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Article Dans Une Revue IEEE Transactions on Automatic Control Année : 2007

Comments on "Control of a Planar Underactuated Biped on a Complete Walking Cycle"

Résumé

The above paper [1] possesses several approximations and flaws, which we try to explain. Roughly, the topic concerns the problem of trajectory tracking for a class of mechanical Lagrangian systems subject to unilateral constraints on the generalized position (q) 0, (q) 2 IR m. Such multibody mechanical systems also involve a com-plementarity relation between the constraint and a Lagrange multiplier 0 (q) ? 0 (1), and generalized velocity jumps (impacts). The complementarity relations and the velocity jump law, form a specific contact model. A contact model is necessary for the chosen model to be meaningful from a mechanical point of view. When dealing with systems of rigid bodies, the complementarity conditions are the simplest way to deal with the contact dynamics: they state that adhesion or magnetic forces are excluded from the model. Such nonsmooth mechanical systems form a special class of complementarity systems, but other formalisms exist [7]. It is worth noting that the complementarity conditions are not included in the model presented in [1], which is therefore incomplete. Specifically, the authors deal with a particular biped robots model that fits within a class of impulsive ODEs, or measure differential equations. We will come back on this later in this note. The tracking problem is examined when the system undergoes an infinity of cycles, each cycle being composed of three phases of motion: single-support phase, double-support phase, and the impact when the feet hit the ground. Apart from possible underactuation, the problem is quite similar to what is tackled in [2]–[5], that concerns fully actuated Lagrangian systems undergoing cycles which consist of free motion phases, constrained motion phases, and transition phases with impacts. The effects of the impacts and of the complementarity relations do not change from one problem to the other one. This is why it is worth understanding the simplest case before tackling more sophisticated control problems (underactuated systems, flexible joint manipulators, to cite a few). It is worth noting that the infinity of cycles (and consequently Manuscript 1 The symbol ? means that (q) and have to be orthogonal one to each other. Since they are both non-negative, this is equivalent to the componentwise relation 0 (q), 0 , (q) = 0 for all 1 i m.
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Dates et versions

hal-01633240 , version 1 (12-11-2017)

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Bernard Brogliato. Comments on "Control of a Planar Underactuated Biped on a Complete Walking Cycle". IEEE Transactions on Automatic Control, 2007, 52 (5), pp.961-964. ⟨10.1109/TAC.2007.895947⟩. ⟨hal-01633240⟩
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