Optimized design method of vibration isolation system in mechanically dithered RLG POS based on motion decoupling

Measurement 48 (2014) 314–324

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Optimized design method of vibration isolation system in mechanically dithered RLG POS based on motion decoupling Junchao Cheng ⇑, Jiancheng Fang, Weiren Wu, Jianli Li Science & Technology on Inertial Laboratory, Key Laboratory of Fundamental Science for National Defense-Novel Inertial Instrument & Navigation System Technology, Beihang University, Beijing 100191, PR China

a r t i c l e

i n f o

Article history: Received 27 January 2013 Received in revised form 8 November 2013 Accepted 11 November 2013 Available online 21 November 2013 Keywords: Dithered ring laser gyroscope Motion decoupling Optimized design Position and orientation system Vibration isolation system

a b s t r a c t The mechanically dithered Ring Laser Gyroscope (RLG) based Position and Orientation System (POS) is a crucial equipment providing high accuracy exterior orientation elements for airborne remote sensors. However, the strenuous dithering motion of RLG causes adverse vibratory disturbance to the inertial measurement unit (IMU) and limits its applications in POS. A Vibration Isolation System (VIS) should be employed inside dithered RLG IMU to attenuate such disturbance but induces additional attitude measurement errors. To solve this problem, a new design method of VIS based on motion decoupling is proposed in this paper. First, structural design of VIS is developed to decouple established six-degreeof-freedom dynamic model to eliminate the motion coupling errors. Based on this, parameters including the natural frequencies and attenuation coefficients of each six-degree-offreedom dynamic model are optimized to constrain the transmitted measurement error of inertial sensors. The designed VIS is implemented and applied in the dithered RLG POS. And the finite element simulation results prove the effectiveness of the proposed method. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction As an inertial navigation system and global navigation satellite system integrated measurement equipment, the Position and Orientation System (POS) is specially designed to measure the dynamic information of position, velocity and attitude to compensate the motion errors of mapping sensors in air remote sensing applications [1–5]. POS could evidently improve the imaging quality and efficiency for digital camera, lidar, synthetic aperture radar etc., and therefore become an increasingly popular equipment employed in the remote sensing community [6,7]. The hardware of POS consists of an Inertial Measurement Unit (IMU), a global navigation satellite system receiver and a POS Computer System [1,8]. The IMU used in POS is quite different from that used in traditional inertial navigation system, as it emphasizes on the high accuracy ⇑ Corresponding author. Tel.: +86 010 8233 9550. E-mail address: [email protected] (J. Cheng). 0263-2241/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.measurement.2013.11.019

measurement of attitude, as well as small volume and lightweight [21]. Attitude measurement accuracy of POS is chiefly determined by the gyroscope in IMU [9–13]. Compared with other types of gyroscope, dithered Ring Laser Gyroscope (RLG) prevails with the advantage of high precision and hence becomes an optimal angular sensor for high performance POS [14,15]. However, a Vibration Isolation System (VIS) must be included in IMU to attenuate the adverse vibratory disturbance aroused by dithering motion of RLG which is a necessity to avoid the lock-in effect [15]. On the other hand, the improperly designed VIS inevitably gives rise to the corrupted measurement errors to gyros and accelerometers, which severely deteriorates the attitude measurement precision of IMU [23]. Therefore, the proper design method of VIS should be studied to eliminate the induced error and guarantee the precision of POS. Many researchers attach great importance to the VIS design of dithered RLG IMU. Some devised a complex VIS

