CHAPTER 2 PROJECT

The Apollo Lunar Module

The Lunar Module (LM) was a small spacecraft that detached from the Apollo Command Module and was designed to land on the Moon. Fast and accurate computations were needed to bring the LM from an orbiting speed of about 5500 ft/s to a speed slow enough to land it within a few feet of a designated target on the Moon's surface. The LM carried a 70-lb computer to assist in guiding it successfully to its target. The approach to the target was split into three phases, each of which followed a reference trajectory specified by NASA engineers.

The position and velocity of the LM were monitored by sensors that tracked its deviation from the preassigned path at each moment. Whenever the LM strayed from the reference trajectory, control thrusters were fired to reposition it. In other words, the LM's position and velocity were adjusted by changing its acceleration.

The reference trajectory for each phase was specified by the engineers to have the form

r_ref(t) = R_T + V_Tt + 1/2 A_Tt^2 + 1/6 J_Tt^3 + 1/24 S_Tt^4 (1)

The variable r_ref represents the intended position of the LM at time t before the end of the landing phase. The engineers specified the end of the landing phase to take place at t = 0, so that during the phase, t was always negative. Note that the LM was landing in three dimensions, so there were actually three equations like (1). Since each of those equations had this same form, we will work in one dimension, assuming, for example, that r represents the distance of the LM above the surface of the Moon.

1. If the LM follows the reference trajectory, what is the reference velocity v_ref(t)?

2. What is the reference acceleration a_ref(t)?

3. The rate of change of acceleration is called jerk. Find the reference jerk J_ref(t).

4. The rate of change of jerk is called snap. Find the reference snap S_ref(t).

5. Evaluate r_ref(t), v_ref(t), a_ref(t) J_ref(t), and S_ref(t) when t = 0.

The reference trajectory given in equation (1) is a fourth-degree polynomial, the lowest degree polynomial that has enough free parameters to satisfy all the mission criteria. Now we see that the parameters R_r = r_ref(0), V_r = v_ref(0), A_r = a_ref(0), J_r = J_ref(0), S_r = S_ref(0). The five parameters in equation (1) are referred to as the target parameters since they provide the path the LM should follow.

But small variations in propulsion, mass, and countless other variables cause the LM to deviate from the predetermined path. To correct the LM's position and velocity, NASA engineers apply a force to the LM using rocket thrusters. That is, they changed the acceleration. (Remember Newton's second law, F = ma.) Engineers modeled the actual trajectory of the LM by

r(t) = R_T + V_Tt + 1/2 A_Tt^2 + 1/6 J_At^3 + 1/24 S_At^4 (2)

We know the target parameters for position, velocity, and acceleration. We need to find the actual parameters for jerk and snap to know the proper force (acceleration) to apply.

6. Find the actual velocity v = v(t) of the LM.

7. Find the actual acceleration a = a(t) of the LM.

8. Use equation (2) and the actual velocity found in Problem 6 to express J_A and S_A in terms of R_T, V_T, A_T, r(t), and v(t).

9. Use the results of Problems 7 and 8 to express the actual acceleration a = a(t) in terms of R_T, V_T, A_T, r(t), and v(t).

The result found in Problem 9 provides the acceleration (force) required to keep the LM in its reference trajectory.

10. When riding in an elevator, the sensation one feels just before the elevator stops at a floor is jerk. Would you want jerk to be small or large in an elevator? Explain. Would you want jerk to be small or large on a roller coaster ride? Explain. How would you explain snap?