Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

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alt="images"/>Column matrix of the end‐effector position in the task space σA general sign variable; σ = ± 1σ_k, Sign variables that indicate multiple solutionsσ_ijkCross product sign variable defined for the indices i, j, and kEffective orientation matrix of ; End‐effector angular velocity influence coefficient due to ; Angular velocity of an implied body or frame (in a general use)Angular velocity of the end‐effector with respect to the base frame; Column matrix representation of in the base frame; Angular velocity of or with respect to an implied frameAngular velocity of or with respect to Angular velocity of with respect to the base frame:

      Abbreviations

ang()Angle between the vectors and atan₂(y, x)Double argument arctangent function; θ ≡ atan₂(r sin θ, r cos θ), r > 0colm()Column matrix generator from a skew symmetric matrix; cpm()Cross product matrix generator from a column matrix; dir()Direction of a vector mag()Magnitude of a vector ; rot(a, b)Rotation operator that rotates a frame into another frame rot()Rotation operator of an angle θ about an axis parallel to a unit vector ssm()Skew symmetric matrix generator from a column matrix;

      Acronyms

      CCylindrical JointCTMComponent Transformation MatrixCPMCross Product MatrixDCMDirection Cosine MatrixDoFDegree of FreedomD‐HDenavit‐HartenbergHTMHomogeneous Transformation MatrixIFBInitial Frame BasedIKLIndependent Kinematic LoopMSFKMotion Singularity of Forward KinematicsMSIKMotion Singularity of Inverse KinematicsPPrismatic JointPMPosture ModePMLPosture Mode of a LegPMCPPosture Mode Changing PosePMCPLPosture Mode Changing Pose of a LegPMFKPosture Multiplicity of Forward KinematicsPMIKPosture Multiplicity of Inverse KinematicsPSFKPosition Singularity of Forward KinematicsPSIKPosition Singularity of Inverse KinematicsRRevolute JointRFBRotated Frame BasedSSpherical JointSSMSkew Symmetric MatrixTMTransformation MatrixUUniversal Joint

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      This book is accompanied by a companion website:

      www.wiley.com/go/ozgoren/spatialmechanicalsystems

      The website includes:

      1 ‐ A communication medium with the readers

      2 ‐ Solved problems as additional examples

      3 ‐ Unsolved problems as typical exercises

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1 Vectors and Their Matrix Representations in Selected Reference Frames

      Synopsis

The main purpose of this chapter is to review the mathematics associated with the vectors and their matrix representations in selected reference frames. This review is expected to be beneficial for the efficient readability of this book. It will also familiarize the reader with the special notation that is used throughout this book. This notation is suitable not only because it can distinguish vectors from their matrix representations, but it can also be used conveniently in both printed texts and handwritten work.

      This chapter also explains why and shows how the vectors are treated in this book as mathematical objects that are distinct from the column matrices that represent them in selected reference frames. As the main distinction, the vectors are independent of any reference frame, whereas their matrix representations are necessarily dependent on the selected reference

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