of fab(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; jkNumber 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 matrixLkkth 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 ; nLNumber of legs (or limbs) of a parallel manipulatorniklNumber of independent kinematic loopsnkpmNumber of distinct posture modes of the leg LknmNumber of moving or movable bodiesnpvNumber of primary variables; npv = μnpmNumber of distinct posture modes of a manipulatornsvNumber of secondary variables; nsv = λniklnvNumber of variables needed to describe the pose of a mechanical systemCommon normal between the axes of and OOrigin of the base frame ; O = O0OkOrigin of the reference frame PAn arbitrary point (in a general use)PTip point of a manipulator; P = Om for a serial manipulatorTip point position vector with respect to the base frame; Column matrix representation of in the base frame; QabContact point on the surface Column matrix of the joint variablesqkGeneralized joint variable of ; qk = θk or qk = skRWrist point of a manipulator; R = Om − 1 for a serial manipulatorMatrix representation of the rotation operator rot(
