Centroidal Dynamics

The centroidal frame is a coordinate system fixed at the robot's center of mass (CoM) and aligned with the principal axes of its inertia. All dynamic equations are most concisely expressed in this frame. The key dynamic quantity is the centroidal angular momentum, which is the angular momentum of the entire multi-body system about its CoM. A fundamental principle is that the rate of change of this angular momentum equals the net external moment acting on the system about the CoM.

The dynamics of the CoM are governed by the Newton-Euler equations, which separate linear and angular momentum:

• Linear Dynamics: The total mass multiplied by CoM acceleration equals the sum of all external forces (primarily ground reaction forces).
• Angular Dynamics: The rate of change of the centroidal angular momentum equals the sum of external moments about the CoM.

These equations form the core constraint for any dynamically feasible motion. For a legged robot, the external forces are the ground reaction forces (GRFs) at each foot in contact.

The centroidal momentum matrix (CMM), often denoted as A_g, is a Jacobian-like matrix that maps the robot's joint velocities to its centroidal linear and angular momentum. It is a function of the robot's configuration (joint angles) and its inertial parameters. This matrix is crucial because it allows controllers to relate high-level momentum objectives to low-level joint motions. The equation is: h_G = A_g(q) * q_dot, where h_G is the centroidal momentum and q_dot is the vector of joint velocities.

A motion is dynamically feasible only if there exists a set of contact forces that satisfy the Newton-Euler equations. This introduces critical constraints:

• Unilateral Contact: Feet can only push, not pull, on the ground (normal force > 0).
• Friction Cone: The tangential force must lie within the friction cone to prevent slipping.
• Torque Limits: The required joint torques, computed via inverse dynamics, must be within actuator capabilities.

Planners and controllers must solve for GRFs that satisfy these constraints while achieving the desired CoM and angular momentum trajectories.

Centroidal dynamics provides the high-level objective for Whole-Body Control (WBC). A typical WBC hierarchy is:

1. Priority 1 (Centroidal): Track desired CoM acceleration and centroidal angular momentum rate.
2. Priority 2: Track end-effector (foot/swing leg) trajectories.
3. Priority 3: Posture/configuration optimization.

The WBC solver uses the CMM and the equations of motion to compute the required joint torques and GRFs in real-time, often formulated as a Quadratic Program (QP). This ensures the entire robot's motion is coordinated to achieve dynamic tasks like walking or jumping.

For real-time planning, the full centroidal dynamics are often approximated by reduced-order models (ROMs) that capture essential behavior:

• Linear Inverted Pendulum (LIP): Assumes constant CoM height, decoupling angular dynamics. Foundation for Zero-Moment Point (ZMP) control.
• Angular Momentum LIP: Extends LIP to include centroidal angular momentum regulation.
• Single Rigid Body (SRB): Models the entire robot as one rigid body with its total mass and inertia. This is a direct application of centroidal dynamics and is highly effective for dynamic motion planning, as used in Boston Dynamics' Atlas robot.

These ROMs enable fast trajectory optimization for the CoM and footsteps, which is then tracked by the whole-body controller.

Centroidal dynamics is the cornerstone for generating physically feasible motions. It ensures that the planned motion of the robot's center of mass (CoM) and its angular momentum are consistent with the ground reaction forces (GRFs) its feet can produce. Without this, a motion plan might be kinematically possible but dynamically impossible to execute, causing the robot to fall. This is solved by formulating a trajectory optimization problem that uses the centroidal dynamics equations as constraints, producing CoM trajectories and contact forces that the full robot can realistically track.

For legged robots, maintaining balance is a continuous challenge. Centroidal dynamics enables real-time balance control by relating the net external wrench (force and torque) to the acceleration of the CoM and the rate of change of centroidal angular momentum. Controllers use this to compute the required adjustment to the center of pressure (CoP) or to generate angular momentum to counteract a push. For example, a humanoid robot uses a model predictive control (MPC) scheme based on centroidal dynamics to preemptively shift its CoM and plan foot placements to recover from disturbances.

