/
ETS.m
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ETS.m
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%ETS Elementary Transform Sequence class
%
% Manipulate a sequence (vector) of elementary transformations
% - ETS.TX
% - ETS.TY
% - ETS.TZ
% - ETS.RX
% - ETS.RY
% - ETS.RZ
%
% Methods::
% ETS Construct a sequence from string
% isrot True if rotational transform
% istrans True if translational transform
% isjoint Is ETS a function of qj
% njoints Maximum joint variable index
% axis Axis of translation or rotation
% find Find ETS that is a function of qj
% subs Substitute element of sequence
%-
% eval Evaluate ETS
% jacobian Compute Jacobian of ETS
%-
% display Display a sequence in human readable form
% char Convert sequence to a string
%
% Example::
% ets = ETS('Rx(q1)Tx(a1)Ry(q2)Ty(a3)Rz(q3)Rx(pi/2)')
% ets.eval([1 2 3]);
%
% Notes::
% - Still experimental
%
% See also trchain, trchain2.
% TODO:
% - handle 2D case
% - do DHFactor
% - accept parameters from a passed struct rather as well as workspace
% Copyright (C) 1993-2017, by Peter I. Corke
%
% This file is part of The Robotics Toolbox for MATLAB (RTB).
%
% RTB is free software: you can redistribute it and/or modify
% it under the terms of the GNU Lesser General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% RTB is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU Lesser General Public License for more details.
%
% You should have received a copy of the GNU Leser General Public License
% along with RTB. If not, see <http://www.gnu.org/licenses/>.
%
% http://www.petercorke.com
classdef ETS < handle
properties
type % ETS.TX, ETS.TY, ETS.TZ, ETS.RX, ETS.RY, ETS.RZ
val
joint % joint number, if a joint, else 0
constant % eg. 90, for angles
sign
symconstant % eg. L1, for lengths
% DH parameters
theta
D
A
alpha
offset
prismatic
end
properties (Constant)
TX = 0;
TY = 1;
TZ = 2;
RX = 3;
RY = 4;
RZ = 5;
DH = 6;
DHM = 7;
names = {'Tx', 'Ty', 'Tz', 'Rx', 'Ry', 'Rz', 'DH', 'DHm' };
end
% Element.java
% // one of ETS.TX, ETS.TY ... ETS.RZ, DH_STANDARD/MODIFIED
% int type;
%
% // transform parameters, only one of these is set
% String var; // eg. q1, for joint var ETS.TYpes
% String symconst; // eg. L1, for lengths
% int constant; // eg. 90, for angles
%
% // DH parameters, only set if ETS.TYpe is DH_STANDARD/MODIFIED
% int theta,
% alpha;
% String A,
% D;
% int prismatic;
% int offset;
methods
function ets = ETS(s, varargin)
%ETS.ETS Construct elementary transform element or sequence
%
% e = ETS() is a new ETS object.
%
% e = ETS(t) is a clone of the ETS object t and all properties are copied.
%
% e = ETS(op, v) is a new ETS object of type op and value v. OP can be any
% of
% 'Rx' rotation about the x-axis
% 'Ry' rotation about the y-axis
% 'Rz' rotation about the z-axis
% 'Tx' translation along the x-axis
% 'Ty' translation along the y-axis
% 'Tz' translation along the z-axis
% 'transl' sequence of finite translations along the x-, y- and z-directions.
% 'rpy' sequence of finite rotations about the x-, y- and z-directions.
%
% e = ETS(str) is a sequence of ETS objects, each described by a
% subexpression in the string STR. Each subexpression comprises an
% operation as per the table above followed by parentheses and a value.
