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A Multi-Model Combined Filter with Dual Uncertainties for Data Fusion of MEMS Gyro Array Qiang Shen, Jieyu Liu *, Xiaogang Zhou and Lixin Wang Xi’an Research Institute of Hi-Tech, Hongqing Town, Xi’an 710025, China; [email protected] (Q.S.); [email protected] (X.Z.); [email protected] (L.W.) * Correspondence: [email protected]; Tel.: +86-138-9283-5001 Received: 22 November 2018; Accepted: 21 December 2018; Published: 27 December 2018







Abstract: The gyro array is a useful technique in improving the accuracy of a micro-electro-mechanical system (MEMS) gyroscope, but the traditional estimate algorithm that plays an important role in this technique has two problems restricting its performance: The limitation of the stochastic assumption and the influence of the dynamic condition. To resolve these problems, a multi-model combined filter with dual uncertainties is proposed to integrate the outputs from numerous gyroscopes. First, to avoid the limitations of the stochastic and set-membership approaches and to better utilize the potentials of both concepts, a dual-noise acceleration model was proposed to describe the angular rate. On this basis, a dual uncertainties model of gyro array was established. Then the multiple model theory was used to improve dynamic performance, and a multi-model combined filter with dual uncertainties was designed. This algorithm could simultaneously deal with stochastic uncertainties and set-membership uncertainties by calculating the Minkowski sum of multiple ellipsoidal sets. The experimental results proved the effectiveness of the proposed filter in improving gyroscope accuracy and adaptability to different kinds of uncertainties and different dynamic characteristics. Most of all, the method gave the boundary surrounding the true value, which is of great significance in attitude control and guidance applications. Keywords: MEMS gyroscope; gyro array; set-membership filter; combined filter; multiple models; data fusion

1. Introduction The advent of micro-electro-mechanical system (MEMS) gyroscopes have added motion sensing in consumer devices and special military applications [1–3]. This is because it has many excellent properties, including being lightweight and having a compact size, low power consumption, low cost, and ease of mass production. However, when compared to traditional gyroscopes, the MEMS gyroscopes are still not sufficiently accurate for many applications, such as space vehicles. To enlarge the application field of MEMS gyroscopes, it is necessary to improve their accuracy further while not restricting their advantages (e.g., some effective modeling and processing methods were proposed in References [4,5] to reduce temperature drift). Due to the large increase in cost and time consumption of the previous methods focusing on the device itself [6,7], the gyro array was presented with the development of multisensor information fusion technology. The gyro array technique was first proposed by Bayard and Ploen [8], who integrated four gyroscope outputs to obtain an optimal rate signal by data fusion. The gyro array technique made rapid progress after that [9–11]. The key to the gyro array technique is the estimate method. A Kalman filter and its variants and extensions are always used, which have been proven effective to some extent. The fusion method used in the works of Bayard and Pleon was the Kalman filter. Then, an extended Kalman filter to fuse multiple gyroscopes for accuracy improvement was presented by Stubberud [12]. After that, Chang, Sensors 2019, 19, 85; doi:10.3390/s19010085

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Xue, and Jiang at Northwestern Polytechnical University played an important role in promoting the development of gyro arrays. They constructed an array with six gyroscopes [13–16] and adopted an optimal Kalman filter to process their signal. Meanwhile, the performance of the gyro array was analyzed in detail. A 3 × 3 planar array configuration of MEMS gyroscopes was developed by Heera, and Kalman filtering with a constant gain matrix for a minimum rate estimate was implemented in Reference [17]. A new statistic, called the “Allan covariance” between two gyros, was introduced by Richard, and the gyro array model could be used to obtain the Kalman filter-based (KFB) virtual gyro [18]. The fusion methods mentioned above have been proven to improve the accuracy of data effectively. However, some difficulties still exist. On the one hand, the Gaussian noise assumption in these classical techniques may render them invalid in some practical applications. When the Kalman filters are used for a gyro array, the uncertainty of noise statistical character may lead to degeneration in accuracy. An innovative alternative is to use the set-membership estimation approach, in which noises are assumed to be unknown but bounded [19–21]. For the set-membership approach, the estimate results are expressed as state feasible sets. Especially, the ellipsoidal set is an important way to describe the set-membership uncertainty [22,23]. Due to the complexity of the environment and the variation of the motion status, the stochastic uncertainty and the set-membership uncertainty always exist simultaneously in the outputs of the gyroscopes. However, the existence of Gaussian noise always makes the choice of the noise bounds too conservative and leads to accuracy degeneration in the set-membership estimations. In order to take advantage of each type of uncertainty, a combined filter with dual uncertainties is necessary. On the other hand, the motion status of the carrier is always changing in applications, and the dynamic characteristic cannot be well represented by a single model. This may also impact the performance of a gyro array badly. To ensure reliable estimation results in the case of complex motion of the carrier, several models are used to describe the various motion statuses of the carrier. Therefore, the motivation of our research was to design an effective estimate fusion method that could solve the above two difficulties. To achieve this goal, a dual-noise acceleration model is presented to describe the rate signal instead of a random walk, and a multi-model combined filter with dual uncertainties is proposed to combine the MEMS gyro array readings. The proposed method in this paper has two advantages, which are the main contributions of this paper: (1) With two types of uncertainty, the proposed combined filter based on sets of densities can not only capture the true mean and error statistics, but can also obtain the guaranteed estimation bounds; and (2) the subfilters in the proposed algorithm can adaptively switch between a known set of models at each sampling instant to respond to the dynamic changes of the system in time. In this paper, the gyroscope error model with dual uncertainties used in this research is established in Section 2. A brief introduction to the ellipsoidal sets and relevant properties is given in Section 3. Then, the multi-model combined algorithm is deduced in Section 4. Numerical experimental results are presented in Section 5. Conclusions are drawn in Section 6. 2. The Dual Uncertainties Model of a Gyro Array The most common model for a target maneuver is the white-noise acceleration model [24]. It assumes that the target acceleration is an independent process, which is strictly white noise. It differs from the non-maneuver model only in the noise level: The white noise process w used to model the effect of the control input u has a much higher intensity than the one used in a non-maneuver model. Considering that the MEMS gyroscopes always have tiny sampling internally, this kind of model can be used to model attitude maneuver of the gyro or the gyro-installed carrier. The magnitude of the process noise is used to adjust the dynamic characteristics in the model. Due to the fact that both stochastic and set-membership uncertainties always exist simultaneously in the outputs of the gyroscopes, the angular acceleration is assumed to be an independent process with dual uncertainties. Thus, we call it a dual-noise acceleration model.

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Sensors 19, 85 of 17 of the2019, gyroscopes, the angular acceleration is assumed to be an independent process with 3dual uncertainties. Thus, we call it a dual-noise acceleration model.

