💻 computer science

Surrogate Model-Based Near-Optimal Gain Selection for Approach-Angle-Constrained Two-Phase Pure Proportional Navigation

This paper proposes a neural network-based surrogate model to efficiently determine near-optimal navigation gains for two-phase Pure Proportional Navigation, enabling the minimization of guidance effort while satisfying approach-angle constraints for aerodynamically driven vehicles.

Original authors: Abhigyan Roy, Shreeya Padte, Abel Viji George, Vivek A, Satadal Ghosh

Published 2026-04-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Abhigyan Roy, Shreeya Padte, Abel Viji George, Vivek A, Satadal Ghosh

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

Imagine you are driving a self-driving car that needs to park itself perfectly. But there's a catch: it doesn't just need to stop at the spot; it needs to stop facing a very specific direction (like perfectly parallel to the curb) to fit into a tight space.

This paper is about teaching a drone (or any flying vehicle) how to do exactly that: fly to a target and stop at a precise angle, while using the least amount of fuel and effort possible.

Here is the breakdown of the problem and the clever solution the authors came up with, explained through simple analogies.

The Problem: The "Two-Step" Dance

The authors are using a standard flight rule called Proportional Navigation (PPN). Think of this as a simple rule: "If the target moves to the left, I turn left." It's great for hitting a target, but it's rigid. If you just use this simple rule, you can only hit the target from a few specific angles. You can't always make the drone turn sharply enough to face the exact direction you want.

To fix this, they use a Two-Phase strategy (2pPPN):

Phase 1 (The Orientation): The drone does a specific maneuver to get itself into a "good position" where it can easily turn to the final angle.
Phase 2 (The Final Approach): The drone uses the standard rule to lock onto the target and hit it.

The Catch: There are many different ways to do Phase 1. You could turn slowly, turn fast, or take a wide loop. Each way uses a different amount of energy. The goal is to find the perfect way to do Phase 1 so the drone uses the absolute minimum energy to get the job done.

The Challenge: The "Impossible" Math

The authors realized that figuring out the perfect turn for every single possible starting position is a mathematical nightmare. It's like trying to calculate the perfect route for a delivery driver for every single possible traffic jam, weather condition, and road closure in the world. There are too many variables, and the math is too complex to solve instantly on a computer while the drone is flying.

If you tried to calculate the perfect answer every time, the drone would be too slow to react.

The Solution: The "Smart GPS" (Neural Network)

Instead of solving the hard math every time, the authors decided to train a "Smart GPS" (a Neural Network) to guess the answer.

Here is how they built it:

The Practice Run (Data Generation): They ran thousands of computer simulations. They told the drone, "Start here, aim for that angle," and then used a super-computer to find the perfect energy-saving path for that specific situation. They did this for thousands of different scenarios.
The Learning (Training): They fed all these "perfect paths" into a computer program (the Neural Network). The program looked at the starting position and the desired angle, and memorized the pattern of the best turn.
The Result (The Surrogate Model): Now, instead of doing the hard math, the drone just asks the "Smart GPS": "I'm starting here and want to end up facing there. What's the best turn?" The GPS instantly gives the answer.

The Analogy: Learning to Ride a Bike

Imagine you are learning to ride a bike up a hill.

The Old Way (Analytical Math): You try to calculate the physics of friction, wind resistance, your leg strength, and the slope angle every single time you want to pedal. It takes forever, and you might fall before you finish the calculation.
The New Way (This Paper): You ride the hill 1,000 times. You notice that when the hill is steep, you lean forward a bit more. When it's flat, you sit up. You don't calculate the physics; your brain (the Neural Network) learns the feeling of the right move.
The Payoff: The next time you see a hill, you don't think; you just do the right thing instantly.

Why This Matters

Speed: The computer can give the answer in a fraction of a millisecond. The drone can react instantly.
Efficiency: The drone saves battery life because it always picks the most efficient path.
Versatility: It works for almost any starting position and any target angle.

The Bottom Line

The authors created a system that teaches a drone how to "feel" the best way to fly to a target at a specific angle. By using a "Smart GPS" trained on thousands of perfect examples, the drone can make near-perfect decisions instantly, saving energy and ensuring it arrives exactly where and how it needs to.

In short: They turned a super-hard math problem into a simple "lookup" task using a computer brain that learned from experience.

1. Problem Statement

The paper addresses the challenge of terminal approach angle control for aerodynamically driven autonomous vehicles (specifically UAVs) using Pure Proportional Navigation (PPN).

Context: While standard PPN is robust and simple, a single-phase PPN with a constant navigation gain ( $N$ ) can only achieve a limited subset of terminal approach angles (typically restricted to an angular half-space where $N \geq 2$ ).
Existing Solution: A Two-Phase PPN (2pPPN) strategy was previously proposed, consisting of an Orientation Phase (to adjust the pursuer's heading) and a Final Phase (standard PPN for interception). This allows achieving any desired approach angle within a half-space.
The Gap: Previous 2pPPN implementations used fixed or specific rules for the Orientation phase gain ( $N_{ori}$ ), ignoring the fact that multiple feasible trajectories exist for the same terminal condition. This creates an opportunity to minimize the total guidance effort (control energy) by optimally selecting the navigation gains for both phases.
The Challenge: Determining the optimal gain pair ( $N_{ori}, N_f$ ) analytically for arbitrary initial and terminal engagement geometries is mathematically intractable. Numerical optimization is possible but computationally expensive for real-time online implementation.

