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宋体;">一个数控系统除了能实现被控运动对象的精确定位以外，还必须实现被控对象以给定的速度沿着指定的路径运动。为了使机械平滑稳定的运动，</span><span style="text-indent: 21pt;"> </span><span style="text-indent: 21pt; font-family: 宋体;">还需要对驱动电机的速度进行控制，</span><span style="text-indent: 21pt;"> </span><span style="text-indent: 21pt; font-family: 宋体;">本论文先讨论了一般的逐点比较法插补算法，</span><span style="text-indent: 21pt;"> </span><span style="text-indent: 21pt; font-family: 宋体;">然后建立了基本的直线和圆弧插补，</span><span style="text-indent: 21pt;"> </span><span style="text-indent: 21pt; font-family: 宋体;">最后进行了误差分析，</span><span style="text-indent: 21pt;"> </span><span style="text-indent: 21pt; font-family: 宋体;">并针对步进电机容易失步的问题，</span><span style="text-indent: 21pt;"> </span><span style="text-indent: 21pt; font-family: 宋体;">引入了电机的速度控制算法。</span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span lang="EN-US">1 </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">逐点比较法实现的插补算法</span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">被控对象的运动轨迹是由各种线型构成，</span> <span style="font-family:宋体; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">如直线、</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">圆弧、</span> <span style="font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">抛物线等，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">最主要的便是直线和圆弧。为了实现轨迹控制必须进行插补，</span> <span style="font-family: 宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">计算出中间点，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">插补的任务实际上就是从轨迹起点到终点之间的数据密化工作。逐点比较法就是每走一步都要将工作点的瞬间坐标与规定的运动轨迹进行比较，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">判断偏差，</span> <span style="font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">根据偏差确定下一步的进给方向，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">得到一个非常近似于规定轨迹的路线，</span> <span style="font-family:宋体; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">且最大误差当量不超过一个脉冲当量［</span><span lang="EN-US">1</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">］。</span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span lang="EN-US">1</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">．</span><span lang="EN-US"> 1 </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">逐点比较法实现直线插补</span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">逐点比较法在实现直线插补时，</span> <span style="font-family:宋体; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">需要先确定起点和终点，根据起点和终点的相对位置来判断象限，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">并据此来判断进给方向。假设点在第一象限，</span> <span style="font-family:宋体; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">第一象限内的直线插补如图</span><span lang="EN-US">1 </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">所示。</span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span lang="EN-US"><img src="/upload/images/201406/14033647691024.jpg" alt="" /><br /> <!--[endif]--></span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span lang="EN-US">1</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">．</span><span lang="EN-US"> 2 </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">逐点比较法实现圆弧插补</span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">逐点比较法也能很方便地实现圆弧插补。与直线插补类似，</span> <span style="font-family: 宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">圆弧插补是将加工点到圆心的距离与被加工圆弧的名义半径进行比较，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">并根据偏差大小确定进给方向［</span><span lang="EN-US">2</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">］。其在四个象限内分为顺圆弧和逆圆弧插补八种类型。典型的圆弧插补分为偏差判别、</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">坐标进给、</span> <span style="font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">偏差计算、</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">终点判别四个步骤。第一象限内逆圆弧插补如图</span><span lang="EN-US">2 </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">所示。</span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span lang="EN-US">2 </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">步进电机的速度控制算法</span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">引起步进电机失步的主要原因有以下两种</span><span lang="EN-US">: </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">一种是由于系</span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span lang="EN-US"><img src="/upload/images/201406/14033647811245.jpg" alt="" /><br /> <!--[endif]--></span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">统过载，</span> <span style="font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">解决该问题在于是系统负载力矩不超过步进电机的矩频特性曲线并需要保留一定的余量</span><span lang="EN-US">; </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">另一种便是启停过程中的升降速运动变换，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">解决该问题在于必须对步进电机进给脉冲频率进行加减速。当加速时，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">保证进给脉冲频率逐渐增大</span><span lang="EN-US">; </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">当减速时，</span> <span style="font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">保证进给脉冲逐渐减小。在目前的数控系统中，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">常用的加减速算法有直线加减速控制算法和</span><span lang="EN-US"> S </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">曲线加减速控制算法等。</span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span lang="EN-US">2</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">．</span><span lang="EN-US"> 1 </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">直线加减速控制算法</span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">直线加减速控制又称梯形速度曲线控制，</span> <span style="font-family:宋体; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">它使机床启动时，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">速度按照一定斜率的直线上升，</span> <span style="font-family:宋体; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">在停止时，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">速度沿一定斜率的直线下降。如图</span><span lang="EN-US">3 </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">所示。</span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span lang="EN-US"><img src="/upload/images/201406/14033647917744.jpg" alt="" /><br /> <!