Attitude Estimation By Separate-Bias Kalman Filter-Based Data Fusion

THE JOURNAL OF NAVIGATION (2004), 57, 261–273. f The Royal Institute of Navigation DOI: 10.1017/S037346330400270X Printed in the United Kingdom

Attitude Estimation By Separate-Bias Kalman Filter-Based Data Fusion Peyman Setoodeh, Alireza Khayatian and Ebrahim Farjah (Shiraz University, Iran) (Email : [email protected])

Attitude estimation systems often use two or more different sensors to increase reliability and accuracy. Although gyroscopes do not have problems like limited range, interference, and line of sight obscuration, they suffer from slow drift. On the other hand, inclinometers are drift-free but they are sensitive to transverse accelerations and have slow dynamics. This paper presents an extended Kalman filter (EKF)-based data fusion algorithm which utilizes the complementary noise profiles of these two types of sensors to extend their limits. To avoid complexities of dynamic modelling of the platform and its interaction with the environment, gyro modelling will be used to implement indirect (error state) form of the Kalman filter. The great advantage of this approach is its independence from the structure of the platform and its applicability to any system with a similar set of sensors. Separate bias formulation of the Kalman filter will be used to reduce the computational complexity of the algorithm. In addition, a systematic approach based on wavelet decomposition will be utilized to estimate noise covariances used in the Kalman filter formulation. This approach solves many of the convergence problems encountered in the implementation of EKF due to the choice of covariance matrices. Experimental implementation of the estimator shows the excellent performance of the filter. KEY WORDS 1. Attitude Estimation.

2. Extended Kalman Filter (EKF). 3. Strapdown Inertial Navigation System (INS). 4. Wavelet.

1. I N T R O D U C T I O N. Attitude estimation is applied in such diverse fields as vehicle navigation, robotics, virtual environments, surveillance, monitoring and so on. Methods of sensing attitude can be categorized on the basis of the reference system with respect to which the orientation is determined (Doeblin, 1990) : ’ ’ ’

Inertial reference sensing. Gravity reference sensing. Magnetic reference sensing.

Orientation determination of a moving object using gyroscopes, which are inertial reference sensors, has been well developed in the field of Inertial Navigation Systems (INS) (Broxmeyer, 1964; O’Donnel, 1964 ; Britting, 1971 ; Farrell, 1976 ; Titterton

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and Weston, 1997; Lawrence, 1998). Strapdown rate gyros measure the angular rate of the body with respect to the inertial frame in body frame coordinates. They can therefore be used to derive attitudes by integrating the rigid body kinematic equations, starting from a known initial orientation (Foxlin, 1996). All attitude determination systems that use gyros suffer from bias or drift, which is a low frequency noise component. Accurate estimates could be achieved over longer periods of time by employing high quality gyros and good initial guesses of the attitude. However, high quality gyros are too expensive for many applications and access to such sensors can be limited. Even with perfect gyros, accumulated estimation errors cannot be recovered (Rehbinder and Hu, 2000). Due to time dependent errors in their principal operation, the accuracy of strapdown systems degrade quickly if they operate in an unaided mode. So, attitude estimation is usually performed by combining measurements from rate gyros and aiding sensors. One solution is to utilize gravity and magnetic reference sensors for tilt and pan angle measurement, respectively. Inclinometers provide an absolute reference of the attitude by relating the body orientation to the gravity vector. They provide very accurate measurements of the attitude but suffer from low bandwidth and are sensitive to translational accelerations. So, it is not recommended that inclinometers are used alone and they should only be used during low acceleration phases (Foxlin, 1996 ; Rehbinder and Hu, 2000). Inclinometers can be used as tilt sensors and magnetic compass employed for yaw angle measurement. In this paper we use sensor fusion to overcome the drawbacks of the available sensors by combining information from independent sensors with complementary noise profiles to give better accuracy and reliability. The main contribution is the simple formulation of the estimator, which allows the achievement of more accurate estimates by increasing the order of the filter in a straightforward and simple manner. We use Donoho’s proposed approach (Donoho and Johnstone, 1994) to estimate noise covariances based on the finest level of the wavelet decomposition. The structure of this paper is as follows. Section 2 first formulates the problem and then develops a data fusion algorithm based on an extended Kalman filter (EKF) to fuse three strapdown rate gyro measurements with measurements of two inclinometers and a compass for roll, pitch, and yaw angle estimation, respectively. The algorithm is validated experimentally in Section 3 and the paper concludes in Section 4.

