NewtonEuler.md

Newton-Euler Dynamics

Overview & Motivation

The Newton-Euler formulation describes the dynamics of a single rigid body by directly applying Newton's second law for translational motion and Euler's equation for rotational motion. Unlike the Euler-Lagrange approach which works in generalized (joint-space) coordinates, Newton-Euler operates in Cartesian space, computing forces and torques on individual bodies.

In the body-fixed reference frame, the equations are:

$$F = m(\dot{v} + \omega \times v)$$ $$\tau = I\dot{\omega} + \omega \times (I\omega)$$

Where:

• $F \in \mathbb{R}^3$ — applied force in the body frame
• $\tau \in \mathbb{R}^3$ — applied torque in the body frame
• $m$ — body mass (scalar)
• $I \in \mathbb{R}^{3 \times 3}$ — inertia tensor in the body frame
• $v, \dot{v} \in \mathbb{R}^3$ — linear velocity and acceleration in the body frame
• $\omega, \dot{\omega} \in \mathbb{R}^3$ — angular velocity and acceleration in the body frame
• $\omega \times v$ — Coriolis term due to body-frame formulation
• $\omega \times (I\omega)$ — gyroscopic coupling term

This supports two computations:

• Forward dynamics: given forces/torques, compute linear and angular accelerations
• Inverse dynamics: given desired accelerations, compute required forces/torques

Mathematical Theory

Newton's Second Law (Translation)

In an inertial frame, Newton's second law is simply $F = ma$. In a body-fixed (rotating) frame, the transport theorem introduces an additional term:

$$F = m(\dot{v} + \omega \times v)$$

The $\omega \times v$ term is the Coriolis acceleration that arises because the reference frame itself is rotating.

Forward: $\dot{v} = F/m - \omega \times v$

Inverse: $F = m(\dot{v} + \omega \times v)$

Euler's Equation (Rotation)

Euler's equation governs rotational dynamics:

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

The $\omega \times (I\omega)$ term is the gyroscopic torque — it couples different rotational axes when the inertia tensor is non-spherical. For a body spinning about a non-principal axis, this term causes precession even without external torque.

Forward: $\dot{\omega} = I^{-1}(\tau - \omega \times (I\omega))$

This requires solving the $3 \times 3$ linear system $I\dot{\omega} = \tau - \omega \times (I\omega)$ via Gaussian elimination.

Inverse: $\tau = I\dot{\omega} + \omega \times (I\omega)$

This is a direct matrix-vector multiply plus cross product — no linear solve needed.

Properties of the Inertia Tensor

The inertia tensor $I$ is:

• Symmetric: $I = I^T$ (6 independent components, not 9)
• Positive definite: $\omega^T I \omega > 0$ for $\omega \neq 0$
• Constant in body frame: For a rigid body, $I$ does not change with the body's orientation when expressed in the body frame

When $I$ is diagonal (principal axes frame), components are the principal moments of inertia $I_{xx}, I_{yy}, I_{zz}$, and Euler's equation simplifies to the well-known form:

$$\tau_x = I_{xx}\dot{\omega}_x + (I_{zz} - I_{yy})\omega_y\omega_z$$ $$\tau_y = I_{yy}\dot{\omega}_y + (I_{xx} - I_{zz})\omega_z\omega_x$$ $$\tau_z = I_{zz}\dot{\omega}_z + (I_{yy} - I_{xx})\omega_x\omega_y$$

Complexity Analysis

Operation	Time	Space	Notes
Forward dynamics	$O(1)$	$O(1)$	Fixed 3×3 system; Gaussian elimination on 3×3 matrix
Inverse dynamics	$O(1)$	$O(1)$	Matrix-vector multiply + cross products
Cross product	$O(1)$	$O(1)$	6 multiplications and 3 subtractions

All operations are constant-time since the spatial dimension is fixed at 3. The $3 \times 3$ Gaussian elimination is effectively a small constant cost.

