Real-Time TECS Gain Tuning Using Steepest Descent Method for Post-Transition Stability in Unmanned Tilt-Rotor eVTOLs
Abstract
1. Introduction
- The development of a real-time adaptive TECS framework that dynamically tunes gains to improve post-transition stability in unmanned tilt-rotor eVTOLs.
- A simulation-based demonstration showing that SD-TECS outperforms PX4 TECS in altitude and airspeed control under various conditions.
- Validation of the proposed method’s practical applicability through sensitivity and computational complexity analyses, ensuring robustness and feasibility for real-world applications.
2. Preliminaries
2.1. PX4 VTOL Control Architecture Overview
2.2. PX4 Transition Logic for Tilt-Rotor eVTOLs
- Open-loop tilt-angle control: Front rotors tilt incrementally based on fixed airspeed thresholds (e.g., BLENDED_ASPD for initiating control output blending, and TRANSITION_ASPD for full fixed-wing mode engagement).
- Linear control blending: PX4 linearly interpolates between multicopter and fixed-wing attitude controllers based on airspeed, ignoring aerodynamic nonlinearity.
2.3. Total Energy Control System (TECS)
Gain Nomenclature Clarification
3. Literature Review
3.1. Control Strategies for Urban Air Mobility (UAM)
3.2. PX4-Based Transition Control and Limitations
3.3. TECS and Energy-Based Control Advances
3.4. Gradient-Based Adaptive Control Techniques
3.5. Summary
4. Methodology
4.1. Adaptive Gain Tuning via Steepest Descent
4.1.1. Motivation for the Steepest Descent Method
4.1.2. Steepest Descent Gain-Update Formulation
4.2. Implementation and Validation
5. Simulation Model
5.1. Aircraft Configuration and Longitudinal Dynamics
Symbol | Value | Description |
---|---|---|
m | 5.22 kg | Total mass of the aircraft |
1.229 kg·m2 | Moment of inertia about the x-axis | |
0.1702 kg·m2 | Moment of inertia about the y-axis | |
0.8808 kg·m2 | Moment of inertia about the z-axis | |
0.9343 kg·m2 | Product of inertia | |
0.75 m2 | Wing surface area | |
b | 2.10 m | Wingspan |
0.3571 m | Mean aerodynamic chord |
Symbol | Value | Description |
---|---|---|
0.0867 | Lift coefficient at zero angle of attack | |
4.02 | Lift coefficient per radian of angle of attack | |
3.8954 | Lift coefficient per unit pitch rate | |
0.278 | Lift coefficient per unit elevator deflection | |
0.0197 | Drag coefficient at zero angle of attack | |
0.0791 | Drag coefficient per radian of angle of attack | |
1.06 | Drag coefficient per radian squared of angle of attack | |
0.0 | Drag coefficient per unit pitch rate | |
0.0633 | Drag coefficient per unit elevator deflection | |
0.0302 | Moment coefficient at zero angle of attack | |
−0.126 | Moment coefficient per radian of angle of attack | |
−1.3047 | Moment coefficient per unit pitch rate | |
−0.206 | Moment coefficient per unit elevator deflection |
5.2. Control Architecture
- Multicopter Flight Mode:
- –
- Attitude Stabilization: A cascaded PID controller regulates the roll, pitch, and yaw rates using angular rate feedback.
- –
- Altitude Hold: A proportional–integral (PI) controller tracks the desired altitude by modulating the vertical thrust.
- Fixed-Wing Flight Mode:
- –
- Attitude Control: A pitch attitude controller maintains longitudinal stability using elevator surface deflection.
- –
- Energy Management: TECS regulates the total and balanced energy rates by coordinating the throttle and pitch commands.
6. Simulation Results
6.1. Simulation Setup
6.2. TECS State Response
6.3. Energy-Rate Error Response
6.4. TECS Control Outputs
6.5. TECS Gain Tuning
6.6. Performance Analysis
6.6.1. Transient Response Metrics
6.6.2. Quantitative Evaluation: MAE, IAE, and ITAE
- MAE (Mean Absolute Error): ;
- IAE (Integral of Absolute Error): ;
- ITAE (Integral of Time-weighted Absolute Error): .