2. S T A T E M E N T O F T H E P R O B L E M. 2.1. System modelling. The first step in modelling is selection of the variables that should be included in the state and measurement vectors. Since the Kalman filter is supposed to estimate orientation, it is reasonable that the state vector includes it. According to its physical interpretation, simplicity, and allowing for smaller state vector dimension, Euler angle representation is a good choice among the various attitude representations. In Euler angle representation, rotations about the x, y, and z body axes are called roll, pitch and yaw and shown by the symbols; W, h, and Y respectively where the positive direction of x, y, and z axes respectively point forward, right and down. Although there is a singularity at h=t90x, this would not be a problem for many applications. The continuous-time nonlinear Euler angle integration

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kinematics is as follows : 2 3 2 _ 1 sin W tan h W 4 h_ 5=4 0 cos W _ 0 sin W=cos h Y

32 3 cos W tan h vx x sin W 54 vy 5 vz cos W=cos h

263

(1)

where H(t)=[W h Y]T is the attitude vector and V(t)=[vx vy vz]T represents the angular rate vector of the body frame with respect to the inertial frame in body frame coordinates, which is exactly what the strapped-down triad of rate gyros aligned with the body axes measure (Foxlin, 1996). The attitude can be obtained by integration of (1), starting from a known initial orientation (Titterton and Weston, 1997). In most references the angular rates are usually included in the state vector and rate gyro measurements in the measurement vector (Koifman and Merhav, 1990). There is a serious drawback to this approach for attitude estimation because the motion of the body should be included in the system dynamic model to define the derivatives of the angular velocities. There are also other alternatives ; for example some authors assume that the angular acceleration is just derived by some process noise to account for non-constant angular rates (Emura and Tachi, 1994a, b). Since this is not a realistic assumption, the model is not very optimal. Other authors try to increase accuracy by augmenting the state vector with higher order derivatives of angular rates (Barshan and Durrant-Whyte, 1995), which leads to larger state vectors. A good literature survey could be found in (Foxlin, 1996). In our approach we avoid dynamic modelling for the following reasons (Roumeliotis et al., 1999a, b) : ’

’

’

’

The dynamic modelling should be redone for any modification made to the platform i.e. a slightly different platform would require a new estimator. Dynamic modelling would require a very large number of states, thus increasing the computational burden. Dynamic modelling and the added complexity do not always produce the expected results. Precise modelling of the interaction of the platform and the environment is sometimes impossible.

The cumbersome dynamic modelling of the system under study and its interaction with environment is replaced by gyro modelling in our state equations. Treating the gyro measurement as an exogenous input for a single-axis gyro, measurement of angular rate, vm, is related to the attitude, h, by (Roumeliotis et al., 1999a) : h_ =vm +dv+nr dv_ =nw y=h+v

(2) (3) (4)

The model assumes that the gyro noise is composed of two independent elements ; rate noise nr(t), which is additive white noise and rate random walk nw(t), generated when white noise passes through an integrator (Roumeliotis, 2000). Since the time varying gyro bias, dv, is the largest source of error, state vector is augmented with the three gyro biases dvx, dvy, and dvz. The filter then estimates the bias terms as well as the attitude to reduce the estimation errors. So, the gyro signals appear in the system equations instead of the measurement equations and the formulation of the problem

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Figure 1. Schematic of the sensor assembly (Lawrence, 1998).

