%% 机械臂PTP轨迹控制
clear; clc; close all;
% ===========================================
% 机械臂参数定义 (与之前一致)
% ===========================================
a = [0, 0, 500, 500, 0, 0]; % 连杆长度(mm)
alpha = [0, -pi/2, 0, -pi/2, pi/2, -pi/2]; % 连杆扭角(rad)
d = [400, 0, 0, 0, 0, 100]; % 连杆偏移(mm)
joint_limits = [
-pi, pi; % 关节1
-pi/2, pi/2; % 关节2
-pi/2, pi/2; % 关节3
-pi, pi; % 关节4
0, pi; % 关节5
-pi, pi % 关节6
];
% ===========================================
% 轨迹参数设置
% ===========================================
% 工作空间中随机选择两点
P_start = [randi([-800, 800]); randi([-800, 800]); randi([0, 1000])];
P_end = [randi([-800, 800]); randi([-800, 800]); randi([0, 1000])];
% 目标方向矩阵 (保持垂直向下)
R = [1, 0, 0; 0, 1, 0; 0, 0, 1];
% 创建目标位姿矩阵
T_start = [R, P_start; 0 0 0 1];
T_end = [R, P_end; 0 0 0 1];
% 运动参数
total_time = 5; % 总运动时间(秒)
time_step = 0.05; % 时间步长(秒)
time_vector = 0:time_step:total_time;
num_points = length(time_vector);
% ===========================================
% 逆运动学求解起点和终点关节角度
% ===========================================
q_start = inverse_kinematics(T_start, a, alpha, d, joint_limits);
q_end = inverse_kinematics(T_end, a, alpha, d, joint_limits);
% 选择最优解
if ~isempty(q_start) && ~isempty(q_end)
q_start = select_solution(q_start, [0,0,0,0,0,0], joint_limits);
q_end = select_solution(q_end, q_start, joint_limits);
fprintf('起点位置: [%.1f, %.1f, %.1f] mm\n', P_start(1), P_start(2), P_start(3));
fprintf('终点位置: [%.1f, %.1f, %.1f] mm\n', P_end(1), P_end(2), P_end(3));
fprintf('起点关节角度 (度): %s\n', num2str(rad2deg(q_start)));
fprintf('终点关节角度 (度): %s\n', num2str(rad2deg(q_end)));
else
error('无法找到有效的逆运动学解');
end
% ===========================================
% PTP轨迹规划 (关节空间五次多项式插值)
% ===========================================
% 初始化轨迹矩阵: 6关节 x 时间点
q_trajectory = zeros(6, num_points);
dq_trajectory = zeros(6, num_points); % 关节速度
ddq_trajectory = zeros(6, num_points); % 关节加速度
% 计算五次多项式系数 (保证起点和终点的位置、速度、加速度连续)
for j = 1:6
q0 = q_start(j);
qf = q_end(j);
% 边界条件: 起点和终点速度为0，加速度为0
v0 = 0; vf = 0;
a0 = 0; af = 0;
% 求解五次多项式系数: q(t) = a0 + a1*t + a2*t^2 + a3*t^3 + a4*t^4 + a5*t^5
A = [1, 0, 0, 0, 0, 0;
0, 1, 0, 0, 0, 0;
0, 0, 2, 0, 0, 0;
1, total_time, total_time^2, total_time^3, total_time^4, total_time^5;
0, 1, 2*total_time, 3*total_time^2, 4*total_time^3, 5*total_time^4;
0, 0, 2, 6*total_time, 12*total_time^2, 20*total_time^3];
b = [q0; v0; a0; qf; vf; af];
coeffs = A \ b;
% 计算轨迹点
for i = 1:num_points
t = time_vector(i);
q_trajectory(j, i) = coeffs(1) + coeffs(2)*t + coeffs(3)*t^2 + ...
coeffs(4)*t^3 + coeffs(5)*t^4 + coeffs(6)*t^5;
dq_trajectory(j, i) = coeffs(2) + 2*coeffs(3)*t + 3*coeffs(4)*t^2 + ...
