模拟工件静态定位时，如果定位爪边长小于工件边长，那么工件会有旋转，本文正是求这极限情形。

输入与输出

  1. 输入
  • 工件的边长
  • 定位爪的边长
  1. 输出
  • 工件的极限状态
变量说明
变量说明

算法

逻辑

  1. 工件某一边紧贴定位爪的某一边（点）
  2. 计算工件相邻边是否接触定位爪
  3. 工件以定位爪的某一点旋转角度 α
    α=α+1 再判断 Step 2
flowchart LR
    A[Step 1] --> B{Step 2}
    B -->|No| C[Step 3]
    C --> B
    B -->|Yes| E[Step 4]

代码

以下代码经过简化，完整代码请看最后。

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import math

# 工件
class Device(object):
    def __init__(self, x, y):
        self.x = float(x)
        self.y = float(y)
        self.area = self.x * self.y
        self.diagonal = math.sqrt(self.x**2+self.y**2)

# 定位爪
class Position(object):
    def __init__(self, x, y, dx, dy):
        self.x = float(x)
        self.y = float(y)
        self.dx = float(dx)
        self.dy = float(dy)
        self.area = self.x * self.y
        self.diagonal = min(math.sqrt(self.dx**2+self.y**2),math.sqrt(self.x**2+self.dy**2))

# 点到直线距离
def p2l(p=(0, 0), l=(0, 0, 0)):
    x, y = p[0], p[1]
    a, b, c = l[0], l[1], l[2]
    return abs(a * x + b * y + c) / math.sqrt(a**2 + b**2)

# 两直线交点
def l2l(l1=(0, 0, 0), l2=(0, 0, 0)):
    a1, b1, c1 = l1[0], l1[1], l1[2]
    a2, b2, c2 = l2[0], l2[1], l2[2]
    return (
        (c2 * b1 - c1 * b2) / (a1 * b2 - a2 * b1),
        (c1 * a2 - c2 * a1) / (a1 * b2 - a2 * b1),
    )

# 形状检查
def sharp_check(d,p):
    if p.x <= d.x or p.y <= d.y:
        return True
    elif p.dx >= p.x or p.dy >= p.y:
        return True
    elif d.diagonal < p.diagonal:
        return True
    else:
        return False

# 计算多边形的面积
def calculate_polygon_area(*coords):
    n = len(coords)
    area = 0
    j = n - 1
    for i in range(0, n):
        area += (coords[j][0] + coords[i][0]) * (coords[j][1] - coords[i][1])
        j = i
    return abs(area / 2)

