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机械臂正向与逆向运动学求解

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        机械臂的正运动学求解即建立DH参数表，然后计算出各变换矩阵以及最终的变换矩阵。逆运动学求解，即求出机械臂各关节θ角与px,py,pz的关系，建立θ角与末端姿态之间的数学模型，在这里以IRB6700为例，对IRB6700进行正逆运动学求解和验证。

目录

正运动学求解

        首先使用DH法建立坐标系如下：

         查阅IRB6700的参数如下表

连杆i

a_{i-1}/mm

 \alpha_{i-1}/° d_{i}/mm \theta_{i}/° 关节角\theta_{i}范围/°
1 0 0 780 \theta_{1} +170 ——  (-170)
2 320 -90 0 \theta_{2} +85 —— (-65)
3 1125 0 0 \theta_{3} +70 —— (-180)
4 200 -90 1142.5 \theta_{4} +300 —— (-300)
5 0 90 0 \theta_{5} +130 —— (-130)
6 0 -90 200 \theta_{6} +360 —— (-360)

根据变换矩阵公式编写matlab代码计算得到正运动学的各变换矩阵

_{i-1}^{i}\textrm{T}=\; \begin{bmatrix} c_{i} &-s_{i} & 0 & a_{i-1} \\ s_{i}c_{\alpha _{i-1}} & c_{i}c_{\alpha _{i-1}} & -s_{\alpha_{i-1}} &-s_{\alpha_{i-1}}d_i \\ s_{i}s_{\alpha _{i-1}} & c_{i}s_{\alpha _{i-1}} & c_{\alpha _{i-1}} & s_{\alpha_{i-1}}d_i{}\\ 0 & 0 & 0 & 1 \end{bmatrix}

上式中的s_{i}为sin(\theta_{i}),s_{\alpha_{i}}为sin(\alpha_{i})，c为cos

 计算变换矩阵的MATLAB代码如下

clear;clc
%%
%导入参数
%注，这里的a,afa的下标在实际使用时均需要减去1
syms a afa d theta [1 6]
a = [0 a2 a3 a4 0 0];
afa = [0 -90 0 -90 90 -90];
d = [d1 0 0 d4 0 d6];
theta = [theta1 theta2 theta3 theta4 theta5 theta6];
syms T [4 4 6]
%%
%正运动学模型求解
%计算各变换矩阵及总的变换矩阵
for i = 1:6
   T(:,:,i) =  trans_cal(  afa(i), a(i), d(i), theta(i)*180/pi );
          if(i == 1)
              trans_matrix = T(:,:,i);
          else
              trans_matrix = trans_matrix*T(:,:,i);
          end
end

 用到的funcion函数如下

function T = trans_cal(afa_ii,a_ii,d_i,theta_i)
%%
%计算变换矩阵函数T_{i-1,i}
%输入的参数为，afa_{i-1},a_{i-1},d_i,theta_i,与DH表达法的参数表对应
%ii 为i-1
%注意，这里输入的角度，均采用角度制，不采用弧度制

T = [cosd(theta_i)              -sind(theta_i)             0               a_ii
     sind(theta_i)*cosd(afa_ii)  cosd(theta_i)*cosd(afa_ii)  -sind(afa_ii)    -sind(afa_ii)*d_i
    sind(theta_i)*sind(afa_ii)   cosd(theta_i)*sind(afa_ii)  cosd(afa_ii)     cosd(afa_ii)*d_i
     0                         0                         0                1               ];
 
end

 各个变换矩阵存储在T中，其中T(:,:,i)即为变换矩阵_{i}^{i-1}\textrm{T},_{6}^{0}\textrm{T}存储在trans_matrix中。

逆运动学求解

设总的变换矩阵设为下数：

_{6}^{0}\textrm{T} = \begin{bmatrix} a_{11} &a_{12} &a_{13} & p_{x}\\ a_{21} &a_{22} &a_{23} & p_{y}\\ a_{31} &a_{32} &a_{33} & p_{z} \\ 0 & 0 & 0 &1 \end{bmatrix}

有等式

_{6}^{0}\textrm{T} = _{1}^{0}\textrm{T}_{2}^{1}\textrm{T}_{3}^{2}\textrm{T}_{4}^{3}\textrm{T}_{5}^{4}\textrm{T}_{6}^{5}\textrm{T}