ω((tt)) is Assume is the the angular angular velocity velocity of of the the carrier carrier in in aa certain certain coordinate coordinate direction: direction: The Assume that that ω The kinematic acceleration model model is is then then given given as as kinematic equation equation of of the the dual-noise dual-noise acceleration .ω  (t) = w(t) + d(t) ω (t) = w(t) + d(t,),

(1) (1)

where w(t ) is a zero-mean Gaussian noise (stochastic uncertainty), and d(t) is the bounded noise where w(t) is a zero-mean Gaussian noise (stochastic uncertainty), and d(t) is the bounded noise (set-membership (set-membership uncertainty). uncertainty). In In fact, fact, the the dual dual uncertainties uncertainties can can be be interpreted interpreted as as aa set set bounded bounded uncertainty uncertainty incorporated incorporated into into the prior stochastic uncertainty [25]. For the dual-noise acceleration model given in Equation (1), the prior stochastic uncertainty [25]. For the dual-noise acceleration model given in Equation (1), the the . 2 w σ2 (0,σ variance ) with angular velocity velocityacceleration accelerationωωis the is the the zero-mean Gaussian sumsum of theofzero-mean Gaussian noise wnoise ∼ N (0, ) with . 2 σ2 and theσbounded uncertainty ∈ G ⊂ R. Then regarded as a random variable mean ω can variance and the bounded duncertainty . Then be regarded as with a random d ∈ ω⊂can  be 2 )| d ∈ G }), d. The special thing is that it cannot be seen as a single distribution, but a set (i.e., {N ( d, σ . The special thing is that it cannot be seen as a single distribution, but a set variable with mean d 2 Figure 1. as illustrated in (i.e., { (d , σ ) d ∈ } ), as illustrated in Figure 1.

Figure 1. 1. The Themodel modelwith withdual dualuncertainties uncertainties can regarded a set of Gaussian densities Figure can be be regarded as aasset of Gaussian densities (e.g.,(e.g., the the dual-noise acceleration model): (a) Stochastic model (w); (b) set-membership model (d); (c) dual dual w d dual-noise acceleration model): (a) Stochastic model ( ); (b) set-membership model ( ); (c) . uncertainties model model ((ω). ω ). uncertainties

In the multidimensional case, it can be expressed as {N (d, C)|d ∈ G n n⊂ Rn , C ∈ Rn×n }. Then the In the multidimensional case, it can be expressed as {  (d , C ) d ∈  ⊂  n , C ∈  n × n } . Then the state estimate is characterized by a set of probability densities through the combination of stochastic state estimate is characterized by a set of probability densities through the combination of stochastic and set-membership uncertainties. and set-membership uncertainties. Our proposed method poses the gyroscope filtering problem as a hidden Markov model (HMM). T Ourthat proposed method poses filtering problem hidden Markov model (HMM). Assume the state vector is x(the t) =gyroscope [θ (t), ω (t)] , where θ (t) is as thea attitude angle. Considering the T (t) gyroscope [θ (t ), ω(t )] model Assume that the state is x(t ) =state-space , where is the attitude the dual uncertainties, the vector discrete-time forθthe arrayangle. in theConsidering carrier can then be described as dual uncertainties, the discrete-time state-space model for the gyroscope array in the carrier can then (i ) (i ) (i ) (i ) xk = Fk xk −1 + Gk (wk + dk ), (2) be described as (i ) (i ) (i ) (i ) (i ) (i )(i ) xk z=k F xkk−1x+k G =k H +kv(kw+ ekdk, ) , k +

with the mode transition governed by a Markov chain with probabilities zk = H(ki) xk + v(ki) + e(ki ) , j P(mik mchain p ji , probabilities k −1 ) = with the mode transition governed by a Markov with

(2) (3) (3) (4)

P(i()m ki (mi)kj −1 ) = p ji(,i) (4) where zk is the measurement vector, and Fk , G k , and Hk are the state transition matrix, the noise i matrix, and the observation matrix corresponding model mk (i = 1, 2, · · · , r ) at time (i ) (i ) to the active (i ) and Fk , Gk , and Hk are the state transition matrix, the where zk is the measurement (i ) (ivector, ) (i ) (i ) k, respectively; wk ∼ N (0, Qk ) is the stochastic part of the process noise; vki ∼ N (0, Rk ) is the mk (i = 1,2, , r) time noise matrix, themeasurement observation matrix corresponding to the active stochastic partand of the noise, which are both assumed to bemodel white Gaussian; and dkatand ek (i ) )  N(0, Q(ki) ) are v(ki)bounded  N(0, R(kiby ) is the respectively; wkparts, is the stochastic of the process noise; but k , set-membership are which assumed to bepart of unknown distribution, (i )

(i )

(i ) T

( i ) −1 ( i ) dk

D k = { dk : (dk ) (Dk )

≤ 1},

(5)

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(i )

(i ) T

(i )

( i ) −1 ( i ) ek

E k = { ek : (ek ) (Ek )

≤ 1}.

(6)

In this research, six separate gyroscopes were taken to construct a gyroscope array. According to the model shown in Equation (1), the magnitude of the process noise directly affects the ability of the model to reproduce the dynamic characteristics of the input signal. In view of this situation, this paper designed a combined filter using multiple models with different process noise magnitudes and making the noise magnitude cover a certain range by interaction: Thereby the optimality of the estimation and the stability of the filter were guaranteed. Therefore, the model set with dual uncertainties designed in this paper is given by " (i ) Fk

=

(i )

1 0

∆T 1

#

" (i ) , Gk

=

(i )

0 ∆T

#

" (i ) , Hk +1

(i )

=

0 1

0 1

0 1

0 1

0 1

0 1

# , (7)

(i )

Qk = Qi , Rk = r2 CorrM, Dk = Di , Ek = e2 CorrM i = 1, 2, 3 · · · , r

where CorrM is a 6 × 6 correlation matrix of gyro array, r2 is the Gaussian measurement noise variance of component gyroscopes, e2 is a coefficient in the matrix defining the shape of the measurement noise set, and ∆T is the sampling interval. Obviously the differences between the models in Equation (7) are the values of Qi and Di , which represent the level of the stochastic part and set-membership part of the process noise. 3. Ellipsoidal Set and Relevant Properties When considering the set-membership uncertainty, the set operation is involved. In order to simplify the calculations, we confined the bounding feasible sets to the ellipsoids. Several definitions and lemmas about ellipsoid sets are given below, which were used in the derivation of the multi-model combined filter with dual uncertainties. Definition 1. A bounded ellipsoid E of Rn is defined by [23]

E (a, M) = {x ∈ Rn : (x − a)T M−1 (x − a) ≤ 1},

(8)

where a ∈ Rn is the center of E , and M ∈ Rn×n is a positive-definite matrix that specifies its size and orientation. (i )

Then dk and ek in Equations (5) and (6) are actually bounded by the ellipsoids E (0, Dk ) and (i )

E (0, Ek ), respectively. Definition 2. Given an ellipsoid E (a, M)(a1 ∈ Rn ) and a full rank matrix A ∈ Rn×n , then {Ax : x ∈ E (a, M)} can be expressed as AE (a, M). Lemma 1. Given an ellipsoid E (a, M)(a1 ∈ Rn ) and a matrix A ∈ Rn×n , then AE (a, M) = E (Aa, AMAT ). Proof. According to Definition 1, AE (a, M) = {Ax : x ∈ E (a, M)}.

(9)

Assume y = Ax and then x = A−1 y, and thus T

(A−1 y − a) M−1 (A−1 y − a) ≤ 1.

(10)

After some manipulations, Equation (10) becomes

(y − Aa)T (AMAT )

−1

(y − Aa) ≤ 1.

(11)

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(y − Aa)T (AMAT )−1 (y − Aa) ≤ 1 .

