Sensitivity Analysis of a Crawl Gait Multi-objective Optimization System Miguel Oliveira Industrial Electronics Department, School of Engineering, University of Minho 4800-058 Guimaraes, Portugal mcampos@dei.uminho.pt Pedro Silva Industrial Electronics Department, School of Engineering, University of Minho 4800-058 Guimaraes, Portugal psilva@dei.uminho.pt Cristina Santos Industrial Electronics Department, School of Engineering, University of Minho 4800-058 Guimaraes, Portugal cristina@dei.uminho.pt Lino Costa Production Systems Department, School of Engineering,University of Minho 4800-058 Guimaraes, Portugal lac@dps.uminho.pt ABSTRACT This paper describes the analysis of a crawl gait multi-objective optimization system that combines bio-inspired Central Pat- terns Generators (CPGs) and a multi-objective evolutionary algorithm. In order to optimize the crawl gait, a multi- objective problem, an optimization system based on NSGA- II allows to find a set of non-dominated solutions that cor- respond to different motor solutions The experimental results highlight the effectiveness of this multi-objective approach. Categories and Subject Descriptors I.2.9 [Robotics]; G.1.6 [Optimization]: Keywords Evolutionary robotics, Multi-objective optimization, Genetic algorithms 1. INTRODUCTION Legged robot locomotion is a complex problem that usu- ally implies the control and coordination of multiple degrees of freedom. One possible bio-inspired solution is to apply nonlinear dynamical equations of high order with a multidi- mensional and irregular set of parameters, that model the biological nervous system. The lack of knowledge about the dynamical behavior of the robot platform, the environment and the interaction Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. GECCO’13, July 6-10, 2013, Amsterdam, The Netherlands. Copyright 2013 ACM TBA ...$10.00. among both; and the very large number of parameters makes the hand-tuning very hard to achieve and may not ensure the best results. So, biological inspired Evolutionary Computa- tion (EC) appears as a possible choice for gait optimization of legged robots. In [4] we have combined a proposed Central Pattern Gen- erator (CPG) approach with a multi-objective optimization strategy, the elitist Nondominated Sorting Genetic Algo- rithm [2] (NSGA-II), that searches for the best set of CPG parameters that result in the desired crawl gait. Speed, sta- bility and behavioral diversity were the evaluated criteria used to explore the parameter space of the network of CPGs. In this work, we are particularly interested in studying the compromise solutions obtained to see if they reveal interest- ing features of the CPG parameters and in their relation to the addressed objectives. Thus, we proceed to a sensitivity analysis to assess the relationships between the parameters and the objectives and between the objectives themselves. Results demonstrate the performance of the achieved robot walking. 2. OPTIMIZATION SYSTEM The proposed network of CPGs generates trajectories for the robot hip pitch joints, and therefore to the robot limbs. The x i solutions of a modified Landau-Stuart oscillator, di- rectly control each hip pitch joint of a limb, i, as follows ˙ x i = α ( μ - r 2 i ) (x i - O i ) - ω i z i , (1) ˙ z i = α ( μ - r 2 i ) z i + ω i (x i - O i ) , (2) where ri = √ (xi - Oi ) 2 + z 2 i , ωi specifies the oscillation fre- quency (in rad s -1 ) and μ value specifies the oscillation am- plitude A, where μ = A 2 . This oscillator bifurcates from a stable fixed point at (xi ,zi )=(Oi , 0) (when μ< 0) to a structurally stable, harmonic limit cycle, for μ> 0, with off- set Oi . Despite phase relationships, left and right legs have similar trajectories. This enables to minimize the decision vector (x), see Table 1.