% For example:
%
% ets = ETS('Rx(q1)Tx(a1)Ry(q2)Ty(a3)Rz(q3)Rx(pi/2)')
if nargin == 0
ets.joint = false;
ets.constant = [];
return;
end
if ischar(s) || isstring(s)
s = string(s);
if strfind(s, '(')
% string of tokens
tokens = regexp(s, '\s*(?<op>R.?|T.)\(\s*(?<arg>[^)]*)\s*\)\s*', 'names');
joint = 1;
ets = [];
for token = tokens
% get the argument for this transform element
% deal with case of symbolic arg or workspace expression
if token.arg.startsWith("q") || token.arg.startsWith("*")
x = ETS(token.op, "");
x.joint = joint;
joint = joint+1;
else
x = ETS(token.op, string(token.arg));
x.joint = 0;
end
ets = [ets x]; % append to the list
end
else
% ETS('transl', [v])
% ETS('rpy', [v]);
switch s
case 'transl'
v = varargin{1};
ets = [];
if v(1) ~= 0
ets = [ets ETS('Tx', v(1))];
end
if v(2) ~= 0
ets = [ets ETS('Ty', v(2))];
end
if v(3) ~= 0
ets = [ets ETS('Tz', v(3))];
end
case 'rpy'
v = varargin{1};
ets = [];
if v(1) ~= 0
ets = [ets ETS('Rx', v(1))];
end
if v(2) ~= 0
ets = [ets ETS('Ry', v(2))];
end
if v(3) ~= 0
ets = [ets ETS('Rz', v(3))];
end
case 'Rx'
ets = ETS(ETS.RX, varargin{1});
case 'Ry'
ets = ETS(ETS.RY, varargin{1});
case 'Rz'
ets = ETS(ETS.RZ, varargin{1});
case 'Tx'
ets = ETS(ETS.TX, varargin{1});
case 'Ty'
ets = ETS(ETS.TY, varargin{1});
case 'Tz'
ets = ETS(ETS.TZ, varargin{1});
otherwise
error('RTB:trchain:badarg', 'unknown operator/option %s', s);
end
end
elseif isnumeric(s)
ets = ETS();
ets.type = s;
ets.val = varargin{1};
elseif isa(s, 'ETS')
% clone the object
ets.type = s.type;
ets.joint = s.joint;
ets.val = s.val;
ets.symconstant = s.symconstant;
if nargin > 1
ets.type = varargin{1};
end
if nargin > 2
if varargin{2} < 0
ets = ets.negate();
end
end
end
end
function s = op(ets)
s = ets.names{ets.type+1};
end
function s = char(ets)
s = '';
for x=ets
e = x.op;
if isjoint(x)
e = " " + e + "(*q" + x.joint + x.val + ")";
else
e = e + "(" + x.val + ")";
end
s = strcat(s, e);
end
end
function display(ets)
%ETS.display Display parameters
%
% ETS.display() displays the transform parameters in compact single line format.
%
% Notes::
% - This method is invoked implicitly at the command line when the result
% of an expression is a Link object and the command has no trailing
% semicolon.
%
% See also Link.char, Link.dyn, SerialLink.showlink.
