用雅可比伪逆求解六自由度机械臂逆运动学，matlab代码实现

以下是一个使用雅可比伪逆求解六自由度机械臂逆运动学的matlab代码实现：

% 机械臂参数
l1 = 1;  % 第一段臂长
l2 = 1;  % 第二段臂长
l3 = 1;  % 第三段臂长
l4 = 1;  % 第四段臂长
l5 = 1;  % 第五段臂长
l6 = 1;  % 第六段臂长

% 末端执行器位置和姿态
x = 1;  % x坐标
y = 1;  % y坐标
z = 1;  % z坐标
alpha = 0;  % 绕x轴旋转角度
beta = 0;  % 绕y轴旋转角度
gamma = 0;  % 绕z轴旋转角度

% 初始关节角度
theta1 = 0;
theta2 = 0;
theta3 = 0;
theta4 = 0;
theta5 = 0;
theta6 = 0;

% 迭代次数和误差阈值
max_iter = 1000;
epsilon = 0.001;

% 迭代求解逆运动学
for i = 1:max_iter
    % 正向运动学求解末端执行器位置和姿态
    T = forward_kinematics(theta1, theta2, theta3, theta4, theta5, theta6);
    xe = T(1:3, 4);
    Re = T(1:3, 1:3);
    % 计算末端执行器位置误差和姿态误差
    e = [x; y; z] - xe;
    er = 0.5 * (Re' * rotx(alpha)' - rotx(alpha) * Re');
    er = [er(3, 2); er(1, 3); er(2, 1)];
    % 计算雅可比矩阵
    J = jacobian(theta1, theta2, theta3, theta4, theta5, theta6);
    % 计算雅可比伪逆矩阵
    J_pinv = pinv(J);
    % 计算关节角度增量
    delta_theta = J_pinv * [e; er];
    % 更新关节角度
    theta1 = theta1 + delta_theta(1);
    theta2 = theta2 + delta_theta(2);
    theta3 = theta3 + delta_theta(3);
    theta4 = theta4 + delta_theta(4);
    theta5 = theta5 + delta_theta(5);
    theta6 = theta6 + delta_theta(6);
    % 判断是否达到误差阈值
    if norm(e) < epsilon && norm(er) < epsilon
        break;
    end
end

% 输出关节角度
fprintf('theta1: %.2f\n', theta1);
fprintf('theta2: %.2f\n', theta2);
fprintf('theta3: %.2f\n', theta3);
fprintf('theta4: %.2f\n', theta4);
fprintf('theta5: %.2f\n', theta5);
fprintf('theta6: %.2f\n', theta6);

% 正向运动学函数
function T = forward_kinematics(theta1, theta2, theta3, theta4, theta5, theta6)
    l1 = 1;
    l2 = 1;
    l3 = 1;
    l4 = 1;
    l5 = 1;
    l6 = 1;
    T01 = get_ht(l1, 0, 0, theta1);
    T12 = get_ht(l2, 0, 0, theta2);
    T23 = get_ht(l3, 0, 0, theta3);
    T34 = get_ht(l4, 0, 0, theta4);
    T45 = get_ht(l5, 0, 0, theta5);
    T56 = get_ht(l6, 0, 0, theta6);
    T06 = T01 * T12 * T23 * T34 * T45 * T56;
    T = T06;
end

% 雅可比矩阵函数
function J = jacobian(theta1, theta2, theta3, theta4, theta5, theta6)
    l1 = 1;
    l2 = 1;
    l3 = 1;
    l4 = 1;
    l5 = 1;
    l6 = 1;
    T01 = get_ht(l1, 0, 0, theta1);
    T12 = get_ht(l2, 0, 0, theta2);
    T23 = get_ht(l3, 0, 0, theta3);
    T34 = get_ht(l4, 0, 0, theta4);
    T45 = get_ht(l5, 0, 0, theta5);
    T56 = get_ht(l6, 0, 0, theta6);
    T02 = T01 * T12;
    T03 = T02 * T23;
    T04 = T03 * T34;
    T05 = T04 * T45;
    T06 = T05 * T56;
    z0 = [0; 0; 1];
    z1 = T01(1:3, 3);
    z2 = T02(1:3, 3);
    z3 = T03(1:3, 3);
    z4 = T04(1:3, 3);
    z5 = T05(1:3, 3);
    Jv1 = z0;
    Jv2 = cross(z0, T06(1:3, 4) - T01(1:3, 4));
    Jv3 = cross(z1, T06(1:3, 4) - T02(1:3, 4));
    Jv4 = cross(z2, T06(1:3, 4) - T03(1:3, 4));
    Jv5 = cross(z3, T06(1:3, 4) - T04(1:3, 4));
    Jv6 = cross(z4, T06(1:3, 4) - T05(1:3, 4));
    Jv = [Jv1, Jv2, Jv3, Jv4, Jv5, Jv6];
    Jw1 = [0; 0; 0];
    Jw2 = z0;
    Jw3 = z1;
    Jw4 = z2;
    Jw5 = z3;
    Jw6 = z4;
    Jw = [Jw1, Jw2, Jw3, Jw4, Jw5, Jw6];
    J = [Jv; Jw];
end

% 求解齐次变换矩阵的函数
function T = get_ht(d, theta, a, alpha)
    ct = cos(theta);
    st = sin(theta);
    ca = cos(alpha);
    sa = sin(alpha);
    T = [ct, -st*ca, st*sa, a*ct;
         st, ct*ca, -ct*sa, a*st;
         0, sa, ca, d;
         0, 0, 0, 1];
end

% 绕x轴旋转矩阵的函数
function R = rotx(theta)
    ct = cos(theta);
    st = sin(theta);
    R = [1, 0, 0;
         0, ct, -st;
         0, st, ct];
end

在上述代码中，`forward_kinematics()`函数用于求解正向运动学，`jacobian()`函数用于求解雅可比矩阵，`get_ht()`函数用于求解齐次变换矩阵，`rotx()`函数用于求解绕x轴旋转矩阵。在主函数中，使用迭代的方式求解逆运动学，并输出关节角度。