Step-by-Step Walkthrough

Example: Uniform Sphere

Parameters: mass $m = 2,\text{kg}$, radius $r = 0.5,\text{m}$

Inertia tensor: $I = \frac{2}{5}mr^2 \cdot \mathbf{I}_3 = 0.2 \cdot \mathbf{I}_3$

Forward dynamics with $F = [6, 0, 0]^T,\text{N}$, $\tau = [0, 0, 1]^T,\text{N·m}$, $v = \omega = 0$:

$$\dot{v} = F/m - 0 = [3, 0, 0]^T ,\text{m/s}^2$$ $$\dot{\omega} = I^{-1}\tau = [0, 0, 5]^T ,\text{rad/s}^2$$

Example: Gyroscopic Precession

Asymmetric body with $I = \text{diag}(1, 2, 3),\text{kg·m}^2$

Spinning at $\omega = [1, 1, 0]^T,\text{rad/s}$ with no external torque:

$$I\omega = [1, 2, 0]^T$$ $$\omega \times (I\omega) = [1,1,0] \times [1,2,0] = [0, 0, 1]^T$$ $$\dot{\omega} = I^{-1}(0 - [0,0,1]) = [0, 0, -\tfrac{1}{3}]^T ,\text{rad/s}^2$$

The body accelerates about the z-axis despite no external torque — this is the gyroscopic coupling effect due to asymmetric inertia.

Pitfalls & Edge Cases

• Singular inertia: An inertia tensor must be positive definite. Zero or negative eigenvalues indicate a non-physical body definition.
• Body-frame vs. inertial frame: The equations assume all quantities (forces, torques, velocities) are expressed in the body-fixed frame. Users must transform between frames externally.
• Spherical inertia: When $I = c \cdot \mathbf{I}_3$, the gyroscopic term $\omega \times (I\omega) = c(\omega \times \omega) = 0$, simplifying Euler's equation to $\tau = I\dot{\omega}$.
• Fixed-point arithmetic: Like Euler-Lagrange, Newton-Euler involves physical quantities (forces in Newtons, torques in N·m) that typically exceed Q15/Q31 range. Use float.
• Energy conservation: For torque-free motion, kinetic energy $T = \frac{1}{2}\omega^T I \omega$ and angular momentum $L = I\omega$ magnitude should be conserved. Numerical integration may violate this.

Variants & Generalizations

Variant	Key Difference	Use Case
Newton-Euler (this)	Single rigid body, Cartesian space	Spacecraft attitude, single-body simulation
Recursive Newton-Euler (RNEA)	Multi-body recursive algorithm, $O(n)$	Robot inverse dynamics, real-time control
Euler-Lagrange	Generalized coordinates, joint space	Control design, analytical derivations
Spatial vector algebra	6D twist/wrench representation	Compact multi-body formulations

Applications

• Spacecraft attitude control: Reaction wheel and thruster control using Euler's equations
• Drone / UAV dynamics: Quadrotor translational and rotational dynamics
• Projectile dynamics: Spinning projectile with aerodynamic forces
• Gyroscope modeling: Precession and nutation analysis
• Foundation for RNEA: Body-level Newton-Euler equations form the building block for recursive multi-body algorithms

Connections to Other Algorithms

• Gaussian Elimination (GaussianElimination.md): Used to solve the $3 \times 3$ inertia system $I\dot{\omega} = \tau_{net}$ in forward dynamics
• Euler-Lagrange (EulerLagrange.md): Joint-space counterpart; Euler-Lagrange derives the same dynamics from energy principles in generalized coordinates
• Recursive Newton-Euler (RNEA): Extends single-body Newton-Euler to kinematic chains by propagating velocities/accelerations forward and forces/torques backward through the chain

References & Further Reading

• Siciliano, B., Sciavicco, L., Villani, L., & Oriolo, G. (2009). Robotics: Modelling, Planning and Control. Springer. Chapter 7.
• Goldstein, H., Poole, C., & Safko, J. (2002). Classical Mechanics (3rd ed.). Chapters 4–5.
• Hughes, P. C. (1986). Spacecraft Attitude Dynamics. Dover Publications.
• Featherstone, R. (2008). Rigid Body Dynamics Algorithms. Springer. Chapter 2.