J. Cheng et al. / Measurement 48 (2014) 314–324

to avoid structure-borne resonance noise caused by dithering motion of RLG [17,18]. And some other focused on the method to improve the performance of attenuating the adverse dithering noise [19,20]. However, all the VISs are designed to control the vibratory disturbance of RLG without considering the associated attitude measurement error of IMU. The demand on high accuracy attitude measurement of POS makes VIS quite different from that used in traditional inertial navigation system as the attitude error induced by VIS must be taken into account and reduced as much as possible. Such attitude error consists of the motion coupling error and the transmitted measurement error [16]. With improper structural configuration of VIS, the motion coupling errors refer to the bogus outputs of inertial sensors resulting from the coupled linear and rotational movements of inertial sensors assembly (ISA) in IMU. And the transmitted measurement errors represent the unexpectedly distorted output errors of gyros and accelerometers due to improper design of dynamic parameters of VIS. To the best of the authors’ knowledge, this case remains an open problem and hence it is our motivation to find a practical and effective solution to properly design the VIS for dithered RLG POS. In the following sections, we will firstly establish the coupled dynamic model of VIS and discuss the sources of motion coupling problem in Section 2. Based on the model, a new design method for VIS used in dithered RLG POS will be proposed to eliminate the motion coupling errors by structural design and restrain the transmitted measurement errors by parameters optimization in Section 3. Then, in Section 4, a VIS used in dithered RLG POS is developed as an instance, and the finite element simulation result is given, which demonstrates the validity of proposed method. At last, a conclusion can be found in Section 5. 2. Mechanical model of vibration isolation system Firstly, we give a brief description on the IMU used in dithered RLG POS. As shown in Fig. 1 the DRLG IMU is

Fig. 2. Schematic diagram of mechanical structure inside DRLG IMU.

composed of an ISA, a VIS, a support frame structure, and three circuit modules. Three DRLGs and three quartz accelerometers are orthogonally mounted on the precisely milled ISA structure to measure the angular and linear motional information of the IMU in Cartesian coordinate frame. Then, the assembled ISA is mounted onto the support frame structure through VIS which is composed of several absorbers. Although the dithering motions of DRLGs could be attenuated by VIS, the angular velocities and specific forces measured by the inertial sensors have to be transmitted and thus are distorted by VIS at the same time, which inevitably deteriorates the attitude measurement precision of the system. To illustrate the mechanical working principle, a schematic diagram of the structure inside the IMU is shown in Fig. 2. In this figure, O is the geometry center of ISA and is regarded as the origin of reference frame with the reference axes X, Y and Z parallel with the edges of ISA structure; Gx, Gy and Gz denote the DRLGs mounted on ISA structure along the dithering rotation axes bx, by and bz respectively; a1, a2, . . . ,aN denote absorbers securely installed between ISA and support frame structure, and the support frame structure is firmly fixed to the mounting base. As the accelerometers are directly mounted on ISA mechanical structure without any moving part, they are not shown in the figure for the sake of simplicity. From Fig. 2, it can be seen that the motion of ISA has six degree-of-freedom (DOF), which are three linear displacements along axes X, Y and Z and three angular rotations around the same axes. Assuming that the mass of mounting base of IMU is infinite, and the VIS is simplified as an ideal spring damping system, the general mechanical model of the VIS can be expressed as [22]

€ þ C R_ þ KR ¼ T þ C U_ þ KU MR

Fig. 1. The hardware composing of DRLG IMU.

315

ð1Þ

where M represents the inertial matrix of ISA, R and U denote the displacement vector of ISA and support frame structure, C and K denote the damping coefficient matrix and stiffness coefficient matrix of VIS respectively, and T is the dithering torque vector of DRLGs. Next, we give the specific forms of vectors and matrices in Eq. (1) which are the bases to look into the motion coupling problem.

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Firstly, three vectors in Eq. (1) can be expressed as

8  T > R ¼ Rx Ry Rz Ra Rb Rc > <  T U ¼ Ux Uy Uz Ua Ub Uc > >  T : T ¼ 0 0 0 Ta Tb Tc

where the subscripts x, y, z represent the directions of linear displacements, and the subscripts a, b, c denote the directions of rotating shaft which are corresponding to X, Y and Z axes. Then, the inertial matrix M can be split into four submatrices as

where subscript i denotes the order of absorbers, subscripts p, q and r denote three orthogonal stiffness spindles of absorbers, hj, uj and uj(j = p, q, r) stand for the angles between the stiffness spindles defined by the subscripts and the X, Y and Z axes in reference frame, and (w, l, h) are the installation coordinates of absorber. On the other hand, elements in C could be deduced by analogy with elements in K. Obviously, the nonzero off-diagonal elements in matrices M, C and K are the sources leading to the motion coupling problem, which means any DOF movement of IMU will stimulate the other DOF motions of the ISA through the VIS. 3. Optimized design method of vibration isolation system

where M11 denotes the mass effect of ISA whose total mass is m, M21 denotes the eccentric torque effect of ISA with rx, ry and rz being the eccentric distances between the mass center and the geometry center of ISA, M22 denotes the matrix of inertia moment, in which Jxx, Jyy, Jzz are the moments of inertia of ISA and Jxy, Jxz, Jyz are the cross coupled moments of inertia with the symmetry relationship of Jxy = Jyx, Jxz = Jzx, Jyz = Jzy. Finally, C and K are both six order symmetry matrices and we will just display K for instance.