loose = strcmp( get(0, 'FormatSpacing'), 'loose');
if loose
disp(' ');
end
disp([inputname(1), ' = '])
disp( char(ets) );
end % display()
% function disp(ets)
% disp( show(ets) );
% end
%-------------------------------------------------------------
function T = eval(ets, q)
T = eye(4,4);
for x = ets
% get the argument for this transform element
if x.isjoint()
% from the passed in vector q
arg = q(x.joint);
else % or the workspace
try
arg = evalin('base', x.val);
catch
error('RTB:ETS:badarg', 'variable %s does not exist', x.val);
end
end
% now evaluate the element and update the transform chain
switch x.type
case ETS.RX
T = T * trotx(arg, 'deg');
case ETS.RY
T = T * troty(arg, 'deg');
case ETS.RZ
T = T * trotz(arg, 'deg');
case ETS.TX
T = T * transl(arg, 0, 0);
case ETS.TY
T = T * transl(0, arg, 0);
case ETS.TZ
T = T * transl(0, 0, arg);
otherwise
error('RTB:ETS:badarg', 'unknown operator %s', x.op);
end
end
T = trnorm(T);
% if isa(q, 'symfun')
% T = formula(T);
% end
end
function J = jacobian(ets, q)
n = ets.njoints();
J = zeros(6, n);
for j=1:n
% find derivative with respect to q_j
k = ets.find(j);
deriv = ets(k).deriv(q(j));
pre = eval( ets(1:k-1), q );
post = eval( ets(k+1:end), q );
Td = trnorm(pre)*deriv*trnorm(post);
J(:,j) = [ Td(1:3,4); vex(t2r(Td)) ];
end
end
function T = deriv(ets, q)
switch ets.type
case ETS.RX
T = r2t( skew([1 0 0])*rotx(q) );
case ETS.RY
T = r2t( skew([0 1 0])*roty(q) );
case ETS.RZ
T = r2t( skew([0 0 1])*rotz(q) );
case ETS.TX
T = transl(1, 0, 0);
case ETS.TY
T = transl(0, 1, 0);
case ETS.TZ
T = transl(0, 0, 1);
end
T(4,4) = 0;
end
%********************************************************************
% D H F A C T O R
%********************************************************************
function xo = subs(ets, k, x)
fprintf(' Subs: %s := %s\n', char(ets(k)), char(x));
xo = [ets(1:k-1) x ets(k+1:end)];
end
function k = find(ets, j)
[~,k] = find([ets.joint] == j);
end
function n = njoints(ets)
n = max([ets.joint]');
end
function b = isjoint(ets, j)
b = [ets.joint] ~= 0;
end
function b = istrans(ets)
b = ismember(ets.type, [ETS.TX ETS.TY ETS.TZ]);
end
function b = isrot(ets)
b = ismember(ets.type, [ETS.RX ETS.RY ETS.RZ]);
end
function b = axis(ets)
b = char( mod([ets.type],3) + 'x' );
end
function v = sametype(ets, next)
v = ets.type == next.type;
end
function s = show(ets)
for i=1:numel(ets)
e = ets(i);
s = sprintf('%s: type=%d, joint %d, val=%g. const=%s. sign=%d, sym=%s\n', ...
char(e), e.type, e.joint, e.val, e.constant, e.sign, e.symconstant);
if nargout == 0
disp(s);
end
end
end
function negate(ets)
ets.constant = -ets.constant;
s = ets.symconst;
% add initial sign char if none
if s(1) ~= '+' && s(1) ~= '-'
s = ['+' s];
end
% go through the string and flip all sign chars
kp = strfind(s, '+');
kn = strfind(s, '-');
s(kp) = '-';
s(kn) = '+';
% if inital sign is + remove it
if s(1) == '+';
s = s(2:end);
end
ets.symconst = s;
end
% methods from Element.java
% * public boolean istrans()
% * public boolean isrot()
% * public int axis()
% * public boolean isjoint() {
% * public boolean factorMatch(int dhWhich, int i, int verbose) {
% * public void add(Element e) {
% * public Element(int ETS.TYpe, int constant) {
% * public Element(int ETS.TYpe) // new of specified ETS.TYpe
% *
% * public static String toString(Element [] e) {
% * public String argString() {
% * public String toString() {
% *
% * Constructors:
% * Element(Element e) // clone of argument
% * Element(Element e, int ETS.TYpe, int sign) // clone of argument with new ETS.TYpe
% * Element(Element e, int ETS.TYpe) // clone of argument with new ETS.TYpe
% * Element(String s)
function v = plus(ets, e)
if isstring(ets) && isa(e, 'ETS')
% string concatenate
v = ets + char(e);
end
end
function symPlus(ets, e)
end
function new = merge(a, b)
if isjoint(a)
new = ETS(a);
new.val = num2str(eval(new.val + "+" + b.val));
elseif isjoint(b)
new = ETS(b);
new.val = num2str(eval(new.val + "+" + a.val));
else
% two constants
new = ETS(a);
new.val = num2str(eval(new.val + "+" + b.val));
new.symconstant = new.symconstant + "+" + b.symconstant;
if eval(new.val) == 0
new = [];
end
end
if isempty(new)
fprintf(' Merging: %s %s -> (nil)\n', char(a), char(b));
else
fprintf(' Merging: %s %s -> %s\n', char(a), char(b), char(new));
end
end
function b = swap(ets, next, dhWhich)
b = false
% don't swap if both are joint variables
if ets.isjoint() && next.isjoint()
return
end
switch (dhWhich)
case 'standard'
% we want to sort terms into the order: RZ TX TZ RX
if ((ets.type == ETS.TZ) && (next.type == ETS.TX)) || ...
