Close tracking of equilibrium paths
Corresponding Author
Cornel Sultan
Kevin T. Crofton Department of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA
Correspondence
Cornel Sultan, Kevin T. Crofton Department of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0203, USA.
Email: [email protected]
Search for more papers by this authorCorresponding Author
Cornel Sultan
Kevin T. Crofton Department of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA
Correspondence
Cornel Sultan, Kevin T. Crofton Department of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0203, USA.
Email: [email protected]
Search for more papers by this authorSummary
A method to control a generic system of nonlinear ordinary differential equations between equilibrium states is analyzed. The objective is to ensure that the system's state space trajectory closely tracks an equilibrium path. The control law is obtained via time parameterization of the corresponding equilibrium control path. Conditions which guarantee that the system's state space trajectory closely tracks the equilibrium path are proved using two approaches. One approach uses the mean value theorem, and the other uses the slowly time-varying systems theory. Importantly, both methods provide relationships between the control rate norm and the tracking error norm. These allow computation of upper bounds on the control rate norm which guarantee a desired upper bound on the tracking error norm. They also enable computation of upper bounds on the tracking error norm for a given upper bound on the control rate norm. Examples illustrate the theoretical results.
REFERENCES
- 1Sultan C, Skelton RE. Deployment of tensegrity structures. Int J Solids Struct. 2003; 40(18): 4637-4657.
- 2Mallikarachchi HMYC, Pellegrino S. Quasi-static folding and deployment of ultrathin composite tape-spring hinges. J Spacecr Rocket. 2011; 48(1): 187-198.
- 3Rhode-Barbarigos L, Schulin C, Bel Hadj Ali N, Motro R, Smith IFC. Mechanism-based approach for the deployment of a tensegrity-ring module. ASCE J Struct Eng. 2012; 138(4): 539-548.
- 4Mankala KK, Agrawal SK. Equilibrium-to-equilibrium maneuvers of flexible electrodynamic tethers in equatorial orbits. J Spacecr Rocket. 2006; 43(3): 651-658.
- 5Fehse W. Automated Rendezvous and Docking of Spacecraft. Cambridge Aerospace Series. vol 16. Cambridge, UK: Cambridge University Press; 2003.
10.1017/CBO9780511543388 Google Scholar
- 6Stephane R, Luu T, Berube A. Space shuttle testing of the TriDAR 3D rendezvous and docking sensor. J Field Robot. 2012; 29(4): 535-553.
- 7Stephens JP, Vos GA, Bilimoria KD, Mueller ER, Brazzel J, Spehar P. Orion handling qualities during international space station proximity operations and docking. J Spacecr Rocket. 2013; 50(2): 449-457.
- 8Anshelevich E, Owens S, Lamiraux F, Kavraki LE. Deformable volumes in path planning applications. Paper presented at: Proceedings of the IEEE International Conference on Robotics and Automation; 2000; San Francisco, CA.
- 9Lamiraux F, Kavraki LE. Planning paths for elastic objects under manipulation constraints. Int J Robot Res. 2001; 20(3): 188-208.
- 10Moll M, Kavraki LE. Path planning for deformable linear objects. IEEE Trans Robot. 2006; 22(4): 625-636.
- 11Rentschler ME, Farritor SM, Iagnemma KD. Mechanical design of robotic in vivo wheeled mobility. J Mech Des. 2007; 129(10): 1037-1045.
- 12Wang X, Sliker LJ, Qi HJ, Rentschler ME. A quasi-static model of wheel–tissue interaction for surgical robotics. Med Eng Phys. 2013; 35(9): 1368-1376.
- 13Alderliesten T, Konings MK, Niessen WJ. Simulation of minimally invasive vascular interventions for training purposes. Comput Aided Surg. 2004; 9(1-2): 3-15.
- 14Baer SM, Erneux T, Rinzel J. The slow passage through a Hopf bifurcation: delay, memory effects, and resonance. SIAM J Appl Math. 1989; 49(1): 55-71.
- 15Pontryagin LS, Boltyasnskii VG, Gamkrelidze RV, Mishchenko EF. The Mathematical Theory of Optimal Processes. New York, NY: Interscience Publishers; 1963.
- 16Slotine JJ, Sastry SS. Tracking control of non-linear systems using sliding surfaces with application to robot manipulators. Int J Control. 1989; 38(2): 465-492.
