Multidisciplinary Design Optimization of a Low-Thrust Asteroid Orbit Insertion Using Electric Propulsion

This paper presents a multidisciplinary design optimization framework using OpenMDAO and Dymos that simultaneously optimizes low-thrust trajectories and spacecraft power systems for asteroid orbit insertion, explicitly coupling variable-specific impulse Hall thruster performance with time-varying solar power availability to overcome the limitations of traditional simplifying assumptions.

The Gist

Imagine you are planning a road trip across a vast, empty desert. You have a car with a very special engine: it's incredibly fuel-efficient, but it runs entirely on solar power. The catch? You are driving so far away from the sun that the sunlight is weak, like a dim flashlight compared to the blazing noon sun back home.

This paper is about figuring out the perfect way to drive this car to get from a high mountain pass down to a valley floor as quickly as possible, without running out of power or crashing.

Here is the breakdown of their solution, using simple analogies:

1. The Old Way: Driving Blindfolded

Traditionally, engineers would plan the route (the trajectory) and design the car (the power system) separately.

• The Route Planner would say: "I need to drive at a constant speed of 60 mph."
• The Car Designer would say: "Okay, I'll build a solar panel that gives us exactly enough power for 60 mph."

The Problem: In deep space, the sun isn't constant. As you get farther away, the solar panels get weaker. If you plan for a constant speed but the sun dims, your car slows down, or worse, the plan becomes impossible. It's like planning a cross-country drive assuming you'll always have a full tank of gas, ignoring the fact that gas stations are 500 miles apart in the desert.

2. The New Way: The "Super-Optimizer"

The authors built a Multidisciplinary Design Optimization (MDO) framework. Think of this as a super-smart GPS that doesn't just look at the road; it looks at the car, the weather, the fuel, and the driver all at the same time.

Instead of treating the engine and the solar panels as separate things, this system realizes they are best friends.

• If you want to go faster, you need more power.
• To get more power, you need bigger solar panels.
• But bigger solar panels make the car heavier.
• A heavier car needs more power to move.

The computer has to solve this "chicken and egg" puzzle instantly. It asks: "Is it better to carry a heavy solar panel to go fast, or carry a light one and go slow?"

3. The Specific Mission: The "Psyche" Descent

The test case for this paper is a mission to 16-Psyche, a giant metal asteroid.

• The Goal: A tiny probe needs to spiral down from a high orbit (750 km up) to a low orbit (200 km up) to take pictures.
• The Challenge: Psyche is so far from the Sun (2.9 times farther than Earth) that the solar power is very weak. It's like trying to power a high-performance sports car with a single AA battery.

4. How They Solved the Math Puzzle

Optimizing this is incredibly hard because the math is "stiff." Imagine trying to draw a perfect circle by connecting dots. If you only use 50 dots, the circle looks like a jagged star. If you use 1,000 dots, it looks smooth.

• The "Collocation Defect": The computer tried to solve the path with too few "dots" (mathematical steps). The result was a mathematically "correct" answer that looked physically impossible (like a car driving through a wall).
• The Fix: They used a Fast Fourier Series method. Think of this as sketching a rough, smooth curve first to give the computer a good idea of where to start. Then, they used a supercomputer to add thousands of "dots" to the path, ensuring the car actually follows the laws of physics.

5. The Results: Heavy but Fast

When they let the computer optimize everything together, it made a surprising decision:

• The Decision: It chose to make the solar panels much bigger (from 50 m² to almost 80 m²).
• The Consequence: This made the spacecraft heavier (adding about 11% more weight).
• The Reward: Because the bigger panels generated more power, the engine could push harder. The trip took 20% less time.

The Analogy: It's like deciding to carry a heavy backpack full of extra batteries. Even though the backpack slows you down slightly, the extra power lets you run so much faster that you still finish the race 20% sooner than if you had run light but slow.

Summary

This paper proves that in deep space, you can't design the road and the car separately. You have to design them together. By using a smart computer system that understands how solar power, engine performance, and spacecraft weight are all linked, they found a way to get a probe to its destination significantly faster, even in the dim light of deep space.

The Big Takeaway: In the future of space travel, the most efficient path isn't just about the best engine or the biggest solar panel; it's about finding the perfect balance between the two.

Technical Summary

Here is a detailed technical summary of the paper "Multidisciplinary Design Optimization of a Low-Thrust Asteroid Orbit Insertion Using Electric Propulsion."

1. Problem Statement

The paper addresses a critical gap in deep-space mission design: the traditional decoupling of trajectory optimization and spacecraft subsystem design. In low-thrust electric propulsion (SEP) missions, particularly in deep space (e.g., near asteroid 16-Psyche at 2.9 AU), solar irradiance is significantly reduced.

• The Core Issue: Standard design approaches often assume constant thrust or fixed specific impulse ($I_{sp}$), neglecting the strong coupling between trajectory dynamics, available solar power, and propulsion performance.
• Consequences: Decoupled designs can lead to physically infeasible trajectories where the spacecraft demands more power than the solar arrays can generate, or suboptimal designs that fail to account for the mass penalty of larger solar arrays required to compensate for power degradation.
• Specific Challenge: Optimizing a low-thrust spiral descent around a small body (weak gravity) involves high-revolution trajectories that create stiff optimal control problems, often leading to "collocation defects" where numerical solvers fail to capture the continuous physical path.

