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            "1376": {
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                "title": "Richard M. Murray",
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                        "*": "[[Image:Murray-sp07.jpg|left]]\nRichard M. Murray received the B.S. degree in Electrical Engineering from California Institute of Technology in 1985 and the M.S. and Ph.D. degrees in Electrical Engineering and Computer Sciences from the University of California, Berkeley, in 1988 and 1991, respectively.  He is currently a Professor of Control & Dynamical Systems and Bioengineering at the California Institute of Technology, Pasadena.\n\nProfessor Murray's research is in the application of feedback and control to mechanical, information, and biological systems.  Current projects include integration of control, communications, and computer science in multi-agent systems, information dynamics in networked feedback systems, analysis of insect flight control systems, and biological circuit design.  Professor Murray has recently developed a new course at Caltech that is aimed at teaching the principles and tools of control to a broader audience of scientists and engineers, with particular emphasis on applications in biology and computer science.\n\n* [http://www.cds.caltech.edu/~murray Richard Murray's homepage]"
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                "title": "Rigid Body Motion",
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                        "*": "{{chapter header|Introduction|Rigid Body Motion|Manipulator Kinematics}}\n\nA rigid motion of an object is a motion which preserves distance\nbetween points.  The study of robot kinematics, dynamics, and control\nhas at its heart the study of the motion of rigid objects.  In this\nchapter, we provide a description of rigid body motion using the tools\nof linear algebra and screw theory.\n\n== Chapter Summary ==\nThe following are the key concepts covered in this chapter:\n<ol>\n<li>\nRotational motion of a rigid body is represented by an element of\nthe special orthogonal group\n<center><math>\nSO(3) = \\{ R \\in {\\mathbb R}^{3 \\times 3} \\mid R^T R = I, \\det R = 1 \\}.\n</math></center>\n<!--\nwhich is often parameterized  by the exponential map\n<center><amsmath>\n\\exp: so(3) \\longrightarrow SO(3): \\widehat{\\omega}\\theta \\mapsto\ne^{\\widehat{\\omega} \\theta}.\n</amsmath></center>\n-->\nOther parameterizations of SO(3) include fixed and Euler angle sets,\nand unit quaternions.\n</li>\n\n<li> The ''configuration'' of a rigid body is\nrepresented as an element <amsmath>g \\in\n\\text{SE}(3)</amsmath>.  An element <amsmath>g \\in \\text{SE}(3)</amsmath> may also be viewed as a mapping\n<amsmath>g:{\\mathbb R}^3 \\to {\\mathbb R}^3</amsmath> which preserves distances and angles between\npoints.  In homogeneous coordinates, we write\n<center><amsmath>\n  g = \\begin{bmatrix} R & p \\\\ 0 & 1 \\end{bmatrix} \\qquad\n  \\aligned\n    R &\\in SO(3) \\\\\n    p &\\in {\\mathbb R}^3.\n  \\endaligned\n</amsmath></center>\nThe same representation can also be\nused for a rigid body transformation\nbetween two coordinate frames.