# 3D Soft-Landing Dynamic Theoretical Model of Legged Lander: Modeling and Analysis

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## Abstract

**:**

## 1. Introduction

**δ**:

**F**= K ×

**δ**

^{n}, where K represents the stiffness parameter, and the exponent n depends on the topological properties of the contacting surfaces; (3) the simplified dynamic bearing model in the form of the function on the force versus depth and velocity [18]. Moreover, many studies of fixed-shape (non-locomoting) objects impacting and penetrating dry granular media have revealed reaction forces that can be described by

_{p}(z) is a depth-dependent force. α is the inertial drag coefficient [19]. However, during the impact, several different mechanical phenomena can occur. Tension and shear failure, localized deformation, effects of adiabatic shear, and crack propagation are only some of the important phenomena that may occur individually or simultaneously. Then, there is no uniform view on the footpad–ground bearing model when the bearing model considers other items such as penetration speed (linear and nonlinear relation), the shape of the footpad, contact type, loading weight, etc.

## 2. Soft-Landing Model

#### 2.1. Model Definition

_{i}, b

_{i}, and c

_{i}in the No. i landing gear are the universal hinge, while the joints located at the points d

_{i}, e

_{i}and f

_{i}are the ball hinge.

_{g}-x

_{g}y

_{g}z

_{g}and O

_{si}-x

_{si}y

_{si}z

_{si}, are used to define the base reference coordinate system and the position and angle of the local lunar surface, respectively; (2) the lander coordination system, O

_{L}-x

_{L}y

_{L}z

_{L}, is used to define the structural distribution and motion state of the lander, where O

_{L}is the geometry center of the lander; (3) the No. I landing gear coordinate system, O

_{L}-x

_{ilg}y

_{ilg}z

_{ilg}, is used to define the motion and load of the No. i landing gear; (4) the No. i primary strut coordinate system with origin point a

_{i}, a

_{i}-x

_{ilg}y

_{ilg}z

_{ilg}, is used to define the position and velocity of the primary strut.

_{L}to use in Section 2.3, the touchdown point d

_{i}in the No. i landing gear to use in Section 2.5, and the origin point O

_{Lgi}of the No. i landing gear to use in Section 2.4. Moreover, Section 2.3 is used to calculate the landing gear’s velocity and length in the No. i landing gear coordinate system under the buffer and friction effect described in Section 2.6 and Section 2.7. Section 2.5 aims to calculate the binding force of the ground on the footpad, which is passed to the main body of the lander after being buffered by the landing gear.

#### 2.2. Position and Velocity Definition of the Lander

_{g}y

_{g}z

_{g}, can be denoted as follows using the calculation equations of the translation matrix in Equation (2).

_{g}-x

_{g}y

_{g}z

_{g}, can be expressed by the Jacobian matrix and the generalized velocity, $\dot{x},\dot{y},\dot{z},\dot{\psi},\dot{\theta},\dot{\varphi},{\dot{\theta}}_{1}^{i},{\dot{\theta}}_{2}^{i},{\dot{d}}_{1}^{i}$, which are as follows.

_{iLg}in the No. i landing gear coordinate system, O

_{lgi}-x

_{lgi}y

_{lgi}z

_{lgi}, can be expressed by the Jacobian matrix and the generalized coordinate velocity, ${\dot{d}}_{1}^{i},{\dot{\theta}}_{1}^{i},{\dot{\theta}}_{2}^{i}$, which is as follows:

**F**, can be obtained by the Jacobian matrix.

_{Lgi}#### 2.3. Dynamics Model of the Simplified Base Model of the Lander

**mg**, engine thrust force

**T**, and the forces transferred from the No. i landing gear ${F}_{buf}^{i}$. These force and moment matrices, F and M, acting on the lander can be denoted as follows.

#### 2.4. Dynamic Model of Landing Gear

**Φ**cannot be invalidated or deleted and is changed with the time variable during the soft landing, the

**Φ**and

_{qt}**Φ**also equal zero, thus the constraint matrix

_{tt}**γ**is listed as follows.

#### 2.5. Footpad–Ground Bearing Model

#### 2.6. Dynamic Model of the Buffers

#### 2.7. Correction Coefficient η

**η**was only used in the contact force calculation between the outer and inner tubes in the primary strut. Figure 5 illustrates the force relation among the outer, inner, and secondary struts in one landing gear. Based on the structural characteristics and the contact behavior of the primary strut, two assumptions were given: (1) the deflection and angle of each cross-section at the landing gear are consistent with the deformation coordination relationship; (2) the contact pressure p upon the outer tube is distributed in the form of a cosine function by the inner tube, p = p

_{m}× cos(θ), where p

_{m}is the maximum pressure acting on the outer tube and with the same direction as N

_{E/F}; (3) the contact angle θ between the outer and inner tubes is assumed to be π/2 since the inner diameter of the outer tube is approximately equal to the outer diameter of the inner tube. The force diagrams among the outer, inner, and secondary struts in one landing gear are shown in Figure 4.

_{O}, L

_{1}, and L

_{L}are the lengths of the different areas in one landing gear. A detailed introduction is shown in Figure 5.

