目录
0 专栏介绍1 Bresenham算法介绍2 图解Bresenham算法3 算法流程4 仿真实现4.1 ROS C++实现4.2 Python实现4.3 Matlab实现
0 专栏介绍
?课设、毕设、创新竞赛必备！?本专栏涉及更高阶的运动规划算法轨迹优化实战，包括：曲线生成、碰撞检测、安全走廊、优化建模(QP、SQP、NMPC、iLQR等)、轨迹优化(梯度法、曲线法等)，每个算法都包含代码实现加深理解
?详情：运动规划实战进阶：轨迹优化篇
1 Bresenham算法介绍
Bresenham视线生成算法是一种高效的算法，用于在二维网格上绘制直线。它是由Jack Bresenham在1962年提出的，广泛应用于计算机图形学和游戏开发中。该算法的主要优点是只使用整数运算，因此速度较快且易于实现。下面是该算法的动图案例
Bresenham算法可以巧妙地应用在栅格地图中进行一维碰撞检测：首先确定起点和终点，然后使用Bresenham算法绘制从起点到终点的线段，接着检查该路径上每个栅格点是否与障碍物重叠。
2 图解Bresenham算法
Bresenham碰撞测试在三种类型的移动中访问单元格：
x x x方向移动 y y y方向移动对角线移动在栅格地图中，碰撞检测点连线经过若干离散栅格，因此每次移动都将产生非连续误差，Bresenham算法要求下一个移动偏差最小。通过迭代即可访问检测线经过的所有栅格，判断这些栅格的代价是否超过阈值即可完成碰撞检测。
具体地，设需要检测节点 v v v、 w w w间的连线是否经过障碍物，定义缩放误差分别为 δ x = ∣ w . x − v . x ∣ \delta _x=\left| w.x-v.x \right| δx=∣w.x−v.x∣、 δ y = ∣ w . y − v . y ∣ \delta _y=\left| w.y-v.y \right| δy=∣w.y−v.y∣；扩展误差分别为 e x e_x ex、 e y e_y ey；方向增量分别为 Δ x = s g n ( w . x − v . x ) \varDelta x=sgn\left( w.x-v.x \right) Δx=sgn(w.x−v.x)、 Δ y = s g n ( w . y − v . y ) \varDelta y=sgn \left( w.y-v.y \right) Δy=sgn(w.y−v.y)。下面考虑 δ x > δ y \delta _x>\delta _y δx>δy的情形， δ x ⩽ δ y \delta _x\leqslant \delta _y δx⩽δy时可对称导出。
如图所示，根据三角形相似关系可得
{ e y e x = ∣ Δ y ∣ t t δ y = ∣ Δ x ∣ δ x ⇒ e y e x = ∣ Δ y ∣ ∣ Δ x ∣ ⋅ δ x δ y = δ x δ y \begin{cases} \frac{e_y}{e_x}=\frac{\left| \varDelta y \right|}{t}\\ \frac{t}{\delta _y}=\frac{\left| \varDelta x \right|}{\delta _x}\\\end{cases}\Rightarrow \frac{e_y}{e_x}=\frac{\left| \varDelta y \right|}{\left| \varDelta x \right|}\cdot \frac{\delta _x}{\delta _y}=\frac{\delta _x}{\delta _y} {exey=t∣Δy∣δyt=δx∣Δx∣⇒exey=∣Δx∣∣Δy∣⋅δyδx=δyδx
不妨令 e y = δ x e_y=\delta _x ey=δx、 e x = δ y e_x=\delta _y ex=δy，则沿 x x x方向移动将产生负向偏差 δ y \delta _y δy，沿 y y y方向移动将产生正向偏差 δ x \delta _x δx，根据最小化偏差原则选择移动方向
( x , y ) ← { ( x + Δ x , y ) , i f ∣ e + δ x ∣ > ∣ e − δ y ∣ ( x , y + Δ y ) , i f ∣ e + δ x ∣ < ∣ e − δ y ∣ ( x + Δ x , y + Δ y ) , i f ∣ e + δ x ∣ = ∣ e − δ y ∣ \left( x,y \right) \gets \begin{cases} \left( x+\varDelta x,y \right) , \mathrm{if} \left| e+\delta _x \right|>\left| e-\delta _y \right|\\ \left( x,y+\varDelta y \right) , \mathrm{if} \left| e+\delta _x \right|<\left| e-\delta _y \right|\\ \left( x+\varDelta x,y+\varDelta y \right) , \mathrm{if} \left| e+\delta _x \right|=\left| e-\delta _y \right|\\\end{cases} (x,y)←⎩⎪⎨⎪⎧(x+Δx,y),if∣e+δx∣>∣e−δy∣(x,y+Δy),if∣e+δx∣<∣e−δy∣(x+Δx,y+Δy),if∣e+δx∣=∣e−δy∣
下图为Bresenham扩展节点的实例
3 算法流程
算法流程如下所示
4 仿真实现
4.1 ROS C++实现
核心代码如下所示
