The employment of machine learning algorithms to the data provided by wearable movement sensors is one of the most common methods to detect pets' behaviors and monitor their well-being. However, defining features that lead to highly accurate behavior classification is quite challenging. To address this problem, in this study we aim to classify six main dog activities (standing, walking, running, sitting, lying down, and resting) using high-dimensional sensor raw data. Data were received from the accelerometer and gyroscope sensors that are designed to be attached to the dog's smart costume. Once data are received, the module computes a quaternion value for each data point that provides handful features for classification. Next, to perform the classification, we used several supervised machine learning algorithms, such as the Gaussian naïve Bayes (GNB), Decision Tree (DT), K-nearest neighbor (KNN), and support vector machine (SVM). In order to evaluate the performance, we finally compared the proposed approach's F-score accuracies with the accuracy of classic approach performance, where sensors' data are collected without computing the quaternion value and directly utilized by the model. Overall, 18 dogs equipped with harnesses participated in the experiment. The results of the experiment show a significantly enhanced classification with the proposed approach. Among all the classifiers, the GNB classification model achieved the highest accuracy for dog behavior. The behaviors are classified with F-score accuracies of 0.94, 0.86, 0.94, 0.89, 0.95, and 1, respectively. Moreover, it has been observed that the GNB classifier achieved 93% accuracy on average with the dataset consisting of quaternion values. In contrast, it was only 88% when the model used the dataset from sensors' data.