Haptics and Virtual Reality

1. Haptics and Virtual Reality Lecture 11: • Haptic Control M. Zareinejad

2. why do instabilities occur? • 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)

3. Passivity • 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.

4. Passive system properties A passive system is stable

5. Z-width • “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

6. How do you improve Z-width? • 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)

7. stability of thevirtual wall sampled-data system example

10. 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.

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

12. 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

13. Direct coupling – network model

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

15. Interaction behavior of ideal mass (inertia)

16. Interaction behavior of ideal spring

17. Interaction behavior of ideal dashpot

18. Impedance behavior of typical mechanical system

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

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

32. Virtual coupling – closed loop model

33. Virtual coupling – network model where:

34. 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.

35. 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.

36. Teleoperation Unilateral: Transmit operator command. Bilateral: Feel environment response.

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

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

39. General bilateral teleoperator

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

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

42. 4 channels teleoperation for haptics • Transparency requirement: • Stiffness rendering (perfect transparency): • Inertia rendering (intervening impedance): • Control/VE design separated.

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

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

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

46. Stability analysis with Zh and Ze • 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. • m-analysis: • Accounts for model uncertainties.

47. Stability analysis without Zh and Ze • Absolute stability: network is stable for all possible passive terminations. • Llewellyn’s criterion: • h11(s) and h22(s) have no poles in RHP. • Poles of h11(s) and h22(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.,

48. Stability analysis without Zh and Ze (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:

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

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