Model Identification

  • John HollerbachEmail author
  • Wisama Khalil
  • Maxime Gautier
Part of the Springer Handbooks book series (SHB)


This chapter discusses how to determine the kinematic parameters and the inertial parameters of robot manipulators. Both instances of model identification are cast into a common framework of least-squares parameter estimation, and are shown to have common numerical issues relating to the identifiability of parameters, adequacy of the measurement sets, and numerical robustness. These discussions are generic to any parameter estimation problem, and can be applied in other contexts.

For kinematic calibration, the main aim is to identify the geometric Denavit–Hartenberg (DH ) parameters, although joint-based parameters relating to the sensing and transmission elements can also be identified. Endpoint sensing or endpoint constraints can provide equivalent calibration equations. By casting all calibration methods as closed-loop calibration, the calibration index categorizes methods in terms of how many equations per pose are generated.

Inertial parameters may be estimated through the execution of a trajectory while sensing one or more components of force/torque at a joint. Load estimation of a handheld object is simplest because of full mobility and full wrist force-torque sensing. For link inertial parameter estimation, restricted mobility of links nearer the base as well as sensing only the joint torque means that not all inertial parameters can be identified. Those that can be identified are those that affect joint torque, although they may appear in complicated linear combinations.




best linear unbiased estimator




degree of freedom


instrumental variable


linear variable differential transformer


root mean square


  1. 6.1
    P.Z. Marmarelis, V.Z. Marmarelis: Analysis of Physiological Systems (Plenum, London 1978)CrossRefzbMATHGoogle Scholar
  2. 6.2
    C.W. Wampler, J.M. Hollerbach, T. Arai: An implicit loop method for kinematic calibration and its application to closed-chain mechanisms, IEEE Trans. Robotics Autom. 11, 710–724 (1995)CrossRefGoogle Scholar
  3. 6.3
    J.P. Norton: An Introduction to Identification (Academic, London 1986)zbMATHGoogle Scholar
  4. 6.4
    G.H. Golub, C.F. Van Loan: Matrix Computations (Johns Hopkins Univ. Press, Baltimore 1989)zbMATHGoogle Scholar
  5. 6.5
    H. West, E. Papadopoulos, S. Dubowsky, H. Cheah: A method for estimating the mass properties of a manipulator by measuring the reaction moments at its base, Proc. IEEE Int. Conf. Robotics Autom. (ICRA) (1989) pp. 1510–1516Google Scholar
  6. 6.6
    R.P. Paul: Robot Manipulators: Mathematics, Programming, and Control (MIT Press, Cambridge 1981)Google Scholar
  7. 6.7
    S.A. Hayati, M. Mirmirani: Improving the absolute positioning accuracy of robot manipulators, J. Robotics Syst. 2, 397–413 (1985)CrossRefGoogle Scholar
  8. 6.8
    J.M. Hollerbach, C.W. Wampler: The calibration index and taxonomy of kinematic calibration methods, Int. J. Robotics Res. 15, 573–591 (1996)CrossRefGoogle Scholar
  9. 6.9
    A. Goswami, A. Quaid, M. Peshkin: Identifying robot parameters using partial pose information, IEEE Control Syst. 13, 6–14 (1993)CrossRefGoogle Scholar
  10. 6.10
    M.R. Driels, W.E. Swayze: Automated partial pose measurement system for manipulator calibration experiments, IEEE Trans. Robotics Autom. 10, 430–440 (1994)CrossRefGoogle Scholar
  11. 6.11
    G.-R. Tang, L.-S. Liu: Robot calibration using a single laser displacement meter, Mechatronics 3, 503–516 (1993)CrossRefGoogle Scholar
  12. 6.12