requires an indirect (error-state) Kalman filter approach (Foxlin, 1996; Roumeliotis et al., 1999a, b). 2.2. Indirect versus direct Kalman filter. There are two approaches for implementing a Kalman filter in conjunction with inertial navigation systems (INS); the indirect (error state) and the direct (total state) formulation. The indirect form estimates errors in orientation using the difference between the INS and the aiding sensor data as measurement. On the other hand, the direct form estimates the orientation using the INS outputs as measurement. While the total state requires a dynamic model, the indirect formulation allows for removal of the angular rates from the state vector. So, the indirect filter has a superior performance from the viewpoint of computational complexity. Since the indirect filter is out of the INS loop, it requires a much lower sampling rate and guarantees that the rapid dynamic response of the inertial system will not be compromised by the Kalman filter. In addition, in case of the filter failure, the estimator still acts as an integrator on the INS data. In the direct filter structure, the INS is useless without the filter and filter failure leads to failure of the entire navigation algorithm, which is another disadvantage for the direct form compared to the indirect one (Foxlin, 1996; Roumeliotis et al., 1999a, b). The following sections will develop an extended Kalman filter to estimate
 dH x= =[ dW dh dY dvx dvy dvz ] T (5) dV using ^ ^ hinclinometer x h y= [Winclinometer x W

^]T Ycompass x Y

(6)

as measurements for the sensor assembly shown in Figure 1, where dH represents the error in the output of the attitude computer and dV represents the gyro biases. The schematic diagram of the indirect Kalman filter is depicted in Figure 2. In Foxlin’s approach (Foxlin, 1996), which was followed in (Setoodeh et al., 2002), first the continuous-time nonlinear attitude integration kinematics (1) was discretized,

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Rate Gyros

Ω + δΩ

Gyro Error Compensation

Ωˆ

δΩˆ Inclinometers & Compass

Θ m - Θˆ

Attitude Computation

Θˆ

δΘˆ

Kalman Filter Error Estimator

Figure 2. Indirect Kalman filter for attitude estimation (Foxlin, 1996).

and then the nonlinear discrete-time attitude algorithm obtained was linearized to achieve the error dynamics. In our derivation here we will follow a different approach ; the continuous-time nonlinear attitude integration kinematics will be linearized first and then the discrete-time error dynamics will be obtained by discretizing the resulting linear continuous-time equation. Experiments showed that the two methods yield the same results and the discretization and linearization are commutative. Although the two methods are equal, the second method has some advantages compared with the first one. Retaining higher order terms in the Taylor series expansion to achieve acceptable error rates in the second method is much easier and cumbersome computation of derivatives can be avoided. Also, a closed form solution for the discrete-time linear equation is achieved by following the second method. 2.3. Continuous and discrete time linearized error dynamics. The attitude computer must integrate the continuous time nonlinear differential equation given in (1). The system error dynamics can be obtained by linearizing this equation about a nominal path. To obtain an extended Kalman filter, which can estimate both orientation errors and gyro biases, Equation (3) is used to write the augmented linearized system as: 3 2 tan h(cos W v x sin W v ) (1+ tan2 h)(sinW v + cos W v ) 0 1 y z y z _ dW 0 0 0 x(sin W vy +cos W vz ) 6 dh_ 7 6 7 6 6 1 sin h 6 dY _ 7 6 6 (sin W v x sin W v ) + cos W v ) 0 0 (cos W v y z y z 7 6 cos h cos2 h 6 dv_ x 7=6 7 6 6 0 0 0 0 6 4 dv_ y 5 4 0 0 0 0 dv_ z 0 0 0 0 2

3 sin W tan h cos W tan h 2 dW 3 cos W x sin W 7 76 dh 7 7 sin W cos W 76 dY 7 76 7 76 cos h cos h dvx 7 76 7 0 0 76 4 dvy 5 5 0 0 dv z 0 0

(7) Regarding the state vector x in (5), the above equation can be written in the following compact form : x_ (t)=A(t) x(t)

