The inertia tensor is a 3x3 symmetric matrix that characterises how a rigid body’s mass is distributed about a reference point, relating the body’s angular velocity to its angular momentum. Its diagonal entries are the moments of inertia about the coordinate axes and its off-diagonal entries are the products of inertia. It is a foundational quantity in rigid-body dynamics, enabling computation of rotational acceleration under applied torques.
Overview
- Encodes the rotational mass distribution of a body in a single coordinate-dependent matrix.
- Transforms predictably under rotation and parallel-axis shifts, allowing reuse across reference frames.
- Sits at the heart of the Newton-Euler and Lagrangian formulations of robot motion.
Key aspects
- Diagonal moments of inertia quantify resistance to angular acceleration about each axis.
- Off-diagonal products of inertia vanish when axes align with the principal axes.
- The parallel-axis theorem shifts the tensor between the centre of mass and other points.
- Symmetry and positive-definiteness constrain physically valid tensors.
- Link inertia tensors compose into the joint-space mass matrix of a manipulator.
Applications
- Computing forward and inverse dynamics for robot arms and legged platforms.
- Parameterising rigid bodies in physics-simulation engines.
- System identification of unknown payload inertial parameters.
- Spacecraft and drone attitude dynamics modelling.