall structure of the high–low temperature test chamber is made of (width × length × thickness). To simulate the change in environment temperature realistically, acrylic plates consisting of a closed structure with an internal size of 150 cm × 120 cm × 90 cm (width × length × thickness). To simulate the change in environment temperature realistically, the high–low
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temperature test chambertest is composed acrylic plates, temperature sensors, andsensors, an air conditioner the high–low temperature chamber isofcomposed of acrylic plates, temperature and an air (Figure 3). (Figure 3). conditioner Air Out Fan High-Low Temperature Test Chamber INS Temperature Control System Temperature Sensors
Inertial Sensors Platform Air In
Outer
Gimbal
HB 2500 Heat Pump
Air In Air Conditioner
Figure 3. Configuration of the evaluation system. Figure 3. Configuration of the evaluation system.
In Figure 3, the blue area represents the temperature control system of INS. The INS comprises two dual-axis high-precision a drift the ratetemperature of 0.001◦ /h control mounted on a platform. Beyond the blue In Figure 3, the bluegyros area with represents system of INS. The INS comprises region, the high–low temperature testwith chamber constructed. Itsmounted main components comprise an HB two dual-axis high-precision gyros a driftisrate of 0.001°/h on a platform. Beyond the 2500 air conditioner, an exhaust fan, temperature sensors, and a box made of thermal insulation foam. blue region, the high–low temperature test chamber is constructed. Its main components comprise The cost of economical an high–low test system is approximately $1000. of Thethermal green an total HB 2500 airthe conditioner, exhausttemperature fan, temperature sensors, and a box made area represents made thermal insulation foam.temperature Three DS18B20 digitalisthermometers, insulation foam.the Thebox total cost of the economical high–low test system approximately ◦ C, are distributed which as the area temperature sensors with a temperature 0.0625 $1000.serve The green represents the box made of thermal resolution insulation of foam. Three DS18B20 digital on the internal face of serve the box. The air enters from thewith air conditioning duct on the bottom thermometers, which as the temperature sensors a temperature resolution of 0.0625and °C, exits by the exhaust faninternal on the crown The temperature sensor displayed duct and stored are distributed on the face ofwall. the box. The air enters from values the airare conditioning on the simultaneously with a real-time acquisition system developed using Microsoft Visual bottom and exits by the exhaust temperature fan on the crown wall. The temperature sensor values are displayed C++ Given that the temperatures the temperature controlacquisition system andsystem gyros reflect those ofusing the and 6.0. stored simultaneously with aofreal-time temperature developed gimbals and inertial sensors, respectively, these values are used to evaluate the performance of the Microsoft Visual C++ 6.0. Given that the temperatures of the temperature control system and gyros temperature system. and inertial sensors, respectively, these values are used to evaluate the reflect thosecontrol of the gimbals Using theof above mentioned temperature chamber, we can simulate the high and low environment performance the temperature control system. temperatures of an actual outdoor navigation economically Therefore, we and needlow to Using the above mentioned temperature chamber, and we accurately. can simulate the high achieve the heading rotationofonly for simulating the complete experimental affect environment temperatures an actual outdoor navigation economically andconditions accurately.that Therefore, the of the control system. The following section describes the method of weperformance need to achieve thetemperature heading rotation only for simulating the complete experimental conditions achieving rotation indoors. that affectheading the performance of the temperature control system. The following section describes the method of achieving heading rotation indoors. 3.2. Heading Rotation Test Method
3.2. Heading Rotation Test Method 3.2.1. Derivation of the Principle of Heading Rotation 3.2.1. of heading the Principle of Heading Rotation ToDerivation simulate the rotation of the carrier, the mathematical relationship between the heading and the gimbal angles of the navigation system is deduced as follows. Here, the four-gimbal platform To simulate the heading rotation of the carrier, the mathematical relationship between the INS is taken as an example. As shown in Figure 4, the four-gimbal platform INS consists of the outer heading and the gimbal angles of the navigation system is deduced as follows. Here, the four-gimbal (color red), middle, inner, and platform gimbals. The motion of the carrier is isolated using servo platform INS is taken as an example. As shown in Figure 4, the four-gimbal platform INS consists of control of the four gimbals. the outer (color red), middle, inner, and platform gimbals. The motion of the carrier is isolated using To establish the kinematics model of the four-gimbal platform INS, the coordinate systems servo control of the four gimbals. connected to each part are introduced separately. Five coordinate systems are involved, namely, carrier body coordinate system (b-frame), outer gimbal coordinate system (o-frame), middle gimbal coordinate system (m-frame), inner gimbal coordinate system (i-frame), and platform coordinate system (p-frame). The geometric relationship among the coordinate systems is shown in Figure 5.