根据上面的正运动学求解可以得到矩阵中的a_{11}等各元素的值分别为什么

使用matlab的实时计算脚本可以较具体地看到

 然后通过封闭解法求逆，将上述等式_{6}^{0}\textrm{T} = _{1}^{0}\textrm{T}_{2}^{1}\textrm{T}_{3}^{2}\textrm{T}_{4}^{3}\textrm{T}_{5}^{4}\textrm{T}_{6}^{5}\textrm{T}中右边地变换矩阵乘到左边，然后观察等式两边矩阵中地元素，使等式两边的矩阵中的某个元素相等，不断分离变量，然后求解得到\theta角

 首先求解\theta_{1}，构造等式

 

 观察等式，使等式两边矩阵的第三行第四列元素相等

0=\mathrm{PY}_Y\,\cos \left( \theta _1 \right) -\mathrm{PX}_X\,\sin \left( \theta _1 \right) -d_6\,r_{2,3}\,\cos \left( \theta _1 \right) +d_6\,r_{1,3}\,\sin \left( \theta _1 \right)

上述等式只含有一个未知数，一个等式一个未知数，求解过程如下：

注：为了便于计算，{\color{Red}a _{i-1} }均用{\color{Red}a _{i} }代替，{\color{Red}\alpha_{i-1} }也用{\color{Red}\alpha_{i} }代替，即下列式子中的{\color{Red}a _{1} }实际上为{\color{Red}a _{0} }，其他的{\color{Red}a }和{\color{Red}\alpha}亦然如此

 用同样的方式求解\theta_{2}

 令等式左右两边的第一行第四列，和第三行第四列分别相等，求解过程如下

 

然后是\theta_{3}

 令等式两边第一行第四列，第二行第四列相等

求解过程如下：

 再然后是\theta_{4}的求解

令左右两边第三行第三列相等

求解过程如下

 最后是\theta_{5}和\theta_{6}的求解

对\theta_{5} 构造，使等式左右第一行第三列，和第三行第三列相等

对\theta_{6}构造，使等式左右第三行第一列，和第三行第二列相等

 求解过程如下

 至此完成了所有\theta角的求解

 再次注名：上述推导过程中，为了便于计算，{\color{Red}a _{i-1} }均用{\color{Red}a _{i} }代替，{\color{Red}\alpha_{i-1} }也用{\color{Red}\alpha_{i} }代替，即上述式子中的{\color{Red}a _{1} }实际上为{\color{Red}a _{0} }，其他的{\color{Red}a }和{\color{Red}\alpha}亦然如此

正逆运动学模型的验证

正运动学验证

随机选定一组关节角\theta_{1} \sim \theta_{6}进行验证，

关节角 \theta_{1} \theta_{2} \theta_{3} \theta_{4} \theta_{5} \theta_{6}
度数 22.12 -81.4 21.25 -84 19.14 275.13

使用之前求解的正运动学模型求解得到末端姿态矩阵_{6}^{0}\textrm{T}如下

使用RobotStudio软件导入IRB6700机器人，输入选择的\theta_{1} \sim \theta_{6}角

注意：由连杆参数可知，RobotStudio软件中的\theta_{2}角度为原角度加上\frac{\pi }{2}，\theta_{6}为原角度减去\pi，所以RobotStudio中输入的角度应如下：

关节角 \theta_{1} \theta_{2} \theta_{3} \theta_{4} \theta_{5} \theta_{6}
度数 22.12 8.6 21.25 -84 19.14 95.13

 

得到的TCP末端xyz坐标       

 

对比RobotStudio软件中的末端姿态[x,y,z]值与正解求得的矩阵中末端姿态左边[px,py,pz]可得，两者相等，因此正运动学模型正确。

逆运动学验证

由上述的正运动学验证继续计算，上述正运动学得到的末端姿态矩阵如下：

根据此末端姿态矩阵对逆运动学进行求解，可得到八种不同的\theta角组合，使末端姿态为[1635.672,594.531,1396.902]

求解的matlab代码较长，共一个主m文件，以及5个function文件，放在本文最后。

求解得到共八种组合的\theta角结果，根据IRB6700机器人的各关节角工作范围减去4个不符合的解后，得到四个符合的\theta角组合，然后保持RobotStudio软件中的\theta角参数不变，调整RobotStudio软件中的轴配置参数，将逆运动学模型求解得到的结果与RobotStudio软件中的结果进行对比如下表

