From Torque to Angular Acceleration… Correctly.

Introduction

I’ve been seeing the relationship between torque and angular acceleration stated as

\alpha = I^{-1}\tau.

Although this has a nice analogy with the force, mass and acceleration of translational dynamics (a=m^{-1} F), it is actually incorrect. There is an extra term missing from the right-hand side. The correct version is

\alpha = I^{-1}(\tau-\omega\times L).

This extra \omega\times L term is sometimes called the gyroscopic term. It is non-zero when the angular velocity and angular momentum have different directions. Unfortunately there are some lecture notes, slides and articles that make no mention of this extra term, nor do they explain why it has been omitted.

This is particularly the case with material aimed at the development of real-time simulations of rigid-body dynamics, such as game development.

Looking at open source physics libraries, you most likely won’t find this missing term there either, so where has it gone?

The reason for the omission is that the extra term causes instability in numerical simulations with general-purpose solvers. There’s no physical reason to omit it — indeed the extra term has real physical consequences — but where physical accuracy isn’t paramount it is simpler to just remove the term than to employ a special-purpose solver.

This article will derive the torque-angular acceleration relationship in full.

Derivation

From basic rigid-body dynamics we know that the net torque, \tau\in\mathbb R^3, is the time derivative of the angular momentum, L\in\mathbb R^3,

\tau = \dot L.

The moment of inertia tensor, I\in\mathbb R^3\otimes\mathbb R^3\cong\mathbb R^{3\times 3}, is defined as the linear relationship between angular velocity \omega\in\mathbb R^3 and angular momentum,

L = I\omega.

Combining these two equations gives

\tau = \frac{\mathrm d}{\mathrm dt}(I\omega) = I\dot\omega + \dot I\omega.

The key thing to note is that the moment of inertia tensor is a time-dependent quantity — it changes as the body rotates. Only when described in a body-fixed rotating frame is it constant.

Now \dot\omega is just the definition of angular acceleration, \alpha, so this becomes

\tau = I\alpha + \dot I\omega

The tricky bit is the second term, which involves the time derivative of the inertia tensor. If \tilde I is the constant inertia tensor in the body-fixed frame and R\in\mathrm{SO}(3) is the current rotation matrix for the body (the body-to-world rotation), then

I = R \tilde I R^{\mathrm T}

and

\dot I = \dot R \tilde I R^{\mathrm T} + R \tilde I \dot R^{\mathrm T}.

The rotation matrix evolves due to angular velocity via

\dot R = \omega_{\times} R,

where \omega_{\times}\in\mathfrak{so}(3) is the anti-symmetric cross-product matrix generated by \omega, giving

\dot I = \omega_{\times} I - I\omega_{\times}

and

\dot I\omega = \omega_{\times} I\omega - I\omega_{\times}\omega

The second term is null,

I\omega_{\times}\omega = I(\omega\times\omega) = 0,

and the remaining term is

\dot I\omega = \omega_{\times} I\omega = \omega\times L.

Therefore the torque is

\tau = I\alpha + \omega\times L

and the angular acceleration is

\alpha = I^{-1}(\tau - \omega\times L).

The Gyroscopic Term

In the absence of torque the gyroscopic term means the angular acceleration can still be non-zero. This is the cause of a phenomenum called torque-free precession, where the angular velocity vector gradually rotates around the angular momentum.

The reason for the gyroscopic term’s mysterious absence from numerical simulations is that it can cause instability when used with general-purpose solvers. A full treatment of this instability is beyond the scope of this short article, but suffice it to say that numerical instability is not acceptable. A special solver may be needed to safely include this term, and so as long as a physically accurate result isn’t required, it is simpler to omit it completely. This is a simple and cheap solution to the problem, but it means that the simulation isn’t accurate enough for the scientific level as it cannot exhibit phenomena such as torque-free precession. However, for non-scientific applications, such as games, it can still give a physically-plausible result without risk of instabilities nor cost of special solvers.

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