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Even for this simple application, both approaches implement an EKF due to the nonlinearity imposed by the scale factor error In the approach of Section 71 the nonlinearity is induced by the accelerometer scale factor compensation in the time update In the approach of Section 74 the nonlinearity appears in the accelerometer portion of the measurement update In more realistic applications, the kinematics typically involve nonlinearities, so an EKF will be required in either approach In the approach of Section 71, there are no free tuning parameters All the parameter values are determined from the characteristics of the instruments Based on the results of observability, covariance, or simulation analysis, the designer may choose to combine or remove states or to tune the parameters determined from the sensor characteristics; however, there is a rigorous basis for the initial settings and subsequent tuning The basic design model is a Markov process driven by stationary white noise processes and therefore satis es the derivation assumptions for the Kalman lter The approach of Section 74 starts with a model hypothesis for the acceleration state In most applications, there is no time invariant Markov model for this state because the signal characteristics are determined by the end user and usually cannot be reliably predicted during the system design phase The design does involve free tuning parameters and does not satisfy the basic modeling assumptions of the Kalman lter Because both approaches implement EKF s the P matrix is not, strictly speaking, the error covariance matrix in either case However, as long as x is kept small, in the approach of Section 71 the linearization error is small, and P is typically a reasonable representation of the accuracy of x In the case of the approach of Section 74 the matrix P cannot be expected to be a reasonable characterization of the accuracy of x unless the assumed model for the acceleration turns out to be accurate Similarly, the EKF measurement update in Section 71 is the optimal linear correction The EKF measurement update in Section 74 is only optimal if the assumed model for the acceleration turns out to be accurate The computational load of aided navigation systems is typically dominated by the time update of the P matrix For the two algorithms, the computational load for the P matrix time and measurement propagation over each second of operation are compared in the Table 72 Eqn (721).

The Model Advisor, new to Version 83, will attempt to point out design issues with your model, thus helping you to produce a model that is designed well and thus more likely to produce predictable, correct results for your users Consider it validation as opposed to

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The spatial curvature caused by intense acceleration or gravitation produces an effective slowing down of time Remember the fundamental axiom of special relativity: The speed of light is constant no matter what the point of view The laser beam traveling across the space ship, as shown in many of the illustrations in this chapter, always moves at the same speed This is one thing about which all observers, in all reference frames, must agree The path of the light ray, as it travels from the laser to the screen, is longer in the situation shown by Fig 16-7 than in the situation shown by Fig 16-6 This is so in part because the ray takes a diagonal path rather than traveling straight across In addition, however, the path is curved This increases the time interval even more From the vantage point of a passenger in the space ship, the curved path shown in Fig 16-7 represents the shortest possible path the light ray can take across the vessel between the point at which it leaves the laser and the point at which it strikes the screen The laser device itself can be turned slightly, pointing a little bit toward the front of the ship; this will cause the beam to arrive at the center of the screen (Fig 16-9) instead of off-center However, the path of the beam is still curved and is still longer than its path when the ship is not accelerating (see Fig 16-6) The laser represents the most accurate possible timepiece, because it is based on the speed of light, which is an absolute constant Thus time dilation is produced by acceleration not only as seen by observers looking at the ship from the outside but also for passengers within the vessel itself In this respect, acceleration and gravitation are more powerful time dilators than relative motion Suspending our disbelief again, and assuming that we could experience such intense acceleration force (or gravitation) without being physically crushed, we will actually perceive time as slowing down inside the vessel.

Table 72: General comparison of computational load for the navigation systems of Sections 71 and 74 Typically, ne < nt and f2 << f1 Section 71 State dimension Time Update Measurement Update 5 O(125)

Part IV:

O(21, 600) O(21, 600)

under conditions such as those that produce spatial curvature, as shown in Fig 16-7 or Fig 16-9 Clocks will run more slowly even from reference frames inside the ship In addition, everything inside the ship will appear warped out of shape If the acceleration or gravitation becomes far more powerful still (Fig 16-10), the spatial curvature and the time dilation will be considerable You will look across the ship at your fellow travelers and see grotesquely elongated or foreshortened faces (depending on which way you are oriented inside the vessel) Your voices will deepen It will be like a science-fiction movie You and all the other passengers in the ship will know that something extraordinary is happening This same effect will be observed by people foolish enough to jump into a black hole (yet again ignoring the fact that they would be stretched and crushed at the same time by the force)

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