DIME

Despite an increase in the use of exoskeletons, particularly for medical and occupational applications, few studies have focused on the wrist, even though it is the fourth most common site of musculoskeletal pain in the upper limb. The first part of this paper will present the key challenges to be addressed to implement wrist exoskeletons as wearable devices for novel rehabilitation practices and tools in the occupational/industrial sector. Since the wrist is one of the most complex joints in the body, an understanding of the bio-mechanics and musculo-skeletal disorders of the wrist is essential to extracting design requirements. Depending on the application, each wrist exoskeleton has certain specific design requirements. These requirements have been categorized into six sections: purpose, kinematics, dynamics, rigidity, ergonomics, and safety. These form the driving factors behind the choice of a design depending on the objectives. Different design architectures are explored, form...

A geometric setup for constrained variational calculus is presented. The analysis deals with the study of the extremals of an action functional defined on piecewise differentiable curves, subject to differentiable, non-holonomic constraints. Special attention is paid to the tensorial aspects of the theory. As far as the kinematical foundations are concerned, a fully covariant scheme is developed through the introduction of the concept of infinitesimal control. The standard classification of the extremals into normal and abnormal ones is discussed, pointing out the existence of an algebraic algorithm assigning to each admissible curve a corresponding abnormality index, related to the co-rank of a suitable linear map. Attention is then shifted to the study of the first variation of the action functional. The analysis includes a revisitation of Pontryagin's equations and of the Lagrange multipliers method, as well as a reformulation of Pontryagin's algorithm in Hamiltonian term...

This paper is a direct continuation of arXiv:0705.2362 . The Hamiltonian aspects of the theory are further developed. Within the framework provided by the first paper, the problem of minimality for constrained calculus of variations is analyzed among the class of differentiable curves. A necessary and sufficient condition for minimality is proved.

A geometric setup for control theory is presented. The argument is developed through the study of the extremals of action functionals defined on piecewise differentiable curves, in the presence of differentiable non-holonomic constraints. Special emphasis is put on the tensorial aspects of the theory. To start with, the kinematical foundations, culminating in the so called variational equation, are put on

A new variational principle for General Relativity, based on an action functional I(Φ, ∇) involving both the metric Φ and the connection ∇ as independent, unconstrained degrees of freedom is presented. The extremals of I are seen to be pairs (Φ, ∇) in which Φ is a Ricci flat metric, and ∇ is the associated Riemannian connection. An application to Kaluza's theory of interacting gravitational and electromagnetic fields is discussed.

An axiomatic approach to the study o/ relative continuum mechanics in curved space-time is proposed. The explicit assumptions are: a) existence o] the energy.momentum tensor T iJ, satis]ying the eguations o] motion Ti~//~ ~ O, and b) existence o] the congruence o] stream.lines o] the given continuum. The argument relies on a relativistic extension o] the Lagrangian viewpoint, and involves the analysis o/ the relative dynamical behaviour o] an arbitrary infinitesimal globule A o] continuum in a given ]tame o] re]erence [/~]. The ptan is ]ul]illed in two steps: 1) geometrical theory o/ the Lagrangian viewpoint, valid /or any type o] continua satis]ying the stated requirements; 2) physical applications, illustra~ng the general theory in the case o] an energy-momentum tensor o] the ]orm T ij-~. I~oViV j-S ij.

The role of co-moving atlases is discussed in connection with a possible formulation of the problem of motion in General Relativity. The concept of co-moving scheme is defined and applied to various cases of physical interest. In particular in the Einstein-Max well case, we derive a general uniqueness proof for the Maxwell equations. The dynamical meaning of the equation T 1-7 '^-= 0 is proved, and a scheme for the solution of the problem of motion in co-moving coordinates is proposed. * Lavoro eseguito nell'ambito dell'attivita dei Gruppi di Ricerca Matematica del Consiglio Nazionale delle Ricerche. 1 The notation C n stands for "piecewise C π "; the notation (C m , C n)(m<n) for "C m , piecewise C B ".

It is shown that, also in the mixed initial and boundary value problem, Einstein's equations may be replaced by the two subsystems Tι m jj m = 0 and Rκβ=κ I T a β-$-Tg^βj , provided that the initial data verify the consistency conditions Gf =-κTf and that the analogous relations (G lm + κT lm)N™=0 are imposed on the boundaries of the given domain.

