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The redefined output approach is used by feeding back this output to guarantee the minimum phase behavior of the resulting closed-loop system. No a priori knowledge about the nonlinearities of the system is needed and the payload mass is also assumed to be unknown. The network weights are adjusted using a modified online error backpropagation algorithm that is based on the propagation of output tracking error, derivative of that error and the tip deflection of the manipulator.

The real-time controller is implemented on an experimental test bed. The results achieved by the proposed NN-based controller are compared experimentally with conventional proportional-plus-derivative-type and standard inverse dynamics controls to substantiate and verify the advantages of our proposed scheme and its promising potential in identification and control of nonlinear systems.

Article :. Date of Publication: Dec DOI: Need Help? Hirschorn extended the procedures earlier developed for linear system inversion to nonlinear systems.

Control of Flexible-link Manipulators Using Neural Networks

Key concerns in the development of an inverse model are the existence of the inverse and its stability. The aforementioned classical techniques, though very effective, were limited to minimum phase systems. For non-minimum phase systems, the yielded inverse were unstable. Devasia [ 19 ], an author who has done remarkable research in inversion theory, especially for non-minimum phase systems, successfully managed to invert a non minimum phase system by isolating the internal from the external dynamic then decomposing them into stable and unstable dynamics.

He used the preview technique to solve for the unstable internal dynamics. Detailed mathematical presentation of the preview based technique can be found in [ 20 , 21 , 22 ]. Other inversion techniques can be found in [ 23 , 24 ] and the references therein.

In this article, we developed an inverse controller and proposed a controller to stabilize the internal dynamics of the otherwise unstable inverse model. To limit operation speeds to safe levels, the stable inverse model was augmented with lowpass filters and used as a feedforward controller to a two link, 3D flexible manipulator. The main difference between the proposed approach and input shaping is that whereas the trajectories are fixed in the latter method, different joint trajectories can be used with inverse controller. Controller design requires the knowledge of the plant to be controlled.

Consequently, accurate modelling forms a prerequisite for a controller design. A lot of research has been done in the accurate mathematical modelling of flexible manipulators, for example [ 25 , 26 , 27 ]. This involves the application of Lagrangian mechanics or the Euler—Newton formulation which are derived from energy principles. Since these techniques involves the solution of differential equations, the solution are truncated using either finite element method FEM or the assumed modes methods AMM.

Mathematical modelling other than being very tedious and prone to errors especially with growing number of links and joints, also fails to capture all the details of a plant. Alternative solutions include system identification and symbolic modelling. In system identification, input—output data and previous knowledge of the system are used to develop a statistical or neural network model of the dynamic system whose characteristics match in one form or another the input—output relationship. The main limitation of system identification is the fact that some of the phenomenon of the original plant cannot be deduced from the input output relationship.

Control of Flexible-Link Manipulators Using Neural Networks - Semantic Scholar

In the same respect, behaviours outside the test data cannot be identified. Symbolic modelling on the other hand involve the use of computer applications to model and simulate the plant. Mathematical representation of the plant is then obtained, either in state space or in differential algebraic equations DAEs.

The strength of this method lies in its accuracy owing to the fact that all technical complexities, interactions and aspects of the plant that cannot be captured mathematically are put into consideration [ 28 , 29 ].

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The plant presented in this article is a two link, 3D flexible manipulator with a weight attached at the distal end and structured as in Fig. It has three rotary joints driven by dc servomotors and two flexible links assumed to have damping of the Kelvin—Voigt type both in the lateral and the torsional senses. Measurement of angular position and velocity is achieved using encoders coupled to the servomotors while link strain measurement is done by strain gauges positioned at the bottom of each link.

Maplesim employs the Rayleigh beam theory which incorporates the rotary inertia effect over and above the kinetic and potential energies of the bending effects considered in the popular Euler—Bernoulli beam theory [ 30 ]. Graph theory is used in the formulation of the governing dynamic algebraic equations DAE of the flexible manipulator which will take the form of a system of PDEs and boundary conditions in form of ODEs.

Simulations with such infinite dimensional PDEs would require impractical resources in terms of computer memory and would take a long time. To solve this, the truncation of the elastic coordinates for the deformation along each axis amongst the in-plane, out-of-plane, torsional deflections and longitudinal elongation is done using the assumed mode method. In this work, the manipulator was truncated to order 2. In this work, modeling in Maplesim resulted in the model shown in Fig. Manipulator model The manipulator comprises of two flexible links whose joints are driven by dc servomotors and sitting on a rotary joint giving it a 3D motion.

In the modelling of the manipulator in maplesim, the lengths of links 1 and 2 are broken into two to accommodate an instrument to measure the strain. Except for having twice as many flexure variables as the number of links, breaking the links does not affect the performance of the model. To validate the Maplesim model, Figs. We can see perfect agreement between joint angles in 3a—3c, torsional and links strain in 4a—4c.

Journal of Applied Research and Technology. JART

From this observation, we can deduce that the linear model represented an accurate model of the manipulator. Further, inverse model developed from this model was an accurate inverse of the model and that of the actual manipulator. Validation of joint angles. This figure shows the joint trajectories of the nonlinear model and the linearized model against the actual manipulator.

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Validation of links and torsional strain. This figure shows the link and torsional strain of the nonlinear model and the linearized model against the actual manipulator. It is assumed that the system is stable or stabilized by negative feedback. Further, if this holds true in the entire domain in the states, then we say the system has a well defined relative degree.

Repeating this for all the rows and having the resulting expressions in vector form, we have. From Eq. Replacing x t in 4 with the transformed dynamics, the control law to maintain the exact tracking can be written as. Equation 7 together with Eq. Poles and zeros of the plant and its inverse. A closer look at the pole-zero map of the inverse system, and also from the knowledge of the zeros location, the inverse system was unstable. This meant that the internal dynamics i.

This also meant that the control law would contain this non-decaying variables to the plant. To solve this, a feedback controller of the form v is proposed to stabilize the internal dynamics. Consequently, the state Eqs. The system described in Eqs. Block diagram of the inverse system. This figure presents the state space Eqs. With all the poles arbitrarily placed on the left hand side of the s-plane, stability of the internal dynamics and consequently of the inverse controller is assured. The inverse controller required that the joint angles follow the input trajectory.

Step or square wave trajectories for example meant that the joint would have infinite velocities during the rising and falling edges and this may lead to mechanical breakdown of the manipulator. To prevent this, the manipulator is excited through lowpass filters as in the internal model architecture [ 32 , 33 ]. The filter takes the form. The order of the filter n was chosen to be 2. Filtered inverse control system This figure shows the filtered inverse control system comprising of the filter, the inverse controller and the plant.

The experimental setup is as shown in Fig.

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