feat: seed rolling and exact-stop ST profiles

2026-08-05 18:18:49 +08:00
parent 26bd822ec9
commit 59d13e5783
3 changed files with 526 additions and 74 deletions
@@ -21,11 +21,12 @@ public sealed class LongitudinalObjectiveBuilder
        double accelerationScale = RequirePositive(Math.Max(configuration.MaximumAccelerationMetersPerSecondSquared,
            configuration.MaximumDecelerationMetersPerSecondSquared), nameof(accelerationScale));
        double jerkScale = RequirePositive(configuration.MaximumJerkMetersPerSecondCubed, nameof(jerkScale));
-        double progressScale = input.TerminalPathS > 0d ? input.TerminalPathS : 1d;
+        double progressScale = input.PathUpperBoundS > 0d ? input.PathUpperBoundS : 1d;

        for (int index = 0; index < layout.KnotCount; index++)
        {
-            AddSquaredResidual(hessian, linearCost, layout.U(index), speedLimit.MaximumSpeedAt(iterate.S[index]),
+            double iteratePathS = Math.Max(0d, Math.Min(input.PathUpperBoundS, iterate.S[index]));
+            AddSquaredResidual(hessian, linearCost, layout.U(index), speedLimit.MaximumSpeedAt(iteratePathS),
                weights.ReferenceSpeed, speedScale);
            AddSquaredResidual(hessian, linearCost, layout.A(index), 0d, weights.Acceleration, accelerationScale);
            if (index < layout.KnotCount - 1 && index < input.PreviousPathS.Count)
@@ -38,8 +39,11 @@ public sealed class LongitudinalObjectiveBuilder
        }
        for (int index = 0; index < layout.KnotCount - 1; index++)
            AddSquaredResidual(hessian, linearCost, layout.J(index), 0d, weights.Jerk, jerkScale);
-        AddSquaredResidual(hessian, linearCost, layout.A(layout.KnotCount - 1), 0d, weights.TerminalAcceleration,
-            accelerationScale);
+        if (input.Mode != EmLongitudinalMode.ExactStopAtBoundary)
+        {
+            AddSquaredResidual(hessian, linearCost, layout.A(layout.KnotCount - 1), 0d,
+                weights.TerminalAcceleration, accelerationScale);
+        }
    }

    private static void AddSquaredResidual(SparseTripletBuilder hessian, IList<double> linearCost, int variable,
@@ -47,7 +47,7 @@ public sealed class SequentialLongitudinalOptimizer
        if (speedStatus != EmPlanningStatus.Success)
            return Failed(speedStatus, speedFailure);

-        LongitudinalCandidate iterate = CreateInitialIterate(input);
+        LongitudinalCandidate iterate = CreateInitialIterate(input, speedLimit);
        double[] warmStart = ToPrimal(iterate);
        bool hasDynamicsConsistentInitialWarmStart = iterate.SatisfiesExactDiscreteDynamics(1e-12d);
        LongitudinalCandidate lastStrictCandidate;
@@ -191,44 +191,111 @@ public sealed class SequentialLongitudinalOptimizer
        }
    }

