An angular velocity of a rigid body is governed by

where $\bm{\Omega}$, $\bm{J}_{\rm b}$, and $\bm{\tau}_{\rm a}$ denote an angular velocity, body inertia, and action torque, and $S(\cdot)$ is a skew-symmetric matrix such that $S(x)y=x\times y\ (x,\ y \in \mathbb{R}^3)$ holds. Let $\bm{\tau}$ be the sum of a control input $\bm{\tau}_{\rm c}$ and disturbance $\bm{\tau}_{\rm d}$, i.e.,

A disturbance observer estimates $\bm{\tau}_{\rm d}$ and enable us suppress that effect. To construct the observer, let us introduce extended-state-space representation

where

and $\bm{v}$ denotes a process noise that includes perturbation and the derivative of $\bm{\tau}_{\rm d}$. The observable matrix of this extended system is full rank and the observer is able to be constructed:

Since $\bm{J}_{\rm b}$ is known parameter and $\bm{\Omega}$ is measurable, $S(\bm{J}_{\rm b}\bm{\Omega})\bm{\Omega}$ can be regarded as an input to the extended system. Following to the Gopinath's method, the minimal-order state observer is obtained as

where $\bm{Q}$ and $\bm{\omega}_{\rm g}\equiv{\rm diag}\{\omega_{\rm g,i}\}\ (i=1,2,3)$ represent the observer gain and relating parameters that specify the cutoff frequencies of disturbance observation. An observation error converges to 0 when the derivative of $\bm{\tau}_{\rm d}$ is 0, namely, the observer correctly estimates step disturbance and estimates other disturbance with the variation of a gain and phase.

The acceleration control is established using $\hat{\bm{\tau}}_{\rm d}$. Let $\bm{\tau}_{\rm c}$ be composed of a reference input $\bm{\tau}^{\rm ref}$ and $\hat{\bm{\tau}}_{\rm d}$ as

With this rule, the governing equations are expressed as

and we obtain

where

Herein, let us design a reference torque as

where $\bm{\alpha}^{\rm ref}$ is an angular acceleration reference. The dynamics is expressed as

where

Within the bandwidth of $\omega_{\rm gm}=\min(\omega_{\rm g,i})$, $\bm{H}\approx\bm{I}$ and $\bm{T}\approx\bm{0}$ hold and hence the acceleration control is achieved.

An acceleration controller using a disturbance observer enables to control angular acceleration. Therefore, a controller design is boil down to calculate a proper acceleration reference. To control an attitude, a feedforward controller and feedback controller that work as

is adopted, where $\bm{\Phi}= \begin{bmatrix} \phi & \theta & \psi \end{bmatrix}^{\mathrm T}$ and $K$ denote a state that includes the roll, pitch, and yaw angles of a rigid body and command smoother. The smoother $K$ should have a relative order of 2 to make the controller be proper. Assuming that $\bm{\tau}_{\rm e}$ is quite small enough to be ignored, a relation between $\bm{\Phi}^{\rm cmd}$ and $\bm{\Phi}$ is expressed as

Since a tracking error cannot be removed due to uncertainty and disturbance, $\bm{K}_{\rm p}$ and $\bm{K}_{\rm d}$ should get the form of a diagonal matrix to decouple the dynamics on each axis. Let us use the generalized form of a gain set that determines the natural frequency and attenuation rate of the closed-loop system:

where $\omega_{\rm n,i}$ and $\zeta_{\rm i}$ (i=1,2,3) denote the nominal natural frequency and attenuation rate of the controllers on each axis. In an ideal condition, model fluctuation is small and there is no delay in both thrust control and state observation, the open-loop transfer functions of each axis $L_{\rm i}$ (i=1,2,3) are expressed as

Here, the gain crossover frequencies and the phase margins of each axis $\omega_{\rm g, i},\ p_{\rm i}$ (i=1,2,3) are expected to be

A phase margin gets almost 76.35 deg for $\zeta=1.0$ and 65.53 deg for $\zeta=0.707$. The total phase lag of thrust control and state observation around a frequency band of $\omega_{\rm g}$ is permissible in the range shown above.