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现有模型因忽略三维几何约束与动态交互效应, 难以准确模拟复杂楼梯场景下的群体行为. 本文提出一种双层运动模型, 通过分层建模方法融合三维元胞离散化空间、双足步态动力学及接触力扰动分析. 模型将行人抽象为“双足一点”多节点系统, 构建上层质心运动空间与下层双足支撑平面. 模型下层基于元胞路径规划约束跨步运动, 并设计准同步状态切换机制保障群体时空一致性; 上层采用几何检测算法识别行人物理接触, 结合碰撞动力学模型, 量化接触冲突对行人稳定性的影响. 仿真实验表明, 模型能够有效模拟行人上下楼运动轨迹、动态平衡维持机制及失稳事件演化过程. 研究采用稳定裕度评估行人间接触力的扰动效应, 揭示了密度对失稳风险的正向影响, 为楼梯场景下的安全评估与疏散优化提供了高效仿真工具.
This study addresses the critical challenge of simulating pedestrian crowd dynamics in staircase environments, where existing models often neglect three-dimensional geometric constraints and dynamic interactions. We propose a novel dual-layer motion model (DLM) that integrates a hierarchical kinematic-dynamic coupling framework, geometric discretization methods, and crowd interaction mechanisms. The model abstracts pedestrians as a multi-node “bipedal single-point” system, distinguishing between an upper-layer centroid motion plane and a lower-layer dual-foot support space. This method combines spatiotemporal modeling and contact mechanics to address the complexity of stairwell dynamics. The lower layer uses cellular path planning to constrain stepping motions and ensures spatiotemporal consistency of the crowd through a quasi-synchronous state transition mechanism. The upper layer uses an ellipse-projection-based separating axis algorithm to detect collision conflicts and quantifies contact effects by using collision dynamics. Additionally, a quasi-synchronous state migration mechanism is introduced within a hybrid discrete-continuous time framework to coordinate gait cycles in large-scale multi-agent simulations and solve the problem of temporal asynchrony. Based on the stability control principle of inverted pendulum dynamics and combined with biomechanical regulation capabilities and motion threshold constraints, the perturbation effects of contact forces on pedestrian balance are quantified, enabling individual dynamic stability analysis. To validate the model, a parameterized stairwell scenario (step height: 0.15 m, tread depth: 0.26 m) is constructed to simulate the motion of heterogeneous pedestrians (mass: (65 ± 5) kg, height: (1.70 ± 0.2) m). The simulation results show that the model accurately captures the dynamic features of pedestrian movement in stairwells: the centroid displacement ratio is very close to the theoretical staircase slope, and the deviation between the crowd’s average speed and empirical data is less than 6%. Dynamic stability analysis reveals the evolution from individual local imbalance to group instability. Further parametric studies indicate that balancing target attraction weight (α) and repulsion weight (β) can regulate the cohesion of crowd behavior, while increasing the collision recovery coefficient (e) can amplify contact force fluctuations. In conclusion, the dual-layer model links motion planning and dynamic stability in the stairwell environments, providing high-fidelity insights into pedestrian safety. The results emphasize the interdependence between geometric constraints, biomechanical adjustments, and density-driven instability. Future research may extend the model to irregular stair geometries and incorporate heterogeneous pedestrian parameters to improve the predictive accuracy of evacuation optimization and architectural safety design. -
Keywords:
- staircase pedestrian dynamics /
- dual-layer motion model /
- crowd simulation /
- contact mechanics
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图 3 行人运动-群体交互仿真系统框架
Fig. 3. Framework of pedestrian motion-crowd interaction simulation system.
图 5 行人双层运动模型系统组成 (图中, 橙色框(a)内为接触冲突模块框架, 绿色框(b)内为自主运动模块组成, 蓝色框(c)内为准同步机制流程)
Fig. 5. System architecture of DLM (In the figure, the orange box (a) is the contact conflict module framework, the green box (b) is the autonomous motion module composition, and the blue box (c) is the quasi-synchronous mechanism process)
图 6 基于元胞迁移移动规划(上楼)
Fig. 6. Moving planning based on 3D cell migration (ascending scenario).
图 7 楼梯上下行路径的几何计算
Fig. 7. Geometric theoretical calculation of ascending and descending paths.
图 8 不同状态下的行人运动示意图 (a) 平稳运动状态; (b) 失衡前倾状态
Fig. 8. Pedestrian motion in different states: (a) Stable motion state; (b) unbalanced leaning state
图 9 基于倒立摆的失稳动力学模型示意图
Fig. 9. Schematic of inverted pendulum-based instability dynamic model.
图 10 行人下肢与质心的三维运动轨迹分析
Fig. 10. 3 D trajectory analysis of lower limbs and center of mass.
图 12 步长调节与速度波动
Fig. 12. Impact mechanism of step length adjustment on speed fluctuations.
图 13 群体密度与速度的仿真-观测数据对比(时序分析)) (a)个体的上、下层速度时序; (b)人群密度变化; (c)人群移动速度变化
Fig. 13. Comparison of crowd density and speed between simulation and observation data: (a) Time series of the speed of each individual on the upper and lower floors; (b) change of crowd density; (c) change of crowd moving speed
图 14 失稳行人局部区域可视化 (a) 几何可视化; (b) 元胞占用状况
Fig. 14. Visualization of localized region of unstable pedestrian dynamics: (a) Geometric trajectories; (b) cellular occupancy status.
