Abstract
We propose nonholonomic passive decomposition, which enables us to decompose the Lagrange–D’Alembert dynamics of multiple (or a single) nonholonomic mechanical systems with a formation-specifying (holonomic) map <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$h$</tex></formula> into 1) shape system, describing the dynamics of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$h(q)$</tex></formula> (i.e., formation aspect), where <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$q \in \Re^n$</tex></formula> is the systems’ configuration; 2) locked system, describing the systems’ motion on the level set of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$h$</tex></formula> with the formation aspect <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$h(q)$</tex></formula> being fixed (i.e., maneuver aspect); 3) quotient system, whose nonzero motion perturbs both the formation and maneuver aspects simultaneously; and 4) energetically conservative inertia-induced coupling among them. All the locked, shape, and quotient systems individually inherit Lagrangian dynamics-like structure and passivity, which facilitate their control design/analysis. Canceling out the coupling, regulating the quotient system, and controlling the locked and shape systems individually, we can drive the formation and maneuver aspects simultaneously and separately. Notions of formation/maneuver decoupled controllability are introduced to address limitations imposed by the nonholonomic constraint, along with passivity-based formation/maneuver control design examples. Numerical simulations are performed to illustrate the theory. Extension to kinematic nonholonomic systems is also presented.
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