    D.E. Whitney, C.A. Lozinski, J.M. Rourke: Industrial robot forward calibration method and results, ASME J. Dyn. Syst. Meas. Control 108, 1–8 (1986)CrossRefzbMATHGoogle Scholar
  13. 6.13
    B.W. Mooring, Z.S. Roth, M.R. Driels: Fundamentals of Manipulator Calibration (Wiley, New York 1991)Google Scholar
  14. 6.14
    K. Lau, R. Hocken, L. Haynes: Robot performance measurements using automatic laser tracking techniques, Robotics Comput. Manuf. 2, 227–236 (1985)CrossRefGoogle Scholar
  15. 6.15
    C.H. An, C.H. Atkeson, J.M. Hollerbach: Model-Based Control of a Robot Manipulator (MIT Press, Cambridge 1988)Google Scholar
  16. 6.16
    M. Vincze, J.P. Prenninger, H. Gander: A laser tracking system to measure position and orientation of robot end effectors under motion, Int. J. Robotics Res. 13, 305–314 (1994)CrossRefGoogle Scholar
  17. 6.17
    J.M. McCarthy: Introduction to Theoretical Kinematics (MIT Press, Cambridge 1990)Google Scholar
  18. 6.18
    D.J. Bennet, J.M. Hollerbach: Autonomous calibration of single-loop closed kinematic chains formed by manipulators with passive endpoint constraints, IEEE Trans. Robotics Autom. 7, 597–606 (1991)CrossRefGoogle Scholar
  19. 6.19
    W.S. Newman, D.W. Osborn: A new method for kinematic parameter calibration via laser line tracking, Proc. IEEE Int. Conf. Robotics Autom. (ICRA) (1993) pp. 160–165CrossRefGoogle Scholar
  20. 6.20
    X.-L. Zhong, J.M. Lewis: A new method for autonomous robot calibration, Proc. IEEE Int. Conf. Robotics Autom. (ICRA) (1995) pp. 1790–1795Google Scholar
  21. 6.21
    J.M. Hollerbach, D.M. Lokhorst: Closed-loop kinematic calibration of the RSI 6-DOF hand controller, IEEE Trans. Robotics Autom. 11, 352–359 (1995)CrossRefGoogle Scholar
  22. 6.22
    A. Nahvi, J.M. Hollerbach, V. Hayward: Closed-loop kinematic calibration of a parallel-drive shoulder joint, Proc. IEEE Int. Conf. Robotics Autom. (ICRA) (1994) pp. 407–412Google Scholar
  23. 6.23
    O. Masory, J. Wang, H. Zhuang: On the accuracy of a Stewart platform – Part II Kinematic calibration and compensation, Proc. IEEE Int. Conf. Robotics Autom., Piscataway (1994) pp. 725–731Google Scholar
  24. 6.24
    B. Armstrong-Helouvry: Control of Machines with Friction (Kluwer, Boston 1991)CrossRefzbMATHGoogle Scholar
  25. 6.25
    B. Armstrong-Helouvry, P. Dupont, C. de Canudas Wit: A survey of models, analysis tools and compensation methods for the control of machines with friction, Automatica 30, 1083–1138 (1994)CrossRefzbMATHGoogle Scholar
  26. 6.26
    F. Aghili, J.M. Hollerbach, M. Buehler: A modular and high-precision motion control system with an integrated motor, IEEE/ASME Trans. Mechatron. 12, 317–329 (2007)CrossRefGoogle Scholar
  27. 6.27
    W.S. Newman, J.J. Patel: Experiments in torque control of the Adept One robot, Proc. IEEE Int. Conf. Robotics Autom., Piscataway (1991) pp. 1867–1872Google Scholar
  28. 6.28
    W. Khalil, E. Dombre: Modeling, Identification and Control of Robots (Taylor Francis, New York 2002)zbMATHGoogle Scholar
  29. 6.29
    M. Gautier: Dynamic identification of robots with power model, Proc. IEEE Int. Conf. Robotics Autom. (ICRA) (1997) pp. 1922–1927CrossRefGoogle Scholar
  30. 6.30
    W. Khalil, M. Gautier, P. Lemoine: Identification of the payload inertial parameters of industrial manipulators, Proc. IEEE Int. Conf. Robotics Autom. (ICRA) (2007) pp. 4943–4948Google Scholar
  31. 6.31
    J. Swevers, W. Verdonck, B. Naumer, S. Pieters, E. Biber: An Experimental Robot Load Identification Method for Industrial Application, Int. J. Robotics Res. 21(8), 701–712 (2002)CrossRefGoogle Scholar
  32. 6.32
    P.P. Restrepo, M. Gautier: Calibration of drive chain of robot joints, Proc. IEEE Int. Conf. Robotics Autom. (ICRA) (1995) pp. 526–531Google Scholar