## 3. Simulation and Model Validation

#### 3.1. Program Flow

_{0}= 0, define the initial values of the designed variable:

_{n}, identify whether the footpad of the No. i landing gear touched down on the relative local surface or not:

_{n}) of the lander, calculate the position and velocity matrix of point d

_{i}, ${\dot{d}}_{i}$ in the No. i local lunar surface coordinate system using the Jacobian matrix in Equation (5) and the translation matrix in Equation (2). If d

_{lociz}< 0, the transmission force of the No. i landing gear is equal to zero, then perform step 5; if touch down happens, obtain the footpad–ground bearing force vector ${\mathit{F}}_{\mathit{l}\mathit{u}\mathit{n}}^{\mathit{i}}$ using Equations (17) and (18). Based on the Jacobian matrix in Equations (7) and (8), calculate the equivalent dynamic force matrix ${F}_{\mathrm{lgi}}$ in the No. i landing gear and the component force in each strut.

_{n}, identify whether the buffer of the No. i landing gear is crushed or not:

_{n}, obtain the crushing force of the Al-honeycomb buffer of each strut using Equations (21)–(26). According to Equations (12)–(13), the transmission force in the struts and the driving force of the relative struts can be obtained. Using the driving force of the relative strut, the dynamic model of one landing gear is calculated using Equations (14)–(16). In addition, using the transmission force of the relative strut, the forces and moments acting on the lander can be obtained from the No. i landing gear using Equation (11). When the equivalent dynamic force is no less than the defined crush force of the Al-honeycomb buffer of the strut, the buffer begins to crush. The transmission force in the strut is equal to the crushing force of the relative strut. Moreover, the driving force in the relative strut is equal to the remaining force. However, if the equivalent force is less than the crushing force, the buffer does not crush. The driving force in the relative strut is equal to zero, and the transmission force in the strut in the No. i landing gear is equal to the equivalent dynamic force.

_{n}, calculate the dynamic model of the landing gear i and check the constraint stabilization:

_{n}, calculate the dynamic model of the lander:

_{n+}

_{1}) of the lander in the lander coordinate system in turn using the direct integration. Then, based on the transient matrix theory, calculate the position and velocity value of the lander in the global coordinate system.

#### 3.2. Program Flow

## 4. Discussions

#### 4.1. Different Kinds of the Footpad–Ground Bearing Models

_{s}= u

_{d}= 0.4.

#### 4.2. Friction Analysis

#### 4.3. Correction Coefficient η

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${}_{\mathit{i}}{}^{\mathit{i}-1}T$ | = | The transformation matrix from the No. i-1 coordinate system to the No. i coordinate system. |

$\mathit{I}$ | = | Unit matrix. |

${R}_{i-1,i}$ | = | Rotation matrix from the No. i-1 coordinate system to the No. i coordinate system in terms of Z-Y-X axis in the Euler angle way $\psi ,\theta ,\phi $. ${R}_{i-1,i}={R}_{Z}\left(\psi \right){R}_{{Y}^{\prime}}\left(\theta \right){R}_{{X}^{\u2033}}\left(\phi \right)$; ${R}_{Z}\left(\psi \right)=\left[\begin{array}{cc}\begin{array}{cc}\begin{array}{c}c\psi \\ s\psi \end{array}& \begin{array}{c}-s\psi \\ c\psi \end{array}\end{array}& \begin{array}{c}0\\ 0\end{array}\\ \begin{array}{cc}0& 0\end{array}& 1\end{array}\right]$; ${R}_{Y}\left(\theta \right)=\left[\begin{array}{cc}\begin{array}{cc}\begin{array}{c}c\theta \\ 0\end{array}& \begin{array}{c}0\\ 1\end{array}\end{array}& \begin{array}{c}-s\theta \\ 0\end{array}\\ \begin{array}{cc}s\theta & 0\end{array}& c\theta \end{array}\right]$; ${R}_{X}\left(\phi \right)=\left[\begin{array}{cc}\begin{array}{cc}\begin{array}{c}1\\ 0\end{array}& \begin{array}{c}0\\ c\phi \end{array}\end{array}& \begin{array}{c}0\\ -s\phi \end{array}\\ \begin{array}{cc}0& -s\phi \end{array}& c\phi \end{array}\right];s\phi =sin\phi ;c\phi =cos\phi $; |

${P}_{i}$ | = | Translation matrix relative to the No. i coordinate system. ${P}_{i}=\left[{x}_{i};{y}_{i};{z}_{i}\right]$ |

${l}_{{a}_{iLgi}}^{{O}_{LLgi}}$ | = | Translation vector $\overrightarrow{{O}_{L}{a}_{i}}$ relative to the No. i landing gear coordinate system. |

${l}_{{b}_{iLgi}}^{{O}_{LLgi}}$ | = | Translation vector $\overrightarrow{{O}_{L}{b}_{i}}$ relative to the No. i landing gear coordinate system. |

${l}_{{c}_{iLgi}}^{{O}_{LLgi}}$ | = | Translation vector $\overrightarrow{{O}_{L}{c}_{i}}$ relative to the No. i landing gear coordinate system. |

${\theta}_{1}^{i}$ | = | yaw(z) angle of the primary strut in the Euler angle form under the No. i landing gear coordinate system, a_{iLg}-x_{iLg}y_{iLg}z_{iLg}. |

${\theta}_{2}^{i}$ | = | pitch(y) angle of the primary strut in the Euler angle form under the No. i landing gear coordinate system, a_{iLg}-x_{iLg}y_{iLg}z_{iLg}. |

${d}_{1}^{i}$ | = | x-component value in the No. i primary strut coordinate system, a_{iLg}-x_{iLg}y_{iLg}z_{iLg}. |

${\theta}_{b1}^{i},{\theta}_{c1}^{i}$ | = | The yaw(z) angle of the primary strut in the Euler angle form under the No. i landing gear coordinate system, b_{iLg}-x_{iLg}y_{iLg}z_{iLg} and c_{iLg}-x_{iLg}y_{iLg}z_{iLg}_{,} respectively. |

