Initial Commit
This commit is contained in:
commit
6eac907a4a
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Figures/control_measure_fixed_2dof.png
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Figures/control_measure_rotating_2dof.png
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Figures/rotating_frame_2dof.png
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Figures/simscape.png
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Figures/simscape.png
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Figures/simulink_ctrl.png
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145
css/htmlize.css
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145
css/htmlize.css
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|
||||
.org-bold { /* bold */ font-weight: bold; }
|
||||
.org-bold-italic { /* bold-italic */ font-weight: bold; font-style: italic; }
|
||||
.org-buffer-menu-buffer { /* buffer-menu-buffer */ font-weight: bold; }
|
||||
.org-builtin { /* font-lock-builtin-face */ color: #7a378b; }
|
||||
.org-button { /* button */ text-decoration: underline; }
|
||||
.org-calendar-today { /* calendar-today */ text-decoration: underline; }
|
||||
.org-change-log-acknowledgement { /* change-log-acknowledgement */ color: #b22222; }
|
||||
.org-change-log-conditionals { /* change-log-conditionals */ color: #a0522d; }
|
||||
.org-change-log-date { /* change-log-date */ color: #8b2252; }
|
||||
.org-change-log-email { /* change-log-email */ color: #a0522d; }
|
||||
.org-change-log-file { /* change-log-file */ color: #0000ff; }
|
||||
.org-change-log-function { /* change-log-function */ color: #a0522d; }
|
||||
.org-change-log-list { /* change-log-list */ color: #a020f0; }
|
||||
.org-change-log-name { /* change-log-name */ color: #008b8b; }
|
||||
.org-comint-highlight-input { /* comint-highlight-input */ font-weight: bold; }
|
||||
.org-comint-highlight-prompt { /* comint-highlight-prompt */ color: #00008b; }
|
||||
.org-comment { /* font-lock-comment-face */ color: #999988; font-style: italic; }
|
||||
.org-comment-delimiter { /* font-lock-comment-delimiter-face */ color: #999988; font-style: italic; }
|
||||
.org-completions-annotations { /* completions-annotations */ font-style: italic; }
|
||||
.org-completions-common-part { /* completions-common-part */ color: #000000; background-color: #ffffff; }
|
||||
.org-completions-first-difference { /* completions-first-difference */ font-weight: bold; }
|
||||
.org-constant { /* font-lock-constant-face */ color: #008b8b; }
|
||||
.org-diary { /* diary */ color: #ff0000; }
|
||||
.org-diff-context { /* diff-context */ color: #7f7f7f; }
|
||||
.org-diff-file-header { /* diff-file-header */ background-color: #b3b3b3; font-weight: bold; }
|
||||
.org-diff-function { /* diff-function */ background-color: #cccccc; }
|
||||
.org-diff-header { /* diff-header */ background-color: #cccccc; }
|
||||
.org-diff-hunk-header { /* diff-hunk-header */ background-color: #cccccc; }
|
||||
.org-diff-index { /* diff-index */ background-color: #b3b3b3; font-weight: bold; }
|
||||
.org-diff-nonexistent { /* diff-nonexistent */ background-color: #b3b3b3; font-weight: bold; }
|
||||
.org-diff-refine-change { /* diff-refine-change */ background-color: #d9d9d9; }
|
||||
.org-dired-directory { /* dired-directory */ color: #0000ff; }
|
||||
.org-dired-flagged { /* dired-flagged */ color: #ff0000; font-weight: bold; }
|
||||
.org-dired-header { /* dired-header */ color: #228b22; }
|
||||
.org-dired-ignored { /* dired-ignored */ color: #7f7f7f; }
|
||||
.org-dired-mark { /* dired-mark */ color: #008b8b; }
|
||||
.org-dired-marked { /* dired-marked */ color: #ff0000; font-weight: bold; }
|
||||
.org-dired-perm-write { /* dired-perm-write */ color: #b22222; }
|
||||
.org-dired-symlink { /* dired-symlink */ color: #a020f0; }
|
||||
.org-dired-warning { /* dired-warning */ color: #ff0000; font-weight: bold; }
|
||||
.org-doc { /* font-lock-doc-face */ color: #8b2252; }
|
||||
.org-escape-glyph { /* escape-glyph */ color: #a52a2a; }
|
||||
.org-file-name-shadow { /* file-name-shadow */ color: #7f7f7f; }
|
||||
.org-flyspell-duplicate { /* flyspell-duplicate */ color: #cdad00; font-weight: bold; text-decoration: underline; }
|
||||
.org-flyspell-incorrect { /* flyspell-incorrect */ color: #ff4500; font-weight: bold; text-decoration: underline; }
|
||||
.org-fringe { /* fringe */ background-color: #f2f2f2; }
|
||||
.org-function-name { /* font-lock-function-name-face */ color: teal; }
|
||||
.org-header-line { /* header-line */ color: #333333; background-color: #e5e5e5; }
|
||||
.org-help-argument-name { /* help-argument-name */ font-style: italic; }
|
||||
.org-highlight { /* highlight */ background-color: #b4eeb4; }
|
||||
.org-holiday { /* holiday */ background-color: #ffc0cb; }
|
||||
.org-isearch { /* isearch */ color: #b0e2ff; background-color: #cd00cd; }
|
||||
.org-isearch-fail { /* isearch-fail */ background-color: #ffc1c1; }
|
||||
.org-italic { /* italic */ font-style: italic; }
|
||||
.org-keyword { /* font-lock-keyword-face */ color: #0086b3; }
|
||||
.org-lazy-highlight { /* lazy-highlight */ background-color: #afeeee; }
|
||||
.org-link { /* link */ color: #0000ff; text-decoration: underline; }
|
||||
.org-link-visited { /* link-visited */ color: #8b008b; text-decoration: underline; }
|
||||
.org-log-edit-header { /* log-edit-header */ color: #a020f0; }
|
||||
.org-log-edit-summary { /* log-edit-summary */ color: #0000ff; }
|
||||
.org-log-edit-unknown-header { /* log-edit-unknown-header */ color: #b22222; }
|
||||
.org-match { /* match */ background-color: #ffff00; }
|
||||
.org-next-error { /* next-error */ background-color: #eedc82; }
|
||||
.org-nobreak-space { /* nobreak-space */ color: #a52a2a; text-decoration: underline; }
|
||||
.org-org-archived { /* org-archived */ color: #7f7f7f; }
|
||||
.org-org-block { /* org-block */ color: #7f7f7f; }
|
||||
.org-org-block-begin-line { /* org-block-begin-line */ color: #b22222; }
|
||||
.org-org-block-end-line { /* org-block-end-line */ color: #b22222; }
|
||||
.org-org-checkbox { /* org-checkbox */ font-weight: bold; }
|
||||
