System Error Compensation Methodology Based ... - Semantic Scholar

sensors Article

System Error Compensation Methodology Based on a Neural Network for a Micromachined Inertial Measurement Unit Shi Qiang Liu and Rong Zhu * State Key Laboratory of Precision Measurement Technology and Instrument, Department of Precision Instruments, Tsinghua University, Beijing 100084, China; [email protected] * Correspondence: [email protected]; Tel./Fax: +86-10-6278-8935 Academic Editor: Vittorio M. N. Passaro Received: 23 November 2015; Accepted: 27 January 2016; Published: 29 January 2016

Abstract: Errors compensation of micromachined-inertial-measurement-units (MIMU) is essential in practical applications. This paper presents a new compensation method using a neural-network-based identification for MIMU, which capably solves the universal problems of cross-coupling, misalignment, eccentricity, and other deterministic errors existing in a three-dimensional integrated system. Using a neural network to model a complex multivariate and nonlinear coupling system, the errors could be readily compensated through a comprehensive calibration. In this paper, we also present a thermal-gas MIMU based on thermal expansion, which measures three-axis angular rates and three-axis accelerations using only three thermal-gas inertial sensors, each of which capably measures one-axis angular rate and one-axis acceleration simultaneously in one chip. The developed MIMU (100 ˆ 100 ˆ 100 mm3 ) possesses the advantages of simple structure, high shock resistance, and large measuring ranges (three-axes angular rates of ˘4000˝ /s and three-axes accelerations of ˘10 g) compared with conventional MIMU, due to using gas medium instead of mechanical proof mass as the key moving and sensing elements. However, the gas MIMU suffers from cross-coupling effects, which corrupt the system accuracy. The proposed compensation method is, therefore, applied to compensate the system errors of the MIMU. Experiments validate the effectiveness of the compensation, and the measurement errors of three-axis angular rates and three-axis accelerations are reduced to less than 1% and 3% of uncompensated errors in the rotation range of ˘600˝ /s and the acceleration range of ˘1 g, respectively. Keywords: micromachined inertial measurement unit; inertial sensor; thermal gas sensor; error compensation; neural network

1. Introduction With the advancements of micromachined accelerometers and gyroscopes, miniaturized inertial measurement unit (MIMU) has attracted more and more attentions in recent years due to its small size, batch fabrication, low cost, and low power consumption. Nowadays, MIMU plays an important role in civil and military applications (e.g., smartphones, wearable equipment, motion tracking equipment, vehicle navigation, etc.) [1–4]. In general, conventional MIMU with six degrees of freedom is a three-dimensional integrated system consisting of three uniaxial micromachined accelerometers (or a three-axis accelerometer) and three uniaxial micromachined gyroscopes (or a three-axis gyroscope) mounted orthogonally in a Cartesian coordinate system. The accelerometers and gyroscopes measure the accelerations and angular rates around corresponding three-axes, respectively. Theoretically, each output of the inertial sensors is proportional to the angular rate or acceleration around the corresponding axis of the system and, thus, three orthogonal angular

Sensors 2016, 16, 175; doi:10.3390/s16020175

www.mdpi.com/journal/sensors

Sensors 2016, 16, 175

2 of 14

rates and three orthogonal accelerations can be measured directly from the sensor outputs in MIMU. As a matter of fact, however, the outputs of the inertial sensors may be coupled with each other, containing deterministic errors and stochastic errors, all of which corrupt the system accuracy [5]. Stochastic errors are usually represented as bias instability and random walk which can be restrained with filtering technologies, such as prediction error minimization (PEM) method, autoregressive (AR) processes, wavelet multi-resolution techniques, and so on [6,7]. Deterministic errors in a MIMU system are mainly induced by cross-coupling, misalignment, and eccentric problems when assembling multiple sensors into a three-dimensional system, and it has been experimentally demonstrated that they result in navigation errors with quadratic or cubic growth in time [8]. Compensation, often termed MIMU calibration, is generally used to eliminate the deterministic errors by using error-model-based calibration. Presently, many methods of MIMU calibration have been reported, such as multi-position static calibration [9–11], fast field calibration [12], and optical alignment calibration [13]. These methods usually need to establish a physical error model prior to the calibration process through an analytical method. For example, the model of misalignment error can be formulated by a direction cosine matrix [13]. However, some errors, such as cross-coupling, are generally difficult to be physically modeled due to their complexity, multivariability, coupling, and nonlinearity. Therefore, an effective system identification method needs to be investigated to solve this problem and, consequently, eliminate the system errors to the greatest extent. Conventional micromachined inertial sensors, such as vibrating shell gyroscopes [14], tuning-forks gyroscopes [15], and vibrating beams gyroscopes [16], need a mechanical proof mass suspended on springs, which raises complexities of structure and fabrication, and also restricts the high-shock resistance of the device [1]. Another kind are fluidic inertial sensors, which use a fluidic medium instead of mechanical proof mass as the key moving and sensing elements. According to the driving principle of the fluidic medium, the sensor can be categorized into the jet flow gyroscope based on jet flow [17,18], the thermal accelerometer, and gyroscope based on thermal convection [19–22]. Another gas inertial sensor based on thermal expansion was recently reported and demonstrated simple structure, high shock resistance, and wide measuring range [23], which is able to measure one-axis angular rate and one-axis acceleration simultaneously in one chip. However, the thermal inertial sensors exhibit obvious cross-coupling errors among axes and between rotation and acceleration measurements because of the complex thermal fluid motion inside the sensor chamber [23,24], such as the influence of the gravitational acceleration on the gyro output at a vertical position of the sensor. In this paper, we present a thermal gas MIMU system using only three thermal gas inertial sensors to realize measurements of three-axis angular rates in the range of ˘4000˝ /s and three-axis accelerations in the range of ˘10 g. To overcome the complex cross-coupling errors and other system errors of the thermal gas MIMU, we propose a compensation method using a three-layer back propagation (BP) neural network to model the complex and nonlinear error system for eliminating system errors of a MIMU. The parameters of the neural network are estimated by using experiment-based identification. Experiments validate the effectiveness of the compensation. Results also indicate that the neural network-based compensation method is effective for eliminating not only the cross-coupling error but also the errors induced from other various sources, such as misalignment and eccentricity. Compared with other compensation methods, the proposed compensation method is easily operated, physical-model-free, universal, and with good feasibility. 2. Design of Thermal Gas MIMU 2.1. Thermal Gas Inertial Sensor Used in the Gas MIMU The core components of the developed thermal gas MIMU are the micromachined thermal gas inertial sensors based on thermal expansion shown in Figures 1 and 2 [23,25]. A sensor’s body coordinate frame x-y-z is defined, where the x-y plane is the working plane of the sensor and the z-axis is perpendicular to the working plane. The configuration of the thermal gas inertial sensor

