# The analysis and synthesis of spatial compliance

#### Abstract

The spatial force-deflection behavior of an elastically suspended body is characterized by a 6 x 6 positive semidefinite matrix, the stiffness matrix. A better understanding of general compliant behavior is of general interest and is particularly important in understanding the behavior of robotic systems. The ability to specify and realize desired compliant behavior is an important issue in providing a robot the ability to react to contact forces in a prescribed beneficial way. The generation of procedures to realize an arbitrary stiffness matrix using springs with the simplest possible construction is the main component of this work. In the dissertation, the following issues are addressed. (1) The identification of the realizable space of spatial stiffnesses achieved with a parallel connection of "simple springs" (conventional line springs and conventional torsional springs). (2) The synthesis of a simple spring realizable stiffness matrix. An algorithm is developed based on the decomposition of the stiffness matrix into rank-1 components with a specific form. This procedure requires no more than seven simple springs to realize an arbitrary simple spring realizable full-rank stiffness matrix. (3) The realization of an arbitrary stiffness matrix. To overcome the limitation of simple springs, a new type of elastic device that couples the spring force and torque along the same axis must be used. These devices are defined as "screw springs". Procedures to synthesize any stiffness matrix with a parallel connection of simple springs and screw springs are developed. (4) A physical interpretation of spatial stiffness behavior. Based on the stiffness eigen-screw problem, a new decomposition of a stiffness matrix is established. This decomposition is shown to be independent of coordinate frame and to have extremal properties. With this decomposition, some physical insight into the behavior associated with a general spatial stiffness matrix is provided. (5) A classification of spatial stiffness matrices. To evaluate the translational-rotational coupling aspects of a stiffness matrix, an invariant, referred to as the "degree of translational-rotational coupling" (DTRC), is defined. The DTRC is based on the minimum number of "coupled" components in all decompositions of a stiffness matrix. A classification of spatial stiffness matrices based on this number is established.

#### Recommended Citation

Shuguang Huang,
"The analysis and synthesis of spatial compliance"
(January 1, 1998).
*Dissertations (1962 - 2010) Access via Proquest Digital Dissertations.*
Paper AAI9842432.

http://epublications.marquette.edu/dissertations/AAI9842432