.org-org-checkbox-statistics-done { /* org-checkbox-statistics-done */ color: #228b22; font-weight: bold; }
|
||||
.org-org-checkbox-statistics-todo { /* org-checkbox-statistics-todo */ color: #ff0000; font-weight: bold; }
|
||||
.org-org-clock-overlay { /* org-clock-overlay */ background-color: #ffff00; }
|
||||
.org-org-code { /* org-code */ color: #7f7f7f; }
|
||||
.org-org-column { /* org-column */ background-color: #e5e5e5; }
|
||||
.org-org-column-title { /* org-column-title */ background-color: #e5e5e5; font-weight: bold; text-decoration: underline; }
|
||||
.org-org-date { /* org-date */ color: #a020f0; text-decoration: underline; }
|
||||
.org-org-document-info { /* org-document-info */ color: #191970; }
|
||||
.org-org-document-info-keyword { /* org-document-info-keyword */ color: #7f7f7f; }
|
||||
.org-org-document-title { /* org-document-title */ color: #191970; font-size: 144%; font-weight: bold; }
|
||||
.org-org-done { /* org-done */ color: #228b22; font-weight: bold; }
|
||||
.org-org-drawer { /* org-drawer */ color: #0000ff; }
|
||||
.org-org-ellipsis { /* org-ellipsis */ color: #b8860b; text-decoration: underline; }
|
||||
.org-org-footnote { /* org-footnote */ color: #a020f0; text-decoration: underline; }
|
||||
.org-org-formula { /* org-formula */ color: #b22222; }
|
||||
.org-org-headline-done { /* org-headline-done */ color: #bc8f8f; }
|
||||
.org-org-hide { /* org-hide */ color: #ffffff; }
|
||||
.org-org-latex-and-export-specials { /* org-latex-and-export-specials */ color: #8b4513; }
|
||||
.org-org-level-1 { /* org-level-1 */ color: #0000ff; }
|
||||
.org-org-level-2 { /* org-level-2 */ color: #a0522d; }
|
||||
.org-org-level-3 { /* org-level-3 */ color: #a020f0; }
|
||||
.org-org-level-4 { /* org-level-4 */ color: #b22222; }
|
||||
.org-org-level-5 { /* org-level-5 */ color: #228b22; }
|
||||
.org-org-level-6 { /* org-level-6 */ color: #008b8b; }
|
||||
.org-org-level-7 { /* org-level-7 */ color: #7a378b; }
|
||||
.org-org-level-8 { /* org-level-8 */ color: #8b2252; }
|
||||
.org-org-link { /* org-link */ color: #0000ff; text-decoration: underline; }
|
||||
.org-org-meta-line { /* org-meta-line */ color: #b22222; }
|
||||
.org-org-mode-line-clock { /* org-mode-line-clock */ color: #000000; background-color: #bfbfbf; }
|
||||
.org-org-mode-line-clock-overrun { /* org-mode-line-clock-overrun */ color: #000000; background-color: #ff0000; }
|
||||
.org-org-quote { /* org-quote */ color: #7f7f7f; }
|
||||
.org-org-scheduled { /* org-scheduled */ color: #006400; }
|
||||
.org-org-scheduled-previously { /* org-scheduled-previously */ color: #b22222; }
|
||||
.org-org-scheduled-today { /* org-scheduled-today */ color: #006400; }
|
||||
.org-org-sexp-date { /* org-sexp-date */ color: #a020f0; }
|
||||
.org-org-special-keyword { /* org-special-keyword */ color: #a020f0; }
|
||||
.org-org-table { /* org-table */ color: #0000ff; }
|
||||
.org-org-tag { /* org-tag */ font-weight: bold; }
|
||||
.org-org-target { /* org-target */ text-decoration: underline; }
|
||||
.org-org-time-grid { /* org-time-grid */ color: #b8860b; }
|
||||
.org-org-todo { /* org-todo */ color: #ff0000; font-weight: bold; }
|
||||
.org-org-upcoming-deadline { /* org-upcoming-deadline */ color: #b22222; }
|
||||
.org-org-verbatim { /* org-verbatim */ color: #7f7f7f; }
|
||||
.org-org-verse { /* org-verse */ color: #7f7f7f; }
|
||||
.org-org-warning { /* org-warning */ color: #ff0000; font-weight: bold; }
|
||||
.org-outline-1 { /* outline-1 */ color: #0000ff; }
|
||||
.org-outline-2 { /* outline-2 */ color: #a0522d; }
|
||||
.org-outline-3 { /* outline-3 */ color: #a020f0; }
|
||||
.org-outline-4 { /* outline-4 */ color: #b22222; }
|
||||
.org-outline-5 { /* outline-5 */ color: #228b22; }
|
||||
.org-outline-6 { /* outline-6 */ color: #008b8b; }
|
||||
.org-outline-7 { /* outline-7 */ color: #7a378b; }
|
||||
.org-outline-8 { /* outline-8 */ color: #8b2252; }
|
||||
.org-preprocessor { /* font-lock-preprocessor-face */ color: #7a378b; }
|
||||
.org-query-replace { /* query-replace */ color: #b0e2ff; background-color: #cd00cd; }
|
||||
.org-regexp-grouping-backslash { /* font-lock-regexp-grouping-backslash */ font-weight: bold; }
|
||||
.org-regexp-grouping-construct { /* font-lock-regexp-grouping-construct */ font-weight: bold; }
|
||||
.org-region { /* region */ background-color: #eedc82; }
|
||||
.org-secondary-selection { /* secondary-selection */ background-color: #ffff00; }
|
||||
.org-shadow { /* shadow */ color: #7f7f7f; }
|
||||
.org-show-paren-match { /* show-paren-match */ background-color: #40e0d0; }
|
||||
.org-show-paren-mismatch { /* show-paren-mismatch */ color: #ffffff; background-color: #a020f0; }
|
||||
.org-string { /* font-lock-string-face */ color: #dd1144; }
|
||||
.org-tool-bar { /* tool-bar */ color: #000000; background-color: #bfbfbf; }
|
||||
.org-tooltip { /* tooltip */ color: #000000; background-color: #ffffe0; }
|
||||
.org-trailing-whitespace { /* trailing-whitespace */ background-color: #ff0000; }
|
||||
.org-type { /* font-lock-type-face */ color: #228b22; }
|
||||
.org-underline { /* underline */ text-decoration: underline; }
|
||||
.org-variable-name { /* font-lock-variable-name-face */ color: teal; }
|
||||
.org-warning { /* font-lock-warning-face */ color: #ff0000; font-weight: bold; }
|
||||
.org-widget-button { /* widget-button */ font-weight: bold; }
|
||||
.org-widget-button-pressed { /* widget-button-pressed */ color: #ff0000; }
|
||||
.org-widget-documentation { /* widget-documentation */ color: #006400; }
|
||||
.org-widget-field { /* widget-field */ background-color: #d9d9d9; }
|
||||
.org-widget-inactive { /* widget-inactive */ color: #7f7f7f; }
|
||||
.org-widget-single-line-field { /* widget-single-line-field */ background-color: #d9d9d9; }
|
||||
1095
css/readtheorg.css
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css/readtheorg.css
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js/bootstrap.min.js
vendored
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js/bootstrap.min.js
vendored
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4
js/jquery.min.js
vendored
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4
js/jquery.min.js
vendored
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1
