{"id":102644,"date":"2026-09-09T09:57:03","date_gmt":"2026-09-09T01:57:03","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/102644.html"},"modified":"2026-09-09T09:57:03","modified_gmt":"2026-09-09T01:57:03","slug":"07-%c2%b7-ikpy-%e5%85%a5%e9%97%a8","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/102644.html","title":{"rendered":"07 \u00b7 ikpy \u5165\u95e8"},"content":{"rendered":"

\u8fd9\u4e00\u7ae0\u8981\u89e3\u51b3\u4ec0\u4e48\u95ee\u9898：\u4e0d\u60f3\u81ea\u5df1\u5199 DH\u3001\u4e0d\u60f3\u81ea\u5df1\u5b9e\u73b0 DLS，\u6709\u6ca1\u6709\u73b0\u6210\u7684\u5e93\u80fd\u7528？<\/p>\n

\u6709，\u5c31\u662f ikpy\u3002\u672c\u7ae0\u628a\u5b83\u62c6\u5f00\u8bb2\u6e05\u695a：\u4e09\u4e2a\u7c7b\u3001\u4e00\u5957\u53c2\u6570\u3001\u4ee5\u53ca 4.0 \u7248\u672c\u6539\u4e86\u54ea\u4e9b API（\u7f51\u4e0a\u5927\u91cf\u6559\u7a0b\u5df2\u8fc7\u65f6）\u3002<\/p>\n

\u914d\u5957\u4ee3\u7801：[code\/ch07_ikpy_intro.py]<\/p>\n

"""
\n\u7b2c 07 \u7ae0\u914d\u5957\u4ee3\u7801：ikpy \u5165\u95e8\u3002<\/p>\n

\u8fd0\u884c：
\n D:\\\\\\\\Environment\\\\\\\\dm_control_env\\\\\\\\python.exe ch07_ikpy_intro.py<\/p>\n

\u5185\u5bb9：
\n 1. \u73af\u5883\u68c0\u67e5（ikpy + sympy）
\n 2. \u7b2c\u4e00\u4e2a\u7a0b\u5e8f：2 \u8fde\u6746\u5e73\u9762\u81c2（FK + IK）
\n 3. active_links_mask \u7684\u4e09\u79cd\u8bbe\u7f6e\u65b9\u5f0f（\u5b9e\u6d4b\u884c\u4e3a）
\n 4. \u5173\u8282\u9650\u4f4d bounds \u9a8c\u8bc1
\n 5. \u56fa\u5b9a\u8fde\u6746\u5fc5\u987b\u5199 joint_type="fixed"（\u9519\u8bef\u6f14\u793a）
\n 6. DHLink \u4e0e\u7b2c 05 \u7ae0\u624b\u5199 DH \u5bf9\u7167
\n 7. \u52a8\u624b\u7ec3\u7b54\u6848
\n"""<\/span>
\nimport<\/span> warnings
\nimport<\/span> numpy as<\/span> np<\/p>\n

np.<\/span>set_printoptions(<\/span>precision=<\/span>6<\/span>,<\/span> suppress=<\/span>True<\/span>)<\/span><\/p>\n

# ============================================================<\/span>
\n# 7.2 \u73af\u5883\u68c0\u67e5<\/span>
\n# ============================================================<\/span>
\nprint<\/span>(<\/span>"="<\/span> *<\/span> 66<\/span>)<\/span>
\nprint<\/span>(<\/span>"7.2 \u73af\u5883\u68c0\u67e5"<\/span>)<\/span>
\nprint<\/span>(<\/span>"="<\/span> *<\/span> 66<\/span>)<\/span><\/p>\n

import<\/span> importlib<\/p>\n

for<\/span> pkg in<\/span> (<\/span>"ikpy"<\/span>,<\/span> "sympy"<\/span>,<\/span> "numpy"<\/span>,<\/span> "scipy"<\/span>)<\/span>:<\/span>
\n try<\/span>:<\/span>
\n mod =<\/span> importlib.<\/span>import_module(<\/span>pkg)<\/span>
\n print<\/span>(<\/span>f" [OK] <\/span>{<\/span>pkg:<\/span><8<\/span>}<\/span><\/span> <\/span>{<\/span>getattr<\/span>(<\/span>mod,<\/span> '__version__'<\/span>,<\/span> '?'<\/span>)<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span>
\n except<\/span> ImportError as<\/span> e:<\/span>
\n print<\/span>(<\/span>f" [MISSING] <\/span>{<\/span>pkg:<\/span><8<\/span>}<\/span><\/span> <\/span>{<\/span>e}<\/span><\/span>"<\/span><\/span>)<\/span>
\n if<\/span> pkg ==<\/span> "sympy"<\/span>:<\/span>
\n print<\/span>(<\/span>" -> ikpy \u4f9d\u8d56 sympy \u4f46\u4e0d\u4f1a\u81ea\u52a8\u5b89\u88c5，\u8bf7\u6267\u884c: pip install sympy"<\/span>)<\/span><\/p>\n

import<\/span> ikpy
\nfrom<\/span> ikpy.<\/span>chain import<\/span> Chain
\nfrom<\/span> ikpy.<\/span>link import<\/span> OriginLink,<\/span> URDFLink,<\/span> DHLink<\/p>\n

print<\/span>(<\/span>f"\\\\nikpy \u7248\u672c: <\/span>{<\/span>ikpy.<\/span>__version__}<\/span><\/span> (\u672c\u4e66\u6309 4.0 \u5b9e\u6d4b\u7f16\u5199)"<\/span><\/span>)<\/span><\/p>\n

# ============================================================<\/span>
\n# 7.4 \u7b2c\u4e00\u4e2a\u7a0b\u5e8f：2 \u8fde\u6746\u5e73\u9762\u81c2<\/span>
\n# ============================================================<\/span>
\nprint<\/span>(<\/span>"\\\\n"<\/span> +<\/span> "="<\/span> *<\/span> 66<\/span>)<\/span>
\nprint<\/span>(<\/span>"7.4 \u7b2c\u4e00\u4e2a\u7a0b\u5e8f：2 \u8fde\u6746\u5e73\u9762\u81c2"<\/span>)<\/span>
\nprint<\/span>(<\/span>"="<\/span> *<\/span> 66<\/span>)<\/span><\/p>\n

def<\/span> make_two_link<\/span>(<\/span>)<\/span>:<\/span>
\n """\u521b\u5efa\u4e00\u6761 2 \u8fde\u6746\u5e73\u9762\u81c2（\u5728 XZ \u5e73\u9762\u5185\u6446\u52a8，\u4e0e\u672c\u4e66 Z-up \u5750\u6807\u7cfb\u4e00\u81f4）\u3002<\/p>\n

ikpy \u6700\u5173\u952e\u7684\u5efa\u6a21\u7ea6\u5b9a（\u6781\u5176\u5bb9\u6613\u7406\u89e3\u9519）：<\/p>\n

link[i] \u7684 origin_translation = \u4ece\u7236\u5173\u8282\u5230\u672c\u5173\u8282\u7684\u504f\u79fb
\n \u6700\u540e\u4e00\u4e2a link \u7684 origin_translation = \u4ece\u6700\u540e\u5173\u8282\u5230\u672b\u7aef\u7684\u8ddd\u79bb<\/p>\n

\u53ea\u6709"\u5173\u8282\u540e\u9762\u8fd8\u6709\u504f\u79fb"，\u8fd9\u4e2a\u5173\u8282\u7684\u65cb\u8f6c\u624d\u4f1a\u6539\u53d8\u672b\u7aef\u4f4d\u7f6e\u3002
\n \u6240\u4ee5\u4e00\u53f0 2 \u8fde\u6746\u81c2\u9700\u8981 4 \u4e2a\u5143\u7d20：
\n OriginLink + \u5173\u82821 + \u5173\u82822 + \u672b\u7aef\u504f\u79fb(\u56fa\u5b9a)
\n \u82e5\u53ea\u5199 OriginLink + 2 \u4e2a\u5173\u8282，\u7b2c\u4e8c\u4e2a\u5173\u8282\u540e\u9762\u6ca1\u6709\u504f\u79fb，\u5b83\u7684\u65cb\u8f6c
\n \u5b8c\u5168\u4e0d\u5f71\u54cd\u672b\u7aef\u4f4d\u7f6e \u2014\u2014 IK \u4f1a\u89e3\u51fa\u4efb\u610f\u89d2\u5ea6\u5374"\u770b\u8d77\u6765\u6b63\u786e"，\u6781\u5177\u8ff7\u60d1\u6027\u3002
\n """<\/span>
\n return<\/span> Chain(<\/span>
\n name=<\/span>"two_link_arm"<\/span>,<\/span>
\n links=<\/span>[<\/span>
\n OriginLink(<\/span>)<\/span>,<\/span> # \u5730\u57fa<\/span>
\n URDFLink(<\/span>name=<\/span>"shoulder"<\/span>,<\/span> # \u5173\u8282 1（\u57fa\u5ea7\u5728\u539f\u70b9）<\/span>
\n origin_translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n rotation=<\/span>[<\/span>0<\/span>,<\/span> 1<\/span>,<\/span> 0<\/span>]<\/span>)<\/span>,<\/span> # \u7ed5 +Y \u8f6c（XZ \u5e73\u9762\u5185\u4fef\u4ef0）<\/span>
\n URDFLink(<\/span>name=<\/span>"elbow"<\/span>,<\/span> # \u5173\u8282 2（\u8ddd\u5173\u8282 1 \u4e3a L1）<\/span>
\n origin_translation=<\/span>[<\/span>–<\/span>0.30<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n rotation=<\/span>[<\/span>0<\/span>,<\/span> 1<\/span>,<\/span> 0<\/span>]<\/span>)<\/span>,<\/span>
\n URDFLink(<\/span>name=<\/span>"tip"<\/span>,<\/span> # \u672b\u7aef\u504f\u79fb（\u8ddd\u5173\u8282 2 \u4e3a L2）<\/span>
\n origin_translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0.30<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n rotation=<\/span>None<\/span>,<\/span> joint_type=<\/span>"fixed"<\/span>)<\/span>,<\/span>
\n ]<\/span>,<\/span>
\n active_links_mask=<\/span>[<\/span>False<\/span>,<\/span> True<\/span>,<\/span> True<\/span>,<\/span> False<\/span>]<\/span>,<\/span> # \u5730\u57fa\u4e0e\u672b\u7aef\u4e0d\u53c2\u4e0e\u4f18\u5316<\/span>
\n )<\/span><\/p>\n

# ============ \u4e24\u4e2a\u53cd\u9762\u6559\u6750 ============<\/span>
\nwith<\/span> warnings.<\/span>catch_warnings(<\/span>)<\/span>:<\/span>
\n warnings.<\/span>simplefilter(<\/span>"ignore"<\/span>)<\/span><\/p>\n

# \u53cd\u9762\u6559\u6750 A：\u8fde\u6746\u6cbf X\u3001\u5374\u7ed5 X \u8f6c \u2014\u2014 \u8fde\u6746\u65b9\u5411\u4e0e\u65cb\u8f6c\u8f74\u5e73\u884c，\u65cb\u8f6c\u4e0d\u4ea7\u751f\u4f4d\u79fb<\/span>
\n chain_bad_axis =<\/span> Chain(<\/span>name=<\/span>"bad_axis"<\/span>,<\/span> links=<\/span>[<\/span>
\n OriginLink(<\/span>)<\/span>,<\/span>
\n URDFLink(<\/span>name=<\/span>"l1"<\/span>,<\/span> origin_translation=<\/span>[<\/span>–<\/span>0.30<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>–<\/span>1<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>)<\/span>,<\/span>
\n URDFLink(<\/span>name=<\/span>"l2"<\/span>,<\/span> origin_translation=<\/span>[<\/span>–<\/span>0.30<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>–<\/span>1<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>)<\/span>,<\/span>
\n ]<\/span>,<\/span> active_links_mask=<\/span>[<\/span>False<\/span>,<\/span> True<\/span>,<\/span> True<\/span>]<\/span>)<\/span>
\n T_badA =<\/span> chain_bad_axis.<\/span>forward_kinematics(<\/span>[<\/span>0<\/span>,<\/span> np.<\/span>radians(<\/span>30<\/span>)<\/span>,<\/span> np.<\/span>radians(<\/span>–<\/span>45<\/span>)<\/span>]<\/span>)<\/span>
\n print<\/span>(<\/span>f"\u53cd\u9762\u6559\u6750A \u8fde\u6746\u6cbf-X + \u7ed5-X\u8f6c: q=[0,30deg,-45deg] -> \u672b\u7aef <\/span>{<\/span>np.<\/span>round<\/span>(<\/span>T_badA[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span>,<\/span> 4<\/span>)<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span>
\n print<\/span>(<\/span>" \u65e0\u8bba\u5173\u8282\u8f6c\u591a\u5c11，\u672b\u7aef\u90fd\u505c\u5728 (-0.6,0,0) \u2014\u2014 \u65cb\u8f6c\u6ca1\u8d77\u4f5c\u7528！\\\\n"<\/span>)<\/span><\/p>\n

# \u53cd\u9762\u6559\u6750 B：\u5c11\u4e86\u672b\u7aef\u504f\u79fb \u2014\u2014 \u6700\u540e\u4e00\u4e2a\u5173\u8282\u4e0d\u5f71\u54cd\u672b\u7aef\u4f4d\u7f6e<\/span>
\n chain_no_tip =<\/span> Chain(<\/span>name=<\/span>"no_tip"<\/span>,<\/span> links=<\/span>[<\/span>
\n OriginLink(<\/span>)<\/span>,<\/span>
\n URDFLink(<\/span>name=<\/span>"j1"<\/span>,<\/span> origin_translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0.00<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>0<\/span>,<\/span> 1<\/span>,<\/span> 0<\/span>]<\/span>)<\/span>,<\/span>
\n URDFLink(<\/span>name=<\/span>"j2"<\/span>,<\/span> origin_translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0.30<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>0<\/span>,<\/span> 1<\/span>,<\/span> 0<\/span>]<\/span>)<\/span>,<\/span>
\n ]<\/span>,<\/span> active_links_mask=<\/span>[<\/span>False<\/span>,<\/span> True<\/span>,<\/span> True<\/span>]<\/span>)<\/span>
\n T_B1 =<\/span> chain_no_tip.<\/span>forward_kinematics(<\/span>[<\/span>0<\/span>,<\/span> np.<\/span>radians(<\/span>30<\/span>)<\/span>,<\/span> np.<\/span>radians(<\/span>–<\/span>45<\/span>)<\/span>]<\/span>)<\/span>
\n T_B2 =<\/span> chain_no_tip.<\/span>forward_kinematics(<\/span>[<\/span>0<\/span>,<\/span> np.<\/span>radians(<\/span>30<\/span>)<\/span>,<\/span> np.<\/span>radians(<\/span>0<\/span>)<\/span>]<\/span>)<\/span>
\n print<\/span>(<\/span>f"\u53cd\u9762\u6559\u6750B \u7f3a\u672b\u7aef\u504f\u79fb: q2=-45deg -> <\/span>{<\/span>np.<\/span>round<\/span>(<\/span>T_B1[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span>,<\/span> 5<\/span>)<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span>
\n print<\/span>(<\/span>f" q2= 0deg -> <\/span>{<\/span>np.<\/span>round<\/span>(<\/span>T_B2[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span>,<\/span> 5<\/span>)<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span>
\n print<\/span>(<\/span>" \u4e24\u4e2a\u5b8c\u5168\u4e0d\u540c\u7684\u5173\u8282\u89d2，\u672b\u7aef\u4f4d\u7f6e\u5374\u4e00\u6837 \u2014\u2014 \u56e0\u4e3a j2 \u540e\u9762\u6ca1\u6709\u504f\u79fb！\\\\n"<\/span>)<\/span><\/p>\n

with<\/span> warnings.<\/span>catch_warnings(<\/span>)<\/span>:<\/span>
\n warnings.<\/span>simplefilter(<\/span>"ignore"<\/span>)<\/span>
\n chain =<\/span> make_two_link(<\/span>)<\/span><\/p>\n

print<\/span>(<\/span>"links:"<\/span>,<\/span> [<\/span>l.<\/span>name for<\/span> l in<\/span> chain.<\/span>links]<\/span>)<\/span>
\nprint<\/span>(<\/span>"active_links_mask:"<\/span>,<\/span> chain.<\/span>active_links_mask)<\/span><\/p>\n

# \u6b63\u8fd0\u52a8\u5b66 FK<\/span>
\n# \u6ce8\u610f：forward_kinematics \u9700\u8981\u3010\u5b8c\u6574\u3011\u6570\u7ec4，\u957f\u5ea6 = len(chain.links)<\/span>
\n# \u8fd9\u91cc = base(1) + shoulder(1) + elbow(1) + tip(1) = 4 \u4e2a<\/span>
\nq =<\/span> [<\/span>0.0<\/span>,<\/span> np.<\/span>radians(<\/span>30<\/span>)<\/span>,<\/span> np.<\/span>radians(<\/span>–<\/span>45<\/span>)<\/span>,<\/span> 0.0<\/span>]<\/span>
\nT =<\/span> chain.<\/span>forward_kinematics(<\/span>q)<\/span>
\nprint<\/span>(<\/span>f"\\\\nFK: \u5173\u8282\u89d2 [30\u00b0, -45\u00b0] -> \u672b\u7aef <\/span>{<\/span>T[<\/span>:<\/span>3, 3]<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span><\/p>\n

# \u9006\u8fd0\u52a8\u5b66 IK<\/span>
\nq_sol =<\/span> chain.<\/span>inverse_kinematics(<\/span>target_position=<\/span>T[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span>)<\/span>
\nprint<\/span>(<\/span>f"IK: \u89e3(\u5b8c\u6574\u6570\u7ec4) = <\/span>{<\/span>np.<\/span>round<\/span>(<\/span>q_sol,<\/span> 6<\/span>)<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span>
\nprint<\/span>(<\/span>f" \u5173\u8282\u89d2 = <\/span>{<\/span>np.<\/span>round<\/span>(<\/span>np.<\/span>degrees(<\/span>q_sol[<\/span>1<\/span>:<\/span>3<\/span>]<\/span>)<\/span>,<\/span> 3<\/span>)<\/span>}<\/span><\/span>\u00b0 # \u5e94\u56de\u5230 30\u00b0 \/ -45\u00b0"<\/span><\/span>)<\/span><\/p>\n

# \u5f80\u8fd4\u9a8c\u8bc1<\/span>
\nT_back =<\/span> chain.<\/span>forward_kinematics(<\/span>q_sol)<\/span>
\nprint<\/span>(<\/span>f"\u5f80\u8fd4\u8bef\u5dee = <\/span>{<\/span>np.<\/span>linalg.<\/span>norm(<\/span>T_back[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span> –<\/span> T[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span>)<\/span>:<\/span>.2e<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span><\/p>\n

# ============================================================<\/span>
\n# 7.5 active_links_mask \u7684\u5b9e\u6d4b\u884c\u4e3a<\/span>
\n# ============================================================<\/span>
\nprint<\/span>(<\/span>"\\\\n"<\/span> +<\/span> "="<\/span> *<\/span> 66<\/span>)<\/span>
\nprint<\/span>(<\/span>"7.5 active_links_mask \u5b9e\u6d4b\u884c\u4e3a"<\/span>)<\/span>
\nprint<\/span>(<\/span>"="<\/span> *<\/span> 66<\/span>)<\/span><\/p>\n

links_2 =<\/span> [<\/span>
\n OriginLink(<\/span>)<\/span>,<\/span>
\n URDFLink(<\/span>name=<\/span>"shoulder"<\/span>,<\/span> origin_translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>0<\/span>,<\/span> 1<\/span>,<\/span> 0<\/span>]<\/span>)<\/span>,<\/span>
\n URDFLink(<\/span>name=<\/span>"elbow"<\/span>,<\/span> origin_translation=<\/span>[<\/span>–<\/span>0.30<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>0<\/span>,<\/span> 1<\/span>,<\/span> 0<\/span>]<\/span>)<\/span>,<\/span>
\n URDFLink(<\/span>name=<\/span>"tip"<\/span>,<\/span> origin_translation=<\/span>[<\/span>–<\/span>0.30<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>None<\/span>,<\/span> joint_type=<\/span>"fixed"<\/span>)<\/span>,<\/span>
\n]<\/span><\/p>\n

print<\/span>(<\/span>" — \u60c5\u51b5 1：\u4e0d\u4f20 mask（\u4f1a\u8b66\u544a）—"<\/span>)<\/span>
\nwith<\/span> warnings.<\/span>catch_warnings(<\/span>record=<\/span>True<\/span>)<\/span> as<\/span> w:<\/span>
\n warnings.<\/span>simplefilter(<\/span>"always"<\/span>)<\/span>
\n c1 =<\/span> Chain(<\/span>links=<\/span>links_2)<\/span>
\n print<\/span>(<\/span>f" mask = <\/span>{<\/span>c1.<\/span>active_links_mask}<\/span><\/span>"<\/span><\/span>)<\/span>
\n for<\/span> ww in<\/span> w:<\/span>
\n print<\/span>(<\/span>f" Warning: <\/span>{<\/span>str<\/span>(<\/span>ww.<\/span>message)<\/span>[<\/span>:<\/span>80]<\/span>}<\/span><\/span>…"<\/span><\/span>)<\/span><\/p>\n

print<\/span>(<\/span>"\\\\n — \u60c5\u51b5 2：\u4f20 [False, True, True, False]（\u63a8\u8350\u5199\u6cd5）—"<\/span>)<\/span>
\nc2 =<\/span> Chain(<\/span>links=<\/span>links_2,<\/span> active_links_mask=<\/span>[<\/span>False<\/span>,<\/span> True<\/span>,<\/span> True<\/span>,<\/span> False<\/span>]<\/span>)<\/span>
\nprint<\/span>(<\/span>f" mask = <\/span>{<\/span>c2.<\/span>active_links_mask}<\/span><\/span> # \u539f\u6837\u4fdd\u7559，\u65e0\u8b66\u544a"<\/span><\/span>)<\/span><\/p>\n

print<\/span>(<\/span>"\\\\n — \u60c5\u51b5 3：\u4f20 [False, True, False, False]（\u9501\u6b7b\u5173\u8282 2）—"<\/span>)<\/span>
\nc3 =<\/span> Chain(<\/span>links=<\/span>links_2,<\/span> active_links_mask=<\/span>[<\/span>False<\/span>,<\/span> True<\/span>,<\/span> False<\/span>,<\/span> False<\/span>]<\/span>)<\/span>
\nprint<\/span>(<\/span>f" mask = <\/span>{<\/span>c3.<\/span>active_links_mask}<\/span><\/span>"<\/span><\/span>)<\/span>
\nq_locked =<\/span> c3.<\/span>inverse_kinematics(<\/span>target_position=<\/span>[<\/span>0.0<\/span>,<\/span> 0.0<\/span>,<\/span> 0.42<\/span>]<\/span>)<\/span>
\nprint<\/span>(<\/span>f" IK \u89e3 = <\/span>{<\/span>np.<\/span>round<\/span>(<\/span>q_locked,<\/span> 4<\/span>)<\/span>}<\/span><\/span> # \u7b2c 2 \u4e2a\u5173\u8282\u4fdd\u6301 0，\u88ab\u9501\u6b7b"<\/span><\/span>)<\/span><\/p>\n

print<\/span>(<\/span>"\\\\n \u89c4\u5f8b: OriginLink -> False；\u771f\u5b9e\u5173\u8282 -> True；\u56fa\u5b9a\u504f\u79fb\u8fde\u6746 -> False"<\/span>)<\/span><\/p>\n

