Jack and Box Simulation 弹簧玩偶盒仿真

Physics simulation using Lagrangian mechanics and impact dynamics 使用拉格朗日力学和碰撞动力学的物理仿真

Python, SymPy, Dynamics

Authors: Allen Liu

GitHub: View this Project on GitHub

Project Description

Physics-based simulation of a jack-in-the-box system using Lagrangian mechanics and impact dynamics. The project models the coupled dynamics of a spring-loaded jack mechanism inside a box, including collision detection and impulse-based contact resolution.

System Overview

graph TB
    subgraph Model["Physical System"]
        JACK[Jack Mechanism<br/>4 Point Masses]
        BOX[Box<br/>Rigid Body]
        SPRING[Spring Force]
        GRAVITY[Gravitational Force]
    end

    subgraph Simulation["Simulation Engine"]
        LAGRANGE[Lagrangian Formulation<br/>Kinetic & Potential Energy]
        EOM[Equations of Motion<br/>Euler-Lagrange]
        IMPACT[Impact Detection<br/>& Resolution]
    end

    subgraph Computation["SymPy + Python"]
        SYMBOLIC[Symbolic Math<br/>SymPy]
        NUMERIC[Numerical Integration<br/>SciPy ODE Solver]
        VIS[Visualization<br/>Matplotlib Animation]
    end

    JACK --> LAGRANGE
    BOX --> LAGRANGE
    SPRING --> LAGRANGE
    GRAVITY --> LAGRANGE

    LAGRANGE --> EOM
    EOM --> IMPACT
    IMPACT --> NUMERIC

    EOM --> SYMBOLIC
    SYMBOLIC --> NUMERIC
    NUMERIC --> VIS

    style LAGRANGE fill:#e1f5ff
    style IMPACT fill:#fff4e1
    style VIS fill:#d4edda

Amination

The result of the jack and box simulation can be shown in the video below

Structure

Rigit Body Transformation

To model the dynamics about the jack and the box, we defined all necessary frames as shown in figure below:

graph TB
    subgraph World["World Frame"]
        W["w<br/>Origin"]
    end

    subgraph Box["Box (Diamond Shape)"]
        A["a, b<br/>Box Center"]
        C["c<br/>Top Corner"]
        D["d<br/>Left Corner"]
        E["e<br/>Bottom Corner"]
        F["f<br/>Right Corner"]
    end

    subgraph Jack["Jack (X Shape)"]
        G["g, h<br/>Jack Center"]
        I["i<br/>Arm 1"]
        J["j<br/>Arm 2"]
        K["k<br/>Arm 3"]
        L["l<br/>Arm 4"]
    end

    W --> A
    A --> C
    A --> D
    A --> E
    A --> F

    W --> G
    G --> I
    G --> J
    G --> K
    G --> L

    style W fill:#e1f5ff
    style A fill:#fff4e1
    style G fill:#d4edda

Then we can calculate the transformation between all frames:

Box Frame Transformations:

\[\begin{align*} g_{wa} &= G(0, x_b, y_b) \\ g_{ab} &= G(\theta_b, 0, 0) \\ g_{bc} &= G\left(0, \frac{L_b}{2}, \frac{L_b}{2}\right) \\ g_{cd} &= G\left(0, -\frac{L_b}{2}, \frac{L_b}{2}\right) \\ g_{de} &= G\left(0, -\frac{L_b}{2}, -\frac{L_b}{2}\right) \\ g_{ef} &= G\left(0, \frac{L_b}{2}, -\frac{L_b}{2}\right) \end{align*}\]

Jack Frame Transformations:

\[\begin{align*} g_{wg} &= G(0, x_j, y_j) \\ g_{gh} &= G(\theta_j, 0, 0) \\ g_{hi} &= G\left(0, \frac{L_j}{2}, 0\right) \\ g_{hj} &= G\left(0, 0, \frac{L_j}{2}\right) \\ g_{hk} &= G\left(0, -\frac{L_j}{2}, 0\right) \\ g_{hl} &= G\left(0, 0, -\frac{L_j}{2}\right) \end{align*}\]

Euler-Language Equation Formulation

In this project, to simplify the problem, we model the jack as 4 point mass with same mass, $m_j$, and model the box with the mass $m_b$ and moment of intertia of $I_b$, so that the total kinetic energy and potential energy can be easily obtained by this

\[\begin{align*} I &= \frac{1}{3}ML^2 \\ I^{**} &= \begin{bmatrix} M & 0 & 0 & 0 & 0 & 0 \\ 0 & M & 0 & 0 & 0 & 0 \\ 0 & 0 & M & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\cal I} \\ \end{bmatrix} \\ V^b &= \check{\left( g^{-1} \dot{g} \right)} \\ K &= \sum_i{V_{ib}^T I_i^{**}V_i^b} \\ P &= \sum_i{ m_igh_i }\\ L &= K - P \end{align*}\]

Impact Dynamics

The collision between jack and box is resolved using impulse-momentum equations.

flowchart TD
    START([Time Integration]) --> DETECT{Contact<br/>Detected?}

    DETECT -->|No| INTEGRATE[Continue Integration<br/>EOM]
    INTEGRATE --> START

    DETECT -->|Yes| COMPUTE_VEL[Compute Pre-Impact<br/>Velocities]
    COMPUTE_VEL --> IMPULSE[Calculate Impulse<br/>λ Δφ]
    IMPULSE --> UPDATE_VEL[Update Post-Impact<br/>Velocities]
    UPDATE_VEL --> CONSERVE[Verify Energy<br/>Constraint]
    CONSERVE --> INTEGRATE

    style DETECT fill:#fff4e1
    style IMPULSE fill:#e1f5ff
    style CONSERVE fill:#d4edda

Impact Equations:

\[\begin{align*} P \big |^{\tau_+}_{\tau_-} &= \lambda \Delta \phi \quad \text{(Momentum jump)}\\ {\cal H} \big |^{\tau_+}_{\tau_-} &= 0 \quad \text{(Hamiltonian conservation)}\\ {\cal H} &= \frac{d {\cal L}}{\dot{q}} \cdot \dot{q} - {\cal L} \quad \text{(Hamiltonian definition)} \end{align*}\]

Where:

  • $ P $: Generalized momentum
  • $ \lambda $: Impact impulse magnitude
  • $ \Delta \phi $: Contact constraint gradient
  • $ {\cal H} $: Hamiltonian (total energy)

By solving these equations, we simulate the complete dynamics of the jack-in-the-box system including realistic collision behavior.

Python, SymPy, 动力学

作者: Allen Liu

GitHub: 在 GitHub 上查看此项目

项目描述

使用拉格朗日力学和碰撞动力学对弹簧玩偶盒系统进行物理仿真。该项目对盒子内弹簧加载的玩偶机构的耦合动力学进行建模，包括碰撞检测和基于冲量的接触求解。

动画演示

弹簧玩偶盒仿真的结果可以在下面的视频中看到

结构

欧拉-拉格朗日方程推导

在这个项目中，为了简化问题，我们将玩偶建模为4个具有相同质量 $m_j$ 的质点，将盒子建模为质量 $m_b$ 和转动惯量 $I_b$ 的刚体，这样可以轻松获得总动能和势能。

碰撞动力学

玩偶和盒子之间的碰撞使用冲量-动量方程求解。

通过求解这些方程，我们模拟了弹簧玩偶盒系统的完整动力学，包括真实的碰撞行为。