\theta_{i} 1 2 3 4 5 6
逆解1 -157.88 -41.51 -134.76 -30.62 -39.82 -144.08
逆解2 -157.88 -41.51 -134.76 149.38 39.82 35.92
逆解3 22.12 8.6 21.25 96 -19.14 -84.87
逆解4 22.12 8.6 21.25 -84 19.14 95.13

轴配置

（-2,-1,-2,-7）

 -157.88 -41.51 -134.76 -30.62 -39.82 -144.07

轴配置

（-2,1,0,6）

 -157.88 -41.51 -134.76 149.38 39.82 35.93

轴配置

（0,1,-1,1）

 22.12 8.6 21.25 96.01 -19.14 -84.87

轴配置

（0,-1,1,0）

 22.12 8.6 21.25 -84 19.14 95.13

由上表的结果对比可知，除了计算精度问题导致的，部分结果小数点后两位与RobotStudio存在一个单位的差异外，结果一致，因为可以验证逆运动学模型正确。

总的Matlab代码，包含正逆运动学求解和验证

clear;clc
%%
%导入参数
%注，这里的a,afa的下标在实际使用时均需要减去1
syms a afa d theta [1 6]
a = [0 a2 a3 a4 0 0];
afa = [0 -90 0 -90 90 -90];
d = [d1 0 0 d4 0 d6];
theta = [theta1 theta2 theta3 theta4 theta5 theta6];
syms T [4 4 6]
%%
%正运动学模型求解
%计算各变换矩阵及总的变换矩阵
for i = 1:6
   T(:,:,i) =  trans_cal(  afa(i), a(i), d(i), theta(i)*180/pi );
          if(i == 1)
              trans_matrix = T(:,:,i);
          else
              trans_matrix = trans_matrix*T(:,:,i);
          end
end


%%
%逆运动学计算求解
syms r [3 3]
syms PX_X PY_Y PZ_Z
T_60 = [r1_1 r1_2 r1_3 PX_X;
        r2_1 r2_2 r2_3 PY_Y;
        r3_1 r3_2 r3_3 PZ_Z;
        0    0    0    1];
    
left_1 = simplify( inv( T(:,:,2) ) * inv( T(:,:,1) ) * T_60 * inv( T(:,:,6) ) );
right_1 = simplify( T(:,:,3)*T(:,:,4)*T(:,:,5) );


left_2 =  simplify( inv( T(:,:,1) )* T_60 * inv( T(:,:,6) ) );
right_2 = simplify( T(:,:,2)*T(:,:,3)*T(:,:,4)*T(:,:,5) );

left_3 =  simplify( inv(   T(:,:,1)*T(:,:,2)   ) * T_60 * inv( T(:,:,6) ) );
right_3 = simplify( T(:,:,3)*T(:,:,4)*T(:,:,5) );

left_4 =  simplify(  inv(   T(:,:,1)*T(:,:,2)*T(:,:,3)*T(:,:,4)*T(:,:,5)   ) * T_60 );
right_4 = simplify(  T(:,:,6)  );

left_5 =  simplify(  inv(   T(:,:,1)*T(:,:,2)*T(:,:,3)*T(:,:,4) ) * T_60 );
right_5 = simplify(  T(:,:,5)*T(:,:,6)  );

left_6 =  simplify(  inv(   T(:,:,1)*T(:,:,2)*T(:,:,3)*T(:,:,4)*T(:,:,5)   ) * T_60 );
right_6 = simplify(  T(:,:,6)  );


%%
%逆运动学验算

%末端姿态
T_ni = [-0.5    0   0.866   1635.672;
        0       1   0       594.531;
        -0.866  0   -0.5    1396.902;
        0       0   0       1];
    
    
r1_1 = T_ni(1,1);r1_2 = T_ni(1,2);r1_3 = T_ni(1,3);PX_X   = T_ni(1,4);
r2_1 = T_ni(2,1);r2_2 = T_ni(2,2);r2_3 = T_ni(2,3);PY_Y   = T_ni(2,4);
r3_1 = T_ni(3,1);r3_2 = T_ni(3,2);r3_3 = T_ni(3,3);PZ_Z   = T_ni(3,4);

syms a d [1 6]
d1 = 780;d4 = 1142.5;d6 = 200;
a2 = 320;a3 = 1125  ;a4 = 200; 

syms A B C D E F [1 6]

