无锡网站建设 网站制作,网站建设 内容缺乏,博宇娱乐网站建设,wordpress主题生成器梯度下降法求六轴机械臂逆向解
一、几何基础
对于上述六轴机械臂的数学建模来说#xff0c;可以构建一个六轴机械臂的运动学正逆解的数学模型#xff0c;在一个直角坐标系中有如下旋转矩阵#xff1a;
绕x轴旋转 R x ( θ x ) [ 1 0 0 0 cos θ x sin θ x 0 − …梯度下降法求六轴机械臂逆向解
一、几何基础
对于上述六轴机械臂的数学建模来说可以构建一个六轴机械臂的运动学正逆解的数学模型在一个直角坐标系中有如下旋转矩阵
绕x轴旋转 R x ( θ x ) [ 1 0 0 0 cos θ x sin θ x 0 − sin θ x cos θ x ] R_x(\theta_x) \begin{bmatrix} 1 0 0 \\ 0 \cos\theta_x \sin\theta_x \\ 0 -\sin\theta_x \cos\theta_x \end{bmatrix} Rx(θx) 1000cosθx−sinθx0sinθxcosθx 式3-1
绕y轴旋转 R y ( θ y ) [ cos θ y 0 − sin θ y 0 1 0 sin θ y 0 cos θ y ] R_y(\theta_y) \begin{bmatrix} \cos\theta_y 0 -\sin\theta_y \\ 0 1 0 \\ \sin\theta_y 0 \cos\theta_y \end{bmatrix} Ry(θy) cosθy0sinθy010−sinθy0cosθy 式3-2
绕z轴旋转 R z ( θ z ) [ cos θ z sin θ z 0 − sin θ z cos θ z 0 0 0 1 ] R_z(\theta_z) \begin{bmatrix} \cos\theta_z \sin\theta_z 0 \\ -\sin\theta_z \cos\theta_z 0 \\ 0 0 1 \end{bmatrix} Rz(θz) cosθz−sinθz0sinθzcosθz0001 式3-3
在几何学中任何一个坐标系 可以通过三个以上基础旋转唯一变换到另一个同原点的坐标系 两个关系可以用按照变换的先后顺序左乘表示如下所示。 [ X w ′ Y w ′ Z w ′ ] R x ( θ x ) R y ( θ y ) R z ( θ z ) [ X w Y w Z w ] \begin{bmatrix} X_{w} \\ Y_{w} \\ Z_{w} \end{bmatrix} R_x(\theta_x) R_y(\theta_y) R_z(\theta_z) \begin{bmatrix} X_w \\ Y_w \\ Z_w \end{bmatrix} Xw′Yw′Zw′ Rx(θx)Ry(θy)Rz(θz) XwYwZw 式3-4
二、六轴机械臂正解模型
在三维空间里如果有两个坐标系F0F6如果想知道P在F6系中的向量或者点坐标在F0系中是多少可以表示如下 P 0 R 06 P 6 t 0 P_0 R_{06} P_6 t_0 P0R06P6t0 式3-5 [ X 0 Y 0 Z 0 ] [ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ] [ X 6 Y 6 Z 6 ] [ d X 0 d Y 0 d Z 0 ] \begin{bmatrix} X_0 \\ Y_0 \\ Z_0 \end{bmatrix} \begin{bmatrix} * * * \\ * * * \\ * * * \end{bmatrix} \begin{bmatrix} X_6 \ Y_6 \ Z_6 \end{bmatrix} \begin{bmatrix} d_{X_0} \\ d_{Y_0} \\ d_{Z_0} \end{bmatrix} X0Y0Z0 ∗∗∗∗∗∗∗∗∗ [X6 Y6 Z6] dX0dY0dZ0 式3-6
化为齐次形式为 P 0 T P 6 P_0 T P_6 P0TP6 式3-7 [ X 0 Y 0 Z 0 1 ] [ ∗ ∗ ∗ d X 0 ∗ ∗ ∗ d Y 0 ∗ ∗ ∗ d Z 0 0 0 0 1 ] [ X 6 Y 6 Z 6 1 ] \begin{bmatrix} X_0 \\ Y_0 \\ Z_0 \\ 1 \end{bmatrix} \begin{bmatrix} * * * d_{X_0} \\ * * * d_{Y_0} \\ * * * d_{Z_0} \\ 0 0 0 1 \end{bmatrix} \begin{bmatrix} X_6 \\ Y_6 \\ Z_6 \\ 1 \end{bmatrix} X0Y0Z01 ∗∗∗0∗∗∗0∗∗∗0dX0dY0dZ01 X6Y6Z61 式3-8
如上机械臂的末端位姿有两部分组成 ( R 06 , t 0 ) (R_{06}, t_0) (R06,t0) 式3-9
可以理解为一个是末端原点的坐标 也就是式3-7中的T中的最后一列前三个一个是旋转量 或者可以用欧拉角序列、翻滚角-俯仰角-偏航角序列等表示在机械臂中可以控制的只有各个关节的电机角度六轴就是六个电机的角度 通过设定六个电机角度可以把机械臂的末端送到目标的位置。这个过程数学上相当于机械臂基座的坐标系通过一些列的旋转平移变换到了末端由如下的这种机制 T 60 T 65 T 54 T 43 T 32 T 21 T 10 T_{60} T_{65} T_{54} T_{43} T_{32} T_{21} T_{10} T60T65T54T43T32T21T10 式3-10
两边取逆得到 T 60 − 1 T 10 − 1 T 21 − 1 T 32 − 1 T 43 − 1 T 54 − 1 T 65 − 1 T_{60}^{-1} T_{10}^{-1} T_{21}^{-1} T_{32}^{-1} T_{43}^{-1} T_{54}^{-1} T_{65}^{-1} T60−1T10−1T21−1T32−1T43−1T54−1T65−1 式3-11
即 T 06 T 01 T 12 T 23 T 34 T 45 T 56 T_{06} T_{01} T_{12} T_{23} T_{34} T_{45} T_{56} T06T01T12T23T34T45T56 式3-12
机械臂的建模就是基于类似这样的机制学者Denavit和Hartenberg经过研究简化后提出了机器人的DHDenavit-Hartenberg建模理论[44]。于是DH建模所用到的基础旋转矩阵就变为
绕x轴旋转 R x ( θ x ) [ 1 0 0 0 cos θ x − sin θ x 0 sin θ x cos θ x ] R_x(\theta_x) \begin{bmatrix} 1 0 0 \\ 0 \cos\theta_x -\sin\theta_x \\ 0 \sin\theta_x \cos\theta_x \end{bmatrix} Rx(θx) 1000cosθxsinθx0−sinθxcosθx 式3-13
绕y轴旋转 R y ( θ y ) [ cos θ y 0 sin θ y 0 1 0 − sin θ y 0 cos θ y ] R_y(\theta_y) \begin{bmatrix} \cos\theta_y 0 \sin\theta_y \\ 0 1 0 \\ -\sin\theta_y 0 \cos\theta_y \end{bmatrix} Ry(θy) cosθy0−sinθy010sinθy0cosθy 式3-14
绕z轴旋转 R z ( θ z ) [ cos θ z − sin θ z 0 sin θ z cos θ z 0 0 0 1 ] R_z(\theta_z) \begin{bmatrix} \cos\theta_z -\sin\theta_z 0 \\ \sin\theta_z \cos\theta_z 0 \\ 0 0 1 \end{bmatrix} Rz(θz) cosθzsinθz0−sinθzcosθz0001 式3-15
三、六轴机械臂DH建模
在机器人学和机械臂设计领域有两种常用的建模方法标准Denavit-HartenbergSDH和改进Denavit-HartenbergMDH。为了简化建模过程并提高效率本文采用MDH规则进行模型构建。MDH方法提供了一种优化的坐标系定义方式它将坐标系定位在每个连杆的近端而不是远端这样做有助于简化计算和减少误差[45]其主要的原理是这样的以遨博AUBO-i7H协作机器人为例 先确定Z轴各关节的电机转轴 逐步从固定支座坐标系F0确定X轴的方向原则是X轴要是本坐标系Z轴和下一个坐标系Z轴的公垂线。如下图X0轴应该垂直于Z0和Z1那么可以这么定X0垂直于轴Z0和Z1X1轴垂直于Z1和Z2X2应该垂直于Z2和Z3… 图3-6 六轴机械臂各坐标系