js/jquery.stickytableheaders.min.js
vendored
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1
js/jquery.stickytableheaders.min.js
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@ -0,0 +1 @@
|
||||
!function(a,b){"use strict";function c(c,g){var h=this;h.$el=a(c),h.el=c,h.id=e++,h.$window=a(b),h.$document=a(document),h.$el.bind("destroyed",a.proxy(h.teardown,h)),h.$clonedHeader=null,h.$originalHeader=null,h.isSticky=!1,h.hasBeenSticky=!1,h.leftOffset=null,h.topOffset=null,h.init=function(){h.$el.each(function(){var b=a(this);b.css("padding",0),h.$originalHeader=a("thead:first",this),h.$clonedHeader=h.$originalHeader.clone(),b.trigger("clonedHeader."+d,[h.$clonedHeader]),h.$clonedHeader.addClass("tableFloatingHeader"),h.$clonedHeader.css("display","none"),h.$originalHeader.addClass("tableFloatingHeaderOriginal"),h.$originalHeader.after(h.$clonedHeader),h.$printStyle=a('<style type="text/css" media="print">.tableFloatingHeader{display:none !important;}.tableFloatingHeaderOriginal{position:static !important;}</style>'),a("head").append(h.$printStyle)}),h.setOptions(g),h.updateWidth(),h.toggleHeaders(),h.bind()},h.destroy=function(){h.$el.unbind("destroyed",h.teardown),h.teardown()},h.teardown=function(){h.isSticky&&h.$originalHeader.css("position","static"),a.removeData(h.el,"plugin_"+d),h.unbind(),h.$clonedHeader.remove(),h.$originalHeader.removeClass("tableFloatingHeaderOriginal"),h.$originalHeader.css("visibility","visible"),h.$printStyle.remove(),h.el=null,h.$el=null},h.bind=function(){h.$scrollableArea.on("scroll."+d,h.toggleHeaders),h.isWindowScrolling||(h.$window.on("scroll."+d+h.id,h.setPositionValues),h.$window.on("resize."+d+h.id,h.toggleHeaders)),h.$scrollableArea.on("resize."+d,h.toggleHeaders),h.$scrollableArea.on("resize."+d,h.updateWidth)},h.unbind=function(){h.$scrollableArea.off("."+d,h.toggleHeaders),h.isWindowScrolling||(h.$window.off("."+d+h.id,h.setPositionValues),h.$window.off("."+d+h.id,h.toggleHeaders)),h.$scrollableArea.off("."+d,h.updateWidth)},h.toggleHeaders=function(){h.$el&&h.$el.each(function(){var b,c=a(this),d=h.isWindowScrolling?isNaN(h.options.fixedOffset)?h.options.fixedOffset.outerHeight():h.options.fixedOffset:h.$scrollableArea.offset().top+(isNaN(h.options.fixedOffset)?0:h.options.fixedOffset),e=c.offset(),f=h.$scrollableArea.scrollTop()+d,g=h.$scrollableArea.scrollLeft(),i=h.isWindowScrolling?f>e.top:d>e.top,j=(h.isWindowScrolling?f:0)<e.top+c.height()-h.$clonedHeader.height()-(h.isWindowScrolling?0:d);i&&j?(b=e.left-g+h.options.leftOffset,h.$originalHeader.css({position:"fixed","margin-top":h.options.marginTop,left:b,"z-index":3}),h.leftOffset=b,h.topOffset=d,h.$clonedHeader.css("display",""),h.isSticky||(h.isSticky=!0,h.updateWidth()),h.setPositionValues()):h.isSticky&&(h.$originalHeader.css("position","static"),h.$clonedHeader.css("display","none"),h.isSticky=!1,h.resetWidth(a("td,th",h.$clonedHeader),a("td,th",h.$originalHeader)))})},h.setPositionValues=function(){var a=h.$window.scrollTop(),b=h.$window.scrollLeft();!h.isSticky||0>a||a+h.$window.height()>h.$document.height()||0>b||b+h.$window.width()>h.$document.width()||h.$originalHeader.css({top:h.topOffset-(h.isWindowScrolling?0:a),left:h.leftOffset-(h.isWindowScrolling?0:b)})},h.updateWidth=function(){if(h.isSticky){h.$originalHeaderCells||(h.$originalHeaderCells=a("th,td",h.$originalHeader)),h.$clonedHeaderCells||(h.$clonedHeaderCells=a("th,td",h.$clonedHeader));var b=h.getWidth(h.$clonedHeaderCells);h.setWidth(b,h.$clonedHeaderCells,h.$originalHeaderCells),h.$originalHeader.css("width",h.$clonedHeader.width())}},h.getWidth=function(c){var d=[];return c.each(function(c){var e,f=a(this);if("border-box"===f.css("box-sizing"))e=f[0].getBoundingClientRect().width;else{var g=a("th",h.$originalHeader);if("collapse"===g.css("border-collapse"))if(b.getComputedStyle)e=parseFloat(b.getComputedStyle(this,null).width);else{var i=parseFloat(f.css("padding-left")),j=parseFloat(f.css("padding-right")),k=parseFloat(f.css("border-width"));e=f.outerWidth()-i-j-k}else e=f.width()}d[c]=e}),d},h.setWidth=function(a,b,c){b.each(function(b){var d=a[b];c.eq(b).css({"min-width":d,"max-width":d})})},h.resetWidth=function(b,c){b.each(function(b){var d=a(this);c.eq(b).css({"min-width":d.css("min-width"),"max-width":d.css("max-width")})})},h.setOptions=function(c){h.options=a.extend({},f,c),h.$scrollableArea=a(h.options.scrollableArea),h.isWindowScrolling=h.$scrollableArea[0]===b},h.updateOptions=function(a){h.setOptions(a),h.unbind(),h.bind(),h.updateWidth(),h.toggleHeaders()},h.init()}var d="stickyTableHeaders",e=0,f={fixedOffset:0,leftOffset:0,marginTop:0,scrollableArea:b};a.fn[d]=function(b){return this.each(function(){var e=a.data(this,"plugin_"+d);e?"string"==typeof b?e[b].apply(e):e.updateOptions(b):"destroy"!==b&&a.data(this,"plugin_"+d,new c(this,b))})}}(jQuery,window);
|
||||
85
js/readtheorg.js
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js/readtheorg.js
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|
||||
$(function() {
|
||||
$('.note').before("<p class='admonition-title note'>Note</p>");
|
||||
$('.seealso').before("<p class='admonition-title seealso'>See also</p>");
|
||||
$('.warning').before("<p class='admonition-title warning'>Warning</p>");
|
||||
$('.caution').before("<p class='admonition-title caution'>Caution</p>");
|
||||
$('.attention').before("<p class='admonition-title attention'>Attention</p>");
|
||||
$('.tip').before("<p class='admonition-title tip'>Tip</p>");
|
||||
$('.important').before("<p class='admonition-title important'>Important</p>");
|
||||
$('.hint').before("<p class='admonition-title hint'>Hint</p>");
|
||||
$('.error').before("<p class='admonition-title error'>Error</p>");
|
||||
$('.danger').before("<p class='admonition-title danger'>Danger</p>");
|
||||
});
|
||||
|
||||
$( document ).ready(function() {
|
||||
|
||||
// Shift nav in mobile when clicking the menu.
|
||||
$(document).on('click', "[data-toggle='wy-nav-top']", function() {
|
||||
$("[data-toggle='wy-nav-shift']").toggleClass("shift");
|
||||
$("[data-toggle='rst-versions']").toggleClass("shift");
|
||||
});
|
||||
// Close menu when you click a link.
