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preferred representation for strapdown coordinate transformations All these methods are explained in Appendix C, and we explain here how these methods are applied to maintain the transformations between sensor- xed coordinates and the navigation coordinates used for integrating the sensed accelerations 6453 Attitude Rate Integration To resolve sensed accelerations into navigation coordinates, strapdown systems need to keep track of where the navigation coordinates are pointed relative to body- xed sensor coordinates This can be done by rotating the row vectors of Cbody using the quaternion rotation formula of nav Eq C254 and the sensed and error-compensated angular rates in body- xed coordinates The three row vectors of Cbody are the direction cosine vectors of the respective nav navigation coordinate axes in body- xed coordinates.

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R = L(p n P) D I K All of the lower-case letters are xed numbers. Lower-case p is the xed price of the contracts between the retailer and the customer base. Lower-case n is the xed network fee in $/MWH that the ISO charges for its grid services. All of the capitalized variables are random variables. R is the composite value of the retail position in dollars over the delivery month. L is the monthly load obligation of the customer base in MWH. P is the load-weighted day-ahead ISO index in dollars per MWH. D is the intraday costs of adjusting the portfolio in dollars. I is the imbalance penalty in dollars. K is the cost of capacity meeting the ISO capacity tag requirement. Note that K depends on price and the peak hour load, both of which are uncertain. Again we emphasize that none of these variables is known on signing the contract. The retailer can only estimate probability distributions for these variables. By load-weighted day-ahead ISO index, we mean that each hourly ISO dayahead settlement price is weighted by the percentage of load demand for that hour and then the result is averaged. P= Lh Ph h =1 L

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The sampled rotation rate vectors vk , composed of the error-compensated gyroscope outputs, de ne a series of incremental rotation vectors rk vk Dt in body coordinates, where Dt is the sampling interval However, to an observer standing on a rotating earth, the sun appears to be rotating about the earth in the direction opposite earth rotation The same is true of navigation coordinates as viewed in body- xed coordinates When the sensed rotation rate is v, the directions of the navigation coordinates in body- xed coordinates appear to be rotating in the direction of v Therefore, the row vectors of Cbody need to be rotated through the incremental rotations r vk Dt in bodynav xed coordinates The implementation equations for maintaining the coordinate transformation between sensor coordinates and navigation coordinates are given in Eqs C221 C225 (for implementation as direction cosine matrices) or Eqs C252 C.

Before we can state clearly the problems underlying the computation of projections from a database of sample cases, it is helpful to define formally what we understand by database . We distinguish two cases: databases with precise tuples and databases with imprecise tuples. However, since the latter are a generalization of the former, it suffices to consider the latter case. To define the notion of a database of imprecise tuples, we start by extending the definitions of a tuple and of a relation (cf. Definition 3.2.1 on page 63) to capture imprecision w.r.t. the values of the attributes underlying them. Definition 5.1.1 Let U = {A1 , . . . , An } be a (finite) set of attributes with respective domains dom(Ai ), i = 1, . . . , n. A tuple over U is a mapping tU : U

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