6.7. Robustness Analysis
6.8. Sensitivity Analysis
6.9. Computational Complexity Analysis
7. Discussion and Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
SD-TECS | Steepest Descent-based Total Energy Control System |
TECS | Total Energy Control System |
PX4 | PX4 Autopilot |
eVTOL | Electric Vertical Take-Off and Landing |
UAM | Urban Air Mobility |
IAE | Integral of Absolute Error |
ITAE | Integral of Time-weighted Absolute Error |
MAE | Mean Absolute Error |
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Parameter | Value |
---|---|
VTOL Parameters | |
Critical Tilt Angle | 50 degrees |
Transition Thrust | 0.35 |
Blended Airspeed (BLENDED_ASPD) | 6 m/s |
Transition Airspeed (TRANSITION_ASPD) | 15 m/s |
TECS Parameters | |
(STE Learning Rate) | 0.000001 |
(SBE Learning Rate) | 0.000001 |
(STE Sigmoid Parameter) | 0.3 |
(SBE Sigmoid Parameter) | 0.2 |
Initial (STE Proportional Gain) | 0.8 |
Initial (STE Integral Gain) | 0.02 |
Initial (SBE Proportional Gain) | 1.2 |
Initial (SBE Integral Gain) | 0.20 |
Default (Default STE Proportional Gain) | 0.8 |
Default (Default STE Integral Gain) | 0.02 |
Default (Default SBE Proportional Gain) | 1.2 |
Default (Default SBE Integral Gain) | 0.20 |
Default (Default SBE Feedforward Gain) | 1.0 |
Initial Conditions | |
Initial Altitude | 10 m |
Initial Airspeed | 0.1 m/s |
Metric | PX4 Default | Tuned TECS | SD-TECS (Proposed) |
---|---|---|---|
Settling Time | |||
Altitude (s) | 44.31 | 35.31 | 28.56 |
Improvement | - | 20.31% | 35.55% |
Airspeed (s) | 40.20 | 26.75 | 23.14 |
Improvement | - | 45.90% | 57.30% |
Overshoot | |||
Max Altitude Loss (m) | 1.62 | 0.87 | 0.83 |
Improvement | - | 46.3% | 48.9% |
Airspeed Overshoot (m/s) | 1.16 | 0.63 | 0.61 |
Improvement | - | 33.46% | 42.44% |
Metric | SD-TECS | TECS |
---|---|---|
Mean Total Time (s) | 0.7188 | 0.6500 |
Standard Deviation (s) | 0.0062 | 0.0082 |
Mean Time per Call (μs) | 6.95 | 6.29 |
Increase in Simulation Time (%) | 10.58 |
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Lee, C.; Nguyen, N.P.; Bae, S.; Hong, S.K. Real-Time TECS Gain Tuning Using Steepest Descent Method for Post-Transition Stability in Unmanned Tilt-Rotor eVTOLs. Drones 2025, 9, 414. https://doi.org/10.3390/drones9060414
Lee C, Nguyen NP, Bae S, Hong SK. Real-Time TECS Gain Tuning Using Steepest Descent Method for Post-Transition Stability in Unmanned Tilt-Rotor eVTOLs. Drones. 2025; 9(6):414. https://doi.org/10.3390/drones9060414
Chicago/Turabian StyleLee, Choonghyun, Ngoc Phi Nguyen, Sangjun Bae, and Sung Kyung Hong. 2025. "Real-Time TECS Gain Tuning Using Steepest Descent Method for Post-Transition Stability in Unmanned Tilt-Rotor eVTOLs" Drones 9, no. 6: 414. https://doi.org/10.3390/drones9060414
APA StyleLee, C., Nguyen, N. P., Bae, S., & Hong, S. K. (2025). Real-Time TECS Gain Tuning Using Steepest Descent Method for Post-Transition Stability in Unmanned Tilt-Rotor eVTOLs. Drones, 9(6), 414. https://doi.org/10.3390/drones9060414