CHAPTER 4 TEST OF THE IMU DESIGN
4.5 Discussion
The uncertainties of experiment scheme may cause the difference between information from encoder and states of algorithm. The discussion of experimental error below will be shown to emphasize the importance of three features; linear vibration of the optical table, rotational vibration of the optical table and DC-drift of accelerometer output or other noise.
Linear vibrations of the optical table
The first question to be discussed is linear vibration of the optical table. Fig4.5.1 illustrates that the frequency of linear vibration is the same as the motion in the experiment.
Amplitude of this vibration is almost 200mm/s2 in the X-axis and Y-axis, and 50mm/s2 in the Z-axis as shown in Fig.4.5.2. According these conditions, we simulate how this error influences physical quantities in our algorithm.
0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 -150
-100 -50 0 50 100 150
accelerometers output
mm/s2
x-axis y-axis z-axis
Fig.4.5.2 Vibration of tri-axes
Fig.4.5.3 shows that the vibration makes converging rate of the angular rate to be slow, but it does not change the accuracy when it converges. The amplitude will increase by 200mm/s2 in the X-axis and Y-axis, as shown in the Fig.4.5.4. It follows from what has been said that the error of linear vibration affects only the accuracy in the linear acceleration and converging rate of the angular rate in the rotation frame.
0 1 2 3 -0.8
-0.6 -0.4 -0.2 0
rad/s
w1 in rotated frame
0 1 2 3
-0.2 0 0.2 0.4 0.6
rad/s
w2 in rotated frame
0 0.05 0.1 0.15 0.2 0.25
0 0.2 0.4 0.6 0.8 1
rad/s
w3 in rotated frame measured real
measured real
measured real
Fig.4.5.3 Angular rate in the rotation frame (with linear vibration)
0.05 0.1 0.15
-2000 -1000 0 1000 2000
mm/s2
Fr1
0 0.05 0.1
-2000 -1000 0 1000 2000
mm/s2
Fr2
-150 -100 -50 0 50 100 150
mm/s2
Fr3
measured real
measured real
measured real
Rotational vibration of the optical table
Because the angular rate can not be sensed by our experiments, we just discuss this question without experimental verification. We assumed that frequency of rotational vibration is equal to actual motion and the amplitude is about 0.1rad/s2. Fig.4.5.5 and Fig.4.5.6 show that the result of this simulation is contrary to the above simulation.
Rotational vibration affects the accuracy of angular rate in the rotation frame and it doesn’t affect linear acceleration. The influence is the same to rotational vibration we designed. Thus, we see that linear vibration affects linear acceleration and rotational vibration affects angular rate respectively.
0 1 2 3
-0.5 0 0.5 1 1.5
rad/s
w1 in rotated frame
0 1 2 3
0 0.2 0.4 0.6 0.8
rad/s
w2 in rotated frame
0 0.1 0.2 0.3 0.4
0 0.2 0.4 0.6 0.8 1
rad/s
w3 in rotated frame measured real
measured real
measured real
Fig.4.5.5 Angular rate in the rotation frame (with rotational vibration)
0 0.02 0.04 0.06 0.08 0.1 -2000
-1000 0 1000 2000
mm/s2
Fr1
0 0.02 0.04 0.06 0.08
-2000 -1000 0 1000 2000
mm/s2
Fr2
0 1 2 3
-100 -50 0 50 100
mm/s2
Fr3
measured real
measured real
measured real
Fig.4.5.6 Linear accelerations in the rotation frame (with rotational vibration)
DC-drift or other noise
We assumed that there is 5Hz, 100mm/s2 DC-drift in the nine output signals of accelerometers. Fig.4.5.7 gives a good account of the most serious error in transient time. To put it more precisely, DC-drift affects the converging rate more seriously than we discussed above. This kind of error doesn’t affect linear acceleration seriously, as shown in Fig.4.5.8. Fig.4.5.9 shows that the influence of random frequency (1~17HZ) of other noises (backlash of screw, vibration of stage and etc.) on converging rate of angular rate in the rotation frame has slight variations in this two simulations.
