Automatically Computing the Derivatives of a Vector Field using Dual Numbers

Dual numbers are 2-dimensional numbers that can be regarded as an extension of the real numbers. Along with the complex and split-complex numbers, they are one of the three 2-dimensional unital algebras.

Each dual number can be described as a linear combination of the real unit $1$ and the dual unit $\varepsilon$. Whereas the complex numbers feature an imaginary unit that squares to $-1$, the dual unit squares to zero, $\varepsilon^2=0$. This leads to the following algebraic operations.

We will write a dual number $x+\varepsilon y$ as the pair $(x,y)$.

$(x,y) \pm (a,b) = (x\pm a,y\pm b),$
with identity $(0,0)$.

Multiplication is
$(x,y) (a,b) = (xa,ya+xb),$
which is commutative and has identity $(1,0)$.

Division is therefore
$\frac{(x,y)}{(a,b)}= (\frac xa,\frac{ya-xb}{a^2}).$

From this we can see that
$(x,y)^n = (x^n,nx^{n-1} y),\quad n\in\mathbb N_1.$

Extending Real Functions to the Dual Plane

We can use this algebraic structure to extend any real analytic function $f:\mathbb R\to\mathbb R$ to the dual plane, $\tilde f:\mathbb R^2\to\mathbb R^2$ by taking its Taylor series and substituting a dual argument.

$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x-a)^n$

$\tilde f(x,y) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x-a,y)^n$

$\tilde f(x,y) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} \left((x-a)^n,n(x-a)^{n-1} y\right)$

$\tilde f(x,y) = \left(\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n,y\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}n(x-a)^{n-1}\right)$

$\tilde f(x,y) = (f(x),yf'(x)).$

We can use this rule to uniquely extend any differentiable function to the dual plane, or indeed any suitable subset of the real line to the corresponding subset of the dual plane. This extension is compatible with the composition of functions, i.e. the extension of a composition is the composition of the extensions,

$\widetilde{f\circ g} = \tilde f\circ\tilde g.$

This is useful as it means we can extend complicated functions by composing elementary functions.

$f(x)$ $f'(x)$ $\tilde f(x,y)$
$x+c$ $1$ $(x+c,y)$
$cx$ $c$ $(cx,cy)$
$x^n$ $nx^{n-1}$ $(x^n,nx^{n-1}y)$
$e^x$ $e^x$ $(e^x,e^xy)$
$\log x$ $1/x$ $(\log x,y/x)$
$\sin x$ $\cos x$ $(\sin x,y\cos x)$

For example, to extend $\sin^2 x$ we extend $\sin x$ to give $(\sin x,y\cos x)$, then square the result to give $(\sin^2 x,2y\sin x\cos x)$.

In other words, we don’t need to know the derivative of the complicated function in order to extend it, we only need to know the derivatives of the elementary functions of which it is composed.

Automatic Differentiation

A real function’s derivative at a given point $x\in\mathbb R$ can be obtained by applying its extension to the dual number $(x,1)$ and then taking the dual part of the result,

$\tilde f(x,1) = (f(x),f'(x)).$

This is called automatic differentiation. This is particularly useful when we don’t necessarily need to know $f'$ as a function, we just want to obtain its value at a particular point $x$. If the function $f$ is a complicated combination of many elementary functions, then computing the derivative on paper can be both time-consuming and error-prone. Instead of finding $f'$ in full and then evaluating it, we can just evaluate the original function $f$ with a dual number.

This technique can be applied to multivariable functions, such as vector fields, $V:\mathbb R^n\to\mathbb R^n$, giving

$\tilde V(x+\varepsilon y) = V(x) + \varepsilon (y\cdot\nabla) V|_{x}.$

The dual part is the directional derivative of the vector field $V$ in the direction $y$, evaluated at $x$. By choosing the direction to be the $j$th basis vector, $e_j$, and then taking the $i$th component of the result, we can automatically compute the partial derivatives of the vector field,

$\tilde V^i(x+\varepsilon e_j) = V^i(x) + \varepsilon \partial_j V^i|_{x}.$

If we denote the dual part as $\mathfrak D(x,y) = y$ then the Jacobian is given by

$\left[\begin{array}{ccc}\mathfrak D\tilde V(x,e_1)&\cdots&\mathfrak D\tilde V(x,e_n)\end{array}\right].$

This means we can compute the derivatives of an $n$-dimensional vector field by evaluating the vector field $n$ times with dual numbers.

In a practical implementation it is more convenient to think of $(x,y)$ as a vector of dual numbers, rather than a pair of real vectors. This allows for a single implementation of the vector field to be evaluated with a vector of either real or dual elements, for example, by templating the scalar type in C++:

template<typename Scalar>
Vector<Scalar> MyVectorField( const Vector<Scalar>& x ) {
// evaluate the vector field
}


The version of automatic differentation discussed here (using dual numbers) is called forward automatic differentiation. There is an alternate version called reverse automatic differentiation, and also extensions to the higher derivatives. Wikipedia has a decent article on the topic.

