Merge pull request #145 from f1tenth/dev-dynamics-frenet-fix

Fix Cartesian to Frenet Conversion by migrating to ND Cubic Spline

Author: AhmadAmine998

f1tenth_gym/envs/track/cubic_spline.py

@@ -27,16 +27,98 @@ class CubicSpline2D:
         cubic spline for y coordinates.
     """
 
-    def __init__(self, x, y):
-        self.points = np.c_[x, y]
-        if not np.all(self.points[-1] == self.points[0]):
+    def __init__(self, x, y,
+        psis: Optional[np.ndarray] = None,
+        ks: Optional[np.ndarray] = None,
+        vxs: Optional[np.ndarray] = None,
+        axs: Optional[np.ndarray] = None,
+        ss: Optional[np.ndarray] = None,
+    ):
+        self.xs = x
+        self.ys = y
+        input_vals = [x, y, psis, ks, vxs, axs, ss]
+
+        # Only close the path if for the input values from the user,
+        # the first and last points are not the same => the path is not closed
+        # Otherwise, the constructed values can mess up the s calculation and closure
+        need_closure = False
+        for input_val in input_vals:
+            if input_val is not None:
+                if not (input_val[-1] == input_val[0]):
+                    need_closure = True
+                    break
+
+        def close_with_constructor(input_val, constructor, closed_path):
+            '''
+            If the input value is not None, return it.
+            Otherwise, return the constructor, with closure if necessary.
+
+            Parameters
+            ----------
+            input_val : np.ndarray | None
+                The input value from the user.
+            constructor : np.ndarray
+                The constructor to use if the input value is None.
+            closed_path : bool
+                Indicator whether the orirignal path is closed.
+            '''
+            if input_val is not None:
+                return input_val 
+            else:
+                temp_ret = constructor
+                if closed_path:
+                   temp_ret[-1] = temp_ret[0]
+                return temp_ret
+            
+        self.psis = close_with_constructor(psis, self._calc_yaw_from_xy(x, y), not need_closure)
+        self.ks = close_with_constructor(ks, self._calc_kappa_from_xy(x, y), not need_closure)
+        self.vxs = close_with_constructor(vxs, np.ones_like(x), not need_closure)
+        self.axs = close_with_constructor(axs, np.zeros_like(x), not need_closure)
+        self.ss = close_with_constructor(ss, self.__calc_s(x, y), not need_closure)
+        psis_spline = close_with_constructor(psis, self._calc_yaw_from_xy(x, y), not need_closure)
+
+        # If yaw is provided, interpolate cosines and sines of yaw for continuity
+        cosines_spline = np.cos(psis_spline)
+        sines_spline = np.sin(psis_spline)
+        
+        ks_spline = close_with_constructor(ks, self._calc_kappa_from_xy(x, y), not need_closure)
+        vxs_spline = close_with_constructor(vxs, np.zeros_like(x), not need_closure)
+        axs_spline = close_with_constructor(axs, np.zeros_like(x), not need_closure)
+
+        self.points = np.c_[self.xs, self.ys, 
+                            cosines_spline, sines_spline, 
+                            ks_spline, vxs_spline, axs_spline]
+        
+        if need_closure:
             self.points = np.vstack(
                 (self.points, self.points[0])
             )  # Ensure the path is closed
-        self.s = self.__calc_s(self.points[:, 0], self.points[:, 1])
+
+        if ss is not None:
+            self.s = ss
+        else:
+            self.s = self.__calc_s(self.points[:, 0], self.points[:, 1])
+        self.s_interval = (self.s[-1] - self.s[0]) / len(self.s)
+
         # Use scipy CubicSpline to interpolate the points with periodic boundary conditions
-        # This is necessary to ensure the path is continuous
+        # This is necesaxsry to ensure the path is continuous
         self.spline = interpolate.CubicSpline(self.s, self.points, bc_type="periodic")
+        self.spline_x = np.array(self.spline.x) 
+        self.spline_c = np.array(self.spline.c)
+
+
+    def find_segment_for_s(self, x):
+        # Find the segment of the spline that x is in
+        return (x / (self.spline.x[-1] + self.s_interval) * (len(self.spline_x) - 1)).astype(int)
+    
+    def predict_with_spline(self, point, segment, state_index=0):
+        # A (4, 100) array, where the rows contain (x-x[i])**3, (x-x[i])**2 etc.
+        # exp_x = (point - self.spline.x[[segment]])[None, :] ** np.arange(4)[::-1, None]
+        exp_x = ((point - self.spline.x[segment % len(self.spline.x)]) ** np.arange(4)[::-1])[:, None]
+        vec = self.spline.c[:, segment % self.spline.c.shape[1], state_index]
+        # Sum over the rows of exp_x weighted by coefficients in the ith column of s.c
+        point = vec.dot(exp_x)
+        return np.asarray(point)
 
