A Unified Approach to Consensus Control of Three-Link Manipulators

Abstract

We consider a consensus control problem for a set of three-link manipulators connected by digraphs. Assume that the control inputs of each manipulator are the torques on its links and they are generated by adjusting the weighted difference between the manipulator’s states and those of its neighbor agents. Then, we propose a condition for adjusting the weighting coefficients in the control inputs, so that full consensus is achieved among the manipulators. By designing complex Hurwitz polynomials, we obtain a necessary and sufficient condition for achieving the consensus. Moreover, the discussion is extended to the case of designing convergence rate of consensus. Numerical examples are provided to illustrate the condition and the design conditions.

Appendix: Matrices in Dynamical System (3)

By using the joint angles 𝜃₁, 𝜃₂, 𝜃₃ to compute h₁(x), h₂(x),h₃(x) in V (x) and then substituting $T(x, \dot {x})$, V (x) into the Lagrange’s equation, we obtain the dynamical (3).

To describe the matrices M(x), $V(x, \dot {x})$ and G(x) in detail, we use the following shorthand notations

$$ c_{i} {=} \cos \theta_{i} , \quad c_{ij} {=} \cos (\theta_{i} + \theta_{j}) , s_{i} = \sin \theta_{i} , \quad s_{ij} = \sin (\theta_{i} + \theta_{j}) , $$

and let I_xi, I_yi, I_zi be the moments of inertia about the x −, y −, and z −axes of the i th link frame. Then, M(x), $V(x, \dot {x})$ and G(x) are as follows [25].

Manipulator inertia matrix M(x):

$$ \begin{array}{@{}rcl@{}} M(x) = \left[ \begin{array}{rrr} M_{11} & M_{12} & M_{13}\\ M_{21} & M_{22} & M_{23} \\ M_{31} & M_{32} & M_{33} \end{array}\right] \end{array} $$

where

$$ \begin{array}{rcl} M_{11} &=& I_{y2}{s_{2}^{2}} +I_{y3} s_{23}^{2} +I_{z1} + I_{z2} {c_{2}^{2}} +I_{z3} c_{23}^{2} +m_{2}{r_{1}^{2}} {c_{2}^{2}} \\ &&+m_{3}(l_{1} c_{2} + r_{2} c_{23})^{2} \\ M_{22} &=& I_{x2} + I_{x3} + m_{3} {l_{1}^{2}} +m_{2} {r_{1}^{2}} +m_{3} {r_{2}^{2}} + 2 m_{3} l_{1} r_{2} c_{3} \\ M_{23} &=& I_{x3} +m_{3} {r_{2}^{2}}+m_{3} l_{1} r_{2} c_{3} \\ M_{32} &=& I_{x3} + m_{3} {r_{2}^{2}}+m_{3} l_{1} r_{2} c_{3} \\ M_{33} &=& I_{x3} + m_{3} {r_{2}^{2}} \\ M_{12} &=& M_{13}= M_{21}= M_{31}= 0 . \end{array} $$

Coriolis matrix $V(x,\dot {x})$:

$$ V(x, \dot{x}) = \left[ \begin{array}{rrr} V_{11} & V_{12} & V_{13}\\ V_{21} & V_{22} & V_{23}\\ V_{31} & V_{32} & V_{33} \end{array}\right] $$

where $V_{ij} = v_{ij1}\dot {\theta }_{1} + v_{ij2} \dot {\theta }_{2} + v_{ij3} \dot {\theta }_{3}$ and

$$ \begin{array}{rcl} v_{112} &=& (I_{y2}-I_{z2}-m_{2} {r_{1}^{2}} ) c_{2} s_{2} + (I_{y3}-I_{z3})c_{23} s_{23} \\ &&-m_{3} (l_{1} c_{2} + r_{2} c_{23})(l_{1} s_{2} + r_{2} s_{23})\\ v_{113} &=& (I_{y3}-I_{z3})c_{23} s_{23} -m_{3} r_{2} s_{23}(l_{1} c_{2} + r_{2} c_{23})\\ v_{121} &=& (I_{y2}-I_{z2}-m_{2} {r_{1}^{2}} ) c_{2} s_{2} + (I_{y3}-I_{z3})c_{23} s_{23} \\ &&-m_{3} (l_{1} c_{2} + r_{2} c_{23})(l_{1} s_{2} + r_{2} s_{23})\\ v_{131} &=& (I_{y3} - I_{z3})c_{23} s_{23} - m_{3} r_{2} s_{23} (l_{1} c_{2} + r_{2} c_{23})\\ v_{211} &=& (I_{z2}-I_{y2}+ m_{2} {r_{1}^{2}} ) c_{2} s_{2} + (I_{z3}-I_{y3}) c_{23} s_{23} \\ &&+m_{3} (l_{1} c_{2} + r_{2} c_{23})(l_{1} s_{2} + r_{2} s_{23})\\ v_{223} &=& v_{232} = v_{233}= -l_{1} m_{3} r_{2} s_{3} \\ v_{311} &=& (I_{z3}-I_{y3}) c_{23} s_{23} + m_{3} r_{2} s_{23} (l_{1} c_{2} + r_{2} c_{23})\\ v_{322} &=& l_{1} m_{3} r_{2} s_{3} \\ v_{111} &=& v_{122} = v_{123} = v_{132} = v_{133} = v_{212} = v_{213} =v_{221} = v_{222} \\ &{=}& v_{231}{=}v_{312} {=} v_{313} = v_{321}= v_{323} = v_{331} = v_{332} = v_{333} = 0 . \end{array} $$

Gravity terms etc G(x):

$$ G(x)=\left[ \begin{array}{c} 0 \\ -(m_{2} g r_{1} + m_{3} g l_{1})c_{2} -m_{3} g r_{2} c_{23} \\ -m_{3} g r_{2} c_{23} \\ \end{array}\right] . $$