${\theta}_{b2}^{i},{\theta}_{c2}^{i}$ | = | pitch(y) angle of the primary strut in the Euler angle form under the No. i landing gear coordinate system, b_{iLg}-x_{iLg}y_{iLg}z_{iLg} and c_{iLg}-x_{iLg}y_{iLg}z_{iLg,} respectively. |

${d}_{b1}^{i},{d}_{c1}^{i}$ | = | x-component value in the No. i secondary strut coordinate system, b_{i}-x_{ilg}y_{ilg}z_{ilg} and c_{i}-x_{ilg}y_{ilg}z_{ilg}, respectively. |

${\nu}_{{M}_{L}}^{{O}_{g}},{\nu}_{{d}_{i}}^{{O}_{g}},{\nu}_{{e}_{i}}^{{O}_{g}},{\nu}_{{f}_{i}}^{{O}_{g}}$ | = | The velocity matrix of the points M_{L}, d_{i}, e_{i}, f_{i} in the global coordinate system, O_{g}-x_{g}y_{g}z_{g.} |

$J,{J}_{1},{J}_{2}$ | = | The Jacobian matrices for different mapping relationships. |

F_{Lgi} | = | The equivalent dynamic force matrix in the No. i landing gear coordinate system, O_{Lgi}-x_{Lgi}y_{Lgi}z_{Lgi}. |

F_{g} | = | The footpad–ground force matrix_{.} |

F_{XLgi} | = | The x component of the footpad–ground force in the No. i landing gear coordinate system, O_{Lgi}-x_{Lgi}y_{Lgi}z_{Lgi}. |

F_{YLgi} | = | The y component of the footpad–ground force in the No. i landing gear coordinate system, O_{Lgi}-x_{Lgi}y_{Lgi}z_{Lgi}. |

F_{ZLgi} | = | The z component of the footpad–ground force in the No. i landing gear coordinate system, O_{Lgi}-x_{Lgi}y_{Lgi}z_{Lgi}. |

${F}_{pri}^{i}$ | = | The equivalent dynamic forces in the primary strut of the No. i landing gear coordinate system. |

${F}_{\mathrm{sec}\_\mathrm{L}}^{\mathrm{i}}$ | = | The equivalent dynamic forces in the left secondary strut of the No. i landing gear coordinate system. |

${F}_{\mathrm{sec}\_\mathrm{R}}^{\mathrm{i}}$ | = | The equivalent dynamic forces in the right secondary strut of the No. i landing gear coordinate system. |

$\dot{u},\mathrm{v}\dot{},\dot{w}$ | = | x, y, z component of the translation acceleration vector of mass center of the lander in the lander coordinate system. |

$\dot{p},\mathrm{q}\dot{},\dot{r}$ | = | x, y, z component of the angle acceleration vector of mass center of the lander in the lander coordinate system. |

$\dot{x},\mathrm{y}\dot{},\dot{z}$ | = | x, y, z component of the translation velocity vector of mass center of the lander in the global coordinate system. |

$\dot{\psi},\dot{\theta},\dot{\varphi}$ | = | The Euler rates in terms of Euler angles (Z-Y-X) from the global coordinate system to the lander coordinate system. |

${F}_{x},{F}_{y},{F}_{z}$ | = | The x, y, z component of the force acting on the mass center of the lander in the lander coordinate system. |

${M}_{x},{M}_{y},{M}_{z}$ | = | The x, y, z component of the moment acting on the mass center of the lander in the lander coordinate system. |

${H}_{x},{H}_{y},{H}_{z}$ | = | The x, y, z scale component for the moment of momentum in the lander coordinate system. |

${I}_{x},{I}_{y},{I}_{z}$ | = | Mass moments of inertia of the lander about x, y, and z axes in the lander coordinate system. |

${I}_{xy},{I}_{yz},{I}_{xz}$ | = | The products of the inertia of the lander in the lander coordinate system. |

${N}_{pri}^{i}$ | = | The transmission force in the primary strut of the No. i landing gear coordinate system. |

${N}_{\mathrm{sec}\_\mathrm{L}}^{\mathrm{i}}$ | = | The transmission force in the left secondary strut of the No. i landing gear coordinate system. |

${N}_{\mathrm{sec}\_\mathrm{R}}^{\mathrm{i}}$ | = | The transmission force in the right secondary strut of the No. i landing gear coordinate system. |

${F}_{pri\_crush}$ | = | The crushing force of buffer in the primary strut. |

${F}_{Ten\_crush\_\mathrm{sec}}$ | = | The tensile crushing force of buffer in the secondary strut. |

${F}_{C\mathrm{om}\_crus\mathrm{h}\_\mathrm{sec}}$ | = | The compression crushing force of buffer in the secondary strut. |

${F}_{driving\_pri}^{i}$ | = | The remaining driving force in the primary strut of the No. i landing gear coordinate system. |

${F}_{driving\_\mathrm{sec}\_\mathrm{L}}^{i}$ | = | The remaining driving force in the left secondary strut of the No. i landing gear coordinate system. |

${F}_{driving\_\mathrm{sec}\_\mathrm{R}}^{i}$ | = | The remaining driving force in the left secondary strut of the No. i landing gear coordinate system. |

