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Abstract-Numerous robotic tasks associated with underactuation have been studied in the literature. For a large number of these in the plane, the mechanical models have a cyclic variable, the cyclic variable is unactuated, and all shape variables are independently actuated. This paper formulates and solves two control problems for this class of models. If the generalized momentum conjugate to the cyclic variable is not conserved, conditions are found for the existence of a set of outputs that yields a system with a one-dimensional exponentially stable zero dynamics -i.e. an exponentially minimum-phase system -along with a dynamic extension that renders the system locally input-output decouplable. If the generalized momentum conjugate to the cyclic variable is conserved, a reduced system is constructed and conditions are found for the existence of a set of outputs that yields an empty zero dynamics, along with a dynamic extension that renders the system feedback linearizable. A common element in these two feedback problems is the construction of a scalar function of the configuration variables that has relative degree three with respect to one of the input components. The function arises by partially integrating the conjugate momentum. The results are illustrated on two balancing tasks and on a ballistic flip motion.

A new glucose-insulin model is introduced which fits with the clinical data from in- and outpatients for two days. Its stability property is consistent with the glycemia behavior for type 1 diabetes. This is in contrast to traditional glucose-insulin models. Prior models fit with clinical data for a few hours only or display some nonnatural equilibria. The parameters of this new model are identifiable from standard clinical data as continuous glucose monitoring, insulin injection, and carbohydrate estimate. Moreover, it is shown that the parameters from the model allow the computation of the standard tools used in functional insulin therapy as the basal rate of insulin and the insulin sensitivity factor. This is a major outcome as they are required in therapeutic education of type 1 diabetic patients.

However, it is straightforward to verify that there exist constants u1 > 0 and u2 > 0 such that we can write 01(1111(.)11; + ll.r(.)ll; + ~~llE~(.)ll;, and moreover, c0[11.1-011' + do + E:=, TJ,) 5 n[11.r011' + E,"=, d,]. Hence, by combining these inequalities with (4.14), it follows that 111((.)112' + ll.r(.)[lg + E,"=, llEJ(.)llg 5 m/fl1[11.~011~ + E,"=, d,]. That is, condition iii) of Definition 2.2 is satisfied. Conditions ii) and iv) of Definition 2.2 follows directly from the stability of the matrix P. Thus, we can now conclude that the system (2.1), (2.3) is absolutely stabilizable via the control (4.4). 0 Remark: The above theorem gives a necessary and sufficient condition for absolute stabilizability in terms of the solution to a corresponding H m control problem. It is well known that such an H m control problem can be solved in terms of two algebraic Riccati equations; e.g., see [lo]. The following corollary is an immediate consequence of the above theorem. Corollary 4.1: If the uncertain system (2.1), (2.3) satisfies Assumptions 4.1 t 4. 6) and is absolutely stabilizable via nonlinear control, then it will be absolutely stabilizable via a linear controller of the form (4.4). E;=, llEa(-)ll;) 5 (1 / 2 C l)lI4.)lK + ;ll.4.)ll; + Ilso~.

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Algebraic methods for nonlinear control systems. -2nd ed. -(Communications and control engineering) 1. Nonlinear control theory I.