The Dynamics of a Transform Hierarchy

Transforms are used in game development to describe the position, rotation and scale of objects in a virtual world. These properties are encoded in a coordinate transformation, which is a function that transforms the object’s local coordinates into the world’s coordinates. These functions are invertible affine maps, consisting of an invertible linear map and a translation.

Modern games often feature complex objects, each made up of several sub-objects. To manage this complexity it has become common to organize the game objects into a hierarchy, where each object is assigned a parent, which is either another object or the world itself. This forms a tree structure with the world as the root. In such a hierarchy, each object’s transform describes its position, rotation and scale relative to its parent. The textbook example is a tank and its turret — the turret has a constant position and scale relative to its parent tank, but can rotate of its own accord.

In order to determine the transform of an object relative to the world, the individual transforms are composed by walking through the tree from the object to the root. Each transform in the sequence transforms from child to parent and the result of the composition is a single transform that describes the object relative to the world.

Composition of transforms

However, when dealing with dynamics we must acknowledge that these transforms are changing over time. In order to advance a transform forward in time, we could do everything in global terms — applying global velocities and accelerations and then converting the new global transform back into a local transform. This approach is easier, but it effectively bypasses the hierarchy as far as dynamics is concerned.

An interesting alternative is to use local velocities and accelerations, which allows the local transforms to be updated directly. In order take this approach, we need to understand how to convert these velocities and accelerations between the different frames of reference. This is especially important when the dynamics is being driven by physics, such as Newtonian mechanics. Physical forces produce acceleration, and given the acceleration of an object relative to the world, we must determine the acceleration of the object relative to its parent. This will involve taking into account the dynamics of each object in the tree between it and the root. This article explores the mathematics needed to do this.

We begin by expressing the algebraic structure of transforms in terms that keep the translation, rotation and scale components separate (rather than mixing them into a single matrix). Then we discuss our preferred description of the dynamics of a transform using linear and angular velocities and accelerations. The key step is then to extend the algebraic structure to include these velocities and accelerations, which provides the machinery to convert the dynamics between different frames of reference. This is used to derive local-to-global and global-to-local conversions of velocity and acceleration. These results are applied to instantaneous modifications of the dynamics (e.g. by impulses) to gain corresponding conversions for discrete changes in velocity and acceleration.

Finally, we look at a couple of example applications. The first shows how to apply Newtonian forces to generate local accelerations, which results in the well-known inertial forces arising. The second application is for linear impulses, where we also discuss how to counteract inherited changes, which is important for correct collision response.

Transforms

To begin, let’s define more precisely what kind of transforms we are dealing with. Nothing in this article is specific to 3-dimensional space, so we will work in $n$-dimensional space. Let a transform be a triple $(T,R,S)$ consisting of a translation vector $T\in\mathbb R^n$, a rotation matrix $R\in\mathrm{SO}(n,\mathbb R)$, and a positive scale factor $S\in\mathbb R$. As a function $A:\mathbb R^n\to\mathbb R^n$, the transform acts on a coordinate vector $x$ by

$x\mapsto A(x) = R S x + T,$

i.e. the scale is applied first, then the rotation, and finally the translation. We will write the linear part as $J = RS$. The linear part coincides with the Jacobian of the map (hence the notation) and its action performs a change of basis. If $A$ is a transform from child to parent then $J$ is a change of basis from the child basis to the parent basis.

Transforms form a group under composition. We can express the group structure (i.e. composition, identity, inverse) in terms of the individual translation, rotation and scale components.

Composition of Transforms

The composition of two transforms is a third transform. The resulting transform can be expressed with separate translation, rotation and scale components:

$(T_3,R_3,S_3) = (T_2,R_2,S_2)\circ (T_1,R_1,S_1),$

where

$S_3 = S_2 S_1$
$R_3 = R_2 R_1$
$T_3 = R_2 S_2 T_1 + T_2 = A_2(T_1).$

Note that the composition of functions $g\circ f$ means that $f$ is applied first, then $g$.

Some implementations allow for multiple scale factors, one for each axis, which are usually stored as a vector. The use of a vector for this can be rather misleading — the correct interpretation is a diagonal $n\times n$ matrix. When composing such transforms, the scaling matrix loses its diagonality. It becomes a mixture of diagonal scalings and rotations, which requires the full $n\times n$ matrix to describe. For this reason we will restrict ourselves to a single scale factor.

Identity Transform

The identity transform does nothing, $x\mapsto x$. It is given by $(0,\mathrm{id},1)$: the null vector $0\in\mathbb R^n$, the identity matrix $\mathrm{id}\in\mathrm{SO}(n,\mathbb R)$, and the unit scalar $1\in\mathbb R$.

