Finat, Javier 
  
provides the out-of-phase criteria for the reflections associated to the coordinated movements between legs. We interpret 
the body (including the other leg along the navigation phase) as a flexible plate which contributes to the stability of the 
global system. Symmetries appearing in gait are globally controlled by polygonal groups (depending on the number of 
legs and the gait type), which provide a switching process between opossite legs. So, it suffices to analyze the behavior 
along a complete cycle for each leg and then to apply the action of the hexagonal group to control the distribution process 
between different legs. 
3 ALGORITHMS FOR MOTION PLANNING 
The motion planning problem is old and well-known in Robotics. Different approaches to the algorithms design have been 
developed by using techniques arising from Computational Geometry (roadmap, cell decomposition and potential fields, 
e.g.) Probabilistic Modeling (local methods including evaluation and inference models), and Artificial Neural Networks 
(supervised vs. unsupervised learning procedures, self-organising maps) depending on the task and the environment 
(Latombe, 1991), (Berg and O.Schwarzkopf, 1997). In all cases, one uses different optimization criteria based on least 
squares method (LSM), maximum likelihood (ML) and genetic algorithms (GAs). The increasing complexity of tasks and 
the need for adaptability to different environments, require a combination of the above catalogue of techniques and the 
development of hybrid methods to achieve a better performance. In our case, the locomotion is by itself a enough complex 
task, and hence the emphasis for the planning is put onto the propioceptive system, rather than in the scene characteristics 
which we shall suppose free obstacle (otherwise, one could apply the above methods for the center of gravity). 
3.1 General remarks about algorihtms for locomotion 
The planning and generation of movements for kinematic multichains of changing shapes corresponding to articulated 
components (arms, legs, body or hands) is a difficult task even in structured scenes with static objects (Wilfong, 1988) is 
one of the first references for algorithms relative to mobile data. The most difficult problem is linked to the management 
of different kinds of models and constraints (Canny, 1987). The compatibility between constraints requires the incor- 
poration of mobile data structures and fast updating procedures. The updating involves to the design of mobile kinetic 
data structures (Guibas, 1998); two neck bottles of this approach arise with the difficulties to an efficient management 
of spatial data (a common problem of available software in Computational Geometry), and the lack of structural models 
arising from changing vector fields. There is no a general answer to these problems, still; for the locomotion case in the 
meantime one can develop good graphic simulations and deduce mechanical models controlled by known vector fields. 
This is exactly our strategy. 
There are several approaches to the algorithms design depending on the framework and the complexity of the task. Some 
criteria giving crossed classifications are related to the learning processes (supervised and unsupervised), or with model- 
based character (structural and randomized algorithms), depending on the emphasis onto trial-and-error procedures or the 
availability of models. 
Obviously, the best solution for unstructured scenes will be a hybrid combination of both approaches, but in this note 
I shall put emphasis on those based on mechanical models and enough information about the scene. Models can be 
biologically inspired or mechanically based; the biological inspiration provides criteria for the human-based expected 
behavior, but this gives a coarse approach: each human presents some special gait, and makes difficult the transference 
of biological principles to control modules. Instead, one can use mechanical models obtained from the Lagrangian's 
formulation of the motion equations for each leg, which provides a decoupled input to be used in another parts of the 
process. As always, the best solution would must be a hybrid one, but the integration is not easy to perform it. 
Main issues of algorithms for hierarchised models in parallel robots concern to 
e Perception: capture, preprocessing and fusion of information processes 
e Mechanics: including geometric, kinematic and dynamic aspects of movements 
e Motion planning, simulation and evaluation and their symbolic representation in terms of oriented graphs with tran- 
sition, patching and folding phenomena. 
e Adaptive Control: Static, gradient, marginal and dynamic, including an strategy for switching and tuning. 
e Execution and monitoring: Generation of impulses, evaluation, comparison between current and desired trajectories, 
correction of errors. 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 241