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SCIENCE CHINA Information Sciences, Volume 62, Issue 10: 202203(2019) https://doi.org/10.1007/s11432-018-9726-3

Parameter estimation survey for multi-joint robot dynamic calibration case study

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  • ReceivedSep 29, 2018
  • AcceptedDec 12, 2018
  • PublishedAug 21, 2019

Abstract

Accurate model parameters are the basis of robot dynamics. Many linear and nonlinear models have been proposed to calibrate the inertial parameters and friction parameters of multi-joint robots. However, methods of choosing a model and calculating its parameters still have few summaries. This paper reviews typical linear/nonlinear models and different calculation methods for robot dynamic calibration. Through simulations, the features of different methods are analyzed, including torque error, parameter error, model adaptability, solution time, and anti-interference ability of the calibration results. Finally, an experiment performed on a six-degree-of-freedom industrial manipulator is used as an example to illustrate how to select the model for a specified robot. These comparisons and experiments provide references for the parameter calibration of multi-joint robots.


Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. U1713222, 61773378, 61421004, U1806204), Beijing Science and Technology Project (Grant No. Z181100003118006), and Youth Innovation Promotion Association CAS.


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  • Figure 1

    (Color online) The Efort ER20-C10 robot and D-H frames.

  • Figure 3

    The simulation platform for point sampling.

  • Figure 4

    (Color online) A comparison of the Kalman filter and exponential forgetting method. (a) Kalman filter (slope change); (b) exponential forgetting (slope change); (c) Kalman filter (second-order change); (d) exponential forgetting (second-order change).

