## 2 Materials and Methods
### Data Processing and Reduction  
Marker and ground reaction force data were post-processed and trials were trimmed to a single obstacle-crossing gait cycle in Vicon Nexus 2.6.1 (see [53] for more details). Six representative obstacle crossing trials (three right and three left) were selected and exported to Visual3D (C-Motion, Inc., Germantown, MD, USA) for the dynamic stability analysis. The primary variables of interest included a spatial inclination angle (COM-COP angle) and the M/L distances between COM and COP (COP-COM M/L), XCOM and COP (COP-XCOM M/L), and XCOM and the most lateral position of the BOS, i.e., U\({}_{\text{MAX}}\) (U\({}_{\text{MAX}}\)-XCOM M/L) (Figure 4). These variables were measured to determine whether a difference existed between PD and CON participants during the stance phase at five specific time points: first double support (FDS1) and midstance (MST1) for the trailing limb (1 st step); and first double support (FDS2), midstance (MST2), and second double support (SDS2) for the leading limb (2 nd step) (Figure 3).  
Figure 4: Diagram of dynamic stability variables illustrating relationships between the center of mass (COM), the extrapolated center of mass (XCOM), the center of pressure (COP), and the mediolateral  
(M/L) limit of the base of support (U\({}_{\text{MAX}}\), i.e., head of the 5 th metatarsal). Shown are (**A**) a posterior view of the COM-COP inclination angle (\(\theta\)) and the M/L distance between the vertical projection on the ground of the COM and the location of the COP (COP-COM M/L); (**B**) the M/L distance between the COP and XCOM (COP-XCOM M/L); (**C**) the M/L distance between U\({}_{\text{MAX}}\) and XCOM (U\({}_{\text{MAX}}\)-XCOM M/L).  
Using a 15-segment model, total body COM was calculated as the weighted average of all segment COMs; segments were modeled as geometric shapes with masses were expressed as percentages of total body mass [60,61]. The velocity of the COM (\(V_{COM}\)) was calculated with a first-order central difference formula [60]. XCOM was then found using the following equation:  
\[XCOM=COM+(V_{COM}/\omega_{0}), \tag{1}\]  
where \(\omega_{0}\) = angular eigenfrequency of the pendulum;  
\[\omega_{0}=\sqrt{g/l}, \tag{2}\]  
where \(g\) = 9.81 m/s\({}^{2}\) and \(l=leg\)\(length\) in m (distance between COM and ankle joint) [48].  
The toe and fifth metatarsal markers defined the anterior and lateral limits of the BOS (A/P and M/L U\({}_{\text{MAX}}\)), respectively. The M/L direction was the focus of this study; therefore, the MOS during obstacle crossing was determined by the distance between the XCOM and M/L U\({}_{\text{MAX}}\).  
\[MOS=U_{MAX}-XCOM \tag{3}\]  
The other dynamic stability variables, COP-COM and COP-XCOM, were also calculated in the M/L direction (i.e., along the \(x\)-axis) of the lab coordinate system. The COM-COP inclination angle was calculated by finding the spatial angle between the vector from the COM to the vertical projection of the COM on the ground and the vector from the COM to the COP [53].  
The secondary variables of interest were the spatiotemporal gait parameters: stance time, swing time, and step length for both steps and stride length, stride width, and velocity for the gait cycle (averaged across the three right and three left gait cycles). For example, a left obstacle-crossing gait cycle consisted of a right leading limb and a left trailing limb (Figure 3). In this case, the spatiotemporal gait parameters include (1) first stance time, i.e., the time spent from left (trailing limb) heel strike to left toe-off; (2) first swing time, i.e., the time spent from right (leading limb) toe-off to right heel strike; (3) first step length, i.e., the distance from the left heel to the right heel; (4) second stance time, i.e., the time spent from right heel strike to right toe-off; (5) second swing time, i.e., the time spent from left toe-off to left heel strike; and (6) second step length, i.e., the distance from the right heel to the left heel.