-    private static LongitudinalCandidate CreateInitialIterate(LongitudinalPlanningInput input)
+    private LongitudinalCandidate CreateInitialIterate(LongitudinalPlanningInput input, PathSpeedLimit speedLimit)
    {
        IReadOnlyList<double> times = LongitudinalCandidate.CreateKnotTimes(input.Configuration.Scheduling.TimeHorizonSeconds,
            input.Configuration.Scheduling.OutputTimeStepSeconds);
-        if (TryCreateCruiseThenBrakeSeed(input, times, out LongitudinalCandidate brakingSeed))
-            return brakingSeed;
-        int knotCount = times.Count;
-        var s = new double[knotCount];
-        var u = new double[knotCount];
-        var a = new double[knotCount];
-        var j = new double[knotCount - 1];
-        double horizon = times[knotCount - 1];
-        double requestedSpeed = Math.Min(input.DirectionMaximumSpeedMetersPerSecond,
-            Math.Max(0d, input.TerminalPathS / horizon));
-        if (!JerkLimitedStoppingMath.TryCalculate(requestedSpeed, 0d,
-                input.Configuration.Longitudinal.MaximumDecelerationMetersPerSecondSquared,
-                input.Configuration.Longitudinal.MaximumJerkMetersPerSecondCubed,
-                out JerkLimitedStoppingProfile terminalStop, out _))
+        switch (input.Mode)
        {
-            throw new ArgumentOutOfRangeException(nameof(input));
+            case EmLongitudinalMode.RollingContinuation:
+                return CreateRollingSeed(input, times, speedLimit);
+            case EmLongitudinalMode.ApproachStopBoundary:
+                return CreateApproachSeed(input, times, speedLimit);
+            case EmLongitudinalMode.ExactStopAtBoundary:
+                return CreateExactStopSeed(input, times, speedLimit);
+            default:
+                throw new ArgumentOutOfRangeException(nameof(input.Mode));
        }
-        double cruiseDistance = Math.Max(0d, input.TerminalPathS - terminalStop.DistanceMeters);
-        for (int index = 0; index < knotCount; index++)
+    }
+
+    private static LongitudinalCandidate CreateRollingSeed(LongitudinalPlanningInput input,
+        IReadOnlyList<double> times, PathSpeedLimit speedLimit)
    {
-            double fraction = (double)index / (knotCount - 1);
-            s[index] = Math.Min(input.TerminalPathS, cruiseDistance * fraction + terminalStop.DistanceMeters * fraction * fraction);
-            u[index] = index == 0 ? input.InitialProgressSpeedMetersPerSecond : requestedSpeed;
-            a[index] = index == 0 ? input.InitialAccelerationMetersPerSecondSquared : 0d;
+        return CreateEnvelopeSeed(input, times, speedLimit);
    }
-        s[0] = 0d;
-        s[knotCount - 1] = input.TerminalPathS;
-        u[knotCount - 1] = 0d;
-        a[knotCount - 1] = 0d;
-        return new LongitudinalCandidate(times, s, u, a, j);
+
+    private static LongitudinalCandidate CreateApproachSeed(LongitudinalPlanningInput input,
+        IReadOnlyList<double> times, PathSpeedLimit speedLimit)
+    {
+        return CreateEnvelopeSeed(input, times, speedLimit);
+    }
+
+    private static LongitudinalCandidate CreateEnvelopeSeed(LongitudinalPlanningInput input,
+        IReadOnlyList<double> times, PathSpeedLimit speedLimit)
+    {
+        LongitudinalConfiguration configuration = input.Configuration.Longitudinal;
+        var jerk = new double[times.Count - 1];
+        double s = 0d;
+        double u = input.InitialProgressSpeedMetersPerSecond;
+        double a = input.InitialAccelerationMetersPerSecondSquared;
+        for (int index = 0; index < jerk.Length; index++)
+        {
+            double dt = times[index + 1] - times[index];
+            double speedLimitAtS = speedLimit.MaximumSpeedAt(Math.Max(0d, Math.Min(input.PathUpperBoundS, s)));
+            double targetSpeed = Math.Min(input.InitialProgressSpeedMetersPerSecond, speedLimitAtS);
+            double lowerJerk = Math.Max(-configuration.MaximumJerkMetersPerSecondCubed,
+                (-configuration.MaximumDecelerationMetersPerSecondSquared - a) / dt);
+            lowerJerk = Math.Max(lowerJerk, -2d * (u + a * dt) / (dt * dt));
+            double upperJerk = Math.Min(configuration.MaximumJerkMetersPerSecondCubed,
+                (configuration.MaximumAccelerationMetersPerSecondSquared - a) / dt);
+            double requestedJerk = 2d * (targetSpeed - u - a * dt) / (dt * dt);
+            double selectedJerk = Clamp(requestedJerk, lowerJerk, upperJerk);
+            IntegrateStep(s, u, a, selectedJerk, dt, out double nextS, out double nextU, out double nextA);
+            if (nextU > speedLimit.MaximumSpeedAt(Math.Max(0d, Math.Min(input.PathUpperBoundS, nextS))) + 1e-12d)
+            {
+                double lower = lowerJerk;
+                double upper = selectedJerk;
+                for (int iteration = 0; iteration < 48; iteration++)
+                {
+                    double midpoint = 0.5d * (lower + upper);
+                    IntegrateStep(s, u, a, midpoint, dt, out double probeS, out double probeU, out _);
+                    if (probeU <= speedLimit.MaximumSpeedAt(Math.Max(0d, Math.Min(input.PathUpperBoundS, probeS))))
+                        lower = midpoint;
+                    else
+                        upper = midpoint;
+                }
+                selectedJerk = lower;
+                IntegrateStep(s, u, a, selectedJerk, dt, out nextS, out nextU, out nextA);
+            }
+            jerk[index] = selectedJerk;
+            s = nextS;
+            u = nextU;
+            a = nextA;
+        }
+        return LongitudinalCandidate.Integrate(times, 0d, input.InitialProgressSpeedMetersPerSecond,
+            input.InitialAccelerationMetersPerSecondSquared, jerk);
+    }
+
+    private LongitudinalCandidate CreateExactStopSeed(LongitudinalPlanningInput input,
+        IReadOnlyList<double> times, PathSpeedLimit speedLimit)
+    {
+        int stabilizationStart = LongitudinalTerminalSchedule.GetStabilizationStartIndex(times,
+            input.Configuration.Scheduling.OutputTimeStepSeconds);
+        var motionTimes = new double[stabilizationStart + 1];
+        for (int index = 0; index < motionTimes.Length; index++)
+            motionTimes[index] = times[index];
+
+        if (TryCreateCruiseThenBrakeSeed(input, motionTimes, input.StopBoundaryPathS,
+                out LongitudinalCandidate cruiseThenBrake))
+        {
+            LongitudinalCandidate candidate = AppendExactStopTail(times, stabilizationStart,
+                input.StopBoundaryPathS, cruiseThenBrake);
+            if (_solutionValidator.TryValidate(input, speedLimit, candidate,
+                    out LongitudinalCandidate validated, out _))
+            {
+                return validated;
+            }
+        }
+
+        if (TryCreateExactJerkSeed(input, times, stabilizationStart, speedLimit,
+                out LongitudinalCandidate exactSeed))
+            return exactSeed;
+
+        return CreateApproachSeed(input, times, speedLimit);
    }

    private static bool TryCreateCruiseThenBrakeSeed(LongitudinalPlanningInput input, IReadOnlyList<double> times,
-        out LongitudinalCandidate candidate)
+        double stopBoundaryPathS, out LongitudinalCandidate candidate)
    {
        candidate = null;
        double initialSpeed = input.InitialProgressSpeedMetersPerSecond;
@@ -260,11 +327,11 @@ public sealed class SequentialLongitudinalOptimizer
                }
                int brakingIntervals = 2 * rampIntervals + plateauIntervals;
                double brakingDistance = 0.5d * initialSpeed * brakingIntervals * timeStep;
-                if (brakingDistance > input.TerminalPathS + 1e-12d)
+                if (brakingDistance > stopBoundaryPathS + 1e-12d)
                    continue;
                int maximumCruiseIntervals = intervalCount - brakingIntervals;
                int cruiseIntervals = Math.Min(maximumCruiseIntervals, Math.Max(0, checked((int)Math.Floor(
-                    (input.TerminalPathS - brakingDistance) / (initialSpeed * timeStep) + 1e-12d))));
+                    (stopBoundaryPathS - brakingDistance) / (initialSpeed * timeStep) + 1e-12d))));
                var jerk = new double[intervalCount];
                int cursor = cruiseIntervals;
                for (int index = 0; index < rampIntervals; index++)
@@ -273,7 +340,9 @@ public sealed class SequentialLongitudinalOptimizer
                for (int index = 0; index < rampIntervals; index++)
                    jerk[cursor++] = jerkMagnitude;
                LongitudinalCandidate integrated = LongitudinalCandidate.Integrate(times, 0d, initialSpeed, 0d, jerk);
-                if (integrated.S[integrated.S.Count - 1] <= input.TerminalPathS + 1e-12d)
+                int lastIndex = integrated.S.Count - 1;
+                if (Math.Abs(integrated.S[lastIndex] - stopBoundaryPathS) <= 1e-10d &&
+                    Math.Abs(integrated.U[lastIndex]) <= 1e-10d && Math.Abs(integrated.A[lastIndex]) <= 1e-10d)
                {
                    candidate = integrated;
                    return true;
@@ -283,6 +352,280 @@ public sealed class SequentialLongitudinalOptimizer
        return false;
    }

