LSTM-Kalman Filter-Based Multi-Sensor Signal Fusion for UAV Altitude Prediction in Non-Gaussian Environments

The reliable operation of UAVs in complex environments, such as executing obstacle avoidance in urban areas or stabilizing attitude during disaster response operations, places stringent demands on the accurate prediction of their altitude information. Precise altitude prediction is a critical factor in ensuring mission success and flight safety. However, achieving high-accuracy, high-robustness altitude prediction in practical applications faces numerous challenges, particularly at low altitudes or in signal-degraded environments. Traditional physics-based modeling approaches, such as Kalman filtering [1], provide a rigorous framework for state estimation but often exhibit performance degradation when confronted with non-Gaussian noise distributions induced by factors like low-altitude turbulence [2], or spatiotemporal mismatches caused by effects such as multipath [3]. These phenomena lead to systematic deviations between theoretical models and actual dynamic characteristics, potentially compromising altitude estimation accuracy during critical operations.

In recent years, deep learning techniques have offered new avenues for addressing complex problems in sensor data processing. For instance, spatiotemporal convolutional networks enhance nonlinear modeling capabilities through multi-level feature extraction [4], while improved LSTM architectures demonstrate significant advantages in temporal dependency modeling. Nevertheless, existing purely data-driven methods still exhibit limitations when applied to UAV altitude prediction. On one hand, the "black-box" nature of network structures may overlook sensor physical constraints, posing risks of violating kinematic principles [5]. On the other hand, end-to-end training paradigms can disrupt causal relationships within signal processing chains, exacerbating spectral distortions in dynamic environments [6]. These issues are particularly pronounced in UAV altitude prediction tasks, which demand simultaneous adherence to physical consistency constraints and environmental adaptability.

To leverage the complementary strengths of model-driven and data-driven approaches, researchers have proposed various hybrid architectures. Among these, differentiable Kalman filtering frameworks optimize noise parameters via gradient propagation [7], while physics-embedded neural networks encode sensor characteristics as network priors [5]. Although these hybrid methods have achieved certain progress, several key challenges remain in enhancing UAV altitude prediction performance, such as effectively handling non-Gaussian noise and sensor spatiotemporal mismatches in complex environments, and improving the model's adaptive capability while ensuring physical consistency.

Addressing the aforementioned challenges, this study focuses on enhancing the accuracy and robustness of UAV altitude prediction in non-Gaussian environments. It proposes a novel LSTM-Kalman collaborative optimization architecture designed to deeply integrate physical constraints with data-driven optimization. The main contribution of this paper lies in constructing a fusion framework capable of effectively addressing sensor spatiotemporal mismatches and complex noise interference, thereby achieving more precise and reliable altitude state estimation. Theoretical analysis and experimental validation demonstrate that the proposed architecture effectively balances physical interpretability and environmental adaptability, offering a new technical pathway for improving the reliability of UAV navigation systems, particularly the altitude prediction subsystem.

The remainder of this paper is organized as follows: Section 2 reviews advancements and challenges in hybrid fusion architectures. Section 3 elaborates on the design principles and algorithmic implementation of the proposed collaborative optimization framework. Section 4 validates the method’s effectiveness through multidimensional experiments. Section 5 concludes the study and outlines future research directions.

To address the challenge of inaccurate UAV altitude prediction under non-Gaussian noise, this paper proposes a novel architecture featuring deep collaboration between LSTM and a Kalman filter (the overall logic of which is illustrated in Figure 1). This method aligns multi-source sensor data using manifold geometry and leverages bidirectional information interaction between LSTM's data-driven analysis capabilities and the Kalman filter's physical model constraints. This achieves adaptive suppression of complex noise and precise modeling of system dynamics. Experimental results validate the effectiveness of this architecture in enhancing altitude prediction accuracy and robustness compared to traditional methods.

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Figure 1. Flowchart of the LSTM-Kalman collaborative architecture

3.1 Generalized manifold interpolation compensation mechanism

3.1.1 $\mathrm{SE}(3) \times \mathbb{R}^4$ composite manifold construction

To fuse multi-source, heterogeneous sensor information (e.g., IMU, GPS, barometer with different sampling rates and noise characteristics) within a unified geometric framework for UAV systems, this paper proposes constructing a Lie group-based composite manifold. This manifold is designed to integrate the rigid-body motion state of the UAV with its spatiotemporal coordinates. We define this composite manifold space $\mathcal{M}$ as:

$\mathcal{M}=S O(3) \propto \mathbb{R}^3 \times \mathbb{R}^4$ (1)

where, $\mathrm{SE}(3)=\mathrm{SO}(3) \ltimes \mathbb{R}^3$ is the special Euclidean group, representing rotation (SO(3)) and translation $\left(\mathbb{R}^3\right)$ in 3D space. The additional $\mathbb{R}^4$ space encodes altitude (h), latitude $(\varphi)$, longitude $(\lambda)$, and time $(t)$.