图 15 行人失稳前后时段(6 s)内的关键指标变化
Fig. 15. Temporal evolution of key parameters before and after instability (6 s).
图 18 失稳效应下的运动状态转移概率
Fig. 18. Probability of motion state transition under instability effect.
图 19 失稳事件中密度-失稳概率关系
Fig. 19. Density-instability probability relationship in instability events.
图 20 不同参数组合的个体移动速度差异 (a) α = 0.3和β=0.7; (b) α = 0.5和β = 0.5; (c) α = 0.7和β = 0.3
Fig. 20. Differences in individual movement speeds for different parameter combinations: (a) α = 0.3 and β = 0.7; (b) α = 0.5 and β = 0.5; (c) α = 0.7 and β = 0.3.
图 21 不同人数与参数的接触冲突分布
Fig. 21. Distribution of contact conflicts under different crowd sizes and parameter (α/β).
图 22 不同人群规模-总接触时序
Fig. 22. Temporal evolution of total contact conflicts under different crowd sizes.
图 23 不同恢复系数的接触力分布状况 (a)平均接触力; (b)最大接触力
Fig. 23. Distribution of contact forces under different restitution coefficient e: (a) Mean contact force; (b) maximum contact force.
图 24 接触力对恢复系数变化的动态响应
Fig. 24. Dynamic response of contact force to changes in restitution coefficient e.
图 25 接触力与稳定裕度关联分析
Fig. 25. Correlation analysis between contact force and stability margin $\varPhi $.
表 1 双层运动模型关键符号说明
Table 1. Description of key symbols of DLM.
符号 说明 ${{p}^{\text{m}}}$ 行人移动足的位置坐标 ${{p}^{\text{m}}} = \left( {{x_{\text{m}}}, {y_{\text{m}}}, {z_{\text{m}}}} \right)$ ${{p}^{\text{s}}}$ 行人支撑足的位置坐标 ${{p}^{\text{s}}} = \left( {{x_{\text{s}}}, {y_{\text{s}}}, {z_{\text{s}}}} \right)$ ${{p}^{\text{c}}}$ 行人质心的位置坐标 ${{p}^{\text{c}}} = \left( {{x_{\text{c}}}, {y_{\text{c}}}, {z_{\text{c}}}} \right)$ ${{p}^{\text{B}}}$ 行人支撑区域中心在下层空间的位置坐标 ${{p}^{\text{B}}} = $$ ({x_{\text{B}}}, {y_{\text{B}}})$ $v$ 行人速度向量 $v = \left[ {{v_x}, {v_y}, {v_z}} \right]$ $\hat o$ 行人躯干朝向单位向量 $\left\| {\hat o} \right\| = \sqrt {{o_x}^2 + {o_y}^2} = 1$ $ E(\cdot) $ 行人椭圆(Ellipse)形态几何方程 $ R(\cdot) $ 行人矩形(Rectangle)形态几何方程 $\psi $ 行人运动状态, 包括平稳步进(S1)、失衡倾斜(S2)
与失稳倒地(S3), $\psi \in \left\{ {S1, S2, S3} \right\}$$\zeta (\psi )$ 行人所处运动状态$\psi $的持续时长(单位: 秒) $M$ 行人下肢支撑状态标记, 单支撑(摆动)状态为0,
双支撑状态为1.$\theta $ 行人躯干倾斜角度(单位: 弧度) $\omega $ 行人失衡倾倒角速度(单位: 弧度/秒) ${T^n}$ 第n步运动开始时刻(单位: 秒) ${T^{n + 1}}$ 第n+1步运动开始时刻, 即第n步运动结束时刻
(单位: 秒)${\Delta ^n}$ 第n步运动时间长度(单位: 秒) 
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表 2 模型仿真参数设置说明
Table 2. Description of simulation parameters.
参数 数值 参数 数值 行人质量
$m$/kg65+5.0×rand[31] 行人足部宽度
${W_{{\text{foot}}}}$/m0.15[31] 重力加速度
$g$/(m·s–2)9.80[31] 台阶踏板深度
${d_{{\text{stair}}}}$/m0.26[32] 行人的身高
$H$/m1.70+0.2×rand[31] 单层台阶高度
${h_{{\text{stair}}}}$/m0.15[32] 行人质心
高度${H_{\text{c}}}$0.77×$H$[31] 台阶踏板宽度
${w_{{\text{stair}}}}$/m1.0[32] 行人肩宽
${W_{\text{b}}}$/m0.4[31] 楼梯坡面倾斜角
${\varphi _{{\text{stair}}}}$/rad0.5233 行人躯干
厚度${D_{\text{b}}}$/m0.2[31] 绕质心转动惯量
${I_c}$/(kg·m2)9.83[31] 松弛时间
$\varepsilon $/s0.3[30] 倾斜阈值${\theta _{\text{T}}}$/rad 0.75[30] 注: rand表示[0, 1]区间均匀分布的随机数, 用于模拟个体差异; 楼梯坡面倾角由台阶几何计算得出${\varphi _{{\text{stair}}}} = {\tan ^{ - 1}}\left( h_{{\text{stair}}}/ $$ {d_{{\text{stair}}}} \right)$.
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