  • Table 1   The levels of torque error
    Level A Level B Level C
    ${e_{T}}$ textless0.01 textless0.1 textgreater0.1
  • Table 2   The levels of parameter errors
    Level A Level B Level C
    ${e_{d}}$ textless0.2 textless1 textgreater1
    ${e_{f}}$ textless0.2 textless1 textgreater1
    $J_e$ textless1 textless10 textgreater10
    Follow changing parameter Slope change Step change None
  • Table 3   The levels of solution time of offline methods
    Level A Level B Level C
    Solution time textless1 min textless1 h textgreater1 h
  • Table 4   The levels of anti-interference ability
    Level A Level B Level C
    $\Delta~{J_e}$ textless0.1 textless1 textgreater1
  • Table 5   Linear calibration result of joint 3
    ParameterSet
    Pseudo-
    inverse
    Kalman
    Index
    forgetting
    Optimization
    Gradient
    descent
    Genetic
    algorithm
    Particle
    filter
    Mass (kg) $m$ 25.6083 17.7803 17.7782 17.7793 24.5913 24.5699 24.5250
    Center (m)$X$ 0.0938 $-$0.1811 $-$0.1807 $-$0.1809 0.0672 0.0994 0.1208
    $Y$ 0.1383 $-$0.1121 $-$0.1140 $-$0.1132 0.1107 0.1657 0.1384
    $Z$ 0.1164 0.0085 0.0082 0.0083 0.0970 0.1159 0.1148
    Inertia
    (${\rm~kg}~\cdot~{{\rm~m}^2}$)
    $I_{11}$ 0.1706 3.3342 3.3206 3.3262 0.1287 0.1783 0.2189
    $I_{12}$ $-$0.1498 0.5166 0.5146 0.5151 $-$0.2175 $-$0.1258 $-$0.1829
    $I_{13}$ 0.2140 0.6943 0.6915 0.6928 0.1790 0.1615 0.2486
    $I_{22}$ 0.3493 $-$3.5211 $-$3.5359 $-$3.5298 0.4161 0.2925 0.4160
    $I_{23}$ $-$0.1320 $-$0.2403 $-$0.2808 $-$0.2641 $-$0.1994 $-$0.0933 $-$0.1646
    $I_{33}$ 0.2363 $-$0.3693 $-$0.3691 $-$0.3693 0.3062 0.3046 0.3063
    Friction$F_c$ 50.0000 50.0000 49.7802 49.8704 50.2227 50.2716 50.4040
    $\beta$ 40.0000 40.0000 40.0250 40.0161 39.7539 39.6028 39.8216
    Torque
    error (Nm)
    Mean $-$0.0001 $-$0.0066 0.0323 0.2568 0.2725 0.0029
    Std 0.0001 0.2903 0.2210 0.6332 0.8545 0.8825
  • Table 6   Nonlinear calibration result of joint 3
    ParameterSet
    Extended
    Kalman
    Index
    forgetting
    Optimization
    Gradient
    descent
    Genetic
    algorithm
    Particle
    filter
    Mass (kg) $m$ 25.6083 18.0803 18.0862 25.5425 29.5065 29.1977
    Center (m)$X$ 0.0938 $-$0.1405 $-$0.1401 0.075 0.0964 0.1141
    $Y$ 0.1383 $-$0.0374 $-$0.0377 0.1107 0.1156 0.1526
    $Z$ 0.1164 0.0179 0.0175 0.0931 0.1143 0.1158
    Inertia
    (${\rm~kg}~\cdot~{{\rm~m}^2}$)
    $I_{11}$ 0.1706 3.9916 4.0685 0.2302 0.1447 0.2406
    $I_{12}$ $-$0.1498 0.2146 0.2089 $-$0.0886 $-$0.2168 $-$0.097
    $I_{13}$ 0.214 0.5786 0.595 0.265 0.2368 0.1683
    $I_{22}$ 0.3493 $-$3.1141 $-$3.0855 0.4044 0.2933 0.2909
    $I_{23}$ $-$0.132 $-$0.2575 $-$0.2548 $-$0.1994 $-$0.1421 $-$0.0809
    $I_{33}$ 0.2363 0.5709 0.5297 0.2828 0.302 0.2335
    Friction$F_c$ 50 51.7727 52.183 50.625 49.9266 50.2767
    $F_s$ 60 52.1061 52.2013 60 59.8052 60.0016
    $q_s$ 0.1 5.5823 6.3761 0.08 0.1142 0.0996
    $\beta$ 40 38.2327 38.1439 39.8125 39.8125 39.7704
    Torque
    error (Nm)
    Mean 0.0355 0.0314 2.8462 $-$0.0209 0.0062
    Std 1.9688 1.9705 1.2125 0.6137 0.2182
  • Table 7   The standard deviation of noise added to ${\boldsymbol~\tau}~_j$
    Joint 1 Joint 2 Joint 3 Joint 4 Joint 5 Joint 6
    Std (Nm) 10 10 10 1.78 1.78 1.78
  • Table 8   The comparison of $J_e$ with and without noise (linear methods)
    Pseudo-
    inverse
    Kalman
    Exponential
    forgetting
    Optimization
    Gradient
    descent
    Genetic
    algorithm
    Particle
    filter
    Without
    noise
    $e_d$ 279.8642 256.6049 255.1294 0.3839 0.8657 0.6336
    $e_f$ 0.0000 0.0135 0.0080 0.0032 0.0061 0.0185
    $e_T$ 0.0000 0.0000 0.0001 0.0001 0.0006 0.0001
    $e_T\_{\rm~Std}$ 0.0000 0.0007 0.0005 0.0006 0.0012 0.0010
    $J_e$ 279.8642 256.6191 255.1380 0.3879 0.8736 0.6532
    With
    noise
    $e_d$ 289.2396 261.2449 260.0939 0.9258 1.0587 0.9754
    $e_f$ 0.0585 0.0289 0.0314 0.1136 0.1147 0.1258
    $e_T$ 0.0016 0.0005 0.0005 0.0013 0.0008 0.0010
    $e_T\_{\rm~Std}$ 0.0219 0.0203 0.0203 0.0203 0.0209 0.0204
    $J_e$ 289.3216 261.2945 260.1460 1.0610 1.1951 1.1225
    Anti-interference $\Delta~J_e$ 0.0338 0.0182 0.0196 1.7352 0.3680 0.7184
  • Table 9   The comparison of $J_e$ with and without noise (nonlinear methods)
    Extended
    Kalman
    Exponential
    forgetting
    Optimization
    Gradient
    descent
    Genetic
    algorithm
    Particle
    filter
    Without
    noise
    $e_d$ 71.7454 72.6643 0.2243 0.8152 0.7340
    $e_f$ 3.0663 3.4116 0.0681 0.0376 0.0182
    $e_T$ 0.0001 0.0001 0.0014 0.0001 0.0001
    $e_T\_{\rm~Std}$ 0.0061 0.0060 0.0019 0.0013 0.0007
    $J_e$ 74.8180 76.0820 0.2956 0.8541 0.7530
    With
    noise
    $e_d$ 74.0046 74.4194 0.8248 0.8351 0.8438
    $e_f$ 48.4565 54.6694 0.1287 0.1776 0.1493
    $e_T$ 0.0004 0.0004 0.0005 0.0008 0.0006
    $e_T\_{\rm~Std}$ 0.0212 0.0211 0.0197 0.0207 0.0197
    $J_e$ 122.4827 129.1102 0.9737 1.0343 1.0135
    Anti-interference $\Delta~J_e$ 0.6371 0.6970 2.2936 0.2110 0.3459
  • Table 10   The comparison between the IDIM-IV method and Pseudo-inverse method for model (7)
    Pseudo-inverse IDIM-IV
    $J_e$ (with filter) 411.5920 412.3433
    $J_e$ (without filter) 1911.7927 446.4756
    $\Delta~J_e$ 3.6449 0.0828
  • Table 11   The feature levels of different methods
    Pseudo-
    inverse
    Kalman
    Extended
    Kalman
    Exponential
    forgetting
    Optimization
    Gradient
    descent
    Genetic
    algorithm
    Particle
    filter
    Error$e_d$ C C C C B B B
    $e_f$ A A B B A A A
    $e_T$ A A A A A A A
    $J_e$
    (without noise)
    C C C C A A A
    Follow changing
    parameter
    C B B A C C C
    Solution time OfflineA Online Online Online OfflineB OfflineC OfflineB
    Linear/
    nonlinear
    Linear Linear Nonlinear
    Linear/
    nonlinear
    Linear/
    nonlinear
    Linear/
    nonlinear
    Linear/
    nonlinear
    Anti-interference $\Delta~J_e$ A A B B C B B
  • Table 12   The experiment result of torque errors
    Linear modelsNonlinear models
    Eq. (7) Eq. (8) Eq. (9) Eq. (4) Eq. (5) Eq. (6)
    Mean (Nm) 0.4478 0.0469 0.4509 0.4085 2.7903 1.2232
    Std (Nm) 44.7068 30.8389 44.417 19.1073 51.8087 25.5865

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