Sensors 2016, 16, 175

3 of 14

Sensors 2016, 16, 175

3 of 14

comprises three heater wires (ht1, ht2, ht3) and four thermistor wires (R1, R2, R3, R4) which are and coolinginthe heater (ht2) the two over side heaters and ht3) alternately, seismic gas distributed thecentral working plane andand suspended a sealed(ht1 micro chamber filled withtwo a gas medium. streams in and opposite directions along the(ht2) x-axis in the (ht1 working plane and the By heating cooling the central heater andare thegenerated two side heaters and ht3) alternately, Sensors 2016, 16, 175 3 of 14 temperature profiles illustrated as red solid lines Figure are generated formed ininthe y-axis two seismic gas streams in opposite directions alonginthe x-axis2 are thechamber. working The plane and linear acceleration deflects the temperature profiles the two of the y-axis theseismic sameThe direction and cooling the central heater (ht2) twolines sideon heaters (ht1 and ht3) alternately, two gasy-axis the temperature profiles illustrated asand redthe solid in Figure 2sides are formed in thein chamber. opposite directions arethegenerated inofz-axis the and the alongstreams y-axis in (shown in Figure 2a),along whilethe thex-axis rotation around the deflects the temperature linear acceleration deflects the temperature profiles on two sides the working y-axis in plane the same direction temperature profiles illustrated as red solid lines inaround Figure 2 arez-axis formed in the chamber. profiles on the two sides of the in the opposite direction along the y-axis due toThe they-axis opposite along y-axis (shown in Figure 2a),y-axis while the rotation the deflects the temperature profiles linear acceleration deflects the temperature profiles on the two sides of the y-axis in the same direction Coriolis accelerations on the two sides (shown in Figure 2b). These deflections of the temperature on the two sides of the y-axis in the opposite direction along the y-axis due to the opposite Coriolis along y-axis (shown in Figure 2a), while the rotation around the z-axis deflects the temperature profile induced y-axis linearinacceleration z-axis rotation are detected by using the accelerations on by the either two sides (shown Figure 2b). or These deflections of the temperature profile profiles on the two sides of the y-axis in the opposite direction along the y-axis due to the opposite distributed four thermistors, of which are used to deduce the y-axis acceleration and z-axis induced by either y-axis linearoutputs acceleration or z-axis rotation are detected by using the distributed four Coriolis accelerations on the two sides (shown in Figure 2b). These deflections of the temperature angular rate induced [23]. Therefore, they-axis thermal gasdeduce inertialthe sensor hasacceleration anrotation inherent ability to measure thermistors, outputs of which are used to y-axis and z-axis angular rate [23]. profile by either linear acceleration or z-axis are detected by usingthe they-axis acceleration and the z-axis rotation simultaneously in one chip. The structure of the sensor has been Therefore, the thermal gas inertial sensor has an inherent ability to measure the y-axis acceleration distributed four thermistors, outputs of which are used to deduce the y-axis acceleration and z-axis and optimized inrate our[23]. previous work to restrain thermal convection andability enhance thermal the z-axis rotation simultaneously in one chip. The structure of inherent the sensor hastobeen optimized in our angular Therefore, the[23] thermal gas inertial sensor has an measure theexpansion, y-axis resulting in merits of simple structure, high shock resistance, large-range rotation gauge, and low acceleration andto the z-axis rotation simultaneously in one chip. thermal The structure of the sensor has been previous work [23] restrain thermal convection and enhance expansion, resulting in merits optimized in ourhigh previous work [23] to restrain thermal convection and and enhance expansion, cross-coupling effects. of simple structure, shock resistance, large-range rotation gauge, low thermal cross-coupling effects. resulting in merits of simple structure, high shock resistance, large-range rotation gauge, and low cross-coupling effects. Upper silicon cavity Upper silicon cavity

Chamber filled with gas

z y R2

R4z

y ht2 R4 ht1 R2 R1 ht1

ht2 R1

x ht3 xR3 ht3 R3

Chamber filled with gas

Lower silicon cavity

Lower silicon cavity

Figure 1. Schematic diagram of the thermal gas inertial sensor. Figure 1. Schematic thermalgas gasinertial inertial sensor. Figure 1. Schematicdiagram diagram of of the the thermal sensor.

Figure 2. Temperature distribution deflected by a linear acceleration (a) and Coriolis acceleration (b).

Figure (a) and and Coriolis Coriolisacceleration acceleration(b). (b). Figure2.2.Temperature Temperaturedistribution distributiondeflected deflected by by aa linear linear acceleration acceleration (a)

2.2. Design of the Thermal Gas MIMU

2.2. Design of the Thermal Gas MIMU

In this paper three thermal gas inertial sensors are integrated together to construct a MIMU.

In paper three thermal inertial sensors are 3.integrated Thethis developed prototype of thegas MIMU is shown in Figure Each sensortogether is mounted a PCB, where to on construct a MIMU. the conditioning circuit of the sensor is located. The three PCBs are assembled orthogonally to form The The developed developed prototype prototypeof ofthe the MIMU MIMU is shown shown in in Figure Figure 3. 3. Each Each sensor sensor is mounted mounted on on a PCB, where a three-dimensional frame. A rectangular coordinate frame X-Y-Z defined as the MIMU’s body the conditioning circuit of the sensor is The three PCBs are orthogonally to form sensor is located. located. areisassembled assembled orthogonally form frame, where X, Y, and Z axes are served as sensing axes of the MIMU. On each axis of X-Y-Z, there a three-dimensional frame. A rectangular coordinate frame X-Y-Z is defined as the MIMU’s body three-dimensional X-Y-Z is located a thermal gas inertial sensor, the outputs of which correspond to one-axis acceleration and frame, On each axis of of X-Y-Z, there is frame, where where X, X,Y, Y,and andZZaxes axesare areserved servedasassensing sensingaxes axesofofthe theMIMU. MIMU. On each axis X-Y-Z, there one-axis angular rate. Specifically, the outputs of the sensor on the X-axis ( , ) corresponds to Y-axis located a thermal gas inertial is located a thermal gas inertialsensor, sensor,the theoutputs outputsofofwhich whichcorrespond correspondto toone-axis one-axisacceleration acceleration and and acceleration and X-axis angular rate ; that of the sensor on the Y-axis ( , ) corresponds to one-axis angular rate.Specifically, Specifically, outputs of the sensor onthe thesensor X-axis ,ωˆ Z-axis angular rate. thethe outputs of the sensor on the X-axis ( ,on(aˆ)the X ) corresponds Ycorresponds Z-axis acceleration and Y-axis angular rate and that of ( ,to Y-axis ) to ˆ ˆ Y-axis acceleration a and X-axis angular rate ω ; that of the sensor on the Y-axis ( a , ω ) corresponds acceleration and X-axis angular rate ; that of the sensor on the Y-axis ( , ) corresponds to X Z Y Y corresponds to X-axis acceleration and Z-axis angular rate . ˆ ˆ to Z-axis acceleration a and Y-axis angular rate ω and that of the sensor on the Z-axis a , ω Z-axis acceleration and Y-axis angular rate and that of the sensor on the Z-axis ( A diagram of theZ measurement and control circuit Y of the MIMU is illustrated in Figure 4, whereX Z ) corresponds to acceleration a X and Z-axis angular ωZ .demonstrated. the conditioning of the thermal gas inertial sensorrate is also The sensor circuit corresponds to X-axis X-axiscircuit acceleration and Z-axis angular rate . consisting two bridges, amplifying circuit, demodulation circuit, A diagramofof theWheatstone measurement andtwo-level control differential circuit of the MIMU is illustrated in Figure 4, where the conditioning circuit of the thermal gas inertial sensor is also demonstrated. The sensor circuit consisting of two Wheatstone bridges, two-level differential amplifying circuit, demodulation circuit,

Sensors 2016, 16, 175

4 of 14

A diagram of the measurement and control circuit of the MIMU is illustrated in Figure 4, where the conditioning circuit of the thermal gas inertial sensor is also demonstrated. The sensor circuit consisting of two Wheatstone bridges, two-level differential amplifying circuit, demodulation circuit, Sensors 2016, 16, 175 circuit are used to implement the conversions from the deflections 4 of 14 and low-pass filtering of the Sensors 2016, 16, 175 4 of 14 temperature profile to the quantities of acceleration and rotation. Each sensor has one 16-bit A/D and low-pass filtering circuit are used to implement the conversions from the deflections of the converter temperature andlow-pass one micro control unit to extractand one-axis acceleration and one-axis angular rate profile to the quantities of to acceleration Eachfrom sensor one 16-bitofA/D and filtering circuit are(MCU) used implement therotation. conversions thehas deflections the converter one micro control unit (MCU) to extract one-axis acceleration and one-axis rate temperature profile to the quantities of acceleration and rotation. Each has oneangular 16-bit A/D which makes eachand circuit board independent and avoids crosstalk. A sensor high-performance MCU acts as which makes circuitcontrol board independent avoids crosstalk. A high-performance actsrate as converter andeach one micro unit (MCU) toand extract one-axis acceleration and one-axisMCU angular a master for operating compensation algorithm. awhich master for operating compensation algorithm. makes each circuit board independent and avoids crosstalk. A high-performance MCU acts as a master for operating compensation algorithm.

Figure 3. Configuration of the thermal gas MIMU.

Figure 3. Configuration of the thermal gas MIMU. Figure 3. Configuration of the thermal gas MIMU.

R

R1

R BridgeR1 1 Vcc

1 RBridgeR2

Vcc

R R

R2 R3

R BridgeR3 2 2 RBridgeR4 R

Differential amplifier Differential amplifier Differential amplifier Differential amplifier

R4

Summing amplifier Summing amplifier Reference signal Reference signal Differential amplifier Differential amplifier

LP filter LP filter

Demodulation Demodulation

X-axis thermal X-axisgas sensorgas thermal

X-axis signal conditioning X-axis signal circuit conditioning

sensor

circuit

Y-axis thermal Y-axisgas sensorgas thermal

Y-axis signal conditioning Y-axis signal circuit conditioning

sensor

circuit

Z-axis thermal Z-axisgas sensorgas thermal

Z-axis signal conditioning Z-axis signal circuit conditioning

sensor

circuit

X-axis demodulation X-axis & low-pass filtering demodulation & circuit low-pass filtering

A/D conversion A/D conversion

circuit Y-axis demodulation Y-axis & low-pass filtering demodulation & circuit low-pass filtering

Slave MCU Slave MCU SPI

A/D conversion A/D

SPI Master MCU Master

conversion

MCU SPI

A/D conversion A/D conversion

SPI Slave MCU Slave

circuit Z-axis demodulation Z-axis & low-pass filtering demodulation & circuit low-pass filtering circuit

MCU

Figure 4. Schematic diagram of conditioning circuit of the thermal gas MIMU.

