Presentation Acc 2012
Transcript of Presentation Acc 2012
-
8/12/2019 Presentation Acc 2012
1/19
INSTITUTEFOR
SYSTEMSND
ROBOTICS
INSTITUTEFOR
SYSTEMSND
ROBOTICS
Set-Valued Observers for Attitude
and Rate Gyro Bias Estimation
2012 American Control Conference
Montral, Canada
S. Brs P. Rosa C. Silvestre P. Oliveira
Institute for Systems and Robotics
Instituto Superior Tcnico
Lisbon, Portugal
-
8/12/2019 Presentation Acc 2012
2/19
2
Introduction
Some estimators rely on stochastic descriptions of the exogenousdisturbances and measurement noise to provide accurate estimates.
However, in many situations no probability information about
disturbances and noise is available and only magnitude bounds are
known.
In such circumstances, an estimator that computes the set of possible
states given the sensor information is more suitable.
Objective: Simultaneously estimate the smallest set containing the attitude
(rotation matrix) and rate gyro bias.
-
8/12/2019 Presentation Acc 2012
3/19
3
Introduction
Discrete attitude kinematic model
That is an exact description of the physical
quantities involved. No model identification isrequired.
Angular velocity measurements
-
8/12/2019 Presentation Acc 2012
4/19
4
Introduction
Assumption: there are at least three non-coplanar vector
observations (noisy) Magnetometer
Accelerometer
Star-tracker
Sun sensor
The manufactures often provide the maximum noise levels in each
axis of the sensor.
-
8/12/2019 Presentation Acc 2012
5/19
5
Set-Valued Observers
There is uncertainty on the state and on the
measurements.
Uncertainty is described by means of polytopes. The dynamic model may be uncertain.
-
8/12/2019 Presentation Acc 2012
6/19
6
Set-Valued Observers
Prediction Update
(model) (measurements)
-
8/12/2019 Presentation Acc 2012
7/19
7
In our approach we map:
i.e.,
Then we have
Attitude Estimator
-
8/12/2019 Presentation Acc 2012
8/19
8
Attitude Estimator
The solution is then the intersection of state polytope in R9with
SO(3).
Summarizing:
SO(3)
k k+1
SO(3)
R9 R9
No conservatism is added by this operation
-
8/12/2019 Presentation Acc 2012
9/19
9
The uncertainty in the model is associated with the bias and noise in
the rate gyro measurements
Attitude Estimator
-
8/12/2019 Presentation Acc 2012
10/19
10
Attitude Estimator
Main advantages:
The state is guaranteed to be inside the polytope.
There are no singularities.
The solution is global, in the sense that it converges for any initial
conditions.
The estimates are stable, i.e. the set containing the state does not grow
unbounded.
It is based solely on the kinematics, thus it is platform independent and it
is suitable even if a dynamic model is unknown or inaccurate .
Main disadvantages:
Computational cost.
-
8/12/2019 Presentation Acc 2012
11/19
11
Rate Gyro Bias Estimator
Goal: Reduce the uncertainty in the bias to improve the attitude
estimates
Strategy: to design an SVO for the rate gyro bias
Polytope: known from the rate gyro
properties
Polytope: Computed using R(k) and R(k-1)
Vector: sensor measurement
-
8/12/2019 Presentation Acc 2012
12/19
12
Increasing the Convergence Speed
Divide-to-conquer strategy which consists on:
Divide the polytope containing the bias, B(k), into sub-polytopes, so
that only one sub-polytope contains the true bias
Design SVOs initialized with each of the sub-polytopes
The SVOs initialized with polytopes not containing the bias will, at
some time instant, degenerate into empty sets
When all but one SVO have degenerated, the remaining sub-
polytope is divided and the process repeats itself
-
8/12/2019 Presentation Acc 2012
13/19
13
Increasing the Convergence Speed
With this strategy we bring together the SVO methodology and the ideaof model falsification
Disadvantages:
Higher computational cost
Multi-core and multi-processor systems can be exploited for implementing
this strategy since each core can be assign to one SVO
-
8/12/2019 Presentation Acc 2012
14/19
14
Simulation Results
Angular velocity:
Noise bounded by 0.115 deg/s
Vector measurements:
Noise bounded by 0.01Trajectory
-
8/12/2019 Presentation Acc 2012
15/19
15
Simulation Results
The representation of the uncertainty in R9does not provide an intuitive
measure of the uncertainty on the attitude estimate.
To obtain uncertainty bounds on more suitable attitude
representations, different approaches can be explored:
Exploit the rotation matrix polytope to obtain another polytope containingthe rotation vector.
To compute bounds on the Euler angles using nonlinear optimization.
Remark: these operations are conservative but do not add conservatism
to the estimator.
-
8/12/2019 Presentation Acc 2012
16/19
16
Simulation Results
Upper and lower bounds on the attitude estimates
-
8/12/2019 Presentation Acc 2012
17/19
17
Simulation Results
Bias upper and lower bounds
-
8/12/2019 Presentation Acc 2012
18/19
18
Conclusions
A solution for the problem of attitude estimation is proposed based on set-
valued observers.
The observer has no singularities, since the attitude is given by the rotation
matrix and is global, in the sense that, it is valid for any initial conditions.
The nonlinearities of the plant are tackled by adding conservatism to the
estimates.
The online estimation of the rate gyro bias reduces the uncertainty in the
angular velocity measurements and consequently in the model.
Parallel processing can be exploited to increase the estimation rate.
-
8/12/2019 Presentation Acc 2012
19/19
19
Thank you