Analysis of Cartesian stiffness and compliance with applications

Many elastic systems can be modelled by a 6 x 6 Cartesian stiffness or compliance matrix. Using spatial vector (screw) algebra, spatial stiffness and compliance are defined. Investigation of the linear elastic behavior is achieved by analyzing the geometric and constitutive properties of the stiffness and compliance matrices. The results are applicable in the analysis, design and control of elastic systems such as serial and parallel robotic manipulators, robotic grasp problems, assembly automation devices, spatial structures, and so on. The geometric and constitutive properties of an elastic system can be understood in terms of suitable eigenvalue problems. However, construction of physically and geometrically intuitive eigenvalue problems for stiffness and compliance in screw space is neither unique nor straightforward. First, a set of singular eigenvalue problems from earlier studies is shown to be related to free-vectors. Closed form solutions for the location of centers of elasticity, stiffness and compliance are found in terms of quantities related to free-vector eigenvalue problems. Then, the constitutive nature and other properties of the centers of stiffness and compliance are presented, which were previously unknown. The centers of elasticity, stiffness and compliance are shown to be geometrically related. Considering line-vectors, instead of free-vectors, a new set of singular eigenvalue problems is proposed and solved. Every point in space generates a distinct set. Similar to the free-vector case, line-vector decompositions of stiffness and compliance are found and co-centers of elasticity are identified. The free-vector and line-vector results lead to generalized definitions of compliant axes and a refined compliance hierarchy. The stiffness matrix of parallel spatial connections with line and torsional springs is found in closed form. The skew-symmetric part of stiffness for line springs is described completely, which explains previously observed asymmetry. The observation in earlier studies that the stiffness of line springs is symmetric in a special reference frame is explained. There exist infinitely many such frames forming a 2-parameter family. In contrast, there is no such frame for torsional springs. A theory is developed to determine orthogonal sets of isotropic vectors of a symmetric matrix, which, together with the stiffness equation for spring systems, leads to the synthesis of stiffness by springs. A general synthesis solution had not been found until now. The necessary and sufficient condition is that the off-diagonals of stiffness matrix have a zero trace. Algorithms and examples support the theory. The free-vector and line-vector results are applied to rotational symmetry devices such as the remote center of compliance (RCC) device used in automated assembly operations. Previously unavailable and more accurate design equations are determined. Optimum device configurations are demonstrated. The conditions for the construction of RCC devices with beams and springs are found. Definitions of RCC-like devices are generalized. The theory for the elastic systems is shown to be applicable in the dynamics of single rigid body. The mass matrix replaces the stiffness matrix. The free-vector and line-vector eigenvalue problems are explicitly solved for the mass matrix. Special axes resulting from the line-vector case explains the center of percussion phenomenon. A practical optimum design of sport equipment involving the center of percussion is presented. Combination of the elastic and kinetic cases leads to the determination of the necessary and sufficient conditions for the existence of special free vibration modes. Numerical examples are provided for each topic to verify the theoretical results.