  33. 6.33
    P. Corke: In situ measurement of robot motor electrical constants, Robotica 23(14), 433–436 (1996)CrossRefGoogle Scholar
  34. 6.34
    S. Van Huffel, J. Vandewalle: The Total Least Squares Problem: Computational Aspects and Analysis (SIAM, Philadelphia 1991)CrossRefzbMATHGoogle Scholar
  35. 6.35
    M. Gautier, S. Briot: Global identification of drive gains parameters of robots using a known payload, ASME J. Dyn. Syst. Meas. Control 136(5), 051026 (2014)CrossRefGoogle Scholar
  36. 6.36
    W. Khalil, O. Ibrahim: General solution for the dynamic modeling of parallel robots, J. Intell. Robotics Syst. 49, 19–37 (2007)CrossRefGoogle Scholar
  37. 6.37
    S. Guegan, W. Khalil, P. Lemoine: Identification of the dynamic parameters of the Orthoglide, Proc. IEEE Int. Conf. Robotics Autom. (ICRA) (2003) pp. 3272–3277Google Scholar
  38. 6.38
    K. Schröer: Theory of kinematic modelling and numerical procedures for robot calibration. In: Robot Calibration, ed. by R. Bernhardt, S.L. Albright (Chapman Hall, London 1993) pp. 157–196Google Scholar
  39. 6.39
    H. Zhuang, Z.S. Roth: Camera-Aided Robot Calibration (CRC, Boca Raton 1996)Google Scholar
  40. 6.40
    M. Gautier: Numerical calculation of the base inertial parameters, J. Robotics Syst. 8, 485–506 (1991)CrossRefzbMATHGoogle Scholar
  41. 6.41
    J.J. Dongarra, C.B. Mohler, J.R. Bunch, G.W. Stewart: LINPACK User's Guide (SIAM, Philadelphia 1979)CrossRefGoogle Scholar
  42. 6.42
    W. Khalil, F. Bennis: Symbolic calculation of the base inertial parameters of closed-loop robots, Int. J. Robotics Res. 14, 112–128 (1995)CrossRefGoogle Scholar
  43. 6.43
    W. Khalil, S. Guegan: Inverse and direct dynamic modeling of Gough–Stewart robots, IEEE Trans. Robotics Autom. 20, 754–762 (2004)CrossRefGoogle Scholar
  44. 6.44
    W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery: Numerical Recipes in C (Cambridge Univ. Press, Cambridge 1992)zbMATHGoogle Scholar
  45. 6.45
    C.L. Lawson, R.J. Hanson: Solving Least Squares Problems (Prentice Hall, Englewood Cliffs 1974)zbMATHGoogle Scholar
  46. 6.46
    P.R. Bevington, D.K. Robinson: Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York 1992)Google Scholar
  47. 6.47
    B. Armstrong: On finding exciting trajectories for identification experiments involving systems with nonlinear dynamics, Int. J. Robotics Res. 8, 28–48 (1989)CrossRefGoogle Scholar
  48. 6.48
    J. Fiefer, J. Wolfowitz: Optimum designs in regression problems, Ann. Math. Stat. 30, 271–294 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 6.49
    Y. Sun, J.M. Hollerbach: Observability index selection for robot calibration, Proc. IEEE Int. Conf. Robotics Autom. (ICRA), Piscataway (2008) pp. 831–836Google Scholar
  50. 6.50
    J.H. Borm, C.H. Menq: Determination of optimal measurement configurations for robot calibration based on observability measure, Int. J. Robotics Res. 10, 51–63 (1991)CrossRefGoogle Scholar
  51. 6.51
    C.H. Menq, J.H. Borm, J.Z. Lai: Identification and observability measure of a basis set of error parameters in robot calibration, ASME J. Mech. Autom. Des. 111(4), 513–518 (1989)CrossRefGoogle Scholar
  52. 6.52
    M. Gautier, W. Khalil: Exciting trajectories for inertial parameter identification, Int. J. Robotics Res. 11, 362–375 (1992)CrossRefGoogle Scholar
  53. 6.53
    M.R. Driels, U.S. Pathre: Significance of observation strategy on the design of robot, J. Robotics Syst. 7, 197–223 (1990)CrossRefGoogle Scholar
  54. 6.54
    A. Nahvi, J.M. Hollerbach: The noise amplification index for optimal pose selection in robot calibration, Proc. IEEE Int. Conf. Robotics Autom. (ICRA) (1996) pp. 647–654CrossRefGoogle Scholar
