Linear Inverted Pendulum Model (LIPM)

The LIPM reduces the entire robot to a single point mass located at its Center of Mass (CoM). All limbs and links are abstracted away. The leg is modeled as a massless, telescoping strut connecting this point mass to a point contact on the ground. This drastic simplification allows the dynamics to be described by simple Newtonian equations, focusing solely on the CoM trajectory.

A critical linearizing assumption is that the point mass (CoM) moves in a horizontal plane at a fixed height (z_c) above the ground. This constraint eliminates the vertical dynamics, turning the 3D problem into a 2D one. The motion of the CoM is thus governed by:

[ \ddot{x} = \frac{g}{z_c} (x - p_x) ]

where (x) is the CoM position, (p_x) is the foot placement (the pivot), and (g) is gravity. This equation describes a diverging exponential, capturing the inherent instability of walking.

The model assumes the supporting foot can be placed instantaneously and without dynamics at any desired Foot Placement point within kinematic limits. The leg is treated as an ideal, frictionless pin joint at this point. This abstracts away the swing leg dynamics, joint torques, and finite swing time, allowing planners to focus on the strategic question: "Where should the next foot go to maintain stability?" This leads directly to concepts like the Capture Point.

The LIPM assumes the robot has zero centroidal angular momentum. All mass is concentrated at the CoM point, so the system cannot generate or have rotational inertia about its center. This ignores the dynamic effects of swinging arms or leg swing, which in real robots are used to manage angular momentum and improve stability. Control strategies based purely on the LIPM must ensure the full-body controller regulates this momentum to near zero for the model to remain valid.

The model is fundamentally analyzed in the single support phase, where only one foot is on the ground. The contact is modeled as a frictionless point contact (a pin joint). This means:

• No slipping: The foot does not slide.
• No torque: The foot cannot exert a moment about the vertical axis (no ankle torque).
• Double support (both feet on the ground) is treated as an instantaneous transition between single support phases or handled via other methods like the 3D LIPM with flywheel.

These assumptions are not flaws but design choices to create a Reduced-Order Model (ROM) that is:

• Analytically solvable: Enables closed-form solutions for foot placement (Capture Point).
• Computationally trivial: Allows real-time predictive control (e.g., Model Predictive Control).
• Intuitively powerful: Provides deep insight into the fundamental dynamics of walking.

Limitations arise when these assumptions break down: on very uneven terrain (constant height invalid), during highly dynamic motions (angular momentum significant), or for robots with large feet (point contact invalid). Advanced controllers use the LIPM for high-level planning but rely on Whole-Body Control to account for the full dynamics.

The LIPM's linear dynamics allow for the closed-form calculation of future Center of Mass (CoM) trajectories. This enables planners to rapidly evaluate potential footstep locations for stability. Key applications include:

• Predicting the Capture Point: Calculating the point on the ground where the robot can step to bring its CoM to a stop.
• Generating the Divergent Component of Motion (DCM): A derived state variable that captures the unstable part of the CoM dynamics, used to plan stable walking sequences.
• Online gait adjustment: Reactively shifting foot placements in response to pushes or uneven terrain by solving simple linear equations.

The LIPM is the predominant dynamic model inside real-time Model Predictive Control (MPC) loops for bipedal robots. Its simplicity allows the MPC to solve the optimization problem (often a Quadratic Program) within the stringent cycle times of a locomotion controller (e.g., 1-5 ms). The MPC uses the LIPM to:

• Predict CoM motion over a receding horizon.
• Optimize future Ground Reaction Forces (GRFs) and footstep locations.
• Enforce constraints like friction cones and kinematic limits.
• This application is central to the dynamic walking of robots like Boston Dynamics' Atlas and many research humanoids.

The LIPM provides the theoretical foundation for modern dynamic stability criteria beyond static balance. It directly relates to:

• Zero-Moment Point (ZMP) Tracking: Under the LIPM's constant CoM height assumption, controlling the ZMP (the point where net horizontal moment is zero) is equivalent to controlling CoM acceleration. The ZMP must be kept within the support polygon for stability.
• Orbital Energy Analysis: The LIPM's dynamics can be described in terms of orbital energy, which remains constant in the absence of control. Stepping injects or removes energy to regulate walking speed.
• Push Recovery Strategies: By modeling the effect of an external push as an instantaneous change in CoM velocity, the LIPM allows for the calculation of required ankle torque (for small pushes) or necessary stepping location (for large pushes) to regain balance.

