Adaptive Kinematic Control of Redundant Manipulators

Abstract

Redundancy resolution is of great importance in the control of manipulators. Among the existing results for handling this issue, the quadratic program approaches, which are capable of optimizing performance indices subject to physical constraints, are widely used. However, the existing quadratic program approaches require exactly knowing all the physical parameters of manipulators, the condition of which may not hold in some practical applications. This fact motivates us to consider the application of adaptive control techniques for simultaneous parameter identification and neural control. However, the inherent nonlinearity and non-smoothness of the neural model prohibits direct applications of adaptive control to this model and there has been no existing result on adaptive control of robotic arms using projection neural network (PNN) approaches with parameter convergence. Different from conventional treatments in joint angle space, we investigate the problem from the joint speed space and decouple the nonlinear part of the Jacobian matrix from the structural parameters that need to be learnt. Based on the new representation, we establish the first adaptive PNN with online learning for the redundancy resolution of manipulators with unknown physical parameters, which tackles the dilemmas in existing methods. The presented method is capable of simultaneously optimizing performance indices subject to physical constraints and handling parameter uncertainty. Theoretical results are presented to guarantee the performance of the presented neural network. Besides, simulations based on a PUMA 560 manipulator with unknown physical parameters together with the comparison with an existing PNN substantiate the efficacy and superiority of the presented neural network, and verify the theoretical results.

Appendix

According to the D-H convention [41], based on Table 6.1, the forward kinematics of the PUMA 560 manipulator is described as (6.1), where $\mathbf {r}=[r_\text {x},r_\text {y},r_\text {z}]^\text {T}$, and $f(\theta )=[f_1(\theta ),f_2(\theta ),f_3(\theta )]^\text {T}$ with

$$\begin{aligned} \begin{aligned} f_1(\theta )&=(-((\cos \theta _1\cos ^2\theta _2-\cos \theta _1\sin \theta _2\sin \theta _3)\cos \theta _4-\sin \theta _1\sin \theta _4)\sin \theta _5\\&~~-(\cos \theta _1\cos \theta _2\sin \theta _3{+}\cos \theta _1\sin \theta _2 \cos \theta _3)\cos \theta _5)d_6{+}a_2\cos \theta _1\cos \theta _2\cos \theta _3\\&~~-a_2\cos \theta _1\sin \theta _2 \sin \theta _3+d_3\sin \theta _1+a_3\cos \theta _1\cos \theta _2,\\ f_2(\theta )&= (-((\sin \theta _1\cos ^2\theta _2-\sin \theta _1\sin \theta _2\sin \theta _3)\cos \theta _4 +\cos \theta _1\sin \theta _4)\sin \theta _5\\&~~-(\sin \theta _1\cos \theta _2\sin \theta _3+\sin \theta _1\sin \theta _2\cos \theta _3)\cos \theta _5)d_6+a_2\sin \theta _1\cos \theta _2\cos \theta _3\\&~~-a_2\sin \theta _1\sin \theta _2\sin \theta _3-d_3\cos \theta _1+a_3\sin \theta _1\cos \theta _2,\\ f_3(\theta )&=d_1+(-(\sin \theta _2\cos \theta _2+\cos \theta _2\sin \theta _3)\cos \theta _4\sin \theta _5 +(-\sin \theta _2\sin \theta _3\\&~~+\cos \theta _2\cos \theta _3)\cos \theta _5)d_6+a_2\sin \theta _2\cos \theta _3+a_2\cos \theta _2\sin \theta _3+a_3\sin \theta _2. \end{aligned} \end{aligned}$$

Then, the Jacobian matrix $J(\theta )=\partial f(\theta )/\partial \theta$ is derived, which is described as (6.2) with the nonzero elements of $W\in \mathbb {R}^{3\times 11}$ being $w_{11}=d_6$, $w_{12}=a_2$, $w_{13}=d_3$, $w_{14}=a_3$, $w_{25}=d_6$, $w_{26}=a_2$, $w_{27}=d_3$, $w_{28}=a_3$, $w_{39}=d_6$, $w_{3,10}=a_2$, and $w_{3,11}=a_3$. The nonzero elements of the corresponding $\phi (\theta )\in \mathbb {R}^{11\times 6}$ are