((ets.type == ETS.TX) && (next.type == ETS.RX) && next.isjoint()) || ... % push constant translations through rotational joints of the same ETS.TYpe
((ets.type == ETS.TY) && (next.type == ETS.RY)) && next.isjoint() || ...
((ets.type == ETS.TZ) && (next.type == ETS.RZ)) && next.isjoint() || ...
(~ets.isjoint() && (this.type == ETS.RX) && (next.type == ETS.TX)) || ...
(~ets.isjoint() && (this.type == ETS.RY) && (next.type == ETS.TY)) ||....
(~ets.isjoint() && ~next.isjoint() && (this.type == ETS.TZ) && (next.type == ETS.RZ)) || ...
((ets.type == ETS.TY) && (next.type == ETS.TZ)) || ... % move ETS.TY terms to the right
((ets.type == ETS.TY) && (next.type == ETS.TX))
fprintf(['Swap: ' char(ets) ' <-> ' char(next)] );
b = true;
end
case 'modified'
if ((ets.type == ETS.RX) && (next.type == ETS.TX)) || ...
((ets.type == ETS.RY) && (next.type == ETS.TY)) || ...
((ets.type == ETS.RZ) && (next.type == ETS.TZ)) || ...
((ets.type == ETS.TZ) && (next.type == ETS.TX))
fprintf(['Swap: ' char(ets) ' <-> ' char(next)] );
b = true;
end
otherwise
error('bad DH ETS.TYpe');
end
b = false;
end
% /**
% * Substitute this transform for a triple of transforms
% * that includes an ETS.RZ or ETS.TZ.
% *
% * @return - null if no substituion required
% * - array of Elements to substitute
% */
function s = substituteToZ(ets, prev)
switch ets.type
case ETS.TX
s(1) = ETS(ETS.RY, 90);
s(2) = ETS(ets, ETS.TZ);
s(3) = ETS(ETS.RY, -90);
case ETS.TY
% there are two options here
if prev.type == ETS.RZ
s(1) = ETS(ETS.RZ, 90);
s(2) = ETS(ets, ETS.TX);
s(3) = ETS(ETS.RZ, -90);
else
s(1) = ETS(ETS.RX, -90);
s(2) = ETS(ets, ETS.TZ);
s(3) = ETS(ETS.RX, 90);
end
case ETS.RX
s(1) = ETS(ETS.RY, 90);
s(2) = ETS(ets, ETS.RZ);
s(3) = ETS(ETS.RY, -90);
case ETS.RY
s(1) = ETS(ETS.RX, -90);
s(2) = ETS(ets, ETS.RZ);
s(3) = ETS(ETS.RX, 90);
case ETS.RY
s(1) = ETS(ETS.RZ, 90);
s(2) = ETS(ets, ETS.RX);
s(3) = ETS(ETS.RZ, -90);
otherwise
s = [];
end
end
function s = substituteTY(ets, k)
pref = -1;
try
if ets(k+1).axis == 'x'
pref = ETS.RX;
end
end
try
if ets(k+1).axis == 'z'
pref = ETS.RZ;
end
end
assert(pref >= 0, 'substituteTY, can''t decide');
if pref == ETS.RX
s(1) = ETS(ETS.RX, -90);
s(2) = ETS(ets(k), ETS.TZ);
s(3) = ETS(ETS.RX, 90);
else
s(1) = ETS(ETS.RZ, 90);
s(2) = ETS(ets(k), ETS.TX);
s(3) = ETS(ETS.RZ, -90);
end
end
function s = substituteRY(ets)
s(1) = ETS(ETS.RZ, 90);
s(2) = ETS(ets, ETS.RX);
s(3) = ETS(ETS.RZ, -90);
end
% /**
% * Simple rewriting rule for adjacent transform pairs. Attempt to
% * eliminate ETS.TY and ETS.RY.