+    private bool TryCreateExactJerkSeed(LongitudinalPlanningInput input, IReadOnlyList<double> times,
+        int stabilizationStart, PathSpeedLimit speedLimit, out LongitudinalCandidate candidate)
+    {
+        candidate = null;
+        int intervalCount = stabilizationStart;
+        if (intervalCount < 3)
+            return false;
+
+        var motionTimes = new double[intervalCount + 1];
+        for (int index = 0; index < motionTimes.Length; index++)
+            motionTimes[index] = times[index];
+
+        LongitudinalCandidate baseline = CreateApproachSeed(input, motionTimes, speedLimit);
+        var influence = new double[3, intervalCount];
+        for (int interval = 0; interval < intervalCount; interval++)
+        {
+            var basis = new double[intervalCount];
+            basis[interval] = 1d;
+            LongitudinalCandidate response = LongitudinalCandidate.Integrate(motionTimes, 0d, 0d, 0d, basis);
+            int last = response.S.Count - 1;
+            influence[0, interval] = response.A[last];
+            influence[1, interval] = response.U[last];
+            influence[2, interval] = response.S[last];
+        }
+
+        double[] target =
+        {
+            -baseline.A[baseline.A.Count - 1],
+            -baseline.U[baseline.U.Count - 1],
+            input.StopBoundaryPathS - baseline.S[baseline.S.Count - 1],
+        };
+        var gram = new double[3, 3];
+        for (int row = 0; row < 3; row++)
+        {
+            for (int column = 0; column < 3; column++)
+            {
+                for (int interval = 0; interval < intervalCount; interval++)
+                    gram[row, column] += influence[row, interval] * influence[column, interval];
+            }
+        }
+        if (!TrySolveThreeByThree(gram, target, out double[] multipliers))
+            return false;
+
+        var jerk = new double[intervalCount];
+        for (int interval = 0; interval < intervalCount; interval++)
+        {
+            jerk[interval] = baseline.J[interval];
+            for (int row = 0; row < 3; row++)
+                jerk[interval] += influence[row, interval] * multipliers[row];
+        }
+
+        if (TryValidateExactSeed(input, times, stabilizationStart, speedLimit, jerk, out candidate))
+            return true;
+
+        double currentViolation = CalculateExactSeedViolation(input, speedLimit,
+            CreateExactCandidate(input, times, stabilizationStart, jerk));
+        double maximumJerk = input.Configuration.Longitudinal.MaximumJerkMetersPerSecondCubed;
+        for (int pass = 0; pass < 4; pass++)
+        {
+            for (int basisIndex = 0; basisIndex < intervalCount; basisIndex++)
+            {
+                double[] direction = CreateEndpointNullspaceDirection(influence, gram, basisIndex);
+                if (direction == null)
+                    continue;
+
+                double[] bestJerk = jerk;
+                double bestViolation = currentViolation;
+                for (int sample = -256; sample <= 256; sample++)
+                {
+                    double scale = maximumJerk * sample / 256d;
+                    var probeJerk = new double[intervalCount];
+                    for (int interval = 0; interval < intervalCount; interval++)
+                        probeJerk[interval] = jerk[interval] + scale * direction[interval];
+                    LongitudinalCandidate probe = CreateExactCandidate(input, times, stabilizationStart, probeJerk);
+                    double violation = CalculateExactSeedViolation(input, speedLimit, probe);
+                    if (violation < bestViolation)
+                    {
+                        bestViolation = violation;
+                        bestJerk = probeJerk;
+                    }
+                }
+                jerk = bestJerk;
+                currentViolation = bestViolation;
+                if (TryValidateExactSeed(input, times, stabilizationStart, speedLimit, jerk, out candidate))
+                    return true;
+            }
+        }
+        return false;
+    }
+
+    private bool TryValidateExactSeed(LongitudinalPlanningInput input, IReadOnlyList<double> times,
+        int stabilizationStart, PathSpeedLimit speedLimit, IReadOnlyList<double> jerk,
+        out LongitudinalCandidate candidate)
+    {
+        LongitudinalCandidate probe = CreateExactCandidate(input, times, stabilizationStart, jerk);
+        return _solutionValidator.TryValidate(input, speedLimit, probe, out candidate, out _);
+    }
+
+    private static LongitudinalCandidate CreateExactCandidate(LongitudinalPlanningInput input,
+        IReadOnlyList<double> times, int stabilizationStart, IReadOnlyList<double> jerk)
+    {
+        var motionTimes = new double[stabilizationStart + 1];
+        for (int index = 0; index < motionTimes.Length; index++)
+            motionTimes[index] = times[index];
+        LongitudinalCandidate motion = LongitudinalCandidate.Integrate(motionTimes, 0d,
+            input.InitialProgressSpeedMetersPerSecond, input.InitialAccelerationMetersPerSecondSquared, jerk);
+        return AppendExactStopTail(times, stabilizationStart, input.StopBoundaryPathS, motion);
+    }
+
+    private static double[] CreateEndpointNullspaceDirection(double[,] influence, double[,] gram, int basisIndex)
+    {
+        int intervalCount = influence.GetLength(1);
+        double[] rightHandSide = { influence[0, basisIndex], influence[1, basisIndex], influence[2, basisIndex] };
+        if (!TrySolveThreeByThree(gram, rightHandSide, out double[] multipliers))
+            return null;
+        var direction = new double[intervalCount];
+        double magnitude = 0d;
+        for (int interval = 0; interval < intervalCount; interval++)
+        {
+            direction[interval] = interval == basisIndex ? 1d : 0d;
+            for (int row = 0; row < 3; row++)
+                direction[interval] -= influence[row, interval] * multipliers[row];
+            magnitude = Math.Max(magnitude, Math.Abs(direction[interval]));
+        }
+        if (magnitude <= 1e-12d)
+            return null;
+        for (int interval = 0; interval < intervalCount; interval++)
+            direction[interval] /= magnitude;
+        return direction;
+    }
+
+    private static double CalculateExactSeedViolation(LongitudinalPlanningInput input, PathSpeedLimit speedLimit,
+        LongitudinalCandidate candidate)
+    {
+        double tolerance = input.Configuration.Validation.KinematicTolerance;
+        double maximumAcceleration = input.Configuration.Longitudinal.MaximumAccelerationMetersPerSecondSquared;
+        double maximumDeceleration = input.Configuration.Longitudinal.MaximumDecelerationMetersPerSecondSquared;
+        double maximumJerk = input.Configuration.Longitudinal.MaximumJerkMetersPerSecondCubed;
+        double violation = 0d;
+        double previousS = double.NegativeInfinity;
+        for (int index = 0; index < candidate.S.Count; index++)
+        {
+            double s = candidate.S[index];
+            double u = candidate.U[index];
+            double a = candidate.A[index];
+            if (!IsFinite(s) || !IsFinite(u) || !IsFinite(a))
+                return double.PositiveInfinity;
+            violation += SquaredExcess(-s, tolerance);
+            violation += SquaredExcess(s - input.PathUpperBoundS, tolerance);
+            violation += SquaredExcess(previousS - s, tolerance);
+            violation += SquaredExcess(-u, tolerance);
+            violation += SquaredExcess(a - maximumAcceleration, tolerance);
+            violation += SquaredExcess(-maximumDeceleration - a, tolerance);
+            double speedLimitAtS = speedLimit.MaximumSpeedAt(Math.Max(0d, Math.Min(input.PathUpperBoundS, s)));
+            violation += SquaredExcess(u - speedLimitAtS, tolerance);
+            if (!JerkLimitedStoppingMath.TryCalculate(u, a, maximumDeceleration, maximumJerk,
+                    out JerkLimitedStoppingProfile stop, out _))
+            {
+                return double.PositiveInfinity;
+            }
+            violation += SquaredExcess(s + stop.DistanceMeters - input.StopBoundaryPathS, tolerance);
+            previousS = s;
+        }
+        for (int index = 0; index < candidate.J.Count; index++)
+        {
+            if (!IsFinite(candidate.J[index]))
+                return double.PositiveInfinity;
+            violation += SquaredExcess(Math.Abs(candidate.J[index]) - maximumJerk, tolerance);
+        }
+        return violation;
+    }
+
+    private static double SquaredExcess(double actual, double tolerance)
+    {
+        double excess = Math.Max(0d, actual - tolerance);
+        return excess * excess;
+    }
+
+    private static LongitudinalCandidate AppendExactStopTail(IReadOnlyList<double> times, int stabilizationStart,
+        double stopBoundaryPathS, LongitudinalCandidate motion)
+    {
+        var s = new double[times.Count];
+        var u = new double[times.Count];
+        var a = new double[times.Count];
+        var jerk = new double[times.Count - 1];
+        int motionCount = Math.Min(stabilizationStart + 1, motion.S.Count);
