# Parameterized Design and Dynamic Analysis of a Reusable Launch Vehicle Landing System with Semi-Active Control

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Working Principles of the RLV Landing System

#### 2.1. Overall Scheme of the RLV Landing System with MRF Dampers

_{1}, the length of the magnetic flux density lines in the gap is h, and the length of the arched magnetic flux density lines out the cylinder is L

_{2}. Based on the Maxwell equation and Ampere circuit rule,

_{1}is the magnetic flux density in the magnetic core area, B

_{2}is the magnetic flux density in the gap, B

_{3}is the magnetic flux density out the cylinder, and μ

_{m}is the magnetic constant. The magnetic flux density B is in T, the magnetic field intensity H is in A/m, the electric current I is in A. Due to the magnetic field intensity in the magnetic core area and gap is much larger than that out the cylinder, the magnetic field intensity in the gap is

#### 2.2. Working Principles of the RLV Landing System

#### 2.3. Control Approach of the RLV Landing System

_{max}, 2a

_{max}]. The pitch angle and roll angle are set as [–3, 3], due to landing angles of the current RLVs are from −3° to 3° [2]. The output yield stresses of MRF in four primary strut dampers are set as [0, 100% Maximum] to adapt to different landing conditions. Considering the control accuracy and efficiency, the acceleration a is divided into four equal fuzzy sets (Z, S, M, B). The jerk da is divided into two equal fuzzy sets (Z, B). Both the pitch angle alpha and roll angle beta are divided into three equal fuzzy sets (N, Z, P). The output yield stresses are divided into seven equal fuzzy sets (Z, S, SM, M, SB, MB, B). The membership affiliations between physical parameters and fuzzy sets for the inputs are shown in Figure 3. The membership affiliations between physical parameters and fuzzy sets for the outputs are shown in Figure 4.

## 3. Landing Dynamic Analysis and Parameterized Design of RLV Landing System

#### 3.1. Landing Dynamic Analysis of the RLV Landing System

_{A}, y

_{A}). The sphere joint between the primary strut and auxiliary strut is B (x

_{B}, y

_{B}). The projection of the revolute joint between the auxiliary strut and rocket is C (x

_{C}, y

_{C}). The horizontal distance x

_{C}between the origin and C is R. The angle between the primary strut and ground is α. The angle between the auxiliary strut and ground is θ. The vertical distance y

_{A}between the origin and A is H

_{1}. The mass center of the elastic part is P

_{1}(0, H

_{1}+ H

_{2}). Due to auxiliary struts occupy most mass of non-elastic parts, whose center can be simplified as the mass center of non-elastic parts. It is 0.5(x

_{B}+ x

_{C},y

_{B}+ y

_{C}), which is shown as

_{p}is the damping force of the primary strut, F

_{a}is the damping force of the auxiliary strut. The landing dynamic models of non-elastic parts are

_{n}is the contact force between the pad and ground, shown as follows [26]

_{0}is the trigger distance of the impact function. k is the stiffness, e is the contact force exponent, c is the contact damping, and d is the penetration depth.

_{p}is

_{u}is the uncontrollable damping force of the MRF damper. F

_{c}is the controllable damping force of the MRF damper. F

_{Ni}is the air-spring force caused by the accumulator [27,28,29]. F

_{entry}is local resistance caused by the abrupt enlargement, and F

_{exit}is local resistance caused by the abrupt contraction. ρ is the density of MRF. K

_{entry}is the local resistance coefficient of the entry, and K

_{exit}is the local resistance coefficient of the exit. v is the piston velocity. A

_{p}is the piston area. A

_{gap}is the gap area between the master cylinder and piston, and A

_{n}is the cross-section area of the master cylinder. P

_{0}is the initial pressure of the accumulator. V

_{0}is the initial volume of the accumulator, and V is the volume of the accumulator during the landing.