2

K xx

K xy

K xz

K xa

K xb

K xc

6 K yx 6 6 6 K zx K ¼6 6K 6 ax 6 4 K bx

K yy

K yz

K ya

K yb

K cx

3

K zy

K zz

K za

K zb

K ay

K az

K aa

K ab

K by

K bz

K ba

K bb

K yc 7 7 7 K zc 7 7 K ac 7 7 7 K bc 5

K cy

K cz

K ca

K cb

K cc

The diagonal elements Kuu(u = x, y, z, a, b, c) represent the principle linear (denoted by x, y, x) and rotational (denoted by a, b, c) stiffness coefficients of VIS, and the off-diagonal elements Kuv(v = x, y, z, a, b, c, u – v) are the coupled stiffness coefficients with the subscripts indicating their motion coupling relationships. Some typical elements in K are listed as follows, while the other ones could be deduced based on them. 8 N N X X   > > > K xx ¼ kxxi ¼ kpi cos2 hpi þ kqi cos2 hqi þ kri cos2 hri > > > > i¼1 i¼1 > > ! > > N N X X > kpi cos hpi cos /pi þ > > > K xy ¼ kxyi ¼ > > kqi cos hqi cos /qi þ kri cos hri cos /ri > > i¼1 i¼1 > ! > > N N > X X kpi cos hpi cos upi þ > > > K ¼ k ¼ > xzi > xz > kqi cos hqi cos uqi þ kri cos hri cos uri > i¼1 i¼1 > > > > N N N > X X X > 2 2 > > kyyi hi þ kzzi li  2 kyzi hi li > K aa ¼ < i¼1

i¼1

i¼1

N N N N > X X X X > > > K cb ¼ kxzi li wi þ kyxi hi wi  kxxi hi li  kzyi w2i > > > > i¼1 i¼1 i¼1 i¼1 > > > > N N X X > > > > K xa ¼ kyxi hi  kxzi li > > > i¼1 i¼1 > > > > N N > X X > > > K yb ¼ kzyi wi  kyxi hi > > > > i¼1 i¼1 > > > > N N X X > > > > kxzi li  kzyi wi : K zc ¼ i¼1

i¼1

ð2Þ

As mentioned above, the demand of attitude measurement precision for IMU used in POS is very high. However, considering the motion coupling problem induced by VIS, the attitude of ISA will become inconsistent with that of the IMU shell (support structure frame) under dynamic situations. It means that the IMU measurements actually represent the attitude of ISA, and the measurement precision of IMU is below the desired quality. In this section, a new design method of VIS is presented to circumvent this problem in two parts. First, to eliminate the motion coupling errors, the structural design of VIS is modified to decouple the dynamic model with six DOF into six single DOF models. Then, to guarantee the accurate transmitting of sensors measurement, the natural frequencies and attenuation coefficients of each single DOF dynamic model are optimized to constrain the transmitted measurement error. 3.1. Motion decoupling design of VIS Based on the analysis in Section 2, the specific sources leading to the motion coupling problem are analyzed here. As for the inertial matrix M, the elements in M21 represent the effect of eccentric torque and the off-diagonal elements in M22 are induced by misalignment between inertial principal axes of ISA and the reference axes. Meanwhile, elements in K and C are determined by the structural factors of absorbers, such as absorber quantity, the installation positions, as well as the stiffness and damping parameters along each spindle. To sum up, the target of motion decoupling is to convert the off-diagonal elements in M, K and C into zeros through proper structural design method which is described as follows. Firstly, to remove the impact of inertial matrix M, several design principles for ISA structure are summarized as  The outline of ISA structure is designed to be cuboid to orthogonalize the reference coordinate axes.  The mass center and geometry center of ISA are trimmed to be overlapped to make rx = ry = rz = 0.  The principal axes of inertial of ISA are adjusted to be parallel with the reference axes to make Jxy = Jxz = Jyz = 0.