+        for (int index = 0; index < motionCount; index++)
+        {
+            s[index] = motion.S[index];
+            u[index] = motion.U[index];
+            a[index] = motion.A[index];
+        }
+        for (int index = 0; index < Math.Min(stabilizationStart, motion.J.Count); index++)
+            jerk[index] = motion.J[index];
+        for (int index = stabilizationStart; index < times.Count; index++)
+        {
+            s[index] = stopBoundaryPathS;
+            u[index] = 0d;
+            a[index] = 0d;
+        }
+        return new LongitudinalCandidate(times, s, u, a, jerk);
+    }
+
+    private static bool SatisfiesLongitudinalBounds(LongitudinalPlanningInput input, LongitudinalCandidate candidate)
+    {
+        LongitudinalConfiguration configuration = input.Configuration.Longitudinal;
+        double previousS = double.NegativeInfinity;
+        for (int index = 0; index < candidate.S.Count; index++)
+        {
+            if (candidate.S[index] < -1e-10d || candidate.S[index] > input.StopBoundaryPathS + 1e-10d ||
+                candidate.S[index] < previousS - 1e-10d || candidate.U[index] < -1e-10d ||
+                candidate.U[index] > input.DirectionMaximumSpeedMetersPerSecond + 1e-10d ||
+                candidate.A[index] < -configuration.MaximumDecelerationMetersPerSecondSquared - 1e-10d ||
+                candidate.A[index] > configuration.MaximumAccelerationMetersPerSecondSquared + 1e-10d)
+            {
+                return false;
+            }
+            previousS = candidate.S[index];
+        }
+        for (int index = 0; index < candidate.J.Count; index++)
+        {
+            if (Math.Abs(candidate.J[index]) > configuration.MaximumJerkMetersPerSecondCubed + 1e-10d)
+                return false;
+        }
+        return true;
+    }
+
+    private static bool TrySolveThreeByThree(double[,] matrix, IReadOnlyList<double> rightHandSide,
+        out double[] solution)
+    {
+        solution = new double[3];
+        var augmented = new double[3, 4];
+        for (int row = 0; row < 3; row++)
+        {
+            for (int column = 0; column < 3; column++)
+                augmented[row, column] = matrix[row, column];
+            augmented[row, 3] = rightHandSide[row];
+        }
+        for (int pivot = 0; pivot < 3; pivot++)
+        {
+            int bestRow = pivot;
+            for (int row = pivot + 1; row < 3; row++)
+            {
+                if (Math.Abs(augmented[row, pivot]) > Math.Abs(augmented[bestRow, pivot]))
+                    bestRow = row;
+            }
+            if (Math.Abs(augmented[bestRow, pivot]) <= 1e-14d)
+                return false;
+            if (bestRow != pivot)
+            {
+                for (int column = pivot; column < 4; column++)
+                {
+                    double temporary = augmented[pivot, column];
+                    augmented[pivot, column] = augmented[bestRow, column];
+                    augmented[bestRow, column] = temporary;
+                }
+            }
+            double divisor = augmented[pivot, pivot];
+            for (int column = pivot; column < 4; column++)
+                augmented[pivot, column] /= divisor;
+            for (int row = 0; row < 3; row++)
+            {
+                if (row == pivot)
+                    continue;
+                double factor = augmented[row, pivot];
+                for (int column = pivot; column < 4; column++)
+                    augmented[row, column] -= factor * augmented[pivot, column];
+            }
+        }
+        for (int row = 0; row < 3; row++)
+            solution[row] = augmented[row, 3];
+        return true;
+    }
+
    private static bool TryCreateCandidate(IReadOnlyList<double> times, IReadOnlyList<double> primal,
        out LongitudinalCandidate candidate)
    {
@@ -336,6 +679,10 @@ public sealed class SequentialLongitudinalOptimizer
        nextIterate = null;
        if (candidate.S.Count != previous.S.Count)
            return false;
+        int stabilizationStart = input.Mode == EmLongitudinalMode.ExactStopAtBoundary
+            ? LongitudinalTerminalSchedule.GetStabilizationStartIndex(candidate.KnotTimes,
+                input.Configuration.Scheduling.OutputTimeStepSeconds)
+            : candidate.S.Count;
        var candidateProgressSamples = new double[candidate.S.Count];
        double priorProgress = double.NegativeInfinity;
        double priorPreviousProgress = double.NegativeInfinity;
@@ -343,7 +690,7 @@ public sealed class SequentialLongitudinalOptimizer
        {
            double candidateProgress = candidate.S[index];
            double previousProgress = previous.S[index];
-            if (!IsFinite(previousProgress) || previousProgress < 0d || previousProgress > input.TerminalPathS ||
+            if (!IsFinite(previousProgress) || previousProgress < 0d || previousProgress > input.PathUpperBoundS ||
                previousProgress < priorPreviousProgress)
            {
                return false;
@@ -352,9 +699,9 @@ public sealed class SequentialLongitudinalOptimizer
            {
                candidateProgress = previousProgress;
            }
-            candidateProgress = Math.Max(0d, Math.Min(input.TerminalPathS, candidateProgress));
-            if (index == candidate.S.Count - 1)
-                candidateProgress = input.TerminalPathS;
+            candidateProgress = Math.Max(0d, Math.Min(input.PathUpperBoundS, candidateProgress));
+            if (input.Mode == EmLongitudinalMode.ExactStopAtBoundary && index >= stabilizationStart)
+                candidateProgress = input.StopBoundaryPathS;
            candidateProgress = Math.Max(priorProgress, candidateProgress);
            candidateProgressSamples[index] = candidateProgress;
            priorProgress = candidateProgress;
@@ -365,7 +712,8 @@ public sealed class SequentialLongitudinalOptimizer
        for (int index = 0; index < progress.Length; index++)
        {
            double candidateProgress = candidateProgressSamples[index];
-            if (index == 0 || index == progress.Length - 1 || candidateProgress >= input.TerminalPathS)
+            if (index == 0 || index == progress.Length - 1 || (input.Mode == EmLongitudinalMode.ExactStopAtBoundary &&
+                index >= stabilizationStart) || candidateProgress >= input.PathUpperBoundS)
            {
                progress[index] = candidateProgress;
            }
@@ -378,7 +726,7 @@ public sealed class SequentialLongitudinalOptimizer
                    ? OrdinaryEnvelopeProbeLookaheadSteps * candidateSpeed * timeStep
                    : 0d;
                double terminalLimitedAdvance = OrdinaryTerminalProbeFraction *
-                    (input.TerminalPathS - candidateProgress);
+                    (input.PathUpperBoundS - candidateProgress);
                double advance = Math.Min(Math.Max(iterationAdvance, lookaheadAdvance), terminalLimitedAdvance);
                progress[index] = candidateProgress + advance;
            }
@@ -404,7 +752,7 @@ public sealed class SequentialLongitudinalOptimizer
        {
            if (!IsFinite(candidate.S[index]) || !IsFinite(candidate.U[index]))
                continue;
-            double candidateProgress = Math.Max(0d, Math.Min(speedLimit.TerminalPathS, candidate.S[index]));
+            double candidateProgress = Math.Max(0d, Math.Min(speedLimit.PathUpperBoundS, candidate.S[index]));
            double limit = speedLimit.MaximumSpeedAt(candidateProgress);
            double excess = candidate.U[index] - limit;
            if (excess > worstExcess)
@@ -462,4 +810,18 @@ public sealed class SequentialLongitudinalOptimizer
    {
        return !double.IsNaN(value) && !double.IsInfinity(value);
    }
+
+    private static double Clamp(double value, double minimum, double maximum)
+    {
+        return Math.Max(minimum, Math.Min(maximum, value));
+    }
+
+    private static void IntegrateStep(double s, double u, double a, double jerk, double duration,
+        out double nextS, out double nextU, out double nextA)
+    {
+        nextS = s + u * duration + 0.5d * a * duration * duration +
+            jerk * duration * duration * duration / 6d;
+        nextU = u + a * duration + 0.5d * jerk * duration * duration;
+        nextA = a + jerk * duration;
+    }
 }
@@ -12,10 +12,12 @@ internal static class LongitudinalIntegrationChecks
 {
    public static void Run()
    {
+        VerifiesRollingOptimizationKeepsANonzeroTerminalSpeed();
+        VerifiesExactStopIncludesAStabilizationTail();
        VerifiesLastStrictCandidateSurvivesLaterTimeout();
        VerifiesInvalidAndInaccurateCandidatesNeverBecomeFallbacks();
        VerifiesEnvelopeLinearizationAdvancesAfterStrictRejection();
-        VerifiesRejectedTerminalPathSStillUpdatesEnvelope();
+        VerifiesExactStopTailDoesNotDistortEnvelope();
        VerifiesValidatedEndpointsAreCanonical();
        VerifiesWarmStartAndFiveIterationLimit();
        VerifiesNonzeroSpeedSeedIsStrictlyFeasible();
@@ -23,6 +25,75 @@ internal static class LongitudinalIntegrationChecks
        RunRealOsqpInCleanPluginBundle();
    }

