- Lateral stability of the spring-mass hopper suggests a two-step control strategy for running

State space and trajectories of the SLIP with motion constrained to a vertical line. The state variables under this constraint are vertical position of the body (vertical axis) and vertical velocity of the body (horizontal axis). The four quadrants represent the four phases of motion of the SLIP (in order from apex: descent, compression, decompression, and ascent). The hopper is in flight when the vertical position of the body is greater than one (the uncompressed leg length). The quadrant boundaries are the event conditions (touchdown, bottom, lift-off, and apex, respectively). The thick vertical line (in flight, vertical velocity 0) is the Poincaré section defining apex, parametrized by a single variable (vertical body position at apex or hopping height). We assume that trajectories start and stop on this section. In this example, the controller acts by changing the decompression spring constant at the bottom. Trajectories with different possible controls diverge from this point. Some reach subsequent apex states that are too low (red), some that are two high (blue), and for one particular value of the control (the deadbeat control), the subsequent apex state reaches its desired height (green). If the initial height were different (but within some range), the deadbeat control would also be different, but would still exist (not shown).