M | = | Generalized mass matrix of each landing gear system. |

$q$ | = | Generalized coordination vectors matrix, ${\left[\begin{array}{ccccccccc}{d}_{1}^{i}& {\theta}_{1}^{i}& {\theta}_{2}^{i}& {d}_{b1}^{i}& {\theta}_{b1}^{i}& {\theta}_{b2}^{i}& {d}_{c1}^{i}& {\theta}_{c1}^{i}& {\theta}_{c2}^{i}\end{array}\right]}^{T}$ |

$\ddot{q}$ | = | The generalized acceleration vectors matrix, ${\left[\begin{array}{ccccccccc}\ddot{{d}_{1}^{i}}& \ddot{{\theta}_{1}^{i}}& \ddot{{\theta}_{2}^{i}}& \ddot{{d}_{b1}^{i}}& \ddot{{\theta}_{b1}^{i}}& \ddot{{\theta}_{b2}^{i}}& \ddot{{d}_{c1}^{i}}& \ddot{{\theta}_{c1}^{i}}& \ddot{{\theta}_{c2}^{i}}\end{array}\right]}^{T}$ |

Φ_{q} | = | The Jacobi matrix of the constraint equation (Equation (7)). |

λ | = | Lagrange multiplier column matrix. |

γ | = | Constraint matrix. |

${Q}_{driving\_pri}^{i}$ | = | The remaining driving force matrix in the generalized coordinate system. |

F_{lun} | = | Footpad–ground contact force. |

n | = | Exponential coefficient of the penetration depth. |

µ | = | Frictional coefficient of the contact force. |

K_{g}, C_{g} | = | Penetration stiffness and damping coefficient of the footpad–ground model. |

${d}_{\mathrm{locix}},{d}_{\mathrm{lociy}},{d}_{\mathrm{lociz}}$ | = | x, y, z component of the displacement of the No. i footpad in the lunar surface coordinate system. |

${\dot{d}}_{\mathrm{locix}},{\dot{d}}_{\mathrm{lociy}},{\dot{d}}_{\mathrm{lociz}}$ | = | x, y, z component of the translation velocity of the No. i footpad in the lunar surface coordinate system. |

D, D_{1}, D_{2} | = | Defined displacement parameters in contact model. |

$\kappa $ | = | Contact parameter. |

$\ddot{s}$,$\mathrm{s}\dot{}$,$\mathrm{s}$ | = | Crushing acceleration, velocity, and displacement of the buffer. |

a, b, c, d | = | Coefficients of inertia, viscosity, stiffness, and constant resistance of the buffer. |

F_{1}, F_{2} | = | Crush forces of the first and second step of the buffer in the primary strut. |

F_{3} | = | Transmission force after the AL-honeycomb is crushed. |

s_{0} | = | Initial length of the primary strut. |

s_{1}, s_{2} | = | Lengths of the primary strut when the first and second step foam crush are done. |

s_{hismin} | = | Minimum length of the primary strut before the currently measured time. |

s_{c1}, s_{t1} | = | Length value of the secondary strut when the compaction buffer or the tension buffer crush are done. |

F_{com}, F_{Ten} | = | Compaction and tension force of the buffer in the secondary struts. |

F_{com1}, F_{Ten1} | = | Compaction and tension transmission force of the buffer in the secondary struts after the AL-honeycomb is crushed. |

${s}_{his\mathrm{max}}^{\mathrm{s}}$ | = | Maximum length of the secondary strut before the currently measured time. |

${s}_{his\mathrm{min}}^{\mathrm{s}}$ | = | Minimum length of the secondary strut before the currently measured time. |

${N}_{E/F}^{i}$ | = | Total normal contact force acted upon the No. i primary strut by the relative secondary struts. |

${F}_{Ycon}^{i}$ | = | y component of the contact force matrix in the No. i primary strut coordinate system. |

${F}_{Zcon}^{i}$ | = | z component of the contact force matrix in the No. i primary strut coordinate system. |

${F}_{con}^{i}$ | = | Contact force matrix acted upon the No. i primary strut and by the secondary struts in the No. i primary strut coordinate system, d_{i}-x_{ilg}y_{ilg}z_{ilg.} |

η | = | Correction coefficient of the contact between the outer and inner tube in the primary strut. |

µ_{1} | = | Friction factor of the contact between the outer and inner tube in the primary strut. |

η_{1} | = | Correction factor accounts for the changing length of the landing gear during the soft-landing process. |

η_{2} | = | Correction factor takes into consideration the effect on the contact pressure distribution. |

N_{A} | = | Lateral force caused by the upper contact action of the tubes. |

N_{B} | = | Lateral force caused by the below contact action of the tubes. |

μ_{k} | = | Kinetic coefficient of friction. |

F_{C} | = | The magnitude of friction force. |

F_{S} | = | Static friction force. |

F_{D} | = | Dynamic friction force. |

F_{e} | = | External tangential force. |

v | = | The relative tangential velocity of the contacting surfaces. |

v_{o} | = | Stiction velocity. |

p | = | Contact pressure. |

p_{m} | = | Maximum contact pressure. |

${\theta}_{c}$ | = | Contact angle between the outer and inner tube in the primary strut. |

l | = | Arc length of the contact between the outer and inner tube in the primary strut. |

l_{R} | The inner radius of the outer tube in the primary strut. | |

${\Phi}_{i}$ | = | Constraint function in the No. i landing gear. |

${\widehat{q}}^{n+1},{\widehat{\dot{q}}}^{n+1}$ | = | Generalized coordinate displacement and velocity by numerical integration at time step t^{n+1}. |

$\Delta {q}^{n+1},\Delta {\dot{q}}^{n+1}$ | Correction displacement and velocity item at time step t^{n+1}. | |

Subscripts | ||

Soft landing | Any type of spacecraft landing that does not result in significant damage to or destruction of the vehicle or its payload. | |

MC | Center of mass. | |

C.L. | Coordinate location. | |

No. | Number order. | |

C.S. | Coordinate system. | |

L.G. | Landing gear. | |

LC | Load case. | |

Pri. Strut | Primary strut. | |

Sec. Strut | Secondary strut. |

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**Figure 1.**A soft-landing dynamic model of a legged lander. (

**a**) A simplified soft-landing dynamic model; (

**b**) No. i landing gear (L.G.) sketch description. Red arrows are for coordinate system. Blue arrows are for labelling.

