Advancing Control for Shield Tunneling Machine by Backstepping Design with LuGre Friction Model

Table 1

Notations in modeling.

Symbols	Physical quantities
𝑚	Mass of shield machine
𝑓	Lumped force of friction force and viscous force
𝐹	Earth pressure force exerted on cutter head
𝑧	Average deflection of the bristles
𝑣	Relative velocity between two friction surfaces
𝜎0	Stiffness of the bristles
𝜎1	Damping coefficient
𝜎2	Viscous coefficient
𝐹𝐶	Coulomb friction force
𝐹𝑆	Stiction force
𝑣𝑠	Stribeck velocity
𝑥1	Displacement of shield machine
𝑥2	Velocity of shield machine
𝑥1𝑑	Desired displacement
𝑥2𝑑	Desired velocity
𝑒1	Tracking error of displacement
𝑒2	Tracking error of velocity
𝑉𝑖	Lyapunov funcitons
•̇	First order derivative
•̈	second order derivative
•	Estimated value
•	•--•, error between estimated value and true value
𝑘𝑖,	Feedback gain
𝜆𝑖	Adaptation gain