#### 3.2. Parameterized Design of the RLV Landing System

_{1}, α, and θ determine the buffer effects of F

_{a}and F

_{p}, and the efficiency and performance of landing systems [30]. Hence, a parameterized design of the RLV landing system is proposed according to these three parameters to get an effective landing system. The lower and upper limits of these three parameters are given in Table 3. The rocket acceleration, compression strokes of dampers, and the distance between the rocket and ground are the most important indexes for the design of a landing system [31]. A large rocket acceleration will damage structures and instruments [32]. Large compressions of primary strut dampers will cause the rocket to incline or tip over. The distance between the rocket and the ground should be large enough for a safe landing [33]. The mass is also an important index for spacecraft, a lighter landing system means a lower launch cost. Hence, these four design targets of the landing system are selected as responses, as shown in Table 4. The parameterized design principle based on the response surface methodology (RSM) is shown as follows

_{P1}is the initial distance between the mass center of the rocket and the ground.

_{i}. The function combines these four responses in a non-dimensional way. Its design goal is the smallest rocket acceleration, compression stroke, the mass of a set of landing gear, and the largest distance between the rocket and ground.

^{n}, which is the independent variable of its response function y. Their relationship is y = f(x). Based on lots of simulation data, an approximate function of the response $\tilde{y}$ is obtained by the undetermined coefficient method. Considering the efficiency and accuracy, a quadratic function with cross terms is used, which is shown as follows

_{0}is the undetermined coefficient of the constant term, a

_{j}is the undetermined coefficient of the one-degree term, and a

_{ij}is the undetermined coefficient of the quadratic term. $\tilde{y}$ is close to y by keeping their sum of error squares smallest via the least square principle [36].

_{1}is C > B > A > BC > AB > A

^{2}> C

^{2}> AC > B

^{2}. The influences of three codes and their extended codes on R

_{2}is B

^{2}> A

^{2}> A > AC > C

^{2}> BC > C > B > AB. The influences of three codes and their extended codes on R

_{3}is A > C > AC > C

^{2}> B

^{2}> A

^{2}> AB > BC > B. The influences of three codes and their extended codes on R

_{4}is C > A > AC > A

^{2}, B

^{2}, C

^{2}> BC > AB > B.

## 4. Landing Dynamic Simulations

#### 4.1. Highest Rocket Acceleration Condition

_{1}is taken as an example, whose damping forces and damper compression strokes are shown in Figure 15 and Figure 16, respectively.

^{2}. During the landing, the controllable damping forces F

_{c}of four primary strut dampers belong to Z and S. Their uncontrollable damping forces F

_{u}slowly decrease versus time due to compression velocity decrease. The highest rocket acceleration of the entire rigid model is 25.95 m/s

^{2}. The highest rocket acceleration of the entire rigid model with friction is 27.57 m/s

^{2}, which is the largest in these four situations. The highest rocket acceleration of the model with flexible primary struts is 12.73 m/s

^{2}, which is the smallest in these four situations. Additionally, structural flexibility causes fluctuations in acceleration and damping force. Adding friction to the flexible model, the highest rocket acceleration increases to 13.13 m/s

^{2}, and the fluctuations of the rocket acceleration and damping force also increase. The highest rocket accelerations for these two flexible situations decrease by about 51% at the touch down moment. At the same time, the damping force peaks of L

_{1}decrease by about 5%, because the flexible structures absorb parts of the impact energy. However, after the instantaneous contact, rocket accelerations and damping forces of these four situations are close to each other.

^{2}under the highest acceleration landing condition [2,17]. Compared to this, the highest rocket acceleration of the proposed landing system with semi-active control decrease about 30.2%. By controlling the damping forces of the four primary strut dampers, the RLV has much lower rocket accelerations and impact forces, which can protect the structures and instruments better during rocket recycle. As shown in Figure 14 and Figure 16, the distance between the rocket and the ground and the compression strokes of L

_{1}are close to each other in these four situations. In conclusion, friction has little influence on landing performance. However, structural flexibility has a strong influence on rocket acceleration and the damping forces of primary struts.

#### 4.2. Greatest Damper Compressions Condition

_{3}touches the ground first. Second, L

_{2}and L

_{4}touch the ground together. Finally, L

_{1}touches the ground. In brief, it is a kind of 1–2–1 landing condition. The rocket accelerations and the distances between the rocket and the ground are shown in Figure 17 and Figure 18. Because L

_{3}touches the ground first, L

_{3}is taken as an example. The damping forces and compression strokes of L

_{3}are shown in Figure 19 and Figure 20.

