domain: stat info

text:
Genane Youness

Nu Uyen

Thi Phan

Benjamin Cohen Boulakia

BootBOGS: Hands-on optimizing Grid Search in hyperparameter tuning of MLP

Keywords: Grid Search, Bayesian Search, hyperparameter tuning, MLP, Bootstrap

Neural networks are widely used in the literature in a variety of fields and for a large number of applications. A major challenge in their use is the need to identify and process hyperparametric values. Grid Search is a widely used technique for meeting this task. It systematically searches for values in a predefined range of hyperparameters. However, selecting the appropriate range of hyperparameters can be difficult, as the search space can be vast, resulting in an extensive number of combinations to be tested. It is more suited to short, fast searches for hyperparameter values, within ranges that are known to be generally efficient. In this paper, we present an improvement to Grid Search using BootBOGS, a bootstrap-based approach to hyperparameter optimization. BootBOGS is a hybrid approach that combines bootstrap and Bayesian Search with the Grid Search technique to perform an efficient search in hyperparameter space. Bayesian Search is used to initialize hyperparameter ranges. Bootstrap is used to explore the distribution of model performance for each hyperparameter combination and to reduce its variance, enabling us to better understand the margins of these hyperparameters and to reduce these ranges. Grid Search is then used to refine the selection of hyperparameters. To evaluate the effectiveness of the proposed approach, a set of computational experiments are carried out on four different datasets from classification problems, for which we compared BootBOGS to several other strategies: Grid Search, Random Search, and Bayesian Optimization. The results show that our method is able to find better hyperparameter configurations in terms of predictive quality with a reasonable runtime and lead to more robust and reliable hyperparameter tuning processes.

I. INTRODUCTION

In many areas of machine learning, neural networks and more specifically multi-layer perceptron neural networks are proving effective in solving the problem of classification. MLP classifiers require the use of hyperparameters that must be defined beforehand. The setting of these hyperparameters is a crucial phase in their development. It significantly influences a model's performance and its ability to generalize to new data. Among these techniques, Grid Search has always played a predominant role (Figure 1) and is widely used. However, it does have some limitations. As models become more complex and datasets increase in size, the limits of traditional Grid Search have become obvious. Despite its conceptual simplicity, Grid Search struggles with the challenges posed by intricate models operating in high-dimensional parameter spaces. The manual specification of hyperparameters and the lack of adaptability hinder its efficiency in finding optimal configurations.

To address this issue, the main contribution of this work is to propose a novel hybrid method called BootBOGS. BootBOGS combines bootstrapping on hyperparameters with Bayesian optimization and Grid search. On the basis of empirical performance models, BootBOGS is evaluated alongside established and widely used strategies, including Grid search, Random search, and Bayesian optimization, across four datasets. This evaluation encompasses various metrics, such as accuracy, AUC, G-mean, and F2. This comprehensive assessment reveals both the strengths and limitations of BootBOGS, offering valuable insights to inform future advancements and explorations in the field. Furthermore, we have made the code for this experiment open-source, with the aim of contributing to future developments.

II. CONTEXT AND RELATED WORK

Hyperparameter tuning for neural networks is a key area of research aimed at optimizing the performance and improving the efficiency of neural network models. Research continues to develop in this area, presenting innovative methods and tools to meet this challenge. [START_REF] Bergstra | Random search for hyper-parameter optimization[END_REF] presents Random Search as a simple and more efficient approach in hyperparameter optimization, compared to the Grid Search experiments of (2). ( 3) have developed the Hyperband algorithm, which uses a combination of random sampling and successive halving to efficiently allocate resources to different hyperparameter configurations.

(4) combines Hyperband with Bayesian Optimization in a multi-fidelity optimization method BHBO. Predictive hyperparameter optimization is used by [START_REF] Marinov | Hyperparameter optimisation with early termination of poor performers 1[END_REF] to predict the final performance of a machine learning system based on the early termination of poor performance. ( 6) presents an approach based on Bayesian Optimization with Gaussian Processes for selecting MLP model hyperparameters for state estimation purposes.

A. Problem of search space

The parameters are the weights of the connections in the neural network that are learned during the training stage. Hyperparameters are external configurations that cannot be estimated from data, and which are set by trainers and refined by tuning (( 7)). They must be carefully optimized to obtain the best possible performance. The choice can be made on the basis of previously published recommendations in the literature for the same data set, expertise or trial-and-error that can guide the selection of hyperparameters, particularly with limited training data.

Recent advancements in hyperparameter tuning have spurred the development of advanced strategies that enhance Grid Search techniques. To perform a search in the hyperparameters space, Grid Search builds a model for each combination of hyperparameter values and evaluates each model. However, it makes no assumptions about the search space; the user must specify a fixed range of values to search in. This selection plays a key role in the quality of the resulting model, as well as in the efficiency and convergence speed of the development process. The problem arises when dealing with hyperparameters which domains are continuous, leading to exponential growth in the combinations of possible values. The vast multidimensional space resulting from combinations of many hyperparameters presents an even greater challenge. The challenge is to select the right ranges for each hyperparameter. The most used optimization methods, namely Grid Search, Random Search, and Bayesian Search are described below.

B. Grid Search

Grid Search (GS) is a commonly used hyperparameter optimization technique that involves testing different combinations of hyperparameters in a cartesian way to determine the best one for a given problem. All combinations have the same probability of impact on the process. The range for each hyperparameter is generally determined by the developer, along with a step for moving from one hyperparameter value to another. The number of hyperparameter combinations grows exponentially with the search space ((1)), resulting in heavy computational costs. In addition, Grid Search may exploit many insignificant hyperparameter areas or fail to find the best hyperparameters if it is not included in the selected search space.

C. Random Search

Random search (RS) (8) is a hyperparameter tuning technique that involves randomly sampling hyperparameters from a defined search space and evaluating the algorithm's performance. It is a simple and efficient method of hyperparameter tuning (9) allowing a large hyperparameter search space to be explored with fewer iterations than Grid Search. Compared to Grid Search, Random search is particularly useful when the search space is large and the relationship between hyperparameters and model performance is complex (1).

D. Bayesian Search

Bayesian Optimization (BO) is an algorithm that effectively optimizes objective functions with high evaluation costs based on Bayes' Theorem ((10), [START_REF] Wu | Hyperparameter optimization for machine learning models based on bayesian optimization[END_REF]). BO significantly reduces the number of trials [START_REF] Yamashita | Crystal structure prediction accelerated by bayesian optimization[END_REF] and errors when tuning hyperparameters compared to Grid Search or Random Search [START_REF] Snoek | Practical bayesian optimization of machine learning algorithms[END_REF]. The key idea behind Bayesian optimization is to build a probabilistic model of the objective function [START_REF] Daulton | Bayesian optimization over discrete and mixed spaces via probabilistic reparameterization[END_REF], which is typically the validation accuracy or loss of the model. This probabilistic model is updated as new evaluations of the objective function are made, allowing the algorithm to intelligently choose the next set of hyperparameters on the basis of prior samples ((15)). ( 16) introduce a model-based sequential Bayesian optimization approach, called Tree-structured Parzen Estimator (TPE), It uses a tree-structured search process to iteratively split the space of hyperparameters into regions with a high and low probability of containing good hyperparameters and then evaluates the objective function at the best hyperparameters within the high-probability regions.

