{"id":1392,"date":"2023-09-06T10:54:58","date_gmt":"2023-09-06T02:54:58","guid":{"rendered":"https:\/\/www.mustenaka.cn\/?p=1392"},"modified":"2023-11-16T18:09:00","modified_gmt":"2023-11-16T10:09:00","slug":"pbd-method-learn-01","status":"publish","type":"post","link":"https:\/\/www.mustenaka.cn\/index.php\/2023\/09\/06\/pbd-method-learn-01\/","title":{"rendered":"PBD method\u5b66\u4e60\uff08Position Based Dynamics method learn\uff09-01"},"content":{"rendered":"<h1>\u9996<\/h1>\n<p>\u7531\u4e8e\u9700\u8981\u5b9e\u73b0\u8bf8\u591a<span>deformable body\uff08\u53ef\u5f62\u53d8\u7269\u4f53\uff09\u7684\u4ea4\u4e92\u5904\u7406\uff0c\u5305\u62ecSoftbody\uff08\u67d4\u4f53\uff09\u3001\u5e03\u6599\uff08Cloth\uff09\u3001\u7ef3\u7d22\uff08Rope\uff09\u4ee5\u53ca\u57fa\u4e8e\u4ed6\u4eec\u4e4b\u4e0a\u7684\u4ea4\u4e92\u5de5\u4f5c\uff0c\u5728\u7ecf\u8fc7\u5b66\u4e60\u4e4b\u540e\uff08\u4e5f\u8bb8\u662f\u98d8\u4e86\uff09\uff0c\u6253\u7b97\u81ea\u5df1\u8bb2\u5b66\u4e60\u8fc7\u7a0b\u4e2d\u9047\u5230\u7684\u601d\u60f3\u3001\u95ee\u9898\u70b9\uff0c\u89e3\u51b3\u65b9\u6848\u9010\u6b65\u6c47\u603b\u8d77\u6765\u5199\u4e00\u4e2a\u5b66\u4e60\u7cfb\u5217\u7684\u6587\u7ae0\uff0c\u8fd9\u4e2a\u65b9\u5411\u786e\u5b9e\u6709\u96be\u5ea6\uff0c\u4e5f\u8bb8\u5199\u7684\u4e0d\u662f\u5f88\u6b63\u786e\uff0c\u672c\u4eba\u540c\u6837\u4e00\u767e\u5206\u7684\u6b22\u8fce\u5c0f\u4f19\u4f34\u6279\u8bc4\uff0c\u8fd9\u5bf9\u6211\u7684\u5de5\u4f5c\u4e5f\u6709\u5f88\u5927\u7684\u5e2e\u52a9\u3002<\/span><\/p>\n<p>\u53e6\u5916\u9644\u4e00\u4e2aUnity\u76f8\u5173\u7684\u6210\u719f<span>deformable body\u5e93\u6280\u672f\u63a2\u8ba8\u7814\u7a76\u7684\u6c47\u603b\u7bc7\uff0c\u6b63\u5728\u5236\u4f5cing\uff0c\u5b8c\u6210\u540eurl\u4f1a\u9644\u4e0a\uff1a<\/span><\/p>\n<h1>\u7b80\u4ecb<\/h1>\n<p>PBD\uff08Position Based Dynamics\uff09\u7b97\u6cd5\u662f\u4e00\u79cd\u7528\u4e8e\u6a21\u62df\u7269\u4f53\u7684\u7269\u7406\u884c\u4e3a\u548c\u8fd0\u52a8\u7684\u7b97\u6cd5\u3002\u5176\u6838\u5fc3\u601d\u60f3\u662f\u79bb\u6563\u5316\u7269\u4f53\u7684\u8fd0\u52a8\u65b9\u7a0b\uff0c\u5c06\u5176\u5206\u89e3\u4e3a\u4e00\u7cfb\u5217\u8fed\u4ee3\u6b65\u9aa4\uff0c\u6bcf\u4e00\u6b65\u90fd\u7528\u6765\u66f4\u65b0\u7269\u4f53\u7684\u4f4d\u7f6e\u548c\u901f\u5ea6\u3002\u8fd9\u4e9b\u8fed\u4ee3\u6b65\u9aa4\u53ef\u4ee5\u6a21\u62df\u591a\u79cd\u7269\u7406\u73b0\u8c61\uff0c\u5305\u62ec\u5f39\u6027\u3001\u78b0\u649e\u3001\u6db2\u4f53\u6d41\u52a8\u7b49\u3002<\/p>\n<p>PBD\u7b97\u6cd5\u662f\u4e00\u79cd\u7b97\u6cd5\u601d\u60f3\uff0c\u4e3a\u7269\u7406\u6a21\u62df\u63d0\u51fa\u4e86\u8bf8\u591a\u7684\u6b65\u9aa4\u53bb\u8ba1\u7b97\uff0c\u7b97\u6cd5\u4e5f\u53ef\u4ee5\u548c\u8bf8\u591a\u7684\u8ba1\u7b97\u5206\u914d\u505a\u7ed3\u5408\uff0c\u6bd4\u5982\u5229\u7528Unity\u7684DOTS\u6846\u67b6\u4f7f\u5176\u505a\u597d\u591a\u7ebf\u7a0b\u8ba1\u7b97\u5206\u914d\uff0c\u6216\u8005\u5229\u7528CUDA\u8ba1\u7b97\u5355\u5143+FEM\uff08\u6709\u9650\u5143\uff09\u5c06\u95ee\u9898\u77e9\u9635\u5316\u5904\u7406\u4ee5\u5b9e\u73b0\u5927\u89c4\u6a21\u6a21\u62df\u3002<\/p>\n<p>\u76f8\u6bd4\u76f4\u63a5\u8ba1\u7b97\u529b\u5b66\uff0c\u5176\u4e5f\u62e5\u6709\u4e24\u4e2a\u4f18\u70b9<\/p>\n<ol>\n<li data-pid=\"63jBmh9c\">\u76f4\u63a5\u57fa\u4e8e\u4f4d\u7f6e\u800c\u975e<strong>\u53d7\u529b\u60c5\u51b5<\/strong>\u6765\u8ba1\u7b97\u4f4d\u7f6e\u3002\u8ba1\u7b97\u7684\u7ed3\u679c\u867d\u7136\u5e76\u4e0d\u662f\u5b8c\u5168\u771f\u5b9e\u7684\u7269\u7406\u6a21\u62df\uff0c\u4f46\u662f\u53ef\u4fe1\u5ea6\u5f88\u9ad8\u3002<\/li>\n<li data-pid=\"63jBmh9c\">\u6027\u80fd\u66f4\u597d\uff0c\u800c\u4e14\u6027\u80fd\u6d88\u8017\u7684\u529b\u5ea6\u53ef\u63a7\uff0c\u53ef\u4ee5\u6839\u636e\u5b9e\u9645\u5e94\u7528\u4e2d\u5bf9\u771f\u5b9e\u6027\u7684\u9700\u6c42\u5f3a\u5ea6\u6765\u51b3\u5b9a\u8ba1\u7b97\u91cf\u3002<\/li>\n<li data-pid=\"63jBmh9c\">\u5b9e\u65f6\u8ba1\u7b97<\/li>\n<\/ol>\n<h1>\u7b97\u6cd5\u8fc7\u7a0b<\/h1>\n<h2>\u7b26\u53f7\u8bf4\u660e<\/h2>\n<p>\u4eff\u771f\u7269\u4f53\u5305\u62ec <span>\\(N\\)<\/span>\u4e2a\u8282\u70b9\u548c <span>\\(M\\)<\/span>\u4e2a\u7ea6\u675f\u3002<\/p>\n<p>\u6bcf\u4e00\u4e2a\u8282\u70b9<span>\\(i \\in [1, &#8230;, N]\\)\u81f3\u5c11\u5305\u542b\u7684\u53c2\u6570\u6709<\/span><\/p>\n<ul>\n<li><span>\u8d28\u91cf\\(m_i\\)<\/span><\/li>\n<li><span>\u4f4d\u7f6e\\(p_i\\) \u4e5f\u6709\u5f88\u591a\u4eba\u7684\u6587\u7ae0\u559c\u6b22\u7528\\(x_i\\)\u8868\u793a\uff0c\u6211\u559c\u6b22\u7528p\uff0c\u6709position\u4e4b\u610f<\/span><\/li>\n<li><span>\u901f\u5ea6\\(v_i\\)<\/span><\/li>\n<\/ul>\n<p>\u6bcf\u4e00\u4e2a\u7ea6\u675f<span>\\(j \\in [1, &#8230;, M]\\)\u81f3\u5c11\u5305\u542b\u7684\u53c2\u6570\u6709<\/span><\/p>\n<ul>\n<li>\u7ea6\u675f\u57fa\u6570<span>\\(n_j\\) \uff1a\u7ea6\u675f\u51fd\u6570\u4e2d\u4f8b\u5b50\u7684\u4e2a\u6570<\/span><\/li>\n<li>\u7ea6\u675f\u51fd\u6570<span>\\(C_j\\)\uff1a\u4e09\u4e2a\u7ef4\u5ea6\uff08x,y,z\uff09\u5206\u522b\u90fd\u6709\u7ea6\u675f\\(R^{3nj} -&gt; R\\)<\/span><\/li>\n<li>\u7ea6\u675f\u53c2\u6570\n<ul>\n<li>\u521a\u5ea6\uff08stiffness\uff09\u53c2\u6570\uff1a<span>\\(k_j \\in [0,1]\\)<\/span><\/li>\n<li>\u963b\u5c3c\uff08damping\uff09\u53c2\u6570\uff1a<span>\\(d_j \\in [0,1]\\)<\/span><\/li>\n<li>and more\u2026\u2026<\/li>\n<\/ul>\n<\/li>\n<li>\u7ea6\u675f\u7c7b\u578b\n<ul>\n<li>\u7b49\u5f0f\u8868\u793a <span>\\(C_j(x_{i1}, x_{in2}, &#8230;, x_{inj}) = 0\\)\u00a0<\/span><\/li>\n<li>\u4e0d\u7b49\u5f0f\u8868\u793a<span>\\(C_j(x_{i1}, x_{in2}, &#8230;, x_{inj}) &gt;= 0\\)\u00a0<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h2>\u7b97\u6cd5\u6d41\u7a0b<\/h2>\n<h3>\u521d\u59cb\u5316\u9636\u6bb5Start\uff1a<\/h3>\n<p>\u521d\u59cb\u5316\u8282\u70b9\u53c2\u6570<\/p>\n<ul>\n<li>\u8d28\u91cf\u8bbe\u7f6e\uff1a<span>\\(m_i\\)<\/span><\/li>\n<li>\u4f4d\u7f6e\u8bbe\u7f6e\uff1a<span>\\(p_i\\)<\/span><\/li>\n<li>\u901f\u5ea6\u8bbe\u7f6e\uff08\u8fd9\u4e2a\u4e00\u822c\u521d\u59cb\u5316\u7684\u65f6\u5019\u90fd\u662f\u8bbe\u7f6e\u4e3a0\uff09<\/li>\n<\/ul>\n<p>\u521d\u59cb\u5316\u7ea6\u675f\u53c2\u6570<\/p>\n<ul>\n<li>\u521a\u5ea6\u8bbe\u7f6e\uff5c\u68c0\u67e5\uff1a<span>\\(k_j \\in [0,1]\\)<\/span><\/li>\n<li>\u963b\u5c3c\u8bbe\u7f6e\uff5c\u68c0\u67e5\uff1a<span>\\(d_j \\in [0,1]\\)<\/span><\/li>\n<li>and more<\/li>\n<\/ul>\n<p>\u53e6\u5916\u5728\u771f\u6b63\u7684\u8ba1\u7b97\u673a\u6a21\u62df\u5b9e\u73b0\u7684\u65f6\u5019\uff0c\u5fc5\u987b\u3001\u4e00\u5b9a\u8981\u6ce8\u610f\u65f6\u95f4\u5fae\u5206\u5c3a\u5ea6\u7684\u8bbe\u7f6e\uff0c\u4e0d\u80fd\u65e0\u8111\u5229\u7528\u5f15\u64ce\u7684deltaTime( <span>\\(\\text{d}t\\) ) \u6216\u8005\\(\\Delta t\\) <\/span>)\uff0c\u5fc5\u987b\u8981\u5f62\u6210\u81ea\u5df1\u7684\u65f6\u95f4\u5468\u671f\uff0c\u8fd9\u4e5f\u662f\u4e3a\u4ec0\u4e48\u5f88\u591a\u6210\u719f\u7684\u67d4\u4f53\u5e93\u4f1a\u6709\u4e00\u4e2aSolve\u8282\u70b9\u7684\u539f\u56e0\uff0c\u901a\u8fc7\u8fd9\u4e2a\u8282\u70b9\u53bb\u7ea6\u675f\u8ba1\u7b97\u65f6\u95f4\u5e76\u4e14\u6c47\u603b\u8ba1\u7b97\u5904\u7406\uff0c\u751a\u81f3\u8fd8\u8981\u9632\u6b62\u6b7b\u4ea1\u5faa\u73af\uff08\u53c2\u8003\uff1a<a href=\"https:\/\/johnaustin.io\/articles\/2019\/fix-your-unity-timestep\">https:\/\/johnaustin.io\/articles\/2019\/fix-your-unity-timestep<\/a>\uff09<span>\\Delta<\/span><\/p>\n<h3>\u6267\u884c\u9636\u6bb5Update\uff1a<\/h3>\n<p>\u6309\u7167\u6267\u884c\u987a\u5e8f\u5206\u4e3a\u5982\u4e0b\u6b65\u9aa4<\/p>\n<ul>\n<li>\u5904\u7406\u5916\u529b\uff1a<span>\\(v_{i next} = v_i + \\Delta t \\frac{F}{m}\\)\u00a0 F\u8868\u793a\u5916\u529b\uff08\u663e\u793a\u6b27\u62c9\uff09<\/span><\/li>\n<li>\u901f\u5ea6\u963b\u5c3c\uff1a<span>\\(dampV elocities(v_1, &#8230;, v_N)\\)\u00a0 <\/span><\/li>\n<li>\u4e0b\u4e00\u4e2a\u4f4d\u7f6e\u9884\u6d4b\uff1a<span>\\(p_{i next} =p_i + \\Delta t v_i\\)\u00a0<\/span><\/li>\n<li>\u78b0\u649e\u68c0\u6d4b\uff0c\u751f\u6210\u78b0\u649e\u7ea6\u675f\uff1a<span class=\"mord mathdefault\">g<\/span><span class=\"mord mathdefault\">e<\/span><span class=\"mord mathdefault\">n<\/span><span class=\"mord mathdefault\">e<\/span><span class=\"mord mathdefault\">r<\/span><span class=\"mord mathdefault\">a<\/span><span class=\"mord mathdefault\">t<\/span><span class=\"mord mathdefault\">e<\/span><span class=\"mord mathdefault\">C<\/span><span class=\"mord mathdefault\">o<\/span><span class=\"mord mathdefault\">l<\/span><span class=\"mord mathdefault\">l<\/span><span class=\"mord mathdefault\">i<\/span><span class=\"mord mathdefault\">s<\/span><span class=\"mord mathdefault\">i<\/span><span class=\"mord mathdefault\">o<\/span><span class=\"mord mathdefault\">n<\/span><span class=\"mord mathdefault\">C<\/span><span class=\"mord mathdefault\">o<\/span><span class=\"mord mathdefault\">n<\/span><span class=\"mord mathdefault\">s<\/span><span class=\"mord mathdefault\">t<\/span><span class=\"mord mathdefault\">r<\/span><span class=\"mord mathdefault\">a<\/span><span class=\"mord mathdefault\">i<\/span><span class=\"mord mathdefault\">n<\/span><span class=\"mord mathdefault\">t<\/span><span class=\"mord mathdefault\">s<\/span><\/li>\n<li>\u8bfa\u5e72\u6b21\u8fed\u4ee3\uff08\u8ba1\u7b97\u7ed3\u679c\uff09\uff1a\u6bcf\u6b21\u8fed\u4ee3\u5904\u7406\u4e00\u904d\u6240\u6709\u7ea6\u675f<span class=\"mord mathdefault\">p<\/span><span class=\"mord mathdefault\">r<\/span><span class=\"mord mathdefault\">o<\/span><span class=\"mord mathdefault\">j<\/span><span class=\"mord mathdefault\">e<\/span><span class=\"mord mathdefault\">c<\/span><span class=\"mord mathdefault\">t<\/span><span class=\"mord mathdefault\">C<\/span><span class=\"mord mathdefault\">o<\/span><span class=\"mord mathdefault\">n<\/span><span class=\"mord mathdefault\">s<\/span><span class=\"mord mathdefault\">t<\/span><span class=\"mord mathdefault\">r<\/span><span class=\"mord mathdefault\">a<\/span><span class=\"mord mathdefault\">i<\/span><span class=\"mord mathdefault\">n<\/span><span class=\"mord mathdefault\">t<\/span><span class=\"mord mathdefault\">s<span>\\(C_j(x_{i1}, x_{in2}, &#8230;, x_{inj}) \\) <\/span>\u00a0 \u6ce8\u610f\u8fd9\u91cc\u6700\u5bb9\u6613\u9020\u6210\u5361\u987f\u95ee\u9898<\/span><\/li>\n<li>\u901f\u5ea6\u3001\u4f4d\u7f6e\u66f4\u65b0\uff08\u6570\u636e\u5e94\u7528\uff09\uff1aupdate v and p<\/li>\n<li>\u78b0\u649e\u8282\u70b9\u7684\u901f\u5ea6\u66f4\u65b0\uff08\u6570\u636e\u8868\u8fbe\uff09\uff1aupdateVelocity<span>\\(v_1, &#8230;, v_N\\)<\/span><\/li>\n<\/ul>\n<p>\u6ce8\u610f\uff0c\u8fed\u4ee3\u5904\u7406\u9636\u6bb5\u53ef\u4ee5\u91c7\u7528\u4ee5\u4e0b\u4e00\u4e9b\u65b9\u6cd5\u7684\u53d8\u79cd\uff0c\u590d\u6742\u5ea6\u7ea6\u540e\u8d8a\u9ad8\uff0c\u6548\u679c\u4e5f\u8d8a\u597d\uff0c\u5bf9\u7b97\u529b\u8981\u6c42\u4e5f\u8d8a\u9ad8<\/p>\n<ol>\n<li>\u6b27\u62c9\uff08\u5305\u542b\u663e\u793a\u6b27\u62c9Explicit\uff08Forward\uff09Euler&#8217;s Method\uff5c\u9690\u5f0f\u6b27\u62c9Implicit\uff08Backward\uff09Euler&#8217;s Method\uff5c\u534a\u9690\u5f0f\u6b27\u62c9Semi-implicit Euler&#8217;s Method\uff09<\/li>\n<li>\u4e8c\u9636\u6cf0\u52d2\u5c55\u5f00<\/li>\n<li>Verlet\u79ef\u5206\uff08\u8fd9\u4e2a\u597d\u5904\u5c31\u5728\u53ef\u4ee5\u77e5\u9053\u524d\u540e\u65f6\u95f4\uff09<\/li>\n<li>RK4\uff08\u56db\u9636\u4f26\u54e5\u5e93\u5854\uff09Rounge-Kutta Method | \u57fa\u4e8e\u56db\u9636\u6cf0\u52d2\u5c55\u5f00\u7684<\/li>\n<\/ol>\n<h2>\u8be6\u7ec6\u8fc7\u7a0b<\/h2>\n<h3>\u53c2\u6570\u6269\u5c55<\/h3>\n<p>\u5728\u5f00\u5c55\u8be6\u7ec6\u7684\u8ba1\u7b97\u4e4b\u524d\uff0c\u7531\u4e8e\u7269\u4f53\u6709\u5185\u5916\u52a8\u9759\u6469\u64e6\uff0c\u7c98\u6027\uff0c\u963b\u5c3c\u7b49\u7b49\u56e0\u7d20\uff0c\u751a\u81f3\u73b0\u5b9e\u751f\u6d3b\u4e2d\u8fd8\u8981\u8003\u8651\u7a7a\u6c14\u963b\u529b\uff0c\u56e0\u6b64\u5f00\u5c55\u66f4\u52a0\u8be6\u7ec6\u7684\u8ba1\u7b97\u4e4b\u524d\u9700\u8981\u5bf9\u53c2\u6570\u6269\u5c55\u3002<\/p>\n<h4>\u8d28\u5fc3\u3001\u89d2\u901f\u5ea6\u7b49\u53c2\u6570<\/h4>\n<ul>\n<li>\u8d28\u5fc3\uff1a<span>\\(p_c = \\frac{\\sum_i x_i m_i}{\\sum_i m_i}\\) <\/span>\n<ul>\n<li>\u53d6\u5de7\u505a\u6cd5\uff1a\u76f4\u63a5\u63d0\u524d\u8bbe\u7f6e\u4e00\u4e2a\u7279\u6b8a\u7684\u8d28\u5fc3\u8282\u70b9\uff0c\u8bb2\u6240\u9700\u8981\u53c2\u6570\u63d0\u524d\u8f93\u5165\u5904\u7406<\/li>\n<\/ul>\n<\/li>\n<li>\u8d28\u5fc3\u901f\u5ea6\uff1a<span>\\(v_c = \\frac{\\sum_i v_i m_i}{\\sum_i m_i}\\)<\/span>\n<ul>\n<li>\u53d6\u5de7\u505a\u6cd5\uff1a\u540c\u4e0a\uff0c\u8d28\u5fc3\u7684\u5750\u6807\u548c\u4f4d\u7f6e\u4e22\u53bb\u5f53\u4f5c\u4e00\u822c\u8282\u70b9\u8ba1\u7b97\u5373\u53ef<\/li>\n<\/ul>\n<\/li>\n<li>\u7cfb\u7edf\u89d2\u52a8\u91cf\uff1a<span>\\(L = \\sum_i r_i (m_i v_i)\\) \u5176\u4e2d \\(r_i = p_i &#8211; p_c\\) \u6bcf\u4e00\u4e2a\u8282\u70b9\u4f4d\u7f6e\u51cf\u53bb\u8d28\u5fc3\u53c2\u6570<\/span><\/li>\n<li>\u7cfb\u7edf\u8f6c\u52a8\u60ef\u91cf\uff1a<span>\\(I = \\sum_i r_i^2\\) \u7528\u8fd9\u4e2a\u5c31\u884c\u4e86\uff0c\u4e0d\u9700\u8981\u4e09\u91cd\u79ef\u5206\u90a3\u79cd\u65b9\u6cd5<\/span><\/li>\n<li>\u7cfb\u7edf\u89d2\u901f\u5ea6\uff1a<span>\\(\\omega = I^{-1} L\\) <\/span><\/li>\n<\/ul>\n<h4>\u963b\u5c3c\u53c2\u6570<\/h4>\n<p>\u7136\u540e\u8bbe\u5b9a\u963b\u5c3c\u53c2\u6570<span>\\(k_j \\in [0,1]\\)\uff0c\u5bf9\u4e8e\u6bcf\u4e00\u4e2a\u8282\u70b9\u7684\u901f\u5ea6\\(v_i\\)\uff0c\u9700\u8981\u505a\u5982\u4e0b\u7684\u53d8\u6362\uff1a<\/span><\/p>\n<p><span>\\[\\Delta v_i = v_c + \\omega r_i &#8211; v_i\\]<\/span><\/p>\n<p>\u7136\u540e\u6c42\u7684\u901f\u5ea6\u7ecf\u8fc7\u53d8\u6362\u540e\u7684\u5177\u4f53\u503c<\/p>\n<p><span>$$v_{i next} = v_i + k \\Delta v_i$$<\/span><\/p>\n<p>\u963b\u5c3ck\u53ea\u662f\u4e00\u4e2a0\uff5e1\u7684\u7cfb\u6570\uff0c\u5982\u679ck=1\uff0c\u90a3\u4e48\u7cfb\u7edf\u53ef\u4ee5\u89c6\u4f5c\u6ca1\u5f62\u53d8\uff08\u521a\u4f53\uff09\uff0c\u8db3\u591fk=0\uff0c\u90a3\u5c31\u7eaf\u81ea\u7531\u8fd0\u52a8\uff0c\u4e0d\u5b58\u5728\u963b\u5c3c\u3002<\/p>\n<h5>\u6ce8\u610f\u521a\u5ea6\u7cfb\u6570\u548c\u963b\u5c3c\u7cfb\u6570\u533a\u522b<\/h5>\n<p>\u521a\u5ea6\u7cfb\u6570\uff08Stiffness Coefficient\uff09\uff1a<\/p>\n<ol>\n<li>\u521a\u5ea6\u7cfb\u6570\u8861\u91cf\u4e86\u7269\u4f53\u5bf9\u5916\u90e8\u529b\u91cf\u7684\u62b5\u6297\u7a0b\u5ea6\uff0c\u5373\u7269\u4f53\u5728\u53d7\u5230\u5916\u529b\u65f6\u53d8\u5f62\u7684\u96be\u6613\u7a0b\u5ea6\u3002<\/li>\n<li>\u8f83\u9ad8\u7684\u521a\u5ea6\u7cfb\u6570\u610f\u5473\u7740\u7269\u4f53\u66f4\u96be\u53d8\u5f62\uff0c\u66f4\u575a\u786c\u3002\u4f8b\u5982\uff0c\u5f39\u7c27\u7684\u521a\u5ea6\u7cfb\u6570\u8d8a\u9ad8\uff0c\u5f39\u7c27\u53d7\u529b\u65f6\u53d8\u5f62\u8d8a\u5c0f\u3002<\/li>\n<li>\u521a\u5ea6\u7cfb\u6570\u901a\u5e38\u4e0e\u5f39\u6027\u6750\u6599\u7684\u7279\u6027\u6709\u5173\uff0c\u5b83\u51b3\u5b9a\u4e86\u7269\u4f53\u56de\u5f39\u7684\u901f\u5ea6\u548c\u5e45\u5ea6\u3002<\/li>\n<\/ol>\n<p>\u963b\u5c3c\u7cfb\u6570\uff08Damping