A Brief History of Passivity-Based Control in Robotics

1. A Brief History of Passivity-Based Control in Robotics Mark W. Spong, Dean Erik Jonsson School of Engineering and Computer Science Lars Magnus Ericsson Chair in Electrical Engineering Excellence in Education Chair University of Texas at Dallas 800 W. Campbell Road Richardson, Texas 75080 mspong@utdallas.edu

2. About the Jonsson School The Erik Jonsson School of Engineering and Computer Science at the University of Texas at Dallas was established in 1986 with two departments • Electrical Engineering • Computer Science • Degree programs in Electrical Engineering, Computer Science, Computer Engineering, Telecommunications Engineering and Software Engineering

3. About the Jonsson School • I left UIUC in 2008 to become the fourth dean of the Jonsson School • Since 2008 we have more than doubled the size of the School with four new departments and nine new degree programs in: • Materials Science & Engineering (MS, PhD) • Mechanical Engineering (BS, MS, PhD) • Bioengineering (BS, MS, PhD) • Systems Engineering and Management (MS)

4. About the Jonsson School • Overall, UT Dallas has: • 26,000 students • 80% in Science, Engineering, and Business • Among universities younger than 50 years • old, UT Dallas ranks #1 in Texas, #3 in the • United States, and #16 in the world Fall 2008 • 90 Tenure-Track faculty • 2,700 students (60% undergrad; 40% grad) Fall 2016 • 165 Tenure-Track Faculty • 7,000 students (60% undergrad; 40% grad)

5. CCST researchers are actively engaged in theory and applications of control. Application areas include robotics, healthcare systems, energy systems, automotive systems, biomedical systems research, operations research and optimization.  Mark Spong Reza Moheimani Mario Rotea M. Vidyasagar Steve Yurkovich, Director Bob Hunt Suresh Sethi Farokh Bastani Babak Fahimi Bobby Gregg Roozbeh Jafari Nick Gans Ozap Ozer Vish Ramakrishnan Raimund Ober Alain Benssousain B Prabhakaran Yaoyu Li

6. Outline • Historical Perspective • The Passivity Paradigm • Passivity-Based Robust and Adaptive Control • Scattering Theory and Teleoperation • Hybrid and Switching Control • Passivity-Based Control of Bipedal Locomotion • Control of Networked Systems • Feedback Passivation • Small Gain Theorem and Input/Output Stability • IDC-PBA, Controlled Lagrangians, Geometric Reduction

7. Outline • Historical Perspective • The Passivity Paradigm • Passivity-Based Robust and Adaptive Control • Scattering Theory and Teleoperation • Hybrid and Switching Control • Passivity-Based Control of Bipedal Locomotion • Control of Networked Systems • Feedback Passivation • Small Gain Theorem and Input/Output Stability • IDC-PBA, Controlled Lagrangians, Geometric Reduction

8. HistoricalPerspective The term Passivity-Based Control was introduced in R. Ortega and M.W. Spong, “Adaptive Control of Robot Manipulators: A Tutorial,” Proc. 27th IEEE Conference on Decision and Control,” Vol. 2, pp.1575 – 1584, Austin, TX, Dec. 1988. R. Ortega and M.W. Spong, “Adaptive Control of Robot Manipulators: A Tutorial,” Automatica, Vol. 25, No. 6, pp. 877-888, 1989. (1130 citations on google scholar) The first known reference of this term from these papers is L. Liu, Y. Han, R. Lingarkar, N.K. Sinha and M.A. Elbestawi, “On Adaptive Force/Motion Control of Constrained Robots,” 15th Annual Conference of  Industrial Electronics Society, (IECON '89), Vol. 2, pp. 433 - 438, Philadelphia, PA, 1989. This talk is an attempt to recount, from a personal perspective, a small portion of the history of Passivity-Based Control,from the 1980s forward, specific to problems in Robotics.