根据右乘原理MDH中规定当前关节i的参数先后的顺序是 以上面的坐标系F1为例四个参数代表是坐标F0系先以X0轴旋转Alpha再沿着X0轴方向移动a负方向移动为负数再以Z0为轴旋转Theta最后沿着Z0方向移动d最后得到F1坐标系。此处Alpha0X0不需要旋转a0沿着X0没有平移Theta 沿着Z0逆时针旋转145度X0轴转到了X1轴的位置然后d122mm沿着Z0正方向移动了122mm这样经过连续的旋转加平移后就完成了关节1的参数建模。以此类推逐步确定其余5组参数。所有的参数确定如下图所示 图3-7 六轴机械臂的坐标系变换传递
可得到MDH的基本变换矩阵为 A i M D H [ cos θ i − sin θ i 0 a i sin θ i cos α i cos θ i cos α i − sin α i − d i sin α i sin θ i sin α i cos θ i sin α i cos α i d i cos α i 0 0 0 1 ] A_iMDH \begin{bmatrix} \cos\theta_i -\sin\theta_i 0 a_i \\ \sin\theta_i\cos\alpha_i \cos\theta_i\cos\alpha_i -\sin\alpha_i -d_i\sin\alpha_i \\ \sin\theta_i\sin\alpha_i \cos\theta_i\sin\alpha_i \cos\alpha_i d_i\cos\alpha_i \\ 0 0 0 1 \end{bmatrix} AiMDH cosθisinθicosαisinθisinαi0−sinθicosθicosαicosθisinαi00−sinαicosαi0ai−disinαidicosαi1 式3-16
即可完成正向求解所谓正向模型就是输入6个电机的旋转角度及各机械臂段的固定段长可以得出末端的位姿这里可以用位姿矩阵表示。我们根据机械臂实际的尺寸推导出的正向模型 A M D H A 1 M D H ⋅ A 2 M D H ⋅ A 3 M D H ⋅ A 4 M D H ⋅ A 5 M D H ⋅ A 6 M D H A_{MDH} A_1MDH \cdot A_2MDH \cdot A_3MDH \cdot A_4MDH \cdot A_5MDH \cdot A_6MDH AMDHA1MDH⋅A2MDH⋅A3MDH⋅A4MDH⋅A5MDH⋅A6MDH 式3-17
根据参数可求得正向解模型如下q1、q2、q3、q4、q5、q6为六个电机角度
Matrix([[6.12323399573677e-17*(-(-1.0*(-(-6.12323399573677e-17*sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) ((-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949))cos(q4 1.5707963267949) 1.22464679914735e-16sin(q1 - 4.71238898038469)sin(q4 1.5707963267949))sin(q5 1.5707963267949) 6.12323399573677e-17(-1.0(-(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - ((-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949))sin(q4 1.5707963267949) 1.22464679914735e-16sin(q1 - 4.71238898038469)cos(q4 1.5707963267949))cos(q5 1.5707963267949) - 1.0(-1.22464679914735e-16(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) 1.22464679914735e-16(-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))cos(q3 - 1.5707963267949) 1.0sin(q1 - 4.71238898038469))cos(q5 1.5707963267949))sin(q6) ((-1.0(-(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) ((-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949))cos(q4 1.5707963267949) 1.22464679914735e-16sin(q1 - 4.71238898038469)sin(q4 1.5707963267949))cos(q5 1.5707963267949) 6.12323399573677e-17(-1.0(-(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - ((-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949))sin(q4 1.5707963267949) 1.22464679914735e-16sin(q1 - 4.71238898038469)cos(q4 1.5707963267949))sin(q5 1.5707963267949) - 1.0(-1.22464679914735e-16(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) 1.22464679914735e-16(-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)cos(q1 - 4.71238898038469))cos(q3 - 1.5707963267949) 1.0sin(q1 - 4.71238898038469))sin(q5 1.5707963267949))cos(q6) 1.0(-1.0(-(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - 1.0((-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949))sin(q4 1.5707963267949) - 7.49879891330929e-33(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) 7.49879891330929e-33(-6.12323399573677e-17sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)cos(q1 - 4.71238898038469))cos(q3 - 1.5707963267949) 1.22464679914735e-16sin(q1 - 4.71238898038469)cos(q4 1.5707963267949) 6.12323399573677e-17sin(q1 - 4.71238898038469))sin(q6), 6.12323399573677e-17(-(-1.0(-(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) ((-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949))cos(q4 1.5707963267949) 1.22464679914735e-16sin(q1 - 4.71238898038469)sin(q4 1.5707963267949))sin(q5 1.5707963267949) 6.12323399573677e-17(-1.0(-(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - ((-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949))sin(q4 1.5707963267949) 1.22464679914735e-16sin(q1 - 4.71238898038469)cos(q4 1.5707963267949))cos(q5 1.5707963267949) - 1.0(-1.22464679914735e-16(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) 