📄 rtjademo.html
字号:
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/REC-html40/loose.dtd"><html><head> <title>Description of rtjademo</title> <meta name="keywords" content="rtjademo"> <meta name="description" content="RTJADEMO Jacobian demo"> <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1"> <meta name="generator" content="m2html © 2003 Guillaume Flandin"> <meta name="robots" content="index, follow"> <link type="text/css" rel="stylesheet" href="../m2html.css"></head><body><a name="_top"></a><div><a href="../index.html">Home</a> > <a href="#">demos</a> > rtjademo.m</div><!--<table width="100%"><tr><td align="left"><a href="../index.html"><img alt="<" border="0" src="../left.png"> Master index</a></td><td align="right"><a href="index.html">Index for ./demos <img alt=">" border="0" src="../right.png"></a></td></tr></table>--><h1>rtjademo</h1><h2><a name="_name"></a>PURPOSE <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2><div class="box"><strong>RTJADEMO Jacobian demo</strong></div><h2><a name="_synopsis"></a>SYNOPSIS <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2><div class="box"><strong>This is a script file. </strong></div><h2><a name="_description"></a>DESCRIPTION <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2><div class="fragment"><pre class="comment">RTJADEMO Jacobian demo</pre></div><!-- crossreference --><h2><a name="_cross"></a>CROSS-REFERENCE INFORMATION <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>This function calls:<ul style="list-style-image:url(../matlabicon.gif)"></ul>This function is called by:<ul style="list-style-image:url(../matlabicon.gif)"></ul><!-- crossreference --><h2><a name="_source"></a>SOURCE CODE <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2><div class="fragment"><pre>0001 <span class="comment">%RTJADEMO Jacobian demo</span>0002 0003 <span class="comment">% Copyright (C) 1993-2002, by Peter I. Corke</span>0004 echo off0005 <span class="comment">% 6/99 change arg to maniplty() to 'yoshikawa'</span>0006 <span class="comment">% $Log: not supported by cvs2svn $</span>0007 <span class="comment">% Revision 1.2 2002-04-01 11:47:17 pic</span>0008 <span class="comment">% General cleanup of code: help comments, see also, copyright, remnant dh/dyn</span>0009 <span class="comment">% references, clarification of functions.</span>0010 <span class="comment">%</span>0011 <span class="comment">% $Revision: 1.1 $</span>0012 echo on0013 <span class="comment">%</span>0014 <span class="comment">% Jacobian and differential motion demonstration</span>0015 <span class="comment">%</span>0016 <span class="comment">% A differential motion can be represented by a 6-element vector with elements</span>0017 <span class="comment">% [dx dy dz drx dry drz]</span>0018 <span class="comment">%</span>0019 <span class="comment">% where the first 3 elements are a differential translation, and the last 3</span>0020 <span class="comment">% are a differential rotation. When dealing with infinitisimal rotations,</span>0021 <span class="comment">% the order becomes unimportant. The differential motion could be written</span>0022 <span class="comment">% in terms of compounded transforms</span>0023 <span class="comment">%</span>0024 <span class="comment">% transl(dx,dy,dz) * trotx(drx) * troty(dry) * trotz(drz)</span>0025 <span class="comment">%</span>0026 <span class="comment">% but a more direct approach is to use the function diff2tr()</span>0027 <span class="comment">%</span>0028 D = [.1 .2 0 -.2 .1 .1]';0029 diff2tr(D)0030 pause <span class="comment">% any key to continue</span>0031 <span class="comment">%</span>0032 <span class="comment">% More commonly it is useful to know how a differential motion in one</span>0033 <span class="comment">% coordinate frame appears in another frame. If the second frame is</span>0034 <span class="comment">% represented by the transform</span>0035 T = transl(100, 200, 300) * troty(pi/8) * trotz(-pi/4);0036 <span class="comment">%</span>0037 <span class="comment">% then the differential motion in the second frame would be given by</span>0038 0039 DT = tr2jac(T) * D;0040 DT'0041 <span class="comment">%</span>0042 <span class="comment">% tr2jac() has computed a 6x6 Jacobian matrix which transforms the differential</span>0043 <span class="comment">% changes from the first frame to the next.</span>0044 <span class="comment">%</span>0045 pause <span class="comment">% any key to continue</span>0046 0047 <span class="comment">% The manipulator's Jacobian matrix relates differential joint coordinate</span>0048 <span class="comment">% motion to differential Cartesian motion;</span>0049 <span class="comment">%</span>0050 <span class="comment">% dX = J(q) dQ</span>0051 <span class="comment">%</span>0052 <span class="comment">% For an n-joint manipulator the manipulator Jacobian is a 6 x n matrix and</span>0053 <span class="comment">% is used is many manipulator control schemes. For a 6-axis manipulator like</span>0054 <span class="comment">% the Puma 560 the Jacobian is square</span>0055 <span class="comment">%</span>0056 <span class="comment">% Two Jacobians are frequently used, which express the Cartesian velocity in</span>0057 <span class="comment">% the world coordinate frame,</span>0058 0059 q = [0.1 0.75 -2.25 0 .75 0]0060 J = jacob0(p560, q)0061 <span class="comment">%</span>0062 <span class="comment">% or the T6 coordinate frame</span>0063 0064 J = jacobn(p560, q)0065 <span class="comment">%</span>0066 <span class="comment">% Note the top right 3x3 block is all zero. This indicates, correctly, that</span>0067 <span class="comment">% motion of joints 4-6 does not cause any translational motion of the robot's</span>0068 <span class="comment">% end-effector.