基于mathmatica的机械运动仿真

于 2024-01-06 22:57:18 发布
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文章标签： 数学建模 线性代数
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本文链接：https://blog.youkuaiyun.com/toris181/article/details/135432737
版权
该博客记录了使用Mathematica进行五轴打印机机械运动仿真的过程。博主因对五轴打印机感兴趣而开展项目，完成打印机建模后对模型瘦身并转换格式。通过代码实现对模型部件的平移和旋转，利用矩阵运算赋予模型动态效果，展示了具体代码及运行结果。

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此博客用于记录与分享自己在使用mathmatica进行机械运动仿真的所学所作。非常简单，有手就行。

基于Mathematica的机器人仿真环境（机械臂篇）_自己搭建机器人仿真软件-优快云博客

此文给我莫大帮助，本文内容是此文的部分展示。

项目源于我对五轴打印机的兴趣：通过赋予传统3D打印机打印件额外两个旋转自由度，可以大幅增加（在没有支撑的情况下）能打印的造型范围。尤其是以旋转对称为主要特征的模型。

本着没有难度也要增加难度的想法，我在参考了open5x的3d打印机构型后从头造了一个打印机（截至目前2024.1.6没有实现全部功能）。下面进入正文：

在完成打印机建模后，为了减少仿真所需计算量，需要对真实模型瘦身并转换为STL格式（使用solidworks转换时记得勾选“不要转换STL输出数据到正的坐标空间”），最终简化到了6个部件，按1~6的顺序命名，分别对应框架，以及x、y、z、u、v五个自由度所需的五个活动部件。

代码如下

SetDirectory["C:\\Users\\Mr.H\\Desktop\\mmaprint"]
partsName = {"1.stl", "2.stl", "3.stl", "4.stl", "5.stl", "6.stl"};
robotPartsGraphics = Import[#, "Graphics3D"] & /@ partsName;
robotParts = robotPartsGraphics[[;; , 1]];
robotParts = 
  Table[GraphicsComplex[robotParts[[i, 1]], robotParts[[i, 2]]], {i, 
    6}];
colors = {Gray, Cyan, Orange, Green, Magenta, Yellow};
robotPartsColored = Transpose@{colors, robotParts};
Graphics3D[robotPartsColored]

S输出效果如下：

程序中的robotParts[[i, 1]]、robotParts[[i, 2]]分别表示第i个3d模型的所有点和三角形面。通过对这些点和面的移动和旋转，得到移动和旋转整体模型的效果。数据格式中并不存在robotParts[[i, 3]]

接下来就是使用GraphicsComplex（和Manipulate），将不同的平移和旋转矩阵与每个部件的坐标相乘，实现平移或者旋转的（动态）效果。

先将第六个部件初始位置调整一下（导出STL模型时疏忽了）。

newpart61 = GraphicsComplex[(\!\(\*
TagBox[GridBox[{
{".010958077"},
{".00415"},
{
RowBox[{"-", ".16525"}]}
},
GridBoxAlignment->{
            "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
             "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, 
             "Items" -> {}, "ItemsIndexed" -> {}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.5599999999999999]}, 
Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, 
             "ItemsIndexed" -> {}}],
Column]\)) + # & /@ robotParts[[6, 1]], robotParts[[6, 2]]];

Graphics3D[{robotParts[[1]], newpart61}]
Manipulate[
 Graphics3D[{robotParts[[1]], 
   GraphicsComplex[RotationMatrix[x, {0, 0, 1}].# & /@ newpart61[[1]],
     newpart61[[2]]]}], {x, 0 Degree, 360 Degree}]

在每个模型的名字后面我用数字区分变换前后的模型，例如61就是6经过一次变换，接下来62就是61再经过一次变换后的模型。

运行结果如下：

ps：可以通过RotationMatrix[u, {1, 0, 0}] // MatrixForm得到相应的具体变换矩阵

接下来就是对每个部件套矩阵了

<

newpart51 = GraphicsComplex[(\!\(\*
TagBox[GridBox[{
{".157658"},
{".00415"},
{
RowBox[{"-", ".12425"}]}
},
GridBoxAlignment->{
            "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
             "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, 
             "Items" -> {}, "ItemsIndexed" -> {}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.5599999999999999]}, 
Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, 
             "ItemsIndexed" -> {}}],
Column]\)) + # & /@ robotParts[[5, 1]], robotParts[[5, 2]]];
newpart52[x_] := 
  GraphicsComplex[RotationMatrix[x , {1, 0, 0}].# & /@ newpart51[[1]],
    newpart51[[2]]];
newpart62[v_, u_] := GraphicsComplex[\!\(\*
TagBox[
RowBox[{
TagBox[
RowBox[{"(", "", GridBox[{
{"1", "0", "0"},
{"0", 
RowBox[{"Cos", "[", "v", "]"}], 
RowBox[{"-", 
RowBox[{"Sin", "[", "v", "]"}]}]},
{"0", 
RowBox[{"Sin", "[", "v", "]"}], 
RowBox[{"Cos", "[", "v", "]"}]}
},
GridBoxAlignment->{
              "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
               "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, 
               "Items" -> {}, "ItemsIndexed" -> {}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.7]}, 
Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, 
               "ItemsIndexed" -> {}}], "", ")"}],
Function[BoxForm`e$, 
MatrixForm[BoxForm`e$]]], ".", 
RowBox[{"(", "", GridBox[{
{
RowBox[{"Cos", "[", "u", "]"}], 
RowBox[{"-", 
RowBox[{"Sin", "[", "u", "]"}]}], "0"},
{
RowBox[{"Sin", "[", "u", "]"}], 
RowBox[{"Cos", "[", "u", "]"}], "0"},
{"0", "0", "1"}
},
GridBoxAlignment->{
             "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
              "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, 
              "Items" -> {}, "ItemsIndexed" -> {}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.7]}, 
Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, 
              "ItemsIndexed" -> {}}], "", ")"}]}],
Function[BoxForm`e$, 
MatrixForm[BoxForm`e$]]]\).# & /@ newpart61[[1]], newpart61[[2]]];

