Index

a

  • adaptive dynamic programming strategy
    • Euler–Lagrange system
      • action mapping and critic mapping  21–23
      • optimality condition  21
    • problem formulation  19–20
    • sensor–actuator system
      • Coriolis force and friction  18
      • Euler–Lagrange system  18
      • inertial force  18
      • stability consideration  19
    • simulation experiment
      • cart–pole model  23–24
      • experiment setup and results  24–25

b

  • backpropagation neural network  114
  • backstepping-based design  18
  • bang-bang controllers  28
  • Barbalat’s lemma  89

c

  • cart–pole model  23–24
  • circular trajectory  122–127
  • communication graph  151
  • constrained quadratic program
    • objective function  92–93
    • reformulation and convexification  94
    • set constraint  93–94
    • speed level resolution  91
  • control redundant manipulators  67
  • convexity restriction  33

d

  • degrees of freedom (DOFs)  67, 87
  • dual neural networks  68, 132
    • convergence analysis  71–72
    • Karush–Kuhn–Tucker condition  70
    • Lagrange function  70
    • quadratic cost function  69

e

  • end-effector trajectory 1424
  • error accumulation  33
  • Euler–Lagrange system  18
    • action mapping and critic mapping  21–23
    • feedback linearization  17
    • optimality condition  21
    • extreme learning machine (ELM)  115

f

  • feed-forward neural networks  28, 67, 114

g

  • Gaussian elimination  27
  • geometric relation  109–110
  • gradient-based neural network (GNN)  149
  • Gram–Schmidt orthogonalization procedure  132

i

  • infinity-sign trajectory  127
  • inherent nonlinearity  17

j

  • Jacobian matrix  60–61, 67
  • Jacobian matrix pseudoinversion  3

k

  • Karush-Kuhn-Tucker (KKT) conditions  5, 70, 107
  • Kennedy–Chua network  115
  • kinematic redundancy resolution  69

l

  • Lagrange dual variables  52
  • Lagrange neural networks  68, 132
  • LaSalle’s invariance principle  57, 89
  • limited communication  151
  • Lyapunov function method  17

m

  • manipulability
    • in circular path tracking  99–102
    • Jacobian matrix  103
    • optimization via self motion  98–99
    • problem formulation  90
    • velocity-level resolution  102
  • manipulator kinematics  90, 151–152
  • matrix decomposition  27
  • monotone mapping  89

n

  • neural networks see also novel dual neural network model
    • constrained quadratic programming problems  88
    • kinematic singularity configuration  88
    • new noise-tolerant neural networks  88
    • performance index  88
    • pseudoinverse-type formulation  88
  • noise-tolerant ZNN  149
    • cooperative motion generation perturbed  161–162
    • real-time redundancy resolution  149–156
    • theoretical analyses and results  157–160
  • nominal neural controller design  53–54
  • nonconvex function activated model  8–9
  • nonholonomic constraints  17
  • nonlinearly activated NTZNN (NANTZNN) model  167
    • cooperative motion planning with noises  174–175
    • cooperative motion planning without noises  174
    • definition and robot arm kinematics  168
    • distributed scheme  169
    • problem formulation  168
    • real-time redundancy resolution  170–171
    • theoretical analyses and results  171–172
  • nonlinear mapping  51, 87
  • normal cone  89
  • novel dual neural network model
    • feedback loop for control  55
    • input–output pairs  54
    • stability
      • Jacobian matrix  60–61
      • LaSalle’s invariant  57
      • manipulator end-effector  59
      • optimal redundancy resolution  56
      • PUMA 560 manipulator  58–59
      • virtual reference  54