Fig.4.5.10 shows that the influence on linear acceleration in the rotation frame is very
0 1 2 3 -0.5
0 0.5 1
rad/s
w1 in rotated frame
0 1 2 3
-0.8 -0.6 -0.4 -0.2 0
rad/s
w2 in rotated frame
0.1 0.2 0.3 0.4
0 0.5 1
rad/s
w3 in rotated frame measured real
measured real
measured real
Fig.4.5.7 Angular rate in the rotation frame (with 5Hz 100mm/s2 DC-drift)
0 0.02 0.04 0.06 0.08 0.1 -2000
-1000 0 1000 2000
mm/s2
Fr1
0 0.02 0.04 0.06 0.08 0.1 -2000
-1000 0 1000 2000
mm/s2
Fr2
0 1 2 3
-100 -50 0 50 100
mm/s2
Fr3
measured real
measured real
measured real
Fig.4.5.8 Linear accelerations in the rotation frame (with 5Hz 100mm/s2 DC-drift)
0 1 2 3 -1
-0.8 -0.6 -0.4 -0.2 0
rad/s
w1 in rotated frame
0 1 2 3
-0.8 -0.6 -0.4 -0.2 0 0.2
rad/s
w2 in rotated frame
0.1 0.2 0.3 0.4
-0.5 0 0.5 1
rad/s
w3 in rotated frame measured real
measured real
measured real
Fig.4.5.9 Angular rate in the rotation frame (with 1~17Hz 100mm/s2 other noise)
0 0.02 0.04 0.06 0.08 0.1 -2000
-1000 0 1000 2000
mm/s2
Fr1
0.02 0.04 0.06 0.08 0.1 -2000
-1000 0 1000 2000
mm/s2
Fr2
-100 -50 0 50 100
mm/s2
Fr3
measured real
measured real
measured real
According to Fig.4.1.2, the experiments error influence directly converging rate and resolution of angular rate, and resolution of linear acceleration in the rotation frame. It follows from what has been said that linear vibration affects linear acceleration, rotational vibration affects angular rate and DC-drift or other noises affect the converging rate more seriously than that we discuss above.
CHAPTER 5
CONCLUSION AND FUTURE WORKS
5.1 Conclusion
The observer-based planar Gyro-free IMU has been proven to be feasible for deriving position information (angular acceleration, angular velocity, linear acceleration and etc.) of an object moving in space by simulation. For the algorithm proposed in this thesis, outputs of 6 linear accelerometers were employed in the state equation and outputs of the other redundant linear accelerometers were used for the output equation; furthermore, Iterated Extended Kalman filter was treated as a nonlinear observer in order to stabilize the nonlinear dynamic equation and estimate precise state (angular rate in the rotation frame).
Euler’s transform is employed to transfer the physical quantities from the basis of rotation frame to the inertial frame. The result (angular rate in the inertial frame) from algorithm undergoes single integration to obtain 3 rotation angles; and result (linear acceleration in the inertial frame) undergoes double integration to obtain 3 coordinates for location.
Because Euler’s angular displacement is obtained from solving differential equation (2.37), this integral operation would accumulate the error and make serious mistake by using Euler’s transform. Algorithm flow (Fig.4.1.2) indicated that the influence of the error due to integration (Euler’s angle) on linear acceleration in the inertial frame is the most serious. For the linear acceleration in the rotation frame obtained by (4.6) carries
It has to be mentioned that the angular rate in the rotation frame is obtained by Kalman filter without white noise, and it is more accurate than the linear acceleration.
Some experimental errors would cause the difference between encoder information and outputs/states of the algorithm. In this thesis, we pointed out that linear vibration affects linear acceleration, rotational vibration affects angular rate respectively, and DC-drift or other noises affect the convergent rate more seriously than the vibrations we discuss above.
5.2 Future works
Although the preliminary simulation/experiment of observer-base planar Gyro-Free IMU is proposed in this work, more complete simulation, experiment and fabrication have to be done in the future, as listed below:
1. To implement observer-base planar Gyro-Free IMU with control circuitry by MEMS process.
2. To improve the resolution for further reducing the restrictions on the distance between accelerometers and origin point in the rotation frame.
3. Replace Euler’s angle by Euler’s parameters in order to reduce the accumulation of error due to integration.
4. To complete the 6-axes experiment and reduce experimental error.
References
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