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: ai=exp(-(y-μi)T(y-yi)/σi2),
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=[qTq’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 kssgn(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+plL,
where P is the vector for payloads, pl is payload at lth link and L
is [l21 l22 lll2
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.
