Chapter 6: Introduction of Uniform Model Structures for Partial, Temporal, and
6.1. The Orthogonal Group as Source of Uniform Structures in CM
6.1.1. Application Example for the Use of the Orthogonal Matrices as Sources
s 1
= s
α
α (6.1.9)
The basic idea of the above approach was investigated in various contexts (e.g. in [C22], [C23], [C26], [C28], [C40]). In the sequel a polishing application is considered in details using the results published in [C24] [C23]. Following that an alternative approach is detailed that considers the quasi-diagonalization of the inertia matrix in an alternative manner.
6.1.1. Application Example for the Use of the Orthogonal Matrices as Sources of Uniform Structures in Classical Mechanics
Surface to be polished The nominal track is in it.
Center of polishing disk
s0 s_o
s
s_o=FtD/Spr_s Xs
XN (Nominal point) XR
(Max. allow. dist. between real point and nominal point) (Real end point)
(Extended Nom.
Point. for desired Contact Force)
Spring nN=nV
Normal unit vector of the surface
s0-s_o
Figure 6.1.1.1. The idea of transforming the force/position/velocity task into pure kinematic problem by using a passive compliance and the proposed control
The here presented figures and conclusions are taken from [C23]. In the present method a 3 DOF SCARA arm having a translational and two rotary joints was completed by a third “link” in the form of a “pipe” parallel with the telescopic shaft and rigidly attached to the end of the second rotary link. The pipe contains a passive elastic component, a spring of a not very large, a priori known stiffness and negligible viscous damping. Consequently, from the purely kinematic data of the spring's deformation its force depending on the contact force prescribed for polishing as a technology requirement can be determined. By knowing the location of the surface to be polished the required contact force can be transformed into the desired location of the endpoint of the last rotary link which otherwise may have arbitrary velocity with respect to the workshop's system of reference. Via applying a cardan link for fixing the polishing disc in the case of mechanical contact the disc will always be located in the tangent plane of the surface of the work-piece at the given
point. In the case of a relatively precise location of the disk the errors in the positioning of the disk will be transformed into a minor error in the contact force originally prescribed (see Fig. 6.1.1.1.).
Due to the flexibility of the cardan shaft in the model of dynamic interaction of polishing -used in the simulation only- an even pressure distribution “p” over the disc's surface was supposed. The small surface element of the disc “dS” gives the following contribution into the torque:
r r rotation far exceeded the translational component vTR. By neglecting the effect of the central part of the disk this yields the normal component of
RF
in which “F” is the absolute value of the contact force pressing the disc against the work-piece, and “µ” denotes the friction coefficient. The net force of friction from the small surface element has an expression to (6.1.10)
Again, by neglecting the effect of the small central part of the disc the term in the parentheses in (6.1.12) corresponds to a rotating unit vector resulting in zero in the integral. Therefore the effect of the contact forces was modeled according to
≡∂ξ in (6.1.6) and in (6.1.7). The estimated inertia was integrated according to these ever varying coefficients. In principle such decomposition can describe the Coriolis forces and other terms quadratic in the angular velocities in (6.1.5). The initial model was a pure diagonal matrix proportional to the identity operator. This was improved step by step by tuning the “gijk” parameters according to the Simplex Algorithm in which the optimum i.e. the difference between the desired and the achieved joint accelerations was minimized. To support this process the following ancillary tools were applied:
• an “Additional Generalized Force” term based on a simple version of regression analysis in which the prediction is “qualified” and adaptivity for slow motion, too;
• a slower external loop simultaneously tuning a “slope” in the parameters according to the Simplex Algorithm in which a function of the difference between the desired (“D”) and the realized (“R”) joint accelerations is minimized:
This cost function is proportional to the relative error in the acceleration for “large”
desired acceleration, and it approximates the absolute error for “small” desired ones (terms large and small are to be understood in comparison with a constant KRel). To support this process further ancillary tools were applied:
• a sigmoid
( )
x =x/(
1+ x)
∈(
−1,1)
function used in stabilization against the effect of extreme noises in the terms( )
ξ sin(
π(
ξ /π) )
(for reducing computational complexity this saturated nature is not taken into account in the calculation of the partial derivatives of H);
• an “additional generalized force” term based on a simple version of regression analysis in which the prediction is “qualified” and suppressed according to the noisiness of the environment it originates from [C23];
• a PID term in which the coefficients of the proportional, derivative and integrated term are tuned as the function of the integrated error in order to keep a prescribed pole-structure in the desired damping of the coordinate errors fixed (described in details e.g. in [9]); in the present version this approach is improved by allowing this feedback increase if the overall torques of the drives are smooth functions of time, and it is decreased in the more "noisy" phases of the motion;
here “noisiness” is determined by the forgetting sum
(
1)
int( ) ( ) (
1)
int t+ = ×c t + t − t−
c α Q Q and a fuzzy membership
function describing the “smoothness” of the torque signal in comparison with a reference value cCoeff.
( )
′ ′
+
=
∫
∞
− 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