我目前正在使用像这这样的显示激光设备。设备接收2D点的列表,然后显示。在内部,有一个电流计控制反射镜投射激光点。
假设我想显示5个激光点(A,B,C,D,E)。由于激光设备不喜欢在较短的时间间隔内进行长距离旅行,因此必须增加中间点,称为消隐点(激光沿这些空白点行进时关闭)。这一程序实现了不过分强调电流计的目的。
我目前正在用一个简单的最近邻算法计算5个点之间的“最短路径”,最后是直线(下图中的红色虚线)。

通过这种优化,我已经取得了相当好的效果。但我想更进一步。电流计在运动时有一些物理动量。当做急转弯时,例如从C->D和D->E出发,它确实会对激光装置产生压力。
所以我想通过引入弯曲的消隐线来吸收一些物理动量,而不是直线(cp )。要在上面的插图“解决方案”中最后显示图像)。
知道怎么做到吗?
指向某些算法资源和/或一些伪代码或C#将是有帮助的。谢谢!
发布于 2018-07-11 17:10:35
正如其他人所提到的,您需要使用某种三次样条插值。
一旦你知道每个关键点的访问时间和每个关键点的速度,你就可以计算出一个分段三次Hermite样条,它以选定的速度通过关键点。请参阅:样条
由于对速度没有任何特殊的要求,所以您可能想使用一个经典的三次样条(是的,这些东西的名称是模棱两可的):http://mathworld.wolfram.com/CubicSpline.html这种形式的样条确定速度,以确保一阶导数(速度)和二阶导数(加速度)沿着整个路径平稳地变化。
由于对于到达每个关键点的确切时间也没有任何特殊要求,所以您可能希望为整个路径设置最大时间,然后选择关键点的时间,以最小化最大加速度或类似的情况。我可没有那么简单的方法。我想尝试的是:
首先,使关键点之间的时间与这些点之间的距离成比例。然后,应用几轮:
但是,如果没有这些优化回合,您可能会非常高兴--最初的猜测不会太糟糕。
发布于 2018-07-11 20:32:24
我认为你要处理的是一个旅行-推销员-问题(TSP) (这里是博士课程的幻灯片,讨论如何解决这个问题。),而最小化激光上的应力的路径是一个最小化力和力的变化的路径,所以它是具有最小曲率的路径,所以我认为最好的方法是在3对点之间用一个圆弧旋转路径。
关于如何计算通过3个点的圆的参数的例子可以找到这里。
我不太精通C#,所以我将在C#中添加一个实现,我希望您也能发现它也很有用。
它的思想是,对于点A,B,C的每一个三重奏,我找到了圆的弧线,它代表了这三个点,而这个弧线将是连接B和C的路径。
我还没来得及测试这个,所以可能有一些错误的迹象。
# Initial points
points = [(1,1),(2,3),(5,3),(-4.1),(12,3)]
#List of point in the order find by the solution of the TSP
spl = tsp_solve(points) # generic function to solve the TSP
# Append the first two point of the list so that I can iterate over the list
# and parse every triplet of points in the same way.
spl = spl + spl[:2]
# The list where will be added every path that connect the points
paths = []
# For each tirplets of sequential points
for A,B,C in zip(spl[:-2],spl[1:-1],spl[2:]):
# Calculate the angular coefficent of the two line that pass on A,B and B,C
coeff_ab = (B[1] - A[1]) / (B[0] - A[0])
coeff_bc = (C[1] - B[1]) / (C[0] - B[0])
# If the two line have the same coeff then all the 3 point are on the same line
# and therfore the best path is that line.
if(coeff_ab == coeff_bc):
offset_y = A[1] - coeff_ab * A[0]
delta_x = C[0] - B[0]
paths.append({"type":"Line","coeff":coeff_ab,"offset_y":offset_y,"deta_x":delta_x})
continue
# Calculate the x of the center of the circle
center_x = coeff_ab *coeff_bc *(C[0]-A[0])
center_x += coeff_ab *(B[0]+C[0])
center_x -= coeff_bc *(A[0]+B[0])
center_x /= 2*(coeff_ab - coeff_bc)
# Calculate the y of the center of the circle
center_y = (A[1]+B[1)/2
center_y -= (center_x - (A[0] + B[0])/2)
center_y /= coeff_bc
radius = sqrt(center_x**2 + center_y**2)
paths.append({"type":"Circle","Radius":radius,"center_x":center_x,"center_y":center_y})
# Function To Calculate the X and Y of the lines and circles.
def calculate_circle_x(circle,time):
"""Function that return the x of a circle at a given time"""
time = time + circle["time_off"]
return circle["radius"] * cos(2*pi*time) + circle["center_x"]
def calculate_circle_y(circle,time):
"""Function that return the y of a circle at a given time"""
time = time + circle["time_off"]
return circle["radius"] * sin(2*pi*time) + circle["center_y"]
def calculate_line_x(line,time):
"""Function that return the x of a line at a given time"""
time = (line['delta_x']*time) + line["time_off"]
return time
def calculate_line_y(line,time):
"""Function that return the y of a line at a given time"""
time = (line['delta_x']*time) + line["time_off"]
return time * line["coeff"] + line['offset_y']
def calculate_x(obj,time):
"""Function that return the x of whatever it's passed"""
if(obj['type'] == 'Circle'):
return calculate_circle_x(obj,time)
else:
return calculate_line_x(obj,time)
def calculate_y(obj,time):
"""Function that return the y of whatever it's passed"""
if(obj['type'] == 'Circle'):
return calculate_circle_y(obj,time)
else:
return calculate_line_y(obj,time)
# Calculate some sample of the global path to plot it or do whatever with it.
number_of_sample = 100000
path_points = []
number_of_paths = len(paths)
# Calculate some time equidistant point's sample
for i in range(number_of_sample):
# Calculate the global time
global_time = i*number_of_paths/number_of_sample
# Calculate in which path the point it is
path_number = int(global_time)
# Calculate which time of the path it is
local_time = global_time - path_number
path = paths[path_number]
# Calculate the sampled point
new_point = (calculate_x(path,local_time),calculate_y(path,local_time))
# Add the sampled point to the path_points list
path_points.append(new_point)
# Print the result of the path point sampled.
print(path_points)现在,您有了点,或者至少有了如何计算这些点的示例,并且可以将其转换为C#。我试着对它进行了大量的评论,这样即使不了解Python,您也可以理解它。
https://stackoverflow.com/questions/51289132
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