Abstract
A robot making contact with an environment or human presents potential safety risks, including excessive collision force. While experiments on the effect of robot inertia, relative velocity, and interface stiffness on collision are in literature, analytical models for maximum collision force are limited to a simplified mass-spring robot model. This simplified model limits the analysis of control (force/torque, impedance, or admittance) or compliant robots (joint and end-effector compliance). Here, the Sobolev norm is adapted to be a system norm, giving rigorous bounds on the maximum force on a stiffness element in a general dynamic system, allowing the study of collision with more accurate models and feedback control. The Sobolev norm can be found through the $\mathcal{H}_2$ norm of a transformed system, allowing efficient computation, connection with existing control theory, and controller synthesis to minimize collision force. The Sobolev norm is validated, first experimentally with an admittance-controlled robot, then in simulation with a linear flexible-joint robot. It is then used to investigate the impact of control, joint flexibility and end-effector compliance on collision, and a trade-off between collision performance and environmental estimation uncertainty is shown.
Highlights
Mechatronic design for position control is largely standardized - every position-controlled robot has high drivetrain stiffness with a controller designed for high-bandwidth tracking performance and disturbance rejection
If a different linear collision model is written with input u(t) ∈ L∞, with transfer function Guf (s) for dynamics u → f, the force can be bounded as f (t) ∞ < guf (t) 1 u(t) ∞ [31], but the 1 norm of an impulse response guf (t) = L−1 {Guf (s)} is not computed, and the conservatism of this norm in collision analysis is noted [7]
When the signal F is a rational polynomial with relative degree two, poles only in the open left half plane, and lims→0 |F (s)| < ∞, the Sobolev norm exists, and can be computed from existing signal theory. These existence conditions restrict the signals which can be analyzed with the Sobolev norm, but when the force of interest is the force through a stiffness element between two inertias, this condition is satisfied
Summary
Mechatronic design for position control is largely standardized - every position-controlled robot has high drivetrain stiffness with a controller designed for high-bandwidth tracking performance and disturbance rejection. Some trade-offs between performance and robustness have been shown [4], and the role of joint compliance on relaxing coupled stability conditions investigated [5], but unified design methods are not yet established. Reshaped by control due to controller bandwidth limitations This makes the intrinsic dynamics of a robot especially important in collision, motivating the need for physical compliance on otherwise high-impedance robots[15]. For contact with a pure-stiffness environment, there is established literature for stability [7], [10], [28], and limiting the peak force of a rigid, perfectly backdriveable robot [8] using inelastic/elastic collision models. The Sobolev norm is shown to be the H2 norm of a transformed system, allowing computation with existing methods This norm is validated experimentally and in simulation, for pure stiffness as well as inertial environments. A trade-off between collision performance and environmental estimation uncertainty is shown
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