**Figure 2.**The translation relations of the coordinate systems used in the soft-landing dynamic model.

**Figure 4.**The free-body diagram of the legged lander. (

**a**) The simplified base model of the lander; (

**b**) No. 1 landing gear (L.G.). Red arrows are for the coordinate system. Black arrows are for the force. Blue arrows are for Blue arrows is for labelling.

**Figure 5.**Force diagrams among the outer, inner, and secondary struts in one landing gear: (

**a**) force diagram between the outer and inner tube; (

**b**) force diagram between the outer and inner tube in cross-section.

**Figure 8.**The error data of the crushing length and the crushing length of each strut of the lander during soft landing under four typical cases. (

**a**) load case−1 results; (

**b**) load case−2 results; (

**c**) load case−3 results; (

**d**) load case−4 results.

**Figure 9.**The stroke of each strut of the lander during landing. (

**a**) The primary strut of the lander in LC-1; (

**b**) the secondary strut of the lander in LC-1; (

**c**) the primary strut of the lander in LC-2; (

**d**) the secondary strut of the lander in LC-2; (

**e**) the primary strut of the lander in LC-3; (

**f**) the secondary strut of the lander in LC-3; (

**g**) the primary strut of the lander in LC-4; (

**h**) the secondary strut of the lander in LC-4.

Mass Property | C.L. of Points | Crush Force of Buffers | Contact Property | ||||||
---|---|---|---|---|---|---|---|---|---|

Pri. Strut | Sec. Strut | ||||||||

I_{x} | 8.18 × 10^{4} | ${a}_{1}$ | (3.89, 0, −2.18) | F_{1} | 78,000 | F_{com} | 43,000 | K_{g} | (1000)^{1.5}×10 ^{5} |

I_{y} | 8.18 × 10^{4} | ${b}_{1}$ | (3.37, 1.23, −1.01) | F_{2} | 156,000 | F_{ten} | 30,000 | C_{g} | 10,000 |

I_{z} | 8.17 × 10^{4} | ${c}_{1}$ | (3.37, −1.23, −1.01) | ${s}_{1}$ | 0.2 | ${s}_{{t}_{1}}$ | 0.27 | ${D}_{1}$ | 0.0001 |

Mass | 1.6 × 10^{4} | ${d}_{1}$ | (5.48, 0, 0.94) | ${s}_{2}$ | 0.475 | ${s}_{{c}_{1}}$ | 0.27 | u_{s} | 0.4/0.4 |

${e}_{1}$ | (4.65, 1.09, −0.505) | ${u}_{\mathit{d}}$ | 0.1 | ||||||

${f}_{1}$ | (4.65, −1.09, 0.505) | ${u}_{1}$ | 0.1 | ||||||

MC | (0, 0, −3.818) | η | 0 | ||||||

v_{s} | 0.1 | ||||||||

v_{d} | 1 |

^{2}; (2) Kg, Cg, ${D}_{1}$, and u

_{s}, ${u}_{\mathit{d}}$, v

_{s}, ${v}_{\mathit{d}}$ are the default values in MSC Adams.

Load Cases | Velocity m/s | Attitude Angles /Deg | Lunar Surface Slope Angle /Deg | |
---|---|---|---|---|

Vertical | Horizontal | |||

LC-1 | 4 | −0 | 0/0/0 | 0 |

LC-2 | 4 | 1 | 45/0/0 | 0 |

LC-3 | 4 | 1 | 45/−4/0 | 8 |

LC-4 | 4 | 1 | 0/4/0 | 8 |

No. | LC-1 | LC-2 | LC-3 | LC-4 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Theory | Adams | Dev | Theory | Adams | Dev | Theory | Adams | Dev | Theory | Adams | Dev | |

1 ^{1} | 287.0 | 288.8 | −1.8 | 314.3 | 319 | −4.7 | 360.3 | 354.1 | 9.8 | 390.0 | 385.0 | 5.0 |

2 ^{2} | 49.8 | 50.3 | −0.5 | 3.9 | 5.1 | −1.2 | −112.1 | −132.2 | 19.1 | 4.4 | 5.0 | −0.6 |

3 ^{3} | 49.8 | 50.7 | −0.5 | 72.6 | 74.5 | −1.9 | 77.0 | 74.2 | 3.8 | 4.4 | 5.1 | −0.7 |

4 ^{4} | 287.0 | 288.7 | −0.7 | 244.0 | 240.0 | 4.0 | 255.3 | 252.7 | 2.6 | 280.6 | 261.0 | 19.6 |

5 ^{5} | 49.8 | 50.0 | −0.2 | 41.2 | 43.5 | −1.8 | 46.1 | 49.4 | −3.3 | 7.3 | 7.3 | 0 |

6 ^{6} | 49.8 | 50.3 | −0.5 | 113.7 | 119.7 | 6.0 | 132.3 | 135.6 | 3.3 | 151.9 | 164.4 | −13.5 |