1.22464679914735e-16(-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))cos(q3 - 1.5707963267949) 1.0sin(q1 - 4.71238898038469))cos(q5 1.5707963267949))cos(q6) - ((-1.0(-(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) ((-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949))cos(q4 1.5707963267949) 1.22464679914735e-16sin(q1 - 4.71238898038469)sin(q4 1.5707963267949))cos(q5 1.5707963267949) 6.12323399573677e-17(-1.0(-(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - ((-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949))sin(q4 1.5707963267949) 1.22464679914735e-16sin(q1 - 4.71238898038469)cos(q4 1.5707963267949))sin(q5 1.5707963267949) - 1.0(-1.22464679914735e-16(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) 1.22464679914735e-16(-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)cos(q1 - 4.71238898038469))cos(q3 - 1.5707963267949) 1.0sin(q1 - 4.71238898038469))sin(q5 1.5707963267949))sin(q6) 1.0(-1.0(-(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - 1.0((-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949))sin(q4 1.5707963267949) - 7.49879891330929e-33(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) 7.49879891330929e-33(-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)cos(q1 - 4.71238898038469))cos(q3 - 1.5707963267949) 1.22464679914735e-16sin(q1 - 4.71238898038469)cos(q4 1.5707963267949) 6.12323399573677e-17sin(q1 - 4.71238898038469))cos(q6), -6.12323399573677e-17(-(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - 6.12323399573677e-17((-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949))sin(q4 1.5707963267949) - 4.59169004331693e-49(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) 4.59169004331693e-49(-6.12323399573677e-17sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)cos(q1 - 4.71238898038469))cos(q3 - 1.5707963267949) 1.0(-1.0(-(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) ((-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949))cos(q4 1.5707963267949) 1.22464679914735e-16sin(q1 - 4.71238898038469)sin(q4 1.5707963267949))sin(q5 1.5707963267949) - 6.12323399573677e-17(-1.0(-(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - ((-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949))sin(q4 1.5707963267949) 1.22464679914735e-16sin(q1 - 4.71238898038469)cos(q4 1.5707963267949))cos(q5 1.5707963267949) 1.0(-1.22464679914735e-16(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) 1.22464679914735e-16(-6.12323399573677e-17sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)cos(q1 - 4.71238898038469))cos(q3 - 1.5707963267949) 1.0sin(q1 - 4.71238898038469))cos(q5 1.5707963267949) 7.49879891330929e-33sin(q1 - 4.71238898038469)cos(q4 1.5707963267949) 3.74939945665464e-33sin(q1 - 4.71238898038469), 102.5(-(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) 102.5((-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949))sin(q4 1.5707963267949) 7.68626888614202e-31(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) 316(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) 316(-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949) - 7.68626888614202e-31(-6.12323399573677e-17sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)cos(q1 - 4.71238898038469))cos(q3 - 1.5707963267949) 94.0(-1.0(-(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) ((-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949))cos(q4 1.5707963267949) 1.22464679914735e-16sin(q1 - 4.71238898038469)sin(q4 1.5707963267949))sin(q5 1.5707963267949) - 5.75583995599256e-15(-1.0(-(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - ((-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) (-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949))sin(q4 1.5707963267949) 1.22464679914735e-16sin(q1 - 4.71238898038469)cos(q4 1.5707963267949))cos(q5 1.5707963267949) 94.0(-1.22464679914735e-16(-6.12323399573677e-17sin(q1 - 4.71238898038469)*sin(q2 - 1.0471975511966) cos(q1 - 4.71238898038469)cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) 1.22464679914735e-16(-6.12323399573677e-17sin(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966) - sin(q2 - 1.0471975511966)cos(q1 - 4.71238898038469))cos(q3 - 1.5707963267949) 1.0sin(q1 - 4.71238898038469))cos(q5 1.5707963267949) - 2.25335011043113e-14sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) - 1.25526296912604e-14sin(q1 - 4.71238898038469)cos(q4 1.5707963267949) - 140.5sin(q1 - 4.71238898038469) 368cos(q1 - 4.71238898038469)cos(q2 - 1.0471975511966)], [(6.12323399573677e-17(-((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*sin(q3 - 1.5707963267949) (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) - 1.0((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*cos(q3 - 1.5707963267949) - (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*sin(q3 - 1.5707963267949))cos(q4 1.5707963267949) - 1.22464679914735e-16cos(q1 - 4.71238898038469)*cos(q4 1.5707963267949))*sin(q5 1.5707963267949) (((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*sin(q3 - 1.5707963267949) (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - 1.0((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*cos(q3 - 1.5707963267949) - (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*sin(q3 - 1.5707963267949))sin(q4 1.5707963267949) - 1.22464679914735e-16sin(q4 1.5707963267949)cos(q1 - 4.71238898038469))cos(q5 1.5707963267949) - 1.0(1.22464679914735e-16(-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) - 1.22464679914735e-16(sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949) - 1.0cos(q1 - 4.71238898038469))sin(q5 1.5707963267949))cos(q6) 6.12323399573677e-17(6.12323399573677e-17(-((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*sin(q3 - 1.5707963267949) (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) - 1.0((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*cos(q3 - 1.5707963267949) - (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*sin(q3 - 1.5707963267949))cos(q4 1.5707963267949) - 1.22464679914735e-16cos(q1 - 4.71238898038469)*cos(q4 1.5707963267949))*cos(q5 1.5707963267949) - (((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*sin(q3 - 1.5707963267949) (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - 1.0((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*cos(q3 - 1.5707963267949) - (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*sin(q3 - 1.5707963267949))sin(q4 1.5707963267949) - 1.22464679914735e-16sin(q4 1.5707963267949)cos(q1 - 4.71238898038469))sin(q5 1.5707963267949) - 1.0(1.22464679914735e-16(-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) - 1.22464679914735e-16(sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949) - 1.0cos(q1 - 4.71238898038469))cos(q5 1.5707963267949))sin(q6) 1.0(-1.0((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*sin(q3 - 1.5707963267949) (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) - 1.0((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*cos(q3 - 1.5707963267949) - (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*sin(q3 - 1.5707963267949))cos(q4 1.5707963267949) 7.49879891330929e-33(-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) - 7.49879891330929e-33(sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949) - 1.22464679914735e-16cos(q1 - 4.71238898038469)cos(q4 1.5707963267949) - 6.12323399573677e-17cos(q1 - 4.71238898038469))sin(q6), -(6.12323399573677e-17(-((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*sin(q3 - 1.5707963267949) (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) - 1.0((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*cos(q3 - 1.5707963267949) - (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*sin(q3 - 1.5707963267949))cos(q4 