Landing control method of a lightweight fourlegged landing and walking robot
Ke YIN, Chenkun QI, Yue GAO, Qiao SUN, Feng GAO
Landing control method of a lightweight fourlegged landing and walking robot
The prober with an immovable lander and a movable rover is commonly used to explore the Moon’s surface. The rover can complete the detection on relatively flat terrain of the lunar surface well, but its detection efficiency on deep craters and mountains is relatively low due to the difficulties of reaching such places. A lightweight fourlegged landing and walking robot called “FLLWR” is designed in this study. It can take off and land repeatedly between any two sites wherever on deep craters, mountains or other challenging landforms that are difficult to reach by direct ground movement. The robot integrates the functions of a lander and a rover, including folding, deploying, repetitive landing, and walking. A landing control method via compliance control is proposed to solve the critical problem of impact energy dissipation to realize buffer landing. Repetitive landing experiments on a fivedegreeoffreedom lunar gravity testing platform are performed. Under the landing conditions with a vertical velocity of 2.1 m/s and a loading weight of 140 kg, the torque safety margin is 10.3% and 16.7%, and the height safety margin is 36.4% and 50.1% for the cases with or without an additional horizontal disturbance velocity of 0.4 m/s, respectively. The study provides a novel insight into the nextgeneration lunar exploration equipment.
landing and walking robot / lunar exploration / buffer landing / compliance control
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Abbreviations  
5DoFLGTP  Fivedegreeoffreedom lunar gravity testing platform 
COM  Center of mass 
DoF  Degreeoffreedom 
FLLWR  Fourlegged landing and walking robot 
Geninv  Generalized inverse based on Cholesky factorization 
Ginv  Generalized inverse computation method 
IDU  Integrated drive unit 
IMqrg  Improved generalized inverse computation method based on QR factorization 
IMU  Inertial measurement unit 
LRV  Lunar roving vehicle 
PFBM  Parallel fivebar mechanism 
Qrg  Generalized inverse computation method based on QR factorization 
RP  Revolute pair 
SVD  Singular value decomposition 
TPM  Tensor product matrix 
VTLSA  Virtual threeleg supporting algorithm 
Variables  
a_{0}, a_{1}  Initial and target acceleration 
a_{b}  Body linear acceleration 
B, B_{0}, B_{1}, B_{2}, B_{3}  Active damping coefficients 
B_{g}  Ground damping coefficient 
B_{virtual}  Virtual damping coefficient 
c_{i}  Coefficient of interpolation trajectory 
coe_{interp}, coe_{0}, coe_{1}, coeV_{0}, coeV_{1}  Current interpolation coefficient, initial coefficient, target coefficient, initial coefficient velocity, and target coefficient velocity in compliance planning, respectively 
d  Active compression distance 
E_{ps}  Elastic potential energy of passive spring 
F_{i}  ith tiptoe force from ground 
F_{iz}, F_{iz}  Vector and the third component of the ith tiptoe force from ground in the z direction 
${F}_{i\mathit{\text{z}}}^{kmn}$  ith (i = k, m, or n) vertical force in leg kmn supporting 
F_{tip}, F_{x}, F_{y}, F_{z}  Tiptoe force vector and its components 
F_{virtual}  Virtual tiptoe force 
F_{vr}  Virtual resultant force in the z direction 
g, g  Vector and the third component of gravitational acceleration 
g_{e}  Gravitational acceleration on the Earth 
g_{m}  Gravitational accelerations on the Moon 
H  Correction vector 
I_{b}, I_{bx}, I_{by}  Body inertia vector and its components 
j_{0}, j_{1}  Initial and target jerk 
J_{v} (q)  Velocity Jacobian matrix 
