⚡ electrical engineering

Delta-Position Estimation-Based IMU Odometry: A Comparison of MLP and Kolmogorov-Arnold Networks

This paper demonstrates that a Kolmogorov-Arnold Network (KAN) with learnable B-spline activations significantly outperforms a standard Multi-Layer Perceptron (MLP) in IMU odometry by achieving 44% lower cumulative drift with 6.9 times fewer parameters when trained to estimate incremental displacement rather than absolute position.

Original authors: Osman Tokluoglu, Emin Keresteci

Published 2026-06-25

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CC BY 4.0

Original authors: Osman Tokluoglu, Emin Keresteci

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Problem: The "Drunk Walk" of Robots

Imagine a robot flying inside a room without GPS (like a drone in a warehouse). To know where it is, it relies on an IMU (Inertial Measurement Unit), which is basically a high-tech accelerometer and gyroscope. It's like a blindfolded person trying to walk in a straight line by only feeling their own steps.

The problem? These sensors are imperfect. They have tiny "jitters" and biases. If you just add up these tiny errors over time, the robot thinks it's moving in a straight line, but it's actually drifting off course. In math terms, this error grows quadratically—meaning the longer the robot flies, the more wildly it gets lost. It's like a "drunk walk" that gets worse the longer you walk.

The Old Way vs. The New Idea

Traditionally, engineers tried to fix this by fusing the IMU with cameras or lasers (like giving the blindfolded person a flashlight). But this paper asks: Can we teach a computer to be a better "step-counter" just by looking at the IMU data itself?

Instead of asking the computer, "Where are you right now?" (which is hard and leads to big mistakes), the researchers asked, "How far did you move in the last 50 milliseconds?"

The Strategy: Break the flight into tiny 50-millisecond chunks. Predict the tiny movement for each chunk. Then, stitch all those tiny movements together to build the full path.

The Contest: The Heavyweight vs. The Acrobat

The paper compares two different types of "brain" (neural networks) to see which one is better at this step-counting task:

The MLP (Multi-Layer Perceptron): Think of this as a heavyweight bodybuilder. It's a standard, very common type of AI. It has a lot of muscles (parameters)—about 57,859 of them. It's powerful but bulky.
The KAN (Kolmogorov–Arnold Network): Think of this as a flexible acrobat. It's a newer, experimental type of AI. Instead of using fixed "rules" for how it processes data, it uses learnable B-splines.
- The Analogy: Imagine the MLP is a rigid ladder with fixed rungs. The KAN is a rubber band or a slingshot that can stretch and bend to fit the shape of the data perfectly. It has far fewer muscles—only 8,444 parameters (about 7 times smaller than the bodybuilder).

The Results: Who Won?

The researchers let both "brains" try to predict the drone's path on a test flight.

Short-Term: In the very first split second, both brains were almost equally good at guessing the tiny movement. They were neck-and-neck.
Long-Term (The Real Test): This is where the rubber band (KAN) shined.
- The Bodybuilder (MLP) made tiny errors that added up. By the end of the flight, the robot was 17.23 meters off course. That's like missing the target by the length of a bus!
- The Acrobat (KAN) made tiny errors that seemed to cancel each other out better. By the end, it was only 9.61 meters off course.
- The Win: The KAN was 44% more accurate in the long run, despite having much less "brain power" (fewer parameters).

The Catch: Training Time

There is one downside. While the KAN is smaller and more accurate, it was slower to train.

The Bodybuilder (MLP) learned its lesson in about 41 seconds.
The Acrobat (KAN) took about 460 seconds (over 7 minutes) to learn.
Analogy: The KAN is like a student who takes longer to study for the exam but ends up getting a better grade and remembering it longer.

The Bottom Line

The paper claims that using this new "flexible" network (KAN) is a promising way to help drones and robots navigate without GPS. Even though it takes longer to teach the robot, the result is a much more stable path that doesn't drift as wildly as the old, heavier methods. The "learnable rubber bands" inside the KAN seem to handle the messy, real-world physics of motion better than the rigid, standard networks.