  55. 6.55
    D. Daney, B. Madeline, Y. Papegay: Choosing measurement poses for robot calibration with local convergence method and Tabu search, Int. J. Robotics Res. 24(6), 501–518 (2005)CrossRefGoogle Scholar
  56. 6.56
    Y. Sun, J.M. Hollerbach: Active robot calibration algorithm, Proc. IEEE Int. Conf. Robotics Autom. (ICRA), Piscataway (2008) pp. 1276–1281Google Scholar
  57. 6.57
    T.J. Mitchell: An algorithm for the construction of D-Optimal experimental designs, Technometrics 16(2), 203–210 (1974)MathSciNetzbMATHGoogle Scholar
  58. 6.58
    J. Swevers, C. Ganseman, D.B. Tukel, J. De Schutter, H. Van Brussel: Optimal robot excitation and identification, IEEE Trans. Robotics Autom. 13, 730–740 (1997)CrossRefGoogle Scholar
  59. 6.59
    P.O. Vandanjon, M. Gautier, P. Desbats: Identification of robots inertial parameters by means of spectrum analysis, Proc. IEEE Int. Conf. Robotics Autom. (ICRA) (1995) pp. 3033–3038Google Scholar
  60. 6.60
    E. Walter, L. Pronzato: Identification of Parametric Models from Experimental Data (Springer, London 1997)zbMATHGoogle Scholar
  61. 6.61
    D.G. Luenberger: Optimization by Vector Space Methods (Wiley, New York 1969)zbMATHGoogle Scholar
  62. 6.62
    H.W. Sorenson: Least-squares estimation: from Gauss to Kalman, IEEE Spectr. 7, 63–68 (1970)CrossRefGoogle Scholar
  63. 6.63
    Z. Roth, B.W. Mooring, B. Ravani: An overview of robot calibration, IEEE J. Robotics Autom. 3, 377–386 (1987)CrossRefGoogle Scholar
  64. 6.64
    A.E. Bryson Jr., Y.-C. Ho: Applied Optimal Control (Hemisphere, Washington 1975)Google Scholar
  65. 6.65
    M. Gautier, P. Poignet: Extended Kalman fitering and weighted least squares dynamic identification of robot, Control Eng. Pract. 9, 1361–1372 (2001)CrossRefGoogle Scholar
  66. 6.66
    D.J. Bennet, J.M. Hollerbach, D. Geiger: Autonomous robot calibration for hand-eye coordination, Int. J. Robotics Res. 10, 550–559 (1991)CrossRefGoogle Scholar
  67. 6.67
    D.J. Bennet, J.M. Hollerbach, P.D. Henri: Kinematic calibration by direct estimation of the Jacobian matrix, Proc. IEEE Int. Conf. Robotics Autom. (ICRA) (1992) pp. 351–357Google Scholar
  68. 6.68
    P.T. Boggs, R.H. Byrd, R.B. Schnabel: A stable and efficient algorithm for nonlinear orthogonal distance regression, SIAM J. Sci. Stat. Comput. 8, 1052–1078 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  69. 6.69
    W.A. Fuller: Measurement Error Models (Wiley, New York 1987)CrossRefzbMATHGoogle Scholar
  70. 6.70
    J.-M. Renders, E. Rossignol, M. Becquet, R. Hanus: Kinematic calibration and geometrical parameter identification for robots, IEEE Trans. Robotics Autom. 7, 721–732 (1991)CrossRefGoogle Scholar
  71. 6.71
    G. Zak, B. Benhabib, R.G. Fenton, I. Saban: Application of the weighted least squares parameter estimation method for robot calibration, J. Mech. Des. 116, 890–893 (1994)CrossRefGoogle Scholar
  72. 6.72
    M. Gautier, A. Janot, P.O. Vandanjon: A New Closed-Loop Output Error Method for Parameter Identification of Robot Dynamics, IEEE Trans. Control Syst. Techn. 21, 428–444 (2013)CrossRefGoogle Scholar
  73. 6.73
    A. Janot, P.O. Vandanjon, M. Gautier: A Generic Instrumental Variable Approach for Industrial Robots Identification, IEEE Trans. Control Syst. Techn. 22, 132–145 (2014)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • John Hollerbach
    • 1
    Email author
  • Wisama Khalil
    • 2
  • Maxime Gautier
    • 2
  1. 1.School of ComputingUniversity of UtahSalt Lake CityUSA
  2. 2.IRCCyN, ECNUniversity of NantesNantesFrance

Personalised recommendations