9. Historical Perspective Passivity concepts are familiar and well documented in control and will not be repeated here. Consider a dynamical system represented by the state model with the same number of inputs and outputs, where f(x,u) is locally Lipschitz, h(x,u) is continuous with , The above system is Passive if there is scalar function satisfying or equivalently

10. HistoricalPerspective Passivity-Based Controltraces its roots toCircuits and Systems Theory and Passive Network Synthesis Aizerman’s Conjecture, Absolute Stability, Circle Criterion, Popov Criterion, Small-Gain Theorem, Input/Output Stability D. C. Youla, L. J. Castriota, and H. J. Carlin, “Bounded real scattering matrices and the foundations of linear passive network theory,” IRE Trans., Circuit Theory. vol. CT-6, pp. 102-124, March 1959. R. E. Kalman, “On a New Characterization of Linear Passive Systems,” Proc. Allerton Conference on Communication, Control, and Computing, University of Illinois, 1963. R.E. Kalman, “Lyapunov Functions for the Problem of Lur`e in Automatic Control, Proc Nat Acad Sci, 49 (2): 201–205, 1963. B.D.O. Anderson, “The small-gain theorem, the passivity theorem and their equivalence,” Journal of the Franklin Institute, Volume 293, Issue 2, February 1972, Pages 105–115.

11. HistoricalPerspective The traditional signal-processing or frequency domain approach to control design works well for linear systems. Typical robot control problems involve highly complex, nonlinear systems for which such methods do not work as well. Passivity-Based Control is Energy-Based as opposed to Signal-Based and views control in terms of Energy Balance of Interconnections

12. HistoricalPerspective Motivation for Energy/Passivity methods can be traced back to Lagrange (1788) and Dirichlet (1830) who showed that the stable equilibria of mechanical systems correspond to the minima of the potential function. This gives rise to modern energy-shaping plus damping injection methods that first modify the potential energy to introduce a unique global minimum and then modify the damping properties to make the system strictly passive – and the equilibrium asymptotically stable.

13. HistoricalPerspective An early use of Potential Energy Shaping and Damping Injection, and a foreshadowing of passivity-based control in robotics was M. Takegaki and S. Arimoto, “A New Feedback Method for Dynamic Control of Manipulators,” J. Dyn. Sys. Meas. and Cont., Vol 102, pp. 119-125, 1981. The work was fundamental in that it showed rigorously how a relatively simple control, based on energy shaping, can stabilize the complex nonlinear robot dynamics. It also used a Hamiltonian formulation of the robot dynamics thus also foreshadowing IDA-PBC A later refinement of this idea of potential energy shaping was in D. Koditschek, “Natural Motion of Robot Arms,” IEEE Conf. Decision and Control, Las Vegas, 1984.

14. HistoricalPerspective Passivity-Based Control is now applied in many areas such as: • Mechanical systems: walking robots, bilateral teleoperators, pendular systems. • Chemical processes: mass–balance systems, inventory control, reactors • Electrical systems: power systems, power converters. • Electromechanical systems: motors, magnetic levitation systems • Transportation systems: underwater vehicles, surface vessels, aircraft, spacecraft. • Control over networks: formation control, synchronization, consensus problems, drones. • Hybrid systems: switched systems, hybrid passivity A search for Passivity Based Control yields: 200,000+ hits on google scholar 1,665 hits on ieeexplore

15. The Passivity Paradigm A passive system may be stabilized by output feedback Since then we have • Under additional conditions (Detectability), asymptotic stability follows. • Generally, convergence is to an Invariant Manifold. • Parallel and Feedback Interconnections of Passive Systems are Passive.

16. The Passivity Paradigm Moreover, passive-based control can handle things like: Input saturation Block, D.J., Astrom, K.J., and Spong, M.W., The Reaction Wheel Pendulum, Morgan & Claypool Publishers, 2007. Passive-based control can also deal with: Time Delays (Lyapunov-Kraskovsky) Control Switching (Common Lyapunov Functions)

17. The Passivity Paradigm The fundamental unifying paradigm: • Identify the appropriate passive output and feed it back or • Create new passive I/O pairs and feedback the passive output The control is then simple. The problem is shifted to finding the right Passive I/O relationships.