--[endif]--></span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">以加速阶段分析</span><span lang="EN-US">: </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">其关系式可表示为</span><span lang="EN-US"> V = a* t</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">。在给定最高速度</span><span lang="EN-US"> Vm </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">的情况下，</span> <span style="font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">到达时间</span><span lang="EN-US"> Tm = Vm/a</span><span style="font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">一般的实现方法是时间</span><span lang="EN-US">t </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">从</span><span lang="EN-US">0 </span><span style="font-family: 宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">开始递增，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">对应每个</span><span lang="EN-US">t </span><span style="font-family: 宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">代入式</span><span lang="EN-US">v = a* t </span><span style="font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">中计算出</span><span lang="EN-US">v</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">，</span> <span style="font-family: 宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">这个方法可行但是计算量太多，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">涉及到浮点运算。不利于在嵌入式系统中运用。为了提高运算的效率，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">减小浮点运算量，</span> <span style="font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">利用插补算法，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">以数字方式实现运动过程。采用最小偏差法，</span> <span style="font-family:宋体; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">来实现直线运动，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">软件实现其过程的流程图如图</span><span lang="EN-US">4 </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">所示。</span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span lang="EN-US">2</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">．</span><span lang="EN-US"> 2 </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">指数加减速控制算法</span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span lang="EN-US"><img src="/upload/images/201406/14033648012067.jpg" alt="" /><br /> <!--[endif]--></span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">指数加减速曲线如图</span><span lang="EN-US"> 5 </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">所示。它的加速和减速曲线是对称的。下面以加速阶段研究指数运行曲线。其运行速度公式为</span><span lang="EN-US"> v( t) = vc ( 1 </span><span style="font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">－</span><span lang="EN-US"> e</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">－</span><span lang="EN-US"> t /) </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">，</span> <span style="font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">其中</span><span lang="EN-US"> Vc </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">代表终点速度或是频率，</span><span lang="EN-US"> t </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">代表时间，代表调节系统时间常数。时间常数</span> <span style="font-family:宋体; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">反映了系统从速度</span><span lang="EN-US">0 </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">变化到给定的最高速度的变化效率，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">加速过程的时间受该常数的约束，</span> <span style="font-family:宋体; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">所以采用指数曲线进行加减速要根据系统选好时间常数</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">［</span><span lang="EN-US">3</span><span style="font-family: 宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">］。</span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">将上式分解得</span><span lang="EN-US">:</span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span lang="EN-US"><img src="/upload/images/201406/14033648109233.jpg" alt="" /><br /> <!--[endif]--></span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">因此只需给出每个采样间隔</span><span lang="EN-US"> T </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">时间内在速度或频率上需要的进给量，</span> <span style="font-family:宋体; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">就使算法实现数字化。程序实现流程如图</span><span lang="EN-US"> 6</span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">所示</span><span lang="EN-US">:</span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span lang="EN-US">3 </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">两种加减速算法的比较由上述的两种加减速算法可以看出</span><span lang="EN-US">: </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">在给定一样的加减速时间和最高速度的情况下</span><span lang="EN-US">( </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">运动轨迹相同</span><span lang="EN-US">) </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">，</span> <span style="font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">梯形速度发生曲线是一个恒加速过程，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">它的快速性比较好。但它的加速度有突变</span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span lang="EN-US">4 </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">结论</span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">数控系统的任务就是控制电机的位置和速度，</span> <span style="font-family:宋体; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">如何使机床按照规定的直线和曲线运动，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">怎样使电机平稳的运行，</span> <span style="font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">是数控技术的核心问题。本文对数控技术中的关键算法</span><span lang="EN-US">( </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">即插补算法和速度控制算法</span><span lang="EN-US">) </span><span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">进行了完整的描述和分析，</span> <span style="font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">并比较了其优劣，提出了相应实现的方法和编写软件的流程图。并针对大量使用的嵌入式系统进行了算法的优化，</span> <span style="font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">使之能满足嵌入式系统对速度位置控制的要求。</span></p> <h1 align="center" style="margin-top:4.5pt;margin-right:0cm;margin-bottom:7.5pt; 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