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Secondly, according to the elements displayed in K, the motion coupling patterns of VIS could be classified into three categories shown in Table 1. To decouple the linear & linear patterns, the stiffness spindles of absorbers must be parallel with the reference coordinates axes (edges of ISA structure), which means the angles h, u and u should satisfy the following equation.

Table 1 Motion coupling pattern of ISA.

2

considering the cross coupling disturbance. For each single DOF model in Eq. (6), it could be treated as a standard second order linear spring damping vibration system. The angular motion around X reference axis is displayed in a linear form in Fig. 4 as an instance. The corresponding dynamic model is rewritten as

hpi 6/ 4 pi

3 2 hri 90 7 6 /ri 5 ¼ 4 0

hqi /qi

upi uqi uri

0

0

0

7 0 5  90

90



0

3



ð3Þ

To decouple the linear & angular patterns and angular & angular patterns, the necessary conditions should be satisfied as follows.

8 kpi ¼ kp0 ; kqi ¼ kq0 ; kri ¼ kr0 > > > > N N N > X X X > > < li wi ¼ hi li ¼ hi wi ¼ 0

ð4Þ

i¼1 i¼1 i¼1 > > > N N N X X X > > > > li ¼ hi ¼ wi ¼ 0 : i¼1

i¼1

6 6 6 6 6 K ¼6 6 6 6 4

Nkp0 0 0 Nkq0

Kxy, Kyz, Kzx Kxa, Kxb, Kxc, Kya, Kyb, Kyc, Kza, Kzb, Kzc Kcb, Kba, Kac

ð7Þ

Suppose

T a ¼ Ua sinðxga tÞ U a ¼ Wa sinðx0a tÞ

i¼1

0 0

0 0

0 0

0 0

0

0

3 7 7 7 7 7 7 7 7 7 5

0

0

Nkr0

0

0

0

0

Nkq0 h0 þ Nkr0 l0

0

0

0

0

0

0

Nkp0 h0 þ Nkr0 w20

0

0

0

0

0

0

Nkp0 l0 þ Nkq0 w20

2

Coupling factor

Linear & Linear Linear & Angular Angular & Angular

8 € _ _ > > < J xx Z a þ C aa Z a þ K aa Z a ¼ T a þ C aa U a þ K aa U a 2 2 K aa ¼ Nkq0 h0 þ Nkr0 l0 > > : 2 2 C aa ¼ Ncq0 h0 þ Ncr0 l0 

In practice, the absorbers must be identical in stiffness characteristics, N has to be a multiple of four, and the installation positions should be symmetrically located on both sides of the XOY, YOZ and ZOX planes of reference frame. Hence, we define klik = l0, kwik = w0 and khik = h0. A typical configuration of absorbers is shown in Fig. 3, where eight black spots denote the mounting positions of absorbers. Substituting Eqs. (3) and (4) to Eq. (2) yields 2

Pattern

2 2

2

where Ua and xga represent the amplitude and frequency of dithering torque of DRLG in the direction of X; Wa and x0a represent the amplitude and frequency of angular motion of flying carrier around the direction of X, which is to be measured by IMU. Then, the steady state analytical solution of Za is

Z a ðtÞ ¼ Z ga ðtÞ þ Z 0a ðtÞ

Ua =Jxx Z ga ðtÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinðxga t  Dga Þ  2

þ ð2la xga Þ2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u x4na þ ð2la x0a Þ2 sinðx0a t  D0a Þ Z 0a ðtÞ ¼ Wa t  x2na  x20a 2 þ ð2la x0a Þ2

x2na  x2ga

ð8Þ

ð5Þ Accordingly, the six DOF dynamic model shown in Eq. (1) is decomposed into six single DOF dynamic models and hence the motion coupling error induced by VIS is eliminated. Then, the obtained dynamic models of decoupled VIS can be expressed in Eq. (6), where the first three equations represent the linear motions and the other ones indicate the rotational movements.