To define distances and perform geometric operations on this manifold, we introduce a Riemannian metric tensor g. Considering the physical meaning of the subspaces and computational feasibility, we design g with a block-diagonal structure:

$g=\left(\begin{array}{lll}\omega_1 I_3 & 0 & 0 \\ 0 & \omega_2 I_3 & 0 \\ 0 & 0 & \omega_3 \operatorname{diag}\left(1, \cos ^2 \varphi, 1,1\right)\end{array}\right)$      (2)

Here, $I_3$ is the $3 \times 3$ identity matrix. This structure assumes that the metric contributions of rotation, translation, and the other state components are decoupled. The $\mathbb{R}^4$ component includes the $\cos ^2 \varphi$ term to properly handle the geometric properties associated with the geographic coordinate system (latitude $\varphi$, longitude $\lambda$). The scalar weights $\omega_1, \omega_2, \omega_3>0$ are used to adjust the relative scales of these subspaces within the manifold geometry. The choice of scalar weights, rather than full matrix blocks, primarily aims to simplify computations and parameter optimization, facilitating embedded implementation, while also intuitively reflecting the relative importance assigned to each subspace.

3.1.2 Adaptive metric weight optimization via LSTM-Kalman

A static manifold metric cannot adapt to dynamically changing sensor noise and operating environments. To enhance robustness, this paper introduces an adaptive metric weight optimization method based on an LSTM-Kalman collaborative mechanism. The weights $\omega_i$ are not fixed values but are dynamically adjusted based on real-time data.

This mechanism combines the strengths of data-driven and model-driven approaches:

• LSTM Analyzes Historical Residuals: A Long Short-Term Memory (LSTM) network analyzes historical state estimation residuals $\Delta \mathrm{x}_{1: \mathrm{k}-1}$ to learn potential temporal correlations, nonlinearities, and non-Gaussian noise patterns present in the sensor data.
• Kalman Provides Real-time Uncertainty: The Kalman filter (or its variants), during its recursive process, provides quantitative information about the current state estimation uncertainty, typically embodied in its state covariance matrix $P_k$ (or innovation covariance $S_k$).

Information from these two sources is fused via a mapping function $f(\cdot)$ to update the metric weights $\omega_i$ online:

$\omega_i(k)=f\left(\operatorname{LSTM}\left(\Delta \mathrm{x}_{1: \mathrm{k}-1}\right), \operatorname{Kalman}\left(P_k\right)\right)$  (3)

where, Kalman $\left(P_k\right)$ represents a function extracting scalar information related to the uncertainty in each subspace (rotation, translation, etc.) from $P_k$ (the specific form depends on the implementation).

This adaptive weight adjustment allows the manifold metric g to dynamically reflect the current confidence in different sensor readings. For instance, if IMU noise increases (reflected in the corresponding blocks of $P_k$ and potentially detected by LSTM from residuals), the weights $\omega_1, \omega_2$ might decrease. This effectively "stretches" the distance associated with rotation and translation on the manifold, causing subsequent operations based on this metric (such as the manifold interpolation described in Section 3.1.3) to automatically down-weight the influence of this currently less reliable information. In this way, the adaptive metric mechanism ensures geometric consistency and robustness in state estimation under complex environmental perturbations (e.g., time-varying noise, intermittent sensor failures) by adjusting the manifold's own geometric structure to reflect data quality [13].

3.1.3 Geodesic interpolation for asynchronous sensors

To handle asynchronous data from multi-rate sensors, geodesic interpolation is performed on the $\mathrm{SE}(3) \times \mathbb{R}^4$ manifold, inherently preserving kinematic constraints. Utilizing the Lie group exponential (exp) and logarithm (log) maps, the state $\gamma(\mathrm{t})$ at intermediate time t between measurements $T_i$ and $T_{i+1}$ is estimated via:

$\gamma(\mathrm{t})=\mathrm{T}_{\mathrm{i}} \exp (\mathrm{k}(\mathrm{t}) \cdot \frac{\mathrm{t}-\mathrm{t}_{\mathrm{i}}}{\mathrm{t}_{\mathrm{i}+1}-\mathrm{t}_{\mathrm{i}}} \log \left(\mathrm{T}_{\mathrm{i}}^{-1} \mathrm{~T}_{\mathrm{i}+1})+\eta\right. \left.\cdot \operatorname{LSTM}\left(\Delta \mathrm{T}_{\text {hist}}\right)\right)$  (4)