1.5707963267949) - 1.22464679914735e-16cos(q1 - 4.71238898038469)*cos(q4 1.5707963267949))*sin(q5 1.5707963267949) (((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*sin(q3 - 1.5707963267949) (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - 1.0((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*cos(q3 - 1.5707963267949) - (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*sin(q3 - 1.5707963267949))sin(q4 1.5707963267949) - 1.22464679914735e-16sin(q4 1.5707963267949)cos(q1 - 4.71238898038469))cos(q5 1.5707963267949) - 1.0(1.22464679914735e-16(-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) - 1.22464679914735e-16(sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949) - 1.0cos(q1 - 4.71238898038469))sin(q5 1.5707963267949))sin(q6) 6.12323399573677e-17(6.12323399573677e-17(-((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*sin(q3 - 1.5707963267949) (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) - 1.0((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*cos(q3 - 1.5707963267949) - (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*sin(q3 - 1.5707963267949))cos(q4 1.5707963267949) - 1.22464679914735e-16cos(q1 - 4.71238898038469)*cos(q4 1.5707963267949))*cos(q5 1.5707963267949) - (((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*sin(q3 - 1.5707963267949) (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - 1.0((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*cos(q3 - 1.5707963267949) - (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*sin(q3 - 1.5707963267949))sin(q4 1.5707963267949) - 1.22464679914735e-16sin(q4 1.5707963267949)cos(q1 - 4.71238898038469))sin(q5 1.5707963267949) - 1.0(1.22464679914735e-16(-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) - 1.22464679914735e-16(sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949) - 1.0cos(q1 - 4.71238898038469))cos(q5 1.5707963267949))cos(q6) 1.0(-1.0((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*sin(q3 - 1.5707963267949) (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) - 1.0((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*cos(q3 - 1.5707963267949) - (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*sin(q3 - 1.5707963267949))cos(q4 1.5707963267949) 7.49879891330929e-33(-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) - 7.49879891330929e-33(sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949) - 1.22464679914735e-16cos(q1 - 4.71238898038469)cos(q4 1.5707963267949) - 6.12323399573677e-17cos(q1 - 4.71238898038469))cos(q6), -6.12323399573677e-17((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*sin(q3 - 1.5707963267949) (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) - 6.12323399573677e-17((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*cos(q3 - 1.5707963267949) - (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*sin(q3 - 1.5707963267949))cos(q4 1.5707963267949) 4.59169004331693e-49(-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) - 4.59169004331693e-49(sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949) - 6.12323399573677e-17(-((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*sin(q3 - 1.5707963267949) (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) - 1.0((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*cos(q3 - 1.5707963267949) - (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*sin(q3 - 1.5707963267949))cos(q4 1.5707963267949) - 1.22464679914735e-16cos(q1 - 4.71238898038469)*cos(q4 1.5707963267949))cos(q5 1.5707963267949) 1.0(((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*sin(q3 - 1.5707963267949) (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - 1.0((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*cos(q3 - 1.5707963267949) - (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*sin(q3 - 1.5707963267949))sin(q4 