18. Robust and Adaptive Control Passivity was used in adaptive controlof manipulators in: Slotine, J.J.E., and Li, W., “On the Adaptive Control of Robot Manipulators,” Int. J. Robotics Res., Vol 6, No. 3, pp. 147-157, Fall, 1987. I. Landau and R.Horowitz, "Synthesis of Adaptive Controllers for robot manipulators using a passive feedback systems approach." Proc. IEEE conf. on Robotics and Automation. Philadelphia, 1988. Passivity was used in robust control of manipulators in: C.-Y. Su , T. P. Leung and Q.-J. Zhou  "A novel variable structure control scheme for robot trajectory control,”  11th IFAC World Congress,  vol. 9,  pp.121 -124 1990.  M.W. Spong, “On the Robust Control of Robot Manipulators,” IEEE Trans. Aut. Cont., Vol. 37, No. 11, pp. 1782-1786, 1992.

19. Robust and Adaptive Control The key properties of skew symmetry and linearity-in-the-parameters were the key enablers for the problem of robust and adaptive control of Lagrangian systems. M.W. Spong, S. Hutchinson, M. Vidyasagar, Robot Modeling and Control, John Wiley & Sons, Inc., 2006. The condition is equivalent to passivity from input torque to output velocity of the Lagrangian dynamics. Skew symmetry of implies Passivity and that for any vector r This distinction is important.

20. Robust and Adaptive Control With the Lagrangian dynamics expressed as The passivity-based control law is given as where resulting in

21. Robust and Adaptive Control In the case of robust control, one assumes knowledge of a nominal parameter vector so that Setting One can use, for example, Lyapunov Redesign(Leitmann) or Sliding Mode Designto choose In general, one achieves uniform ultimate boundedness of the tracking error with a smooth control law.

22. Robust and Adaptive Control In the case of adaptive control, one assume no knowledge of the parameter vector … One then sets The parameter update law achieves global asymptotic stability of the tracking error and boundedness of the parameter estimates (with the usual caveats of adaptive control).

23. Scattering Theory and Teleoperation Bilateral Teleoperators can be modeled as in interconnection of 2-port networks B. Hannaford, “A Design Framework for Teleoperators with Kinesthetic Feedback,” IEEE Transactions on Robotics and Automation, Vol 5 No 4, pp. 426–434 1989 Velocity Human Master Comm Slave Environ Force The interconnection of passive 2-ports is passive

24. Scattering Theory and Teleoperation Velocity Human Master Comm Slave Environ • Master and Slave Robots = Passive • Environment = Passive (mass, spring, damper) • Human (under some assumptions) = passive • Communication Block = pure time delay = Not Passive! Force

25. Scattering Theory and Teleoperation The communication introduces a time delay, T, and is made passive by the well-known Scattering Transformation approach [Anderson and Spong, 1989], where the scattering variables Master Scattering Transformation Delay Scattering Transformation Slave are transmitted across the delay line instead of the original velocities and forces The cascade connections of the Scattering Transformations and time delay becomes a Lossless Transmission Line = Passive Independent of the Time Delay.

26. Scattering Theory and Teleoperation R. Anderson and M. W. Spong, “Bilateral Teleoperation with Time Delay,” IEEE Trans. Aut. Cont., Vol. 34, No. 5, pp. 494-501, May, 1989. Anderson, R.J., and Spong, M.W., “Asymptotic Stability for Force Reflecting Teleoperators with Time Delay,” The International Journal on Robotics Research, Vol.11, No.2, pp.135-149, Apr.,1992. Niemeyer and Slotine reformulated this as Wave Variablesin G. Niemeyer and J.-J. E. Slotine, “Stable Adaptive Teleoperation,” IEEE Journal of Oceanographic Engineering, pp Vol. 16, No. 1, pp. 152–162, 1991. and introduced Impedance Matching to avoid Reflections

27. Hybrid and Switching Control Passivity is also useful for stabilization of hybrid and switched systems. Mareczek, Joerg, Buss, M., and Spong, M.W., "Invariance Control for a Class of Cascade Nonlinear Systems," IEEE Trans. Aut. Control, Vol 47, No.4, pp. 636-640, April, 2002.