Inverse Transform

The inverse of a transform $(T,R,S)$ is a transform denoted $(T,R,S)^{-1}$ that is defined by

$(T,R,S)\circ (T,R,S)^{-1} = (0,\mathrm{id},1) = (T,R,S)^{-1} \circ (T,R,S),$

i.e. the inverse undoes the action of the transform, resulting in the identity transform. It is given by

$(T',R',S') = (T,R,S)^{-1},$

where

$S' = 1 / S$
$R' = R^{\mathrm T}$
$T' = R^{\mathrm T}( -T/S ) = -S' R' T = -J'T$

and $R^{\mathrm T}$ is the matrix transpose of $R$, which describes the inverse rotation. $J'$ is the linear part of the inverse, which is the inverse of the linear part,

$J'= J^{-1} = R'S' = R^{\mathrm T}/S.$

Dynamics of the Transform

To describe how a transform $(T,R,S)$ is changing over time we need to consider the time derivatives of the three components. In this article we are going to assume that the scale factor $S$ is constant. It’s unusual to have a time-varying scale factor, and taking it to be constant simplifies some of the following equations.

For the translational and rotational parts, we will consider the first two time derivatives (velocity and acceleration). Dynamics in this domain is often specified by second-order differential equations, such as Newton’s second law, and it is unusual to encounter derivatives of third-order or higher, so two should be sufficient for most applications.

Describing Translational Dynamics

Since the translation $T$ describes the relative position of the object to its parent, the relative velocity is its first derivative and the relative acceleration is its second derivative,

$v = \dot T,$
$a = \dot v = \ddot T.$

Describing Rotational Dynamics

The dynamics of the rotation $R$ is given by the derivatives $\dot R$ and $\ddot R$. However, it is not convenient to use these quantities directly. We are using $n\times n$ matrices to describe rotations, but only a relatively small number of matrices are valid rotation matrices. The valid matrices form a smooth geometric object — a manifold — within the ambient space of matrices. The time derivative $\dot R$ is a tangent vector to this manifold. Due to the curvature of this manifold, the types of matrix that describe valid tangent vectors is dependent on the current rotation, $R$. This means we cannot choose $\dot R$ independently of $R$.

Angular velocity as a tangent vector to the rotation group at the identity.

What we would like is a way to specify the time derivative in terms independent of the current rotation, analogous to the way we can choose a velocity independently of the current position. We do this by describing the time derivative using a vector in the tangent space at the identity rotation, $\omega\in\mathfrak{so}(n,\mathbb R)$. This tangent vector can be associated with a tangent vector at $R$ by

$\dot R = \omega R.$

We call the matrix $\omega$ an angular velocity. The tangent space at the identity is denoted $\mathfrak{so}(n,\mathbb R)$ and consists of $n\times n$ skew-symmetric matrices, i.e.

$\omega^{\mathrm T} = -\omega.$

This follows straightforwardly from the definition of a rotation matrix (by differentiating the orthogonality constraint). The skew-symmetry is preserved by addition and scalar multiplication, which isn’t surprising given that it characterizes a vector space. Although individual matrix multiplications do not preserve skew-symmetry, the space does feature a Lie bracket given by the matrix commutator, $[\alpha,\beta] = \alpha\beta - \beta\alpha$.

Although the angular velocities are $n\times n$ matrices, we do not need $n^2$ scalars to describe them. The dimension of the rotation group (and hence its tangent spaces) is $n(n-1)/2$. This means for 2-dimensional space we need only a single scalar,

$\left[ \begin{array}{cc} 0 & -\omega\\\omega & 0 \end{array} \right] \in\mathfrak{so}(2,\mathbb R).$

The action of this matrix on a 2-vector $v$ corresponds to scalar multiplication of the perpendicular vector,

$\left[ \begin{array}{cc} 0 & -\omega\\\omega & 0 \end{array} \right] v = \omega v^\perp$

where the perpendicular vector is $(x,y)^\perp = (-y,x)$. The commutator vanishes for the 2-dimensional case, $[\alpha,\beta]=0\ \forall\alpha,\beta\in\mathfrak{so}(2,\mathbb R)$.

For 3-dimensional space we need 3 scalars,

$\left[\begin{array}{ccc}0 & -\omega_3 & \omega_2\\\omega_3 & 0 & -\omega_1\\-\omega_2 & \omega_1 & 0\end{array}\right] \in\mathfrak{so}(3,\mathbb R).$

This allows for an angular velocity to be interpreted as a 3-vector $(\omega_1,\omega_2,\omega_3)\in\mathbb R^3$. Under this identification, the action of the matrix on a 3-vector $v$ corresponds to the cross product $\omega\times v$. Similarly, the commutator of the matrix representations corresponds to the cross product of the vector representations, $[\alpha,\beta]=\alpha\times\beta$. Since this vector interpretation of angular velocity is unique to the 3-dimensional case, we will keep things in the matrix form. The matrix form is also a bit nicer to work with, since matrix multiplication is associative, while the cross product is not.

The second derivative is then given by

$\ddot R = (\alpha + \omega^2) R,$

where we have defined the angular acceleration,

$\alpha = \dot\omega.$

The angular accelerations are also skew-symmetric matrices. This allows us to choose $R$, $\omega$ and $\alpha$ independently of each other while still providing a complete description of $\dot R$ and $\ddot R$.

So a transform $A=(T,R,S)$ and its dynamics can be described in terms of the quantities $(A,v,a,\omega,\alpha)$. Now that we have decided how to describe the dynamics of individual transforms, we can extend the transforms’ group structure to include the velocities and accelerations. This algebraic structure will be the means by which we convert between local and global descriptions of an object’s dynamics.