_{3}touches the ground, and rocket accelerations and damping forces increase vertically. At about 0.085 s, L

_{2}and L

_{4}touch the ground at the same time. The rocket accelerations increase vertically again. At about 0.167 s, L

_{1}touches the ground, and the rocket accelerations increase vertically for a third time. From 0.167 s to 0.310 s, the controllable damping force F

_{c}of L

_{1}belongs to Z, and the controllable damping forces F

_{c}of L

_{2}, L

_{3}, and L

_{4}belong to S. Their uncontrollable damping forces F

_{u}decrease versus time slowly due to the decrease of compression velocities. The air-spring forces F

_{Ni}increase because damper compressions increase. Hence, their resultant forces remain basically stable. Additionally, the pitch angle gradually decreases to 0 due to the horizontal velocity and control of the damping forces. The rocket accelerations increase slightly during this time. After 0.310 s, four controllable damping forces F

_{c}belong to Z together, and the rocket accelerations have a small vertical decrease.

^{2}[2]. The highest rocket acceleration of the proposed landing system with a rigid model is 5.37 m/s

^{2}, which is a decrease of about 76.1%. Additionally, the highest rocket accelerations also decrease by about 54% at the three touch down moments in these two flexible situations. At the same time, damping forces decrease by about 11% in these two flexible situations. Except for the three touch down moments, the rocket accelerations and damping forces of flexible situations are a little higher than those of rigid situations. Structural flexibility also causes an approximately 0.005 s delay, and fluctuations of rocket accelerations and damping forces. The compression strokes of primary strut dampers and distances between the rocket and ground are also close in these four situations.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Notation

α | Angle between the primary strut and ground |

θ | Angle between the auxiliary strut and ground |

μ | Friction coefficient |

τ | Maximum yield stress of MRF |

ρ | Density of MRF |

η | Viscosity of MRF |

a_{0} | undetermined coefficient of the constant term |

a_{j} | the undetermined coefficient of one-degree term |

a_{ij} | the undetermined coefficient of the quadratic term. |

c | Contact damping of the impact function |

d | Penetration depth of the impact function |

e | Contact force exponent of the impact function |

k | Stiffness of the impact function |

q | Distance function of the impact function |

q_{0} | Trigger distance of the impact function |

v | Piston velocity |

A_{p} | Piston area |

A_{gap} | The gap area between the master cylinder and piston |

A_{n} | Cross-section area of the master cylinder |

D | Diameter of piston |

D | Diameter of piston rod |

F_{a} | Force of auxiliary strut acting at point C |

F_{c} | Controllable damping force of MRF damper |

F_{p} | Force of primary strut acting at point A |

F_{u} | Uncontrollable damping force of MRF damper |

F_{Ni} | Air-spring force caused by the accumulator |

K_{entry} | Local resistance coefficient of the entry |

K_{entry} | Local resistance coefficient of the exit |

H_{1} | Vertical distance between the origin and point A |

H_{2} | Vertical distance between the mass center P1 of elastic parts and point A |

L | Length of coils |

R | Horizontal distance between the origin of rocket coordinate system and point C |

