Mobile robot path planning: artificial potential field method [easy to understand]

Hello everyone , I meet you again , I'm your friend, Quan Jun .

The artificial potential field method is a simple method Move robot Path planning algorithm , It regards the position of the target point as the lowest point of potential energy , Treat obstacles in the map as potential energy highs , Calculate the potential field diagram of the whole known map , And then, ideally , The robot is like a rolling ball , Automatically avoid all obstacles and roll to the target point .

• Reference resources ： Source code potential_field_planning.py Courseware CMU RI 16-735 Robot path planning 4 speak ： Artificial potential field method

In particular , The potential energy formula of the target point is ：

It reads , In order to prevent the speed from being too fast when it is far away from the target point , It can be described in the form of piecewise function , It will be shown later . The potential energy of the obstacle is expressed as ：

That is, it has high potential energy in a certain range around the obstacle , The influence of out of range visual obstacles is 0. Finally, add the potential energy of the target point and the obstacle , Get the whole potential energy map ：

The following is the source code in the reference link , It's easier to understand ：

""" Potential Field based path planner author: Atsushi Sakai (@Atsushi_twi) Ref: https://www.cs.cmu.edu/~motionplanning/lecture/Chap4-Potential-Field_howie.pdf """

from collections import deque
import numpy as np
import matplotlib.pyplot as plt

# Parameters
KP = 5.0  # attractive potential gain
ETA = 100.0  # repulsive potential gain
AREA_WIDTH = 30.0  # potential area width [m]
# the number of previous positions used to check oscillations
OSCILLATIONS_DETECTION_LENGTH = 3

show_animation = True


def calc_potential_field(gx, gy, ox, oy, reso, rr, sx, sy):
    """  Calculate the potential field diagram  gx,gy:  Target coordinates  ox,oy:  Obstacle coordinate list  reso:  Potential field diagram resolution  rr:  Robot radius  sx,sy:  Starting point coordinates  """
    #  Determine the coordinate range of the potential field diagram ：
    minx = min(min(ox), sx, gx) - AREA_WIDTH / 2.0
    miny = min(min(oy), sy, gy) - AREA_WIDTH / 2.0
    maxx = max(max(ox), sx, gx) + AREA_WIDTH / 2.0
    maxy = max(max(oy), sy, gy) + AREA_WIDTH / 2.0
    #  Determine the number of cells according to the range and resolution ：
    xw = int(round((maxx - minx) / reso))
    yw = int(round((maxy - miny) / reso))

    # calc each potential
    pmap = [[0.0 for i in range(yw)] for i in range(xw)]

    for ix in range(xw):
        x = ix * reso + minx   #  According to index and resolution x coordinate 

        for iy in range(yw):
            y = iy * reso + miny  #  According to index and resolution x coordinate 
            ug = calc_attractive_potential(x, y, gx, gy)  #  Calculate gravity 
            uo = calc_repulsive_potential(x, y, ox, oy, rr)  #  Calculated repulsion 
            uf = ug + uo
            pmap[ix][iy] = uf

    return pmap, minx, miny


def calc_attractive_potential(x, y, gx, gy):
    """  Calculate the gravitational potential energy ：1/2*KP*d """
    return 0.5 * KP * np.hypot(x - gx, y - gy)


def calc_repulsive_potential(x, y, ox, oy, rr):
    """  Calculate the repulsive potential energy ：  If the distance from the nearest obstacle dq In the robot expansion radius rr within ：1/2*ETA*(1/dq-1/rr)**2  otherwise ：0.0 """
    # search nearest obstacle
    minid = -1
    dmin = float("inf")
    for i, _ in enumerate(ox):
        d = np.hypot(x - ox[i], y - oy[i])
        if dmin >= d:
            dmin = d
            minid = i

    # calc repulsive potential
    dq = np.hypot(x - ox[minid], y - oy[minid])

    if dq <= rr:
        if dq <= 0.1:
            dq = 0.1

        return 0.5 * ETA * (1.0 / dq - 1.0 / rr) ** 2
    else:
        return 0.0


def get_motion_model():
    # dx, dy
    #  Jiugonggezhong  8 Possible directions of motion 
    motion = [[1, 0],
              [0, 1],
              [-1, 0],
              [0, -1],
              [-1, -1],
              [-1, 1],
              [1, -1],
              [1, 1]]

    return motion


def oscillations_detection(previous_ids, ix, iy):
    """  Oscillation detection ： avoid “ Repeat the cross jump ” """
    previous_ids.append((ix, iy))

    if (len(previous_ids) > OSCILLATIONS_DETECTION_LENGTH):
        previous_ids.popleft()

    # check if contains any duplicates by copying into a set
    previous_ids_set = set()
    for index in previous_ids:
        if index in previous_ids_set:
            return True
        else:
            previous_ids_set.add(index)
    return False


def potential_field_planning(sx, sy, gx, gy, ox, oy, reso, rr):