Figure 4. Schematic diagram of conditioning circuit of the thermal gas MIMU.

AsFigure mentioned above, each thermalofgas inertial sensor is capable simultaneously measuring 4. Schematic diagram conditioning circuit of theofthermal gas MIMU. one-axis acceleration and one-axis rotation. Three gas inertial sensors are perfectly adequate to As mentioned above, each thermal gas inertial sensor is capable of simultaneously measuring measure accelerations androtation. three-axis rotations, which simplifies theperfectly MIMU structure one-axis three-axis acceleration and one-axis Three gas inertial sensors are adequateand to

As mentioned above, each thermal gas inertial sensor is capable of simultaneously measuring measure three-axis accelerations and three-axis rotations, which simplifies the MIMU structure and one-axis acceleration and one-axis rotation. Three gas inertial sensors are perfectly adequate to measure three-axis accelerations and three-axis rotations, which simplifies the MIMU structure and operating circuit, miniaturizes the system size, and reduces the power consumption and fabrication cost of the MIMU.

Sensors 2016, 16, 175

5 of 14

3. Error Source Analysis of the MIMU In general, error sources of a MIMU mainly consist of cross-coupling, misalignment, eccentricity, noise, etc. Among them, the cross-coupling error is significant for a thermal gas MIMU due to the complex fluidic motion in the sensor chamber. The cross-coupling effects include the coupling errors among the measurements of three-axis accelerations and three-axis angular rates. Although an impressive decrease of the coupling was obtained via using thermal expansion instead of thermal convection [23], the cross-coupling error still cannot be ignored for the gas MIMU in practical applications. According to our previous studies [23,24], there are two types of fluid motions in the gas inertial sensors, thermal convection flow and thermal expansion flow. The fluid motion driven by thermal convection is dependent on the acceleration and, thus, results in cross-coupling effects between acceleration and rotation measurements of the sensor, while the fluid motion driven by thermal expansion is independent of the acceleration. A non-dimensional momentum model characterizing the kinetic process of the three-dimensional flow in a chamber of a thermal gas sensor is formulated as [23]: Ñ ˆ ˙ ˙ ˆ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ BT Ñ BU Ñ U ´ 2Ω ˆ U ´ Ω ˆ Ω ˆ r ` ∇ ¨ τ ` U ¨ ∇U ` U ∇ ¨ U “ Gr∆Θ ` αv Bδ Bδ

(1)

¸ αv pTh ´ T0 q a x H 3 αv pTh ´ T0 q ay H 3 αv pTh ´ T0 q az H 3 , , where the Grashof number Gr “ Gr x , Gry , Grz “ ; v2 v2 v2 ¨ ˛T Ñ ˙T ˆ ˆ ˙T Ñ ´ x y z ¯T Ñ Ñ Ñ Ñ Ñ r B B B U Ux Uy Uz 2 , , , τ “ ∇U ` ∇U ´ σ∇ ¨ U; U “ ´ v ¯ “ ˝ v , v , v ‚ , r “ “ , , , ∇“ Bx By Bz 3 H H H H H H H H Ñ ˆ ˙ Ωy T ´ T0 Ñ Ω Ωx Ωz T ˘ “ Θ “ ,Ω “ ` , , ; δ “ tυ{H 2 ; Th is the heater temperature; Th ´ T0 υ{H 2 υ{H 2 υ{H 2 υ{H 2 Ñ

˜

`

˘

Ñ

T0 refers to the ambient temperature; r “ px, y, zq refers to the position vector in the sensor’s body Ñ ` ˘ Ñ ` ˘T coordinate frame; angular rate vector Ω “ Ω x , Ωy , Ωz ; U “ Ux , Uy , Uz refers to fluid velocity Ñ ` ˘T µ vector f “ a x , ay , az refers to the acceleration vector; υ(υ “ ) is momentum viscosity; H is ρ ˇ ˇ 1 BV ˇˇ 1 Bρ ˇˇ 1 ρ ´ ρ0 the feature dimension; the coefficient of thermal expansion αv “ “ ´ ; p p «´ V BT ˇ ρ BT ˇ ρ T ´ T0 Ñ

∇ P “ ρ0 f ; ρ0 refers to the gas density at the ambient temperature; ρ, P, µ, T are gas density, pressure, dynamic viscosity, and temperature in the sensor chamber. Ñ

In Equation (1), the item of Gr∆Θ is a driving force to generate a thermal convection flow, BT Ñ U is a driving force to generate the thermal expansion flow. Obviously, while the term of αv Bδ Ñ the former is correlative with the Grashof number Gr, which is dependent on the acceleration Ñ ` ˘T f “ a x , ay , az , while the latter is correlative with the transient variation rate of the temperature that is independent of the acceleration. From Equation (1), it is theoretically demonstrated that the thermal convection flow is generally existent in a fluid dynamic system, which unavoidably results in a cross-coupling between acceleration and rotation measurements, such as the influence of the gravitational acceleration on the gyro output at a vertical position of the sensor. In addition, the problems are more complex due to misalignment and asymmetry of the sensor structure induced in fabrication, thermal distortion in the operation of the sensor, all of which will induce the cross-coupling effects. Consequently, the actual couplings exhibit a complex interactive relationship between not only rotation and acceleration measurements, but also rotation measurements among axes and acceleration measurements among axes. In addition, it is seen that other inertial sensors based on thermal principles all suffer cross-coupling effects, such as thermal jet gyroscopes and thermal accelerometers.

Sensors 2016, 16, 175

6 of 14

In addition to the cross-coupling errors, misalignment is another error source of the MIMU. Misalignment errors generally exist when assembling multiple sensors into a three-dimensional system, which bring about negative effects on the system accuracy. In addition, assembly positions of individual sensor deviated from the centroid of the MIMU system will also result in assembling errors (termed as eccentric errors) in the system. In addition of above deterministic errors, there are stochastic errors, which are not a concern of this paper. 4. Error Compensation Method Based on a BP Neural Network 4.1. System Error Modelling The basic idea of the proposed compensation method is to establish a relationship model between the outputs of the sensors and the physical quantities to be measured. For a MIMU, the quantities to be measured are the three-axis accelerations and three-axis angular rates along the X-Y-Z axes denoted as a “ pa X , aY , a Z q and ω “ pωX , ωY , ωZ q, respectively. The outputs of the sensors in the MIMU are the voltage outputs of the sensors representing three-axis acceleration outputs and three-axis angular rate outputs denoted as aˆ “ paˆ X , aˆ Y , aˆ Z q and ω ˆ “ pωˆ X , ωˆ Y , ωˆ Z q, respectively. Obviously, the relationship model of the MIMU is a multi-input and multi-output (MIMO) coupling system, which can be formulated by: ra, ωs “ f pra, ˆ ωsq ˆ (2) In general, the work for error compensation focuses on mapping relationship f which is a multivariate and nonlinear function. Analytical methods are commonly used for approximating f , but are usually too complicated and inefficient, especially for cross-coupling effects as mentioned above. An alternative method is experimental identification, which is more reliable and effective. In this paper, a system identification method based on experiment to model f is proposed using a three-layer back propagation (BP) neural network as the model structure. A BP neural network is simpler and more mature than other neural networks. Moreover, it has been theoretically proven that three layers of a neural network could solve arbitrarily complicated nonlinear mapping problems [26]. ˆ ωs ˆ and six Illustrated in Figure 5, the structure of the BP neural network with six inputs ra, outputs ra, ωs was used to represent the model of f . The network includes several hidden neurons, the number of which was determined by trial. In Figure 5, the transfer function of the hidden layer fhid , the function of output layer fout , and transfer matrix wih and who are the key parameters of the network model structure. These parameters, including the transfer functions fhid and fout , the number of hidden neurons, transfer weights wih and who need to be determined through training using sufficient experimental data, in which the outputs of the gas inertial sensors were used as the inputs of the network, while the corresponding data of the actual accelerations and angular rates were used as the outputs of the network. The training of the neural network is to search for appropriate network parameters to minimize the sum-squared error between the network outputs and actual values by using an iteration algorithm. In the standard back propagation learning algorithm [26], two key parameters, including the learning rate and momentum parameter, need to be configured properly. The learning rate determines the increments of the transfer weights in each iteration step. A large learning rate leads to a fast learning but may result in non-convergence. The additional momentum makes it possible to avoid dropping into local minimum, which improves the efficiency of convergence in training. The BP learning algorithm with a learning rate of 0.005 and a momentum parameter of 0.5 is used in our training. The transfer functions fhid and fout play important roles in the model structure and accuracy. The hidden layer fhid usually uses tan-sigmoid or log-sigmoid function and the output layer fout often uses log-sigmoid or pure-line functions. Through experimental trials, we finally select a tan-sigmoid function of fhid and a pure-line function of fout according to the accuracy of the model. The number of hidden neurons is another key parameter dominating system accuracy and structural simplicity. A small number of hidden neurons will cause large fitting errors because there are not enough to model the MIMU system accurately, while too many hidden neurons will be redundant,

