3.2 State  estimation

A key problem is the need for the real-time computation of state estimation. This includes the helicopter’s attitude (e.g. the Eulerian angles with reference to an Earth frame φ, θ, ψ), its rotational speeds (p, q, r), and its linear velocity and position (u, v, w and x, y, z). Attitudes and rotational speeds are needed for helicopter stabilisation; the position in the form of the relative distances between the members of the flock will be used for regulating flocking.

Given the reduced payload only a very light IMU can be carried on board. We have selected the Memsense nIMU [16] which includes three linear accelerom- eters, three gyros, and three magnetic field sensors. MEMS (Micro-Electro- Mechanical Systems) technology allows the production of inertial sensors of very compact size and reduced weight (the whole IMU has a weight of only 15g); however, their performance in terms of error and temperature bias tends to be significantly worse than alternative navigation grade solutions using opti- cal methods. The dependency on temperature variation is already compensated within the IMU sensor, but the effects of noise and, more importantly, of the drift that affects gyros and accelerometers must still be compensated to allow sufficiently accurate data to be obtained.

The main inputs used for the state estimation are the measured rotational speed and linear acceleration; both are affected by noise, and by a time-variant bias. The magnetometer readings, the ultrasonic sensor values, and possibly the

position obtained from the tracking system will be used to ”correct” the inertial data. The magnetometers and ultrasonic sensors are not affected by time variant bias, and so have error characteristics complementary to those of the inertial sensor; the fusion of the two types of sensor data will allow us to improve the state estimation.

The most commonly used techniques for data fusion rely on Bayesian filtering techniques, among which the best known is probably the EKF (extended Kalman filter). Although it has been successfully applied to helicopter state estimation problems ([17] [18]) the EKF has some weaknesses when compared to other similar approaches. Van der Merwe and Wan conducted a comparison analysis between an EKF and a UKF (unscented Kalman filter) The analysis [19] shows that since the sensor model used in the filter is strongly nonlinear (due to the change of coordinates), the UKF can improve the estimation performance. In addition the implementation of the UKF is comparatively much simpler than that of the EKF since the there is no need to calculate the derivatives of the state equations. Given our limited computational power, it will also be interesting to explore the possibility of implementing the UKF in its square root form, [20] which presents improved numerical stability along with reduced computational complexity.

In order to limit the computational complexity, our system model is simply that of a 6DoF rigid body freely moving in a 3D space. Position, speeds, Euler angle and sensor bias constitute the state estimated by the filter. The system equations are represented by the classic equation of motion of a rigid body  in a 3D space. The acceleration and rotational speed values coming from the IMU are used as control inputs to the model in order to propagate the system state. Readings from the ultrasonics sensors and the magnetometers constitute the measurements that will allow the correction, through the observation model, of the predicted state in the interactive prediction-correction fashion typical of Bayesian filtering.

The system model includes the update equations necessary to estimate the biases of the accelerometers and gyros, and so it will therefore allow them to be compensated.

4 An automated design method

We already mentioned in section 2 the clear advantages offered by using an automated method to deal with the unknown dynamic differences between the aerial vehicles. Such an automated method will be useful in the future as well, as we plan to move the system outdoors, and therefore to use heavier and more complex  helicopters.