%theta1的解
theta1_1 = ( atan2(0,1) - atan2( d6*r2_3-PY_Y , PX_X-d6*r1_3 ) )*180/pi %22.1230
theta1_2 = ( -atan2(0,-1) - atan2( d6*r2_3-PY_Y , PX_X-d6*r1_3 ) )*180/pi %-157.8770



%theta2的解
[theta2_1 , theta2_2] = theta2_calculate(theta1_1)  %theta1为22.123°时
[theta2_3 , theta2_4] = theta2_calculate(theta1_2)  %theta1为-157.8770°时


%theta3的解
theta3_11_21    =   theta3_calculate(theta1_1,theta2_1)  %theta1为22.123°theta2为-81.3955°时
theta3_11_22    =   theta3_calculate(theta1_1,theta2_2)  %theta1为22.123°theta2为22.0674°时
theta3_12_23    =   theta3_calculate(theta1_2,theta2_3)  %theta1为-157.8770°theta2为172.8834°时
theta3_12_24    =   theta3_calculate(theta1_2,theta2_4)  %theta1为-157.8770°theta2为228.4872°时



%theta4的解
[theta4_1_11_21,theta4_2_11_21] = theta4_calculate(theta1_1,theta2_1,theta3_11_21)  %theta1为22.123°theta2为-81.3955°时
[theta4_1_11_22,theta4_2_11_22] = theta4_calculate(theta1_1,theta2_2,theta3_11_22)  %theta1为22.123°theta2为22.0674°时
[theta4_1_12_23,theta4_2_12_23] = theta4_calculate(theta1_2,theta2_3,theta3_12_23)  %theta1为-157.8770°theta2为172.8834°时
[theta4_1_12_24,theta4_2_12_24] = theta4_calculate(theta1_2,theta2_4,theta3_12_24)  %theta1为-157.8770°theta2为228.4872°时


%theta5的解
theta5_11_21_41 = theta5_calculate(theta1_1,theta2_1,theta3_11_21,theta4_1_11_21) %theta1为22.123°theta2为-81.3955°,theta4为第1种
theta5_11_21_42 = theta5_calculate(theta1_1,theta2_1,theta3_11_21,theta4_2_11_21) %theta1为22.123°theta2为-81.3955°,theta4为第2种
theta5_11_22_41 = theta5_calculate(theta1_1,theta2_2,theta3_11_22,theta4_1_11_22) %theta1为22.123°theta2为22.0674°,theta4为第1种
theta5_11_22_42 = theta5_calculate(theta1_1,theta2_2,theta3_11_22,theta4_2_11_22) %theta1为22.123°theta2为22.0674°,theta4为第2种

theta5_12_23_41 = theta5_calculate(theta1_2,theta2_3,theta3_12_23,theta4_1_12_23) %theta1为-157.8770°theta2为172.8834°,theta4为第1种
theta5_12_23_42 = theta5_calculate(theta1_2,theta2_3,theta3_12_23,theta4_2_12_23) %theta1为-157.8770°theta2为172.8834°,theta4为第2种
theta5_12_24_41 = theta5_calculate(theta1_2,theta2_4,theta3_12_24,theta4_1_12_24) %theta1为-157.8770°theta2为228.4872°,theta4为第1种
theta5_12_24_42 = theta5_calculate(theta1_2,theta2_4,theta3_12_24,theta4_2_12_24) %theta1为-157.8770°theta2为228.4872°,theta4为第2种


% theta6的解
theta6_11_21_41 = theta6_calculate(theta1_1,theta2_1,theta3_11_21,theta4_1_11_21) %theta1为22.123°theta2为-81.3955°,theta4为第1种
theta6_11_21_42 = theta6_calculate(theta1_1,theta2_1,theta3_11_21,theta4_2_11_21) %theta1为22.123°theta2为-81.3955°,theta4为第2种
theta6_11_22_41 = theta6_calculate(theta1_1,theta2_2,theta3_11_22,theta4_1_11_22) %theta1为22.123°theta2为22.0674°,theta4为第1种
theta6_11_22_42 = theta6_calculate(theta1_1,theta2_2,theta3_11_22,theta4_2_11_22) %theta1为22.123°theta2为22.0674°,theta4为第2种
theta6_12_23_41 = theta6_calculate(theta1_2,theta2_3,theta3_12_23,theta4_1_12_23) %theta1为-157.8770°theta2为172.8834°,theta4为第1种
theta6_12_23_42 = theta6_calculate(theta1_2,theta2_3,theta3_12_23,theta4_2_12_23) %theta1为-157.8770°theta2为172.8834°,theta4为第2种
theta6_12_24_41 = theta6_calculate(theta1_2,theta2_4,theta3_12_24,theta4_1_12_24) %theta1为-157.8770°theta2为228.4872°,theta4为第1种
theta6_12_24_42 = theta6_calculate(theta1_2,theta2_4,theta3_12_24,theta4_2_12_24) %theta1为-157.8770°theta2为228.4872°,theta4为第2种