# ============================================================<\/span>
\n# 7.6 \u5173\u8282\u9650\u4f4d bounds<\/span>
\n# ============================================================<\/span>
\nprint<\/span>(<\/span>"\\\\n"<\/span> +<\/span> "="<\/span> *<\/span> 66<\/span>)<\/span>
\nprint<\/span>(<\/span>"7.6 \u5173\u8282\u9650\u4f4d bounds"<\/span>)<\/span>
\nprint<\/span>(<\/span>"="<\/span> *<\/span> 66<\/span>)<\/span><\/p>\n

with<\/span> warnings.<\/span>catch_warnings(<\/span>)<\/span>:<\/span>
\n warnings.<\/span>simplefilter(<\/span>"ignore"<\/span>)<\/span>
\n chain_lim =<\/span> Chain(<\/span>
\n name=<\/span>"limited_arm"<\/span>,<\/span>
\n links=<\/span>[<\/span>
\n OriginLink(<\/span>)<\/span>,<\/span>
\n URDFLink(<\/span>name=<\/span>"j1"<\/span>,<\/span> origin_translation=<\/span>[<\/span>–<\/span>0.30<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>0<\/span>,<\/span> 1<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n bounds=<\/span>(<\/span>–<\/span>3.14<\/span>,<\/span> 3.14<\/span>)<\/span>)<\/span>,<\/span>
\n URDFLink(<\/span>name=<\/span>"j2"<\/span>,<\/span> origin_translation=<\/span>[<\/span>–<\/span>0.30<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>0<\/span>,<\/span> 1<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n bounds=<\/span>(<\/span>–<\/span>1.5<\/span>,<\/span> 1.5<\/span>)<\/span>)<\/span>,<\/span> # \u2190 \u9650\u4f4d\u6536\u7d27\u5230 \u00b11.5 rad<\/span>
\n URDFLink(<\/span>name=<\/span>"ee"<\/span>,<\/span> origin_translation=<\/span>[<\/span>–<\/span>0.25<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>None<\/span>,<\/span>
\n joint_type=<\/span>"fixed"<\/span>)<\/span>,<\/span> # \u672b\u7aef\u56fa\u5b9a\u504f\u79fb（\u5c0f\u81c2）<\/span>
\n ]<\/span>,<\/span>
\n active_links_mask=<\/span>[<\/span>False<\/span>,<\/span> True<\/span>,<\/span> True<\/span>,<\/span> False<\/span>]<\/span>,<\/span>
\n )<\/span><\/p>\n

print<\/span>(<\/span>"\u5404\u5173\u8282 bounds:"<\/span>,<\/span> [<\/span>l.<\/span>bounds for<\/span> l in<\/span> chain_lim.<\/span>links]<\/span>)<\/span>
\nprint<\/span>(<\/span>"(\u8fde\u6746\u6cbf -X，\u7ed5 Y \u8f6c -> \u5728 XZ \u5e73\u9762\u8fd0\u52a8；\u603b\u81c2\u957f 0.30+0.25 = 0.55)\\\\n"<\/span>)<\/span><\/p>\n

for<\/span> tgt in<\/span> (<\/span>[<\/span>–<\/span>0.20<\/span>,<\/span> 0.0<\/span>,<\/span> 0.45<\/span>]<\/span>,<\/span> [<\/span>–<\/span>0.10<\/span>,<\/span> 0.0<\/span>,<\/span> 0.50<\/span>]<\/span>,<\/span> [<\/span>–<\/span>0.35<\/span>,<\/span> 0.0<\/span>,<\/span> –<\/span>0.20<\/span>]<\/span>)<\/span>:<\/span>
\n qq =<\/span> chain_lim.<\/span>inverse_kinematics(<\/span>target_position=<\/span>tgt)<\/span>
\n TT =<\/span> chain_lim.<\/span>forward_kinematics(<\/span>qq)<\/span>
\n err =<\/span> np.<\/span>linalg.<\/span>norm(<\/span>TT[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span> –<\/span> np.<\/span>array(<\/span>tgt)<\/span>)<\/span>
\n hit =<\/span> " \u2605\u649e\u9650\u4f4d"<\/span> if<\/span> abs<\/span>(<\/span>abs<\/span>(<\/span>qq[<\/span>2<\/span>]<\/span>)<\/span> –<\/span> 1.5<\/span>)<\/span> <<\/span> 1e-6<\/span> else<\/span> ""<\/span>
\n print<\/span>(<\/span>f" \u76ee\u6807 <\/span>{<\/span>tgt}<\/span><\/span> -> \u89e3=<\/span>{<\/span>np.<\/span>round<\/span>(<\/span>qq[<\/span>1<\/span>:<\/span>3<\/span>]<\/span>,<\/span> 4<\/span>)<\/span>}<\/span><\/span> "<\/span><\/span>
\n f"\u672b\u7aef=<\/span>{<\/span>np.<\/span>round<\/span>(<\/span>TT[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span>,<\/span> 5<\/span>)<\/span>}<\/span><\/span> \u8bef\u5dee=<\/span>{<\/span>err:<\/span>.2e<\/span>}<\/span><\/span>{<\/span>hit}<\/span><\/span>"<\/span><\/span>)<\/span><\/p>\n

print<\/span>(<\/span>"\\\\n >> \u82e5\u76ee\u6807\u9700\u8981\u8d85\u51fa\u9650\u4f4d\u624d\u80fd\u5230\u8fbe，j2 \u4f1a\u88ab\u94b3\u5728 \u00b11.5，\u672b\u7aef\u56e0\u6b64\u6709\u6b8b\u5dee\u3002"<\/span>)<\/span>
\nprint<\/span>(<\/span>" ikpy \u4e0d\u4f1a\u62a5\u9519，\u800c\u662f\u8fd4\u56de'\u5c3d\u529b\u800c\u4e3a'\u7684\u89e3\u3002"<\/span>)<\/span><\/p>\n

# ============================================================<\/span>
\n# 7.5.4 \u56fa\u5b9a\u8fde\u6746\u5fc5\u987b\u5199 joint_type="fixed"（\u9519\u8bef\u6f14\u793a）<\/span>
\n# ============================================================<\/span>
\nprint<\/span>(<\/span>"\\\\n"<\/span> +<\/span> "="<\/span> *<\/span> 66<\/span>)<\/span>
\nprint<\/span>(<\/span>"7.5.4 \u56fa\u5b9a\u8fde\u6746\u7684\u5751（\u9519\u8bef vs \u6b63\u786e\u5199\u6cd5）"<\/span>)<\/span>
\nprint<\/span>(<\/span>"="<\/span> *<\/span> 66<\/span>)<\/span><\/p>\n

# \u9519\u8bef\u5199\u6cd5<\/span>
\ntry<\/span>:<\/span>
\n URDFLink(<\/span>name=<\/span>"bad_ee"<\/span>,<\/span> origin_translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0.045<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>None<\/span>)<\/span> # \u7f3a joint_type<\/span>
\nexcept<\/span> ValueError as<\/span> e:<\/span>
\n print<\/span>(<\/span>f" \u274c \u9519\u8bef\u5199\u6cd5\u62a5\u9519:\\\\n <\/span>{<\/span>type<\/span>(<\/span>e)<\/span>.<\/span>__name__}<\/span><\/span>: <\/span>{<\/span>e}<\/span><\/span>"<\/span><\/span>)<\/span><\/p>\n

# \u6b63\u786e\u5199\u6cd5<\/span>
\ngood =<\/span> URDFLink(<\/span>name=<\/span>"good_ee"<\/span>,<\/span> origin_translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0.045<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>None<\/span>,<\/span>
\n joint_type=<\/span>"fixed"<\/span>)<\/span>
\nprint<\/span>(<\/span>f" \u2705 \u6b63\u786e\u5199\u6cd5 OK: name=<\/span>{<\/span>good.<\/span>name}<\/span><\/span>, joint_type=<\/span>{<\/span>good.<\/span>joint_type}<\/span><\/span>"<\/span><\/span>)<\/span><\/p>\n

# ikpy 3.x \u7684\u65e7\u53c2\u6570\u540d（\u4f1a\u62a5\u9519）<\/span>
\nprint<\/span>(<\/span>"\\\\n \u65e7\u7248(ikpy 3.x)\u5199\u6cd5\u5728 4.0 \u4e0b\u4f1a\u62a5\u9519:"<\/span>)<\/span>
\ntry<\/span>:<\/span>
\n URDFLink(<\/span>name=<\/span>"x"<\/span>,<\/span> translation_vector=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0.3<\/span>]<\/span>,<\/span>
\n orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>0<\/span>,<\/span> –<\/span>1<\/span>,<\/span> 0<\/span>]<\/span>)<\/span>
\nexcept<\/span> TypeError as<\/span> e:<\/span>
\n print<\/span>(<\/span>f" TypeError: <\/span>{<\/span>e}<\/span><\/span>"<\/span><\/span>)<\/span><\/p>\n

# ============================================================<\/span>
\n# 7.7 DHLink \u4e0e\u7b2c 05 \u7ae0\u624b\u5199 DH \u5bf9\u7167<\/span>
\n# ============================================================<\/span>
\nprint<\/span>(<\/span>"\\\\n"<\/span> +<\/span> "="<\/span> *<\/span> 66<\/span>)<\/span>
\nprint<\/span>(<\/span>"7.7 DHLink vs \u7b2c 05 \u7ae0\u624b\u5199 DH"<\/span>)<\/span>
\nprint<\/span>(<\/span>"="<\/span> *<\/span> 66<\/span>)<\/span><\/p>\n

def<\/span> dh_transform<\/span>(<\/span>a,<\/span> alpha,<\/span> d,<\/span> theta)<\/span>:<\/span>
\n """\u7b2c 05 \u7ae0\u624b\u5199\u7684\u6807\u51c6 DH \u53d8\u6362\u77e9\u9635\u3002"""<\/span>
\n ct,<\/span> st =<\/span> np.<\/span>cos(<\/span>theta)<\/span>,<\/span> np.<\/span>sin(<\/span>theta)<\/span>
\n ca,<\/span> sa =<\/span> np.<\/span>cos(<\/span>alpha)<\/span>,<\/span> np.<\/span>sin(<\/span>alpha)<\/span>
\n return<\/span> np.<\/span>array(<\/span>[<\/span>
\n [<\/span>ct,<\/span> –<\/span>st *<\/span> ca,<\/span> st *<\/span> sa,<\/span> a *<\/span> ct]<\/span>,<\/span>
\n [<\/span>st,<\/span> ct *<\/span> ca,<\/span> –<\/span>ct *<\/span> sa,<\/span> a *<\/span> st]<\/span>,<\/span>
\n [<\/span>0.0<\/span>,<\/span> sa,<\/span> ca,<\/span> d]<\/span>,<\/span>
\n [<\/span>0.0<\/span>,<\/span> 0.0<\/span>,<\/span> 0.0<\/span>,<\/span> 1.0<\/span>]<\/span>,<\/span>
\n ]<\/span>)<\/span><\/p>\n

with<\/span> warnings.<\/span>catch_warnings(<\/span>)<\/span>:<\/span>
\n warnings.<\/span>simplefilter(<\/span>"ignore"<\/span>)<\/span>
\n chain_dh =<\/span> Chain(<\/span>
\n name=<\/span>"dh_arm"<\/span>,<\/span>
\n links=<\/span>[<\/span>
\n OriginLink(<\/span>)<\/span>,<\/span>
\n DHLink(<\/span>name=<\/span>"j1"<\/span>,<\/span> d=<\/span>0.20<\/span>,<\/span> a=<\/span>0.30<\/span>,<\/span> alpha=<\/span>0.0<\/span>,<\/span> theta=<\/span>0.0<\/span>)<\/span>,<\/span>
\n DHLink(<\/span>name=<\/span>"j2"<\/span>,<\/span> d=<\/span>0.00<\/span>,<\/span> a=<\/span>0.25<\/span>,<\/span> alpha=<\/span>0.0<\/span>,<\/span> theta=<\/span>0.0<\/span>)<\/span>,<\/span>
\n ]<\/span>,<\/span>
\n active_links_mask=<\/span>[<\/span>False<\/span>,<\/span> True<\/span>,<\/span> True<\/span>]<\/span>,<\/span>
\n )<\/span><\/p>\n

q1,<\/span> q2 =<\/span> np.<\/span>radians(<\/span>30<\/span>)<\/span>,<\/span> np.<\/span>radians(<\/span>45<\/span>)<\/span>
\nT_dh =<\/span> chain_dh.<\/span>forward_kinematics(<\/span>[<\/span>0.0<\/span>,<\/span> q1,<\/span> q2]<\/span>)<\/span>
\nT_manual =<\/span> dh_transform(<\/span>0.30<\/span>,<\/span> 0.0<\/span>,<\/span> 0.20<\/span>,<\/span> q1)<\/span> @ dh_transform(<\/span>0.25<\/span>,<\/span> 0.0<\/span>,<\/span> 0.0<\/span>,<\/span> q2)<\/span><\/p>\n

print<\/span>(<\/span>f" ikpy DHLink = <\/span>{<\/span>np.<\/span>round<\/span>(<\/span>T_dh[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span>,<\/span> 6<\/span>)<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span>
\nprint<\/span>(<\/span>f" \u7b2c05\u7ae0\u624b\u5199 DH = <\/span>{<\/span>np.<\/span>round<\/span>(<\/span>T_manual[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span>,<\/span> 6<\/span>)<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span>
\nprint<\/span>(<\/span>f" \u4e00\u81f4: <\/span>{<\/span>np.<\/span>allclose(<\/span>T_dh[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span>,<\/span> T_manual[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span>)<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span>
\nprint<\/span>(<\/span>"\\\\n >> \u8bc1\u660e\u7b2c 05 \u7ae0\u624b\u5199\u7684 DH \u53d8\u6362\u77e9\u9635\u4e0e ikpy \u7684\u5b9e\u73b0\u5b8c\u5168\u7b49\u4ef7\u3002"<\/span>)<\/span><\/p>\n

# ============================================================<\/span>
\n# 7.10 \u52a8\u624b\u7ec3 \u53c2\u8003\u7b54\u6848<\/span>
\n# ============================================================<\/span>
\nprint<\/span>(<\/span>"\\\\n"<\/span> +<\/span> "="<\/span> *<\/span> 66<\/span>)<\/span>
\nprint<\/span>(<\/span>"7.10 \u52a8\u624b\u7ec3 \u53c2\u8003\u7b54\u6848"<\/span>)<\/span>
\nprint<\/span>(<\/span>"="<\/span> *<\/span> 66<\/span>)<\/span><\/p>\n

# \u7ec3\u4e60 1：2 \u8fde\u6746 FK<\/span>
\nwith<\/span> warnings.<\/span>catch_warnings(<\/span>)<\/span>:<\/span>
\n warnings.<\/span>simplefilter(<\/span>"ignore"<\/span>)<\/span>
\n ch1 =<\/span> make_two_link(<\/span>)<\/span>
\nT1 =<\/span> ch1.<\/span>forward_kinematics(<\/span>[<\/span>0.0<\/span>,<\/span> np.<\/span>radians(<\/span>30<\/span>)<\/span>,<\/span> np.<\/span>radians(<\/span>–<\/span>45<\/span>)<\/span>,<\/span> 0.0<\/span>]<\/span>)<\/span>
\nprint<\/span>(<\/span>f"\u7ec3\u4e601 \u672b\u7aef\u4f4d\u7f6e = <\/span>{<\/span>np.<\/span>round<\/span>(<\/span>T1[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span>,<\/span> 6<\/span>)<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span>
\n# \u624b\u7b97\u4ea4\u53c9\u9a8c\u8bc1: \u672b\u7aef = Ry(q1) @ (t2 + Ry(q2) @ t3)<\/span>
\n# t2=(-0.3,0,0) (elbow\u504f\u79fb), t3=(0,0,0.3) (tip\u504f\u79fb) \u2014\u2014 \u4e0e make_two_link() \u4e00\u81f4<\/span>
\n_th1,<\/span> _th2 =<\/span> np.<\/span>radians(<\/span>30<\/span>)<\/span>,<\/span> np.<\/span>radians(<\/span>–<\/span>45<\/span>)<\/span><\/p>\n

def<\/span> _ry<\/span>(<\/span>a)<\/span>:<\/span>
\n c,<\/span> s =<\/span> np.<\/span>cos(<\/span>a)<\/span>,<\/span> np.<\/span>sin(<\/span>a)<\/span>
\n return<\/span> np.<\/span>array(<\/span>[<\/span>[<\/span>c,<\/span> 0<\/span>,<\/span> s]<\/span>,<\/span> [<\/span>0<\/span>,<\/span> 1<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> [<\/span>–<\/span>s,<\/span> 0<\/span>,<\/span> c]<\/span>]<\/span>)<\/span><\/p>\n

_manual =<\/span> _ry(<\/span>_th1)<\/span> @ (<\/span>np.<\/span>array(<\/span>[<\/span>–<\/span>0.3<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>)<\/span> +<\/span> _ry(<\/span>_th2)<\/span> @ np.<\/span>array(<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0.3<\/span>]<\/span>)<\/span>)<\/span>
\nprint<\/span>(<\/span>f"\u7ec3\u4e601 \u624b\u7b97\u9a8c\u8bc1 = <\/span>{<\/span>np.<\/span>round<\/span>(<\/span>_manual,<\/span> 6<\/span>)<\/span>}<\/span><\/span> \u4e00\u81f4: <\/span>{<\/span>np.<\/span>allclose(<\/span>T1[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span>,<\/span> _manual)<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span><\/p>\n

# \u7ec3\u4e60 2：mask \u5bf9\u6bd4<\/span>
\nprint<\/span>(<\/span>"\u7ec3\u4e602 \u89c1 7.5 \u8282：\u4e0d\u4f20 mask -> \u5168 True + \u8b66\u544a；\u4f20 [F,T,T] -> \u539f\u6837\u4fdd\u7559"<\/span>)<\/span><\/p>\n

# \u7ec3\u4e60 3：bounds=(-1.0, 1.0)<\/span>
\nwith<\/span> warnings.<\/span>catch_warnings(<\/span>)<\/span>:<\/span>
\n warnings.<\/span>simplefilter(<\/span>"ignore"<\/span>)<\/span>
\n ch3 =<\/span> Chain(<\/span>
\n name=<\/span>"tight"<\/span>,<\/span> links=<\/span>[<\/span>
\n OriginLink(<\/span>)<\/span>,<\/span>
\n URDFLink(<\/span>name=<\/span>"j1"<\/span>,<\/span> origin_translation=<\/span>[<\/span>–<\/span>0.30<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>0<\/span>,<\/span> 1<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n bounds=<\/span>(<\/span>–<\/span>3.14<\/span>,<\/span> 3.14<\/span>)<\/span>)<\/span>,<\/span>
\n URDFLink(<\/span>name=<\/span>"j2"<\/span>,<\/span> origin_translation=<\/span>[<\/span>–<\/span>0.30<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>0<\/span>,<\/span> 1<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n bounds=<\/span>(<\/span>–<\/span>1.0<\/span>,<\/span> 1.0<\/span>)<\/span>)<\/span>,<\/span> # \u2190 \u66f4\u7d27\u7684\u9650\u4f4d<\/span>
\n URDFLink(<\/span>name=<\/span>"ee"<\/span>,<\/span> origin_translation=<\/span>[<\/span>–<\/span>0.25<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>None<\/span>,<\/span>
\n joint_type=<\/span>"fixed"<\/span>)<\/span>,<\/span>
\n ]<\/span>,<\/span> active_links_mask=<\/span>[<\/span>False<\/span>,<\/span> True<\/span>,<\/span> True<\/span>,<\/span> False<\/span>]<\/span>)<\/span>
\n# \u7ed9\u4e00\u4e2a\u9700\u8981\u5927\u5e45\u5f2f\u66f2\u624d\u80fd\u5230\u8fbe\u7684\u76ee\u6807（\u9760\u8fd1\u57fa\u5ea7）<\/span>
\nq_tight =<\/span> ch3.<\/span>inverse_kinematics(<\/span>target_position=<\/span>[<\/span>–<\/span>0.10<\/span>,<\/span> 0.0<\/span>,<\/span> 0.10<\/span>]<\/span>)<\/span>
\nprint<\/span>(<\/span>f"\u7ec3\u4e603 \u89e3 = <\/span>{<\/span>np.<\/span>round<\/span>(<\/span>q_tight[<\/span>1<\/span>:<\/span>3<\/span>]<\/span>,<\/span> 4<\/span>)<\/span>}<\/span><\/span> "<\/span><\/span>
\n f"j2 \u662f\u5426\u88ab\u94b3\u5728 \u00b11.0: <\/span>{<\/span>abs<\/span>(<\/span>abs<\/span>(<\/span>q_tight[<\/span>2<\/span>]<\/span>)<\/span> –<\/span> 1.0<\/span>)<\/span> <<\/span> 1e-3<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span><\/p>\n

