we all use GPS and they are useful for many reasons. GPS receivers use signals f
ID: 1808135 • Letter: W
Question
we all use GPS and they are useful for many reasons. GPS receivers use signals from more than three satellites so that cheap quartz clocks can be used instead of expensive atomic clocks. The quartz clocks tend to drift up to + or -1 ns/s (1 part in 109). If the receiver clock were calibrated once a day, what is the approximate maximum position uncertainty if only three satellite signals were used to determine location? Would the GPS system be useful under these conditions?
please answer in details including proper mathimatical steps and reasoning. thanks
Explanation / Answer
Navigation Equations: The receiver uses messages received from satellites to determine the satellite positions and time sent. The x, y, and zcomponents of satellite position and the time sent are designated as [xi, yi, zi, ti] where the subscript i denotes the satellite and has the value 1, 2, ..., n, where n ge 4. When the time of message reception indicated by the on-board clock is , ilde{t}_ ext{r}, the true reception time is , ilde{t}_ ext{r} + b where , b is receiver's clock bias (i.e., clock delay). The message's transit time is , ilde{t}_ ext{r} + b - t_i. Assuming the message traveled at the speed of light , , c , the distance traveled is , left( ilde{t}_ ext{r} + b - t_i ight) c. Knowing the distance from receiver to satellite and the satellite's position implies that the receiver is on the surface of a sphere centered at the satellite's position with radius equal to this distance. Thus the receiver is at or near the intersection of the surfaces of the spheres if it receives signals from more than one satellite. In the ideal case of no errors, the receiver is at the intersection of the surfaces of the spheres. The clock error or bias, b, is the amount that the receiver's clock is off. The receiver has four unknowns, the three components of GPS receiver position and the clock bias [x, y, z, b]. The equations of the sphere surfaces are given by:(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2 = igl([ ilde{t}_ ext{r} + b - t_i]cigr)^2, ; i=1,2,dots,n or in terms of pseudoranges, p_i = left ( ilde{t}_ ext{r} - t_i ight )c, asp_i = sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}- bc, ;i=1,2,...,n. These equations can be solved by algebraic or numerical methods. Multidimensional Newton Ralphson Calculation: Alternatively, multidimensional root finding methods such as the Newton-Raphson method can be used.[88] The approach is to linearize around an approximate solution, say left [x^{(k)}, y ^{(k)}, z^{(k)}, b^{(k)} ight ] from iteration k, then solve the linear equations derived from the quadratic equations above to obtain left [x^{(k+1)}, y^{(k+1)}, z^{(k+1)}, b^{(k+1)} ight ]. Although there is no guarantee that the method always converges due to the fact that multidimensional roots cannot be bounded, when a neighborhood containing a solution is known as is usually the case for GPS, it is quite likely that a solution will be found.[88] It has been shown[89] that results are comparable in accuracy to those of Bancroft's method.
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