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insight - Neural Networks - # Uncertainty Propagation in Guidance and Control Networks

Certifying Guidance & Control Networks: Uncertainty Propagation on an Event Manifold Using Differential Algebra


Core Concepts
This research paper introduces a novel method for certifying the robustness of Guidance & Control Networks (G&CNETs) by propagating uncertainty on an event manifold using Differential Algebra, addressing the limitations of traditional Monte Carlo simulations.
Abstract
  • Bibliographic Information: Origer, S., Izzo, D., Acciarini, G., Biscani, F., Mastroianni, R., Bannach, M., & Holt, H. (2024). Certifying Guidance & Control Networks: Uncertainty Propagation to an Event Manifold. International Astronautical Congress, Milan, Italy, 14-18 October 2024.
  • Research Objective: This paper aims to develop a more robust and efficient method for certifying the performance of G&CNETs, particularly in handling uncertainties and predicting system behavior at specific mission stages defined by event manifolds.
  • Methodology: The researchers employ Differential Algebra to perform high-order Taylor expansions of the final states on an event manifold. They utilize the Cauchy-Hadamard theorem to establish confidence bounds for these expansions and apply moment-generating functions to compute statistical moments of the final states under initial condition uncertainties. This approach is demonstrated on three optimal control problems: interplanetary transfer, asteroid landing, and drone racing.
  • Key Findings: The study demonstrates that uncertainty propagation on an event manifold using Differential Algebra provides a more rigorous and insightful approach to G&CNET certification compared to relying solely on Monte Carlo simulations. The Cauchy-Hadamard theorem offers valuable confidence bounds for the polynomial expansions, ensuring reliable uncertainty analysis within defined limits.
  • Main Conclusions: The proposed methodology effectively assesses the robustness of G&CNETs at specific mission stages, overcoming the limitations of traditional methods focused on locally stable points. This approach holds significant potential for enhancing the certification and, ultimately, the deployment of G&CNETs in real-world space missions.
  • Significance: This research contributes significantly to the field of neural network certification for safety-critical applications like spacecraft guidance and control. The proposed method provides a more rigorous and efficient alternative to traditional techniques, paving the way for increased confidence in G&CNETs for future space missions.
  • Limitations and Future Research: The paper acknowledges the computational limitations associated with high-order polynomial expansions, particularly for complex systems. Future research could explore techniques for optimizing the computational efficiency of the proposed method. Additionally, investigating the application of this approach to more complex scenarios and different types of neural networks would further enhance its applicability.
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Stats
The target body for the interplanetary transfer is in a circular orbit of radius R with an angular velocity Ω = √(𝜇/R^3). The interplanetary transfer spacecraft has a constant acceleration of magnitude Γ = 0.1 mm/s^2. The radius of the sphere of influence for the target planet in the interplanetary transfer problem is R_SOI = 924,000 km. The asteroid landing problem uses a rotating frame R with an angular velocity ω. The drone racing problem uses two coordinate frames: an inertial world frame W and a body frame B attached to the drone. The drone model has 16 states and 4 control inputs, with each control input u_i restricted within u_i ∈ [0, 1]. The G&CNETs for each problem have two hidden layers with 32 neurons each. The interplanetary transfer and asteroid landing G&CNETs were trained on 400,000 and 300,000 optimal trajectories, respectively, generated using the Backward Generation of Optimal Examples (BGOE) technique. The drone racing G&CNET was trained on 10,000 optimal trajectories solved individually using a direct method. The polynomial expansions achieved 8th-order accuracy for the interplanetary transfer and asteroid landing problems and 7th-order accuracy for the drone racing problem. The Monte Carlo analysis for the asteroid landing problem used 10,000 random perturbations per state. The initial state uncertainty in spacecraft mass for the asteroid landing requirement is ±5%, uniformly distributed in m_0 ∈ [0.95m_0, 1.05m_0] ≈ [335, 370] kg.
Quotes
"Simply evaluating the neural network over countless Monte Carlo simulations is not only time-consuming, it also does not provide a rigorous answer to the question: 'Will my G&CNET behave as intended when presented with a state it has never seen before?'" "Unless we revert back to a Monte Carlo-based approach, we cannot verify such a requirement by expanding the neural flow because the time at which the spacecraft will cross this boundary (defined as 1 km in altitude above the asteroid surface) is not known in advance and it is not an equilibrium point." "This work is driven by the recognition that there is a need to increase confidence in neural network certification for future missions."

Deeper Inquiries

How could this methodology be adapted for use in real-time onboard decision-making during space missions?

