Control Allocation Strategy Based on Min–Max Optimization and Simple Neural Network
Abstract
:1. Introduction
2. Dynamics
- The world frame defined under the north–east–down convention.
- The body frame whose origin is at the center of mass. Here, points forward along arm 2, and is perpendicular to the fuselage plane, pointing downward.
- The origin of the static joint frame for the ith arm is located at the center of the revolute joint with its axis always remaining parallel to the axis.
- The rotatable joint frame for the ith arm , which can be obtained by rotating by degree.
2.1. Actuator Dynamics
2.2. Platform Dynamics
3. Nominal Controller
4. Geometric Allocation
4.1. Force Decomposition Method
4.2. Nullspace-Based Method
5. Control Allocation Method Considering Thrust Balance
5.1. Min–Max-Based Control Allocation Strategy
5.2. Control Allocation Strategy Based on Simple Neural Network and Min–Max Method
- Target Mapping for ApproximationThe min–max-based control allocation optimization problem is formulated in Section 5.1, with the objective function, inequality constraints, and equality constraints given by Equations (43)–(45), respectively. Accordingly, the neural network is trained to approximate the mapping from the desired control input to the optimal compensation term :
- Generating the Training DatasetA control allocation training dataset is constructed to train the neural network, where denotes the control input vector representing the desired forces and torques along the three axes, and is the optimal compensation term used to correct the FD result via nullspace projection. To generate the dataset, simulated flight experiments are conducted using the MATLAB R2023b Simscape platform. During each simulation, the high-level controller outputs a sequence of desired control inputs , which are passed to an active-set solver to solve the min–max linear programming problem and obtain the corresponding optimal compensation term .
- Network Architecture DesignA fully connected feedforward neural network (multi-layer perceptron, MLP) is adopted, as it is well suited for small-scale regression tasks involving low-dimensional control inputs. The input layer contains six neurons, corresponding to the control input . The network consists of two hidden layers with 20 and 10 neurons, respectively, both fully connected. The output layer consists of two neurons, representing the elements of the compensation vector . The overall architecture of the designed neural network is illustrated in Figure 2.
- Neural Network TrainingThe neural network is trained using the Levenberg–Marquardt (LM) algorithm, which combines the advantages of gradient descent and Newton’s method to achieve fast convergence and improved numerical stability. The algorithm adaptively adjusts the step size during optimization: it provides strong global search capability when far from the optimum and accelerates convergence as the solution is approached. The LM algorithm is particularly effective for training small- to medium-scale feedforward neural networks, such as the one used in this study. Compared to conventional gradient descent methods, LM converges more rapidly while avoiding the numerical instability associated with inverting the Hessian matrix in classical Newton-based approaches.During training, the number of epochs is set to 500, the convergence tolerance is defined as , and the learning rate is set to 0.01.The LM-based training process ensures efficient and robust convergence of the neural network, effectively avoiding numerical instability and enabling reliable learning of the compensation mapping.
6. Simulation and Experiment
6.1. Simulation
6.2. Experiment
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Method | FD | Nullspace | Min–Max | MN |
---|---|---|---|---|
Avg Time (ms) | <0.1 | 5.22 | 9.43 | 0.12 |
Method | Position RMSE (m) | Attitude RMSE (deg) | Maximum Attitude Error (deg) | ||||||
---|---|---|---|---|---|---|---|---|---|
Roll | Pitch | Yaw | Roll | Pitch | Yaw | ||||
FD | 0.041 | 0.2625 | 0.1571 | 1.785 | 3.211 | 2.609 | 13.620 | 8.384 | 6.977 |
Min–Max | 0.0245 | 0.069 | 0.120 | 1.504 | 1.512 | 2.101 | 6.276 | 2.402 | 3.251 |
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Li, K.; Liu, M.; Li, X.; Yu, X.; Liu, K. Control Allocation Strategy Based on Min–Max Optimization and Simple Neural Network. Drones 2025, 9, 372. https://doi.org/10.3390/drones9050372
Li K, Liu M, Li X, Yu X, Liu K. Control Allocation Strategy Based on Min–Max Optimization and Simple Neural Network. Drones. 2025; 9(5):372. https://doi.org/10.3390/drones9050372
Chicago/Turabian StyleLi, Kaixin, Mei Liu, Xinliang Li, Xiaobin Yu, and Kun Liu. 2025. "Control Allocation Strategy Based on Min–Max Optimization and Simple Neural Network" Drones 9, no. 5: 372. https://doi.org/10.3390/drones9050372
APA StyleLi, K., Liu, M., Li, X., Yu, X., & Liu, K. (2025). Control Allocation Strategy Based on Min–Max Optimization and Simple Neural Network. Drones, 9(5), 372. https://doi.org/10.3390/drones9050372