diff --git a/A/4/kinematics_check.py b/A/4/kinematics_check.py
index 27a03bd..3f61a25 100644
--- a/A/4/kinematics_check.py
+++ b/A/4/kinematics_check.py
@@ -1,2 +1,201 @@
-import loong
-loong.kPointsConsidered = 2 # Just Check the first 2 blocks to see whether it will be stuck
\ No newline at end of file
+from loong import *
+import json
+import numpy as np
+import sys
+import matplotlib.pyplot as plt
+import io
+from PIL import Image
+from matplotlib.patches import Rectangle
+import multiprocessing
+
+
+class GoodOrbit(Orbit):
+
+  def __init__(self):
+    self.kAlpha = mp.mpf("1.7") / (2 * mp.pi)
+    self.kCriticalTheta = 2.86 / ((2 / 3) * self.kAlpha) - 0.1
+    self.r = (1 / 3) * self.kAlpha * mp.sqrt(1 + self.kCriticalTheta**2)
+    self.point_A_cartesian = (
+        self.kAlpha * self.kCriticalTheta * mp.cos(self.kCriticalTheta),
+        self.kAlpha * self.kCriticalTheta * mp.sin(self.kCriticalTheta),
+    )
+    self.point_B_cartesian = (-self.kAlpha * self.kCriticalTheta * mp.cos(self.kCriticalTheta),
+                              -self.kAlpha * self.kCriticalTheta * mp.sin(self.kCriticalTheta))
+    self.kPhi = mp.atan(self.kCriticalTheta)
+    print(f"Phi={self.kPhi}", file=sys.stderr)
+    dx, dy = self.point_A_cartesian[0] - self.point_B_cartesian[0], self.point_A_cartesian[1] - self.point_B_cartesian[1]
+    self.angle = mp.atan2(dy, dx)
+    print(f"angle={self.angle}", file=sys.stderr)
+    self.point_C1_cartesian = (self.point_A_cartesian[0] + 2 * self.r * mp.cos(self.angle + 0.5 * mp.pi + self.kPhi),
+                               self.point_A_cartesian[1] + 2 * self.r * mp.sin(self.angle + 0.5 * mp.pi + self.kPhi))
+    self.point_C2_cartesian = (self.point_B_cartesian[0] + 1 * self.r * mp.cos(self.angle - 0.5 * mp.pi + self.kPhi),
+                               self.point_B_cartesian[1] + 1 * self.r * mp.sin(self.angle - 0.5 * mp.pi + self.kPhi))
+    self.radius_of_C1 = 2 * self.r
+    self.radius_of_C2 = 1 * self.r
+    self.arclength = 6 * self.r * self.kPhi
+    self.edge_k = self.kAlpha * mp.sqrt(1 + self.kCriticalTheta * self.kCriticalTheta)
+    self.n = -1
+    for i in range(3, 20, 2):
+      self.a = (self.arclength - 2 * self.edge_k * self.kCriticalTheta) / (2 * (1 - i) * self.kCriticalTheta**i)
+      self.b = (i * self.arclength - 2 * self.edge_k * self.kCriticalTheta) / (2 * (i - 1) * self.kCriticalTheta)
+      if self.a > 0 and self.b > 0:
+        self.n = i
+        break
+    print(f"arclength={self.arclength}", file=sys.stderr)
+    print(f"edge_k={self.edge_k}", file=sys.stderr)
+    print(f"a={self.a}", file=sys.stderr)
+    print(f"b={self.b}", file=sys.stderr)
+    print(f"n={self.n}", file=sys.stderr)
+    print(f"now k={self.n*self.a*self.kCriticalTheta**(self.n-1)+self.b}", file=sys.stderr)
+    if self.n == -1:
+      raise Exception("n must be set")
+    self.edge_raw_C = self.kAlpha * 0.5 * (
+        self.kCriticalTheta * mp.sqrt(1 + self.kCriticalTheta * self.kCriticalTheta) -
+        mp.log(-self.kCriticalTheta + mp.sqrt(1 + self.kCriticalTheta * self.kCriticalTheta)))
+
+  def InitIdx(self):
+    return mp.mpf("0.0")
+
+  def InitC(self):
+    return mp.mpf("0.0")
+
+  def Idx2C(self, idx):  # this function must be monotonically increasing
+    if idx >= 0:
+      theta = idx + self.kCriticalTheta
+      tmp = mp.sqrt(1 + theta * theta)
+      return self.kAlpha * 0.5 * (theta * tmp - mp.log(-theta + tmp)) - self.edge_raw_C
+    elif idx >= -2 * self.kCriticalTheta:
+      x = idx + self.kCriticalTheta
+      y = (self.a * (x**self.n) + self.b * x) - 0.5 * self.arclength
