提高超越方程解的精度

问题描述

我有一个特定的运动学作为更复杂机器的一部分，需要计算一些物理参数，这些参数非常难(更像是不可能)以适当的准确度测量> 使用我可以使用的仪器

I have a specific kinematics as a part of a more complex machine and need to compute some physical parameters that are very hard (more like impossible) to measure with proper accuracy with instruments I have at my disposal

[运动学]

乍一看是一个简单的1自由度臂(黑色)，可以绕x轴旋转.它有一个重量迫使它总是向上移动，直到它碰到机械端点(角度 a0)或半径为 r0 的某个管子(蓝色).手臂旋转中心在y0.管子可以移动到任何y(t)高度.

At first look it is a simple 1 degree of freedom arm (black) which can rotate around x axis. It has a weight to force it to go always up until it hit the mechanic endpoint (angle a0) or some tube (blue) with radius r0. Arm rotation center is at y0. The tube can be moved to any y(t) height.

[使用]

这用于测量管的半径以进行进一步处理.半径可以计算(通过基本的测角法)，这导致图像底部的方程.常数 a0,y0,z0 很难测量(它在复杂的机械内部)所以距离的测量精度为 min 0.1 mm 和角度 0.1 deg 甚至这是有问题的.

This is used to measure the radius of a tube for further processing. The radius can be computed (by basic goniometry) which leads to equation in the bottom of image. The constants a0,y0,z0 are very hard to measure (it is inside complex machinery) so the measurement accuracy for distances is min 0.1 mm and angle 0.1 deg and even that is questionable.

[校准]

所以我决定尝试从机器本身完成的一组测量中计算这些参数(自动校准).所以我有已知半径 r0 的校准管.所有绿色参数都可以作为常量处理.现在我沿着 y 轴定位管子，以覆盖尽可能多的手臂角度.遗憾的是，范围只有 20 度(对于当前机器设置)记住测量的 a(t) 预设 y(t) ...作为 n 点数据集.这给了我 n 个超越方程的系统.由此我尝试/猜测 a0,y0,z0 记住最佳解决方案的所有"可能性(最接近 r0)

So I decided to try compute these parameters from set of measurements done by the machine itself (auto-calibration). So I have calibration tube with known radius r0. All green parameters can be handled as constants. Now I position the tube along y axis to cover as much angles of arm as I could. Sadly the range is only about 20 degrees (for current machine setup) remembering measured a(t) for preset y(t) ... as n point dataset. This gives me system of n transcendent equations. From this I try/guess "all" possibilities of a0,y0,z0 remembering the best solution (closest to r0)

[a0,y0,z0 的近似值]

近似是基于我的这个类:

approximation is based on this class of mine:

//---------------------------------------------------------------------------
class approx
    {
public:
    double a,aa,a0,a1,da,*e,e0;
    int i,n;
    bool done,stop;

    approx()            { a=0.0; aa=0.0; a0=0.0; a1=1.0; da=0.1; e=NULL; e0=NULL; i=0; n=5; done=true; }
    approx(approx& a)   { *this=a; }
    ~approx()           {}
    approx* operator = (const approx *a) { *this=*a; return this; }
    //approx* operator = (const approx &a) { ...copy... return this; }

    void init(double _a0,double _a1,double _da,int _n,double *_e)
        {
        if (_a0<=_a1) { a0=_a0; a1=_a1; }
        else          { a0=_a1; a1=_a0; }
        da=fabs(_da);
        n =_n ;
        e =_e ;
        e0=-1.0;
        i=0; a=a0; aa=a0;
        done=false; stop=false;
        }
    void step()
        {
        if ((e0<0.0)||(e0>*e)) { e0=*e; aa=a; }         // better solution
        if (stop)                                       // increase accuracy
            {
            i++; if (i>=n) { done=true; a=aa; return; } // final solution
            a0=aa-fabs(da);
            a1=aa+fabs(da);
            a=a0; da*=0.1;
            a0+=da; a1-=da;
            stop=false;
            }
        else{
            a+=da; if (a>a1) { a=a1; stop=true; }       // next point
            }
        }
    };
//---------------------------------------------------------------------------

它通过一些初始步骤搜索单个变量的完整范围，然后找到最小偏差点.之后改变范围和步长以关闭该点的区域并递归提高精度.

It search the full range of single variable by some initial step then find the min deviation point. After that change the range and step to close area of this point and recursively increase accuracy.

