International Journal of Computer Applications (0975 - 8887) Volume 57 - No. 20, November 2012 Multi-Objective Constrained Optimization using Discrete Mechanics and NSGA-II Approach Sneha Desai M.Tech Control System Department of Electrical Engineering, Veermata Jijabai Technological Institute (VJTI), INDIA Sushant Bahadure M.Tech Control System Department of Electrical Engineering, Veermata Jijabai Technological Institute (VJTI), INDIA Faruk Kazi Faculty at Department of Electrical Engineering Veermata Jijabai Technological Institute (VJTI), INDIA Navdeep Singh Faculty at Department of Electrical Engineering Veermata Jijabai Technological Institute (VJTI), INDIA ABSTRACT A novel approach to solve multi-objective optimization problems of complex mechanical systems is proposed based on evolution- ary algorithm. Discrete mechanics derives structure preserving con- straint equations and objective functions. Standard non-linear opti- mization techniques used to obtain optimal solution to these equa- tions fails to find global optimum solution and also requires system satisfying initial guess. Multi-objective optimization technique like non-dominated sorting genetic algorithm-II (NSGA-II) finds global optimal solution without giving any initial guess for multiple con- flicting objectives. This method is numerically illustrated by opti- mizing an underactuated mechanical system called 2D SpiderCrane system. In SpiderCrane, fast and precise payload positioning is to be achieved while keeping payload swing minimum along the tra- jectory. Minimizing the time of operation requires greater amount of force which may lead to unacceptable payload sway, while de- creasing forces increases the time of operation. Proposed control law to optimize this conflicting multi-objectives is validated with simulation results. Keywords: Optimization, Non-dominated sorting genetic algorithm, Dis- crete mechanics optimal control, Bio-inspired 2D Spider- Crane.ifx 1. INTRODUCTION In order to solve multi-objective optimization problem of me- chanical systems, one is often interested in preserving certain properties of the mechanical system for the approximated so- lution and steer a mechanical system from an initial to a fi- nal state under the influence of control forces such that a given quantity, for example control effort or maneuver time is min- imal i.e multiple conflicting objectives are needed to be opti- mized. The presence of these multiple conflicting objectives for- mulates the task as a (global) multi-objective optimization prob- lem (MOP), which resorts to a number of trade-off optimal so- lutions. Classical methods like the objective weighted method, the hierarchical optimization, the constraint method, the goal programming method and many more aggregates the multiple- objective in a single, parametrized objective function. However, for systematically varying the parameters, knowledge of problem is very much necessary which may not be available [1]. Also, there are possibilities of producing biased result by setting pri- orities to objectives and finding one solution in one simulation run. Because of which several optimization runs are required to obtain approximate Pareto-optimal set. Evolutionary algorithms (EAs), on the other hand, can find multiple optimal solutions in one single simulation run due to their population-approach. EAs are ideal for solving multi-objective optimization problems. Although there exist a number of multi-objective evolutionary algorithms (EMO), non-dominated sorting genetic algorithm II (NSGA-II), have gained tremendous popularity in solving differ- ent kinds of engineering problems [1], [2]. NSGA-II implements elitism for multi-objective search which enhances the conver- gence properties towards the true Pareto-optimal set. The con- straint handling method does not make use of penalty parame- ters. The algorithm implements a modified definition of domi- nance in order to solve constrained multi-objective efficiently. Discrete Mechanics and Optimal Control (DMOC) is used to derive structure preserving constraint equations and objective functions. These equations are then used by NSGA-II to obtain global optimum solution. DMOC is introducesd in [4], [5]. In the context of variational integrators [6], the discretization of the Lagrange-d’Alembert principle leads to structure preserving time stepping equations which serve as equality constraints for the resulting finite dimensional non-linear optimization problem. This problem can be solved by standard non-linear optimization techniques such as Sequential Quadratic Programming (SQP) leading to local optimal solutions dependent on the initial guess [7]. Although this method works very successfully in many ap- plications, they fail to find the global optimal solution for MOP without using any initial guess. A remedy for these difficulty is found in the MOEA. In this paper, 2D SpiderCrane mechanism is considered and the objective is to steer its payload from stable equilibrium point with zero initial velocity to other stable equilibrium point with minimum effort in minimum time, i.e. stabilization of the load along with the time and force minimization by keeping the payload swing minimum along the trajectory. This bio-inspired mechanism is proposed by the Laboratory of Automatic Control at ´ Ecole Polytechnique F´ ed´ erale de Lausanne [8], [9], [10]. The main contribution of this work is to provide a methodology for performing multi-objective constrained optimization of com- plex mechanical systems without providing any initial guess and using equations derived from DMOC as objective function and as constraint equations. The paper is organised as follows: Sec- tion II includes brief introduction of DMOC. Section III summa- rizes basic principles of NSGA-II. Section IV includes dynamics and modelling of 2D SpiderCrane. Section V includes applica- tion of NSGA-II to 2D SpiderCrane system and based on the simulation results, the performances of NSGA-II and local op- timization method are compared and discussed. Section VI out- lines the conclusion and future research. 14