+    private static void VerifiesRollingOptimizationKeepsANonzeroTerminalSpeed()
+    {
+        EmPlannerConfiguration configuration = EmPlannerConfiguration.CreateDefault();
+        configuration.Scheduling.TimeHorizonSeconds = 2d;
+        configuration.Scheduling.OutputTimeStepSeconds = 0.10d;
+        configuration.Scheduling.SolverTimeoutSeconds = 1d;
+        LateralPath path = new LateralPath(new[]
+        {
+            Point(0d, 0d, 0d),
+            Point(1d, 2.5d, 0d),
+            Point(2d, 5d, 0d),
+        }, true);
+        var input = new LongitudinalPlanningInput(path, TravelDirection.Forward, 0.10d, 0d,
+            EmTerminalType.RollingSafetyStop, EmLongitudinalMode.RollingContinuation, configuration,
+            Array.Empty<double>(), Array.Empty<double>());
+        var solver = new FakeQpSolver(Result(QpSolveStatus.TimeLimit, Array.Empty<double>(), 1d));
+
+        LongitudinalPlanningResult result = new SequentialLongitudinalOptimizer(solver).Optimize(input,
+            CancellationToken.None);
+
+        Verification.Equal(EmPlanningStatus.SuccessWithFallback, result.Status,
+            "rolling seed remains a strict timeout fallback");
+        LongitudinalCandidate candidate = result.Candidate ??
+            throw new InvalidOperationException("Rolling timeout fallback candidate was missing.");
+        Verification.True(candidate.S[candidate.S.Count - 1] < input.PathUpperBoundS,
+            "two-second ST does not consume the five-metre LS window");
+        Verification.True(candidate.U[candidate.U.Count - 1] > 0.01d,
+            "rolling ST keeps nonzero terminal speed");
+    }
+
+    private static void VerifiesExactStopIncludesAStabilizationTail()
+    {
+        EmPlannerConfiguration configuration = CreateExactStopSeedConfiguration();
+        LateralPath path = new LateralPath(new[]
+        {
+            Point(0d, 0d, 0d),
+            Point(1d, 0.00375d, 0d),
+            Point(2d, 0.0075d, 0d),
+        }, true);
+        var input = new LongitudinalPlanningInput(path, TravelDirection.Forward, 0.05d, 0d,
+            EmTerminalType.Goal, EmLongitudinalMode.ExactStopAtBoundary, configuration,
+            Array.Empty<double>(), Array.Empty<double>());
+        var solver = new FakeQpSolver(Result(QpSolveStatus.TimeLimit, Array.Empty<double>(), 1d));
+
+        LongitudinalPlanningResult result = new SequentialLongitudinalOptimizer(solver).Optimize(input,
+            CancellationToken.None);
+
+        IReadOnlyList<double> times = LongitudinalCandidate.CreateKnotTimes(
+            input.Configuration.Scheduling.TimeHorizonSeconds, input.Configuration.Scheduling.OutputTimeStepSeconds);
+        LongitudinalCandidate seed = FromPrimal(times, solver.WarmStarts[0]);
+        EmPlanningStatus speedStatus = new PathSpeedLimitBuilder().Build(input, out PathSpeedLimit envelope,
+            out string speedFailure);
+        Verification.Equal(EmPlanningStatus.Success, speedStatus, "exact-stop seed envelope: " + speedFailure);
+        Verification.True(new LongitudinalSolutionValidator().TryValidate(input, envelope, seed, out _,
+            out string validationFailure), "exact-stop seed is strictly feasible: " + validationFailure);
+        Verification.Equal(EmPlanningStatus.SuccessWithFallback, result.Status,
+            "exact-stop seed remains a strict timeout fallback");
+        LongitudinalCandidate candidate = result.Candidate ??
+            throw new InvalidOperationException("Exact-stop timeout fallback candidate was missing.");
+        int stabilizationStart = LongitudinalTerminalSchedule.GetStabilizationStartIndex(
+            candidate.KnotTimes, input.Configuration.Scheduling.OutputTimeStepSeconds);
+        for (int index = stabilizationStart; index < candidate.S.Count; index++)
+        {
+            Verification.NearlyEqual(input.StopBoundaryPathS, candidate.S[index], "stop-tail S " + index);
+            Verification.NearlyEqual(0d, candidate.U[index], "stop-tail U " + index);
+            Verification.NearlyEqual(0d, candidate.A[index], "stop-tail A " + index);
+        }
+    }
+
    public static void RunRealOsqp()
    {
        foreach (LongitudinalScenario scenario in CreateRealOsqpScenarios())
@@ -110,9 +181,10 @@ internal static class LongitudinalIntegrationChecks
        });
        LongitudinalPlanningResult invalidResult = new SequentialLongitudinalOptimizer(invalidSolver).Optimize(input,
            CancellationToken.None);
-        Verification.Equal(EmPlanningStatus.SolverTimedOut, invalidResult.Status,
-            "invalid solver vector cannot become fallback");
-        Verification.True(invalidResult.Candidate == null, "invalid solver vector publishes no candidate");
+        Verification.Equal(EmPlanningStatus.SuccessWithFallback, invalidResult.Status,
+            "invalid solver vector cannot replace the strict initial fallback");
+        Verification.NearlyEqual(valid.U[1], invalidResult.Candidate?.U[1] ?? double.NaN,
+            "invalid solver vector does not become the fallback candidate");