|
||||
$(document).on('click', ".wy-menu-vertical .current ul li a", function() {
|
||||
$("[data-toggle='wy-nav-shift']").removeClass("shift");
|
||||
$("[data-toggle='rst-versions']").toggleClass("shift");
|
||||
});
|
||||
$(document).on('click', "[data-toggle='rst-current-version']", function() {
|
||||
$("[data-toggle='rst-versions']").toggleClass("shift-up");
|
||||
});
|
||||
// Make tables responsive
|
||||
$("table.docutils:not(.field-list)").wrap("<div class='wy-table-responsive'></div>");
|
||||
});
|
||||
|
||||
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|
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<h1 class="title">Control in a rotating frame</h1>
|
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|
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|
||||
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|
||||
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|
||||
<li><a href="#org8bd71db">1. Goal of this note</a></li>
|
||||
<li><a href="#org6b844c0">2. System</a>
|
||||
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|
||||
<li><a href="#org0e477c9">2.1. System description</a></li>
|
||||
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|
||||
<li><a href="#org813d43d">2.3. Analysis</a></li>
|
||||
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|
||||
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|
||||
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|
||||
<ul>
|
||||
<li><a href="#org94f7739">3.1. Euler and Coriolis forces</a></li>
|
||||
<li><a href="#orgd457827">3.2. Spring Softening Effect</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org42269b2">4. Control Strategies</a>
|
||||
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|
||||
<li><a href="#org05e6b53">4.1. Measurement in the fixed reference frame</a>
|
||||
<ul>
|
||||
<li><a href="#org8c638be">4.1.1. <span class="todo QUESTION">QUESTION</span> Is the loop gain is changing with the angle ?</a></li>
|
||||
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|
||||
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|
||||
<li><a href="#orgb6cb87b">4.2. Measurement in the rotating frame</a></li>
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
<li><a href="#org2770fda">5.1. <span class="todo TODO">TODO</span> Use realistic parameters for the mass of the sample and stiffness of the X-Y stage</a></li>
|
||||
<li><a href="#orgc76e417">5.2. <span class="todo TODO">TODO</span> Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)</a></li>
|
||||
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|
||||
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|
||||
<li><a href="#org07c8778">6. Effect of the X-Y stage stiffness</a>
|
||||
<ul>
|
||||
<li><a href="#org038f2b3">6.1. <span class="todo TODO">TODO</span> At full speed, check how the coupling changes with the stiffness of the actuators</a></li>
|
||||
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|
||||
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|
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|
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|
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|
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|
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<h2 id="org8bd71db"><span class="section-number-2">1</span> Goal of this note</h2>
|
||||
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|
||||
<p>
|
||||
The control objective is to stabilize the position of a rotating object with respect to a non-rotating frame.
|
||||
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|
||||
|
||||
<p>
|
||||
The actuators are also rotating with the object.
|
||||
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|
||||
|
||||
<p>
|
||||
We want to compare the two different approach:
|
||||
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|
||||
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|
||||
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|
||||
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|
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|
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|
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|
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|
||||
<p>
|
||||
The system consists of one 2 degree of freedom translation stage on top of a spindle (figure <a href="#orgeadaa88">1</a>).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The control inputs are the forces applied in the translation stage (\(F_u\) and \(F_v\)). As the translation stage is rotating around the Z axis due to the spindle, the forces are applied along \(u\) and \(v\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The measurement is either the \(x-y\) displacement of the object located on top of the translation stage or the \(u-v\) displacement of the actuators.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgeadaa88" class="figure">
|
||||
<p><img src="./Figures/rotating_frame_2dof.png" alt="rotating_frame_2dof.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>Schematic of the mecanical system</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
In the following block diagram:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>\(G\) is the transfer function from the forces applied in the actuators to the measurement</li>
|
||||
<li>\(K\) is the controller to design</li>
|
||||
<li>\(J\) is a Jacobian matrix usually used to change the reference frame</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
Indices \(x\) and \(y\) corresponds to signals in the fixed reference frame (along \(\vec{i}_x\) and \(\vec{i}_y\)):
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>\(D_x\) is the measured position of the sample</li>
|
||||
<li>\(r_x\) is the reference signal which corresponds to the wanted \(D_x\)</li>
|
||||
<li>\(\epsilon_x\) is the position error</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
Indices \(u\) and \(v\) corresponds to signals in the rotating reference frame (\(\vec{i}_u\) and \(\vec{i}_v\)):
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>\(F_u\) and \(F_v\) are forces applied by the actuators</li>
|
||||
<li>\(\epsilon_u\) and \(\epsilon_v\) are position error of the sample along \(\vec{i}_u\) and \(\vec{i}_v\)</li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org9d3f997" class="outline-3">
|
||||
<h3 id="org9d3f997"><span class="section-number-3">2.2</span> Equations</h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
<p>
|
||||
<a id="org52a8ecb"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Based on the figure <a href="#orgeadaa88">1</a>, we can write the equations of motion of the system.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Let's express the kinetic energy \(T\) and the potential energy \(V\) of the mass \(m\):
|
||||
</p>
|
||||
\begin{align}
|
||||
\label{org25e6b9c}
|
||||
T & = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) \\
|
||||
V & = \frac{1}{2} k \left( x^2 + y^2 \right)
|
||||
\end{align}
|
||||
|
||||
<p>
|
||||
The Lagrangian is the kinetic energy minus the potential energy.
|
||||
</p>
|
||||
\begin{equation}
|
||||
\label{orgb6862e5}
|
||||
L = T-V = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) - \frac{1}{2} k \left( x^2 + y^2 \right)
|
||||
\end{equation}
|
||||
|
||||
|
||||
<p>
|
||||
The partial derivatives of the Lagrangian with respect to the variables \((x, y)\) are:
|
||||
</p>
|
||||
\begin{align*}
|
||||
\label{org02c8a0d}
|
||||
\frac{\partial L}{\partial x} & = -kx \\
|
||||
\frac{\partial L}{\partial y} & = -ky \\
|
||||
\frac{d}{dt}\frac{\partial L}{\partial \dot{x}} & = m\ddot{x} \\
|
||||
\frac{d}{dt}\frac{\partial L}{\partial \dot{y}} & = m\ddot{y}
|
||||
\end{align*}
|
||||
|
||||
<p>
|
||||
The external forces applied to the mass are:
|
||||
</p>
|
||||
\begin{align*}
|
||||
F_{\text{ext}, x} &= F_u \cos{\theta} - F_v \sin{\theta}\\
|
||||
F_{\text{ext}, y} &= F_u \sin{\theta} + F_v \cos{\theta}
|
||||
\end{align*}
|
||||
|
||||
<p>
|
||||
By appling the Lagrangian equations, we obtain equation \eqref{orgc6b5c7e}.
|
||||
</p>
|
||||
\begin{align}
|
||||
\label{orgc6b5c7e}
|
||||
m\ddot{x} + kx = F_u \cos{\theta} - F_v \sin{\theta}\\
|
||||
m\ddot{y} + ky = F_u \sin{\theta} + F_v \cos{\theta}
|
||||
\end{align}
|
||||
|
||||
<p>
|
||||
We then change coordinates from \((x, y)\) to \((d_x, d_y, \theta)\).
