"use strict"; const {Fragment: _Fragment, jsx: _jsx, jsxs: _jsxs} = arguments[0]; const {useMDXComponents: _provideComponents} = arguments[0]; function _createMdxContent(props) { const _components = { a: "a", annotation: "annotation", code: "code", h1: "h1", img: "img", li: "li", math: "math", mi: "mi", mn: "mn", mo: "mo", mrow: "mrow", msub: "msub", msup: "msup", p: "p", path: "path", pre: "pre", semantics: "semantics", span: "span", strong: "strong", svg: "svg", ul: "ul", ..._provideComponents(), ...props.components }; return _jsxs(_Fragment, { children: [_jsxs(_components.h1, { id: "环境准备", children: [_jsx(_components.a, { className: "heading-link", href: "#环境准备", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "环境准备"] }), "\n", _jsx(_components.pre, { className: "language-python", children: _jsxs(_components.code, { className: "language-python code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token comment", children: "# 导入必要的库 / Import necessary libraries" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "import" }), " mujoco\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "import" }), " numpy ", _jsx(_components.span, { className: "token keyword", children: "as" }), " np\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "import" }), " matplotlib", _jsx(_components.span, { className: "token punctuation", children: "." }), "pyplot ", _jsx(_components.span, { className: "token keyword", children: "as" }), " plt\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "from" }), " matplotlib", _jsx(_components.span, { className: "token punctuation", children: "." }), "animation ", _jsx(_components.span, { className: "token keyword", children: "import" }), " FuncAnimation\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "import" }), " mediapy ", _jsx(_components.span, { className: "token keyword", children: "as" }), " media\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "from" }), " typing ", _jsx(_components.span, { className: "token keyword", children: "import" }), " List", _jsx(_components.span, { className: "token punctuation", children: "," }), " Tuple", _jsx(_components.span, { className: "token punctuation", children: "," }), " Optional", _jsx(_components.span, { className: "token punctuation", children: "," }), " Dict\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "import" }), " time\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "from" }), " IPython", _jsx(_components.span, { className: "token punctuation", children: "." }), "display ", _jsx(_components.span, { className: "token keyword", children: "import" }), " HTML\n"] }), _jsx(_components.span, { className: "code-line", children: "\n" }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token comment", children: "# 设置随机种子 / Set random seed" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: ["np", _jsx(_components.span, { className: "token punctuation", children: "." }), "random", _jsx(_components.span, { className: "token punctuation", children: "." }), "seed", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token number", children: "42" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] })] }) }), "\n", _jsxs(_components.h1, { id: "机械臂设计与建模", children: [_jsx(_components.a, { className: "heading-link", href: "#机械臂设计与建模", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "机械臂设计与建模"] }), "\n", _jsx(_components.p, { children: "机械臂的设计参数由 D-H 表示法给出：" }), "\n", _jsx(_components.img, { src: "https://obsidian2flowershow-inkem.flowershow.me/I-%E9%99%84%E4%BB%B6/Pasted%20image%2020250919125556.png", alt: "Pasted image 20250919125556", className: "internal", "data-fs-resolved-file-path": "/I-附件/Pasted image 20250919125556.png" }), "\n", _jsx(_components.img, { src: "https://obsidian2flowershow-inkem.flowershow.me/I-%E9%99%84%E4%BB%B6/Pasted%20image%2020250919125630.png", alt: "Pasted image 20250919125630", className: "internal", "data-fs-resolved-file-path": "/I-附件/Pasted image 20250919125630.png" }), "\n", _jsx(_components.p, { children: "本节演示使用三自由度机械臂：" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "关节 1：基座旋转" }), "\n", _jsx(_components.li, { children: "关节 2：肩部俯仰" }), "\n", _jsx(_components.li, { children: "关节 3：肘部俯仰" }), "\n"] }), "\n", _jsxs(_components.p, { children: ["在我们的简化模型中，两个连杆是位于同一平面的直杆，因此 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "α" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\alpha_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0037em" }, children: "α" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0037em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " 和 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "d" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "d_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8444em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "d" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " 均为 0（除了 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "α" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msup, { children: [_jsx(_components.mn, { children: "90" }), _jsx(_components.mo, { children: "∘" })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\alpha_0=90^\\circ" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.5806em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0037em" }, children: "α" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0037em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "0" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6741em" } }), _jsx(_components.span, { className: "mord", children: "9" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6741em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mbin mtight", children: "∘" }) })] }) }) }) }) })] })] })] })] }), "），只需给出连杆长度 ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "d" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "d_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8444em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "d" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " 即可。", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "θ" }), _jsx(_components.mi, { children: "i" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\theta_i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8444em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.0278em" }, children: "θ" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0278em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " 作为关节变量需配置其运动范围。"] }), "\n", _jsx(_components.p, { children: "下面这个机械臂参数类除了运动学相关参数，还定义了一些外观参数：" }), "\n", _jsx(_components.pre, { className: "language-python", children: _jsxs(_components.code, { className: "language-python code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token comment", children: "# 定义机械臂参数 / Define arm parameters" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "class" }), " ", _jsx(_components.span, { className: "token class-name", children: "ArmParameters" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token triple-quoted-string string", children: "\"\"\"机械臂参数类 / Arm parameters class\"\"\"" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "def" }), " ", _jsx(_components.span, { className: "token function", children: "__init__" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "self", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "# 连杆长度 / Link lengths" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" self", _jsx(_components.span, { className: "token punctuation", children: "." }), "base_height ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token number", children: "0.1" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" self", _jsx(_components.span, { className: "token punctuation", children: "." }), "link1_length ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token number", children: "0.3" }), " ", _jsx(_components.span, { className: "token comment", children: "# 上臂 / Upper arm" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" self", _jsx(_components.span, { className: "token punctuation", children: "." }), "link2_length ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token number", children: "0.25" }), " ", _jsx(_components.span, { className: "token comment", children: "# 前臂 / Forearm" }), "\n"] }), _jsx(_components.span, { className: "code-line", children: " \n" }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "# 连杆半径 / Link radii" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" self", _jsx(_components.span, { className: "token punctuation", children: "." }), "base_radius ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token number", children: "0.05" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" self", _jsx(_components.span, { className: "token punctuation", children: "." }), "link_radius ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token number", children: "0.03" }), "\n"] }), _jsx(_components.span, { className: "code-line", children: " \n" }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "# 关节限位 (度) / Joint limits (degrees)" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" self", _jsx(_components.span, { className: "token punctuation", children: "." }), "joint_limits ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token punctuation", children: "[" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token operator", children: "-" }), _jsx(_components.span, { className: "token number", children: "180" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token number", children: "180" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token comment", children: "# 基座旋转 / Base rotation" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token operator", children: "-" }), _jsx(_components.span, { className: "token number", children: "90" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token number", children: "90" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token comment", children: "# 肩部俯仰 / Shoulder pitch" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token operator", children: "-" }), _jsx(_components.span, { className: "token number", children: "135" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token number", children: "135" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token comment", children: "# 肘部俯仰 / Elbow pitch" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: "]" }), "\n"] }), _jsx(_components.span, { className: "code-line", children: " \n" }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "# 颜色 / Colors" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" self", _jsx(_components.span, { className: "token punctuation", children: "." }), "base_color ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token punctuation", children: "[" }), _jsx(_components.span, { className: "token number", children: "0.2" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token number", children: "0.2" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token number", children: "0.2" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token comment", children: "# 深灰 / Dark gray" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" self", _jsx(_components.span, { className: "token punctuation", children: "." }), "link1_color ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token punctuation", children: "[" }), _jsx(_components.span, { className: "token number", children: "0.8" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token number", children: "0.2" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token number", children: "0.2" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token comment", children: "# 红色 / Red" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" self", _jsx(_components.span, { className: "token punctuation", children: "." }), "link2_color ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token punctuation", children: "[" }), _jsx(_components.span, { className: "token number", children: "0.2" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token number", children: "0.2" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token number", children: "0.8" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token comment", children: "# 蓝色 / Blue" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" self", _jsx(_components.span, { className: "token punctuation", children: "." }), "joint_color ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token punctuation", children: "[" }), _jsx(_components.span, { className: "token number", children: "0.7" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token number", children: "0.7" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token number", children: "0.7" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " ", _jsx(_components.span, { className: "token number", children: "1" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token comment", children: "# 浅灰 / Light gray" }), "\n"] }), _jsx(_components.span, { className: "code-line", children: "\n" }), _jsxs(_components.span, { className: "code-line", children: ["params ", _jsx(_components.span, { className: "token operator", children: "=" }), " ArmParameters", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] })] }) }), "\n", _jsx(_components.pre, { className: "language-python", children: _jsxs(_components.code, { className: "language-python code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "def" }), " ", _jsx(_components.span, { className: "token function", children: "create_3dof_arm_xml" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "params", _jsx(_components.span, { className: "token punctuation", children: ":" }), " ArmParameters", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token operator", children: "-" }), _jsx(_components.span, { className: "token operator", children: ">" }), " ", _jsx(_components.span, { className: "token builtin", children: "str" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token triple-quoted-string string", children: "\"\"\"\n" })] }), _jsx(_components.span, { className: "code-line", children: _jsx(_components.span, { className: "token triple-quoted-string string", children: " 创建3自由度机械臂的MJCF XML / Create MJCF XML for 3-DOF arm\n" }) }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token triple-quoted-string string", children: " \"\"\"" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" xml ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token string-interpolation", children: _jsx(_components.span, { className: "token string", children: "f\"\"\"\n" }) })] }), _jsx(_components.span, { className: "code-line", children: _jsx(_components.span, { className: "token string-interpolation", children: _jsx(_components.span, { className: "token string", children: " \n" }) }) }), _jsx(_components.span, { className: "code-line", children: _jsx(_components.span, { className: "token string-interpolation", children: _jsx(_components.span, { className: "token string", children: " \n" }) }) }), _jsx(_components.span, { className: "code-line", children: _jsx(_components.span, { className: "token string-interpolation", children: _jsx(_components.span, { className: "token string", children: " \n" }) }) }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token string-interpolation", children: _jsx(_components.span, { className: "token string", children: " \"\"\"" }) }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " xml\n"] }), _jsx(_components.span, { className: "code-line", children: "\n" }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token comment", children: "# 创建模型 / Create model" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: ["arm_xml ", _jsx(_components.span, { className: "token operator", children: "=" }), " create_3dof_arm_xml", _jsx(_components.span, { className: "token punctuation", children: "(" }), "params", _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: ["arm_model ", _jsx(_components.span, { className: "token operator", children: "=" }), " mujoco", _jsx(_components.span, { className: "token punctuation", children: "." }), "MjModel", _jsx(_components.span, { className: "token punctuation", children: "." }), "from_xml_string", _jsx(_components.span, { className: "token punctuation", children: "(" }), "arm_xml", _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: ["arm_data ", _jsx(_components.span, { className: "token operator", children: "=" }), " mujoco", _jsx(_components.span, { className: "token punctuation", children: "." }), "MjData", _jsx(_components.span, { className: "token punctuation", children: "(" }), "arm_model", _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] })] }) }), "\n", _jsxs(_components.h1, { id: "正向运动学分析", children: [_jsx(_components.a, { className: "heading-link", href: "#正向运动学分析", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "正向运动学分析"] }), "\n", _jsxs(_components.p, { children: ["回顾一下【机器人学导论】Ep3.