% * @param previous element in list
% * @return - null if no substituion required
% * - array of Elements to subsitute
% */
function s = substituteY(this, prev, next)
s = [];
if (prev.isjoint() || e.isjoint())
return
end
% note that if rotation is -90 we must make the displacement -ve */
if ((prev.type == ETS.RX) && (this.type == ETS.TY))
% RX.TY -> TZ.RX
s(1) = ETS(this, ETS.TZ, prev.constant);
s(2) = ETS(prev);
elseif ((prev.type == ETS.RX) && (this.type == ETS.TZ))
% RX.TZ -> TY.RX
s(1) = ETS(this, ETS.TY, -prev.constant);
s(2) = ETS(prev);
elseif ((prev.type == ETS.RY) && (this.type == ETS.TX))
% RY.TX-> TZ.RY
s(1) = ETS(this, ETS.TZ, -prev.constant);
s(2) = ETS(prev);
elseif ((prev.type == ETS.RY) && (this.type == ETS.TZ))
% RY.TZ-> TX.RY
s(1) = ETS(this, ETS.TX, prev.constant);
s(2) = ETS(prev);
elseif ((prev.type == ETS.TY) && (this.type == ETS.RX))
% TY.RX -> RX.TZ
s(1) = ETS(this);
s(2) = ETS(prev, ETS.TZ, -this.constant);
%%return s;
s = [];
elseif ((prev.type == ETS.RY) && (this.type == ETS.RX))
% RY(Q).RX -> RX.RZ(-Q)
s(1) = ETS(this);
s(2) = ETS(prev, ETS.RZ, -1);
elseif ((prev.type == ETS.RX) && (this.type == ETS.RY))
% RX.RY -> RZ.RX
s(1) = ETS(this, ETS.RZ);
s(2) = ETS(prev);
elseif ((prev.type == ETS.RZ) && (this.type == ETS.RX))
% RZ.RX -> RX.RY
s(1) = ETS(this);
s(2) = ETS(prev, ETS.RY);
s = [];
end
end
function out = mergeterms(ets)
i = 1;
while i < numel(ets)-1
cur = ets(i);
next = ets(i+1);
if cur.sametype(next)
new = merge(cur, next);
if isempty(new)
% cancelation
ets(i:i+1) = [];
else
ets(i) = new;
ets(i+1) = [];
end
else
i = i+ 1;
end
end
out = ets;
end
function out = ordering(ets)
for i=1:numel(ets)-1
cur = ets(i);
next = ets(i+1);
% TZ RZ -> RZ TZ
if cur.type == ETS.TZ && next.type == ETS.RZ && next.isjoint()
ets(i) = next;
ets(i+1) = cur;
fprintf(' Swap: %s <--> %s\n', char(cur), char(next));
end
% RX TX -> TX RX
if cur.type == ETS.RX && next.type == ETS.TX && ~next.isjoint()
ets(i) = next;
ets(i+1) = cur;
fprintf(' Swap: %s <--> %s\n', char(cur), char(next));
end
end
out = ets;
end
function out = simplify(ets)
fprintf('--------- simplify\n');
ets = mergeterms(ets);
while 1
len = numel(ets);
ets = ordering(ets);
ets = mergeterms(ets);
if numel(ets) == len
break;
end
end
out = ets;
end
function v = contains(ets, which)
w = ETS.name2type(which);
for e = ets
v = e.type == w;
if v
return
end
end
end
function dhfactor(ets)
disp(char(ets))
% find which joint to substitute
fprintf('--------- align joints with z-axis\n');
fixup = find( ets.isjoint & ([ets.type] ~= ETS.RZ) );
for k=fliplr(fixup) % work right to left so indices are not broken
new = substituteToZ(ets(k));
ets = subs(ets, k, new);
end
disp(char(ets))
%
ets = simplify(ets);
