## 2 Methodology
### Implementation of the Methodology  
Test bench: The test bench that was selected for the experiment, to provide a reliable, efficient, and repeatable way of testing the position of the ERMIS, is an anthropomorphic mechanism [36]. This type of mechanism simulates the movements of an arm is fast and has great accessibility and maneuverability, and is a small device considering the work field it is used in. The anthropomorphic 3 DOF mechanism for positioning rehabilitation devices is a structure with three rotational joints (3 DOF or RRR, rotational joints, see Figure 3). The end-effector position is specified in angular coordinates or cartesian coordinates with a transformation of the data. The mechanism has a cardan joint as an end effector. This is a mechanical component that allows the transmission of rotational movement between two non-collinear axes. The purpose of the cardan joints is that the links of the rehabilitation device and the anthropomorphic mechanism can rotate with regard to the bars. The rotation between the non-parallel axes will be transmitted. Mathematical model position: The representation of the position in the workspace of the ERMIS, by means of the anthropomorphic mechanism, is obtained through mathematical expressions from a mathematical model. The success of the model lies in the accuracy with which it can represent the object or phenomenon under study [37]. A mathematical model capable of describing the direct kinematics of the system is developed; this model is used for the location of points in the trajectories generated by the ERMIS movements. Through the analysis of direct kinematics, a set of kinematic equations useful for calculating the position of the end-effector of the anthropomorphic mechanism, using specific values  
Figure 3: Anthropomorphic mechanism of 3 DOF for position detection. of the angles between the links, is obtained. Figure 4 shows the orientation of the reference frame for the anthropomorphic mechanism. The nomenclature used for homogeneous transformation matrices is as follows: \(c_{\theta_{1}}=c_{1}\); \(c_{\theta_{2}-10^{\circ}}=c_{2}\); \(c_{\theta_{3}+10^{\circ}}=c_{3}\); \(s_{\theta_{1}}=s_{1}\); \(s_{\theta_{2}-10^{\circ}}=s_{2}\); \(s_{\theta_{3}+10^{\circ}}=s_{3}\). The Denavit-Hartenberg parameters are shown in Table 2.  
Equation (1) shows the homogeneous transformation matrix that takes us from the origin to the end-effector of the anthropomorphic mechanism. \[H=A_{0}^{1}A_{1}^{2}A_{2}^{3}=\left[ \begin{array}{cccc}c_{2}c_{3}c_{1}-s_{ 3}s_{2}c_{1}&-c_{1}c_{2}s_{3}&-c_{1}s_{2}c_{3}&s_{1}&a_{2}c_{2}c_{1}+a_{3}c_{2} c_{3}c_{1}-a_{3}s_{3}s_{2}c_{1}\\ c_{2}c_{3}s_{1}-s_{1}s_{2}s_{3}&-c_{2}s_{3}s_{1}-c_{3}s_{2}s_{1}&-c_{1}&a_{2}c_ {2}s_{1}+a_{3}c_{2}c_{3}s_{1}-a_{3}s_{3}s_{2}s_{1}\\ c_{2}s_{3+}c_{3}s_{2}&c_{2}c_{3}-s_{2}s_{3}&0&d_{1}+a_{2}s_{2}+a_{3}c_{2}s_{3}+ a_{3}c_{3}s_{2}\\ 0&0&0&1\end{array} \right] \tag{1}\]  
Identification of actual operating parameters: This is the step in the methodology in which an analysis of the case studies is carried out. The case studies analyzed are those corresponding to the trajectories programmed in ERMIS [18], which are the movements used during a physical therapy session:  
* Case study 1. Exclusive series of exercises per joint. They consist of performing one or two anatomical movements to stimulate the affected regions in repetitions of 3 to 5 times. * Case study 2. Exercise with an apparent load. These exercises emulate the lifting of an object by a person planning to deposit it somewhere. * Case study 3. Activation of muscle memory through trajectory tracking. A relatively new rehabilitation technique is muscle memory [38]. * Case study 4. Emulation of shoulder wheel or rudder. The tiller is a mechanical device that facilitates shoulder, arm, elbow, and wrist rehabilitation. It performs a movement of the entire upper limb [33]. Table 3 shows the operating conditions of the system in which the analysis of the position of the ERMIS exoskeleton is performed, showing the different movements and