Sensors 2016, 16, 175

7 of 14

which2016, require more Sensors 16, 175

calculations but do not help to improve the model accuracy. Considering7both of 14 accuracy and simplicity, the number of hidden neurons is finally determined to be 120 through trials the the sum-squared becomes steady. MATLAB is useduntil to develop the neural that decrease graduallyofincrease the numbererror of hidden neurons from a small number the decrease of the network. sum-squared error becomes steady. MATLAB is used to develop the neural network. Sensors 2016, 16, 175 7 of 14

(6) is used to develop the neural Input (6)becomes Hidden (n) Output the decrease of the sum-squared error steady. MATLAB network. fout Input (6)

Hidden fout(6) fhid (n) Output

fhid

fout

fout

fhid

fout

fhid

fout

⁞

fhid

fout fout

⁞

fout

fhid

fout

fout

fout Figure 5. Structure of of three-layer three-layer back back propagation propagation (BP) (BP) neural neural network to to model model the the errors errors of of Figure 5. Structure network the MIMU. the MIMU. Figure 5. Structure of three-layer back propagation (BP) neural network to model the errors of the MIMU. Calibration Method

4.2. Calibration Method 4.2.

Calibration isMethod an experimental experimental method method to to obtain obtain aa group group of of sample sample data data used used for for training training the the 4.2. Calibration Calibration is an above neural network model. The acquired data of the sample should be sufficient, by which the above neural network model. The acquired of the sample shoulddata be sufficient, by which Calibration is an experimental method data to obtain a group of sample used for training the the working conditions of MIMU are working conditions of the the model. MIMUThe are represented. represented. above neural network acquired data of the sample should be sufficient, by which the The calibration experiment conducted, including three multi-position staticmultitests, working conditions of the MIMU are represented. The calibration experiment is is conducted, including three tests:tests: multi-position static tests, multi-position turntable rotation tests, and centrifugation tests. A two-axis turntable (902E-1, AVIC The calibration experiment is conducted, including three tests: multi-position static tests, multiposition turntable rotation tests, and centrifugation tests. A two-axis turntable (902E-1, AVIC Beijing ˝ /s, position turntable rotation tests,Aircraft andAircraft centrifugation tests. A measurement two-axis turntable (902E-1, AVIC Beijing Beijing Precision Engineering Institute Industry range rotation is Precision Engineering Institute Industry the the measurement range ofofrotation is ˘600 ±600°/s, ˝ Precision of Engineering Institute Aircraft Industry the measurement range of rotation is ±600°/s, the resolution is is 0.001 /s, the angle accuracy around inner-axis and outer-axis is ˘5” the resolution ofangular angularrate rate 0.001°/s, the angle accuracy around inner-axis and outer-axis is and ±5″ the resolutionthe of angular rate is 0.001°/s, the angle accuracyaxes around inner-axis and outer-axis is leveling ±5″ ˘8” respectively, perpendicularity between the rotation of both turntables is ˘5”, the and ±8″ respectively, the perpendicularity between the rotation axes of both turntables is ±5″, the and ±8″ respectively, the perpendicularity between the rotation axes of both turntables is ±5″, the accuracy of the rotation is ˘5”) Figurein 6 is used 6inisthe experiment. The turntable leveling accuracy of the axes rotation axesshown is ±5″)inshown Figure used in the experiment. The leveling accuracy of the rotation axes is ±5″) shown in Figure 6 is used in the experiment. The contains an electric slip ring used for signal transmission between the MIMU and other instruments. turntable contains an electric slip ring transmissionbetween between MIMU turntable contains an electric slip ringused usedfor for signal signal transmission thethe MIMU and and otherother A reference coordinate frame X -Y -Z is defined as North-West-Upwards, where the axes of Xi andthe Yi i i i instruments. A reference coordinate frame X i -Y i -Z i is defined as North-West-Upwards, where instruments. A reference coordinate frame Xi-Yi-Zi is defined as North-West-Upwards, where the lie inof a horizontal plane and the axis of Z points to the opposite direction of gravity. i axes X i and Y i lie in a horizontal plane and the axis of Z i points to the opposite direction of gravity. axes of Xi and Yi lie in a horizontal plane and the axis of Zi points to the opposite direction of gravity.

ZZii

θ θ

θX

θ Yi Yi

i

Xi

Figure 6. Two-axis turntable used for calibration.

Figure 6. Two-axis turntable used for calibration.

6. Two-axis turntable to used for calibration. The multi-position Figure static tests are conducted learn the coupling relationship between acceleration measurements X-Y-Z axes. The to developed MIMU is mounted on the turntable The multi-position staticamong teststhe are conducted learn the coupling relationship between The multi-position static tests conducted relationship between and the X-Y-Z axes of the MIMU areare adjusted, as shown inlearn Figurethe 7,MIMU bycoupling rotating the turntable, acceleration measurements among the X-Y-Z axes. Theto developed is mounted on thewhere turntable acceleration amongfrom the X-Y-Z axes. The MIMU is mounted turntable θ is a tiltmeasurements angle which is applied 0° to 360° at 5° perdeveloped step. At each step, the outputs ofon thethe sensors andX-Y-Z actualaxes accelerations along X-Y-Z as axes are recorded. In by this experiment, the actual and the of the MIMU arethe adjusted, shown in Figure 7, rotating the turntable, where

θ is a tilt angle which is applied from 0° to 360° at 5° per step. At each step, the outputs of the sensors and actual accelerations along the X-Y-Z axes are recorded. In this experiment, the actual

Sensors 2016, 16, 175

8 of 14

and the X-Y-Z axes of the MIMU are adjusted, as shown in Figure 7, by rotating the turntable, where θ is a tilt angle which is applied from 0˝ to 360˝ at 5˝ per step. At each step, the outputs of the sensors and actual actual Sensors 2016, 16, 175accelerations along the X-Y-Z axes are recorded. In this experiment, the 8 of 14 accelerations to be measured are the gravity components along the X-Y-Z axes and the actual angular to be measured arezero the gravity components alongof theEarth X-Y-Zrotation. axes and the rates accelerations are approximately equal to without consideration Theactual Earthangular rotation is Sensorsare 2016, 16, 175 8 of 14 rates approximately equal to zero without consideration of Earth rotation. The Earth rotation is ˝ much less than the noise level of the thermal gas inertial sensor (around 1 /s RMS [23]). Therefore, much less than the noise level of the thermal gas inertial sensor (around 1°/s RMS [23]). Therefore, we ignored the Earth inare this accelerations to berotation measured thestudy. gravity components along the X-Y-Z axes and the actual angular we ignored the Earth rotation in this study. rates are approximately equal to zero without consideration of Earth rotation. The Earth rotation is much less than the noise level of the thermal gas inertial sensor (around 1°/s RMS [23]). Therefore, X Z Y X Y study. we ignored the EarthZ rotation in this

θ

X Z

X

Y θ

X

θ

Y

Z

(a)

Y

θ

Y

θ

Z

X θ

Z

(b)

(c)

Figure 7. Processes of multi-position static calibration; (a) the rotation axis X is along Yi; (b) the

