Finat, Javier
They are linked between them, and this makes difficult to give a global overview of all of them. Nevertheless, I shall
paid my attention about some aspects related to mechanical aspects and motion planning, where structural models emerge
in a natural way from the stratified nature of tasks and constraints involving to articulated mechanisms. The basic idea
of this note concerns to the incorporation of hierarchised models in both processes. In both cases, one has a natural
hierarchy which can be expressed in terms of the natural projection 7 : C — W and their successive prolongations j^.
So, for k — 0 we shall have the topological and geometric aspects, for k = 1 the kinematics and for k = 2 the dynamics
(Finat and S.Urbaneja, 1998). A complex task to be performed by a parallel robot composed by several independent
kinematic chains requires a multi-point or a multi-line approach with a strongly hierarchised scheme of switching and
tunning processes.
3.2 Minimal remarks about Symmetries for Optimization and Control
It is very important to remark that all these processes must incorporate some kind of symmetries for transmission, co-
ordination and control to simplify the distribution of processes, the balance between dynamic effects and the design of
accesibility and controllability in locally symmetric terms (Lie algebras) for the control chart.
The algorithm design must incorporate also some evaluation procedure, where one can apply optimization standard proce-
dures to improve the execution of movements. This involves to the choice of objective functions linked to cost functions,
and this is not easy for parallel robots, by virtue of their distributed character, and the multiplicity of control elements
(points or lines, involving to encoders and actuators, e.g.).
By example, a multiobjective nonlinear programming in parallel or sequential form controlled by different Lyapunov
functions (one for each controller which is associated to different phases of a complex task), disregards the different nature
of scalar and vector constraints, and the transference of information (mechanical connections) between them. In simplified
models, this can be performed in terms of usual Moment Map (Finat, 2000), where one has two kinematic scalars relative
to the energy functional and the square of the norm of the angular momentum; both of them are the basic invariants
for the mechanics of rigid bodies, but this is not true for the locomotion of articulated robots. Thus, it is necessary to
construct vector fields able of explaining how these changing quantities are periodically transferred between different
components of articulated mechanism and phases of movement, including sequential phenomena onto spatio-temporal
models linked to the dissipation effects (due to friction and impact phenomena) and delays associated to out-of-phase
between components. In the meantime, I shall give some ideas which work to low-level from the identification of some
elementary symmetric patterns.
3.3 Symmetries for planning
The existence of different kinds of symmetries (rigid movements, e.g.) or alternate periodic processes (translated into
switching and tunning) makes easier a modular design for planning. Spatial symmetries can be described (Chasles, hacia
1860) in terms of reflection groups involving to motion at joints and to the description of translations and rotations as
composition of reflections (Hestenes, 2000). Next, one can add temporal delays to incorporate alternance or periodic
effects onto the artificial mechanism. Thus, I shall restrict myself to situations where symmetries are easily identifiable
(gait tasks, e.g.) in terms of reflections. Terrestrial locomotion is not only the capability of self-propulsion, but also
Most of industrial robots are composed by only one kinematic chain (the 6R would be a typical one) or by a mobile
platform which is stopped before performing another tasks. However, a parallel robot is composed by a finite collection
of kinematic chains (legs of a multilegged robot or fingers of an artificial hand) which are connected to a mobile platform
(Merlet, 1996), and with an alternating character (open and closed loops), depending on the phase of the task to be
performed. This changing character is physically interpreted as a phase transition or as a singularity for some kinematic
or dynamic model. The algorithms design for locomotion must evaluate the current state in VV to prevent the alternance
between open (allowing translations and rotations in VV) and closed (only rotations in YV) chains.
the permissible oscillations of the pressure center Cp along the locomotion. The pressure center is the virtual position
of the center of gravity Cc under the effect of virtual displacements produced by the resultant of forces and momenta
(wrenches) in non-equilibrium positions. In prevision of failures in the kinematic and dynamic data of control points, and
due to the double distributed character as hierarchised and parallel system, we must implement a control chart with the
corresponding optimization criteria for hyperredundant articulated mechanisms.
242 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000.
m
a m pat MA jou — AA
IE e l-— ade fu^ cM us P ud et
Tn an az a x:
cb dab gel) — MÓN Jud 5 FN de sub dade anna. bud
lub — A