28. Hybrid and Switching Control

29. Control Objectives

30. Hybrid and Switching Control

31. Passivity-Based Switching Control Another approach, which we use in the referenced paper, is to exploit passivity and switching control to achieve stability and/or invariance of a given state-space region. The basic idea is:

32. Passivity-Based Switching Control

33. Example

36. Applications This idea was used for the problem of rollover stabilization of heavy vehicles in: D Wollherr, J Mareczek, M Buss, G Schmidt, “Rollover avoidance for steerable vehicles by invariance control,” Proceedings of the European Control Conference, 3522-3527, 2001. A multi-input extension was used for an aircraft autopilot to enforce a no-fly zone in: Cronin, B. and Spong, M.W., “Switching Control for Multi-Input Cascade Nonlinear Systems,'' IEEE CDC 2003, Maui, Hawaii, December, 2003.

37. Controlled Symmetries The next topic of discussion is Controlled Symmetries in Lagrangian Systems. This notion was defined in: M.W. Spong and F. Bullo, “Controlled Symmetries and Passive Walking,” IEEE Transactions on Automatic Control, Vol 50 No 7 pp 1025 Vol. 50, No. 7, pp 1025-1031 2005. It has subsequently been used in: G Russo, JJE Slotine, “Symmetries, stability, and control in nonlinear systems and networks,” Physical Review E 84 (4) L. Gerard and J-J. E. Slotine, “Neuronal Networks and Controlled Symmetries, A Generic Framework,” Research Gate, 2006.

38. Passive Dynamic Walking It is well known that locomotion of simple mechanisms is achievable passively – i.e. without sensing or actuation McGeer, T., “Passsive Dynamic Walking,” Int. J. of Robotics Research, 1990 Goswami, et.al., “A Study of the Passive Gait of a Compass-like Biped Robot: Symmetry and Chaos,” International Journal of Robotics Research, 1998. Collins, SH, Wisse, M, Ruina, A, “A 3-D passive-dynamic walking robot with two legs and knees,” Int. J. of Robotics Research, 2001. Passive walking involves a delicate balance among: • Kinetic energy • Potential energy • Impact energy The Cornell 3D passive walker of Collins, Wisse and Ruina

39. Dynamics and Control • The control problem for biped robots is complicated by • nonlinear dynamics • unilateral constraints (foot/ground) • impacts (impulsive inputs) • We consider a general n-DOF biped in 3D • The Configuration Space is

40. Symmetry in Lagrangian Systems Symmetries give rise to conserved quantities. For example, translational symmetry gives rise to conservation of momentum, etc.

41. Controlled Symmetry

42. Controlled Symmetry Limit cycles for the compass-gait biped for three different ground slopes Compass-Gait Biped on Level Ground

43. Controlled Symmetry The controlled symmetry control can be applied to any biped for which a passive limit cycle exists Biped with knees (above) and with knees and torso (right)

44. Passivity-Based Control

45. Passivity-Based Control

46. Passivity-Based Control Plot of the Storage Function for the Compass-Gait Biped k=0.5 With PBC (right) Without PBC (left)

47. Passivity-Based Control The Passivity-Based Control increases the speed of convergence to the limit cycle

48. Passivity-Based Control The increased basin of attraction and speed of convergence to the limit cycle is important when slope changes are considered. The local slope, determined by the two-point contact, is used in the controlled symmetry and passivity-based control to achieve robustness to the slope change. Animation of the Compass-Gait Biped on a Sinusoidally Changing Ground Slope

49. Consensus Control of Multi-Agent Systems The usual starting point is to consider a first-order model for the dynamics of individual agents together with the usual consensus control 2 1 which is written as 3 4 5 where is the Laplacian of the graph defined by the network (communication, sensing, etc.) This is an example of Passivity-Based Control.

50. Consensus Control of Multi-Agent Systems To see this, note that the system is Passive with Storage function 2 1 since 3 4 5 As a result, the Consensus Protocol amounts to feedback of the difference of the passive outputs