E. Overview

Table I provides an overview of the advantages and disadvantages of these different optimization methods.Each approach has merits that meet specific characteristics of the problem.

TABLE I: Advantages and disadvantages of optimization method.

Method

Advantages Disadvantages

Grid Search

Guaranteed to find the best combination if it exists within the selected search space. Simple and transparent.

Computationally expensive, especially with large search spaces. Inefficient when hyperparameters are not equally important. Limited to predefined hyperparameter values.

Random Search

Efficient in terms of computation time. Better exploration of the search space compared to Grid Search.

No guarantee of finding the best combination. Randomness may lead to suboptimal solutions in some cases

Bayesian Search

Efficient and intelligent exploration of the search space. Typically requires fewer iterations to find good hyperparameters.

More complex to implement than Grid and Random Search. May require tuning additional parameters for the probabilistic model.

In short, Random Search presents itself as a quick and direct approach to explore hyperparametric spaces, but with randomness in optimization. Grid Search offers full coverage but can extend calculation time. Bayesian Optimization establishes a trade-off by using probabilistic models to strategically maneuver the research space and adjust its approach based on historical assessments. The choice of the most appropriate technique depends on factors such as the complexity of the problem, the available computing resources, and the delicate balance between exploration and exploitation.

III. METHODOLOGY

The motivation behind developing a new hyperparameter optimization method is rooted in the limitations of existing methods. This novel approach aims to combine their strengths, and thus optimize classical Grid Search. The method we propose, BootBOGS, uses bootstrap with Bayesian Search to select efficient hyperparameter ranges for Grid Search, in order to perform an efficient search in hyperparameter space. In the following, we provide information and details about the BootBOGS methodology used on the MLP-based model.

A. Selection of MLP's hyperparameters

MLP's hyperparameters are selected from three distinct types: those pertaining to the network's structural configuration, those governing the learning and optimization process, and those responsible for introducing regularization effects. In our study, we deliberately singled out one key hyperparameter from each of these categories. Within the first category, we elected to manipulate the number of nodes in each layer, recognizing that this choice is intrinsically intertwined with the nature of the dataset, including factors like the number of hidden layers and the activation functions applied. In the second category, our focus was on hyperparameters such as the learning rate, loss function, batch size, optimizer, and loss function. Among these, we gave precedence to the learning rate, as the others primarily relate to the dataset's feature correlations. Lastly, for the third category, we considered the dropout rate, primarily due to its linkage with feature correlations within the dataset, denoted as Lambda. These meticulous selections aim to optimize our neural network's performance by fine-tuning critical aspects of its configuration and learning process.

B. Bootstrap Resampling

Bootstrap ( 17) is a statistical inference technique that has become increasingly popular in recent years, due to its versatility and ability to provide reliable estimates of uncertainty in statistical analyses. It is widely used to estimate the sampling distribution of a quantity or to build confidence intervals by generating new data sets through resampling with the replacement of the original data set. Percentile bootstrap confidence intervals are a valuable statistical tool that can be used to estimate the uncertainty or variability of sample statistics such as mean. Achieved through resampling from the original dataset, these intervals come into their own when the true distribution of data remains unknown or when dealing with small sample sizes. One of the remarkable attributes of these intervals is their applicability across a spectrum of scenarios, regardless of the underlying distribution's complexity. In the realm of hyperparameter optimization, these confidence intervals offer an insightful approach to identifying the optimal parameter values.

C. Proposed Method

The algorithm 1 gives us an overview of the step-by-step process of the proposed method. First, after splitting the dataset into trainning and testing sets, Bootstrap resampling is applied, iterating this process n times on the training set. It appears as a dynamic tool, allowing a complete exploration of performance variations through various combinations of hyperparameters. This iterative technique not only provides a holistic understanding of the ranges in which these hyperparameters work but also facilitates the identification of optimal values that give better model performance. Here, We choose n equal 10. At each iteration, Bayesian optimization, and more specifically the tree structure of the Hyperband Parzen (TPE) estimator, is used with HyperOpt, a hyperparametric optimization library, to identify optimal parameter combinations in the resampled training set. HyperOpt TPE efficiently navigates through the hyperparameter space, rapidly converging toward promising parameter regions. Subsequently, this optimal combination is used within the test set to compute the AUC score. From these values, we derive a confidence interval that provides a measure of the statistical significance of our results. At this step, striking the delicate balance between hyperparameters and model performance requires a discerning strategy. A bootstrap confidence interval of the 68% percentile of AUC values is used within a region of ± 1 standard deviation. This confidence interval provides a region within ±1 standard error of the mean. This narrower interval provides a reliable range in which optimal hyperparameters are likely to reside. Expanding the perspective and opting for a broader CI 95% could also be considered, but should not be taken lightly, since it has the potential to inadvertently dilute the precision of the analysis. We proceed with a second round of bootstrap sampling, employing the new search space to iteratively fine-tune the hyperparameters. This dual-iteration process substantially narrows down the scope of exploration, leading to a more focused and efficient search for optimal hyperparameter values. The Grid Search is then used on the new reduced search space to select the best hyperparameters from the search space.

IV. EXPERIMENTAL RESULTS AND EVALUATION

In this section, we detail the experiments used to validate the proposed BootBOGS method on four available real datasets from various application domains. We compare BootBOGS' results to three traditional approaches, classical Grid Search, Random Search, and Bayesian Optimization. The proposed method is also compared with the most recent studies carried out on the same data sets.

A. Experimental Settings

BootBOGS is coded in Python language using the packages sci-kit-learn, TensorFlow, and Keras. The training model was run on a computer with Intel(R) Core(TM) i5 -1.70GHz processor and 8GB RAM. The main hyperparameters of the model are shown in Table II. As shown in Table III, BootBOGS focuses on the value intervals of the following three hyperparameters: dropout, learning rate, and number of neurons in the hidden layer. The number of neurons in the experiments is chosen according to the rule of thumb ( 18) that considers the number of neurons as a power of two, 2 k , and the maximum number of hidden neurons as 2 k-1 , where k is the number of input elements. This approach can help simplify the calculation of the number of neurons and reduce the number of values to consider during hyperparameter tuning. We've established a dropout range between 0.0 and 0.5 to strike a balance between retaining valuable information and preventing overfitting. The code of our experiments can be accessed with the following link: https://github.com/thiphan94/BootBOGS.

B. Datasets

We use four datasets of binary classification tasks in order to validate BootBOGS. The first dataset, known as the Diabetes PIMA dataset [START_REF]Pima Indians Diabetes[END_REF], sourced from Kaggle, comprises 768 instances, characterized by eight characteristics including glucose level, blood pressure, and body mass index (BMI), together with a target variable that denotes the presence or absence of diabetes.

The second dataset, the German Credit dataset [START_REF]South German Credit[END_REF], is related to the evaluation of credit risk. Comprising 1,000 instances, each housing 21 attributes, this dataset is instrumental in determining if a loan applicant is a favorable or risky credit candidate.

