六自由度 机械手臂 python求解

求解六自由度机械臂可以使用逆运动学算法。在Python中，可以使用SymPy库来进行符号计算，从而解出机械臂的逆运动学问题。

首先，需要定义机械臂的DH参数，并在SymPy中定义符号变量。然后，可以使用SymPy的运动学模块计算机械臂的正运动学问题，得到末端执行器的位置和方向。

接下来，可以根据逆运动学的公式，从末端执行器的位置和方向反推出机械臂的关节角度。这个过程可以使用SymPy的符号求解功能来实现。

最后，将求解出的关节角度代入机械臂的正运动学公式中，验证机械臂的运动是否符合要求。

以下是一个简单的六自由度机械臂逆运动学求解的Python代码示例：

from sympy import symbols, cos, sin, pi, simplify
from sympy.matrices import Matrix

# DH parameters
theta1, theta2, theta3, theta4, theta5, theta6, d1, d2, d3, d4, d5, d6, a1, a2, a3, a4, a5, a6, alpha1, alpha2, alpha3, alpha4, alpha5, alpha6 = symbols('theta1:7 d1:7 a1:7 alpha1:7')

# Homogeneous Transforms
s = {alpha1: 0, a1: 0, d1: 0,
     alpha2: -pi/2, a2: 0, d2: 0,
     alpha3: 0, a3: 0, d3: 0,
     alpha4: -pi/2, a4: 0, d4: 0,
     alpha5: pi/2, a5: 0, d5: 0,
     alpha6: 0, a6: 0, d6: 0}

T0_1 = Matrix([[cos(theta1), -sin(theta1)*cos(alpha1), sin(theta1)*sin(alpha1), a1*cos(theta1)],
              [sin(theta1), cos(theta1)*cos(alpha1), -cos(theta1)*sin(alpha1), a1*sin(theta1)],
              [0, sin(alpha1), cos(alpha1), d1],
              [0, 0, 0, 1]])

T1_2 = Matrix([[cos(theta2), -sin(theta2)*cos(alpha2), sin(theta2)*sin(alpha2), a2*cos(theta2)],
              [sin(theta2), cos(theta2)*cos(alpha2), -cos(theta2)*sin(alpha2), a2*sin(theta2)],
              [0, sin(alpha2), cos(alpha2), d2],
              [0, 0, 0, 1]])

T2_3 = Matrix([[cos(theta3), -sin(theta3)*cos(alpha3), sin(theta3)*sin(alpha3), a3*cos(theta3)],
              [sin(theta3), cos(theta3)*cos(alpha3), -cos(theta3)*sin(alpha3), a3*sin(theta3)],
              [0, sin(alpha3), cos(alpha3), d3],
              [0, 0, 0, 1]])

T3_4 = Matrix([[cos(theta4), -sin(theta4)*cos(alpha4), sin(theta4)*sin(alpha4), a4*cos(theta4)],
              [sin(theta4), cos(theta4)*cos(alpha4), -cos(theta4)*sin(alpha4), a4*sin(theta4)],
              [0, sin(alpha4), cos(alpha4), d4],
              [0, 0, 0, 1]])

T4_5 = Matrix([[cos(theta5), -sin(theta5)*cos(alpha5), sin(theta5)*sin(alpha5), a5*cos(theta5)],
              [sin(theta5), cos(theta5)*cos(alpha5), -cos(theta5)*sin(alpha5), a5*sin(theta5)],
              [0, sin(alpha5), cos(alpha5), d5],
              [0, 0, 0, 1]])

T5_6 = Matrix([[cos(theta6), -sin(theta6)*cos(alpha6), sin(theta6)*sin(alpha6), a6*cos(theta6)],
              [sin(theta6), cos(theta6)*cos(alpha6), -cos(theta6)*sin(alpha6), a6*sin(theta6)],
              [0, sin(alpha6), cos(alpha6), d6],
              [0, 0, 0, 1]])

T0_6 = simplify(T0_1 * T1_2 * T2_3 * T3_4 * T4_5 * T5_6)

# solve inverse kinematics
x, y, z = symbols('x y z')
px = T0_6[0, 3]
py = T0_6[1, 3]
pz = T0_6[2, 3]
theta1 = simplify(-sin(theta2)*px + cos(theta2)*py)
theta2 = simplify(cos(theta1)*px + sin(theta1)*py)
theta3 = simplify(pz - d1 - d2)
theta4 = 0
theta5 = -pi/2
theta6 = 0

# substitute known values
theta1 = theta1.subs(s)
theta2 = theta2.subs(s)
theta3 = theta3.subs(s)

# simplify
theta1 = simplify(theta1)
theta2 = simplify(theta2)
theta3 = simplify(theta3)

# print results
print("theta1:", theta1)
print("theta2:", theta2)
print("theta3:", theta3)
print("theta4:", theta4)
print("theta5:", theta5)
print("theta6:", theta6)