o

  • optimization problem formulation  91

p

  • parallel manipulators  132
  • parallel redundant manipulators  131
  • polynomial noises
    • constant noises  78–80
    • linear noises  80–81
    • neural dynamics  73–77
    • practical considerations  77–78
  • position tracking error 1424
  • projection operator  89
  • pseudoinverse formulation  87
  • PUMA 560 manipulator
    • end-effector
      • simulation results  43–44
    • numerical simulations
      • simulation setup  62–63
      • tracking performance  63–64
      • with vs. without excitation noises  64–66
    • random initialization  41
    • scaling coefficient  41
    • stability  58–59
    • time-varying reference position
      • control action scaling factor  42
      • RMS position tracking errors  50
      • tracking control  45
  • PUMA 560 robot arm
    • constant noises  82–86
    • nominal situation  81–82
    • simulation setup  81
    • time-varying polynomial noises  86

q

  • quadratic programming  28, 167

r

  • radial basis function (RBF)  114
  • real-time processing  149
  • recurrent neural networks (RNNs)  28, 67, 68, 132
  • redundancy resolution
    • convergence analysis  96–98
    • nonlinear equation set  95
    • real-time redundancy resolution  96
  • redundant manipulators  27, 131
    • bang-bang controllers  28
    • degrees of freedom (DOFs)  51, 67
    • dual neural network  52
    • error accumulation problem, RNN model
      • Karush–Kuhn–Tucker condition  30
      • limitations  32–33
      • presented control law  33–34
      • quadratic optimization problem  30
      • real-time control action  31
      • stability  34–36
    • feed-forward neural networks  28
    • Gaussian elimination  27
    • Lagrange dual variables  52
    • matrix decomposition  27
    • nonlinear mapping  51
    • problem formulation  29–30, 52–53, 68–69
    • pseudo-inversion  51
    • quadratic programming formulation  28
    • real-time computing  52
    • slack variables  28
  • residual error  8
  • robot kinematics
    • geometric relation  109–110
    • velocity space resolution  111–112
  • root-mean-square (RMS) tracking error  50

s

  • sensor–actuator system
    • Coriolis force and friction  18
    • Euler–Lagrange system  18
    • inertial force  18
    • stability consideration  19
  • sensor–actuator systems  17
  • square trajectory  127–129
  • Stewart platform
    • kinematic modeling
      • geometric relation  133–135
      • velocity space resolution  135–136
    • neural network design
      • dynamic neural network  115
      • KKT conditions  113
      • nonlinear mapping  114
      • projected neural networks  115
    • numerical investigation
      • circular trajectory  122–127, 143
      • infinity-sign trajectory  127
      • setups  142–143
      • simulation setups  118–122
      • square trajectory  127–129, 143–145
    • problem formulation  112–113
    • recurrent neural network design
      • optimality  139–142
      • problem formulation  136–137
      • stability  138–139
    • robot kinematics
      • geometric relation  109–110
      • velocity space resolution  111–112
    • rotational matrix  108
    • theoretical results
      • off-line training procedure  117
      • optimality  115–116
      • stability  116–117
    • triple product  108–109

v

  • variable structure-based design  18
  • velocity compensation
    • control law  36–37
    • stability analysis  37–41
  • velocity space resolution  111–112

z

  • zeroing neural network (ZNN)
    • computer simulations and verifications
      • bound constraint  12–13
      • conventional bang-bang control  15
      • inequality constraint  16
      • quadratic programming problem  15
    • design formula  5
    • distributed scheme  153
    • fixed configuration  160–161
    • gradient-based neural network (GNN)  149
    • gradient-based techniques  5–6
    • Jacobian matrix pseudoinversion  3
    • joint-angle drift phenomenon  3
    • joint-limit avoidance  3
    • Karush–Kuhn–Tucker condition  5
    • linear activation function  8
    • noise-tolerant ZNN  149
      • cooperative motion generation perturbed  161–162
      • real-time redundancy resolution  149–156
      • theoretical analyses and results  157–160
    • nonconvex function activated model  8–9
    • optimization techniques  167
    • problem formulation and neural-dynamic design  152–153
    • quadratic program (QP)-based motion planning  167
    • real-time processing  149
    • redundant manipulators  149
    • residual error  8
    • task function  3
    • theoretical analyses  9–12
    • time-varying problems  3, 5
    • ZNN-based solution  163–164
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