The third dataset, the Taiwan Bankruptcy Prediction dataset [START_REF]Taiwanese Bankruptcy Prediction[END_REF], encapsulates financial ratios extracted from Taiwanese companies' annual reports. With 6819 instances and 96 attributes, this dataset is commonly used to predict whether a company is likely to go bankrupt.

The fourth dataset used is the Polish Bankruptcy dataset from the third year [START_REF] Tomczak | Polish companies bankruptcy data[END_REF]. This compilation of financial indicators is designed to predict corporate bankruptcy. Sourced from the UCI Machine Learning Repository, it encompasses data from Polish companies, featuring 10,503 instances and 64 attributes.

Table IV lists the number of attributes, the number of instances, and the relative size of the majority class for each dataset. The flowchart in Figure 2 gives us a visual representation of the step-by-step process of data preprocessing.

In the context of data preprocessing, missing value imputation is a critical process where missing data points in a dataset are filled using various techniques. In some datasets, we chose to use median imputation as a method to handle missing values. This approach involves replacing missing values with the median value of the corresponding feature. Moving on to another preprocessing step, the identification and management of outliers plays a significant role in the preparation of a data set for machine learning models. Outliers, which are data points substantially different from the majority, can distort relationships between features and the target variable, negatively affecting model performance. To address outliers in the diabetes dataset, we adopted the Interquartile Range (IQR) method. This method involves identifying observations with values greater than 1.5 times the IQR and classifying them as outliers. These outliers are then removed from the dataset to prevent undue influence on the subsequent modeling process.

The next pre-processing step is feature selection, a crucial process to improve the interpretability and efficiency of the model. For this dataset, we used three distinct techniques for feature selection: random variable, permutation, and SHAP. Employing the Random Forest algorithm, a widely used technique for gauging feature importance, we adopted three strategies:

• Built-in Scikit-learn Method with a Random Feature: This approach leverages Scikit-learn's built-in feature selection method using Random Forest. It identifies pertinent features that significantly contribute to the model's predictive capability. A random feature is introduced as a reference point against which the importance of the actual features is assessed. The lower importance of a real feature compared to the random feature might indicate a chance-driven significance, necessitating further scrutiny. • Permutation Feature Importance: This technique involves shuffling the values of a single feature and assessing its impact on the model's performance. Features demonstrating substantial influence on model performance are regarded as important for prediction. • Random Forest Feature Importance with SHAP: SHAP is a comprehensive approach for interpreting machine learning model outputs. It evaluates the importance of features by considering all possible combinations of features. This method provides a nuanced understanding of each feature's contribution to model predictions. Furthermore, since all datasets are imbalanced, meaning that there are significantly more instances of negative cases (class 0) in the datasets than positive ones (class 1), we applied SMOTE-Tomek [START_REF] Batista | Balancing training data for automated annotation of keywords: a case study[END_REF] which combines the strengths of SMOTE and Tomek links. It first applies SMOTE to over-sample the minority class, creating synthetic instances. Then, Tomek links are used to identify and remove ambiguous samples that might cause misclassification. The result is a balanced dataset that preserves the distinctive characteristics of both classes. SMOTE-Tomek helps models generalize better by providing a more balanced training set, reducing bias, and improving their ability to capture the minority class. SMOTE-Tomek is applied to datasets with a high majority class ratio, where the relative size of the majority class exceeds the critical threshold of 70% with scaling features. Note that before applying the resampling technique, each dataset is split into 80% training set and 20% test set.

BootBOGS is evaluated using common metrics: accuracy, AUC, G-Mean, and F β score for imbalanced datasets. Accuracy, calculated as the ratio of correctly classified instances to the total, tends to favor the majority class. As such, it can lead to suboptimal hyperparameter settings that neglect the minority class and fail to capture rare but critical events. So, for an imbalanced dataset, accuracy cannot be considered a reliable measure. Complementary metrics are used, including Area Under the Receiver Operating Characteristic Curve (AUC), Geometric Mean (G-Mean), and another measure F β .

Geometric Mean (G-Mean) is defined as the square root of the product of true positive rate (or recall) and true negative rate (or specificity) and is formulated as follows:

G -M ean = √ T P R × T N R (1)
Where true positive rate (TPR) and true negative rate (TNR) are :

T P R(Recall) = T P T P + F N (2)
T N R(Specif icity) = T N T N + F P

F β score is a weighted harmonic mean of precision and recall. The formula of F β is as follows [START_REF] He | Imbalanced learning: foundations, algorithms, and applications[END_REF]:

F β -score = T P T P + 1 1+β 2 (β 2 F N + F P ) (4)
Where TP, FP, and FN present the counts of true positives, false positives, and false negatives, respectively. In our experiments, we set the beta value beta for the F β score to 2. This choice of F2 score allows us to emphasize the importance of recall over precision. Through the systematic application of these evaluation metrics, we obtained a comprehensive understanding of how well BootBOGS performs hyperparameter optimization. These metrics not only allow us to quantify the quality of the resulting models but also provide insights into their adaptability across various scenarios, including situations with imbalanced class distributions. The insights gained from these metrics play a crucial role in validating the effectiveness of our approach and its potential to redefine the landscape of hyperparameter optimization for machine learning models.

V. RESULTS AND DISCUSSION

In this section, we evaluate the performance of BootBOGS in various datasets using the metrics presented previously in order to see how the proposed approach is an improvement compared to the classical Grid search. The results in TableV show the performances of the Grid Search method, Random Search method, Bayesian method, and BootBOGS for the datasets used in experiments. From the results shown in Table V, we can see that Boot-BOGS always identifies hyperparameter values that yield better results than traditional Grid Search across the four datasets, evaluated based on AUC, Accuracy, and G-mean metrics and F2 score. In the case of the Taiwan Bankruptcy and Polish Bankruptcy datasets, BootBOGS' performance is not the best when compared to Random Search and Bayesian. However, the achieved performance remains very close to theirs. The fact that those datasets are significantly imbalanced class distribution (96.77% for Taiwan Bankruptcy, 95.3% for Polish Bankruptcy, as opposite to Diabetes with 65.1%, and Diabetes with 65.1%) suggests that in scenarios where the majority class dominates to such a degree, alternative optimization methods may be more effective. Conversely, across all four optimization methods, our approach consistently excels when class imbalances are less pronounced.

A common performance statistic for assessing the effectiveness of binary classification models, particularly when the data is unbalanced, is the AUC-PR score (Area Under the Precision-Recall Curve) [START_REF] Davis | The relationship between precision-recall and roc curves[END_REF]. It evaluates a model's ability to distinguish between the positive and negative classes by taking into account precision and recall. The AP score, which we now know also represents the AUC-PR score, is shown in Figure 4. In terms of speed of execution, figure 3 shows that Boot-BOGS always reduces execution compared to Grid Search, although it still takes more time than Random Search and Bayesian. Time gain over Grid Search becomes quite significant for the Diabetes dataset (54.7%) and for the German Credit dataset (37.3%).

A comparison of BootBOGS with previous research is presented in Table VI. It shows the efficiency of our method compared to other studies with the same datasets (information such as hyperparameters values or performance metrics for those works are provided when made available) by authors. BootBOGS improves most of the performance metrics, even though it uses the simple model of a neural network with only one hidden layer and a small search space for hyperparameter tuning.