Here, the factor $\mathrm{k}(\mathrm{t})=1+\frac{\|\omega(\mathrm{t})\|^2}{4}$, derived from the instantaneous IMU angular velocity norm $\|\omega(\mathrm{t})\|$, dynamically adjusts the influence of the geometric interpolation step based on rotational dynamics. The term involving $\log \left(\mathrm{T}_{\mathrm{i}}^{-1} \mathrm{~T}_{\mathrm{i}+1}\right)$ represents the relative transformation in the tangent space, weighted by time and scaled by $k(t)$, before being mapped back to the manifold via $\exp (\cdot)$.

Crucially, this formulation synergistically combines the kinematically consistent geometric interpolation (adaptively scaled by $k(t)$ with data-driven compensation $\eta$. $\operatorname{LSTM}\left(\Delta \mathrm{T}_{\text {hist}}\right)$ derived from LSTM predictions based on historical residuals. This hybrid approach enhances robustness by allowing the interpolation to adapt using both instantaneous rotational dynamics $(\omega)$ and learned temporal patterns (LSTM), providing higher-fidelity input for subsequent fusion steps.

Regarding the mathematical properties of the interpolation operator $\gamma(\mathrm{t})$ defined in Eq. (4), its continuity with respect to inputs (time, measurements $\mathrm{T}_{\mathrm{i}}^{-1} \mathrm{~T}_{\mathrm{i}+1}$, and LSTM outputs) is expected, owing to the continuity of the Lie group exponential map and assuming standard continuous activation functions within the LSTM. While a formal proof of diffeomorphism is complex due to the data-driven term, the operator is anticipated to possess sufficient local smoothness. These properties (continuity and local smoothness) are important for ensuring the well-behaved and stable operation of subsequent steps like the synchronization mechanism, which relies on consistent error evaluation on the manifold.

3.1.4 Bio-inspired synchronization mechanism

Inspired by neural synaptic plasticity [14], a dual-manifold feedback system synchronizes sensor timestamps via the update rule:

$\frac{\mathrm{d} \tau}{\mathrm{dt}}=\alpha \cdot \operatorname{sigmoid}(\beta \cdot\|\nabla \mathrm{J}\|)+\gamma \cdot \operatorname{tr}\left(\mathrm{P}_{\mathrm{k}}\right)$ （5）

where, the alignment error J combines geometric and temporal residuals. Parameters $(\alpha, \beta, \gamma)$ are optimized via Lyapunov stability criteria, enabling adaptive prioritization where increased Kalman covariance $P_k$ amplifies LSTM-based compensation. This proposed scheme achieves significantly improved synchronization accuracy compared to traditional methods based on fixed interpolation coefficients [15], as detailed in Section 4.

3.2 Noise-aware residual propagation network

This module establishes a deep coupling architecture between LSTM and Kalman filtering to achieve dynamic noise feature perception and collaborative suppression. The technical evolution follows the progressive logic of "feature decoupling $\rightarrow$ parameter optimization $\rightarrow$ hardware acceleration", forming a vertically integrated innovation framework from algorithmic theory to engineering implementation.

3.2.1 Bidirectional gated neural differentiator

To address the time-frequency coupling characteristics of non-stationary noise, we propose a Bidirectional Gated Neural Differentiator (Bi-DGND). The core innovation lies in establishing gradient dialogue mechanisms between LSTM and Kalman filtering. The forward gating unit achieves temporal feature extraction through hidden state fusion:

$\Gamma_f^{\mathrm{t}}=\sigma\left(\mathrm{W}_{\mathrm{f}} \cdot\left[\mathrm{h}_{\mathrm{t}-1}, \epsilon_{\text {Kalman }}^{\mathrm{t}}\right]+\mathrm{b}_{\mathrm{f}}\right)$  (6)

where, $\mathrm{h}_{\mathrm{t}-1}$ encodes historical motion trajectories, while $\epsilon_{\text {Kalman}}^{\mathrm{t}}$ injects Kalman residual constraints at current timestep. During backpropagation, the residual gradient $\partial \epsilon_{\text {Kalman }}^{\mathrm{t}} / \partial \mathrm{W}$ modifies LSTM weights through gating channels, forming feedforward-feedback closed loop between physical models and deep learning. This design breaks through the unidirectional modeling limitation of traditional recurrent networks, constructing noise feature decoupling channels in the frequency domain to provide high-purity feature inputs for parameter adaptation [13].

