Multi-objective Optimization Technique for Simulated Active Body Control with Frictional Contacts LIVIA SANGEORZAN 1 , MIRCEA PARPALEA 2 , CEZAR PODASCA 3 , MILAN TUBA 4 1 Department of Computer Science, 3 Department of Medicine Transilvania University of Brasov, ROMANIA 25, Eroilor, blvd, 500030, Brasov, ROMANIA 2 National College “Andrei Saguna”, ROMANIA 2 Megatrend University Belgrade, SERBIA sangeorzan@unitbv.ro, parpalea@yahoo.co, caesarpodaska@yahoo.com, tubamilan@ptt.yu http://www.unitbv.ro Abstract: - The Genetic Algorithm is a stochastic optimization routine based on Darwin’s theory of evolution and genetics. An evolutionary process arrives at the optimized solution over several iterations (generations), by selecting only the best (the fittest) solutions and allowing these to survive and form the basis for calculating the next round of solutions. In this manner the optimization routine evolves the initial solutions to the optimum. The simultaneous optimization of multiple, possibly competing, objective functions deviates from scalar objective optimization. Instead of finding one perfect solution, multi-objective optimization problems tend to be characterized by a family of alternatives that must be considered equivalent in the absence of information concerning the relevance of each objective relative to the others. Therefore, the first objective in multi-objective optimization is to find the Pareto set, and the next is to select a proper solution from the found Pareto solution set. Although standing is easily mastered by humans, it requires careful and deliberate manipulation of contact forces. The variation in contact configuration presents a real challenge for simulations while performing tasks in the presence of external disturbances. An analytic approach for control of standing in three-dimensional simulations is described based upon local optimization. Key-Words: - Computer Graphics, Multi-objective Optimization, Genetic algorithms, Frictional contacts 1 Introduction Dynamic simulation of passive phenomena, such as articulated bodies, has enabled the creation of increasingly complex virtual environments. The simulation of active bodies such as humans and robots allowed a better understanding of the real motion according to the dynamic surroundings. In this paper, an analytic control formulation is described that solves an optimization for the mechanics of active bodies. The optimization adapts the control to the frictional properties of the simulation, the mass properties and the posture of the active body. The constraints imposed by frictional contacts with the environment are also being taken in account. A genetic algorithm that maximizes performance objectives subject to contact forces adapts to changing properties of the body, the environment, and the contact 2 Theoretical Aspects 2.1 Contact Mechanics Motion of a body in contact with the environment is more complex than motion in free space due to the presence of reaction forces that push on the body at each contact point. For the case of sustained contact control the linear relationship between reaction forces and accelerations can be exploited. This relationship can be computed and used to control active bodies. [1] Contact dynamics expresses the relationship between the motion variables ) , , ( q q q    of an articulated body, its internal torques, and external forces. (Figure 1) The contact between two surfaces is modelled by a set of m point contacts ) ( ) ( m C C r r  1 and the matching contact forces ) ( ) 1 ( m C C f f  . Each contact force is restricted by a convex cone ) ( i K according to the standard model of friction. Contacts with environment, as shown in Figure 2, restrict the relative velocity of each contact point 3   ) (i C r , for i = 1 . . . m. In the case of a non-slipping contact, the relative velocity is zero: 0 ) (  i C r  . This condition can also be expressed in terms of joint velocities n q    by using the Jacobian matrix n i G    3 ) ( to compute the body velocity at the point of contact [2]: 0 ) (   i C i r q G   . (1) MATHEMATICAL METHODS AND APPLIED COMPUTING ISSN: 1790-2769 594 ISBN: 978-960-474-124-3