Platform(p)
N D
Outer gimbal(o)
E
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Middle gimbal(m) Inner gimbal(i)
N
Platform(p)
D Carrier
Outer gimbal(o)
E
Earth
Inner gimbal(i)
Middle gimbal(m)
Equator
Carrier Figure 4. High-precision four-gimbal platform INS structure.
To establish the kinematics model of the four-gimbal platform INS, the coordinate systems Earth connected to each part are introduced separately. Five coordinate systems are involved, namely, carrier body coordinate system (b-frame), outer gimbal coordinate system (o-frame), middle gimbal Equator coordinate system (m-frame), inner gimbal coordinate system (i-frame), and platform coordinate system (p-frame). The geometric relationship among the coordinate systems is shown in Figure 5. Figure 4. High-precisionfour-gimbal four-gimbal platform Figure 4. High-precision platformINS INSstructure. structure.
xm xb To establish the xkinematics model of the four-gimbal platform INS, the coordinate systems o xi,xp connected to eachxmpart are introduced separately. Five coordinate systems are involved, namely, q s carrier body coordinate system (b-frame), outer gimbal coordinate system (o-frame), middle gimbal r p coordinate system (m-frame), inner gimbal coordinate system (i-frame), and platform coordinate zm system (p-frame). The geometric relationship among the coordinate systems iszpshown in Figure 5. r s o zo,zb q o p r zi,zm
xm
xb
xo xm
xi,xpq p
s yb s yo,ym
ym
r
yi
q yp
p
Figure5.5.Four-gimbal Four-gimbalplatform coordinatesystem systemdefinition definition[37]. [37]. Figure coordinate zplatform m
s zo,zb angles ofothe platform, q o and s represent InFigure Figure5,5, p, the rotation inner, middle, and outer p inner, p , q,qr, s represent In , r ,. and the rotation angles of the platform, middle, and . . . r gimbals, respectively; and p, q, r, and s are their rotation rates. In this way, we specify the angular rate z i, z m outer gimbals, respectively; and p , q , r , and s are their rotation rates. In this way, we specify of the carrier motion h the angular rate of theas carrier motion as q iT s ωb = ωbx ωby ωbzp , (1) T ωb = ωbx ωybym ωbz , (1) yb where ωbx , ωby , and ωbz are the angular rates components yi in the direction of three axes, respectively. yo,ym ω5, the angular rates of the coordinate yp systems have the following relations ωbx , ωbyto, Figure and bz are the angular rates components in the direction of three axes, whereAccording r
zp
respectively. h 5. Four-gimbal platform i T h coordinate i h definition [37]. i T Figure . T system o ω ωoFigure = ω5,oxtheωangular +C ωbytheωfollowing 0 coordinate 0 s oy ωoz rates=of the bx have bz According to systems relations b h i iT h h iT T . p q m s represent In Figure 5, ω,m = , rω,mx andωmy =the 0rotation + Co ofωthe ωmz r 0 angles ωoy ωoz inner, middle, and ox platform, (2) h i Tr h i h iT , T p q . outer gimbals, respectively; and , , , and s are their irotation rates. In this way, we specify = 0 0 q + Cm ωmx ωmy ωmz ωi = ωix ωiy ωiz h iT h iT the angular rate of the carrier motion as iT h . p ω = ω = p 0 0 +C ω ω ω ω ω p
px
py
pz
T
ωb = ωbx ωby ωbz ,
i
ix
iy
iz
(1)
where ωij is the angular rate, the subscript i represents the coordinate systems fixed with each frame, andωjbxdenotes corresponding axisangular direction.rates The transformation among theof coordinate ωbz are the , ωby ,the and components matrixes in the direction three axes, where systems are
respectively. According to Figure 5, the angular rates of the coordinate systems have the following relations
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Cbo = i = Cm p Cb .