+      return y
+    else:
+      theta = -idx - self.kCriticalTheta
+      tmp = mp.sqrt(1 + theta * theta)
+      return -self.kAlpha * 0.5 * (theta * tmp - mp.log(-theta + tmp)) + self.edge_raw_C - self.arclength
+
+  def Idx2Cartesian(self, idx):
+    if idx >= 0:
+      theta = idx + self.kCriticalTheta
+      return [self.kAlpha * theta * mp.cos(theta), self.kAlpha * theta * mp.sin(theta)]
+    elif idx >= -2 * self.kCriticalTheta:
+      c = self.Idx2C(idx) + self.arclength
+      # if c < 0 or c > self.arclength:
+      # raise Exception(f"idx={idx}, c={c}")
+      if c <= self.arclength / 3:
+        # In C2
+        delta_angle = c / self.radius_of_C2
+        actual_angle = self.angle + 0.5 * mp.pi + self.kPhi - delta_angle
+        return [
+            self.point_C2_cartesian[0] + self.radius_of_C2 * mp.cos(actual_angle),
+            self.point_C2_cartesian[1] + self.radius_of_C2 * mp.sin(actual_angle)
+        ]
+      else:
+        delta_angle = (c - self.arclength / 3) / self.radius_of_C1
+        actual_angle = self.angle - 0.5 * mp.pi - self.kPhi + delta_angle
+        return [
+            self.point_C1_cartesian[0] + self.radius_of_C1 * mp.cos(actual_angle),
+            self.point_C1_cartesian[1] + self.radius_of_C1 * mp.sin(actual_angle)
+        ]
+    else:
+      theta = -idx - self.kCriticalTheta
+      return [-self.kAlpha * theta * mp.cos(theta), -self.kAlpha * theta * mp.sin(theta)]
+
+  def C2Idx(self, C):
+
+    def f(idx):
+      return self.Idx2C(idx) - C
+
+    return mp.findroot(f, (-100*2*mp.pi,100*2*mp.pi), solver='bisect')
+
+  def GenerateNextPointIdx(self, cur_point_idx, expected_distance, guess=None):
+    if guess is None:
+      cur_point_C = self.Idx2C(cur_point_idx)
+      guess = self.C2Idx(cur_point_C + expected_distance)
+    cur_point_dot = self.Idx2Cartesian(cur_point_idx)
+
+    def f(idx):
+      test_point_dot = self.Idx2Cartesian(idx)
+      return mp.sqrt((cur_point_dot[0] - test_point_dot[0])**2 +
+                     (cur_point_dot[1] - test_point_dot[1])**2) - expected_distance
+
+    return mp.findroot(f, guess, solver='secant')
+
+  def GenerateImg(self, node_list):
+    fig = plt.figure(figsize=(12, 12))
+
+    # 绘制轨道线
+    idx_list = np.linspace(-12 * 2 * np.pi, 8 * 2 * np.pi, 10000)
+    x = [float(self.Idx2Cartesian(t)[0]) for t in idx_list]
+    y = [float(self.Idx2Cartesian(t)[1]) for t in idx_list]
+    plt.plot(x, y, color='gray', linewidth=0.5)
+
+    # 绘制节点、连接线和木板
+    for i in range(len(node_list) - 1):
+      x1, y1 = [float(coord) for coord in node_list[i]["node"]]
+      x2, y2 = [float(coord) for coord in node_list[i + 1]["node"]]
+
+      # 绘制红色节点
+      plt.plot(x1, y1, 'ro', markersize=3)
+
+      # 绘制蓝色连接线
+      plt.plot([x1, x2], [y1, y2], 'b-', linewidth=0.5)
+
+      # 计算并绘制木板（长方形）
+      dx = x2 - x1
+      dy = y2 - y1
+      length = np.sqrt(dx**2 + dy**2)
+      angle = np.arctan2(dy, dx)
+
+      rect_length = length + 0.55  # 总长度加上两端各延伸的0.275m
+      rect_width = 0.3
+
+      # 计算长方形的中心点
+      center_x = (x1 + x2) / 2
+      center_y = (y1 + y2) / 2
+
+      # 计算长方形的左下角坐标
+      rect_x = center_x - rect_length / 2 * np.cos(angle) + rect_width / 2 * np.sin(angle)
+      rect_y = center_y - rect_length / 2 * np.sin(angle) - rect_width / 2 * np.cos(angle)
+
+      rect = Rectangle((rect_x, rect_y), rect_length, rect_width, angle=angle * 180 / np.pi, fill=False, edgecolor='g')
+      plt.gca().add_patch(rect)
+
+    # 绘制最后一个节点
+    x, y = [float(coord) for coord in node_list[-1]["node"]]
+    plt.plot(x, y, 'ro', markersize=3)
+
+    plt.axis('equal')
+