print("工件面积：{:.2f}".format(device.area))
print("定位爪区域面积：{:.2f}".format(position.area))
# 工件
cv.create_rectangle((-device.x / 2, device.y / 2), (device.x / 2, -device.y / 2))
# 定位爪
cv.create_rectangle((position.x / 2, position.dy / 2),(position.x / 2 + math.inf, -position.dy / 2))
cv.create_rectangle((-position.dx / 2, position.y / 2 + math.inf),(position.dx / 2, position.y / 2))
cv.create_rectangle((-position.x / 2 - math.inf, position.dy / 2),(-position.x / 2, -position.dy / 2))
cv.create_rectangle((-position.dx / 2, -position.y / 2),(position.dx / 2, -position.y / 2 - math.inf))
# 形状非法
if sharp_check(device,position):
    print("形状非法，请重新输入。\n")
    return
print("生成中，请稍候...")
# 工件的四个顶点坐标
p01=(position.x/2,-position.dy/2)
p02=(position.dx/2,position.y/2)
p03=(-position.x/2,position.dy/2)
p04=(-position.dx/2,-position.y/2)
# 计算 旋转工件四个顶点
for alpha in range(1, 90000):
    alpha = math.radians(alpha/1000)
    k = math.tan(alpha)
    p1 = (device.x/2*math.cos(alpha)+device.y/2*math.sin(alpha),device.x/2*math.sin(alpha)-device.y/2*math.cos(alpha))
    p2 = (device.x/2*math.cos(alpha)-device.y/2*math.sin(alpha),device.x/2*math.sin(alpha)+device.y/2*math.cos(alpha))
    p3 = (-device.x/2*math.cos(alpha)-device.y/2*math.sin(alpha),-device.x/2*math.sin(alpha)+device.y/2*math.cos(alpha))
    p4 = (-device.x/2*math.cos(alpha)+device.y/2*math.sin(alpha),-device.x/2*math.sin(alpha)-device.y/2*math.cos(alpha))
    l1 = (-1/k,-1,1/k*p1[0]+p1[1])
    l2 = (k,-1,-k*p2[0]+p2[1])
    l3 = (-1/k,-1,1/k*p3[0]+p3[1])
    l4 = (k,-1,-k*p4[0]+p4[1])
    # 碰到定位爪停止
    if any((p2l(p01,l1)<=0.001,p2l(p02,l2)<=0.001,p2l(p03,l3)<=0.001,p2l(p04,l4)<=0.001)):
        print("工件极限旋转角：{:.2f}°".format(math.degrees(alpha)))
        # X向定位
        if any((p2l(p01,l1)<=0.001,p2l(p03,l3)<=0.001)):
            p1 = (p1[0],p1[1]-p2l(p02,l2))
            p2 = (p2[0],p2[1]-p2l(p02,l2))
            p3 = (p3[0],p3[1]-p2l(p02,l2))
            p4 = (p4[0],p4[1]-p2l(p02,l2))
        # Y向定位
        else:
            p1 = (p1[0]-p2l(p01,l1),p1[1])
            p2 = (p2[0]-p2l(p01,l1),p2[1])
            p3 = (p3[0]-p2l(p01,l1),p3[1])
            p4 = (p4[0]-p2l(p01,l1),p4[1])
        # 工件的极限旋转情形
        cv.create_polygon(p1,p2,p3,p4)
        break
# 重叠区域计算
# 极限情形的各边
l1 = (-1/k, -1, 1/k*p1[0]+p1[1])
l2 = (k, -1, -k*p2[0]+p2[1])
l3 = (-1/k, -1, 1/k*p3[0]+p3[1])
l4 = (k, -1, -k*p4[0]+p4[1])
# 极限情形与理论情形的各边交点
p11 = l2l(l1,(1,0,-device.x/2))
p12 = l2l(l1,(0,1,-device.y/2))
p13 = l2l(l2,(0,1,-device.y/2))
p14 = l2l(l2,(1,0,device.x/2))
p15 = l2l(l3,(1,0,device.x/2))
p16 = l2l(l3,(0,1,device.y/2))
p17 = l2l(l4,(0,1,device.y/2))
p18 = l2l(l4,(1,0,-device.x/2))
print("重叠部分面积为：{:.2f}\n".format(calculate_polygon_area(p11, p12, p13, p14, p15, p16, p17, p18)))
cv.create_polygon(p11, p12, p13, p14, p15, p16, p17, p18)

成品预览

Python 3.12
Python 3.12

完整代码

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import math
import re
# import time
import tkinter
import tkinter.ttk
from tkinter.scrolledtext import ScrolledText


# 工件
class Device(object):
    def __init__(self, x, y):
        self.x = float(x)
        self.y = float(y)
        self.area = self.x * self.y
        self.diagonal = math.sqrt(self.x**2+self.y**2)


# 定位爪
class Position(object):
    def __init__(self, x, y, dx, dy):
        self.x = float(x)
        self.y = float(y)
        self.dx = float(dx)
        self.dy = float(dy)
        self.area = self.x * self.y
        self.diagonal = min(math.sqrt(self.dx**2+self.y**2),math.sqrt(self.x**2+self.dy**2))


# 点到直线距离
def p2l(p=(0, 0), l=(0, 0, 0)):
    x, y = p[0], p[1]
    a, b, c = l[0], l[1], l[2]
    return abs(a * x + b * y + c) / math.sqrt(a**2 + b**2)


# 两直线交点
def l2l(l1=(0, 0, 0), l2=(0, 0, 0)):
    a1, b1, c1 = l1[0], l1[1], l1[2]
    a2, b2, c2 = l2[0], l2[1], l2[2]
    return (
        (c2 * b1 - c1 * b2) / (a1 * b2 - a2 * b1),
        (c1 * a2 - c2 * a1) / (a1 * b2 - a2 * b1),
    )


