M. Kemal Ozgoren - Kinematics of General Spatial Mechanical Systems

ang( )Angle between the vectors and atan 2( y , x )Double argument arctangent function; θ ≡ atan 2( 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;

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

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

Scan this QR code to visit the companion website.

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 frames. Similarly, a vector equation can be written without indicating any reference frame, whereas the selected reference frame must be indicated for the corresponding matrix equation.

This section gives general information about the special notation that is used throughout the book. This notation is convenient because it can be used not only in printed texts but also in handwritten work. It also has the desirable feature that it can distinguish column matrices from vectors, which are actually different mathematical objects. The main features of the notation are explained below.

A scalar is denoted by a plain letter such as s.

A vector is denoted by a letter with an overhead arrow such as .

A column matrix is denoted by a letter with an overhead bar such as .

A square or a rectangular matrix is denoted by a capital letter with an overhead circumflex (a.k.a. hat) such as .

A skew symmetric matrix is denoted by a letter with an overhead tilde such as .

The transpose of a matrix is denoted by a superscript t such as and .

A vector is defined as an entity that can be described by a magnitude and a direction . In this book, it is assumed that all the vectors belong to the three‐dimensional Euclidean space .

The magnitude of a vector is denoted as shown below.

(1.1)