^{1}

A simple approach to gyro and accelerometer bias estimation is proposed. It does not involve Kalman filtering or similar formal techniques. Instead, it is based on physical intuition and exploits a duality between gimbaled and strapdown inertial systems. The estimation problem is decoupled into two separate stages. At the first stage, inertial system attitude errors are corrected by means of a feedback from an external aid. In the presence of uncompensated biases, the steady-state feedback rebalances those biases and can be used to estimate them. At the second stage, the desired bias estimates are expressed in a closed form in terms of the feedback signal. The estimator has only three tunable parameters and is easy to implement and use. The tests proved the feasibility of the proposed approach for the estimation of low-cost MEMS inertial sensor biases on a moving land vehicle.

In most inertial measurement devices, for example, attitude and heading reference systems (AHRS), the output performance suffers from gyro and accelerometer biases. Uncompensated portions of these biases result in unbounded accumulation of attitude errors. A classical approach to AHRS correction, dated back to the 1930s, is attitude error elimination by means of an external aid that provides vehicle acceleration data [

This approach is also directly applicable to a strapdown AHRS if a notion of a “virtual platform” is introduced [

Even though the correction procedure prevents the accumulation of attitude errors, it cannot completely zero them, as inertial sensor biases remain uncompensated.

In modern studies, Kalman filtering [

Though very popular, Kalman filtering has some inherent drawbacks. First, for any time-varying or nonlinear system (an AHRS being among them) it requires online computation of the error covariance matrix, which squares the number of necessary update operations at each time step. Second, the Jacobians of dynamics and measurement functions must be provided. Third, the optimality and even the convergence of the filter are not guaranteed for nonlinear systems [

As sensor biases are very slow varying quantities with respect to vehicle attitude parameters, a “separate-bias” estimation technique was proposed [

The dissatisfaction with the existing filtering methods leads to the growing interest in estimation approaches which make use of some specific properties of inertial navigation problems [

In this paper the bias estimation problems for gyros and accelerometers are treated in a unified manner. The key idea is that attitude error correction equations of a strapdown AHRS can be considered as the first stage of a “separate-bias” estimator, and the residual “torques” applied to the “virtual platform” can be directly used as measurements for an extremely simple inertial sensor bias estimator.

This treatment provides an intuitive view of the bias estimation problem. While gyro and accelerometer biases tend to increase the attitude errors, the applied “torques” try to rebalance them. Thus, in the steady-state operation, these “torques” are proportional to biases and can be naturally used to estimate them.

The only tunable parameters of the estimator are three time constants: one for the attitude correction, one for the gyro bias filter, and one for the accelerometer bias filter. Each of them has an obvious meaning and allows the AHRS designer to easily determine the dynamic response properties and to obtain well-predictable estimator behavior.

In the proposed approach, sensor bias observability conditions [

Consider a strapdown AHRS that computes the direction cosine matrix

Ideally, the “virtual platform” lies in the local level plane, and its axes coincide with the North-East-Down frame

In the presence of gyro and accelerometer biases,

Equation (

It is the last term

The first term in the right-hand side of (

Finally, (

For a GPS-aided AHRS installed on a land vehicle, some simplifications in (

Equation (

Equation (

When the solution is substituted into (

Equation (

Thus, the applied “torque” merely compensates the gyro bias and can be used to estimate it. Accelerometer biases are not observable in these conditions.

Second, suppose that the gyro bias has already been estimated and compensated, so that

Equations (

Bias estimator block diagram.

The main advantage of this approach over traditional Kalman filter-based methods is its simplicity: while the complete system model contains nonlinear dynamics (

Even though the estimation equations can be used directly, it is likely that the “torque”

The obtained estimates can be used in two different ways. The first way is to augment the attitude estimation part by velocity and position correction blocks and to construct bias estimate feedback loops. In this case, the estimator becomes a self-contained navigation filter with the same capabilities as the Kalman filter. The second way is to use the bias estimator as an independent “black box” whose estimates are subtracted from the input data of a primary navigation filter.

The proposed estimator was implemented by Topcon Positioning Systems, LLC, in the software of a GPS/GLONASS receiver equipped with a built-in low-cost MEMS inertial measurement unit (IMU). The aim was to estimate and compensate roll and pitch gyro biases of the order of 0.1 deg/s and accelerometer biases of the order of 0.2 m/s^{2}.

To obtain true acceleration data, the carrier-phase velocity measurements provided by the receiver were differentiated. The attitude correction time constant was set to 4 s, whereas time constants for both bias filters were set to 40 s.

The tests were conducted on a John Deere 5515 wheel tractor and a Caterpillar Challenger rubber tracked tractor with the receiver units installed on the cabin roofs. Test paths consisted of straight line segments as well as of turn arcs to satisfy the observability conditions for all estimated quantities (Figure

(a) Heading angle (John Deere wheel tractor). (b) Heading angle (Caterpillar tracked tractor).

Gyro bias estimates obtained in the tests are shown in Figure

(a) Gyro bias estimates (John Deere wheel tractor). (b) Gyro bias estimates (Caterpillar tracked tractor).

The gyro bias filter convergence time was about 120 s, which is equal to three time constants, as predicted by filtering theory. The steady-state estimation errors reached 0.01 deg/s (RMS) in the first test and 0.02 deg/s (RMS) in the second one. These results can be compared to the performance of a “symmetry-preserving filter” where bias estimate instabilities were of the order of 0.01 rad/s or 0.5 deg/s [

Accelerometer bias estimation is usually considered as a much more difficult problem because these biases are not observable in the straight motion of a vehicle [^{2}. The true bias values were unknown, as they were undistinguishable from the tractor’s actual attitude and IMU installation errors. Therefore, the estimation accuracy assessment was done indirectly. The precisely known artificial biases were added to the accelerometer measurements, and then the tests were repeated. In Figure

(a) Accelerometer bias estimates (John Deere wheel tractor). (b) Accelerometer bias estimates (Caterpillar tracked tractor).

The accelerometer bias filter convergence time was about 300 s, which was more than twice longer than for the gyro bias filter in spite of the fact that both filters had equal time constants. This reflected the weak observability of accelerometer biases. The steady-state estimation errors were 0.04 m/s^{2} (RMS) in both tests.

The tests also confirm that the estimation technique is not sensitive to IMU installation angles if they do not exceed 10–15 deg. Any installation error is implicitly treated as a redefinition of body frame axes and does not affect the estimation accuracy.

The proposed sensor bias estimator proved its feasibility for the estimation of MEMS gyro and accelerometer biases of a low-cost AHRS installed on a land vehicle. It can be widely used in the fields of precision agriculture, construction engineering, and so forth. The applicability of this approach to tactical grade inertial systems requires further investigation. In this case, the attitude correction gain should be significantly decreased, and the Earth’s shape and rotation should be appropriately taken into account. For navigation grade inertial systems the more general techniques, such as Kalman filtering, are probably still preferable.