## 4 Tracking
### Estimation
#### 4.2.2 EKF  
The first nonlinear filtering algorithm, EKF, was developed on the basis of the theory of Kalman filtering. In this method, the Taylor expansion of a nonlinear function is used to retain the first-order linearization, and other high order terms are omitted to approximate the nonlinear function [78].  
The details of the EKF algorithm are shown in Table 1, compared with a standard Kalman filter. The prediction and update of the state and covariance of both filters are listed. We can see the system to which the EKF is applied is nonlinear with the system models \(f(x(k),w(k))\) and \(h(x(k),w(k))\).  
Owing to the Taylor expansion, EKF can deal with the nonlinear characteristics of the system and has been used in many applications. In Ref. [10], an extended Kalman filter was used to estimate the speed of a motor based on the measured quantities, such as stator currents and direct current link voltage. The estimated speed was used for vector control and overall speed control. To obtain the stable and convergent solutions, a weighted global iteration procedure with an objective function is proposed for stable estimation and is incorporated into the extended Kalman filter algorithm [79].  
Regarding the navigation system, the EKF was designed for the online estimation of the speed and rotor position by only using measurements of the motor voltages and currents [80]. The EKF algorithms allow efficient computation compared with other nonlinear estimation methods. A quaternion based EKF was developed to determine the orientation of a rigid body from the outputs of IMUs [11].  
An EKF is commonly applicable to the system with weak nonlinearity, because of the Taylor expansion only to the first order. If the nonlinearity of the system is strong, there will be a large difference between the approximate linear system of the Taylor expansion and the original nonlinear system. This leads to a decrease of estimation performance or even the filter divergence.  
Because of its small amount of calculation, an EKF is apt for application in a practical system. [PERSON] et al. [81] presented an extended Kalman filter for real-time estimation of rigid body orientation using an IMU. Different than the work in Ref. [11], the Gauss-Newton iteration algorithm was utilized in Ref. [81] to find the best quaternion that relates the measured accelerations and Earth's magnetic field in the body coordinate frame to calculate values in the Earth's coordinate frame. The best  
Figure 6: Illustration of Kalman filter. From an initial value \(\hat{x}(0|0)\), two steps are used to obtain the final estimate: (**1**) Prediction step. The Kalman filter produces prediction estimates \(\hat{x}(k|k-1)\) along with their uncertainties from the last estimate \(\hat{x}(k-1|k-1)\), where \(A(k-1)\) is the process matrix and \(Q(k-1)\) is the covariance of process noise. (**2**) Update step. The estimates \(\hat{x}(k|k)\) are updated using a filter gain \(K(k)\) based on \(\hat{x}(k|k-1)\) and \(z(k)\), where \(C(k)\) is the measurement matrix and \(R(k)\) is the covariance of measured noise.  
quaternion is used as part of the measurements for the Kalman filter. As a result, the measurement equations of the Kalman filter become linear, and the computational requirements are significantly reduced, making it possible to estimate the orientation in real time. [PERSON] and [PERSON] [82] presented a method of tracking an elliptical shape approximation of an extended object based on a varying number of spatially distributed measurements. Based on an explicit nonlinear measurement equation about the kinematic and shape parameters, an EKF was derived for a closed-form recursive measurement update.