解决方案本身如下所示:

The solution itself looks like this:

// (global) input data
#define _irc_calib_n 100
#define _irc_approx_n 5
int    irc_calib_ix; // number of measured points
double irc_calib_y[_irc_calib_n]; // y(t)
double irc_calib_a[_irc_calib_n]; // a(t)
double irc_calib_r; // calibration tube radius + arm radius

// approximation
int ix=0;
double e,a,deg=M_PI/180.0;
approx aa,ay,az;
//           min       max       step     recursions    ErrorOfSolutionVariable
for (aa.init(-90.0*deg,+90.0*deg,10.0*deg,_irc_approx_n,&e);!aa.done;aa.step())
for (ay.init(  0.0    ,200.0    ,10.0    ,_irc_approx_n,&e);!ay.done;ay.step())
for (az.init( 50.0    ,400.0    ,10.0    ,_irc_approx_n,&e);!az.done;az.step())
    {
    for (e=0.0,ix=0;ix<_irc_calib_n;ix++) // test all measured points (e is cumulative error)
        {
        a=irc_calib_a[ix]+aa.a;
        if (a> pi) a-=pi2;
        if (a<-pi) a+=pi2;
        if (fabs(a)>0.5*pi) { e=100.0; break; } // ignore too far angles
        e+=fabs(+(cos(a)*(irc_calib_y[ix]-ay.a))
                -(sin(a)*(az.a))
                -(irc_calib_r));
        }
    }
// here aa.a,ay.a,az.a holds the result

这导致解接近测量值，但在模拟中结果仍然不够准确.根据点数和角度范围，它从 0.1 毫米到 0.5 毫米不等.如果我正确测量 z0 并忽略它的近似值，那么精度会显着提高，使 y0 没有错误(在模拟中)和 a0 的误差约为 0.3学位

This leads to solution close to the measured values but inside simulation the result is still not accurate enough. It is from 0.1 mm to 0.5 mm depending on number of points and angle range. If I measure properly z0 and ignore its approximation then the precision is boosted significantly leaving y0 without error (in simulation) and a0 with error around 0.3 degree

Q1 如何进一步提高解的准确度?

我无法增加角度范围.点数最好在 100 左右，精度越高越好，但超过 150 结果不稳定(对于某些半径，它完全关闭).完全不知道为什么.6 上面的递归数没有太大影响

I cannot increase the angular range. The number of points is best around 100 the more the better accuracy but above 150 the result is unstable (for some radiuses it is completely off). Have absolutely no clue why. The recursions number above 6 has not much effect

能否帮助根据与 0 度 的角距离对偏差进行加权?但遗憾的是 a(t) 范围不一定包括 0 度

Could help weighting the deviations according to angular distance from 0 degree ? But sadly a(t) range does not necessarily include 0 degrees

y0,z0 的0.01 mm 和a0

Q2 有什么我遗漏的地方吗?

比如错误嵌套的近似值或一些数学简化或不同的方法

Like wrongly nested approximations or some math simplification or different approach

[注释]

角度必须是a(t)+a0的形式，因为它是由IRC测量的，软件复位(16000 steps/round).它在 a0 位置时重置，我不计算振动和校准管偏心，它们已经得到处理，我的第一个目标是在没有它们的情况下在模拟中进行这项工作.管子y(t)可以随意定位，a(t)测量可以随意进行.

The angle must be in form of a(t)+a0 because it is measured by IRC with SW reset (16000 steps/round). It is reseted when in a0 position I do not count vibrations and calibration tube eccentricity they are taken care of already and my first goal is to make this work in simulation without them. Tube y(t) can be positioned at free will and the a(t) measurement can be done at will.

现在校准过程沿着 y 轴扫描点(从 a0 向下移动).使用 6 递归计算大约需要 35 秒(所以请耐心等待).5 递归大约需要 22 秒

Right now the calibration process scan points along y axis (movement from a0 down). Computation with 6 recursions take around 35 seconds (so be patient). 5 recursions take around 22 seconds

[edit1] 这里是如何进行模拟

approx aa; double e;
for (aa.init(-90.0*deg,+90.0*deg,10.0*deg,6,&e);!aa.done;aa.step())
 e=fabs(+(cos(aa.a)*(y(t)-y0))
        -(sin(aa.a)*(z0))
        -(irc_calib_r));
if (aa.a<a0) aa.a=a0;

[edit2] 一些值

刚刚意识到我在模拟代码中只有 4 个递归来匹配输入的 IRC 精度，那么必须有 6 个递归.更改后(也在之前的编辑中)这里有一些结果

Just realized that I had only 4 recursions in simulation code to match the input IRC accuracy then there must be 6 recursions. After changing it (also in previous edit) here are some results

                | a0[deg]| y0[mm] | z0[mm] | 
    simulated   | -7.4510|191.2590|225.9000|
    z0 known    | -7.4441|191.1433|225.9000|
    z0 unknown  | -7.6340|191.8074|225.4971|

因此，z0 测量的精度几乎在所需范围内，但 z0 未知，误差仍然比所需的 ~10 大.提高模拟精度对6递归没有影响，也没有意义，因为真实的输入数据也不会更准确.

So the accuracy with z0 measured is almost in desired range but with z0 unknown the error is still ~10 times bigger then needed. Increasing simulation accuracy has no effect above 6 recursions and also no sense because real input data will not be more accurate either.

