Categoría Valores puestos en juego entre profesores 72

In this chapter we proposed several attitude estimation and reconstruction strategies. In one way or another, all of the proposed methods require the use of vector mea- surements. The attitude reconstruction algorithms offer closed form-solutions for the orientation of a rigid-body by using one or more vector measurements. In the case where a single vector measurement is available, to the best of our knowledge it is not possible to reconstruct (in closed-form) the orientation of a rigid-body1. Therefore, the solution provided in Section 3.1.1 does not necessarily fully describe the orien- tation of a rigid-body. However, this result is still useful, especially in the position control of VTOL UAVs, which is discussed later in Section 4.3.

1. Although we are currently unaware of any solutions for the single-vector attitude reconstruction problem (closed-form solutions), there does exist attitude observers which can yield estimates of rigid-body attitude using only a single vector measurement. This problem has been solved, for example see Mahony et al. (2009), by assuming the system body-referenced angular velocity ispersistently excited in order to fully recover the attitude of the rigid-body.

Chapter 3: Attitude Reconstruction and Estimation 72

In most situations involving the attitude estimation of mobile robotics, the mag- netometer and accelerometer sensors have been used to provide vector measurements, in order to provide body-frame coordinates of the Earth’s magnetic field and gravity vectors, respectively. However, since the accelerometer also measures forces due to linear accelerations, that attitude reconstruction algorithms are thought to be more accurate at low frequencies (i.e., when the system is not moving). Alternatively, the gyroscopes are considered to be accurate at higher frequencies, since problems associ- ated with gyroscopic drift occurs due to a constant sensor bias (low frequency). This motivated the research community to combine the use of the attitude reconstructions and the integration of the gyroscope, in order to take advantage of the characteristics of each method (i.e., frequency versus accuracy), which led to the introduction of the complementary filter, for example the observer (or filter) given in Section 3.2. However, in theory, the attitude estimates obtained from a complementary filter are shown to converge to the value obtained from the attitude reconstructions, and not necessarily to the actual attitude. Therefore, when using complementary filters, one must assume the system eventually comes to rest in order to guarantee asymptotic convergence of the estimates to the actual attitude.

Subsequently, new observers were also proposed which did not require the use of the attitude reconstruction algorithms. Instead, these new observers use the vec- tor measurements directly in the attitude estimation laws, for example the observer given in Section 3.3.3. This observer was able to avoid the problem of unwinding, which negatively affected the complementary filter proposed in Section 3.2. However, the new vector measurement based observer is affected by an invariant manifold, which contains a set of trajectories which do not converge to one of the two desired equilibria. Fortunately, the invariance of this manifold can be broken, by increasing the order of the observer, for example the observer given in Section 3.3.4. Another

attractive characteristic of this result is that the vector measurements are applied to a low-pass filter as a part of the observer design, which is desirable in practice (i.e., when the vector measurements are perturbed by noise). Unfortunately, these vector-measurement based attitude observers still depend on an assumption that the inertial vectors are known and constant in the inertial frame, which may be violated especially in the case where the accelerometer is used.

Control Scheme Advantages Disadvantages

Complementary Filter

• Gyroscope bias estima- tion.

•Susceptible to unwinding.

•Requires high gains.

• Sensitive to disturbances and noise. Observers using vector measure- ments •Avoids unwinding.

• Does not require recon- struction of attitude.

•Affected by rigid-body lin- ear accelerations.

Observer using

filtered vector

measurement

•Proves performance of ob- servers which filter sensor data.

•Higher order.

• Trade off between

noise rejection and rate of convergence.

•Affected by rigid-body lin- ear accelerations.

IMU/GPS

Based Ob-

servers

• Greatly simplified proofs of performance (with re- spect to existing observers).

•Dependant on linear veloc- ity measurement.

• Allows the use of ac-

celerometer in the presence of linear accelerations.

Table 3.2: Comparison of Estimation Strategies

This problem has motivated the research community to use the accelerometer in a more realistic manner: to measure the body-referenced system apparent acceler- ation, which is a combination of gravity forces and forces due to linear acceleration of the rigid-body. This problem is complicated by the fact that we don’t know the inertial-frame coordinates of the apparent acceleration vector (we only know body-

Chapter 3: Attitude Reconstruction and Estimation 74

referenced coordinates as obtained by the accelerometer measurement). Instead, we rely on measurements of the system linear velocity which is intrinsically related to the apparent acceleration (i.e., through an integral). Therefore, using the linear ve- locity measurements with a special filter, we obtain information about the apparent acceleration vector, which can be used with the accelerometer measurement to ob- tain information about the system attitude. In theory, we show these velocity-aided attitude observers given in Section 3.3.5 ensure that the attitude estimates converge asymptotically to the actual rigid-body attitude in a specified domain of attraction. This domain of attraction can be arbitrarily increased to contain almost any set, ex- cept in the case where the attitude estimates are rotated 180 degrees away from the actual system attitude. However, despite this limitation, simulation results show that the attitude estimates converge even for this worst case scenario and for any choice of the observer gains. Therefore, future work may involve showing stronger stability results for these observers.

Chapter 4

VTOL UAV Control Design