8 € mZ x þ C xx Z_ x þ K xx Z x ¼ C xx U_ x þ K xx U x > > > > > mZ€y þ C yy Z_ y þ K yy Z y ¼ C yy U_ y þ K yy U y > > > > < mZ€ þ C Z_ þ K Z ¼ C U_ þ K U z zz z zz z zz z zz z € a þ C aa Z_ a þ K aa Z a ¼ T a þ C aa U_ a þ K aa U a > J Z > xx > > > € > J yy Z b þ C bb Z_ b þ K bb Z b ¼ T b þ C bb U_ b þ K bb U b > > > : J zz Z€ c þ C cc Z_ c þ K cc Z c ¼ T c þ C cc U_ c þ K cc U c

where

8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > xna ¼ K aa =J xx > > > > < la ¼ C aa =2J xx D0a ¼ arctan x22laxx02a > na 0a > > > > 2la ðxn2a x20a Þ @ D0a > > D t ¼ ¼ : 2 2 @ x0a ðx2na x20a Þ þð2la x0a Þ

ð6Þ

3.2. Parameters optimization design of VIS Comparing with original dynamic model shown in Eq. (1), the decoupled models obtained in preceding section are greatly simplified, and hence the parameters of each single DOF can be designed independently without

Fig. 3. An example of installation positions of absorbers.

ð9Þ

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J. Cheng et al. / Measurement 48 (2014) 314–324

Fig. 4. Schematic diagram of rotational movement around X axis in a linear form.

As shown in Eqs. (8) and (9), Zga(t) represents the response of Ta, Z0a(t) represents the response of Ua, xna is the natural frequency of VIS, la is the attenuation coefficient, Dga is the response phase delay from Ta, D0a is the phase delay from Ua and t0a represents the corresponding time delay with D0a. Hence, the dithering torque transmitted from ISA to the support frame structure can be deduced by

Ma ¼ C aa Z_ ga þ K aa Z ga vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u x4na þ ð2la xga Þ2 b ga Þ ð10Þ ¼ Ua u sinðxga t  D 2 t x2na  x2ga þ ð2la xga Þ2 According to Eqs. (8) and (10), the absolute transmissibility F of VIS in such DOF is defined as



M a Z 0a



Fðxin ; xna ; la Þ ¼

¼

Ua

Wa

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u u x4na þ 2la xin 2 ¼ t    x2na  x2in 2 þ 2la xin 2

ð11Þ

Therefore, F is determined by three factors including the input frequency xin, the natural frequency xna and the attenuation coefficient la. As mentioned above, the VIS used in DRLG POS is quite different from that used in traditional VIS, for it should satisfy the demands of not only attenuating dithering noise of DRLGs in high frequency band, but also accurately transmitting the valid motional information of IMU to be measured from the support frame structure to ISA both in the aspects of amplitude and time delay. At the same time, the structure resonance phenomenon must be avoided to guarantee the safety of IMU. All necessary requirements for VIS could be summarized as

8 xga > xna > x0a > > > < Fðx ; x ; l Þ < e xin > xga na in a > Fðxin ; xna ; la Þ < 1 þ g xin < x0m > > : Dtðxin ; xna ; la Þ < s xin < x0m

ð12Þ

where e denotes the upper limit of transmissibility for attenuating dithering noise, g and s denote the maximum acceptable amplification error and time delay for transmitting the valid motional information, x0m = max[x0a],

represents the maximum frequency of angular motion of flying carrier to be measured. To simultaneously satisfy all in Eq. (12), the relationship among the transmissibility, natural frequency and attenuation coefficient is investigated as the input frequency varies from low to high frequency band, and the corresponding simulation result are shown in Fig. 5. From Fig. 5(a) and (c), as xin stays at low frequency band (0–100 Hz), larger natural frequency and larger attenuation coefficient produce better transmissibility which is closer to 1 as the amplification error g could be reduced. From Fig. 5(b) and (d), when xin stays at high frequency band (600–800 Hz), smaller natural frequency and smaller attenuation coefficient produce smaller transmissibility as the performance for attenuating dithering noise is better. The selections of natural frequency and attenuation coefficient lead to conflict at different input frequency band. According to Eqs. (11) and (12), we could obtain the value intervals for the natural frequency and attenuation coefficient which are displayed in Eqs. (13) and (14).

