Brief sensor analysis for Borea project

Author: Title: Location: Date: Email:

Jiawei Wu Doc1-Brief Sensor Analysis For Borea Project.Doc Politecnico di Torino 04/12/14 [email protected]

Document History Version Number 0.1 0.2 0.3

Date Updated Oct-27-2014 Oct-28-2014 Oct-29-2014

Author

Brief Description of Changes

Jiawei Wu Jiawei Wu Jiawei Wu

0.4

Oct-30-2014

Jiawei Wu

0.5

Nov-3-2014

Jiawei Wu

Ultrasonic sensor Barometric pressure sensor GPS, gyroscope Accelerometer Complete Modify equation(1.15) Modify the rate of decrease in air temperature L , it is corrected into negative Emphasis that the output of barometric pressure sensor is pressure which is relative to altitude

Author: Title: Location: Date: Email:

Jiawei Wu Doc1-Brief Sensor Analysis For Borea Project.Doc Politecnico di Torino 04/12/14 [email protected]

Table of Contents 1. 

Sensor analysis ................................................................................................................... 2  1.1  Accelerometer .............................................................................................................. 2  1.2  Gyroscope .................................................................................................................... 2  1.2.1  Tayt-Brain 3-2-1 rotation sequence...................................................................... 3  1.3  Global positioning system ........................................................................................... 4  1.4  Ultrasonic sensor ......................................................................................................... 5  1.5  Barometric pressure sensor .......................................................................................... 7  1.6  Magnetometer .............................................................................................................. 8  2.  Reference ............................................................................................................................ 9 

Author: Title: Location: Date: Email:

Jiawei Wu Doc1-Brief Sensor Analysis For Borea Project.Doc Politecnico di Torino 04/12/14 [email protected]

1. Sensor analysis

Inertial measurement units (IMU) are widely used in motion measurement. However, the drift of IMUs results in significant accumulated errors for long-term attitude measurement. Hence, we apply sensor fusion techniques to compensate and correction the measurement. The following content introduces the functions of each sensor, and the output, meanwhile, indicating the usage of those data which is prepared for the sensor fusion algorithm employment. Throughout the following content, we choose different sensors: 1) Accelerometer 2) Gyroscope 3) GPS 4) Sonar 5) Barometer 6) Magnetometer

1.1 Accelerometer Accelerometer relies on earth’s gravitational acceleration to determine attitude are sensitive to disturbances such as linear accelerations and motor vibrations, but the attitude determined does not drift with time, making the accelerometer suitable for long term attitude estimation. The sensor works in body frame, in order to get the value in inertial frame, it requires a transformation. The output of the accelerometer is given by a m  ASFCC abm  abias  acc

(1.1)

Where ASFCC is the errors relative to the scale factor and axes misalignment, and abm is the acceleration in meters per second, abias is a bias term, it strongly dependent on temperature and should be calibrated prior, acc is zero mean white noise. It is important to note that for the quadrotor, the output of accelerometers is independent of angles. This is in contrast to the non-accelerated flight for fixed wing vehicles where the accelerometers are used to measure the gravity vector and thereby extract roll and pitch angles.

1.2 Gyroscope Gyroscope is widely used to measure the angular velocity b with unit [deg./s], because this sensor is mounted on the rigid body, thereby the output value is in body frame which has the origin locating in the centre of mass of quadrotor, it enables to get angular displacement by single integration. 2 Last modified at: 04/12/2014 11:01:51 By author: Jiawei Wu This document can only be used by Space and Precision Automatic research group. It is not allowed to copy, modify, print, post and distribute to other people without the written consent of the author. If you have any doubts, please feel free to contact Jiawei Wu ([email protected]) at any time.

Author: Title: Location: Date: Email:

Jiawei Wu Doc1-Brief Sensor Analysis For Borea Project.Doc Politecnico di Torino 04/12/14 [email protected]

As[1] describes that, angular rate output from gyroscopes are integrated to obtain attitude, but because of the discrete time integration process, and integration of random noise at every time step, the integrated angle drifts with time. The gyroscope is however, largely immune to linear disturbances, thus making it suitable for short term attitude estimation. The output of gyroscope is

mis  b  SFCC  bias   gyro

(1.2)

Where b is the angular velocity, SFCC is the error relative to scale factor and axes misalignment,  gyro is the zero mean white noise, bias is the bias term. The following section describes how to use b get angular displacement by applying rotation with sequence 3-2-1.

