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<!DOCTYPE html>
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<li class="chapter" data-level="1" data-path=""><a href="#bas"><i class="fa fa-check"></i><b>1</b> BAS案例二:龙门起重机运动控制</a><ul>
<li class="chapter" data-level="1.1" data-path=""><a href="#section-1.1"><i class="fa fa-check"></i><b>1.1</b> 问题背景</a></li>
<li class="chapter" data-level="1.2" data-path=""><a href="#section-1.2"><i class="fa fa-check"></i><b>1.2</b> 优化问题抽象</a></li>
<li class="chapter" data-level="1.3" data-path=""><a href="#section-1.3"><i class="fa fa-check"></i><b>1.3</b> 优化理论</a></li>
<li class="chapter" data-level="1.4" data-path=""><a href="#section-1.4"><i class="fa fa-check"></i><b>1.4</b> 优化结果</a></li>
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<h4 class="author"><em>Jiangyu Wang</em></h4>
<h4 class="date"><em>2018骞<b4>9鏈<88>7鏃<a5></em></h4>
</div>
<div id="bas" class="section level1">
<h1><span class="header-section-number">第 1 章</span> BAS案例二:龙门起重机运动控制</h1>
<blockquote>
<p>由群友李晓晓同学提供案例。</p>
</blockquote>
<div id="section-1.1" class="section level2">
<h2><span class="header-section-number">1.1</span> 问题背景</h2>
<p>龙门起重机（gantry crane），如图<a href="#fig:gc">1.1</a>所示，是水平桥架设置在两条支腿上构成门架形状的一种桥架型起重机。起重小车trolley在桥架上运行，利用绳索（cable）一类的柔性体连接负载（load）。在负载的输送过程中，load的摆动问题一直是影响吊车装运效率提高的一大难题。</p>
<div class="figure" style="text-align: center"><span id="fig:gc"></span>
<img src="img/gc.png" alt="龙门起重机示意" width="70%" />
<p class="caption">
图 1.1: 龙门起重机示意
</p>
</div>
<p>模型可以简化为图<a href="#fig:gc2">1.2</a>。重物通过绳索与小车相连，小车在外力的作用下水平运动，小车质量为M（kg），重物的质量为m（kg），绳索的长度为r（m），<span class="math inline">\(\theta\)</span>为摆角，x表示水平方向上的位移。</p>
<div class="figure" style="text-align: center"><span id="fig:gc2"></span>
<img src="img/gc2.png" alt="龙门起重机模型" width="60%" />
<p class="caption">
图 1.2: 龙门起重机模型
</p>
</div>
</div>
<div id="section-1.2" class="section level2">
<h2><span class="header-section-number">1.2</span> 优化问题抽象</h2>
<p>优化的变量为<code>bang-coast-bang</code>控制中，<span class="math inline">\(x = [tp_1,tp_2,tf]^T\)</span>，两个给予脉冲加速度激励的时段以及全部的作业时间，以此来使得负载的摆角（优化目标）最小。</p>
<p>取小车的位置x，摆角<span class="math inline">\(\theta\)</span>作为系统的广义坐标系。小车和负载在t时刻的位置坐标如式<a href="#eq:loadposition">(1.1)</a>所示。</p>
<span class="math display" id="eq:loadposition">\[\begin{equation}
\begin{cases}
x_M(t)&=&x(t)\\
y_M(t)&=&0 \\
x_m(t)&=&x(t)+rsin\theta(t)\\
t_m(t)&=&-rco\theta(t)
\end{cases}
\tag{1.1}
\end{equation}\]</span>
<p>因此，对<span class="math inline">\(t\)</span>求导，得到小车和负载的速度分量如式<a href="#eq:speedp">(1.2)</a>所示。</p>
<span class="math display" id="eq:speedp">\[\begin{equation}
\begin{cases}
\dot{x}_M(t) &=& \dot{x}_t\\
\dot{y}_M(t) &=& 0\\
\dot{x}_m(t) &=& \dot{x}(t)+r\dot{\theta}(t)cos\theta(t)\\
\dot{y}_m(t) &=& r\dot{\theta}(t)sin\theta(t)
\end{cases}
\tag{1.2}
\end{equation}\]</span>
<p>系统动能如式<a href="#eq:kineticenergy">(1.3)</a>：</p>
<span class="math display" id="eq:kineticenergy">\[\begin{equation}
\begin{split}
T = &\frac{1}{2}MV_M^2(t) + \frac{1}{2}mV_m^2(t) \\
=&\frac{1}{2}M(\dot{x}_M^2(t)+\dot{y}_M^2(t))+\frac{1}{2}m(\dot{x}_m^2(t)+\dot{y}_m^2(t))\\
