Three constrained extended Kalman filters (CEKF) are developed by making use of condition equations which equations allows one to predict directly the residuals of all variables. The first one is a general CEKF algorithm in which it is supposed that all of the observation equations, system equations and constraints of a dynamic problem are non-linear functions. Although the constrained Kalman filter was already investigated by a few contributions, they assumed some restrictive conditions such as linearity of constraints and/or equations. Moreover, this generalization helps one to deal with problems which encounter with raw GPS data. In some problems, constrains can be expressed by a quadratic form. Hence, the second algorithm proposes a CEKF solution with quadratic constraints. In this algorithm, the constraints are not linearized. Eventually, in case of using refined GPS data in which quadratic constraints must be imposed to the state vector, the third algorithm is developed. In this algorithm one does not require to linearize any part of the dynamic model. Rigorous prediction of posterior dispersion (variance-covariance) matrix of the unknown parameters is the other attainment of this contribution. A numerical example demonstrates the efficiency of the proposed algorithm.

通过利用条件方程式开发了三个约束扩展卡尔曼滤波器（CEKF），该方程式使人们可以直接预测所有变量的残差。第一个是通用的CEKF算法，其中假定所有观察方程，系统方程和动态问题的约束都是非线性函数。尽管已经通过一些贡献对约束卡尔曼滤波器进行了研究，但是他们假设了一些限制性条件，例如约束和/或方程的线性。此外，这种概括有助于处理原始GPS数据遇到的问题。在某些问题中，约束可以用二次形式表示。因此，第二种算法提出了具有二次约束的CEKF解决方案。在这种算法中，约束没有线性化。最终，在使用必须对状态向量施加二次约束的精确GPS数据的情况下，开发了第三种算法。在这种算法中，不需要线性化动态模型的任何部分。对未知参数的后方离散度（方差-协方差）矩阵的严格预测是这一贡献的另一个成就。数值算例说明了该算法的有效性。