1 Mathematical Preliminaries -- 1.0 Introduction -- 1.1 Vector space, linear dependence and basis of a vector space -- 1.2 Linear transformation and its matrix representation -- 1.3 Range and null space of a linear transformation -- 1.4 Eigenvalues and eigenvectors of a linear transformation -- 1.5 Change of basis -- 1.6 Diagonalization of matrices -- 1.7 Bilinear forms and sign definition of matrices -- 1.8 Norms, isometries, orthogonal and unitary matrices -- 1.9 Properties of unitary and orthogonal matrices -- 1.10 Stationary points of scalar functions of a vector argument -- 1.11 Linear algebraic systems -- 1.12 Numerical solution of linear algebraic systems -- 1.13 Numerical solution of nonlinear algebraic systems -- References -- 2. Fundamentals of Rigid-Body Three-Dimensional Kinematics -- 2.1 Introduction -- 2.2 Motion of a rigid body -- 2.3 The Theorem of Euler and the revolute matrix -- 2.4 Groups of rotations -- 2.5 Rodriguesโ formula and the Cartesian decomposition of the rotation matrix -- 2.6 General motion of a rigid body and Chaslesโ Theorem -- 2.7 Velocity of a point of a rigid body rotating about a fixed point -- 2.8 Velocity of a moving point referred to a moving observer -- 2.9 General motion of a rigid body -- 2.10 Theorems related to the velocity distribution in a moving rigid body -- 2.11 Acceleration distribution in a rigid body moving about a fixed point -- 2.12 Acceleration distribution in a rigid body under general motion -- 2.13 Acceleration of a moving point referred to a moving observer -- References -- 3. Generalities on Lower-Pair Kinematic Chains -- 3.1 Introduction -- 3.2 Kinematic pairs -- 3.3 Degree of freedom -- 3.4 Classification of lower pairs -- 3.5 Classification of kinematic chains -- 3.6 Linkage problems in the Theory of Machines and Mechanisms -- References -- 4. Analysis of Motions of Kinematic Chains -- 4.1 Introduction -- 4.2 The method of Denavit and Hartenberg -- 4.3 An alternate method of analysis -- 4.4 Applications to open kinematic chains -- References -- 5. Synthesis of Linkages -- 5.1 Introduction -- 5.2 Synthesis for function generation -- 5.3 Mechanism synthesis for rigid-body guidance -- 5.4 A different approach to the synthesis problem for rigid-body guidance -- 5.5 Linkage synthesis for path generation -- 5.6 Epilogue -- References -- 6. An Introduction to the Optimal Synthesis of Linkages -- 6.1 Introduction -- 6.2 The optimisation problem -- 6.3 Overdetermined problems of linkage synthesis -- 6.4 Underdetermined problems of linkage synthesis subject to no inequality constraints -- 6.5 Linkage optimisation subject to inequality constraints. Penalty function methods -- 6.6 Linkage optimisation subject to inequality constraints. Direct methods -- References -- Appendix 1 Algebra of dyadics -- Appendix 2 Derivative of a determinant with respect to a scalar argument -- Appendix 4 Synthesis of plane linkages for rigid-body guidance

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