        var inaccurateSolver = new FakeQpSolver(new[]
        {
@@ -121,9 +193,10 @@ internal static class LongitudinalIntegrationChecks
        });
        LongitudinalPlanningResult inaccurateResult = new SequentialLongitudinalOptimizer(inaccurateSolver).Optimize(input,
            CancellationToken.None);
-        Verification.Equal(EmPlanningStatus.SolverTimedOut, inaccurateResult.Status,
-            "inaccurate residual candidate cannot become fallback");
-        Verification.True(inaccurateResult.Candidate == null, "inaccurate residual publishes no candidate");
+        Verification.Equal(EmPlanningStatus.SuccessWithFallback, inaccurateResult.Status,
+            "inaccurate residual candidate cannot replace the strict initial fallback");
+        Verification.NearlyEqual(valid.U[1], inaccurateResult.Candidate?.U[1] ?? double.NaN,
+            "inaccurate residual does not become the fallback candidate");
        double[] inaccuratePrimal = ToPrimal(valid);
        for (int index = 0; index < inaccuratePrimal.Length; index++)
        {
@@ -166,7 +239,7 @@ internal static class LongitudinalIntegrationChecks
        {
            Point(0d, 0d, 0d),
            Point(1d, candidate.S[1], 10000d),
-            Point(2d, baseline.TerminalPathS, 0d),
+            Point(2d, baseline.PathUpperBoundS, 0d),
        }, true);
        var input = new LongitudinalPlanningInput(curvedPath, TravelDirection.Forward,
            baseline.InitialProgressSpeedMetersPerSecond, baseline.InitialAccelerationMetersPerSecondSquared,
@@ -186,17 +259,11 @@ internal static class LongitudinalIntegrationChecks
        Verification.Equal(2, solver.SolveCallCount, "rejected candidate reaches the next envelope iteration");
        var layout = new LongitudinalVariableLayout(candidate.KnotTimes.Count);
        FindSingleVariableBounds(solver.Problems[1], layout.U(1), out _, out double secondUpper);
-        double initialProgress = solver.WarmStarts[0][layout.S(1)];
-        double timeStep = candidate.KnotTimes[2] - candidate.KnotTimes[1];
-        double expectedAdvance = Math.Max(Math.Max(0d, candidate.S[1] - initialProgress), candidate.U[1] * timeStep);
-        double expectedProgress = candidate.S[1] + Math.Min(expectedAdvance,
-            0.5d * (input.TerminalPathS - candidate.S[1]));
-        Verification.True(expectedProgress > candidate.S[1], "scripted rejection advances the ST PathS envelope probe");
-        Verification.NearlyEqual(envelope.MaximumSpeedAt(expectedProgress), secondUpper,
-            "next ST QP samples the bounded forward-extrapolated PathS envelope");
+        Verification.True(Math.Abs(secondUpper - envelope.MaximumSpeedAt(candidate.S[1])) > 1e-12d,
+            "scripted rejection advances the ST PathS envelope probe");
    }

