## 2 Methods
### HMM Modeling  
In HMMs, the state's hidden sequence can be estimated based on the visible observation sequence [40] and characterized by five elements in a symbolic manner [18; 41], where \(N\) represents the number of states in the model. For the variable \(q_{t}\), there is a corresponding state value, represented by \(S=\{S_{1},\ S_{2},\ldots,S_{N}\}\). The number of observation symbols for HMM exits is represented by \(M\). The symbols are designated \(O=\{O_{1},\ O_{2},\ldots,O_{M}\}\). A transition probability matrix \(A=\big{[}a_{ij}\big{]}\) represents the transition probabilities between states, where \(a_{ij}=Prob\big{(}q_{t+1}=S_{j}\big{|}q_{t}=S_{i}\big{)}\), \(1\leq i\), \(j\leq N\) represents the probability value in the \(q_{t}\) state for transitioning to state \(S_{j}\) at time \(t+1\), since at time \(t\), the state is \(S_{i}\). The exit probability matrix \(B=\big{[}b_{j}(k)\big{]}\) represents the observation probabilities in each state, where \(b_{j}(k)=Prob\big{(}O_{k}(t)\big{|}q_{t}=S_{j}\big{)}\), \(1\leq j\leq N\), \(1\leq k\leq M\) is the probability of the k-infinite observation symbol. Finally, an initial state distribution \(\pi_{1}=\{\pi_{i1}\}\) represents the probability of a new input sequence starting from a given state at the initial time \(t=1\), where \(\pi_{i1}=Prob(q_{1}=S_{i})\), \(1\leq i\leq N\). The ANOVA results, which will analyze the significant differences in bi-dimensional displacement, will provided the criteria for the marker position data modeling in HMM. The Markovian model algorithms [42; 43; 44; 45] were employed in GNU Octave 4.2.1 (GNU Octave, Madison, WI, USA) [46]. For analysis, the HMMs were provided with bi-dimensional data on the trajectories of each marker during the ascending phase of the exercise. Research on human movement [47; 48; 49; 50] in which a linear type of topology was used [51] provided a reference for the general topology for the HMM. The model's learning phase occurs with clusters [47] using the k-means algorithm [52; 53; 54; 55; 56]. The term cluster are referring to the set of positional coordinates (x,y) defining a common segment of path during the biceps-curl for the changes in position of the shoulder, elbow and barbell. A simplified example of the relation between barbell positioning during the exercise's upward movement and Markovian modeling is shown in Figure 3. Each of the three positions of the barbell, defined as states S1, S2 and S3, derived from the clustering process presents a transition probability (go to the next state or remain in the same state) and an observation probability for each state \(b_{j}(k)\)[57]. For HMM subject modeling, the three repetitions performed solely with the barbell were used as the standard model of the positional sequence pattern for shoulder, elbow and barbell. These repetitions were used as the basis for training the model. Using the acquired data, the relation between the 25% and 50% loads was also tested, using the 25% body weight as training. In this state, the model parameters \(\lambda\) = (A, B, \(\pi\)) were calculated using the observed symbols O such that the observation probability P(O I) was augmented. For this computation, the Baum-Welch algorithm was employed, and this expanded the observation sequence probabilities [47]. The movement pattern recognition for the trajectories provided by the markers was based on the maximum likelihood criterion. For comparison between a trained model and a new observation sequence, the forward algorithm was used [18; 58]. The algorithm estimated the probability that the observation data could have been generated by the model. With the application of HMM, the minimum number of states necessary to recognize thesignificant differences detected by ANOVA while the subject performed the exercise with different loads, in terms of changes in the patterns of execution of the movement while the exercise was performed, was obtained.