R_{1} | Highest rocket acceleration |

R_{2} | Greatest compression stroke |

R_{3} | Distance between rocket and ground |

R_{4} | Mass of a set of landing gear |

R_{P1} | Initial distance between the mass center of the rocket and the ground |

V | Volume of accumulator |

V_{0} | Initial volume of accumulator |

## Appendix A

**Table A1.**Fuzzy control rules for inputs and outputs [39].

Inputs | Outputs | ||||||
---|---|---|---|---|---|---|---|

Acceleration | Jerk | alpha | beta | τ_{1} | τ_{2} | τ_{3} | τ_{4} |

B | All | All | All | Z | Z | Z | Z |

Z | Z | Z | Z | B | B | B | B |

Z | B | Z | Z | MB | MB | MB | MB |

S | Z | Z | Z | SB | SB | SB | SB |

S | B | Z | Z | M | M | M | M |

M | Z | Z | Z | SM | SM | SM | SM |

M | B | Z | Z | S | S | S | S |

Z | Z | P | Z | S | SB | MB | SB |

Z | B | P | Z | S | M | SB | M |

Z | Z | N | Z | MB | SB | S | SB |

Z | B | N | Z | SB | M | S | M |

Z | Z | Z | P | SB | S | SB | MB |

Z | B | Z | P | M | S | M | SB |

Z | Z | Z | N | SB | MB | SB | S |

Z | B | Z | N | M | SB | M | S |

Z | Z | N | P | MB | SB | SB | MB |

Z | B | N | P | SB | M | M | SB |

Z | Z | P | N | SB | MB | MB | SB |

Z | B | P | Z | M | SB | SB | M |

Z | Z | N | N | SB | MB | MB | SB |

Z | B | N | N | SB | SB | M | M |

Z | Z | P | P | SB | SB | MB | MB |

Z | B | P | P | M | M | SB | SB |

S | Z | P | Z | Z | M | SB | M |

S | B | P | Z | Z | SM | M | SM |

S | Z | N | Z | SB | M | Z | M |

S | B | N | Z | M | SM | Z | SM |

S | Z | Z | P | M | Z | M | SB |

S | B | Z | P | SM | Z | SM | M |

S | Z | Z | N | M | SB | M | Z |

S | B | Z | N | SM | M | SM | Z |

S | Z | P | P | M | M | SB | SB |

S | B | P | P | SM | SM | M | M |

S | Z | N | N | SB | SB | M | M |

S | B | N | N | M | M | SM | SM |

S | Z | P | N | M | SB | SB | M |

S | B | P | N | SM | M | M | SM |

S | Z | N | P | SB | M | M | SB |

S | B | N | P | M | SM | SM | M |

M | Z | P | Z | Z | SM | M | SM |

M | B | P | Z | Z | S | SM | S |

M | Z | N | Z | SM | Z | M | SM |

M | B | N | Z | S | Z | SM | S |

M | Z | Z | N | SM | M | SM | Z |

M | B | Z | N | S | SM | S | Z |

M | Z | Z | P | SM | Z | SM | M |

M | B | Z | N | S | SM | S | Z |

M | Z | N | N | M | M | SM | SM |

M | B | N | N | SM | SM | S | S |

M | Z | N | P | M | SM | SM | M |

M | B | N | P | SM | S | S | SM |

M | Z | P | N | SM | M | M | SM |

M | B | P | N | S | SM | SM | S |

M | Z | P | P | SM | SM | M | M |

M | B | P | P | S | S | SM | SM |

Run | Factor 1: A(θ/°) | Factor 2: B(α/°) | Factor 3: C(H_{1}/mm) | Response 1: R_{1} (a_{max}/m/s^{2}) | Response 2: R_{2} (Strokes/mm) | Response 3: R_{3} (Distance/mm) | Response 4: R_{4} (Mass/kg) |
---|---|---|---|---|---|---|---|

1 | 27 | 42 | 1800 | 23.3495 | 79.5787 | 2240 | 108.446 |

2 | 24 | 42 | 1500 | 17.7913 | 64.4743 | 1112.8 | 91.6878 |

3 | 30 | 42 | 1500 | 33.193 | 113.355 | 2113.5 | 109.83 |

4 | 27 | 42 | 1200 | 20.5145 | 82.2504 | 1448.6 | 89.412 |

5 | 30 | 53 | 1500 | 30.0239 | 46.4953 | 902.999 | 79.913 |

6 | 27 | 47.5 | 1500 | 61.8792 | 21.4722 | 1121 | 85.2949 |

7 | 27 | 47.5 | 1500 | 61.8792 | 21.4722 | 1121 | 85.2949 |

8 | 24 | 47.5 | 1800 | 69.0255 | 18.5479 | 1028.8 | 87.5495 |

9 | 24 | 47.5 | 1200 | 21.6645 | 56.0716 | 894.954 | 85.2949 |

10 | 27 | 47.5 | 1500 | 61.8792 | 21.4722 | 1121 | 85.2949 |

11 | 27 | 53 | 1800 | 29.5464 | 40.7394 | 873.652 | 82.5555 |

12 | 30 | 47.5 | 1200 | 44.3331 | 70.8267 | 1013.2 | 82.452 |

13 | 30 | 47.5 | 1800 | 36.5785 | 62.3615 | 1923.2 | 98.0229 |

14 | 27 | 53 | 1200 | 27.7053 | 43.8536 | 602.336 | 72.0231 |

15 | 24 | 53 | 1500 | 31.3973 | 95.6979 | 375.021 | 75.1617 |

16 | 27 | 47.5 | 1500 | 61.8792 | 21.4722 | 1121 | 85.2949 |

17 | 27 | 47.5 | 1500 | 61.8792 | 21.4722 | 1121 | 85.2949 |

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**Figure 3.**Membership Affiliations of inputs: (

**a**) Membership affiliation of a; (

**b**) Membership affiliation of da; (

**c**) Membership affiliations of alpha and beta.

**Figure 16.**Damper compression strokes of L

_{1}of the RLV under the highest acceleration landing condition.