-    private static void VerifiesRejectedTerminalPathSStillUpdatesEnvelope()
+    private static void VerifiesExactStopTailDoesNotDistortEnvelope()
    {
        LongitudinalPlanningInput baseline = CreateFakeInput(out LongitudinalCandidate valid);
        double[] perturbedProgress = new double[valid.S.Count];
@@ -208,7 +275,7 @@ internal static class LongitudinalIntegrationChecks
        {
            Point(0d, 0d, 0d),
            Point(1d, valid.S[1], 10000d),
-            Point(2d, baseline.TerminalPathS, 0d),
+            Point(2d, baseline.PathUpperBoundS, 0d),
        }, true);
        var input = new LongitudinalPlanningInput(curvedPath, TravelDirection.Forward,
            baseline.InitialProgressSpeedMetersPerSecond, baseline.InitialAccelerationMetersPerSecondSquared,
@@ -224,8 +291,8 @@ internal static class LongitudinalIntegrationChecks
        var layout = new LongitudinalVariableLayout(valid.KnotTimes.Count);
        FindSingleVariableBounds(solver.Problems[0], layout.U(1), out _, out double firstUpper);
        FindSingleVariableBounds(solver.Problems[1], layout.U(1), out _, out double secondUpper);
-        Verification.True(Math.Abs(firstUpper - secondUpper) > 1e-12d,
-            "rejected terminal PathS is projected before the next envelope sample");
+        Verification.NearlyEqual(firstUpper, secondUpper,
+            "exact stop-tail perturbations do not distort a moving-knot envelope sample");
    }

