Finat, Javier
4.2 Algorithms and constraints for Kinematics
Kinematic aspects: They concern to the generation of impulses, the transmission along the truss and the coordination
between legs. To simplify the transference problems it is convenient to adopt from the beginning a formulation in terms
of twists (lines representing the angular momentum and the linear momentum of a line). This program includes
1) The evaluation of desired values of kinematic variables along marginal stable trajectories. This concerns to the resolu-
tion of direct and inverse kinematics given by the Jacobian matrix and its pseudoinverse associated to j!. In this phase
one can apply the Pseudo-inverse algorithm (Kerr and B.Roth, 1986). The simultaneous optimization is performed in
terms of SVD.
2) The adjustment of values for local invariants of each motion phase following the equivariant version of the EMAD.
In this phase, one must use linear programming techniques (simplex method, e.g.) but adapted to multivector quantities
(Finat, 2000), and standard non-linear programming for quadratic distance functionals.
3) The selection of a threshold for transmission phenomena involving to the exchange between scalar and vector quantities
along the architecture (propagation phenomena onto each leg as a soliton, e.g.). This work in progress must combine recent
approaches to the Mechanics with Symmetry (Marsden, 1992) with the implementation of controllers design for changing
dynamics (Zefran and J.W.Burdick, 1998)
4) The introduction of nonlinear constraints for a limited coordination between legs (controlled by polygonal groups
depending on the parallel robot and the task to be performed). Here, one can apply some adaptation to the Locomotion of
the usual Lagrange multipliers method (Nakamura and T.Yoshikawa, 1989).
4.3 Algorithms and constraints for Dynamics
The dynamic aspects concern to the interactions between different components and with the ground. The hyperredun-
dant character of multibody systems gives non-unique solutions for the distribution of forces and moments (grouped as
wrenches in a representation in terms of the geometry of lines). Hence, one must to apply different optimization proce-
dures to evaluate right evaluation of wrenches which are meaningful to achieve the locomotion. The reasoning scheme
is exactly the same than above for kinematics, but by replacing now the twists by wrenches, with the corresponding
interpretation for the dynamical effects which have been disregarded in lower level analysis.
So, a right evaluation of the model would must guarantee an efficient transference between different quantities (even
in presence of singularities), to make corrections of impulse generation, switching and tuning processes for the activa-
tion/inhibition of movements for each component. The algorithm must evaluate the effect of rotational vector fields actin g
simultaneously onto several joints in C to perform a movement in y, and inversely. This involves to the SVD of the
matrix expression, again.
Global constraints are introduced in terms of Lagrange multipliers involving to each phase: so, they arise from relative
location of interaction forces in friction cones in phase transitions for the ground contact (which involves to the transitions
phases in locomotion with an interchange between closed and open loops for kinematic chains in stance and swing phases),
and incidence conditions about action lines (representing forces and momenta for each component) for the swing phase.
The right coordination requires a model able of integrating both of them. The Clifford Calculus (Hestenes and G.Sobczyk,
1984) provides the theoretical framework to perform this integration (Finat, 2000).
5 CONCLUSIONS AND FUTURE DEVELOPMENTS
In this work, I propose a general model for the Mechanics, Motion Planning and Control of Multilegged Robots, with
à biological inspiration arising from the biped humanoid with some restrictive assumptions. This hierarchised model
supports an equivariant structure, to simplify the design of control charts, and to allow the reusability from replication
of elementary patterns by means of different kinds of reflections. Obviously, this model can be extended to another
multilegged or parallel robots, is the next step.
Propagation phenomena are geometrically modeled in terms of reflections. To develop a more dynamical approach, one
needs to control them in terms of vector fields. This problems concerns to the interaction with the environment, and the
identification of geometric changing elements for the gait analysis. In both cases, the reflections play also an important
role, and this presence allow us to develop the same approach. To identify the mirrors, one can adopt some kind of
mid point rule for equilibrium postures or G-average for scalar time-dependent functionals. Both strategies appear in
recent formulations of analytic mechanics (Marsden, 1992), but as a technical trick more than a systematic approach in a
common geometric framework. The main source for errors arises from troubles to estimate changes appearing in phase
transitions. One needs to develop spatio-temporal dynamical models to improve the matching of a changing dynamics.
244 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000.
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