A Physically Consistent Stiffness Formulation for Contact-Rich Manipulation

In the realm of robotics, the concept of "controller design in the physical domain" (Sharon et al., 1989, 1991) and the associated methodology of "control by interconnection" (Stramigioli, 2001; van der Schaft, 2016) emphasize that robot controllers should be more than mere signal processors. Instead, they should have a physical interpretation (Lachner, 2022), which is especially important for robots that physically interact with the environment (Hogan, 1988; Dietrich and Hogan, 2022; Hogan, 2022). This paper delves into impedance control (Hogan, 1984) during physical interaction, specifically focusing on the symmetry of the stiffness matrix and its role in ensuring passive physical equivalent robot controllers. Passivity is a fundamental property for ensuring coupled stability when interacting with arbitrary passive objects (Colgate and Hogan, 1988a). Stability in robotics can be achieved by monitoring and controlling the energy supplied by the controller (Colgate and Hogan, 1987, 1988b,a; Stramigioli, 2015). In impedance-controlled robots, this monitoring is particularly straightforward, as energy is stored in virtual springs (potential energy) and transferred into kinetic energy during movement (Lachner et al., 2021). During physical interaction, stiffness plays a crucial role, as it defines how energy is stored and exchanged between the robot and its environment. Task-space stiffness determines interaction forces due to contact, which is especially important at low frequencies (e.g., steady-state). 1