Structure | Material | Relative Permeability | Conductivity (S/m) | Relative Permittivity |
---|---|---|---|---|

Cylinders | Aluminum | 1 | 3.774 × 10^{7} | 1.000022202 |

Magnetic core | AISI1010 | 500 | 6.452 × 10^{6} | 1 |

Coils | Copper | 1 | 5.998 × 10^{7} | 0.9999935542 |

Gas in the accumulator | Nitrogen | 1 | 0 | 1 |

Parameters | Value |
---|---|

Viscosity | 0.112 Pa s |

Density | 2.95 g/cm^{3} |

Solid content by Weight | 20.98% |

Maximum Yield Stress | 48 kPa |

Operating Temperature | −40∼+130 °C |

Codes | Design Parameters | Lower Limits | Upper Limits |
---|---|---|---|

A | The angle between the auxiliary strut and ground (θ) | 24° | 30° |

B | Angle between the primary strut and ground (α) | 42° | 53° |

C | The vertical distance between point A and C (H_{1}) | 1200 mm | 1800 mm |

Responses | Design Targets | Goal |
---|---|---|

R_{1} | Highest rocket acceleration (m/s^{2}) | Minimize |

R_{2} | Greatest compression stroke (mm) | Minimize |

R_{3} | Distance between rocket and ground (mm) | Maximum |

R_{4} | Mass of a set of landing gear (kg) | Minimize |

Response | Fitted Functions |
---|---|

R_{1} | R_{1} = +61.879 + 0.531A + 2.978B + 5.535C − 4.194AB − 13.779AC − 0.248BC − 8.078A^{2} − 25.670B^{2} − 10.901C^{2} |

R_{2} | R_{2} = +21.472 + 7.281A − 14.109B − 6.472C − 24.521AB + 7.265AC − 0.111BC + 24.440A^{2} + 34.093B^{2} + 6.040C^{2} |

R_{3} | R_{3} = +1121.000 + 317.666A − 520.112B + 263.320C − 118.180AB + 194.039AC − 130.021BC − 35.514A^{2} + 40.594B^{2} + 129.553C^{2} |

R_{4} | R_{4} = +87.578 + 3.815A − 11.215B + 5.924 C − 3.3477AB + 3.329AC − 2.125BC |

Parameters (Design Parameters) | Responses (Design Targets) | |||||
---|---|---|---|---|---|---|

A (θ) | B (α) | C (H_{1}) | Highest rocket acceleration | Greatest compression stroke | Distance between rocket and ground | Mass |

29.58° | 52.42° | 1800 mm | 17.79 m/s^{2} | 47.31 mm | 1286.82 mm | 85.12 Kg |

Landing Condition | Vertical Velocity | Horizontal Velocity | Pitch Angle |
---|---|---|---|

Highest acceleration | −2 m/s | 1 m/s | 0° |

Greatest compression | −2 m/s | 1 m/s | 3° |

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## Share and Cite

**MDPI and ACS Style**

Wang, C.; Chen, J.; Jia, S.; Chen, H.
Parameterized Design and Dynamic Analysis of a Reusable Launch Vehicle Landing System with Semi-Active Control. *Symmetry* **2020**, *12*, 1572.
https://doi.org/10.3390/sym12091572

**AMA Style**

Wang C, Chen J, Jia S, Chen H.
Parameterized Design and Dynamic Analysis of a Reusable Launch Vehicle Landing System with Semi-Active Control. *Symmetry*. 2020; 12(9):1572.
https://doi.org/10.3390/sym12091572

**Chicago/Turabian Style**

Wang, Chen, Jinbao Chen, Shan Jia, and Heng Chen.
2020. "Parameterized Design and Dynamic Analysis of a Reusable Launch Vehicle Landing System with Semi-Active Control" *Symmetry* 12, no. 9: 1572.
https://doi.org/10.3390/sym12091572