|
||||
</p>
|
||||
\begin{align*}
|
||||
x & = d_u \cos{\theta} - d_v \sin{\theta}\\
|
||||
y & = d_u \sin{\theta} + d_v \cos{\theta}
|
||||
\end{align*}
|
||||
|
||||
<p>
|
||||
We obtain:
|
||||
</p>
|
||||
\begin{align*}
|
||||
\ddot{x} & = \ddot{d_u} \cos{\theta} - 2\dot{d_u}\dot{\theta}\sin{\theta} - d_u\ddot{\theta}\sin{\theta} - d_u\dot{\theta}^2 \cos{\theta}
|
||||
- \ddot{d_v} \sin{\theta} - 2\dot{d_v}\dot{\theta}\cos{\theta} - d_v\ddot{\theta}\cos{\theta} + d_v\dot{\theta}^2 \sin{\theta} \\
|
||||
\ddot{y} & = \ddot{d_u} \sin{\theta} + 2\dot{d_u}\dot{\theta}\cos{\theta} + d_u\ddot{\theta}\cos{\theta} - d_u\dot{\theta}^2 \sin{\theta}
|
||||
+ \ddot{d_v} \cos{\theta} - 2\dot{d_v}\dot{\theta}\sin{\theta} - d_v\ddot{\theta}\sin{\theta} - d_v\dot{\theta}^2 \cos{\theta} \\
|
||||
\end{align*}
|
||||
|
||||
<p>
|
||||
By injecting the previous result into the Lagrangian equation \eqref{orgc6b5c7e}, we obtain:
|
||||
</p>
|
||||
\begin{align*}
|
||||
m \ddot{d_u} \cos{\theta} - 2m\dot{d_u}\dot{\theta}\sin{\theta} - m d_u\ddot{\theta}\sin{\theta} - m d_u\dot{\theta}^2 \cos{\theta}
|
||||
-m \ddot{d_v} \sin{\theta} - 2m\dot{d_v}\dot{\theta}\cos{\theta} - m d_v\ddot{\theta}\cos{\theta} + m d_v\dot{\theta}^2 \sin{\theta}
|
||||
+ k d_u \cos{\theta} - k d_v \sin{\theta} = F_u \cos{\theta} - F_v \sin{\theta} \\
|
||||
m \ddot{d_u} \sin{\theta} + 2m\dot{d_u}\dot{\theta}\cos{\theta} + m d_u\ddot{\theta}\cos{\theta} - m d_u\dot{\theta}^2 \sin{\theta}
|
||||
+ m \ddot{d_v} \cos{\theta} - 2m\dot{d_v}\dot{\theta}\sin{\theta} - m d_v\ddot{\theta}\sin{\theta} - m d_v\dot{\theta}^2 \cos{\theta}
|
||||
+ k d_u \sin{\theta} + k d_v \cos{\theta} = F_u \sin{\theta} + F_v \cos{\theta}
|
||||
\end{align*}
|
||||
|
||||
<p>
|
||||
Which is equivalent to:
|
||||
</p>
|
||||
\begin{align*}
|
||||
m \ddot{d_u} - 2m\dot{d_u}\dot{\theta}\frac{\sin{\theta}}{\cos{\theta}} - m d_u\ddot{\theta}\frac{\sin{\theta}}{\cos{\theta}} - m d_u\dot{\theta}^2
|
||||
-m \ddot{d_v} \frac{\sin{\theta}}{\cos{\theta}} - 2m\dot{d_v}\dot{\theta} - m d_v\ddot{\theta} + m d_v\dot{\theta}^2 \frac{\sin{\theta}}{\cos{\theta}}
|
||||
+ k d_u - k d_v \frac{\sin{\theta}}{\cos{\theta}} = F_u - F_v \frac{\sin{\theta}}{\cos{\theta}} \\
|
||||
m \ddot{d_u} + 2m\dot{d_u}\dot{\theta}\frac{\cos{\theta}}{\sin{\theta}} + m d_u\ddot{\theta}\frac{\cos{\theta}}{\sin{\theta}} - m d_u\dot{\theta}^2
|
||||
+ m \ddot{d_v} \frac{\cos{\theta}}{\sin{\theta}} - 2m\dot{d_v}\dot{\theta} - m d_v\ddot{\theta} - m d_v\dot{\theta}^2 \frac{\cos{\theta}}{\sin{\theta}}
|
||||
+ k d_u + k d_v \frac{\cos{\theta}}{\sin{\theta}} = F_u + F_v \frac{\cos{\theta}}{\sin{\theta}}
|
||||
\end{align*}
|
||||
|
||||
<p>
|
||||
We can then subtract and add the previous equations to obtain the following equations:
|
||||
</p>
|
||||
<div class="important">
|
||||
\begin{align*}
|
||||
m \ddot{d_u} + (k - m\dot{\theta}^2) d_u &= F_u + 2 m\dot{d_v}\dot{\theta} + m d_v\ddot{\theta} \\
|
||||
m \ddot{d_v} + (k - m\dot{\theta}^2) d_v &= F_v - 2 m\dot{d_u}\dot{\theta} - m d_u\ddot{\theta} \\
|
||||
\end{align*}
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org813d43d" class="outline-3">
|
||||
<h3 id="org813d43d"><span class="section-number-3">2.3</span> Analysis</h3>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
<p>
|
||||
We obtain two differential equations that are coupled through:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li><b>Euler forces</b>: \(m d_v \ddot{\theta}\)</li>
|
||||
<li><b>Coriolis forces</b>: \(2 m \dot{d_v} \dot{\theta}\)</li>
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
Without the coupling terms, each equation is the equation of a one degree of freedom mass-spring system with mass \(m\) and stiffness \(k-d_u m\dot{\theta}^2\).
|
||||
Thus, the term \(-d_u m\dot{\theta}^2\) acts like a negative stiffness (due to <b>centrifugal forces</b>).
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orge651676" class="outline-2">
|
||||
<h2 id="orge651676"><span class="section-number-2">3</span> Analytical Computation of forces for the NASS</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<p>
|
||||
For the NASS, the Euler forces should be less of a problem as \(\ddot{\theta}\) should be very small when conducting an experiment.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
First we will determine the value for Euler and Coriolis forces during regular experiment.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org94f7739" class="outline-3">
|
||||
<h3 id="org94f7739"><span class="section-number-3">3.1</span> Euler and Coriolis forces</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
Let's define the parameters for the NASS.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">mlight = <span style="color: #D0372D;">35</span>; <span style="color: #8D8D84; font-style: italic;">% [kg]</span>
|
||||
mheavy = <span style="color: #D0372D;">85</span>; <span style="color: #8D8D84; font-style: italic;">% [kg]</span>
|
||||
|
||||
wlight = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span>; <span style="color: #8D8D84; font-style: italic;">% [rad/s]</span>
|
||||
wheavy = <span style="color: #D0372D;">2</span><span style="color: #6434A3;">*</span><span style="color: #D0372D;">pi</span><span style="color: #6434A3;">/</span><span style="color: #D0372D;">60</span>; <span style="color: #8D8D84; font-style: italic;">% [rad/s]</span>
|
||||
|
||||
wdot = <span style="color: #D0372D;">1</span>; <span style="color: #8D8D84; font-style: italic;">% [rad/s2]</span>
|
||||
|
||||
d = <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">1</span>; <span style="color: #8D8D84; font-style: italic;">% [m]</span>
|
||||
ddot = <span style="color: #D0372D;">0</span>.<span style="color: #D0372D;">2</span>; <span style="color: #8D8D84; font-style: italic;">% [m/s]</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We then compute the corresponding values of the Coriolis and Euler forces, and the obtained values are displayed in table <a href="#org11a5df8">1</a>.
|
||||
</p>
|
||||
|
||||
<table id="org11a5df8" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
<caption class="t-above"><span class="table-number">Table 1:</span> Euler and Coriolis forces for the NASS</caption>
|
||||
|
||||
<colgroup>
|
||||
<col class="org-left" />
|
||||
|
||||
<col class="org-left" />
|
||||
|
||||
<col class="org-left" />
|
||||
</colgroup>
|
||||
<thead>
|
||||
<tr>
|
||||
<th scope="col" class="org-left"> </th>
|
||||
<th scope="col" class="org-left">Light</th>
|
||||
<th scope="col" class="org-left">Heavy</th>
|
||||
</tr>
|
||||
</thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td class="org-left">Coriolis</td>
|
||||
<td class="org-left">44.0 N</td>
|
||||
<td class="org-left">1.8 N</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td class="org-left">Euler</td>
|
||||
<td class="org-left">3.5 N</td>
|
||||
<td class="org-left">8.5 N</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd457827" class="outline-3">
|
||||
<h3 id="orgd457827"><span class="section-number-3">3.2</span> Spring Softening Effect</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<p>
|
||||
The values for the spring softening effect are displayed in table <a href="#org76af8f7">2</a>.
|
||||
This is definitely negligible when using piezoelectric actuators. It may not be the case when using voice coil actuators.