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T

y ∈(Aa, AMA ) .The This means   proof is complete. This means y ∈ E Aa , AMAT . TheΨproof is complete.  Definition 3. The Minkowski sum s of the two ellipsoids  ( a1 , M1 ) and  ( a 2 , M 2 ) is defined as Definition 3. The Minkowski sum Ψs of the two ellipsoids E (a1 , M1 ) and E (a2 , M2 ) is defined as Ψ s = {x : x = x1 + x 2 , x1 ∈  ( a1 , M 1 ) , x 2 ∈ ( a 2 , M 2 )} , Ψs = {x : x = x1 + x2 , x1 ∈ E (a1 , M1 ), x2 ∈ (a2 , M2 )}, denoted as denoted as ΨΨs ==  (Ea(1 ,aM, 1M  (a , M ) . )⊕ s 1 1 ) ⊕ E2(a2 , 2M2 ).

(12) (12) (13)(13)

The ellipsoid satisfying  ( as , Ms ) ⊇Ψs is called the outer bounding ellipsoid of the Minkowski The ellipsoid satisfying E (as , Ms ) ⊇ Ψs is called the outer bounding ellipsoid of the Minkowski sum Ψs , as illustrated in Figure 2. sum Ψs , as illustrated in Figure 2.

 ( a1 , M1 )

 (as , M s )

 ( a2 , M 2 ) Figure 2. The Minkowski sum of two ellipsoids. Figure 2. The Minkowski sum of two ellipsoids.

The following lemma is given to calculate the outer bounding ellipsoid of the Minkowski sum of The following lemma is given to calculate the outer bounding ellipsoid of the Minkowski sum two ellipsoids: of two ellipsoids: Lemma 2. ∀ ∈p(∈0,( 0, ++∞ ∞)), E, (a(sa, M ) and s ) ⊇ E (a1 , = aa1 1++ a2 a(2a1(,aa12,∈a2∈n )Rnand 1 ) ⊕ E (a2 , M2 ), with aas = Lemma 2.p
∀ s , M s ) ⊇  ( a1 M1 ) ⊕  ( a2 , M2 ) , with s − 1 n × n Ms = 1 + p M + 1 + p M M , M ∈ R . ( ) ( ) 2 2 1 1 M s = (1 + p −1 ) M 1 + (1 + p ) M 2 ( M 1 , M 2 ∈  n× n ) .

Proof. works of Maksarov Norton [22]. Proof. SeeSee thethe works of Maksarov andand Norton [22]. For a weighted outer bounding boundingellipsoid ellipsoidisisgiven givenbybythe For a weightedMinkowski Minkowskisum sumofofseveral several ellipsoids, ellipsoids, the outer thefollowing followinglemma. lemma.

Lemma 3. Given μk ∈ and the ellipsoids  ( a k , M k ) ( k = 1,2,r, r ∈ ), ∀ α ∈  r , with Lemma 3. Given µk ∈ R and the ellipsoids E (ak , Mk ) (k = 1, 2, · · · r, r ∈ R+ ), ∀α ∈ Rr , with αk > 0 and r r αk ∑> 0α and k = 1 , the = 1,  theαellipsoid E (ellipsoid a, M) with ( a, M) with +

k =1

k

k =1

r

a =r µ a , a = k∑ μak k =k1 k ,

(14) (14)

k =1

r

1 2 M =r ∑ α − k µ k Mk , −1 2 k = 1 M = αk μk Mk , contains µ1 E (a1 , M1 ) ⊕ µ2 E (a2 , M2 ) ⊕ · · · ⊕ µkr=E1 (ar , Mr ).

(15) (15)

contains μ1 ( a1 , M1 ) ⊕ μ2 ( a2 , M2 ) ⊕ ⊕ μr  ( ar , Mr ) . Proof. According to Lemma 1, it can be easily obtained that Proof. According to Lemma 1, it can be easily obtained that µk E (ak , Mk ) = E (µk ak , µ2k Mk ).

(16)

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Thus, the weighted Minkowski sum can be transformed into μk  (ak , Mk ) =  (μk ak , μk2M k).       2 2 ⊕ E can µ2 abe E µr ar , µ2r Mr . E µMinkowski Thus, the weighted into 2 , µtransformed 1 a1 , µ1 M1 sum 2 M2 ⊕ · · · ⊕

(16) (17)

 ( μ1a1 , μwith ⊕  ( μ 2 aconclusions  ( μ r a r , μ r[23], M r )a. parametrized family (17)of 1 M1 ) 2 , μ2 M 2 ) ⊕  On this basis and combined relevant in⊕Reference ellipsoids given byand Equations (14)with and (15) is obtained.  in Reference [23], a parametrized family On this basis combined relevant conclusions 2

2

2

of ellipsoids given by Equations (14) and (15) is obtained. 4. Multi-Model Combined Filtering with Dual Uncertainties (MMCF) 4. Multi-Model Combined Filtering with Dual Uncertainties (MMCF) Following the interacting multiple model (IMM) filter [26] merging strategy, the proposed Following the estimate interacting multiple model (IMM) filter [26] strategy, the proposed multi-model state combining dual uncertainties canmerging be updated through 4 steps: multiFirst, model state estimate combining dual uncertainties can be updated through 4 steps: First, the mixing the mixing probabilities and the reinitialized estimates for the corresponding mode-matched filter probabilities andThen the reinitialized forare theused corresponding mode-matched filter are calculated. are calculated. the mixingestimates estimates to input into the mode-matched filter new Then the mixing estimates are used to input into the mode-matched filter new measurements and to measurements and to update the estimates. Next, the updated mode probability is calculated. Finally, update the estimates estimates.are Next, theusing updated modematching. probability is calculated. Finally, the overall estimates the overall found moment The main difference is that the set operation are found using moment matching. The main difference is that the set operation is involved in the is involved in the proposed algorithm, and the output contains the covariance matrix Pk and an ˆ ( , ) P  x X and an ellipsoidal set proposed algorithm, and the output contains the covariance matrix k k kis ellipsoidal set E (xˆ k , Xk ), which represents a set of means. The architecture of the MMCF algorithm , which represents set of means. The architecture of the MMCF algorithm is illustrated in Figure 3. illustrated in Figurea 3.  (1)  (x (1) k −1 , X k −1 ) P (1)

zk

(1)  (xˆ (1) k , Xk ) (1) Pk

k −1

 (2)  (x (2) k −1 , X k −1 ) P (2)

 (xˆ k , X k ) Pk

(2)  (xˆ (2) k , Xk ) Pk(2)

k −1

 (3)  (x (3) k −1 , X k −1 ) P (3)

(3)  (xˆ (3) k , Xk ) Pk(3)

k −1

Figure 3. Structure of multi-model combined filtering with dual uncertainties (MMCF) algorithm (with Figure 3. Structure of multi-model combined filtering with dual uncertainties (MMCF) algorithm three models). (with three models).

At At time time k, k, the the MMCF MMCF algorithm algorithm can can be be given given as as follows. follows. 4.1. 4.1. Model-Conditioned Model-Conditioned Reinitialization Reinitialization The The predicted predicted probability probability is is given given by by r r ( j) (i ) ( i ) (i )( i ) k− k −11 µk,kμ , P { m } =  P m Z = Z ∑p jipμjikµ( −j )k1 −, 1 ,  , − 1 k k k −1 k

{

}

(18) (18)

1 1 j =j =

( j) is the calculated probability for model m j where μ at time k − 1 , and Z = z , z ,, z . where µk−k −11 is the calculated probability for model mk−k−11 at time k − 1, and Zk−1 = {{z1 1, z22, · · · , zk k−1−}1 }. Then the the mixing mixing weight weight is is calculated calculated by by Then ( j)

j

{

k −1

}

ji ( j ) (i ) ( j) (i ) k −1 j|i μk −1  P ( jm ) −1 m (ik) , Z k −1 = p ji μk −(1j) μk ,k −(1i ). µ k −1 , P{mk−1k m } = p ji µk−1 /µk,k−1 . k ,Z ( j)

j

(19) (19) ( j)

( j)

( j)

Let xˆ k−1 be the state estimate of the subfilter for model mk−1 , and let  (xˆ k −1, Xk −1 ) and Pk−1 be the error characteristics of the state estimate for the set-membership uncertainty and the stochastic uncertainty, respectively.

