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accomodate Allen Back application atlas behavior boundary functions charts and patches components Compression Thrust Cornell University corresponding data structure object decompression phase Definition of Hybrid denote described dstool program DYNAMICAL SIMULATION FAC example flight phase four mechanical phases governing dynamics hopping robot hybrid dynamical system hybrid systems index set initial conditions John Guckenheimer Koditschek legged robot limit cycle list of boundary Mark Myers Mathematical Sciences Institute model for Raibert's nonlinear spring numerical algorithms open set patch boundaries period-doubling bifurcation Periodic hopping periodic orbit phases which comprise physical device pneumatic cylinder procedure Raibert's hopper Raibert's hopping Raibert's robot real numbers REPORT 92-6 REVISED REVISED A DYNAMICAL robot body robot height routine shown in Figure smooth map defined spring constant studying dynamical systems subcomponent SYSTEMS BY Allen target constants target values TECHNICAL REPORT 92-6 thrust and decompression thrust phase trajectory segment transition functions transition set two-dimensional phase portrait vector field