K, K_{0}, K_{1}, K_{2}  Active stiffness coefficients 
k_{dz}  Derivative gain in the z direction 
k_{d,θ}  Derivative gain of roll and pitch angles 
K_{g}  Ground stiffness coefficient 
k_{i,z}  Integral gain in the z direction 
k_{i,θ}  Integral gain of roll and pitch angles 
k_{p,z}  Proportional gain in the z direction 
k_{p,θ}  Proportional gain of roll and pitch angles 
K_{ps}  Stiffness coefficient of passive spring 
K_{virtual}  Virtual stiffness coefficient 
L_{l}, L_{r}  Distances from the intersection point of thigh and shank to the left and right fixed points of passive spring, respectively 
L_{ps}, L_{ps0}  Current and original length of the passive spring, respectively 
L_{s}  Shank length 
L_{t}  Thigh length 
m_{b}  Body mass 
m_{c1}  Counterweight 1 mass 
m_{c2}  Counterweight 2 mass 
m_{t}  System mass of the lander and loads 
O_{b}  Origin of body coordinate frame 
O_{li}  Origin of the ith leg coordinate frame 
O_{w}  Origin of world coordinate frame 
${\mathit{P}}_{\mathrm{t}\mathrm{i}\mathrm{p}}$, ${\dot{\mathit{P}}}_{\mathrm{t}\mathrm{i}\mathrm{p}}$  Tiptoe position and velocity, respectively 
${}^{}\mathrm{b}{\mathit{P}}_{\mathrm{l}i}$  Origin position of ${\mathrm{\Sigma}}_{\mathrm{l}i}$ in the body coordinate frame ${\mathrm{\Sigma}}_{\mathrm{b}}$ 
${}^{}\mathrm{l}i{\mathit{P}}_{\mathrm{t}\mathrm{i}\mathrm{p}}$  Tiptoe position in the leg coordinate frame ${\mathrm{\Sigma}}_{\mathrm{l}i}$ 
${}^{}\mathrm{w}{\mathit{P}}_{\mathrm{b}}$  Body position in the world coordinate frame ${\mathrm{\Sigma}}_{\mathrm{w}}$ 
${}^{}\mathrm{w}{\mathit{P}}_{\mathrm{t}\mathrm{i}\mathrm{p}}$  Tiptoe position in the world coordinate frame ${\mathrm{\Sigma}}_{\mathrm{w}}$ 
$\mathit{q}$, $\dot{\mathit{q}}$  Generalized coordinate and velocity vector of joints, respectively 
${\mathit{r}}_{\mathrm{c}\mathrm{o}\mathrm{m}}$, ${r}_{\mathrm{c}\mathrm{o}\mathrm{m}x}$, ${r}_{\mathrm{c}\mathrm{o}\mathrm{m}y}$  Vector and its components from the COM to the origin O_{b} 
${\mathit{r}}_{i}$, ${r}_{ix}$, ${r}_{iy}$, ${r}_{i\mathit{\text{z}}}$  Vector and its components from the origin ${O}_{\mathrm{b}}$ to the ith tiptoe 
${}^{}\mathrm{b}{\mathit{R}}_{\mathrm{l}i}$  Rotation transformation matrix from ${\mathrm{\Sigma}}_{\mathrm{l}i}$ to ${\mathrm{\Sigma}}_{\mathrm{b}}$ 
${}^{}\mathrm{w}{\mathit{R}}_{\mathrm{b}}$  Rotation transformation matrix from ${\mathrm{\Sigma}}_{\mathrm{b}}$ to ${\mathrm{\Sigma}}_{\mathrm{w}}$ 
$t$, ${t}_{0}$, ${t}_{1}$, ${t}_{\mathrm{r}}$, T  Current time, initial time, end time, duration ratio time, and total duration time, respectively 
${T}_{\mathrm{s}}$, ${T}_{\mathrm{t}}$  Shank and thigh IDUs torques, respectively 
${v}_{0}$, ${v}_{1}$  Initial and target velocity, respectively 
${v}_{\mathrm{h}}$  Horizontal velocity 
${v}_{x}$  Horizontal velocity in the x direction 
${v}_{\mathit{\text{z}}}$  Vertical velocity 
${x}_{\mathrm{b}}$, ${y}_{\mathrm{b}}$, ${\mathit{\text{z}}}_{\mathrm{b}}$  Forward, left, and upper direction, respectively 
${x}_{\mathrm{l}i}$, ${y}_{\mathrm{l}i}$, ${\mathit{\text{z}}}_{\mathrm{l}i}$  Components of tiptoe position in the ith leg coordinate frame, respectively 
${x}_{\mathrm{t}\mathrm{i}\mathrm{p}}$, ${x}_{\mathrm{t}\mathrm{i}\mathrm{p}0}$, ${x}_{\mathrm{t}\mathrm{i}\mathrm{p}1}$  Current, initial position, and terminated position of tiptoe in the x direction, respectively 
${\dot{x}}_{\mathrm{t}\mathrm{i}\mathrm{p}}$, ${\dot{y}}_{\mathrm{t}\mathrm{i}\mathrm{p}}$, ${\dot{\mathit{\text{z}}}}_{\mathrm{t}\mathrm{i}\mathrm{p}}$  Current tiptoe velocity in the x, y and z directions, respectively 