Figure 7. Processes of multi-position static calibration; (a) the rotation axis X is along Yi ; (b) the rotation rotation axis Y is along Y i; (c) the rotation axis Z is along Xi. (a) (b). (c) axis Y is along Yi ; (c) the rotation axis Z is along X i Figure 7. Processes ofturntable multi-position statictests calibration; (a) the rotation axisthe X is along Yi;relationship (b) the The multi-position rotation are conducted to learn coupling rotation axis Y is along Y i; (c) the rotation axis Z is along Xi. between rotation measurements thetests X-Y-Z axes as well as to thelearn coupling rotation and The multi-position turntable among rotation are conducted the among coupling relationship

acceleration Three casesthe of X-Y-Z rotationaxes withasdifferent are conducted in this and between rotationmeasurements. measurements among well astilt theangles coupling among rotation The multi-position turntable rotation tests-axis are which conducted toopposite learn the coupling relationship experiment. The rotation axis is set around is the direction of gravity. As acceleration measurements. Three cases of rotation with anglesamong are conductedand in this between measurements theon X-Y-Z axes as different well asaccuracy thetilt coupling shown inrotation Figure 8, the MIMU is among mounted a wedge (the angle is ±5″) with arotation tilt angle α experiment. The rotation axis is set around Z -axis which is the opposite direction of gravity. As shown acceleration measurements. Three cases ofi rotation withrotation different tilt where angles αare in 45°, this and the wedge is mounted on the turntable to perform tests, is conducted set as 0°, 30°, in Figure 8, the MIMU is mounted onaround a wedge-axis (the which angle is accuracy is ˘5”) with aoftilt angleAs α and experiment. The rotation axis is set the opposite direction gravity. and 60°, respectively, while the rotation axis is around the -axis. In the rotation test, ˝the angular ˝ , 45˝α, and the wedge is mounted on the turntable to perform rotation tests, where α is set as 0 , 30 shown in Figure 8, the MIMU is mounted on a wedge (the angle accuracy is ±5″) with a tilt angle rate is applied from −600°/s up to 600°/s at 50°/s per step. At each step, the outputs of the sensors, and theangular wedge is mounted on theaxis turntable to X-Y-Z perform where α test, is cases setthe asare 0°, 30°, 45°, 60˝ , respectively, while the is around the axes Zirotation -axis. Intests, the rotation angular rate is actual rates, and rotation accelerations along the are recorded. The three shown in ˝ ˝ ˝ and 60°, respectively, while the rotation axis is around the -axis. In the rotation test, the angular applied from 600 /s 50 plane, /s perand step. each step, theMIMU outputs the sensors, actual Figure 9, ´600 where /s theup X-Ytoplane, theatX-Z theAt X-Y plane of the areof adjusted, in turn, rate isparallel applied from −600°/s to 600°/s each step,The the outputs of experiment, theare sensors, angular rates, andwith accelerations theatX-Y-Z axes areAt recorded. three cases shown in to be the slope up ofalong the wedge in50°/s casesper (a),step. (b), and (c), respectively. In this actual angular rates, andare accelerations along the X-Y-Z axesthe areX-Y-Z recorded. The three casesangular are shown in the9,actual accelerations the gravity components along axes the actual rates Figure where the X-Y plane, the X-Z plane, and the X-Y plane of theand MIMU are adjusted, in turn, Figure 9, where the X-Y plane, the X-Z plane, and the X-Y plane of the MIMU are adjusted, in turn, the rotation components the X-Y-Z axes. (a), (b), and (c), respectively. In this experiment, to be are parallel with the slope ofalong the wedge in cases to be parallel with the slope of the wedge in cases (a), (b), and (c), respectively. In this experiment, the actual accelerations areare thethe gravity along theX-Y-Z X-Y-Zaxes axes and actual angular the actual accelerations gravitycomponents components along the and thethe actual angular ratesrates ω are the components along thethe X-Y-Z arerotation the rotation components along X-Y-Zaxes. axes. Z

ω

Y

X Z

Y X

MIMU Wedge MIMU Turntable Wedge

Figure 8. MIMU assembly scheme of multi-position turntable rotation rate calibration. Turntable

The centrifugation tests are conducted to learn more about the coupling relationship between Figurerotation 8. MIMUmeasurements. assembly scheme ofcases multi-position turntable rotation rate calibration. acceleration of rotations with different distance d from Figureand 8. MIMU assembly schemeSix of multi-position turntable rotationeccentric rate calibration. the centroid of the MIMU are conducted in this experiment. The rotation axis is set around the TheAs centrifugation tests10, arethe conducted learn more about the coupling between -axis. shown in Figure MIMU istomounted on the turntable with anrelationship eccentric distance d The centrifugation tests are conducted to learn more about the coupling relationship between acceleration and rotation measurements. Six cases of rotations with different eccentric distance d from varying from −60 mm to 60 mm at an interval of 20 mm. At each eccentric position, the angular rate the centroid of around the MIMU conducted infrom thisof−600°/s experiment. Theatdifferent rotation set at around the acceleration and rotation measurements. Six cases rotations with eccentric distance d from of the turntable the are -axis is applied to 600°/s 50°/s peraxis step,isand each step, -axis. As shown in Figure 10, the MIMU is mounted on the turntable with an eccentric distance d the the centroid of the MIMU are conducted in this The rotation setrecorded. around the outputs of the sensors, actual angular rates andexperiment. accelerations along the X-Y-Zaxis axesisare varying from −60 mm to 60 mm at an interval of 20 mm. At each eccentric position, the angular rate Zi -axis. shownaxis in isFigure 10, X-axis, the MIMU mounted on the turntable withas anshown eccentric distance TheAs eccentric set as the Y-axis,isand Z-axis, respectively, in six cases in Figure of the turntable around the the-axis is applied from −600°/s togravity 600°/s at 50°/s per step, and at each step, 10a–f. In this experiment, actual accelerations are the components combined with the d varying from ´60 mm to 60 mm at an interval of 20 mm. At each eccentric position, the angular the outputsaccelerations, of the sensors,and actual angular rates and accelerations alongrates the˝ X-Y-Z axes are recorded. ˝ ˝ centrifugal the actual angular rates are the angular around the vertical axis. rate of the turntable around the Zi -axis is applied from ´600 /s to 600 /s at 50 /s per step, and at The axis is set asinthe Y-axis, and Z-axis, respectively, in six cases as shown in Figure Eartheccentric rotation is ignored theX-axis, experiment. each step, outputs of the sensors, actual angular are rates accelerations along the X-Y-Z 10a–f.the In this experiment, the actual accelerations theand gravity components combined with axes the are recorded. The eccentric axis is setthe asactual the X-axis, Y-axis, andthe Z-axis, respectively, sixvertical cases as shown centrifugal accelerations, and angular rates are angular rates aroundinthe axis. Earth rotation is ignored in the experiment.

Sensors 2016, 16, 175

9 of 14

in Figure 10a–f. In this experiment, the actual accelerations are the gravity components combined with the centrifugal accelerations, and the actual angular rates are the angular rates around the vertical axis. Earth rotation is ignored in the experiment. Sensors 2016, 16, 175

9 of 14

Sensors 2016, 16, 175

Z ω Z ω X

(a) (a)

Y

X (1)

(b)

Z

Y (1)

X

Z (1)

ω

Y (2) (2) ω

Y

Y ω Z

(c)

(2)ω

XY

Z

Y (1)ω (c)

X (2) X

X ω Y

ω

ZX

Y

X (1)ω (b)

ω

Z

YZ

X

Z (2)

ω

ω

9 of 14

ω Y 60° ω Y 60°

Z Z Y Y ω 45° 30° Z X Z X Y Y 45° 30° X X (3) (4) (3)ω

(4)ω X Z Z ω ω 45° 30° X Y X Y Z Z 45° 30° Y Y (3) (4) X

Z 60° Z 60°

(3) ω

(4) ω X Y X X 60° ω X ω 45° 30° Y Z Y Z X X 60° 45° 30° Z Z (3) (4) Y

Figure 9. Processes of multi-position turntable rotation rate calibration; the tilt angle α is adjusted in

(1) (2) Figure 9. Processes of(a) multi-position turntable rate (3) calibration; the (4) tilt angle α is adjusted in the Y-Z plane X-Z plane; (b) and X-Y plane;rotation (c) respectively. Figure Processes of (b) multi-position rotation rate calibration; the tilt angle α is adjusted in the Y-Z plane (a)9.X-Z plane; and X-Y turntable plane; (c) respectively. the Y-Z planeω(a) X-Z plane; (b) and X-Y plane;ω (c) respectively.

