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fix: certify local G2 curve derivatives

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梁薄云
2026-07-30 17:44:37 +08:00
parent 1662b68062
commit 9edb80e552
1 changed files with 162 additions and 9 deletions
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ClumsyPilot/ParkrobTrajplanner/PathSmoothing/LocalG2/LocalG2CandidateBuilder.cs
+162 -9
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@@ -12,7 +12,9 @@ namespace MultiWheelC.TrajectoryPlanning.PathSmoothing.LocalG2;
internal sealed class LocalG2CandidateBuilder
{
private const double MinimumDerivativeNorm = 1e-10d;
private const int MinimumCurveEvaluationIntervals = 32;
private const int MaximumDerivativeCertificationDepth = 40;
private const int MaximumDerivativeCertificationIntervals = 8192;
private const double DerivativeCertificationMargin = 1e-12d;
private const int MaximumSubdivisionDepth = 32;
private static readonly double[] DerivativeScaleMultipliers = { 1d, 0.85d, 1.15d };
@@ -195,13 +197,9 @@ internal sealed class LocalG2CandidateBuilder
List<SmoothingPoint2D> output,
CancellationToken cancellationToken)
{
// A short endpoint chord does not prove that a fifth-degree curve has no interior cusp.
// Check a deterministic interior grid before the chord-driven subdivision below; every
// subsequently emitted subdivision parameter is checked again by TryEvaluate.
for (int index = 0; index <= MinimumCurveEvaluationIntervals; index++)
{
if (!TryEvaluate(curve, (double)index / MinimumCurveEvaluationIntervals, out _)) return false;
}
// A short endpoint chord and a finite sample grid cannot rule out an interior cusp.
// Certify a derivative-norm lower bound over the whole parameter interval first.
if (!TryCertifyDerivativeLowerBound(curve)) return false;
if (!TryEvaluate(curve, 0d, out CurveSample start) || !TryEvaluate(curve, 1d, out CurveSample end)) return false;
if (output.Count == 0 && !TryAppendPoint(start, left, right, segment, output)) return false;
return TrySubdivide(curve, left, right, segment, outputSpacingMeters, 0d, start, 1d, end, 0, output, cancellationToken);
@@ -278,6 +276,148 @@ internal sealed class LocalG2CandidateBuilder
return true;
}
private static bool TryCertifyDerivativeLowerBound(QuinticHermiteCurve2D curve)
{
if (!TryCreateDerivativeBezierControls(curve, out DerivativeControlPoint d0, out DerivativeControlPoint d1,
out DerivativeControlPoint d2, out DerivativeControlPoint d3, out DerivativeControlPoint d4))
{
return false;
}
int visitedIntervals = 0;
return TryCertifyDerivativeLowerBound(d0, d1, d2, d3, d4, 0, ref visitedIntervals);
}
private static bool TryCreateDerivativeBezierControls(
QuinticHermiteCurve2D curve,
out DerivativeControlPoint d0,
out DerivativeControlPoint d1,
out DerivativeControlPoint d2,
out DerivativeControlPoint d3,
out DerivativeControlPoint d4)
{
d0 = d1 = d2 = d3 = d4 = default;
curve.Evaluate(0d, out _, out _, out double dx0, out double dy0, out double ddx0, out double ddy0);
curve.Evaluate(0.5d, out _, out _, out double dxMiddle, out double dyMiddle, out _, out _);
curve.Evaluate(1d, out _, out _, out double dxEnd, out double dyEnd, out double ddxEnd, out double ddyEnd);
d0 = new DerivativeControlPoint(dx0, dy0);
d1 = new DerivativeControlPoint(dx0 + ddx0 / 4d, dy0 + ddy0 / 4d);
d4 = new DerivativeControlPoint(dxEnd, dyEnd);
d3 = new DerivativeControlPoint(dxEnd - ddxEnd / 4d, dyEnd - ddyEnd / 4d);
d2 = new DerivativeControlPoint(
(16d * dxMiddle - d0.X - 4d * d1.X - 4d * d3.X - d4.X) / 6d,
(16d * dyMiddle - d0.Y - 4d * d1.Y - 4d * d3.Y - d4.Y) / 6d);
return d0.IsFinite && d1.IsFinite && d2.IsFinite && d3.IsFinite && d4.IsFinite;
}
private static bool TryCertifyDerivativeLowerBound(
DerivativeControlPoint d0,
DerivativeControlPoint d1,
DerivativeControlPoint d2,
DerivativeControlPoint d3,
DerivativeControlPoint d4,
int depth,
ref int visitedIntervals)
{
if (++visitedIntervals > MaximumDerivativeCertificationIntervals) return false;
if (!TryGetOriginToConvexHullDistance(d0, d1, d2, d3, d4, out double lowerBound, out double margin))
return false;
if (lowerBound > MinimumDerivativeNorm + margin) return true;
if (depth >= MaximumDerivativeCertificationDepth) return false;
DerivativeControlPoint d01 = Midpoint(d0, d1);
DerivativeControlPoint d12 = Midpoint(d1, d2);
DerivativeControlPoint d23 = Midpoint(d2, d3);
DerivativeControlPoint d34 = Midpoint(d3, d4);
DerivativeControlPoint d012 = Midpoint(d01, d12);
DerivativeControlPoint d123 = Midpoint(d12, d23);
DerivativeControlPoint d234 = Midpoint(d23, d34);
