代码收藏家技术教程 2024-08-22

“华数杯”数学竞赛A题深度解析与代码分享：2024年赛题全攻略

机器臂关节角路径的优化设计

问题一

代码

Matlab

clc
clear
% 参数定义
a = [0, 300, 1200, 300, 0, 0];
alpha = [0, -90, 0, -90, -90, -90];
d = [600, 0, 0, 1200, 0, 0];
theta_min = [-160, -150, -200, -180, -120, -180];
theta_max = [160, 15, 80, 180, 120, 180];
P_target = [1500, 1200, 200]; % 目标位置
lambda = 1; % 权重因子

% 初始种群生成
population_size = 100;
num_generations = 100;
crossover_rate = 0.8;
mutation_rate = 0.1;

% 随机生成初始种群
population = zeros(population_size, 6);
for i = 1:population_size
    for j = 1:6
        population(i, j) = theta_min(j) + (theta_max(j) - theta_min(j)) * rand;
    end
end

% 遗传算法主程序
for generation = 1:num_generations
    % 适应度计算
    fitness_values = zeros(population_size, 1);
    for i = 1:population_size
        fitness_values(i) = fitness(population(i, :), P_target);
    end
    
    % 将适应度值转换为非负值
    min_fitness = min(fitness_values);
    fitness_values = fitness_values - min_fitness + 1; % 确保所有适应度值为正
    
    % 选择（轮盘赌算法）
    probabilities = fitness_values / sum(fitness_values);
    selected_indices = randsample(1:population_size, population_size, true, probabilities);
    selected_population = population(selected_indices, :);
    
    % 交叉
    new_population = zeros(population_size, 6);
    for i = 1:2:population_size
        if rand < crossover_rate
            crossover_point = randi([1, 5]);
            new_population(i, :) = [selected_population(i, 1:crossover_point), selected_population(i + 1, crossover_point + 1:end)];
            new_population(i + 1, :) = [selected_population(i + 1, 1:crossover_point), selected_population(i, crossover_point + 1:end)];
        else
            new_population(i, :) = selected_population(i, :);
            new_population(i + 1, :) = selected_population(i + 1, :);
        end
    end
    
    % 变异
    for i = 1:population_size
        if rand < mutation_rate
            mutation_point = randi([1, 6]);
            new_population(i, mutation_point) = theta_min(mutation_point) + (theta_max(mutation_point) - theta_min(mutation_point)) * rand;
        end
    end
    
    population = new_population;
end

% 输出最优解
fitness_values = zeros(population_size, 1);
for i = 1:population_size
    fitness_values(i) = fitness(population(i, :), P_target);
end

[~, best_index] = min(fitness_values);
best_solution = population(best_index, :)
best_fitness = fitness_values(best_index)
% 正运动学计算函数
function T = forward_kinematics(theta)
    a = [0, 300, 1200, 300, 0, 0];
    alpha = [0, -90, 0, -90, -90, -90];
    d = [600, 0, 0, 1200, 0, 0];
    T = eye(4);
    for i = 1:6
        A = [cosd(theta(i)), -sind(theta(i)) * cosd(alpha(i)), sind(theta(i)) * sind(alpha(i)), a(i) * cosd(theta(i));
             sind(theta(i)), cosd(theta(i)) * cosd(alpha(i)), -cosd(theta(i)) * sind(alpha(i)), a(i) * sind(theta(i));
             0, sind(alpha(i)), cosd(alpha(i)), d(i);
             0, 0, 0, 1];
        T = T * A;
    end
end

% 适应度计算函数
function E_error = fitness(theta, P_target)
    T = forward_kinematics(theta);
    P_actual = T(1:3, 4)';
    E_error = norm(P_target - P_actual);
end

Python

import numpy as np
from scipy.optimize import minimize

# D-H参数
a = [0, 300, 1200, 300, 0, 0]
alpha = [0, -np.pi/2, 0, -np.pi/2, -np.pi/2, -np.pi/2]
d = [600, 0, 0, 1200, 0, 0]
theta_ranges = [(-160, 160), (-150, 15), (-200, 80), (-180, 180), (-120, 120), (-180, 180)]

# 目标位置
target = np.array([1500, 1200, 200])

# 计算D-H矩阵
def dh_matrix(a, alpha, d, theta):
    return np.array([
        [np.cos(theta), -np.sin(theta)*np.cos(alpha), np.sin(theta)*np.sin(alpha), a*np.cos(theta)],
        [np.sin(theta), np.cos(theta)*np.cos(alpha), -np.cos(theta)*np.sin(alpha), a*np.sin(theta)],
        [0, np.sin(alpha), np.cos(alpha), d],
        [0, 0, 0, 1]
    ])

# 计算末端位姿
def end_effector_position(thetas):
    T = np.eye(4)
    for i in range(6):
        T = T @ dh_matrix(a[i], alpha[i], d[i], np.radians(thetas[i]))
    return T[:3, 3]

# 目标函数
def objective(thetas):
    pos = end_effector_position(thetas)
    return np.linalg.norm(pos - target)

# 初始猜测
initial_guess = [0, -90, 0, 180, -90, 0]

# 约束
bounds = [(low, high) for low, high in theta_ranges]

# 优化
result = minimize(objective, initial_guess, bounds=bounds, method='SLSQP')
optimal_thetas = result.x

print("最优关节角度:", optimal_thetas)
print("最小化误差:", result.fun)

# 打印末端位置
end_position = end_effector_position(optimal_thetas)
print("末端位置:", end_position)

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