M. Kemal Ozgoren - Kinematics of General Spatial Mechanical Systems

Readers are encouraged and sincerely welcome to contact me at ozgoren@metu.edu.tr with regard to feedback, suggestions, and questions.

Ankara, March 2019

M. Kemal Ozgoren

This book is based on the graduate courses “Advanced Dynamics” and “Principles of Robotics” that I have been teaching for a long time. Therefore, many students of these courses and several colleagues involved with related subjects have so far indicated their valuable opinions and encouraged me to write such a book. I would like to express my thanks to all of them and also to the Mechanical Engineering Department of Middle East Technical University that has provided the excellent academic medium and all the conveniences for me in developing and teaching the mentioned courses.

I would also like to express my thanks to the reviewers for their constructive comments and to all the related personnel of Wiley for their highly appreciated efforts during all the time from the initiation to the finalization of this book.

Acceleration of an implied point (in a general use) Acceleration of a point P with respect to an implied reference frame Acceleration of a point P with respect to a reference frame ; b kEffective length of between and along A rigid body denoted by the index a and represented by the frame Orientation matrix of an implied body or frame (in a general use) Orientation matrix of the end‐effector ( ) with respect to the base frame; CTM that transforms components from to ; Orientation matrix of with respect to ; Orientation matrix of with respect to Orientation matrix of with respect to ; A general column matrix; A general row matrix, i.e. the transpose of Skew symmetric matrix generated from ; D aVector differentiator with respect to ; d kConstant offset of with respect to if is revolute d mTip point offset; d m= RP = O m − 1 O m E Elbow point of a manipulator Exponential form of Exponential form of Kinematic element of that mates with of f ab( x , y , z )Surface function that describes the surface in the link frame A reference frame with an orientation index a , whose origin is implied A reference frame with a specific origin Q Reference frame attached to Base frame, i.e. the reference frame attached to the base link Gradient vector of the surface Column matrix representation of in the link frame Abbreviation for of the last link HTM that represents the displacement from to HTM that transforms coordinates from to ; Hessian matrix of the surface expressed in the link frame Abbreviation for of the link Identity matrix; j kNumber of joints with k degrees of relative freedom A general Jacobian matrix The joint (i.e. kinematic pair) between and ; Joint between and Tip point Jacobian matrix Wrist point Jacobian matrix L k k th leg (or limb) of a parallel manipulator A link denoted by the index a k th link of a manipulator The last link (i.e. end‐effector) of a serial manipulator with m links A general square matrix m Number of links and joints of a serial manipulator A general unit vector Unit column matrix that represents the twisted axis of ; n LNumber of legs (or limbs) of a parallel manipulator n iklNumber of independent kinematic loops n kpmNumber of distinct posture modes of the leg L k n mNumber of moving or movable bodies n pvNumber of primary variables; n pv= μ n pmNumber of distinct posture modes of a manipulator n svNumber of secondary variables; n sv= λn ikl n vNumber of variables needed to describe the pose of a mechanical system Common normal between the axes of and O Origin of the base frame ; O = O 0 O kOrigin of the reference frame P An arbitrary point (in a general use) P Tip point of a manipulator; P = O mfor a serial manipulator Tip point position vector with respect to the base frame; Column matrix representation of in the base frame; Q abContact point on the surface Column matrix of the joint variables q kGeneralized joint variable of ; q k= θ kor q k= s k R Wrist point of a manipulator; R = O m − 1for a serial manipulator Matrix representation of the rotation operator rot( ) in a frame Matrix representation of the rotation operator rot( a , b ) in a frame k th basic rotation matrix; Augmented position matrix of a point P with respect to a frame Position vector of an implied point (in a general use) Wrist point position vector with respect to the base frame; Column matrix representation of in the base frame; Column matrix representation of a vector in a frame ; Column matrix representation of a vector in a frame ; k th component of a vector in a frame ; Displacement vector from the origin O k − 1to the origin O k Column matrix representation of in the link frame Position vector of a point P with respect to an implied point Position vector of a point P with respect to a point Q ; Displacement vector from a point Q to a point P ; s kJoint variable (sliding displacement) of the prismatic joint s kVariable offset of with respect to if is prismatic S Shoulder point of a manipulator Surface of the kinematic element Joint space of a manipulator Task space of a manipulator Approach vector of the end‐effector; k th basic column matrix; k th unit basis vector of a frame Column matrix representation of in a frame ; Normal vector of the end‐effector; Side vector of the end‐effector; Unit vector along the common normal Unit vector along the axis of Basis vector triad of a reference frame ; Basis vector triad of the link frame ; Tip point velocity influence coefficient due to ; Velocity of an implied point (in a general use) Tip point velocity with respect to the base frame; Column matrix representation of in the base frame; Velocity of a point P with respect to an implied reference frame Velocity of a point P with respect to a reference frame ; Wrist point velocity influence coefficient due to ; Wrist point velocity with respect to the base frame; Column matrix representation of in the base frame; Generatrix line vector of a conical surface Column matrix representation of in the link frame Column matrix of the primary variables Column matrix of the secondary variables Column matrix of the primary and secondary variables