Analysis of Proportional-Derivative and Nonlinear Control of Mechanical Systems with Dynamic Backlash

The Complementarity Class of Hybrid Dynamical Systems

“…Jugglers are a sub-class of complementarity Lagrangian mechanical systems which can be written as follows: Examples of mechanical jugglers are running biped robots, hoppers, controlled structures, manipulators with dynamic passive environment, systems with dynamic backlash or liquid slosh phenomena [69], tethered satellites [60], etc. The analysis and control of jugglers have been investigated in [18,23,68,103,104].…”

“…The aim is not to derive conditions on observability, but to design asymptotically stable observers. A two-degree-of-freedom system is considered in [73] (this is an impacting pair that may model dynamic backlash [69], and whose dynamics fits within jugglers dynamics in (83)). The measured output is assumed to be…”

Feedback control of multibody systems with joint clearance and dynamic backlash: a tutorial

“…The controllability properties and the control of juggling systems have been investigated in [22,57,53,91,20,55], where dead-beat control is a central tool. Most of these results apply to the all-linear case (linear dynamics and linear constraints).…”

“…However if the two constraints are programmed to be hit simultaneously (a multiple, two-impact occurs), then issues related to discontinuity with respect to initial data should be examined as they may influence robustness and stability. Specific controllers are proposed for step 3 in [22,53,55,57], including also step 4. Clearly various strategies may be adopted (impulsive, piecewise-constant, optimal control).…”

Dead‐beat regulation of mechanical juggling systems

“…Hurwitz. According to [1,11], the dynamic state-feedback control law (10) is such that no contact occurs for t ∈ (k T , (k + 1) T ), whereas an impact occurs at t = k T , thus rendering Property 1 satisfied. Moreover, if the continuous-time reference signal y d is such that (7), (8) and (9) hold, witḣy F =Ĵ T −1Ĵ⊤ 2 K D Q N (k), one has that the control input given at each impact time to system (3) is exactly the one given in (4).…”