4*coeffs(5)*t^3 + 5*coeffs(6)*t^4;
ddq_trajectory(j, i) = 2*coeffs(3) + 6*coeffs(4)*t + ...
12*coeffs(5)*t^2 + 20*coeffs(6)*t^3;
end
end
% ===========================================
% 轨迹可视化
% ===========================================
% 1. 关节角度轨迹
figure('Name', '关节角度轨迹', 'Position', [100, 100, 1200, 800]);
titles = {'关节1', '关节2', '关节3', '关节4', '关节5', '关节6'};
for j = 1:6
subplot(2, 3, j);
plot(time_vector, rad2deg(q_trajectory(j, :)), 'LineWidth', 2);
hold on;
% 绘制关节限位
ylim_min = min(joint_limits(j, 1), joint_limits(j, 2)) * 180/pi;
ylim_max = max(joint_limits(j, 1), joint_limits(j, 2)) * 180/pi;
ylim_range = ylim_max - ylim_min;
plot([0, total_time], [rad2deg(joint_limits(j, 1)), rad2deg(joint_limits(j, 1))], 'r--');
plot([0, total_time], [rad2deg(joint_limits(j, 2)), rad2deg(joint_limits(j, 2))], 'r--');
title(titles{j});
xlabel('时间 (秒)');
ylabel('角度 (度)');
grid on;
legend('关节角度', '关节限位', 'Location', 'best');
% 设置合理的Y轴范围
y_min = min(rad2deg(q_trajectory(j, :))) - 0.1*ylim_range;
y_max = max(rad2deg(q_trajectory(j, :))) + 0.1*ylim_range;
ylim([y_min, y_max]);
end
% 2. 关节速度和加速度
figure('Name', '关节速度和加速度', 'Position', [100, 100, 1200, 800]);
for j = 1:6
subplot(2, 3, j);
yyaxis left;
plot(time_vector, rad2deg(dq_trajectory(j, :)), 'b-', 'LineWidth', 2);
ylabel('速度 (度/秒)');
yyaxis right;
plot(time_vector, rad2deg(ddq_trajectory(j, :)), 'r-', 'LineWidth', 2);
ylabel('加速度 (度/秒²)');
title([titles{j} ' 运动状态']);
xlabel('时间 (秒)');
grid on;
legend('速度', '加速度', 'Location', 'best');
end
% 3. 末端轨迹动画
figure('Name', '机械臂运动轨迹动画', 'Position', [100, 100, 800, 600]);
hold on;
grid on;
axis equal;
view(3);
xlabel('X (mm)'); ylabel('Y (mm)'); zlabel('Z (mm)');
title('机械臂PTP运动轨迹');
% 绘制起点和终点
plot3(P_start(1), P_start(2), P_start(3), 'go', 'MarkerSize', 10, 'MarkerFaceColor', 'g');
plot3(P_end(1), P_end(2), P_end(3), 'ro', 'MarkerSize', 10, 'MarkerFaceColor', 'r');
% 绘制轨迹
traj_points = zeros(3, num_points);
for i = 1:num_points
T = forward_kinematics(q_trajectory(:, i), a, alpha, d);
traj_points(:, i) = T(1:3,4);
end
plot3(traj_points(1,:), traj_points(2,:), traj_points(3,:), 'b-', 'LineWidth', 1.5);
% 动画
h_robot = plot_robot(q_trajectory(:, 1), a, alpha, d, 'k', 2);
legend('起点', '终点', '轨迹', '机械臂', 'Location', 'best');
for i = 1:10:num_points
delete(h_robot);
h_robot = plot_robot(q_trajectory(:, i), a, alpha, d, 'k', 2);
drawnow;
pause(0.05);
end
% ===========================================
% 核心函数定义
% ===========================================
%% 正运动学函数 (Modified D-H方法)
function T = forward_kinematics(q, a, alpha, d)
T = eye(4);
for i = 1:6
ct = cos(q(i));
st = sin(q(i));
ca = cos(alpha(i));
sa = sin(alpha(i));
Ti = [ct, -st*ca, st*sa, a(i)*ct;
st, ct*ca, -ct*sa, a(i)*st;
0, sa, ca, d(i);
0, 0, 0, 1 ];
T = T * Ti;
end
end
%% 逆运动学函数 (数值解法)
function solutions = inverse_kinematics(T_target, a, alpha, d, joint_limits)
lb = joint_limits(:, 1);
ub = joint_limits(:, 2);
options = optimoptions('fmincon', 'Display', 'off', ...
