Module 02: Rigid Body Dynamics

Introduction

Understanding rigid body dynamics is fundamental to robotics. This module covers the physics that governs how robots move and interact with their environment, providing the mathematical foundation for control and simulation.

Section 1: Newton-Euler Formulation

1.1 Newton's Laws for Rigid Bodies

Rigid Body: A solid object in which the distance between any two points remains constant regardless of external forces applied.

For a rigid body, Newton's second law extends to:

$\mathbf{F} = m\mathbf{a}_{cm}$

where $\mathbf{F}$ is the net force, $m$ is mass, and $\mathbf{a}_{cm}$ is acceleration of the center of mass.

1.2 Euler's Equations for Rotation

The rotational equivalent involves angular momentum:

$\boldsymbol{\tau} = \mathbf{I}\dot{\boldsymbol{\omega}} + \boldsymbol{\omega} \times (\mathbf{I}\boldsymbol{\omega})$

where:

$\boldsymbol{\tau}$ is the net torque
$\mathbf{I}$ is the inertia tensor
$\boldsymbol{\omega}$ is angular velocity

Section 2: Mass Properties

2.1 Center of Mass

$\mathbf{r}_{cm} = \frac{1}{M}\sum_{i} m_i \mathbf{r}_i$

2.2 Inertia Tensor

The inertia tensor captures rotational inertia about all axes:

def compute_inertia_tensor(masses, positions):
    """Compute inertia tensor for a set of point masses."""
    I = np.zeros((3, 3))
    for m, r in zip(masses, positions):
        r_sq = np.dot(r, r)
        I += m * (r_sq * np.eye(3) - np.outer(r, r))
    return I

Section 3: Equations of Motion

3.1 Free Body Diagrams

Every force and torque acting on a body must be identified:

Gravitational forces
Contact forces (normal and friction)
Joint reaction forces
External applied forces

3.2 Multi-Body Systems

For articulated robots, the dynamics become:

$\mathbf{M}(\mathbf{q})\ddot{\mathbf{q}} + \mathbf{C}(\mathbf{q}, \dot{\mathbf{q}})\dot{\mathbf{q}} + \mathbf{g}(\mathbf{q}) = \boldsymbol{\tau}$

The mass matrix $\mathbf{M}(\mathbf{q})$ is configuration-dependent and must be recomputed at each time step.

Summary

Key takeaways:

Newton-Euler equations govern rigid body motion
Mass properties (COM, inertia) are essential for dynamics
Multi-body dynamics leads to coupled differential equations
Numerical integration enables simulation

Key Concepts

Rigid Body: Idealized solid with fixed shape
Inertia Tensor: 3×3 matrix describing rotational inertia
Equations of Motion: ODEs describing system dynamics
Newton-Euler Method: Algorithm for computing joint torques

Introduction​

Section 1: Newton-Euler Formulation​

1.1 Newton's Laws for Rigid Bodies​

1.2 Euler's Equations for Rotation​

Section 2: Mass Properties​

2.1 Center of Mass​

2.2 Inertia Tensor​

Section 3: Equations of Motion​

3.1 Free Body Diagrams​

3.2 Multi-Body Systems​

Summary​

Key Concepts​

Further Reading​