7 ^{7} | 287 | 289.2 | 2.0 | 244.0 | 242.9 | 1.1 | 255.4 | 251.2 | 4.2 | 277.9 | 261.0 | 16.9 |

8 ^{8} | 49.8 | 50.6 | −0.8 | 113.7 | 121.4 | −7.7 | 132.6 | 135.6 | 3.0 | 98.3 | 121.9 | −20.9 |

9 ^{9} | 49.8 | 50.7 | −0.9 | 41.2 | 45.5 | −4.3 | 46.0 | 49.4 | 3.0 | 98.3 | 121.5 | −20.9 |

10 ^{10} | 287.0 | 287.5 | 2.0 | 314.2 | 319.5 | −5.3 | 360.3 | 354.0 | 6.3 | 280.6 | 261.6 | 19.0 |

11 ^{11} | 49.8 | 51.7 | −1.9 | 72.6 | 75.1 | −2.5 | 77.0 | 74.5 | 2.5 | 151.9 | 165.0 | −14.1 |

12 ^{12} | 49.8 | 51.7 | −1.9 | 3.9 | 5.2 | −1.3 | −114.7 | −132.2 | 18.5 | 7.3 | 7.3 | 0 |

^{1}No. 1 measure option: the crush length of each Al-honeycomb in the primary strut in the No. 1 landing gear.

^{2}No. 2 measure option: the crush length of each Al-honeycomb in the secondary strut (left) in the No. 1 landing gear.

^{3}No. 3 measure option: the crush length of each Al-honeycomb in the secondary strut (right) in the No. 1 landing gear.

^{4}No. 4 measure option: the crush length of each Al-honeycomb in the primary strut in the No. 2 landing gear.

^{5}No. 5 measure option: the crush length of each Al-honeycomb in the secondary strut (left) in the No. 2 landing gear.

^{6}No. 6 measure option: the crush length of each Al-honeycomb in the secondary strut (right) in the No. 2 landing gear.

^{7}No. 7 measure option: the crush length of each Al-honeycomb in the primary strut in the No. 3 landing gear.

^{8}No. 8 measure option: the crush length of each Al-honeycomb in the secondary strut (left) in the No. 3 landing gear.

^{9}No. 9 measure option: the crush length of each Al-honeycomb in the secondary strut (right) in the No. 3 landing gear.

^{10}No. 10 measure option: the crush length of each Al-honeycomb in the primary strut in the No. 4 landing gear.

^{11}No. 11 measure option: the crush length of each Al-honeycomb in the secondary strut (left) in the No. 4 landing gear.

^{12}No. 12 measure option: the crush length of each Al-honeycomb in the secondary strut (right) in the No. 4 landing gear.

Contact Model Types | Equation | Contact Model | Equation |
---|---|---|---|

Hertz contact | $F=K{\delta}^{n}$ | Hertz contact + linear damping | $F=K{\delta}^{n}+D\dot{\delta}$ |

Hertz contact + step damping | $F=\left\{\begin{array}{cc}K{\delta}^{n}+D\dot{\delta}& \delta \le d\\ K{\delta}^{n}& \delta \ge d\end{array}\right.$ | Hertz contact + bilinear damping | $F=K{\delta}^{n}+{D}_{1}\dot{\delta}+{D}_{2}{\dot{\delta}}^{2}$ |

Hertz contact + hysteresis damping ^{1} | $F=K{\delta}^{n}+\chi {\delta}^{{}^{n}}\dot{\delta}$ |

^{1}$\chi =\frac{3\left(1-{c}_{r}\right)}{2{\dot{\delta}}^{\left(-\right)}}K,{c}_{r}=1-\alpha {\dot{\delta}}^{\left(-\right)}.$ $\alpha $ is a constant value ranging between 0.008 and 0.32.

Model Name ^{1} | Model Type | Parameter Defined in the Model |
---|---|---|

Hertz contact | Hertz contact | K = (1000) ^{1.5} × 1.0 × 10^{5}, n = 1.5 |

Kelvin–Voigt model | Hertz contact + linear damping factor | K = (1000)^{1.5} × 1.0 × 10^{5}, D = 10 × 1000, n = 1.5 |

Kelvin–Voigt 1 | Hertz contact + step damping factor | K = (1000)^{1.5} × 1.0 × 10^{5}, D = 10 × 1000, n = 1.5 |

Kelvin–Voigt 2 | Hertz contact + bilinear damping factor | K = (1000)^{1.5} × 1.0 × 10^{5}, D_{1} = 10 × 1000, D_{2} = 10 × 1000, n = 1.5 |

Hunt–Crossley | Hertz contact + hysteresis damping factor | K = (1000)^{1.5} × 1.0 × 10^{5}, n = 1.5 |

^{1}The hertz contact + step damping factor model and the Hertz contact + bilinear damping factor model are the contact models with relation to velocity options, which are similar to the Kelvin–Voigt model. To clearly express the difference between the contact models, the names Kelvin–Voigt 1 and Kelvin–Voigt 2 are defined as the two contact models, respectively.