# \u7ec3\u4e60 4：DHLink \u91cd\u5efa 3 \u81ea\u7531\u5ea6\u81c2（\u542b\u79fb\u52a8\u5173\u8282）<\/span>
\nprint<\/span>(<\/span>"\\\\n\u7ec3\u4e604 3 \u81ea\u7531\u5ea6\u81c2（2 \u8f6c + 1 \u79fb）:"<\/span>)<\/span>
\nwith<\/span> warnings.<\/span>catch_warnings(<\/span>)<\/span>:<\/span>
\n warnings.<\/span>simplefilter(<\/span>"ignore"<\/span>)<\/span>
\n ch4 =<\/span> Chain(<\/span>
\n name=<\/span>"dh_3dof"<\/span>,<\/span> links=<\/span>[<\/span>
\n OriginLink(<\/span>)<\/span>,<\/span>
\n DHLink(<\/span>name=<\/span>"j1"<\/span>,<\/span> d=<\/span>0.20<\/span>,<\/span> a=<\/span>0.30<\/span>,<\/span> alpha=<\/span>0.0<\/span>,<\/span> theta=<\/span>0.0<\/span>)<\/span>,<\/span>
\n DHLink(<\/span>name=<\/span>"j2"<\/span>,<\/span> d=<\/span>0.00<\/span>,<\/span> a=<\/span>0.25<\/span>,<\/span> alpha=<\/span>0.0<\/span>,<\/span> theta=<\/span>0.0<\/span>)<\/span>,<\/span>
\n ]<\/span>,<\/span> active_links_mask=<\/span>[<\/span>False<\/span>,<\/span> True<\/span>,<\/span> True<\/span>]<\/span>)<\/span>
\n# \u6ce8\u610f：ikpy \u7684 DHLink \u4e0d\u76f4\u63a5\u652f\u6301"\u79fb\u52a8\u5173\u8282\u53d8\u91cf d"，<\/span>
\n# \u79fb\u52a8\u5173\u8282\u9700\u8981\u7528 URDFLink(translation=…) \u5b9e\u73b0<\/span>
\nwith<\/span> warnings.<\/span>catch_warnings(<\/span>)<\/span>:<\/span>
\n warnings.<\/span>simplefilter(<\/span>"ignore"<\/span>)<\/span>
\n ch4b =<\/span> Chain(<\/span>
\n name=<\/span>"urdf_3dof"<\/span>,<\/span> links=<\/span>[<\/span>
\n OriginLink(<\/span>)<\/span>,<\/span>
\n # \u4e0e DH \u53c2\u6570\u5bf9\u5e94：j1=d=0.20(\u57fa\u5ea7\u9ad8)+a=0.30(\u81c2\u957f), \u7ed5Z\u8f6c<\/span>
\n URDFLink(<\/span>name=<\/span>"j1"<\/span>,<\/span> origin_translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 1<\/span>]<\/span>)<\/span>,<\/span>
\n URDFLink(<\/span>name=<\/span>"j2"<\/span>,<\/span> origin_translation=<\/span>[<\/span>0.30<\/span>,<\/span> 0<\/span>,<\/span> 0.20<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 1<\/span>]<\/span>)<\/span>,<\/span>
\n URDFLink(<\/span>name=<\/span>"j3"<\/span>,<\/span> origin_translation=<\/span>[<\/span>0.25<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
\n translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 1<\/span>]<\/span>,<\/span> # \u79fb\u52a8\u5173\u8282\u6cbfZ（\u5bf9\u5e94DH\u7684d\u53d8\u91cf）<\/span>
\n joint_type=<\/span>"prismatic"<\/span>)<\/span>,<\/span> # \u2190 \u5fc5\u987b\u663e\u5f0f\u6307\u5b9a！<\/span>
\n ]<\/span>,<\/span> active_links_mask=<\/span>[<\/span>False<\/span>,<\/span> True<\/span>,<\/span> True<\/span>,<\/span> True<\/span>]<\/span>)<\/span>
\nq4 =<\/span> [<\/span>0.0<\/span>,<\/span> np.<\/span>radians(<\/span>30<\/span>)<\/span>,<\/span> np.<\/span>radians(<\/span>45<\/span>)<\/span>,<\/span> 0.1<\/span>]<\/span>
\nT4 =<\/span> ch4b.<\/span>forward_kinematics(<\/span>q4)<\/span>
\nT4_dh =<\/span> (<\/span>dh_transform(<\/span>0.30<\/span>,<\/span> 0<\/span>,<\/span> 0.20<\/span>,<\/span> np.<\/span>radians(<\/span>30<\/span>)<\/span>)<\/span>
\n @ dh_transform(<\/span>0.25<\/span>,<\/span> 0<\/span>,<\/span> 0.0<\/span>,<\/span> np.<\/span>radians(<\/span>45<\/span>)<\/span>)<\/span>
\n @ dh_transform(<\/span>0.0<\/span>,<\/span> 0<\/span>,<\/span> 0.1<\/span>,<\/span> 0.0<\/span>)<\/span>)<\/span>
\nprint<\/span>(<\/span>f" URDFLink \u7248 (\u542b\u79fb\u52a8\u5173\u8282) = <\/span>{<\/span>np.<\/span>round<\/span>(<\/span>T4[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span>,<\/span> 6<\/span>)<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span>
\nprint<\/span>(<\/span>f" \u624b\u5199 DH \u7248 = <\/span>{<\/span>np.<\/span>round<\/span>(<\/span>T4_dh[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span>,<\/span> 6<\/span>)<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span>
\nprint<\/span>(<\/span>f" \u4e00\u81f4: <\/span>{<\/span>np.<\/span>allclose(<\/span>T4[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span>,<\/span> T4_dh[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span>)<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span>
\nprint<\/span>(<\/span>" >> \u79fb\u52a8\u5173\u8282\u7528 URDFLink(translation=\u8f74\u65b9\u5411) \u5b9e\u73b0，\u5bf9\u5e94 DH \u91cc\u7684 d \u53d8\u91cf\u3002"<\/span>)<\/span><\/p>\n

print<\/span>(<\/span>"\\\\n\u7b2c 07 \u7ae0\u793a\u4f8b\u4ee3\u7801\u8fd0\u884c\u5b8c\u6bd5\u3002"<\/span>)<\/span><\/p>\n

\n

\u672c\u7ae0\u5b66\u4e60\u76ee\u6807<\/h3>\n

\u5b66\u5b8c\u672c\u7ae0，\u4f60\u5c06\u80fd\u591f：<\/p>\n

  • \u7406\u89e3 ikpy \u5728\u4eff\u771f\u6808\u4e2d\u7684\u5b9a\u4f4d\u2014\u2014\u5b83\u53ea\u505a\u8fd0\u52a8\u5b66，\u4e0e MuJoCo \u4e92\u8865<\/li>\n
  • \u638c\u63e1 ikpy \u7684\u4e09\u4e2a\u6838\u5fc3\u7c7b：Chain\u3001OriginLink\u3001URDFLink（\u53ca DHLink）<\/li>\n
  • \u6b63\u786e\u6784\u5efa\u4e00\u6761\u8fd0\u52a8\u94fe\u2014\u2014\u7406\u89e3 origin_translation \u7684\u8bed\u4e49，\u907f\u514d"\u6700\u540e\u4e00\u4e2a\u5173\u8282\u4e0d\u8d77\u4f5c\u7528"\u7684\u5751<\/li>\n
  • \u7406\u89e3 active_links_mask \u7684\u4f5c\u7528\u2014\u2014\u54ea\u4e9b\u5173\u8282\u53c2\u4e0e IK \u4f18\u5316<\/li>\n
  • \u8bbe\u7f6e\u5173\u8282\u9650\u4f4d bounds\u2014\u2014\u7406\u89e3 ikpy \u7684"\u5c3d\u529b\u800c\u4e3a"\u884c\u4e3a<\/li>\n
  • \u7528 DHLink \u5efa\u94fe\u2014\u2014\u4e0e\u7b2c 05 \u7ae0\u7684\u624b\u5199 DH \u4ea4\u53c9\u9a8c\u8bc1<\/li>\n
  • \u7406\u89e3 ikpy \u5185\u90e8\u673a\u5236\u2014\u2014\u7b26\u53f7\u77e9\u9635 FK + scipy \u4f18\u5316 IK<\/li>\n
  • \u6392\u67e5 ikpy \u5e38\u89c1\u9519\u8bef\u2014\u2014API \u53d8\u66f4\u3001\u56fa\u5b9a\u8fde\u6746\u7c7b\u578b\u3001mask \u957f\u5ea6\u7b49<\/li>\n

    \u524d\u7f6e\u77e5\u8bc6：\u672c\u7ae0\u9700\u8981\u7b2c 05 \u7ae0\u7684\u6b63\u8fd0\u52a8\u5b66\u6982\u5ff5\u548c\u7b2c 06 \u7ae0\u7684\u9006\u8fd0\u52a8\u5b66\u57fa\u672c\u6982\u5ff5\u3002ikpy \u662f\u8fd9\u4e24\u7ae0\u7406\u8bba\u7684"\u73b0\u6210\u5b9e\u73b0"，\u7406\u89e3\u7406\u8bba\u540e\u518d\u770b ikpy \u4f1a\u975e\u5e38\u987a\u7545\u3002<\/p>\n

    \n

    7.1 ikpy \u5728\u4eff\u771f\u6808\u91cc\u7684\u4f4d\u7f6e<\/h3>\n

    \u56de\u987e\u7b2c 00 \u7ae0\u7684\u5206\u5c42：<\/p>\n

    \u7b2c 3 \u5c42 \u4efb\u52a1\u4e0e\u7b97\u6cd5 \u2192 dm_control
    \n\u7b2c 2 \u5c42 \u7269\u7406\u4eff\u771f \u2192 MuJoCo
    \n\u7b2c 1 \u5c42 \u8fd0\u52a8\u5b66 \u2192 ikpy \u2190 \u672c\u7ae0
    \n\u7b2c 0 \u5c42 \u6570\u5b66 \u2192 NumPy<\/p>\n

    ikpy \u53ea\u505a\u4e00\u4ef6\u4e8b：\u8fd0\u52a8\u5b66\u3002<\/p>\n

    \u5b83\u662f\u7eaf\u51e0\u4f55\u8ba1\u7b97，\u4e0d\u6d89\u53ca\u8d28\u91cf\u3001\u91cd\u529b\u3001\u78b0\u649e\u3001\u6469\u64e6\u3002\u7ed9\u5b83\u4e00\u7ec4\u5173\u8282\u89d2，\u5b83\u7b97\u51fa\u672b\u7aef\u5728\u54ea（FK）；\u7ed9\u5b83\u4e00\u4e2a\u672b\u7aef\u76ee\u6807，\u5b83\u7b97\u51fa\u5173\u8282\u89d2（IK）\u3002\u4ec5\u6b64\u800c\u5df2\u3002<\/p>\n

    💡 \u751f\u6d3b\u7c7b\u6bd4：ikpy \u5c31\u50cf\u8ba1\u7b97\u5668\u91cc\u7684"\u51e0\u4f55\u8ba1\u7b97"\u529f\u80fd\u2014\u2014\u4f60\u8f93\u5165\u8fb9\u957f\u548c\u89d2\u5ea6，\u5b83\u7b97\u51fa\u5bf9\u89d2\u7ebf\u957f\u5ea6\u3002\u5b83\u4e0d\u5173\u5fc3\u8fd9\u4e2a\u7269\u4f53\u662f\u4ec0\u4e48\u6750\u8d28\u3001\u4f1a\u4e0d\u4f1a\u6454\u788e\u3001\u6709\u6ca1\u6709\u649e\u5230\u522b\u7684\u4e1c\u897f\u3002\u90a3\u4e9b\u662f\u7269\u7406\u4eff\u771f\u5668（MuJoCo）\u7684\u4e8b\u3002<\/p>\n

    \u8fd9\u610f\u5473\u7740：<\/p>\n\nikpy \u80fd\u505a\u7684ikpy \u4e0d\u80fd\u505a\u7684<\/tr>\n\n\n\n\n\n
    \u2705 \u6b63\/\u9006\u8fd0\u52a8\u5b66<\/td>\n\u274c \u7269\u7406\u4eff\u771f（\u4f1a\u4e0d\u4f1a\u5012\u3001\u4f1a\u4e0d\u4f1a\u649e）<\/td>\n<\/tr>\n
    \u2705 \u5173\u8282\u9650\u4f4d\u7ea6\u675f<\/td>\n\u274c \u52a8\u529b\u5b66（\u529b\u77e9\u3001\u80fd\u8017）<\/td>\n<\/tr>\n
    \u2705 \u59ff\u6001\u63a7\u5236<\/td>\n\u274c \u78b0\u649e\u68c0\u6d4b<\/td>\n<\/tr>\n
    \u2705 \u53ef\u89c6\u5316\u8fd0\u52a8\u94fe<\/td>\n\u274c \u6e32\u67d3\u771f\u5b9e\u573a\u666f<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    💡 ikpy \u4e0e MuJoCo \u662f\u4e92\u8865\u7684：ikpy \u544a\u8bc9\u4f60"\u5173\u8282\u5e94\u8be5\u8f6c\u5230\u8fd9\u91cc"，MuJoCo \u544a\u8bc9\u4f60"\u5b9e\u9645\u8f6c\u8fc7\u53bb\u4f1a\u53d1\u751f\u4ec0\u4e48"\u3002\u672c\u9879\u76ee\u4e24\u8005\u90fd\u7528\u3002<\/p>\n

    \n

    7.2 \u5b89\u88c5<\/h3>\n

    pip install<\/span> ikpy<\/p>\n

    💡 \u5173\u4e8e\u4f9d\u8d56 sympy：ikpy 4.0 \u5df2\u5728\u5305\u5143\u6570\u636e\u4e2d\u58f0\u660e\u4f9d\u8d56 sympy，\u6b63\u5e38 pip install ikpy \u4f1a\u81ea\u52a8\u628a\u5b83\u5e26\u4e0a；\u53ea\u6709\u5728 –no-deps\u3001\u79bb\u7ebf\u5b89\u88c5 wheel \u7b49\u573a\u666f\u4e0b\u624d\u53ef\u80fd\u7f3a\u5931\u3002\u5982\u679c\u4f60\u770b\u5230：<\/p>\n

    ModuleNotFoundError: No module named 'sympy'<\/p>\n

    \u6267\u884c pip install sympy \u5373\u53ef\u3002ikpy \u7528 sympy \u505a\u7b26\u53f7\u5316\u7684\u53d8\u6362\u77e9\u9635\u63a8\u5bfc，\u8fd9\u662f\u5b83\u7684\u6838\u5fc3\u673a\u5236（\u4e5f\u662f\u5b83\u6bd4\u7eaf\u6570\u503c\u5b9e\u73b0\u66f4\u7075\u6d3b\u7684\u539f\u56e0）\u3002<\/p>\n

    \u9a8c\u8bc1\u5b89\u88c5：<\/p>\n

    import<\/span> ikpy
    \nfrom<\/span> ikpy.<\/span>chain import<\/span> Chain
    \nfrom<\/span> ikpy.<\/span>link import<\/span> OriginLink,<\/span> URDFLink,<\/span> DHLink
    \nprint<\/span>(<\/span>ikpy.<\/span>__version__)<\/span> # \u672c\u4e66\u5b9e\u6d4b：4.0.0<\/span><\/p>\n

    💡 \u4e3a\u4ec0\u4e48 ikpy \u7528 sympy（\u7b26\u53f7\u8ba1\u7b97）：ikpy \u5728\u6784\u5efa Chain \u65f6，\u7528 sympy \u4e3a\u6bcf\u4e2a\u8fde\u6746\u63a8\u5bfc\u51fa\u7b26\u53f7\u5316\u7684\u53d8\u6362\u77e9\u9635（\u5305\u542b\u7b26\u53f7\u53d8\u91cf \u03b81, \u03b82, \u2026）\u3002\u7136\u540e\u5728 FK\/IK \u65f6，\u628a\u5177\u4f53\u6570\u503c\u4ee3\u5165\u7b26\u53f7\u8868\u8fbe\u5f0f\u3002\u8fd9\u6837\u505a\u7684\u597d\u5904\u662f：<\/p>\n

      \n
    • \u63a8\u5bfc\u4e00\u6b21，\u53cd\u590d\u4f7f\u7528（\u6784\u5efa\u94fe\u65f6\u63a8\u5bfc，\u8ba1\u7b97\u65f6\u4ee3\u5165）<\/li>\n
    • \u652f\u6301\u4efb\u610f\u8f74\u3001\u4efb\u610f\u6b27\u62c9\u89d2\u8868\u793a（\u7b26\u53f7\u8ba1\u7b97\u4e0d\u5173\u5fc3\u5177\u4f53\u6570\u503c）<\/li>\n
    • \u53ef\u4ee5\u81ea\u52a8\u8ba1\u7b97\u96c5\u53ef\u6bd4（\u5bf9\u7b26\u53f7\u8868\u8fbe\u5f0f\u6c42\u5bfc）<\/li>\n<\/ul>\n

      \u4ee3\u4ef7\u662f\u6bd4\u7eaf\u6570\u503c\u5b9e\u73b0\u6162\u2014\u2014\u8fd9\u4e5f\u662f\u4e3a\u4ec0\u4e48\u7b2c 06 \u7ae0\u7684\u81ea\u7814 DLS \u7528 MuJoCo \u7684\u89e3\u6790\u96c5\u53ef\u6bd4\u4f1a\u5feb\u7ea6\u4e00\u4e2a\u6570\u91cf\u7ea7（\u89c1 8.7 \u5b9e\u6d4b）\u3002<\/p>\n

      \n

      7.3 \u4e09\u4e2a\u6838\u5fc3\u7c7b<\/h3>\n

      7.3.1 \u7c7b\u7684\u5c42\u6b21<\/h4>\n

      Link（\u62bd\u8c61\u57fa\u7c7b）
      \n \u251c\u2500\u2500 OriginLink \u94fe\u7684\u8d77\u70b9，\u56fa\u5b9a\u4e0d\u52a8，\u4e0d\u63d0\u4f9b\u53d8\u6362
      \n \u251c\u2500\u2500 URDFLink URDF \u98ce\u683c\u7684\u8fde\u6746（\u6700\u5e38\u7528）
      \n \u2514\u2500\u2500 DHLink DH \u53c2\u6570\u98ce\u683c\u7684\u8fde\u6746（\u547c\u5e94\u7b2c 05 \u7ae0）<\/p>\n\n\u7c7b\u7528\u9014\u4ec0\u4e48\u65f6\u5019\u7528<\/tr>\n\n\n\n\n\n
      Chain<\/td>\n\u628a\u4e00\u5806 Link \u4e32\u6210\u4e00\u6761\u94fe，\u63d0\u4f9b FK\/IK<\/td>\n\u6c38\u8fdc\u662f\u5165\u53e3<\/td>\n<\/tr>\n
      OriginLink<\/td>\n\u94fe\u7684"\u5730\u57fa"<\/td>\n\u6bcf\u6761\u94fe\u7684\u7b2c\u4e00\u4e2a\u5143\u7d20<\/td>\n<\/tr>\n
      URDFLink<\/td>\n\u4e00\u4e2a\u5173\u8282 + \u4e00\u6bb5\u504f\u79fb<\/td>\n\u63cf\u8ff0\u5927\u591a\u6570\u673a\u5668\u4eba<\/td>\n<\/tr>\n
      DHLink<\/td>\n\u7528 DH \u56db\u53c2\u6570\u63cf\u8ff0\u4e00\u4e2a\u5173\u8282<\/td>\n\u4f60\u624b\u4e0a\u6709 DH \u8868\u65f6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      💡 \u7c7b\u6bd4：Chain \u662f\u4e00\u4e32\u73cd\u73e0\u9879\u94fe，Link \u662f\u6bcf\u4e00\u9897\u73cd\u73e0\u3002OriginLink \u662f\u9879\u94fe\u7684\u642d\u6263（\u56fa\u5b9a\u7684\u8d77\u70b9），URDFLink \u662f\u666e\u901a\u73cd\u73e0（\u6bcf\u4e2a\u90fd\u6709\u81ea\u5df1\u7684\u4f4d\u7f6e\u548c\u65cb\u8f6c\u8f74），DHLink \u662f\u7528 DH \u53c2\u6570\u63cf\u8ff0\u7684\u7279\u6b8a\u73cd\u73e0\u3002<\/p>\n

      7.3.2 URDFLink \u7684\u53c2\u6570（\u6700\u91cd\u8981）<\/h4>\n

      ikpy 4.0 \u7684\u5b9e\u6d4b\u7b7e\u540d：<\/p>\n

      URDFLink(<\/span>
      \n name:<\/span> str<\/span>,<\/span>
      \n origin_translation:<\/span> np.<\/span>ndarray,<\/span> # \u2190 \u6ce8\u610f！\u4e0d\u662f translation_vector<\/span>
      \n origin_orientation:<\/span> np.<\/span>ndarray,<\/span>
      \n rotation:<\/span> np.<\/span>ndarray =<\/span> None<\/span>,<\/span> # \u65cb\u8f6c\u5173\u8282\u7684\u8f74<\/span>
      \n translation:<\/span> np.<\/span>ndarray =<\/span> None<\/span>,<\/span> # \u79fb\u52a8\u5173\u8282\u7684\u8f74<\/span>
      \n bounds:<\/span> tuple<\/span> =<\/span> None<\/span>,<\/span> # \u5173\u8282\u9650\u4f4d (lower, upper)<\/span>
      \n angle_representation:<\/span> str<\/span> =<\/span> 'rpy'<\/span>,<\/span>
      \n use_symbolic_matrix:<\/span> bool<\/span> =<\/span> True<\/span>,<\/span>
      \n joint_type:<\/span> str<\/span> =<\/span> 'revolute'<\/span>,<\/span> # 'revolute' | 'prismatic' | 'fixed'<\/span>
      \n)<\/span><\/p>\n\n\u53c2\u6570\u5bf9\u5e94 URDF \u7684\u4ec0\u4e48\u8bf4\u660e<\/tr>\n\n\n\n\n\n\n\n
      origin_translation<\/td>\n<origin xyz="…"\/><\/td>\n\u76f8\u5bf9\u7236\u8fde\u6746\u7684\u5e73\u79fb<\/td>\n<\/tr>\n
      origin_orientation<\/td>\n<origin rpy="…"\/><\/td>\n\u76f8\u5bf9\u7236\u8fde\u6746\u7684\u65cb\u8f6c（\u6b27\u62c9\u89d2）<\/td>\n<\/tr>\n
      rotation<\/td>\n<axis xyz="…"\/><\/td>\n\u65cb\u8f6c\u5173\u8282\u7684\u8f6c\u8f74<\/td>\n<\/tr>\n
      translation<\/td>\n<axis xyz="…"\/><\/td>\n\u79fb\u52a8\u5173\u8282\u7684\u79fb\u52a8\u65b9\u5411<\/td>\n<\/tr>\n
      bounds<\/td>\n<limit lower upper\/><\/td>\n\u5173\u8282\u9650\u4f4d (lower, upper)<\/td>\n<\/tr>\n
      joint_type<\/td>\n<joint type="…"\/><\/td>\nrevolute \/ prismatic \/ fixed<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      \u53c2\u6570\u8be6\u89e3：<\/p>\n

        \n
      • origin_translation：\u8fd9\u662f\u6700\u91cd\u8981\u7684\u53c2\u6570\u3002\u5b83\u8868\u793a"\u8fd9\u4e2a\u8fde\u6746\u7684\u5173\u8282\u539f\u70b9，\u76f8\u5bf9\u4e8e\u7236\u8fde\u6746\u5173\u8282\u539f\u70b9\u7684\u504f\u79fb"\u3002\u6ce8\u610f：\u4e0d\u662f"\u8fde\u6746\u957f\u5ea6"，\u800c\u662f"\u5173\u8282\u5230\u5173\u8282\u7684\u8ddd\u79bb"\u3002\u8be6\u89c1 7.4 \u8282\u3002<\/li>\n
      • origin_orientation：\u8fde\u6746\u5750\u6807\u7cfb\u76f8\u5bf9\u4e8e\u7236\u5750\u6807\u7cfb\u7684\u65cb\u8f6c，\u7528 RPY（Roll-Pitch-Yaw）\u6b27\u62c9\u89d2\u8868\u793a\u3002\u5927\u591a\u6570\u60c5\u51b5\u4e0b\u662f [0,0,0]\u3002<\/li>\n
      • rotation：\u65cb\u8f6c\u5173\u8282\u7684\u8f6c\u8f74，\u5355\u4f4d\u5411\u91cf\u3002\u5982 [0,1,0] \u8868\u793a\u7ed5 Y \u8f74\u65cb\u8f6c\u3002\u65cb\u8f6c\u5173\u8282\u5fc5\u987b\u8bbe\u7f6e\u8fd9\u4e2a\u53c2\u6570\u3002<\/li>\n
      • translation：\u79fb\u52a8\u5173\u8282\u7684\u79fb\u52a8\u65b9\u5411，\u5355\u4f4d\u5411\u91cf\u3002\u5982 [0,0,1] \u8868\u793a\u6cbf Z \u8f74\u79fb\u52a8\u3002\u79fb\u52a8\u5173\u8282\u5fc5\u987b\u8bbe\u7f6e\u8fd9\u4e2a\u53c2\u6570，\u4e14 joint_type \u5fc5\u987b\u4e3a 'prismatic'\u3002<\/li>\n
      • bounds：\u5173\u8282\u9650\u4f4d (lower, upper) \u5143\u7ec4，\u5355\u4f4d\u662f\u5f27\u5ea6（\u65cb\u8f6c\u5173\u8282）\u6216\u7c73（\u79fb\u52a8\u5173\u8282）\u3002ikpy \u4f1a\u628a\u5b83\u4f20\u7ed9 scipy \u4f18\u5316\u5668\u7684 bounds \u53c2\u6570\u3002<\/li>\n
      • joint_type：\u5173\u8282\u7c7b\u578b\u3002'revolute'（\u65cb\u8f6c，\u9ed8\u8ba4）\u3001'prismatic'（\u79fb\u52a8）\u3001'fixed'（\u56fa\u5b9a）\u3002\u56fa\u5b9a\u8fde\u6746\u5fc5\u987b\u663e\u5f0f\u5199 joint_type="fixed"\u3002<\/li>\n<\/ul>\n

