Falla gastrointestinal: Transfusión de sangre mayor de 20 ml/Kg en 24 horas

Lindgren [72], with Nardone and Aidala [82] laid some of BOT problem’s founda- tional groundwork in the submarine context by developing criteria for observability. Efforts to increase observability began with “two leg” options, looking for a “lead-lag” trajectory [76], or fixing the heading for the initial leg and optimizing the heading for the second leg [29].

Hammel expanded on this [47, 45], pushed the BOT processing algorithms [46], and investigated the application to trajectory planning by maximizing an analytic approximation of the determinant of the FIM. The FIM provides a measure of the amount of information that is obtained from measurements, and is a function of the geometry of the problem, rather than the estimation method. Maximizing the determinant of the FIM effectively minimizes the volume of the uncertainty ellipsoid around the target position estimate.

Hammel’s method for optimal control problem formulation became the standard approach—the continuous problem was parameterized, assuming the observer to have a constant speed and infinite heading-rate ability. A preset number of equal length, constant heading segments was then assumed, reducing the optimal control to a single sequence of headings to apply to the segments. Note that a constant velocity and fixed final time (indirectly assumed through a fixed number of equal duration segments) are common assumptions made in these techniques for tractability. This represents a major shortcoming—in effect, when solving for the optimal path, the sensitive parameters of path length and the number of measurements must be provided as assumed inputs, though they greatly change the nature of the solution. Figure 7

demonstrates this with plots from Hammel [44] and Oshman [84], where both 𝑉 𝑇 /𝑟0

and 𝐾 represent a required solution input parameter of the ratio of total path length to initial (unknown) range—essentially a fixed final time for the constant velocity observer. A ratio of one or greater results in collision and a singularity.

(a) Families of Solutions Varying 𝑉 𝑇 /𝑟0[44] (b) Families of Solutions Varying 𝐾 [84]

Figure 7. Effect of Specifying Path Length on Localization

Passerieux followed Hammel with much of the same approach, but instituted a numeric solution for the actual optimization [87]. Oshman did likewise, comparing the optimization of a direct gradient-based method (collocation), an orthogonal func- tion parameterization method (still direct collocation, but performed with fewer pa- rameters by approximating the control vector with orthogonal basis functions), and with a differential inclusion method (removing control by replacing it with a state constraint, such as an equation for constant velocity) [85]. Liu used a suboptimal approach, analytically maximizing a lower bound on the determinant of the FIM, vice the determinant itself [73].

Faced with problems that stem from compression of the information metric into a scalar, Helferty moved away from the determinant of the FIM [50]. Minimizing the scalar uncertainty volume (or the area, for this particular 2-D case) was found to produce solutions that may favor highly eccentric confidence ellipsoids. This is

especially problematic for localization problems, where the largest ambiguity axis often corresponds to the unknown range variable, where most of our attention is needed. Minimizing the trace of the CRLB was suggested instead. The CRLB is a lower bound on the error covariance of the estimation problem. It represents the best certainty attainable from measurements along that path, not necessarily what could be obtained by some other path, and by definition is the inverse of the FIM for an efficient estimator. With each eigenvalue corresponding to the square of one axis of the confidence ellipsoid, the trace (sum of the eigenvalues) yields the sum of the squares of each axis. Therefore, minimizing it penalizes solutions with a large axis of uncertainty resulting in less ambiguity of optimal solutions [49]. Logothetis developed a similar “mutual information metric,” the maximization of which was equivalent to the minimization of the CRLB determinant [74].

The trace of the FIM has at times been selected as the metric of choice and efficient to calculate, but has also been shown to be unstable and potentially singular [88]. Le Cadre created an approximation of the FIM that was additively monotonic, and then took the trace of the approximation [67]. He later followed the concept, allowing for maneuver of the target using a hidden Markov model (HMM), and determined the optimal heading sequence with classical dynamic programming [68]. More recently, Per Skoglar used a steepest descent method for the optimization and a particle filter for the estimation. For the Gaussian case, he showed that trajectory planning with the determinant of the FIM was equivalent to using the differential entropy of the posterior target density [102]. In the context of multiple robots using Model Predictive Control (MPC), Leung chose to maximize the minimum eigenvalue of the FIM for localization [70].

Ponda compared solutions using several of the most popular FIM metrics (deter- minant, maximum eigenvalue of the inverse, negative trace, and trace of the inverse)

in the context of the same problem—determining the location of a ground target optically with a sUAS, allowing 100 measurements in a fixed path length [88]. Un- surprisingly, the determinant of the FIM was found to no longer contain information about the angular dependence between the measurements (compression). Maximizing the trace of the FIM was better, and avoided some local minimum problems along a single path, but found to be unstable and have the potential to result in a singu- larity. The largest eigenvalue of the inverse of the FIM (minimizing the largest axis of the uncertainty ellipsoid), and the trace of the inverse of the FIM (minimizing the average variance of the estimates) yielded similar results, with faster convergence and higher stability in the optimization. The final metric was preferred. In simulation, Ponda found that increasing the allowed number of measurements led to a growing number of local extrema with severe sensitivities to initialization. As must often be done in the world of optimization, impractical results were avoided by initializing the optimization close to the global minimum, which, of course, is problematic for real applications.

Note that a common thread in all of these cases is that a scalar approximation of the information metric is the cost functional that is optimized. Regardless of which particular metric is used, all of them suffer from the loss of some directional infor- mation when a scalar is produced from a multi-dimensional information matrix. The effort to minimize this unavoidable effect is one reason for the variety of approaches. Another common theme is that bearing-only tracking and localization techniques se- lect guidance purely for better estimation of the target location. The actual path that is selected is of no consequence, excepting that the path must be restricted from reaching the target, else the optimal information gathering technique becomes collision (information from bearing measurements will be shown to be inversely pro- portional to the square of range). The UAV scenarios accomplish this by mandating a

fixed, planar altitude above the target and optimizing over a receding horizon, and the submarine and robot scenarios typically choose a fixed final time indirectly, short of that required to reach the target of interest. Unfortunately, such solutions are highly dependent on the time horizon selected, making “optimality” more of a mathematical construct than a practical reality.