Dynamics of the Composition

To compute the dynamics of the composition, we take the expression for the composition of two transforms and compute the derivatives of its translation and rotation components. The definitions of velocity and acceleration are then used to identify $v$, $a$, $\omega$ and $\alpha$ for the composition.

$A_3 = A_2\circ A_1$
$v_3 = v_2 + J_2 v_1 + \omega_2 J_2 T_1$
$a_3 = a_2 + J_2 a_1 + \alpha_2 J_2 T_1 + (\omega_2)^2 J_2 T_1 + 2\omega_2 J_2 v_1$
$\omega_3 = \omega_2 + R_2 \omega_1 R_2^{\mathrm T}$
$\alpha_3 = \alpha_2 + R_2 \alpha_1 R_2^{\mathrm T} + [\omega_2,\omega_3]$.

Dynamics of the Inverse

The inverse converts a child-to-parent transform into a parent-to-child transform. The inverse transform describes the parent relative to the child. By computing the time derivative of the expression for the inverse, we can express the dynamics of the parent relative to the child.

$A' = A^{-1}$
$v' = J^{-1}( \omega T - v )$
$a' = J^{-1}( \alpha T - \omega^2 T + 2\omega v - a )$
$\omega' = -R^{\mathrm T}\omega R$
$\alpha' = -R^{\mathrm T}\alpha R$.

Dynamics of the Identity

The identity transform describes an object relative to itself. The dynamics of such a transform is null:

$A = (0,\mathrm{id},1)$
$v = 0$
$a = 0$
$\omega = 0$
$\alpha = 0$.

Note that $v$ and $a$ are null vectors, while $\omega$ and $\alpha$ are zero matrices (the zero matrix is skew-symmetric).

The interpretation of this is that an object is always at rest relative to itself, regardless of its dynamics relative to other objects/the world.

Local and Global Dynamics

We now have enough machinery to convert between local and global descriptions of an object’s dynamics.

For each object we can describe its velocities and accelerations either locally (relative to the parent) or globally (relative to the world). For objects that are parented to the world these notions coincide, but in general they do not.

A hierarchy of objects related by transforms. Each object has exactly one parent.

Let an object have transform $A_1$ relative to its parent, and the parent have transform $A_2$ relative to the world.

Converting Local Dynamics To Global Dynamics

The composition $A_3=A_2\circ A_1$ describes the object relative to the world. Given its local dynamics, and the global dynamics of the parent, the object’s global dynamics is given by the dynamics of the composition, given above.

In the case where there are many levels in the hierarchy between the root and the object, each node’s global dynamics depends on the global dynamics of its parent. This approach can be applied recursively, in exactly the same way that a transform hierarchy usually operates. In iterative terms this means starting at the root and walking through the tree towards the object, computing the global transform (and dynamics) of each node in sequence.

Converting Global Dynamics To Local Dynamics

If we are given the dynamics of $A_3$ (the global dynamics of the object), and want to find the dynamics of $A_1$ (the local dynamics of the object), we can express $A_1$ in terms of $A_2$ and $A_3$ via

$A_1 = A_2^{-1}\circ A_3.$

The dynamics of $A_1$ is then given by combining the dynamics of the inverse of $A_2$ with the dynamics of the composition with $A_3$:

$A_1 = A_2^{-1}\circ A_3$
$v_1 = J_2^{-1}( v_3-v_2-\omega_2(T_3-T_2) )$
$a_1 = J_2^{-1}( a_3-a_2-(\alpha_2-(\omega_2)^2)(T_3-T_2) - 2\omega_2(v_3-v_2) )$
$\omega_1 = R_2^{\mathrm T}( \omega_3-\omega_2 ) R_2$
$\alpha_1 = R_2^{\mathrm T}( \alpha_3-\alpha_2 + [\omega_3,\omega_2] ) R_2.$

The velocity and acceleration can also be written as

$v_1 = J_2^{-1}( v_3-v_2-\omega_2J_2T_1 )$
$a_1 = J_2^{-1}( a_3-a_2-(\alpha_2+(\omega_2)^2)J_2T_1 - 2\omega_2J_2v_1 ).$

Local and Global Deltas

There are occasions when it is necessary to make instantaneous modifications to the velocity and acceleration of an object. A relevant example is modifying an object’s velocity in response to a collision. We can use the above results to convert the changes in velocities and accelerations between local and global forms.

As before, $A_1$ is the child to parent transform, $A_2$ is the parent to world transform, and $A_3=A_2\circ A_1$ is the child to world transform. These transforms are assumed to be unchanged by the interaction, and the velocity/accel is modified instantaneously, meaning no time elapses between the old and new velocities/accelerations.

Global change in child due to local change in child

$\Delta v_3 = J_2\Delta v_1$
$\Delta a_3 = J_2\Delta a_1 + 2\omega_2\Delta v_3$
$\Delta\omega_3 = R_2 \Delta\omega_1 R_2^{\mathrm T}$
$\Delta\alpha_3 = R_2 \Delta\alpha_1 R_2^{\mathrm T} + [\omega_2,\Delta\omega_3]$

Local change in child required to produce the given global change in child

This is just the inverse of the above.

