M. Zareinejad 1

 fundamentally, instability has the potential to occur because real-world interactions are only approximated in the virtual world  although these approximation errors are small, their potentially non-passive nature can have profound effects, notably:  instability  limit cycle oscillations (which can be just as bad as instability) 2

 a useful tool for studying the stability and performance of haptic systems a one-port is passive if the integral of power extracted over time does not exceed the initial energy stored in the system. 3

4 A passive system is stable

 “Z-width” is the dynamic range of impedances that can be rendered with a haptic display while maintaining passivity we want a large z-width, in particular:  zero impedance in free space  large impedance during interactions with highly massive/viscous/stiff objects 5

 lower bound depends primarily on mechanical design (can be modified through control)  upper bound depends on sensor quantization, sampled data effects, time delay (in teleoperators), and noise (can be modified through control)  in a different category are methods that seek to create a perceptual effect (e.g., event-based rendering) 6

sampled-data system example 7

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Direct coupling – limitations in 1DOF Direct coupling – limitations in 1DOF  No distinction between simulation and control: ◦ VE used to close the force feedback loop.  In 1DOF: ◦ Energy leaks.  ZOH.  Asynchronous switching.

Passivity and stability Passivity and stability  A passive system is stable.  Any interconnection of passive systems (feedforward or feedback) is stable.

 Real contact ◦ No contact oscillations. ◦ Wall does not generate energy (passive).  Haptic contact ◦ Contact oscillations possible. ◦ Wall may generate energy (may be active). Haptic vs. physical interaction

Direct coupling – network model Direct coupling – network model

Impedance –Measures how much a system impedes motion. –Input: velocity. –Output: force. Varies with frequency.  Interaction behavior = dynamic relation among port variables (effort & flow). Admittance –Measures how much a system admits motion. –Input: force. –Output: velocity.

Interaction behavior of ideal mass (inertia) Interaction behavior of ideal mass (inertia)

Interaction behavior of ideal spring Interaction behavior of ideal spring

Interaction behavior of ideal dashpot Interaction behavior of ideal dashpot

Impedance behavior of typical mechanical system Impedance behavior of typical mechanical system

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Direct coupling – network model (ctd. II) Direct coupling – network model (ctd. II)  Continuous time system:  Sampled data system – never unconditionally stable!

Virtual coupling – mechanical model Virtual coupling – mechanical model Rigid Compliant connection between haptic device and virtual tool

Virtual coupling – closed loop model Virtual coupling – closed loop model

Virtual coupling – network model Virtual coupling – network model where:

Virtual coupling – absolute stability Virtual coupling – absolute stability Llewellyn’s criterion: Need: Physical damping. “Give” during rigid contact. Larger coupling damping for larger contact stiffness. Worst-case for stability: loose grasp during contact. Not transparent.

Virtual coupling – admittance device Virtual coupling – admittance device Impedance / admittance duality: Spring inertia. Damper damper. Need: “Give” during contact. Physical damping. Larger coupling damping for larger contact stiffness. Worst case for stability: rigid grasp during free motion.

Teleoperation Teleoperation  Unilateral: ◦ Transmit operator command.  Bilateral: ◦ Feel environment response.

Ideal bilateral teleoperator Ideal bilateral teleoperator Position & force matching: Impedance matching: Intervening impedance: Position & force matching impedance matching.

Bilateral teleoperation architectures Bilateral teleoperation architectures Position/Position (P/P). Position/Force (P/F). Force/Force (F/F). 4 channels (PF/PF). Local force feedback.

General bilateral teleoperator General bilateral teleoperator

General bilateral teleoperator (ctd.) General bilateral teleoperator (ctd.) Impedance transmitted to user: Perfect transparency: Trade-off between stability & transparency.

Haptic teleoperation Haptic teleoperation  Teleoperation: ◦ Master robot. ◦ Slave robot. ◦ Communication (force, velocity).  Haptics: ◦ Haptic device. ◦ Virtual tool. ◦ Communication (force, velocity).

4 channels teleoperation for haptics 4 channels teleoperation for haptics Transparency requirement: –Stiffness rendering (perfect transparency): –Inertia rendering (intervening impedance): Control/VE design separated.

Direct coupling as bilateral teleoperation Direct coupling as bilateral teleoperation Perfect transparency. Potentially unstable.

Virtual coupling as bilateral teleoperation Virtual coupling as bilateral teleoperation Not transparent. Unconditionally stable.

Stability analysis Stability analysis Questions: Is the system stable? How stable is the system? Analysis methods: Linear systems: Analysis including Z h and Z e : Non-conservative. Need user & virtual environment models. Analysis without Z h and Z e : Z h and Z e restricted to passive operators. Conservative. No need for user & virtual environment models. Nonlinear systems – based on energy concepts (next lecture): Lyapunov stability. Passivity.

Stability analysis with Z h and Z e Stability analysis with Z h and Z e Can incorporate time delays. Methods: Routh-Hurwitz: Stability given as a function of multiple variables. Root locus: Can analyze performance. Stability given as function of single parameter. Nyquist stability. Lyapunov stability: Stability margins not available. Cannot incorporate time delay. Small gain theorem: Conservative: considers only magnitude of OL.  -analysis: Accounts for model uncertainties.

Stability analysis without Z h and Z e Stability analysis without Z h and Z e Absolute stability: network is stable for all possible passive terminations. Llewellyn’s criterion: h 11 (s) and h 22 (s) have no poles in RHP. Poles of h 11 (s) and h 22 (s) on imaginary axis are simple with real & positive residues. For all frequencies: Network stability parameter (equivalent to last 3 conditions): Perfect transparent system is marginally absolutely stable, i.e.,

Stability analysis without Z h and Z e (ctd. I) Stability analysis without Z h and Z e (ctd. I) Passivity: Raisbeck’s passivity criterion: h-parameters have no poles in RHP. Poles of h-parameters on imaginary axis are simple with residues satisfying: For all frequencies:

Stability analysis without Z h and Z e (ctd. II) Stability analysis without Z h and Z e (ctd. II) Passivity: Scattering parameter S: As function of h-parameters: Perfect transparent system is marginally passive, i.e.,

Passivity – absolute stability – potential instability [Haykin ’70] Passivity – absolute stability – potential instability [Haykin ’70]