cos s
sin s
0
cos r
− sin r
0
m 1 0 − sin s cos s 0 , Co = 0 sin r 0 cos r 0 0 1 . 1 0 0 cos q sin q 0 p sin p − sin q cos q 0 , Ci = 0 cos p 0 − sin p cos p 0 0 1
(3)
The transformation matrix between the carrier and platform coordinate systems is denoted as Thus, p p i m o Cb = Ci Cm Co Cb . (4)
From Equations (1)–(4), the mathematical relationship among the angular motion of the platform, the gimbals, and the carrier are obtained as follows: p
ω p = Cb ωb + C
.
h
.
.
.
iT
,
(5)
− cos q sin r cos p sin q sin r + sin p cos r , − sin p sin q sin r + cos pcosr
(6)
p
q
r
s
where ω p is the angular rate of the platform;
1 C= 0 0
p Cb =
sin q cos p cos q − sin p cos q
0 sin p cos p
cos q cos r cos s − sin s sin q
cos q cos r sin s + sin q cos s
− sin r cos q
cos s(− cos p sin q cos r + sin p sin r )
sin s(− cos p sin q cos r + sin p sin r )
cos p sin q sin r
− sin s cos p cos q
+ cos s cos p cos q
+ sin p cos r
cos s(sin p sin q cos r + cos p sin r )
sin s(sin p sin q cos r + cos p sin r )
− sin p sin q sin r
+ sin s sin p cos q
− cos s sin p cos q
+ cos p cos r
.
(7)
Our aim is to simulate the heading rotation, which is defined as the angle at which the carrier rotates counterclockwise around the z axis in the horizontal plane. Thus, we only need to discuss the angular rate vector component along the z axis. From Equation (5), we obtain p ω pz = Cb
iT
h 3i {i =1,2,3}
ωbx
ωby
ωbz
+ C3j{ j=1,2,3,4}
h
.
p
.
q
.
r
.
s
iT
.
(8)
Given that high-performance inertial sensors are often used for the platform INS, the physical platform changes caused by gyro drift are neglected. We obtain ωp =
iT
h ω px
ω py
ω pz
=
h
0
0
0
iT
.
(9)
To simplify Equation (8), we set the platform and inner gimbals to certain positions. The test can . . be conducted in a static horizontal base. Thus, p, q and p, q are set to 0. Then, Equations (6) and (7) can be simplified as 1 0 0 − sin r C= 0 0 1 (10) 0 , 0 1 0 cosr cos r cos s cos r sin s − sin r p Cb = − sin s (11) cos s 0 . cos s sin r sin s sin r cos r
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Substituting Equations (9)–(11) into Equation (8) yields .
ωbx cos s sin r + ωby sin s sin r + ωbz cos r + scosr = 0.
(12)
Given that the INS is placed in a horizontal plane, the roll and pitch angles of the carrier remain unchanged. Thus, we can obtain ωb =
iT
h ωbx
ωby
ωbz
=
h
0
0
iT ωbz
.
(13)
Then, Equation (12) can be decreased into .
ωbz cos r + scosr = 0,
(14)
where r is the relative rotation of the middle gimbal and is related to the local latitude of the carrier and the pitch value. It is a stationary base; thus, cos r is a fixed and a nonzero value. ωbz is the angular rate of the vehicle heading in the static horizontal plane. Equation (14) can be written as the following final form: . . ψ = ωbz = −s, (15) .