disp(char(ets))
% now eliminate all Ry
if ets.contains('Ry')
fprintf('--------- eliminate RY\n');
fixup = find( [ets.type] == ETS.RY );
for k=fliplr(fixup) % work right to left so indices are not broken
new = substituteRY(ets(k));
ets = subs(ets, k, new);
end
disp(char(ets))
ets = simplify(ets);
end
% now eliminate all Ty
if ets.contains('Ty')
fprintf('--------- eliminate TY\n');
fixup = find( [ets.type] == ETS.TY );
for k=fliplr(fixup) % work right to left so indices are not broken
new = substituteTY(ets, k);
ets = subs(ets, k, new);
end
disp(char(ets))
ets = simplify(ets);
end
ets = ordering(ets);
disp(char(ets))
factorize(ets)
end
function nfactors = factorize(ets, mdh)
if nargin < 2
mdh = false;
end
verbose = true;
nfactors = 0;
out = [];
for i=1:length(ets)
j = i;
jvars = 0; match = 0;
% scan next 4 terms
if verbose
fprintf("start at " + ets(i));
end
match = 0; jvars = 0; j = i;
for f=1:4
if j > length(ets)
break;
else
e = ets(j);
end
if e.factormatch(f, mdh)
j = j + 1;
match = match + 1;
if (e.isjoint())
jvars = jvars + 1;
end
if jvars > 1 % can only have 1 joint var per DH
break;
end
end
end
if match == 0 || jvars == 0
continue; % no DH subexpression found, keep looking
end
if verbose
fprintf(" found subexpression " + match + " " + jvars + "\n");
end
if jvars == 0
continue;
end
first = i;
last = j;
if jvars > 1
last = last - 1;
end
[first last]
% dh = ETS();
% dh.type = ETS.DH;
%
% for j=first:last
% dh.add( this.get(i) );
% this.remove(i);
% end
%
% this.add(i, dh);
% nfactors = nfactors + 1;
% if verbose
% fprintf(" result: " + dh);
% end
end
end
function match = factormatch(ets, i, mdh)
if nargin < 2
mdh = false;
end
if mdh
dhFactors = [
ETS.RX 0
ETS.TX 0
ETS.RZ 1
ETS.RZ 1 ];
else
dhFactors = [
ETS.RZ 1
ETS.TX 0
ETS.TZ 1
ETS.RX 0 ];
end
match = (ets.type == dhFactors(i,1)) && ...
~((dhFactors(i,2) == 0) && ets.isjoint());
fprintf(" matching " + ets + " (i=" + i + ") " + " to " + ...
op(dhFactors(i,1)) + "<" + dhFactors(i,2) + ">" + ...
" -> " + match + "\n");
end
function new = add(e1, e2)
assert(e1.type == DH || e1.type == DHM, 'wrong element type');
fprintf(" adding: " + e1 + " += " + e2);
switch e2.type
case RZ
if e2.isjoint()
e1.prismatic = 0;
e1.var = e1.var;
e1.offset = e1.constant;
e1.theta = 0.0;
else
e1.theta = e1.constant;
end
case TX
e1.A = e2.symconst;
case TZ
if e1.isjoint()
e1.prismatic = 1;
e1.var = e2.var;
e1.D = null;
else
e1.D = e2.symconst;
end
case RX
e1.alpha = e2.constant;
otherwise
error("cant factorize " + e);
end
end
end
methods (Static)
function t = name2type(name)
switch lower(name)
case 'rx', t = ETS.RX;
case 'ry', t = ETS.RY;
case 'rz', t = ETS.RZ;
case 'tx', t = ETS.TX;
case 'ty', t = ETS.TY;
case 'tz', t = ETS.TZ;
end
end
end
end