1. State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150001, China 2. Robotic System Department, Jiangsu Jitri-Hust Intelligent Equipment Technology, Wuxi 214000, China
When free-floating space robots perform space tasks, the satellite base attitude is disturbed by the dynamic coupling. The disturbance of the base orientation may affect the communication between the space robot and the control center on earth. In this paper, the enhanced bidirectional approach is proposed to plan the manipulator trajectory and eliminate the final base attitude variation. A novel acceleration level state equation for the nonholonomic problem is proposed, and a new intermediate variable-based Lyapunov function is derived and solved for smooth joint trajectory and restorable base trajectories. In the method, the state equation is first proposed for dual-arm robots with and without end constraints, and the system stability is analyzed to obtain the system input. The input modification further increases the system stability and simplifies the calculation complexity. Simulations are carried out in the end, and the proposed method is validated in minimizing final base attitude change and trajectory smoothness. Moreover, the minute internal force during the coordinated operation and the considerable computing efficiency increases the feasibility of the method during space tasks.
State matrices of the free-end system and constraint-end system in the EBA
A1, A2
State matrices of the real and virtual robots in the free-end system in the EBA
AL1, AL2
State matrices of the real and virtual robots in the constraint-end system in the EBA
B, BL
Input matrices of the free-end system and constraint-end system in the EBA
B1, B2
Input matrices of the real and virtual robots in the free-end system in the EBA
BL1, BL2
Input matrices of the real and virtual robots in the constraint-end system in the EBA
h
Number of the Fourier orthogonal basis
H
Coefficient of the geometric constraints in coordinated operation
I
Identity matrix
JGva, JGvb
Velocity general-Jacobian matrix of arms A and B
Angular velocity general-Jacobian matrix of arm i
Jsα,
Analytical and geometric Base-Jacobian
k
Arbitrary positive number
kij, kcij, ksij
Coefficients of the near-optimal control approach
ki
Coefficients of the 5-degree-polynomial
Kp
Proportional parameter in the closed-loop PD inverse dynamic control method
Kd
Differential parameter in the closed-loop PD inverse dynamic control method
L
Null space of H
m
Arbitrary positive number
N
Joint number of space robot
Og
Inertial coordinate system
P
Undetermined intermediate matrix that unifies the dimensions of Δx and z
Q
Arbitrary symmetric positive-definite matrix
ra, rb
End vectors of arms A and B, respectively
rab
Vector pointing from arm B end to arm A end
s
Combined variable used for Lyapunov function
t0
Time when the joint velocities are desired to be zero
tm
Meeting time
t*
Initial time of trajectory planning
T
Total planning time of the near-optimal method
u
System input in the BA
u1, u2
Inputs of the real and virtual robots in the BA, respectively
Augmented input composed by u1 and u2, and T = []T
U
System input in the EBA
U1, U2
Inputs of the real and virtual robots in the EBA, respectively
Augmented input composed by U1 and U2, and T = []T
va, vb
End velocity of arms A and B, respectively
V
Lyapunov function of the system
W
Input matrix of the robot system in the BA
W1, W2
Input matrices of the real and virtual robot systems in the BA, respectively
Augmented input matrix of the robot system in BA, and = [W1, −W2]
WL
Mapping matrix from variable z to variable of the constraint-end robot system
L
Augmented mapping matrix in the constraint-end system
x
State variable of robot in the BA
x1, x2
System state variables of the real and virtual robots in the BA, respectively
Δx
System state error defined by Δx =x1 − x2
X
State variable of robot in the EBA
X1, X2
System state variable of the real and virtual robots in the EBA, respectively
z
Joint angular velocity in the EBA and is a component of the system state variable in the EBA
α
Vector of satellite base roll-pitch-yaw (RPY) angle, rad
αx, αy, αz
x, y, z terms of the satellite base RPY angle, respectively
Δα
Base RPY angle error, rad
θ
Vector of joint angle, rad
Δθ
Joint angle error, rad
End angular velocities of arms A and B, respectively
λ
Damping factor
ξ
Arbitrary vector
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