Problem 4: Save the Satellite! Your autonomous space startup, SpaceY, has just launched its first satellite! Everything is going great when all of a sudden, it loses orientation data and begins spinning wildly out of control. Let's see what we can do to recover our satellite's orientation data and put our mission back on track. (a) Although our onboard sensors have failed, we still have a set of n sensors on the ground that we can use to help locate our satellite. The only problem is - each sensor only reports a scalar distance d; to the satellite rather than an XYZ position. Find an expression for the XYZ position of the satellite in terms of dy, ..., d,, and vy, ..., v1, R3, the vectors pointing from Sensor 1 to the other n-1 sensors. Your solution should assume that each sensor points in a different direction and that no three sensors are collinear. It should be specified in the frame of Sensor 1. Hint: How can we use least-squares to help solve this problem? (b) Using our solution above, our tracking station, which has an orthonormal frame A, has picked up the XYZ positions of four points py, p2, p3, ps R? on the satellite's body. A Choosing p; as the origin of the body frame B, find an expression for three vectors vy, vpo. tp3 R3 with respect to frame A that form an orthonormal basis on the body of the satellite. You may leave your answer in terms of py, p2, p3, pa. Hint: What procedure from linear algebra can we use to make a set orthonormal? (c) Find the rotation matrix R, from the body frame B of the satellite to the base frame A in terms of vy, vp0, vp3. Congratulations, you've successfully recovered the orientation data of your satellite