</span>0069 pause <span class="comment">% any key to continue</span>0070 0071 <span class="comment">%</span>0072 <span class="comment">% Many control schemes require the inverse of the Jacobian. The Jacobian</span>0073 <span class="comment">% in this example is not singular</span>0074 det(J)0075 <span class="comment">%</span>0076 <span class="comment">% and may be inverted</span>0077 Ji = inv(J)0078 pause <span class="comment">% any key to continue</span>0079 <span class="comment">%</span>0080 <span class="comment">% A classic control technique is Whitney's resolved rate motion control</span>0081 <span class="comment">%</span>0082 <span class="comment">% dQ/dt = J(q)^-1 dX/dt</span>0083 <span class="comment">%</span>0084 <span class="comment">% where dX/dt is the desired Cartesian velocity, and dQ/dt is the required</span>0085 <span class="comment">% joint velocity to achieve this.</span>0086 vel = [1 0 0 0 0 0]'; <span class="comment">% translational motion in the X direction</span>0087 qvel = Ji * vel;0088 qvel'0089 <span class="comment">%</span>0090 <span class="comment">% This is an alternative strategy to computing a Cartesian trajectory</span>0091 <span class="comment">% and solving the inverse kinematics. However like that other scheme, this</span>0092 <span class="comment">% strategy also runs into difficulty at a manipulator singularity where</span>0093 <span class="comment">% the Jacobian is singular.</span>0094 0095 pause <span class="comment">% any key to continue</span>0096 <span class="comment">%</span>0097 <span class="comment">% As already stated this Jacobian relates joint velocity to end-effector</span>0098 <span class="comment">% velocity expressed in the end-effector reference frame. We may wish</span>0099 <span class="comment">% instead to specify the velocity in base or world coordinates.</span>0100 <span class="comment">%</span>0101 <span class="comment">% We have already seen how differential motions in one frame can be translated</span>0102 <span class="comment">% to another. Consider the velocity as a differential in the world frame, that</span>0103 <span class="comment">% is, d0X. We can write</span>0104 <span class="comment">% d6X = Jac(T6) d0X</span>0105 <span class="comment">%</span>0106 T6 = fkine(p560, q); <span class="comment">% compute the end-effector transform</span>0107 d6X = tr2jac(T6) * vel; <span class="comment">% translate world frame velocity to T6 frame</span>0108 qvel = Ji * d6X; <span class="comment">% compute required joint velocity as before</span>0109 qvel'0110 <span class="comment">%</span>0111 <span class="comment">% Note that this value of joint velocity is quite different to that calculated</span>0112 <span class="comment">% above, which was for motion in the T6 X-axis direction.</span>0113 pause <span class="comment">% any key to continue</span>0114 <span class="comment">%</span>0115 <span class="comment">% At a manipulator singularity or degeneracy the Jacobian becomes singular.</span>0116 <span class="comment">% At the Puma's `ready' position for instance, two of the wrist joints are</span>0117 <span class="comment">% aligned resulting in the loss of one degree of freedom. This is revealed by</span>0118 <span class="comment">% the rank of the Jacobian</span>0119 rank( jacobn(p560, qr) )0120 <span class="comment">%</span>0121 <span class="comment">% and the singular values are</span>0122 svd( jacobn(p560, qr) )0123 pause <span class="comment">% any key to continue</span>0124 <span class="comment">%</span>0125 <span class="comment">% When not actually at a singularity the Jacobian can provide information</span>0126 <span class="comment">% about how `well-conditioned' the manipulator is for making certain motions,</span>0127 <span class="comment">% and is referred to as `manipulability'.</span>0128 <span class="comment">%</span>0129 <span class="comment">% A number of scalar manipulability measures have been proposed. One by</span>0130 <span class="comment">% Yoshikawa</span>0131 maniplty(p560, q, <span class="string">'yoshikawa'</span>)0132 <span class="comment">%</span>0133 <span class="comment">% is based purely on kinematic parameters of the manipulator.</span>0134 <span class="comment">%</span>0135 <span class="comment">% Another by Asada takes into account the inertia of the manipulator which</span>0136 <span class="comment">% affects the acceleration achievable in different directions. This measure</span>0137 <span class="comment">% varies from 0 to 1, where 1 indicates uniformity of acceleration in all</span>0138 <span class="comment">% directions</span>0139 maniplty(p560, q, <span class="string">'asada'</span>)0140 <span class="comment">%</span>0141 <span class="comment">% Both of these measures would indicate that this particular pose is not well</span>0142 <span class="comment">% conditioned.</span>0143 pause <span class="comment">% any key to continue</span>0144 0145 <span class="comment">% An interesting class of manipulators are those that are redundant, that is,</span>0146 <span class="comment">% they have more than 6 degrees of freedom. Computing the joint motion for</span>0147 <span class="comment">% such a manipulator is not straightforward. Approaches have been suggested</span>0148 <span class="comment">% based on the pseudo-inverse of the Jacobian (which will not be square) or</span>0149 <span class="comment">% singular value decomposition of the Jacobian.</span>0150 <span class="comment">%</span>0151 echo off</pre></div><hr><address>Generated on Sun 15-Feb-2009 18:09:29 by <strong><a href="http://www.artefact.tk/software/matlab/m2html/">m2html</a></strong> © 2003</address></body></html>⌨️ 快捷键说明
复制代码
Ctrl + C
搜索代码
Ctrl + F
全屏模式
F11
切换主题
Ctrl + Shift + D
显示快捷键
?
增大字号
Ctrl + =
减小字号
Ctrl + -