# Machine Learning Sequential Methodology for Robot Inverse Kinematic Modelling
## 1 Introduction  
Robot configuration is based on several actuated joints that link the rigid elements in a series-wise, parallel, or hybrid approach for a general-purpose manipulation task. For precise motion tracking, the focus is on the relation between the base frame, located at the robot ground, and the end-effector frame, that defines the physical interaction with the external environment. Multi degree-of-freedom (DOF) robots exhibit a working volume that enables the pose reaching in different configurations of the kinematic chain, except for the singularity positions. The robot's kinematics formulate the transfer functions between the joint actuators' values and the corresponding Cartesian coordinates of the end-effector with respect to the base frame. The forward kinematics (FK) equations describe the robotic manipulator, from joints to Cartesian coordinates, computing the position and orientation of an element (e.g., tool center point--TCP) and defining the required motion of the robot actuators. The function is unique, a given joint configuration corresponds to a specific Cartesian positionand orientation. The robot control and the trajectory planning task require the robot joint space information. In the literature, a significant amount of research work has been presented dealing with this set of equations through matrix multiplications and mathematical procedures [1; 2]. Most industrial robots are developed with simple-form kinematics to obtain analytic closed-form solutions. Nevertheless, several factors, such as manufacturing tolerances, wear, transmission errors, and set-up errors, compromise the internal design model used in the robot controller, affecting the transformation accurately and deviating from the desired locations. For these reasons, the inverse kinematic (IK) formulation may be an alternative approach to obtain joint variables for any cartesian coordinate information in reaching a required point. The inverse transformation may be a computationally intensive procedure for general-form multi-link manipulators, depending on the modeling, formulation used for the kinematics, robot structure, and workspace. In IK, for all the poses within the workspace volume exist \(n\) potential joint configurations, where \(n\) is the redundancy degree. Robots capable of reaching a point in the cartesian space with multiple configurations are known as redundant robots since they have more degrees of freedom than required to perform the point-reaching task [3]. The closed-form analytical IK solution becomes more complex as the DOF increases, requiring optimizations and constraints [4; 5; 6]. There is a set of conventional techniques to solve IK problems such as algebraic, geometric, and iterative methods [1; 7; 8]. Nevertheless, these techniques show several issues for robots with complex structures, such as closed-form solutions not guaranteed, high computation costs, or results depending on the start point. For these types of manipulators, optimization methods, such as particle swarm optimization (PSO) [9; 10; 11] or genetic algorithm (GA) [12; 13], have been adopted to avoid high-order polynomials. Nevertheless, the impact of the initial setting and the inadequate convergence are constraints that need to be regulated [14]. Recently, machine learning (ML) procedures have been used to model the IK, as in other industrial applications [15; 16; 17; 18; 19; 20; 21]. The predictions are promptly obtained by comparing numerical and optimization methods, using data from the offline training phase. One of the most recognized applications of ML schemes observed in literature is based on artificial neural networks (ANN). A planar ANN scheme was proposed by [PERSON] in [22], creating a dataset from the FK of a 3-DOF planar robot, used to train a shallow network with 100 neurons in its hidden layer. The results showed a mean square error (MSE) equal to 0.0054387 rad in the joint space.