The dynamical meaning of the equations T 1-^ = 0 is derived as a consequence of the mathematical structure of Einstein's equations. A generalization of Lichnerowicz's analysis of the gravitational equations is proposed. * Lavoro eseguito nel centro di Matematica e Fisica Teorica del C.N.R. presso ΓUniversita di Genova. 1 Throughout the paper, Latin indices will run from 1 to 4, and Greek indices from 1 to 3, unless otherwise stated. The metric is assumed to be of normal hyperbolic type with signature (+ + +-). Partial derivatives will usually be indicated by a comma; covariant derivatives by a double vertical stroke (like in Eq. (1.4)).

In recent years, following an earlier result of C. Lanczos concerning the representation of the Weyl tensor in arbitrary space-times, it has been conjectured that the Riemann tensor itself admits a linear representation in terms of the covariant derivatives of a suitable "potential" tensor of rank 3. This conjecture is shown to be false, at least for a class of spacetime geometries including several physically significant ones.

A self-consistent theory of spatial differential forms over a pair (M,F) is proposed. The operators a (spatial exterior differentiation), a T (temporal Lie derivative) and ~ (spatial Lie derivative) are defined, and their properties are discussed. These results are then applied to the study of the torsion and curvature tensor fields determined by an arbitrary spatial tensor analysis (?~T) over (M~F). The structural equations of (V~T) and the corresponding spatial Bianchi identities are discussed. The special case (~,VT) = (9',9T*) is examined in detail. The spatial resolution of the Riemann tensor of the manifold M is finally analysed; the resulting structure of Einstein's equations over a pair (Vd~F) is established. An application to the study of the problem of motion in terms of co-moving atlases is proposed.

The general theory of space tensors is applied to the study of a space-time manifold ~24 carrying a distinguished timelike congruence F. The problem is to determine a physically relevant spatial tensor analysis (~,gT) over (~4,F), in order to proceed to a correct formulation of Relative Kinematics and Dynamics. This is achieved by showing that each choice of (~,9 T) gives rise to a corresponding notion of 'frame of reference' associated with the congruence F. In particular, the frame of reference (F,~Te) determined by the standard spatial tensor analysis (9n,9~ T) is shown to provide the most natural generalization of the concept of frame of reference in Classical Physics. The previous arguments are finally applied to the study of geodesic motion in ~4" As a result, the general structure of the gravitational fields in the frame of reference

A pair (M,F) is defined as a Riemannian manifold M of normal hyperbolic type carrying a distinguished time-like congruence r. The spatial tensor algebra ~ associated with the pair (M,F) is discussed. A general definition of the concept of spatial tensor analysis over (M,r) is then proposed. Basically, this includes a spatial covariant differentiation ~ and a time-derivative 9 T, both acting on ~ and commuting with the process of raising and lowering the tensor indices. The torsion tensor fields of the pair (V,VT) are discussed, as well as the corresponding structural equations. The existence of a distinguished spatial tensor analysis over (M,r) is finally established, and the resulting mathematical structure is examined in detail.

A revisitation of the Legendre transformation in the context of affine principal bundles is presented. The argument, merged with the gauge-theoretical considerations developed in [1], provides a unified representation of Lagrangian and Hamiltonian Mechanics, extending to arbitrary non-autonomous systems the symplectic approach of W.M. Tulczyjew.

II concerto di sistema di riferimeuto fisico in uoo spazio-tempo Oa viene applicato allo studio del moto relativo di un giroscopio puntiforme in Relativitgz Generale. La precessione di Thomas e/a precessione di Fokker souo ricavate come casi particolari de/risultato generate.

In a recent paper [1, 2], a new mathematical setting for the formulation of Classical Mechanics, automatically embodying the gauge invariance of the theory under arbitrary transformations L ! L +,&lt;SUB&gt;dt of the Lagrangian has been proposed. The construction relies on the introduction of a principal fiber bundle ? : P ! V&lt;SUB&gt;n+1 over the configuration space-time V&lt;SUB&gt;n+1, with structural

A unified formulation of rigid body dynamics based on Gauss principle is proposed. The Lagrange, Kirchhoff and Newton-Euler equations are seen to arise from different choices of the quasi-coordinates in the velocity space. The group-theoretical aspects of the method are discussed.