=&\frac{1}{2}M\dot{x}_M^2(t) + \frac{1}{2}m\dot{x}_m^2(t) + \\
&\frac{1}{2}mr^2\dot{\theta}^2(t) + mr\dot{x}(t)\dot{\theta}(t)cos\theta(t)
\end{split}
\tag{1.3}
\end{equation}\]</span>
<p>势能如式<a href="#eq:potentialenergy">(1.4)</a>：</p>
<span class="math display" id="eq:potentialenergy">\[\begin{equation}
P = -mgrcos\theta(t)
\tag{1.4}
\end{equation}\]</span>
<p>通过式<a href="#eq:potentialenergy">(1.4)</a>和式<a href="#eq:kineticenergy">(1.3)</a>，我们可以得到拉格朗日方程，即<span class="math inline">\(L = T -P\)</span>。根据欧拉-拉格朗日方程(Euler-Lagrangian equation)，龙门起重机的动力学模型如式<a href="#eq:EL">(1.5)</a>所示。</p>
<span class="math display" id="eq:EL">\[\begin{equation}
\frac{d}{dt}[\frac{\partial{L}}{\partial{\dot{q_k}}}] - \frac{\partial{L}}{\partial{q_k}}=Q_k, \quad k=1,\cdots,n
\tag{1.5}
\end{equation}\]</span>
<p>其中，<span class="math inline">\(n\)</span>表示系统的自由度，<span class="math inline">\(\{q_1,\cdots,q_n\}\)</span>表示广义坐标集合，<span class="math inline">\(\{Q_1,\cdots,Q_n\}\)</span>表示广义的驱动力（对应于各自的广义坐标）。把摆角作为广义坐标，可以得到式<a href="#eq:thetaEL1">(1.6)</a>。</p>
<span class="math display" id="eq:thetaEL1">\[\begin{equation}
\frac{d}{dt}[\frac{\partial{L}}{\partial{\dot{\theta}}}] - \frac{\partial{L}}{\partial{\theta}}=0
\tag{1.6}
\end{equation}\]</span>
<p>第一项为<span class="math inline">\(mr^2\ddot{\theta}(t)+mr\ddot{x}(t)cos\theta(t)\)</span>，第二项为<span class="math inline">\(-mgrsin\theta(t)\)</span>，代入式<a href="#eq:thetaEL1">(1.6)</a>中可以得到<span class="math inline">\(\theta\)</span>的求解方程，即式<a href="#eq:thetaEL2">(1.7)</a>。</p>
<span class="math display" id="eq:thetaEL2">\[\begin{equation}
\ddot{\theta}(t)=-\frac{\ddot{x}(t)cos\theta(t)+gsin\theta(t)}{r}
\tag{1.7}
\end{equation}\]</span>
<p>通过求解该方程，我们可以得到我们的优化目标。</p>
</div>
<div id="section-1.3" class="section level2">
<h2><span class="header-section-number">1.3</span> 优化理论</h2>
<p>优化理论为二阶BAS算法。李晓晓同学给出的参数是，<span class="math inline">\(w_0 = 0.7\)</span>， <span class="math inline">\(w_1 = 0.2\)</span>。当然，文档中对于不同的目标函数可能有不同的调参结果，大家可以自行调试。</p>
</div>
<div id="section-1.4" class="section level2">
<h2><span class="header-section-number">1.4</span> 优化结果</h2>
<table class="table table-striped" style="width: auto !important; margin-left: auto; margin-right: auto;">
<caption>
<span id="tab:example2parms">表 1.1: </span>实验参数取值
</caption>
<thead>
<tr>
<th style="text-align:left;">
Parameter
</th>
<th style="text-align:left;">
Description
</th>
<th style="text-align:left;">
Values
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<span class="math inline">\(c\)</span>
</td>
<td style="text-align:left;">
Constant
</td>
<td style="text-align:left;">
5
</td>
</tr>
<tr>
<td style="text-align:left;">
<span class="math inline">\(\lambda\)</span>
</td>
<td style="text-align:left;">
Step length ratio
</td>
<td style="text-align:left;">
0.95
</td>
</tr>
<tr>
<td style="text-align:left;">
<span class="math inline">\(\gamma\)</span>
</td>
<td style="text-align:left;">
Penalty coefficient
</td>
<td style="text-align:left;">
<span class="math inline">\(10^{50}\)</span>
</td>
</tr>
<tr>
<td style="text-align:left;">
<span class="math inline">\(n\)</span>
</td>
<td style="text-align:left;">
Number of iterations
