## 2 Related Work and Contributions
### Neural Networks for Force Estimation in Robotics  
Force estimation in robotics was mostly based on force observers in the past [11; 12; 13; 14; 15]. With knowledge of the control forces and torques, these estimates are deduced using the measured pose data. The multilayer perceptron (MLP) network [16] and recurrent neural networks [17] are two examples of neural networks that have been introduced using certain force observer-based methods. Deep neural networks are making a comeback in a variety of fields, including robotics. Robotics has used deep neural networks to address a variety of issues, including grasping and control [18; 19]. However, force estimation in robotics has not been extensively addressed by deep learning. Still, there are a few interesting studies on the subject. Neural networks were used in [20] as an extension of the force observer, which needed to have available the precise dynamic model of a manipulator, which is usually known but is unreliable, or inaccurate. The results of the force estimation are, therefore, unsatisfactory in that case. The inverse dynamics of a robotic manipulator could be resolved, according to the authors, using neural networks. The developed neural network model, as was demonstrated in the paper, was more precise than classical force observers. Furthermore, it was simpler to implement since understanding the dynamic model of a robotic manipulator was not required. In [17], a deep neural network-based observer was used to estimate non-contact forces arising from inertia and which interfere with measurements of contact forces. It was shown that in a highly dynamic motion, the proposed neural network outperformed the analytical method based on the identified inertial parameters. A similar motivation was behind the work in [21], where a neural network was used to estimate free space motion robot joint torques. These values were then subtracted from the ones measured by joint motors, and, based on that, tool interaction forces were estimated. The proposed method was able to estimate outside forces within 10% normalised root mean squared error (NRMSE) compared to an external force sensor, outperforming model-based approaches. There are also methods based on neural networks developed with particular applications in mind. For instance, in robotic surgery [22], the force was estimated without force sensors using visual feedback and neural networks. There are also approaches, like the one in [23], which combine visual information with robot state information in order to increase neural network estimation accuracy (to 0.488 N root mean squared error (RMSE)) to the point of outperforming physics-based models. There have recently been advances made in neural network-based techniques that can learn the inverse dynamic models of robots [24; 25]. These techniques incorporate prior knowledge from the system being modelled while simultaneously learning it. Energy conservation law establishes constraints that improve performance by formulating prior knowledge for physical systems. It was emphasised that deep learning had excelled in every area of application, with the exception of physical systems (at the time) [24]. So, the Deep Lagrangian Network (DeLaN), a novel model that made use of system prior knowledge, was constructed. Before developing more precise models and guaranteeing physical validity while ensuring that the models can effectively extrapolate new samples, the Euler-Lagrange equation was utilised as the underlying physics prior. Sometimes multilayer perceptrons overfit training data, since they are not required to follow the energy conservation law and because they need more training data to generalise correctly in these situations. On the other hand, the Euler-Lagrange equations were applied as a neural network in [24] to achieve more realistic performance. The Lagrangian neural network (LNN), a more versatile technique for learning arbitrary Lagrangians, was introduced in [25]. The paper also gave an overview of physical dynamics models based on neural networks.