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Constrained Lagrangian dynamics based on reduced quasi-velocities and quasi-forces

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Abstract

This paper presents a formulation of Lagrangian dynamics of constrained mechanical systems in terms of reduced quasi-velocities and quasi-forces that can be used for simulation, analysis, and control purposes. In this formulation, a Cholesky decomposition of the mass matrix in conjunction with adequate orthogonal matrices is used to define reduced-quasi-velocities, input quasi-forces, and constraint quasi-forces which possess natural metric. The new state and input variables always have homogeneous units despite the generalized coordinates may involve in both translational and rotational components and the constraint wrench may involve in both force and moment components. Therefore, this formulation is inherently invariant with respect to changes in dimensional units without requiring weighting matrices. Moreover, in this formulation the equations of motion are completely decoupled from those of the constrained force. This allows the possibility of a simple force control action that is totally independent of the motion control action facilitating a hybrid force/motion control. The properties of the new dynamics formulation are investigated and subsequently force/motion tracking control and regulation of constrained multibody systems based on quasi-velocities and quasi-forces are presented.

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  1. Canadian Space Agency (CSA), Saint-Hubert, Canada

    Farhad Aghili

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  1. Farhad Aghili
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Appendix

Appendix

Consider the following positive–definite function:

$$ V = \frac{1}{2} \| \boldsymbol{\epsilon }_{\theta } \| ^{2}, $$
(51)

whose time-derivative along (39) gives

$$ \dot{V} = - k_{p} \| \boldsymbol{\epsilon }_{\theta } \|^{2} + \boldsymbol{\epsilon }_{ \theta }^{T} \boldsymbol{D}(\boldsymbol{\theta }) \boldsymbol{\epsilon }. $$

From (38) and (40), one can find a bound on \(\dot{V}\) as

$$ \dot{V} \leq - 2 k_{p} V + c_{d} \| \boldsymbol{\epsilon }(0) \| \| \boldsymbol{\epsilon }_{ \theta } \| e^{-k_{p} t}, $$
(52)

which has the form of a Bernoulli differential inequality. The above nonlinear inequality can be linearized by the following change of variable \(U=\sqrt{V}\), i.e.,

$$ \dot{U} \leq - k_{p} U + \frac{c_{d} \| \boldsymbol{\epsilon }(0) \|}{\sqrt{2}} e^{-k_{p} t} .$$
(53)

In view of the comparison lemma [47, p. 222] and (38), one can show that the solution of (53) must satisfy

$$ U \leq \Big( U(0) + \frac{c_{d} \| \boldsymbol{\epsilon }(0) \|}{\sqrt{2}} \Big) e^{- k_{p} t}, $$

which is equivalent to (41).

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Aghili, F. Constrained Lagrangian dynamics based on reduced quasi-velocities and quasi-forces. Multibody Syst Dyn 53, 327–343 (2021). https://doi.org/10.1007/s11044-021-09795-9

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