MoRPI-PINN: Inertial Navigation for Mobile Robots

• The paper presents a physics-informed neural network that embeds reduced 2D INS dynamics to mitigate drift in pure-inertial dead-reckoning.
• It combines supervised trajectory fitting, initial-condition constraints, and physics-based ODE losses to enforce consistency with inertial dynamics.
• Real-world tests on an RC car show sub-meter accuracy with a 94% improvement over 2D-INS and an average trajectory error of 0.8 m, enabling edge deployment.

Searching arXiv for the specified paper and closely related entries to ground the article. MoRPI-PINN is a physics-informed neural network framework for accurate inertial-based mobile robot navigation in scenarios where satellite navigation or cameras are unavailable. It addresses pure-inertial dead-reckoning for a wheeled mobile robot by embedding reduced 2D strap-down INS dynamics, initial conditions, and supervised trajectory information into a single training objective. In the reported real-world experiments, the framework is trained on a snake-like slithering motion and evaluated on four unseen trajectories, where it achieves an average absolute trajectory error of $0.8\ \mathrm{m}$, corresponding to a $94\%$ improvement over 2D-INS and an $85\%$ improvement over MoRPINet; it is also described as lightweight enough for edge deployment (Sahoo et al., 24 Jul 2025).

1. Problem setting and reduction to 2D inertial navigation

The target problem is pure-inertial “dead-reckoning” for a wheeled mobile robot using only IMU data from an accelerometer and gyroscope. In continuous time, the full 3D strap-down INS equations in a local North-East-Down frame are reduced, under the flat-earth, negligible roll/pitch, low-cost-sensor, and short-duration assumptions, to a 2D system consisting of planar position, planar velocity, and yaw dynamics (Sahoo et al., 24 Jul 2025).

The reduced equations are

$\dot{\mathbf p}^n = \mathbf v^n,$

$$\dot{\mathbf v}^n = \mathbf C_b^n\,\mathbf f_{ib}^b + \mathbf g^n,$$

$\dot\psi = \omega_z,$

with $\mathbf p^n=(x,y)^T$, $\mathbf v^n=(v_x,v_y)^T$, $\mathbf f_{ib}^b=(f_x,f_y)^T$, and rotation matrix

$$\mathbf C_b^n(\psi)= \begin{bmatrix} \cos\psi & -\sin\psi\ \sin\psi & \cos\psi \end{bmatrix}.$$

Within this formulation, $94\%$0 is the gyro’s about-vertical rate and $94\%$1 is the constant horizontal gravity vector, often $94\%$2. The reduction is operationally important because the framework is not posed as a full 6-DoF inertial navigation system; it is a planar navigation model specialized to mobile robots with negligible roll and pitch.

The motivation for this reduction is the failure mode of direct inertial integration with low-cost sensors. The reported sensor characteristics include accelerometer noise density of approximately $94\%$3 and bias of approximately $94\%$4, together with gyroscope noise density of approximately $94\%$5 and bias of approximately $94\%$6. Under these conditions, a purely model-based 2D-INS yields errors on the order of tens of meters within minutes, with average ATE approximately $94\%$7 in the reported tests. This framing makes clear that MoRPI-PINN is designed as a drift-mitigation method for low-cost inertial sensing rather than as a general-purpose replacement for high-grade INS (Sahoo et al., 24 Jul 2025).

2. Physics-informed formulation

MoRPI-PINN embeds the 2D INS dynamics as soft constraints in the loss function. The network takes inputs $94\%$8 and outputs $94\%$9. Its objective combines supervised trajectory fitting, enforcement of the known start point, and consistency with the reduced inertial ODEs through automatic differentiation (Sahoo et al., 24 Jul 2025).

The total loss is given in the source as

$85\%$0

This decomposition corresponds to three terms.

Data loss enforces agreement between predicted and ground-truth position and velocity, where $85\%$1 and $85\%$2 come from RTK-GNSS-derived supervision.

Initial-condition loss constrains the first few outputs to match the known start point $85\%$3. The reported implementation uses $85\%$4.

Physics-informed loss enforces the residuals of the kinematic, velocity, and yaw equations at collocation points using automatic differentiation. The reported implementation uses $85\%$5.

The role of the weighting coefficients $85\%$6, $85\%$7, and $85\%$8 is to balance fidelity and physical consistency. A plausible implication is that the framework should be interpreted as a hybrid estimator: it is neither a purely supervised trajectory regressor nor a purely model-based INS, but a PINN in which the inertial equations act as regularizing structure.