VI. CONCLUSION

The proposed method BootBOGS presented in this paper is an improved technique compared to Grid Search. The outcomes of our study underscore the significance of intelligent hyperparameter tuning in enhancing the predictive capabilities of machine learning models. By strategically incorporating bootstrap sampling, we have not only streamlined the search process but also injected diversity into the hyperparameter space, enabling more robust exploration.

Our experimental study has several limitations. One limitation of our study is that our method was developed in the context of binary classification. Acknowledging the binary classification limitation, we fine-tuned our approach to excel

Fig. 1 :

 1 Fig. 1: Popularity index over years 1

Algorithm 1 :

 1 BootBOGS Split dataset, utilize only the train dataset for the bootstrap step Choose the range of search space for hyperparameters repeat 2 times 1) repeat 10 times -Bootstrap sampling train set with Stratified K-Folds cross-validator -Apply HyperOpt TPE and store the combination of hyperparameters -Get the AUC score performance for the test set end 2) Calculate 68% confidence interval from the list of scores (AUC) 3) Remove hyperparameter values combinations that have score out of this confidence interval 4) Create a new search space of HyperOpt TPE based on this new list of hyperparameters end Use the new reduced search space for Grid Search

Fig. 2 :

 2 Fig. 2: Data Preprocessing.

Fig. 3 :

 3 Fig. 3: Execution time (seconds) of four method through various datasets

Fig. 4 :

 4 Fig. 4: Precision Recall curves of Grid Search and BootBOGS through four datasets

TABLE II :

 II Main model hyperparameters.

	Hyperparameter	Hyperparameter Value
	Kernel	normal
	Activation function	Relu
	Activation function in output layer	sigmoid
	Batch size	32
	Epoch	100
	Optimizer	Adam
	Loss function	binary cross entropy

TABLE III :

 III Hyperparameters range for experiments.

	Hyperparameter	Min value Max value
	Dropout rate	0.0	0.5
	Learning rate	0.001	0.1
	number of neurons	4	144

TABLE IV :

 IV Datasets used in our experiments.

	Dataset	Attrs	Size	Majority class(%)
	Diabetes (19)	8	768	65.1
	German Credit (20)	21	1000	70
	Taiwan Bankruptcy (21)	96	6819	96.77
	Polish Bankruptcy (third year) (22)	64	10503	95.3

TABLE V :

 V Evaluation metrics of methods in various dataset.

	Dataset	Method	AUC	Accuracy	G-Mean	F2
		GS	0.877	0.889	0.877	0.837
	Diabetes	RS	0.901	0.913	0.90	0.867
		BO	0.877	0.889	0.877	0.837
	BootBOGS	0.914	0.921	0.914	0.888
	German Credit	GS RS BO	0.71 0.691 0.705	0.77 0.755 0.755	0.704 0.673 0.695	0.591 0.54 0.585
	BootBOGS	0.735	0.79	0.723	0.612
	Taiwan Bankruptcy	GS RS BO	0.64 0.75 0.74	0.942 0.932 0.937	0.554 0.732 0.72	0.301 0.468 0.465
	BootBOGS	0.67	0.944	0.61	0.361
	Polish Bankruptcy	GS RS BO	0.716 0.762 0.77	0.950 0.94 0.955	0.667 0.736 0.742	0.458 0.483 0.514
	BootBOGS	0.757	0.953	0.726	0.556

Note: GS: Grid Search, RS: Random Search, BO: Bayesian search.

TABLE VI :

 VI Results comparison of different approaches using various datasets.

Source: Google Trends for Three Hyperparameter Tuning Methods in Computer Science Worldwide from

to 2023 [accessed 4Sep, 2023]

Another key direction for our work is the extension of BootBOGS to multi-class classification problems. This is a significant step forward in making BootBOGS capable of addressing real-world problems. Its robustness and adaptability allow us to identify the challenges associated with its use in a regression framework. This limitation offers a clear path for future research.

Moreover, the promising results obtained in this work pave the way for several avenues to test the approach on complex data such as images or text.

In our initial efforts, we fine-tuned just three key hyperparameters: the dropout rate, the learning rate, and the number of neurons in the hidden layer. As our research advances and we delve into more complex models, our plan is to broaden our optimization efforts, encompassing a wider array of hyperparameters, in particular the number of hidden layers. This expansion will further refine and enhance the performance of our machine learning models.

And finally, the study focuses on MLP hyperparameters. In terms of future work, a promising direction could be to extend BootBOGS to other machine learning methods.


Atharva Rajesh Peshkar

Danna Gurari

Sergi Pujades

Michael Black

David Thomas

Computer vision assisted alignment for stereotactic body radiation therapy (SBRT)

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Innovation/Impact: Surface Guided Radiation Therapy (SGRT) techniques have been introduced to improve patient alignment without increasing imaging dose. However, current SGRT techniques are limited in their accuracy to track internal tumor/organ motion when the target is not close to the surface. Additionally, higher BMI has been linked to decreased accuracy of SGRT, raising a health-equity issue for patient populations with already reduced outcomes. Current SGRT matching algorithms underestimate motion of internal anatomy for flat chested patients and overestimate motion of internal anatomy for large BMI patients. Realistic body modelling will overcome current issues with lack of anatomic variation which can lead to poor correlation and large random errors in SGRT.

Fig1 shows an example of skeletal misalignment between days of treatment for a pelvic patient based on a typical SGRT matching algorithm, due to the limited field of view available to the SGRT algorithm. Realistic fullbody modelling has the potential to overcome current issues due to lack of surface anatomic variation which can lead to poor correlation and large random errors.

This project consists of two major innovations. Our first innovation is the use of a novel patient-specific skin mesh model to enable marker-less 3D human anatomy tracking. Our patient-specific skin mesh model is based on a recently published Sparse Trained Articulated Human Body Regressor (STAR), which is a realistic 3D model of the human body learned from >10,000 3D body scans. It allows estimation, synthesis, and analysis of 3D human pose and shape with sub-centimeter accuracy (Fig. 2). Our second innovation is an improved internal to external anatomical model that fuses patient specific body models to their internal skeletal anatomy, improving the correspondence between surface imaging and internal anatomy. We have adapted the technique outlined in Keller et al, 2022, using a neural network trained on 1000s of dual-energy X-ray absorptiometry (DXA) scans to predict the skeleton from the body surface when patients are in the treatment position (Fig. 3b-c). Accounting for patient pose and body type will be particularly valuable for overcoming two health-equity issues of current SGRT techniques, Key Results: 3D human pose and shape can be acquired in real time with sub-centimeter accuracy from video inputs (Fig 12). Skeletal anatomy can be predicted from surface imaging to enable alignment to a patient's day-of-treatment conebeam CT (CBCT) and improving the correspondence between surface imaging and internal anatomy (

  Fig 3). This technique can be also used to extract breathing motion for gating purposes as shown in Fig 3(h), with clear differences in the rib positions between deep inhalation and exhalation (XCAT phantom data shown).

Figure 1 .

 1 Figure 1. An example of skeletal misalignment based on current SGRT matching algorithms between days of treatment for a pelvic patient.

Figure 2 .

