Efficient localization methods for a point source and rigid body
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Localization has been a very important and fundamental research topic in GPS, radar, sonar, and especially in mobile communications and sensor networks over the past few years. Localization of a signal source is often accomplished by using a number of sensors that measure the radiated signal from the source, here we consider the range based measurements, including time of arrival (TOA) and time difference of arrival (TDOA). In such study, the object is far away or only the position information is needed, and we refer this as point source localization. For some applications, e.g., robotics, spacecraft, and gaming, orientation information in addition to position is also needed. Although an inertial measurement unit (IMU) can perform such task once the initial state is available, it suffers from long-term performance deviation and requires accurate calibration using additional devices. Here we consider joint position and orientation estimation using the distance or AOA measurements between the fixed sensors on the object and the anchors at fixed locations, and it is called rigid body localization. Our research has two manifolds: First, for the point source localization, the original squared range least squares (SR-LS) admits global and computationally efficient solution using generalized trust region subproblems (GTRS) technique but with non-optimal accuracy, therefore we add proper range weighting (SR-WLS) into it and investigate the resulting performances of mean squared error (MSE) and bias. Its asymptotic efficiency is proven theoretically and validated by simulations. The effects of range weighting on the localization performance under different sensor number, noise correlation, and localization geometry are examined. We also conduct similar range weighting for squared range difference least squares (SRD-LS and SRD-WLS) under TDOA measurements. In addition, the weighting technique is extended to the scenario where the sensor positions are not exactly known. The resultant cost function has the same structure as that without sensor position errors, thereby existing algebraic or exact solutions to the squared measurements can still be used without requiring new optimization method. Second, for the rigid body localization, under distance measurements, the existing method cancels the quadratic term of the sensor position in the squared distance measurement equations, which may cause serious degradation. Our proposed estimators are non-iterative and have two steps: preliminary and refinement. The preliminary step provides a coarse estimate and the refinement step improves the first step estimate to yield an accurate solution. When the rigid body is stationary, we are able to locate the rigid body with accuracy higher than the solutions of comparable complexity found in the literature. When the rigid body is moving, we introduce additional Doppler shift measurements and develop an estimator that contains the additional unknowns of angular and translational velocities. Simulations show that the proposed estimators, in both stationary and moving cases, can approach the Cramer-Rao lower bound (CRLB) performance under Gaussian noise over the small error region. Under AOA measurements, we solve the 3D scenario that is seldom considered before, through estimating its distances to landmarks and contrasting the landmark positions in object local frame and the global frame. Furthermore, we extend it to the scenario where there is more than one AOA sensor on-board, which either increases the robustness and accuracy or decreases the minimum requirement on number of landmarks. And the methods for 2D and 3D are designed respectively. The simulations confirm the effectiveness of proposed methods.