Scale Invariant Features Transform and Speed-up Robust Features

The last category of techniques is based on the SIFT or SURF scheme described in Section 2.2.2. It consists in detecting the SIFT or SURF features in each image. Their descriptors are also computed. The tracking is achieved by matching the features of the previous image with the features in the next. This is done by comparing the descriptors of the features using the Euclidian distance. By nature, SIFT and SURF algorithms are very robust to scale change and affine projection between images. They are then able to track large spacecraft motions. This robustness has the cost of being computational expensive and complex to implement. This approach is also not adapted to track the same features over several images. Compared to the block matching or to the differential approach, each feature of the previous image has a much lower probability of being matched with those in the current image. Consequently, the likelihood that a given feature is still tracked after several frames tends quickly to zero. The SIFT and the SURF methods are then incompatible with the relative navigation method based on the feature position estimation presented in Section 2.5.2.

2.4.

State Estimation

The state estimation regroups the techniques that optimally estimate the state variables of a system in a recursive fashion using incomplete (the system state variables are rarely measured directly) and noisy measurements. These techniques can be used to solve the following high-level problems [75]:

 State estimation: The non-measured states of a given dynamic system are estimated from incomplete and noisy measurements. For instance, in [76] the position, velocity, attitude quaternion and attitude rate of a spacecraft are estimated using inertial and vision sensors.

 Parameter estimation or identification: The unknown parameters of a given dynamic system are estimated. For instance, in [77], the center of gravity and the inertia matrix of a small laboratory satellite are estimated using an estimation filter.

 Dual Estimation: State estimation and parameter estimation are used in parallel (with two separated estimation filters) on the same dynamic system [78].

 Joint Estimation: The state estimation and parameter estimation are used in the same estimation filter [78]. This is achieved by augmenting the state vector with the parameters to estimate. For instance, in [8, 79], the states of a vehicle is estimated at the same time as the bias of the inertial sensor.

This section presents an exhaustive overview of most important estimation techniques available in the literature. These algorithms are summarized in the following table:

Table 2.5: Overview of Main Estimation Algorithms for Linear and Nonlinear Systems

Estimators Descriptions Pros Cons References

Optimal Kalman Filter

(KF)

KF corresponds to the analytic solution of the optimal recursive Bayesian estimator for linear and Gaussian systems.

 It is optimal for linear and Gaussian systems.

 It is computationally efficient.

 It does not manage

nonlinear system. [80-85]

Sub-optimal Extended Kalman Filter

(EKF)

The EKF is based on the same framework as the KF, but it uses a first-order Taylor series expansion to approximate nonlinear models.

 It manages nonlinear and nearly Gaussian systems

 Its computational load is increased by the local linearization.

 It may suffer from numerical instability problems due to ill- conditioned covariance matrix.

 It may diverge if the dynamical model of the system is highly nonlinear and/or estimation error is high.

 It may result in poor performance for non- Gaussian systems.

Sub-optimal Sigma-Point Kalman Filter

(SPKF)

The SPKF avoids the linearization involved in the EKF by using the Sigma-Point Transform (SPT). These Sigma-Points (SP), carefully weighted, are chosen to represent the probability density of the states, which is approximated by a Gaussian function. These SP are propagated through the nonlinear dynamic equations of the systems to compute the first two moments of the estimated states (mean and covariance).

 It manages highly nonlinear and nearly Gaussian systems better than EKF.

 It is robust to numerical instability problems often observed with the EKF.

 Its computational load is increased by the sigma- point propagation.

 It may result in poor performance for highly non-Gaussian systems. [75, 79, 94- 106] Sub-optimal Gaussian Mixture Filter (GMF)

The GMF approximates the non- Gaussian probability density functions by sums of Gaussian densities. Each of the components of this mixture is treated with a traditional filter algorithm (EKF, SPKF). The final estimation of the GMF is a weighted sum of the mixture components. The weights of the components are established according to their measurement sensitivity.

 It is able to deal with highly nonlinear systems.

 It offers an improved performance with non- Gaussian state error, process noise and/or measurement noise.

 Its computational load is increased since it uses a bank of filters.

 Empirical method must be used to maintain bounded the number of mixture components.

[107, 108]

Sub-optimal Particle Filter

(PF)

The PF is based on Monte Carlo Sampling (MCS) and Sequential Importance Sampling (SIS) principles. The density of the states is

represented by using a set of random weighted samples. At each time step, the previous state density is augmented (leading the posterior distribution) using the dynamic model and current measurements. By analysing the characteristics of the posterior distribution, an estimate of the system states is obtained.

 It is perfectly adapted for nonlinear and non- Gaussian systems.

 It requires high computational power to propagate the particles.

[109-114] Sub-optimal Receding- Horizon Estimator (RHE)

The MHE involves the online solving of a finite horizon state estimation problem. As new measurements become available, the old measurements are discarded from the estimation windows and the finite horizon state estimation problem is resolved to obtain a new estimate of the states.

 It is compatible with nonlinear systems.

 It requires very high computational power due to the need of complex nonlinear optimization algorithm.

 It may result in poor performance for highly non-Gaussian systems.

[115]

The following sections give more details on how to implement these filters.