操作比（正向）运动学的内容，对一个具有", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), "个自由度的操作臂来说，它的所有连杆位置可由一组", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }), "个关节变量确定，这样的一组变量被称为", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "×" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n\\times1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6667em", verticalAlign: "-0.0833em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "×" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }), "的关节向量，所有关节向量组成的空间称为", _jsx(_components.strong, { children: "关节空间" }), "。将已知的关节空间描述转化为在", _jsx(_components.strong, { children: "笛卡尔空间" }), "中的描述，称为", _jsx(_components.strong, { children: "正向运动学" }), "，反之则为", _jsx(_components.strong, { children: "逆运动学" }), "。"] }), "\n", _jsx(_components.p, { children: "一般情况下，正向运动学的计算需要根据连杆参数构成连杆变换矩阵。本例机械臂结构简单，且暂不考虑姿态问题，因此可直接求解：" }), "\n", _jsx(_components.pre, { className: "language-python", children: _jsxs(_components.code, { className: "language-python code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "def" }), " ", _jsx(_components.span, { className: "token function", children: "forward_kinematics" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "q", _jsx(_components.span, { className: "token punctuation", children: ":" }), " np", _jsx(_components.span, { className: "token punctuation", children: "." }), "ndarray", _jsx(_components.span, { className: "token punctuation", children: "," }), " params", _jsx(_components.span, { className: "token punctuation", children: ":" }), " ArmParameters", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token operator", children: "-" }), _jsx(_components.span, { className: "token operator", children: ">" }), " np", _jsx(_components.span, { className: "token punctuation", children: "." }), "ndarray", _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token triple-quoted-string string", children: "\"\"\"\n" })] }), _jsx(_components.span, { className: "code-line", children: _jsx(_components.span, { className: "token triple-quoted-string string", children: " 计算正向运动学 / Compute forward kinematics\n" }) }), _jsx(_components.span, { className: "code-line", children: _jsx(_components.span, { className: "token triple-quoted-string string", children: " \n" }) }), _jsx(_components.span, { className: "code-line", children: _jsx(_components.span, { className: "token triple-quoted-string string", children: " Args:\n" }) }), _jsx(_components.span, { className: "code-line", children: _jsx(_components.span, { className: "token triple-quoted-string string", children: " q: 关节角度 [q1, q2, q3] (radians)\n" }) }), _jsx(_components.span, { className: "code-line", children: _jsx(_components.span, { className: "token triple-quoted-string string", children: " params: 机械臂参数 / Arm parameters\n" }) }), _jsx(_components.span, { className: "code-line", children: _jsx(_components.span, { className: "token triple-quoted-string string", children: " \n" }) }), _jsx(_components.span, { className: "code-line", children: _jsx(_components.span, { className: "token triple-quoted-string string", children: " Returns:\n" }) }), _jsx(_components.span, { className: "code-line", children: _jsx(_components.span, { className: "token triple-quoted-string string", children: " 末端执行器位置 [x, y, z]\n" }) }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token triple-quoted-string string", children: " \"\"\"" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" q1", _jsx(_components.span, { className: "token punctuation", children: "," }), " q2", _jsx(_components.span, { className: "token punctuation", children: "," }), " q3 ", _jsx(_components.span, { className: "token operator", children: "=" }), " q\n"] }), _jsx(_components.span, { className: "code-line", children: " \n" }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "# 基座高度 / Base height" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" z0 ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token number", children: "0.42" }), " ", _jsx(_components.span, { className: "token operator", children: "+" }), " params", _jsx(_components.span, { className: "token punctuation", children: "." }), "base_height\n"] }), _jsx(_components.span, { className: "code-line", children: " \n" }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "# 计算各关节位置 / Calculate joint positions" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "# 关节2位置 / Joint 2 position" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" x2 ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token number", children: "0" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" y2 ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token number", children: "0" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" z2 ", _jsx(_components.span, { className: "token operator", children: "=" }), " z0 ", _jsx(_components.span, { className: "token operator", children: "+" }), " params", _jsx(_components.span, { className: "token punctuation", children: "." }), "link1_length\n"] }), _jsx(_components.span, { className: "code-line", children: " \n" }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "# 末端执行器位置 / End effector position" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "# 