Mathematical Problems in Engineering

Volume 2016, Article ID 2962671, 7 pages

http://dx.doi.org/10.1155/2016/2962671

## Rigid Body Inertia Estimation Using Extended Kalman and Savitzky-Golay Filters

^{1}LG Electronics, Seoul 08592, Republic of Korea^{2}Korea Aerospace University, Goyang 10540, Republic of Korea

Received 24 December 2015; Accepted 24 March 2016

Academic Editor: Anton V. Doroshin

Copyright © 2016 Donghoon Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1, 2]. Particularly, the inertia properties are extremely important parameters for unmanned vehicles, which operate automatically in an unexpected environment. In short, accurate inertia properties are requisite for performing efficient attitude control.

Palimaka and Burlton presented the mass property estimation method using the weighted least squares [3]. Bergmann et al. developed the real time estimation method for asymmetrical satellites [4]. Kutlu et al. presented the estimation algorithm of inertia properties including center of mass using the extended Kalman filter (EKF) [5]. Zhao et al. suggested the estimation method using the discrete Kalman filter for the mated flight control of space vehicles [6]. Conti and Souza presented the estimation result of the inertia properties for the satellite attitude control simulator using the recursive least squares (RLS) [7]. Although these researches include inertia properties and center of masses, POI elements, relatively smaller values than MOI elements, need to be considered to avoid the degradation of control accuracy. Yang et al. introduced a regressor matrix including POI elements and suggested the full inertia estimation algorithm based on the RLS [8]. In the estimation process, the angular accelerations must be used and need to be calculated in general [9]. The angular accelerations are usually obtained by the difference method: forward, backward, and central difference. However, these methods are not valid any further when the noise levels in measurements and the sampling time are not small enough [10]. Kim et al. introduced a Savitzky-Golay filter (SGF) to obtain reliable angular accelerations and firstly applied to estimate MOI for spacecraft [10, 11]. The SGF is a simple smoothing and differentiation filter which can be applied to a set of consecutive, uniformly spaced, odd number sampled data [11]. For the application of MOI properties estimation, Kim et al. suggested the estimation algorithm based on the linear least squares (LLS) [12].

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 [1, 13]. Next, the accurate angular acceleration is obtained using the SGF for an even number of sampled data. Last, the full inertia properties are estimated using the LLS based on the suggested regressor matrix in [8]. The performance of the combined method is demonstrated using the design parameters for the one of the Korea Science Technology Satellite, STSAT-3, which has already been developed.

#### 2. Combined Method for Inertia Estimation

##### 2.1. Inertia Estimation Using Linear Least Squares

The rotational dynamics of a rigid body is described as [14]where is the angular velocity vector of the rigid body, is the command torque vector, and is the inertia matrix of the rigid body. The associated measurement equation is expressed aswhere is the measurement vector and is the measurement error vector with zero mean and covariance .

Under the assumption that the inertia vector is the constant during the integration interval, (1) is expressed aswhere

The matrices and in the matrix are defined as follows:

Using the LLS, the estimated inertia vector is obtained as

As shown in (6), the matrix is composed with the angular velocities and accelerations . Therefore, the accurate angular velocities and accelerations lead to the precise estimated inertia vector , and they are obtained using the EKF and the SGF, respectively.

##### 2.2. Noise Reduction Using Extended Kalman Filter

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 [13]. The EKF is well known as one of the best estimators for the state based on reducing the noise level of measurements [13]. The continuous-discrete EKF is selected to handle the nonlinearity inherent in (1).

Equation (1) is transformed into the 1st-order differential equation aswhere is the state vector and is the state noise vector with zero mean and covariance . The filtering process using the EKF is listed in Table 1 [15].