${y}_{\mathrm{t}\mathrm{i}\mathrm{p}}$, ${y}_{\mathrm{t}\mathrm{i}\mathrm{p}0}$, ${y}_{\mathrm{t}\mathrm{i}\mathrm{p}1}$  Current, initial, and terminated positions of tiptoe in the y direction, respectively 
${\mathit{\text{z}}}_{\mathrm{a}\mathrm{b}}$, ${\dot{\mathit{\text{z}}}}_{\mathrm{a}\mathrm{b}}$  Actual body position and velocity in the z direction, respectively 
${\ddot{\mathit{z}}}_{\mathrm{b}}$, ${\ddot{\mathit{\text{z}}}}_{\mathrm{b}}$  Vector and the third component of body linear acceleration in the z direction, respectively 
${\mathit{\text{z}}}_{\mathrm{b}\mathrm{o}\mathrm{d}\mathrm{y}0}$, ${\mathit{\text{z}}}_{\mathrm{b}\mathrm{o}\mathrm{d}\mathrm{y}1}$  Body initial and target position in the z direction, respectively 
${\mathit{\text{z}}}_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{p}}$  Realtime body interpolation trajectory 
${\mathit{\text{z}}}_{\mathrm{r}\mathrm{b}}$, ${\dot{\mathit{\text{z}}}}_{\mathrm{r}\mathrm{b}}$  Reference body position and velocity in the z direction, respectively 
${\mathit{\text{z}}}_{\mathrm{t}\mathrm{i}\mathrm{p}}$, ${\mathit{\text{z}}}_{\mathrm{t}\mathrm{i}\mathrm{p}0}$, ${\mathit{\text{z}}}_{\mathrm{t}\mathrm{i}\mathrm{p}1}$  Current, initial, and target positions of the tiptoe in the z direction, respectively 
$\text{\delta}{\mathit{P}}_{\mathrm{t}\mathrm{i}\mathrm{p}}$  Virtual displacement of tiptoe 
$\text{\delta}\mathit{q}$  Virtual displacement of the generalized coordinate vector $\mathit{q}$ 
$\text{\delta}{W}_{\mathrm{p}\mathrm{s}}$  Virtual work of passive spring 
${\mathit{\theta}}_{\mathrm{a}\mathrm{j}}$, ${\dot{\mathit{\theta}}}_{\mathrm{a}\mathrm{j}}$  Actual joint angle and velocity, respectively 
${\theta}_{\mathrm{a}\mathrm{p}}$, ${\dot{\theta}}_{\mathrm{a}\mathrm{p}}$  Actual pitch angle and velocity, respectively 
${\theta}_{\mathrm{a}\mathrm{r}}$, ${\dot{\theta}}_{\mathrm{a}\mathrm{r}}$  Actual roll angle and velocity, respectively 
${\theta}_{\mathrm{r}}$, ${\dot{\theta}}_{\mathrm{r}}$  Rocker arm angle and velocity, respectively 
${\mathit{\theta}}_{\mathrm{r}\mathrm{j}}$, ${\dot{\mathit{\theta}}}_{\mathrm{r}\mathrm{j}}$  Reference joint angle and velocity, respectively 
${\theta}_{\mathrm{r}\mathrm{p}}$, ${\dot{\theta}}_{\mathrm{r}\mathrm{p}}$  Reference pitch angle and velocity, respectively 
${\theta}_{\mathrm{r}\mathrm{r}}$, ${\dot{\theta}}_{\mathrm{r}\mathrm{r}}$  Reference roll angle and velocity, respectively 
${\theta}_{\mathrm{s}}$, ${\dot{\theta}}_{\mathrm{s}}$  Side angle and velocity, respectively 
${\theta}_{\mathrm{t}}$, ${\dot{\theta}}_{\mathrm{t}}$  Thigh angle and velocity, respectively 
$\mathit{\tau}$, ${\tau}_{\mathrm{s}}$, ${\tau}_{\mathrm{t}}$, ${\tau}_{\mathrm{r}}$  Joint torque vector and its components, respectively 
${\mathit{\tau}}_{\mathrm{a}\mathrm{j}}$, ${\mathit{\tau}}_{\mathrm{c}\mathrm{j}}$, ${\mathit{\tau}}_{\mathrm{r}\mathrm{j}}$  Actual, command, and reference joint torques, respectively 
${\mathit{\tau}}_{\mathrm{v}\mathrm{i}\mathrm{r}\mathrm{t}\mathrm{u}\mathrm{a}\mathrm{l}}$  Virtual joint torque 
${\mathit{\tau}}_{\mathrm{v}\mathrm{r}}$  Virtual resultant torsions in the $x$ and $y$ directions, respectively 
${\mathrm{\Sigma}}_{\mathrm{b}}$  Body coordinate frame 
${\mathrm{\Sigma}}_{\mathrm{i}\mathrm{m}\mathrm{u}}$  IMU coordinate frame 
${\mathrm{\Sigma}}_{\mathrm{l}i}$  ith leg coordinate frame 
${\mathrm{\Sigma}}_{\mathrm{w}}$  World coordinate frame 
${\text{\omega}}_{\mathrm{b}}$  Body angular velocity 
${\dot{\text{\omega}}}_{\mathrm{b}}$, ${\dot{\omega}}_{\mathrm{b}x}$, ${\dot{\omega}}_{\mathrm{b}\mathrm{y}}$  Vector and its components of body angular acceleration, respectively 
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