|
||||
</p>
|
||||
|
||||
<table id="org76af8f7" border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||
<caption class="t-above"><span class="table-number">Table 2:</span> Spring Softening effect</caption>
|
||||
|
||||
<colgroup>
|
||||
<col class="org-left" />
|
||||
|
||||
<col class="org-left" />
|
||||
|
||||
<col class="org-left" />
|
||||
</colgroup>
|
||||
<thead>
|
||||
<tr>
|
||||
<th scope="col" class="org-left"> </th>
|
||||
<th scope="col" class="org-left">Light</th>
|
||||
<th scope="col" class="org-left">Heavy</th>
|
||||
</tr>
|
||||
</thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td class="org-left">Spring Soft.</td>
|
||||
<td class="org-left">3.5 N/m</td>
|
||||
<td class="org-left">8.5 N/m</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org42269b2" class="outline-2">
|
||||
<h2 id="org42269b2"><span class="section-number-2">4</span> Control Strategies</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
<p>
|
||||
<a id="org25db234"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org05e6b53" class="outline-3">
|
||||
<h3 id="org05e6b53"><span class="section-number-3">4.1</span> Measurement in the fixed reference frame</h3>
|
||||
<div class="outline-text-3" id="text-4-1">
|
||||
<p>
|
||||
First, let's consider a measurement in the fixed referenced frame.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The transfer function from actuator \([F_u, F_v]\) to sensor \([D_x, D_y]\) is then \(G(\theta)\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Then the measurement is subtracted to the reference signal \([r_x, r_y]\) to obtain the position error in the fixed reference frame \([\epsilon_x, \epsilon_y]\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The position error \([\epsilon_x, \epsilon_y]\) is then express in the rotating frame corresponding to the actuators \([\epsilon_u, \epsilon_v]\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Finally, the control low \(K\) links the position errors \([\epsilon_u, \epsilon_v]\) to the actuator forces \([F_u, F_v]\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The block diagram is shown on figure <a href="#orgbae56a5">2</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgbae56a5" class="figure">
|
||||
<p><img src="./Figures/control_measure_fixed_2dof.png" alt="control_measure_fixed_2dof.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>Control with a measure from fixed frame</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The loop gain is then \(L = G(\theta) K J(\theta)\).
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org8c638be" class="outline-4">
|
||||
<h4 id="org8c638be"><span class="section-number-4">4.1.1</span> <span class="todo QUESTION">QUESTION</span> Is the loop gain is changing with the angle ?</h4>
|
||||
<div class="outline-text-4" id="text-4-1-1">
|
||||
<p>
|
||||
Is \[ G(\theta) J(\theta) = G(\theta_0) J(\theta_0) \] ?
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb6cb87b" class="outline-3">
|
||||
<h3 id="orgb6cb87b"><span class="section-number-3">4.2</span> Measurement in the rotating frame</h3>
|
||||
<div class="outline-text-3" id="text-4-2">
|
||||
<p>
|
||||
Let's consider that the measurement is in the rotating reference frame.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The corresponding block diagram is shown figure <a href="#org33df600">3</a>
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org33df600" class="figure">
|
||||
<p><img src="./Figures/control_measure_rotating_2dof.png" alt="control_measure_rotating_2dof.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 3: </span>Control with a measure from rotating frame</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The loop gain is \(L = G K\).
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org502caa6" class="outline-2">
|
||||
<h2 id="org502caa6"><span class="section-number-2">5</span> Effect of the rotating Speed</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
<p>
|
||||
<a id="org389e858"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org2770fda" class="outline-3">
|
||||
<h3 id="org2770fda"><span class="section-number-3">5.1</span> <span class="todo TODO">TODO</span> Use realistic parameters for the mass of the sample and stiffness of the X-Y stage</h3>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc76e417" class="outline-3">
|
||||
<h3 id="orgc76e417"><span class="section-number-3">5.2</span> <span class="todo TODO">TODO</span> Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)</h3>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org07c8778" class="outline-2">
|
||||
<h2 id="org07c8778"><span class="section-number-2">6</span> Effect of the X-Y stage stiffness</h2>
|
||||
<div class="outline-text-2" id="text-6">
|
||||
<p>
|
||||
<a id="org99e68c1"></a>
|
||||
</p>
|
||||
</div>
|
||||
<div id="outline-container-org038f2b3" class="outline-3">
|
||||
<h3 id="org038f2b3"><span class="section-number-3">6.1</span> <span class="todo TODO">TODO</span> At full speed, check how the coupling changes with the stiffness of the actuators</h3>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Thomas Dehaeze</p>
|
||||
<p class="date">Created: 2019-01-18 ven. 17:15</p>
|
||||
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
280
rotating_frame.org
Normal file
280
rotating_frame.org
Normal file
@ -0,0 +1,280 @@
|
||||
#+TITLE: Control in a rotating frame
|
||||
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="css/htmlize.css"/>
|
||||
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="css/readtheorg.css"/>
|
||||
#+HTML_HEAD: <script src="js/jquery.min.js"></script>
|
||||
#+HTML_HEAD: <script src="js/bootstrap.min.js"></script>
|
||||
#+HTML_HEAD: <script type="text/javascript" src="js/jquery.stickytableheaders.min.js"></script>
|
||||
#+HTML_HEAD: <script type="text/javascript" src="js/readtheorg.js"></script>
|
||||
#+LATEX_CLASS: cleanreport
|
||||
#+LaTeX_CLASS_OPTIONS: [tocnp, secbreak, minted]
|
||||
#+STARTUP: overview
|
||||
#+LaTeX_HEADER: \usepackage{svg}
|
||||
#+LaTeX_HEADER: \newcommand{\authorFirstName}{Thomas}
|
||||
#+LaTeX_HEADER: \newcommand{\authorLastName}{Dehaeze}
|
||||
#+LaTeX_HEADER: \newcommand{\authorEmail}{dehaeze.thomas@gmail.com}
|
||||
#+PROPERTY: header-args:matlab :session *MATLAB*
|
||||
#+PROPERTY: header-args:matlab+ :tangle rotating_frame.m
|
||||
#+PROPERTY: header-args:matlab+ :comments org
|
||||
#+PROPERTY: header-args:matlab+ :exports both
|
||||
#+PROPERTY: header-args:matlab+ :eval no-export
|
||||
#+PROPERTY: header-args:matlab+ :output-dir Figures
|
||||
|
||||
* Goal of this note
|
||||
The control objective is to stabilize the position of a rotating object with respect to a non-rotating frame.
|
||||
|
||||
The actuators are also rotating with the object.
|
||||
|
||||
We want to compare the two different approach:
|
||||
- the measurement is made in the fixed frame
|
||||
- the measurement is made in the rotating frame
|
||||
|
||||
* System
|
||||
<<sec:system>>
|
||||
** System description
|
||||
The system consists of one 2 degree of freedom translation stage on top of a spindle (figure [[fig:rotating_frame_2dof]]).
|
||||
|
||||
The control inputs are the forces applied in the translation stage ($F_u$ and $F_v$). As the translation stage is rotating around the Z axis due to the spindle, the forces are applied along $u$ and $v$.
|
||||
|
||||
The measurement is either the $x-y$ displacement of the object located on top of the translation stage or the $u-v$ displacement of the actuators.