# 坐标转换
def convert(x, y):
    return (x + 450 / 2, 450 / 2 - y)


# 输入检查
def input_check(*args):
    pattern = r'^[-+]?[0-9]*\.?[0-9]+([eE][-+]?[0-9]+)?$'
    for i in args:
        if not bool(re.match(pattern, i)) or len(i) == 0:
            return True
    return False

# 形状检查
def sharp_check(d,p):
    if p.x <= d.x or p.y <= d.y:
        return True
    elif p.dx >= p.x or p.dy >= p.y:
        return True
    elif d.diagonal < p.diagonal:
        return True
    else:
        return False

# 计算多边形的面积
def calculate_polygon_area(*coords):
    n = len(coords)
    area = 0
    j = n - 1
    for i in range(0, n):
        area += (coords[j][0] + coords[i][0]) * (coords[j][1] - coords[i][1])
        j = i
    return abs(area / 2)

# 生成
def submit():
    cv.delete("all")
    if input_check(
        device_x.get(),
        device_y.get(),
        position_x.get(),
        position_y.get(),
        position_dx.get(),
        position_dy.get(),
    ):
        print_t("输入有误，请重新输入。")
    else:
        calculate(
            Device(device_x.get(), device_y.get()),
            Position(position_x.get(), position_y.get(), position_dx.get(), position_dy.get()),
        )
    # 坐标轴
    cv.create_line(0, 450 / 2, 450, 450 / 2, dash=(1, 1))
    cv.create_line(450 / 2, 0, 450 / 2, 450, dash=(1, 1))