ω

ω

Y MIMU

d

Y

Z

Y

MIMU XTurntable

d Z

ω

Z

d

X

Z

d Y

ω

X

d

X

X

Z d

X

Y Y

Z

Turntable (a)

(b)

(c)

ω (a)

ω (b)

ω (c)

ω d

X

Z

d X (d)

ω

Z

d

Y

Y

X

d

Y

Y (e)

ω

X

Y

d

Z

X

Y

d

Z

X

Z Z

(f)

Figure 10. Processes of centrifugation calibration; (d) (e)the eccentric axis is set as the X-axis (f) (a), (b), Y-axis (c), (d), and Z-axis (e), (f), respectively. Figure 10. Processes of centrifugation calibration; the eccentric axis is set as the X-axis (a), (b), Y-axis 10. (c),Processes of centrifugation calibration; the eccentric axis is set as the X-axis (a), and Z-axis (e), (f), respectively. The (d), sample data acquired in the calibration experiment are used to train the BP neural network

Figure Y-axisto(c), (d), and the Z-axis (e), (f), of respectively. determine parameters the network, where the output voltages of the MIMU are used as

(b),

The sample data acquireddata in the calibration experimentand are angular used to rate trainare theused BP neural network inputs, and the corresponding of the actual acceleration as the outputs to determine the parameters of the network, where the output voltages of the MIMU are used as of the network. The sample acquired in thedata calibration experiment are angular used torate train the BP neural network to inputs, data and the corresponding of the actual acceleration and are used as the outputs determine5.of the parameters of the network, where the output voltages of the MIMU are used as inputs, the network. Experimental Results and Discussions

and the corresponding data of three-axis the actual acceleration and angular rate are used as the outputs of The measurements angular rates ranging from −4000°/s to +4000°/s and three-axis 5. Experimental Resultsofand Discussions the network. accelerations ranging from −10 g to +10 g are conducted for the developed thermal gas MIMU using 5.

The measurements of three-axis angular rates ranging from −4000°/s to +4000°/s and three-axis accelerations ranging from −10 g to +10 g are conducted for the developed thermal gas MIMU using Experimental Results and Discussions

The measurements of three-axis angular rates ranging from ´4000˝ /s to +4000˝ /s and three-axis accelerations ranging from ´10 g to +10 g are conducted for the developed thermal gas MIMU

Sensors 2016, 16, 175

10 of 14

Sensors 2016, 16, 175 10 of 14 using an uniaxial turntable (920ET, AVIC Beijing Precision Engineering Institute Aircraft Industry, the ˝ /s; the resolution of angular rate is 0.001˝ /s). The acceleration measurement range of rotation is ˘4000 an uniaxial turntable (920ET, AVIC Beijing Precision Engineering Institute Aircraft Industry, the measurements of ˘10range g are by using centrifugation the turntable accuracy of the measurement of conducted rotation is ±4000°/s; the resolution of angularof rate is 0.001°/s). The(the acceleration measurements of g are60 conducted using centrifugation of the range). turntable (the of the reference acceleration is ±10 about ppm inbythe full measurement Theaccuracy experimental results reference acceleration is abouta60 ppm measuring in the full measurement range). The experimental results shown in Sensors Figure 11 demonstrate large range of the gyro/rate sensor of the MIMU. 16, 175 10 of 14 shown2016, in Figure 11 demonstrate a large measuring range of the gyro/rate sensor of the MIMU. The The nonlinearity in full-scale range of rotation and acceleration is less than 2%. nonlinearity in full-scale range of rotation and acceleration is less than 2%.

Accelerometer-Output (V) Accelerometer-Output (V)

Gyroscope-Output (V) Gyroscope-Output (V)

an uniaxial turntable (920ET, AVIC Beijing Precision Engineering Institute Aircraft Industry, the measurement range of rotation is ±4000°/s; the resolution of angular rate is 0.001°/s). The acceleration X-axisaccuracy of the measurements1.0of ±10 gX-axis are conducted by using centrifugation of the turntable (the 0.10 Y-axis Y-axis reference acceleration is about 60 ppm in the full measurement range). The experimental results Z-axis Z-axis shown in Figure range of the gyro/rate sensor of the MIMU. The 0.5 11 demonstrate a large measuring 0.05 nonlinearity in full-scale range of rotation and acceleration is less than 2%. 0.0 X-axis Y-axis Z-axis

1.0 -0.5 0.5 -1.0

0.0-4000 -3000 -2000 -1000

0

1000 2000 3000 4000

Angular rate (degree/s)

-0.5

0.00 X-axis Y-axis Z-axis

0.10 -0.05 0.05 -0.10

0.00-98.0 -73.5 -49.0 -24.5 0.0 24.5 49.0 73.5 98.0 2

Acceleration (m/s )

-0.05

(a)

(b)

Figure 11. The experimental measurements of three-axis angular rates (±4000°/s) (a) and three-axis

-1.0 -0.10 Figure 11. The experimental measurements of three-axis angular rates (˘4000˝ /s) (a) and three-axis accelerations (±10 g); (b) using the MIMU. accelerations (˘10 g); (b) using the0MIMU. -4000 -3000 -2000 -1000 1000 2000 3000 4000 -98.0 -73.5 -49.0 -24.5 0.0 24.5 49.0 73.5 98.0

10 10 5 5 0 0 -5 -5 -10 -10

ucompensated results compensated results -10

-5

0

5

10

2

Actual acceleration along X-axis (m/s )

10 5

(a)

0

Errors of measured acceleration Errors (m/s^2)of measured acceleration (m/s^2)

Measured acceleration (m/s^2) Measured acceleration (m/s^2)

2 To test the cross-coupling performance of the MIMU, experimental calibrations and network Angular rate (degree/s) Acceleration (m/s ) training are conducted, as described in Section 4. A neural network model used for compensating (a) (b) To test the cross-coupling performance of the MIMU, experimental calibrations and network system errors of the gas MIMU is therefore established. The established model is then employed in Figure 11. The experimental measurements of three-axis angular rates (±4000°/s) (a) and three-axis training are conducted, as described inaccelerations Section 4. and A neural model used for compensating actual measurements to extract the angular network rates along the X-Y-Z axes using the accelerations (±10 g); (b) using the MIMU. sensorofreadouts the gas MIMU. A validation experiment is further conducted to evaluate theemployed system errors the gasofMIMU is therefore established. The established model is then compensation method following the similar processes of the calibration in Section 4 but with different in actual measurements to extract the accelerations and angular rates along the To test the cross-coupling performance of the MIMU, experimental calibrations andX-Y-Z networkaxes using rotations ranging from −600°/s to 600°/s and different accelerations ranging from −9.8 m/s2 to +9.8 m/s2. are conducted, as described in Section 4. A neural network model used for compensating the sensortraining readouts of the gas MIMU. A validation experiment is further conducted to evaluate The output quantities of the MIMU compensated by using the neural network model (termed system errors of the gasfollowing MIMU is therefore established. The established model is then employed in the compensation method the similar processes of the calibration in Section 4 but compensated results) are compared with the original output quantities of the MIMU without any actual measurements to extract the accelerations and angular rates along the X-Y-Z axes using the ˝ ˝ compensation (termed uncompensated results) the 600 left figures of Figures 12 and accelerations 13. The corresponding with different rotations ranging from ´600 /sin to /s and different ranging from sensor readouts of the gas MIMU. A validation experiment is further conducted to evaluate the measurement errors are shown in the right figures of Figures 12 and 13. Uncompensated resultsthe neural 2 . The ´9.8 m/s2compensation to +9.8 m/s output quantities of the MIMU compensated by using method following the similar processes of the calibration in Section 4 but with different demonstrate significant errors, which are induced from coupling effects, misalignment, and rotations(termed ranging from −600°/s to 600°/sresults) and different m/s2 tooutput +9.8 m/s2.quantities network model compensated are accelerations comparedranging with from the −9.8 original assembling errors. These errors of the MIMU are greatly reduced by using the compensation method The output quantities the MIMU compensated using the neural network model of the MIMU without any ofcompensation (termedby uncompensated results) in (termed the left figures based on a neural network model. compensated results) are compared with the original output quantities of the MIMU without any of Figurescompensation 13 and 15. (termed The corresponding measurement errors are shown in the right figures of uncompensated results) in the left figures of Figures 12 and 13. The corresponding 10 1 Figures 13measurement and 15. Uncompensated demonstrate significant errors, which areresults induced from errors are shown results in the right figures of Figures 12 and 13. Uncompensated 5 0 demonstrate significant errors, which are induced effects, misalignment, and reduced 0 misalignment, coupling effects, and assembling errors.-1 from Thesecoupling errors of the MIMU are greatly -5 assembling errors. These errors of the MIMU are greatlynetwork reduced by using the compensation method compensated results by using the compensation method based on a neural model. -2 compensated results based -10 on a neural network model. -3

1 1 0 0 -1 -1 -2 -2 -3 -3 -10

-5

0

ucompensated results compensated results 5 10

Actual acceleration along X-axis (m/s^2) 1

(b)

0 -1

Figure 12. Cont.