Info. | No. | Base Result | Hertz | Kelvin–Voigt | Kelvin–Voigt 1 | Kelvin–Voigt 2 | Hunt–Crossley |
---|---|---|---|---|---|---|---|

LC-1 | 1 ^{1} | 287.0 | 288.8 | 287.3 | 288.8 | 287.0 | 289.8 |

2 ^{2} | 49.8 | 50.5 | 49.8 | 50.5 | 49.8 | 50.7 | |

3 ^{3} | 49.8 | 50.5 | 49.8 | 50.5 | 49.8 | 50.7 | |

LC-2 | 4 ^{4} | 314.0 | 312.9 | 314.1 | 311.4 | 314.0 | 317.7 |

5 ^{5} | 3.9 | 6.3 | 4.0 | 6.4 | 3.9 | 12.1 | |

6 ^{6} | 41.1 | 41.6 | 41.2 | 41.8 | 41.1 | 41.6 | |

7 ^{7} | 244.0 | 244.5 | 244.1 | 244.9 | 244.0 | 244.1 | |

8 ^{8} | 113.6 | 112.7 | 113.9 | 114.0 | 113.6 | 109.1 | |

9 ^{9} | 73.4 | 74.2 | 72.7 | 75.1 | 73.4 | 74.0 |

^{1}No. 1 measure option: the crush length of each Al-honeycomb in the primary strut in the No. 1 landing gear.

^{2}No. 2 measure option: the crush length of each Al-honeycomb in the secondary strut (left) in the No. 1 landing gear.

^{3}No. 3 measure option: the crush length of each Al-honeycomb in the secondary strut (right) in the No. 1 landing gear.

^{4}No. 4 measure option: the crush length of each Al-honeycomb in the primary strut in the No. 1/4 landing gear.

^{5}No. 5 measure option: the crush length of each Al-honeycomb in the secondary strut (left) in the No. 1 landing gear.

^{6}No. 6 measure option: the crush length of each Al-honeycomb in the secondary strut (left) in the No. 2 landing gear.

^{7}No. 7 measure option: the crush length of each Al-honeycomb in the primary strut in the No. 2/3 landing gear.

^{8}No. 8 measure option: the crush length of each Al-honeycomb in the secondary strut (left) in the No. 3 landing gear.

^{9}No. 9 measure option: the crush length of each Al-honeycomb in the secondary strut (left) in the No. 4 landing gear.

Friction Model Types ^{1} | Calculation Equations ^{2} | The Change Rule Figures |
---|---|---|

The static Coulomb friction model | ${F}_{C}=\left\{\begin{array}{cc}{u}_{s}N\mathrm{sgn}(v)& v\ne 0\\ \mathrm{min}({F}_{e},{u}_{s}N)\mathrm{sgn}(v)& v=0\end{array}\right.$ | |

The static Coulomb friction model with viscous effect | ${F}_{C}=\left\{\begin{array}{cc}{u}_{s}N\mathrm{sgn}(v)+{C}_{v}v& v\ne 0\\ \mathrm{min}({F}_{e},{u}_{s}^{}N)\mathrm{sgn}(v)& v=0\end{array}\right.$ | |

The regularized Coulomb friction model | ${F}_{C}=\left\{\begin{array}{cc}{u}_{s}N\left(\frac{1-{e}^{-\frac{3\left|v\right|}{{v}_{0}}}}{1-{e}^{-3}}\right)\mathrm{sgn}(v)& \left|v\right|\le {v}_{0}\\ {u}_{s}N\mathrm{sgn}(v)& \left|v\right|>{v}_{0}\end{array}\right.$ | |

The stiction + dynamic Coulomb model | ${F}_{C}=\left\{\begin{array}{cc}{u}_{s}N\mathrm{sgn}(v)& v\ne 0\\ \mathrm{min}({F}_{e},{u}_{d}N)\mathrm{sgn}(v)& v=0\end{array}\right.$ | |

The stiction + Stribeck + Coulomb + viscous friction model | ${F}_{f}=\left({F}_{D}^{}+\left({F}_{S}^{}-{F}_{D}^{}\right){e}^{-{\left(\frac{\left|v\right|}{{v}_{s}}\right)}^{\delta}}\right)\mathrm{sgn}(v)$ | |

The stiction + modified Stribeck + Coulomb + viscous friction model | ${F}_{f}=\left\{\begin{array}{cc}\left(-\frac{{F}_{S}}{{v}_{0}^{2}}{\left(\left|v\right|-{v}_{0}\right)}^{2}+{F}_{S}\right)\mathrm{sgn}(v)& \left|v\right|<{v}_{0}\\ \left({F}_{C}^{}+\left({F}_{S}^{}-{F}_{C}^{}\right){e}^{-\epsilon {\left(\frac{\left|v\right|}{{v}_{s}}\right)}^{\delta}}\right)\mathrm{sgn}(v)& \left|v\right|\ge {v}_{0}\end{array}\right.$ |

^{1}Stribeck effect is the effect that describes the decrease in the friction force as the relative tangential velocity increases; the viscous effect can be established as a proportion of the relative tangential velocity of the sliding surfaces, F = u × v; the stiction effect is the manifestation that static friction is greater than dynamic friction mechanics;

^{2}v

_{o}is the stiction velocity, and when v is no greater than v

_{o}, the static force is the main force in friction force.