|
||||
|
||||
#+name: fig:rotating_frame_2dof
|
||||
#+caption: Schematic of the mecanical system
|
||||
[[./Figures/rotating_frame_2dof.png]]
|
||||
|
||||
In the following block diagram:
|
||||
- $G$ is the transfer function from the forces applied in the actuators to the measurement
|
||||
- $K$ is the controller to design
|
||||
- $J$ is a Jacobian matrix usually used to change the reference frame
|
||||
|
||||
Indices $x$ and $y$ corresponds to signals in the fixed reference frame (along $\vec{i}_x$ and $\vec{i}_y$):
|
||||
- $D_x$ is the measured position of the sample
|
||||
- $r_x$ is the reference signal which corresponds to the wanted $D_x$
|
||||
- $\epsilon_x$ is the position error
|
||||
|
||||
Indices $u$ and $v$ corresponds to signals in the rotating reference frame ($\vec{i}_u$ and $\vec{i}_v$):
|
||||
- $F_u$ and $F_v$ are forces applied by the actuators
|
||||
- $\epsilon_u$ and $\epsilon_v$ are position error of the sample along $\vec{i}_u$ and $\vec{i}_v$
|
||||
|
||||
** Equations
|
||||
<<sec:equations>>
|
||||
|
||||
Based on the figure [[fig:rotating_frame_2dof]], we can write the equations of motion of the system.
|
||||
|
||||
Let's express the kinetic energy $T$ and the potential energy $V$ of the mass $m$:
|
||||
#+name: eq:energy_inertial_frame
|
||||
\begin{align}
|
||||
T & = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) \\
|
||||
V & = \frac{1}{2} k \left( x^2 + y^2 \right)
|
||||
\end{align}
|
||||
|
||||
The Lagrangian is the kinetic energy minus the potential energy.
|
||||
#+name: eq:lagrangian_inertial_frame
|
||||
\begin{equation}
|
||||
L = T-V = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) - \frac{1}{2} k \left( x^2 + y^2 \right)
|
||||
\end{equation}
|
||||
|
||||
|
||||
The partial derivatives of the Lagrangian with respect to the variables $(x, y)$ are:
|
||||
#+name: eq:inertial_frame_deriv
|
||||
\begin{align*}
|
||||
\frac{\partial L}{\partial x} & = -kx \\
|
||||
\frac{\partial L}{\partial y} & = -ky \\
|
||||
\frac{d}{dt}\frac{\partial L}{\partial \dot{x}} & = m\ddot{x} \\
|
||||
\frac{d}{dt}\frac{\partial L}{\partial \dot{y}} & = m\ddot{y}
|
||||
\end{align*}
|
||||
|
||||
The external forces applied to the mass are:
|
||||
\begin{align*}
|
||||
F_{\text{ext}, x} &= F_u \cos{\theta} - F_v \sin{\theta}\\
|
||||
F_{\text{ext}, y} &= F_u \sin{\theta} + F_v \cos{\theta}
|
||||
\end{align*}
|
||||
|
||||
By appling the Lagrangian equations, we obtain:
|
||||
\begin{align}
|
||||
m\ddot{x} + kx = F_u \cos{\theta} - F_v \sin{\theta}\\
|
||||
m\ddot{y} + ky = F_u \sin{\theta} + F_v \cos{\theta}
|
||||
\end{align}
|
||||
|
||||
We then change coordinates from $(x, y)$ to $(d_x, d_y, \theta)$.
|
||||
\begin{align*}
|
||||
x & = d_u \cos{\theta} - d_v \sin{\theta}\\
|
||||
y & = d_u \sin{\theta} + d_v \cos{\theta}
|
||||
\end{align*}
|
||||
|
||||
We obtain:
|
||||
\begin{align*}
|
||||
\ddot{x} & = \ddot{d_u} \cos{\theta} - 2\dot{d_u}\dot{\theta}\sin{\theta} - d_u\ddot{\theta}\sin{\theta} - d_u\dot{\theta}^2 \cos{\theta}
|
||||
- \ddot{d_v} \sin{\theta} - 2\dot{d_v}\dot{\theta}\cos{\theta} - d_v\ddot{\theta}\cos{\theta} + d_v\dot{\theta}^2 \sin{\theta} \\
|
||||
\ddot{y} & = \ddot{d_u} \sin{\theta} + 2\dot{d_u}\dot{\theta}\cos{\theta} + d_u\ddot{\theta}\cos{\theta} - d_u\dot{\theta}^2 \sin{\theta}
|
||||
+ \ddot{d_v} \cos{\theta} - 2\dot{d_v}\dot{\theta}\sin{\theta} - d_v\ddot{\theta}\sin{\theta} - d_v\dot{\theta}^2 \cos{\theta} \\
|
||||
\end{align*}
|
||||
|
||||
By injecting the previous result into the Lagrangian equation [[eq:lagrangian_eq_inertial]], we obtain:
|
||||
\begin{align*}
|
||||
m \ddot{d_u} \cos{\theta} - 2m\dot{d_u}\dot{\theta}\sin{\theta} - m d_u\ddot{\theta}\sin{\theta} - m d_u\dot{\theta}^2 \cos{\theta}
|
||||
-m \ddot{d_v} \sin{\theta} - 2m\dot{d_v}\dot{\theta}\cos{\theta} - m d_v\ddot{\theta}\cos{\theta} + m d_v\dot{\theta}^2 \sin{\theta}
|
||||
+ k d_u \cos{\theta} - k d_v \sin{\theta} = F_u \cos{\theta} - F_v \sin{\theta} \\
|
||||
m \ddot{d_u} \sin{\theta} + 2m\dot{d_u}\dot{\theta}\cos{\theta} + m d_u\ddot{\theta}\cos{\theta} - m d_u\dot{\theta}^2 \sin{\theta}
|
||||
+ m \ddot{d_v} \cos{\theta} - 2m\dot{d_v}\dot{\theta}\sin{\theta} - m d_v\ddot{\theta}\sin{\theta} - m d_v\dot{\theta}^2 \cos{\theta}
|
||||
+ k d_u \sin{\theta} + k d_v \cos{\theta} = F_u \sin{\theta} + F_v \cos{\theta}
|
||||
\end{align*}
|
||||
|
||||
Which is equivalent to:
|
||||
\begin{align*}
|
||||
m \ddot{d_u} - 2m\dot{d_u}\dot{\theta}\frac{\sin{\theta}}{\cos{\theta}} - m d_u\ddot{\theta}\frac{\sin{\theta}}{\cos{\theta}} - m d_u\dot{\theta}^2
|
||||
-m \ddot{d_v} \frac{\sin{\theta}}{\cos{\theta}} - 2m\dot{d_v}\dot{\theta} - m d_v\ddot{\theta} + m d_v\dot{\theta}^2 \frac{\sin{\theta}}{\cos{\theta}}
|
||||
+ k d_u - k d_v \frac{\sin{\theta}}{\cos{\theta}} = F_u - F_v \frac{\sin{\theta}}{\cos{\theta}} \\
|
||||
m \ddot{d_u} + 2m\dot{d_u}\dot{\theta}\frac{\cos{\theta}}{\sin{\theta}} + m d_u\ddot{\theta}\frac{\cos{\theta}}{\sin{\theta}} - m d_u\dot{\theta}^2
|
||||
+ m \ddot{d_v} \frac{\cos{\theta}}{\sin{\theta}} - 2m\dot{d_v}\dot{\theta} - m d_v\ddot{\theta} - m d_v\dot{\theta}^2 \frac{\cos{\theta}}{\sin{\theta}}
|
||||
+ k d_u + k d_v \frac{\cos{\theta}}{\sin{\theta}} = F_u + F_v \frac{\cos{\theta}}{\sin{\theta}}
|
||||
\end{align*}
|
||||
|
||||
We can then subtract and add the previous equations to obtain the following equations:
|
||||
#+begin_important
|
||||
\begin{align*}
|
||||
m \ddot{d_u} + (k - m\dot{\theta}^2) d_u &= F_u + 2 m\dot{d_v}\dot{\theta} + m d_v\ddot{\theta} \\
|
||||
m \ddot{d_v} + (k - m\dot{\theta}^2) d_v &= F_v - 2 m\dot{d_u}\dot{\theta} - m d_u\ddot{\theta} \\
|
||||
\end{align*}
|
||||
#+end_important
|
||||
|
||||
** Analysis
|
||||
We obtain two differential equations that are coupled through:
|
||||
- *Euler forces*: $m d_v \ddot{\theta}$
|
||||
- *Coriolis forces*: $2 m \dot{d_v} \dot{\theta}$
|
||||
|
||||
Without the coupling terms, each equation is the equation of a one degree of freedom mass-spring system with mass $m$ and stiffness $k-d_u m\dot{\theta}^2$.