function [theta2_1,theta2_2] = theta2_calculate(theta1)
%%
%导入参数
    syms r [3,3]
    syms a afa d
    syms PX_X PY_Y_Y PZ_Z
    T_ni = [-0.5    0   0.866   1635.672;
        0       1   0       594.531;
        -0.866  0   -0.5    1396.902;
        0       0   0       1];    
    r1_1 = T_ni(1,1);r1_2 = T_ni(1,2);r1_3 = T_ni(1,3);PX_X   = T_ni(1,4);
    r2_1 = T_ni(2,1);r2_2 = T_ni(2,2);r2_3 = T_ni(2,3);PY_Y   = T_ni(2,4);
    r3_1 = T_ni(3,1);r3_2 = T_ni(3,2);r3_3 = T_ni(3,3);PZ_Z   = T_ni(3,4);
    syms a d [1 6]
    d1 = 780;d4 = 1142.5;d6 = 200;
    a2 = 320;a3 = 1125  ;a4 = 200; 
    syms A B C D E F [1 6]
    theta1 = theta1/180*pi;

%%
%theta2计算
    A2 = ( PX_X - d6*r1_3 )*cos(theta1) + ( PY_Y - d6*r2_3 )*sin(theta1) - a2;
    B2 = d1 + d6*r3_3 - PZ_Z;
    C2 = 2*A2*a3;
    D2 = 2*B2*a3;
    E2 = A2^2 + B2^2 + a3^2 - a4^2 - d4^2;
    F2 = sqrt( C2^2 + D2^2 );
    
    theta2_1 = ( -atan2(C2,D2)  + atan2(  E2/F2 ,  sqrt( 1- (E2/F2)^2 ) ) )*180/pi  ;
    theta2_2 = ( -atan2(C2,D2)  + atan2(  E2/F2 , -sqrt( 1- (E2/F2)^2 ) ) )*180/pi  ;

end
function theta3 = theta3_calculate(theta1,theta2)
%%
%导入参数
    syms r [3,3]
    syms a afa d
    syms PX_X PY_Y_Y PZ_Z
    T_ni = [-0.5    0   0.866   1635.672;
        0       1   0       594.531;
        -0.866  0   -0.5    1396.902;
        0       0   0       1];    
    r1_1 = T_ni(1,1);r1_2 = T_ni(1,2);r1_3 = T_ni(1,3);PX_X   = T_ni(1,4);
    r2_1 = T_ni(2,1);r2_2 = T_ni(2,2);r2_3 = T_ni(2,3);PY_Y   = T_ni(2,4);
    r3_1 = T_ni(3,1);r3_2 = T_ni(3,2);r3_3 = T_ni(3,3);PZ_Z   = T_ni(3,4);
    syms a d [1 6]
    d1 = 780;d4 = 1142.5;d6 = 200;
    a2 = 320;a3 = 1125  ;a4 = 200; 
    syms A B C D E F [1 6]
    theta1 = theta1/180*pi;
    theta2 = theta2/180*pi;

%%
%theta3计算
    A3 = d1*sin(theta2) - a2*cos(theta2) - PZ_Z*sin(theta2) + d6*r3_3*sin(theta2) + PX_X*cos(theta1)*cos(theta2) ...
    + PY_Y*cos(theta2)*sin(theta1) - d6*r1_3*cos(theta1)*cos(theta2) - d6*r2_3*cos(theta2)*sin(theta1) - a3;