+    # 创建一个 BytesIO 对象来存储图像
+    buf = io.BytesIO()
+    plt.savefig(buf, format='png')
+    buf.seek(0)
+
+    # 清除当前图形，释放内存
+    plt.close(fig)
+
+    # 返回图像对象
+    return Image.open(buf)
+
+
+if __name__ == "__main__":
+  orbit = GoodOrbit()
+  loong = Loong(orbit, 224, mp.mpf("2.0"), mp.mpf("1e-8"))
+  res_list = []
+  for ti in np.arange(5, 10, 0.025):
+    print(f"calculating time_point={ti}", file=sys.stderr)
+    res_list.append(loong.CalcStatusListByTime(mp.mpf(ti), res_list[-1] if res_list else None))
+    # 转换成内置浮点数并保留6位
+  float_res_list = [[{
+      "idx": round(float(node["idx"]), 6),
+      "node": [
+          round(float(node["node"][0]), 6),
+          round(float(node["node"][1]), 6),
+      ],
+      "C": round(float(node["C"]), 6),
+      "v": round(float(node["v"]), 6),
+  } for node in res] for res in res_list]
+  with open("A4_res.json", "w") as file:
+    json.dump(float_res_list, file, indent=4)
+  img_list = [orbit.GenerateImg(res) for res in res_list]
+  img_list[0].save("A4.gif", save_all=True, append_images=img_list[1:], duration=25, loop=0)
diff --git a/A/4/loong.py b/A/4/loong.py
index 61e7def..9b540c6 100644
--- a/A/4/loong.py
+++ b/A/4/loong.py
@@ -16,7 +16,7 @@ class Orbit:
     raise NotImplementedError
   def C2Idx(self, C:[mp.mpf, mp.mpf]) -> mp.mpf:
     raise NotImplementedError
-  def GenerateNextPointIdx(self, cur_point_idx:mp.mpf, expected_distance:mp.mpf)->mp.mpf:
+  def GenerateNextPointIdx(self, cur_point_idx:mp.mpf, expected_distance:mp.mpf, guess=None)->mp.mpf:
     raise NotImplementedError
 
 class Loong:
@@ -27,7 +27,7 @@ class Loong:
     self.delta_idx = delta_idx
     self.kSegLength1 = mp.mpf('2.86')
     self.kSegLength2 = mp.mpf('1.65')
-  def CalcStatusListByIdx(self, cur_idx:mp.mpf):
+  def CalcStatusListByIdx(self, cur_idx:mp.mpf, last_time_status=None):
     first_node_idx=cur_idx
     first_node_C=self.orbit.Idx2C(first_node_idx)
     first_node_dot = self.orbit.Idx2Cartesian(first_node_idx)
@@ -37,18 +37,18 @@ class Loong:
     node_list = [{"idx": first_node_idx, "node": first_node_dot, "C": first_node_C, "v": self.speed}]
     for i in range(1, self.total_points):
       expected_distance = self.kSegLength1 if i == 1 else self.kSegLength2
-      cur_node_idx = self.orbit.GenerateNextPointIdx(node_list[-1]["idx"], expected_distance)
+      cur_node_idx = self.orbit.GenerateNextPointIdx(node_list[-1]["idx"], expected_distance, guess=last_time_status[i]["idx"] if last_time_status else None)
       cur_node_dot = self.orbit.Idx2Cartesian(cur_node_idx)
       cur_node_C = self.orbit.Idx2C(cur_node_idx)
 
-      virtual_cur_node_idx = self.orbit.GenerateNextPointIdx(virtual_first_node_idx, expected_distance)
+      virtual_cur_node_idx = self.orbit.GenerateNextPointIdx(virtual_first_node_idx, expected_distance, guess=last_time_status[i]["idx"] if last_time_status else None)
       virtual_cur_node_C = self.orbit.Idx2C(virtual_cur_node_idx)
       v = (virtual_cur_node_C - cur_node_C) / delta_T
 
       node_list.append({"idx": cur_node_idx, "node": cur_node_dot, "C": cur_node_C, "v": v})
       virtual_first_node_idx = virtual_cur_node_idx
     return node_list
-  def CalcStatusListByTime(self, time_point:mp.mpf):
+  def CalcStatusListByTime(self, time_point:mp.mpf, last_time_status=None):
     first_node_C = self.orbit.InitC() - time_point * self.speed
     first_node_idx = self.orbit.C2Idx(first_node_C)
-    return self.CalcStatusListByIdx(first_node_idx)
\ No newline at end of file
+    return self.CalcStatusListByIdx(first_node_idx, last_time_status)
\ No newline at end of file