2. Methodology

The authors propose a data-driven framework using a Neural Network (NN) as a surrogate model to approximate the optimal gain manifold.

A. Problem Formulation

The objective is to minimize the total guidance effort, defined as the integral of the squared lateral acceleration:
$J = \int_{0}^{t_f} a_P^2 dt$
Two specific optimization problems are defined:

Problem 1: Given initial conditions ( $\alpha_{P0}$ ), desired terminal angle ( $\alpha_{Pf}^d$ ), and a fixed final-phase gain ( $N_f$ ), find the near-optimal orientation-phase gain ( $N_{ori}^*$ ).
Problem 2: Given only initial and desired terminal conditions, find the near-optimal pair ( $N_{ori}^*, N_f^*$ ) that minimizes $J$ .

B. Data Generation

Simulation: A dataset was generated by numerically simulating 2pPPN engagements for a stationary target.
Parameters:
- Initial heading ( $\alpha_{P0}$ ) and desired terminal angle ( $\alpha_{Pf}^d$ ) were varied across the feasible angular half-space.
- Final gain ( $N_f$ ) was constrained to $[2, 5]$ .
- Orientation gain ( $N_{ori}$ ) was bounded geometrically to ensure the trajectory reaches the "collision isocurve" required for the final phase.
Optimization: For each scenario, a constrained numerical optimization was performed to find the true optimal gains. This resulted in a dataset of ~94,857 valid scenarios, yielding 4,216 optimal cases for Problem 1 and 136 optimal cases for Problem 2.
Observation: The resulting optimal gain surfaces were found to vary smoothly with respect to the engagement geometry, making them suitable for regression.

C. Neural Network Architecture

Two feedforward neural networks were developed to act as surrogates for the optimization process:

Model A (Problem 1):
- Inputs: $\alpha_{P0}, \alpha_{Pf}^d, N_f$ (3 neurons).
- Output: $N_{ori}^*$ (1 neuron).
- Structure: 2 hidden layers (32, 16 neurons) with Tanh activation.
Model B (Problem 2):
- Inputs: $\alpha_{P0}, \alpha_{Pf}^d$ (2 neurons).
- Output: $(N_{ori}^*, N_f^*)$ (2 neurons).
- Structure: 3 hidden layers (32, 64, 16 neurons) with Tanh activation.
Training: The models were trained using the Adam optimizer with Mean Squared Error (MSE) loss. Data was normalized and split 80/20 for training/testing.

3. Key Contributions

Formulation of Gain Optimization: The paper formally defines the problem of minimizing guidance effort in 2pPPN by treating the orientation-phase gain (and potentially the final-phase gain) as decision variables rather than fixed parameters.
Surrogate Modeling: It introduces a lightweight Neural Network-based surrogate to replace computationally intensive online optimization. This enables real-time, single-shot calculation of near-optimal gains.
Smooth Manifold Discovery: The authors demonstrate that the optimal gain manifold is smooth with respect to engagement conditions, validating the use of regression models over discrete lookup tables or complex analytical derivations.
Dual-Model Approach: The development of two distinct models allows flexibility: Model A is useful when the final-phase gain is constrained by hardware (e.g., actuator limits), while Model B provides a fully autonomous solution.

4. Results

Accuracy:
- Model A achieved a coefficient of determination ( $R^2$ ) of 0.932, indicating high accuracy in predicting $N_{ori}^*$ when $N_f$ is provided.
- Model B achieved $R^2$ values of 0.880 (for $N_f^*$ ) and 0.901 (for $N_{ori}^*$ ).
Error Analysis: Predictions were highly accurate for the majority of the domain. Slight deviations were observed at the extreme boundaries of the feasible region (where training data was sparser), but the models remained robust.
Computational Efficiency: The trained NNs can resolve a gain query in approximately $O(10^2)$ microseconds, making them suitable for real-time onboard guidance systems.
Visualization: The NN successfully learned the smooth surfaces of the optimal gains, effectively interpolating between the discrete points generated by the numerical sweep.

5. Significance

Real-Time Implementation: By replacing iterative numerical optimization with a forward pass through a neural network, the paper enables real-time, near-optimal guidance for UAVs with limited onboard computational resources.
Energy Efficiency: The method ensures that autonomous vehicles achieve precise terminal approach angles (critical for docking, inspections, or munitions impact) with minimal control energy, extending mission endurance.
Scalability: The data-driven approach is generalizable. It avoids the need for complex analytical derivations for new engagement scenarios, as the model can be retrained with new simulation data if the vehicle dynamics or constraints change.
Practical Application: This framework bridges the gap between theoretical optimal control (which is often too slow for real-time use) and practical guidance laws (which are often sub-optimal), offering a "best of both worlds" solution for modern autonomous missions.