🎯 基本定义对比

📐 几何与控制能力

Bézier：

• 控制点越多，曲线阶数越高，变得更难控制
• 高阶 Bézier 灵活但非常敏感 ❌
• 分段 Bézier 可局部构造，但连接点的连续性需手动调优（如重复点/对称配置）

B-spline：

• 控制点多少 ≠ 曲线阶数，稳定性更强
• 每段曲线只由 degree + 1 个控制点控制，局部调整容易
• 调节某点时只会影响临近几段 ✅
• 支持非均匀 knot，灵活调整形状与速度剖面

🔄 动态调节与可视化上的优势

🛠️ 实际轨迹规划中常见用途

💡 动画对比

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Circle
from matplotlib.lines import Line2D
from scipy.interpolate import BSpline

# Bézier 曲线（de Casteljau）
def bezier_curve(points, n=300):
    t = np.linspace(0, 1, n)
    n_points = len(points)
    curve = np.zeros((n, 2))
    from math import comb
    for i in range(n_points):
        coeff = comb(n_points - 1, i) * (1 - t)**(n_points - 1 - i) * t**i
        curve += np.outer(coeff, points[i])
    return curve

# B样条构造
def bspline_curve(ctrl_pts, degree=3, n=200):
    n_ctrl = len(ctrl_pts)
    n_knots = n_ctrl + degree + 1
    knots = np.concatenate((
        np.zeros(degree),
        np.linspace(0, 1, n_knots - 2 * degree),
        np.ones(degree)
    ))
    t = np.linspace(knots[degree], knots[-degree - 1], n)
    bx = BSpline(knots, ctrl_pts[:, 0], degree)
    by = BSpline(knots, ctrl_pts[:, 1], degree)
    return np.vstack((bx(t), by(t))).T, knots

# 初始化控制点
ctrl_pts = np.array([
    [0, 0],
    [1.5, 2],
    [3, -1],
    [4.5, 2],
    [6, 0],
    [7, 2],
    [8, 0],
    [9, -1],
])

degree = 3
n_segs = len(ctrl_pts) - degree

# 可视化设置
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))
fig.canvas.manager.set_window_title(" Bézier vs B-Spline - Control Point Locality")
for ax in [ax1, ax2]:
    ax.set_xlim(-1, 10)
    ax.set_ylim(-3, 4)
    ax.set_aspect("equal")
    ax.grid(True)

ax1.set_title("Bézier Curve (Global Influence)
ax2.set_title("B-Spline Curve (Local Influence)")

# Bézier 曲线 + 控制点
bezier_line, = ax1.plot([], [], 'b-', lw=2)
bezier_ctrl_line, = ax1.plot([], [], 'gray', ls='--', lw=1)
bezier_circles = []

# B样条曲线段
bspline_segments = []
bspline_ctrl_line, = ax2.plot([], [], 'gray', ls='--', lw=1)
bspline_circles = []

# 控制点圆圈 + 拖动
class Draggable:
    def __init__(self, artist, index):
        self.artist = artist
        self.index = index
        self.press = None
        self.cid_press = artist.figure.canvas.mpl_connect("button_press_event", self.on_press)
        self.cid_release = artist.figure.canvas.mpl_connect("button_release_event", self.on_release)
        self.cid_motion = artist.figure.canvas.mpl_connect("motion_notify_event", self.on_motion)

    def on_press(self, event):
        if event.inaxes != self.artist.axes: return
        contains, _ = self.artist.contains(event)
        if contains:
            self.press = (event.xdata, event.ydata)
            highlight_control_point(self.index)
            highlight_bspline_segments(self.index)

    def on_motion(self, event):
        if self.press is None or event.inaxes != self.artist.axes: return
        dx = event.xdata - self.press[0]
        dy = event.ydata - self.press[1]
        ctrl_pts[self.index] += np.array([dx, dy])
        self.press = (event.xdata, event.ydata)
        update_all()

    def on_release(self, event):
        self.press = None
        highlight_control_point(-1)
        highlight_bspline_segments(-1)

def update_all():
    # 更新 Bézier
    bez = bezier_curve(ctrl_pts)
    bezier_line.set_data(bez[:, 0], bez[:, 1])
    bezier_ctrl_line.set_data(ctrl_pts[:, 0], ctrl_pts[:, 1])
    for i, c in enumerate(bezier_circles):
        c.center = ctrl_pts[i]

    # 更新 B-Spline
    spline_pts, _ = bspline_curve(ctrl_pts, degree)
    for i, seg in enumerate(bspline_segments):
        t0, t1 = i / n_segs, (i + 1) / n_segs
        t = np.linspace(t0, t1, 50)
        bx = BSpline(knots, ctrl_pts[:, 0], degree)
        by = BSpline(knots, ctrl_pts[:, 1], degree)
        seg.set_data(bx(t), by(t))

    bspline_ctrl_line.set_data(ctrl_pts[:, 0], ctrl_pts[:, 1])
    for i, c in enumerate(bspline_circles):
        c.center = ctrl_pts[i]

    fig.canvas.draw_idle()

def highlight_control_point(idx):
    for i, c in enumerate(bezier_circles + bspline_circles):
        if i == idx:
            c.set_radius(0.15)
            c.set_color('lime')
            c.set_zorder(10)
        else:
            c.set_radius(0.1)
            c.set_color('r' if i < len(bezier_circles) else 'orange')
            c.set_zorder(1)

def highlight_bspline_segments(ctrl_idx):
    for i, seg in enumerate(bspline_segments):
        if i <= ctrl_idx <= i + degree:
            seg.set_color('orange')
            seg.set_alpha(1.0)
            seg.set_linewidth(3)
        else:
            seg.set_color('gray')
            seg.set_alpha(0.4)
            seg.set_linewidth(2)

# 添加 Bézier 控制点图形
draggers = []
for i, pt in enumerate(ctrl_pts):
    c1 = Circle(pt, 0.1, color='r', alpha=0.8)
    c2 = Circle(pt, 0.1, color='orange', alpha=0.8)
    ax1.add_patch(c1)
    ax2.add_patch(c2)
    bezier_circles.append(c1)
    bspline_circles.append(c2)
    draggers.append(Draggable(c1, i))
    draggers.append(Draggable(c2, i))

# 添加 Bézier 控制点编号
for i, pt in enumerate(ctrl_pts):
    ax1.text(pt[0], pt[1]+0.25, f"P{i}", fontsize=9, ha='center', color='black')
    ax2.text(pt[0], pt[1]+0.25, f"P{i}", fontsize=9, ha='center', color='black')

# 初始化 B-Spline 段
_, knots = bspline_curve(ctrl_pts, degree)
for i in range(n_segs):
    line, = ax2.plot([], [], lw=2, alpha=0.5)
    bspline_segments.append(line)

update_all()
plt.tight_layout()
plt.show()

⭐总结

所以总的来说在机器人轨迹规划中，B样条因其局部可调、平滑连续且易于约束控制，更适合动态环境下的高效、安全路径生成