     def __calc_s(self, x: np.ndarray, y: np.ndarray) -> np.ndarray:
         """
@@ -60,6 +142,24 @@ def __calc_s(self, x: np.ndarray, y: np.ndarray) -> np.ndarray:
         s = [0]
         s.extend(np.cumsum(self.ds))
         return np.array(s)
+    
+    def _calc_yaw_from_xy(self, x, y):
+        dx_dt = np.gradient(x, edge_order=2)
+        dy_dt = np.gradient(y, edge_order=2)
+        heading = np.arctan2(dy_dt, dx_dt)
+        return heading
+
+    def _calc_kappa_from_xy(self, x, y):
+        # For more stable gradients, extend x and y by two (edge_order 2) elements on each side
+        # The elements are taken from the other side of the array
+        x_extended = np.concatenate((x[-2:], x, x[:2]))
+        y_extended = np.concatenate((y[-2:], y, y[:2]))
+        dx_dt = np.gradient(x_extended, edge_order=2)
+        dy_dt = np.gradient(y_extended, edge_order=2)
+        d2x_dt2 = np.gradient(dx_dt, edge_order=2)
+        d2y_dt2 = np.gradient(dy_dt, edge_order=2)
+        curvature = (dx_dt * d2y_dt2 - d2x_dt2 * dy_dt) / (dx_dt * dx_dt + dy_dt * dy_dt)**1.5
+        return curvature[2:-2]
 
     def calc_position(self, s: float) -> np.ndarray:
         """
@@ -78,7 +178,10 @@ def calc_position(self, s: float) -> np.ndarray:
         y : float | None
             y position for given s.
         """
-        return self.spline(s)
+        segment = self.find_segment_for_s(s)
+        x = self.predict_with_spline(s, segment, 0)[0]
+        y = self.predict_with_spline(s, segment, 1)[0]
+        return x,y
 
     def calc_curvature(self, s: float) -> Optional[float]:
         """
@@ -95,11 +198,29 @@ def calc_curvature(self, s: float) -> Optional[float]:
         k : float
             curvature for given s.
         """
-        dx, dy = self.spline(s, 1)
-        ddx, ddy = self.spline(s, 2)
-        k = (ddy * dx - ddx * dy) / ((dx**2 + dy**2) ** (3 / 2))
+        segment = self.find_segment_for_s(s)
+        k = self.predict_with_spline(s, segment, 4)[0]
         return k
 
+    def find_curvature(self, s: float) -> Optional[float]:
+        """
+        Find curvature at the given s by the segment.
+
+        Parameters
+        ----------
+        s : float
+            distance from the start point. if `s` is outside the data point's
+            range, return None.
+
+        Returns
+        -------
+        k : float
+            curvature for given s.
+        """
+        segment = self.find_segment_for_s(s)
+        k = self.points[segment, 4]
+        return k
+        
     def calc_yaw(self, s: float) -> Optional[float]:
         """
         Calc yaw angle at the given s.
@@ -114,11 +235,10 @@ def calc_yaw(self, s: float) -> Optional[float]:
         yaw : float
             yaw angle (tangent vector) for given s.
         """
-        dx, dy = self.spline(s, 1)
-        yaw = math.atan2(dy, dx)
-        # Convert yaw to [0, 2pi]
-        yaw = yaw % (2 * math.pi)
-
+        segment = self.find_segment_for_s(s)
+        cos = self.predict_with_spline(s, segment, 2)[0]
+        sin = self.predict_with_spline(s, segment, 3)[0]
+        yaw = (math.atan2(sin, cos) + 2 * math.pi) % (2 * math.pi)
         return yaw
 
