## 2 Methods
### Data Reduction and Analysis  
For the data analysis, the following meaningful events were defined for Walk and Canter: E1 (the moment the right hind foot contacts the ground), E2 (the moment the right front foot contacts the ground), E3 (the moment the left hind foot contacts the ground), E4 (the moment the left front foot contacts the ground), E5 (the moment the right hind foot contacts the ground), E6 (the moment the right front foot contacts the ground), E7 (the moment the left hind foot contacts the ground), and E8 (the moment the left front foot contacts the ground). In addition, seven phases were defined based on the events: P1 (E1-E2), P2 (E2-E3), P3 (E3-E4), P4 (E4-E5), P5 (E5-E6), and P7 (E7-E8). Motion capture data were digitized to track reflective markers with the Kwon3D software. The three-dimensional coordinates of the reflective markers were generated via a DLT algorithm [15] and were smoothed using an 8 Hz low-pass fourth-order Butterworth filter. In this global coordinate system, the y-axis is the forward-moving direction of the horse; the x-axis is the left-right lateral direction, and the z-axis is the vertically upward direction. The kinematic variables were the duration of the phases of gaits, head-neck angles, and center of mass displacement. The duration of the phases of the gait variables was analyzed from the first event (E1) to each event and the total time. The duration of the phases of each gait variable was calculated using the following formula:  
\[t(i\sim f)\Big{(}F_{f}-F_{i}\Big{)}=\frac{1}{Frame\ rate} \tag{1}\]  
where \(t(i\)-\(f)=\) the time required, \(F\!f=\) the frame number of each event ends, and \(F_{i}=\) the frame number of each event start. Additionally, the angular variable was the head-neck angle (A) and head-y-axis angle (B). The angle between two adjacent segments was obtained by using the dot product of the vector. That is, the angle \(\theta\) between two vectors \(U\) (\(Xi\), \(Y\!i\), \(Z\!i\)) and \(V\) (\(X\!j\), \(Y\!j\), \(Z\!j\)) was calculated using the following formula:  
\[\theta=COS^{-1}\frac{U\cdot V}{|U|\cdot|V|},\ \theta=COS^{-1}\Bigg{\{}\frac{a_{ x}b_{x}+a_{y}b_{y}+a_{z}a_{z}}{\sqrt{a^{2_{x}}+a^{2^{y}2}+a^{2^{z}2}}\sqrt{b^{2_{x}}+b^{ 2_{y}2}+b^{2_{z}2}}}\Bigg{\}} \tag{2}\]  
The head-neck angle is shown in Figure 3.  
Figure 3: Head–neck angle. The linear velocity variable was the body center velocity and head center velocity. The velocity was first calculated for each of the \(X\), \(Y\), and \(Z\) components using the following equation, and the \(X\), \(Y\), and \(Z\) components were synthesized to obtain the resultant velocity:  
\[X_{i}=\frac{X_{i+1}-X_{i-1}}{2\Delta t} \tag{3}\]  
where \(X_{i-1}\): \(i-\) first \(X\) coordinates, \(X_{i+1}\): \(i+\) first \(X\) coordinates, and \(X_{i}\): \(i\)th \(X\) component velocity; and  
\[V=\sqrt{X_{i}^{\prime 2}+Y_{i}^{\prime 2}+Z_{i}^{\prime 2}} \tag{4}\]  
where \(V\): \(i\)th resultant velocity, \(X_{i}^{\prime}\): \(i\)th \(X\) component's velocity, \(Y_{i}^{\prime}\): \(i\)th \(Y\) component's velocity, and \(Z_{i}^{\prime}\): \(i\)th \(Z\) component's velocity. The following formula calculated the center of mass coordinate (\(C\)) of Segment \(i\) :  
\[c_{i}=(1-p_{i})\cdot P_{i}+p_{i}\cdot D_{i} \tag{5}\]  
where \(p_{i}\): the ratio of the center of mass displacement from the proximal end to the segment length, and \(P_{i}\), \(D_{i}\): coordinates of segmental proximal and distal ends. The following formula calculated the location of the center of mass:  
\[C=\frac{\sum_{i}(c_{i}\cdot m_{i})}{M} \tag{6}\]  
_mi_: mass of the segment, and \(M\): whole body mass  
The video data collected by 12 cameras were labeled with each marker in the NEXUS program of VICON, and C3D files were extracted for each trial from the pipeline. After calculating raw data using C3D files in Kwon3D, values for each event, mean value, and standard deviation were calculated using Microsoft's Excel program. Finally, the paired t-test was performed with the SPSS ver.18.0 program for statistical processing.