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Inertia properties of rigid body such as ground, aerial, and space vehicles may be changed by several occasions, and this variation of the properties influences the control accuracy of the rigid body. For this reason, accurate inertia properties need to be obtained for precise control. An estimation process is required for both noisy gyro measurements and the time derivative of the gyro measurements. In this paper, an estimation method is proposed for having reliable estimates of inertia properties. First, the Euler equations of motion are reformulated to obtain a regressor matrix. Next, the extended Kalman filter is adopted to reduce the noise effects in gyro angular velocity measurements. Last, the inertia properties are estimated using linear least squares. To achieve reliable and accurate angular accelerations, a Savitzky-Golay filter based on an even number sampled data is utilized. Numerical examples are presented to demonstrate the performance of the proposed algorithm for the case of a space vehicle. The numerical simulation results show that the proposed algorithm provides accurate inertia property estimates in the presence of noisy measurements.

In rotational dynamics of rigid body, appropriate command torque for attitude control is necessary to achieve target orientations. Accordingly, a full component of inertia matrix which consists of moment of inertia (MOI) and product of inertia (POI) elements needs to be considered. There exist various methods to obtain inertia properties of objects: torsion pendulum method, usage of equipment, computer aided design software, and so forth. These methods, however, provide the inertia property information before the operation. For the operating object, inertia properties can be changed by several reasons: fuel consumption, fuel sloshing, connection with other parts, collision with unexpected object, and so forth. This unknown variation of inertia properties affects the performance of attitude control [

Palimaka and Burlton presented the mass property estimation method using the weighted least squares [

In this paper, a combined method is suggested for acquiring full inertia properties. The estimation process consists of the following three steps: noise reduction, calculation of angular acceleration, and inertia estimation. First, the noise in the measurements is filtered using the EKF which has proven to provide the best performance with respect to the noise reduction [

The rotational dynamics of a rigid body is described as [

Under the assumption that the inertia vector

The matrices

Using the LLS, the estimated inertia vector

As shown in (

The angular velocities obtained from the rate gyro sensors include noises caused by various sources, such as the other parts’ vibration and the characteristic of hardware [

Equation (

Noise reduction process using the EKF.

Model | |
---|---|

Initialization | |

| |

Kalman Gain | |

| |

Update State | |

| |

Propagation | |

In Table

Savitzky and Golay introduced the simplified digital filter, known as the SGF, for the purpose of calculating the smoothing and differentiation data by the LLS. The odd number of data points, which are also consecutive and uniformly spaced, is necessary for the SGF to operate appropriately. In [

Let the index of sampled data range from

Table

Convolution coefficients for quartic polynomial.

Number of sampled data | 4 | 6 | 8 | | 2 |
---|---|---|---|---|---|

| | ||||

| | | |||

−3 | −7 | | −7 | ||

−2 | −5 | −5 | | −5 | |

−1 | −3 | −3 | −3 | | −3 |

0 | −1 | −1 | −1 | | −1 |

1 | 1 | 1 | 1 | | 1 |

2 | 3 | 3 | 3 | | 3 |

3 | 5 | 5 | | 5 | |

4 | 7 | | 7 | ||

| | | |||

| | ||||

| |||||

Normalization | 10 | 35 | 84 | | |

The simulation parameters are listed in Table

Simulation parameters.

Final time (sec) | 600 |
---|---|

Covariance matrix | |

| |

Gyroscope noise level (deg/s) | 0.0104 |

| |

Command torque (Nm) | |

| |

Initial state (deg/s) | |

| |

True inertia (kg⋅m^{2}) | |

| |

Nominal inertia (kg⋅m^{2}) | |

| |

Sampled data size | 6 |

| |

Polynomial order | Quartic |

Inertia estimation process.

Model | |
---|---|

Initialization | |

| |

Kalman Gain | |

| |

Update State | |

| |

Propagation | |

| |

Calculate angular acceleration | |

| |

Construct | |

| |

Estimate inertia (every 40 sec) | |

Selection of the optimal sampled data size.

The filtered angular velocities and 3

Calculated angular acceleration error results.

Properties | RMS Error | Difference | |
---|---|---|---|

Backward difference | SGF | ||

| 0.0081 | 0.0058 | 28.28 |

| 0.0081 | 0.0058 | 28.58 |

| 0.0081 | 0.0057 | 29.13 |

Noise reduction results using the EKF.

Angular velocity errors and

Mean square error comparisons of the calculated angular acceleration.

As shown in Figure

Estimation results.

Inertia | True | Estimated | Error |
---|---|---|---|

| 14.2000 | 14.1823 | 0.12 |

| 17.3000 | 17.2876 | 0.07 |

| 20.3000 | 20.2859 | 0.07 |

| 0.0867 | 0.0864 | 0.39 |

| 0.1357 | 0.1350 | 0.48 |

| 0.6016 | 0.6009 | 0.12 |

Estimation results with respect to number of estimation processes.

In this paper, a combined methodology is proposed to estimate full inertia properties which are moment and product of inertia elements. The key idea of this research is to utilize the following three methods: the extended Kalman filter (EKF), the Savitzky-Golay filter (SGF), and the regressor matrix. First, the noise in measured angular velocities is reduced using the EKF. Next, the reliable angular acceleration is calculated using the SGF based on an even number of sampled data. Last, the suggested regressor matrix provides the good estimation result of full inertia properties using the linear least squares. The numerical simulation is performed for evaluating the estimation accuracy of the proposed approach. The result shows that the proposed method is able to achieve the improvement of estimation accuracy with respect to full inertia properties and track the true value of full inertia properties well.

The authors declare that there are no competing interests regarding the publication of this paper.