Miniature Inertial Measurement Unit - IMU - Revision history

LubinKerhuel: /* Acknowledgement */

2010-10-14T06:47:35Z

Acknowledgement

LubinKerhuel at 09:54, 25 April 2010

2010-04-25T09:54:46Z

LubinKerhuel: /* Extended IMU algorithm */

2009-07-07T23:40:51Z

Extended IMU algorithm

LubinKerhuel: /* Extended IMU algorithm */

2009-02-03T00:20:45Z

Extended IMU algorithm

LubinKerhuel: /* Extended IMU algorithm */

2009-02-03T00:20:24Z

Extended IMU algorithm

LubinKerhuel at 23:03, 12 January 2009

2009-01-12T23:03:15Z

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LubinKerhuel: /* Extended IMU algorithm */

2008-12-30T16:01:14Z

Extended IMU algorithm

LubinKerhuel at 15:59, 30 December 2008

2008-12-30T15:59:55Z

LubinKerhuel at 23:49, 29 December 2008

2008-12-29T23:49:12Z

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LubinKerhuel at 14:32, 29 December 2008

2008-12-29T14:32:17Z

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← Older revision		Revision as of 06:47, 14 October 2010
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	[[Category:Example]]		[[Category:Example]]
−
−	~~------~~
−	~~<comments />~~

@@ Line 18: / Line 18: @@
-*<Tex>\Theta</Tex> the absolute pitch angle and '''q''' the pitch angular rate.
+*<math>\Theta</math> the absolute pitch angle and '''q''' the pitch angular rate.
-<Tex>\hat \Theta(t) = \int_{t_0}^t{q(t) dt}+\Theta(t_0)</Tex>
+<math>\hat \Theta(t) = \int_{t_0}^t{q(t) dt}+\Theta(t_0)</math>
-*<Tex>\Phi</Tex> the absolute roll angle and '''p''' the roll angular rate.
+*<math>\Phi</math> the absolute roll angle and '''p''' the roll angular rate.
-<Tex>\hat \Phi(t) = \int_{t_0}^t{p(t) dt} + \Phi(t_0)</Tex>
+<math>\hat \Phi(t) = \int_{t_0}^t{p(t) dt} + \Phi(t_0)</math>
 Unfortunately, the rate gyro sensor is not ideal. Integrating the rate gyro also ''' integrate its DC value'''(bias) or its non zero mean noise. The integration of this error introduced a '''growing error on the estimated angle'''. As the gyro's DC value drifts slowly over time, it is not possible to cancel this bias by subtracting this DC offset value to the gyro output. Thus, Integration of a rate gyro results in an angular drift which is about 1° per minute (strongly depending on the rate gyro used!). It is necessary to use another sensor so as to recover the gyro bias and correcting for the angle error introduced by the integration process.
@@ Line 33: / Line 33: @@
 *az the acceleration measured with the accelerometer on the z axis (vertical)
-Using the acceleration vector (i.e. gravity) measured with the accelerometers, we get the <Tex>\Theta</Tex> and <Tex>\Phi</Tex> angles :
+Using the acceleration vector (i.e. gravity) measured with the accelerometers, we get the <math>\Theta</math> and <math>\Phi</math> angles :
-*<Tex>\hat \Theta= \arctan \left( \frac{a_z}{a_x} \right)</Tex>
+*<math>\hat \Theta= \arctan \left( \frac{a_z}{a_x} \right)</math>
-*<Tex>\hat \Phi = \arctan \left( \frac{a_z}{a_y} \right)</Tex>
+*<math>\hat \Phi = \arctan \left( \frac{a_z}{a_y} \right)</math>
 Note than when the axis az and ay are in the horizontal plane, the result is undefined. But pitch or roll angle is not defined as well (eg: a plane which has its nose in the sky, perpendicular to the horizontal has its roll angle undefined).
@@ Line 184: / Line 184: @@
 * The 0.08Hz first order low pass filter transfer function for the accelerometer is :
-<Tex>Lp(s) = \frac{0.5}{s+0.5}</Tex>
+<math>Lp(s) = \frac{0.5}{s+0.5}</math>
 * The 0.08Hz first order High pass filter transfer function for the rate gyro is :
-<Tex>Hp(s) = \frac{s}{s+0.5}</Tex>
+<math>Hp(s) = \frac{s}{s+0.5}</math>
 [[Image:Bode_1stOrderComplementaryFilter.png|thumb|center|450px|Bode diagram of a 0.08Hz 1st order complementary filter. The blue curve is the low pass filter. The green curve is the high pass filter and the red curve is the addition of both filters. The red curve has a phase of 0° and a gain of 1.]]
 * From the analog device datasheet, the rate gyro has a cut off frequency at 40Hz. Thus, the transfer function of the integration of the rate gyro including the inverse dynamic of the rate gyro is:
-<Tex>Int(s) = \frac{1}{314} * \frac{s+314}{s}</Tex>
+<math>Int(s) = \frac{1}{314} * \frac{s+314}{s}</math>
 *For the rate gyro, we obtain the simplification :
-<Tex>Hp(s) * Int(s) = \frac{s}{s+0.5} * \frac{1}{314} * \frac{s+314}{s} = \frac{1}{314} * \frac{s+314}{s+0.5}</Tex>
+<math>Hp(s) * Int(s) = \frac{s}{s+0.5} * \frac{1}{314} * \frac{s+314}{s} = \frac{1}{314} * \frac{s+314}{s+0.5}</math>
 The upper part of the complementary filter block estimates the Y axis angle computing the arctan of the X and Z accelerometers values.
@@ Line 207: / Line 207: @@
 * The high pass filter is:
-<Tex>Hp_2(s) = \frac{s^2}{(s+0.2)^2}</Tex>
+<math>Hp_2(s) = \frac{s^2}{(s+0.2)^2}</math>
 * The low pass filter is the complementary:
-<Tex>Lp_2(s) = 1 - Hp_2(s) =  \frac{0.4s + 0.02}{(s+0.2)^2}</Tex>
+<math>Lp_2(s) = 1 - Hp_2(s) =  \frac{0.4s + 0.02}{(s+0.2)^2}</math>
 * As for the 1st order complementary filter, the high pass filter can combine with the integration of the rate gyro. We do not take into account the dynamic of the rate gyro:
-<Tex>\frac{1}{s} * \frac{s^2}{(s+0.2)^2} = \frac{s}{(s+0.2)^2}</Tex>
+<math>\frac{1}{s} * \frac{s^2}{(s+0.2)^2} = \frac{s}{(s+0.2)^2}</math>
 This 2nd order complementary filter compensates for the rate gyro bias with no steady state error.
@@ Line 328: / Line 328: @@
 [[Category:Example]]