 2 Figure 2. (a) camera system was synchronized and calibrated using a 50cmx50cm checkerboard to account for intrinsic and extrinsic camera distortions. (b) real-time 3D pose was extracted from multiple 2D images of a healthy volunteer in various treatment positions. (c) each pose was calculated and mapped to a 3D avatar with matching gender and BMI.

Figure 4 .

 4 Figure 4. (a) Flow chart showing the method for inferring the skeleton from surface. (b) example of a surface acquired from CT. (c) model input data. (d) optimization of skeletal position. (e) inferred skeleton from (a). (f-g) method showing patient in treatment position in the virtual treatment room. (h) breathing motion can be extracted from skeletal anatomy; overlap of deep inhalation vs exhalation from the XCAT phantom shown.

Figure 3 .

 3 Figure 3. (a) a virtual patient positioned on the treatment couch in a virtual treatment room environment. (b) one of five virtual camera views monitoring patient pose. (c) the overlay of the 3D avatar on the 2D image in (b). (d) real time tracking of avatar, couch, and gantry motion during treatment.


Mohamad Hachem
email: mohamad.hachem@enac.fr

Thierry Miquel
email: thierry.miquel@enac.fr

Murat Bronz
email: murat.bronz@enac.fr

Clément Roos
email: clement.roos@onera.fr

Trajectory Optimization for Fully Actuated Hexacopters: Enhancing Maneuverability and Applications

Trajectory optimization is a challenging task in the robotics community. Several factors need to be taken into account when generating a feasible optimized trajectory. The optimization process heavily relies on the dynamic model of the system. Currently, there are various drone designs available, categorized based on their actuation status. In this study, we apply a trajectory optimization technique to a fully actuated hexacopter (FA-Hex), which is a new application to the best of our knowledge. This type of vehicle has been successfully integrated into several practical applications. Unlike the under-actuated hexacopter (UA-Hex), the FA-Hex can perform maneuvers with minimal banking angles, significantly enhancing the drone's maneuverability. Our research focuses specifically on trajectory optimization for the FA-Hex and demonstrates the adaptability of our method to different scenarios. We discuss two specific applications: a drone filming without a gimbal joint and a drone with a cable-suspended pendulum. We compare the simulation results with the UA-Hex model to highlight the differences in maneuverability between the two systems. The trajectory optimization is performed offline using CasADi in the MATLAB framework.

INTRODUCTION

Aerial robotic systems have found extensive applications in various fields, including package delivery, photography, rescue operations, and construction inspection [START_REF] Sanchez-Cuevas | Fullyactuated aerial manipulator for infrastructure contact inspection: Design, modeling, localization, and control[END_REF], [START_REF] Chamberlain | Design and implementation of a quadcopter drone control system for photography applications[END_REF]. These missions have led to the development of different types of vehicles tailored to meet specific requirements. Nowadays, aerial systems utilize cable-suspended loads for package delivery in various environments. Additionally, vehicles de-signed for photography and inspection purposes often integrate a gimbal joint to ensure optimal camera positioning. In these cases, minimizing drone weight and optimizing trajectory and energy consumption are essential objectives. However, striking a balance between control and vehicle design is critical to achieving optimal performance.

Aerial vehicle systems can be classified based on their actuation level as under-actuated (UA), fully-actuated (FA), and over-actuated (OV) systems [START_REF] Hamandi | Design of multirotor aerial vehicles: A taxonomy based on input allocation[END_REF]. A quad-rotor drone serves as a common example of an UA system, which relies on total thrust and three torques as virtual control inputs. However, these virtual control inputs are insufficient to directly control all 6 degrees of freedom (DoF) of the system. On the other hand, a FA system possesses a number of virtual controllers equal to the DoF, matching the number of actuators. Conversely, OV drones have an excess number of actuators, enabling them to handle actuator failures using various control allocation techniques. FA vehicles have a more extensive range of maneuvers compared to UA ones. Some drones can adapt to online actuation changes by employing additional servo motors that consider the tilt angle of the motors. Indeed, hexacopters can be classified as either UA or FA systems based on their actuator configuration. Hexacopters with actuators in one plane, which typically means they have fixed propellers, are considered UA systems. On the other hand, hexacopters equipped with tilting actuators, often known as coaxial hexacopters or hexacopters with tilting propellers, are recognized as FA systems. The ability of the latter to tilt their actuators provides them with greater control authority and maneuverability, allowing them to be fully actuated and perform more complex maneuvers compared to the former. This work's primary contribution lies in its flexible approach to solving trajectory optimization problems, which can be adapted to suit a wide range of drone applications. The emphasis lies on scenarios where drones track specific objects through photography. In such cases, it is vital for the drone to maintain a clear field of view of the target by moving with minimal banking angles. Improving the trajectory sent to the low-level controller enhances the drone's ability to accomplish assigned tasks effectively. Additionally, the FA-Hex can be used for package delivery with the help of a cable-suspended pendulum mechanism. This setup enables http://www.imavs.org/ IMAV2023-28 14 th ANNUAL INTERNATIONAL MICRO AIR VEHICLE CONFERENCE AND COMPETITION the drone to reduce oscillations and perform precise maneuvers, even in challenging and static environments.

Design and Control Strategies

Various vehicle design and control strategies have been developed and utilized to manage the dynamics of FA drones. Employing these vehicles offers the key advantage of independent attitude and altitude control by decoupling their dynamics [START_REF] Rashad | Fully actuated multirotor UAVs: A literature review[END_REF].

Non-linear control based on full geometric control, ensuring precise position tracking, is applied by [START_REF] Franchi | Full-pose tracking control for aerial robotic systems with laterally bounded input force[END_REF] and [START_REF] Ryll | 6D physical interaction with a fully actuated aerial robot[END_REF]. Stability of the resulting controller is established using Lyapunov techniques. Furthermore, [START_REF] Rajappa | Modeling, control and design optimization for a fully-actuated hexarotor aerial vehicle with tilted propellers[END_REF] proposes an optimal design for the tilted angle based on the active set of possible wrenches exerted by the FA drone, using a feedback linearization control approach with a PID controller to decouple position and orientation control. Another approach is presented in [START_REF] Odelga | A fully actuated quadrotor UAV with a propeller tilting mechanism: Modeling and control[END_REF], by adding a tilting mechanism to the motors of the UA drone, incorporating two servo actuators to rotate the rotor's angle in radial and tangential directions around the arm. Feedbacklinearization control is employed for managing the drone's dynamics. To handle environmental perturbations, [START_REF] Arizaga | Adaptive control for a tilted-motors hexacopter UAS flying on a perturbed environment[END_REF], uses adaptive control for the FA-Hex drone. Similarly, [START_REF] Arizaga | Payload swing attenuation of a fully-actuated hexacopter via extended high gain observer based adaptive sliding control[END_REF] proposes adaptive sliding mode control for FA-Hex drones with a cable-suspended pendulum, compensating for the inability to measure the load's position by developing an extended gain observer-based adaptive sliding mode approach.