-5

ucompensated results

-10 -10

-5

0

5

10

-2

ucompensated results

-3 -10

2

-5

0

5

Actual acceleration along X-axis (m/s^2)

Actual acceleration along X-axis (m/s )

(a)

(b) Figure 12. Cont.

Figure 12. Cont.

10

Sensors 2016, 16, 175

11 of 14

Sensors 2016, 16, 175

11 of 14 1

10 Sensors 2016, 16, 175

Errors of measured acceleration (m/s^2) Errors of measured acceleration (m/s^2)

-1

0 Measured acceleration (m/s^2) Measured acceleration (m/s^2)

-5 10

1

-2

compensated results

-10 5

compensated results

0

-5 10 ucompensated results

-10 5 0 -10

-5

0

5

10

2

-5

Actual acceleration along Y-axis (m/s ) ucompensated results

-10 -10

0

5

5

0

-10

(d)

-5

Errors of measured acceleration (m/s^2) Errors of measured acceleration (m/s^2)

-2

compensated results

-3

0 compensated results

5-10

0

0

5

10

(d) 1 compensated results

0 -1

1

10 -5

0

2

0

-10 5

10

Actual acceleration along Y-axis (m/s )

-1

-5 10

5 2

Actual acceleration along Y-axis (m/s ) uncompensated results

1

(c)

0

uncompensated results -5

-2

10

2

Actual acceleration along Y-axis (m/s )

10

compensated results

-3

(c)

-5

compensated results

0 -3 -1 1 -2 0 -3 -1 1 -2 0 -3 -10 -1

0

10 -5 5-10

Measured acceleration (m/s^2) Measured acceleration (m/s^2)

11 of 14

0

5

-2

compensated results

-3

-1

-5 10

-10 5

-2

uncompensated results

-3

0 -10

-5

-5

0

5

10

Actual acceleration long Z-axis (m/s^2) uncompensated results

-10 -10

(e)

-5

1 uncompensated results

0 -10 -1

-5

0

-3 0

5

-10

10

5

10

2

Actual acceleration along Z-axis (m/s ) uncompensated results

-2

(f)

-5

0

5

10

2

Actual acceleration long Z-axis (m/s^2)

Actual acceleration along Z-axis (m/s )

Figure 12. The acceleration measurements and the errors of the MIMU system with, and without,

Figure 13. using The acceleration measurements and the errors of the MIMU system with, and without, using the compensation(e)method based on a neural network. (a), (c), and (e) (f)refer to the measured the compensation method based on a neural network. (a), (c), and (e) refer to the acceleration acceleration along the X, Y, and Z axes, respectively, with and without compensation; measured (b), (d), and (f) Figure 12. The acceleration measurements and the errors of the MIMU system with, and without, refer to the measurement errors as functions of the actual accelerations along the X, Y and Z axes, along the X,using Y, and Z axes, respectively, with and without compensation; (b), (d), and (f) refer to the the compensation method based on a neural network. (a), (c), and (e) refer to the measured respectively, and without The asymmetric along uncompensated to the measurement errors with as functions ofcompensation. the actual accelerations X, Yerrors andare Z due axes, respectively, acceleration along the X, Y, and Z axes, respectively, with and withoutthe compensation; (b), (d), and (f) amount of experiment data with negative values of the applied accelerations is more than that with refer to the measurement errors functions of the actual accelerationserrors along the Y and axes, with and without compensation. Theasasymmetric uncompensated areX,due to Zthe amount of positive values of the applied accelerations. and without compensation. The asymmetric uncompensated due with to the positive experimentrespectively, data withwith negative values of the applied accelerations is moreerrors thanarethat amount of experiment data with negative values of 300 the applied accelerations is more than that with values of600 the applied accelerations. 400 Measured angular rate (°/s) Measured angular rate (°/s)

0 600 -200 400 -400 200 -600 0 -200 600 -400 400 -600 200 -800 0 -200 -400 -600 -800

150

Errors of measured angular rate (°/s) Errors of measured angular rate (°/s)

positive values of the applied accelerations.

200 0 -200 600 -400 400 -600 200

0 300

-150 150

compensated results

compensated results

-300 0 300 -150 150

compensated results

-150 150


-400

-200

0

200

400

-300

600

Actual angular rate about X-axis (°/s) uncompensated results -600

-400

(a)

-200

0

200

Actual angular rate about X-axis (°/s)

400

600

compensated results

-300 0 300

0


-400

-200

0

400

600


-150 -300 -600

Figure 13. Cont.

(a)

-400

(b)

-200

Figure 14. Cont.

uncompensated results 0

200


(b) Figure 13. Cont.

200

400

600

Sensors 2016, 16, 175

12 of 14

Sensors 2016, 16, 175

12 of 14 300

600 400 200 0 -200 -400 -600

Errors of measured angular rate (°/s)

Measured angular rate (°/s)

150

compensated results

600 400 200 0 -200 -400 -600


-400

-200

0

200

400

0 -150 -300

compensated results

-450 300 150 0 -150 -300


-450 -600

600

-400

-200

200

400

600

Actual angular rate about Y-axis (°/s)

Actual angular rate about Y-axis (°/s)

(c)

(d)

300

600 400 200 0 -200 -400 -600

0

Errors of measured angular rate (°/s)

150 0

Measured angular rate (°/s)

-150

600 400 200 0 -200 -400 -600

compensated results

-300

compensated results

300 150 0

-150

-400

-200

0

200

400


-300


-600

600

-400

-200

0

200

400

600

Actual angular rate about Z-axis (°/s)

Actual angular rate about Z-axis (°/s)

(e)

(f)

Figure 13. The angular rate measurements and the errors of the MIMU system with, and without,

Figure 15. using The angular rate measurements and the errors of the MIMU system with, and without, the compensation method based on a neural network. (a), (c), and (e) refer to the measured using the compensation method based neural network. (c),compensation; and (e) refer to(d), the measured angular rate around the X, Y, and Z on axes,a respectively, with and (a), without (b), and refer to the as respectively, functions of thewith actualand angular rate around the X, Y, and Z axes, angular rate(f)around themeasurement X, Y, and Zerrors axes, without compensation; (b), (d), and (f) with and without compensation. refer to therespectively, measurement errors as functions of the actual angular rate around the X, Y, and Z axes, respectively, with and without compensation.

The root mean square (RMS) errors of the gas MIMU system are listed in Table 1, which indicated that by using the neural network-based compensation method the measurement errors of three-axis angular rates and three-axis accelerations the gas MIMUsystem are reduced to less than 1% and of theindicated The root mean square (RMS) errors of theofgas MIMU are listed in Table 1,3% which uncompensated errors in the rotation range of ±600°/s and the acceleration range of ±1 g, respectively. that by using the neural network-based compensation method the measurement errors of three-axis According to the component performances of the thermal gas inertial sensor [23], the angular rate angular rates three-axis accelerations of the arenoise reduced to 1less 1% and 3% noiseand of the sensor reaches around 1°/s RMS andgas the MIMU acceleration is around mg than RMS. The ˝ compensated errors of the MIMU have reached the/s noise level of acceleration the individual sensors, of the uncompensated errors in developed the rotation range of ˘600 and the range of ˘1 g, which implies theto system errors are completely eliminatedof by the using the compensation method. respectively. According the component performances thermal gas inertial sensor [23], the

angular rate noise reaches around 1˝ /sand RMS and the is around 1 mg Tableof 1. the RMS sensor errors of the measured accelerations angular rates by acceleration the MIMU systemnoise with, and RMS. The compensated errors of the developed MIMU have reached the noise level of the individual without, using network-based compensation method. sensors, which implies the system errors are completely by using method. 2 (m/s2) ωX (°/s) ωY (°/s) the ωZcompensation (°/s) RMS Error aX (m/s ) aY (m/s2) aZeliminated Compensated

0.0100

0.0085

0.0093

0.44

0.55

0.52

0.5129and 0.3294 88.58 by the 87.43MIMU 62.56 Table 1. RMS errorsUncompensated of the measured 0.5784 accelerations angular rates system with, and Comp./Uncomp. (%) 1.73 1.66 2.83 0.50 0.63 0.84 without, using network-based compensation method.

