Chapter 6: Introduction of Uniform Model Structures for Partial, Temporal, and
6.1. The Orthogonal Group as Source of Uniform Structures in CM
6.1.2. Simulation Results for the Use of Diagonalization of the Inertia Matrix 46
′ ′
+
=
∫
∞
− t
t d t sigmoid
c ε
κ 1 50
2 (6.1.16)
κ κ
κ , 4
, 4
2
2 c
k c c
b c
c′= + ′= + = (6.1.17)
The fraction in ‘c’ can be also interpreted as a fuzzy set describing the
“smoothness” of the control: for small torque derivatives it approaches 1, while for too fast changes in the momentum it converges to zero; this rigid rule means that for strongly varying momentum it is not reasonable to require too strong feedback in order to avoid instabilities and overshoots, but in the “stable phase” of the control an increase in the feedback may improve accuracy.
• “external loop parameters” of slow tuning used as reference values -built in certain fuzzy membership functions- in the “assessment” of several properties of the control; their appropriate value can be set roughly “experimentally”; further slow real-time tuning can help in finding their optimum value; since the optimum setting can change in time, it is expedient to keep them adjusted in real-time.
All the above ancillary tools required minor computational power and also were independent of the particular characteristics of the control problem to be solved.
6.1.2. Simulation Results for the Use of Diagonalization of the Inertia Matrix In the simulations the robot had the task of polishing a strip on a bell-shaped surface. The strip was located at constant distance from the telescopic axis of the robot. The force with which the polishing disk was requested to press the surface was 1200 [N], the spring in the elastic component had the stiffness of Spr=400 [N/m].
Detailed figures are given in Appendix A.3.
As a conclusion of this section it can be stated that it was illustrated via simulations that the proposed method combining an improved version of the classic PID/ST and simple uniform structures with free parameters adjusted by the Simplex Algorithm and with the ancillary tool of regression analysis can co-operate successfully. The synthesis of the individually quite limited methods leads to an efficient control in which the significance of the different components remains comparable and changes according to the task to be executed. The method is free from scaling problems. It can be regarded as a compromise between the traditional Soft Computing and Hard Computing. The introduction of the passive compliant element makes was successful for technological operations.
6.2. The Symplectic Group as Source of Uniform Structures in CM
The phenomenological foundation of any analytical description in Classical Analytical Mechanics is the Lagrangian Model, by the use of which the kinetic energy of the mechanical systems can be formulated in an inertial system of coordinates in the Newtonian sense. The generalized coordinates and generalized forces in the Lagrangian model normally in principle are directly measurable (observable) quantities as rotational angles, angular or linear velocities, and force or torque components. Via introducing the generalized momentums the Hamiltonian Model can be “built up” on the basis of the Lagrangian one for conservative mechanical systems by using the Legendre Transformation. This model considers the set of the possible physical states of the system to be a differentiable manifold for the
description of which different “maps” or abstract systems of coordinates can be applied. Nature distinguishes those maps by the use of which the mathematical form of the state propagation gains the possible simplest form. The coordinates of these special maps are referred to as canonical coordinates, by the use of which the equations of motion take the form of
Free represented by the canonical coordinates consisting of the generalized coordinates q in the first, and the generalized momentums p in the second block of DOF dimensions (DOF = Degrees of Freedom) [qT,pT]T, and the block of the generalized forces ~Free [ T, FreeT]T
Q 0
Q = . It is important to note that the first DOF components of Q~Free
must be zero, this directly follows from the Legendre Transformation and from the Euler-Lagrange equations of motion. From physical point of view the nonzero components of the generalized forces have force dimension [N] for linear, and torque dimensions [Nm] for the rotary joints. They represent the appropriate projections of the external free forces exerted by the environment on the robot or by its own drives.
