Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren

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 impliedA reference frame with a specific origin QReference 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 freedomA general Jacobian matrixThe joint (i.e. kinematic pair) between and ; Joint between and Tip point Jacobian matrixWrist point Jacobian matrixL_kkth leg (or limb) of a parallel manipulatorA link denoted by the index akth link of a manipulatorThe last link (i.e. end‐effector) of a serial manipulator with m linksA general square matrixmNumber of links and joints of a serial manipulatorA general unit vectorUnit column matrix that represents the twisted axis of ; n_LNumber of legs (or limbs) of a parallel manipulatorn_iklNumber of independent kinematic loopsn_kpmNumber of distinct posture modes of the leg L_kn_mNumber of moving or movable bodiesn_pvNumber of primary variables; n_pv = μn_pmNumber of distinct posture modes of a manipulatorn_svNumber of secondary variables; n_sv = λn_ikln_vNumber of variables needed to describe the pose of a mechanical systemCommon normal between the axes of and OOrigin of the base frame ; O = O₀O_kOrigin of the reference frame PAn arbitrary point (in a general use)PTip point of a manipulator; P = O_m for a serial manipulatorTip 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 variablesq_kGeneralized joint variable of ; q_k = θ_k or q_k = s_kRWrist point of a manipulator; R = O_{m − 1} for a serial manipulatorMatrix representation of the rotation operator rot(