    # calc potential field
    pmap, minx, miny = calc_potential_field(gx, gy, ox, oy, reso, rr, sx, sy)

    # search path
    d = np.hypot(sx - gx, sy - gy)
    ix = round((sx - minx) / reso)
    iy = round((sy - miny) / reso)
    gix = round((gx - minx) / reso)
    giy = round((gy - miny) / reso)

    if show_animation:
        draw_heatmap(pmap)
        # for stopping simulation with the esc key.
        plt.gcf().canvas.mpl_connect('key_release_event',
                lambda event: [exit(0) if event.key == 'escape' else None])
        plt.plot(ix, iy, "*k")
        plt.plot(gix, giy, "*m")

    rx, ry = [sx], [sy]
    motion = get_motion_model()
    previous_ids = deque()

    while d >= reso:
        minp = float("inf")
        minix, miniy = -1, -1
        #  seek 8 The direction in which the potential field is the smallest among the three directions of motion 
        for i, _ in enumerate(motion):
            inx = int(ix + motion[i][0])
            iny = int(iy + motion[i][1])
            if inx >= len(pmap) or iny >= len(pmap[0]) or inx < 0 or iny < 0:
                p = float("inf")  # outside area
                print("outside potential!")
            else:
                p = pmap[inx][iny]
            if minp > p:
                minp = p
                minix = inx
                miniy = iny
        ix = minix
        iy = miniy
        xp = ix * reso + minx
        yp = iy * reso + miny
        d = np.hypot(gx - xp, gy - yp)
        rx.append(xp)
        ry.append(yp)
        #  Oscillation detection , To avoid falling into a local minimum ：
        if (oscillations_detection(previous_ids, ix, iy)):
            print("Oscillation detected at ({},{})!".format(ix, iy))
            break

        if show_animation:
            plt.plot(ix, iy, ".r")
            plt.pause(0.01)

    print("Goal!!")

    return rx, ry


def draw_heatmap(data):
    data = np.array(data).T
    plt.pcolor(data, vmax=100.0, cmap=plt.cm.Blues)


def main():
    print("potential_field_planning start")

    sx = 0.0  # start x position [m]
    sy = 10.0  # start y positon [m]
    gx = 30.0  # goal x position [m]
    gy = 30.0  # goal y position [m]
    grid_size = 0.5  # potential grid size [m]
    robot_radius = 5.0  # robot radius [m]
    #  The following obstacle coordinates are my modified , It is used to show the situation that the artificial potential field method is trapped in local optimization ：
    ox = [15.0, 5.0, 20.0, 25.0, 12.0, 15.0, 19.0, 28.0, 27.0, 23.0, 30.0, 32.0]  # obstacle x position list [m]
    oy = [25.0, 15.0, 26.0, 25.0, 12.0, 20.0, 29.0, 28.0, 26.0, 25.0, 28.0, 27.0]  # obstacle y position list [m]

    if show_animation:
        plt.grid(True)
        plt.axis("equal")

    # path generation
    _, _ = potential_field_planning(
        sx, sy, gx, gy, ox, oy, grid_size, robot_radius)

    if show_animation:
        plt.show()


if __name__ == '__main__':
    print(__file__ + " start!!")
    main()
    print(__file__ + " Done!!")

Copy

One of the main disadvantages of the artificial potential field method is that it may fall into the local optimal solution , The following figure is the result of running the source code ：

Here is after I added some obstacles , The artificial potential field method is trapped in the case of local optimal solution ： Although the target point has not been reached , But the potential field determines that the path cannot go any further .

It should be noted that , When calculating the potential field of the target point in the source code , It uses a x,y A term of the distance between the position and the target point , The quadratic term is not used as shown in the courseware , It is also to make the potential field change less quickly . The following is according to the courseware , The result of using the quadratic term of distance , We can see , For normal operation ,KP It needs to be adjusted very low ：

KP = 0.1
def calc_attractive_potential(x, y, gx, gy):
    """  Calculate the gravitational potential energy ：1/2*KP*d^2 """
    return 0.5 * KP * np.hypot(x - gx, y - gy)**2

Copy

The normal operation ：

Trapped in local best ：

You can see from the potential field diagram , Gravity changes much faster than the previous example .

Last , We modify the program into the segmentation function shown in the screenshot of the courseware above ：

KP = 0.25
dgoal = 10
def calc_attractive_potential(x, y, gx, gy):
    """  Calculate gravity ： Such as courseware screenshot  """
    dg = np.hypot(x - gx, y - gy)
    if dg<=dgoal:
        U = 0.5 * KP * np.hypot(x - gx, y - gy)**2
    else:
        U = dgoal*KP*np.hypot(x - gx, y - gy) - 0.5*KP*dgoal
    return U

Copy

The normal operation ：

Trapped in local optima ：

You can see the effect of gravitational potential field segmentation .

Publisher ： Full stack programmer stack length , Reprint please indicate the source ：https://javaforall.cn/148611.html Link to the original text ：https://javaforall.cn