Scalar-first quaternions and attitude integration

Learning outcomes

Use MuJoCo's wxyz quaternion order.
Integrate body angular velocity on the rotation manifold.
Compute a sign-invariant orientation error.

Prerequisites: Linear algebra, ODEs, and earlier lectures in this sequence.

Quaternion convention

We use a unit quaternion $Q=(w,x,y,z)=(w,r)$ , scalar first, representing a body-to-world rotation. Identity is $(1,0,0,0)$ . For axis $n$ and angle $\theta$ ,

Q=(\cos(\theta/2),\ n\sin(\theta/2)).

The Hamilton product is

Q_1\otimes Q_2=(w_1w_2-r_1^\top r_2,\ w_1r_2+w_2r_1+r_1\times r_2).

Both

Q

and

-Q

represent the same rotation. Four unit-norm components describe three degrees of freedom. Uppercase Q distinguishes one attitude from the full configuration q.

Integrating angular velocity

For body angular velocity $\omega^B$ ,

\dot Q=\tfrac12 Q\otimes(0,\omega^B).

A constant body angular velocity over a step gives

Q_{k+1}=Q_k\otimes\operatorname{Exp}_Q(\Delta t\omega^B),

where

\operatorname{Exp}_Q(\phi)=(\cos(\|\phi\|/2),\phi\sin(\|\phi\|/2)/\|\phi\|)

, using the continuous limit at zero. World angular velocity multiplies on the other side.

Normalizing a forward-Euler update repairs quaternion norm but does not make its angular increment exact. Manifold preservation and integration accuracy are different properties.

Worked quarter-turn

Apply $\omega^B=(0,0,\pi/2)$ rad/s for one second from identity. The result is $(\sqrt2/2,0,0,\sqrt2/2)$ , mapping body x to world y. This catches component-order and half-angle mistakes.

For normalized quaternions, orientation distance is $2\arccos(\min(1,|\langle Q_1,Q_2\rangle|))$ . The absolute dot handles sign equivalence. A logarithm-based calculation can be more accurate near zero.

MuJoCo configuration operations

A free joint stores position followed by wxyz, while velocity has linear and angular components. Use mj_integratePos to update the full configuration, and mj_differentiatePos for tangent differences. Do not infer velocity dimension from qpos length.

Practice 2 cross-checks an independently implemented exponential update against MuJoCo and tests sign equivalence. A non-identity initial orientation and nonparallel angular velocity distinguish left from right multiplication.

Exercise

Identity and negative identity have raw quaternion difference norm 2. What is the physical attitude error? Answer: zero. A raw four-component quadratic penalty can punish a representation change without physical motion. Local orientation control and optimization should use a three-dimensional tangent error.

When importing a scalar-last quaternion, reorder explicitly once at the boundary and test the quarter-turn. Avoid scattered reordering throughout the dynamics code.