TE-PINN: Quaternion-Based Orientation Estimation

Table of Contents

• Overview
• Key Features
• Architecture
• Installation
• Usage
• Experimental Results
• Contributing
• Citation

Performance Overview

| Metric | TE-PINN | SOTA | Improvement | |--------|---------|------|-------------| | Mean Euler Error | 0.0195 | 0.0216 | -36.84% | | Dynamic Error | 0.1677 | 0.1242 | -35.04% | | Uncertainty Correlation | 0.0147 | 0.0553 | +73.44% | | Computational Time | 25ms | 33ms | -25.00% | | Memory Usage | 45MB | 62MB | -27.42% |

Overview

TE-PINN represents a significant advancement in orientation estimation by combining transformer architectures with physics-informed neural networks. Our approach uniquely integrates:

Core Components

• Quaternion-based transformer encoder for temporal modeling
• Physics-informed neural network for dynamic constraints
• Adaptive multi-objective loss function
• RK4-based quaternion integration
• Uncertainty quantification through evidential learning

Technical Innovation

1. First implementation of transformer architecture for IMU-based orientation estimation
2. Novel integration of physical constraints in attention mechanism
3. Real-time capable implementation for embedded systems

Architecture

System Overview

```mermaid graph TD A[IMU Data Input] --> B[Quaternion Transformer] B --> C[Multi-Head Attention] A --> D[Physics Layer] D --> E[RK4 Integration] C --> F[Fusion Layer] E --> F F --> G[Attitude Correction] G --> H[Final Quaternion Output]

subgraph Physics Constraints
D
E
end

subgraph Transformer Processing
B
C
end

subgraph Output Processing
F
G
H
end

```

Data Flow

```mermaid sequenceDiagram participant IMU as IMU Sensors participant TE as Transformer Encoder participant PL as Physics Layer participant AC as Attitude Correction participant OUT as Output

IMU->>TE: (t), a(t)
Note over TE: Multi-head Attention
TE->>PL: Initial Estimates
Note over PL: RK4 Integration<br/>Rigid Body Dynamics
PL->>AC: Physics-Constrained Output
Note over AC: Yaw Correction
AC->>OUT: Final Quaternion

```

Mathematical Foundations

1. Input Processing

python x(t) = [(t), a(t)] # IMU measurements

2. Positional Encoding

python P(i,2k) = sin(t/10000^(2k/dmodel)) P(i,2k+1) = cos(t/10000^(2k/dmodel))

3. Transformer Layer

python H = Linear(x) + P Z = LayerNorm(H + MultiHeadAttention(H)) H = LayerNorm(Z + FeedForward(Z))

4. Physics Integration

```python

Rigid Body Dynamics

I d/dt + (I) =

Quaternion Integration

qw = cos(/2)cos(/2) qx = sin(/2)cos(/2) qy = cos(/2)sin(/2) qz = sin(/2)sin(/2) ```

Installation

Prerequisites

• Python 3.8+
• PyTorch 2.0+
• CUDA 11.0+ (for GPU support)

```bash

Clone repository

git clone https://github.com/yourusername/tepinn cd tepinn

Install dependencies

pip install -r requirements.txt

Optional: Install development dependencies

pip install -r requirements-dev.txt ```

Usage

Basic Usage

```python import tepinn

Initialize model

model = tepinn.TEPINN( transformerlayers=6, heads=8, dmodel=256 )

Train model

model.train(imudata, quaterniongt)

Inference

qpred = model.predict(imusequence) ```

Advanced Configuration

```python

Custom physics constraints

model.setphysicsparameters( inertiatensor=custominertia, gravityvector=customgravity )

Uncertainty estimation

qpred, uncertainty = model.predictwithuncertainty(imusequence) ```

Experimental Results

Performance Analysis

1. Error Metrics

| Scenario | Mean Error (deg) | STD | Max Error | |----------|------------------|-----|-----------| | Static | 0.195 | 0.023 | 0.412 | | Dynamic | 0.677 | 0.089 | 1.234 | | High Speed | 1.242 | 0.156 | 2.567 |

2. Ablation Study Results

| Component | Mean Error | Dynamic Error | Compute Time | |-----------|------------|---------------|--------------| | Base Model | 0.0216 | 0.1242 | 33ms | | + Transformer | 0.0205 | 0.1198 | 30ms | | + Physics | 0.0195 | 0.1677 | 25ms | | + Uncertainty | 0.0195 | 0.1677 | 25ms |

Hyper-parameters

| Parameter | Value | Description | |-----------|-------|-------------| | Transformer Layers | 6 | Number of transformer encoder layers | | Attention Heads | 8 | Number of parallel attention heads | | Model Dimension | 256 | Internal representation dimension | | Learning Rate | 1e-4 | Initial learning rate | | Batch Size | 32 | Training batch size | | acc | 1.0 | Accelerometer loss weight | | gyro | 0.5 | Gyroscope loss weight | | dynamics | 0.1 | Dynamics loss weight |

Model Capabilities

1. Real-time Performance

• 25ms average inference time on embedded systems
• 45MB memory footprint
• Support for batch processing

2. Robustness Features

• Handles sensor noise up to 0.1 rad/s for gyroscope
• Manages accelerometer noise up to 0.2 m/s
• Adaptive to varying sampling rates (100-500 Hz)

3. Uncertainty Quantification

• Provides calibrated uncertainty estimates
• Supports multiple uncertainty metrics
• Real-time confidence bounds

Citation

bibtex @article{asgharpoor2024tepinn, title={TE-PINN: Quaternion-Based Orientation Estimation using Transformer-Enhanced Physics-Informed Neural Networks}, author={Asgharpoor Golroudbari, Arman}, journal={arXiv preprint arXiv:2409.16214}, year={2024} }

Future Work

1. Extended Functionality

   • Multi-sensor fusion support
   • Adaptive learning rates
   • Online calibration

2. Performance Optimization

   • Model quantization
   • Sparse attention mechanisms
   • Hardware-specific optimizations

Contributing

We welcome contributions! Please see our Contributing Guidelines for details.

Development Process

1. Fork the repository
2. Create your feature branch
3. Commit your changes
4. Push to the branch
5. Create a Pull Request

License

This project is licensed under the MIT License - see the LICENSE file for details.

Acknowledgments

• Students' Scientific Research Center for research support
• Open-source community for various tools and libraries
• All contributors and testers