        🔥 \u672c\u4e66\u5b9e\u6d4b\u8e29\u5751（\u7b2c 1 \u53f7）：<\/p>\n\n\u7f51\u4e0a\u65e7\u6559\u7a0b（ikpy 3.x）ikpy 4.0 \u6b63\u786e\u5199\u6cd5<\/tr>\n\n\n
        URDFLink(translation_vector=[…])<\/td>\nURDFLink(origin_translation=[…])<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

        \u7528\u65e7\u5199\u6cd5\u4f1a\u76f4\u63a5\u62a5\u9519：<\/p>\n

        TypeError: URDFLink.__init__() got an unexpected keyword argument 'translation_vector'<\/p>\n

        \u672c\u4e66\u6240\u6709\u4ee3\u7801\u90fd\u6309 4.0 \u5b9e\u6d4b\u7f16\u5199\u3002<\/p>\n

        7.3.3 Chain \u7684\u6784\u9020<\/h4>\n

        Chain(<\/span>
        \n links,<\/span> # Link \u5217\u8868<\/span>
        \n active_links_mask=<\/span>None<\/span>,<\/span> # \u54ea\u4e9b\u5173\u8282\u53c2\u4e0e IK \u4f18\u5316<\/span>
        \n name=<\/span>'chain'<\/span>,<\/span>
        \n urdf_metadata=<\/span>None<\/span>,<\/span>
        \n jax_precompile=<\/span>True<\/span>,<\/span>
        \n)<\/span><\/p>\n

        Chain \u6784\u5efa\u6d41\u7a0b\u56fe：<\/p>\n

        #mermaid-svg-jLYbYdI7A7pceNVJ{font-family:\\”trebuchet ms\\”,verdana,arial,sans-serif;font-size:16px;fill:#333;}@keyframes edge-animation-frame{from{stroke-dashoffset:0;}}@keyframes dash{to{stroke-dashoffset:0;}}#mermaid-svg-jLYbYdI7A7pceNVJ .edge-animation-slow{stroke-dasharray:9,5!important;stroke-dashoffset:900;animation:dash 50s linear infinite;stroke-linecap:round;}#mermaid-svg-jLYbYdI7A7pceNVJ .edge-animation-fast{stroke-dasharray:9,5!important;stroke-dashoffset:900;animation:dash 20s linear infinite;stroke-linecap:round;}#mermaid-svg-jLYbYdI7A7pceNVJ .error-icon{fill:#552222;}#mermaid-svg-jLYbYdI7A7pceNVJ .error-text{fill:#552222;stroke:#552222;}#mermaid-svg-jLYbYdI7A7pceNVJ .edge-thickness-normal{stroke-width:1px;}#mermaid-svg-jLYbYdI7A7pceNVJ .edge-thickness-thick{stroke-width:3.5px;}#mermaid-svg-jLYbYdI7A7pceNVJ .edge-pattern-solid{stroke-dasharray:0;}#mermaid-svg-jLYbYdI7A7pceNVJ .edge-thickness-invisible{stroke-width:0;fill:none;}#mermaid-svg-jLYbYdI7A7pceNVJ .edge-pattern-dashed{stroke-dasharray:3;}#mermaid-svg-jLYbYdI7A7pceNVJ .edge-pattern-dotted{stroke-dasharray:2;}#mermaid-svg-jLYbYdI7A7pceNVJ .marker{fill:#333333;stroke:#333333;}#mermaid-svg-jLYbYdI7A7pceNVJ .marker.cross{stroke:#333333;}#mermaid-svg-jLYbYdI7A7pceNVJ svg{font-family:\\”trebuchet ms\\”,verdana,arial,sans-serif;font-size:16px;}#mermaid-svg-jLYbYdI7A7pceNVJ p{margin:0;}#mermaid-svg-jLYbYdI7A7pceNVJ .label{font-family:\\”trebuchet ms\\”,verdana,arial,sans-serif;color:#333;}#mermaid-svg-jLYbYdI7A7pceNVJ .cluster-label text{fill:#333;}#mermaid-svg-jLYbYdI7A7pceNVJ .cluster-label span{color:#333;}#mermaid-svg-jLYbYdI7A7pceNVJ .cluster-label span p{background-color:transparent;}#mermaid-svg-jLYbYdI7A7pceNVJ .label text,#mermaid-svg-jLYbYdI7A7pceNVJ span{fill:#333;color:#333;}#mermaid-svg-jLYbYdI7A7pceNVJ .node rect,#mermaid-svg-jLYbYdI7A7pceNVJ .node circle,#mermaid-svg-jLYbYdI7A7pceNVJ .node ellipse,#mermaid-svg-jLYbYdI7A7pceNVJ .node polygon,#mermaid-svg-jLYbYdI7A7pceNVJ .node path{fill:#ECECFF;stroke:#9370DB;stroke-width:1px;}#mermaid-svg-jLYbYdI7A7pceNVJ .rough-node .label text,#mermaid-svg-jLYbYdI7A7pceNVJ .node .label text,#mermaid-svg-jLYbYdI7A7pceNVJ .image-shape .label,#mermaid-svg-jLYbYdI7A7pceNVJ .icon-shape .label{text-anchor:middle;}#mermaid-svg-jLYbYdI7A7pceNVJ .node .katex path{fill:#000;stroke:#000;stroke-width:1px;}#mermaid-svg-jLYbYdI7A7pceNVJ .rough-node .label,#mermaid-svg-jLYbYdI7A7pceNVJ .node .label,#mermaid-svg-jLYbYdI7A7pceNVJ .image-shape .label,#mermaid-svg-jLYbYdI7A7pceNVJ .icon-shape .label{text-align:center;}#mermaid-svg-jLYbYdI7A7pceNVJ .node.clickable{cursor:pointer;}#mermaid-svg-jLYbYdI7A7pceNVJ .root .anchor path{fill:#333333!important;stroke-width:0;stroke:#333333;}#mermaid-svg-jLYbYdI7A7pceNVJ .arrowheadPath{fill:#333333;}#mermaid-svg-jLYbYdI7A7pceNVJ .edgePath .path{stroke:#333333;stroke-width:2.0px;}#mermaid-svg-jLYbYdI7A7pceNVJ .flowchart-link{stroke:#333333;fill:none;}#mermaid-svg-jLYbYdI7A7pceNVJ .edgeLabel{background-color:rgba(232,232,232, 0.8);text-align:center;}#mermaid-svg-jLYbYdI7A7pceNVJ .edgeLabel p{background-color:rgba(232,232,232, 0.8);}#mermaid-svg-jLYbYdI7A7pceNVJ .edgeLabel rect{opacity:0.5;background-color:rgba(232,232,232, 0.8);fill:rgba(232,232,232, 0.8);}#mermaid-svg-jLYbYdI7A7pceNVJ .labelBkg{background-color:rgba(232, 232, 232, 0.5);}#mermaid-svg-jLYbYdI7A7pceNVJ .cluster rect{fill:#ffffde;stroke:#aaaa33;stroke-width:1px;}#mermaid-svg-jLYbYdI7A7pceNVJ .cluster text{fill:#333;}#mermaid-svg-jLYbYdI7A7pceNVJ .cluster span{color:#333;}#mermaid-svg-jLYbYdI7A7pceNVJ div.mermaidTooltip{position:absolute;text-align:center;max-width:200px;padding:2px;font-family:\\”trebuchet ms\\”,verdana,arial,sans-serif;font-size:12px;background:hsl(80, 100%, 96.2745098039%);border:1px solid #aaaa33;border-radius:2px;pointer-events:none;z-index:100;}#mermaid-svg-jLYbYdI7A7pceNVJ .flowchartTitleText{text-anchor:middle;font-size:18px;fill:#333;}#mermaid-svg-jLYbYdI7A7pceNVJ rect.text{fill:none;stroke-width:0;}#mermaid-svg-jLYbYdI7A7pceNVJ .icon-shape,#mermaid-svg-jLYbYdI7A7pceNVJ .image-shape{background-color:rgba(232,232,232, 0.8);text-align:center;}#mermaid-svg-jLYbYdI7A7pceNVJ .icon-shape p,#mermaid-svg-jLYbYdI7A7pceNVJ .image-shape p{background-color:rgba(232,232,232, 0.8);padding:2px;}#mermaid-svg-jLYbYdI7A7pceNVJ .icon-shape .label rect,#mermaid-svg-jLYbYdI7A7pceNVJ .image-shape .label rect{opacity:0.5;background-color:rgba(232,232,232, 0.8);fill:rgba(232,232,232, 0.8);}#mermaid-svg-jLYbYdI7A7pceNVJ .label-icon{display:inline-block;height:1em;overflow:visible;vertical-align:-0.125em;}#mermaid-svg-jLYbYdI7A7pceNVJ .node .label-icon path{fill:currentColor;stroke:revert;stroke-width:revert;}#mermaid-svg-jLYbYdI7A7pceNVJ :root{–mermaid-font-family:\\”trebuchet ms\\”,verdana,arial,sans-serif;}<\/p>\n

        <\/span><\/p>\n

        <\/span><\/p>\n

        <\/span><\/p>\n

        <\/span><\/p>\n

        <\/span><\/p>\n

        <\/span><\/p>\n

        <\/span><\/p>\n

        <\/p>\n

        \u662f<\/p>\n

        <\/span><\/p>\n

        <\/p>\n

        \u5426<\/p>\n

        <\/span><\/p>\n

        <\/span><\/p>\n

        <\/span><\/p>\n

        <\/span><\/p>\n

        <\/p>\n

        \u5f00\u59cb\u6784\u5efa Chain<\/p>\n

        <\/span><\/p>\n

        <\/p>\n

        \u521b\u5efa OriginLink() \u4f5c\u4e3a\u94fe\u7684\u8d77\u70b9<\/p>\n

        <\/span><\/p>\n

        <\/p>\n

        \u5bf9\u6bcf\u4e2a\u5173\u8282\u521b\u5efa URDFLink<\/p>\n

        <\/span><\/p>\n

        <\/p>\n

        \u8bbe\u7f6e origin_translation (\u76f8\u5bf9\u7236\u7684\u504f\u79fb)<\/p>\n

        <\/span><\/p>\n

        <\/p>\n

        \u8bbe\u7f6e rotation (\u65cb\u8f6c\u8f74) \u6216 translation (\u79fb\u52a8\u8f74)<\/p>\n

        <\/span><\/p>\n

        <\/p>\n

        \u8bbe\u7f6e bounds (\u5173\u8282\u9650\u4f4d)<\/p>\n

        <\/span><\/p>\n

        <\/p>\n

        \u8bbe\u7f6e joint_type (revolute\/prismatic\/fixed)<\/p>\n

        <\/span><\/p>\n

        <\/p>\n

        \u8fd8\u6709\u5173\u8282?<\/p>\n

        <\/span><\/p>\n

        <\/p>\n

        \u521b\u5efa\u672b\u7aef\u56fa\u5b9a\u504f\u79fb URDFLink (joint_type=fixed)<\/p>\n

        <\/span><\/p>\n

        <\/p>\n

        \u8bbe\u7f6e active_links_mask = [False] + [True]*n + [False]<\/p>\n

        <\/span><\/p>\n

        <\/p>\n

        Chain(name=…, links=…, active_links_mask=…)<\/p>\n

        <\/span><\/p>\n

        <\/p>\n

        \u5b8c\u6210!<\/p>\n

        <\/span><\/p>\n

        \n

        7.4 \u7b2c\u4e00\u4e2a\u7a0b\u5e8f：2 \u8fde\u6746\u5e73\u9762\u81c2<\/h3>\n

        7.4.1 \u26a0\ufe0f \u5148\u641e\u61c2\u4e00\u4e2a\u81f4\u547d\u7ea6\u5b9a：origin_translation \u5230\u5e95\u91cf\u7684\u662f\u4ec0\u4e48<\/h4>\n

        \u8fd9\u662f ikpy \u6700\u5bb9\u6613\u7406\u89e3\u9519\u3001\u4e14\u9519\u4e86\u4e4b\u540e\u6781\u96be\u81ea\u67e5\u7684\u5730\u65b9\u3002 \u8bf7\u52a1\u5fc5\u5148\u8bfb\u8fd9\u4e00\u5c0f\u8282\u3002<\/p>\n

        \u4e00\u6761 ikpy \u94fe\u7684\u6574\u4f53\u53d8\u6362\u662f：<\/p>\n

        T_total = T\u2081 \u00b7 T\u2082 \u00b7 … \u00b7 T\u2099
        \n\u672b\u7aef\u4f4d\u7f6e = T_total \u4f5c\u7528\u4e8e\u539f\u70b9 (0,0,0)<\/p>\n

        \u5c55\u5f00\u540e\u4f60\u4f1a\u53d1\u73b0\u4e00\u4e2a\u5173\u952e\u4e8b\u5b9e：<\/p>\n

        \u672b\u7aef = t\u2081 + R\u2081\u00b7( t\u2082 + R\u2082\u00b7( t\u2083 + … ) )
        \n \u2514 \u5173\u82821 \u524d\u7684\u504f\u79fb \u2514 \u5173\u82821\u2192\u5173\u82822 \u2514 \u5173\u82822\u2192\u672b\u7aef<\/p>\n

        \u6240\u4ee5 origin_translation \u7684\u8bed\u4e49\u662f：<\/p>\n\n\u4f4d\u7f6eorigin_translation \u7684\u542b\u4e49<\/tr>\n\n\n\n\n
        links[1]<\/td>\n\u57fa\u5ea7 \u2192 \u5173\u8282 1 \u7684\u504f\u79fb<\/td>\n<\/tr>\n
        links[i]<\/td>\n\u5173\u8282 i-1 \u2192 \u5173\u8282 i \u7684\u504f\u79fb（\u7b2c i-1 \u6bb5\u8fde\u6746\u957f\u5ea6）<\/td>\n<\/tr>\n
        \u6700\u540e\u4e00\u4e2a link<\/td>\n\u6700\u540e\u5173\u8282 \u2192 \u672b\u7aef（TCP） \u7684\u8ddd\u79bb<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

        \u63a8\u8bba（\u6781\u5176\u91cd\u8981）：<\/p>\n

        \u53ea\u6709"\u5173\u8282\u540e\u9762\u8fd8\u6709\u504f\u79fb"，\u8fd9\u4e2a\u5173\u8282\u7684\u65cb\u8f6c\u624d\u4f1a\u6539\u53d8\u672b\u7aef\u4f4d\u7f6e\u3002<\/p>\n

        \u5982\u679c\u6700\u540e\u4e00\u4e2a link \u5c31\u662f\u4f60\u7684\u6700\u540e\u4e00\u4e2a\u5173\u8282，\u90a3\u5b83\u540e\u9762\u6ca1\u6709\u4efb\u4f55\u504f\u79fb\u2014\u2014\u8fd9\u4e2a\u5173\u8282\u8f6c\u591a\u5c11\u5ea6，\u672b\u7aef\u90fd\u7eb9\u4e1d\u4e0d\u52a8！<\/p>\n

        💡 \u751f\u6d3b\u7c7b\u6bd4：\u60f3\u8c61\u4f60\u624b\u91cc\u62ff\u7740\u4e00\u6839\u68cd\u5b50\u3002\u68cd\u5b50\u7684\u4e00\u7aef\u5728\u4f60\u624b\u91cc（\u5173\u8282），\u53e6\u4e00\u7aef\u662f\u68cd\u5b50\u7684\u672b\u7aef\u3002\u5982\u679c\u4f60\u8f6c\u52a8\u624b\u8155（\u5173\u8282\u65cb\u8f6c），\u68cd\u5b50\u7684\u672b\u7aef\u4f1a\u753b\u4e00\u4e2a\u5706\u5f27\u2014\u2014\u56e0\u4e3a\u68cd\u5b50\u6709\u957f\u5ea6（\u5173\u8282\u540e\u9762\u6709\u504f\u79fb）\u3002<\/p>\n

        \u4f46\u5982\u679c\u4f60\u624b\u91cc\u4ec0\u4e48\u90fd\u6ca1\u62ff（\u5173\u8282\u540e\u9762\u6ca1\u6709\u504f\u79fb），\u4f60\u8f6c\u52a8\u624b\u8155，"\u672b\u7aef"\u5c31\u662f\u4f60\u7684\u624b\u8155\u672c\u8eab\u2014\u2014\u5b83\u4e0d\u4f1a\u56e0\u4e3a\u8f6c\u52a8\u800c\u79fb\u52a8\u4f4d\u7f6e\u3002\u8fd9\u5c31\u662f"\u6700\u540e\u4e00\u4e2a\u5173\u8282\u4e0d\u8d77\u4f5c\u7528"\u7684\u539f\u56e0\u3002<\/p>\n

        7.4.2 \u4e24\u4e2a\u53cd\u9762\u6559\u6750（\u5b9e\u6d4b）<\/h4>\n

        \u53cd\u9762\u6559\u6750 A：\u8f6c\u8f74\u4e0e\u8fde\u6746\u540c\u5411<\/p>\n

        # \u8fde\u6746\u6cbf -X \u65b9\u5411\u4f38\u5c55，\u5374\u7ed5 -X \u8f74\u8f6c \u2014\u2014 \u65cb\u8f6c\u5bf9\u8fde\u6746\u4e0d\u4ea7\u751f\u4f4d\u79fb<\/span>
        \nChain(<\/span>links=<\/span>[<\/span>
        \n OriginLink(<\/span>)<\/span>,<\/span>
        \n URDFLink(<\/span>name=<\/span>"l1"<\/span>,<\/span> origin_translation=<\/span>[<\/span>–<\/span>0.30<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
        \n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>–<\/span>1<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>)<\/span>,<\/span>
        \n URDFLink(<\/span>name=<\/span>"l2"<\/span>,<\/span> origin_translation=<\/span>[<\/span>–<\/span>0.30<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
        \n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>–<\/span>1<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>)<\/span>,<\/span>
        \n]<\/span>,<\/span> active_links_mask=<\/span>[<\/span>False<\/span>,<\/span> True<\/span>,<\/span> True<\/span>]<\/span>)<\/span><\/p>\n

        \u5b9e\u6d4b（q=[0, 30\u00b0, -45\u00b0]）：<\/p>\n

        \u53cd\u9762\u6559\u6750A \u8fde\u6746\u6cbf-X + \u7ed5-X\u8f6c: q=[0,30deg,-45deg] -> \u672b\u7aef [-0.6 0. 0.]
        \n \u65e0\u8bba\u5173\u8282\u8f6c\u591a\u5c11，\u672b\u7aef\u90fd\u505c\u5728 (-0.6,0,0) \u2014\u2014 \u65cb\u8f6c\u6ca1\u8d77\u4f5c\u7528！<\/p>\n

        💡 \u4e3a\u4ec0\u4e48\u8f6c\u8f74\u4e0e\u8fde\u6746\u540c\u5411\u540e\u65cb\u8f6c\u4e0d\u4ea7\u751f\u4f4d\u79fb：\u5982\u679c\u8fde\u6746\u6cbf -X \u8f74\u4f38\u5c55，\u5173\u8282\u4e5f\u7ed5 X \u8f74\u65cb\u8f6c，\u90a3\u4e48\u65cb\u8f6c\u5c31\u662f"\u7ed5\u7740\u8fde\u6746\u81ea\u8eab\u7684\u8f74\u8f6c"\u2014\u2014\u5c31\u50cf\u8f6c\u52a8\u87ba\u4e1d\u5200，\u87ba\u4e1d\u5200\u7684\u5c16\u7aef\u4f4d\u7f6e\u4e0d\u53d8，\u53ea\u662f\u5728\u539f\u5730\u8f6c\u3002\u8981\u8ba9\u65cb\u8f6c\u4ea7\u751f\u4f4d\u79fb，\u8f6c\u8f74\u5fc5\u987b\u4e0e\u8fde\u6746\u65b9\u5411\u5782\u76f4（\u6216\u81f3\u5c11\u6709\u5782\u76f4\u5206\u91cf）\u3002<\/p>\n

        \u53cd\u9762\u6559\u6750 B：\u5c11\u4e86\u672b\u7aef\u504f\u79fb<\/p>\n

        # \u53ea\u6709 OriginLink + 2 \u4e2a\u5173\u8282，\u7b2c\u4e8c\u4e2a\u5173\u8282\u540e\u9762\u6ca1\u6709\u504f\u79fb<\/span>
        \nChain(<\/span>links=<\/span>[<\/span>
        \n OriginLink(<\/span>)<\/span>,<\/span>
        \n URDFLink(<\/span>name=<\/span>"j1"<\/span>,<\/span> origin_translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0.00<\/span>]<\/span>,<\/span>
        \n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>0<\/span>,<\/span> 1<\/span>,<\/span> 0<\/span>]<\/span>)<\/span>,<\/span>
        \n URDFLink(<\/span>name=<\/span>"j2"<\/span>,<\/span> origin_translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0.30<\/span>]<\/span>,<\/span>
        \n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>0<\/span>,<\/span> 1<\/span>,<\/span> 0<\/span>]<\/span>)<\/span>,<\/span>
        \n]<\/span>,<\/span> active_links_mask=<\/span>[<\/span>False<\/span>,<\/span> True<\/span>,<\/span> True<\/span>]<\/span>)<\/span><\/p>\n

        \u5b9e\u6d4b：<\/p>\n

        q2 = -45\u00b0 -> \u672b\u7aef [0.15 0. 0.25981]
        \nq2 = 0\u00b0 -> \u672b\u7aef [0.15 0. 0.25981] \u2190 \u5b8c\u5168\u4e00\u6837！<\/p>\n

        \u4e24\u4e2a\u622a\u7136\u4e0d\u540c\u7684\u5173\u8282\u89d2，\u672b\u7aef\u4f4d\u7f6e\u76f8\u540c \u2014\u2014 \u56e0\u4e3a j2 \u540e\u9762\u6ca1\u6709\u504f\u79fb，\u5b83\u7684\u65cb\u8f6c\u88ab"\u6d6a\u8d39"\u6389\u4e86\u3002<\/p>\n