\n</li>\n\n<li> ''Rigid body transformations'' can be represented as the\nexponentials of twists:\n<center><amsmath>\n  g = \\exp(\\widehat{\\xi} \\theta) \\qquad \n  \\widehat{\\xi} = \\begin{bmatrix} \\widehat{\\omega} & v \\\\ 0 & 0 \\end{bmatrix}, \\quad\n  \\aligned\n    \\widehat{\\omega} &\\in so(3), \\\\\n    v &\\in {\\mathbb R}^3, \\theta \\in {\\mathbb R}.\n  \\endaligned\n</amsmath></center>\n\nThe twist coordinates of <amsmath>\\widehat{\\xi}</amsmath> are <amsmath>\\xi = (v,\\omega) \\in\n{\\mathbb R}^6</amsmath>.\n</li>\n\n<li> A twist <amsmath>\\xi = (v, \\omega)</amsmath> is associated with a ''screw''\nmotion having attributes\n<center><amsmath>\n  \\alignedat 2 \n    &\\text{pitch:} &\\qquad h &= \\frac{\\omega^T v}{\\|\\omega\\|^2}; \\\\\n    &\\text{axis:} &\\qquad\n      l &= \\begin{cases}\n        \\{ \\frac{\\omega\\times v}{\\|\\omega\\|^2} + \\lambda\\omega:\n\t  \\lambda \\in{\\mathbb R} \\}, &\\text{if $\\omega\\neq0$} \\\\\n\t\\{ 0 + \\lambda v:\\lambda \\in {\\mathbb R} \\}, &\\text{if $\\omega=0$};\n      \\end{cases} \\\\\n    &\\text{magnitude:} &\\qquad\n      M &= \\begin{cases}\n        \\|\\omega\\|, &\\text{if $\\omega\\neq0$} \\\\\n        \\| v \\|, &\\text{if $\\omega = 0$}.\n      \\end{cases}\n  \\endalignedat\n</amsmath></center>\n\nConversely, given a screw we can write the associated twist.  Two\nspecial cases are ''pure rotation'' about an axis <amsmath>l\n= \\{q + \\lambda \\omega\\}</amsmath> by an amount <amsmath>\\theta</amsmath> and ''pure translation''\nalong an axis <amsmath>l = \\{0 + \\lambda v\\}</amsmath>:\n<center><amsmath>\n  \\xi = \\begin{bmatrix} -\\omega\\times q \\\\ \\omega \\end{bmatrix} \\theta\n    \\quad\\!\\text{(pure rotation)} \n  \\qquad\n  \\xi = \\begin{bmatrix} v \\\\ 0 \\end{bmatrix} \\theta \n    \\quad\\!\\text{(pure translation)}.\n</amsmath></center>\n\n</li>\n\n<li> The ''velocity'' of a rigid motion <amsmath>g(t) \\in \\text{SE}(3)</amsmath> can be\nspecified in two ways.  The ''spatial velocity'',\n<center><amsmath>\n  \\widehat{V}^s = \\dot g g^{-1},\n</amsmath></center>\n\nis a twist which gives the velocity of the rigid body as measured by\nan observer at the origin of the reference frame.  The ''body velocity'',\n<center><amsmath>\n  \\widehat{V}^b = g^{-1} \\dot g,\n</amsmath></center>\n\nis the velocity of the object in the instantaneous body frame.  These\nvelocities are related by the ''adjoint transformation''\n<center><amsmath>\n  V^s = \\operatorname{Ad}_g V^b \\qquad \n  \\operatorname{Ad}_g = \\begin{bmatrix} R & \\widehat{p}R \\\\ 0 & R \\end{bmatrix},\n</amsmath></center>\n\nwhich maps <amsmath>{\\mathbb R}^6 \\to {\\mathbb R}^6</amsmath>.  To transform velocities between\ncoordinate frames, we use the relations\n<center><amsmath>\n  \\aligned\n    V_{ac}^s &= V_{ab}^s + \\operatorname{Ad}_{g_{ab}} V_{bc}^s \\\\\n    V_{ac}^b &= \\operatorname{Ad}_{g_{bc}^{-1}} V_{ab}^b + V_{bc}^b,\n  \\endaligned\n</amsmath></center>\n\nwhere <amsmath>V_{ab}^s</amsmath> is the spatial velocity of coordinate frame <amsmath>B</amsmath>\nrelative to frame <amsmath>A</amsmath> and <amsmath>V_{ab}^b</amsmath> is the body velocity.\n</li>\n\n<li> ''Wrenches'' are represented as a force, moment pair\n<center><amsmath>\n  F = (f, \\tau) \\in {\\mathbb R}^6.