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( j)

( j)

j

( j)

( j)

Let xˆ k−1 be the state estimate of the subfilter for model mk−1 , and let E (xˆ k−1 , Xk−1 ) and Pk−1 be the error characteristics of the state estimate for the set-membership uncertainty and the stochastic uncertainty, respectively. ( j)

Suppose that only the zero-mean Gaussian uncertainty exists (i.e., Xk−1 = 0): Then the mixing estimate and covariance matrix are r

(i )

e xk −1 = e k(i−) 1 = P

r

∑

j =1

h

j |i

∑ µk−1 xˆk−1 , ( j)

(20)

j −1

i ( j) ( j) ( j) ( j) ( j) j |i Pk−1 + ( xek−1 − xˆ k−1 )( xek−1 − xˆ k−1 )0 µk−1 .

(21)

When considering the set-membership uncertainty, the mixing set of means therefore yields the Minkowski sum (i ) (1) (1) (2) (2) (r ) (r ) 1| i 2| i r |i Xek−1 = µk−1 E (xˆ k−1 , Xk−1 ) ⊕ µk−1 E (xˆ k−1 , Xk−1 ) ⊕ · · · ⊕ µk−1 E (xˆ k−1 , Xk−1 ).

(22)

(i ) e (i ) e (i ) Then the object in this step is to find an optimal ellipsoid E (e xk −1 , X k −1 ) ⊇ Xk −1 . According to (i )

Lemma 3, there is a family of ellipsoids that meet this condition with a center e xk−1 and the matrix defining the shape r

e (ki−) 1 = X

j |i

∑ ( µ k −1 )

j =1

(i )

where the parameter α j (i )

r

2 ( j) (i ) Xk−1 /α j ,

(i )

(23) (i )

> 0 and ∑ α j = 1. The parameters α j are chosen to minimize the size j =1

(i )

e k−1 ) by using the volume minimization criterion or trace minimization criterion. In this of E (e xk −1 , X

e (ki−) 1 was minimized due to the explicit expression of the parameter and avoidance paper, the trace of X of the nonlinear equation solution. Obviously, this criterion is favorable to computation efficiency. Then the parameter optimization problem is given by (i )

e k −1 ) Min: tr(X  r (i ) (i )   ∑ α j = 1, α j > 0  j =1 S.T. r 2  e (ki−) 1 = ∑ (µ j|i ) X( j) /α(i)   X j k −1 k −1

(24)

j =1

(i )

Based on the Lagrangian multiplier method, the optimal value of α j is calculated by (i ) αj

=

j |i µ k −1

q

( j) tr(Xk−1 )

r

∑

j =1

j |i µ k −1

q

! −1 ( j) tr (Xk−1 )

.

(25)

4.2. Model-Conditioned Filtering Then each subfilter for model mik updates state estimates with the new measurement zk . From Equations (2) and (5), the prediction set of means can be given by (i )

Xk,k−1

(i )

(i )

(i )

(i )

(i )

e k −1 ) ⊕ G D = Fk E (e xk −1 , X k k T

T

(i ) (i ) (i ) e (i ) (i ) (i ) (i ) (i ) = E (Fk e xk −1 , Fk X k −1 (Fk ) ) ⊕ E (0, Gk Dk (Gk ) )

(26)

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(i )

(i )

The ellipsoid E (xˆ k,k−1 , Xk,k−1 ) externally approximates the sum with (i )

(i ) (i )

xˆ k,k−1 = Fk e xk −1 and

(27)

T

T

(i ) (i ) e (i ) (i ) (i ) (i ) (i ) Xk,k−1 = (1 + p−1 )Fk X k −1 (Fk ) + (1 + p )Gk Dk (Gk ) .

(28)

Equations (27) and (28) are obtained on the basis of Lemma 2. The parameter p that optimizes the (i )

trace of Xk,k−1 is v  u (i ) e (i ) (i ) T u u tr(Fk Xk−1 (Fk ) u p=t . (i ) (i ) (i ) T tr(Gk Dk (Gk ) )

(29)

The associated covariance matrix is stated by T

T

(i ) (i ) (i ) (i ) (i ) (i ) e (i ) Pk,k−1 = Fk P k −1 (Fk ) + Gk Qk (Gk ) .

(30)

In the case of vanishing bounded perturbations, the state estimate is updated by means of the Kalman filter, as below: (i ) (i ) (i ) (i ) (i ) xˆ k = xˆ k,k−1 + Kk (zk − Hk xˆ k,k−1 ) (31) (i ) (i ) (i ) (i ) = (I − Kk Hk )xˆ k,k−1 + Kk zk (i )

(i )

(i )

(i )

Pk = (I − Kk Hk )Pk,k−1 ,

(32)

(i )

where the gain Kk is given by (i )

(i ) T

(i )

(i ) T

(i ) (i )

(i )

−1

Kk = Pk,k−1 (Hk ) (Hk Pk,k−1 (Hk ) + Rk )

.

(33)

In the presence of set-membership uncertainty, the measurement zk combined with the ellipsoidal error can be regarded as an ellipsoid set, (i ) (i ) (i ) (i ) (i ) Zk = {zk − ek ek ∈ Ek } = E (zk , Ek ). (i )

(34)

(i )

Then the predicted set E (xˆ k,k−1 , Xk,k−1 ) is updated through (i )

Xk

(i )

(i )

(i )

(i )

(i )

(i )

(i )

(i )

(i )

(i )

(i )

= (I − Kk Hk )E (xˆ k,k−1 , Xk,k−1 ) + Kk Zk

(i )

= (I − Kk Hk )E (xˆ k,k−1 , Xk,k−1 ) + Kk E (zk , Ek ) =

(35)

(i ) (i ) (i ) (i ) (i ) (i ) (i ) (i ) T (i ) (i ) (i ) (i ) T E ((I − Kk Hk )xˆ k,k−1 , (I − Kk Hk )Xk,k−1 (I − Kk Hk ) ) + E (Kk zk , Kk Ek (Kk ) )

(i )

(i )

(i )

(i ) T

(i )

(i )

(i )

It is easily deduced that the shaping matrix of E (xˆ k , Xk ), satisfying E (xˆ k , Xk ) ⊇ Xk , is (i )

(i )

(i )

(i )

(i ) (i )

(i ) T

Xk = (1 + q−1 )(I − Kk Hk )Xk,k−1 (I − Kk Hk ) + (1 + q)Kk Ek (Kk ) ,

(36)

(i )

with the parameter q chosen to minimize the trace of Xk : This is v  u (i ) (i ) (i ) (i ) (i ) T u tr (( I − K H ) X ( I − K H ) u k k k,k −1 k k u q=t . (i ) (i ) (i ) T tr(Kk Ek (Kk ) )

(37)

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4.3. Mode Probability Update The mode likelihood is obtained as (i ) (i ) (i ) Lk , p[e zk mk , Zk−1 ] (i ) (i )

(i )

assume

=

(i ) T

(i ) (i )

(i )

N (e zk ; 0, Sk ),

(38)

(i )

where e zk = zk − Hk xˆ k,k−1 , Sk = Hk Pk,k−1 (Hk ) + Rk . On this basis, we can calculate the model probability by (i )

(i ) µk

=

(i )

µ k | k −1 L k ( j)

( j)

∑ j µ k | k −1 L k

.