No. | Friction Model | Parameter |
---|---|---|

TY-1 | Static Coulomb friction | ${u}_{s}$ = 0.4 |

TY-2 | Static Coulomb friction model with viscous effect | ${u}_{s}$ = 0.4; ${C}_{v}$ = 100 |

TY-3 | Regularized Coulomb friction | ${v}_{0}$ = 0.1 m/s; ${u}_{s}$ = 0.4 |

TY-4 | Stiction + dynamic Coulomb | ${u}_{d}$ = 0.4 |

TY-5 | Stiction + Stribeck + Coulomb + viscous friction | ${v}_{s}$ = 1 m/s; $\delta =0.85$ |

TY-6 | Stiction + modified Stribeck + Coulomb + viscous friction | ${v}_{0}={v}_{s}=\frac{0.1m}{s};\epsilon =0.85$ $\delta =0.85$ |

Measure Option | Base Result | TY-1 | TY-2 | TY-3 | TY-4 | TY-5 | TY-6 | |
---|---|---|---|---|---|---|---|---|

LC-1 | 1 ^{1} | 287.0 | 295.1 | 295.0 | 290.6 | 295.1 | 295.1 | 290.2 |

2 ^{2} | 49.8 | 48.4 | 48.4 | 49.3 | 48.4 | 48.4 | 49.5 | |

3 ^{3} | 49.8 | 48.4 | 48.4 | 49.3 | 48.4 | 48.4 | 49.5 | |

LC-2 | 4 ^{4} | 314.0 | 368.3 | 324.4 | 316.0 | 368.3 | 323.7 | 313.4 |

5 ^{5} | 4.0 | 10.2 | 7.4 | 7.2 | 10.2 | 7.4 | 6.8 | |

6 ^{6} | 41.1 | 39.6 | 39.8 | 40.9 | 39.6 | 39.8 | 41.2 | |

7 ^{7} | 244.0 | 252.7 | 249.0 | 245.3 | 252.7 | 249.1 | 245.5 | |

8 ^{8} | 113.6 | 92.4 | 106.0 | 109.9 | 92.4 | 107.0 | 111.8 | |

9 ^{9} | 73.4 | 62.0 | 71.8 | 75.7 | 62.0 | 73.2 | 76.4 |

^{1}No. 1 measure option: the crush length of each Al-honeycomb in the primary strut in the No. 1 landing gear.

^{2}No. 2 measure option: the crush length of each Al-honeycomb in the secondary strut (left) in the No. 1 landing gear.

^{3}No. 3 measure option: the crush length of each Al-honeycomb in the secondary strut (right) in the No. 1 landing gear.

^{4}No. 4 measure option: the crush length of each Al-honeycomb in the primary strut in the No. 1/4 landing gear.

^{5}No. 5 measure option: the crush length of each Al-honeycomb in the secondary strut (left) in the No. 1 landing gear.

^{6}No. 6 measure option: the crush length of each Al-honeycomb in the secondary strut (left) in the No. 2 landing gear.

^{7}No. 7 measure option: the crush length of each Al-honeycomb in the primary strut in the No. 2/3 landing gear.

^{8}No. 8 measure option: the crush length of each Al-honeycomb in the secondary strut (left) in the No. 3 landing gear.

^{9}No. 9 measure option: the crush length of each Al-honeycomb in the secondary strut (left) in the No. 4 landing gear.

Info. | Measure Options | $\mathbf{Without}\mathit{\eta}$/mm | $\mathbf{Within}\mathit{\eta}$/mm | Error/% |
---|---|---|---|---|

LC-1 | 1 ^{1} | 287.0 | 276.9 | 3.6 |

2 ^{2} | 49.8 | 48.2 | 3.3 | |

3 ^{3} | 49.8 | 48.2 | 3.3 | |

LC-2 | 4 ^{4} | 314.0 | 300.9 | 4.4 |

5 ^{5} | 4.0 | 6.6 | −39.0 | |

6 ^{6} | 41.1 | 40.0 | 2.7 | |

7 ^{7} | 243.7 | 235.2 | 3.6 | |

8 ^{8} | 113.6 | 109.9 | 3.4 | |

9 ^{9} | 73.4 | 73.3 | 0.1 |

^{1}No. 1 measure option: the crush length of each Al-honeycomb in the primary strut in the No. 1 landing gear.

^{2}No. 2 measure option: the crush length of each Al-honeycomb in the secondary strut (left) in the No. 1 landing gear.

^{3}No. 3 measure option: the crush length of each Al-honeycomb in the secondary strut (right) in the No. 1 landing gear.

^{4}No. 4 measure option: the crush length of each Al-honeycomb in the primary strut in the No. 1/4 landing gear.

^{5}No. 5 measure option: the crush length of each Al-honeycomb in the secondary strut (left) in the No. 1 landing gear.

^{6}No. 6 measure option: the crush length of each Al-honeycomb in the secondary strut (left) in the No. 2 landing gear.

^{7}No. 7 measure option: the crush length of each Al-honeycomb in the primary strut in the No. 2/3 landing gear.

^{8}No. 8 measure option: the crush length of each Al-honeycomb in the secondary strut (left) in the No. 3 landing gear.

^{9}No. 9 measure option: the crush length of each Al-honeycomb in the secondary strut (left) in the No. 4 landing gear.

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© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, Z.; Chen, C.; Chen, J.; Zheng, G.
3D Soft-Landing Dynamic Theoretical Model of Legged Lander: Modeling and Analysis. *Aerospace* **2023**, *10*, 811.
https://doi.org/10.3390/aerospace10090811

**AMA Style**

Wang Z, Chen C, Chen J, Zheng G.
3D Soft-Landing Dynamic Theoretical Model of Legged Lander: Modeling and Analysis. *Aerospace*. 2023; 10(9):811.
https://doi.org/10.3390/aerospace10090811

**Chicago/Turabian Style**

Wang, Zhiyi, Chuanzhi Chen, Jinbao Chen, and Guang Zheng.
2023. "3D Soft-Landing Dynamic Theoretical Model of Legged Lander: Modeling and Analysis" *Aerospace* 10, no. 9: 811.
https://doi.org/10.3390/aerospace10090811