Coefficient\uff09\uff1a<\/p>\n<ol>\n<li>\u963b\u5c3c\u7cfb\u6570\u63cf\u8ff0\u4e86\u7269\u4f53\u5728\u632f\u52a8\u6216\u8fd0\u52a8\u8fc7\u7a0b\u4e2d\u635f\u5931\u52a8\u80fd\u7684\u901f\u5ea6\uff0c\u5373\u51cf\u5c0f\u632f\u5e45\u7684\u901f\u5ea6\u3002<\/li>\n<li>\u963b\u5c3c\u7cfb\u6570\u7684\u5b58\u5728\u53ef\u4ee5\u51cf\u5c11\u632f\u52a8\u7684\u6301\u7eed\u65f6\u95f4\uff0c\u4f7f\u632f\u52a8\u9010\u6e10\u51cf\u5c0f\u76f4\u81f3\u505c\u6b62\u3002<\/li>\n<li>\u8f83\u9ad8\u7684\u963b\u5c3c\u7cfb\u6570\u901a\u5e38\u4f1a\u5bfc\u81f4\u632f\u52a8\u8870\u51cf\u5f97\u66f4\u5feb\uff0c\u800c\u8f83\u4f4e\u7684\u963b\u5c3c\u7cfb\u6570\u53ef\u80fd\u5bfc\u81f4\u632f\u52a8\u6301\u7eed\u65f6\u95f4\u8f83\u957f\u3002<\/li>\n<\/ol>\n<h3>\u78b0\u649e\u8fd0\u52a8<\/h3>\n<p>\u78b0\u649e\u8fd0\u52a8\u5305\u542b\u78b0\u649e\u68c0\u6d4b\u3001\u78b0\u649e\u5904\u7406\u4e24\u4e2a\u90e8\u5206<\/p>\n<h4>\u78b0\u649e\u68c0\u6d4b<\/h4>\n<p>\u78b0\u649e\u6709\u4e09\u79cd\u72b6\u6001\uff0c\u672a\u78b0\u649e\u3001\u8868\u9762\u78b0\u649e\u3001\u7a7f\u900f\u78b0\u649e\uff08\u5361\u5230\u53e6\u4e00\u4e2a\u7269\u4f53\u7684\u5185\u90e8\u53bb\u4e86\uff09\uff0c\u9700\u8981\u5904\u7406\u7684\u6709\u4e24\u79cd\uff0c\u5206\u522b\u6709\u5982\u4e0b\u7ea6\u675f\u65b9\u6cd5\uff08\u76f8\u5bf9\u6700\u7b80\u5355\u7684\u65b9\u6cd5\uff09<\/p>\n<ol>\n<li>\u8868\u9762\u78b0\u649e(static collision)\uff1a<span>\\(C(p) = (p &#8211; q_s) n_s \\)\uff0c\u521a\u5ea6\u53c2\u6570k=1<\/span><\/li>\n<li>\u7a7f\u900f\u78b0\u649e(continue collision)\uff1a<span>\\(C(p) = (p &#8211; q_c) n_c \\)\uff0c\u521a\u5ea6\u53c2\u6570k=1<\/span><\/li>\n<\/ol>\n<h4>\u78b0\u649e\u5904\u7406<\/h4>\n<p>\u78b0\u649e\u53d1\u751f\u540e\u7684\u5904\u7406\u65b9\u6848\u65e0\u975e\u4e5f\u5c31\u662f\u6324\u538b\u63a8\u79bb\u5f00\uff0c\u6700\u7b80\u5355\u7684\u529e\u6cd5\u5c31\u662f\u6839\u636e\u6bcf\u4e00\u4e2a\u8282\u70b9\uff08\u4e00\u4e2a\u5c0f\u5706\u7403\uff09\u7684\u534a\u5f84\u505a\u6324\u538b\uff0c\u8fd9\u4e5f\u5c5e\u4e8e\u5206\u5b50\u52a8\u529b\u5b66\u7684\u4e00\u79cd\u601d\u60f3\u3002<\/p>\n<p>\u5176\u4ed6\u7684\u5904\u7406\u8fd8\u6709\u6bd4\u5982SAT\uff0cEPA\uff08GJK\u68c0\u6d4b\uff0c\u4e0d\u9002\u7528\u4e8e\u53ef\u5f62\u53d8\u7269\u4f53\uff09\uff0c\u51e0\u4f55\u53cd\u9988\uff0c\u6df1\u5ea6\u56fe\uff0c\u516b\u53c9\u6811\u7a7a\u95f4\u6784\u9020\u7b49\u7b49\u2026\u2026<\/p>\n<h3>\u8fed\u4ee3\uff08\u903c\u8fd1\uff09<\/h3>\n<p>PBD\u7684\u8fed\u4ee3\u672c\u8eab\u5c5e\u4e8e\u725b\u987f\u8fed\u4ee3\u6cd5\u601d\u60f3<\/p>\n<blockquote><p><span>\u725b\u987f\u8fed\u4ee3\u6cd5\uff08Newton&#8217;s Method\uff09\u662f\u4e00\u79cd\u7528\u4e8e\u6c42\u89e3\u65b9\u7a0b\u6839\u7684\u8fed\u4ee3\u6570\u503c\u65b9\u6cd5\uff0c\u5176\u6838\u5fc3\u601d\u60f3\u662f\u901a\u8fc7\u4e0d\u65ad\u903c\u8fd1\u51fd\u6570\u7684\u6839\u6765\u5bfb\u627e\u65b9\u7a0b\u7684\u89e3\u3002<\/span><\/p>\n<p>&nbsp;<\/p>\n<ol>\n<li><strong>\u9009\u62e9\u521d\u59cb\u8fd1\u4f3c\u503c\uff1a<\/strong> \u9996\u5148\uff0c\u9009\u62e9\u4e00\u4e2a\u521d\u59cb\u7684\u8fd1\u4f3c\u89e3\uff08\u901a\u5e38\u8bb0\u4e3ax\u2080\uff09\uff0c\u8fd9\u4e2a\u503c\u53ef\u4ee5\u662f\u65b9\u7a0b\u89e3\u7684\u4efb\u4f55\u4f30\u8ba1\u503c\uff0c\u4f46\u901a\u5e38\u9009\u62e9\u8d8a\u63a5\u8fd1\u771f\u5b9e\u89e3\u7684\u503c\uff0c\u8fed\u4ee3\u8fc7\u7a0b\u4f1a\u66f4\u5feb\u6536\u655b\u3002<\/li>\n<li><strong>\u8ba1\u7b97\u51fd\u6570\u503c\u548c\u5bfc\u6570\u503c\uff1a<\/strong> \u5728\u5f53\u524d\u8fd1\u4f3c\u89e3x\u2080\u5904\uff0c\u8ba1\u7b97\u65b9\u7a0b\u5de6\u8fb9\u7684\u51fd\u6570\u503cf(x\u2080)\u548c\u5176\u5bfc\u6570\uff08\u6216\u68af\u5ea6\uff09\u503cf'(x\u2080)\u3002\u8fd9\u662f\u4e3a\u4e86\u6784\u5efa\u5207\u7ebf\uff0c\u5e76\u901a\u8fc7\u5207\u7ebf\u6765\u4f30\u8ba1\u4e0b\u4e00\u4e2a\u8fd1\u4f3c\u89e3\u3002<\/li>\n<li><strong>\u66f4\u65b0\u8fd1\u4f3c\u89e3\uff1a<\/strong> \u4f7f\u7528\u4ee5\u4e0b\u516c\u5f0f\u6765\u8ba1\u7b97\u4e0b\u4e00\u4e2a\u8fd1\u4f3c\u89e3x\u2081\uff1ax\u2081 = x\u2080 &#8211; f(x\u2080) \/ f'(x\u2080)\u8fd9\u4e2a\u516c\u5f0f\u662f\u901a\u8fc7\u5728\u70b9(x\u2080, f(x\u2080))\u5904\u4f7f\u7528\u5207\u7ebf\u6765\u4f30\u8ba1\u65b9\u7a0b\u7684\u6839\u3002\u65b0\u7684\u8fd1\u4f3c\u89e3x\u2081\u901a\u5e38\u66f4\u63a5\u8fd1\u5b9e\u9645\u89e3\u3002<\/li>\n<li><strong>\u91cd\u590d\u8fed\u4ee3\uff1a<\/strong> \u91cd\u590d\u6b65\u9aa42\u548c\u6b65\u9aa43\uff0c\u4e0d\u65ad\u66f4\u65b0\u8fd1\u4f3c\u89e3\uff0c\u76f4\u5230\u6ee1\u8db3\u4e00\u5b9a\u7684\u6536\u655b\u6761\u4ef6\uff0c\u4f8b\u5982\uff0c\u8fbe\u5230\u9884\u5b9a\u7684\u8fed\u4ee3\u6b21\u6570\u6216\u8005\u8fd1\u4f3c\u89e3\u7684\u53d8\u5316\u5f88\u5c0f\u3002<\/li>\n<li><strong>\u8f93\u51fa\u7ed3\u679c\uff1a<\/strong> \u5f53\u6ee1\u8db3\u6536\u655b\u6761\u4ef6\u65f6\uff0c\u505c\u6b62\u8fed\u4ee3\uff0c\u5e76\u5c06\u6700\u7ec8\u7684\u8fd1\u4f3c\u89e3\u4f5c\u4e3a\u65b9\u7a0b\u7684\u6839\u7684\u4f30\u8ba1\u503c\u3002<\/li>\n<\/ol>\n<\/blockquote>\n<p>\u6bcf\u4e00\u6b21\u8fed\u4ee3\u91c7\u7528\u9ad8\u65af\u7b97\u6cd5\u7684\u601d\u60f3\uff0c\u72ec\u7acb\u5904\u7406\u6bcf\u4e00\u4e2a\u7ea6\u675f\uff0c\u6bd4\u5982\u6c42\u89e3<span class=\"mord mathdefault\"><span>\\(Ax = b\\)\u7684\u95ee\u9898\u65f6\uff0c\u4e00\u4e2a\u4e2a\u9010\u6b65\u6c42\\(x_i\\)\uff0c\u5e76\u4e14\u5f53\u524d\u6c42\u5f97\u7684\\(x_i\\)\u4f1a\u7528\u5230\u540e\u9762\u7684\u95ee\u9898\u6c42\u89e3\u4e2d\uff0c\u8fd9\u6837\u7b97\u6cd5\u5c31\u9010\u6b65\u9010\u6b65\u8fed\u4ee3\u8fc7\u53bb\u4e86<\/span><\/span><\/p>\n<blockquote><p>\u5907\u6ce8\uff1a<\/p>\n<p>\u201c<strong><span class=\"mord mathdefault\">\u5f53\u524d\u6c42\u5f97\u7684\\(x_i\\)\u4f1a\u7528\u5230\u540e\u9762\u7684\u95ee\u9898\u6c42\u89e3\u4e2d<\/span><\/strong>\u201d\uff0c\u8fd9\u53e5\u8bdd\u7684\u542b\u4e49\u4e0d\u662f\u5f53\u524d\u6279\u6b21\u4ea7\u751f\u5f71\u54cd\uff0c\u8fd9\u6837\u4f1a\u4ea7\u751f\u87ba\u65cb\u7684\uff0c\u800c\u662f\u6bcf\u4e00\u4e2a\u6279\u6b21Batch\u7684\u7ed3\u679c\u8f93\u51fa\u4f1a\u4f5c\u4e3a\u4e0b\u4e00\u4e2a\u6279\u6b21Batch\u5904\u7406\u7684\u8f93\u5165\uff0c\u5e76\u4e14\u9010\u6b65\u8d8b\u5411\u95ee\u9898\u7b54\u6848<\/p><\/blockquote>\n<p>\u8fed\u4ee3\u4e2d\u7ea6\u675f\u7684\u5904\u7406\uff0c\u9700\u8981\u5206\u7c7b\u4e24\u79cd\u5904\u7406<\/p>\n<ul>\n<li>\u7b49\u5f0f\u8868\u793a <span>\\(C_j(x_{i1}, x_{in2}, &#8230;, x_{inj}) = 0\\)\u00a0<\/span><\/li>\n<li>\u4e0d\u7b49\u5f0f\u8868\u793a<span>\\(C_j(x_{i1}, x_{in2}, &#8230;, x_{inj}) &gt;= 0\\)\u00a0<\/span><\/li>\n<\/ul>\n<h4>\u7b49\u5f0f\u7ea6\u675f<\/h4>\n<p>\u5bf9\u4e8e\u4e00\u4e2a\u7b49\u5f0f\u7ea6\u675f<span>\\(C(p) = C(p_1, &#8230;, p_n) = 0\\)\uff0c\u5bf9\u4e8e\u5f53\u524d\u7684\u4f4d\u7f6e\\(p\\)\u6211\u4eec\u9700\u8981\u8ba1\u7b97\u4e00\u4e2a\\(\\Delta p\\)\uff0c\u4f7f\u5f97\u8d28\u70b9\u5728\u5f53\u524d\u4f4d\u7f6e\u79fb\u52a8\\(\\Delta p\\)\u540e\u6ee1\u8db3\u7ea6\u675f\uff08\\(p\\)\u4e3a0\u5411\u91cf\u65f6\uff0c\u5f53\u524d\u4f4d\u7f6e\u5c31\u662f\u6ee1\u8db3\u6761\u4ef6\u7684\uff09<\/span><\/p>\n<p><span>$$C(p + \\Delta p) \\approx C(p) + \\nabla _p C(p) \\Delta p = 0$$<\/span><\/p>\n<blockquote><p>\u5907\u6ce8<\/p>\n<p>\u590d\u4e60\u4e00\u4e0b\u6cf0\u52d2\u7ea7\u6570\uff0c\u51fd\u6570\u5728a\u5904\u7684\u5c55\u5f00\u5f0f\u5982\u4e0b\uff1a<\/p>\n<p><span>$$f(a + h) = f(a) + f^`(a)h + o(h)$$<\/span><\/p><\/blockquote>\n<p>\u7531\u4e8e<span>\\(\\Delta p\\)\u5728\u68af\u5ea6\u7684\u65b9\u5411\u4e0a\uff0c\u76f4\u63a5\u6309\u7167\u68af\u5ea6\u65b9\u5411\u8bbe\u7f6e\u6b65\u957f\u65b9\u5411\u8bbe\u7f6e\u7cfb\u6570\\(\\lambda\\)\u53bb\u8fed\u4ee3\u76ee\u6807\u503c\u5373\u53ef<\/span><\/p>\n<p><span>$$ \\Delta p = \\lambda \\nabla _p C(p) $$<\/span><\/p>\n<p>\u5e26\u5165\u4e0a\u9762\u7684\u516c\u5f0f\u5c31\u53ef\u4ee5\u5f97\u5230<\/p>\n<p><span>$$C(p + \\Delta p) \\approx C(p) + \\nabla _p C(p) \\lambda \\nabla _p C(p) = 0$$<\/span><\/p>\n<p><span>$$\\lambda =\u00a0 &#8211; \\frac{C(p)}{\\left\\Vert \\nabla _p C(p) \\right\\Vert ^2}$$<\/span><\/p>\n<p><span>$$\\Delta p = &#8211; \\frac{C(p)}{\\left\\Vert \\nabla _p C(p) \\right\\Vert ^2} \\nabla _p C(p)$$<\/span><\/p>\n<h4>\u4e0d\u7b49\u5f0f\u7ea6\u675f<\/h4>\n<p>\u5bf9\u4e8e\u4e0d\u7b49\u5f0f\u7ea6\u675f\uff0c\u53ea\u9700\u8981\u5728<span>\\(C(p) &lt; 0\\)\u65f6\u8fdb\u884c\u8fed\u4ee3\u53d8\u6362\u5373\u53ef\uff0c\u5982\u679c\u518d\u8003\u8651\u521a\u5ea6\u3001\u963b\u5c3c\u7b49\u53c2\u6570\uff0c\u5728\u5904\u7406\u7ea6\u675f\u51fd\u6570\u7684\u65f6\u5019\u7b97\u4e0a\u5c31\u884c\uff0c\u53e6\u5916\u4e5f\u4e0d\u8981\u5ffd\u7565\u4e86\u7ebf\u5f62\u8bef\u5dee\\(\\Delta p(1 &#8211; k)\\)<\/span><\/p>\n<h4>\u7ea6\u675f\u51fd\u6570<span>\\(C(p)\\)\u51fd\u6570\u542b\u4e49<\/span><\/h4>\n<h5>\u8ddd\u79bb\u7ea6\u675f<\/h5>\n<p>\u4e24\u4e2a\u8d28\u70b9\u5f39\u7c27\u9700\u8981\u8d8b\u5411\u6052\u5b9a\uff0c\u9700\u8981\u53cc\u5411\u5904\u7406\uff0c\u56e0\u6b64\u5982\u679c\u662f\u8ddd\u79bb\u7ea6\u675f\u7684\u8bdd\\(C(p)\\)\u662f\uff1a<\/p>\n<p><span>$$C(\\overrightarrow{p1}, \\overrightarrow{p2}) = \\left\\Vert \\overrightarrow{p1} &#8211; \\overrightarrow{p2} \\right\\Vert &#8211; d = 0$$<\/span><\/p>\n<blockquote><p>PS\uff1a\u597d\u50cf\u4efb\u4f55\u5f39\u7c27-\u8d28\u70b9\u7684\u53ef\u5f62\u53d8\u7cfb\u7edf\u90fd\u662f\u9700\u8981\u6709\u8ddd\u79bb\u7ea6\u675f\uff0c\u56e0\u6b64\u57fa\u7840\u5b66\u79d1\u5f88\u91cd\u8981\u5440<\/p>\n<p>PPS\uff1a\u80e1\u514b\u5b9a\u5f8b\u5728\u8fd9\u4e2a\u65b9\u9762\u7528\u7684\u5f88\u91cd\uff0c\\(F = kx\\)\uff0cF\u8868\u793a\u529b\uff0cK\u8868\u793a\u6bd4\u4f8b\u5e38\u6570\uff0cx\u8868\u793a\u5e94\u53d8\u8ddd\u79bb<\/p><\/blockquote>\n<h5>\u5f2f\u66f2\u7ea6\u675f<\/h5>\n<p>\u4e24\u4e2a\u8d28\u70b9\u5f39\u7c27\u9700\u8981\u8d8b\u5411\u6052\u5b9a\uff0c\u9700\u8981\u53cc\u5411\u5904\u7406\uff0c\u56e0\u6b64\u5982\u679c\u662f\u8ddd\u79bb\u7ea6\u675f\u7684\u8bdd\\(C(p)\\)<\/p>\n<p>\u4e24\u4e2a\u5171\u8f6d\u9876\u70b9\u7684Mesh\u53d1\u751f\u5f2f\u66f2\uff0c\u56e0\u6b64\u4e24\u4e2aMesh\u5305\u542b\u56db\u4e2a\u9876\u70b9\u7ec4\u6210\u4e00\u4e2a\u7ea6\u675f\uff0c\u5b9a\u4e49\u4e3ap1,p2,p3,p4\uff0c\u5176\u4e2dp1,p2\u4e3a\u4e24\u4e2aMesh\u7684\u5171\u8f6d\u9876\u70b9\uff0c\u7ea6\u675f\u51fd\u6570\u4e3a\uff1a<\/p>\n<p><span>$$C_{bend}(p1, p2, p3, p4) = \\arccos(\\frac{(p2 &#8211; p1) \\times (p3 &#8211; p1)}{\\left\\vert(p2 &#8211; p1) \\times (p3 &#8211; p1)\\right\\vert} \\ast \\frac{(p2 &#8211; p1) \\times (p3 &#8211; p1)}{\\left\\vert(p2 &#8211; p1) \\times (p3 &#8211; p1)\\right\\vert}) &#8211; \\psi _0 =0$$<\/span><\/p>\n<p>\u592a\u96be\u4e86\u770b\u4e0d\u61c2\uff0c\u76f4\u63a5\u622a\u56fe\u5427<\/p>\n<p><img fetchpriority=\"high\" decoding=\"async\" src=\"https:\/\/www.mustenaka.cn\/wp-content\/uploads\/2023\/09\/1641694068004_.pic_-229x300.png\" alt=\"\" width=\"618\" height=\"809\" class=\" wp-image-1468 aligncenter\" srcset=\"https:\/\/www.mustenaka.cn\/wp-content\/uploads\/2023\/09\/1641694068004_.pic_-229x300.png 229w, https:\/\/www.mustenaka.cn\/wp-content\/uploads\/2023\/09\/1641694068004_.pic_-782x1024.png 782w, https:\/\/www.mustenaka.cn\/wp-content\/uploads\/2023\/09\/1641694068004_.pic_-768x1006.png 768w, https:\/\/www.mustenaka.cn\/wp-content\/uploads\/2023\/09\/1641694068004_.pic_.png 1055w\" sizes=\"(max-width: 618px) 100vw, 618px\" \/><\/p>\n<h5>and more \u7ea6\u675f&#8230;<\/h5>\n<p>\u8fd8\u6709\u5f88\u591a\u7ea6\u675f\uff0c\u53ef\u4ee5\u53c2\u8003\u5404\u79cd\u5404\u6837\u7684\u8bba\u6587<\/p>\n<h3>\u66f4\u65b0\u53c2\u6570<\/h3>\n<p>\u76f4\u63a5\u66f4\u65b0\u53c2\u6570\u5373\u53ef<\/p>\n<p>\u5982\u679c\u4e00\u4e2a\u8282\u70b9\u53d1\u751f\u4e86\u78b0\u649e\uff0c\u5047\u8bbe\u78b0\u649e\u4f4d\u7f6e\u7684\u6cd5\u5411\u91cf\u4e3a n \\boldsymbol{n}n \uff0c\u90a3\u4e48\u8be5\u8282\u70b9\u901f\u5ea6\u5782\u76f4\u4e8e n \\boldsymbol{n}n \u7684\u5206\u91cf\u5c06\u56e0\u4e3a\u6469\u64e6\u529b\u800c\u51cf\u5c11\uff0c\u5e76\u4e14\u7531\u4e8e\u53cd\u5f39\u4f1a\u4ea7\u751f\u4e0e n \\boldsymbol{n}n \u540c\u65b9\u5411\u7684\u901f\u5ea6\u5206\u91cf\u3002<\/p>\n<h1>\u6269\u5c55<\/h1>\n<h2>\u52a8\u529b\u5b66<\/h2>\n<p>\u9664\u4e86PBD\u4ee5\u5916\uff08PBD\u57fa\u4e8e\u8d28\u70b9\uff09\uff0c\u9700\u8981\u8865\u5145\u4e00\u4e0b\u7c92\u5b50\u52a8\u529b\u5b66\uff08Particle Dynamics\uff09\u548c\u521a\u4f53\u52a8\u529b\u5b66\u57fa\u672c\u53c2\u6570\uff08Rigid Body Dynamics\uff09\u7684\u57fa\u672c\u53c2\u6570\u6982\u5ff5<\/p>\n<h3>\u7c92\u5b50\u52a8\u529b\u5b66<\/h3>\n<div class=\"supsystic-table-loader spinner\"style=\"background-color:#000000\"><\/div><div id=\"supsystic-table-22_36141\" class=\"supsystic-tables-wrap \" style=\" width:100%; visibility: hidden; \" data-table-width-fixed=\"100%\" data-table-width-mobile=\"100%\" ><table id=\"supsystic-table-22\" class=\"supsystic-table border lightboxImg cell-border\" data-id=\"22\" data-view-id=\"22_36141\" data-title=\"\u7c92\u5b50\u52a8\u529b\u5b66\" data-currency-format=\"$1,000.00\" data-percent-format=\"10.00%\" data-date-format=\"DD.MM.YYYY\" data-time-format=\"HH:mm\" data-features=\"[&quot;after_table_loaded_script&quot;]\" data-search-value=\"\" data-lightbox-img=\"\" data-head-rows-count=\"1\" data-pagination-length=\"50,100,All\" data-auto-index=\"off\" data-searching-settings=\"{&quot;columnSearchPosition&quot;:&quot;bottom&quot;,&quot;minChars&quot;:&quot;0&quot;}\" data-lang=\"default\" data-override=\"{&quot;emptyTable&quot;:&quot;&quot;,&quot;info&quot;:&quot;&quot;,&quot;infoEmpty&quot;:&quot;&quot;,&quot;infoFiltered&quot;:&quot;&quot;,&quot;lengthMenu&quot;:&quot;&quot;,&quot;search&quot;:&quot;&quot;,&quot;zeroRecords&quot;:&quot;&quot;,&quot;exportLabel&quot;:&quot;&quot;,&quot;file&quot;:&quot;default&quot;}\" data-merged=\"[]\" data-responsive-mode=\"0\" data-from-history=\"0\" ><thead><tr><th class=\"\" style=\"width:; padding: 0 !important;\"><\/th><th class=\"\" style=\"width:; padding: 0 !important;\"><\/th><\/tr><\/thead><tbody><tr style=\"height:px\" ><td data-cell-id=\"A1\" data-x=\"0\" data-y=\"1\" data-db-index=\"1\" class=\"\" data-cell-type=\"text\" data-original-value=\"Position\" data-order=\"Position\" style=\"min-width:66.8675%; \" >Position <\/td><td data-cell-id=\"B1\" data-x=\"1\" data-y=\"1\" data-db-index=\"1\" class=\"\" data-cell-type=\"text\" data-original-value=\"\u4f4d\u7f6e\" data-order=\"\u4f4d\u7f6e\" style=\"min-width:33.1325%; \" >\u4f4d\u7f6e <\/td><\/tr><tr style=\"height:px\" ><td data-cell-id=\"A2\" data-x=\"0\" data-y=\"2\" data-db-index=\"2\" class=\"\" data-cell-type=\"text\" data-original-value=\"Linear Velocity\" data-order=\"Linear Velocity\" >Linear Velocity <\/td><td data-cell-id=\"B2\" data-x=\"1\" data-y=\"2\" data-db-index=\"2\" class=\"\" data-cell-type=\"text\" data-original-value=\"\u901f\u5ea6\" data-order=\"\u901f\u5ea6\" >\u901f\u5ea6 <\/td><\/tr><tr style=\"height:px\" ><td data-cell-id=\"A3\" data-x=\"0\" data-y=\"3\" data-db-index=\"3\" class=\"\" data-cell-type=\"text\" data-original-value=\"Acceleration\" data-order=\"Acceleration\" >Acceleration <\/td><td data-cell-id=\"B3\" data-x=\"1\" data-y=\"3\" data-db-index=\"3\" class=\"\" data-cell-type=\"text\" data-original-value=\"\u52a0\u901f\u5ea6\" data-order=\"\u52a0\u901f\u5ea6\" >\u52a0\u901f\u5ea6 <\/td><\/tr><tr style=\"height:px\" ><td data-cell-id=\"A4\" data-x=\"0\" data-y=\"4\" data-db-index=\"4\" class=\"\" data-cell-type=\"text\" data-original-value=\"Mass\" data-order=\"Mass\" >Mass <\/td><td data-cell-id=\"B4\" data-x=\"1\" data-y=\"4\" data-db-index=\"4\" class=\"\" data-cell-type=\"text\" data-original-value=\"\u8d28\u91cf\" data-order=\"\u8d28\u91cf\" >\u8d28\u91cf <\/td><\/tr><tr style=\"height:px\" ><td data-cell-id=\"A5\" data-x=\"0\" data-y=\"5\" data-db-index=\"5\" class=\"\" data-cell-type=\"text\" data-original-value=\"Momentum\" data-order=\"Momentum\" >Momentum <\/td><td data-cell-id=\"B5\" data-x=\"1\" data-y=\"5\" data-db-index=\"5\" class=\"\" data-cell-type=\"text\" data-original-value=\"\u52a8\u91cf\" data-order=\"\u52a8\u91cf\" >\u52a8\u91cf <\/td><\/tr><tr style=\"height:px\" ><td data-cell-id=\"A6\" data-x=\"0\" data-y=\"6\" data-db-index=\"6\" class=\"\" data-cell-type=\"text\" data-original-value=\"Force\" data-order=\"Force\" >Force <\/td><td data-cell-id=\"B6\" data-x=\"1\" data-y=\"6\" data-db-index=\"6\" class=\"\" data-cell-type=\"text\" data-original-value=\"\u51b2\u529b\" data-order=\"\u51b2\u529b\" >\u51b2\u529b <\/td><\/tr><\/tbody><\/table><!-- \/#supsystic-table-22.supsystic-table --><\/div><!-- \/.supsystic-tables-wrap --><!-- Tables Generator by Supsystic --><!-- Version:1.11.0 --><!-- http:\/\/supsystic.com\/ --><a title=\"Table Plugin\" style=\"display:none;\" href=\"https:\/\/supsystic.com\/plugins\/wordpress-data-table-plugin\/?utm_medium=love_link\" target=\"_blank\">Table Plugin<\/a>\n<p>\\(\\overrightarrow{x}\\)<\/p>\n<p>\\(\\overrightarrow{v} = \\frac{d \\overrightarrow{x}}{dt} \\)<\/p>\n<p>\\(\\overrightarrow{a} = \\frac{d \\overrightarrow{v}}{dt} = \\frac{d^2 \\overrightarrow{x}}{dt^2} \\)<\/p>\n<p>\\(M\\)<\/p>\n<p>\\(p = M \\overrightarrow{v} \\)<\/p>\n<p>\\(\\overrightarrow{F} = \\frac{d \\overrightarrow{p}}{dt} = M \\overrightarrow{a}\\)<\/p>\n<h3>\u521a\u4f53\u52a8\u529b\u5b66<\/h3>\n<div class=\"supsystic-table-loader spinner\"style=\"background-color:#000000\"><\/div><div id=\"supsystic-table-23_89373\" class=\"supsystic-tables-wrap \" style=\" width:100%; visibility: hidden; \" data-table-width-fixed=\"100%\" data-table-width-mobile=\"100%\" ><table id=\"supsystic-table-23\" class=\"supsystic-table border lightboxImg cell-border\" data-id=\"23\" data-view-id=\"23_89373\" data-title=\"\u521a\u4f53\u52a8\u529b\u5b66\" data-currency-format=\"$1,000.00\" data-percent-format=\"10.00%\" data-date-format=\"DD.MM.YYYY\" data-time-format=\"HH:mm\" data-features=\"[&quot;after_table_loaded_script&quot;]\" data-search-value=\"\" data-lightbox-img=\"\" data-head-rows-count=\"1\" data-pagination-length=\"50,100,All\" data-auto-index=\"off\" data-searching-settings=\"{&quot;columnSearchPosition&quot;:&quot;bottom&quot;,&quot;minChars&quot;:&quot;0&quot;}\" data-lang=\"default\" data-override=\"{&quot;emptyTable&quot;:&quot;&quot;,&quot;info&quot;:&quot;&quot;,&quot;infoEmpty&quot;:&quot;&quot;,&quot;infoFiltered&quot;:&quot;&quot;,&quot;lengthMenu&quot;:&quot;&quot;,&quot;search&quot;:&quot;&quot;,&quot;zeroRecords&quot;:&quot;&quot;,&quot;exportLabel&quot;:&quot;&quot;,&quot;file&quot;:&quot;default&quot;}\" data-merged=\"[]\" data-responsive-mode=\"0\" data-from-history=\"0\" ><thead><tr><th class=\"\" style=\"width:; padding: 0 !important;\"><\/th><th class=\"\" style=\"width:; padding: 0 !important;\"><\/th><\/tr><\/thead><tbody><tr style=\"height:px\" ><td data-cell-id=\"A1\" data-x=\"0\" data-y=\"1\" data-db-index=\"1\" class=\"\" data-cell-type=\"text\" data-original-value=\"Orientation\" data-order=\"Orientation\" style=\"min-width:45.8689%; \" >Orientation <\/td><td data-cell-id=\"B1\" data-x=\"1\" data-y=\"1\" data-db-index=\"1\" class=\"\" data-cell-type=\"text\" data-original-value=\"\u59ff\u6001\" data-order=\"\u59ff\u6001\" style=\"min-width:54.1311%; \" >\u59ff\u6001 <\/td><\/tr><tr style=\"height:px\" ><td data-cell-id=\"A2\" data-x=\"0\" data-y=\"2\" data-db-index=\"2\" class=\"\" data-cell-type=\"text\" data-original-value=\"Angular Velocity\" data-order=\"Angular Velocity\" >Angular Velocity <\/td><td data-cell-id=\"B2\" data-x=\"1\" data-y=\"2\" data-db-index=\"2\" class=\"\" data-cell-type=\"text\" data-original-value=\"\u89d2\u901f\u5ea6\" data-order=\"\u89d2\u901f\u5ea6\" >\u89d2\u901f\u5ea6 <\/td><\/tr><tr style=\"height:px\" ><td data-cell-id=\"A3\" data-x=\"0\" data-y=\"3\" data-db-index=\"3\" class=\"\" data-cell-type=\"text\" data-original-value=\"Angular Acceleration\" data-order=\"Angular Acceleration\" >Angular Acceleration <\/td><td data-cell-id=\"B3\" data-x=\"1\" data-y=\"3\" data-db-index=\"3\" class=\"\" data-cell-type=\"text\" data-original-value=\"\u89d2\u52a0\u901f\u5ea6\" data-order=\"\u89d2\u52a0\u901f\u5ea6\" >\u89d2\u52a0\u901f\u5ea6 <\/td><\/tr><tr style=\"height:px\" ><td