(39)

4.4. Estimation Fusion Using the matching model, the overall set of means can be expressed as (1)

(i )

(i )

(2)

(i )

(i )

(r )

(i )

(i )

Xk = µk E (xˆ k , Xk ) ⊕ µk E (xˆ k , Xk ) ⊕ · · · ⊕ µk E (xˆ k , Xk ).

(40)

Obviously the problem here is similar to that in Section 4.1, and thus the minimal-trace ellipsoid E (xˆ k , Xk ) containing the sum Xk is obtained based on Equations (20)–(25): xˆ k =

∑ xˆk

(i ) (i ) µk ,

i

r

Xk =

∑ j

(i ) µk

q

! (i ) tr(Xk )

r

∑ j

(i ) (i ) µ k Xk /

(41)

q

! (i ) tr(Xk )

.

(42)

It should be noted that this solution is optimal only in the family of ellipsoids, as in Equation (23). The overall covariance is stated as h i (i ) (i ) (i ) (i ) Pk = ∑ Pk + (xˆ k − xˆ k )(xˆ k − xˆ k )0 µk . (43) i

Remark 1. Finally, the proposed algorithm is implemented as below: Step 1. Initialize xˆ 0 , P0 , X0 , and set k ← 1 .

j |i

Step 2. Model-conditioned reinitialization: Compute the mixing weight µk−1 using Equations (18) and (19), (i )

(i )

(i )

e k−1 , and X e k−1 using Equations (20), (21), and (23), respectively. and then for each sum filter, compute e xk −1 , P (i )

(i )

(i )

Step 3. Model-conditioned filtering: For each subfilter, update xˆ k , Pk−1 , and Xk−1 using Equations (27)–(33) and (36). Step 4. Mode probability update: Calculate the model probability using Equation (39). Step 5. Estimation fusion: Using the matching model, calculate the overall estimate results xˆ k , Xk , and Pk using Equations (41)–(43), respectively. Step 6. Set k ← k + 1 and return to Step 2. 5. Experiments and Results 5.1. Experimental Setup In this section, experiments were implemented to demonstrate the performance of the proposed filter. Six MEMS gyroscope chips (ADXRS300 [27]) with external circuits were soldered onto the same printed circuit board (PCB) to construct an array. The prototype of the gyro array is shown in Figure 4. The output signals of the gyroscopes were collected through the data acquisition module

5. Experiments and Results 5.1. Experimental Setup In this section, experiments were implemented to demonstrate the performance of the proposed Sensors Six 2019,MEMS 19, 85 gyroscope chips (ADXRS300 [27]) with external circuits were soldered onto the10 of 17 filter. same printed circuit board (PCB) to construct an array. The prototype of the gyro array is shown in Figure 4. The output signals of the gyroscopes were collected through the data acquisition module constructed by a PCI (peripheral component interconnection) extension for instrumentation (PXI) constructed by a PCI (peripheral component interconnection) extension for instrumentation (PXI) system. The bandwidth of an individual gyroscope was 40 Hz, so the sampling rate was set to be system. The bandwidth of an individual gyroscope was 40 Hz, so the sampling rate was set to be 200 200 Hz to satisfy the Nyquist theorem. Hz to satisfy the Nyquist theorem.

Figure Figure 4. 4. Prototype Prototype of of the the gyro gyro array. array. Sensors x FOR PEER the REVIEW 10 offor 17 In2018, the 18, experiment, gyro array was put on a horizontal turntable. The experiment lasted 180 s, during which the turntable was set to rotate as follows: In the experiment, the gyro array was put on a horizontal turntable. The experiment lasted for 180 •s, during 15–40which s: Swing 10◦ angle amplitude 0.5 Hz frequency; the with turntable was set to rotateand as follows: ◦ ° amplitude 15–40 s: s: Rotate Swing with a1040 angle •• 45–65 /s constant rate; and 0.5 Hz frequency; ° /◦ /s s constant 45–65 s: s: Rotate with a − 4020 •• 70–90 constant rate; rate; ◦ ° / s constant rate;and 0.25 Hz frequency; • 70–90 s: Rotate with a −20 • 100–136 s: Swing with 10 angle amplitude ° angleamplitude amplitudeand and0.5 0.25 frequency; 100–136 s: s: Swing Swing with with 20 10◦ angle •• 145–172 HzHz frequency. • 145–172 s: Swing with 20 ° angle amplitude and 0.5 Hz frequency. During the the other other times, times, except except for for the theabove abovetime timeperiods, periods,the theturntable turntable kept kept still. still. It It should should be be noted that itit took tookaaperiod periodofoftime timetotoadjust adjust the motion state each time turntable. Thus, the motion state each time forfor thisthis turntable. Thus, bothboth the the phases each motion state hadtransition transitionperiods, periods,and andonly onlythe themiddle middle part part was in startstart andand endend phases of of each motion state had accordance accordance with with the the set set conditions conditions above. above. The The true angular rate of the turntable is shown in Figure 5.

Angular Rate (°/s)

100 50 0 -50 -100

0

20

40

60

80 100 Time (s)

120

140

160

180

Figure Figure 5. 5. The true angular rate.

Then the the rate rate signals signals of of the the gyroscopes gyroscopes in in the the array array were were gathered gatheredand andprocessed. processed. Then Through multiple tests, the correlation factor between the component gyroscopes was obtained Through multiple tests, the correlation factor between the component gyroscopes was obtained and is shown in Table 1: It played an important role in accuracy improvement of the gyro array [28]. and is shown Table 1: Itinplayed important role correlation in accuracy factor improvement of the gyro This meant theinelements CorrMan were given. The was analyzed with array static [28]. rate This meant elements in CorrMmethod were given. The correlation factor was analyzed with static rate signals for 1the h, and the computing of the correlation factor followed [14].

signals for 1 h, and the computing method of the correlation factor followed [14]. Table 1. Correlation factor between the component gyroscopes.

Gyro 1 Gyro 2

Gyro 1 1.0000 0.0016

Gyro 2 0.0016 1.0000

Gyro 3 −0.0047 0.0595

Gyro 4 0.0297 −0.0095

Gyro 5 0.0467 0.0591

Gyro 6 0.0451 0.0542

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Table 1. Correlation factor between the component gyroscopes.