The model is used to synthesize periodic walking patterns (gait generation) and to enable precise tracking of desired velocities. Key mechanisms include:

• 3D Linear Inverted Pendulum Mode: Extending the LIPM into 3D allows for the independent generation of sagittal (forward/back) and lateral (side-to-side) walking motions.
• Analytical CoM Trajectories: The solution to the LIPM differential equations produces hyperbolic CoM trajectories. By planning the Capture Point at each step, a controller can dictate the robot's walking speed and direction.
• This application allows a high-level planner to command velocity setpoints (e.g., walk at 0.6 m/s) which the low-level LIPM-based controller translates into dynamically consistent footstep plans.

The LIPM is rarely used in isolation. Its primary role is to serve as a planning layer whose outputs are tracked by a whole-body controller (WBC) that manages the robot's full dynamics. This hierarchical approach is standard:

1. LIPM/MPC Layer: Plans the overall CoM trajectory, footstep locations, and net GRFs.
2. Whole-Body Control Layer: Uses inverse dynamics or QP optimization to distribute the desired net wrenches into individual joint torques, while satisfying all constraints of the floating base dynamics.

• This separation of concerns makes the complex problem of bipedal control tractable, with the LIPM handling the high-level "where to step" problem.

Due to its computational simplicity, the LIPM is an indispensable tool in simulation for:

• Algorithm Development: Researchers prototype and test new footstep planners, MPC formulations, and stability criteria without the overhead of a full-body simulator.
• Teaching Core Concepts: It serves as the canonical model for explaining the fundamental dynamics of bipedal walking in academia.
• Initial Benchmarking: New control policies can be first validated on the LIPM before the costly and complex process of sim-to-real transfer to a full-body simulation or physical robot. This accelerates the development cycle for new locomotion algorithms.

The Zero-Moment Point (ZMP) is the primary criterion for dynamic stability in legged locomotion. It is defined as the point on the ground where the net moment of the inertial forces and gravitational forces acting on the robot has no horizontal component. If the ZMP remains within the support polygon, the robot is dynamically stable. The ZMP is closely related to the Center of Pressure (CoP) and is a central concept in the ZMP-based walking control paradigm used by many humanoid robots.

The Capture Point is a point on the ground where a legged robot can place its foot to come to a complete stop in a single step. It is derived directly from the Linear Inverted Pendulum Model (LIPM) dynamics. The calculation uses the robot's current center of mass (CoM) position and velocity to project the Divergent Component of Motion (DCM). Planning foot placements at or beyond the instantaneous capture point is a core strategy for push recovery and reactive locomotion.

The Divergent Component of Motion (DCM) is a state variable that decomposes the LIPM dynamics into stable and unstable parts. It is defined as ξ = x + (ẋ / ω), where x is the CoM position, ẋ is its velocity, and ω is the pendulum's natural frequency. The DCM captures the unstable dynamics of the system. Controlling the DCM (often by placing the Capture Point) is equivalent to controlling the system's orbital energy and is a modern method for planning stable walking trajectories.

A Reduced-Order Model (ROM) is a simplified dynamic representation that captures the essential behavior of a complex system for control and planning. The Linear Inverted Pendulum Model (LIPM) is a quintessential ROM for bipedal walking, simplifying the multi-body robot to a point mass. Other key ROMs include:

• Spring-Loaded Inverted Pendulum (SLIP) for running/hopping.
• Rimless Wheel for passive walking. Using ROMs allows for real-time trajectory generation and high-level planning, which is then tracked by a full-body controller.

Centroidal dynamics describes the relationship between the net external wrenches (forces and moments) acting on a robot and the motion of its center of mass (CoM) and its centroidal angular momentum. It is the dynamics of the robot's composite rigid body as seen at its CoM. This framework is crucial for whole-body control (WBC) as it provides a direct link between high-level LIPM-based CoM planning and the low-level joint torque commands required to realize those plans while managing angular momentum.

Model Predictive Control (MPC) is an advanced control method that uses an internal dynamic model (often the LIPM) to predict future system behavior over a finite time horizon. At each control cycle, it solves an optimization problem to determine the optimal sequence of control inputs (e.g., ground reaction forces or footstep locations) that minimizes a cost function (e.g., tracking error, energy) while satisfying constraints (e.g., friction cones, ZMP stability). This allows legged robots to proactively plan motions and recover from disturbances.