Manipulate[
 Graphics3D[{LightGray, robotParts[[1]],
   Blue, GraphicsComplex[(\!\(\*
TagBox[GridBox[{
{"x"},
{"0"},
{"0"}
},
GridBoxAlignment->{
             "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
              "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, 
              "Items" -> {}, "ItemsIndexed" -> {}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.5599999999999999]}, 
Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, 
              "ItemsIndexed" -> {}}],
Column]\)) + # & /@ robotParts[[2, 1]], robotParts[[2, 2]]],
   Red, GraphicsComplex[(\!\(\*
TagBox[GridBox[{
{"x"},
{"y"},
{"0"}
},
GridBoxAlignment->{
             "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
              "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, 
              "Items" -> {}, "ItemsIndexed" -> {}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.5599999999999999]}, 
Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, 
              "ItemsIndexed" -> {}}],
Column]\)) + # & /@ robotParts[[3, 1]], robotParts[[3, 2]]],
   Green, GraphicsComplex[(\!\(\*
TagBox[GridBox[{
{"0"},
{"0"},
{"z"}
},
GridBoxAlignment->{
             "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
              "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, 
              "Items" -> {}, "ItemsIndexed" -> {}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.5599999999999999]}, 
Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, 
              "ItemsIndexed" -> {}}],
Column]\)) + # & /@ robotParts[[4, 1]], robotParts[[4, 2]]],
   Magenta, GraphicsComplex[(\!\(\*
TagBox[GridBox[{
{
RowBox[{"-", ".157658"}]},
{
RowBox[{"-", ".00415"}]},
{
RowBox[{".12425", "+", "z"}]}
},
GridBoxAlignment->{
             "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
              "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, 
              "Items" -> {}, "ItemsIndexed" -> {}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.5599999999999999]}, 
Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, 
              "ItemsIndexed" -> {}}],
Column]\)) + # & /@ newpart52[u ][[1]], newpart52[u ][[2]]], Yellow, 
   GraphicsComplex[(\!\(\*
TagBox[GridBox[{
{
RowBox[{"-", ".157658"}]},
{
RowBox[{"-", ".00415"}]},
{
RowBox[{".12425", "+", "z"}]}
},
GridBoxAlignment->{
             "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
              "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, 
              "Items" -> {}, "ItemsIndexed" -> {}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.5599999999999999]}, 
Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, 
              "ItemsIndexed" -> {}}],
Column]\)) + \!\(\*
TagBox[
RowBox[{"(", "", GridBox[{
{"1", "0", "0"},
{"0", 
RowBox[{"Cos", "[", "u", "]"}], 
RowBox[{"-", 
RowBox[{"Sin", "[", "u", "]"}]}]},
{"0", 
RowBox[{"Sin", "[", "u", "]"}], 
RowBox[{"Cos", "[", "u", "]"}]}
},
GridBoxAlignment->{
              "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
               "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, 
               "Items" -> {}, "ItemsIndexed" -> {}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.7]}, 
Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, 
               "ItemsIndexed" -> {}}], "", ")"}],
Function[BoxForm`e$, 
MatrixForm[BoxForm`e$]]]\).\!\(\*
TagBox[
RowBox[{"(", "", 
TagBox[GridBox[{
{"0.14669992299999998`"},
{"0.`"},
{"0.04100000000000001`"}
},
GridBoxAlignment->{
               "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, 
                "Rows" -> {{Baseline}}, "RowsIndexed" -> {}, 
                "Items" -> {}, "ItemsIndexed" -> {}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.5599999999999999]}, 
Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}, "RowsIndexed" -> {}, "Items" -> {}, 
                "ItemsIndexed" -> {}}],
Column], "", ")"}],
Function[BoxForm`e$, 


MatrixForm[BoxForm`e$]]]\) + # & /@ newpart64[u, v][[1]], 
    newpart62[u, v][[2]]]
   }], {x, 0, .12}, {y, 0, .12}, {z, 0, .12}, {u, 0 Degree, 
  90 Degree }, {v, 0 Degree, 90 Degree }]

>

效果如下：

toris181
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