Adaptive Neural Network Control of Robot Manipulators in Task Space

Shuzhi S. Ge, C.C. Huang, L.C. Woon

IEEE Transactions on Industrial Electronics, VOL.44, NO.6, December 1997

Summary

                  Flexibility is one of the main issues in a production facility and several researches have been done to enable it in control system. Computed Torque Control is an intuitive scheme which has the objective of cancelling non linear dynamics of the manipulator system, but it requires the exact dynamic system which mean that is wouldn’t be flexible enough, so this is why adaptive control methods have been searched to overcome the problem of requiring a priori knowledge. Interesting is the application of Neural Networks, they work well in many systems and if use with parameterized variables, they can be used in different environments. The problem is extended I Task Space (Cartesian Space) and controlled must be the end effector position and it’s force. In order to create the model the GL (Ge-Lee) Matrix and its operator are introduced (page 746-747).

·       Model

In the field of control engineering, neural networks are used for approximating a given non linear function f(y) up to a small error tolerance. The neural network is based on 3 layers: input layer (n nodes), hidden layer (l nodes) and output layer (m nodes). For each hidden layer a Gaussian function is defined: a_i=exp(-(y-μ_i)^T(y-y_i)/σ_i²), with μ being the center vector and σ² being the variance of the gaussian distibution. The output appears to be the Gaussian Functin coming out from the hidden layer weigheted by W. It has been proven that any continuos function can uniformly be approximated by a linear combination of Gaussians.

In modelling a robot’s manipulator the kinematics would be described in the following manner: D(q)q’’+C(q,q’)q’+G(q)=τ, where D os the symmetric positive definite inertia matrix, C is the Coriolis matrix, G is the vector for gravitational forces and τ is the joint torque vector. D(q) and G(q) are function of only q, therefore the are considered as static neural networks and the equation previously introduced can be adapt for their description (page 748), while C is described as a dynamic neural network and therefore q’ is needed to model it, a parameter z=[q^Tq’^T]∈R²ⁿ is used. A general controller can then be constructed demonstrating that no Jacobian matrix is required (for the function please refer to page 749); Kr is introduced to give the proportional derivative type control and k_ssgn(r) is indicating the tracking error. In a closed loop system with positive derivative and tracking signal error, e and e’ tend to be stably 0 with t going to infinite, the presence of sgn(.) function denotates chattering which needs to be minimized.

Key Concepts

Artificial Neural Networks, Control Systems

Key Results

The model of a Neural Network for robot controlling has been applied in simulation in comparison with a basic PD controller (non adaptive control). The simulation involved a two link manipulator and vector M, P and L where introduced. The vector M is: M=P+p_lL, where P is the vector for payloads, p_l is payload at lth link and L is [l²₁ l²₂ l_ll_{2 l1} l₂]^T. For D and G function a 100-node static neural network is used, for C we use a 200-node dynamic neural network.

In the simulation non adaptive control appears to have a significant tracking error and cannot handle changes, using adaptive control allows to reduce a lot the tracking error thanks to the learning mechanism which is provided by the neural network methodology.

Actual D and actual C are shown to not converge to optimal D and C, which actual G converges to its optimal value, this is due to the fact that the desired trajectory is not persistently exciting as for real-world applications.

In conclusion the model allows a good control system without the need of time-consuming computations for obtaining the Jacobian, which describes the necessary inverce kinematics for traditional control systems.