Finite-Time Task-Space Coordinated Tracking of Networked Robotic Manipulators With Uncertain Dynamics and Disturbances

A unified inner‐/outer‐loop distributed design for task‐space robust cooperative tracking of uncertain networked manipulators under directed graphs

“…Inspired by the design in the previous work, 26 the following finite-time distributed nonlinear estimator is developed under a directed graph with a spanning tree to estimate and reconstruct the states (x d , ̇xd ) for each robot i ∈  in a finite time, where only a subset of the followers can have access to the desired reference states.…”

“…where Ĵ+ i (q i ) = ĴT i (q i )( Ĵi (q i ) ĴT i (q i )) −1 is a generalized inverse of the approximate Jacobian matrix for redundant robots (or Ĵ+ i (q i ) = Ĵ−1 i (q i ) for nonredundant robots), 9,11,13,15,17,[19][20][21]26 𝜗 i denotes the estimated task-space reference velocity obtained from the estimator in ( 7) and ( 8) and e i is the cross-coupling error defined in (14).…”

“…Step (i): we prove Ĵi s i ∈  ∞ and lim t→∞ Ĵi s i = 0 n , i ∈ . For the cascade closed-loop system in (26), we first construct the following Lyapunov function candidate:…”

“…In the presence of uncertain kinematics, the Jacobian matrix $$$ {J}_i\left({q}_i\right) $$$ becomes unknown. By Property 3, an estimated Cartesian‐space velocity under uncertain kinematics is expressed as 14,20,21,26 where $$$ {\hat{a}}_i $$$ is an estimated kinematic parameter to be designed later, and $$$ {\hat{J}}_i\left({q}_i\right) $$$ is an estimated Jacobian matrix.…”

“…Remark Without loss of generality, the manipulator is assumed to be operating in the workspace such that the approximate Jacobian matrix is of full rank to avoid singularities, that is, $$$ {\hat{J}}_i\left({q}_i\right) $$$ is nonsingular 9,11,13,15,17,19‐21,26 . In fact, $$$ {x}_d $$$ is commonly given entirely inside the workspace of manipulator so that it operates in a reachable finite task‐space without encountering the kinematic singularity problem.…”