考虑所有旋转 / Consider all rotations" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" l2 ", _jsx(_components.span, { className: "token operator", children: "=" }), " params", _jsx(_components.span, { className: "token punctuation", children: "." }), "link2_length\n"] }), _jsxs(_components.span, { className: "code-line", children: [" l3 ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token number", children: "0.05" }), " ", _jsx(_components.span, { className: "token comment", children: "# 包括末端大小 / Including end" }), "\n"] }), _jsx(_components.span, { className: "code-line", children: " \n" }), _jsxs(_components.span, { className: "code-line", children: [" x ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "l2 ", _jsx(_components.span, { className: "token operator", children: "*" }), " np", _jsx(_components.span, { className: "token punctuation", children: "." }), "sin", _jsx(_components.span, { className: "token punctuation", children: "(" }), "q2", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token operator", children: "+" }), " l3 ", _jsx(_components.span, { className: "token operator", children: "*" }), " np", _jsx(_components.span, { className: "token punctuation", children: "." }), "sin", _jsx(_components.span, { className: "token punctuation", children: "(" }), "q2 ", _jsx(_components.span, { className: "token operator", children: "+" }), " q3", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token operator", children: "*" }), " np", _jsx(_components.span, { className: "token punctuation", children: "." }), "cos", _jsx(_components.span, { className: "token punctuation", children: "(" }), "q1", _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" y ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token punctuation", children: "(" }), "l2 ", _jsx(_components.span, { className: "token operator", children: "*" }), " np", _jsx(_components.span, { className: "token punctuation", children: "." }), "sin", _jsx(_components.span, { className: "token punctuation", children: "(" }), "q2", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token operator", children: "+" }), " l3 ", _jsx(_components.span, { className: "token operator", children: "*" }), " np", _jsx(_components.span, { className: "token punctuation", children: "." }), "sin", _jsx(_components.span, { className: "token punctuation", children: "(" }), "q2 ", _jsx(_components.span, { className: "token operator", children: "+" }), " q3", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token operator", children: "*" }), " np", _jsx(_components.span, { className: "token punctuation", children: "." }), "sin", _jsx(_components.span, { className: "token punctuation", children: "(" }), "q1", _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" z ", _jsx(_components.span, { className: "token operator", children: "=" }), " z2 ", _jsx(_components.span, { className: "token operator", children: "+" }), " l2 ", _jsx(_components.span, { className: "token operator", children: "*" }), " np", _jsx(_components.span, { className: "token punctuation", children: "." }), "cos", _jsx(_components.span, { className: "token punctuation", children: "(" }), "q2", _jsx(_components.span, { className: "token punctuation", children: ")" }), " ", _jsx(_components.span, { className: "token operator", children: "+" }), " l3 ", _jsx(_components.span, { className: "token operator", children: "*" }), " np", _jsx(_components.span, { className: "token punctuation", children: "." }), "cos", _jsx(_components.span, { className: "token punctuation", children: "(" }), "q2 ", _jsx(_components.span, { className: "token operator", children: "+" }), " q3", _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsx(_components.span, { className: "code-line", children: " \n" }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "return" }), " np", _jsx(_components.span, { className: "token punctuation", children: "." }), "array", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token punctuation", children: "[" }), "x", _jsx(_components.span, { className: "token punctuation", children: "," }), " y", _jsx(_components.span, { className: "token punctuation", children: "," }), " z", _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] })] }) }), "\n", _jsx(_components.p, { children: "通过与MuJoCo仿真结果进行误差比较，我们可以验证运动学计算公式是否准确：" }), "\n", _jsx(_components.pre, { className: "language-python", children: _jsxs(_components.code, { className: "language-python code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token comment", children: "# 测试运动学 / Test kinematics" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: ["test_angles ", _jsx(_components.span, { className: "token operator", children: "=" }), " np", _jsx(_components.span, { className: "token punctuation", children: "." }), "array", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token punctuation", children: "[" }), _jsx(_components.span, { className: "token number", children: "0" }), _jsx(_components.span, { className: "token punctuation", children: "," }), " np", _jsx(_components.span, { className: "token punctuation", children: "." }), "pi", _jsx(_components.span, { className: "token operator", children: "/" }), _jsx(_components.span, { 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children: "," }), " params", _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsx(_components.span, { className: "code-line", children: "\n" }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token