这里是使用上述模拟设置进行测试的模拟/测量点:

Here the simulated/measured points for testing with above simulated settings:

 ix   a [deg]    y [mm]
  0   -0.2475 +105.7231 
  1   -0.4500 +104.9231 
  2   -0.6525 +104.1231 
  3   -0.8550 +103.3231 
  4   -1.0575 +102.5231 
  5   -1.2600 +101.7231 
  6   -1.4625 +100.9231 
  7   -1.6650 +100.1231 
  8   -1.8675  +99.3231 
  9   -2.0700  +98.5231 
 10   -2.2725  +97.7231 
 11   -2.4750  +96.9231 
 12   -2.6775  +96.1231 
 13   -2.8575  +95.3077 
 14   -3.0600  +94.5154 
 15   -3.2625  +93.7231 
 16   -3.4650  +92.9308 
 17   -3.6675  +92.1385 
 18   -3.8700  +91.3462 
 19   -4.0725  +90.5538 
 20   -4.2750  +89.7615 
 21   -4.4877  +88.9692 
 22   -4.6575  +88.1769 
 23   -4.8825  +87.3615 
 24   -5.0850  +86.5154 
 25   -5.2650  +85.7000 
 26   -5.4675  +84.9077 
 27   -5.6700  +84.1154 
 28   -5.8725  +83.3231 
 29   -6.0750  +82.5308 
 30   -6.2775  +81.7000 
 31   -6.5025  +80.8462 
 32   -6.6825  +80.0462 
 33   -6.8850  +79.2538 
 34   -7.0875  +78.4615 
 35   -7.2900  +77.6538 
 36   -7.5159  +76.7692 
 37   -7.6725  +75.9769 
 38   -7.8750  +75.1846 
 39   -8.1049  +74.3692 
 40   -8.2800  +73.5000 
 41   -8.4825  +72.7077 
 42   -8.6850  +71.9154 
 43   -8.9100  +71.0308 
 44   -9.0900  +70.2231 
 45   -9.2925  +69.4308 
 46   -9.5175  +68.5462 
 47   -9.6975  +67.7462 
 48   -9.9000  +66.9462 
 49  -10.1025  +66.0615 
 50  -10.3148  +65.2692 
 51  -10.4850  +64.3769 
 52  -10.6875  +63.5846 
 53  -10.9125  +62.7462 
 54  -11.0925  +61.9077 
 55  -11.2950  +61.0846 
 56  -11.4975  +60.2231 
 57  -11.7000  +59.3923 
 58  -11.9025  +58.5308 
 59  -12.1288  +57.6692 
 60  -12.3075  +56.8385 
 61  -12.5100  +55.9462 
 62  -12.7125  +55.1538 
 63  -12.9150  +54.2615 
 64  -13.1175  +53.4000 
 65  -13.2975  +52.5769 
 66  -13.5000  +51.6846 
 67  -13.7025  +50.7923 
 68  -13.9050  +50.0000 
 69  -14.1075  +49.1077 
 70  -14.3100  +48.2154 
 71  -14.5350  +47.3615 
 72  -14.7150  +46.5308 
 73  -14.9175  +45.6385 
 74  -15.1200  +44.7462 
 75  -15.3225  +43.8538 
 76  -15.5250  +42.9615 
 77  -15.7490  +42.0692 
 78  -15.9075  +41.2769 
 79  -16.1100  +40.3846 
 80  -16.3125  +39.4923 
 81  -16.5150  +38.6000 
 82  -16.7175  +37.7077 
 83  -16.9200  +36.8154 
 84  -17.1225  +35.9231 
 85  -17.3250  +34.9308 
 86  -17.5275  +34.0385 
 87  -17.7300  +33.1462 
 88  -17.9325  +32.2538 
 89  -18.1350  +31.3615 
 90  -18.3405  +30.4692 
 91  -18.5175  +29.4769 
 92  -18.7200  +28.5846 
 93  -18.9225  +27.6923 
 94  -19.1250  +26.8000 
 95  -19.3275  +25.8077 
 96  -19.5300  +24.9154 
 97  -19.7325  +23.9231 
 98  -19.9350  +23.0308 
 99  -20.1375  +22.1385 

[edit3] 进度更新

对@Ben 的一些说明

工作原理

第一张图片下的彩色方程给出了半径 r0 它由 2 个连接的 90 度 三角形(基本三角学)组成

the colored equation under the first image gives you the radius r0 it is made from 2 joined 90 degree triangles (basic trigonometry)

红色的东西:

• y(t) 是电机位置，它是已知的
• a(t) 也是已知的 IRC 状态

• y(t) is motor position and it is known
• a(t) is IRC state also known

绿色的东西:

• a0,y0,z0 是机械尺寸，已知但不精确，所以我测量了 y(t) 不同位置的许多 a(t) 使用已知的校准管 r0 并从中计算出更高精度的 a0,y0,z0

    ;      computed     simulated  |delta|
    a0=  -6.915840 ;  -6.916710   +0.000870 deg
    y0=+186.009765 ;+186.012822   +0.003057 mm
    y1=+158.342452 ;+158.342187   +0.000264 mm
    z0=+228.102470 ;+228.100000   +0.002470 mm