## 3 Results and Discussion
### Lateral-Wedge Hardness Effect  
Lateral stability, as a crucial biomechanical parameter in sports science, is defined as the capacity of the ankle complex to resist excessive pronation or inversion under dynamic loading conditions [12, 34], with its primary function lying in the prevention of ankle injuries during directional changes [12, 35]. This stability mechanism is intrinsically modulated by footwear design parameters, particularly lateral-wedge hardness, which refers to the stiffness of the lateral side of a shoe's midsole, influencing foot-ground interaction during weight-bearing movements. In sports requiring frequent lateral shifts, such as badminton, this parameter affects foot stability, energy transfer, and impact attenuation. A harder lateral wedge is generally associated with reduced midsole deformation, potentially enhancing foot alignment and responsiveness during rapid directional changes [23, 27, 30]. However, excessive stiffness may restrict foot flexibility, alter joint loading patterns, and influence both performance and injury risk [27, 30, 31].  
[PERSON] et al. [12] investigated the effects of varying lateral-wedge hardness levels on lower-limb biomechanics during badminton-specific movements, including jump landing, cross-step, sidestep, and lunges. The study quantified joint kinematics and kinetics under different stiffness conditions, aiming to understand how changes in lateral-wedge hardness affect footwork mechanics and injury risk. The results showed that increased lateral-wedge hardness shortened contact times by 8.9-13.5% and increased joint stiffness in the foot-ankle complex during sidestep maneuvers, suggesting improved stability and agility. Time-varying differences were noted in the initial landing and push-off phases of cross- and sidesteps, as well as in the drive-off phase of lunges, indicating that stiffer lateral wedges facilitate footwork execution by improving ground reaction efficiency.  
These findings align with the \"shoe midsole stiffness-energy loss\" theory proposed by [PERSON] and [PERSON] [23], which suggests that stiffer midsole materials minimize energy dissipation at the metatarasophalangeal joint, thereby improving propulsion efficiency during takeoff. However, the study also observed that a 70D lateral wedge, while reducing ankle inversion angles, may excessively limit ankle dorsiflexion and knee joint range of motion in the coronal plane. This reflects a non-linear relationship between stiffness and joint freedom, similar to findings on forefoot bending stiffness [14], highlighting the need to balance rigidity with functional mobility in footwear design.  
While [PERSON] et al. [12] provides valuable insights into badminton shoe development, certain limitations must be acknowledged. Laboratory conditions may not fully replicate real-game scenarios, where fatigue accumulation and unpredictable movement patterns can influence biomechanical responses. Additionally, individual differences such as body weight, technical habits, and foot morphology may modulate the effects of lateral-wedge hardness, underscoring the need for further research on personalized footwear optimization.