1.5707963267949) - 1.22464679914735e-16sin(q4 1.5707963267949)cos(q1 - 4.71238898038469))sin(q5 1.5707963267949) 1.0(1.22464679914735e-16(-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) - 1.22464679914735e-16(sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949) - 1.0cos(q1 - 4.71238898038469))cos(q5 1.5707963267949) - 7.49879891330929e-33cos(q1 - 4.71238898038469)cos(q4 1.5707963267949) - 3.74939945665464e-33cos(q1 - 4.71238898038469), 102.5((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*sin(q3 - 1.5707963267949) (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) 102.5((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*cos(q3 - 1.5707963267949) - (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*sin(q3 - 1.5707963267949))cos(q4 1.5707963267949) 316(-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))sin(q3 - 1.5707963267949) - 7.68626888614202e-31(-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) 7.68626888614202e-31(sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949) 316(sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))cos(q3 - 1.5707963267949) - 5.75583995599256e-15(-((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*sin(q3 - 1.5707963267949) (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) - 1.0((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*cos(q3 - 1.5707963267949) - (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*sin(q3 - 1.5707963267949))cos(q4 1.5707963267949) - 1.22464679914735e-16cos(q1 - 4.71238898038469)*cos(q4 1.5707963267949))cos(q5 1.5707963267949) 94.0(((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*sin(q3 - 1.5707963267949) (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))*cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - 1.0((-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)*cos(q2 - 1.0471975511966))*cos(q3 - 1.5707963267949) - (sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)*cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949))sin(q4 1.5707963267949) - 1.22464679914735e-16sin(q4 1.5707963267949)cos(q1 - 4.71238898038469))sin(q5 1.5707963267949) 94.0(1.22464679914735e-16(-sin(q1 - 4.71238898038469)sin(q2 - 1.0471975511966) 6.12323399573677e-17cos(q1 - 4.71238898038469)cos(q2 - 1.0471975511966))cos(q3 - 1.5707963267949) - 1.22464679914735e-16(sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 6.12323399573677e-17sin(q2 - 1.0471975511966)cos(q1 - 4.71238898038469))sin(q3 - 1.5707963267949) - 1.0cos(q1 - 4.71238898038469))cos(q5 1.5707963267949) 368sin(q1 - 4.71238898038469)cos(q2 - 1.0471975511966) 2.25335011043113e-14sin(q2 - 1.0471975511966)cos(q1 - 4.71238898038469) 1.25526296912604e-14cos(q1 - 4.71238898038469)cos(q4 1.5707963267949) 140.5cos(q1 - 4.71238898038469)], [6.12323399573677e-17(-(-1.0(1.0sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.0cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) (-1.0sin(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 1.0sin(q3 - 1.5707963267949)cos(q2 - 1.0471975511966))cos(q4 1.5707963267949) - 7.49879891330929e-33sin(q4 1.5707963267949))sin(q5 1.5707963267949) 6.12323399573677e-17(-1.0(1.0sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.0cos(q2 - 1.0471975511966)*cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - (-1.0sin(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 1.0sin(q3 - 1.5707963267949)cos(q2 - 1.0471975511966))sin(q4 1.5707963267949) - 7.49879891330929e-33cos(q4 1.5707963267949))cos(q5 1.5707963267949) - 1.0(1.22464679914735e-16sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.22464679914735e-16cos(q2 - 1.0471975511966)*cos(q3 - 1.5707963267949) - 6.12323399573677e-17)cos(q5 1.5707963267949))sin(q6) ((-1.0(1.0sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.0cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) (-1.0sin(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 