Trajectory Optimization

Generating a reference trajectory to be tracked by the lower-level controllers is a demanding task when working with FA vehicles. Currently, the literature review on trajectory optimization primarily focuses on UA vehicles. Therefore, it becomes a challenging problem to develop an optimal controller that generates desired positions and orientations for FA vehicles. The research in this area can be categorized into two main approaches. Firstly, a differential flatness approach is utilized to obtain a smooth and feasible trajectory [START_REF] Mellinger | Minimum snap trajectory generation and control for quadrotors[END_REF][START_REF] Hehn | Quadrocopter trajectory generation and control[END_REF][START_REF] Zeng | Differential flatness based path planning with direct collocation on hybrid modes for a quadrotor with a cable-suspended payload[END_REF][START_REF] Sreenath | Geometric control and differential flatness of a quadrotor UAV with a cable-suspended load[END_REF]. Secondly, the dynamical model is incorporated as a constraint in the optimization problem [START_REF] Falanga | Fast trajectory optimization for agile quadrotor maneuvers with a cable-suspended payload[END_REF][START_REF] Hashemi | Trajectory planning of quadrotor UAV with maximum payload and minimum oscillation of suspended load using optimal control[END_REF][START_REF] Potdar | Online trajectory planning and control of a MAV payload system in dynamic environments[END_REF]. Utilizing the differential flatness approach in formulating the optimization problem has shown to reduce computational time. It has been demonstrated by [START_REF] Sreenath | Geometric control and differential flatness of a quadrotor UAV with a cable-suspended load[END_REF] that the UA quad-rotor drones are differential flat systems, which can be characterized by the following flat output vector: position (x, y, z) and heading (ψ), of the system. In contrast, FA vehicles are considered to be differential flat based on the full dynamics of the drone. In our approach, we have chosen to incorporate the dynamical system as a constraint in the optimization problem. Differential flatness is utilized to generate the trajectory of an UA vehicle, which is then used as an initial condition for the optimization problem of the FA-Hex vehicle. This overview is the basis of our contribution to the study of a trajectory optimization problem for FA vehicles, taking into account specific objectives.

We organize the remainder of the paper as follows. In Sections 2.1 and 2.2, we present the mathematical model of the FA-Hex and the extended model that incorporates a cablesuspended pendulum, and in Section 3, we formulate the proposed trajectory optimization problem. In Section 4 the simulation results are presented based on the defined optimization problem.

𝑥 𝐼 𝑦 𝐼 𝑧 𝐼 𝑥 𝐵 𝑦 𝐵 𝑧 𝐵 𝛼 𝐿 𝛽 𝐿

MODEL AND SYSTEM VARIABLES

The importance of the dynamical system in achieving workable results, as mentioned earlier, makes it necessary to define the mathematical model of the system we're looking at. In this section, we present the dynamic model of the FA-Hex and its extension to include the cable-suspended pendulum. The model is derived from Newton's Euler equation, while taking into account the following assumptions:

• The drone is assumed to be a rigid body.

• The drone is assumed to be symmetric with respect to its axis, and its inertia matrix is diagonal and denoted as

I B =   I xx 0 0 0 I yy 0 0 0 I zz   .

FA-Hex Modeling

The modeling approach of the drone involves two main frames: the inertial frame (F I ) defined by (x I , y I , z I ) axis, and the body frame (F B ) defined by (x B , y B , z B ) axis, as illustrated in Figure 1. The rotation between F I and F B is defined by the Euler angles µ = [ϕ, θ, ψ] T and the corresponding rotational matrix R(µ) where: http://www.imavs.org/ IMAV2023-28

14 th ANNUAL INTERNATIONAL MICRO AIR VEHICLE CONFERENCE AND COMPETITION R(µ) =   C ψ C θ C ψ S θ S ϕ -S ψ C ϕ C ψ S θ C ϕ + S ψ S θ S ψ C θ S ψ S θ S ϕ + C ψ C ϕ S ψ S θ C ϕ -C ψ S ϕ -S θ C θ S ϕ C θ C ϕ  
with the notations C * = cos( * ) and S * = sin( * ).

The position of the drone in the inertial frame is denoted as ξ = [x, y, z] T , and its translation velocity is represented by v = [v x , v y , v z ] T . The angular velocity in the body frame is given by Ω = [p, q, r] T . The relationship between the Euler rates μ and the angular body rates Ω is expressed by the transformation matrix W (µ), where μ = W (µ

) -1 Ω such that   φ θ ψ  =   1 S ϕ T θ C ϕ T θ 0 C ϕ -S ϕ 0 S ϕ /C θ C ϕ /C θ     p q r  
where T * = tan( * ). The states of the FA-Hex are defined as x = [ξ T , v T , µ T , Ω T ] T , and the system dynamics are expressed as follows:

ξ = v (1)
m ξ = mg + R(µ) 6 i=1 F Bi (2)
I B Ω = 6 i=1 τ Bi -Ω × I B Ω (3)
Here, × is the cross product operator, whereas

F Bi = [F xi F yi F zi ] T and τ Bi = [τ xi τ yi τ zi ]
T represent the forces and torques exerted by the i-th motor expressed in the body frame, respectively. F Bi and τ Bi depend on the propeller's angular speed ω i , as well as the motor's orientation α i within the drone's geometric frame, and the tangential orientation β defined as the angle between the motor's axis z mi and the drone's z B axis. This is represented in Figure 2, where

F i = F 2 xi + F 2 yi
is the projection of F Bi in the (x B ,y B ) plane. The expressions of F Bi and τ Bi are given in Table 1. It can be checked that F Bi = F thrusti , where ∥F thrusti ∥ = k t ω 2 i , while τ Bi = τ thrusti + τ dragi , where ∥τ thrusti ∥ = k t lω 2 i and ∥τ dragi ∥ = k q ω 2 i . k t , k q and l stand for the thrust coefficient, the drag coefficient, and the distance between the i-th motor and the drone's center of mass respectively.

FA-Hex Modeling with Cable Suspended Pendulum

The FA-Hex system with the cable-suspended payload operates in a hybrid mode, which relies on the tension force in the cable, as discussed in [START_REF] Sreenath | Geometric control and differential flatness of a quadrotor UAV with a cable-suspended load[END_REF]. The payload's behavior is modeled under two main scenarios: taut and slack cables. The transition between these modes adds intricacy to the modeling approach, thus augmenting the complexity of the optimization function. Nevertheless, by introducing a linear complementary constraint into the optimization problem, it is feasible to eliminate the necessity for the hybrid model [START_REF] Falanga | Fast trajectory optimization for agile quadrotor maneuvers with a cable-suspended payload[END_REF].