@@ Line 316: / Line 316: @@
 All files of this 3 DoF IMU are provided as is.
-'''[http://www.kerhuel.eu/download/IMU_2008_10_as_is.zip IMU_2008_10_as_is.rar]:  All Simulink and Matlab file used with data logged for the 6DoF IMU.''' (''Updated on 30 December 2008'')
+'''[http://www.kerhuel.eu/download/IMU_2008_10_as_is.zip IMU_2008_10_as_is.zip]:  All Simulink and Matlab file used with data logged for the 6DoF IMU.''' (''Updated on 30 December 2008'')
 This IMU is being tested onboard a Remote Control (RC) flying wing. It is destined to become an important part of the autopilot currently under development.  For more information, see page :

@@ Line 316: / Line 316: @@
 All files of this 3 DoF IMU are provided as is.
-'''[[http://www.kerhuel.eu/download/IMU_2008_10_as_is.zip IMU_2008_10_as_is.rar]]:  All Simulink and Matlab file used with data logged for the 6DoF IMU.''' (''Updated on 30 December 2008'')
+'''[http://www.kerhuel.eu/download/IMU_2008_10_as_is.zip IMU_2008_10_as_is.rar]:  All Simulink and Matlab file used with data logged for the 6DoF IMU.''' (''Updated on 30 December 2008'')
 This IMU is being tested onboard a Remote Control (RC) flying wing. It is destined to become an important part of the autopilot currently under development.  For more information, see page :

