Finat, Javier 
  
2.1 A hierarchised support for mechanics 
If we add tasks and constraints to the well-known transference between small movements at joints and motions in ambient 
working space (Finat and S.Urbaneja, 1999), one can represent it by means of a general commutative Extended Main 
Analytic Diagram (EMAD in the successive): 
PR A PC uM. JR? 
i + i i 
PR A PC WE JB? 
i i i i 
JURA.» JOC WE OR? 
where each arrow corresponds to geometric, kinematic and dynamic aspects of the mechanics in terms of the configuration 
C and working W spaces related through the natural projection z : C — Y and their kth order prolongations given by 
the k-jets j^ : J^C — J*W. So, the horizontal lines concern to relations between forward and inverse geometry, the 
kinematics (for the first prolongation j'7 is represented by the usual jacobian matrix), and the dynamics (for j?7 giving 
the Euler-Lagrange formulation). 
Onto this diagram, one represents the tasks as paths v : R® — C which can be composed with z to give tasks on working 
space or lifted to successive prolongations, obviously giving the left part of the EMAD. Similarly, the constraints can be 
represented as maps j^g : J^W — J*RP depending if they involve to the geometry, the kinematics or the dynamics, 
giving the right part of the EMAD. Their lifting to J^C is performed thanks to the local triviality conditions for the 
topological fibrations j^. 
Furthermore these horizontal relations, one has the canonical projections p J*X — J*'-1x, which forget the gen- 
eralized coordinates corresponding to the k-th order derivatives, where .Y represents C or W. So, the vertical arrows 
concern to the transference between geometric, kinematic and dynamic principles for a parallel robot. This transference 
is performed in terms of vector fields which (due to the singularities) can present singularities at the adherences of strata. 
To simplify the spatio-temporal matching, I shall consider only vector fields which can be replicated at such boundaries 
by means the application of reflections (this can be justified in terms of an equivariant stratification). 
This general hierarchised model has its corresponding version for stability and control chart for locomotion, which are 
not considered here due to the lack of space. 
2.2 Symmetries for Mechanical Models 
Any model is a simplified version of the real world, where one must specify assumptions in order to capture invariant 
facts which can be repeated, simulated and generated to improve his execution. The existence of any kind of symmetries 
allow us to apply replication procedures for different phases, and simplify the distributed design of algorithms. The key 
fact is that above EMAD gives equivariant stratifications for the representation of groups generated by reflections (Finat, 
2000) 
In the classical case (rigid bodies, e.g.) constraints are linked to the preservation of scalar quantities in the kinematic 
(energy functionals) or the dynamic framework (virtual work), or to the preservation of vector quantities (angular mo- 
menta) by vector fields, which are considered in an independent way. In locomotion tasks of multibody systems instead 
of looking at the preservation of above scalar or vector quantities (which is not true), we must look at the preservation 
of the global structure supporting the dynamics, which is given by a Clifford algebra (a symplectic or a contact structure 
in the classical case). The specification of the mathematical model is some cumbersome and I shall restrict myself to 
comment that the emphasis is put on the preservation of global mechanical constraints involving to any prolongation by 
means invariant differential operators defined onto the jets space. 
Indeed, geometric, kinematic and dynamic constraints are also modifying along the locomotion, and there exists a feed- 
back between scalar and (multi)vector constraints: thus, we must introduce a formulation to control the exchange between 
scalar and vector quantities with changing constraints depending on the phase of the motion. The periodicity conditions 
about these phases involve also to optimization criteria which must have also present as a periodic or at least an alternating 
behavior. 
Locomotion tasks for biped or multilegged robots involve to the generation of periodic patterns including stance, land- 
off, navigation and ground impact phases for each leg. The concatenation of all these phases is performed in terms 
of flexion/extensions operations allowing shape changes in articulated mechanisms. 
To the lowest level, this agonis- 
tic/antagonistic behavior for articulations is modelled in terms of reflection groups. T 
he coupling between components 
240 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 
  
  
  
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