Conventional compensation methods are used to deal with specific error problem, such as

˝ ˝ /s) 2) RMS Errorerrors, misalignment ωY (are ωZ aX (m/s2 ) errorsaYof(m/s aZ (m/s2 )theseω nonlinear the MIMU. However, existing invalid to (˝ /s) X ( /s)methods

compensate complex and nonlinear cross-coupling errors exhibited in the proposed MIMU. It is the

Compensated 0.0100 0.0085 0.0093 0.44 0.55 0.52 first time that a neural network is used to comprehensively compensate the system errors of the Uncompensated 0.5784 0.5129 0.3294 88.58 87.43 62.56 thermal gas MIMU, which are complex multivariate and nonlinear coupling, including Comp./Uncomp. (%) 1.73 1.66 2.83 0.50 0.63 0.84 cross-coupling errors, misalignment errors, eccentric errors, and any other deterministic errors. The

Conventional compensation methods are used to deal with specific error problem, such as nonlinear errors, misalignment errors of the MIMU. However, these existing methods are invalid to compensate complex and nonlinear cross-coupling errors exhibited in the proposed MIMU. It is the first time that a neural network is used to comprehensively compensate the system errors of the thermal gas MIMU, which are complex multivariate and nonlinear coupling, including cross-coupling errors, misalignment errors, eccentric errors, and any other deterministic errors. The proposed error compensation method is easily operated, physical-model-free and universal, which is applicable to not

Sensors 2016, 16, 175

13 of 14

only compensate the system errors of the gas MIMU, but also overcome various deterministic errors in general integrated sensor systems. 6. Conclusions A gas MIMU is developed by integrating only three thermal gas inertial sensors to measure three-axis accelerations (˘10 g) and three-axis angular rates (˘4000˝ /s). The errors of the gas MIMU are analyzed, which mainly contain cross-coupling errors, misalignment errors, and eccentric errors. A universal compensation method by using a neural network to model system errors is proposed to eliminate system errors of an integrated sensor system. The compensation method is applied to compensate the errors of the thermal gas MIMU. The experiments validate the effectiveness of the method and demonstrate that the measurement errors of three-axis angular rates and three-axis accelerations are reduced to less than 1% and 3% of uncompensated errors in the rotation range of ˘600˝ /s and the acceleration range of ˘1 g, respectively, by using the neural network-based compensation method, due to the limitation of our available experimental facility. It is evident that the compensation methodology can be easily scaled up to the full measurement range of the MIMU when using larger-range equipment. Acknowledgments: This work has been supported by the National High-tech Program “863” of China under the grants 2012AA02A604. Author Contributions: Shi Qiang Liu developed the MIMU system, conducted experiments. Shi Qiang Liu and Rong Zhu proposed the original idea and wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11. 12. 13.

Barbour, N.; Schmidt, G. Inertial Sensor Technology Trends. IEEE Sens. J. 2001, 1, 332–339. [CrossRef] Hoflinger, F.; Muller, J.; Zhang, R.; Reindl, L.M.; Burgard, W. A Wireless Micro Inertial Measurement Unit (IMU). IEEE Trans. Instrum. Meas. 2013, 62, 2583–2595. [CrossRef] Brown, A.K.; Lu, Y. Performance test results of an integrated GPS/MEMS inertial navigation package. In Proceedings of the ION GNSS 2004, Long Beach, CA, USA, 21–24 September 2004; pp. 825–832. Hanse, J.G. Honeywell MEMS inertial technology product status. In Proceedings of the Plans 2004 Position Location and Navigation Symposium, Minneapolis, MN, USA, 26–29 April 2004; pp. 43–48. Aggarwal, P.; Syed, Z.; Niu, X.; El-Sheimy, N. A standard testing and calibration procedure for low cost MEMS inertial sensors and units. J. Navig. 2008, 61, 323–336. [CrossRef] Jafari, M.; Najafabadi, T.A.; Moshiri, B.; Tabatabaei, S.S.; Sahebjameyan, M. PEM Stochastic Modeling for MEMS Inertial Sensors in Conventional and Redundant IMUs. IEEE Sens. J. 2014, 14, 2019–2027. [CrossRef] Naseri, H.; Homaeinezhad, M.R. Improving measurement quality of a MEMS-based gyro-free inertial navigation system. Sens. Actuators A Phys. 2014, 207, 10–19. [CrossRef] El-Sheimy, N. Inertial Techniques and INS/DGPS Integration. In Engo 623-Course Notes; Department of Geomatics Engineering, University of Calgary: Calgary, AB, Canada, 2006. Chatfield, A.B. Fundamentals of High Accuracy Inertial Navigation; American Institute of Aeronautics and Astronautics: Reston, VA, USA, 1997. Nieminen, T.; Kangas, J.; Suuriniemi, S.; Kettunen, L. An enhanced multi-position calibration method for consumer-grade inertial measurement units applied and tested. Meas. Sci. Technol. 2010, 21, 1069–1076. [CrossRef] Zhang, H.L.; Wu, Y.X.; Wu, W.Q.; Wu, M.P.; Hu, X.P. Improved multi-position calibration for inertial measurement units. Meas. Sci. Technol. 2009, 21, 209–213. [CrossRef] Ma, L.; Chen, W.W.; Li, B.; You, Z.; Chen, Z.G. Fast Field Calibration of MIMU Based on the Powell Algorithm. Sensors 2014, 14, 16062–16081. [CrossRef] [PubMed] Zhu, R.; Zhou, Z.Y. Calibration of three-dimensional integrated sensors for improved system accuracy. Sens. Actuators A Phys. 2006, 127, 340–344. [CrossRef]

Sensors 2016, 16, 175

14. 15.

16. 17.

18. 19. 20.

21. 22. 23. 24. 25.

26.

14 of 14

Ayazi, F.; Najafi, K. A Harpss polysilicon vibrating ring gyroscope. J. Microelectromechan. Syst. 2001, 10, 169–179. [CrossRef] Bernstein, J.; Cho, S.; King, A.T.; Kourepenis, A.; Maciel, P.; Weinberg, M. A micromachined comb-drive tuning fork rate gyroscope. In Proceedings of the IEEE Micro Electro Mechanical Systems (MEMS), Cambridge, MA, USA, 7–10 February 1993; pp. 143–148. Maenaka, K.; Shiozawa, T. A study of silicon angular rate sensors using anisotropic etching technology. Sens. Actuators A Phys. 1994, 43, 72–77. [CrossRef] Dao, D.V.; Dau, V.T.; Dinh, T.X.; Sugiyama, S. A fully integrated MEMS-based convective 3-DOF gyroscope. In Proceedings of the International Transducers 2007 Solid-State Sensors, Actuators and Microsystems Conference, Lyon, France, 10–14 June 2007; pp. 1211–1214. Dau, V.T.; Tomonori, O.; Dinh, T.X.; Dao, D.V.; Sugiyama, S. A multi axis fluidic inertial sensor. In Proceedings of the 2008 IEEE Sensors, Lecce, Italy, 26–29 October 2008; pp. 666–669. Bahari, J.; Feng, R.; Leung, A.M. Robust MEMS Gyroscope Based on Thermal Principles. J. Microelectromech. Syst. 2014, 23, 100–116. [CrossRef] Garraud, A.; Combette, P.; Gosalbes, J.M.; Charlot, B.; Giani, A. First high-g measurement by thermal accelerometers. In Proceedings of the 16th International Solid-State Sensors, Actuators and Microsystems Conference, Beijing, China, 5 June 2011; pp. 84–87. Feng, R.; Bahari, J.; Jones, J.D.; Leung, A.M. MEMS thermal gyroscope with self-compensation of the linear acceleration effect. Sens. Actuators A Phys. 2013, 203, 413–420. [CrossRef] Garraud, A.; Giani, A.; Combette, P.; Charlot, B.; Richard, M. A dual axis CMOS micromachined convective thermal accelerometer. Sens. Actuators A Phys. 2011, 170, 44–50. [CrossRef] Zhu, R.; Cai, S.L.; Ding, H.G.; Yang, Y.J.; Su, Y. A micromachined gas inertial sensor based on thermal expansion. Sens. Actuators A Phys. 2014, 212, 173–180. [CrossRef] Zhu, R.; Ding, H.G.; Su, Y.; Zhou, Z.Y. Micromachined gas inertial sensor based on convection heat transfer. Sens. Actuators A Phys. 2006, 130, 68–74. [CrossRef] Cai, S.L.; Zhu, R.; Ding, H.G.; Yang, Y.J.; Su, Y. A micromachined integrated gyroscope and accelerometer based on gas thermal expansion. In Proceedings of the 2013 Transducers & Eurosensors XXVII: The 17th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS & EUROSENSORS XXVII, Barcelona, Spain, 16–20 June 2013; pp. 50–53. Lau, C. Neural Networks: Theoretical Foundations and Analysis; IEEE Press: Piscataway, NJ, USA, 1991. © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).