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 na

x >

x40m  ð2la x0m Þ2 þ x20m g2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð13Þ

e2 x2ga þ e2 x4ga þ 4l2a x2ga ð1  e2 Þ x2na < 1  e2 



qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2na  x20m 1  1  4s2 x20m 0 < la < 4sx20m

ð14Þ

It could be found that, the lower and upper limits of xna and la are coupled together. Hence, we use the iterative calculation to optimize the estimation of such two coefficients. Also, the parameters optimization for the other 5 DOF models shown in Eq. (6) can be deduced by analogy with the procedures described above. 4. Implementation and simulation To demonstrate the validity of the proposed method, a VIS for DRLG POS is designed and implemented as an instance. Then, the finite element simulation experiments are carried out to evaluate the performance of the designed VIS. 4.1. Implementation of VIS Firstly, the detailed structural design of VIS is presented to realize the motion decoupling. A three-dimension digital model established by using SolidWorks software is shown in Fig. 6. The outline of ISA structure constructed beforehand is a standard cuboid with all inertial sensors being wrapped. The actual dimension of ISA is 155 mm  155 mm  89 mm. And the total weight is 3.85 kg. After elaborately trimming out the mechanical counterweight and installation position of sensors, the eccentric distance between the mass center and the geometric center of ISA are reduced to 1.05 mm, 1.93 mm and 2.12 mm along X, Y and Z

J. Cheng et al. / Measurement 48 (2014) 314–324

Fig. 5. Relationship among transmissibility, natural frequency and attenuation coefficient at different frequency bands.

axes respectively. In addition, the matrix of moment of inertia is 2

Jxx

J xy J xz

3

2

1:02e  2 2:49e  4 1:98e  4

3

7 6 7 6 J 4 yx Jyy J yz 5 ¼ 4 2:49e  4 8:9e  3 4:32e  4 5kg m2 ; Jzx J zy Jzz 1:98e  4 4:32e  4 1:46e  2

where the principal moments of inertial on diagonal positions of the matrix are 20 times larger than the cross coupled moments. Therefore the matrix can be approximated as a diagonal one. On the other hand, to satisfy the requirements for motion decoupling, we adopt eight absorbers to constitute the VIS with the same configuration shown in Fig. 3. Meanwhile, the stiffness spindles of absorbers are parallel with the reference axes, as can be seen in Fig. 6. And for each absorber, the stiffness and damping coefficients along three spindles are designed to have the same value, i.e. kp0 = kq0 = kr0 = k0 and cp0 = cq0 = cr0 = c0. At last, to enhance the stability and reliability of ISA, and considering the convenience of assembling, the installation coordinates of

Fig. 6. Explosion figure of developed IMU and absorber.

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J. Cheng et al. / Measurement 48 (2014) 314–324

absorbers are l0 = 78 mm, w0 = 56 mm and h0 = 34 mm respectively. With a step mentioned above, it is easy to achieve the motion decoupling design of VIS and eliminate the motion coupling error of inertial sensors. Then, we focus on the parameters optimization to meet the attitude measurement demands of POS. The conditional parameters for VIS design are listed in Table 2. Such parameters are brought into Eqs. (13) and (14), and the iterative calculation is performed to optimize the natural frequencies and attenuation coefficients of each DOF models. It has been proved that the vibration modes of former six order natural frequencies of VIS correspond to its six single DOF motions along/around the reference axes. Then, the calculated results for each order and the corresponding vibration modes are listed in Table 3. The stiffness and damping coefficients of absorber are calculated and set to be k0 = 9.0  105 N/m and c0 = 7.0468 N s/m. In the following part, we will carry out finite element simulations to evaluate the performance of designed VIS.

4.2. Simulation and analysis The finite element model of developed IMU was built by using ANSYS Workbench software and the material parameters and connection relationships are set. After that, the simulations are carried out in two steps. Firstly, the natural frequencies of VIS are tested to validate the design parameters in Table 3. Secondly, the transmissibility and time delay parameters are examined under different kinds of load to evaluate the dynamic performance of designed VIS.

