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健身网站模板,五种网络广告形式,网站设计的一般流程,second是什么意思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​) ​100​0cosθx​−sinθx​​0sinθx​cosθx​​ ​ 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​) ​cosθy​0sinθy​​010​−sinθy​0cosθy​​ ​ 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\thetaz 0 \ 0 0 1 \end{bmatrix} Rz​(θz​) ​cosθz​−sinθz​0​sinθz​cosθz​0​001​ ​ 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​) ​Xw​Yw​Zw​​ ​ 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 P0 R{06} P_6 t_0 P0​R06​P6​t0​ ​ 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 \ Z6 \end{bmatrix} \begin{bmatrix} d{X0} \ d{Y0} \ d{Z_0} \end{bmatrix} ​X0​Y0​Z0​​ ​ ​∗∗∗​∗∗∗​∗∗∗​ ​[X6​ Y6​ Z6​​] ​dX0​​dY0​​dZ0​​​ ​ 式3-6 化为齐次形式为 P 0 T P 6 P_0 T P_6 P0​TP6​ 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 \ Z0 \ 1 \end{bmatrix} \begin{bmatrix} * * * d{X0} \ * * * d{Y0} \ * * * d{Z_0} \ 0 0 0 1 \end{bmatrix} \begin{bmatrix} X_6 \ Y_6 \ Z6 \ 1 \end{bmatrix} ​X0​Y0​Z0​1​ ​ ​∗∗∗0​∗∗∗0​∗∗∗0​dX0​​dY0​​dZ0​​1​ ​ ​X6​Y6​Z6​1​ ​ 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}, t0) (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} T60​T65​T54​T43​T32​T21​T10​ 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} T60−1​T10−1​T21−1​T32−1​T43−1​T54−1​T65−1​ 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} T06​T01​T12​T23​T34​T45​T56​ 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​) ​100​0cosθx​sinθx​​0−sinθx​cosθx​​ ​ ​ 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​) ​cosθy​0−sinθy​​010​sinθy​0cosθy​​ ​ 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​) ​cosθz​sinθz​0​−sinθz​cosθz​0​001​ ​ ​ 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\alphai \ 0 0 0 1 \end{bmatrix} Ai​MDH ​cosθi​sinθi​cosαi​sinθi​sinαi​0​−sinθi​cosθi​cosαi​cosθi​sinαi​0​0−sinαi​cosαi​0​ai​−di​sinαi​di​cosαi​1​ ​ 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 AMDH​A1​MDH⋅A2​MDH⋅A3​MDH⋅A4​MDH⋅A5​MDH⋅A6​MDH 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 q1​2.1372radq2​0.2629radq3​0.2045radq4​0.8672radq5​2.1497radq6​−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。