        🔥 \u8fd9\u4e2a\u9519\u8bef\u6781\u5177\u8ff7\u60d1\u6027：IK \u7167\u6837\u80fd\u8dd1\u3001\u7167\u6837"\u6536\u655b"\u3001\u8bef\u5dee\u7167\u6837\u63a5\u8fd1 0，\u4f46\u4f60\u7684\u673a\u68b0\u81c2\u5c11\u4e86\u4e00\u4e2a\u6709\u6548\u81ea\u7531\u5ea6\u3002\u5982\u679c\u4f60\u53d1\u73b0\u67d0\u4e2a\u5173\u8282\u600e\u4e48\u52a8\u90fd\u4e0d\u5f71\u54cd\u672b\u7aef，\u5148\u68c0\u67e5\u5b83\u540e\u9762\u6709\u6ca1\u6709\u504f\u79fb\u3002<\/p>\n

        7.4.3 \u6b63\u786e\u7684 2 \u8fde\u6746\u81c2<\/h4>\n

        \u4e00\u53f0\u771f\u6b63\u7684 2 \u8fde\u6746\u81c2\u9700\u8981 4 \u4e2a\u5143\u7d20：<\/p>\n

        import<\/span> numpy as<\/span> np
        \nfrom<\/span> ikpy.<\/span>chain import<\/span> Chain
        \nfrom<\/span> ikpy.<\/span>link import<\/span> OriginLink,<\/span> URDFLink<\/p>\n

        chain =<\/span> Chain(<\/span>
        \n name=<\/span>"two_link_arm"<\/span>,<\/span>
        \n links=<\/span>[<\/span>
        \n OriginLink(<\/span>)<\/span>,<\/span> # \u2460 \u5730\u57fa<\/span>
        \n URDFLink(<\/span>name=<\/span>"shoulder"<\/span>,<\/span> # \u2461 \u5173\u8282 1（\u57fa\u5ea7\u5728\u539f\u70b9）<\/span>
        \n origin_translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0.00<\/span>]<\/span>,<\/span>
        \n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
        \n rotation=<\/span>[<\/span>0<\/span>,<\/span> 1<\/span>,<\/span> 0<\/span>]<\/span>)<\/span>,<\/span> # \u7ed5 Y \u8f6c<\/span>
        \n URDFLink(<\/span>name=<\/span>"elbow"<\/span>,<\/span> # \u2462 \u5173\u8282 2（\u8ddd\u5173\u8282 1 \u4e3a L1=0.30，\u6cbf -X）<\/span>
        \n origin_translation=<\/span>[<\/span>–<\/span>0.30<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
        \n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
        \n rotation=<\/span>[<\/span>0<\/span>,<\/span> 1<\/span>,<\/span> 0<\/span>]<\/span>)<\/span>,<\/span>
        \n URDFLink(<\/span>name=<\/span>"tip"<\/span>,<\/span> # \u2463 \u672b\u7aef\u504f\u79fb（\u8ddd\u5173\u8282 2 \u4e3a L2=0.30，\u6cbf +Z）<\/span>
        \n origin_translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0.30<\/span>]<\/span>,<\/span>
        \n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
        \n rotation=<\/span>None<\/span>,<\/span> joint_type=<\/span>"fixed"<\/span>)<\/span>,<\/span>
        \n ]<\/span>,<\/span>
        \n active_links_mask=<\/span>[<\/span>False<\/span>,<\/span> True<\/span>,<\/span> True<\/span>,<\/span> False<\/span>]<\/span>,<\/span> # \u2464 \u5730\u57fa\u4e0e\u672b\u7aef\u4e0d\u53c2\u4e0e\u4f18\u5316<\/span>
        \n)<\/span><\/p>\n

        # \u6b63\u8fd0\u52a8\u5b66：\u5173\u8282\u89d2 \u2192 \u672b\u7aef\u4f4d\u59ff<\/span>
        \n# \u6ce8\u610f：forward_kinematics \u9700\u8981\u3010\u5b8c\u6574\u3011\u6570\u7ec4，\u957f\u5ea6 = len(chain.links) = 4<\/span>
        \nq =<\/span> [<\/span>0.0<\/span>,<\/span> np.<\/span>radians(<\/span>30<\/span>)<\/span>,<\/span> np.<\/span>radians(<\/span>–<\/span>45<\/span>)<\/span>,<\/span> 0.0<\/span>]<\/span>
        \nT =<\/span> chain.<\/span>forward_kinematics(<\/span>q)<\/span>
        \nprint<\/span>(<\/span>"\u672b\u7aef\u4f4d\u7f6e:"<\/span>,<\/span> T[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span>)<\/span><\/p>\n

        # \u9006\u8fd0\u52a8\u5b66：\u672b\u7aef\u4f4d\u7f6e \u2192 \u5173\u8282\u89d2<\/span>
        \nq_sol =<\/span> chain.<\/span>inverse_kinematics(<\/span>target_position=<\/span>T[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span>)<\/span><\/p>\n

        2 \u8fde\u6746\u81c2\u7ed3\u6784\u793a\u610f\u56fe（\u96f6\u4f4d\u59ff\u6001\u4fa7\u89c6\u56fe，X \u5411\u53f3\u3001Z \u5411\u4e0a；\u4e24\u4e2a\u5173\u8282\u90fd\u7ed5 Y \u8f6c，\u5728 XZ \u5e73\u9762\u5185\u6446\u52a8）：<\/p>\n

        \u672b\u7aef (tip)
        \n \u25cf
        \n \u2502
        \n \u2502 L2 = 0.30 (origin_translation of tip, \u6cbf +Z)
        \n \u2502
        \n \u57fa\u5ea7 \u25cf\u2500\u2500 J1 (shoulder) \u25cf\u2190\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u25cf J2 (elbow) \u2190 \u90fd\u7ed5 Y \u65cb\u8f6c
        \n (shoulder \u7684 origin_translation \u2502
        \n = 0, \u57fa\u5ea7\u5728\u539f\u70b9) \u2502
        \n L1 = 0.30 (origin_translation of elbow, \u6cbf -X)<\/p>\n

        \u5b9e\u6d4b\u8f93\u51fa：<\/p>\n

        FK: \u5173\u8282\u89d2 [30\u00b0, -45\u00b0] -> \u672b\u7aef [-0.337453 0. 0.439778]
        \nIK: \u89e3(\u5b8c\u6574\u6570\u7ec4) = [ 0. 0.523599 -0.785398 0. ]
        \n \u5173\u8282\u89d2 = [ 30. -45.]\u00b0 \u2190 \u6b63\u597d\u56de\u5230\u6211\u4eec\u8f93\u5165\u7684 30\u00b0 \/ -45\u00b0
        \n\u5f80\u8fd4\u8bef\u5dee = 5.08e-09<\/p>\n

        💡 IK \u4ece\u5168\u96f6\u79cd\u5b50\u51fa\u53d1，\u6536\u655b\u56de\u4e86\u8f93\u5165\u7684 [30\u00b0, -45\u00b0]（\u5f80\u8fd4\u8bef\u5dee 5.08e-09）\u2014\u2014\u6ce8\u610f\u9006\u89e3\u4e0d\u552f\u4e00（\u7b2c 06 \u7ae0）\u5e76\u4e0d\u610f\u5473\u7740\u6bcf\u6b21\u90fd\u4f1a\u8fd4\u56de\u4e0d\u540c\u7684\u89e3：\u4f18\u5316\u5668\u4ece\u79cd\u5b50\u51fa\u53d1"\u5c31\u8fd1"\u6536\u655b\u3002\u6362\u7528\u5176\u4ed6\u79cd\u5b50，\u53ef\u80fd\u6536\u655b\u5230\u522b\u7684\u7b49\u4ef7\u6784\u578b，\u4e5f\u53ef\u80fd\u5361\u5728\u4e0d\u597d\u7684\u8c37\u91cc\u2014\u2014\u7b2c 09 \u7ae0 9.3 \u8282\u4f1a\u4e13\u95e8\u5b9e\u6d4b initial_position \u5bf9\u6536\u655b\u7684\u5f71\u54cd\u3002<\/p>\n

        7.4.4 \u4ee3\u7801\u9010\u884c\u89e3\u8bfb<\/h4>\n\n\u884c\u8bf4\u660e<\/tr>\n\n\n\n\n\n\n
        \u2460 OriginLink()<\/td>\n\u6bcf\u6761\u94fe\u5fc5\u987b\u4ee5\u5b83\u5f00\u5934，\u4ee3\u8868\u56fa\u5b9a\u7684\u57fa\u5ea7<\/td>\n<\/tr>\n
        \u2461 \u5173\u8282 1<\/td>\norigin_translation = \u57fa\u5ea7\u5230\u5173\u8282 1 \u7684\u504f\u79fb（\u8fd9\u91cc\u662f 0）<\/td>\n<\/tr>\n
        \u2462 \u5173\u8282 2<\/td>\norigin_translation = \u7b2c 1 \u6bb5\u8fde\u6746\u957f\u5ea6 L\u2081 = 0.30<\/td>\n<\/tr>\n
        \u2463 \u672b\u7aef tip<\/td>\norigin_translation = \u7b2c 2 \u6bb5\u8fde\u6746\u957f\u5ea6 L\u2082 = 0.30，\u5fc5\u987b joint_type="fixed"<\/td>\n<\/tr>\n
        \u2464 active_links_mask<\/td>\nOriginLink \u4e0e\u56fa\u5b9a\u672b\u7aef\u90fd\u662f False，\u53ea\u6709\u771f\u5b9e\u5173\u8282\u662f True<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

        \u624b\u7b97\u4ea4\u53c9\u9a8c\u8bc1（\u9a8c\u8bc1\u4f60\u771f\u7684\u7406\u89e3\u4e86）：<\/p>\n

        \u672b\u7aef =<\/span> Ry(<\/span>q1)<\/span> @ (<\/span>t2 +<\/span> Ry(<\/span>q2)<\/span> @ t3)<\/span>
        \n =<\/span> Ry(<\/span>30<\/span>\u00b0)<\/span> @ (<\/span> [<\/span>–<\/span>0.3<\/span>,<\/span>0<\/span>,<\/span>0<\/span>]<\/span> +<\/span> Ry(<\/span>–<\/span>45<\/span>\u00b0)<\/span> @ [<\/span>0<\/span>,<\/span>0<\/span>,<\/span>0.3<\/span>]<\/span> )<\/span>
        \n =<\/span> [<\/span>–<\/span>0.337453<\/span>,<\/span> 0<\/span>,<\/span> 0.439778<\/span>]<\/span> # \u4e0e ikpy \u5b8c\u5168\u4e00\u81f4 \u2713<\/span><\/p>\n

        💡 \u624b\u7b97\u516c\u5f0f\u7684\u542b\u4e49：<\/p>\n

          \n
        • t2 = [-0.3,0,0] \u662f\u5173\u8282 1 \u5230\u5173\u8282 2 \u7684\u504f\u79fb（\u7b2c\u4e00\u6bb5\u8fde\u6746\u957f\u5ea6，\u6cbf -X）<\/li>\n
        • t3 = [0,0,0.3] \u662f\u5173\u8282 2 \u5230\u672b\u7aef\u7684\u504f\u79fb（\u7b2c\u4e8c\u6bb5\u8fde\u6746\u957f\u5ea6，\u6cbf +Z）<\/li>\n
        • Ry(q2) @ t3：\u7b2c\u4e8c\u6bb5\u8fde\u6746\u5148\u88ab\u5173\u8282 2 \u7684\u65cb\u8f6c\u6539\u53d8\u65b9\u5411<\/li>\n
        • t2 + Ry(q2) @ t3：\u4e24\u6bb5\u504f\u79fb\u5728\u5173\u8282 2 \u7684\u5750\u6807\u7cfb\u4e2d\u76f8\u52a0<\/li>\n
        • Ry(q1) @ (…)：\u6574\u4f53\u518d\u88ab\u5173\u8282 1 \u7684\u65cb\u8f6c\u6539\u53d8\u65b9\u5411<\/li>\n<\/ul>\n

          \u8fd9\u5c31\u662f"\u5148\u5e73\u79fb\u540e\u65cb\u8f6c"\u7684\u8bed\u4e49，\u4e0e\u7b2c 05 \u7ae0\u8bb2\u7684 MuJoCo \u8bed\u4e49\u4e00\u81f4\u3002<\/p>\n

          \n

          7.5 \u26a0\ufe0f active_links_mask：ikpy \u6700\u5bb9\u6613\u641e\u9519\u7684\u5730\u65b9<\/h3>\n

          \u8fd9\u4e2a\u53c2\u6570\u51b3\u5b9a\u54ea\u4e9b\u5173\u8282\u4f1a\u88ab IK \u4f18\u5316\u5668\u6539\u53d8\u3002<\/p>\n

          7.5.1 \u5b9e\u6d4b\u884c\u4e3a<\/h4>\n

          links =<\/span> [<\/span>OriginLink(<\/span>)<\/span>,<\/span> shoulder,<\/span> elbow,<\/span> tip]<\/span> # 4 \u4e2a\u5143\u7d20<\/span><\/p>\n

          Chain(<\/span>links=<\/span>links)<\/span> # \u4e0d\u4f20 mask<\/span>
          \n# \u2192 active_links_mask = [True, True, True, True]<\/span><\/p>\n

          Chain(<\/span>links=<\/span>links,<\/span> active_links_mask=<\/span>[<\/span>False<\/span>,<\/span> True<\/span>,<\/span> True<\/span>,<\/span> False<\/span>]<\/span>)<\/span>
          \n# \u2192 active_links_mask = [False, True, True, False] # \u539f\u6837\u4fdd\u7559<\/span><\/p>\n

          Chain(<\/span>links=<\/span>links,<\/span> active_links_mask=<\/span>[<\/span>False<\/span>,<\/span> True<\/span>,<\/span> False<\/span>,<\/span> False<\/span>]<\/span>)<\/span>
          \n# \u2192 active_links_mask = [False, True, False, False] # \u539f\u6837\u4fdd\u7559<\/span><\/p>\n

          \u89c4\u5219\u5f88\u7b80\u5355：<\/p>\n

            \n
          • \u4e0d\u4f20 \u2192 \u5168\u90e8 True（\u5305\u62ec OriginLink \u548c\u56fa\u5b9a\u672b\u7aef）<\/li>\n
          • \u4f20\u4e86 \u2192 \u539f\u6837\u4fdd\u7559，ikpy \u4e0d\u4f1a\u5e2e\u4f60\u6539<\/li>\n<\/ul>\n

            7.5.2 \u4e0d\u4f20\u4f1a\u600e\u6837<\/h4>\n

            fixed \u7c7b\u578b\u7684 link（OriginLink \u548c\u672b\u7aef\u504f\u79fb）\u4e0d\u63d0\u4f9b\u4efb\u4f55\u53d8\u6362，\u5374\u88ab\u6807\u8bb0\u4e3a"\u53ef\u4f18\u5316"，ikpy \u4f1a\u5bf9\u6bcf\u4e00\u4e2a\u53d1\u51fa\u8b66\u544a：<\/p>\n

            UserWarning: Link Base link (index: 0) is of type 'fixed' but set as active
            \nUserWarning: Link tip (index: 3) is of type 'fixed' but set as active<\/p>\n

            \u867d\u7136\u53ea\u662f\u8b66\u544a\u3001\u7ed3\u679c\u901a\u5e38\u4ecd\u6b63\u786e，\u4f46\u5e94\u5f53\u663e\u5f0f\u8bbe\u7f6e\u4ee5\u907f\u514d\u6b67\u4e49\u3002<\/p>\n

            \u26a0\ufe0f \u6ce8\u610f：mask \u7684\u957f\u5ea6\u5fc5\u987b\u4e0e links \u4e00\u81f4，\u5426\u5219 ikpy \u4f1a\u62a5\u9519\u3002\u8fd9\u662f\u6539\u94fe\u7ed3\u6784\u65f6\u6700\u5e38\u89c1\u7684\u4f4e\u7ea7\u9519\u8bef\u3002<\/p>\n

            7.5.3 \u6b63\u786e\u7684\u8bbe\u7f6e\u65b9\u6cd5<\/h4>\n

            # \u6700\u7b80\u5355\u60c5\u5f62：\u9664 OriginLink \u5916\u5168\u662f\u771f\u5b9e\u5173\u8282（\u6700\u540e\u4e00\u4e2a link \u4e0d\u662f\u56fa\u5b9a\u504f\u79fb）<\/span>
            \nn =<\/span> len<\/span>(<\/span>links)<\/span>
            \nactive_links_mask =<\/span> [<\/span>False<\/span>]<\/span> +<\/span> [<\/span>True<\/span>]<\/span> *<\/span> (<\/span>n –<\/span> 1<\/span>)<\/span><\/p>\n

            \u66f4\u7cbe\u7ec6\u7684\u60c5\u51b5（\u6bd4\u5982\u672b\u7aef\u6709\u4e2a\u56fa\u5b9a\u7684\u5939\u722a\u8fde\u6746）\u2014\u2014\u5224\u65ad\u4f9d\u636e\u662f"\u8fd9\u4e2a link \u6709\u6ca1\u6709\u5173\u8282"，\u800c\u4e0d\u662f\u5b83\u7684\u4f4d\u7f6e：<\/p>\n

            # OriginLink + 6 \u4e2a\u5173\u8282 + 2 \u4e2a\u56fa\u5b9a\u8fde\u6746（mount \/ ee）<\/span>
            \nactive_links_mask =<\/span> [<\/span>False<\/span>]<\/span> +<\/span> [<\/span>True<\/span>]<\/span> *<\/span> 6<\/span> +<\/span> [<\/span>False<\/span>,<\/span> False<\/span>]<\/span><\/p>\n

            📌 \u89c4\u5f8b：\u6709\u65cb\u8f6c\/\u79fb\u52a8\u5173\u8282\u7684 link \u4e3a True，OriginLink \u4e0e\u672b\u7aef\u56fa\u5b9a link \u4e3a False\u3002<\/p>\n

              \n
            • OriginLink \u2192 \u6c38\u8fdc False<\/li>\n
            • \u771f\u5b9e\u5173\u8282 \u2192 True<\/li>\n
            • \u56fa\u5b9a\u504f\u79fb\u8fde\u6746（joint_type="fixed"）\u2192 False<\/li>\n<\/ul>\n

              \u26a0\ufe0f \u6240\u4ee5\u672c\u7ae0\u7684 4 \u5143\u7d20\u94fe（OriginLink + 2 \u5173\u8282 + \u672b\u7aef\u56fa\u5b9a）\u5e94\u5199 [False, True, True, False]\u2014\u2014\u4e0d\u80fd\u7167\u642c\u4e0a\u9762\u7684\u901a\u5f0f [False] + [True] * (n – 1)，\u5426\u5219\u672b\u7aef\u56fa\u5b9a link \u4f1a\u88ab\u8bef\u8bbe\u4e3a True \u5e76\u89e6\u53d1\u8b66\u544a\u3002<\/p>\n

              7.5.4 \u26a0\ufe0f \u56fa\u5b9a\u8fde\u6746\u8fd8\u5fc5\u987b\u5199 joint_type="fixed"<\/h4>\n

              \u672c\u4e66\u5b9e\u6d4b\u8e29\u5751（\u7b2c 2 \u53f7）：\u5982\u679c\u4f60\u521b\u5efa\u4e86\u4e00\u4e2a\u65e2\u65e0 rotation \u4e5f\u65e0 translation \u7684\u8fde\u6746（\u7eaf\u504f\u79fb），\u5fc5\u987b\u663e\u5f0f\u5199 joint_type="fixed"：<\/p>\n

              # \u274c \u9519\u8bef：\u9ed8\u8ba4 joint_type='revolute'，\u4f46\u6ca1\u6709 rotation \u8f74<\/span>
              \nURDFLink(<\/span>name=<\/span>"ee"<\/span>,<\/span> origin_translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0.045<\/span>]<\/span>,<\/span>
              \n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>None<\/span>)<\/span>
              \n# ValueError: Joint type is 'revolute' but rotation axis = False and translation axis = False<\/span><\/p>\n

              # \u2705 \u6b63\u786e<\/span>
              \nURDFLink(<\/span>name=<\/span>"ee"<\/span>,<\/span> origin_translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0.045<\/span>]<\/span>,<\/span>
              \n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>None<\/span>,<\/span> joint_type=<\/span>"fixed"<\/span>)<\/span><\/p>\n

              💡 \u4e3a\u4ec0\u4e48 ikpy \u4e0d\u81ea\u52a8\u68c0\u6d4b：\u56e0\u4e3a rotation=None \u4e5f\u53ef\u80fd\u662f"\u7528\u6237\u5fd8\u4e86\u8bbe\u7f6e"，\u800c\u4e0d\u662f"\u8fd9\u662f\u4e00\u4e2a\u56fa\u5b9a\u8fde\u6746"\u3002ikpy \u9009\u62e9\u62a5\u9519\u800c\u4e0d\u662f\u731c\u6d4b，\u907f\u514d\u9759\u9ed8\u9519\u8bef\u3002\u6240\u4ee5\u4f60\u5fc5\u987b\u663e\u5f0f\u58f0\u660e joint_type="fixed"\u3002<\/p>\n

              \n

              7.6 \u5173\u8282\u9650\u4f4d：bounds<\/h3>\n

              bounds \u662f\u4e00\u4e2a (lower, upper) \u5143\u7ec4，ikpy \u4f1a\u628a\u5b83\u4f20\u7ed9\u5e95\u5c42\u4f18\u5316\u5668（scipy.optimize.least_squares \u7684 bounds \u53c2\u6570），\u4fdd\u8bc1\u89e3\u4e00\u5b9a\u5728\u9650\u4f4d\u5185\u3002<\/p>\n

              URDFLink(<\/span>name=<\/span>"j2"<\/span>,<\/span>
              \n origin_translation=<\/span>[<\/span>–<\/span>0.06<\/span>,<\/span> 0.0<\/span>,<\/span> 0.144<\/span>]<\/span>,<\/span>
              \n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
              \n rotation=<\/span>[<\/span>0<\/span>,<\/span> 1<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
              \n bounds=<\/span>(<\/span>–<\/span>2.705260<\/span>,<\/span> 1.169371<\/span>)<\/span>)<\/span> # \u672c\u4e66 6 \u8f74\u81c2\u7684 J2 \u9650\u4f4d（\u5f27\u5ea6）<\/span><\/p>\n

              bounds \u7684\u5de5\u4f5c\u539f\u7406：<\/p>\n

              scipy.optimize.least_squares(
              \n fun=residual_function, # \u672b\u7aef\u8bef\u5dee
              \n x0=initial_position, # \u521d\u59cb\u5173\u8282\u89d2
              \n bounds=(q_min, q_max), # \u2190 ikpy \u628a bounds \u4f20\u8fd9\u91cc
              \n …
              \n)<\/p>\n

              scipy \u7684 least_squares \u4f1a\u5728\u4f18\u5316\u8fc7\u7a0b\u4e2d\u81ea\u52a8\u628a\u89e3\u9650\u5236\u5728 bounds \u8303\u56f4\u5185，\u4f7f\u7528\u7684\u662f"\u4fe1\u4efb\u533a\u57df\u53cd\u5c04\u7b97\u6cd5"（Trust Region Reflective）\u3002<\/p>\n

              \u5b9e\u6d4b\u9a8c\u8bc1（2 \u8fde\u6746：L\u2081=0.30\u3001L\u2082=0.25，\u8fde\u6746\u6cbf -X \u4f38\u5c55\u3001\u7ed5 Y \u8f6c，\u5728 XZ \u5e73\u9762\u8fd0\u52a8；J2 \u9650\u4f4d \u00b11.5）：<\/p>\n