where ψ is the angular rate of the heading. In accordance with Equation (16), the heading rotation of the INS can be simulated by rotating the outer gimbal. 3.2.2. Outer Gimbal Rotation Test Procedure Before the heading rotation test, we need to consider whether there is singularity in the whole rotation process. The singular points can cause unpredictable consequences of the system. i and In the outer gimbal rotation test, firstly, p and q are set equal to 0 and unchanged, thus, Cm p Ci are identity matrices; Secondly, r is the relative rotation of the middle gimbal and is determined by the local latitude of the carrier and the pitch value. It is a stationary base indoors, thus, cos r is a fixed and nonzero value, except for the situation that systems are placed at the south pole and north pole points of earth and the pitch value is 90◦ ; Finally, neither s nor the matrix containing s does not appear on the denominators in mathematical relations. Thus, there will be no singularities. The heading rotation of the INS can be simulated by rotating the outer gimbal with angle range 0~360◦ . A rotation test can be initially conducted to identify the maximum temperature fluctuation regions for testing the performance of the temperature control system. The heading angle can be expressed as: ψ(t) = ψ(0) + k × t,
(16)
where ψ(t) is the heading at time t, ψ(0) is the heading angle at the initial time, and k is the heading angle rotation rate. After the maximum temperature fluctuation point at heading angle ψc is obtained, the periodic heading rotation test can be conducted. Then, ψ(t) can be expressed as ψ(t) = ψc + A × sin(2π f h × t),
(17)
where f h is the frequency of heading, ψc is the mean value of the harmonic oscillation, and A is the swing amplitude value. Using the continuous and periodic heading rotation test, the performance of the temperature control system can be evaluated quickly.
ψ ( t ) =ψ c + A× sin ( 2πfh × t ) ,
(17)
where f h is the frequency of heading, ψ c is the mean value of the harmonic oscillation, and A is the swing amplitude value. the continuous and periodic heading rotation test, the performance of the temperature SensorsUsing 2018, 18, 3967 9 of 18 control system can be evaluated quickly. 4. Evaluation Evaluation Process Process for for aa High-Precision High-Precision Platform Platform INS INS 4. The proposed proposed system system can can simulate simulate the the heading heading (azimuth) (azimuth) changes changes of of the the carrier carrier and and the the The environment temperature changes of the INSs. Figure 6 shows a typical evaluation process, which is environment temperature changes of the INSs. Figure 6 shows a typical evaluation process, which is conducted in in the thesubsequent subsequentsections. sections.The The high–low temperature system gimbals of conducted high–low temperature testtest system andand the the gimbals of the the platform are combined to build the high–low temperature and heading rotation test system. platform are combined to build the high–low temperature and heading rotation test system. The The high–low temperature is done to evaluate temperature differenceofofthe thegyros gyrosin in different different high–low temperature test test is done to evaluate thethe temperature difference environment temperatures. temperatures. The The continuous continuous rotation rotation test test is is adopted adopted to to identify identify the the large large temperature temperature environment fluctuation field of the temperature control system, and the sinusoidal rotation test is performed to fluctuation field of the temperature control system, and the sinusoidal rotation test is performed to analyze this large temperature fluctuation field further. The temperature data are recorded in the tests, analyze this large temperature fluctuation field further. The temperature data are recorded in the and the performance of the of temperature controlcontrol systemsystem can then bethen evaluated. In the following section, tests, and the performance the temperature can be evaluated. In the following practical tests are conducted to verify the proposed method. section, practical tests are conducted to verify the proposed method. High-low temperature test system + Platform INS
Economic indoor evaluation system
Low-temperature test
Continuous rotation test
High-temperature test
Maximum temperature fluctuation region analysis
Sinusoidal rotation
Temperature control system evaluation
Figure 6. Flowchart of the evaluation method.
