Taylor Series Iteration in Localization Algorithms

In localization algorithms, Taylor series iteration serves as a numerical approximation-based optimization method. Its core principle involves linearizing nonlinear problems using Taylor series expansion. This algorithm typically starts with initial values provided by the least squares method and progressively approaches the optimal solution through iterative steps. Fundamental Principles Initial Estimation: The least squares method provides preliminary position estimates, serving as the starting point for Taylor series expansion. In code implementation, this often involves solving a linear system using matrix operations like the pseudoinverse (pinv() in MATLAB). Linearization Process: First-order Taylor expansion is applied to nonlinear observation equations near the initial point, transforming the problem into a system of linear equations. Algorithm implementation requires calculating partial derivatives (Jacobian matrix) of observation equations with respect to position parameters. Iterative Correction: Through residual calculation and parameter updates, the positioning results are gradually refined until convergence meets predetermined accuracy. The iterative process typically involves while/for loops with convergence checks using norm-based thresholds. Advantages and Limitations Advantages: Fast convergence speed, suitable for localization problems with moderate nonlinearity. The algorithm efficiency can be optimized by implementing adaptive step size control. Limitations: Dependent on good initial estimates; may diverge with large initial value deviations. Code implementation should include divergence detection mechanisms using maximum iteration counters. This method finds extensive applications in GPS positioning and wireless sensor networks, effectively balancing computational complexity with positioning accuracy requirements. Practical implementation often involves hybrid approaches combining Taylor iteration with other optimization techniques for robustness.