## 1 Introduction  
Badminton, one of the most popular sports globally [1; 2; 3; 4], is the fastest non-contact racket sport, and requires a combination of strength, speed, and stamina. While playing this sport, players must repetitively lunge, jump, and quickly change direction from a wide variety of positions. A previous epidemiology study conducted in Ireland reported that badminton was the sport with the most injuries [5]. Lower extremity injuries accounted for 43 to 86% of all injuries regardless of the nationality [6], and overuse is considered the major reason [7; 8]. Ankles and knees are the most injured sites [1; 7; 9], particularly the knee joint [7; 8]. Patellar tendinopathy [7] is the most common type of knee injury. As a specific and often-used example of footwork [10; 11], the repetitive lunge is a likely cause of patellar tendinopathy, particularly for teenager players [3; 12]. Moreover, epidemiology studies have also shown that a higher badminton injury rate was found at the end of a match or training [13], and higher rates have been found during training [7; 14]. This could be explained by a higher intensity of training routine [7], which may induce fatigue. Because fatigue reduces the capacity of muscles to generate force, it may be an important factor causing injury. In badminton, increased ankle sprain injuries were found to occur at the end of a training or match session due to the accumulation of fatigue [6]. Due to fatigue caused by repeating the forward lunge, the activity of the vastus lateralis, vastus medialis, and biceps femoris showed a significant change [15], and knee injury risk was increased [16]. A number of studies [10; 17; 18; 19; 20; 21] have investigated the biomechanical characteristics of the lunge, particularly its stance phase, which is defined as the period of time from initial contact to final lift-off from the ground by the dominant limb [10]. Ankle sprain [6] and patellar tendinopathy [14] may occur during the stance phase. However, the relationship between the repetitive lunge and injury is still not clear. A previous study examined the lunge under the condition of exhaustion to investigate the fatigue effect on the knee [16]. A limitation of this study is that the changes of lunge motion during the fatiguing process were ignored. Injury usually occurs at a certain time, instantaneously, and the factors inducing injury are prolonged. However, we are unaware of any studies that have investigated the biomechanical changes during the fatiguing process in badminton. Therefore, there is a lack of objective data on lower joint kinematics and kinetics of the lunge during the fatiguing process, which may provide essential insight into the understanding of the mechanism of injury. Additionally, discrete data (related, for example, to the peak angle) has traditionally been used for statistical analysis to confirm the differences outlined above. However, it is worth noting that all kinematic and kinetic variables are continuous variables with time. Furthermore, as mentioned above, injury occurs partly due to the accumulation of changes. Thus, continuous data analysis may provide other useful information. In recent years, one-dimensional statistical parametric mapping (SPM (1D)) has been accepted as an effective method to analyze waveform data [22]. Kinematic waveform data of hip, knee, and ankle joints in three planes, measured in players with different levels, have been analyzed by SPM (1D) [18; 19]. However, to the best of the authors' knowledge, no research has investigated the fatigue response by analyzing the biomechanics waveform data of the lunge. Consequently, the purpose of this study was to investigate the changes of lunge biomechanical characteristics during the fatiguing process, from normal to fatigued states. To achieve this objective, a repetitive forehand forward lunge to exhaustion was proposed as the fatigue protocol. The fatiguing process was divided into sub-stages.