---
# System prepended metadata

title: Magnetic Control Two-step Rotations
tags: [Internship, NSPO]

---

# Magnetic Control Two-step Rotations
###### tags: `Internship`,`NSPO`
[TOC]

## 工作環境
- 軌道：低軌道(Low Earth Orbit)
- 適用衛星：nano satellites & micro satellites
- 特色：Low cost, High reliability, Ease of development
## 常見力矩
1. **The B-dot attitude stabilization method**
2. The cross product method
3. Reaction wheels (RW)
(in a near magnetic equatorial orbit)
4. MTQs
(change four parameters for attitude control – enough to control three axes.)
## 研究步驟
- **First** Shows the feasibility of using two rotations for achieving three-axis attitude.
- **Second** The paper presents a method for choosing the two rotations.
- **Third** Presents a method for control attitude using a PD controller.
- **Forth** Simulation results.

### First
![](https://i.imgur.com/THW0yLe.png)
Fig. 1註解：*The z-axis is deﬁned as the geomagnetic ﬁeld vector. The x- and y-axes have arbitrary directions. And, the quaternions qs and qt represent the initial quaternion and target quaternion, respectively.*

:::success
As shown in Fig.1.This subsection shows that the satellite can achieve any attitude using two-step rotations.
:::
### Second
First rotation
![](https://i.imgur.com/cgGMoIX.png)
Second rotation
![](https://i.imgur.com/mbBmh2N.png)

- The initial quaternion to the target quaternion
![](https://i.imgur.com/492TECx.png)
Be written as follow
![](https://i.imgur.com/uD5rng3.png)
And, the represent
![](https://i.imgur.com/yad8699.png)

Show Examples of objective functions in the two-step rotations
![](https://i.imgur.com/8OOndXr.png)

And, the firsr rotation is constant.

### Third:Magnetic attitude control using a PD controller(On-board calculation)
The required torque vector T in the body coordinate system in a rotation can be calculated as follows:
![](https://i.imgur.com/YOS6IwW.png)
*the normalized torque vector t should be transformed using DCM (Direction Cosine Matrix) ![](https://i.imgur.com/0AxYuGe.png) to obtain the torque vector in the body coordinate system. BY IMU*



The rotation angle θ1 for the ﬁrst rotation can be written as follows with the forth component of the quaternion:
![](https://i.imgur.com/4SKpeYQ.png)
The vector t 1 representing the ﬁrst rotation center is also calculated as follows:
![](https://i.imgur.com/prKgZLc.png)
The angular velocity around the torque vector can be obtained directly from the gyro measurements, by taking the inner product of the angular velocity vector of the satellite and the normalized torque vector as follows:
![](https://i.imgur.com/YuvFTyr.png)
In this proposed method, the geomagnetic ﬁeld vector, B, is perpendicular to the torque vector, T. Therefore, the magnetic moment can be written as follows:
![](https://i.imgur.com/6hXcouf.png) -> ![](https://i.imgur.com/LksepMq.png)
### Forth-Simulation
- Parameters
![](https://i.imgur.com/vfjHJ86.png)

Simulation results representing a 90° rotation about the direction:
![](https://i.imgur.com/f4BYdgE.png)

### 問題
1. 似乎因為翻兩次的關係，會過久才收斂
![](https://i.imgur.com/ZQvdHAu.png)
2. 還未確定pattern的取得方式


# Magnetic coils
:::info
Magnetic three axis stabilization of a spacecraft can be achieved by using feedback from magnetic field measurements and angular velocity.
:::

## A. Energy considerations
the potential energy due to the revolution of the satellite about the Earth is given by
![](https://i.imgur.com/kzkfmZi.png)

## B.Detumbling
:::info
When the satellite is released from the launcher it will have an initial angular velocity. Before the boom can be deployed, angular velocity must be reduced and the body frame must be aligned with the orbit frame.
:::

The only sensor available in this mode will be the magnetometer. After the rate detumbling phase the satellite may have an arbitrary attitude. Before the boom can be deployed we must ensure that the body zb -axis is aligned with the orbit zo -axis.

Proposition 1: The control law
![](https://i.imgur.com/TUnfO9K.png)

The control torque
![](https://i.imgur.com/bBy2hMr.png)
![](https://i.imgur.com/RtTAZVh.png)
be written as:
![](https://i.imgur.com/xo4wxcT.png)

## B. Stabilization
Proposition 2: The control law
![](https://i.imgur.com/XO0Zyd2.png)
bring back to torque equ.
![](https://i.imgur.com/yCkOYwk.png)

## C. Simulation
Detumble mode simulation
![](https://i.imgur.com/yaC7zjD.png)
The stabilized attitude of nCube
![](https://i.imgur.com/b73RUvo.png)



































# Principle of Magnetic Torquers
## Magnetorquing
Two torquers aligned with X- and Y-axes will be used primarily for attitude control in the IAA. The vector dipole moment M of magnetic torquer rod will interact with the geomagnetic ﬁeld vector B to generate a magnetic torque N

![](https://i.imgur.com/sN9GC06.png)

where M is determined by on-board controller and geomagnetic ﬁeld vector B is measured through the on-board magnetometer.

**IAA controller design**

The objective of IAA controller is to reduce the angular rate of spacecraft and to align its +Z -axis with the Earth’s magnetic ﬁeld. To accomplish this, detumbling the following control torque is proposed:

![](https://i.imgur.com/7e1vjV8.png)


**Angular rate reduction**
Where is the spacecraft body rate.
![](https://i.imgur.com/qadbF2v.png)
**Align with geomagnetic ﬁeld**
The control dipole M align for aligning the +Z spacecraft body axis with the Earth’s magnetic ﬁeld.
![](https://i.imgur.com/JUTj0E4.png)
## Simulation
![](https://i.imgur.com/9zggqlu.png)
![](https://i.imgur.com/OMcG7TZ.png)








