**Introduction**

The dimensions of the Apollo 11 Lunar Module (LM). To control its landing, the engineers needed to control the acceleration (and therefore also the deceleration) of the craft as well as its orientation. Linear thrusters are standard for controlling the linear acceleration. If the thruster is aligned perfectly with the centre of mass (also known as the centre of gravity) of the spacecraft with mass m (Fig. 2), then when it is fired, the ensuing force F will only cause the vehicle to accelerate in the same direction as the applied force according to:

If there is any misalignment, then a turning moment is produced, and the vehicle is turned in addition to accelerating linearly. To control rotations, thrusters on spacecraft are always installed as pairs (Fig. 3). Then, if a linear acceleration is desired, the pair of thrusters are activated, potentially with slightly different thrust to overcome any misalignments that might create a turning motion. In other words, if a pair of forces F1 and F2 are applied parallel to each other and out of alignment with the centre of gravity of an object with mass m (Fig. 3), then the object’s linear acceleration a will be in the same direction as the applied forces according to:

F1 + F2 = ma . (2)

In addition, the object will rotate about the (positive) x-axis according to:

F2r2 âˆ’ F1r1 = IxxÎ±x (3)

where Ixx is the mass moment of inertia about the x-x axes and Î±x is the angular acceleration about the x-axis. The mass moment of inertia has dimensions of mass multiplied by distance- squared. In Eq. (3), only the magnitude of the forces is important (in conjunction with the orthogonal distance between their vectors and the centre of gravity).

Your task in this question is to determine the forces necessary to produce the desired ac- celerations in various manoeuvres undertaken by the LM using the system of thrusters shown in Fig. 4. Each cluster has four thrusters: two in the x-direction and one each in the y- and z-directions. The inertial properties of the LM are recorded in Table 1.

For the more complex manoeuvres, for your assignment the following values are to be used and not shared with anyone else:

ax = [(1 + 1.2617) Ã· 10] m/s

ay = [(1 + 8.0252) Ã· 10] m/s

az = [(1 + 5.7996) Ã· 10] m/s

Î±x = [(1 + 1.8749) Ã· 500] rad/s2

Î±y = [(1 + 8.335) Ã· 500] rad/s2

Î±z = [(1 + 2.0413) Ã· 500] rad/s2

Locations of the Apollo 11 Lunar Module (LM) thrusters in the x-y and y-z planes relative to the centre of gravity [NASA (1969)]. In the upper image, the positive x-axis is in the downwards direction, while the positive y-axis is in the left direction. The upper image is a different scale to the lower image (the thrusters are shown to be larger); the lower image is a schematic (dimensions are not to scale and this is not a plan to be used for building the system). The blue lines link the same thruster between the two views. All dimensions for this configuration are fully specified on this image (you need to use some geometry).

Requirements

For this assessment item, you must perform hand calculations using data from Table 1 and Fig. 4:

1. Calculate F using Eq. (1) for an acceleration of 0.5 m/s2.

2. Calculate F1 and F2 using Eqs. (2)–(3) for an acceleration in the positive y-direction of

0.5 m/s2 and an angular acceleration in the x-direction of 0.25°/s2. Ensure you convert everything (be careful about every variable!) to SI units before performing any calcula- tions.

3. Express the system of equations in Requirement 2 in matrix form, and therefore calculate F1 and F2 for the same conditions as Requirement 2. Verify1 your answer using the result from Requirement 2.

You must also produce MATLAB code which (if any system is under-determined, then specify that Clusters 2 and 3 have the same z-force and the same x-force as each other):

4. Repeats Requirements 1 and 3. Reports and verifies the results.

5. *Repeats Requirement 3 using your values of ay and Î±x. Reports which thruster(s) are used and the corresponding force for each thruster.

6. **For your values of ay, az and Î±x, calculates the forces for the required thrusters, and reports which thruster(s) are used (by referring to the Cluster number and direction of the thruster) and the corresponding force for each thruster. If you obtain a negative thrust force, note in your report that the corresponding thruster opposite the chosen one is to be used.

7. ***For your values of ax, ay, az, Î±x, Î±y and Î±z, calculates the forces for the required thrusters, and reports which thruster(s) are used (by referring to the Cluster number and direction of the thruster) and the corresponding force for each thruster.

8. ***Repeat Requirement 7, but assume that the thrusters have not been installed correctly so that each thruster in Cluster 1 is 1 cm further away from the centre of gravity than specified in Fig. 4, each thruster in Cluster 2 is 2 cm further away from the center of gravity, etc. for all Clusters. Assume that there is a negligible effect on the values in Table 1 by this misalignment.

9. Has appropriate comments throughout.

The projected difficulty of a Requirement is indicated by the number of * at the start. All students are expected to be able to complete Requirements that do not have an *.

