# Data-Driven Identification of Crane Dynamics Using Regularized Genetic Programming
## 1 Introduction
In recent research works, a double-layer neural network structure has been used to estimate unknown nonlinear parts of ship-mounted crane dynamics [20], a container crane's dynamics have been approximated using a radial basis function network (RBFN) with Gaussian activation functions [21], and the ANN model has been trained offline using the Bayesian regularization method to predict the parameters of the time-optimal obstacle avoidance trajectory function for a rotary crane [22]. In [23], the adaptive control scheme for a rubber-tired gantry container crane was tested on the laboratory stand, with the ANN models used to estimate the velocities of actuated and underactuated coordinates in three-dimensional spaces, friction parameters, and unknown disturbances. The offline-trained multi-layer perceptron neural network model has been identified to predict the sway of a quay crane's spreader in [24]. The paper reports the data acquisition equipment and experiments carried out on a physical system and analyses the effectiveness of different learning methods to develop the structure and adjust the weights of the neural network predictor of the spreader's sway velocity. Different structures of ANN models of an overhead crane's inverse dynamics were trained and validated using the Levenberg-Marquardt algorithm with data computed in simulations carried out using a model derived analytically [25]. Some works have studied crane modeling and the control problem using metaheuristic algorithms, e.g., genetic algorithm (GA) [26; 27]. In [28], an overhead crane's dynamics were approximated by adaptive neuro-fuzzy inference systems (ANFISs) identified using hairpin RNA GA. A planar crane's dynamics were decoupled and presented using two ANFIS empirical models of actuated trolley motion and unactuated pendulum motion. By analyzing the aforementioned works dealing with crane dynamics' identification, it can be concluded that most of the approaches are based on ANN and TSF models. The majority of works available in the literature focuses on the parameter identification of a predefined model, with only a few works focusing on model structure optimization. Model structure optimization is composed of model term construction and parameter estimation. Model term construction can be formulated as a symbolic regression problem in which a space of mathematical functions is searched to find a set of model terms that minimize a problem-dependent loss function. Genetic programming (GP) is an effective symbolic regression tool to identify both dynamic and static relationships between input-output pairs that can be modeled in a more interpretable form than using the black-box modeling approaches. Multi-dimensional genetic programming (MGP), such as multi-gene genetic programming (MGGP) [29], the feature engineering automation tool (FEAT) [30], and the proposed G3P-SR, naturally tackle both problems simultaneously since each individual is composed of a set of programs whose outputs can then be fed into a deterministic algorithm for parameter estimation. In [31], the MGGP technique was used to establish a dynamic model of an overhead crane. The initial gene weights were estimated using the least squares in the serial-parallel configuration after which the Levenberg-Marquardt algorithm was used to find the local optimum of the simulation error by changing the model to a parallel configuration; however, both the evolutionary process of searching for the best model structure and parameter estimation were guided by the model performance criteria, without taking into account the complexity of the model. To address this issue, this paper proposes the use of sparse regression on a fixed set of candidate model terms that are obtained from a biased search of the function space. Regularization has been successfully applied in the field of system identification to overcome model complexity and prevent overfitting problems. \(l_{0}\) regularization is used for promoting sparse solutions, although an \(l_{0}\)-regularized problem is nonconvex and proven to be NP-hard.