# 计算+绘图
def calculate(device, position):
    zoom = 360 / max(device.x, device.y, position.x, position.y)
    print_t("工件面积：{:.2f}".format(device.area))
    print_t("定位爪区域面积：{:.2f}".format(position.area))
    # 工件
    cv.create_rectangle(
        convert(-device.x / 2 * zoom, device.y / 2 * zoom),
        convert(device.x / 2 * zoom, -device.y / 2 * zoom),
        fill="#696969",
        outline="",
    )
    # 定位爪
    cv.create_rectangle(
        convert(position.x / 2 * zoom, position.dy / 2 * zoom),
        convert(position.x / 2 * zoom + 999, -position.dy / 2 * zoom),
        fill="#00BFFF",
        outline="",
    )
    cv.create_rectangle(
        convert(-position.dx / 2 * zoom, position.y / 2 * zoom + 999),
        convert(position.dx / 2 * zoom, position.y / 2 * zoom),
        fill="#00BFFF",
        outline="",
    )
    cv.create_rectangle(
        convert(-position.x / 2 * zoom - 999, position.dy / 2 * zoom),
        convert(-position.x / 2 * zoom, -position.dy / 2 * zoom),
        fill="#00BFFF",
        outline="",
    )
    cv.create_rectangle(
        convert(-position.dx / 2 * zoom, -position.y / 2 * zoom),
        convert(position.dx / 2 * zoom, -position.y / 2 * zoom - 999),
        fill="#00BFFF",
        outline="",
    )
    # 形状非法
    if sharp_check(device,position):
        print_t("形状非法，请重新输入。\n")
        return
    print_t("生成中，请稍候...")
    p01=(position.x/2,-position.dy/2)
    p02=(position.dx/2,position.y/2)
    p03=(-position.x/2,position.dy/2)
    p04=(-position.dx/2,-position.y/2)
    # 计算 旋转工件四个顶点
    for alpha in range(1, 90000):
        alpha = math.radians(alpha/1000)
        k = math.tan(alpha)
        # 旧算法，有bug
        # (-position.dx/2,-position.y/2)
        # (-position.dx/2-math.cos(alpha)*device.y,-position.y/2+math.sin(alpha)*device.y)
        # y-y1=k(x-x1)
        # kx-y-kx1+y1=0
        # p = (position.dx / 2, position.y / 2)
        # l = (k,-1,-k * (-position.dx / 2 - math.cos(90 - alpha) * device.y) - position.y / 2 + math.sin(90 - alpha) * device.y,
        # )
        # if p2l(p, l) < 0.001:
        #     break
        # print_t("工件最大旋转角：{:.2f}°".format(alpha))
        # # (-position.x/2,position.dy/2)
        # p1 = l2l(l, (-1 / k, -1, 1 / k * (-position.x / 2) + position.dy / 2))
        # p2 = (
        #     p1[0] + math.cos(alpha) * device.x,
        #     p1[1] + math.sin(alpha) * device.x,
        # )
        # p3 = (
        #     p2[0] + math.sin(alpha) * device.y,
        #     p2[1] - math.cos(alpha) * device.y,
        # )
        # p4 = (
        #     p3[0] - math.cos(alpha) * device.x,
        #     p3[1] - math.sin(alpha) * device.x,
        # )
        p1 = (device.x/2*math.cos(alpha)+device.y/2*math.sin(alpha),device.x/2*math.sin(alpha)-device.y/2*math.cos(alpha))
        p2 = (device.x/2*math.cos(alpha)-device.y/2*math.sin(alpha),device.x/2*math.sin(alpha)+device.y/2*math.cos(alpha))
        p3 = (-device.x/2*math.cos(alpha)-device.y/2*math.sin(alpha),-device.x/2*math.sin(alpha)+device.y/2*math.cos(alpha))
        p4 = (-device.x/2*math.cos(alpha)+device.y/2*math.sin(alpha),-device.x/2*math.sin(alpha)-device.y/2*math.cos(alpha))
        l1 = (-1/k,-1,1/k*p1[0]+p1[1])
        l2 = (k,-1,-k*p2[0]+p2[1])
        l3 = (-1/k,-1,1/k*p3[0]+p3[1])
        l4 = (k,-1,-k*p4[0]+p4[1])
        # 碰到定位爪停止
        if any((p2l(p01,l1)<=0.001,p2l(p02,l2)<=0.001,p2l(p03,l3)<=0.001,p2l(p04,l4)<=0.001)):
            print_t("工件极限旋转角：{:.2f}°".format(math.degrees(alpha)))
            # X向定位
            if any((p2l(p01,l1)<=0.001,p2l(p03,l3)<=0.001)):
                p1 = (p1[0],p1[1]-p2l(p02,l2))
                p2 = (p2[0],p2[1]-p2l(p02,l2))
                p3 = (p3[0],p3[1]-p2l(p02,l2))
                p4 = (p4[0],p4[1]-p2l(p02,l2))
            # Y向定位
            else:
                p1 = (p1[0]-p2l(p01,l1),p1[1])
                p2 = (p2[0]-p2l(p01,l1),p2[1])
                p3 = (p3[0]-p2l(p01,l1),p3[1])
                p4 = (p4[0]-p2l(p01,l1),p4[1])
            cv.create_polygon(
                convert(p1[0] * zoom, p1[1] * zoom),
                convert(p2[0] * zoom, p2[1] * zoom),
                convert(p3[0] * zoom, p3[1] * zoom),
                convert(p4[0] * zoom, p4[1] * zoom),
                fill="",
                outline="#FF4500",
                width=2,
                dash=(10,5),
            )
            break
    # 重叠区域计算
    # 极限情形的各边
    l1 = (-1/k,-1,1/k*p1[0]+p1[1])
    l2 = (k,-1,-k*p2[0]+p2[1])
    l3 = (-1/k,-1,1/k*p3[0]+p3[1])
    l4 = (k,-1,-k*p4[0]+p4[1])
    p11 = l2l(l1,(1,0,-device.x/2))
    p12 = l2l(l1,(0,1,-device.y/2))
    p13 = l2l(l2,(0,1,-device.y/2))
    p14 = l2l(l2,(1,0,device.x/2))
    p15 = l2l(l3,(1,0,device.x/2))
    p16 = l2l(l3,(0,1,device.y/2))
    p17 = l2l(l4,(0,1,device.y/2))
    p18 = l2l(l4,(1,0,-device.x/2))
    print_t("重叠部分面积为：{:.2f}\n".format(calculate_polygon_area(p11, p12, p13, p14, p15, p16, p17, p18)))
    # 极限情形与理论情形的各边交点
    cv.create_polygon(
        convert(p11[0] * zoom, p11[1] * zoom),
        convert(p12[0] * zoom, p12[1] * zoom),
        convert(p13[0] * zoom, p13[1] * zoom),
        convert(p14[0] * zoom, p14[1] * zoom),
        convert(p15[0] * zoom, p15[1] * zoom),
        convert(p16[0] * zoom, p16[1] * zoom),
        convert(p17[0] * zoom, p17[1] * zoom),
        convert(p18[0] * zoom, p18[1] * zoom),
        fill="",
        outline="",
        activefill="#FF0000",
    )


