Method for controlling vibration of flexible mechanical arm based on cooperative tracking

A method for controlling vibration of a plurality of flexible mechanical arms based on cooperative tracking, the method comprising:

• kinematic each of establishing a dynamic model equation for each of the plurality of flexible mechanical arms, according to characteristics of the flexible mechanical arms including kinetic energy, potential energy, and virtual work done by non-conservative force acting on the flexible mechanical arm;
• assigning one of the plurality of flexible mechanical arms as a leader and the rest ones as followers which need to track the leader's motion trajectory;
• designing cooperative tracking-based boundary controllers based on the plurality of flexible mechanical arms;
• constructing a Lyapunov function for each of the plurality of flexible mechanical arms based on the dynamic model equation and the boundary controllers;
• and
• validating stability of each of the plurality of flexible mechanical arms, according to the Lyapunov function,
• wherein the dynamic model equation is obtained by substituting the kinetic energy, the potential energy, and the virtual work into Hamilton's principle, and the kinematic model equation is as follows:
• ρ
• ⁢
• w
• ¨
• i
• +
• Elw
• i
• ′′′
• -
• Tw
• i
• ′′
• +
• γ
• ⁢
• w
• .
• i
• =
• -
• (
• r
• +
• x
• )
• ⁢
• (
• ρ
• ⁢
• θ
• ¨
• i
• +
• γ
• ⁢
• θ
• .
• i
• )
• ,
• and
• ⁢
• I
• h
• ⁢
• θ
• ¨
• =
• -
• γ
• ⁢
• Elw
• i
• ′′′′
• (
• 0
• ,
• t
• )
• +
• Elw
• i
• ′′
• (
• 0
• ,
• t
• )
• +
• Tw
• i
• (
• l
• ,
• t
• )
• +
• u
• 2
• ⁢
• l
• ,
• where wi(x,t) represents a vibration offset of an ith flexible mechanical arm in xoy coordinate system, {dot over (w)}i(x,t) and {umlaut over (w)}i(x,t) represent the first and second derivative of time and are abbreviated as {dot over (w)}i and {umlaut over (w)}i respectively, w′i(x,t), w″i(x,t), w′″i(x,t) and w″″i(x,t) represent the first, second, third and fourth derivatives of wi(x,t) with respect to x and are abbreviated as w″i, w″i, w′″i and w″″i respectively, ρ represents a uniform mass per unit length of the flexible mechanical arm, m represents a tip mass of the flexible mechanical arm, l represents a length of the mechanical arm, r represents a radius of a rigid hub, lh represents a hub inertia, θi represents an attitude angle of the ith flexible mechanical arm, {dot over (θ)}i and {umlaut over (θ)}i represent the first and second derivative of θi with respect to time respectively, T represents a tension, EI represents a bending stiffness, y represents a viscous damping coefficient, w′″i(0,t) represents a value of w′″i(x,t) at x=0, w″″i(x,t) represents a value of w″″i(x,t) at x=0, wi(l,t) represents a value of wi(x,t) at x=l, and u2l represents a controller at a fixed end position of the flexible mechanical arm,
• wherein a boundary condition is as follows:
• m
• ⁢
• w
• ¨
• i
• (
• l
• ,
• t
• )
• =
• EIw
• i
• ′′′
• (
• 0
• ,
• t
• )
• -
• m
• ⁡
• (
• r
• +
• l
• )
• ⁢
• θ
• ¨
• i
• -
• Tw
• i
• ′
• (
• l
• ,
• t
• )
• +
• u
• 1
• ⁢
• i
• ,
• and
• w
• i
• (
• 0
• ,
• t
• )
• =
• w
• i
• ′
• (
• 0
• ,
• t
• )
• =
• w
• i
• ′′
• (
• l
• ,
• t
• )
• =
• 0
• ,
• where wi(l,t) and wi(l,t) represent values of wi(x,t) and wi(x,t) at x=l respectively, w′i(l,t) and w″i(l,t) represent values of w″i(x,t) and w″i(x,t) at x=l respectively, wi(0,t) and w″i(0,t) represent values of wi(x,t) and w″i(x,t) at x=0 respectively, and u1i represents a controller at a tip position of the flexible mechanical arm, and
• wherein the boundary controllers are configured to suppress vibration of the plurality of flexible mechanical arms, and to perform cooperative control in such a way that the flexible mechanical arm as one of the followers tracks the motion trajectory of the flexible mechanical arm as the leader.