Spring-Loaded Inverted Pendulum

弹簧负载倒立摆SLIPAdvanced

Models running and hopping with a point mass on top of a single massless spring leg.

The spring-loaded inverted pendulum is a classic simplified model of running and hopping: the body collapses into a single point mass, and the leg is a massless spring. During the stance phase, when the foot is on the ground, the spring first compresses and stores energy, then extends and launches the body back up; during the flight phase, when the foot is off the ground, the point mass simply undergoes projectile motion under gravity, and the two phases alternate. Biomechanics research — such as work by Blickhan and Full in 1993 — has found that the center-of-mass motion of running animals ranging from insects to humans can be described by this model. It differs from an ordinary inverted pendulum in that the leg's length can change and it can store and release elastic energy, so it can capture running and hopping with an airborne phase, whereas the linear inverted pendulum better describes walking. In robotics, SLIP is often used as a 'template': footfall angle, leg stiffness, and other parameters are first planned on this low-dimensional model and then mapped onto the real robot; legged robots such as RHex were designed with reference to SLIP dynamics.

ExampleWhen a person runs, the knee and ankle bend and the body's center of mass dips after the foot lands (equivalent to the spring compressing), then extends to launch the body into the air. SLIP reproduces this rise-and-fall trajectory of the center of mass using just a handful of parameters: mass, leg length, spring stiffness, and touchdown angle.

Also called: SLIP, Spring-Mass Model
Related: Inverted Pendulum Model (IPM) · Linear Inverted Pendulum Model (LIPM) · Bipedal Locomotion · Flight Phase (Aerial Phase) · Gait · Passive Dynamic Walking
Sources: Truax et al., Optimizing Design and Control of Running Robots Abstracted as TD-SLIP (arXiv:2407.12120)
Chen, Wensing, Zhang, Optimal Control of a Differentially Flat 2D Spring-Loaded Inverted Pendulum Model (arXiv:1911.07168)