Читать книгу Kinematics of General Spatial Mechanical Systems - M. Kemal Ozgoren - Страница 11
ОглавлениеList of Commonly Used Symbols, Abbreviations, and Acronyms
Symbols Based on the Latin Alphabet
Acceleration of an implied point (in a general use)Acceleration of a point P with respect to an implied reference frameAcceleration of a point P with respect to a reference frame ; bkEffective 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 ; DaVector differentiator with respect to ; dkConstant offset of with respect to if is revolutedmTip point offset; dm = RP = Om − 1OmEElbow point of a manipulatorExponential form of Exponential form of Kinematic element of that mates with 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() in a frame Matrix representation of the rotation operator rot(a, b) in a frame kth 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 ; kth component of a vector in a frame ; Displacement vector from the origin Ok − 1 to the origin OkColumn matrix representation of in the link frame Position vector of a point P with respect to an implied pointPosition vector of a point P with respect to a point Q; Displacement vector from a point Q to a point P; skJoint variable (sliding displacement) of the prismatic joint skVariable offset of with respect to if is prismaticSShoulder point of a manipulatorSurface of the kinematic element Joint space of a manipulatorTask space of a manipulatorApproach vector of the end‐effector; kth basic column matrix; kth 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 frameVelocity 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 variablesColumn matrix of the secondary variablesColumn matrix of the primary and secondary variables
Symbols Based on the Greek Alphabet
Angular acceleration of an implied body or frame (in a general use)Angular acceleration of or with respect to an implied frameAngular acceleration of or with respect to βkTwist angle of with respect to γkCumulative twist angle of with respect to ; γk = β1 + β2 + … + βkModified gradient of a conical surface expressed in the link frame δijKronecker Delta function of the indices i and jδkConstant rotation angle of with respect to if is prismaticεijkLevi‐Civita Epsilon function of the indices i, j, and kColumn matrix of the end‐effector velocity state in the task space θkJoint variable (angular displacement) of the revolute joint θkVariable rotation angle of with respect to if is revoluteλDoF of the working or operational space of a mechanical systemλabGradient ratio between and ; μMobility (DoF) of a system in its working or operational spaceμabRelative mobility (DoF) of with respect to ; μba = μabColumn matrix of the end‐effector position in the task space σA general sign variable; σ = ± 1σk, Sign variables that indicate multiple solutionsσijkCross 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