By applying some different map for describing the same physical system another canonical coordinates x'(x) can be introduced leaving the form of the equation of motion exactly the same as in (6.2.1). These transformations are defined by the restriction that their Jacobian must be Symplectic, that is
1
and lead to the “transformation rule” for the generalized momentums as
∑
′ described by the Orthogonal Transformations leaving the scalar product of two arbitrary vectors unchanged both in form and in numerical value, the Symplectic Transformations can also be considered as mathematical tools describing the internal symmetry of the Classical Mechanical Systems. They leave both the numerical value and the form of the quadratic Symplectic structure∑
ℑj v are two arbitrary vectors of the tangent space of the system's physical states. The geometry based on the Symplectic structure is called the Symplectic Geometry (see details in Appendix A.10. for the analogies between various geometries occurring in Natural Sciences).
In close analogy with the idea applied by Lajos Jánossy the Symplectic Transformations given in (6.2.2) can be also interpreted in an alternative manner that offers the possibility for using them in modeling Classical Mechanical Systems for control purposes. Jánossy studied the Lorentz Transformations for the four-component space-time and other physically important vectors x’(x)=ΛΛΛx being the Λ internal symmetry of Maxwell’s Electrodynamics (see details in Appendix A.10.) and applied the following observation: a given Lorentz transformation may have two
kinds of physical interpretation: a) the x’ coordinates serve as new coordinates on the same physical system for the description of which the original coordinates x were used; b) the x’ coordinates may be interpreted as the coordinates of a different physical system (the “deformed” version of the original one) that behaves similar manner as the original physical system since it obeys the same symmetry restriction.
Jánossy called this latter interpretation as the “Deformation Principle”.
According to the Deformation Principle the Canonical Transformations may be also interpreted as deformations of the original mechanical system, and on this basis local canonical transformations may be regarded as mathematical possibilities for describing adaptive control by modifying the free parameters of these local transformations. Geometric, group-theoretic and algorithmic aspects of the method were analyzed in details in [J2]. However, for correct phenomenological interpretability the restriction guaranteeing the structure ~Free [ T, FreeT]T
Q 0
Q = in
(6.2.3) has to be kept in mind, too.
As it will be discussed in details in paragraph “6.2.2. Complementary Tuning Possibilities in the Cumulative Control”, for keeping the first n (n is equal to the Degree of Freedom of the mechanical system) components of Q~Free
zero even in the case of external perturbations in general block-diagonal symplectic transformations are needed. (Though these special block-diagonal transformations do not “mix” the q and p components, some coupling between them still remains: if q is shrunk/stretched then p has to suffer stretching/shrinking. The dimension of the Symplectic transformations not mixing the q and p components is n2.)
If we apply a very rough approximate dynamic system model, it can be characterized by a constant inertia matrix M, a constant gravitational term h, and a Lagrangian defined as
( )
q q qTMq hTqL & ≡ & & +
2
, 1 (6.2.4)
The appropriate restrictions to be imposed for the purposes of the deformation principle are as follows: In the canonical map directly deduced from the Lagrangian model the generalized force vectors have only 1×DOF non-zero components. Since a general canonical transformation can combine all the 2×DOF components of the transformed generalized force vectors, a considerable part of the canonical transformations cannot be applied for deformation purposes. Only those solutions can be accepted for which the necessarily “truncated”, phenomenologically non-interpretable components of the generalized force vector are negligible in comparison with the interpretable parts.
On the basis of simple geometric and algebraic considerations (using the Symplectizing Algorithm operating with the concept of the Antiorthogonal Subspaces detailed in Table A.10.1. of Appendix A.10.) in the first step of the control a
“drastic” Symplectic transformation can be introduced.
The effect of this transformation can either be “refined” by applying further Symplectic deformations in the consecutive control steps, or it can be started from the initial rough model immediately in each step.
In the first case, due to the group properties of the symplectic matrices the expected result is a symplectic matrix expressed as a product of the first matrix and many other, near unity symplectic transformations, assuming that the method converges. Since the effects of the step by step deformations are “accumulated” in this product, this solution is referred to as the “cumulative approach”. The latter one
using drastic deformations in each single step is called the “non-cumulative approach”. Of course, when the result of the cumulative approach starts to become
“extreme”, casually the identification can be started the onslaught again.
Further possibility independent of this cumulative/non-cumulative approach is tuning the parameters of the Symplectic Group. For each Lie Group in a manner similar to that of the Orthogonal Matrices uniform structures can be introduced for describing its elements. In the sequel the use of these two kinds of transformations will be discussed.