4.2.1. Natural frequency simulation Based on the established finite element model, the modal function of ANSYS Workbench software is applied to simulate natural frequencies of VIS. The original form of ISA, the simulated natural frequencies of former six orders and the corresponding vibration modes are shown in Fig. 7 respectively. Compared with the original form of ISA in Fig. 7(a), the vibration modes of each order could be distinguished by the directions of linear/rotational displacements of ISA. From Fig. 7(b)–(d), the vibration modes of former three natural frequencies are linear displacements along Z, Y and X directions, respectively. Meanwhile, from Fig. 7(e)–(g), the vibration modes of latter three natural frequencies are rotational displacement around X, Z and Y directions, respectively. Considering that the motional directions displayed above are perpendicular with one

Table 2 Conditional parameters for VIS design. Parameter

Value

xga xgb xgc x0m e g s

614 Hz 670 Hz 722 Hz 50 Hz 0.3 10% 5  105 s

Table 3 Calculated parameters of each degree of freedom motions. Order

Calculated inherent frequency

Calculated attenuation coefficient

Vibration mode

1

217

7.3124

2

217

7.3124

3

217

7.3124

4

277

0.0314

5

339

0.0675

6

385

0.0530

Linear vibration along X axis Linear vibration along Y axis Linear vibration along Z axis Rotational vibration along X axis Rotational vibration along Z axis Rotational vibration along Y axis

another, it could be concluded that the vibration mode of each order is decoupled from each other. The comparisons between the calculated and simulated natural frequencies are shown in Fig. 8. We can find that the simulated frequencies and their vibration modes at each order are perfectly matched with the calculated ones, and the maximum frequency relative error are less than 10%. In addition, the seventh order inherent frequency of VIS is 934.83 Hz, which is larger than the highest dithering frequency of DRLGs (722 Hz). Therefore, the natural frequencies of VIS have just kept away from the dithering frequency band of DRLGs; and the structure resonance aroused by dithering has been avoided successfully. 4.2.2. Dynamic performance simulation The dynamic performance simulations are carried out with three different types of loads to imitate the mechanical environment of ISA in reality. The mechanical loads of ISA could be classified into three kinds, including the rotational dithering of DRLGs, the linear displacement and the rotational displacement of mounting base. The Harmonic Response function of ANSYS Workbench software is applied and the simulated results of the transmissibility and time delay are examined. 4.2.2.1. Case 1: Dithering of DRLG. According to the specification of DRLGs used in our system, the amplitude of dithering torques inspiring the mechanical dithering are Ta = 4.5581 N m, Tb = 5.3146 N m and Tc = 6.3026 N m. And the support frame structure is fixed to simulate the aroused angular displacement Zga, Zgb and Zgc of ISA. The simulated result of ISA inspired by the dithering of DRLG in the direction of X at a frequency of 614 Hz is shown in Fig. 9 as an instance. As shown in this figure, the ISA rotates around the X axis, and the maximum linear displacement emerges at the cuboid corner indicated by a red label, whose value is 0.0051 mm. The corresponding angular displacement could be approximated as

kZ ga k ¼

0:0051 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  360 ¼ 0:0033 2p ð155=2Þ2 þ ð89=2Þ2

J. Cheng et al. / Measurement 48 (2014) 314–324

321

Fig. 7. Natural frequency simulation result of VIS.

and the simulated displacement could be expressed as Zga = 0.0033°  sin (2p  614 + Dga). Then, we could calcu-

late the amplitude of torque transmitted to the mounting base through

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2.14

400 Calculated Frequencies Simulation Frequencies

350

2.12

Amplitude (mm)

Frequency (Hz)

300 250 200 150

2.08 2.06 2.04

100

2.02

50 0

2.1

2 1

2

3

4

5

6

5

10

15

20

25

30

35

40

45

50

40

45

50

Frequency (Hz)

Modal Order Fig. 8. Comparison between calculated and natural inherent frequencies.

0.45 0.4

Phase Angle (°)

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

5

10

15

20

25

30

35

Frequency (Hz) Fig. 9. Displacement of ISA inspired by the dithering of DRLG around X axis.