1.2.1 Tayt-Brain 3-2-1 rotation sequence Body-to-fixed transformation and rotation axes are defined as following          Ri1  C , i11 , i21 , i3  , Ri 2  C , b1 , i21 , i32 , Rb  C , b1 , b2 , b3







R     Z   Y   X         k1 , k2 , k3  i3 , i21 , b1



i b

 

 (1.3)



The relation between three frames is indicated in the following figure Fixed pole Body pole

Line of nodes

Figure 1. Relation between three frames in spatial

The body representation K becomes   K   b1 X    i21

 X    Y    i3  

(1.4)

Considering the existence of variables b and b , we get

3 Last modified at: 04/12/2014 11:01:51 By author: Jiawei Wu This document can only be used by Space and Precision Automatic research group. It is not allowed to copy, modify, print, post and distribute to other people without the written consent of the author. If you have any doubts, please feel free to contact Jiawei Wu ([email protected]) at any time.

Author: Title: Location: Date: Email:

Jiawei Wu Doc1-Brief Sensor Analysis For Borea Project.Doc Politecnico di Torino 04/12/14 [email protected]

bx  b  0 0     0   X     X  Y   0   b b  b  b   by    bz   0   b   0 

(1.5)

Expanding the equation above, we get

0 bx  b  1     0    0 c b  by      bz   0   0  sb

0   0  1 0    sb  b   0 cb   cb   0  0  sb

0  cb  sb   0  cb   sb

0  sb   0   1 0   0  0 cb   b 

(1.6)

Computing the final equation, we get 0 bx  1     0 c b  by   bz   0  s b 

 sb  b   cb sb   b   cb cb   b  sb tb cb tb  bx  b  1        0  c s   by  b   b b      b   0 sb secb cb secb  bz 

(1.7)

According the above equation, we can get angular displacement from the data got by gyroscope, but meanwhile, we need to know two angles first, in other document, we will introduce the technique of quaternion transformation.

1.3 Global positioning system GPS receivers are widely used in navigation system design, currently, the GPS system has 31 active satellites in orbits inclined 55 degrees to the equator. The satellites orbit about 20,000 km from the earth’s surface and make two orbits per day. The orbits are designed so that there are always 6 satellites in view, from most places on the earth. The GPS receiver gets a signal from each GPS satellite. The satellites transmit the exact time the signals are sent. By subtracting the time the signal was transmitted from the time it was received, the GPS enables to show the distance from each satellite. So given the travel time of the GPS signals from three satellites and their exact position in the sky, the GPS receiver can determine the position in three dimensions, latitude, longitude and attitude. If the GPS receiver is only able to get signals from 3 satellites, it will be less accurate, so in order to precisely acquire the position in 3-dimensions, GPS receiver needs 4 satellites at least. Hence, we add a GPS receiver on our platform, it provides three dimensional velocity VGPS and position PGPS , and meanwhile, it has a long-time error stability and high accuracy level. This sensor works in inertial frame which the origin locates in the centre of Earth, the output value position is in meter unit, it is also known as the attitude of quadrotor, and also outputs the velocity is in the meter per second unit.

4 Last modified at: 04/12/2014 11:01:51 By author: Jiawei Wu This document can only be used by Space and Precision Automatic research group. It is not allowed to copy, modify, print, post and distribute to other people without the written consent of the author. If you have any doubts, please feel free to contact Jiawei Wu ([email protected]) at any time.

Author: Title: Location: Date: Email:

Jiawei Wu Doc1-Brief Sensor Analysis For Borea Project.Doc Politecnico di Torino 04/12/14 [email protected]

1.4 Ultrasonic sensor Ultrasonic sensors are widely used to identify the quadrotor attitude and its height from the ground through time of flight, however, the performance issues related to reflections, occlusions, and maximum emitting angles limit independent use of ultrasonic sensors. To avoid drifts and integration errors, we propose to use complementary ultrasonic sensors to provide redundant position information to calibrate the accelerometers. As[2] describes, the ultrasonic sensor measures the time delay between signal sending and reflecting, distance between the emitter and the touch point. Although ultrasonic sensors have no accumulated errors and are quite accurate in position determination, however, the sampling rate of ultrasonic sensors is rather low due to the delay between sending and receiving. In addition, the movement of the emitter may render obstacles appearing on the path while reflecting back, resulting in blind areas and thus occasional or frequent signal loss. Therefore it is  mounted on the bottom of the quadrotor, pointing downwards (it has the same direction as k p  axis in the body frame). Generally, this direction is not the same as ki (measured in the inertial frame), because it depends on the attitude of quadrotor.