3. Network architecture and training pipeline

The architecture is a fully-connected feed-forward DNN with 10 hidden layers, each of width 128, with layer normalization after each layer and a nonlinear activation named “SinTanh,” defined as

$85\%$9

The input is $\dot{\mathbf p}^n = \mathbf v^n,$0, and the output is $\dot{\mathbf p}^n = \mathbf v^n,$1. The final network is described as only approximately $\dot{\mathbf p}^n = \mathbf v^n,$2 layers $\dot{\mathbf p}^n = \mathbf v^n,$3 $\dot{\mathbf p}^n = \mathbf v^n,$4 neurons and using only basic tensor operations (Sahoo et al., 24 Jul 2025).

The training data come from a single $\dot{\mathbf p}^n = \mathbf v^n,$5 run in which the robot is driven in a “snake-like” slithering pattern. RTK GNSS at $\dot{\mathbf p}^n = \mathbf v^n,$6 with $\dot{\mathbf p}^n = \mathbf v^n,$7 accuracy provides ground-truth $\dot{\mathbf p}^n = \mathbf v^n,$8, and $\dot{\mathbf p}^n = \mathbf v^n,$9 are numerically differentiated from RTK data. The IMU is a Movella DOT operating at $$\dot{\mathbf v}^n = \mathbf C_b^n\,\mathbf f_{ib}^b + \mathbf g^n,$$0 and supplying $$\dot{\mathbf v}^n = \mathbf C_b^n\,\mathbf f_{ib}^b + \mathbf g^n,$$1. Overlapping windows of 10 IMU samples with stride 12 preserve temporal context, and 200 repeated start-point IMU readings generate the initial-condition samples.

Optimization uses Adam with learning rate $$\dot{\mathbf v}^n = \mathbf C_b^n\,\mathbf f_{ib}^b + \mathbf g^n,$$2, weight decay $$\dot{\mathbf v}^n = \mathbf C_b^n\,\mathbf f_{ib}^b + \mathbf g^n,$$3, batch size 32, and dropout 0.2. A learning-rate scheduler halves the learning rate on a 50-epoch plateau, and early stopping with patience is used. Training is reported on an NVIDIA RTX 4090 with $$\dot{\mathbf v}^n = \mathbf C_b^n\,\mathbf f_{ib}^b + \mathbf g^n,$$4 for 1,000 epochs.

Several design choices are explicitly linked to performance. The source attributes improved ODE consistency to SinTanh activations and layer normalization, and attributes part of the overall effectiveness to the periodic excitation induced by snake motion, which raises the IMU signal-to-noise ratio and makes system identification easier. This suggests that the framework depends not only on the PINN formulation but also on the excitation profile used during data collection.

4. Experimental platform and quantitative performance

The experimental platform is an RC car, specifically the STORM Electric 4WD Climbing Car, with dimensions $$\dot{\mathbf v}^n = \mathbf C_b^n\,\mathbf f_{ib}^b + \mathbf g^n,$$5, wheelbase $$\dot{\mathbf v}^n = \mathbf C_b^n\,\mathbf f_{ib}^b + \mathbf g^n,$$6, and tire diameter $$\dot{\mathbf v}^n = \mathbf C_b^n\,\mathbf f_{ib}^b + \mathbf g^n,$$7. The IMU is a Movella DOT, and ground truth is provided by a Javad SIGMA-3N RTK GNSS at $$\dot{\mathbf v}^n = \mathbf C_b^n\,\mathbf f_{ib}^b + \mathbf g^n,$$8 with approximately $$\dot{\mathbf v}^n = \mathbf C_b^n\,\mathbf f_{ib}^b + \mathbf g^n,$$9 accuracy. Evaluation is conducted on four unseen trajectories of $\dot\psi = \omega_z,$0–$\dot\psi = \omega_z,$1 each, totaling approximately $\dot\psi = \omega_z,$2 (Sahoo et al., 24 Jul 2025).

The reported baselines are 2D-INS with no learning and MoRPINet, described as “1D-CNN + Madgwick + dead-reckoning.” The average ATE values are $\dot\psi = \omega_z,$3 for 2D-INS, $\dot\psi = \omega_z,$4 for MoRPINet, and $\dot\psi = \omega_z,$5 for MoRPI-PINN.

