数学建模及其应用

本文以2025年“高教社杯”全国大学生数学建模竞赛E题为背景，针对立定跳远过程，构建了融合运动者体质信息与姿态特征的综合运动模型.通过质心计算、关键点约束与机器学习验证，准确识别了滞空阶段并拟合出质心的抛物线轨迹.在影响因素分析中，结合姿态特征与体质数据，确定了包括起跳初速度在内的关键影响变量.然后采用多种机器学习方法对跳远成绩进行拟合与预测，结果显示随机森林回归模型具有最佳性能.为提升数据质量，提出了符合人体运动学规律的数据增强与补帧方法，进一步对性能进行了优化.最后，基于随机森林的特征重要性分析与差分进化方法，获得了关键动作特征及其最优调整方案，从而为运动员提供了科学的训练建议并可预测理想成绩.

Based on the 2025 CUMCM Problem E, this study models the standing long jump process by integrating athletes' physical and postural features. Through center-of-mass calculation, key-point constraints, and machine-learning validation, the suspension phase was accurately identified and its parabolic trajectory fitted. Combining posture and physical data, key factors such as take-off velocity were determined. Multiple machine-learning models were employed to fit jump performance, among which Random Forest regression achieved the best outcome. A kinematics-based data-augmentation and frame-interpolation method further enhanced performance. Finally, feature-importance analysis and differential-evolution optimization revealed key motion features and their optimal adjustments, providing scientific training recommendations for athletes and enabling prediction of ideal performance outcomes.