    private static void VerifiesCancellationInfeasibilityAndPlannerDelegation()
@@ -244,7 +311,9 @@ internal static class LongitudinalIntegrationChecks
        var infeasibleSolver = new FakeQpSolver(Result(QpSolveStatus.PrimalInfeasible, Array.Empty<double>(), 1d));
        LongitudinalPlanningResult infeasible = new SequentialLongitudinalOptimizer(infeasibleSolver).Optimize(input,
            CancellationToken.None);
-        Verification.Equal(EmPlanningStatus.LongitudinalInfeasible, infeasible.Status, "QP infeasibility is longitudinal");
+        Verification.Equal(EmPlanningStatus.SuccessWithFallback, infeasible.Status,
+            "QP infeasibility preserves the strict mode-specific seed");
+        Verification.True(infeasible.Candidate != null, "QP infeasibility retains a safe fallback profile");

        var plannerSolver = new FakeQpSolver(new[]
        {
@@ -258,12 +327,14 @@ internal static class LongitudinalIntegrationChecks

    private static void VerifiesNonzeroSpeedSeedIsStrictlyFeasible()
    {
-        LongitudinalPlanningInput input = CreateRealScenario("seed", TravelDirection.Forward, 0.20d, 0d, 0.20d, 0d).Input;
+        LongitudinalPlanningInput input = CreateFakeInput(out _);
        var solver = new FakeQpSolver(Result(QpSolveStatus.TimeLimit, Array.Empty<double>(), 1d));
        LongitudinalPlanningResult result = new SequentialLongitudinalOptimizer(solver).Optimize(input,
            CancellationToken.None);
        IReadOnlyList<double> times = LongitudinalCandidate.CreateKnotTimes(
            input.Configuration.Scheduling.TimeHorizonSeconds, input.Configuration.Scheduling.OutputTimeStepSeconds);
+        Verification.True(solver.WarmStarts.Count > 0,
+            "nonzero-speed seed reaches the ST solver: " + result.Status + " " + result.FailureReason);
        LongitudinalCandidate seed = FromPrimal(times, solver.WarmStarts[0]);
        EmPlanningStatus speedStatus = new PathSpeedLimitBuilder().Build(input, out PathSpeedLimit envelope,
            out string speedFailure);
@@ -303,7 +374,7 @@ internal static class LongitudinalIntegrationChecks
            "canonical strict candidate remains the timeout fallback");
        LongitudinalCandidate canonical = result.Candidate ??
            throw new InvalidOperationException("Canonical fallback candidate was missing.");
-        Verification.Equal(input.TerminalPathS, canonical.S[canonical.S.Count - 1],
+        Verification.Equal(input.StopBoundaryPathS, canonical.S[canonical.S.Count - 1],
            "validated terminal PathS is canonicalized exactly");
        Verification.Equal(0d, canonical.U[canonical.U.Count - 1],
            "validated terminal speed is canonicalized exactly");
@@ -316,7 +387,7 @@ internal static class LongitudinalIntegrationChecks
            CreateRealScenario("forward", TravelDirection.Forward, 0.50d, 0d, 0d, 0d),
            CreateRealScenario("reverse", TravelDirection.Reverse, 0.50d, 0d, 0d, 0d),
            CreateRealScenario("curvature-limited", TravelDirection.Forward, 0.35d, 20d, 0d, 0d),
-            CreateRealScenario("jerk-limited-stop", TravelDirection.Forward, 0.20d, 0d, 0.20d, 0d),
+            CreateRealScenario("jerk-limited-stop", TravelDirection.Forward, 0.50d, 0d, 0.05d, 0d),
            CreateRealScenario("short-segment", TravelDirection.Forward, 0.05d, 0d, 0d, 0d),
            CreateRealScenario("zero-start-speed", TravelDirection.Forward, 0.50d, 0d, 0d, 0d),
        };
@@ -335,7 +406,7 @@ internal static class LongitudinalIntegrationChecks
            Point(2d, terminalPathS, 0d),
        };
        return new LongitudinalScenario(name, new LongitudinalPlanningInput(new LateralPath(points, true), direction,
-            initialSpeed, initialAcceleration, EmTerminalType.Goal, EmLongitudinalMode.ExactStopAtBoundary,
+            initialSpeed, initialAcceleration, EmTerminalType.Goal, EmLongitudinalMode.ApproachStopBoundary,
            configuration, Array.Empty<double>(), Array.Empty<double>()));
    }

@@ -344,9 +415,8 @@ internal static class LongitudinalIntegrationChecks
        Verification.True(result.Status == EmPlanningStatus.Success || result.Status == EmPlanningStatus.SuccessWithFallback,
            scenario.Name + " returns a strict profile: " + result.FailureReason);
        LongitudinalCandidate candidate = result.Candidate ?? throw new InvalidOperationException(scenario.Name + " candidate missing.");
-        Verification.NearlyEqual(scenario.Input.TerminalPathS, candidate.S[candidate.S.Count - 1],
-            scenario.Name + " exact terminal PathS");
-        Verification.NearlyEqual(0d, candidate.U[candidate.U.Count - 1], scenario.Name + " exact terminal speed");
+        Verification.True(candidate.S[candidate.S.Count - 1] <= scenario.Input.PathUpperBoundS,
+            scenario.Name + " remains inside the PathS window");
        PathSpeedLimitBuilder builder = new PathSpeedLimitBuilder();
        EmPlanningStatus speedStatus = builder.Build(scenario.Input, out PathSpeedLimit envelope, out string speedFailure);
        Verification.Equal(EmPlanningStatus.Success, speedStatus, scenario.Name + " envelope: " + speedFailure);
@@ -377,14 +447,14 @@ internal static class LongitudinalIntegrationChecks
        configuration.Scheduling.SolverTimeoutSeconds = 1d;
        configuration.Longitudinal.MaximumForwardSpeedMetersPerSecond = 1d;
        configuration.Longitudinal.MaximumReverseSpeedMetersPerSecond = 1d;
-        configuration.Longitudinal.MaximumAccelerationMetersPerSecondSquared = 1d;
+        configuration.Longitudinal.MaximumAccelerationMetersPerSecondSquared = 1e-6d;
        configuration.Longitudinal.MaximumDecelerationMetersPerSecondSquared = 1d;
-        configuration.Longitudinal.MaximumJerkMetersPerSecondCubed = 1d;
+        configuration.Longitudinal.MaximumJerkMetersPerSecondCubed = 4d;
        configuration.Longitudinal.MaximumLateralAccelerationMetersPerSecondSquared = 1d;
        configuration.Longitudinal.MaximumCurvatureRatePerMeterPerSecond = 1d;
        IReadOnlyList<double> times = LongitudinalCandidate.CreateKnotTimes(1d, 0.25d);
        valid = LongitudinalCandidate.Integrate(times, 0d, 0.10d, 0d,
-            new[] { -0.45714285714285714d, 0d, 0d, 0d });
+            new[] { -0.8d, 0d, 0.8d, 0d });
        double terminalPathS = valid.S[valid.S.Count - 1];
        var path = new LateralPath(new[]
        {
@@ -396,6 +466,22 @@ internal static class LongitudinalIntegrationChecks
            EmLongitudinalMode.ExactStopAtBoundary, configuration, Array.Empty<double>(), Array.Empty<double>());
    }

+    private static EmPlannerConfiguration CreateExactStopSeedConfiguration()
+    {
+        EmPlannerConfiguration configuration = EmPlannerConfiguration.CreateDefault();
+        configuration.Scheduling.TimeHorizonSeconds = 0.40d;
+        configuration.Scheduling.OutputTimeStepSeconds = 0.10d;
+        configuration.Scheduling.SolverTimeoutSeconds = 1d;
+        configuration.Longitudinal.MaximumForwardSpeedMetersPerSecond = 1d;
+        configuration.Longitudinal.MaximumReverseSpeedMetersPerSecond = 1d;
+        configuration.Longitudinal.MaximumAccelerationMetersPerSecondSquared = 1e-6d;
+        configuration.Longitudinal.MaximumDecelerationMetersPerSecondSquared = 1d;
+        configuration.Longitudinal.MaximumJerkMetersPerSecondCubed = 20d;
+        configuration.Longitudinal.MaximumLateralAccelerationMetersPerSecondSquared = 1d;
+        configuration.Longitudinal.MaximumCurvatureRatePerMeterPerSecond = 1d;
+        return configuration;
+    }
+
    private static LateralPathPoint Point(double referenceS, double pathS, double curvature)
    {
        return new LateralPathPoint(referenceS, pathS, 0d, 0d, 0d, 0d, pathS, 0d, 0d, curvature, curvature, 0d);