Abstract
In this study, we derive sensitivity equations for the problem of optimization-based motion prediction of a mechanical system using the inverse recursive Lagrangian formulation. The simulation and sensitivity formulations are based on Denavit–Hartenberg transformation matrices. External forces and moments are taken into account in the formulation. The sensitivity information is needed in the optimization-based simulation process. The proposed formulation is demonstrated by calculating sensitivities for the optimal time trajectory planning problem of a two-link manipulator. In addition, sensitivities obtained using the proposed algorithm are compared to those obtained using the closed-form equations of motion. The two sensitivities match quite closely. The lifting motion of the two-link manipulator with external loads is also optimized by using the algorithm developed in this paper. More complex applications of the proposed formulation to digital human motion prediction are presented elsewhere.
Similar content being viewed by others
References
Anderson K, Hsu Y (2002) Analytical fully-recursive sensitivity analysis for multibody dynamic chain systems. Multibody Syst Dyn 8:1–27
Anderson FC, Pandy MG (2001) Dynamic optimization of human walking. J Biomech Eng 123(5):381–390
Armstrong WM (1979) Recursive solution to the equations of motion of an N-link manipulator. In: Proceedings of the 5th world congress on theory of machines and mechanisms, vol. 2, pp 1343–1346
Bae DS, Haug EJ (1987) A recursive formulation for constrained mechanical system dynamics: part I. Open loop systems. Mechan Struct Mach 15(3):359–382
Bae DS, Cho H, Lee S, Moon W (2001) Recursive formulas for design sensitivity analysis of mechanical systems. Comput Methods Appl Mech Eng 190:3865–3879
Barthelemy JFM, Hall LE (1995) Automatic differentiation as a tool in engineering design. Struct Optim 9:76–82
Bessonnet G, Lallemand JP (1990) Optimal trajectories of robot arms minimizing constrained actuators and travelling time. IEEE Int Conf Robot Autom 1:112–117
Chung HJ, Xiang Y, Mathai A, Rahmatalla S, Kim J, Marler T, Beck S, Yang J, Arora JS, Abdel-Malek K, Obuseck J (2007) A robust formulation for prediction of human running. 2007 Digital human modeling for design and engineering symposium, Seattle, Washington, 16–18 June
Denavit J, Hartenberg RS (1955) A kinematic notation for lower-pair mechanisms based on matrices. J Appl Mech 77:215–221
Dissanayake MWMG, Goh CJ, Phan-Thien N (1991) Time-optimal trajectories for robot manipulators. Robotica 9:131–138
Eberhard P, Bischof C (1999) Automatic differentiation of numerical integration algorithms. Math Comput 68:717–731
Eberhard P, Schiehlen W (2006) Computational dynamics of multibody systems: history, formalisms, and applications. J Comput Nonlinear Dyn 1:1–12
Featherstone R (1987) Robot dynamics algorithms. Kluwer, Boston
Fu KS, Gonzalez RC, Lee CSG (1987) Robotics: control, sensing, vision, and intelligence. McGraw-Hill, New York
Furukawa T (2002) Time-subminimal trajectory planning for discrete non-linear systems. Eng Optim 34:219–243
Gill PE, Murray W, Saunders MA (2002) SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM J Optim 12:979–1006
Hollerbach JM (1980) A recursive Lagrangian formulation of manipulator dynamics and a comparative study of dynamics formulation complexity. IEEE Trans Syst Man Cybern 11(10):730–736
Hsieh CC, Arora JS (1984) Design sensitivity analysis and optimization of dynamic response. Comput Methods Appl Mech Eng 43(2):195–219
Kim JG, Baek JH, Park FC (1999) Newton-type algorithms for robot motion optimization. Proceedings of IEEE/RSJ international conference on intelligent robots and systems 3:1842–1847
Lo J, Huang G, Metaxas D (2002) Human motion planning based on recursive dynamics and optimal control techniques. Multibody Syst Dyn 8(4):433–458
Luh J, Walker M, Paul R (1980) On-line computational scheme for mechanical manipulators. J Dyn Syst Meas Control 102(2):69–76
Orin D, McGhee R, Vukobratovic M, Hartoch G (1979) Kinematic and kinetic analysis of open-chain linkages utilizing Newton–Euler methods. Math Biosci 43(1–2):107–130
Park FC, Bobrow JE, Ploen SR (1995) A lie group formulation of robot dynamics. Int J Robot Res 14(6):609–618
Rein U (1995) Efficient object-oriented programming of multibody dynamics formalisms. Computational dynamics in multibody systems. Kluwer, Dordrecht, The Netherlands, pp 37–47
Rodríguez JI, Jiménez JM, Funes FJ, Jalón JGD (2004) Recursive and residual algorithms for the efficient numerical integration of multi-body systems. Multibody Syst Dyn 11:295–320
Schiehlen W (1997) Multibody system dynamics: roots and perspectives. Multibody Syst Dyn 1:149–188
Serban R, Haug EJ (1998) Kinematic and kinetic derivatives in multibody system analysis. Mech Struct Mach 26(2):145–173
Snyman JA, Berner DF (1999) A mathematical optimization methodology for the optimal design of a planar robotic manipulator. Int J Numer Methods Eng 44:535–550
Sohl GA, Bobrow JE (2001) A recursive multibody dynamics and sensitivity algorithm for branched kinematic chains. J Dyn Syst Meas Control 123:391–399
Toogood RW (1989) Efficient robot inverse and direct dynamics algorithms using micro-computer based symbolic generation. IEEE Int Conf Robot Autom 3:1827–1832
Uicker JJ (1965) On the dynamic analysis of spatial linkages using 4 × 4 matrices. Ph.D. thesis, Northwestern University, Evanston, IL, USA
Wang Q, Xiang Y-J, Kim H-J, Arora JS, Abdel-Malek K (2005) Alternative formulations for optimization-based digital human motion prediction. Paper 2005-01-2691, 2005 digital human modeling for design and engineering symposium, Iowa City, IA, 14–16 June
Wang Q, Xiang Y-J, Arora JS, Abdel-Malek K (2007) Alternative formulations for optimization-based human gait planning. 48th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference, Honolulu, Hawaii, 23–26 April
Xiang Y, Chung HJ, Mathai A, Rahmatalla S, Kim J, Marler T, Beck S, Yang J, Arora JS, Abdel-Malek K, Obuseck J (2007) Optimization-based dynamic human walking prediction. 2007 digital human modeling for design and engineering symposium, Seattle, Washington, 11–14 June
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xiang, Y., Arora, J.S. & Abdel-Malek, K. Optimization-based motion prediction of mechanical systems: sensitivity analysis. Struct Multidisc Optim 37, 595–608 (2009). https://doi.org/10.1007/s00158-008-0247-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-008-0247-2