     def calc_arclength(
@@ -145,15 +265,15 @@ def calc_arclength(
         """
 
         def distance_to_spline(s):
-            x_eval, y_eval = self.spline(s)[0]
+            x_eval, y_eval = self.spline(s)[0, :2]
             return np.sqrt((x - x_eval) ** 2 + (y - y_eval) ** 2)
 
         output = so.fmin(distance_to_spline, s_guess, full_output=True, disp=False)
         closest_s = float(output[0][0])
         absolute_distance = output[1]
         return closest_s, absolute_distance
 
-    def calc_arclength_inaccurate(self, x: float, y: float) -> tuple[float, float]:
+    def calc_arclength_inaccurate(self, x: float, y: float, s_inds=None) -> tuple[float, float]:
         """
         Fast calculation of arclength for a given point (x, y) on the trajectory.
         Less accuarate and less smooth than calc_arclength but much faster.
@@ -173,15 +293,16 @@ def calc_arclength_inaccurate(self, x: float, y: float) -> tuple[float, float]:
         ey : float
             lateral deviation for given x, y.
         """
+        if s_inds is None:
+            s_inds = np.arange(self.points.shape[0])
         _, ey, t, min_dist_segment = nearest_point_on_trajectory(
-            np.array([x, y]), self.points
+            np.array([x, y]).astype(np.float32), self.points[s_inds, :2]
         )
-        # s = s at closest_point + t
+        min_dist_segment_s_ind = s_inds[min_dist_segment]
         s = float(
-            self.s[min_dist_segment]
-            + t * (self.s[min_dist_segment + 1] - self.s[min_dist_segment])
+            self.s[min_dist_segment_s_ind]
+            + t * (self.s[min_dist_segment_s_ind + 1] - self.s[min_dist_segment_s_ind])
         )
-
         return s, ey
 
     def _calc_tangent(self, s: float) -> np.ndarray:
@@ -199,7 +320,7 @@ def _calc_tangent(self, s: float) -> np.ndarray:
         tangent : float
             tangent vector for given s.
         """
-        dx, dy = self.spline(s, 1)
+        dx, dy = self.spline(s, 1)[:2]
         tangent = np.array([dx, dy])
         return tangent
 
@@ -218,6 +339,6 @@ def _calc_normal(self, s: float) -> np.ndarray:
         normal : float
             normal vector for given s.
         """
-        dx, dy = self.spline(s, 1)
+        dx, dy = self.spline(s, 1)[:2]
         normal = np.array([-dy, dx])
         return normal

f1tenth_gym/envs/track/track.py

@@ -355,20 +355,19 @@ def cartesian_to_frenet(self, x, y, phi, use_raceline=False, s_guess=0):
             ephi: heading deviation
         """
         line = self.raceline if use_raceline else self.centerline
-        s, ey = line.spline.calc_arclength_inaccurate(x, y)
+        # s, ey = line.spline.calc_arclength_inaccurate(x, y) # inaccurate, but much faster
+        s, ey = line.spline.calc_arclength(x, y, s_guess)
         # Wrap around
         s = s % line.spline.s[-1]
 
         # Use the normal to calculate the signed lateral deviation
-        normal = line.spline._calc_normal(s)
+        yaw = line.spline.calc_yaw(s)
+        normal = np.asarray([-np.sin(yaw), np.cos(yaw)])
         x_eval, y_eval = line.spline.calc_position(s)
         dx = x - x_eval
         dy = y - y_eval
         distance_sign = np.sign(np.dot([dx, dy], normal))
         ey = ey * distance_sign
 
-        ephi = phi - line.spline.calc_yaw(s)
-        # ephi is unbouded, so we need to wrap it to [-pi, pi]
-        ephi = (ephi + np.pi) % (2 * np.pi) - np.pi
-
-        return s, ey, ephi
+        phi = phi - yaw
+        return s, ey, np.arctan2(np.sin(phi), np.cos(phi))