In contrast to existing hybrid architectures like differentiable Kalman filter variants, which primarily leverage gradient-based optimization to adapt the filter's internal parameters (e.g., noise covariance matrices), the proposed Bi-DGND establishes a deeper, structural coupling between the LSTM and the Kalman filter. The key distinction lies in the bidirectional gradient dialogue: Kalman residuals $\left(\epsilon_{\text {Kalman }}^{\mathrm{t}}\right)$ directly inform LSTM's forward feature extraction, while gradients derived from these residuals $\partial \epsilon_{\text {Kalman }}^{\mathrm{t}} / \partial W$ are propagated back to explicitly shape the LSTM's learned representation. This bidirectional flow forms a closed loop between the physical model and the deep learning component, yielding enhanced synergy. This mechanism enables the LSTM to learn features more attuned to physical state estimation while being regularized by the Kalman constraints, resulting in improved noise decoupling and robustness against complex dynamics compared to approaches focusing solely on filter parameter adaptation.

3.2.2 Kalman gain adaptation mechanism

Based on fractal dimension analysis of LSTM hidden states, we establish a dynamic gain adjustment model to handle time-varying noise environments. As shown in Eq. (2), noise statistical characteristics are quantified through hidden state autocorrelation coefficients:

$\mathrm{D}_{\mathrm{f}}=\frac{\log (\mathrm{N})}{\log \left(1 / \rho\left(\mathrm{h}_{\mathrm{t}}\right)\right)}$ (7)

When $\mathrm{D}_{\mathrm{f}}>1.6$ (indicating impulse noise dominance), the system triggers spectral norm scaling of observation noise covariance matrix. This process dynamically generates scaling factor $\beta$ through nonlinear mapping of LSTM outputs, enabling $\mathrm{R}_{\mathrm{t}}^{\text {eff }}$ to adaptively adjust within [0.15, 0.35]. This hidden-state-driven parameter optimization strategy maintains the theoretical completeness of Kalman filtering while endowing the algorithm with strong robustness against complex noise [16].

3.2.3 Fractal noise suppression theory

To suppress fractal noise with long-range correlation, we propose a gradient-coupled Hurst index estimator. As shown in Eq. (3), this method combines rescaled range analysis (R/S) with LSTM gradient feedback:

$\mathrm{H}_{\mathrm{t}}=\frac{1}{2} \cdot \frac{\log \left(\frac{\mathrm{R}}{\bar{S}}\right)_{\mathrm{t}}}{\log (\mathrm{N})}+\lambda \cdot \frac{\partial \mathcal{L}_{\mathrm{LSTM}}}{\partial \mathrm{H}_{\mathrm{t}}}$  （8）

When $\mathrm{H}_{\mathrm{t}}>0.7$, a third-order recursive fractal filter is activated, with coefficients designed following fractional calculus theory. Through CUDA kernel fusion technology, this module achieves instruction-level optimization: (1) Pipeline integration of DMA transfers and computing kernels eliminates memory access latency; (2) Shared memory reuse of LSTM weight caches reduce bandwidth occupancy; (3) Dynamic allocation of stream processor resources through warp schedulers. This hardware-software co-design reduces single-filter latency to 18 clock cycles, compared to the CPU baseline implementation with OpenCV optimization (single-filter latency: 98 clock cycles), our hardware-software co-design reduces latency to 18 clock cycles, achieving a 5.4:1 speedup ratio (equivalent to 4.4× real-time performance improvement) under identical input conditions.

While a formal convergence proof for the specific recursive structure involving LSTM gradients in Eq. (7) remains complex and beyond the scope of this work's theoretical analysis, we note that rigorous convergence analysis, often employing stochastic process theory, has been successfully applied to parameter estimation for other classes of complex nonlinear feedback systems [17]. This highlights the feasibility of such analyses in related domains, further motivating future theoretical investigation for the proposed Hurst index estimator. Nevertheless, the stability and effectiveness of the proposed method are empirically supported by the experimental results presented in Section 4.

3.3 LSTM-Kalman collaborative architecture with bidirectional optimization

3.3.1 Collaborative architecture design

This architecture innovatively constructs a feature-state bidirectional closed-loop system, overcoming the fragmentation between data-driven and model-driven modules in traditional approaches. The system achieves dynamic collaboration through dual-channel mechanisms of forward feature transmission and reverse gradient coupling:

Forward Channel

The LSTM network extracts physically meaningful residual features from raw sensor data. These features reconstruct the Kalman filter's observation matrix via a differentiable mapping function $\Phi(\cdot)$, extending the traditional observation equation H to:

$H_{\text {eff }}=H+\Phi\left(\Delta x_t^{\text {LSTM }}\right)$  (9)

This design draws inspiration from de Curtò and de Zarzà’s [18] hybrid architecture but innovatively introduces Lie group constraints to ensure kinematic compliance during feature mapping.

