The most general motion of a rigid body B in a global frame G is made by a rotation ϕ about an axis \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat {u}$$ \end{document}, plus a displacement d. Kinematically, the rigid body motion may be expressed by a 3 × 3 rotation matrix plus a 3 × 1 displacement vector. It may also be expressed by a 4 × 4 homogenous transformation matrix. This chapter will review all methods to express rigid body motions. Every rigid body represents a link of a robot.
Arbitrary motion of a body B with respect to another body G is called rigid body motion and can be expressed by a rotation GRBBr plus a translation Gd. The vector Gd is translation of B with respect to G, and GRB is the rotation transformation matrix to map Br to Gr when Gd = 0. The screw motion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\check {s}(h,\phi ,\hat {u},\mathbf {s})$$ \end{document} is indicated by screw parameters: a unit vector on the axis of rotation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat {u}$$ \end{document}, a location vector s, a twist angle ϕ, and a translation h (or pitch p). The location vector s indicates the global position of a point on the screw axis. When s = 0, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat {u}$$ \end{document} passes through the origin of the coordinate frame and the screw motion is called a central screw \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\check {s}(h,\phi ,\hat {u})$$ \end{document}. Every screw motion can be decomposed into three principal central screws about the three axes of the coordinate frame G.
Motion Kinematics
Theory of Applied Robotics ; Chapter : 4 ; 149-224
2021-12-08
76 pages
Article/Chapter (Book)
Electronic Resource
English
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