Figure 2. Ultrasonic sensor with respect to inertial and body frame

   Where Rp  C p , i p , j p , k p represents the inertial frame with the origin in the centre of Earth,    Rb  C , ib , jb , kb indicates the body frame with the origin in the centre of mass of quadrotor,   rp is the position of sonar in body frame, rm is the distance which is acquired by sonar, h represents the height of sensor.









In order to consider the generic problem, assuming the ultrasonic sensor is mounted on the  rigid body, aligned with the body vertical axis kb , the output value is in the body frame with meter unit, hereby, and it requires a transformation from body-to-inertial frame. Firstly, we 5 Last modified at: 04/12/2014 11:01:51 By author: Jiawei Wu This document can only be used by Space and Precision Automatic research group. It is not allowed to copy, modify, print, post and distribute to other people without the written consent of the author. If you have any doubts, please feel free to contact Jiawei Wu ([email protected]) at any time.

Author: Title: Location: Date: Email:

Jiawei Wu Doc1-Brief Sensor Analysis For Borea Project.Doc Politecnico di Torino 04/12/14 [email protected]

transform the position of sonar from body-to-inertial frame by rotating with sequence 1-2-3, the equation is as following:  rpix   rpbx   rp   X  Y  Z   rp         by   iy   rpiz   rpbz  1 0    0 c  0 s 

0   c   s   0 c    s

0 s   c 1 0   s 0 c   0

 s c

0

0   rpbx  0   rpby  1   rpbz 

(1.8)

The final compact equation is

c c s   rpbx   c s  rpix     rp    s s c  c s  (1.9)  iy         s s s  c c  s c   rpby   rpiz    c s c  s s c s s  s c c c   rpbz   While we get the output rm of sonar, it requires a transformation from body-to-inertial frame by rotating with sequence 1-2-3, the equation is as following:  rmix   0   rm   X  Y  Z   0          iy   rmiz   rmbz   c c    s s c  c s   c s c  s s 

 c s  s s s  c c c s s  s c

s   0    s c   0  c c   rmbz 

(1.10)

The following simplified figure represents the computation of height of quadrotor.

Figure 3. Relation between sonar position and quadrotor height

6 Last modified at: 04/12/2014 11:01:51 By author: Jiawei Wu This document can only be used by Space and Precision Automatic research group. It is not allowed to copy, modify, print, post and distribute to other people without the written consent of the author. If you have any doubts, please feel free to contact Jiawei Wu ([email protected]) at any time.

Author: Title: Location: Date: Email:

Jiawei Wu Doc1-Brief Sensor Analysis For Borea Project.Doc Politecnico di Torino 04/12/14 [email protected]

Where rp represents the position of sensor in body frame, the equation is as following: rp    c s c  s s

c s s  s c

 rpbx  c c   rpby   rpbz 

(1.11)

From the above figure (Figure 4), where hi is the height of sonar in inertial frame, but sensors outputs the height of sonar in body frame hb , these two values have the following relation:

hi  Rbi hb  i 1 hb   Rb  hi

(1.12)

According to equation(1.12), we can derive that:

hb 

hi c c

(1.13)

Finally, the real height value of centre of mass of quadrotor equals to measured distance subtracting the sonar position, considering the errors of sensor and other disturbance, we add the third component d sonar indicating the disturbance from sensor, hereby, we can write the completely equation as following:

yiz  rmiz  rpiz  d sonar  rmbz c c    c s c  s s

c s s  s c

 rpbx  c c   rpby   d sonar  rpbz 

(1.14)

Where yiz is the height of centre of mass of quadrotor in inertial frame.

1.5 Barometric pressure sensor Here, it refers to barometer which is widely used to compensate the measurement of altitude by measuring the absolute pressure of the environment. Variations of pressure are used to estimate variations in the attitude of system. It measures the pressure at local position, outputs data with mbar unit, considering to use this data to correct the attitude measurement, we need to convert it into attitude. As[3]represents, based on the standard atmosphere, it allows to calculate altitude at a certain point by using the atmospheric pressure at that point. LR   T0   P  gM  H 1   L   P0    

(1.15)

7 Last modified at: 04/12/2014 11:01:51 By author: Jiawei Wu This document can only be used by Space and Precision Automatic research group. It is not allowed to copy, modify, print, post and distribute to other people without the written consent of the author. If you have any doubts, please feel free to contact Jiawei Wu ([email protected]) at any time.

