Robust Time-Optimal Path Tracking Control of Robots: Theory and Experiments Implementation Details Aidan James Cahill B.Sc. (Hons) Mathematics, Northumbria (UK) December 1995 An addendum to a thesis submitted for the degree of Doctor of Philosophy of the Australian National University Department of Systems Engineering Research School of Information Sciences and Engineering The Australian National University
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R ob u st T im e-O p tim al P a th Tracking
C ontrol o f R obots:
T heory and E xp erim en ts
Im plem entation D etails
A idan Jam es CahillB.Sc. (Hons) M athem atics, Northum bria (UK)
December 1995
An addendum to a thesis submitted for the degree of
Doctor of Philosophy of the Australian National University
D epartm ent of Systems Engineering
Research School of Information Sciences and Engineering
The Australian National University
A b strac t
This document supplements the information contained in the thesis “R obust Time-
O ptim al P a th Tracking Control of Robots: Theory and Experim ents” . Specifically, it
deals w ith im plem entation of the theory described in C hapters 3, 4 and 5 of the m ain
tex t which describes a m ethod for the planning of tim e-optim al trajectories th a t are
robust to plant uncertainties and which ensures the tracking of paths to a specified
tolerance.
l
C ontents
A b stra c t
1 In tro d u c tio n ................................................................................................................
2 Identification of the Robot M o d e l.......................................................................
3 Real-Time Control L a w s ........................................................................................
4 Im plem enting the Acceleration D isturbance T h e o ry .....................................
4.1 Calculation of the Acceleration D isturbance S ig n a l ........................
4.2 Selecting the Controller Gains to Control the E r r o r s ....................
4.3 Calculating the Com pensation T o rques................................................
5 Offline Trajectory P l a n n in g .................................................................................
6 Closed-Loop Trajectory G e n e ra tio n ....................................................................
6.1 Calculation of the W orst-Case Controllable Region Boundary . .
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6.2 Online Trajectory Generation 13
1 In tro d u ctio n 1
1 In trod u ction
This docum ent has been w ritten as a supplem ent to the thesis “R obust Tim e-O ptim al
P a th Tracking Control of Robots: Theory and Experim ents” , subm itted for the degree
of Doctor of Philosophy of the A ustralian National University.
The m ain purpose of the document is to provide additional inform ation regarding
the im plem entation of the theory described in C hapters 3, 4 and 5 in the m ain text.
The theory presented therein describes a m ethod for the planning of tim e-optim al
trajectories th a t are robust to plant uncertainties and which ensures the tracking of
paths to a specified tolerance. The inform ation presented herein focusses m ainly on
the m athem atical and com putational aspects of the im plem entation.
This docum ent does not consider either the work involving the dynamic program
ming solution to the tim e-optim al trajectory planning problem (C hapter 2) or the work
involving the use of singular configurations for high-speed pick and place operations
(C hapter 6). The formulation and com putation of the dynamic program m ing solution
are described adequately in the main tex t, and the details of the im plem entation of
the solution are covered by the work described herein. Similarly, the algorithm s for the
high-speed pick and place operations are based on a simple adap tation of the shooting
m ethods of Bobrow et al. [1] discussed herein.
2 Id en tifica tio n o f th e R ob ot M od el 2
2 Id en tifica tion o f th e R ob ot M od el
In our experim ents, we utilise 2 degrees of freedom of the 4 degree of freedom SCARA
m anipulator th a t we have in our laboratory. For the identification of the model param
eters of these 2 degrees of freedom, a standard least-squares identification technique is
employed.
The technique is based on noting th a t whilst the dynamic equations
M(q(*))q(t) + n(q(t), q(t)) = T (1)
are highly nonlinear in the joint positions and velocities, they are linear in the param
eters. Thus, equations (1) may be re-expressed as
w(q(£) ,q ( t ) ,q ( f ) )$ = r{ t ) , ( 2)
where <I> = ( / n , / 12, •••, ^2j>, n)T is the vector containing the model param eters.
An experim ent is run in which known and discrete torque values T ^ , k = 1 to kf ,
are applied as input and the resulting joint positions q ^ , k = 1 to kf, are measured.
Here, kf = ^ + 1 , where ty is the tim e over which the experim ent is to be run and A t
is the sample period.
Estim ates of joint velocities q ^ and accelerations q(*:} are taken using the Euler-
step approxim ations
n (fc) _ n (fc—l ) a (*0 _ / , (£ - ! )q(*> = q - q and q<*> = - S ------ , k = 1 to kf , (3)M At At 1
where q(°) = q(0) and q(°) = 0, and then the values of q ^ :̂ , q ^ , q ^ and T ^ are
inserted into the discrete equivalent of equations (2)
w ( q W , q W , q W ) $ = T<*> . (4)
Next, bo th sides of equations (4) are low-pass filtered to reduce the effects of noise
introduced by taking the numerical derivatives (3). This yields
(w(qW, qW, qW)) i $ = ( r W ) i , (5)
2 Identification o f the R obot M odel 3
where (•)/ represents the low-pass filter w ith cut-off frequency lrad /s , and where the
constant model param eter vector <I> is unaffected by the filtering.
Finally, equations (5) are inverted as
$ = ( w ( q W ,q < ‘),q<*>))| , (6)
to provide a least-squares fit to the model param eters <I>. Here, ^ w ( q ^ , q(fc) ,
is the pseudo-inverse of ^ w (q (^ , q ^ ) , q (D )J^
In our case, we differ from this procedure only in th a t we identify the model param
eters in two stages. Firstly, we identify the friction param eters by using the “slow”
joint position tra jectory
sin(O.lf) + cos(0.25t) \q r = 60000 t = 0 : A t : 12s (7)sin(0.25t) + sin(0.4f) I
as reference input to the PD control law
r<*> = kf, (q(‘> - q<*>) + k„q(t> , 1 to (8)
Secondly, we identify the remaining param eters by using the “fast” jo in t position t r a