In some geometric control papers, the author usually defines the real-valued error function to be:
$\Psi(R,R_d)$ = $\frac{1}{2} Trace[I - R_d^T R ]$. (1)
where $R_d$ is the arbitrary smooth attitude tracking command.
My question is what is the physical meaning of Eq.(1).
There is some physical intuition behind the formula. Set $M = R_d^\top R$. By Rodrigues's Rotation Formula, there exists a unit vector $a\in\mathbb{R}^3$ and angle $\theta \in (-\pi, \pi]$ so that,
$$M = I_{3\times 3} + (\sin\theta)\,S(a) + (1 - \cos\theta)\,S(a)^2$$
where $S(a)\in\mathrm{Skew}(3)$ is the skew symmetric matrix constructed from the vector $a.$ This is just the axis-angle representation of the error rotation $M.$ The angle $\theta$ essentially is an angular error. Verify that,
$$\mathrm{Trace}(I_{3\times 3} - M) = -(\sin\theta)\mathrm{Trace}(S(a)) - (1 - \cos\theta) \mathrm{Trace}(S(a)^2).$$
The trace of a skew symmetric matrix is zero so,
$$\mathrm{Trace}(I_{3\times 3} - M) = -(1 - \cos\theta) \mathrm{Trace}(S(a)^2).$$
I'll implore you to do the computation to verify that, using $\|a\|=1,$ $\mathrm{Trace}(S(a)^2)=-2.$ Deduce from this that,
$$\mathrm{Trace}(I_{3\times 3} - M) = 2\left(1 - \cos(\theta)\right).$$
Therefore the expression
$$\frac{1}{2}\,\mathrm{Trace}(I_{3\times 3} - R_d^\top R) = 1 - \cos(\theta) = 2\,\sin^2\left(\frac{\theta}{2}\right). $$
is really just a measure of positive angular deviation from the current orientation to the desired orientation.