Conditions for symmetric running in single- and double-support

Even and odd functions of time - symmetric functions - have been used to simplify the control of running robots and have been observed in the body and leg motion of running animals. This paper explores the relationship between symmetric motion of the body and symmetric use of the legs in three planar models. The algebra of symmetric functions is applied to the equations of motion to show that symmetric leg actuation is required for symmetric body motion in the single-support case, but not in the double-support case unless additional constraints are imposed.