Spring-Loaded Inverted Pendulum
- Also called: SLIP, spring-mass model
- Category: Control › Locomotion control
- Tags: legged-locomotion, reduced-order-models, running, hopping, compliance
A reduced-order model of running and hopping: a point mass bouncing on a massless spring leg, alternating between stance on the spring and ballistic flight.
Definition
The spring-loaded inverted pendulum (SLIP) models a running or hopping body as a point mass at the center of mass, carried on a massless leg that behaves like a linear spring. Motion alternates between two phases. In stance, the foot is pinned to the ground and the mass compresses and then extends the spring while rotating over the foot, like an inverted pendulum whose length can change. In flight, the foot leaves the ground and the mass follows a ballistic arc under gravity while the massless leg is swung to a new touchdown angle. The model came from biomechanics, where Blickhan and McMahon and Cheng used it around 1990 to describe the bouncing center-of-mass motion of running humans and animals.
Because touchdown and liftoff switch the equations of motion, SLIP is a hybrid system. The stance dynamics are nonlinear and have no general closed-form solution, so controllers rely on numerical integration, approximate analytical solutions or precomputed lookup tables. The main control input is the leg angle at touchdown; actuated variants also change leg stiffness or rest length during stance to add or remove energy. Studies of the model showed that a suitable choice of touchdown angle can make running stable without any feedback during stance, which helped explain why spring-like legs are attractive in robot design.
In robotics, SLIP serves both as a design target and as a planning template. Marc Raibert's hopping machines of the 1980s regulated forward speed by placing the foot relative to the point under the center of mass at touchdown, a foot-placement rule that works on spring-mass dynamics of this kind. Later bipeds such as ATRIAS were built so that their natural dynamics resemble SLIP. In planners, the model predicts how the body will move between touchdown and liftoff, so a controller can choose foot positions that add, keep or remove forward velocity, for example to bring a robot to rest when it lands from a jump. A whole-body or joint-level controller then makes the real robot, with its leg mass and joint limits, approximate that motion.
Why it matters
Running, hopping and jumping involve flight phases and impacts that are hard to plan with full robot models in real time. SLIP captures the bouncing center-of-mass motion with a few parameters, giving controllers a fast way to reason about touchdown angle, foot placement and energy storage, and giving designers a reference behavior for compliant legs.
Key applications
- Running and hopping control for bipeds, quadrupeds and single-legged hoppers
- Foot placement for speed regulation and stopping
- Landing and jump planning for legged robots
- Design of compliant legs and series elastic actuators
- Biomechanical analysis of human and animal running
In practice
A team programming a quadruped to land safely from a jump models each landing as a SLIP stance phase, with leg stiffness set from the gains of their leg impedance controller. Given the body velocity estimated just before touchdown, they evaluate candidate foot positions relative to the center of mass, integrating the SLIP stance dynamics for each, and pick the one that brings horizontal velocity closest to zero by the end of the stance phase. The chosen foot positions go to the swing-leg controller, and once the feet touch down the stance legs run an impedance controller that makes them act like damped springs, absorbing the vertical impact. Logs from real landings are compared with the SLIP prediction to retune stiffness and damping.
Common misconceptions
- SLIP is about foot slip — SLIP is the spring-loaded inverted pendulum model of running; slip is unwanted sliding of a wheel or foot over the ground, which the SLIP model assumes does not happen
- SLIP is the same as the linear inverted pendulum — the LIP keeps the center of mass at constant height and suits walking, while SLIP has a compliant leg, varying height and a flight phase, which suits running and hopping
- A robot must have physical springs to use SLIP — many robots with stiff actuators emulate spring-like legs through impedance control, and the model is still useful as a planning template
Who should know this
- Legged locomotion control engineers
- Designers of compliant and elastic robot legs
- Biomechanics researchers studying running and hopping
- Students learning reduced-order models of legged robots
Related technologies
- Linear Inverted Pendulum Model
- Raibert Heuristic
- Impedance Control
- Series Elastic Actuators
- Hybrid Zero Dynamics