$\Delta v_1 = J_2^{-1}\Delta v_3$
$\Delta a_1 = J_2^{-1}( \Delta a_3 - 2\omega_2\Delta v_3 )$
$\Delta\omega_1 = R_2^{\mathrm T} \Delta\omega_3 R_2$
$\Delta\alpha_1 = R_2^{\mathrm T}( \Delta\alpha_3 + [\Delta\omega_3,\omega_2] ) R_2$

Global change in child due to global change in parent

A child’s global dynamics depends on both its local dynamics and its parent’s global dynamics. If the child is locally unmodified but the parent is globally modified then the child’s global dynamics changes according to the following.

$\Delta v_3 = \Delta v_2 + \Delta\omega_2 J_2T_1$
$\Delta a_3 = \Delta a_2 + (\Delta\alpha_2 + \Delta\omega_2\omega_2 + \omega_2\Delta\omega_2 + (\Delta\omega_2)^2) J_2T_1 + 2\Delta\omega_2 J_2v_1$
$\Delta\omega_3 = \Delta\omega_2$
$\Delta\alpha_3 = \Delta\alpha_2 + [\Delta\omega_2,\omega_3-\omega_2]$

Example: Newtonian Mechanics — Force to Acceleration

In Newtonian mechanics, the dynamics of an object is driven by forces, which are influences that determine its acceleration via $F=ma$, where $F\in\mathbb R^n$ is the total force, and $m>0$ is the mass of the object. However, this is only valid in what physicists call an inertial frame, which is a frame of reference that is either stationary or moving with constant (translational) velocity — no acceleration or rotation. This corresponds to our world frame, the root of the hierarchy.

A transform hierarchy of two objects in the world.

Consider a non-inertial frame, such as the Earth, which has translational velocity and acceleration due to its orbit around the Sun, angular velocity due to its rotation around its axis, and even some angular acceleration that causes its axis to precess. Let us also have an object that is parented to the non-inertial frame, such as a vehicle on the Earth.

If the Earth has transform $A_2$ and the vehicle has transform $A_1$ then we can convert the acceleration of the object relative to the world to acceleration relative to the Earth by using the global-to-local acceleration equation above,

$a_1 = J_2^{-1}( a_3-a_2-(\alpha_2+(\omega_2)^2)J_2T_1 - 2\omega_2J_2v_1 ).$

We can write this in terms of the applied force $F$ and mass $m$,

$a_1 = \frac{1}{m} J_2^{-1} F - J_2^{-1}(a_2 + (\alpha_2+(\omega_2)^2)J_2T_1 + 2\omega_2J_2v_1 ).$

Recall that the role of $J_2$ is only to change between bases; it does not affect the meaning of the vector it acts on. Therefore let’s introduce a bit of notation:

• Applied force expressed in the Earth basis: $\tilde F = J_2^{-1} F$.
• Acceleration of the Earth relative to the world expressed in the Earth basis: $\tilde a_2 = J_2^{-1} a_2$.
• Angular velocity of the Earth relative to the world expressed in the Earth basis: $\tilde\omega_2 = J_2^{-1} \omega_2 J_2$.
• Angular acceleration of the Earth relative to the world expressed in the Earth basis: $\tilde\alpha_2 = J_2^{-1} \alpha_2 J_2$.

With this notation, the local acceleration becomes much cleaner:

$a_1 = \frac{1}{m}\tilde F - \tilde a_2 - \tilde\alpha_2 T_1 - (\tilde\omega_2)^2 T_1 - 2\tilde\omega_2v_1.$

The local acceleration is therefore given not only by the applied force, but by four additional contributions due to the global dynamics of the parent. The additional four terms are conventionally interpreted as extra forces (by $F=ma$), and are known to physicists as inertial forces. They are regarded as fictitious’ forces, as they do not result from physical interactions.

• The linear inertial acceleration: $-\tilde a_2$
• The Centrifugal acceleration: $-(\tilde\omega_2)^2 T_1$
• The Coriolis acceleration: $-2\tilde\omega_2v_1$
• The Euler acceleration: $-\tilde\alpha_2T_1$

This illustrates that in the absence of applied forces an object may still experience local accelerations due to the dynamics of its parent.

Since we have kept this in the $n$-dimensional matrix form, it is now easy to go into the 2- and 3-dimensional specializations.

In 3D

In 3D, the skew-symmetric matrices are identified with 3-vectors, and their action on a vector becomes the cross product:

• Centrifugal: $-\tilde\omega_2\times(\tilde\omega_2\times T_1)$
• Coriolis: $-2\tilde\omega_2\times v_1$
• Euler: $-\tilde\alpha_2\times T_1$

Since physicists usually operate in 3-dimensions using the vector version of angular velocity, this form may be more familiar.