Adapting this methodology for real-time onboard decision-making during space missions presents a significant challenge due to the computational demands of high-order polynomial expansions. However, several potential adaptations could bridge the gap between theoretical certification and practical implementation: Offline Pre-computation and Online Interpolation: The most computationally intensive step, generating the high-order Taylor maps and confidence bounds, could be performed offline before the mission. This would generate a "look-up table" of pre-computed uncertainty propagation results for a range of anticipated initial conditions and potential event manifolds. During the mission, the spacecraft could then use interpolation techniques to quickly estimate uncertainty propagation for its current state, leveraging the pre-computed data. Reduced-Order Modeling: For highly complex systems, exploring model reduction techniques could prove beneficial. This might involve simplifying the system dynamics or using lower-order approximations for specific mission phases where real-time performance is critical. Techniques like Proper Orthogonal Decomposition (POD) or balanced truncation could be investigated to reduce model complexity while preserving essential dynamics. Hybrid Approaches: Combining this methodology with other faster but less rigorous techniques could offer a practical solution. For instance, one could use Monte Carlo simulations during the initial mission phases to quickly explore a wider range of scenarios. As the spacecraft approaches a critical event manifold, the onboard system could switch to the more precise uncertainty propagation using Taylor maps and moment-generating functions to ensure robust decision-making during these critical phases. Hardware Acceleration: Investigating the use of specialized hardware, such as Field-Programmable Gate Arrays (FPGAs) or Graphics Processing Units (GPUs), could significantly accelerate the computation of polynomial expansions. These hardware platforms excel at parallel processing, potentially enabling real-time or near-real-time evaluation of uncertainty propagation even for complex systems. Adaptive Order Selection: Instead of using a fixed polynomial order throughout the mission, one could dynamically adjust the order based on the required accuracy and available computational resources. For instance, during nominal mission phases or when far from the event manifold, a lower-order approximation might suffice. As the spacecraft approaches the event, the system could increase the polynomial order to enhance the accuracy of uncertainty propagation. By carefully considering these adaptations, it might be possible to leverage the strengths of this methodology for real-time onboard decision-making, enhancing the reliability and robustness of G&CNETs in future space missions.

Could the reliance on high-order polynomial expansions, which can be computationally expensive, limit the scalability of this approach for highly complex systems or real-time applications?

Yes, the reliance on high-order polynomial expansions does indeed pose a scalability challenge for this approach, particularly for highly complex systems or real-time applications. The computational cost of calculating and storing these expansions grows significantly with increasing system dimensionality and polynomial order. Several factors contribute to this limitation: Curse of Dimensionality: As the number of states in the system increases, the number of terms in the polynomial expansion grows exponentially. This "curse of dimensionality" makes it computationally intractable to compute and store high-order expansions for very high-dimensional systems. Polynomial Order: Higher-order expansions offer increased accuracy but come at the cost of significantly higher computational demands. Determining the appropriate order for a given application involves a trade-off between accuracy and computational feasibility. Real-Time Constraints: Real-time applications impose strict timing requirements on computations. The time required to evaluate high-order polynomial expansions might exceed these constraints, especially for complex systems or when using high polynomial orders. Addressing these scalability limitations requires exploring alternative strategies or combining this methodology with other techniques: Model Reduction: Simplifying the system dynamics through model reduction techniques can significantly reduce the dimensionality of the problem, making high-order expansions more manageable. Sparse Representations: Exploiting potential sparsity patterns in the Taylor maps could lead to more efficient storage and computation. If many of the terms in the expansion are negligible, using sparse matrix representations could significantly reduce memory requirements and computation time. Hybrid Methods: Combining this approach with other faster but less rigorous techniques could offer a practical compromise. For instance, one could use Monte Carlo simulations for initial exploration and switch to Taylor map-based uncertainty propagation for critical decision points. Parallel Computing: Leveraging parallel computing architectures, such as GPUs or FPGAs, can significantly accelerate the computation of polynomial expansions, potentially enabling real-time or near-real-time performance even for complex systems. In conclusion, while the computational cost of high-order polynomial expansions poses a scalability challenge, exploring these alternative strategies and hybrid approaches can mitigate these limitations and potentially pave the way for applying this methodology to more complex systems and real-time scenarios.

If we view the evolution of a G&CNET as a form of artificial intelligence development, what ethical considerations arise from applying these certification methods, particularly regarding accountability and unintended consequences?

Viewing the evolution of G&CNETs through the lens of artificial intelligence development raises important ethical considerations, particularly concerning accountability and unintended consequences: Accountability: Attribution of Failure: If a G&CNET makes a decision that leads to a mission failure, attributing responsibility becomes complex. Is it the fault of the network itself, the training data, the certification process, or the engineers who designed and deployed the system? Establishing clear lines of accountability is crucial for learning from failures and improving future systems. Human Oversight and Intervention: Determining the appropriate level of human oversight and intervention in G&CNET decision-making is crucial. Should humans have veto power over critical decisions, or should the system be allowed a degree of autonomy? Balancing autonomy with accountability is essential. Unintended Consequences: Bias in Training Data: G&CNETs are trained on large datasets of optimal trajectories. If these datasets contain biases or reflect limited scenarios, the network might make biased or unsafe decisions when encountering situations outside its training domain. Emergent Behavior: As G&CNETs become more complex, they might exhibit emergent behavior not explicitly programmed by the designers. This could lead to unintended consequences, especially in unforeseen situations. Over-Reliance and Deskilling: Over-reliance on certified G&CNETs could lead to a decline in human expertise in guidance and control. Maintaining a balance between automation and human skill is crucial for long-term mission success and safety. Addressing these ethical considerations requires a multi-faceted approach: Transparent Development and Certification: The development and certification processes for G&CNETs should be transparent and well-documented, allowing for independent audits and scrutiny. Robustness and Uncertainty Quantification: Certification methods should rigorously assess the robustness of G&CNETs to uncertainties and unforeseen situations, going beyond nominal performance evaluation. Ethical Frameworks and Guidelines: Developing ethical frameworks and guidelines specifically for AI-enabled space systems is crucial. These frameworks should address issues of accountability, transparency, and unintended consequences. Ongoing Monitoring and Adaptation: Continuous monitoring of G&CNET performance during missions is essential. The system should be designed for adaptability and updates based on lessons learned and evolving ethical considerations. By proactively addressing these ethical considerations, we can foster the responsible development and deployment of G&CNETs, ensuring that these powerful technologies contribute to safe and successful space exploration while upholding ethical principles.
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