<p>Distributed end-effector formation control for networked underactuated manipulators remains an open problem, primarily due to their second-order nonholonomic constraints that fundamentally limit motion feasibility and complicate stability analysis. Existing works are limited to manipulators with artificially introduced torsional springs at the joints or to manipulators with fully integrable constraints; however, a general framework that accommodates heterogeneous second-order nonholonomic behaviors remains largely unexplored. In light of these motivations, this paper investigates distributed end-effector formation control for a network of two representative underactuated manipulators operating in a gravity-free plane: (i) the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\hbox {PA}^{n-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mtext>PA</mtext> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> manipulator (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n \ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>), with a passive first joint and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> actuated joints, whose second-order nonholonomic constraint is only partially integrable; and (ii) the AP manipulator, a two-link system with an active first joint and a passive second joint, whose second-order nonholonomic constraint is non-integrable. For each <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\hbox {PA}^{n-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mtext>PA</mtext> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> manipulator, we extend our previous work by exploiting the partial integrability of its second-order nonholonomic constraint, which allows the passive-joint dynamics to be eliminated at the velocity level and enables the design of a distributed formation controller using only active joint variables. For each AP manipulator, a novel two-step distributed control strategy is proposed. Specifically, the concept of the virtual end-effector is introduced, whose position serves as an intermediate formation variable: a formation controller first steers the virtual end-effector to its corresponding position in the desired formation, and an iterative steering controller subsequently aligns the actual end-effector with its virtual counterpart. The analysis and simulations show that although the switching between these two control steps prevents networked agents from achieving asymptotic convergence, the residual convergence error can be arbitrarily small.</p>

Distributed end-effector formation control of heterogeneous underactuated manipulators with two representative actuator configurations

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发表日期：

2026年4月20日

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Zhiyu Peng,
Bayu Jayawardhana,
Xin Xin

摘要

Distributed end-effector formation control for networked underactuated manipulators remains an open problem, primarily due to their second-order nonholonomic constraints that fundamentally limit motion feasibility and complicate stability analysis. Existing works are limited to manipulators with artificially introduced torsional springs at the joints or to manipulators with fully integrable constraints; however, a general framework that accommodates heterogeneous second-order nonholonomic behaviors remains largely unexplored. In light of these motivations, this paper investigates distributed end-effector formation control for a network of two representative underactuated manipulators operating in a gravity-free plane: (i) the $\hbox {PA}^{n-1}$ ${PA}^{n - 1}$ manipulator ( $n \ge 3$ $n \geq 3$ ), with a passive first joint and $n-1$ $n - 1$ actuated joints, whose second-order nonholonomic constraint is only partially integrable; and (ii) the AP manipulator, a two-link system with an active first joint and a passive second joint, whose second-order nonholonomic constraint is non-integrable. For each $\hbox {PA}^{n-1}$ ${PA}^{n - 1}$ manipulator, we extend our previous work by exploiting the partial integrability of its second-order nonholonomic constraint, which allows the passive-joint dynamics to be eliminated at the velocity level and enables the design of a distributed formation controller using only active joint variables. For each AP manipulator, a novel two-step distributed control strategy is proposed. Specifically, the concept of the virtual end-effector is introduced, whose position serves as an intermediate formation variable: a formation controller first steers the virtual end-effector to its corresponding position in the desired formation, and an iterative steering controller subsequently aligns the actual end-effector with its virtual counterpart. The analysis and simulations show that although the switching between these two control steps prevents networked agents from achieving asymptotic convergence, the residual convergence error can be arbitrarily small.