In 2D

In 2D, the skew-symmetric matrices are identified with scalars, and their action on a vector becomes scalar multiplication with the perpendicular vector, $(x,y)^\perp = (-y,x)$:

• Centrifugal: $(\omega_2)^2 T_1$
• Coriolis: $-2\omega_2 v_1^\perp$
• Euler: $-\alpha_2 T_1^\perp$

Example: Newtonian Mechanics — Linear Impulses

An impulse is the integral of an applied force over a given time interval. In an inertial frame this is equal to the overall change in momentum produced by that force acting over the time interval. However, when a large force acts over a very small time interval it is often not feasible to simulate the action of the force itself. Instead, the corresponding impulse can be applied as an instantaneous modification to the object’s dynamics. A relevant example is collision response.

An impulse corresponds to a change in velocity, $\Delta v = \frac 1m\Delta p$, in the same way that a force corresponds to an acceleration. In order to convert the global change in velocity into a local change in velocity, we use the global-to-local conversion of delta.

Let the global delta be $\Delta v_3 = \Delta p/m$ then, applying the formulae above, the local deltas are

$\Delta v_1 = \frac 1m J_2^{-1} \Delta p$
$\Delta a_1 = -\frac 2m J_2^{-1} \omega_2\Delta p.$

This shows that applying an impulse can causes changes in both the local velocity and the local acceleration of an object. The change in acceleration is due to the Coriolis term, which is dependent on velocity.

Counteracting the Inherited Effect from the Parent

When the parent’s global dynamics have been modified, all of its children (and childrens’ children etc.) will automatically inherit this modification to their own global dynamics thanks to the hierarchy. There may be some applications where this is undesirable and you want to modify an object in isolation. An example is a car crash — the dynamic child objects inside the car should continue globally moving forward when the car crashes, otherwise they will appear to be fixed to the car.

To do this we need to add the opposite global effect to the immediate children of the object that has been modified to counter the inherited changes to their dynamics. The childrens’ children will inherit this correction automatically and do not need to be corrected further. If $\Delta v$ is the global change in velocity that has been added to the parent, then for each child the local velocity and acceleration need to be updated by

$\Delta v_1 = - J_2^{-1} \Delta v$
$\Delta a_1 = 2 J_2^{-1} \omega_2\Delta v.$

It is especially important to do this when a child collides with its parent (or the parent’s parent etc.) to get the correct collision response.

Implementation

To implement this approach, you will need not only a Transform class, but also a DynamicTransform class. Both of these will have the group structure of composition, identity and inverse derived in this article. A hierarchy can then be made in the usual way, just using DynamicTransforms in place of Transforms.

Here is a sketch to illustrate.

template<typename T,int N>
struct Transform {
Vec<T,N> translation;
Rotation<T,N> rotation;
T scale;

// Function evaluation
Vec<T,N> operator()( const Vec<T,N>& rhs ) const;

// Composition
Transform<T,N> operator*( const Transform<T,N>& rhs ) const;
Transform<T,N>& operator*=( const Transform<T,N>& rhs );

Transform<T,N> Inverse() const;

static const Transform<T,N>& Identity();

// Linear part
Transform<T,N> LinearTransform() const;

// Inverse transpose of linear part
Transform<T,N> NormalTransform() const;

// To augmented matrix representation
Mat<T,N+1,N+1> ToMatrix() const;
};

template<typename T,int N>
struct DynamicTransform {
Transform<T,N> transform;
Vec<T,N> velocity, accel;
Skew<T,N> angularVelocity, angularAccel;

// Composition
DynamicTransform<T,N> operator*( const DynamicTransform<T,N>& rhs ) const;
DynamicTransform<T,N>& operator*=( const DynamicTransform<T,N>& rhs );

DynamicTransform<T,N> Inverse() const;

static const DynamicTransform<T,N>& Identity();

// Transform deltas using this dynamic transform
Vec<T,N> TransformDeltaVel( const Vec<T,N>& dv ) const;
Vec<T,N> TransformDeltaAcc( const Vec<T,N>& dv, const Vec<T,N>& da ) const;
Skew<T,N> TransformDeltaAngVel( const Skew<T,N>& domega ) const;
Skew<T,N> TransformDeltaAngAcc( const Skew<T,N>& domega, const Skew<T,N>& dalpha ) const;
};


A Quick Note On Rigid Body Dynamics

For application to rigid bodies, the positions’ of objects here correspond to the centre of mass of the rigid body. This is because rigid body motion is decomposed into rotations about the centre of mass and translations of the centre of mass.

Conclusion

We have discussed the algebraic structure of $n$-dimensional transforms consisting of translation, rotation and scale components, and described their dynamics in terms of linear and angular velocities and accelerations. By extending the algebraic structure to include these velocities and accelerations, we can convert between local and global descriptions of an object’s dynamics. The resulting equations apply to any number of dimensions, to any source of dynamics, and to a hierarchy of arbitrary depth. This in turn allowed us to convert instantaneous changes in velocity and acceleration between local and global frames of reference. We applied these results to the example of Newtonian mechanics, where the global-to-local conversion of acceleration produces the well-known inertial forces, and to the example of linear impulses.