Determining how much force each foot should apply during multi-contact scenarios (e.g., a quadruped trotting or a humanoid taking a step) is a core application. The centroidal dynamics equations act as equality constraints in a quadratic program (QP). The optimizer solves for the optimal distribution of ground reaction forces that:

• Satisfy the net wrench needed for the desired CoM motion.
• Stay within friction cones to prevent slipping.
• Minimize a cost like energy expenditure or joint torque. This force distribution is then passed to a whole-body controller to compute the necessary joint torques.

High-level planners for walking, running, or climbing over rough terrain rely on simplified models derived from centroidal dynamics. The most common is the Linear Inverted Pendulum Model (LIPM), which assumes constant CoM height and zero angular momentum about the CoM. More advanced planners use the Variable Height LIPM or the Angular Momentum LIPM to plan motions involving crouching, jumping, or swinging the torso. These reduced-order models, grounded in centroidal dynamics, allow for fast, receding-horizon planning of footstep locations and CoM trajectories in complex environments.

Centroidal dynamics is not limited to ground contact. For flying robots like quadrotors equipped with a manipulator arm, the dynamics of the total system (drone + arm) are described by its centroidal dynamics. Moving the arm changes the system's center of mass and generates reaction moments. The flight controller must account for these changes to maintain stable flight. By formulating the control problem using the centroidal dynamics of the combined system, the controller can precisely compensate for the arm's motion, enabling stable hovering and trajectory tracking during manipulation tasks.

High-fidelity physics simulators for robotics use centroidal dynamics algorithms to efficiently compute forward dynamics for complex, floating-base systems. Furthermore, centroidal dynamics is the basis for creating template models used in control design. Models like the Spring-Loaded Inverted Pendulum (SLIP) for running or the Dynamically Extended LIPM are all abstractions that capture the essential centroidal behavior of a much more complex robot. These models are tractable for real-time control and provide deep insights into the principles of dynamic locomotion.

Floating base dynamics formulates the equations of motion for a multi-body system (like a legged robot) whose base link is not fixed to the world. This introduces six unactuated degrees of freedom (three for position, three for orientation) that must be controlled indirectly through contact forces. The dynamics are typically expressed in a centroidal frame, directly linking them to centroidal dynamics. Solving these equations is fundamental for whole-body control and inverse dynamics computations.

Whole-Body Control (WBC) is a hierarchical control framework that coordinates all of a robot's degrees of freedom to execute multiple tasks simultaneously. It uses centroidal dynamics as a high-priority task to ensure dynamic balance, while lower-priority tasks might include swinging a leg or moving an arm. WBC typically solves a Quadratic Program (QP) in real-time to compute optimal joint torques that satisfy dynamics, contact, and actuation constraints.

The Ground Reaction Force (GRF) is the force vector exerted by the ground on a robot's foot during contact. It is the primary mechanism through which a robot influences its centroidal dynamics. The GRF has:

• A normal component that supports the robot's weight.
• A frictional component that propels and stabilizes the robot. Optimal distribution and modulation of GRFs across multiple contacts is a core problem in legged locomotion planning and control.

A Reduced-Order Model (ROM) is a simplified dynamic representation that captures the essential locomotion dynamics while ignoring the full complexity of the robot. Key ROMs used with centroidal dynamics include:

• Linear Inverted Pendulum Model (LIPM): Assumes constant center of mass height for walking.
• Spring-Loaded Inverted Pendulum (SLIP): Models the leg as a spring for running/hopping. These models enable tractable motion planning and gait generation by abstracting the full dynamics into the motion of the center of mass.

Model Predictive Control (MPC) is an advanced control method that uses an internal dynamic model (often a centroidal dynamics ROM) 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 (like footstep locations and GRFs) that satisfies constraints and minimizes a cost function (e.g., energy, tracking error). This allows the robot to proactively plan for disturbances and uneven terrain.

A Quadratic Program (QP) Formulation is the standard mathematical optimization framework for solving many real-time robotics control problems. It involves minimizing a quadratic cost function subject to linear equality and inequality constraints. In legged locomotion, QPs are used to:

• Solve the inverse dynamics problem using centroidal dynamics.
• Implement whole-body control.
• Distribute optimal ground reaction forces. The linear constraints encode physical laws (dynamics), contact conditions (no slipping), and actuator limits.