Landing control method of a lightweight four-legged landing and walking robot

Abstract

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 four-legged 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 five-degree-of-freedom 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 next-generation lunar exploration equipment.

Graphical abstract

Keywords

landing and walking robot / lunar exploration / buffer landing / compliance control

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Nomenclature

Abbreviations
5-DoF-LGTP	Five-degree-of-freedom lunar gravity testing platform
COM	Center of mass
DoF	Degree-of-freedom
FLLWR	Four-legged 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 five-bar mechanism
Qrg	Generalized inverse computation method based on QR factorization
RP	Revolute pair
SVD	Singular value decomposition
TPM	Tensor product matrix
VTLSA	Virtual three-leg supporting algorithm
Variables
a0, a1	Initial and target acceleration
ab	Body linear acceleration
B, B0, B1, B2, B3	Active damping coefficients
Bg	Ground damping coefficient
Bvirtual	Virtual damping coefficient
ci	Coefficient of interpolation trajectory
coeinterp, coe0, coe1, coeV0, coeV1	Current interpolation coefficient, initial coefficient, target coefficient, initial coefficient velocity, and target coefficient velocity in compliance planning, respectively
d	Active compression distance
Eps	Elastic potential energy of passive spring
Fi	ith tiptoe force from ground
Fiz, Fiz	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 k-m-n supporting
Ftip, Fx, Fy, Fz	Tiptoe force vector and its components
Fvirtual	Virtual tiptoe force
Fvr	Virtual resultant force in the z direction
g, g	Vector and the third component of gravitational acceleration
ge	Gravitational acceleration on the Earth
gm	Gravitational accelerations on the Moon
H	Correction vector
Ib, Ibx, Iby	Body inertia vector and its components
j0, j1	Initial and target jerk
Jv (q)	Velocity Jacobian matrix
K, K0, K1, K2	Active stiffness coefficients
kdz	Derivative gain in the z direction
kd,θ	Derivative gain of roll and pitch angles
Kg	Ground stiffness coefficient
ki,z	Integral gain in the z direction
ki,θ	Integral gain of roll and pitch angles
kp,z	Proportional gain in the z direction
kp,θ	Proportional gain of roll and pitch angles
Kps	Stiffness coefficient of passive spring
Kvirtual	Virtual stiffness coefficient
Ll, Lr	Distances from the intersection point of thigh and shank to the left and right fixed points of passive spring, respectively
Lps, Lps0	Current and original length of the passive spring, respectively
Ls	Shank length
Lt	Thigh length
mb	Body mass
mc1	Counterweight 1 mass
mc2	Counterweight 2 mass
mt	System mass of the lander and loads
Ob	Origin of body coordinate frame
Oli	Origin of the ith leg coordinate frame
Ow	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 Ob
{\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}}	Real-time 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{δ}{\mathit{P}}_{\mathrm{t}\mathrm{i}\mathrm{p}}	Virtual displacement of tiptoe
\text{δ}\mathit{q}	Virtual displacement of the generalized coordinate vector \mathit{q}
\text{δ}{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{ω}}_{\mathrm{b}}	Body angular velocity
{\dot{\text{ω}}}_{\mathrm{b}}, {\dot{\omega }}_{\mathrm{b}x}, {\dot{\omega }}_{\mathrm{b}\mathrm{y}}	Vector and its components of body angular acceleration, respectively

Acknowledgements

This work was funded by the National Key R&D Program of China (Grant No. 2021YFF0307905).

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