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interaction_stab

# interaction_stab - Contact instability Problem Contact and...

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Contact instability Problem: Contact and interaction with objects couples their dynamics into the manipulator control system This change may cause instability Example: integral-action motion controller coupling to more mass evokes instability Impedance control affords a solution: Make the manipulator impedance behave like a passive physical system Hogan, N. (1988) On the Stability of Manipulators Performing Contact Tasks, IEEE Journal of Robotics and Automation, 4: 677-686. Mod. Sim. Dyn. Sys. Interaction Stability Neville Hogan page 1

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Example: Integral-action motion controller System: (ms 2 + bs + x k) = f - cu Mass restrained by linear spring & x c = damper, driven by control actuator & u ms 2 + bs + k external force g Controller: u = (r x) Integral of trajectory error s System + controller: (ms 3 + bs 2 + ks + x cg) = - cgr f s x cg = 3 r ms + bs 2 + ks + cg s: Laplace variable bk Isolated stability: x: displacement variable f: external force variable Stability requires upper bound on > g u: control input variable controller gain cm r: reference input variable m: mass constant b: damping constant k: stiffness constant c: actuator force constant g: controller gain constant Mod. Sim. Dyn. Sys. Interaction Stability Neville Hogan page 2
Example (continued) m e : object mass constant e Object mass: f = s m 2 x Coupled system: [(m + )s m 3 + bs 2 + ks + x cg] = cgr e x = cg r (m + m )s 3 + bs 2 + ks + cg e Coupled stability: bk > cg(m + m ) e Choose any positive controller gain bk > g that will ensure isolated stability: cm That controlled system is m e > bk m destabilized by coupling to a cg sufficiently large mass Mod. Sim. Dyn. Sys. Interaction Stability Neville Hogan page 3

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Problem & approach Problem: Find conditions to avoid instability due to contact & interaction Approach: Design the manipulator controller to impose a desired interaction-port behavior Describe the manipulator and its controller as an equivalent physical system Find an (equivalent) physical behavior that will avoid contact/coupled instability Use our knowledge of physical system behavior and how it is constrained Mod. Sim. Dyn. Sys. Interaction Stability Neville Hogan page 4
General object dynamics * e , ± Assume: L ( q q ) = E ( q q ) E p ( q ) e k e , ± e e Lagrangian dynamics dt q ± e q L = P e q q D ) Passive d L e ( e , ± e Stable in isolation e p = L q ± e = E * k q ± e e t * Legendre transform: E k ( q p ) = q p e , ± e ± e E k ( q q ) e , e e Kinetic co-energy to kinetic H e ( q p ) = q p e , ± t energy e , e e ± e L ( q q e ) Lagrangian form to Hamiltonian q ± e = ∂ H e p e form p ± = H e q D e + P e e e q e : (generalized) coordinates Hamiltonian = total system energy L: Lagrangian E k * : kinetic co-energy H e ( q p ) = E ( q p ) + E p ( q ) E p : potential energy e , e k e , e e D e : dissipative (generalized) forces P e : exogenous (generalized) forces H e : Hamiltonian Mod. Sim. Dyn. Sys. Interaction Stability Neville Hogan page 5

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Sir William Rowan Hamilton
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