@@ Line 316: / Line 316: @@
 All files of this 3 DoF IMU are provided as is.
-'''[[Media: IMU_2008_10_as_is.rar|IMU_2008_10_as_is.rar]]:  All Simulink and Matlab file used with data logged for the 6DoF IMU.''' (''Updated on 30 December 2008'')
+'''[[http://www.kerhuel.eu/download/IMU_2008_10_as_is.zip IMU_2008_10_as_is.rar]]:  All Simulink and Matlab file used with data logged for the 6DoF IMU.''' (''Updated on 30 December 2008'')
 This IMU is being tested onboard a Remote Control (RC) flying wing. It is destined to become an important part of the autopilot currently under development.  For more information, see page :
 =Acknowledgement=

← Older revision		Revision as of 16:01, 30 December 2008
Line 318:		Line 318:
	All files of this 3 DoF IMU are provided as is.		All files of this 3 DoF IMU are provided as is.

−	'''[[Media: IMU_2008_10_as_is.rar\|IMU_2008_10_as_is.rar: All Simulink and Matlab file used with data logged for the 6DoF IMU.''' (''Updated on 30 December 2008'')	+	'''[[Media: IMU_2008_10_as_is.rar\|IMU_2008_10_as_is.rar]]: All Simulink and Matlab file used with data logged for the 6DoF IMU.''' (''Updated on 30 December 2008'')

@@ Line 286: / Line 286: @@
-=IMU based on a 6 DoF sensor board (3 gyro, 3 accelero)=
+=IMU based on a 6 DoF sensor board (3 rate gyro, 3 accelerometers)=
 ==Limitations the IMU based on a 5DoF sensor board==
-The IMU based on the 5DoF sensor board have no rate gyro for the yaw axis. If the angular yaw rate is not null and when system is not horizontal, the pitch and roll angle of the system change despite the angular pitch rate and roll rate is null! Thus, the pitch and roll angle estimation is false, until their correct value are corrected thanks to accelerometers information.
+The IMU based on the 5DoF sensor board have no rate gyro on the yaw axis. Thus, when the angular '''yaw rate''' is not null while the system is '''not horizontal''', the pitch and roll angle of the system are modified despite the angular '''pitch and roll rate is 0'''! Thus, the pitch and roll angle estimation based on pitch and roll rate gyro is false, until the accelerometers information is taken into account and correct for the error (like a bias correction).
-This problem is of particular concern when the IMU is placed onboard a plane. If we decompose a plane change of direction: it first takes a roll angle; then, it rotates about its pitch and yaw rate so as to keep its velocity vector in the horizontal plane. Thus, the pitch rate which tends to make the plane climbing is compensated for by the yaw rate. If the IMU do not take into account the yaw rate, it estimates a growing pitch angle (the plane is climbing) despite the still horizontal velocity of the plane.
-==Log raw data from the 6 DoF sensor board==
+The added rate gyro had an unexpected effect; il changes most of the gain of the others rate gyro. Thus, the calibration was done again starting from scratch. Hopefully, the calibration had to be done only once because the sensors parameters do not changed.
-One rate gyro has been added on the sensor presented above to measure the yaw rate.
+===Extended IMU algorithm===
-* Problem of logging data is now different
-* the two gyro gain change (unexpected!)
+The algorithm used is an extension of the previous complementary filter algorithm. I used the state space approach.
-* data analysis
+Two version of the extended algorithm were done. The simpler one simply used exactly the same state space subsystem, but modified the two gyro input for both the pitch and roll input. The new pitch and roll input are a combination of both pitch and yaw or roll and yaw. The new pitch and roll input value for the subsystem (state space model) are the pitch and roll rotational rate in the earth frame of reference (in opposition to the previous plane frame of reference).  The two rate angle were combined using trigonometric relation (cosine or sine) based on the current estimated angle.  This method improved the results. However, the state space model estimation of the gyro bias is difficult as its input is now a combination of two rate gyro.
-==State space approach with non linearity==
+All files of this 3 DoF IMU are provided as is.
-* biais analysis is different : develop (do not use magnetic field)
+'''[[Media: IMU_2008_10_as_is.rar|IMU_2008_10_as_is.rar:  All Simulink and Matlab file used with data logged for the 6DoF IMU.''' (''Updated on 30 December 2008'')
-Point to clarify, error, remarks ? Please, leave your comment on the forum!
+Point to clarify, error, remarks? Please, leave your comment on the forum!
 [[Category:Example]]