Numerical Integration for Rotational Dynamics

In order to advance a simulation forward in time, a numerical integration scheme is needed to evaluate the continuous part of the dynamics, which is specified by a differential equation. For translational dynamics, this is usually of the form

$\begin{array}{l}\dot x = v\\\dot v=a(x,v),\end{array}$

where $x$ is the position, and $v$ is the velocity. The acceleration, defined as $\dot v$, is a function of the position and velocity. Under the assumption that acceleration is constant, the differential equations have a closed-form solution, giving the update formula

$\begin{array}{l}\Delta v = a \Delta t\\ \Delta x = \frac 12 (v + v') \Delta t,\end{array}$

where $x' = x + \Delta x$ and $v' = v + \Delta v$.

In this article, we will derive an analogous result for rotational dynamics — the numerical integration scheme for the differential equations describing rotational motion under the assumption of constant angular acceleration.

The assumption of constant acceleration is not always appropriate for translational dynamics, particularly for systems like springs that have rapidly varying acceleration. For rotational dynamics, a proper discussion of the appropriateness of constant angular acceleration is best left to a dedicated article, as it would involve the details of rigid body dynamics, which is not the concern of this article.

Translational and Rotational Analogies in 3D

Whereas translational motion describes the change in an object’s position over time, rotational motion describes the change in an object’s orientation. Positions are described relative to an origin by a displacement vector, with the null vector describing the origin. In an analogous fashion, orientations are described relative to a standard orientation by a rotation, with the identity rotation describing the standard.

Rotations in 3D can be quantified by:

• an axis and an angle;
• by virtue of being linear maps: a rotation matrix — a real 3-by-3 matrix with determinant 1 and inverse coinciding with its transpose;
• a unit quaternion.

We will assume that the reader is already familiar with rotations and their various descriptions.

In 3-dimensional space, the space of rotations also possesses a 3-dimensional structure. This is not the case in general — for example, in 2D the space of rotations is 1-dimensional. Because of this, there are many analogies between translational and rotational dynamics in 3D.

Translational Rotational
position, $x\in\mathbb R^3$ rotation matrix, $R\in\mathrm{SO}(3,\mathbb R)$, or unit quaternion, $q\in S^3\subset\mathbb H$
velocity, $v=\dot x\in\mathbb R^3$ angular velocity, $\omega\in\mathbb R^3$
acceleration, $a=\dot v\in\mathbb R^3$ angular acceleration, $\alpha=\dot\omega\in\mathbb R^3$
mass, $m\in\mathbb R$ moment of inertia tensor, $I\in\mathbb R^3\otimes\mathbb R^3\cong\mathbb R^{3\times 3}$
momentum, $p=mv\in\mathbb R^3$ angular momentum, $L=I\omega\in\mathbb R^3$
force, $F=\dot p\in\mathbb R^3$ torque, $\tau=\dot L\in\mathbb R^3$

The aim of this article is not to discuss how accelerations are produced by forces, nor how angular accelerations are produced by torques, and therefore we will be focusing on the top half of this table.

The main reason that rotational motion is more complicated than translational motion is that the space of rotations is not a linear space — it is curved’. In a linear space, a constant velocity moves you along a straight line; in the space of rotations, a constant angular velocity moves you along a curve.

The Rotational Equations

The key piece of information missing from the table above is the relation between the angular velocity, $\omega$, and the time derivative of the rotation, $\dot R$ (or $\dot q$).

For rotation matrices this is
$\dot R=\omega_{\times} R,$

where $\omega_{\times}$ is an anti-symmetric matrix describing the linear map $x\mapsto\omega\times x$, called the cross-product matrix,
$\omega_{\times} = \left[\begin{array}{ccc}0 & -\omega_3 & \omega_2\\\omega_3 & 0 & -\omega_1\\-\omega_2 & \omega_1 & 0\end{array}\right].$

For the mathematically literate, $\omega_{\times}$ is an element of the Lie algebra $\mathfrak{so}(3,\mathbb R)$, the tangent space to $\mathrm{SO}(3,\mathbb R)$ at the identity. The conversion $\omega\mapsto\omega_{\times}$ is an isomorphism between our physical space $\mathbb R^3$ and $\mathfrak{so}(3,\mathbb R)$, which is possible as they are of the same dimension. Under this identification, the cross product coincides with the matrix commutator, which is the Lie bracket for the algebra. The time derivative $\dot R$ is a tangent vector at $R$, which is given by the pushforward of $\omega_{\times}$ by right translation.

The quaternionic version of this is
$\dot q=\frac 12\omega q,$

where $\omega$ is interpreted as an imaginary quaternion and the multiplication is quaternionic. The identification of vectors with imaginary quaternions is analogous to the identification of vectors with anti-symmetric matrices, as the imaginary quaternions form the tangent space to the unit quaternions at the identity, 1. The factor of $\frac 12$ comes from the property that each rotation is described by two unique unit quaternions, whereas there is only one rotation matrix.

Integrating the Rotational Equations

The easy part of the integration is the angular velocity, which, assuming constant angular acceleration gives

$\omega'=\omega + \alpha\Delta t.$

However the other equation is not so straightforward.

