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Locomotion Foundations: ZMP, CoM, and Linear Inverted Pendulum

Duration: 60 min · Level: Advanced · Module: 3. Bipedal Locomotion & Whole-Body Control · Focus: ZMP, locomotion, balance, theory

Learning objectives

By the end of this lesson you will be able to explain and apply:

  • Zero Moment Point
  • Linear Inverted Pendulum Model (LIPM)
  • ZMP limitation
  • Center of Mass (CoM) trajectory planning
  • ASIMO (Honda) used ZMP control

Why this matters​

Zero Moment Point (ZMP) theory, developed by Vukobratović in 1969, was the theoretical foundation of humanoid locomotion for 30 years.

Overview​

Zero Moment Point (ZMP) theory, developed by Vukobratović in 1969, was the theoretical foundation of humanoid locomotion for 30 years. ASIMO, HRP-2, and early ATLAS all used ZMP-based controllers. While modern robots have moved beyond it, understanding ZMP is prerequisite to understanding why we moved on.

Key concepts​

Key idea

Zero Moment Point: the point on the ground where the net moment of inertial and gravitational forces has no component along horizontal axes; ZMP inside support polygon → stable

  • Linear Inverted Pendulum Model (LIPM): approximates the robot as a point mass on a massless leg; enables real-time trajectory planning with convex optimization
  • ZMP limitation: requires slow, flat-footed gait to maintain ZMP inside support polygon; prevents dynamic maneuvers like running or jumping
  • Center of Mass (CoM) trajectory planning: plan CoM trajectory such that ZMP constraint is satisfied; solved as QP (quadratic program) in real-time
  • ASIMO (Honda) used ZMP control: could walk at 2.7 km/h on flat surfaces, completely failed on any unexpected perturbation or terrain variation
  • Capture Point theory (Pratt 2006): extends ZMP to handle single-step recovery; still used as fallback in modern robots when RL policy fails

Check your understanding​

Try to recall each answer before expanding it.

Q1. What do you know about Zero Moment Point?

the point on the ground where the net moment of inertial and gravitational forces has no component along horizontal axes; ZMP inside support polygon → stable

Q2. What do you know about Linear Inverted Pendulum Model (LIPM)?

approximates the robot as a point mass on a massless leg; enables real-time trajectory planning with convex optimization

Q3. What do you know about ZMP limitation?

requires slow, flat-footed gait to maintain ZMP inside support polygon; prevents dynamic maneuvers like running or jumping

Q4. What do you know about Center of Mass (CoM) trajectory planning?

plan CoM trajectory such that ZMP constraint is satisfied; solved as QP (quadratic program) in real-time

Q5. What do you know about ASIMO (Honda) used ZMP control?

could walk at 2.7 km/h on flat surfaces, completely failed on any unexpected perturbation or terrain variation

References​

  • Biped Stability Conditions with Ankle, Hip and Foot Rotation Indicators — Vukobratović and Borovac (2004). International Journal of Humanoid Robotics
  • Capture Point: A Step toward Humanoid Push Recovery — Pratt et al. (2006). Humanoids 2006

Next: 3.2 Model Predictive Control for Dynamic Walking →

Part of Module 3: Bipedal Locomotion & Whole-Body Control.