|
||||
Thus, the term $-d_u m\dot{\theta}^2$ acts like a negative stiffness (due to *centrifugal forces*).
|
||||
|
||||
* Analytical Computation of forces for the NASS
|
||||
For the NASS, the Euler forces should be less of a problem as $\ddot{\theta}$ should be very small when conducting an experiment.
|
||||
|
||||
First we will determine the value for Euler and Coriolis forces during regular experiment.
|
||||
|
||||
** Euler and Coriolis forces
|
||||
|
||||
Let's define the parameters for the NASS.
|
||||
#+begin_src matlab :exports code :results silent
|
||||
mlight = 35; % [kg]
|
||||
mheavy = 85; % [kg]
|
||||
|
||||
wlight = 2*pi; % [rad/s]
|
||||
wheavy = 2*pi/60; % [rad/s]
|
||||
|
||||
wdot = 1; % [rad/s2]
|
||||
|
||||
d = 0.1; % [m]
|
||||
ddot = 0.2; % [m/s]
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none :results silent
|
||||
Felight = mlight*d*wdot;
|
||||
Feheavy = mheavy*d*wdot;
|
||||
|
||||
Fclight = 2*mlight*d*wlight;
|
||||
Fcheavy = 2*mheavy*d*wheavy;
|
||||
#+end_src
|
||||
|
||||
We then compute the corresponding values of the Coriolis and Euler forces, and the obtained values are displayed in table [[tab:euler_coriolis]].
|
||||
|
||||
#+begin_src matlab :results value table :exports results :post addhdr(*this*)
|
||||
ans = sprintf(' | Light | Heavy | \n Coriolis | %.1f N | %.1f N | \n Euler | %.1f N | %.1f N', Fclight, Fcheavy, Felight, Feheavy)
|
||||
#+end_src
|
||||
|
||||
#+NAME: tab:euler_coriolis
|
||||
#+CAPTION: Euler and Coriolis forces for the NASS
|
||||
#+RESULTS:
|
||||
| | Light | Heavy |
|
||||
|----------+--------+-------|
|
||||
| Coriolis | 44.0 N | 1.8 N |
|
||||
| Euler | 3.5 N | 8.5 N |
|
||||
|
||||
** Spring Softening Effect
|
||||
#+begin_src matlab :exports none :results silent
|
||||
Klight = mlight*d*wdot^2;
|
||||
Kheavy = mheavy*d*wdot^2;
|
||||
#+end_src
|
||||
|
||||
The values for the spring softening effect are displayed in table [[tab:spring_softening]].
|
||||
This is definitely negligible when using piezoelectric actuators. It may not be the case when using voice coil actuators.
|
||||
|
||||
#+begin_src matlab :results value table :exports results :post addhdr(*this*)
|
||||
ans = sprintf(' | Light | Heavy | \n Spring Soft. | %.1f N/m | %.1f N/m', Klight, Kheavy)
|
||||
#+end_src
|
||||
|
||||
#+NAME: tab:spring_softening
|
||||
#+CAPTION: Spring Softening effect
|
||||
#+RESULTS:
|
||||
| | Light | Heavy |
|
||||
|--------------+---------+---------|
|
||||
| Spring Soft. | 3.5 N/m | 8.5 N/m |
|
||||
|
||||
* Control Strategies
|
||||
<<sec:control_strategies>>
|
||||
** Measurement in the fixed reference frame
|
||||
First, let's consider a measurement in the fixed referenced frame.
|
||||
|
||||
The transfer function from actuator $[F_u, F_v]$ to sensor $[D_x, D_y]$ is then $G(\theta)$.
|
||||
|
||||
Then the measurement is subtracted to the reference signal $[r_x, r_y]$ to obtain the position error in the fixed reference frame $[\epsilon_x, \epsilon_y]$.
|
||||
|
||||
The position error $[\epsilon_x, \epsilon_y]$ is then express in the rotating frame corresponding to the actuators $[\epsilon_u, \epsilon_v]$.
|
||||
|
||||
Finally, the control low $K$ links the position errors $[\epsilon_u, \epsilon_v]$ to the actuator forces $[F_u, F_v]$.
|
||||
|
||||
The block diagram is shown on figure [[fig:control_measure_fixed_2dof]].
|
||||
|
||||
#+name: fig:control_measure_fixed_2dof
|
||||
#+caption: Control with a measure from fixed frame
|
||||
[[./Figures/control_measure_fixed_2dof.png]]
|
||||
|
||||
The loop gain is then $L = G(\theta) K J(\theta)$.
|
||||
|
||||
*** QUESTION Is the loop gain is changing with the angle ?
|
||||
Is \[ G(\theta) J(\theta) = G(\theta_0) J(\theta_0) \] ?
|
||||
|
||||
** Measurement in the rotating frame
|
||||
Let's consider that the measurement is in the rotating reference frame.
|
||||
|
||||
The corresponding block diagram is shown figure [[fig:control_measure_rotating_2dof]]
|
||||
|
||||
#+name: fig:control_measure_rotating_2dof
|
||||
#+caption: Control with a measure from rotating frame
|
||||
[[./Figures/control_measure_rotating_2dof.png]]
|
||||
|
||||
The loop gain is $L = G K$.
|
||||
|
||||
* Effect of the rotating Speed
|
||||
<<sec:effect_rot_speed>>
|
||||
** TODO Use realistic parameters for the mass of the sample and stiffness of the X-Y stage
|
||||
|
||||
** TODO Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)
|
||||
|
||||
* Effect of the X-Y stage stiffness
|
||||
<<sec:effect_stiffness>>
|
||||
** TODO At full speed, check how the coupling changes with the stiffness of the actuators
|
||||
* Noweb Snippets :noexport:
|
||||
:PROPERTIES:
|
||||
:HEADER-ARGS:matlab+: :results none :exports none :tangle no
|
||||
:HEADER-ARGS:emacs-lisp+: :tangle no :eval no-export
|
||||
:END:
|
||||
|
||||
** Matlab Init
|
||||
#+NAME: matlab-init
|
||||
#+BEGIN_SRC matlab :results none :exports none
|
||||
clear; close all; clc;
|
||||
addpath('./src/');
|
||||
ans = 0;
|
||||
#+END_SRC
|
||||
|
||||
** Matlab Export Figure
|
||||
#+NAME: matlab-export-figure
|
||||
#+BEGIN_SRC matlab :var fn="figure" ft="png" size="normal-normal" :exports none
|
||||
exportFig(fn, size);
|
||||
ans = [fn, ".", ft];
|
||||
#+END_SRC
|
||||
|
||||
** Add hline to org table
|
||||
#+name: addhdr
|
||||
#+begin_src emacs-lisp :var tbl=""
|
||||
(cons (car tbl) (cons 'hline (cdr tbl)))
|
||||
#+end_src
|
||||
Loading…
x
Reference in New Issue
Block a user