You must submit a short video (10–30 seconds) where you discuss:

1. the part of the code of which you are most proud; and

2. the part of the code which you found most difficult to get working correctly.

Your video must show the relevant part (s) of the code while you are discussing them. Your video will be marked based on whether one is submitted or not: full marks for this item if a video is submitted; zero marks if a video is not submitted.

2 (worth 100 marks)

Introduction

Using the data from your data file from Assignment 1, build models for the descent of the Apollo 11 Lunar Module (LM), and predict at what range and when the LM landed. Also predict the range and when the LM reached the following altitude (this altitude is not to be shared with anyone else):

he = (10 + 3.9718) Ã— 100 ft .

Convert the time data into seconds after 102 hrs G.E.T. The function seconds will be useful when performing calculations using the time data in ‘Apollo 11 landing.csv’.

Requirements

For this assessment item, you must perform hand calculations using range as a function of altitude:

1. Take the first two values of altitude and range where the alarm “1201” was active and estimate the coefficients of the three standard curve-fitting functions. These data points will provide a qualitative representation of the overall trend.

You must also produce MATLAB code which:

2. Repeats Requirement 1 and verifies the results.

3. Performs curve-fits of all the data2 for range as a function of altitude.

4. Validates the three standard curve-fitting functions obtained in Requirement 3 by comparing them with the parameters obtained in Requirement 1. Given the limited data used in Requirement 1, don’t expect the values to be very close. There will also be some variations between Requirements 1 and 3 if the data was normalized for one calculation, but not the other.

5. Determines which curve-fit is the best.

6. Demonstrates that the chosen curve-fit is the best both graphically and numerically, showing both the data and the relevant curve-fit.

7. Displays a message in the Command Window stating which type of curve-fit was chosen, stating the parameters of the curve-fit and the result of the numerical test of the curve-fit.

8. *Uses the best curve-fit to estimate: where the LM landed, and the range at altitude he. Depending on the curve-fit, the first estimate may need to be for just before the LM landed. Compares the estimates with the entries in the data log that have the closest altitudes to landing and he.

9. Plots the best curve-fit along with the data and the points calculated in Requirement 8 in a separate figure with normal-scale axes.

10. **Performs the necessary curve-fit analysis to estimate when the LM landed and when it was at altitude he. Plot altitude as a function of time for the equivalent of Requirement 9.

11. Discusses the quality of the curve-fits.

12. Has appropriate comments throughout.

The projected difficulty of a Requirement is indicated by the number of * at the start. All students are expected to be able to complete Requirements which do not have an *.

You must submit a short video (10–30 seconds) where you discuss:

1. the part of the code of which you are most proud; and

2. the part of the code which you found most difficult to get working correctly.

Your video must show the relevant part (s) of the code while you are discussing them. Your video will be marked based on whether one is submitted or not: full marks for this item if a video is submitted; zero marks if a video is not submitted.

**3 (worth 100 marks)**

Introduction

This question provides an alternative methodology to the problem in Question 2. In this question, you will use linear and spline interpolation to provide the guesses for range and time at altitude. You will also use one of the following other methods for comparison:

Ë† If Q3 â‰¤ 5, use the nearest neighbor method.

Ë† If Q3 > 5, use the shape-preserving piecewise cubic interpolation method.

For your assignment, the following value is to be used and not shared with anyone else:

Q3 = 3.8112 .

Requirements

For this assessment item, you must perform hand calculations:

1. Estimate the range at altitude 2500 ft using linear interpolation.

2. Estimate the time the LM was at altitude 2500 ft using linear interpolation. You must also produce MATLAB code which:

3. Repeats Requirement 1, reporting the result to the Command Window. You will need to perform some filtering of the data because the altitude fluctuated close to landing.

4. Verifies Requirement 3.

5. Repeats Requirement 1 using cubic splines and the method determined by Q3. Reports the value of Q3. Additionally validates the results with Requirement 3.

6. **Determines the range at altitude he using the same three methods used in Requirements 3 and 5.

7. **Repeats the calculations of Requirements 3–6 for time instead of range3.

8. Compares and discusses the answers for range and time at he from Question 2 and the answers from this question.

9. ***Solves the following problem. Assume that the data recording permanently failed at some point during the landing and you have to use the successfully-recorded data to estimate where the LM landed. What is the earliest moment at which the recorder could fail, and linear extrapolation can provide an estimate of range that is within 1% of the actual value? Report if it is impossible to achieve this with the existing data.

10. ***Repeats Requirement 9, providing an estimate of landing time that is within 1% of the actual value.

11. Has appropriate comments throughout.

The projected difficulty of a Requirement is indicated by the number of * at the start. All students are expected to be able to complete Requirements which do not have an *.

You must submit a short video (10–30 seconds) where you discuss:

1. the part of the code of which you are most proud; and

2. the part of the code which you found most difficult to get working correctly.