              \u5404\u5173\u8282 bounds: [(-inf, inf), (-3.14, 3.14), (-1.5, 1.5), (-inf, inf)]
              \n(\u8fde\u6746\u6cbf -X，\u7ed5 Y \u8f6c -> \u5728 XZ \u5e73\u9762\u8fd0\u52a8；\u603b\u81c2\u957f 0.30+0.25 = 0.55)<\/p>\n

              \u76ee\u6807 [-0.2, 0.0, 0.45] -> \u89e3=[1.2693 1.1593] \u672b\u7aef=[-0.2 0. 0.45] \u8bef\u5dee=1.16e-08
              \n \u76ee\u6807 [-0.1, 0.0, 0.5] -> \u89e3=[1.7647 0.4111] \u672b\u7aef=[-0.1 0. 0.5] \u8bef\u5dee=1.70e-09
              \n \u76ee\u6807 [-0.35, 0.0, -0.2] -> \u89e3=[-0.6603 -1.5 ] \u672b\u7aef=[-0.39797 0. -0.39181] \u8bef\u5dee=1.98e-01 \u2605\u649e\u9650\u4f4d<\/p>\n

              \u524d\u4e24\u4e2a\u76ee\u6807\u5728\u9650\u4f4d\u5185\u53ef\u8fbe，\u8bef\u5dee\u5728 1e-8 \u91cf\u7ea7；\u7b2c 3 \u4e2a\u76ee\u6807\u7684 j2 \u88ab\u6b7b\u6b7b\u94b3\u5728 -1.5（\u9650\u4f4d\u503c），\u672b\u7aef\u56e0\u6b64\u7559\u4e0b\u4e86\u660e\u663e\u6b8b\u5dee（19.8 cm）\u3002<\/p>\n

              💡 \u9650\u4f4d vs \u53ef\u8fbe\u6027：\u5982\u679c\u76ee\u6807\u9700\u8981\u8d85\u51fa\u9650\u4f4d\u624d\u80fd\u5230\u8fbe，ikpy \u4e0d\u4f1a\u62a5\u9519，\u800c\u662f\u8fd4\u56de\u4e00\u4e2a\u5c3d\u529b\u800c\u4e3a\u7684\u89e3（\u8d34\u5728\u9650\u4f4d\u4e0a，\u672b\u7aef\u6709\u6b8b\u5dee）\u3002\u8fd9\u548c\u7b2c 06 \u7ae0\u81ea\u7814 DLS \u7684\u884c\u4e3a\u4e00\u81f4\u3002<\/p>\n

              \u5de5\u7a0b\u505a\u6cd5：\u8c03\u7528\u5b8c IK \u540e，\u4e00\u5b9a\u8981\u81ea\u5df1\u68c0\u67e5 \u2016FK(q) – target\u2016\u3002ikpy \u4e0d\u4f1a\u544a\u8bc9\u4f60\u5b83\u5931\u8d25\u4e86\u3002<\/p>\n

              \n

              7.7 DHLink：\u76f4\u63a5\u7528 DH \u53c2\u6570\u5efa\u94fe<\/h3>\n

              \u7b2c 05 \u7ae0\u6211\u4eec\u624b\u5199\u4e86 DH \u53d8\u6362\u77e9\u9635\u3002ikpy \u5185\u7f6e\u4e86 DHLink，\u53c2\u6570\u5c31\u662f DH \u56db\u53c2\u6570：<\/p>\n

              DHLink(<\/span>name=<\/span>None<\/span>,<\/span> d=<\/span>0<\/span>,<\/span> a=<\/span>0<\/span>,<\/span> alpha=<\/span>0<\/span>,<\/span> theta=<\/span>0<\/span>,<\/span> bounds=<\/span>None<\/span>,<\/span>
              \n use_symbolic_matrix=<\/span>True<\/span>,<\/span> length=<\/span>0<\/span>)<\/span><\/p>\n

              DHLink \u53c2\u6570\u4e0e\u7b2c 05 \u7ae0 DH \u53c2\u6570\u7684\u5bf9\u5e94：<\/p>\n\nDHLink \u53c2\u6570\u7b2c 05 \u7ae0 DH \u53c2\u6570\u542b\u4e49<\/tr>\n\n\n\n\n\n
              d<\/td>\nd\u1d62<\/td>\n\u8fde\u6746\u504f\u79fb（\u79fb\u52a8\u5173\u8282\u7684\u53d8\u91cf）<\/td>\n<\/tr>\n
              a<\/td>\na\u1d62<\/td>\n\u8fde\u6746\u957f\u5ea6<\/td>\n<\/tr>\n
              alpha<\/td>\n\u03b1\u1d62<\/td>\n\u8fde\u6746\u626d\u89d2<\/td>\n<\/tr>\n
              theta<\/td>\n\u03b8\u1d62<\/td>\n\u5173\u8282\u89d2（\u65cb\u8f6c\u5173\u8282\u7684\u53d8\u91cf）<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

              \u7528\u5b83\u91cd\u5efa\u7b2c 05 \u7ae0\u90a3\u53f0 3 \u81ea\u7531\u5ea6\u81c2\u7684\u524d\u4e24\u8282：<\/p>\n

              chain =<\/span> Chain(<\/span>name=<\/span>"dh_arm"<\/span>,<\/span> links=<\/span>[<\/span>
              \n OriginLink(<\/span>)<\/span>,<\/span>
              \n DHLink(<\/span>name=<\/span>"j1"<\/span>,<\/span> d=<\/span>0.20<\/span>,<\/span> a=<\/span>0.30<\/span>,<\/span> alpha=<\/span>0.0<\/span>,<\/span> theta=<\/span>0.0<\/span>)<\/span>,<\/span>
              \n DHLink(<\/span>name=<\/span>"j2"<\/span>,<\/span> d=<\/span>0.00<\/span>,<\/span> a=<\/span>0.25<\/span>,<\/span> alpha=<\/span>0.0<\/span>,<\/span> theta=<\/span>0.0<\/span>)<\/span>,<\/span>
              \n]<\/span>,<\/span> active_links_mask=<\/span>[<\/span>False<\/span>,<\/span> True<\/span>,<\/span> True<\/span>]<\/span>)<\/span><\/p>\n

              T =<\/span> chain.<\/span>forward_kinematics(<\/span>[<\/span>0<\/span>,<\/span> np.<\/span>radians(<\/span>30<\/span>)<\/span>,<\/span> np.<\/span>radians(<\/span>45<\/span>)<\/span>]<\/span>)<\/span><\/p>\n

              \u5b9e\u6d4b\u5bf9\u6bd4（q\u2081=30\u00b0, q\u2082=45\u00b0）：<\/p>\n

              ikpy DHLink = [0.324512 0.391481 0.200000]
              \n\u7b2c 05 \u7ae0\u624b\u5199 DH = [0.324512 0.391481 0.200000]
              \n\u4e00\u81f4: True<\/p>\n

              \u2705 \u8fd9\u8bc1\u660e\u4e86\u7b2c 05 \u7ae0\u6211\u4eec\u624b\u5199\u7684 DH \u53d8\u6362\u77e9\u9635\u4e0e ikpy \u7684\u5b9e\u73b0\u5b8c\u5168\u7b49\u4ef7\u3002<\/p>\n

              \u4ec0\u4e48\u65f6\u5019\u7528 DHLink vs URDFLink：<\/p>\n\n\u573a\u666f\u7528\u54ea\u4e2a<\/tr>\n\n\n\n\n
              \u624b\u4e0a\u6709 DH \u53c2\u6570\u8868（\u8bba\u6587\u3001\u8001\u624b\u518c）<\/td>\nDHLink<\/td>\n<\/tr>\n
              \u624b\u4e0a\u6709 URDF \/ MJCF \u6a21\u578b<\/td>\nURDFLink（\u53c2\u6570\u53ef\u76f4\u63a5\u6284）<\/td>\n<\/tr>\n
              \u673a\u68b0\u81c2\u8f74\u7cfb\u6df7\u5408\u591a\u6837（\u5982\u672c\u4e66 J1 \u7ed5 Z\u3001J2\/J3 \u7ed5 Y\u3001J4\/J6 \u7ed5 X）<\/td>\nURDFLink（\u66f4\u76f4\u89c2）<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

              \u672c\u4e66\u5b9e\u6218\u7528 URDFLink，\u56e0\u4e3a\u53c2\u6570\u80fd\u4ece MJCF \u76f4\u63a5\u6284\u3002<\/p>\n

              \u26a0\ufe0f \u6ce8\u610f：DHLink \u4e0d\u652f\u6301\u79fb\u52a8\u5173\u8282\u3002\u5982\u679c\u4f60\u7684\u673a\u68b0\u81c2\u6709\u79fb\u52a8\u5173\u8282（prismatic），\u9700\u8981\u7528 URDFLink(translation=\u8f74\u65b9\u5411, joint_type="prismatic")\u3002\u8fd9\u662f\u56e0\u4e3a DH \u53c2\u6570\u6cd5\u4e2d\u79fb\u52a8\u5173\u8282\u7684\u53d8\u91cf\u662f d，\u4f46 DHLink \u7684\u5b9e\u73b0\u628a theta \u4f5c\u4e3a\u552f\u4e00\u53d8\u91cf（\u65cb\u8f6c\u5173\u8282），\u4e0d\u652f\u6301\u628a d \u4f5c\u4e3a\u53d8\u91cf\u3002<\/p>\n

              \n

              7.8 ikpy \u7684"\u9ed1\u7bb1"\u91cc\u662f\u4ec0\u4e48<\/h3>\n

              \u77e5\u9053 ikpy \u5185\u90e8\u600e\u4e48\u505a，\u624d\u80fd\u8c03\u597d\u5b83\u3002<\/p>\n

              7.8.1 FK：\u7b26\u53f7\u77e9\u9635 + \u6570\u503c\u4ee3\u5165<\/h4>\n

              ikpy \u7528 sympy \u4e3a\u6bcf\u4e2a\u8fde\u6746\u63a8\u5bfc\u51fa\u7b26\u53f7\u5316\u7684\u53d8\u6362\u77e9\u9635，\u7136\u540e\u5728 forward_kinematics \u65f6\u628a\u5177\u4f53\u89d2\u5ea6\u4ee3\u8fdb\u53bb\u3002<\/p>\n

                \n
              • \u4f18\u70b9：\u63a8\u5bfc\u4e00\u6b21，\u53cd\u590d\u4f7f\u7528；\u652f\u6301\u4efb\u610f\u8f74\u3001\u4efb\u610f\u6b27\u62c9\u89d2\u8868\u793a\u3002<\/li>\n
              • \u7f3a\u70b9：\u6bd4\u7eaf\u6570\u503c\u6162（\u8fd9\u4e5f\u662f\u4e3a\u4ec0\u4e48\u7b2c 06 \u7ae0\u81ea\u7814 DLS \u7528 MuJoCo \u7684 mj_jacSite \u4f1a\u5feb\u7ea6\u4e00\u4e2a\u6570\u91cf\u7ea7，\u89c1 8.7 \u5b9e\u6d4b）\u3002<\/li>\n<\/ul>\n

                FK \u5185\u90e8\u6d41\u7a0b：<\/p>\n

                \u6784\u5efa Chain \u65f6：
                \n \u5bf9\u6bcf\u4e2a Link：
                \n \u7528 sympy \u63a8\u5bfc\u7b26\u53f7\u53d8\u6362\u77e9\u9635 T_i(\u03b8_i)
                \n \u4fdd\u5b58\u7b26\u53f7\u8868\u8fbe\u5f0f<\/p>\n

                \u8c03\u7528 forward_kinematics(q) \u65f6：
                \n \u628a q \u7684\u6570\u503c\u4ee3\u5165\u7b26\u53f7\u8868\u8fbe\u5f0f
                \n \u8ba1\u7b97 T_total = T_1(\u03b81) \u00b7 T_2(\u03b82) \u00b7 … \u00b7 T_n(\u03b8n)
                \n \u8fd4\u56de T_total<\/p>\n

                7.8.2 IK：scipy \u6570\u503c\u4f18\u5316<\/h4>\n

                ikpy \u7684 IK \u672c\u8d28\u662f\u6700\u5c0f\u5316\u672b\u7aef\u8bef\u5dee：<\/p>\n

                minimize \u2016FK(q) – target\u2016\u00b2
                \n q
                \ns.t. q_min \u2264 q \u2264 q_max<\/p>\n

                \u7528\u7684\u662f scipy.optimize.least_squares（\u9ed8\u8ba4）\u6216 scipy.optimize.minimize\u3002<\/p>\n

                \u4e24\u79cd\u4f18\u5316\u5668：<\/p>\n

                chain.<\/span>inverse_kinematics(<\/span>target_position=<\/span>.<\/span>.<\/span>.<\/span>,<\/span> optimizer=<\/span>"least_squares"<\/span>)<\/span> # \u9ed8\u8ba4<\/span>
                \nchain.<\/span>inverse_kinematics(<\/span>target_position=<\/span>.<\/span>.<\/span>.<\/span>,<\/span> optimizer=<\/span>"scalar"<\/span>)<\/span> # \u65e7\u7248\u9ed8\u8ba4<\/span><\/p>\n

                least_squares \u901a\u5e38\u66f4\u5feb\u66f4\u7a33，\u4fdd\u6301\u9ed8\u8ba4\u5373\u53ef\u3002<\/p>\n

                💡 least_squares vs scalar \u7684\u533a\u522b：<\/p>\n

                  \n
                • least_squares：\u4e13\u95e8\u9488\u5bf9\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898\u4f18\u5316，\u4f7f\u7528\u4fe1\u4efb\u533a\u57df\u7b97\u6cd5，\u6536\u655b\u66f4\u5feb<\/li>\n
                • scalar：\u901a\u7528\u7684 scipy.optimize.minimize，\u4f7f\u7528 BFGS \u6216\u5176\u4ed6\u901a\u7528\u7b97\u6cd5，\u66f4\u6162\u4f46\u66f4\u901a\u7528<\/li>\n<\/ul>\n

                  ikpy 4.0 \u9ed8\u8ba4\u7528 least_squares，\u8fd9\u662f\u6b63\u786e\u7684\u9009\u62e9\u2014\u2014IK \u672c\u8d28\u5c31\u662f\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898\u3002<\/p>\n

                  7.8.3 \u4e0e\u81ea\u7814 DLS \u7684\u672c\u8d28\u533a\u522b<\/h4>\n\n\u81ea\u7814 DLS（\u7b2c 06 \u7ae0）ikpy<\/tr>\n\n\n\n\n\n\n
                  \u65b9\u6cd5<\/td>\n\u6cbf\u96c5\u53ef\u6bd4\u8fed\u4ee3\u903c\u8fd1<\/td>\n\u5168\u5c40\u4f18\u5316（\u6700\u5c0f\u5316\u8bef\u5dee）<\/td>\n<\/tr>\n
                  \u6bcf\u6b65<\/td>\n\u7528\u96c5\u53ef\u6bd4\u4fe1\u606f，\u65b9\u5411\u660e\u786e<\/td>\n\u6570\u503c\u8bd5\u63a2，\u9760 scipy \u6536\u655b<\/td>\n<\/tr>\n
                  \u901f\u5ea6<\/td>\n\u5feb（\u53ef\u7528\u89e3\u6790\u96c5\u53ef\u6bd4）<\/td>\n\u8f83\u6162<\/td>\n<\/tr>\n
                  \u7075\u6d3b\u6027<\/td>\n\u56fa\u5b9a\u516c\u5f0f<\/td>\n\u53ef\u52a0\u6b63\u5219\u9879\u3001\u59ff\u6001\u7ea6\u675f\u7b49<\/td>\n<\/tr>\n
                  \u591a\u89e3<\/td>\n\u7531\u521d\u59cb\u503c\u51b3\u5b9a<\/td>\n\u7531\u521d\u59cb\u503c + \u6b63\u5219\u9879\u51b3\u5b9a<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                  7.8.4 ikpy \u7b26\u53f7\u8ba1\u7b97\u7684\u6df1\u5165\u7406\u89e3<\/h4>\n

                  ikpy \u7684\u6838\u5fc3\u7ade\u4e89\u529b\u5728\u4e8e\u5b83\u7684\u7b26\u53f7\u5316\u53d8\u6362\u77e9\u9635\u63a8\u5bfc\u3002\u8ba9\u6211\u4eec\u6df1\u5165\u7406\u89e3\u8fd9\u662f\u5982\u4f55\u5de5\u4f5c\u7684\u3002<\/p>\n

                  \u6784\u5efa Chain \u65f6\u53d1\u751f\u4e86\u4ec0\u4e48：<\/p>\n

                  chain =<\/span> Chain(<\/span>name=<\/span>"arm"<\/span>,<\/span> links=<\/span>[<\/span>OriginLink(<\/span>)<\/span>,<\/span> URDFLink(<\/span>.<\/span>.<\/span>.<\/span>)<\/span>,<\/span> .<\/span>.<\/span>.<\/span>]<\/span>)<\/span><\/p>\n

                  \u5f53\u8fd9\u884c\u4ee3\u7801\u6267\u884c\u65f6，ikpy \u5185\u90e8\u4f1a：<\/p>\n

                • \u4e3a\u6bcf\u4e2a\u8fde\u6746\u521b\u5efa\u7b26\u53f7\u53d8\u91cf：\u03b81, \u03b82, …, \u03b8n（sympy \u7684 Symbol \u5bf9\u8c61）<\/li>\n
                • \u63a8\u5bfc\u6bcf\u4e2a\u8fde\u6746\u7684\u7b26\u53f7\u53d8\u6362\u77e9\u9635：\u7528 origin_translation\u3001origin_orientation\u3001rotation \u7b49\u53c2\u6570，\u6784\u9020\u51fa\u5305\u542b\u7b26\u53f7\u53d8\u91cf\u7684 4\u00d74 \u77e9\u9635<\/li>\n
                • \u8fde\u4e58\u5f97\u5230\u603b\u53d8\u6362\u77e9\u9635：T_total = T1(\u03b81) \u00b7 T2(\u03b82) \u00b7 … \u00b7 Tn(\u03b8n)，\u7ed3\u679c\u662f\u4e00\u4e2a\u5305\u542b\u6240\u6709\u7b26\u53f7\u53d8\u91cf\u7684 4\u00d74 \u77e9\u9635<\/li>\n
                • \u7f16\u8bd1\u4e3a\u6570\u503c\u51fd\u6570（\u53ef\u9009）：\u7528 sympy.lambdify \u628a\u7b26\u53f7\u8868\u8fbe\u5f0f\u7f16\u8bd1\u4e3a\u9ad8\u6548\u7684\u6570\u503c\u51fd\u6570<\/li>\n

                  \u7b26\u53f7\u77e9\u9635\u7684\u6837\u5b50（\u7b80\u5316\u7684 2 \u8fde\u6746\u793a\u4f8b）：<\/p>\n

                  T_total = [
                  \n [cos(\u03b81+\u03b82), -sin(\u03b81+\u03b82), 0, L1\u00b7cos(\u03b81) + L2\u00b7cos(\u03b81+\u03b82)],
                  \n [sin(\u03b81+\u03b82), cos(\u03b81+\u03b82), 0, L1\u00b7sin(\u03b81) + L2\u00b7sin(\u03b81+\u03b82)],
                  \n [0, 0, 1, 0],
                  \n [0, 0, 0, 1]
                  \n]<\/p>\n

                  \u8fd9\u662f\u4e00\u4e2a\u7b26\u53f7\u8868\u8fbe\u5f0f\u2014\u2014\u03b81 \u548c \u03b82 \u4e0d\u662f\u6570\u5b57，\u800c\u662f sympy \u7684\u7b26\u53f7\u53d8\u91cf\u3002\u5f53\u8c03\u7528 forward_kinematics([0, 0.5, 0.3, 0]) \u65f6，ikpy \u628a \u03b81=0.5, \u03b82=0.3 \u4ee3\u5165\u7b26\u53f7\u8868\u8fbe\u5f0f，\u5f97\u5230\u6570\u503c\u7ed3\u679c\u3002<\/p>\n

                  \u4e3a\u4ec0\u4e48\u7b26\u53f7\u8ba1\u7b97\u6bd4\u7eaf\u6570\u503c\u6162：<\/p>\n\n\u64cd\u4f5c\u7eaf\u6570\u503c（NumPy）\u7b26\u53f7\u8ba1\u7b97（sympy）<\/tr>\n\n\n\n\n\n
                  \u77e9\u9635\u4e58\u6cd5<\/td>\nC \u5b9e\u73b0，\u5fae\u79d2\u7ea7<\/td>\nPython \u7b26\u53f7\u8fd0\u7b97，\u6beb\u79d2\u7ea7<\/td>\n<\/tr>\n
                  \u4e09\u89d2\u51fd\u6570<\/td>\n\u76f4\u63a5\u8c03\u7528 np.cos<\/td>\n\u7b26\u53f7\u5316\u7b80\u540e\u518d\u4ee3\u5165<\/td>\n<\/tr>\n
                  \u5185\u5b58<\/td>\n\u5b58\u50a8\u6570\u503c\u6570\u7ec4<\/td>\n\u5b58\u50a8\u7b26\u53f7\u8868\u8fbe\u5f0f\u6811<\/td>\n<\/tr>\n
                  \u4f18\u70b9<\/td>\n\u5feb<\/td>\n\u7075\u6d3b\u3001\u53ef\u6c42\u5bfc\u3001\u53ef\u5316\u7b80<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                  \u7b26\u53f7\u8ba1\u7b97\u7684\u4f18\u52bf：<\/p>\n

                • \u81ea\u52a8\u6c42\u96c5\u53ef\u6bd4：\u5bf9\u7b26\u53f7\u77e9\u9635\u6c42\u504f\u5bfc，\u53ef\u4ee5\u5f97\u5230\u7cbe\u786e\u7684\u89e3\u6790\u96c5\u53ef\u6bd4，\u4e0d\u9700\u8981\u6570\u503c\u5dee\u5206<\/li>\n
                • \u652f\u6301\u4efb\u610f\u8f74\/\u6b27\u62c9\u89d2：\u7b26\u53f7\u8ba1\u7b97\u4e0d\u5173\u5fc3\u5177\u4f53\u6570\u503c，\u4efb\u4f55\u65cb\u8f6c\u8868\u793a\u90fd\u80fd\u5904\u7406<\/li>\n
                • \u53ef\u5316\u7b80：sympy \u53ef\u4ee5\u81ea\u52a8\u5316\u7b80\u590d\u6742\u7684\u4e09\u89d2\u8868\u8fbe\u5f0f，\u51cf\u5c11\u8ba1\u7b97\u91cf<\/li>\n
                • \u53ef\u5bfc\u51fa：\u7b26\u53f7\u8868\u8fbe\u5f0f\u53ef\u4ee5\u5bfc\u51fa\u4e3a C \u4ee3\u7801\u6216 MATLAB \u4ee3\u7801，\u7528\u4e8e\u5d4c\u5165\u5f0f\u7cfb\u7edf<\/li>\n