Heading rotation conducted to to verify thethe performance of the rotation and andhigh–low high–lowtemperature temperaturetests testsare are conducted verify performance of proposed economical test method. The INS two dual-axis gyros with lowwith driftlow ratesdrift mounted the proposed economical test method. Thecomprises INS comprises two dual-axis gyros rates on a four-gimbal platform. The high–low temperature chamber istest produced in accordance with mounted on a four-gimbal platform. The high–low test temperature chamber is produced in Figure 3. The angle of the3.outer gimbalofand thetemperatures inertial sensors, temperature control accordance with Figure The angle thetemperatures outer gimbalofand of the inertial sensors, system and environment are retained. The performance of the temperature control of system is evaluated temperature control system and environment are retained. The performance the temperature using the test results. On the basis of the test results, the temperature control system can be improved control system is evaluated using the test results. On the basis of the test results, the temperature and its performance be re-verified. control system can becan improved and its performance can be re-verified. 4.1. Long-Term Long-Term High–Low 4.1. High–Low Temperature Temperature Test Test The high–low high–low temperature temperature test test is is performed performed to to simulate simulate the the change change in in the the external external environment environment The temperature. The The environment environment temperature temperature changes changes slowly slowly in in practical practical use use in in aa short In this this temperature. short time. time. In case, the temperature fluctuation of temperature control system is very small. Thus, in the high–low case, the temperature fluctuation of temperature control system is very small. Thus, in the high–low temperature test, test,only onlythe thetemperature temperaturefluctuation fluctuation inertial sensors is calculated. In high the high temperature ofof thethe inertial sensors is calculated. In the and ◦ ◦ and low temperature tests, the environment temperatures are set to 29 C and 19 C, respectively. low temperature tests, the environment temperatures are set to 29 °C and 19 °C, respectively. The The fluctuations of the gyroscopes retained after subtracting constantvalue valueatathigh high and and low low fluctuations of the twotwo gyroscopes areare retained after subtracting a aconstant temperatures in in 24 24 h. temperatures h. Figure 7a shows the temperature temperature fluctuations fluctuations of the two two gyros gyros in in the the high–low high–low temperature temperature tests tests Figure 7a shows the of the in 24 h. Figure 7b indicates the temperature calculated difference of the same gyros between the highin 24 h. Figure 7b indicates the temperature calculated difference of the same gyros between the high◦ C level, and low-temperature low-temperaturetests. tests.The Theaverage average temperature differences of both gyros at °C 1.0level, and temperature differences of both gyros are are at 1.0 and ◦ andhighest the highest temperature difference reaches 1.51 C. findings These findings imply the temperature the temperature difference reaches 1.51 °C. These imply that thethat temperature control control of system of the platform INS be should be improved system the platform INS should improved further.further.
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4.2. Short-Term Heading Rotation Test 4.2. Short-Term Heading Rotation Test The heading rotation test is realized by rotating the outer gimbal. The rotation test of outer gimbal The heading rotation test istemperature realized byfield rotating outer gimbal. Thesystem, rotation test of outer will directly change the internal of thethe temperature control which changes gimbal will directly change the internal temperature field of the temperature control system, which the temperature of the temperature control system quickly. However, the former change slightly affects changes the isolated temperature of the in temperature control system only quickly. However, the former change the internal gyroscopes a short time. Accordingly, the temperature fluctuation of the slightly affects the internal isolated gyroscopes in a short time. Accordingly, only the temperature temperature control system is evaluated in the heading rotation test. fluctuation of the temperature control system isfluctuation evaluated in the heading To determine the maximum temperature region quickly,rotation we findtest. a relatively wide To determine the maximum temperature fluctuation region quickly, we find a relativelyInwide range of excessive temperature fluctuations, firstly, by rotating the outer gimbal uniformly. this range of excessive temperature fluctuations, firstly, by rotating the outer gimbal uniformly. In and this part, we propose a fast rotation method that utilizes the principle of four-equal-division of cycle part, we propose a fast rotation method that utilizes the principle of four-equal-division of cycle and symmetry of the structure. The rotation process of the outer gimbal is given in Table 1. symmetry of the structure. The rotation process of the outer gimbal is given in Table 1. Table 1. Uniform rotation process of the outer gimbal for determining the regions with excessive Table 1. Uniform rotation process of the outer gimbal for determining the regions with excessive temperature fluctuation of the temperature control system. temperature fluctuation of the temperature control system. Rotation Process Total Angle (◦ ) Excessive Fluctuation