In the special case that $\alpha=0$, the angular velocity is constant and the solution is given by the exponential map

$R'=\exp(\omega_{\times}\Delta t) R$
$q'=\exp(\frac 12 \omega\Delta t) q.$

But when $\alpha\neq 0$ and $\omega$ varies with time, this becomes much more complicated. The solution can be written as an infinite series via the Magnus expansion,

$R'=\exp( \Omega_{\times} ) R$
$\Omega=\Omega_1 + \Omega_2 + \cdots.$

For constant angular acceleration, we can compute the first few of these terms by hand,

$\Omega_1 = \frac 12 (\omega+\omega')\ \Delta t$
$\Omega_2 = \frac{1}{12}\ \alpha\times\omega\ \Delta t^3 = \frac{1}{12}\ \omega'\times\omega\ \Delta t^2$
$\Omega_3 = \frac{1}{240}\ \alpha\times(\alpha\times\omega)\ \Delta t^5.$

This series is guaranteed to converge as long as the angle rotated per time-step is less than $\pi/\sqrt{2}$ (which is about 127 degrees). Typically the first one or two terms are plenty. Note that we have given $\Omega$ in vector form, but for use in the matrix exponential the sum should be converted to the cross-product matrix form. We can also reverse-engineer the quaternion version from this,

$q'=\exp( \frac 12 \Omega ) q,$

where $\Omega$ is now an imaginary quaternion.

A Geometric Interpretation

We can think of the Magnus expansion $\Omega$ as a polynomial in $\Delta t$ with values in $\mathfrak{so}(3,\mathbb R)$, the tangent space to $\mathrm{SO}(3,\mathbb R)$ at the identity. In other words it is a path in the tangent space that begins at the null vector $\Omega(0)=0$ and ends at some vector $\Omega(\Delta t)$.

The exponential map maps this tangent space to the manifold, $\mathfrak{so}(3,\mathbb R)\to\mathrm{SO}(3,\mathbb R)$, in a way that identifies the null vector with the identity. Therefore it wraps’ the path around the manifold so the starting point is the identity, $\exp(\Omega(0))=\exp(0)=\mathrm{id}$, and the end point is $\exp(\Omega(\Delta t))$.

The act of multiplying this path with the current rotation on the right $\exp(\Omega)\mapsto \exp(\Omega)R$ is called right translation. It `moves’ the path across the manifold so that the starting point is now at $R$, and the end point is the new rotation $R'$. Therefore the integration moves us along paths that are constrained to rotation group.

Evaluating the Exponential Map

For both matrices and quaternions, the exponential map is defined by the series

$\exp( X ) = \sum_{n=0}^\infty \frac{X^n}{n!}.$

However, for the case that $X$ is an anti-symmetric matrix, the exponential map corresponds to the conversion from axis-angle to rotation matrix (the length of the associated vector is the angle, the direction is the axis). The analogous statement applies for the quaternionic version: the exponential map acting on imaginary quaternions corresponds to the axis-angle to unit quaternion conversion,

$\exp( X ) = \cos\left(\left\lVert X\right\rVert\right) + \frac{X}{\left\lVert X\right\rVert}\sin\left(\left\lVert X\right\rVert\right).$

However, performing such a conversion involves evaluating trig functions, which are expensive. Instead, we can truncate the series expansion of the exponential map to get an approximation. This is appropriate as we will be evaluating it with $\Delta t$, so the higher powers will be negligible.

For an imaginary quaternion, $X^2 = -\left\lVert X\right\rVert^2$. This means the quaternion exponential map to third-order is
$\exp( X ) \approx (1 - \frac 12 \left\lVert X\right\rVert^2) + (1 - \frac 16 \left\lVert X\right\rVert^2) X.$

The full exponential map produces unit quaternions from imaginary quaternions, so we should also build a normalization into this approximation,
$\exp( X ) \approx \frac{(1 - \frac 12 \left\lVert X\right\rVert^2) + (1 - \frac 16 \left\lVert X\right\rVert^2) X }{ \sqrt{(1 - \frac 12 \left\lVert X\right\rVert^2)^2 + (1 - \frac 16 \left\lVert X\right\rVert^2)^2 \left\lVert X\right\rVert^2}}.$

This approximation is good for small rotations per time-step. The error is less than 1% for rotations up to 90 degrees per time-step. For faster rotations, the full exponential map should be applied.

Unfortunately, rotation matrices are not so easily (ortho)normalizable, and so you’re best off using the full axis-angle to rotation matrix conversion to implement the exponential map for matrices. For this reason the quaternion description is preferable to rotation matrices.

Conclusion

We have derived a method for time-stepping rotational dynamics that expresses each update as a rotation. The update rotation is determined in vector form (axis-angle) by truncating the Magnus expansion and then converted to the underlying algebraic representation (rotation matrix/unit quaternion) by applying the exponential map.

For the quaternion version, we also approximated the exponential map in a way that avoids evaluating trig functions while preserving normalization, so the resulting quaternion is always a valid rotation.