This book establishes recursive relations concerning kinematics and dynamics of constrained robotic systems. It uses matrix modeling to determine the connectivity conditions on the relative velocities and accelerations in order to compare two efficient energetic ways in dynamics modeling: the principle of virtual work, and the formalism of Lagrange's equations.
First, a brief fundamental theory is presented on matrix mechanics of the rigid body, which is then developed in the following five chapters treating matrix kinematics of the rigid body, matrix kinematics of the composed motion, kinetics of the rigid body, dynamics of the rigid body, and analytical mechanics. By using a set of successive mobile frames, the geometrical properties and the kinematics of the vector system of velocities and accelerations for each element of the robot are analysed. The dynamics problem is solved in two energetic ways: using an approach based on the principle of virtual work and applying the formalism of Lagrange's equations of the second kind. These are shown to be useful for real-time control of the robot's evolution. Then the recursive matrix method is applied to the kinematics and dynamics analysis of five distinct case studies: planar parallel manipulators, spatial parallel robots, planetary gear trains, mobile wheeled robots and, finally, two-module hybrid parallel robots.
1 Introduction.- 1.1 Robotic systems.- 1.2 Historical development.- 1.3 Mechanics of robots.- 2 Matrix kinematics of the rigid body.- 2.1 Position and orientation of a rigid body.- 2.2 Velocity field.- 2.3 Acceleration field.- 2.4 Twist of velocity field of a rigid body.- 2.5 Types of rigid body motions.- 3 Matrix kinematics of composed motion.- 3.1 Kinematics of composed motion of a point.- 3.2 Kinematics of composed motion of a rigid body.- 3.3 Application to kinematics analysis of mechanisms.- 4 Kinetics of the rigid body.- 4.1 Centre of mass and tensor of static moments of a rigid body.- 4.2 Moments of inertia of a rigid body.- 4.3 Kinetic impulse of a system of particles.- 4.4 Kinetic moment of a rigid body.- 4.5 Kinetic energy of a rigid body.- 4.6 Power and work of the forces acting on a system of particles.- 4.7 Power and work of the forces acting on a rigid body.- 5 Dynamics of the rigid body.- 5.1 Fundamental system of differential equations of motion for a system of particles.- 5.2 Theorem of kinetic impulse.- 5.3 Theorem of kinetic moment.- 5.4 Theorem of kinetic moment with respect to a translating frame.- 5.5 Theorem of kinetic energy.- 5.6 Conservation of mechanical energy.- 5.7 Theorem of kinetic energy with respect to a translating frame.- 5.8 Equations of motion in dynamics of the rigid body.- 6 Analytical Mechanics.- 6.1 Principle of virtual work.- 6.2 D'Alembert principle.- 6.3 Lagrange equations.- 6.4 Canonical Hamiltonian equations.- 7 Dynamics of constrained robotic systems.- 7.1 Geometric model of the robot.- 7.2 Velocities and accelerations.- 7.3 Equations of motion.- 7.4 Advantages of the present method.- 7.5 Application to dynamics analysis of mechanisms.- 8 Planar parallel robots.- 8.1 Power requirement comparison in dynamics of the 3-PRR planar parallel robot.- 8.2 Internal reaction joint forces in dynamics of the 3-RRR planar parallel robot.- 8.3 Inverse kinematics and dynamics of a 3-PRP planar parallel robot.- 9 Spatial parallel robots.- 9.1 Dynamics modelling of Delta translational parallel robot.- 9.2 Inverse dynamics of Agile Wrist spherical parallel robot.- 9.3 Dynamics of the 6-6 Stewart parallel manipulator.- 9.4 Internal joint forces in dynamics of a 3-RPS parallel manipulator.- 10 Geared parallel mechanisms.- 10.1 Kinematics and dynamics analysis of the Minuteman cover drive.- 10.2 Inverse dynamics of a 2-DOF orienting gear train.- 10.3 Dynamics analysis of the Cincinnati-Milacron wrist robot.- 11 Mobile wheeled robots.- 11.1 Kinematics and dynamics of a mobile robot provided with caster wheel.- 11.2 Dynamics of the non-holonomic two-wheeled pendulum robot.- 12 Kinematics and dynamics of a hybrid parallel manipulator.- 12.1 Structural description of the hybrid parallel manipulator.- 12.2 Kinematics analysis.- 12.3 Inverse dynamics model.- References.