1.0sin(q3 - 1.5707963267949)cos(q2 - 1.0471975511966))cos(q4 1.5707963267949) - 7.49879891330929e-33sin(q4 1.5707963267949))cos(q5 1.5707963267949) 6.12323399573677e-17(-1.0(1.0sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.0cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - (-1.0sin(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 1.0sin(q3 - 1.5707963267949)cos(q2 - 1.0471975511966))sin(q4 1.5707963267949) - 7.49879891330929e-33cos(q4 1.5707963267949))sin(q5 1.5707963267949) - 1.0(1.22464679914735e-16sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.22464679914735e-16cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 6.12323399573677e-17)sin(q5 1.5707963267949))cos(q6) 1.0(-1.0(1.0sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.0cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - 1.0(-1.0sin(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 1.0sin(q3 - 1.5707963267949)cos(q2 - 1.0471975511966))sin(q4 1.5707963267949) 7.49879891330929e-33sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 7.49879891330929e-33cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 7.49879891330929e-33cos(q4 1.5707963267949) - 3.74939945665464e-33)sin(q6), 6.12323399573677e-17(-(-1.0(1.0sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.0cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) (-1.0sin(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 1.0sin(q3 - 1.5707963267949)cos(q2 - 1.0471975511966))cos(q4 1.5707963267949) - 7.49879891330929e-33sin(q4 1.5707963267949))sin(q5 1.5707963267949) 6.12323399573677e-17(-1.0(1.0sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.0cos(q2 - 1.0471975511966)*cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - (-1.0sin(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 1.0sin(q3 - 1.5707963267949)cos(q2 - 1.0471975511966))sin(q4 1.5707963267949) - 7.49879891330929e-33cos(q4 1.5707963267949))cos(q5 1.5707963267949) - 1.0(1.22464679914735e-16sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.22464679914735e-16cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 6.12323399573677e-17)cos(q5 1.5707963267949))cos(q6) - ((-1.0(1.0sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.0cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) (-1.0sin(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 1.0sin(q3 - 1.5707963267949)cos(q2 - 1.0471975511966))cos(q4 1.5707963267949) - 7.49879891330929e-33sin(q4 1.5707963267949))cos(q5 1.5707963267949) 6.12323399573677e-17(-1.0(1.0sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.0cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - (-1.0sin(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 1.0sin(q3 - 1.5707963267949)cos(q2 - 1.0471975511966))sin(q4 1.5707963267949) - 7.49879891330929e-33cos(q4 1.5707963267949))sin(q5 1.5707963267949) - 1.0(1.22464679914735e-16sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.22464679914735e-16cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 6.12323399573677e-17)sin(q5 1.5707963267949))sin(q6) 1.0(-1.0(1.0sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.0cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - 1.0(-1.0sin(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 1.0sin(q3 - 1.5707963267949)cos(q2 - 1.0471975511966))sin(q4 1.5707963267949) 7.49879891330929e-33sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 7.49879891330929e-33cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 7.49879891330929e-33cos(q4 1.5707963267949) - 3.74939945665464e-33)cos(q6), -6.12323399573677e-17(1.0sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.0cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - 6.12323399573677e-17(-1.0sin(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 1.0sin(q3 - 1.5707963267949)cos(q2 - 