Fx 1 = -ktS β Sα 1 ω 2 1 τx 1 = (-ktC β Sα 1 l + kqS β Sα 1 )ω 2 1 Fy 1 = ktS β Cα 1 ω 2 1 τy 1 = (ktC β Cα 1 l -kqS β Cα 1 )ω 2 1 Fz 1 = -ktC β ω 2 1 τz 1 = (ktS β l -kqC β )ω 2 1 Fx 2 = ktS β Sα 2 ω 2 2 τx 2 = (-ktC β Sα 2 l + kqS β Sα 2 )ω 2 2 Fy 2 = ktS β Cα 2 ω 2 2 τy 2 = (ktC β Cα 2 l + kqS β Cα 2 )ω 2 2 Fz 2 = -ktC β ω 2 2 τ z2 = (-ktS β l + kqC β )ω 2 2 Fx 3 = -ktS β Sα 3 ω 2 3 τx 3 = (-ktC β Sα 3 l + kqS β Sα 3 )ω 2 3 Fy 3 = -ktS β Cα 3 ω 2 3 τy 3 = (-ktC β Cα 3 l + kqS β Cα 3 )ω 2 3 Fz 3 = -ktC β ω 2 3 τz 3 = (ktS β l -kqC β )ω 2 3 Fx 4 = -ktS β Sα 4 ω 2

Table 1: Forces and Torques Projections

The state vector of the model is expanded and modified to incorporate both the position and velocity of the load. This state is denoted as

x = [ξ T , v T , µ T , Ω T , ξ T L , v T L ] T , where ξ L = [x L , y L , z L ]
T represents the load's position in the inertial frame, and v L = [v Lx , v Ly , v Lz ] T represents the load's velocity. With the presence of the load, a new tension force T comes into play, affecting the drone's dynamics. Furthermore, due to the cable's rotation around the inertial axis, a unit vector describing a rotation R L (η) is introduced to take account of the drone's orientation in relation to the load. Here, η = [α L , β L ] T , as depicted in Figure 1.

R L (η) =   sin(α L ) cos(β L ) sin(α L ) sin(β L ) cos(α L )  
The extended dynamic model is then represented as follows:

ξ = v (4
(m + m L ) ξ = (m + m L )g + R(µ) 6 i=1 F Bi + T R L (η) (5)
I B Ω = 6 i=1 τ Bi -Ω × I B Ω (6)
ξ L = ξ + R L (η)L (7) ξL = v L (8) ξL = m L g -T R L (η) ( 9
)
where T is the tension in the cable, L is the length of the cable, and m L is the mass of the load respectively.

TRAJECTORY OPTIMIZATION PROBLEM

Optimization methods are employed to address trajectory optimization problems by minimizing a cost function, while taking into account different constraints. To achieve a viable and physically relevant solution, the optimization problem needs to encompass the system dynamics along with the practical restrictions on the states and inputs. Broadly, optimization problems fall into two primary categories: direct methods [START_REF] Posa | Direct Trajectory Optimization of Rigid Body Dynamical Systems through Contact[END_REF] and indirect methods [START_REF] Hashemi | Trajectory planning of quadrotor UAV with maximum payload and minimum oscillation of suspended load using optimal control[END_REF].

The indirect method involves formulating the optimization problem as a boundary value problem (BVP) and solving it by defining the co-state vector. It is worth mentioning that the model of the system is incorporated into the Hamiltonian-Jacobian-Bellman equation, which is solved using the BVP approach. On the other hand, the direct optimization problem takes into account a discrete dynamical model along with its constraints. The main distinction between the two methods lies in the integration aspect. Direct methods are known for incorporating integrated solutions for the dynamical model within their approach. It is worth noting that indirect methods generally tend to be more accurate than direct methods, provided that a good initial guess can be obtained, which can be challenging in practice. However, in the realm of robotics applications, direct methods are widely utilized for designing optimal controllers. This preference is largely attributed to their expansive region of convergence [START_REF] Bordalba | Direct collocation methods for trajectory optimization in constrained robotic systems[END_REF].

In this work, our focus is on utilizing a direct method to obtain the optimal trajectory. Direct methods can be formulated using shooting and collocation methods, with the choice depending on the specific dynamical equation. When an explicit model is available, the shooting method is often employed. This method involves decomposing the trajectory into sub-optimal intervals and calculating the spline between each interval. The formulation can be done through single shooting, where only the state variables are optimized, or multiple shooting, where both control inputs and states are optimized at each control interval. The resulting discrete problem is then addressed using non-linear programming methods [START_REF] Posa | Direct Trajectory Optimization of Rigid Body Dynamical Systems through Contact[END_REF].

General Problem Formulation

We pose our trajectory optimization problem as follows :

min x,u J(x, u) s.t. g(x, u) = 0 f (x, u) ≥ 0 (10)
In this formulation, x represents the state vector of the system, and u denotes the angular speed of each motor. The equality constraints g(x, u) encompass various constraints, including the dynamical constraint of the system, while the inequality constraints f (x, u) define additional inequality restrictions. To solve the optimization problem, we divide the trajectory into N control sub-intervals. For this purpose, we employ the direct multiple shooting optimization formulation, which offers several advantages over other techniques [START_REF] Posa | Direct Trajectory Optimization of Rigid Body Dynamical Systems through Contact[END_REF].

FA-Hex

First, we will define the optimization problem for the FA-Hex targeting photography application. a) Cost Function: The cost function to be minimized is defined as follows:

min x J =S f t f + t f 0 x T Qxdt (11)
Here, x = [ξ T , v T , µ T , Ω T ] T and Q ∈ R 12×12 represents the weight cost assigned to the states of the system, t f is the time allotted to the whole scenario, and S f denotes the slack weight used to handle the priority of the specified time.

b) Dynamical Model Constraint: The dynamical model, defined in Section 2.1, is discretized using fourth-order Runge-Kutta. This model is employed to ensure the attainment of dynamically feasible states by solving x k+1 = h(x k , u k ), where h is a vector field representing the discretized dynamical model of the system. c) Multiple-Shooting Constraint: The method involves computing the next state using the discretized model. The computed next state is constrained to align with the optimized parameter, which can be achieved by ensuring that x nextx k = 0. Here, x k represents the system state at the k-th interval, and x next corresponds to the state calculated using the discretized dynamical model.

d) Boundary Conditions:

To ensure that the optimization problem begins and ends within feasible constraints, additional boundary constraints are included. The variables x initial and x final are defined and incorporated into the equality constraint g(x, u) as x 0x initial = 0 and x Nx final = 0. The states and inputs of the system should not exceed their respective boundaries.

To enforce this requirement, two additional equations [START_REF] Hehn | Quadrocopter trajectory generation and control[END_REF][START_REF] Zeng | Differential flatness based path planning with direct collocation on hybrid modes for a quadrotor with a cable-suspended payload[END_REF] are introduced to the function f (x, u).

x min ≤ x k ≤ x max (12)
u min ≤ u k ≤ u max (13)
f) Waypoint Navigation Constraint: The primary objective of this work is to navigate and fly over specific points. To achieve this goal, an additional constraint is introduced: w pδ < x k < w p + δ, where w p represents the specified waypoint. The inclusion of δ in this equation ensures the robustness of the algorithm and helps avoid excessive equality constraints. It is important to note that this problem does not aim to optimize time since such optimization conflicts with the defined constraint. For more detailed information on time-optimal trajectory optimization the reader can refer to [START_REF] Foehn | Time-optimal planning for quadrotor waypoint flight[END_REF].