Gyro 1 Gyro 2 Gyro 3 Gyro 4 Gyro 5 Gyro 6

Gyro 1

Gyro 2

Gyro 3

Gyro 4

Gyro 5

Gyro 6

1.0000 0.0016 −0.0047 0.0297 0.0467 0.0451

0.0016 1.0000 0.0595 −0.0095 0.0591 0.0542

−0.0047 0.0595 1.0000 −0.0939 −0.0381 −0.0688

0.0297 −0.0095 −0.0939 1.0000 0.0169 0.0946

0.0467 0.0591 −0.0381 0.0169 1.0000 0.0501

0.0451 0.0542 −0.0688 0.0946 0.0501 1.0000

Four models were used in the experiment, and other parameters in the model were given as ∆T = 0.005, r = 0.6, and e = 0.6. Qi and Di for different models are shown in Table 2. Table 2. Qi and Di for different models. Models

Model 1

Model 2

Model 3

Model 4

Parameters

Q1 = 8 D1 = 6

Q2 = 80 D2 = 60

Q3 = 800 D3 = 600

Q4 = 8000 D4 = 6000

5.2. Results

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For comparison purposes, the stand IMM estimator using a Kalman filter and the combined filter gyroscopes analyzed. Bymodel using were a sliding window with a width of 50,errors the means over in Section were 4.2 using a single implemented. First, the output of oneofofthe theerror individual gyroscopes analyzed. By using aassliding with width 50,that the means of mean the error over different timewere periods were calculated, shownwindow in Figure 6. Ita can be of seen the error value different time periods were calculated, as shown in Figure 6. It can be seen that the error mean value of the gyroscope changed significantly with time due to the dynamic change of input. This indicated of the gyroscope changed significantly with time due to the dynamic change of input. This indicated that the assumption of two random variables as noise representation in this paper was reasonable, that the assumption of two random variables as noise representation in this paper was reasonable, because a set ofof means was thethe main feature distinguishing this assumption from the single stochastic because a set means was main feature distinguishing this assumption from the single stochastic uncertainty assumption. In In addition, it can bebe seen that thethe means of of different time periods were allall uncertainty assumption. addition, it can seen that means different time periods were within the hypothetical set constructed by the parameters given in Section 5.1. within the hypothetical set constructed by the parameters given in Section 5.1.

Error Mean (°/s)

0.4 0.2 0 -0.2 -0.4

0

20

40

60

80 100 Time (s)

120

140

160

180

Figure 6. 6. The error mean over different time periods. Figure The error mean over different time periods.

Based onon this assumption, the proposed multi-model combined filter could get ananellipsoidal Based this assumption, the proposed multi-model combined filter could get ellipsoidal ) feasible setset of of means at at each moment (i.e., ), while the standard IMM filter with respect toto feasible means each moment (i.e., (Exˆ (k xˆ, X , X ) ), while the standard IMM filter with respect k k k Gaussian noise could only obtain a point estimate. addition, center of the ellipsoid xˆ kxˆcould Gaussian noise could only obtain a point estimate. In In addition, thethe center of the ellipsoid k could be considered to be the point estimate when needed. Figure 7 shows the state ellipsoidal estimates be considered to be the point estimate when needed. Figure 7 shows the state ellipsoidal estimates from s to s in phase plane using proposed algorithm. It should noted that, clarity, from 2525 s to 3030 s in thethe phase plane using thethe proposed algorithm. It should be be noted that, forfor clarity, only part of the ellipsoids were selected to be shown in the figure, and the time interval between two only part of the ellipsoids were selected to be shown in the figure, and the time interval between two adjacent ellipsoids was 0.05 s. The center of each ellipsoid is highlighted with “*”, and the true state adjacent ellipsoids was 0.05 s. The center of each ellipsoid is highlighted with “*”, and the true state at with “+”. It can be be seen from Figure 7 that all ofall theofvalues at the thecorresponding correspondingmoment momentisishighlighted highlighted with “+”. It can seen from Figure 7 that the of the true state at every moment lay within the corresponding ellipsoid. On this basis, the upper values of the true state at every moment lay within the corresponding ellipsoid. On this basis, the and lower boundaries of the estimation were acquired by by using thethe shape matrices ofofellipsoids, upper and lower boundaries of the estimation were acquired using shape matrices ellipsoids,as as shown in Figure 8. It can be guaranteed that the angular rate estimates are within a hard boundary and that the bounded sets contain the true value, which is meaningful for the robustness of the system in control and guidance applications. 30

from 25 s to 30 s in the phase plane using the proposed algorithm. It should be noted that, for clarity, only part of the ellipsoids were selected to be shown in the figure, and the time interval between two adjacent ellipsoids was 0.05 s. The center of each ellipsoid is highlighted with “*”, and the true state at the corresponding moment is highlighted with “+”. It can be seen from Figure 7 that all of the Sensors 2019, 19, 85 12 of 17 values of the true state at every moment lay within the corresponding ellipsoid. On this basis, the upper and lower boundaries of the estimation were acquired by using the shape matrices of ellipsoids, as shown Figure8.8.ItItcan canbe beguaranteed guaranteedthat thatthe the angular angular rate rate estimates estimates are shown ininFigure are within within aahard hardboundary boundary and that the bounded sets contain the true value, which is meaningful for the robustness and that the bounded sets contain the true value, which is meaningful for the robustnessof ofthe thesystem system in control and guidance applications. in control and guidance applications. 30 True state MMCF Ellipsoidal bounds

20

ω (°/s)

10 0 -10 -20 -30

-20

-40 -10

-5

0

θ (°)

5 -25

10 -5

15

-4

-3

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Angular Rate (°/s)

Figure Figure7.7.Phase-plane Phase-planeestimation estimationusing usingthe theMMCF. MMCF. Up bound

50

Lower bound

True rate

0 40.5 40 39.5

-50 0

20

-15 40

66 60

-16 67 80 121 100 122 120 123140 Time (s)

160

180

Figure 8. The estimated bounds by MMCF.

Below, Below, the the center center of of the the ellipsoid ellipsoid was was used used as as the the point point estimate, estimate, and and the the accuracy accuracy of of the the gyro gyro array was analyzed. The original rate signals of the individual gyroscopes and processing results array was analyzed. The original rate signals of the individual gyroscopes and processing results by by different and estimated errors areare shown in Figure 10. different methods methodsare aregiven givenininFigure Figure9.9.The Theoutput outputerrors errors and estimated errors shown in Figure In order to demonstrate the effect of the above filters clearly, the root mean square error (RMSE) of the 10. In order to demonstrate the effect of the above filters clearly, the root mean square error (RMSE) estimated results results was calculated and is presented in Table in 3. For further the tests, errorsthe were analyzed of the estimated was calculated and is presented Table 3. Fortests, further errors were through Allan the deviation method, and the results are shown inshown Figurein 11.Figure Besides, the angular analyzedthe through Allan deviation method, and the results are 11. Besides, the random walk (ARW), rate random walk (RRW), and bias instability (BI) were calculated from an Allan angular random walk (ARW), rate random walk (RRW), and bias instability (BI) were calculated from deviation and are alsoand presented in Table 3. In the test, the measurements an Allan plot, deviation plot, are also presented in Allan Tabledeviation 3. In the Allan deviation test,were the recorded for 1 h at a sampling rate of 200 Hz. Before that, the gyroscopes were powered on for 0.5 h. measurements were recorded for 1 h at a sampling rate of 200 Hz. Before that, the gyroscopes were The experiments were performed in a temperature control turntable, and the temperature was set to powered on for 0.5 h. The experiments were performed in a temperature control turntable, and the be 25 ◦ C. Other experimental such as hardware setup, such were as thehardware same as described in temperature was set to be 25 configurations, °C. Other experimental configurations, setup, were Section 5.1. The turntable kept still during most times, except for 1 min. The turntable was set to swing the same as described in Section 5.1. The turntable kept still during most times, except for 1 min. The with 10◦ angle amplitude 0.510° Hzangle frequency in thisand minute to test the dynamic of the turntable was set to swingand with amplitude 0.5 Hz frequency in thisperformance minute to test the proposed method. Thenofthe theThen output the gyroscopes (array) the actual dynamic performance thedifference proposedbetween method. theofdifference between the and output of the rotational of the analyzed as the error. was analyzed as the output error. gyroscopesspeed (array) andturntable the actualwas rotational speed of output the turntable Table 3. The overall results of the gyro array. IMM: Interacting multiple model; RMSE: Root mean square error; IF: Improve factor; ARW: Angular random walk; RRW: Rate random walk; BI: Bias instability. Terms

