In practice, the cases are not uncommon when rare jumps (of a step type) can occur in the errors of sensors (accelerometers or gyroscopes). Then, in particular, the system is not subject to calibration and must be brought to a standard condition. So, the presence of jumps is important for making a decision about the admittance of SDINS to the operation. Therefore, the problem of such failures detection is topical. Usually, the navigation algorithms (including calibration algorithms) are based on the Kalman filtering theory. Their distinctive feature is that they resolve quadratic optimization problems and therefore are very convenient from the computational point of view. Because of the jump variations of signals the estimates obtained by standard quadratic methods are not sufficiently accurate. Thus one needs to elaborate the estimation algorithms that are robust with respect to the jumps in the errors of sensors. The paper is devoted to a nonstandard method for processing the output signals of SDINS. We propose to use an 1l-norm approximation based approach (cf. (16)) that allows us to estimate efficiently the state of dynamic sys-tems even under anomalous variation of the errors (see figures 1-3).