DerivativeControlPoint d0123 = Midpoint(d012, d123);
DerivativeControlPoint d1234 = Midpoint(d123, d234);
DerivativeControlPoint middle = Midpoint(d0123, d1234);
if (!d01.IsFinite || !d12.IsFinite || !d23.IsFinite || !d34.IsFinite || !d012.IsFinite || !d123.IsFinite ||
!d234.IsFinite || !d0123.IsFinite || !d1234.IsFinite || !middle.IsFinite)
{
return false;
}
return TryCertifyDerivativeLowerBound(d0, d01, d012, d0123, middle, depth + 1, ref visitedIntervals) &&
TryCertifyDerivativeLowerBound(middle, d1234, d234, d34, d4, depth + 1, ref visitedIntervals);
}
private static bool TryGetOriginToConvexHullDistance(
DerivativeControlPoint d0,
DerivativeControlPoint d1,
DerivativeControlPoint d2,
DerivativeControlPoint d3,
DerivativeControlPoint d4,
out double distance,
out double margin)
{
distance = 0d;
margin = 0d;
var controls = new[] { d0, d1, d2, d3, d4 };
double maximumCoordinate = 0d;
for (int index = 0; index < controls.Length; index++)
{
if (!controls[index].IsFinite) return false;
maximumCoordinate = Math.Max(maximumCoordinate, Math.Max(Math.Abs(controls[index].X), Math.Abs(controls[index].Y)));
}
if (!NumericGuard.IsFinite(maximumCoordinate)) return false;
margin = DerivativeCertificationMargin * Math.Max(1d, maximumCoordinate);
for (int first = 0; first < controls.Length - 2; first++)
{
for (int second = first + 1; second < controls.Length - 1; second++)
{
for (int third = second + 1; third < controls.Length; third++)
{
if (ContainsOrigin(controls[first], controls[second], controls[third], margin)) return true;
}
}
}
distance = double.PositiveInfinity;
for (int first = 0; first < controls.Length; first++)
{
distance = Math.Min(distance, controls[first].Norm);
for (int second = first + 1; second < controls.Length; second++)
distance = Math.Min(distance, DistanceToSegment(controls[first], controls[second]));
}
return NumericGuard.IsFinite(distance);
}
private static bool ContainsOrigin(DerivativeControlPoint first, DerivativeControlPoint second,
DerivativeControlPoint third, double margin)
{
double crossFirstSecond = Cross(first, second);
double crossSecondThird = Cross(second, third);
double crossThirdFirst = Cross(third, first);
if (!NumericGuard.IsFinite(crossFirstSecond) || !NumericGuard.IsFinite(crossSecondThird) || !NumericGuard.IsFinite(crossThirdFirst))
return true;
double areaMargin = margin * Math.Max(1d, Math.Max(first.Norm, Math.Max(second.Norm, third.Norm)));
return (crossFirstSecond >= -areaMargin && crossSecondThird >= -areaMargin && crossThirdFirst >= -areaMargin) ||
(crossFirstSecond <= areaMargin && crossSecondThird <= areaMargin && crossThirdFirst <= areaMargin);
}
private static double DistanceToSegment(DerivativeControlPoint start, DerivativeControlPoint end)
{
double dx = end.X - start.X;
double dy = end.Y - start.Y;
double denominator = dx * dx + dy * dy;
if (!NumericGuard.IsFinite(denominator)) return double.NaN;
if (denominator == 0d) return start.Norm;
double parameter = -(start.X * dx + start.Y * dy) / denominator;
if (!NumericGuard.IsFinite(parameter)) return double.NaN;
parameter = Math.Max(0d, Math.Min(1d, parameter));
double x = start.X + parameter * dx;
double y = start.Y + parameter * dy;
return Math.Sqrt(x * x + y * y);
}
private static DerivativeControlPoint Midpoint(DerivativeControlPoint left, DerivativeControlPoint right) =>
new DerivativeControlPoint((left.X + right.X) / 2d, (left.Y + right.Y) / 2d);
private static double Cross(DerivativeControlPoint left, DerivativeControlPoint right) => left.X * right.Y - left.Y * right.X;
private static bool IsValidInput(PreparedDirectionSegment segment, LocalG2SmoothingRegion region,
double outputSpacingMeters, LocalG2OptionsSnapshot options)
{
@@ -356,6 +496,19 @@ internal sealed class LocalG2CandidateBuilder
internal double Dy { get; }
}
private readonly struct DerivativeControlPoint
{
internal DerivativeControlPoint(double x, double y)
{
X = x;
Y = y;
}
internal double X { get; }
internal double Y { get; }
internal bool IsFinite => NumericGuard.IsFinite(X) && NumericGuard.IsFinite(Y);
internal double Norm => Math.Sqrt(X * X + Y * Y);
}
/// <summary>反射脚本使用的窄范围候选与拼接场景入口。</summary>
public static class TestHooks
{
@@ -537,7 +690,7 @@ internal sealed class LocalG2CandidateBuilder
private static CandidateTestSnapshot BuildInteriorStationaryCurve()
{
const double endpointX = 7d / 1500d;
const double endpointX = 0.00325d;
if (!QuinticHermiteCurve2D.TryCreate(0d, 0d, 0.01d, 0d, 0d, 0d,
endpointX, 0d, 0.01d, 0d, 0d, 0d, out QuinticHermiteCurve2D curve, out string reason))
{