Single Gyroscope

Model 1

Model 2

Model 3

RMSE( ° s )

0.4916

8.3886

0.9663

0.1828

0.2009

0.1628

0.1478

IF

1

0.0586

0.5087

2.6893

2.4470

3.0197

3.3261

11.4362

/

/

4.8251

4.8458

4.0748

3.4395

1/2

ARW( ° h

)

Model 4

IMM

MMCF

Angular AngularRate Rate(°(/s) °/s)

Sensors Sensors 2019, 2018, 19, 18, 85 x FOR PEER REVIEW Sensors 2018, 18, x FOR PEER REVIEW

13 of of 17 13 17 13 of 17

16 16

50 50

14 14 119 119

0 0

-50 -50 0 0

Gyro 1 Gyro 1 Gyro 2 Gyro 2 20 40 20 40

120 120

121 121

Gyro 3 Gyro 5 Gyro 3 Gyro 5 Gyro 4 Gyro 6 Gyro 4 Gyro 6 60 80 100 120 60 80 100 120 Time (s) Time (s)

140 140

160 160

180 180

140 140

160 160

180 180

(a) (a) 16 16 Angular AngularRate Rate(°(/s) °/s)

50 50

14 14 119 120 121 119 120 121

0 0

-50 -50 0 0

Model1 Model1 Model2 Model2 20 40 20 40

Model3 IMM Model3 IMM Model4 MMCF Model4 MMCF 60 80 100 120 60 80 100 120 Time (s) Time (s)

(b) (b) Figure 9. 9. The The overall overall experiment experiment outputs: outputs: (a) The outputs of individual gyroscopes; gyroscopes; (b) (b) the the angular angular Figure (a)The Theoutputs outputsof ofindividual individual Figure 9. The overall experiment outputs: (a) gyroscopes; (b) the angular rate of the gyro array estimated by using different methods. rateof ofthe thegyro gyroarray array estimated estimated by by using using different different methods. methods. rate

Error Error(°(/s) °/s)

2 2

Gyro 1 Gyro 1 Gyro 2 Gyro 2

Gyro 3 Gyro 3 Gyro 4 Gyro 4

Gyro 5 Gyro 5 Gyro 6 Gyro 6

1 1 0 0 -1 -1 -2 -2 0 0

20 20

40 40

60 60

80 100 80 100 Time (s) Time (s)

1 1

Error Error(°(/s) °/s)

50 50

140 140

160 160

180 180

(a) (a)

0 0 -1 -1 119.5120120.5 119.5120120.5

0 0

-50 -50 0 0

120 120

Model1 Model1 Model2 Model2 20 20

40 40

Model3 Model3 Model4 Model4 60 60

IMM IMM MMCF MMCF

80 100 80 100 Time (s) Time (s)

120 120

140 140

160 160

180 180

(b) (b) Figure 10. Errors of the the overall overall outputs: outputs: (a) (a)The Theoutput output errors errors of of individual individual gyroscopes; gyroscopes; (b) the angular Figure 10. Errors of the overall outputs: (a) The output errors of individual gyroscopes; (b) the angular rate errors of the gyro gyro array array estimated estimated by by using using different different methods. methods. rate errors of the gyro array estimated by using different methods.

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0

10

10

Allan Deviation (°/s)

Allan Deviation (°/s)

0

-1

10

-2

10

Gyro 1 Gyro 2

-3

10

-1

Gyro 3 Gyro 4

0

10

1

Gyro 5 Gyro 6 2

10 10 10 Integration Times (s)

10

-1

10

-2

10

Model1 Model2

-3

3

10

10

-1

10

Model3 Model4

0

1

IMM MMCF 2

3

10 10 10 Integration Times (s)

(a)

10

(b)

Figure Allan deviation results: Allan deviation of the errors from individual Figure 11. 11. TheThe Allan deviation testtest results: (a) (a) TheThe Allan deviation of the errors from individual gyroscopes; (b) the Allan deviation of the errors from array. gyroscopes; (b) the Allan deviation of the errors from the the gyrogyro array. Table 3. The overall results of the gyro array. IMM: Interacting multiple model; RMSE: Root mean square It can be seen from these results that both the multi-model methods and the combined filter using error; IF: Improve factor; ARW: Angular random walk; RRW: Rate random walk; BI: Bias instability.

some single models (Models 3 and 4) were effective in combining numerous gyroscopes, and the Single

Terms Modelbetter. 1 Model Model 3 4 multi-model methods performed When2 using Model 1 Model or Model 2, IMM the errorsMMCF could be well Gyroscope

suppressed input signal constant. However, when RMSE (◦ /s)by the filter 0.4916when the 8.3886 0.9663was 0 or 0.1828 0.2009 0.1628the turntable 0.1478 swung, IF led to amplitude 1 0.0586 0.5087 demonstrated 2.6893 2.4470 3.0197could not 3.3261 the filter attenuation, which that this model accurately ◦ 1/2 ARW ( /h

)

11.4362

/

/

4.8251

4.8458

4.0748

3.4395

reproduce the of the/ input signal. using Model 3 or Model 4, the result 435.2964characteristics / 192.5314When198.8246 178.2495 156.2842 RRW (◦ /h3/2 ) dynamic BI (◦ /h)

63.2521

/

/

41.5448

33.1668

28.9512

21.0312

was the opposite: The errors were greatly reduced with sinusoidal rate input, but the filter had poor performance with constant rate input. Through switching the models according to different situations, In Table 3, IF is the improve factor, defined as the multi-model methods could reflect the dynamic characteristic of the input signal more accurately. Table 3 shows that the RMSE was reduced froms /RMSE 0.4916 °as, to 0.1628 ° s and 0.1478 ° s by the IMM IF = RMSE (44)

Error (°/s)

Error (°/s)

filter and MMCF, respectively. It reveals that the performance of the MMCF was higher than that of where RMSEs refers to the RMSE of the rate signal for the single gyroscopes in the array, and RMSEa is the IMM filter. From the Allan deviation plot and the results in Table 3, the ARW noise, RRW noise, that of the gyro array. and biasbeinstability reduced by using the multi-model methods combined with It can seen from were these all results that both the multi-model methods and theand combined filterfilter using Models 4 to (Models fuse measurements fromeffective the gyroin array. In addition, it is obvious that theand reduction some single3 and models 3 and 4) were combining numerous gyroscopes, the multi-model methods performed better. When using Model 1 or Model 2, the errors could be well factor by the MMCF was greater than that by other methods. As for the filter using Models 1 and 2, suppressed the filter the were input difficult signal was 0 or constant. However, when the turntable the abovebythree kindswhen of noise to calculate due to large sinusoidal errors. The swung, reason for the filter led to amplitude attenuation, which demonstrated that this model could not accurately the failure of Models 1–3 in Figure 11 was that these models could not accurately reproduce the reproduce the dynamic characteristics of the input signal. When using Model 3 or Model 4, the result characteristics the greatly input signal, and thesinusoidal filters using led tohad amplitude wasdynamic the opposite: The errorsofwere reduced with rate these input,models but the filter poor attenuation when the turntable swung. Thisswitching means large errors were added to the output performance with constant rate input. Through thesinusoidal models according to different situations, thenoise, multi-model methods couldin reflect the10. dynamic characteristic of the input signal more accurately. similarly to the errors Figure Table 3 shows that the RMSE was reduced from 0.4916◦ /s to 0.1628◦ /s and 0.1478◦ /s by the IMM filter To further analyze the performance of the proposed method, the estimated errors over different and MMCF, respectively. It reveals that the performance of the MMCF was higher than that of the time periods are shown in Figure 12. The detailed results are illustrated in Table 4. IMM filter. From the Allan deviation plot and the results in Table 3, the ARW noise, RRW noise, and bias instability were all reduced by using the multi-model and combined filter with Models 3 Single gyroscope methods MMCF and 4 to fuse measurements from the gyro array. In addition, it is obvious that the reduction factor 2 2 by the MMCF was greater than that by other methods. As for the filter using Models 1 and 2, the 1 1 above three kinds of noise were difficult to calculate due to large sinusoidal errors. The reason for the 0 1–3 in Figure 11 was that these models could 0 failure of Models not accurately reproduce the dynamic characteristics of the input signal, and the filters using these models led to amplitude attenuation when -1 -1 -2

