The accumulation of error over time is an inherent limitation of Inertial Navigation Systems (INS), typically necessitating correction via external sensors. Underwater, Doppler Velocity Logs (DVL) provide velocity observations, while on land, odometers (OD) facilitate dead reckoning. Although these approaches appear distinct, they are unified under the mathematical framework of "kinematic constraint-aided filtering." This paper provides an in-depth comparison of the technical characteristics of INS/DVL and INS/OD navigation systems, focusing on their error models and Kalman filter fusion strategies, while revealing how environmental factors—specifically "ocean current disturbances" in underwater settings and "anomalous measurements" in land-based environments—have driven divergent algorithmic evolutions.
With its autonomous operation, an inertial navigation system (INS) plays a central role in GPS denied environments, such as underwater locations and tunnels. However, its navigation errors grow through integration. Let the velocity computed by the INS be vins and the true velocity be v. The velocity error δv =vins− v satisfies the following equation: δ= f×∅+∇
Here, f is the specific force, ∅ is the misalignment angle, and ∇ is the accelerometer bias. Introducing an external aiding source is the only way to suppress this error. Underwater, a DVL is commonly used to obtain the vehicle’s velocity relative to the ground; on land, an odometer is commonly used to obtain its forward velocity. The state equations of the two systems are largely identical, with the main differences lying in the formulation of their measurement equations and their noise characteristics.
Both underwater and land based integrated navigation systems commonly use an error state Kalman filter (ESKF). The system state vector x generally includes the velocity error δvn, attitude misalignment angle ∅, position error δp, and the biases b and scale factor errors s of the inertial sensors. The state equation takes the following unified form: = Fx + Gw
At the measurement level, both systems share the same objective: to construct the measurement equation z = Hx + v from the difference between the external sensor measurement zaux and the INS predicted value zins.
A DVL uses the acoustic Doppler effect to measure the vehicle’s three dimensional velocity in its own coordinate frame. After this velocity is transformed into the navigation frame (n frame) and compared with the INS velocity, the measurement equation is:
This measurement equation is straightforward. However, when the DVL is affected by ocean currents or loses beam lock, its measurement noise vdvl may be accompanied by severe outliers.
The motion of a land vehicle is subject to nonholonomic constraints (NHC), meaning that it has no lateral slip or vertical jumping motion. Although the odometer provides only the forward velocity vod, these constraints can be used to construct a three dimensional velocity observation. Let the odometer output in the vehicle body frame be =[0 , 0 ,vod]T. The measurement equation is then:
In addition, the odometer scale factor error sod is a key state variable that must be estimated in real time within the filter or calibrated offline.
Core Challenges and Diverging Algorithms Under Environmental Constraints
Underwater ocean currents introduce unknown biases into DVL measurements, so an interacting multiple model (IMM) approach uses soft switching to adapt to the environment. On land, tire slip causes abrupt changes in odometer measurements, so robust filtering adaptively inflates the noise covariance to reduce the impact of anomalies. These approaches use multiple models and robust estimation, respectively, to address persistent uncertainty and sudden disturbances, reflecting the fundamental constraints that the environment imposes on algorithm design.
Underwater Navigation Using IMM to Address Ocean Currents
In the presence of ocean currents, a condition described as “stationary on the seabed” appears as constant velocity motion in the navigation frame, causing conventional zero velocity updates (ZUPT) to fail. An IMM aided ZUPT method has been proposed to address this problem.
The IMM algorithm runs several models in parallel, such as an INS/DVL model and an INS/ZUPT model, and uses model probabilities for soft switching. When the vehicle’s velocity relative to the water is very small, the weight of the ZUPT model increases, effectively suppressing DVL errors. When appreciable ocean currents are present, the algorithm switches back to the INS/DVL model.
Land Navigation Using Robust Filtering to Address Wheel Slip
A critical limitation of odometers is the mismatch between wheel derived velocity and ground velocity during wheel slip. In this case, vod in the measurement equation contains large errors and no longer satisfies the Gaussian white noise assumption.
The response is to introduce robust estimation. By constructing an equivalent weighting function or reformulating the measurement noise covariance matrix R, the filter automatically inflates R when an abnormally large innovation k is detected, reducing the influence of the anomalous measurement on the filter estimate.
Conclusion
Underwater INS/DVL navigation and land based INS/odometer navigation are, in mathematical terms, both velocity observation fusion problems based on an error state Kalman filter. Their central difference lies in the sources of error: underwater, complex ocean currents invalidate the constraints; on land, nonideal road conditions cause abrupt changes in measurements.
This difference leads to distinct algorithmic strategies. Underwater navigation relies more heavily on multiple model soft switching (IMM) to adapt to unknown hydrodynamic conditions, whereas land navigation relies more heavily on robust estimation within a single model to resist sudden measurement anomalies. Understanding this shared mathematical structure and its different error sources is essential to designing highly reliable and robust integrated navigation systems.
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