(8)

where A(t) refers to the first matrix in the right hand side of the equality sign in (7). Discretizing Equation (8) yields the discrete-time error dynamics as follows : x(t+Dt)=F(t) x(t)

(9)

F(t)=eA(t) Dt

(10)

where

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One method to compute F, is to approximate the exponential term by its Taylor series expansion : F(t)=I+A(t)Dt+12A2 (t)Dt2 +




(11)

Comparison of this equation with the Taylor series approximation used in (Foxlin, 1996), reveals the benefits of this approach especially when we need to retain higher order terms to increase accuracy. We should do just matrix multiplication which is much easier than calculating high order time derivatives of Equation (1). Another advantage of this method is that a closed form solution for F can be obtained from : F(t)=Lx1 { [sIxA(s)]x1 }

(12)

So, changing the order of discretization and linearization operations allows for a straightforward and much simpler formulation of the problem in order to achieve more accurate results. The linear error propagation model of (9) provides the basis for an extended Kalman filter to estimate the error states. The model has been manipulated into a form in which non-constant drifts of gyros are derived by some process noise. xk+1 =Fk xk +vk

(13)

yk =Hk xk +wk

(14)

where F is given in (10) and Hk=[I3r3 03r3], v and w are process and measurement noise with covariance matrices Q and R respectively. 2.4. Separate-bias Kalman filter. The Kalman filter is a recursive algorithm for computing an estimate of state, xˆ, which is optimal in the sense of least square error under certain circumstances (Brown and Hwang, 1997). The Kalman filter recursive equations are summarized in Figure 3. The method of augmenting the state vector with bias terms is reasonably effective when the number of bias terms is small relative to the state variables of the original problem and the dimension of the state vector is not significantly increased. When the number of bias terms is comparable to the number of state variables of the original problem, the computation required by the filtering algorithm may become excessive and numerical inaccuracies are introduced by computations with large vectors and matrices (Friedland, 1969). In 1969, Friedland introduced a two-stage Kalman filter algorithm applicable to the case in which the state vector is composed of a set of dynamic states and a set of constant but unknown bias states. By this separate-bias Kalman algorithm, a single calculation involving large matrices is reduced to a sequence of calculations involving smaller matrices to avoid the mentioned problems. Separate-bias Kalman estimators have received considerable attention since 1969 because of their usefulness in many practical problems. Alternative derivations and realizations of the estimator have been presented by different authors to deal with cases in which the bias is stochastic and the bias state noise is correlated with that of the dynamic state (Ignagni, 1981, 1990, 2000; Alouani, 1993 ; Hsieh, 1999). Switching to Friedland’s notation, an augmented state vector zk=[xTk bTk ]T is defined which is composed of an error state vector xk=dH(tk) and a bias state vector bk=dV(tk)

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ATTITUDE ESTIMATION SYSTEMS Enter prior estimate xˆ 0− & its error covariance P0−

Compute Kalman gain: K k = Pk− H kT ( H k Pk− H kT + Rk ) −1

y 0 , y1 , ...

Update estimate with measurement y : k

Project ahead:

xˆ k−+1 = Fk xˆ k

xˆ k = xˆ k− + K k ( y k − H k xˆ k− )

Pk−+1 = Fk Pk FkT + Q k

Compute error covariance for updated estimate: Pk = ( I − K k H k ) Pk−

xˆ 0 , xˆ1 , ...

