The Global Positioning System (GPS) - a type of Global Navigation Satellite System (GNSS) - uses trilatera- tion to estimate the position of a receiver with respect to the position of satellites, as depicted in the image below*. 1 4- 3 2 Three satellites would normally be sufficient for localisation but a fourth satellite allows for corrections due to clock errors in the receiver. With tuned clocks in satellites and receiver, the distance between satellite and the user / receiver is given by dk = ctk = c(Tk - Tu). Here c = 299792.458 km/s is the speed of light, and the signal flight time t = Tk – T₁ is the difference between the respective clock times T₁ and T₁ (in seconds) on satellite and the receiver. Errors in the receiver clock make this formula unsuitable, however. Instead we use the usual 2-norm distance formula modified with a clock correction term. Using Carte- sian coordinates in units of kilometres, the distance between satellite at (ak, yk, 2k) and the user/receiver at (x, y, z) is d = √(x − xk)² + (y − Yk )² + (z − zk)² + ct, where t (in seconds) incorporates the clock differences. k = 1,2,3,4. With apparent distance dk = ctk and measurements (k, Yk, 2k, tk), the values (x, y, z, t) can be deter- mined by solving the system of equations √√(x − xk)² + (y — Yk)² + (z − zk)² + c(t − tk) = 0, k = 1,2,3,4. (1) Newton's method, in its multi-function, multi-variable form, provides an ideal solution method for this. Taking f(x, y, z, t)=√√(x − xk)² + (y − Yk)² + (z − zk)² + c(t − tk), we solve the system f = 0 in the iterative, linearised Newton form, k = 1,2,3,4, (2) J(m) (m) = f(m) (3) Recalling that the Jacobian matrix J(m) is a matrix of partial derivatives, the form of ƒ in equation (2) makes for unappealing partial derivatives, especially with computational errors in mind. 1. A different version of fk can be obtained by modifying equation (1), leading to the more palatable partial derivatives, afı дх =2(x-x1), an მყ =2(y-y1), af Əz afı =2(2-21), =2c² (t1-t). Ət Determine f(x, y, z, t). 2. Construct the Jacobian matrix J(m) and the function vector f(m). 3. With the correction vector defined as c(m) = x(m+1) = x(m) and x being the ordered list (x, y, z, t), describe the algorithmic steps required to solve for the user / receiver position.