Finat, Javier 
  
are translated into twists and wrenches in W. They follow biologically inspired rules for binary activation/inhibition 
patterns to perform the planned tasks which are translated into switching and tuning behaviors in control processes. 
These fields can be used for any other parallel mechanism, s.t. a multifingered hand where one uses a symbolic represen- 
tation based on cubes with binary vertices to identify typical postures of an artificial hand (Gonzalo-Tasis and D.Sanchez, 
2000) ; this representation can support also kinematic and dynamic information as paths onto unions of cubes with variable 
weights playing the role of local varying curvatures. 
Boissonnat et al (1995) develop algorithms for motion planning based on accessible and stable configurations of legged 
robots (Boissonnat and S.Lazard, 1995). Nevertheless its local efficiency, to update this information in the working space 
it is necessary to have to our disposal a cellular decomposition of a highly structured scene, and transfer the information 
relative to possible swings of each leg. Even in this case, the update of this information is too casuistic and some expensive 
from the computational viewpoint, due to the need of verifying a complex system of constraints. Furthermore, it is not 
clear how to evaluate different kinds of unstability in legged mechanisms by using only information about the geometry 
of configurations and working space. The source of unstability in locomotion tasks arises from kinematics and dynamic 
of multilegged robots, independently of the scene. One needs to design a hierarchised system to diminish the number of 
verifications, and the introduction of local symmetries to make easier the propagation of elementary patterns along the 
truss. 
There exist another low-level approaches based on ANN (Artificial Neural Networks) or the Fuzzy-Logic (FL) Algo- 
rithms, based on logic rules which are very useful for unstructured environments or in absence of a better knowledge of 
the mechanics of multilegged robots. However, after the training process, the ANN is not reusable when the conditions 
are changing. The FL-approach does not allow to incorporate mechanical properties of locomotion phases to control, and 
it requires strong compensating mechanisms due to its ignorance about the mechanical model. Finally, it is difficult to 
find convergence criteria for optimization processes developed from genetic algorithms. 
Thus, in this note, I shall paid attention only to structural models. My approach is related to the adjustment postural 
strategy (Gorce, 1998). It allows to integrate local and global aspects of the mechanical model, and modify it in terms of 
vector fields which update mobile information. I shall put more emphasis onto some hierarchies of mechanical aspects 
which allows us an integrated treatment of kinematic and dynamical constraints in articular coordinates to assert a marginal 
and dynamic behavior around stable trajectories in locomotion. 
To solve the geometric representation problem of multibody systems in working space WW, one can introduce adapted co- 
ordinated frames associated to mobile references following a hierarchised model associated to multi-points or alternately 
to configurations of lines which can be extended to motion planning and control. Anyway, the articular and (multi)vector 
representation are related through the representation of elementary reflections and this provides an easy description of 
transmission and propagation phenomena along the mechanical architecture going from the configuration space C to the 
working space WV (and inversely). The spatio-temporal matching of these mobile references is performed in terms of 
vector fields, by providing in the way the necessary feedback for the force-position traditional control. 
2 THE MECHANICAL MODEL 
The general ingredients for the mechanical model are related with the support (configurations and working spaces), the 
tasks to be performed, variable constraints acting along different phases and devices to simplify the analysis (invariants to 
identify, cost-benefit functions for optimization processes, etc). 
Any multibody system is a collection of kinematic chains linked to a mobile platform. In our case, each kinematic chain 
is given by a leg with a planar motion, after selecting the value of the corresponding Euler angle at spherical joint at the 
hip. Each leg is modeled as the coupling of three pendula with two rotational joints (for the knee and heel), and a third 
spherical joint in the hip for the configurations space C. Onto the working space )V one can introduce mixed optimization 
criteria given by a homotopy between a general quadratic functional plus a smoothing operator. 
Nevertheless the generality of this approach, their meaning is different depending on the context. So, the quadratic 
functional corresponds in the visual case to relative distance functionals (differences between desired and current position- 
orientation in model-based vision or disparity in Stereo Vision, e.g.). In the mechanical case it corresponds to the ordinary 
total energy function if we take control nodes or to the weighted squared norm of the corresponding 6-dimensional twist 
(angular momenta and linear momenta) if we take the geometry of lines as the framework to develop the kinematics; 
difficulties appearing with odometry and calibration suggest to use lines instead of nodes for relative positioning. In the 
same way one can introduce similar quadratic functionals for screws in Geometry and for wrenches in Dynamics. So, the 
conclusions can be extended for any model based on the geometry of lines (the right framework is the Clifford algebra, 
but I have no space for this more general approach, here). The analysis for smoothing operator is similar. 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part BS. Amsterdam 2000. 239