    B3 = d1*cos(theta2) - PZ_Z*cos(theta2) + a2*sin(theta2) + d6*r3_3*cos(theta2) - PX_X*cos(theta1)*sin(theta2) ...
    - PY_Y*sin(theta1)*sin(theta2) + d6*r2_3*sin(theta1)*sin(theta2) + d6*r1_3*cos(theta1)*sin(theta2);

    C3 = ( a4*B3-d4*A3 ) / ( a4^2 + d4^2 );
    
    D3 = ( a4*A3+d4*B3 ) / ( a4^2 + d4^2 );
    
    theta3 = atan2( C3,D3 )*180/pi;


end
function [theta4_1,theta4_2] = theta4_calculate(theta1,theta2,theta3)
%%
%导入参数
    syms r [3,3]
    syms a afa d
    syms PX_X PY_Y_Y PZ_Z
    T_ni = [-0.5    0   0.866   1635.672;
        0       1   0       594.531;
        -0.866  0   -0.5    1396.902;
        0       0   0       1];    
    r1_1 = T_ni(1,1);r1_2 = T_ni(1,2);r1_3 = T_ni(1,3);PX_X   = T_ni(1,4);
    r2_1 = T_ni(2,1);r2_2 = T_ni(2,2);r2_3 = T_ni(2,3);PY_Y   = T_ni(2,4);
    r3_1 = T_ni(3,1);r3_2 = T_ni(3,2);r3_3 = T_ni(3,3);PZ_Z   = T_ni(3,4);
    syms a d [1 6]
    d1 = 780;d4 = 1142.5;d6 = 200;
    a2 = 320;a3 = 1125  ;a4 = 200; 
    syms A B C D E F [1 6]
    theta1 = theta1/180*pi;
    theta2 = theta2/180*pi;
    theta3 = theta3/180*pi;

%%
%theta4计算

A4 = r3_3*cos(theta2)*sin(theta3) + r3_3*cos(theta3)*sin(theta2) - r1_3*cos(theta1)*cos(theta2)*cos(theta3) ...
    - r2_3*cos(theta2)*cos(theta3)*sin(theta1) + r1_3*cos(theta1)*sin(theta2)*sin(theta3) ...
    + r2_3*sin(theta1)* sin(theta2)*sin(theta3);  
B4 = r2_3*cos(theta1) - r1_3*sin(theta1);
theta4_1 = ( atan2( 0,1 ) + atan2( B4,A4 ) )*180/pi;
theta4_2 = ( atan2( 0,-1 ) + atan2( B4,A4 ) )*180/pi;

end
function theta5 = theta5_calculate(theta1,theta2,theta3,theta4)
%%
%导入参数
    syms r [3,3]
    syms a afa d
    syms PX_X PY_Y_Y PZ_Z
    T_ni = [-0.5    0   0.866   1635.672;
        0       1   0       594.531;
        -0.866  0   -0.5    1396.902;
        0       0   0       1];    
    r1_1 = T_ni(1,1);r1_2 = T_ni(1,2);r1_3 = T_ni(1,3);PX_X   = T_ni(1,4);
    r2_1 = T_ni(2,1);r2_2 = T_ni(2,2);r2_3 = T_ni(2,3);PY_Y   = T_ni(2,4);
    r3_1 = T_ni(3,1);r3_2 = T_ni(3,2);r3_3 = T_ni(3,3);PZ_Z   = T_ni(3,4);
    syms a d [1 6]
    d1 = 780;d4 = 1142.5;d6 = 200;
    a2 = 320;a3 = 1125  ;a4 = 200; 
    syms A B C D E F [1 6]
    theta1 = theta1/180*pi;
    theta2 = theta2/180*pi;
    theta3 = theta3/180*pi;
    theta4 = theta4/180*pi;

%%
%theta4计算

A5 = r1_3*sin(theta1)*sin(theta4) - r2_3*cos(theta1)*sin(theta4) - r3_3*cos(theta2)*cos(theta4)*sin(theta3) ...
    - r3_3*cos(theta3)*cos(theta4)*sin(theta2) + r1_3*cos(theta1)*cos(theta2)*cos(theta3)*cos(theta4) + ...
    r2_3*cos(theta2)*cos(theta3)*cos(theta4)*sin(theta1) - r1_3*cos(theta1)*cos(theta4)*sin(theta2)*sin(theta3) ...
    - r2_3*cos(theta4)*sin(theta1)*sin(theta2)*sin(theta3);
B5 = r3_3*sin(theta2)*sin(theta3) - r3_3*cos(theta2)*cos(theta3) - r1_3*cos(theta1)*cos(theta2)*sin(theta3) ... 
    - r1_3*cos(theta1)*cos(theta3)*sin(theta2) - r2_3*cos(theta2)*sin(theta1)*sin(theta3) - ... 
    r2_3*cos(theta3)*sin(theta1)*sin(theta2);
theta5 = atan2(-A5,B5)*180/pi;