Rotation Process
Total Angle (°) Excessive Fluctuation 90 No 90 0 No No 0 90 No Yes 0 No 90 Yes 180 Yes 0 No 180–0° 180 Yes As shown in Table 1, the gimbal first rotates from 360◦ to 270◦ with a uniform rotation rate of ◦ ◦ for first As The shown in Table 1, the gimbal rotates fromto 360° to 270°the with a uniform rotation rate of 0.1 /s. gimbal pauses at 270 several minutes decrease effect of previous rotation to ◦ ◦ 0.1°/s. The gimbal pauses at 270° for several minutes to decrease the effect of previous rotation to the the next one. Then, the gimbal rotates from 270 to 180 . A large temperature fluctuation is observed. ◦ toto0◦180°. next one. Then, the gimbal rotates rotates from from180 270° A large temperature fluctuation Thereafter, the outer directly because the internal structure is of observed. the INS is Thereafter, the outer gimbalThe rotates from 180° with to 0° excessive directly because the internal structure of the INS approximately symmetric. second region temperature fluctuation is observed as is approximately symmetric. The second region with excessive temperature fluctuation is observed expected (Figure 8). as expected Figure 8(Figure shows8). the temperature fluctuation curve of the temperature control system, which is obtained by rotating the outer gimbal uniformly in the entire cycle. As shown in the figure, two regions have excessive temperature fluctuation. The symmetry of the two regions can be observed clearly in the partially enlarged Figure 9a,b, which indicates that the entire structure of the INS is symmetric. The maximum temperature fluctuation reaches 0.68 ◦ C. After the two regions with excessive temperature fluctuation are obtained, we make the outer gimbal rotate sinusoidally to simulate the heading change of the ship in actual navigation. To further analyze the fluctuation caused by the actual heading rotation, some outer gimbal rotations with 360–270◦ 360–270° Pause ◦ 270–180Pause Pause 270–180° 180–0◦Pause
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Figure 8. Temperature fluctuation of the temperature control system. 1
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Table 2 lists the temperature fluctuation values of0 the two regions with excessive temperature 180 0 100 two 200 regions 300 400 500 600 700 800 temperature 900 1000 0 100 200 are 300 obtained, 400 500 600 700 800the 900outer 1000 After the with excessive fluctuation we make fluctuation in different heading temperature fluctuation Time (s) rotation processes, in which the maximum Time (s) gimbal rotate sinusoidally to simulate the heading change of the ship in actual navigation. To further◦ and its corresponding mean (a) angle of the heading rotation test are in bold.(b)Regions of around 64.5 analyze the fluctuation caused by the actual heading rotation, some outer gimbal rotations with with excessive temperature fluctuation and are symmetric in structure are found. The maximum sinusoidal period are carried out near the two with excessive temperature fluctuation. The ◦ C.regions temperature point reachesof0.8125 The temperature fluctuations in angle otherranges regions Figure 9.fluctuation Temperature fluctuation the temperature control system in the gimbal of are (a) less rotation process and relevant parameters of two platform INSs are recorded in Table 2. ◦ 270–246°C. and (b) 92–54°. than 0.4500 We take the maximum temperature fluctuation of INS 1# as an example. The mean value, After the two regions with excessive fluctuation are obtained,Given we make the outer rotation amplitude, and rotation period aretemperature 64.5◦ , 30◦ , and 1200 s, respectively. the structural gimbal rotate sinusoidally toonly simulate the headingresult change of the ship in actual further symmetry of the two regions, the experimental conducted in one regionnavigation. (92–54◦ ) is To presented analyze the fluctuation caused by the actual heading rotation, some outer gimbal rotations with in Figure 10. sinusoidal arethat carried out near the two regions with excessive fluctuation. The Figureperiod 10 shows excessive temperature fluctuations are againtemperature observed when the heading rotation process and relevant parameters of two platform INSs are recorded in Table 2. is set as a sinusoidal rotation. The test result is in line with the previous uniform rotation test. The maximum temperature fluctuation point reaches 0.8125 ◦ C. The temperature control system of the INS should be further improved on the basis of the results of the high–low temperature test (Section 4.1) and rotation test (Section 4.2). In the following part,
Table 2. Sinusoidal periodic rotation process and relevant parameters of the outer gimbal. Temperature Maximum Excessive Amplitude (°) Period (s) Fluctuation (°C) Fluctuation Sensors 2018, 18, 3967 12 of 18 64.5 −0.8125 Yes −0.7813 64.5 + 180 the temperature is optimized using the evaluation INS 1# control system of the high-precision platform INS0.5625 results above. 64.5 + 90