data-cell-id=\"A4\" data-x=\"0\" data-y=\"4\" data-db-index=\"4\" class=\"\" data-cell-type=\"text\" data-original-value=\"Inertia tensor\" data-order=\"Inertia tensor\" >Inertia tensor <\/td><td data-cell-id=\"B4\" data-x=\"1\" data-y=\"4\" data-db-index=\"4\" class=\"\" data-cell-type=\"text\" data-original-value=\"\u8f6c\u52a8\u60ef\u91cf\uff08\u8f6c\u52a8\u60ef\u91cf\u5f20\u91cf\uff09\" data-order=\"\u8f6c\u52a8\u60ef\u91cf\uff08\u8f6c\u52a8\u60ef\u91cf\u5f20\u91cf\uff09\" >\u8f6c\u52a8\u60ef\u91cf\uff08\u8f6c\u52a8\u60ef\u91cf\u5f20\u91cf\uff09 <\/td><\/tr><tr style=\"height:px\" ><td data-cell-id=\"A5\" data-x=\"0\" data-y=\"5\" data-db-index=\"5\" class=\"\" data-cell-type=\"text\" data-original-value=\"Angular Momentum\" data-order=\"Angular Momentum\" >Angular Momentum <\/td><td data-cell-id=\"B5\" data-x=\"1\" data-y=\"5\" data-db-index=\"5\" class=\"\" data-cell-type=\"text\" data-original-value=\"\u89d2\u52a8\u91cf\" data-order=\"\u89d2\u52a8\u91cf\" >\u89d2\u52a8\u91cf <\/td><\/tr><tr style=\"height:px\" ><td data-cell-id=\"A6\" data-x=\"0\" data-y=\"6\" data-db-index=\"6\" class=\"\" data-cell-type=\"text\" data-original-value=\"Torque\" data-order=\"Torque\" >Torque <\/td><td data-cell-id=\"B6\" data-x=\"1\" data-y=\"6\" data-db-index=\"6\" class=\"\" data-cell-type=\"text\" data-original-value=\"\u529b\u77e9\" data-order=\"\u529b\u77e9\" >\u529b\u77e9 <\/td><\/tr><\/tbody><\/table><!-- \/#supsystic-table-23.supsystic-table --><\/div><!-- \/.supsystic-tables-wrap --><!-- Tables Generator by Supsystic --><!-- Version:1.11.0 --><!-- http:\/\/supsystic.com\/ --><a title=\"Table Plugin\" style=\"display:none;\" href=\"https:\/\/supsystic.com\/plugins\/wordpress-data-table-plugin\/?utm_medium=love_link\" target=\"_blank\">Table Plugin<\/a>\n<p>\\(R\\)<\/p>\n<p>\\(\\overrightarrow{\\omega} = \\frac{\\overrightarrow{v} \\times \\overrightarrow{r}}{\\left\\Vert \\overrightarrow{v} \\right\\Vert}\\)<\/p>\n<p>\\(\\overrightarrow{\\alpha} = \\frac{d \\overrightarrow{\\omega}}{dt} = \\frac{\\overrightarrow{a} \\times \\overrightarrow{r}}{\\left\\Vert \\overrightarrow{r}^2 \\right\\Vert}\\)<\/p>\n<p>\\(I = RI_0R^T\\)<\/p>\n<p>\\(L = I\\overrightarrow{\\omega}\\)<\/p>\n<p>\\(\\overrightarrow{\\tau} = \\frac{d \\overrightarrow{L}}{dt}\\)<\/p>\n<h2>\u590d\u4e60\u4e00\u4e0b\u725b\u987f\u5b9a\u5f8b<\/h2>\n<h3>\u725b\u987f\u7b2c\u4e00\u5b9a\u5f8b<\/h3>\n<p><span>\u4e00\u5207\u7269\u4f53\u4fdd\u6301\u5176\u539f\u6765\u7684\u72b6\u6001\uff0c\u5373\u4fdd\u6301\u9759\u6b62\u7684\u7269\u4f53\u4fdd\u6301\u9759\u6b62\uff0c\u5300\u901f\u8fd0\u52a8\u7684\u7269\u4f53\u4fdd\u6301\u5300\u901f\u8fd0\u52a8\uff0c\u9664\u975e\u6709\u5916\u529b\u4f5c\u7528\u4e8e\u5b83\u4eec\u3002<\/span><\/p>\n<h3>\u725b\u987f\u7b2c\u4e8c\u5b9a\u5f8b<\/h3>\n<p><span>\u7269\u4f53\u7684\u52a0\u901f\u5ea6\uff08a\uff09\u4e0e\u4f5c\u7528\u5728\u7269\u4f53\u4e0a\u7684\u603b\u5916\u529b\uff08F\uff09\u4e4b\u95f4\u6210\u6b63\u6bd4\uff0c\u4e14\u4e0e\u7269\u4f53\u7684\u8d28\u91cf\uff08m\uff09\u6210\u53cd\u6bd4\uff0c\u8868\u8fbe\u5f0f\u4e3a\uff1a<\/span>\\(F = ma\\)<\/p>\n<h1>\u672c\u6587\u53c2\u8003<\/h1>\n<p>https:\/\/zhuanlan.zhihu.com\/p\/338474692<\/p>\n<p>https:\/\/blog.csdn.net\/dragonylee\/article\/details\/121360710<\/p>\n<p>https:\/\/johnaustin.io\/articles\/2019\/fix-your-unity-timestep<\/p>\n<p>https:\/\/www.xianlongok.site\/post\/d6327b48\/#rigid-body-dynamics<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u9996 \u7531\u4e8e\u9700\u8981\u5b9e\u73b0\u8bf8\u591adeformable body\uff08\u53ef\u5f62\u53d8\u7269\u4f53\uff09\u7684\u4ea4\u4e92\u5904\u7406\uff0c\u5305\u62ecSoftbody\uff08\u67d4 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5,118],"tags":[123,116,6],"class_list":["post-1392","post","type-post","status-publish","format-standard","hentry","category-algorithm","category-unity3d","tag-pbd","tag-unity3d","tag-algorithm"],"views":3424,"_links":{"self":[{"href":"https:\/\/www.mustenaka.cn\/index.php\/wp-json\/wp\/v2\/posts\/1392","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mustenaka.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.mustenaka.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.mustenaka.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mustenaka.cn\/index.php\/wp-json\/wp\/v2\/comments?post=1392"}],"version-history":[{"count":59,"href":"https:\/\/www.mustenaka.cn\/index.php\/wp-json\/wp\/v2\/posts\/1392\/revisions"}],"predecessor-version":[{"id":1473,"href":"https:\/\/www.mustenaka.cn\/index.php\/wp-json\/wp\/v2\/posts\/1392\/revisions\/1473"}],"wp:attachment":[{"href":"https:\/\/www.mustenaka.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1392"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.mustenaka.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1392"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.mustenaka.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1392"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}