Interaction ControlObject BehaviorAlternative: control port behavior Impedance & AdmittanceImpedance & Admittance (continued)Impedance as dynamic stiffnessInteraction control: causal considerationsRobot Impedance ControlOSCAR the robotNetwork modeling perspective on interaction control Equivalent networksNonlinear equivalent networksNonlinear equivalent network for interaction controlVirtual trajectorySuperposition of “impedance forces”One application: collision avoidanceHigh-speed collision avoidanceImpedance Control ImplementationRobot ModelSimple Impedance ControlMechanism singularitiesMechanism junction structureControl at mechanism singularitiesGeneralized coordinatesInverse kinematicsIntrinsically variable impedanceIntrinsically variable stiffnessOpposing actuators at a jointConfiguration-dependent moment armsThis is the “tent-pole” effectIntrinsically variable inertiaCausal analysisMod. Sim. Dyn. Sys. Interaction Control Neville Hogan page 1Interaction Control• Manipulation requires interaction– object behavior affects control of force and motion• Independent control of force and motion is not possible– object behavior relates force and motion• contact a rigid surface: kinematic constraint• move an object: dynamic constraint• Accurate control of force or motion requires detailed models of• manipulator dynamics • object dynamics – object dynamics are usually known poorly, often not at allMod. Sim. Dyn. Sys. Interaction Control Neville Hogan page 2Object Behavior• Can object forces be treated as external (exogenous) disturbances?– the usual assumptions don’t apply:• “disturbance” forces depend on manipulator state• forces often aren’t small by any reasonable measure• Can forces due to object behavior be treated as modeling uncertainties?– yes (to some extent) but the usual assumptions don’t apply:• command and disturbance frequencies overlap• Example: two people shaking hands– how each person moves influences the forces evoked• “disturbance” forces are state-dependent– each may exert comparable forces and move at comparable speeds• command & “disturbance” have comparable magnitude & frequencyMod. Sim. Dyn. Sys. Interaction Control Neville Hogan page 3Alternative: control port behavior• Port behavior: – system properties and/or behaviors “seen” at an interaction port• Interaction port: – characterized by conjugate variables that define power flow• Key point:[][]⎪⎪⎩⎪⎪⎨⎧===s)(velocitie flows (forces) efforts in power 11tntnffeePLLfefetport behavior is unaffected by contact and interactionMod. Sim. Dyn. Sys. Interaction Control Neville Hogan page 4Impedance & Admittance• Impedance and admittance characterize interaction– a dynamic generalization of resistance and conductance• Usually introduced for linear systems but generalizes to nonlinear systems– state-determined representation:– this form may be derived from or depicted as a network model()()⎪⎪⎩⎪⎪⎨⎧ℜ∈ℜ∈ℜ∈ℜ∈===PVFzVFPVzZFVzZzmmntos,,,,,&()()()()()()sLsisesZCssisesZ====)( inductor electrical1 capacitor electrical()xfvxΦ==&(spring)element elastic1Dnonlinear State equationsOutput equationsConstraint on input & outputMod. Sim. Dyn. Sys. Interaction Control Neville Hogan page 5Impedance & Admittance (continued)• Admittance is the causal dual of impedance– Admittance: flow out, effort in– Impedance: effort out, flow in• Linear system: admittance is the inverse of impedance• Nonlinear system: – causal dual is well-defined:– but may not correspond to any impedance• inverse may not exist ()()()()()CssesisYsZsY===−capacitor electrical1()()⎪⎪⎩⎪⎪⎨⎧ℜ∈ℜ∈ℜ∈ℜ∈===PVFyVFPFyYVFyYymmntos,,,,,&()pvfpΨ==&(mass)element inertial1DnonlinearMod. Sim. Dyn. Sys. Interaction Control Neville Hogan page 6Impedance as dynamic stiffness• Impedance is also loosely defined as a dynamic generalization of stiffness– effort out, displacement in• Most useful for mechanical systems – displacement (or generalized position) plays a key role()()⎪⎪⎩⎪⎪⎨⎧ℜ∈ℜ∈ℜ∈ℜ∈===PXFzdXFdWXzZFXzZzmmntos,,,,,&Mod. Sim. Dyn. Sys. Interaction Control Neville Hogan page 7Interaction control: causal considerations• What’s the best input/output form for the manipulator?• The set of objects likely to be manipulated includes–inertias • minimal model of most movable objects– kinematic constraints• simplest description of surface contact• Causal considerations:–inertias prefer admittance causality– constraints require admittance causality – compatible manipulator behavior should be an impedance• An ideal controller should make the manipulator behave as an impedance• Hence impedance control– Hogan 1979, 1980, 1985, etc.Mod. Sim. Dyn. Sys. Interaction Control Neville Hogan page 8Robot Impedance Control• Works well for interaction tasks:– Automotive assembly • (Case Western Reserve University, US)– Food packaging • (Technical University Delft, NL)– Hazardous material handling • (Oak Ridge National Labs, US)– Automated excavation • (University of Sydney, Australia)– … and many more• Facilitates multi-robot / multi-limb coordination• Schneider et al., Stanford• Enables physical cooperation of robots and humans• Kosuge et al., Japan• Hogan et al., MITMod. Sim. Dyn. Sys. Interaction Control Neville Hogan page 9OSCAR the robotE.D.Fasse & J.F.Broenink, U. Twente, NLPhotograph removed due to copyright restrictions.Mod. Sim. Dyn. Sys. Interaction Control Neville Hogan page 10Network modeling perspective on interaction control• Port concept– control interaction port behavior– port behavior is unaffected by contact and interaction• Causal analysis– impedance and admittance characterize interaction– object is likely an admittance– control manipulator impedance• Model structure– structure is important– power sources are commonly modeled as equivalent networks• Thévenin equivalent• Norton equivalent• Can equivalent network structure be applied to interaction control?Mod. Sim. Dyn. Sys. Interaction Control Neville Hogan page 11Equivalent networks• Initially applied to networks of static linear elements• Sources & linear resistors– Thévenin equivalent network– M. L. Thévenin, Sur un nouveau théorème d’électricité dynamique.Académie des Sciences, Comptes Rendus 1883,