Presentation Acc 2012

8/12/2019 Presentation Acc 2012

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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


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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.


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Introduction

Discrete attitude kinematic model

That is an exact description of the physical

quantities involved. No model identification isrequired.

Angular velocity measurements


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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.


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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.


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Set-Valued Observers

Prediction Update

(model) (measurements)


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In our approach we map:

i.e.,

Then we have

Attitude Estimator


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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


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The uncertainty in the model is associated with the bias and noise in

the rate gyro measurements

Attitude Estimator


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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.


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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


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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


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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


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Simulation Results

Angular velocity:

Noise bounded by 0.115 deg/s

Vector measurements:

Noise bounded by 0.01Trajectory


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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.


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Simulation Results

Upper and lower bounds on the attitude estimates


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Simulation Results

Bias upper and lower bounds


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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.


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Thank you

Presentation Acc 2012

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