Reverse Channel

The Kalman filter's posterior covariance matrix is transformed into a regularization term, backpropagated to the LSTM network through differentiable pathways. This physics-constrained gradient transfer mechanism effectively suppresses overfitting risks in dynamic systems:

$\nabla_{\mathrm{W}} \mathcal{L}=\frac{\partial\left\|\mathrm{X}_{\mathrm{t}}^{\mathrm{GT}}-\hat{\mathrm{x}}_{\mathrm{t}}\right\|^2}{\partial \mathrm{~W}}+\lambda \cdot \operatorname{Tr}\left(\frac{\partial \mathrm{P}_{\mathrm{t}}^{+}}{\partial \mathrm{W}}\right)$    （10）

where, $\lambda$ is an adaptive coupling strength coefficient that enhances robustness in noisy environments.

3.3.2 Parameter Alternation Optimization Algorithm

To address the coupled optimization challenge between manifold interpolation parameters $\left(\Theta_{\mathrm{M}}\right)$ and residual network parameters $\left(\Theta_R\right)$, we propose the Manifold-Aware Alternating Training (MAT) algorithm. Its core idea is to alternately optimize between manifold-constrained LSTM training and residual-corrected manifold projection until parameter convergence. The two alternating phases are:

1. Manifold Constraint Phase

Fix the manifold interpolation parameters $\left(\Theta_M\right)$ and optimize the LSTM network within the $\mathrm{SE}(3) \times \mathbb{R}^4$ manifold space. The loss function is defined as:

$\mathcal{L}_{\mathrm{R}}=\mathbb{E}\left[\mathrm{d}_{\text {geo }}\left(\hat{\mathrm{x}}_{\mathrm{t}}, \mathrm{x}_{\mathrm{t}}^{\mathrm{GT}}\right)\right]$    (11)

Here, $\mathrm{d}_{\mathrm{geo}}$ represents the geodesic distance metric, which is adapted from Tang et al.'s [13] manifold learning methodology but incorporates temporal sliding-window averaging to enhance convergence speed.

2. Residual Correction Phase

Fix the LSTM network parameters $\Theta_{\mathrm{R}}$ and optimize the Lie group projection parameters of the manifold interpolation operator. This phase employs covariance-driven adaptive learning rates:

$\eta_{\mathrm{M}}=\frac{\alpha}{\sqrt{\operatorname{Tr}\left(\mathrm{P}_t^{+} \mathrm{P}_{\mathrm{t}}^{-}\right)}}$    （12）

This design ensures automatic adjustment of parameter update steps in high-noise scenarios to prevent manifold structure distortion. The two phases are alternated at a fixed frequency, and experimental results demonstrate that this strategy achieves significantly faster parameter convergence and superior generalization capability compared to traditional joint training methods.

3.3.3 Dynamic resource allocation theory

To meet real-time requirements on embedded platforms, we propose a Lyapunov stochastic optimization-based dynamic scheduling framework. The resource allocation problem is formulated as:

$\min _{\mu_t}=\mathbb{E}\left[\mathcal{L}_t\right] \quad$ s.t. $\quad Q_{\mathrm{CPU}}(\mathrm{t}) \leq \mathrm{Q}_{\max }$ (13)

Let $\mathrm{Q}_{\mathrm{CPU}}$ denote the task queue length and $\mu_{\mathrm{t}}$ represent the resource allocation decision. By constructing a drift-pluspenalty function:

$\Delta\left(\mu_t\right)=\mathbb{E}[\mathrm{L}(\mathrm{t}+1)-\mathrm{L}(\mathrm{t})]+\operatorname{VE}\left[\mathcal{L}_t\right]$   (14)

The optimal resource allocation policy is derived as:

$\mu_{\mathrm{t}}^*=\frac{\mathrm{V} \cdot \partial \mathcal{L}_{\mathrm{t}} / \partial \mu_{\mathrm{t}}-\mathrm{Q}_{\mathrm{CPU}}(\mathrm{t})}{2 \mathrm{c}}$    （15）

where, c is the platform's computational capability coefficient. This theory extends edge computing framework, achieving the first real-time resource coordination between LSTM and Kalman filtering in navigation systems.