# 输出
def print_t(message):
    output.config(state="normal")
    output.insert("end", f"{message}\n")
    output.see("end")
    output.config(state="disabled")


# GUI
root = tkinter.Tk()
root.title("四边定位的旋转量模拟")
root.geometry("650x480")
root.resizable(False, False)

# 画布
cv = tkinter.Canvas(
    root, width=450, height=450, highlightbackground="gray", highlightthickness=1
)
cv.create_line(0, 450 / 2, 450, 450 / 2, dash=(1, 1))
cv.create_line(450 / 2, 0, 450 / 2, 450, dash=(1, 1))
cv.place(x=5, y=5)

# 输入
frame1 = tkinter.ttk.LabelFrame(root, text="输入", height=240, width=200)
frame1.place(x=470, y=2)

tkinter.ttk.Label(frame1, text="工件x").grid(column=0, row=0, sticky=tkinter.W)
device_x = tkinter.ttk.Entry(
    frame1,
    width=10,
    validate="key",
    validatecommand=(root.register(lambda x: x.isdigit() or x == "" or x.count(".") == 1),"%P",),
)
device_x.grid(column=1, row=0)

tkinter.ttk.Label(frame1, text="工件y").grid(column=0, row=1, sticky=tkinter.W)
device_y = tkinter.ttk.Entry(
    frame1,
    width=10,
    validate="key",
    validatecommand=(root.register(lambda x: x.isdigit() or x == "" or x.count(".") == 1),"%P",),
)
device_y.grid(column=1, row=1)

tkinter.ttk.Label(frame1, text="定位爪x").grid(column=0, row=2, sticky=tkinter.W)
position_x = tkinter.ttk.Entry(
    frame1,
    width=10,
    validate="key",
    validatecommand=(root.register(lambda x: x.isdigit() or x == "" or x.count(".") == 1),"%P",),
)
position_x.grid(column=1, row=2)

tkinter.ttk.Label(frame1, text="定位爪y").grid(column=0, row=3, sticky=tkinter.W)
position_y = tkinter.ttk.Entry(
    frame1,
    width=10,
    validate="key",
    validatecommand=(
        root.register(lambda x: x.isdigit() or x == "" or x.count(".") == 1),
        "%P",
    ),
)
position_y.grid(column=1, row=3)

tkinter.ttk.Label(frame1, text="定位爪Dx").grid(column=0, row=4, sticky=tkinter.W)
position_dx = tkinter.ttk.Entry(
    frame1,
    width=10,
    validate="key",
    validatecommand=(root.register(lambda x: x.isdigit() or x == "" or x.count(".") == 1),"%P",),
)
position_dx.grid(column=1, row=4)

tkinter.ttk.Label(frame1, text="定位爪Dy").grid(column=0, row=5, sticky=tkinter.W)
position_dy = tkinter.ttk.Entry(
    frame1,
    width=10,
    validate="key",
    validatecommand=(root.register(lambda x: x.isdigit() or x == "" or x.count(".") == 1),"%P",),
)
position_dy.grid(column=1, row=5)

# 生成
button1 = tkinter.ttk.Button(frame1, text="生成", command=submit)
button1.grid(column=0, row=6, columnspan=2)

# 输出
frame2 = tkinter.ttk.LabelFrame(root, text="输出", height=240, width=170)
frame2.place(x=470, y=230)
output = ScrolledText(frame2, width=20, height=15)
output.pack()

root.mainloop()

代码下载请见 Malvern's File Server

许可协议

本文采用 CC BY-NC-SA 4.0 许可协议，转载请注明出处，谢谢！

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