Author: Title: Location: Date: Email:

Jiawei Wu Doc1-Brief Sensor Analysis For Borea Project.Doc Politecnico di Torino 04/12/14 [email protected]

Where H is estimated altitude, P represents the measured atmospheric pressure, P0 is the atmospheric pressure at sea level, the sea level air temperature T0 , the rate of decrease in air temperature L ,noticing that, when using this expression that L near the ground is a negative number, normal gravity g, gas constant R , and the molecular mass of air at the sea surface M as constant. In practice, the values of R and M are fixed while the other values vary from day to day with atmospheric conditions and location. Therefore, the values for the current time and current location need to be measured or estimated. In order to simplify the computation of the above equation, we define the constant values in the following table:

No. Parameters

Description

1

P0

Sea level standard atmospheric pressure

2

T0

Sea level standard temperature

3 4 5 6 7 13

L

Temperature lapse rate Earth surface gravitational acceleration Universal gas constant Molar mass of dry air Altitude Measured atmospheric pressure

g

R M H P

Unit

Values

[hPa]

1013.25

[K]

288.15

[K/m] [ ] [J/(mol∙K)] [Kg/mol] [m] [hPa]

-0.0065 9.80665 8.31447 0.0289644 Calculated Measured

Table 1. Parameters and variables definition

According to the datasheet of barometric pressure sensor, the output data is in mbar unit, thus, it requires to convert to hPa. Here, as well-known, the atmospheric pressure is a physical property strongly related with the altitude and the height above a certain level. We can use barometric sensors to sense the value of the pressure and then transform the pressure to the altitude[4], hereby, we can compute the pressure at certain altitude and height g M

 HL  LR P  P0  1   T0  

(1.16)

1.6 Magnetometer The magnetometer is widely used to measure the earth magnetic field strength along the three Euclidian axes. The strength is defined by the location and orientation. This sensor gives us the addition of the last three degrees of freedom. The magnetometer along with the gyroscope and accelerometer allows us to have a 9 degrees of freedom. The output is limited to only affect the yaw axis, as this is the one area the accelerometer proves insufficient. The magnetometers are highly sensitive to the electromagnetic and pure magnetic fields. Especially the electromagnetic fields produces by the motors and their power cables affect the output of these sensors. The output of magnetometer is B m  BSFCC  Bb  Bbias  mag

(1.17)

8 Last modified at: 04/12/2014 11:01:51 By author: Jiawei Wu This document can only be used by Space and Precision Automatic research group. It is not allowed to copy, modify, print, post and distribute to other people without the written consent of the author. If you have any doubts, please feel free to contact Jiawei Wu ([email protected]) at any time.

Author: Title: Location: Date: Email:

Jiawei Wu Doc1-Brief Sensor Analysis For Borea Project.Doc Politecnico di Torino 04/12/14 [email protected]

Where Bb is the earth’s magnetic field, BSFCC is the error relative to scale factor and axes misalignment, mag is the zero mean white noise, Bbias is the bias term.

2. Reference [1] [2] [3] [4]

A.-L. Chan, S.-L. Tan, and C.-L. Kwek, "Sensor data fusion for attitude stabilization in a low cost Quadrotor system," in Consumer Electronics (ISCE), 2011 IEEE 15th International Symposium on, 2011, pp. 34-39. H. Zhao and Z. Wang, "Motion measurement using inertial sensors, ultrasonic sensors, and magnetometers with extended kalman filter for data fusion," Sensors Journal, IEEE, vol. 12, pp. 943-953, 2012. W. Namiki, M. Ichino, and H. Yoshiura, "Altitude estimation using mobile terminal's pressure sensor and external weather information," in Consumer Electronics (ISCE 2014), The 18th IEEE International Symposium on, 2014, pp. 1-3. H. Wang, H. Lenz, A. Szabo, U. D. Hanebeck, and J. Bamberger, "Fusion of barometric sensors, WLAN signals and building information for 3-D indoor/campus localization," in proceedings of International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2006), S, 2006, pp. 426-432.

9 Last modified at: 04/12/2014 11:01:51 By author: Jiawei Wu This document can only be used by Space and Precision Automatic research group. It is not allowed to copy, modify, print, post and distribute to other people without the written consent of the author. If you have any doubts, please feel free to contact Jiawei Wu ([email protected]) at any time.