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 1Example: Integral-action motion controller • System: (ms2 + bs + x k) = f -cu – Mass restrained by linear spring & x c = damper, driven by control actuator & u ms2 + bs + k external force g• Controller: u = (r − x)– Integral of trajectory error s • System + controller: (ms3 + bs2 + ks + x cg) = -cgr f s x cg= 3r ms + bs2 + ks + cg s: Laplace variable bk • Isolated stability: x: displacement variable f: external force variable – Stability requires upper bound on > g u: control input variablecontroller gain cm r: reference input variable m: mass constantb: damping constant k: stiffness constantc: actuator force constant g: controller gain constant Mod. Sim. Dyn. Sys. Interaction Stability Neville Hogan page 2Example (continued) me: object mass constant e• Object mass: f = s m 2 x • Coupled system: [(m + )s m 3 + bs2 + ks + x cg] = cgre x = cg r (m + m )s3 + bs2 + ks + cge • Coupled stability: bk > cg(m + m )e • Choose any positive controller gain bk > gthat will ensure isolated stability: cm • That controlled system is me > bk − mdestabilized by coupling to a cgsufficiently large massMod. Sim. Dyn. Sys. Interaction Stability Neville Hogan page 3Problem & 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 4General object dynamics * e, • Assume: L( q q )= E ( q q )− Ep ( 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  ee t *• Legendre transform: Ek ( q p )= q p e, e  e − Ek ( q q )e,e e – Kinetic co-energy to kinetic He ( q p )= q p e, t energy e, e e  e − L( q q e ) – Lagrangian form to Hamiltonian q  e =∂ He ∂ p eform p  ∂ − = He ∂ q − D e + P ee e q e: (generalized) coordinates • Hamiltonian = total system energy L: Lagrangian Ek*: kinetic co-energy He ( q p )= E ( q p )+ Ep ( q ) Ep: potential energy e,e k e,e e D e: dissipative (generalized) forces P e: exogenous (generalized) forces He: Hamiltonian Mod. Sim. Dyn. Sys. Interaction Stability Neville Hogan page 5Sir William Rowan Hamilton • William Rowan Hamilton – Born 1805, Dublin, Ireland – Knighted 1835 – First Foreign Associate elected to U.S. National Academy of Sciences– Died 1865 • Accomplishments – Optics – Dynamics – Quaternions – Linear operators – Graph theory – …and more – http://www.maths.tcd.ie/pub/ HistMath/People/Hamilton/ Mod. Sim. Dyn. Sys. Interaction Stability Neville Hogan page 6Passivity • Basic idea: system cannot supply power indefinitely – Many alternative definitions, the best are energy-based • Wyatt et al. (1981) Wyatt, J. L., Chua, L. O., Gannett, J. W., Göknar, I. C. and Green, D. N. (1981) Energy Concepts in the State-Space Theory • Passive: total system energy is lower-bounded of Nonlinear n-Ports: Part I — Passivity. – More precisely, available energy is lower-bounded IEEE Transactions on Circuits and Systems, Vol. CAS-28, No. 1, pp. 48-61. • Power flux may be positive or negative • Convention: power positive in – Power in (positive)—no limit – Power out (negative)—only until stored energy exhausted • You can store as much energy as you want but you can withdraw only what was initially stored (a finite amount) • Passivity ≠ stability – Example: • Interaction between oppositely charged beads, one fixed, on free to move on a wire Mod. Sim. Dyn. Sys. Interaction Stability Neville Hogan page 7Stability • Stability: – Convergence to equilibrium • Use Lyapunov’s second method – A generalization of energy-based analysis – Lyapunov function: positive-definite non-decreasing state function – Sufficient condition for asymptotic stability: Negative semi-definitive rate of change of Lyapunov function • For physical systems total energy may be a useful candidate Lyapunov function – Equilibria are at an energy minima – Dissipation ⇒ energy reduction ⇒ convergence to equilibrium – Hamiltonian form describes dynamics in terms of total energy Mod. Sim. Dyn. Sys. Interaction Stability Neville Hogan page 8Steady state & equilibrium • Steady state: – Kinetic energy is a positive-definite non-decreasing function of generalized momentum • Assume: – Dissipative (internal) forces vanish in steady-state • Rules out static (Coulomb) friction – Potential energy is a positive-definite non-decreasing function of generalized displacement • Steady-state is a unique equilibrium configuration • Steady state is equilibrium at the origin of the state space {p e,q e} 0 q ∂ = H= ∂ p =∂ Ek ∂ p e e e e =⇒=∂∂ eekE 0 p 0 p 0 p − = ∂ H= ∂ q − D e + P e e e e ( ,Assume q 0 D )= 0e e =Isolated ⇒ 0 P e ∂ E∂ He ∂ Ek + p= ∂ q ∂ q ∂ qe 0 p = e 0 p = e e e ∂ E∂ Ek ∂ He = p= 0 ∴∂ q ∂ q ∂ qe 0 p = e 0 p = e e e =⇒=∂∂ eepE 0 q 0 q Mod. Sim. Dyn. Sys. Interaction Stability Neville Hogan page 9Notation • Represent partial derivatives using ∂ He eqsubscripts H =∂ q e • H is a scalar ∂ Hee – the Hamiltonian state function H ep =∂ p e • H is a vectoreq – Partial derivatives of the Hamiltonian q  = H ep ( q p )w.r.t. each element of q e e,e e e (p  e − = H eq ( q p )− q