简介

作业1简单实现了一个以一定初始速度和角速度的模型和墙壁碰撞的效果.
总共讲解了三种算法

1. impulse (脉冲法)

2. Shape Matching(基于形状保持的算法, 不包含物理特性)

3. Penalty methods

Shape Matching 可以说是最简单的方法之一

因为完全不涉及角速度. 基础逻辑理论就是, 当模型和墙壁发生碰撞的时候.这些碰撞的粒子会产生相反的形变但是. 我们强制将其整理保持形状的一致性. 就是先形变. 再变回来.
由于形变的步骤. 是中间步骤. 不会展示出来. 所以效果还是挺好的.

Image

TIPS

code

using System.Collections;
using System.Collections.Generic;
using UnityEngine;

public class Rigid_Bunny_by_Shape_Matching : MonoBehaviour
{
	public bool launched = false;
	Vector3[] X; // world position
	Vector3[] Y; // temp world position
	Vector3[] Q; // Local coordinates
	Vector3[] V; // speed
	Vector3[] OldX;
	Matrix4x4 QQt = Matrix4x4.zero;
	Vector3 G = new Vector3(0.0f, -9.8f, 0.0f);
	float linear_decay = 0.999f;
	Vector3 ground = new Vector3(0, 0.01f, 0);
	Vector3 groundNormal = new Vector3(0, 1, 0);
	Vector3 wall = new Vector3(2.01f, 0, 0);
	Vector3 wallNormal = new Vector3(-1, 0, 0);
	float mu_T = 0.5f;  // μ_T may be coefficient of air resistance
	float mu_N = 5.0f; // μ_N may be Coefficient of Restitution
	float m_timer = 0;


	// Start is called before the first frame update
	void Start()
    {
    	Mesh mesh = GetComponent().mesh;
        V = new Vector3[mesh.vertices.Length];
        X = mesh.vertices;
		Y = mesh.vertices;
		OldX = mesh.vertices;
        Q = mesh.vertices;

        //Centerizing Q.
        Vector3 c=Vector3.zero;
        for(int i=0; i 1.0f) value= 1.0f;
	        float phi = Mathf.Acos(value);
	        float lambda2=(I_c+2*k_root*Mathf.Cos(phi/3))/3.0f;
	        float lambda=Mathf.Sqrt(lambda2);
	        
	        float III_u = Mathf.Sqrt(III_c);
	        if(det<0)   III_u=-III_u;
	        float I_u = lambda + Mathf.Sqrt(-lambda2 + I_c + 2*III_u/lambda);
	        float II_u=(I_u*I_u-I_c)*0.5f;
	        
	        
	        float inv_rate, factor;
	        inv_rate=1/(I_u*II_u-III_u);
	        factor=I_u*III_u*inv_rate;
	        
	       	Matrix4x4 U = Matrix4x4.zero;
			U[0,0]=factor;
	        U[1,1]=factor;
	        U[2,2]=factor;
	        
	        factor=(I_u*I_u-II_u)*inv_rate;
	        for(int i=0; i<3; i++)
	        for(int j=0; j<3; j++)
	            U[i,j]+=factor*C[i,j]-inv_rate*C2[i,j];
	        
	        inv_rate=1/III_u;
	        factor=II_u*inv_rate;
	        inv_U[0,0]=factor;
	        inv_U[1,1]=factor;
	        inv_U[2,2]=factor;
	        
	        factor=-I_u*inv_rate;
	        for(int i=0; i<3; i++)
	        for(int j=0; j<3; j++)
	            inv_U[i,j]+=factor*U[i,j]+inv_rate*C[i,j];
	    }
	    
	    Matrix4x4 R=Matrix4x4.zero;
	    for(int ii=0; ii<3; ii++)
	    for(int jj=0; jj<3; jj++)
	    for(int kk=0; kk<3; kk++)
	        R[ii,jj]+=F[ii,kk]*inv_U[kk,jj];
	    R[3,3]=1;
	    return R;
	}

	// Update the mesh vertices according to translation c and rotation R.
	// It also updates the velocity.
	void Update_Mesh(Vector3 c, Matrix4x4 R, float inv_dt)
   	{
   		for(int i=0; i().mesh;
		mesh.vertices=X;
   	}

	void Collision(float inv_dt)
	{
		for(int i=0; i