'Algorithm', 'sqp', ...
'MaxIterations', 1000, ...
'MaxFunctionEvaluations', 5000);
solutions = [];
num_attempts = 10;
for attempt = 1:num_attempts
q0 = lb + (ub - lb) .* rand(6,1);
[q_opt, ~, exitflag] = fmincon(@(q)ik_objective(q, T_target, a, alpha, d), ...
q0, [], [], [], [], lb, ub, [], options);
if exitflag > 0
T_achieved = forward_kinematics(q_opt, a, alpha, d);
pos_error = norm(T_achieved(1:3,4) - T_target(1:3,4));
if pos_error < 10 % 位置误差小于10mm
solutions = [solutions; q_opt'];
end
end
end
if isempty(solutions)
fprintf('警告: 未找到有效的逆运动学解\n');
end
end
%% 逆运动学目标函数
function cost = ik_objective(q, T_target, a, alpha, d)
T_achieved = forward_kinematics(q, a, alpha, d);
% 位置误差
pos_error = norm(T_achieved(1:3,4) - T_target(1:3,4));
% 方向误差 (简化处理)
rot_error = norm(T_achieved(1:3,1:3) - T_target(1:3,1:3), 'fro');
% 总成本
cost = pos_error + 0.1 * rot_error;
end
%% 最优解选择函数
function best_solution = select_solution(solutions, reference, joint_limits)
if size(solutions, 1) == 1
best_solution = solutions(1, :);
return;
end
min_cost = Inf;
best_idx = 1;
for i = 1:size(solutions, 1)
% 1. 关节角度变化最小化
joint_change = norm(solutions(i, :) - reference);
% 2. 关节限位惩罚
joint_penalty = 0;
for j = 1:6
margin = min(solutions(i,j) - joint_limits(j,1), ...
joint_limits(j,2) - solutions(i,j));
if margin < deg2rad(5)
joint_penalty = joint_penalty + 10 * (deg2rad(5) - margin);
end
end
% 3. 总成本
total_cost = joint_change + joint_penalty;
if total_cost < min_cost
min_cost = total_cost;
best_idx = i;
end
end
best_solution = solutions(best_idx, :);
end
%% 机械臂可视化函数
function h = plot_robot(q, a, alpha, d, color, linewidth)
if nargin < 5, color = 'k'; end
if nargin < 6, linewidth = 2; end
% 计算各关节位置
T = cell(1, 7);
T{1} = eye(4);
for i = 1:6
T{i+1} = T{i} * dh_transform(a(i), alpha(i), d(i), q(i));
end
% 提取位置
positions = zeros(3, 7);
for i = 1:7
positions(:, i) = T{i}(1:3, 4);
end
% 绘制连杆
h = plot3(positions(1,:), positions(2,:), positions(3,:), ...
'o-', 'Color', color, 'LineWidth', linewidth, ...
'MarkerFaceColor', color, 'MarkerSize', 6);
end
%% DH变换矩阵
function T = dh_transform(a, alpha, d, theta)
T = [cos(theta), -sin(theta)*cos(alpha), sin(theta)*sin(alpha), a*cos(theta);
sin(theta), cos(theta)*cos(alpha), -cos(theta)*sin(alpha), a*sin(theta);
0, sin(alpha), cos(alpha), d;
0, 0, 0, 1];
end
（阅读代码，不修改）
本文介绍了一个名为QF05的随机数值生成器函数组，它包括了生成随机数、整数随机数及保存初始值等功能模块。与F052函数组类似，QF05为开发者提供了灵活的随机数生成解决方案。
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