5

10 Time (s)

15

-2 30

32

34 36 Time (s)

38

40

Models 3 and 4 to fuse measurements from the gyro array. In addition, it is obvious that the reduction factor by the MMCF was greater than that by other methods. As for the filter using Models 1 and 2, the above three kinds of noise were difficult to calculate due to large sinusoidal errors. The reason for the failure Sensors 2019, 19,of85Models

1–3 in Figure 11 was that these models could not accurately reproduce 15 ofthe 17

dynamic characteristics of the input signal, and the filters using these models led to amplitude attenuation theThis turntable This means largewere sinusoidal were added the output the turntablewhen swung. meansswung. large sinusoidal errors added errors to the output noise,tosimilarly to noise, similarly to the the errors in Figure 10.errors in Figure 10. To To further further analyze analyze the the performance performance of of the the proposed proposedmethod, method,the theestimated estimatederrors errorsover overdifferent different time periods are shown in Figure 12. The detailed results are illustrated in Table 4. time periods are shown in Figure 12. The detailed results are illustrated in Table 4. MMCF 2

1

1

Error (°/s)

Error (°/s)

Single gyroscope 2

0 -1 -2

5 10 Sensors 2018, 18, x FOR PEER REVIEW Time (s)

0 -1 -2 30

15

32

2

2

1

1

0 -1 -2 55

60 Time (s)

-2 80

65

82

1

1

Error (°/s)

Error (°/s)

84 86 Time (s)

88

90

(d) 2

0 -1 124 126 Time (s)

15 of 17

-1

2

122

40

0

(c)

-2 120

38

(b)

Error (°/s)

Error (°/s)

(a)

34 36 Time (s)

128

130

0 -1 -2 155

(e)

160 Time (s)

165

(f)

Figure12. 12.Estimated Estimatederror errorwith withMMCF MMCFover overdifferent differenttime timeperiods: periods:(a) (a)at at5–15 5–15s,s,the theinput inputsignal signalwas was Figure ω = 10 π sin( π t ) ° s ω = 0 ; (b) at 30–40 s, the input signal was ; (c) at 55–65 s, the input signal was ◦ ω = 0; (b) at 30–40 s, the input signal was ω = 10π sin(πt) /s; (c) at 55–65 s, the input signal was ◦ ◦ 40 °/s; s ; (d) −20 s ; (e) (d) at at 80–90 80–90 s,s, the theinput inputsignal signalwas wasω ω==− 120–130 s, s, the the input inputsignal signalwas was ωω== 40 20° /s; (e) at at 120–130 ωω== 5π sin(0.5πt and(f) (f)atat155–165 155–165s,s,the theinput inputsignal signalwas was ω ω ==20 20π sinπ(tπt 5π sin(0.5 π t ) )°◦s/s; π sin( ) ° )s◦./s. ; and Table 4. The test results over different time periods (different input).

Figure 12 shows that the proposed method could effectively reduce the errors of the gyroscope in Single Gyroscope Gyro Array any kind of motion. Inputresults (◦ /s) in Table 4 also demonstrate the performance improvements. Time (s) The statistics RMSE (◦ /s)

RMSE (◦ /s)

IF

It can be seen from Table 4 that the RMSEs of the estimation were obviously reduced compared to the 5–15 0 0.4939 0.1120 4.4098 30–40 ω = 10π sin(πt) 0.4775 0.2081 2.2946 40 0.5190 0.1212 The reduction 4.2822 factor with method had 55–65 to do with the dynamic characteristics of the input signal. 80–90 −20 0.4320 0.0850 5.0824 respect to 0 input and constant input is about 4–5, whereas the corresponding factor for sinusoidal 120–130 ω = 5π sin(0.5πt) 0.4778 0.1686 2.8339 input was about 2–3, which the 0 input and0.2200 constant input. Additionally, it 155–165 ω =was 20πlower sin(πt)than that of 0.4797 2.1805

original rate signals of the individual gyroscopes. Furthermore, the performance of the proposed

t refers frequency to the time. and amplitude with respect to the swing indicated an increase in the RMSE with Note: increasing

rate signal. Table 4. The test results over different time periods (different input).

Time (s)

Input ( ° s )

Single Gyroscope RMSE ( ° s )

Gyro Array RMSE ( ° s )

IF

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Figure 12 shows that the proposed method could effectively reduce the errors of the gyroscope in any kind of motion. The statistics results in Table 4 also demonstrate the performance improvements. It can be seen from Table 4 that the RMSEs of the estimation were obviously reduced compared to the original rate signals of the individual gyroscopes. Furthermore, the performance of the proposed method had to do with the dynamic characteristics of the input signal. The reduction factor with respect to 0 input and constant input is about 4–5, whereas the corresponding factor for sinusoidal input was about 2–3, which was lower than that of the 0 input and constant input. Additionally, it indicated an increase in the RMSE with increasing frequency and amplitude with respect to the swing rate signal. 6. Conclusions In this paper, a multi-model combined filter allowing for simultaneous treatment of stochastic and set-membership uncertainties was proposed to combine multiple MEMS gyroscopes to improve overall accuracy. The dual uncertainties model of a gyroscope array was established. To reproduce the dynamic characteristics of the rate signal, multiple models with different process noises were involved in the filter. Combined with the IMM update process and the calculation method for a Minkowski sum of multiple ellipsoidal sets, the multi-model state estimates integrating dual uncertainties was updated. The experimental results showed that the RMSE of the estimated rate signal by the proposed method was reduced from 0.4916◦ /s to 0.1478◦ /s, which performed better than the IMM filter and combined filters with single models. This proved that the proposed method is efficient in improving the system overall performance and that it produces better adaptability to different kinds of uncertainties and different dynamic characteristics. Moreover, the proposed method can provide the bounds of the rate signal estimates. This property is meaningful for the robustness of the system and benefits attitude control and guidance. Author Contributions: Conceptualization, Q.S.; formal analysis, Q.S. and X.Z.; funding acquisition, Q.S.; investigation, X.Z. and L.W.; methodology, Q.S. and J.L.; project administration, L.W.; software, Q.S.; supervision, J.L.; validation, Q.S., X.Z., and L.W.; writing–original draft, Q.S.; writing–review and editing, J.L. and L.W. Funding: This research was funded by the National Natural Science Foundation of China, grant numbers 61503390 and 61503392. Acknowledgments: The authors of this paper would like to gratefully thank all the referees for their constructive and insightful comments. Conflicts of Interest: The authors declare no conflict of interest.

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