</td>
<td style="text-align:left;">
1000
</td>
</tr>
<tr>
<td style="text-align:left;">
<span class="math inline">\(r\)</span>
</td>
<td style="text-align:left;">
Height of crane
</td>
<td style="text-align:left;">
20(m)
</td>
</tr>
<tr>
<td style="text-align:left;">
<span class="math inline">\(h\)</span>
</td>
<td style="text-align:left;">
Load displacement
</td>
<td style="text-align:left;">
40(m)
</td>
</tr>
</tbody>
</table>
<table class="table table-striped" style="width: auto !important; margin-left: auto; margin-right: auto;">
<caption>
<span id="tab:example2result">表 1.2: </span>优化结果对比
</caption>
<thead>
<tr>
<th style="text-align:left;">
Variables
</th>
<th style="text-align:right;">
fmincon
</th>
<th style="text-align:right;">
BAS
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<span class="math inline">\(tp_1\)</span>
</td>
<td style="text-align:right;">
8.9624000
</td>
<td style="text-align:right;">
6.9082000
</td>
</tr>
<tr>
<td style="text-align:left;">
<span class="math inline">\(tp_2\)</span>
</td>
<td style="text-align:right;">
8.9803000
</td>
<td style="text-align:right;">
6.9741000
</td>
</tr>
<tr>
<td style="text-align:left;">
<span class="math inline">\(tf\)</span>
</td>
<td style="text-align:right;">
19.8346000
</td>
<td style="text-align:right;">
15.8866000
</td>
</tr>
<tr>
<td style="text-align:left;">
<span class="math inline">\(f\)</span>
</td>
<td style="text-align:right;">
0.0029586
</td>
<td style="text-align:right;">
0.0001415
</td>
</tr>
</tbody>
</table>
<p>从表<a href="#tab:example2result">1.2</a>中可以看到，BAS能够让起重机小车在7秒内加速到最大速度，在16秒内就能到达终点并消除摆角。动图<a href="#fig:example2fminconbob">1.3</a>至<a href="#fig:example2bastrace">1.6</a>分别为采用<code>fmincon</code>和<code>BAS</code>做优化的结果，可以让我们有更为直观的认识。</p>
<div class="figure" style="text-align: center"><span id="fig:example2fminconbob"></span>
<img src="img/fminconbob.gif" alt="fmincon摆锤" />
<p class="caption">
图 1.3: fmincon摆锤
</p>
</div>
<div class="figure" style="text-align: center"><span id="fig:example2basbob"></span>
<img src="img/BASbob.gif" alt="BAS摆锤" />
<p class="caption">
图 1.4: BAS摆锤
</p>
</div>
<div class="figure" style="text-align: center"><span id="fig:example2fmincontrace"></span>
<img src="img/fmincontrace.gif" alt="fmincon轨迹" />
<p class="caption">
图 1.5: fmincon轨迹
</p>
</div>
<div class="figure" style="text-align: center"><span id="fig:example2bastrace"></span>
<img src="img/BAStrace.gif" alt="BAS轨迹" />
<p class="caption">
图 1.6: BAS轨迹
</p>
</div>
<div class="figure" style="text-align: center"><span id="fig:ex2accel"></span>
<img src="img/ex2_1.png" alt="起重小车加速度曲线" width="70%" />
<p class="caption">
图 1.7: 起重小车加速度曲线
</p>
</div>
<div class="figure" style="text-align: center"><span id="fig:ex2velocity"></span>
<img src="img/ex2_2.png" alt="起重小车速度曲线" width="70%" />
<p class="caption">
图 1.8: 起重小车速度曲线
</p>
</div>
<div class="figure" style="text-align: center"><span id="fig:ex2position"></span>
<img src="img/ex2_3.png" alt="起重小车位置曲线" width="70%" />
<p class="caption">
图 1.9: 起重小车位置曲线
</p>
</div>
</div>
</div>
</section>
</div>
</div>
</div>
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