comment", children: "# 使用MuJoCo验证 / Verify with MuJoCo" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: ["arm_data", _jsx(_components.span, { className: "token punctuation", children: "." }), "qpos", _jsx(_components.span, { className: "token punctuation", children: "[" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), _jsx(_components.span, { className: "token number", children: "3" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token operator", children: "=" }), " test_angles\n"] }), _jsxs(_components.span, { className: "code-line", children: ["mujoco", _jsx(_components.span, 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"code-line", children: [_jsx(_components.span, { className: "token keyword", children: "print" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token string-interpolation", children: _jsx(_components.span, { className: "token string", children: "f\"正向运动学结果 / Forward kinematics results:\"" }) }), _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "print" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsxs(_components.span, { className: "token string-interpolation", children: [_jsx(_components.span, { className: "token string", children: "f\"自定义计算 / Custom calculation: " }), _jsxs(_components.span, { className: "token interpolation", children: [_jsx(_components.span, { className: "token punctuation", children: "{" }), "ee_pos_fk", _jsx(_components.span, { className: "token punctuation", children: "}" })] }), _jsx(_components.span, { className: "token string", children: "\"" })] }), _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "print" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsxs(_components.span, { className: "token string-interpolation", children: [_jsx(_components.span, { className: "token string", children: "f\"MuJoCo计算 / MuJoCo calculation: " }), _jsxs(_components.span, { className: "token interpolation", children: [_jsx(_components.span, { className: "token punctuation", children: "{" }), "ee_pos_mj", _jsx(_components.span, { className: "token punctuation", children: "}" })] }), _jsx(_components.span, { className: "token string", children: "\"" })] }), _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "print" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsxs(_components.span, { className: "token string-interpolation", children: [_jsx(_components.span, { className: "token string", children: "f\"误差 / Error: " }), _jsxs(_components.span, { className: "token interpolation", children: [_jsx(_components.span, { className: "token punctuation", children: "{" }), "np", _jsx(_components.span, { className: "token punctuation", children: "." }), "linalg", _jsx(_components.span, { className: "token punctuation", children: "." }), "norm", _jsx(_components.span, { className: "token punctuation", children: "(" }), "ee_pos_fk ", _jsx(_components.span, { className: "token operator", children: "-" }), " ee_pos_mj", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), _jsx(_components.span, { className: "token format-spec", children: ".6f" }), _jsx(_components.span, { className: "token punctuation", children: "}" })] }), _jsx(_components.span, { className: "token string", children: "\"" })] }), _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] })] }) }), "\n", _jsx(_components.pre, { children: _jsxs(_components.code, { className: "language-auto code-highlight", children: [_jsx(_components.span, { className: "code-line", children: "正向运动学结果 / Forward kinematics results:\n" }), _jsx(_components.span, { className: "code-line", children: "自定义计算 / Custom calculation: [0.1767767 0. 1.0467767]\n" }), _jsx(_components.span, { className: "code-line", children: "MuJoCo计算 / MuJoCo calculation: [0.1767767 0. 1.0467767]\n" }), _jsx(_components.span, { className: "code-line", children: "误差 / Error: 0.000000\n" })] }) }), "\n", _jsxs(_components.h1, { id: "工作空间分析", children: [_jsx(_components.a, { className: "heading-link", href: "#工作空间分析", children: _jsx(_components.svg, { xmlns: "http:www.w3.org/2000/svg", fill: "#ab2b65", viewBox: "0 0 20 20", className: "w-5 h-5", children: _jsx(_components.path, { fillRule: "evenodd", clipRule: "evenodd", d: "M9.493 2.853a.75.75 0 00-1.486-.205L7.545 6H4.198a.75.75 0 000 1.5h3.14l-.69 5H3.302a.75.75 0 000 1.5h3.14l-.435 3.148a.75.75 0 001.486.205L7.955 14h2.986l-.434 3.148a.75.75 0 001.486.205L12.456 14h3.346a.75.75 0 000-1.5h-3.14l.69-5h3.346a.75.75 0 000-1.5h-3.14l.435-3.147a.75.75 0 00-1.486-.205L12.045 6H9.059l.434-3.147zM8.852 7.5l-.69 5h2.986l.69-5H8.852z" }) }) }), "工作空间分析"] }), "\n", _jsx(_components.p, { children: "研究笛卡尔空间中点的位置在操作臂的关节空间中是否有相应的解的问题，即分析操作臂的工作空间。回顾一下【机器人学导论】Ep4.操作臂逆运动学的内容：" }), "\n", _jsx(_components.img, { src: "https://obsidian2flowershow-inkem.flowershow.me/I-%E9%99%84%E4%BB%B6/Pasted%20image%2020250921113648.png", alt: "Pasted image 20250921113648", className: "internal", "data-fs-resolved-file-path": "/I-附件/Pasted image 20250921113648.png" }), "\n", _jsx(_components.p, { children: "工作空间可以由直接计算求解，而在仿真软件下，为了进行可视化，我们可以通过蒙特卡洛方法在关节空间中随机采样，由MuJoCo模拟出采样点在笛卡尔空间中的位置，得到工作空间点云图：" }), "\n", _jsx(_components.pre, { className: "language-python", children: _jsxs(_components.code, { className: "language-python code-highlight", children: [_jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token keyword", children: "def" }), " ", _jsx(_components.span, { className: "token function", children: "visualize_workspace" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "model", _jsx(_components.span, { className: "token punctuation", children: ":" }), " mujoco", _jsx(_components.span, { className: "token punctuation", children: "." }), "MjModel", _jsx(_components.span, { className: "token punctuation", children: "," }), " params", _jsx(_components.span, { className: "token punctuation", children: ":" }), " ArmParameters", _jsx(_components.span, { className: "token punctuation", children: "," }), " samples", _jsx(_components.span, { className: "token punctuation", children: ":" }), " ", _jsx(_components.span, { className: "token builtin", children: "int" }), " ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token number", children: "1000" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token triple-quoted-string string", children: "\"\"\"\n" })] }), _jsx(_components.span, { className: "code-line", children: _jsx(_components.span, { className: "token triple-quoted-string string", children: " 可视化机械臂工作空间 / Visualize robot workspace\n" }) }), _jsxs(_components.span, { className: "code-line", children: [_jsx(_components.span, { className: "token triple-quoted-string string", children: " \"\"\"" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" data ", _jsx(_components.span, { className: "token operator", children: "=" }), " mujoco", _jsx(_components.span, { className: "token punctuation", children: "." }), "MjData", _jsx(_components.span, { className: "token punctuation", children: "(" }), "model", _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" workspace_points ", _jsx(_components.span, { className: "token operator", children: "=" }), " ", _jsx(_components.span, { className: "token punctuation", children: "[" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), "\n"] }), _jsx(_components.span, { className: "code-line", children: " \n" }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "# 随机采样关节角度 / Random sample joint angles" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "for" }), " _ ", _jsx(_components.span, { className: "token keyword", children: "in" }), " ", _jsx(_components.span, { className: "token builtin", children: "range" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), "samples", _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "# 在关节限位内随机采样 / Random sample within joint limits" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" q ", _jsx(_components.span, { className: "token operator", children: "=" }), " np", _jsx(_components.span, { className: "token punctuation", children: "." }), "zeros", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token number", children: "3" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token keyword", children: "for" }), " i ", _jsx(_components.span, { className: "token keyword", children: "in" }), " ", _jsx(_components.span, { className: "token builtin", children: "range" }), _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token number", children: "3" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" q", 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children: "1" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsx(_components.span, { className: "code-line", children: " \n" }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "# 设置关节角度 / Set joint angles" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" data", _jsx(_components.span, { className: "token punctuation", children: "." }), "qpos", _jsx(_components.span, { className: "token punctuation", children: "[" }), _jsx(_components.span, { className: "token punctuation", children: ":" }), _jsx(_components.span, { className: "token number", children: "3" }), _jsx(_components.span, { className: "token punctuation", children: "]" }), " ", _jsx(_components.span, { className: "token operator", children: "=" }), " q\n"] }), _jsxs(_components.span, { className: "code-line", children: [" mujoco", _jsx(_components.span, { className: "token punctuation", children: "." }), "mj_forward", _jsx(_components.span, { className: "token punctuation", children: "(" }), "model", _jsx(_components.span, { className: "token punctuation", children: "," }), " data", _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsx(_components.span, { className: "code-line", children: " \n" }), _jsxs(_components.span, { className: "code-line", children: [" ", _jsx(_components.span, { className: "token comment", children: "# 获取末端位置 / Get end effector position" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" ee_pos ", _jsx(_components.span, { className: "token operator", children: "=" }), " data", _jsx(_components.span, { className: "token punctuation", children: "." }), "site", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token string", children: "'ee_site'" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), _jsx(_components.span, { className: "token punctuation", children: "." }), "xpos", _jsx(_components.span, { className: "token punctuation", children: "." }), "copy", _jsx(_components.span, { className: "token punctuation", children: "(" }), _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsxs(_components.span, { className: "code-line", children: [" workspace_points", _jsx(_components.span, { className: "token punctuation", children: "." }), "append", _jsx(_components.span, { className: "token punctuation", children: "(" }), "ee_pos", _jsx(_components.span, { className: "token punctuation", children: ")" }), "\n"] }), _jsx(_components.span, { className: "code-line", children: " \n" }), _jsxs(_components.span, { 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