Figure 3. Kalman filter loop (Brown and Hwang, 1997).

where tk is the time at the kth iteration of the filtering algorithm. The attitude estimation problem is defined by the discretized equation set : xk =Ak xkx1 +Bk bkx1 +jxk bk =bkx1 +jbk yk =xk +gk where the matrix Fk in (13) is composed of
A Fk = k 0

(15)

Bk I

 (16)

The following partitioned form specializes Kalman estimator to the problem at hand :


 Px (k) Pxb (k) Kx (k) Qx 0 K(k)= P(k)= T (17) Q= Pxb (k) Pb (k) Kb (k) 0 Qb Diagonal matrices Qx, Qb and R are respectively the covariance matrices of noise vectors jx, jb, and g. Utilizing the separate-bias estimator, mathematical operations on 6-by-6 matrices are replaced by a somewhat greater number of operations on 3-by3 matrices. This reduces the number of flops per iteration of the filtering algorithm from 2648 to 1425, which is more suitable for real-time implementation. It should be stressed that even though the extended Kalman filter often can be used successfully for nonlinear state estimation, there are no general convergence or stability guarantees. Also, its convergence strongly depends on choosing Q and R covariance matrices. Diagonal elements of Qx and R, which are noise variances, can be estimated using wavelet decomposition of the related signals. Since in the wavelet decomposition of a signal, the detail coefficients at the finest resolution contain the highest frequency band within that signal, they include highest number of noise coefficients. So, the standard deviation of the noise could be computed from the

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Angle (˚)

Yaw (˚) Pitch(˚) Roll(˚)

268

(a)

(b)

Figure 4. Results of 10 minute drift integration tests for (a) integrating each channel separately, (b) integrating all three channels jointly.

median of the magnitudes of all of these coefficients (Donoho and Johnstone, 1994) : ^ =Median(jDj) s

(18)

where D contains the detail coefficients at the finest decomposition resolution. This approach exhibits a systematic method for estimating noise covariance matrices and avoids the troubles caused by adjusting the covariance values by trial and error. Regarding the fact that the noise characteristics of inertial sensors depend on the motion they experience, this method was expected to improve the accuracy of the estimates; this is shown in the next section. 3. E X P E R I M E N T A L R E S U L T S. An experimental setup was developed to study the algorithm in the real world. The prototype consisted of a platform with three single-axis orthogonal angular rate gyros together with two inclinometers for roll and pitch drift compensation and a magnetic compass for yaw drift compensation. Low-cost low-accuracy RG18A rate gyros and PENDL2A inclinometers were employed in this prototype (Setoodeh, 2002). The measurement errors associated with inertial sensors are dependent on the physical operational principle of the sensor itself. Temperature and memory effect play a significant role in the performance of low cost inertial sensors and, in mechanical sensors, noise may have peaks at particular frequencies depending on the sensor construction. For these reasons, not all the values for the sensor terms mentioned in the datasheet are valid and testing is required on the application at hand. To extract the statistical characteristics of gyro drifts, which are needed for Kalman filter implementation, a tombstone test was performed by keeping the sensor assembly still and recording the outputs periodically at a 10 Hz sampling rate for a specified time interval. _ =vx, h_ =vy and When the initial attitude is zero, Equation (1) reduces to : W _ Y=vz. Using similar gyros, it is expected that drift rates be comparable in three Euler angles but it is possible that gyros of the same model have slightly different performances. Regarding the non-symmetric structure of Equation (1) with respect to Euler angles, this leads to difference in drift accumulation in different Euler angles. The result of drift recordings, plotted for a 10-minute period, is shown in Figure 4.

ATTITUDE ESTIMATION SYSTEMS

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Roll angle (˚)

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Figure 5. Test run with extended Kalman filter for roll angle estimation.