$$\begin{aligned} \begin{aligned} \phi _{11}(\theta )&=\sin \theta _5(\cos \theta _1\sin \theta _4 + cos^2\theta _2\cos \theta _4\sin \theta _1 - \cos \theta _4\sin \theta _1\sin \theta _2\sin \theta _3) \\&~~+ \sin (\theta _2+ \theta _3)\cos \theta _5\sin \theta _1,\\ \phi _{12}(\theta )&=\cos \theta _1\cos \theta _2\cos \theta _4\sin \theta _5(2\sin \theta _2 + \sin \theta _3)-\cos (\theta _2+\theta _3)\cos \theta _1\cos \theta _5,\\ \phi _{13}(\theta )&=\cos \theta _5(\cos \theta _1\sin \theta _2\sin \theta _3 - \cos \theta _1\cos \theta _2\cos \theta _3)+ \cos \theta _1\cos \theta _3\\&~~\cdot \cos \theta _4\sin \theta _2\sin \theta _5,\\ \phi _{14}(\theta )&=\sin \theta _5(\sin \theta _4(\cos \theta _1\cos ^2\theta _2 - \cos \theta _1\sin \theta _2\sin \theta _3)+ \cos \theta _4\sin \theta _1),\\ \phi _{15}(\theta )&=\sin \theta _5(\cos \theta _1\cos \theta _2\sin \theta _3 + \cos \theta _1\cos \theta _3\sin \theta _2)- \cos \theta _5(\cos \theta _4\\&~~\cdot (\cos \theta _1\cos ^2\theta _2 - \cos \theta _1\sin \theta _2\sin \theta _3)\sin \theta _1\sin \theta _4),\\ \phi _{21}(\theta )&=-\cos (\theta _2 + \theta _3)\sin \theta _1,\\ \phi _{22}(\theta )&=\phi _{23}(\theta )=-\sin (\theta _2 + \theta _3)\cos \theta _1,\\ \phi _{31}(\theta )&=\cos \theta _1,\\ \phi _{41}(\theta )&=-\sin \theta _1\cos \theta _2,\\ \phi _{42}(\theta )&=-\cos \theta _1\sin \theta _2,\\ \phi _{51}(\theta )&=\sin \theta _5(\sin \theta _1\sin \theta _4 - \cos \theta _1\cos ^2\theta _2\cos \theta _4 + \cos \theta _1\cos \theta _4\sin \theta _2\sin \theta _3) \\&~~- \sin (\theta _2 + \theta _3)\cos \theta _1\cos \theta _5,\\ \phi _{52}(\theta )&=\cos \theta _2\cos \theta _4\sin \theta _1\sin \theta _5(2\sin \theta _2 + \sin \theta _3) - \cos (\theta _2+ \theta _3)\cos \theta _5\sin \theta _1,\\ \phi _{53}(\theta )&=\cos \theta _5(\sin \theta _1\sin \theta _2\sin \theta _3 - \cos \theta _2\cos \theta _3\sin \theta _1)+ \cos \theta _3\cos \theta _4\sin \theta _1\\&~~\cdot \sin \theta _2\sin \theta _5,\\ \phi _{54}(\theta )&=-\sin \theta _5(\cos \theta _1\cos \theta _4 -\sin \theta _4(\cos ^2\theta _2\sin \theta _1 - \sin \theta _1\sin \theta _2\sin \theta _3)),\\ \phi _{55}(\theta )&=\sin (\theta _2 + \theta _3)\sin \theta _1\sin \theta _5 - \cos \theta _5(\cos \theta _1\sin \theta _4+ \cos ^2\theta _2\cos \theta _4\sin \theta _1 \\&~~- \cos \theta _4\sin \theta _1\sin \theta _2\sin \theta _3),\\ \phi _{61}(\theta )&=\cos (\theta _2 + \theta _3)\cos \theta _1,\\ \phi _{62}(\theta )&=\phi _{63}(\theta )=-\sin (\theta _2 + \theta _3)\sin \theta _1,\\ \phi _{71}(\theta )&=\sin \theta _1,\\ \phi _{81}(\theta )&=\cos \theta _1\cos \theta _2,\\ \phi _{82}(\theta )&=-\sin \theta _1\sin \theta _2,\\ \phi _{92}(\theta )&=\cos \theta _4\sin \theta _5(\sin \theta _2\sin \theta _3 + 2\sin ^2\theta _2 - 1) - \sin (\theta _2+ \theta _3)\cos \theta _5,\\ \phi _{93}(\theta )&=- \sin (\theta _2 + \theta _3)\cos \theta _5 - \cos \theta _2\cos \theta _3\cos \theta _4\sin \theta _5,\\ \phi _{94}(\theta )&=\cos \theta _2\sin \theta _4\sin \theta _5(\sin \theta _2 + \sin \theta _3),\\ \phi _{95}(\theta )&=- \cos (\theta _2 + \theta _3)\sin \theta _5 - \cos \theta _2\cos \theta _4\cos \theta _5(\sin \theta _2+ \sin \theta _3),\\ \phi _{10,2}(\theta )&=\phi _{10,3}(\theta )=\cos (\theta _2 + \theta _3),\\ \phi _{11,2}(\theta )&=\cos \theta _2. \end{aligned} \end{aligned}$$