Method	T1–T4 ATE (m)	Avg ATE (m)
2D-INS	11.2, 7.1, 11.3, 27.9	14.3
MoRPINet	5.8, 5.1, 5.4, 6.6	5.7
MoRPI-PINN	1.1, 0.9, 0.9, 0.4	0.8

These results are summarized in the source as a $\dot\psi = \omega_z,$6 improvement over 2D-INS and an $\dot\psi = \omega_z,$7 improvement over MoRPINet. Velocity quality is also reported: MoRPI-PINN attains velocity NVRMSEs in the range $\dot\psi = \omega_z,$8–$\dot\psi = \omega_z,$9, whereas plain 2D-INS exceeds $\mathbf p^n=(x,y)^T$0; MoRPINet does not estimate velocity. Figures 6–9 in the source are identified as trajectory overlays and ATE time-series.

The significance of these numbers lies in the contrast between sub-meter dead-reckoning and the unbounded drift of pure inertial integration under low-cost sensing. Within the confines of the reported experimental design, MoRPI-PINN functions as a learned correction mechanism that remains tethered to inertial physics.

5. Computational profile and deployment claims

The implementation is described as lightweight and suitable for edge devices. The source states that the final network uses only basic tensor operations and is suitable for ARM-based MCUs or mobile GPUs. Inference latency on small benchmarks is reported as less than $\mathbf p^n=(x,y)^T$1 per time step, enabling $\mathbf p^n=(x,y)^T$2 real-time operation (Sahoo et al., 24 Jul 2025).

These claims are narrower than a blanket statement of universal deployability. The source also notes that further compression or pruning may be needed for deployment on severely resource-limited microcontrollers. A plausible implication is that the framework occupies an intermediate deployment regime: lightweight relative to larger deep models, but still potentially requiring model reduction for the most constrained platforms.

The edge-device discussion is closely tied to the architecture choice. A fully connected model of approximately $\mathbf p^n=(x,y)^T$3 layers by $\mathbf p^n=(x,y)^T$4 neurons is materially simpler than multimodal or heavy convolutional designs, and the absence of camera processing is consistent with the intended GNSS-denied, vision-denied operating scenario. In that sense, the deployment argument is integral to the problem formulation rather than an incidental implementation remark.

6. Interpretation, limitations, and nomenclatural ambiguity

The source identifies several reasons for the reported performance gains. The physics loss anchors predictions to the known 2D-INS ODEs and is said to prevent unbounded drift; the initial-condition term stabilizes early-time errors; SinTanh activations and layer normalization improve network smoothness and thus aid ODE consistency; and periodic snake-motion excitation raises the IMU signal-to-noise ratio, making system identification easier (Sahoo et al., 24 Jul 2025).

The same source also states several limitations. In extremely data-rich, non-periodic motions, strict physics constraints may underfit abrupt patterns, suggesting a need for adaptive $\mathbf p^n=(x,y)^T$5 scheduling. The current formulation is 2D, so extending it to full 3D, including roll and pitch, will require richer sensor and terrain models. Online adaptation of IMU biases and handling of non-Gaussian noise remain open. These caveats are important because they delimit the framework’s present scope: it is a planar, offline-trained PINN for low-cost inertial navigation under specific excitation conditions, not a fully general inertial navigation solution.

A common misconception is to treat the method as removing the need for informative motion. The reported workflow instead depends on a “snake-like” slithering pattern during training, and the source explicitly credits this periodic excitation as part of the framework’s effectiveness. Another common misconception is to interpret “MoRPI-PINN” as an unambiguous acronym. The same acronym is also used by “A Multimodal Physics-Informed Neural Network Approach for Mean Radiant Temperature Modeling” (Shaeri et al., 11 Mar 2025), where it denotes a multimodal PINN for $\mathbf p^n=(x,y)^T$6 estimation rather than mobile robot navigation. In arXiv usage, therefore, the term is context-dependent.

Taken together, these points position MoRPI-PINN, in the mobile-robot sense, as a specific synthesis of model-based INS and data-driven learning for GNSS-denied dead-reckoning. Its reported contribution is not merely improved regression accuracy but the use of physical residuals to constrain a navigation network so that low-cost inertial sensing becomes substantially more usable over short mobile-robot trajectories (Sahoo et al., 24 Jul 2025).