- 17Suykens JAK, De Moor BLR, Vandewalle J. Static and dynamic stabilizing neural controllers applicable to transition between equilibrium points. Neural Netw. 1994; 7(5): 819-831.
- 18Camacho EF, Bordons C. Model Predictive Control. London, UK: Springer; 2007.
10.1007/978-0-85729-398-5 Google Scholar
- 19Spong MW, Hutchinson SR, Vidyasagar M. Robot Modeling and Control. New York, NY: Wiley; 2006.
- 20Yoshikawat T, Hosodat K, Doit T, Murakami H. Quasi-static trajectory tracking control of flexible manipulator by macro-micro manipulator system. Paper presented at: Proceedings IEEE Conference on Robotics and Automation; 1993; Atlanta, GA.
- 21Dhanda A, Franklin G. Real-time generation of time-optimal commands for rest-to-rest motion of flexible systems. IEEE Trans Control Syst Technol. 2013; 21(3): 958-963.
- 22Sultan C. Tensegrity deployment using infinitesimal mechanisms. Int J Solids Struct. 2014; 51(21): 3653-3668.
- 23Yang S, Sultan C. Control-oriented modeling and deployment of tensegrity–membrane system. Int J Robust Nonlinear Control. 2017; 27(16): 2722-2748. https://doi.org/10.1002/rnc.3708
- 24Silvestre C, Pascoal A, Kaminer I. On the design of gain-scheduled trajectory tracking controllers. Int J Robust Nonlinear Control. 2002; 12(9): 797-839.
- 25Balas GJ. Linear parameter-varying control and its application to a turbofan engine. Int J Robust Nonlinear Control. 2002; 12(9): 763-796.
- 26Shu Y, Sultan C. LPV control of a tensegrity-membrane system. Mech Syst Signal Process. 2017; 95: 397-424.
- 27Rugh WJ, Shamma JS. Research on gain scheduling. Automatica. 2000; 36: 1401-1425.
- 28Shamma JS. Overview of LPV systems. In: Control of Linear Parameter Varying Systems with Applications. J Mohammadpour, CW Scherer, eds. New York, NY: Springer; 2012: 3-26.
10.1007/978-1-4614-1833-7_1 Google Scholar
- 29Khalil HK. Nonlinear Systems. Upper Saddle River, NJ: Prentice Hall; 2002.
- 30Kelemen M. A stability property. IEEE Trans Autom Control. 1986; 31(8): 766-768.
- 31Lawrence DA, Rugh WJ. On a stability theorem for nonlinear systems with slowly varying inputs. IEEE Trans Autom Control. 1990; 35(7): 860-864.
- 32Sultan C. Nonlinear systems control using equilibrium paths. Paper presented at: Proceedings IEEE Conference on Decision and Control; 2007; New Orleans, LA.
- 33Hoppensteadt F. Stability in systems with parameter. J Math Anal Appl. 1967; 18: 129-134.
- 34Khalil HK, Kokotovic P. On stability properties of nonlinear systems with slowly varying inputs. IEEE Trans Autom Control. 1991; 36(2): 229.
- 35Sultan C. Advances of oscillations in Hopf bifurcation revisited. Paper presented at: Proceedings of the American Control Conference; 2015; Chicago, IL.
- 36Sultan C. Stiffness formulations and necessary and sufficient conditions for exponential stability of prestressable structures. Int J Solids Struct. 2003; 50(14): 2180-2195.
- 37Sultan C. On the nonlinear dynamic stability of prestressable structures. Paper presented at: Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference; 2015; Boston, MA
- 38Wang Y, Ameer GA, Sheppard BJ, Langer R. A tough biodegradable elastomer. Nat Biotechnol. 2002; 20: 602-606.
- 39Lee LY, Wu SC, Fu SS, Zeng SY, Leong WS, Tan LP. Biodegradable elastomer for soft tissue engineering. Eur Polym J. 2009; 45(11): 3249-3256.
- 40Sonnenschein MF, Ginzburg VV, Schiller KS, Wendt BL. Design, polymerization, and properties of high performance thermoplastic polyurethane elastomers from seed-oil derived soft segments. Polym. 2013; 54(4): 1350-1360.