\n</amsmath></center>\n\nIf <amsmath>B</amsmath> is a coordinate frame attached to a rigid body, then we write\n<amsmath>F_b = (f_b, \\tau_b)</amsmath> for a wrench applied at the origin of <amsmath>B</amsmath>, with\n<amsmath>f_b</amsmath> and <amsmath>\\tau_b</amsmath> specified with respect to the <amsmath>B</amsmath> coordinate frame.\nIf <amsmath>C</amsmath> is a second coordinate frame, then we can write <amsmath>F_b</amsmath> as an\n''equivalent wrench'' applied at <amsmath>C</amsmath>:\n<center><amsmath>\n  F_c = \\operatorname{Ad}_{g_{bc}}^T F_b.\n</amsmath></center>\n\nFor a rigid body with configuration <amsmath>g_{ab}</amsmath>, <amsmath>F^s := F_a</amsmath> is called\nthe ''spatial'' wrench and <amsmath>F^b := F_b</amsmath> is called the ''body''\nwrench.\n</li>\n\n<li> A wrench <amsmath>F = (f, \\tau)</amsmath> is associated with a screw having attributes\n<center><amsmath>\n  \\alignedat 2 \n    &\\text{pitch:} &\\qquad h &= \\frac{f^T \\tau}{\\|f\\|^2}; \\\\\n    &\\text{axis:} &\\qquad\n      l &= \\begin{cases}\n        \\{ \\frac{f\\times \\tau}{\\|f\\|^2} + \\lambda f:\n\t  \\lambda \\in{\\mathbb R} \\}, &\\text{if $f\\neq0$} \\\\\n\t\\{ 0 + \\lambda \\tau:\\lambda \\in {\\mathbb R} \\}, &\\text{if $f=0$};\n      \\end{cases} \\\\\n    &\\text{magnitude:} &\\qquad\n      M &= \\begin{cases}\n        \\|f\\|, &\\text{if $f\\neq0$} \\\\\n        \\| \\tau \\|, &\\text{if $f = 0$}.\n      \\end{cases}\n  \\endalignedat\n</amsmath></center>\n\nConversely, given a screw we can write  the associated wrench.\n</li>\n\n<li> A wrench <amsmath>F</amsmath> and a twist <amsmath>V</amsmath> are ''reciprocal'' if <amsmath>F \\cdot V = 0</amsmath>.\nTwo screws <amsmath>S_1</amsmath> and <amsmath>S_2</amsmath> are reciprocal if the twist <amsmath>V_1</amsmath>\nabout <amsmath>S_1</amsmath> and the wrench <amsmath>F_2</amsmath> along <amsmath>S_2</amsmath> are reciprocal.  The\n''reciprocal product'' between two screws is given by\n<center><amsmath>\n  S_1 \\odot S_2 = V_1 \\cdot F_2 = V_1 \\odot V_2 = v_1 \\cdot \\omega_2\n  + v_2^T \\omega_1\n</amsmath></center>\n\nwhere <amsmath>V_i = (v_i, \\omega_i)</amsmath> represents the twist associated with\nthe screw <amsmath>S_i</amsmath>.  Two screws are reciprocal if the reciprocal product\nbetween the screws is zero.\n</li>\n\n<li> A ''system of screws'' <amsmath>\\{S_1, \\dots, S_k\\}</amsmath> describes the\nvector space of all linear combinations of the screws <amsmath>\\{S_1,\n\\dots, S_k\\}</amsmath>.  A ''reciprocal screw system'' is the set of all\nscrews which are reciprocal to <amsmath>S_i</amsmath>.  The dimensions of a screw\nsystem and its reciprocal system sum to 6 (in <amsmath>se(3)</amsmath>).</li>\n\n<!--\n<li>\nA Lie subgroup of SE(3) is used to model constrained rigid motions.\nThe three one dimensional Lie subgroups <amsmath>R, T</amsmath> and <amsmath>H_{\\rho}</amsmath>\nare used to  model rigid motions generated by  primitive joints, and\nthe  Lie subgroups <amsmath> T(3), SO(3)</amsmath>, <amsmath>SE(2), X </amsmath> and\n<amsmath>SE(3)</amsmath> are often\nused to model a manipulator's end-effector motions.</li>\n-->\n\n<!--\n<li>\nTwo special families of regular submanifolds of SE(3), category I\nand category II submanifolds, are used to model constrained rigid\nmotions that lack a group structure.\n</li>\n-->\n</ol>\n\n== Additional Information =="
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