# Machine Learning Sequential Methodology for Robot Inverse Kinematic Modelling
## 1 Introduction
[PERSON] and [PERSON], in [23], implemented an orientation function based on the cosine for a shallow network with 12 neurons, obtaining a validation of MSE equal to 0.00038523 rad. [PERSON] et al. deployed a 5-layer ANN based on an identical planar manipulator. The proposed scheme was designed with the PSO algorithm, obtaining a final validation of MSE close to 0.00013 rad in joint space [24]. However, the inverse kinematic computation was simplified in the planar motion configuration due to the dimensional reduction. [PERSON] et al. presented an interesting study in [25], increasing the analysis complexity. Compared to previous works, the authors modeled an industrial robot with 6-DOFs (RS10N), using a shallow single hidden layer back-propagation neural network with 30 hidden neurons, applied in a constrained workspace and validated on a rectangle trajectory. The obtained results showed an error lower than \(\pm 0.05^{\circ}\) with a maximum deviation of 1.00 mm from the horizontal axis. A further method was developed by [PERSON] et al. [26], proposing the ANFIS (adaptive neuro-fuzzy inference systems) technique to predict the IK of a 4-DOF SCARA robot in a reduced workspace. The maximum observed deviation of the end-effector was 0.3775 mm in X-direction, 0.4135 mm in Y-direction, and 0.0027 mm in Z-direction, respectively. Nevertheless, these applications considered limited workspaces with planar or highly constrained robot motions, facilitating the corresponding inverse kinematic computation and reducing the effect of the estimated errors. To avoid the workspace limitations, an alternative approach was presented by [PERSON] et al. [27], dividing the spatial workspace into quadrants and applying a 5-layer ANN scheme. The study aimed to model a 5-DOFs robot. The application was validated for a single quadrant, obtaining a joint error with a range equal to \(\pm\)0.003 rad. [PERSON] et al. [28] suggested a further method to address the spatial complexity in modeling the inverse kinematics of a 6-DOF robot (Denso VP6242). Inner feedback data for the current joints' values was included as an extra input to the 10-hidden layers ANN. The process validation was performed using a spring curve trajectory demonstrating a maximum error in the Z-direction lower than 0.35 mm. Despite the promising results, the inclusion of the joint values may increase the complexity in the scheme generalization since it is based on the trajectory as an input of the ANN requirement. [PERSON] et al. [29] proposed a long short-term memory (LSTM) method regarding a recurrent neural network model, with two hidden layers of 10 LSTM neurons. The aim was to represent the inverse kinematics of the left arm of the Baxter Robot (7 DOFs). The study showed a final MSE lower than 0.02, obtained from the test data set. A tree-based algorithm was implemented by [PERSON] et al. [30] to model the inverse kinematics of a 6-DOF parallel manipulator. The dataset was generated by recording the end-effector poses with a camera and the corresponding joint values with ultrasonic sensors. The random forest (RF) algorithm showed the best training results, with a determination coefficient equal to 94.61% and a root-mean-square error (RMSE) close to 0.036 m.  
[PERSON] et al. [31] proposed the committee schemes as an alternative solution to represent a 6-DOF redundant robot, Hitachi M6100. The concept was based on 10 parallel shallow networks, simplified to 6 networks, with variable hidden neurons. The scheme output was selected from the parallel networks based on the minimization of the Cartesian error, obtained using the forward kinematics. Compared to a global ANN approach, the proposed method showed a significant reduction of the final error range, from 5.76 to 13.41 mm to 0.39 to 0.74 mm. Nevertheless, the multiple model training time and the higher amount of data required significant computational cost due to the high complexity.