# American Institute of Mathematical Sciences

December  2018, 10(4): 467-502. doi: 10.3934/jgm.2018018

## A coordinate-free theory of virtual holonomic constraints

 1 Dipartimento di Ingegneria dell'Informazione, Università di Parma, Parco Area delle Scienze 181/a, 43124 Parma, Italy 2 Department of Electrical and Computer Engineering, University of Toronto, 10 King's College Road, Toronto, Ontario, M5S 3G4, Canada

* Corresponding author

Some of the ideas of this paper appeared in preliminary form in [11]

Received  September 2017 Revised  September 2018 Published  November 2018

This paper presents a coordinate-free formulation of virtual holonomic constraints for underactuated Lagrangian control systems on Riemannian manifolds. It is shown that when a virtual constraint enjoys a regularity property, the constrained dynamics are described by an affine connection dynamical system. The affine connection of the constrained system has an elegant relationship to the Riemannian connection of the original Lagrangian control system. Necessary and sufficient conditions are given for the constrained dynamics to be Lagrangian. A key condition is that the affine connection of the constrained dynamics be metrizable. Basic results on metrizability of affine connections are first reviewed, then employed in three examples in order of increasing complexity. The last example is a double pendulum on a cart with two different actuator configurations. For this control system, a virtual constraint is employed which confines the second pendulum to within the upper half-plane.

Citation: Luca Consolini, Alessandro Costalunga, Manfredi Maggiore. A coordinate-free theory of virtual holonomic constraints. Journal of Geometric Mechanics, 2018, 10 (4) : 467-502. doi: 10.3934/jgm.2018018
##### References:

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##### References:
Transversality condition in the definition of regular VHC
The vector bundle map $\sigma: T {\cal Q}|_{\cal C} \to T {\cal C}$
Coordinate systems used in Section 4.3
The set ${\cal C}$ in Example 1 and its parametrization
The VHC ${\cal C}$ in Example 2 and its parametrization
Illustration of the case when the control accelerations are orthogonal to ${\cal C}$
The parallel transport map at the north pole of the unit sphere in $\mathbb{R}^3$, with Riemannian connection induced by the Euclidean metric in $\mathbb{R}^3$. The loop $\gamma_q$ is a triangle on the sphere
The double pendulum on a cart of Example 3. Case (a): control force on the cart. Case (b): control torque on the last joint. The orthogonal frame in the figure is the inertial reference frame
Configurations of the double pendulum on the VHC ${\cal C}$ of Example 3. The missing configurations on the right-hand side are deduced by symmetry with respect to the vertical axis
Parallel transport on $\mathbb{R} \times \mathbb{S}^1$ from $(0, 0)$ to $(s^1, s^2)$
$(q_2, \dot q_2)$ orbits of a few solutions of the double pendulum on a cart subject to the VHC $q_3 = \rho(q_2)$. On the left, case (a) (force on cart). On the right, case (b) (torque on last joint)
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