Learning the Kinematics of a Manipulator Based on VQTAM
Abstract
:1. Introduction
2. Research Methodology
3. Kinematics of the Robotic Manipulator
 (1)
 The 6DOF parameter is used to describe the pose relationship between two adjacent coordinate systems completely, thus avoiding the lack of completeness in the DH model.
 (2)
 The global coordinate system is used to describe the motion state of a rigid body, which overcomes the singularity of the DH model.
 (3)
 The motion characteristics of rigid bodies can be clearly described from a global perspective, thus simplifying the analysis of complex mechanisms and avoiding the abstraction of mathematical symbols.
4. System Identification of a Nonlinear Dynamical System Using VQTAM
4.1. Learning Strategy of VQTAM
Algorithm 1: VQTAM Algorithm: 
Begin 
(training part) 
1 Input: X^{in} and X^{out} in training set 
2 Search activating neuron according to X^{in} 
$i*=\underset{i\in A}{\mathrm{arg}\mathrm{min}}\left\{\Vert {\mathit{X}}^{in}{\omega}_{i}^{in}\Vert \right\}$ 
3 Update the weight vector of the neuron 
${\omega}_{i}^{in}\Leftarrow \alpha (t)h(i*,i;t)[{X}^{in}{\omega}_{i}^{in}]$ 
${\omega}_{i}^{out}\Leftarrow \alpha (t)h(i*,i;t)[{\mathit{X}}^{out}{\omega}_{i}^{out}]$ 
4 Continue execution until termination conditions are met 
(testing part) 
5 Input: X_{test}^{in} in testing set 
6 Search activating neuron according to X_{test}^{in} 
$i*=\underset{i\in A}{\mathrm{arg}\mathrm{min}}\left\{\Vert {\mathit{X}}_{test}^{in}{\omega}_{i}^{in}\Vert \right\}$ 
7 Output: 
${\widehat{\mathit{X}}}_{test}^{out}={\omega}_{i*}^{out}$ 
4.2. Searching the Activated Neuron by the Priority Search KMeans Tree Algorithm
Algorithm 2: Kmeans tree data structure building Algorithm [32]: 
Begin 
1 Input: weight vector ω_{i}^{in} as search data set D, branch parameter B, maximum iteration number I_{max}, center selection algorithm using C_{alg} 
2 Compare size D of data set D with branch parameter B 
3 If D<B: Create leaf nodes from data sets 
4 else, P ← uses C_{alg} algorithm to select B points from data set D 
5 start loop: 
6 C ← Clustering Data in D Centered on P 
7 P_{new} ← Finding the Mean Value of Group C Data after Clustering 
8 if P = P_{new}, P_{new} is the nonleaf node, and terminate the loop 
9 These processes are executed on the subregions C until all leaf nodes are created 
10 Output: the entire Kmeans tree data structure 
Algorithm 3: PSKMT Algorithm [32]: 

5. VQTAM Local Linear Improvement Algorithms
5.1. Improvement Algorithm of VQTAM with LLR
5.2. Improvement Algorithm of VQTAM with LWR
5.3. Improvement Algorithm of VQTAM with LLE
6. Simulation Results and Discussion
6.1. Standard VQTAM Network Test Results
6.2. VQTAM Local Linear Improvement Algorithms
6.3. Overall Examples and Test Results
7. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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parameter  x_{2} (mm)  x_{3} (mm)  x_{4} (mm)  x_{5} (mm)  x_{6} (mm) 
value  175  175  175  1445  1445 
parameter  z_{2} (mm)  z_{3} (mm)  z_{4} (mm)  z_{5} (mm)  z_{6} (mm) 
value  495  1590  1765  1765  1765 
θ  g_{st}(θ)  

θ_{1}  θ_{2}  θ_{3}  θ_{4}  θ_{5}  θ_{6}  x (mm)  y (mm)  z(mm)  u  v  w 
0  0  0  0  0  0  1580  0  1765  −2.81  −1.57  2.81 
−3.14  0.39  −3.14  0.03  −1.41  −4.81  717.58  9.07  1715.51  −1.44  −0.54  30.09 
−2.71  1.58  −1.11  7.78  0.81  3.25  −2292.10  −1151.06  23.08  2.63  0.24  −1.95 
−0.63  −1.53  −3.1  −1.46  −1.82  −5.04  −597.84  594.06  −706.84  −2.67  −0.73  1.24 
−2.24  −0.95  −1.55  7.43  2.22  6.30  1152.39  1311.719  1743.21  2.40  −0.15  −1.67 
2.81  −0.42  −2.98  −2.90  2.01  6.95  1304.39  −420.67  912.84  2.02  0.54  2.64 
3.12  0.79  −0.24  −7.64  −2.34  −7.61  −2057.54  −44.10  822.60  −2.44  −0.52  −0.97 
n_{q}  n_{p}  epoch  α_{0}  α_{M}  σ_{0}  σ_{M} 

3  3  5000  0.8  0.001  15  0.001 
θ_{1}  θ_{2}  θ_{3}  θ_{4}  θ_{5}  θ_{6}  

RMSE  0.1871  0.1663  0.0895  0.2189  0.2459  0.5708 
R^{2}  0.9931  0.9808  0.9972  0.9984  0.9808  0.9894 
RMAE  0.3914  0.6660  0.4840  0.1734  0.9828  0.5477 
MLP  VQTAM (60 × 60)  VQTAM (50 × 50)  
RMSE  1.5935  0.2901  0.7823 
VQTAM with LLR (50 × 50)  VQTAM with LWR (50 × 50)  VQTAM with LLE (50 × 50)  
RMSE  0.5163  0.6792  0.3018 
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Lan, L.; Li, H.; Yang, W.; Yongqiao, W.; Qi, Z. Learning the Kinematics of a Manipulator Based on VQTAM. Symmetry 2020, 12, 519. https://doi.org/10.3390/sym12040519
Lan L, Li H, Yang W, Yongqiao W, Qi Z. Learning the Kinematics of a Manipulator Based on VQTAM. Symmetry. 2020; 12(4):519. https://doi.org/10.3390/sym12040519
Chicago/Turabian StyleLan, Luo, Hou Li, Wu Yang, Wei Yongqiao, and Zhang Qi. 2020. "Learning the Kinematics of a Manipulator Based on VQTAM" Symmetry 12, no. 4: 519. https://doi.org/10.3390/sym12040519