dynamics. M ARY Engineering Department, Bilkent ElectricalSUM and Electronics University, 06800 Ankara, Turkey 2. Approximate Analytical Solution (AAS): This option 2 Even though dynamic models for which we have a Department of Computer Engineering, Middle East Technical University, 06800 Ankara, Turkey uses analytic differentiation of AAS derived in [2]. sufficiently good analytical understanding can support physically relevant controller designs, the measurement İsmail Uyanık and Ömer Morgül and estimation of particularly the dynamic parameters, Fig Eng., 2. illustrates the block diagram for the adaptive Abstract – Practical realizationDept. ofof Electrical model-based & Electronics Bilkent University, Turkey such as spring and damping constants for flexible parameter correction scheme we propose. Our method dynamic legged behaviors is substantially more components of a robotic platform, is still a challenging Uluç Saranlı relies on the availability of an approximate return map g challenging than statistically stable behaviors due to their problem. Fortunately, this issueDept. is not confined to the of Computer Eng., Middle East University, Turkey that canTechnical predict the apex state outcome of a single stride. heavy dependence on second order system dynamics. In control of legged locomotion and received considerable In this study, we consider two alternatives for this this paper, weadaptive presentalgorithm an on-line, model-based adaptive • A novel from to achieve gait control system identification for running with a planar hopper (SLIP) attention the adaptive controlaccurate community [1]. and approximate predictor model: control method for running with a planar spring-mass model. by work in this area, this study presents a new Motivated 1. Exact SLIP Model (ESM): This option predicts the hopper based on once-per-step scheme. • Model-based estimation ofcorrection legmethod spring and either be used to eliminate tracking errors or to achieve model-based adaptive control fordamping runningconstants with canoutcome through numerical simulation of SLIP system identification. theaccurate well-known Spring-Loaded Inverted Pendulum dynamics. Fig 2. The in proposed adaptive control Prediction SUM M ARY • Comparison a non-adaptive shows substantial both tracking accuracy andstrategy. parameter estimation (SLIP) modelwith (see Fig. 1), approach emphasizing on-line 2.improvements Approximate Analytical Solution (AAS): This errors of an approximate plant model g areoption used to Even though dynamic models for which we have a performance. estimation of unknown or miscalibrated dynamic system uses analytic differentiation of AAS derived in [2]. dynamically adjust parameter estimates. sufficiently good analytical understanding can support parameters. physically relevant controller designs, the measurement In order to test our algorithm, we run a large number of and estimation of particularly the dynamic parameters, simulations using different ranges of system parameters. such as spring and damping constants for flexible We then define three error measures where SSE k & SSEd components of a robotic platform, is still a challenging capture system identification performance and SSE a problem. Fortunately, this issue is not confined to the characterizes the tracking performance of the adaptive control of legged locomotion and received considerable controller. Table I summarizes the average apex state attention from the adaptive control community [1]. tracking and parameter estimation errors and their Motivated by work in this area, this study presents a new standard deviations across all simulations. model-based adaptive control method for running with the well-known Spring-Loaded Inverted Pendulum TheProposed proposed adaptive control strategy. Prediction Fig 1.Spring-Loaded The Spring-Loaded Inverted Pendulum (SLIP) Fig 2.Table Inverted (SLIP) model Adaptive Gait Controller I . Percentage apex tracking and parameter (SLIP) model (see Fig. 1), Pendulum emphasizing on-line model. Dashed curve illustrates a single stride, defining errors of an approximate plant model g are used to 1 Adaptive Control of A Spring-Mass Hopper