Ma ¼ kC aa Z_ ga þ K aa Z ga k ¼ 0:83N m The final transmissibility is Fga = 0.182. As for the dithering load raised by DRLGs in the directions of Y and Z axes, the calculated transmissibility are Fgb = 0.231 and Fgc = 0.208. Therefore, all the transmissibility results in three orthogonal directions satisfy the design requirement of F < e = 0.3. 4.2.2.2. Case 2: Linear displacement of mounting base. According to the dynamic environment of the flying carrier, the amplitude and frequency of linear displacement of the mounting base vary within a large range. Motions with small amplitude and high frequency are especially hard to be precisely measured by IMU [16]. Suppose the amplitude of linear displacement of mounting base is 2 mm, and the frequency varies from 0 to 50 Hz to simulate the displacement amplitude and phase delay of ISA and then the transmissibility and time delay are calculated. Such simulations are carried out along X, Y and Z axes respectively. The amplitude and phase angle response curve along X axis are plotted in Fig. 10 as an instance. We can see that the induced motional amplitude and phase angle of ISA monotonically increase as the input frequency varies from low to high band. The maximum

Fig. 10. Frequency response along X axis.

displacement amplitude is 2.12 mm at 50 Hz, which produces 6% amplification error. And the maximum phase angle is 0.4185° at 50 Hz, which equals to 4.65  105 s. For the simulated results of the other two directions, the maximum amplitudes are both 2.13 mm, and the time delay are 4.13  105 s and 4.51  105 s. Hence, the simulated results of transmissibility and time delay under the load of linear displacement of the mounting base satisfy the requirements given in Table 2. 4.2.2.3. Case 3: Rotational displacement of mounting base. Suppose the amplitude of angular motion of mounting base is 2°, and the frequency also varies from 0 to 50 Hz. Then the induced displacement amplitude and phase delay of ISA are simulated around X, Y and Z axes respectively. The rotational amplitude and the phase angle response curve around X axis are plotted in Fig. 11. The varying rule of curves displayed in above figures exhibit the same characteristic as those in Fig. 10. As the input frequency increases up to 50 Hz, the maximum amplitude of induced angular displacement of ISA is 2.175° and the maximum phase angle shift is 0.3996°. The corresponding amplification error and time delay are 8.75% and 4.44  105 s respectively. For the simulated

J. Cheng et al. / Measurement 48 (2014) 314–324

modes of natural frequencies at each order have been decoupled from one another, and the transmissibility as well as the time delay under different types of mechanical loads satisfy the measurement requirements of POS. However, owing to the lack of vibration experimental conditions, the real effect of VIS in manufactured IMU hasn’t been tested, which could be the future work of our research.

2.18 2.17 2.16

Amplitude (°)

323

2.15 2.14 2.13 2.12

Acknowledgements

2.11 2.1 2.09

5

10

15

20

25

30

35

40

45

50

Frequency (Hz)

0.4

This work was supported in part by the National Natural Science Foundation of China (Nos. 60825305, 61121003, and 60904093) and in part by the National Program on key Basic Research Projects of China (No. 2009CB724002). References

Phase Angle (°)

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 5

10

15

20

25

30

35

40

45

50

Frequency (Hz)

Fig. 11. Angular amplitude frequency response around X axis.

results around Y and Z axes, the maximum amplitudes are 2.168° and 2.184°, and the time delay are 4.63  105 s and 4.27  105 s. Hence, the simulated results of transmissibility and time delay under the load of rotational displacement of the mounting base satisfy the requirements of POS. 5. Conclusion In this paper, a new design method of VIS used in DRLG POS is proposed to not only attenuate dithering noise of DRLGs, but also accurately transmit the valid motional information of IMU from the support frame structure to ISA. The proposed method consists of two steps. Firstly, aiming at eliminating the motion coupling error, several design principles, such as cuboid structural design of ISA as well as the symmetrical configuration design of absorbers, are presented to decouple the established six DOF dynamic model of VIS into six single DOF models. Secondly, the natural frequencies and attenuation coefficients of each single DOF models are optimized to satisfy the transmissibility and time delay requirements of POS. A design example is presented and the corresponding finite element simulation results are given to validate the effectiveness of proposed method. It is demonstrated that, the vibration

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