end
function theta6 = theta6_calculate(theta1,theta2,theta3,theta4)
%%
%导入参数
    syms r [3,3]
    syms a afa d
    syms PX_X PY_Y_Y PZ_Z
    T_ni = [-0.5    0   0.866   1635.672;
        0       1   0       594.531;
        -0.866  0   -0.5    1396.902;
        0       0   0       1];    
    r1_1 = T_ni(1,1);r1_2 = T_ni(1,2);r1_3 = T_ni(1,3);PX_X   = T_ni(1,4);
    r2_1 = T_ni(2,1);r2_2 = T_ni(2,2);r2_3 = T_ni(2,3);PY_Y   = T_ni(2,4);
    r3_1 = T_ni(3,1);r3_2 = T_ni(3,2);r3_3 = T_ni(3,3);PZ_Z   = T_ni(3,4);
    syms a d [1 6]
    d1 = 780;d4 = 1142.5;d6 = 200;
    a2 = 320;a3 = 1125  ;a4 = 200; 
    syms A B C D E F [1 6]
    theta1 = theta1/180*pi;
    theta2 = theta2/180*pi;
    theta3 = theta3/180*pi;
    theta4 = theta4/180*pi;

%%
%theta4计算

A6 = r2_1*cos(theta1)*cos(theta4) - r1_1*cos(theta4)*sin(theta1) - r3_1*cos(theta2)*sin(theta3)*sin(theta4) ...
    - r3_1*cos(theta3)*sin(theta2)*sin(theta4) + r1_1*cos(theta1)*cos(theta2)*cos(theta3)*sin(theta4) + ...
    r2_1*cos(theta2)*cos(theta3)*sin(theta1)*sin(theta4) - r1_1*cos(theta1)*sin(theta2)*sin(theta3)*sin(theta4) ...
    - r2_1*sin(theta1)*sin(theta2)*sin(theta3)*sin(theta4);
B6 = r2_2*cos(theta1)*cos(theta4) - r1_2*cos(theta4)*sin(theta1) - r3_2*cos(theta2)*sin(theta3)*sin(theta4) ... 
    - r3_2*cos(theta3)*sin(theta2)*sin(theta4) + r1_2*cos(theta1)*cos(theta2)*cos(theta3)*sin(theta4)  ...
    + r2_2*cos(theta2)*cos(theta3)*sin(theta1)*sin(theta4) - r1_2*cos(theta1)*sin(theta2)*sin(theta3)*sin(theta4) ...
    - r2_2*sin(theta1)*sin(theta2)*sin(theta3)*sin(theta4);

theta6 = atan2(-A6,-B6)*180/pi;

end
function T = trans_cal(afa_ii,a_ii,d_i,theta_i)
%%
%计算变换矩阵函数T_{i-1,i}
%输入的参数为，afa_{i-1},a_{i-1},d_i,theta_i,与DH表达法的参数表对应
%ii 为i-1
%注意，这里输入的角度，均采用角度制，不采用弧度制

T = [cosd(theta_i)              -sind(theta_i)             0               a_ii
     sind(theta_i)*cosd(afa_ii)  cosd(theta_i)*cosd(afa_ii)  -sind(afa_ii)    -sind(afa_ii)*d_i
    sind(theta_i)*sind(afa_ii)   cosd(theta_i)*sind(afa_ii)  cosd(afa_ii)     cosd(afa_ii)*d_i
     0                         0                         0                1               ];
 
end

参考文献

[1] 尹绪伟. 打磨机器人不同位姿下的刚度特性研究[D]. 武汉: 武汉理工大学, 2019.

[2] 王宇迪. 基于加工姿态最优的发动机飞轮壳机器人铣削修边空间路径规划[D]. 武汉: 武汉理工大学, 2022.

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原文链接：https://blog.csdn.net/piginthesouth/article/details/127521366

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