2. Methodology

The authors propose a Multidisciplinary Design Optimization (MDO) framework that simultaneously optimizes the spacecraft trajectory, solar array sizing, and electric propulsion performance.

A. Framework Architecture

• Tools: The framework is built on OpenMDAO (for system integration and analytic derivatives) and Dymos (for trajectory optimization using direct collocation).
• Solver: The resulting Nonlinear Programming (NLP) problem is solved using IPOPT.
• Objective: Time-optimal control (minimizing time of flight, $t_f$) for a spiral descent from a 750 km parking orbit to a 200 km science orbit around asteroid 16-Psyche.

B. Key Modeling Components

1. Trajectory Dynamics: Modeled in a 2D inertial frame using polar coordinates ($r, \theta$) and velocity components ($v_r, v_\theta$). The equations of motion include variable thrust magnitude ($T$) and mass flow rate ($\dot{m}$) as functions of time-varying electrical power.
2. Solar Power Generation:
   • Models solar array output ($P_{SA}$) based on heliocentric distance ($r_s$), array area ($A_{SA}$), and a time-dependent degradation factor ($\sigma \approx 3\%/yr$).
   • Includes a high-fidelity polynomial correction for temperature and efficiency variations in the asteroid belt.
   • Uses an $L_2$-norm smoothing technique to handle the logical minimum function between available power and Power Processing Unit (PPU) saturation limits, ensuring differentiability for gradient-based optimization.
3. Electric Propulsion (VSI Model):
   • Utilizes a Variable Specific Impulse (VSI) model based on the SPT-140 Hall thruster.
   • Instead of fixed parameters, the model uses a discrete throttle table (21 modes) with specific power, thrust, mass flow, and efficiency values.
   • Smooth Mode Selection: A differentiable, sigmoid-based switching logic maps continuous commanded power ($P_E$) to a weighted average of discrete throttle modes, allowing the optimizer to dynamically throttle the engine as power availability changes.
4. Mass-Area Coupling:
   • A critical novelty is the dynamic calculation of initial wet mass ($m_{initial}$) based on the design variable $A_{SA}$.
   • $m_{dry} = m_{bus} + (N_{eng} \times m_{eng}) + (\rho_{SA} \times A_{SA})$.
   • This forces the optimizer to weigh the benefit of increased power (larger arrays) against the kinetic penalty of increased spacecraft mass.

C. Numerical Strategies

• Initial Guess Generation: To overcome sensitivity to starting parameters, a Fast Fourier Series (FFS) shape-based method is used to generate dynamically feasible initial trajectories before optimization.
• Resolving Collocation Defects: High-revolution spirals often cause collocation defects with coarse grids. The authors utilized High-Performance Computing (HPC) to refine the transcription grid to over 100 segments and enforced control rate continuity to achieve zero-defect solutions.

3. Key Contributions

1. Fully Coupled MDO Framework: Demonstrates a control co-design approach that integrates trajectory dynamics, power subsystem sizing, and propulsion performance into a single optimization loop, moving beyond decoupled design paradigms.
2. Differentiable VSI Model: Introduces a smooth, gradient-compatible method for modeling discrete Hall thruster throttle modes within a continuous optimization framework.
3. Dynamic Mass-Power Trade-off: Explicitly models the structural mass penalty of solar arrays, allowing the optimizer to determine the optimal array size that balances power generation needs with mass constraints.
4. Numerical Robustness: Addresses the specific numerical challenges of low-thrust, high-revolution trajectories around small bodies using FFS initialization and HPC-based grid refinement.

4. Results

The framework was tested on a 16-Psyche orbit insertion scenario, comparing a Baseline (Trajectory-Only) approach against the Fully Coupled MDO approach.

• Performance Gains:
  • Time of Flight: The coupled MDO reduced the mission time by 20.07% (from ~20.78 hours to ~16.61 hours).
  • Solar Array Sizing: The optimizer increased the solar array area from a fixed 50 $m^2$ to an optimized 79.79 $m^2$.
  • Mass Penalty: This resulted in an 11.02% increase in initial wet mass (from 418 kg to 464 kg) due to the heavier solar arrays.
  • Propellant Usage: Propellant consumption increased by 113.6% (0.66 kg to 1.41 kg), but this was a worthwhile trade-off for the significant time reduction.
• Optimization Behavior:
  • The coupled solution produced smooth, physically realistic control profiles where commanded power tracked the dynamic available power limit perfectly.
  • The baseline trajectory-only solution exhibited highly oscillatory controls and significant collocation defects (divergence between discrete nodes and continuous integration), highlighting the necessity of the proposed methodology.

5. Significance

This paper underscores that for deep-space electric propulsion missions, treating propulsion and power subsystems as fixed parameters leads to suboptimal or infeasible designs. By integrating these disciplines:

• Mission Efficiency: Significant reductions in time of flight can be achieved even in power-starved environments by intelligently sizing power subsystems.
• Realism: The framework captures the fundamental trade-off between mass and power, ensuring designs are physically realizable.
• Scalability: The use of OpenMDAO and Dymos provides a scalable, open-source foundation for future mission designs involving complex subsystems (e.g., nozzle geometry, multiple thrusters) and different objective functions (e.g., fuel-optimal transfers).

In conclusion, the study validates that integrated multidisciplinary optimization is essential for next-generation deep-space missions, particularly those operating in low-irradiance environments where the coupling between power availability and trajectory dynamics is non-negligible.