FA-Hex With Cable Suspended Pendulum

In this section, we will introduce the extended constraints and objective function that are used to optimize the trajectory for FA-Hex with a cable-suspended pendulum. a) Cost Function: The extended cost function to be minimized on the variables is defined as follows:

min x,T,L J = S f t f + t f 0 x T ext Q L x ext dt + t f 0 ( R0 T + R1 (L 0 -L))dt (14)
Here,

x ext = [ξ T , v T , µ T , Ω T , ξ T L , v T L ] T , Q L ∈ R 18×18
and the added terms R0 T and R1 (L 0 -L) represent the cost associated with the taut and slack modes of the cable, where R0 and R1 are scalar weights, and L 0 is the initial length of the load in taut mode. These terms are included in the objective function to address the hybrid modeling problem of the pendulum. The trajectory is generated with a corresponding slack and taut mode, based on the weighting values. b) Dynamical Model Constraint: The extended dynamical model introduced in Section 2.2 is also discretized using fourth-order Runge-Kutta and incorporated into the equality constraints g(x, u).

c) Linear Complementary Constraint: This constraint is added to eliminate the hybrid dynamical model, assuming that either the slack mode or the taut mode is present. It is represented by T (L 0 -L), where L 0 is the cable length when taut, and T represents the tension in the cable.

d) Tension Boundaries Constraint:

To ensure a realistic and feasible solution, it is important to impose bounds on the tension. This can be achieved by incorporating the constraint 0 ≤ T ≤ T max , where T max represents the maximum allowable tension. The value of T max can be determined based on prior information or known limitations of the system. e) Cable Length Constraint: In addition to the complementary constraint, the length of the cable should be positive, L ≥ 0.

f) Payload Swung Angle Constraint: To prevent swinging and potential collisions with the drone, a constraint is introduced as α L ≥ α Lmin and α L ≤ α Lmax .

The other constraints defined in the FA-Hex drone are primarily utilized as constraints for this particular application.

SIMULATIONS

The defined optimization problem is formulated as a nonlinear optimization problem using the CasADi Toolbox [START_REF] Joel A E Andersson | CasADi -A software framework for nonlinear optimization and optimal control[END_REF]. As mentioned earlier, this study considers two primary scenarios: simple FA-Hex and FA-Hex with a cable-suspended pendulum. A comparison between FA-Hex and UA-Hex is accomplished showing the enhancing of maneuverability of the FA-Hex. The drone parameters used to simulate the drone model are presented in Table 2. The scenario involves the drone passing through multiple waypoints. To assess the algorithm's ability to achieve minimal banking angles, several test cases were conducted. A trajectory comprising four points is shown in Figure 4. The time allotted for this scenario is t f = 8s, with N = 100 control intervals. It's worth noting that, as classified by [START_REF] Franchi | Full-pose tracking control for aerial robotic systems with laterally bounded input force[END_REF], FA vehicles belong to the category of Lateral Bounded Force vehicles (LBF). In these vehicles, lateral acceleration is constrained to be smaller than the vertical acceleration of the drone. The UA-Hex shares the same parameters as the FA-Hex, except for having a tilting rotor angle β = 0. The achieved velocity of the FA-Hex is approximately 1.5m/s when flying without banking, while the velocity of the UA-Hex can reach 2.5m/s.

FA-Hex With Cable Suspended Pendulum

In another application, the optimization problem is expanded, as explained in Section 3.3, to be solved across the optimization variables x, u, T, L, and q. As in Section 4.1, the scenario involves flying between predetermined points. Utilizing linear complementary constraints, it becomes feasible to generate a trajectory with minimal slack, indicating that the cable length remains close to L 0 for the majority of the trajectory. The simulation results are depicted in Figure 5, illustrating both the payload and drone trajectories.

Initial Guess

The initial guess plays a pivotal role in ensuring the optimization problem converges to its optimal solution. Providing a feasible initial guess that closely approximates the expected solution is essential. This step is critical for achiev- ing convergence and reducing computational time needed to solve the problem. In this context, it is important to note that the drone is initially treated as a UA system. The differential flatness approach is employed to generate an initial guess for the problem. Furthermore, motion planning algorithms grounded in graph theory, such as the A * and RT T algorithms, can be employed. These algorithms facilitate the creation of a global path, effectively aiding in solving the trajectory planning problem.

CONCLUSION AND EXTENSIONS

In this work, we have proposed a trajectory optimization formulation for FA-Hex systems. The simulations demonstrate the feasibility of this approach by generating trajectories with minimal banking angles. However, the algorithms used in this study have certain constraints when it comes to defining the nodes and time. The algorithm shows fast performance when when the trajectory is divided into N = 100 control intervals, taking only a few seconds to converge to the optimal solution. Future work could involve extending the problem to incorporate avoidance of static defined obstacles by implementing specific maneuvers.

As an extension of this paper, we are currently working on the identification and control of the real FA-Hex system. Our objective is to conduct real experimental flights to test the optimized trajectory in the mentioned scenarios. The planned tests will utilize the Paparazzi autopilot [START_REF] Gorraz | Using the paparazzi UAV system forscientific research[END_REF] with a modified INDI controller or feedback linearization. http://www.imavs.org/ IMAV2023-28

Figure 1 :

 1 Figure 1: Fully Actuated Hexacopter attached to a payload with Inertial (F I ) and Body (F B ) Frames.

Figure 2 :

 2 Figure 2: Forces projection in the lateral plane with the motor tilting angles β between z B and z mi the axis of the i-th motor.

4 τx 4 4 Fy 4 = 4 τy 4 4 Fz 4 = -ktC β ω 2 4 τz 4 4 Fx 5 = 5 τx 5 = 5 Fy 5 = 5 τx 5 = 5 Fz 5 = -ktC β ω 2 5 τz 5 = 5 Fx 6 = 6 τx 6 = 6 Fy 6 = 6 τy 6 = 6 Fz 6 = -ktC β ω 2 6 τz 6

 4444444444455555555555566666666666 = (ktC β Sα 4 l -kqS β Sα 4 )ω 2 ktS β Cα 4 ω 2 = (-ktC β Cα 4 l + kqS β Cα 4 )ω 2 = (-ktS β l + kqC β )ω 2 ktS β Sα 5 ω 2 (ktC β Sα 5 l -kqS β Sα 5 )ω 2 ktS β Cα 5 ω 2 (ktC β Cα 5 l -kqS β Cα 5 )ω 2 (ktS β l -kqC β )ω 2 -ktS β Sα 6 ω 2 (ktC β Sα 6 l -kqS β Sα 6 )ω 2 -ktS β Cα 6 ω 2 (ktC β Cα 6 l -kqS β Cα 6 )ω 2 = (-ktS β l + kqC β )ω 2 6

m

  = 0.6656 kg l = 0.15 m g = 9.81 m/sec 2 kt = 3.4 × 10 -5 N/ω 2 kq = 3.4 × 10 -6 Nm/ω 2 Ixx = 0.0411 kgm 2 Iyy = 0.0478 kgm 2 Izz = 0.0599 kgm 2

Figure 3 Figure 3 :Figure 4 :

 334 Figure 3: Euler angles comparison between FA-Hex and UA-Hex.

Figure 5 :

 5 Figure 5: Projection in the horizontal plane of the optimized trajectory for FA-Hex with cable suspended pendulum.

  IMAV2023-28 14 th ANNUAL INTERNATIONAL MICRO AIR VEHICLE CONFERENCE AND COMPETITION

) http://www.imavs.org/

  IMAV2023-28 14 th ANNUAL INTERNATIONAL MICRO AIR VEHICLE CONFERENCE AND COMPETITION e) State and Input Constraint:

http://www.imavs.org/

Table 2 :

 2 Drone Simulation Parameters 4.1 FA-Hex and UA-Hex Trajectory Generation