Linear Inverted Pendulum

• Also called: LIP, LIPM, linear inverted pendulum model, cart-table model
• Category: Control › Locomotion control
• Tags: bipedal-locomotion, balance, reduced-order-models, humanoids, legged-robots

A simplified model of a walking or balancing robot as a point mass at a constant height on a massless leg, which makes its center-of-mass dynamics linear and easy to control.

Definition

Linear inverted pendulum (LIP) is a model that treats a legged robot, or a standing human, as a single point mass at the center of mass (CoM) supported by a massless, telescoping leg that touches the ground at a pivot point. The pivot corresponds to the center of pressure, or zero moment point (ZMP), under the feet. The key assumption is that the CoM stays at a constant height, which is what makes the dynamics linear.

Under that assumption, the horizontal motion of the CoM follows a simple equation: its acceleration is proportional to its distance from the pivot, ẍ = ω²(x − p), where p is the pivot position and ω = √(g/z), with g gravitational acceleration and z the constant CoM height. The CoM is pushed away from the pivot, which is why balance requires moving the pivot by shifting pressure within the foot or by stepping. The two horizontal directions decouple, so each can be handled separately.

The model is associated with Shuuji Kajita and colleagues, who used it for biped walking. It underlies ZMP preview control, in which a reference ZMP trajectory along planned footsteps is turned into a CoM trajectory, and it is the basis of the capture point and the divergent component of motion. An equivalent formulation is the cart-table model.

The simplicity has costs. The LIP ignores angular momentum about the CoM, vertical CoM motion, leg mass and impacts. Extensions add a flywheel to represent angular momentum or allow varying height, while whole-body controllers track the LIP-based CoM plan with the full robot model.

Why it matters

The linear inverted pendulum turns the hard problem of keeping a multi-joint robot balanced into a linear problem that can be solved quickly and analyzed in closed form. It remains the starting point for much humanoid walking control and for balance analysis in biomechanics and wearable robotics.

Key applications

• CoM trajectory generation for humanoid and biped walking
• ZMP preview control and model predictive control of walking
• Push recovery and step placement decisions
• Balance analysis for exoskeletons and assistive devices

In practice

A humanoid walking stack commonly plans footsteps first, then uses the LIP to compute a CoM trajectory that keeps the ZMP inside each support foot, often with preview control or model predictive control. A whole-body controller then tracks that CoM trajectory and the swing-foot motions with the full dynamics model. Teams pick the model's constant height from the robot's nominal standing posture and rely on feedback to absorb the effects the model leaves out.

Common misconceptions

• The robot actually walks at constant height — the constant height is a modeling assumption; the full robot's CoM varies, and a lower-level controller handles the difference
• The LIP is the same as the spring-loaded inverted pendulum — the SLIP model includes a compliant leg and is used for running and hopping, while the LIP keeps height fixed
• A LIP-based controller accounts for upper-body rotation — the basic model has no angular momentum about the CoM; extensions such as a flywheel are needed for that

Who should know this

• Humanoid and biped locomotion engineers
• Exoskeleton and rehabilitation robotics researchers
• Control students learning reduced-order models of legged robots

Related technologies

• Zero Moment Point (ZMP)
• Capture Point
• Divergent Component of Motion
• Model Predictive Control
• Whole-Body Control
• Spring-Loaded Inverted Pendulum (SLIP)

Related terms

• Gait
• Humanoid Robot
• Divergent Component of Motion
• Centroidal Dynamics

Related projects

• Extreme Parkour with Legged Robots: RL system for legged robot parkour trained in under 20 hours
• PACE Sim2Real: Systematic sim-to-real transfer via actuator dynamics identification
• Zeroth Bot: Low-cost 3D-printed humanoid robot platform for sim-to-real reinforcement learning