                  💡 ikpy \u7684 use_symbolic_matrix \u53c2\u6570：URDFLink \u6709\u4e00\u4e2a use_symbolic_matrix=True \u53c2\u6570\u3002\u5982\u679c\u8bbe\u4e3a False，ikpy \u4f1a\u7528\u7eaf\u6570\u503c\u8ba1\u7b97（\u66f4\u5feb，\u4f46\u7075\u6d3b\u6027\u964d\u4f4e）\u3002\u9ed8\u8ba4\u662f True，\u56e0\u4e3a\u7b26\u53f7\u8ba1\u7b97\u7684\u7075\u6d3b\u6027\u66f4\u91cd\u8981\u3002<\/p>\n

                  \u5728\u6027\u80fd\u654f\u611f\u7684\u573a\u666f\u4e0b，\u53ef\u4ee5\u5c1d\u8bd5\u8bbe\u4e3a False，\u4f46\u8981\u6ce8\u610f\u67d0\u4e9b\u529f\u80fd（\u5982\u81ea\u52a8\u96c5\u53ef\u6bd4）\u53ef\u80fd\u4e0d\u53ef\u7528\u3002<\/p>\n

                  7.8.5 \u4ece\u96f6\u6784\u5efa Chain \u7684\u5b8c\u6574\u793a\u4f8b<\/h4>\n

                  \u8ba9\u6211\u4eec\u901a\u8fc7\u4e00\u4e2a\u5b8c\u6574\u7684\u793a\u4f8b，\u7406\u89e3\u6784\u5efa Chain \u7684\u6bcf\u4e00\u6b65\u3002<\/p>\n

                  \u76ee\u6807：\u6784\u5efa\u4e00\u4e2a 3 \u81ea\u7531\u5ea6\u673a\u68b0\u81c2\u2014\u2014\u57fa\u5ea7\u65cb\u8f6c（J1，\u7ed5 Z）\u3001\u5927\u81c2\u4fef\u4ef0（J2，\u7ed5 Y）\u3001\u672b\u7aef\u504f\u79fb（\u56fa\u5b9a）\u3002<\/p>\n

                  \u7b2c 1 \u6b65：\u786e\u5b9a\u8fd0\u52a8\u5b66\u53c2\u6570<\/p>\n

                  \u57fa\u5ea7 \u2192 J1: \u504f\u79fb (0, 0, 0.1)，\u7ed5 Z \u65cb\u8f6c
                  \nJ1 \u2192 J2: \u504f\u79fb (0, 0, 0.3)，\u7ed5 Y \u65cb\u8f6c
                  \nJ2 \u2192 \u672b\u7aef: \u504f\u79fb (0, 0, 0.25)，\u56fa\u5b9a<\/p>\n

                  \u7b2c 2 \u6b65：\u786e\u5b9a\u5173\u8282\u9650\u4f4d<\/p>\n

                  J1: \u00b1180\u00b0 (\u00b1\u03c0)
                  \nJ2: -90\u00b0 ~ +90\u00b0 (-\u03c0\/2 ~ \u03c0\/2)<\/p>\n

                  \u7b2c 3 \u6b65：\u7f16\u5199\u4ee3\u7801<\/p>\n

                  import<\/span> numpy as<\/span> np
                  \nfrom<\/span> ikpy.<\/span>chain import<\/span> Chain
                  \nfrom<\/span> ikpy.<\/span>link import<\/span> OriginLink,<\/span> URDFLink<\/p>\n

                  # \u6784\u5efa Chain：4 \u4e2a\u5143\u7d20（OriginLink + 2\u5173\u8282 + \u672b\u7aef\u56fa\u5b9a）<\/span>
                  \nchain =<\/span> Chain(<\/span>
                  \n name=<\/span>"3dof_arm"<\/span>,<\/span>
                  \n links=<\/span>[<\/span>
                  \n # \u2460 \u57fa\u5ea7（\u56fa\u5b9a）<\/span>
                  \n OriginLink(<\/span>)<\/span>,<\/span>
                  \n # \u2461 J1：\u57fa\u5ea7\u65cb\u8f6c，\u7ed5 Z \u8f74<\/span>
                  \n URDFLink(<\/span>
                  \n name=<\/span>"j1"<\/span>,<\/span>
                  \n origin_translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0.1<\/span>]<\/span>,<\/span> # \u57fa\u5ea7\u9ad8 0.1m<\/span>
                  \n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
                  \n rotation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 1<\/span>]<\/span>,<\/span> # \u7ed5 Z \u8f74<\/span>
                  \n bounds=<\/span>(<\/span>–<\/span>np.<\/span>pi,<\/span> np.<\/span>pi)<\/span>,<\/span> # \u00b1180\u00b0<\/span>
                  \n )<\/span>,<\/span>
                  \n # \u2462 J2：\u5927\u81c2\u4fef\u4ef0，\u7ed5 Y \u8f74<\/span>
                  \n URDFLink(<\/span>
                  \n name=<\/span>"j2"<\/span>,<\/span>
                  \n origin_translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0.3<\/span>]<\/span>,<\/span> # \u5927\u81c2\u957f 0.3m<\/span>
                  \n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
                  \n rotation=<\/span>[<\/span>0<\/span>,<\/span> 1<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> # \u7ed5 Y \u8f74<\/span>
                  \n bounds=<\/span>(<\/span>–<\/span>np.<\/span>pi\/<\/span>2<\/span>,<\/span> np.<\/span>pi\/<\/span>2<\/span>)<\/span>,<\/span> # \u00b190\u00b0<\/span>
                  \n )<\/span>,<\/span>
                  \n # \u2463 \u672b\u7aef\u56fa\u5b9a\u504f\u79fb<\/span>
                  \n URDFLink(<\/span>
                  \n name=<\/span>"tip"<\/span>,<\/span>
                  \n origin_translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0.25<\/span>]<\/span>,<\/span> # \u672b\u7aef\u957f 0.25m<\/span>
                  \n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
                  \n rotation=<\/span>None<\/span>,<\/span>
                  \n joint_type=<\/span>"fixed"<\/span>,<\/span> # \u5fc5\u987b\u663e\u5f0f\u58f0\u660e！<\/span>
                  \n )<\/span>,<\/span>
                  \n ]<\/span>,<\/span>
                  \n # active_links_mask：\u57fa\u5ea7=False, J1=True, J2=True, \u672b\u7aef=False<\/span>
                  \n active_links_mask=<\/span>[<\/span>False<\/span>,<\/span> True<\/span>,<\/span> True<\/span>,<\/span> False<\/span>]<\/span>,<\/span>
                  \n)<\/span><\/p>\n

                  # \u9a8c\u8bc1\u7ed3\u6784<\/span>
                  \nprint<\/span>(<\/span>"links:"<\/span>,<\/span> [<\/span>l.<\/span>name for<\/span> l in<\/span> chain.<\/span>links]<\/span>)<\/span>
                  \nprint<\/span>(<\/span>"active:"<\/span>,<\/span> chain.<\/span>active_links_mask)<\/span><\/p>\n

                  # FK \u6d4b\u8bd5：J1=30\u00b0, J2=45\u00b0<\/span>
                  \nq_full =<\/span> [<\/span>0<\/span>,<\/span> np.<\/span>radians(<\/span>30<\/span>)<\/span>,<\/span> np.<\/span>radians(<\/span>45<\/span>)<\/span>,<\/span> 0<\/span>]<\/span>
                  \nT =<\/span> chain.<\/span>forward_kinematics(<\/span>q_full)<\/span>
                  \nprint<\/span>(<\/span>"\u672b\u7aef\u4f4d\u7f6e:"<\/span>,<\/span> T[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span>)<\/span><\/p>\n

                  # IK \u6d4b\u8bd5：\u6c42\u89e3\u76ee\u6807\u4f4d\u7f6e [0.3, 0, 0.4]<\/span>
                  \nq_sol =<\/span> chain.<\/span>inverse_kinematics(<\/span>target_position=<\/span>[<\/span>0.3<\/span>,<\/span> 0<\/span>,<\/span> 0.4<\/span>]<\/span>)<\/span>
                  \nprint<\/span>(<\/span>"IK \u89e3(\u5ea6):"<\/span>,<\/span> np.<\/span>degrees(<\/span>q_sol[<\/span>1<\/span>:<\/span>3<\/span>]<\/span>)<\/span>)<\/span><\/p>\n

                  \u9010\u6bb5\u8bb2\u89e3：<\/p>\n\n\u6b65\u9aa4\u8bf4\u660e\u6ce8\u610f\u4e8b\u9879<\/tr>\n\n\n\n\n\n\n\n
                  \u2460 OriginLink()<\/td>\n\u94fe\u7684\u8d77\u70b9，\u4ee3\u8868\u56fa\u5b9a\u57fa\u5ea7<\/td>\n\u5fc5\u987b\u662f\u7b2c\u4e00\u4e2a\u5143\u7d20<\/td>\n<\/tr>\n
                  \u2461 J1 URDFLink<\/td>\n\u7b2c\u4e00\u4e2a\u5173\u8282：\u57fa\u5ea7\u65cb\u8f6c<\/td>\norigin_translation \u662f\u57fa\u5ea7\u5230 J1 \u7684\u504f\u79fb，rotation \u662f\u65cb\u8f6c\u8f74<\/td>\n<\/tr>\n
                  \u2462 J2 URDFLink<\/td>\n\u7b2c\u4e8c\u4e2a\u5173\u8282：\u5927\u81c2\u4fef\u4ef0<\/td>\norigin_translation \u662f J1 \u5230 J2 \u7684\u504f\u79fb（\u5373\u5927\u81c2\u957f\u5ea6）<\/td>\n<\/tr>\n
                  \u2463 tip URDFLink<\/td>\n\u672b\u7aef\u56fa\u5b9a\u504f\u79fb<\/td>\n\u5fc5\u987b\u5199 joint_type="fixed"，\u5426\u5219\u62a5\u9519<\/td>\n<\/tr>\n
                  active_links_mask<\/td>\n\u6807\u8bb0\u54ea\u4e9b\u5173\u8282\u53c2\u4e0e IK<\/td>\n\u957f\u5ea6\u5fc5\u987b\u7b49\u4e8e len(links)，\u57fa\u5ea7\u548c\u56fa\u5b9a\u672b\u7aef\u4e3a False<\/td>\n<\/tr>\n
                  q_full<\/td>\nFK \u7684\u8f93\u5165\u6570\u7ec4<\/td>\n\u957f\u5ea6\u5fc5\u987b\u7b49\u4e8e len(links)，\u56fa\u5b9a link \u586b 0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                  \u5e38\u89c1\u9519\u8bef\u6392\u67e5：<\/p>\n

                  \u9519\u8bef 1: ValueError: Joint type is 'revolute' but rotation axis = False
                  \n\u539f\u56e0: \u672b\u7aef\u56fa\u5b9a\u8fde\u6746\u6ca1\u5199 joint_type="fixed"
                  \n\u89e3\u51b3: \u6dfb\u52a0 joint_type="fixed"<\/p>\n

                  \u9519\u8bef 2: UserWarning: Link Base link (index: 0) is of type 'fixed' but set as active
                  \n\u539f\u56e0: \u6ca1\u8bbe active_links_mask，\u6216\u8bbe\u9519\u4e86
                  \n\u89e3\u51b3: active_links_mask=[False, True, True, False]<\/p>\n

                  \u9519\u8bef 3: \u672b\u7aef\u4f4d\u7f6e\u4e0d\u968f J2 \u53d8\u5316
                  \n\u539f\u56e0: J2 \u540e\u9762\u6ca1\u6709\u672b\u7aef\u504f\u79fb（\u7f3a tip link）
                  \n\u89e3\u51b3: \u6dfb\u52a0\u672b\u7aef\u56fa\u5b9a\u504f\u79fb URDFLink<\/p>\n

                  \u9519\u8bef 4: IK \u7ed3\u679c\u79bb\u76ee\u6807\u5f88\u8fdc
                  \n\u539f\u56e0: \u76ee\u6807\u4e0d\u53ef\u8fbe，\u6216\u649e\u5230\u5173\u8282\u9650\u4f4d
                  \n\u89e3\u51b3: \u68c0\u67e5\u76ee\u6807\u662f\u5426\u5728\u5de5\u4f5c\u7a7a\u95f4\u5185；\u56de\u4ee3 FK \u9a8c\u8bc1\u672b\u7aef\u8bef\u5dee<\/p>\n

                  \n

                  7.9 \u5e38\u89c1\u9519\u8bef\u901f\u67e5<\/h3>\n\n\u62a5\u9519\u539f\u56e0\u89e3\u51b3<\/tr>\n\n\n\n\n\n\n\n\n\n\n
                  No module named 'sympy'<\/td>\n\u7f3a\u4f9d\u8d56<\/td>\npip install sympy<\/td>\n<\/tr>\n
                  unexpected keyword argument 'translation_vector'<\/td>\n\u7528\u4e86 3.x \u7684 API<\/td>\n\u6539\u6210 origin_translation<\/td>\n<\/tr>\n
                  Joint type is 'revolute' but rotation axis = False<\/td>\n\u56fa\u5b9a\u8fde\u6746\u6ca1\u5199\u7c7b\u578b<\/td>\n\u52a0 joint_type="fixed"<\/td>\n<\/tr>\n
                  UserWarning: Link Base link … is of type 'fixed'<\/td>\n\u6ca1\u8bbe active_links_mask<\/td>\n\u8bbe [False] + [True]*(n-1)<\/td>\n<\/tr>\n
                  IK \u7ed3\u679c\u79bb\u76ee\u6807\u5f88\u8fdc<\/td>\n\u76ee\u6807\u8d85\u51fa\u5de5\u4f5c\u7a7a\u95f4，\u6216\u649e\u5230\u5173\u8282\u9650\u4f4d<\/td>\n\u68c0\u67e5\u76ee\u6807\u534a\u5f84\u4e0e bounds；ikpy \u4f1a"\u5c3d\u529b\u800c\u4e3a"\u4e0d\u62a5\u9519<\/td>\n<\/tr>\n
                  \u67d0\u4e2a\u5173\u8282\u600e\u4e48\u52a8，\u672b\u7aef\u90fd\u4e0d\u53d8<\/td>\n\u8be5\u5173\u8282\u540e\u9762\u6ca1\u6709\u504f\u79fb（\u7f3a\u672b\u7aef link），\u6216\u8f6c\u8f74\u4e0e\u8fde\u6746\u540c\u5411<\/td>\n\u89c1 7.4.1 \/ 7.4.2，\u8865\u672b\u7aef\u504f\u79fb\u6216\u6539\u8f6c\u8f74<\/td>\n<\/tr>\n
                  ValueError: Your target must be a 4×4<\/td>\n\u7528\u4e86 inverse_kinematics_frame \u4f46\u76ee\u6807\u4e0d\u662f 4\u00d74<\/td>\n\u7528 inverse_kinematics \u4f20 3 \u7ef4\u4f4d\u7f6e<\/td>\n<\/tr>\n
                  mask \u957f\u5ea6\u62a5\u9519<\/td>\nactive_links_mask \u957f\u5ea6\u4e0e links \u4e0d\u4e00\u81f4<\/td>\n\u4e24\u8005\u957f\u5ea6\u5fc5\u987b\u76f8\u7b49<\/td>\n<\/tr>\n
                  Initial guess is outside of provided bounds<\/td>\ninitial_position \u8d85\u51fa bounds<\/td>\n\u5148 np.clip \u5230\u9650\u4f4d\u5185<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                  \n

                  7.10 \u52a8\u624b\u7ec3<\/h3>\n
                • \n

                  \u6700\u5c0f\u94fe：\u7528 OriginLink + 2 \u4e2a\u5173\u8282 + 1 \u4e2a\u672b\u7aef\u504f\u79fb（L1=L2=0.3）\u5efa\u4e00\u6761 2 \u8fde\u6746\u81c2，\u8ba1\u7b97 q=[30\u00b0, -45\u00b0] \u7684\u672b\u7aef\u4f4d\u7f6e，\u5e76\u7528 7.4.4 \u7684\u624b\u7b97\u516c\u5f0f\u4ea4\u53c9\u9a8c\u8bc1\u3002 \u63d0\u793a：\u5982\u679c\u53ea\u5199 2 \u4e2a URDFLink \u4e0d\u52a0\u672b\u7aef\u504f\u79fb，\u4f60\u4f1a\u53d1\u73b0\u7b2c\u4e8c\u4e2a\u5173\u8282\u8f6c\u591a\u5c11\u5ea6\u672b\u7aef\u90fd\u4e0d\u52a8\u3002<\/p>\n<\/li>\n

                • \n

                  active_links_mask \u5b9e\u9a8c：\u5206\u522b\u7528"\u4e0d\u4f20 mask"\u548c"\u4f20 [False,True,True,False]"\u5efa\u540c\u4e00\u6761\u94fe，\u89c2\u5bdf\u8b66\u544a\u6570\u91cf\u4e0e FK \u7ed3\u679c\u662f\u5426\u4e00\u81f4\u3002<\/p>\n<\/li>\n

                • \n

                  \u9650\u4f4d\u9a8c\u8bc1：\u7ed9\u7b2c 2 \u4e2a\u5173\u8282\u8bbe bounds=(-1.0, 1.0)，\u6c42\u4e00\u4e2a\u9700\u8981\u5927\u5e45\u5f2f\u66f2\u624d\u80fd\u5230\u8fbe\u7684\u76ee\u6807（\u5982 [0.10, 0, 0.10]），\u9a8c\u8bc1\u89e3\u786e\u5b9e\u88ab\u94b3\u5728\u9650\u4f4d\u5185\u3002<\/p>\n<\/li>\n

                • \n

                  DHLink \u5bf9\u7167：\u7528 DHLink \u91cd\u5efa\u7b2c 05 \u7ae0\u7684 3 \u81ea\u7531\u5ea6\u81c2（\u542b\u79fb\u52a8\u5173\u8282），\u4e0e\u624b\u5199 DH \u5bf9\u6bd4\u3002 \u63d0\u793a：\u79fb\u52a8\u5173\u8282\u9700\u8981\u7528 URDFLink(translation=\u8f74\u65b9\u5411, joint_type="prismatic")，DHLink \u4e0d\u652f\u6301\u3002<\/p>\n<\/li>\n

                  \u7ec3\u4e60\u53c2\u8003\u7b54\u6848\u4e0e\u63d0\u793a<\/h4>\n

                  \u7ec3\u4e60 1 \u7b54\u6848：<\/p>\n

                  # \u5efa\u94fe（4\u4e2a\u5143\u7d20：OriginLink + 2\u5173\u8282 + \u672b\u7aef\u504f\u79fb）<\/span>
                  \nchain =<\/span> Chain(<\/span>name=<\/span>"two_link"<\/span>,<\/span> links=<\/span>[<\/span>
                  \n OriginLink(<\/span>)<\/span>,<\/span>
                  \n URDFLink(<\/span>name=<\/span>"j1"<\/span>,<\/span> origin_translation=<\/span>[<\/span>0<\/span>,<\/span>0<\/span>,<\/span>0<\/span>]<\/span>,<\/span> origin_orientation=<\/span>[<\/span>0<\/span>,<\/span>0<\/span>,<\/span>0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>0<\/span>,<\/span>1<\/span>,<\/span>0<\/span>]<\/span>)<\/span>,<\/span>
                  \n URDFLink(<\/span>name=<\/span>"j2"<\/span>,<\/span> origin_translation=<\/span>[<\/span>–<\/span>0.3<\/span>,<\/span>0<\/span>,<\/span>0<\/span>]<\/span>,<\/span> origin_orientation=<\/span>[<\/span>0<\/span>,<\/span>0<\/span>,<\/span>0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>0<\/span>,<\/span>1<\/span>,<\/span>0<\/span>]<\/span>)<\/span>,<\/span>
                  \n URDFLink(<\/span>name=<\/span>"tip"<\/span>,<\/span> origin_translation=<\/span>[<\/span>0<\/span>,<\/span>0<\/span>,<\/span>0.3<\/span>]<\/span>,<\/span> origin_orientation=<\/span>[<\/span>0<\/span>,<\/span>0<\/span>,<\/span>0<\/span>]<\/span>,<\/span> rotation=<\/span>None<\/span>,<\/span> joint_type=<\/span>"fixed"<\/span>)<\/span>,<\/span>
                  \n]<\/span>,<\/span> active_links_mask=<\/span>[<\/span>False<\/span>,<\/span> True<\/span>,<\/span> True<\/span>,<\/span> False<\/span>]<\/span>)<\/span><\/p>\n

                  q =<\/span> [<\/span>0<\/span>,<\/span> np.<\/span>radians(<\/span>30<\/span>)<\/span>,<\/span> np.<\/span>radians(<\/span>–<\/span>45<\/span>)<\/span>,<\/span> 0<\/span>]<\/span>
                  \nT =<\/span> chain.<\/span>forward_kinematics(<\/span>q)<\/span>
                  \nprint<\/span>(<\/span>T[<\/span>:<\/span>3<\/span>,<\/span> 3<\/span>]<\/span>)<\/span> # \u5b9e\u6d4b\u5f97\u5230 [-0.337453, 0, 0.439778]<\/span><\/p>\n

                  \u624b\u7b97\u9a8c\u8bc1：Ry(30\u00b0) @ ([-0.3,0,0] + Ry(-45\u00b0) @ [0,0,0.3]) = [-0.337453, 0, 0.439778] \u2713（\u4e0e 7.4.4 \u7684\u624b\u7b97\u516c\u5f0f\u4e00\u81f4）<\/p>\n

                  \u7ec3\u4e60 2 \u63d0\u793a：\u4e0d\u4f20 mask \u65f6，ikpy \u4f1a\u5bf9 OriginLink \u548c\u672b\u7aef\u56fa\u5b9a\u8fde\u6746\u5404\u53d1\u4e00\u6761 UserWarning（\u5171 2 \u6761），\u4f46 FK \u7ed3\u679c\u4e0e\u4f20 mask \u65f6\u5b8c\u5168\u4e00\u81f4\u3002\u8fd9\u8bf4\u660e\u8b66\u544a\u4e0d\u5f71\u54cd\u8ba1\u7b97\u7ed3\u679c，\u4f46\u5e94\u8be5\u663e\u5f0f\u8bbe\u7f6e mask \u4ee5\u4fdd\u6301\u4ee3\u7801\u6e05\u6670\u3002<\/p>\n

                  \u7ec3\u4e60 3 \u63d0\u793a：\u8bbe\u7f6e bounds=(-1.0, 1.0) \u540e，\u6c42\u89e3\u9700\u8981\u5927\u5e45\u5f2f\u66f2\u7684\u76ee\u6807\u65f6，j2 \u4f1a\u88ab\u94b3\u5728 \u00b11.0 rad（\u7ea6 \u00b157\u00b0），\u672b\u7aef\u4f1a\u6709\u6b8b\u5dee\u3002\u68c0\u67e5 np.abs(q_sol[2]) <= 1.0 + 1e-6 \u9a8c\u8bc1\u9650\u4f4d\u751f\u6548\u3002<\/p>\n

                  \u7ec3\u4e60 4 \u5b8c\u6574\u53c2\u8003\u7b54\u6848：\u7528 DHLink \u91cd\u5efa\u7b2c 05 \u7ae0\u7684 3 \u81ea\u7531\u5ea6\u81c2（2 \u8f6c + 1 \u79fb），\u4e0e\u624b\u5199 DH \u5bf9\u6bd4\u3002<\/p>\n

                  \u5173\u952e\u77e5\u8bc6\u70b9：\u79fb\u52a8\u5173\u8282\u4e0d\u80fd\u7528 DHLink<\/h5>\n