Technical Summary: Delta-Position Estimation-Based IMU Odometry: A Comparison of MLP and Kolmogorov–Arnold Networks

Problem Statement
Accurate position estimation for autonomous aerial vehicles in GPS-denied environments remains a critical challenge. While Inertial Measurement Units (IMUs) provide high-frequency data, classical dead reckoning suffers from quadratic error growth due to the double integration of noisy measurements and biases. Traditional solutions rely on sensor fusion (e.g., Visual-Inertial Odometry) and model-based filtering (e.g., EKF, Factor Graphs), which often degrade when sensor models mismatch or kinematic assumptions fail. Conversely, learning-based approaches, such as those using Multi-Layer Perceptrons (MLPs) for local displacement regression, offer data-driven alternatives but may still suffer from systematic error accumulation over time. This study investigates whether Kolmogorov–Arnold Networks (KANs), a newer architecture utilizing learnable activation functions, can outperform standard MLPs in inertial odometry by reducing long-term drift while maintaining a compact model size.

Methodology
The study utilizes the EuRoC MAV benchmark dataset (specifically the V1_01_easy sequence), featuring 200 Hz IMU data and Vicon ground truth. The problem is formulated as delta-position estimation rather than absolute position regression to avoid inherent drift in the target formulation.

Data Processing: Raw IMU measurements (angular velocity and specific force) are organized into sliding windows of $W=10$ samples (50 ms). The target is the incremental displacement ( $\Delta p$ ) between the start and end of each window.
Architectures:
- MLP Baseline: A standard feedforward network ( $\mathbb{R}^{60} \to 256 \to 128 \to 64 \to \mathbb{R}^3$ ) with Batch Normalization, ReLU, and Dropout. It contains 57,859 trainable parameters.
- KAN: A network with learnable univariate B-spline functions on edges instead of fixed node activations. The architecture is defined as $[60, 7, 5, 3]$ with cubic B-splines ( $k=3$ ) and 5 grid points. It contains 8,444 trainable parameters (approximately 6.9 times fewer than the MLP).
Training: Both models were trained using the Adam optimizer with early stopping. Trajectories were reconstructed via numerical integration of predicted $\Delta p$ vectors.
Evaluation: Performance was measured using per-step displacement errors (dRMSE, dMAE) and cumulative drift metrics (Median $P_{50}$ , 90th percentile $P_{90}$ , and Final Drift) on a chronologically split test set.

Key Contributions

Empirical Comparison: The study provides a controlled comparison demonstrating that a KAN with significantly fewer parameters (8,444 vs. 57,859) achieves a 44% reduction in final cumulative drift compared to a standard MLP on the same delta-position estimation task.
Systematic Ablation: The authors conducted ablation studies regarding IMU sampling frequency and window size to identify optimal operating conditions for each architecture.
Interpretability Analysis: The paper includes an analysis of the learned KAN spline functions, revealing physically meaningful channel preferences and locality biases consistent with inertial sensor physics, suggesting the model learns interpretable representations of sensor dynamics.

Results

Short-Term Accuracy: Both models exhibited comparable short-term prediction accuracy, with a negligible difference in dRMSE (0.023 m for MLP vs. 0.022 m for KAN).
Long-Term Stability: The KAN model demonstrated superior stability in error accumulation.
- Final Drift: MLP resulted in 17.23 m, whereas KAN achieved 9.61 m.
- Percentile Drift: KAN showed lower median ( $P_{50}$ : 3.22 m vs. 5.18 m) and 90th percentile ( $P_{90}$ : 6.96 m vs. 10.06 m) cumulative drift values.
Computational Cost: The KAN model required significantly more training time (approx. 460 seconds) compared to the MLP (approx. 41 seconds), despite its smaller parameter count.

Significance and Claims
The paper claims that the learnable B-spline-based activation structure of KANs offers a structural advantage for inertial odometry by producing error components that partially compensate for one another during numerical integration, thereby reducing systematic drift. The authors posit that KANs are a promising alternative for compact motion-estimation applications where long-term stability is prioritized over training speed. However, the study modestly concludes that while the results are encouraging, the generalization performance of KANs across different window lengths, sampling frequencies, and flight sequences requires further investigation in future work. The findings suggest that the KAN architecture's ability to model complex nonlinearities with fewer parameters makes it a viable candidate for reducing error accumulation in learning-based inertial odometry.