1.0471975511966))sin(q4 1.5707963267949) 1.0(-1.0(1.0sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.0cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) (-1.0sin(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 1.0sin(q3 - 1.5707963267949)cos(q2 - 1.0471975511966))cos(q4 1.5707963267949) - 7.49879891330929e-33sin(q4 1.5707963267949))sin(q5 1.5707963267949) - 6.12323399573677e-17(-1.0(1.0sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.0cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - (-1.0sin(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 1.0sin(q3 - 1.5707963267949)cos(q2 - 1.0471975511966))sin(q4 1.5707963267949) - 7.49879891330929e-33cos(q4 1.5707963267949))cos(q5 1.5707963267949) 1.0(1.22464679914735e-16sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.22464679914735e-16cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 6.12323399573677e-17)cos(q5 1.5707963267949) 4.59169004331693e-49sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 4.59169004331693e-49cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 4.59169004331693e-49cos(q4 1.5707963267949) - 2.29584502165847e-49, 102.5(1.0sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.0cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) 102.5(-1.0sin(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 1.0sin(q3 - 1.5707963267949)cos(q2 - 1.0471975511966))sin(q4 1.5707963267949) 94.0(-1.0(1.0sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.0cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949))sin(q4 1.5707963267949) (-1.0sin(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 1.0sin(q3 - 1.5707963267949)cos(q2 - 1.0471975511966))cos(q4 1.5707963267949) - 7.49879891330929e-33sin(q4 1.5707963267949))sin(q5 1.5707963267949) - 5.75583995599256e-15(-1.0(1.0sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.0cos(q2 - 1.0471975511966)*cos(q3 - 1.5707963267949))cos(q4 1.5707963267949) - (-1.0sin(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 1.0sin(q3 - 1.5707963267949)cos(q2 - 1.0471975511966))sin(q4 1.5707963267949) - 7.49879891330929e-33cos(q4 1.5707963267949))cos(q5 1.5707963267949) 94.0(1.22464679914735e-16sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 1.22464679914735e-16cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 6.12323399573677e-17)cos(q5 1.5707963267949) - 7.68626888614202e-31sin(q2 - 1.0471975511966)sin(q3 - 1.5707963267949) - 316.0sin(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) - 368.0sin(q2 - 1.0471975511966) - 316.0sin(q3 - 1.5707963267949)cos(q2 - 1.0471975511966) 7.68626888614202e-31cos(q2 - 1.0471975511966)cos(q3 - 1.5707963267949) 7.68626888614202e-31cos(q4 1.5707963267949) 122.0], [0, 0, 0, 1]])
四、机械臂逆向解
六轴机械臂的逆向解是指给定机械臂末端执行器例如夹爪或工具头在空间中的位置和姿态位姿求解机械臂各关节的角度使得末端执行器能够达到这个特定的位姿。简而言之逆向解就是从末端执行器的位姿反推关节角度的过程。在实际应用中逆向解非常重要因为通常我们关心的是机械臂末端执行器需要到达的位置和姿态而不是关节的具体运动。逆向解使得机械臂能够根据任务需求来规划其运动路径。逆向解的求解其实就是正向模型设置6个未知数然后根据正向方程解未知数过程通常涉及到线性方程组或非线性方程组的求解。逆向解的求解可以是解析解或数值解。解析解是指通过数学公式直接求解得到的精确解而数值解则需要通过迭代算法如梯度下降法逐步逼近解。解析解通常更稳定但并不是所有机械臂模型都能得到解析解有时需要使用数值方法来求解。这里用python开源库pytorch的梯度下降法来求解可以很方便地构建张量计算图并运用反向传播的思想调用电脑GPU加快求解速度。 图3-8 使用梯度下降法求六轴机械臂的逆解误差图 图3-9 六个电机角度逆解
可以求得机械臂六个电机的转动角度 q 1 2.1372 r a d q 2 0.2629 r a d q 3 0.2045 r a d q 4 0.8672 r a d q 5 2.1497 r a d q 6 − 0.2768 r a d q_12.1372rad\\ q_20.2629rad\\ q_30.2045rad\\ q_40.8672rad\\ q_52.1497rad\\ q_6-0.2768rad q12.1372radq20.2629radq30.2045radq40.8672radq52.1497radq6−0.2768rad 据此求出与目标点的误差为 d x 0.042694091796875 m m d y 0.00885009765625 m m d z 0.01141357421875 m m dx0.042694091796875mm\\ dy0.00885009765625mm\\ dz0.01141357421875mm\\ dx0.042694091796875mmdy0.00885009765625mmdz0.01141357421875mm 可以看出误差很小基本可以满足大部分工业场景在实际应用中逆向解的求解可能面临多个解的问题因为机械臂可能有多个不同的关节角度配置可以使得末端执行器达到相同的位姿。此外逆向解的求解还可能遇到奇异点问题即某些关节角度配置使得机械臂的某些部分相互交叉或无法达到目标位姿。逆向解在机器人学、自动化、制造业等领域有着广泛的应用是机械臂控制和路径规划的基础。