Figure 4(a) depicts the output of gyros, which were integrated separately for each channel. Recording all three axes simultaneously and integrating Equation (1) results in Figure 4(b). In spite of the steady monotonic drifts of Figure 4(a), which are based on the assumption of three independent random walks, the joint drift experiment of Figure 4(b) exhibits some peculiar correlations. Q and R covariance matrices were selected based on experimental tests and the filter remains stable indefinitely. After solving the convergence problem, the estimator performance was evaluated by collecting several datasets. In the first dataset during the test period of approximately 100 seconds, the platform was repeatedly turned through x15x about the roll axis, left to rest, then returned to rest in its horizontal orientation. The roll Euler angle is plotted against time in Figure 5, which demonstrates the problem with unaided inertial integration – accumulated error by the end of the run was about 50x. As shown in the figure a certain amount of error accumulates each time the sensor is fixed. In the second dataset during the test period of approximately 110 seconds, the platform was turned through x40x about the pitch axis, returned to rest in its horizontal orientation, turned through +30x about the pitch axis and then returned to rest in its horizontal orientation. The pitch Euler angle is plotted against time in Figure 6(a), which again demonstrates the problem with unaided inertial integration, accumulated error by the end of the run was about 50x. In x40x and +30x orientation, the platform was vibrated with a low amplitude high frequency motion to evaluate filter performance from another point of view. Results are shown in Figure 6(b), which are close-ups of the relevant portions of Figure 6(a). Finally, in order to study how sudden impacts could affect the performance, the setup was allowed to experience such impacts in the first portion of the second dataset. The difference between Kalman filter estimates and the aiding sensor measurements is

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Lower plots are contained within these regions.

Pitch angle (˚)

Pitch angle (˚)

(a)

Pitch angle (˚)

(b)

(c) Figure 6. Test run with extended Kalman filter for pitch angle estimation: (a) orientation estimation, (b) close-ups relevant to fast motion, (c) difference between estimate and aiding sensor measurement.

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Orientation (˚)

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Orientation (˚)

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Orientation (˚)

Orientation (˚)

(a)

(b) Figure 7. Orientation covariances for (a) roll angle estimation and, (b) pitch angle estimation. Left-hand plots are related to estimates using constant Q are R matrices and right-hand plots are related to state estimates using Q are R matrices obtained from the detail coefficients at the finest decomposition resolution using Daubechies 2 wavelets.

plotted against time in Figure 6(c). The difference increases whenever the setup moves fast, which shows the drawbacks of an inclinometer in following fast motions. As is obvious from the test results, the filter incorporates the drift-free but noisy measurements of inclinometers and effectively compensates the drift of the inertial system without compromising the rapid dynamic response of inertial tracking technique. The implemented filter is reasonably fast and is suitable for real-time applications. Since the covariance captures the uncertainty, it can be used as a measure of the accuracy of the estimation. To highlight the improvements caused by using the wavelet-based covariance estimation method, covariance of the orientation state is plotted against time for the two collected datasets in Figure 7. Left-hand plots are related to estimates using constant Q are R matrices obtained based on the tombstone test. Right-hand plots are typical plots related to state estimates using Q are R matrices obtained from the detail coefficients at the finest decomposition resolution using Daubechies 2 wavelets on a moving window of 100 samples. Comparing left-hand and right-hand plots reveals that accuracy can be increased by the proposed method because the smaller covariances express the less amount of uncertainty or in other words, more information is contained in the estimates.

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4. C O N C L U S I O N. This paper presented an attitude estimation algorithm based on an extended Kalman filter to fuse measurements of three strapdown rate gyros and some drift-free aiding sensors to achieve the excellent dynamic response of an inertial system without drift and without acceleration sensitivity. Sensor fusion was used to overcome the drawbacks of available sensors and extend their limits. We focused on obtaining a good attitude estimate without dynamic modelling. Dynamic modelling was replaced by gyro modelling. Separate-bias formulation of the Kalman filter was used to reduce the computational burden. The implementation is more suitable for real-time applications. By employing a waveletbased approach to estimate the required covariances in the Kalman filter formulation, trial and error adjustments were avoided and no convergence problems were encountered. A setup was implemented and the algorithm was tested experimentally with satisfactory results.

ACKNOWLEDGEMENT This work was partially supported by Engineering Research Centre of Fars (Shiraz, IRAN), which the authors appreciate. The authors would also like to thank Dr Eric Foxlin, Interscience Inc. (MA, USA) and Dr Stergios Roumeliotis, Department of Computer and Information Sciences, University of Minnesota (USA).

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