template <typename Point, typename F_is_obs>static bool BresenhamCollisionDetection(const Point& pt1, const Point& pt2, F_is_obs func_is_obs){ int s_x = (pt1.x() - pt2.x() == 0) ? 0 : (pt1.x() - pt2.x()) / std::abs(pt1.x() - pt2.x()); int s_y = (pt1.y() - pt2.y() == 0) ? 0 : (pt1.y() - pt2.y()) / std::abs(pt1.y() - pt2.y()); int d_x = std::abs(pt1.x() - pt2.x()); int d_y = std::abs(pt1.y() - pt2.y()); // check if any obstacle exists between pt1 and pt2 if (d_x > d_y) { int tau = d_y - d_x; int x = pt2.x(), y = pt2.y(); int e = 0; while (x != pt1.x()) { if (e * 2 > tau) { x += s_x; e -= d_y; } else if (e * 2 < tau) { y += s_y; e += d_x; } else { x += s_x; y += s_y; e += d_x - d_y; } if (func_is_obs(Point(x, y))) // obstacle detected return true; } } else { // similar. swap x and y int tau = d_x - d_y; int x = pt2.x(), y = pt2.y(); int e = 0; while (y != pt1.y()) { if (e * 2 > tau) { y += s_y; e -= d_x; } else if (e * 2 < tau) { x += s_x; e += d_y; } else { x += s_x; y += s_y; e += d_y - d_x; } if (func_is_obs(Point(x, y))) // obstacle detected return true; } } return false;}4.2 Python实现
核心代码如下所示
def BresenhamCollisionDetection(self, node1: Node, node2: Node) -> bool:if node1.current in self.obstacles or node2.current in self.obstacles: return Falsex1, y1 = node1.currentx2, y2 = node2.currentif x1 < 0 or x1 >= self.env.x_range or y1 < 0 or y1 >= self.env.y_range: return Falseif x2 < 0 or x2 >= self.env.x_range or y2 < 0 or y2 >= self.env.y_range: return Falsed_x = abs(x2 - x1)d_y = abs(y2 - y1)s_x = 0 if (x2 - x1) == 0 else (x2 - x1) / d_xs_y = 0 if (y2 - y1) == 0 else (y2 - y1) / d_yx, y, e = x1, y1, 0# check if any obstacle exists between node1 and node2if d_x > d_y: tau = (d_y - d_x) / 2 while not x == x2: if e > tau: x = x + s_x e = e - d_y elif e < tau: y = y + s_y e = e + d_x else: x = x + s_x y = y + s_y e = e + d_x - d_y if (x, y) in self.obstacles: return False# swap x and yelse: tau = (d_x - d_y) / 2 while not y == y2: if e > tau: y = y + s_y e = e - d_x elif e < tau: x = x + s_x e = e + d_y else: x = x + s_x y = y + s_y e = e + d_y - d_x if (x, y) in self.obstacles: return Falsereturn True4.3 Matlab实现
核心代码如下所示
function flag = BresenhamCollisionDetection(map, node1, node2)% @breif: Judge collision when moving from node1 to node2 using Bresenham. if (map(node1(1), node1(2)) == 2) || (map(node2(1), node2(2)) == 2) flag = true; return end x1 = node1(1); y1 = node1(2); x2 = node2(1); y2 = node2(2); d_x = abs(x2 - x1); d_y = abs(y2 - y1); if (x2 - x1) == 0 s_x = 0; else s_x = (x2 - x1) / d_x; end if (y2 - y1) == 0 s_y = 0; else s_y = (y2 - y1) / d_y; end x = x1; y = y1; e = 0; % check if any obstacle exists between node1 and node2 if d_x > d_y tao = (d_y - d_x) / 2; while x ~= x2 if e > tao x = x + s_x; e = e - d_y; elseif e < tao y = y + s_y; e = e + d_x; else x = x + s_x; y = y + s_y; e = e + d_x - d_y; end if map(x, y) == 2 flag = true; return; end end % swap x and y else tao = (d_x - d_y) / 2; while y ~= y2 if e > tao y = y + s_y; e = e - d_x; elseif e < tao x = x + s_x; e = e + d_y; else x = x + s_x; y = y + s_y; e = e + d_y - d_x; end if map(x, y) == 2 flag = true; return; end end end flag = false;end完整工程代码请联系下方博主名片获取
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