                  \u7b2c 05 \u7ae0\u7684 3 \u81ea\u7531\u5ea6\u81c2\u6709\u4e00\u4e2a\u79fb\u52a8\u5173\u8282（J3，\u6cbf Z \u8f74\u4f38\u7f29）\u3002DHLink \u7684\u5b9e\u73b0\u628a theta \u4f5c\u4e3a\u552f\u4e00\u53d8\u91cf（\u65cb\u8f6c\u5173\u8282），\u4e0d\u652f\u6301\u628a d \u4f5c\u4e3a\u53d8\u91cf（\u79fb\u52a8\u5173\u8282）\u3002\u6240\u4ee5\u79fb\u52a8\u5173\u8282\u5fc5\u987b\u7528 URDFLink \u5b9e\u73b0：<\/p>\n

                  URDFLink(<\/span>
                  \n name=<\/span>"j3"<\/span>,<\/span>
                  \n origin_translation=<\/span>[<\/span>0.25<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> # \u7236\u5173\u8282\u5230\u672c\u5173\u8282\u7684\u504f\u79fb<\/span>
                  \n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
                  \n translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 1<\/span>]<\/span>,<\/span> # \u2190 \u79fb\u52a8\u65b9\u5411：\u6cbf Z \u8f74<\/span>
                  \n joint_type=<\/span>"prismatic"<\/span>,<\/span> # \u2190 \u5fc5\u987b\u663e\u5f0f\u6307\u5b9a！<\/span>
                  \n)<\/span><\/p>\n\n\u53c2\u6570\u65cb\u8f6c\u5173\u8282\u79fb\u52a8\u5173\u8282<\/tr>\n\n\n\n\n
                  \u8fd0\u52a8\u8f74<\/td>\nrotation=[x, y, z]<\/td>\ntranslation=[x, y, z]<\/td>\n<\/tr>\n
                  joint_type<\/td>\n"revolute"（\u9ed8\u8ba4）<\/td>\n"prismatic"（\u5fc5\u987b\u663e\u5f0f\u5199）<\/td>\n<\/tr>\n
                  \u53d8\u91cf\u542b\u4e49<\/td>\n\u5173\u8282\u89d2（\u5f27\u5ea6）<\/td>\n\u79fb\u52a8\u91cf（\u7c73）<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                  \u26a0\ufe0f \u5e38\u89c1\u9519\u8bef：\u79fb\u52a8\u5173\u8282\u53ea\u5199\u4e86 translation \u4f46\u5fd8\u4e86\u5199 joint_type="prismatic"\u3002ikpy \u9ed8\u8ba4 joint_type="revolute"，\u4f1a\u5bfc\u81f4\u53c2\u6570\u6821\u9a8c\u5931\u8d25\u6216\u884c\u4e3a\u5f02\u5e38\u3002<\/p>\n

                  \u5b8c\u6574\u4ee3\u7801：\u6df7\u5408\u4f7f\u7528 DHLink \u548c URDFLink<\/h5>\n

                  from<\/span> ikpy.<\/span>chain import<\/span> Chain
                  \nfrom<\/span> ikpy.<\/span>link import<\/span> OriginLink,<\/span> URDFLink,<\/span> DHLink
                  \nimport<\/span> numpy as<\/span> np<\/p>\n

                  # \u65b9\u6848 A：\u524d\u4e24\u4e2a\u65cb\u8f6c\u5173\u8282\u7528 DHLink，\u7b2c\u4e09\u4e2a\u79fb\u52a8\u5173\u8282\u7528 URDFLink<\/span>
                  \nchain_dh =<\/span> Chain(<\/span>
                  \n name=<\/span>"dh_3dof"<\/span>,<\/span>
                  \n links=<\/span>[<\/span>
                  \n OriginLink(<\/span>)<\/span>,<\/span>
                  \n DHLink(<\/span>name=<\/span>"j1"<\/span>,<\/span> d=<\/span>0.20<\/span>,<\/span> a=<\/span>0.30<\/span>,<\/span> alpha=<\/span>0.0<\/span>,<\/span> theta=<\/span>0.0<\/span>)<\/span>,<\/span>
                  \n DHLink(<\/span>name=<\/span>"j2"<\/span>,<\/span> d=<\/span>0.00<\/span>,<\/span> a=<\/span>0.25<\/span>,<\/span> alpha=<\/span>0.0<\/span>,<\/span> theta=<\/span>0.0<\/span>)<\/span>,<\/span>
                  \n ]<\/span>,<\/span>
                  \n active_links_mask=<\/span>[<\/span>False<\/span>,<\/span> True<\/span>,<\/span> True<\/span>]<\/span>,<\/span>
                  \n)<\/span><\/p>\n

                  # \u65b9\u6848 B：\u5168\u90e8\u7528 URDFLink（\u542b\u79fb\u52a8\u5173\u8282）<\/span>
                  \nchain_full =<\/span> Chain(<\/span>
                  \n name=<\/span>"urdf_3dof"<\/span>,<\/span>
                  \n links=<\/span>[<\/span>
                  \n OriginLink(<\/span>)<\/span>,<\/span>
                  \n # \u4e0e DH \u53c2\u6570\u5bf9\u5e94：j1 = d=0.20(\u57fa\u5ea7\u9ad8) + a=0.30(\u81c2\u957f)，\u7ed5 Z \u8f6c<\/span>
                  \n URDFLink(<\/span>name=<\/span>"j1"<\/span>,<\/span> origin_translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
                  \n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 1<\/span>]<\/span>)<\/span>,<\/span>
                  \n URDFLink(<\/span>name=<\/span>"j2"<\/span>,<\/span> origin_translation=<\/span>[<\/span>0.30<\/span>,<\/span> 0<\/span>,<\/span> 0.20<\/span>]<\/span>,<\/span>
                  \n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> rotation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 1<\/span>]<\/span>)<\/span>,<\/span>
                  \n URDFLink(<\/span>name=<\/span>"j3"<\/span>,<\/span> origin_translation=<\/span>[<\/span>0.25<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
                  \n origin_orientation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span>
                  \n translation=<\/span>[<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 1<\/span>]<\/span>,<\/span> # \u79fb\u52a8\u65b9\u5411：\u6cbf Z \u8f74<\/span>
                  \n joint_type=<\/span>"prismatic"<\/span>)<\/span>,<\/span> # \u5fc5\u987b\u663e\u5f0f\u6307\u5b9a！<\/span>
                  \n ]<\/span>,<\/span>
                  \n active_links_mask=<\/span>[<\/span>False<\/span>,<\/span> True<\/span>,<\/span> True<\/span>,<\/span> True<\/span>]<\/span>,<\/span>
                  \n)<\/span><\/p>\n

                  # \u6d4b\u8bd5：q = [J1=30\u00b0, J2=45\u00b0, J3=0.1m]<\/span>
                  \nq =<\/span> [<\/span>0.0<\/span>,<\/span> np.<\/span>radians(<\/span>30<\/span>)<\/span>,<\/span> np.<\/span>radians(<\/span>45<\/span>)<\/span>,<\/span> 0.1<\/span>]<\/span>
                  \nT_urdf =<\/span> chain_full.<\/span>forward_kinematics(<\/span>q)<\/span><\/p>\n

                  # \u624b\u5199 DH \u9a8c\u8bc1（\u7b2c 05 \u7ae0\u7684 dh_transform）<\/span>
                  \nT_dh_manual =<\/span> (<\/span>
                  \n dh_transform(<\/span>0.30<\/span>,<\/span> 0<\/span>,<\/span> 0.20<\/span>,<\/span> np.<\/span>radians(<\/span>30<\/span>)<\/span>)<\/span>
                  \n @ dh_transform(<\/span>0.25<\/span>,<\/span> 0<\/span>,<\/span> 0.0<\/span>,<\/span> np.<\/span>radians(<\/span>45<\/span>)<\/span>)<\/span>
                  \n @ dh_transform(<\/span>0.0<\/span>,<\/span> 0<\/span>,<\/span> 0.1<\/span>,<\/span> 0.0<\/span>)<\/span> # \u79fb\u52a8\u5173\u8282：d=0.1<\/span>
                  \n)<\/span><\/p>\n

                  \u5b9e\u6d4b\u8f93\u51fa<\/h5>\n

                  URDFLink \u7248 (\u542b\u79fb\u52a8\u5173\u8282) = [0.324512 0.391481 0.3 ]
                  \n\u624b\u5199 DH \u7248 = [0.324512 0.391481 0.3 ]
                  \n\u4e00\u81f4: True<\/p>\n

                  \u2705 \u79fb\u52a8\u5173\u8282\u7528 URDFLink(translation=\u8f74\u65b9\u5411, joint_type="prismatic") \u5b9e\u73b0，\u5bf9\u5e94 DH \u91cc\u7684 d \u53d8\u91cf\u3002\u4e24\u8005\u7ed3\u679c\u5b8c\u5168\u4e00\u81f4\u3002<\/p>\n

                  \u6df7\u5408\u4f7f\u7528\u7684\u6ce8\u610f\u4e8b\u9879<\/h5>\n
                • \n

                  \u5750\u6807\u7cfb\u4e00\u81f4\u6027：DHLink \u7528\u6807\u51c6 DH \u7ea6\u5b9a（Rz(\u03b8)\u00b7Tz(d)\u00b7Tx(a)\u00b7Rx(\u03b1)），\u800c URDFLink \u7528"\u5148\u5e73\u79fb\u540e\u65cb\u8f6c"\u7ea6\u5b9a\u3002\u4e24\u8005\u5728\u540c\u4e00\u6761\u94fe\u4e2d\u6df7\u7528\u65f6，\u5fc5\u987b\u786e\u4fdd\u6bcf\u4e2a link \u7684 origin_translation \u4e0e DH \u53c2\u6570\u7684\u51e0\u4f55\u542b\u4e49\u4e00\u81f4\u3002\u6700\u7b80\u5355\u7684\u505a\u6cd5\u662f：\u8981\u4e48\u5168\u7528 DHLink（\u53ea\u6709\u65cb\u8f6c\u5173\u8282\u65f6），\u8981\u4e48\u5168\u7528 URDFLink（\u6709\u79fb\u52a8\u5173\u8282\u65f6）\u3002<\/p>\n<\/li>\n

                • \n

                  active_links_mask \u957f\u5ea6：\u5fc5\u987b\u4e0e links \u5217\u8868\u957f\u5ea6\u4e00\u81f4\u3002\u65b9\u6848 A \u6709 3 \u4e2a\u5143\u7d20（OriginLink + 2 DHLink），\u65b9\u6848 B \u6709 4 \u4e2a\u5143\u7d20（OriginLink + 3 URDFLink）\u3002<\/p>\n<\/li>\n

                • \n

                  \u79fb\u52a8\u5173\u8282\u7684 bounds：\u5355\u4f4d\u662f\u7c73（\u4e0d\u662f\u5f27\u5ea6）\u3002\u4f8b\u5982 bounds=(0.0, 0.5) \u8868\u793a\u79fb\u52a8\u8303\u56f4 0~0.5m\u3002<\/p>\n<\/li>\n

                  \u53c2\u8003\u7b54\u6848\u89c1 code\/ch07_ikpy_intro.py\u3002<\/p>\n

                  \n

                  7.11 \u5c0f\u7ed3<\/h3>\n

                  \u6838\u5fc3\u8981\u70b9\u56de\u987e<\/h4>\n
                    \n
                  • ikpy \u53ea\u505a\u8fd0\u52a8\u5b66（FK\/IK），\u4e0d\u505a\u7269\u7406\u2014\u2014\u4e0e MuJoCo \u4e92\u8865\u3002<\/li>\n
                  • \u56db\u4e2a\u7c7b：Chain（\u5165\u53e3）\u3001OriginLink（\u5730\u57fa）\u3001URDFLink（\u5173\u8282\u8fde\u6746）\u3001DHLink（DH \u53c2\u6570\u8fde\u6746）\u3002<\/li>\n
                  • 🔥 origin_translation \u91cf\u7684\u662f"\u5173\u8282\u5230\u5173\u8282"\u7684\u8ddd\u79bb，\u672b\u7aef\u5fc5\u987b\u5355\u72ec\u52a0\u4e00\u4e2a\u56fa\u5b9a\u504f\u79fb link\u2014\u2014\u5426\u5219\u6700\u540e\u4e00\u4e2a\u5173\u8282\u5b8c\u5168\u4e0d\u8d77\u4f5c\u7528（7.4.1，\u6700\u96be\u81ea\u67e5\u7684\u5751）\u3002<\/li>\n
                  • 🔥 \u8f6c\u8f74\u4e0d\u80fd\u4e0e\u8fde\u6746\u540c\u5411，\u5426\u5219\u65cb\u8f6c\u4e0d\u4ea7\u751f\u4f4d\u79fb（7.4.2 \u53cd\u9762\u6559\u6750 A）\u3002<\/li>\n
                  • 🔥 ikpy 4.0 \u7528 origin_translation，\u4e0d\u662f translation_vector\u3002<\/li>\n
                  • 🔥 \u56fa\u5b9a\u8fde\u6746\u5fc5\u987b\u5199 joint_type="fixed"；\u79fb\u52a8\u5173\u8282\u5fc5\u987b\u5199 joint_type="prismatic"\u3002<\/li>\n
                  • active_links_mask：\u4e0d\u4f20\u5219\u5168 True（\u4f1a\u5bf9\u6bcf\u4e2a fixed link \u53d1\u8b66\u544a），\u4f20\u4e86\u5219\u539f\u6837\u4fdd\u7559；OriginLink \u548c\u56fa\u5b9a\u8fde\u6746\u5e94\u4e3a False\u3002<\/li>\n
                  • \u5173\u8282\u9650\u4f4d\u7528 bounds=(lo, hi)，ikpy \u4fdd\u8bc1\u89e3\u5728\u9650\u4f4d\u5185；\u8d85\u51fa\u53ef\u8fbe\u8303\u56f4\u65f6\u8fd4\u56de"\u8d34\u5728\u9650\u4f4d\u4e0a"\u7684\u5c3d\u529b\u89e3，\u4e14\u4e0d\u62a5\u9519\u2014\u2014\u5fc5\u987b\u81ea\u5df1\u68c0\u67e5\u672b\u7aef\u8bef\u5dee\u3002<\/li>\n
                  • IK \u672c\u8d28\u662f scipy.optimize.least_squares \u6700\u5c0f\u5316\u672b\u7aef\u8bef\u5dee；\u89e3\u53ef\u80fd\u4e0d\u662f\u4f60\u8f93\u5165\u7684\u90a3\u7ec4（\u591a\u89e3，\u89c1\u7b2c 09 \u7ae0）\u3002<\/li>\n
                  • DHLink \u7684 FK \u4e0e\u7b2c 05 \u7ae0\u624b\u5199 DH \u5b8c\u5168\u4e00\u81f4（\u5b9e\u6d4b [0.324512, 0.391481, 0.2]）\u3002<\/li>\n<\/ul>\n

                    \u5173\u952e API \u901f\u67e5\u8868<\/h4>\n\nAPI\u4f5c\u7528<\/tr>\n\n\n\n\n\n\n\n\n
                    Chain(links=…, active_links_mask=…)<\/td>\n\u6784\u5efa\u8fd0\u52a8\u94fe<\/td>\n<\/tr>\n
                    OriginLink()<\/td>\n\u94fe\u7684\u8d77\u70b9（\u57fa\u5ea7）<\/td>\n<\/tr>\n
                    URDFLink(name, origin_translation, rotation, bounds, joint_type)<\/td>\nURDF \u98ce\u683c\u8fde\u6746<\/td>\n<\/tr>\n
                    DHLink(name, d, a, alpha, theta)<\/td>\nDH \u53c2\u6570\u98ce\u683c\u8fde\u6746<\/td>\n<\/tr>\n
                    chain.forward_kinematics(q_full)<\/td>\n\u6b63\u8fd0\u52a8\u5b66（\u8f93\u5165\u5b8c\u6574\u957f\u5ea6\u6570\u7ec4）<\/td>\n<\/tr>\n
                    chain.inverse_kinematics(target_position=…, initial_position=…)<\/td>\n\u9006\u8fd0\u52a8\u5b66<\/td>\n<\/tr>\n
                    chain.plot(q, ax, target=…)<\/td>\n\u4e09\u7ef4\u53ef\u89c6\u5316<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                    \u5728\u672c\u9879\u76ee\u4e2d\u7684\u5e94\u7528<\/h4>\n
                      \n
                    • \u7b2c 08 \u7ae0\u7528 ikpy \u6784\u5efa\u672c\u4e66 6 \u8f74\u673a\u68b0\u81c2\u7684\u5b8c\u6574 Chain<\/li>\n
                    • \u7b2c 09 \u7ae0\u7528 ikpy \u505a\u59ff\u6001\u63a7\u5236\u3001\u5197\u4f59\u81ea\u7531\u5ea6\u7ba1\u7406\u3001\u907f\u969c<\/li>\n
                    • \u7b2c 19 \u7ae0\u7528 ikpy \u505a\u4ea4\u53c9\u9a8c\u8bc1（\u4e0e\u81ea\u7814 DLS \u5bf9\u6bd4）<\/li>\n
                    • pick_and_place.py \u4e3b\u8981\u7528\u81ea\u7814 DLS（\u5b9e\u65f6\u6027\u66f4\u597d），\u4f46 ikpy \u7528\u4e8e\u79bb\u7ebf\u9a8c\u8bc1\u548c\u5feb\u901f\u539f\u578b<\/li>\n<\/ul>\n

                      \u6269\u5c55\u9605\u8bfb\u65b9\u5411<\/h4>\n
                    • ikpy \u5b98\u65b9\u6587\u6863：https:\/\/github.com\/Phylliade\/ikpy ，\u67e5\u770b\u6700\u65b0 API \u548c\u793a\u4f8b<\/li>\n
                    • URDF \u89c4\u8303：\u7406\u89e3 URDF \u7684 <joint>\u3001<link>\u3001<origin>\u3001<axis>\u3001<limit> \u7b49\u5143\u7d20，\u6709\u52a9\u4e8e\u7406\u89e3 ikpy \u7684\u53c2\u6570\u8bbe\u8ba1<\/li>\n
                    • sympy \u7b26\u53f7\u8ba1\u7b97：ikpy \u7684\u6838\u5fc3\u673a\u5236，\u4e86\u89e3 sympy \u7684 Matrix\u3001Symbol\u3001lambdify \u7b49\u529f\u80fd<\/li>\n
                    • scipy.optimize.least_squares：ikpy IK \u7684\u5e95\u5c42\u4f18\u5316\u5668，\u4e86\u89e3\u4fe1\u4efb\u533a\u57df\u7b97\u6cd5\u548c bounds \u5904\u7406<\/li>\n
                    • PyKDL \/ trac_ik：\u5176\u4ed6\u4e3b\u6d41\u8fd0\u52a8\u5b66\u5e93，\u4e0e ikpy \u5bf9\u6bd4\u5b66\u4e60\u3002PyKDL \u662f ROS \u751f\u6001\u7684\u6807\u51c6，trac_ik \u662f\u66f4\u5feb\u901f\u7684 IK \u6c42\u89e3\u5668<\/li>\n

                      \u4e0b\u4e00\u7ae0\u7528 ikpy \u5b8c\u6574\u9a71\u52a8\u672c\u4e66\u7684 6 \u8f74\u673a\u68b0\u81c2\u3002<\/p>\n

                      \n

                      \u4e0a\u4e00\u7ae0：06 \u00b7 \u9006\u8fd0\u52a8\u5b66 IK ｜ \u4e0b\u4e00\u7ae0：08 \u00b7 ikpy FK\/IK \u5b9e\u6218<\/p>\n","protected":false},"excerpt":{"rendered":"

                      \u8fd9\u4e00\u7ae0\u8981\u89e3\u51b3\u4ec0\u4e48\u95ee\u9898：\u4e0d\u60f3\u81ea\u5df1\u5199 DH\u3001\u4e0d\u60f3\u81ea\u5df1\u5b9e\u73b0 DLS，\u6709\u6ca1\u6709\u73b0\u6210\u7684\u5e93\u80fd\u7528？ \u6709，\u5c31\u662f ikpy\u3002\u672c\u7ae0\u628a\u5b83\u62c6\u5f00\u8bb2\u6e05\u695a：\u4e09\u4e2a\u7c7b\u3001\u4e00\u5957\u53c2\u6570\u3001\u4ee5\u53ca 4.0 \u7248\u672c\u6539\u4e86\u54ea\u4e9b API（\u7f51\u4e0a\u5927\u91cf\u6559\u7a0b\u5df2\u8fc7\u65f6）\u3002 \u914d\u5957\u4ee3\u7801：[code\/ch07_ikpy_intro.py]
                      \n\\”\\”\\”
                      \n\u7b2c 07 \u7ae0\u914d\u5957\u4ee3\u7801：ikpy \u5165\u95e8\u3002\u8fd0\u884c：D:\\\\\\\\Environ<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[81],"topic":[],"class_list":["post-102644","post","type-post","status-publish","format-standard","hentry","category-server","tag-python"],"yoast_head":"\n07 \u00b7 ikpy \u5165\u95e8 - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.wsisp.com\/helps\/102644.html\" \/>\n<meta property=\"og:locale\" content=\"zh_CN\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"07 \u00b7 ikpy \u5165\u95e8 - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3\" \/>\n<meta property=\"og:description\" content=\"\u8fd9\u4e00\u7ae0\u8981\u89e3\u51b3\u4ec0\u4e48\u95ee\u9898：\u4e0d\u60f3\u81ea\u5df1\u5199 DH\u3001\u4e0d\u60f3\u81ea\u5df1\u5b9e\u73b0 DLS，\u6709\u6ca1\u6709\u73b0\u6210\u7684\u5e93\u80fd\u7528？ \u6709，\u5c31\u662f ikpy\u3002\u672c\u7ae0\u628a\u5b83\u62c6\u5f00\u8bb2\u6e05\u695a：\u4e09\u4e2a\u7c7b\u3001\u4e00\u5957\u53c2\u6570\u3001\u4ee5\u53ca 4.0 \u7248\u672c\u6539\u4e86\u54ea\u4e9b API（\u7f51\u4e0a\u5927\u91cf\u6559\u7a0b\u5df2\u8fc7\u65f6）\u3002 \u914d\u5957\u4ee3\u7801：[code\/ch07_ikpy_intro.py] """ \u7b2c 07 \u7ae0\u914d\u5957\u4ee3\u7801：ikpy \u5165\u95e8\u3002\u8fd0\u884c：D:\\\\Environ\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.wsisp.com\/helps\/102644.html\" \/>\n<meta property=\"og:site_name\" content=\"\u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3\" \/>\n<meta property=\"article:published_time\" content=\"2026-09-09T01:57:03+00:00\" \/>\n<meta name=\"author\" content=\"admin\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u4f5c\u8005\" \/>\n\t<meta name=\"twitter:data1\" content=\"admin\" \/>\n\t<meta name=\"twitter:label2\" content=\"\u9884\u8ba1\u9605\u8bfb\u65f6\u95f4\" \/>\n\t<meta name=\"twitter:data2\" content=\"24 \u5206\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.wsisp.com\/helps\/102644.html\",\"url\":\"https:\/\/www.wsisp.com\/helps\/102644.html\",\"name\":\"07 \u00b7 ikpy \u5165\u95e8 - 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