ardupilot/libraries/AP_Math/control.cpp

/*
 * control.cpp
 * Copyright (C) Leonard Hall 2020
 *
 * This file is free software: you can redistribute it and/or modify it
 * under the terms of the GNU General Public License as published by the
 * Free Software Foundation, either version 3 of the License, or
 * (at your option) any later version.
 *
 * This file is distributed in the hope that it will be useful, but
 * WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
 * See the GNU General Public License for more details.
 *
 * You should have received a copy of the GNU General Public License along
 * with this program.  If not, see <http://www.gnu.org/licenses/>.
 */

/*
 *  this module provides common controller functions
 */
#include "AP_Math.h"
#include "vector2.h"
#include "vector3.h"
#include <AP_InternalError/AP_InternalError.h>

// control default definitions
#define CORNER_ACCELERATION_RATIO   1.0/safe_sqrt(2.0)   // acceleration reduction to enable zero overshoot corners

// Normalised cross-track reference magnitude below which limit_accel_xy() fades
// out cross-track prioritisation. The reference is expected normalised by the
// maximum speed, so this is a fraction of max speed (0.25 -> 25%). When the
// reference is small its direction is ill-defined, so prioritising "cross-track"
// acceleration re-projects a saturated command onto a rapidly rotating axis and
// injects a lateral acceleration spike (e.g. the roll wobble seen at the
// zero-crossing of a hard Loiter stick reversal). Below this magnitude we fade to
// an isotropic magnitude limit that preserves the commanded direction.
#define LIMIT_ACCEL_XY_MIN_REF   0.25f

// Projects velocity forward in time using acceleration, constrained by directional limit.
// - If `limit` is non-zero, it defines a direction in which acceleration is constrained.
// - The `vel_error` value defines the direction of velocity error (its sign matters, not its magnitude).
// - When `limit` is active, velocity is only updated if doing so would not increase the error in the limited direction.
// - If velocity is currently opposing the limit direction, the update is clipped to avoid crossing zero.
// This prevents unwanted acceleration in the direction of a constraint when it would worsen velocity error.
void update_vel_accel(float& vel, float accel, float dt, float limit, float vel_error)
{
    float delta_vel = accel * dt;
    // do not add delta_vel if it will increase the velocity error in the direction of limit
    // unless adding delta_vel will reduce vel towards zero
    if (is_positive(delta_vel * limit) && is_positive(vel_error * limit)) {
        if (is_negative(vel * limit)) {
            delta_vel = constrain_float(delta_vel, -fabsf(vel), fabsf(vel));
        } else {
            delta_vel = 0.0;
        }
    }
    vel += delta_vel;
}

// Projects position and velocity forward in time using acceleration, constrained by directional limit.
// - `limit` defines the constrained direction of motion.
// - `pos_error` and `vel_error` define the sign of error in position and velocity respectively (magnitude is ignored).
// - If the update would increase position error in the constrained direction, the position update is skipped.
// - The velocity update then proceeds with directional limit handling via `update_vel_accel()`.
// This prevents motion in a constrained direction if it would worsen the position or velocity error.
void update_pos_vel_accel(postype_t& pos, float& vel, float accel, float dt, float limit, float pos_error, float vel_error)
{
    // move position and velocity forward by dt if it does not increase error when limited.
    float delta_pos = vel * dt + accel * 0.5f * sq(dt);
    // do not add delta_pos if it will increase the velocity error in the direction of limit
    if (is_positive(delta_pos * limit) && is_positive(pos_error * limit)) {
        delta_pos = 0.0;
    }
    pos += delta_pos;

    update_vel_accel(vel, accel, dt, limit, vel_error);
}

// Projects velocity forward in time using acceleration, constrained by directional limits.
// - If the `limit` vector is non-zero, it defines a direction in which acceleration is constrained.
// - The `vel_error` vector defines the direction of velocity error (its magnitude is unused).
// - When `limit` is active, velocity is only updated if doing so would not increase the error in the limited direction.
// This function prevents the system from increasing velocity along the limit direction
// if doing so would worsen the velocity error.
void update_vel_accel_xy(Vector2f& vel, const Vector2f& accel, float dt, const Vector2f& limit, const Vector2f& vel_error)
{
    // increase velocity by acceleration * dt if it does not increase error when limited.
    // unless adding delta_vel will reduce the magnitude of vel
    Vector2f delta_vel = accel * dt;
    if (!limit.is_zero() && !delta_vel.is_zero()) {
        // check if delta_vel will increase the velocity error in the direction of limit
        if (is_positive(delta_vel * limit) && is_positive(vel_error * limit) && !is_negative(vel * limit)) {
            delta_vel.zero();
        }
    }
    vel += delta_vel;
}

// Projects position and velocity forward in time using acceleration, constrained by directional limits.
// - The `limit` vector defines a directional constraint: if non-zero, motion in that direction is restricted.
// - The `pos_error` and `vel_error` vectors represent the direction of error (magnitude is not used).
// - If a motion step would increase error along a limited axis, it is suppressed.
// This function avoids changes to position or velocity in the direction of `limit`
// if those changes would worsen the position or velocity error respectively.
void update_pos_vel_accel_xy(Vector2p& pos, Vector2f& vel, const Vector2f& accel, float dt, const Vector2f& limit, const Vector2f& pos_error, const Vector2f& vel_error)
{
    // move position and velocity forward by dt.
    Vector2f delta_pos = vel * dt + accel * 0.5f * sq(dt);

    if (!is_zero(limit.length_squared())) {
        // zero delta_pos if it will increase the velocity error in the direction of limit
        if (is_positive(delta_pos * limit) && is_positive(pos_error * limit)) {
            delta_pos.zero();
        }
    }

    pos += delta_pos.topostype();

    update_vel_accel_xy(vel, accel, dt, limit, vel_error);
}

// Applies jerk-limited shaping to the acceleration value to gradually approach a new target.
// - Constrains the rate of change of acceleration to be within ±`jerk_max` over time `dt`.
// - The current acceleration value is modified in-place.
// Useful for ensuring smooth transitions in thrust or lean angle command profiles.
void shape_accel(float accel_desired, float& accel,
                 float jerk_max, float dt)
{
    // sanity check jerk_max
    if (!is_positive(jerk_max)) {
        INTERNAL_ERROR(AP_InternalError::error_t::invalid_arg_or_result);
        return;
    }

    // jerk limit acceleration change
    if (is_positive(dt)) {
        float accel_delta = accel_desired - accel;
        accel_delta = constrain_float(accel_delta, -jerk_max * dt, jerk_max * dt);
        accel += accel_delta;
    }
}

// Applies jerk-limited shaping to a 2D acceleration vector.
// - Constrains the rate of change of acceleration to a maximum of `jerk_max` over time `dt`.
// - The current acceleration vector is modified in-place to approach `accel_desired`.
/// Ensures smooth acceleration transitions in both axes simultaneously.
void shape_accel_xy(const Vector2f& accel_desired, Vector2f& accel,
                    float jerk_max, float dt)
{
    // sanity check jerk_max
    if (!is_positive(jerk_max)) {
        INTERNAL_ERROR(AP_InternalError::error_t::invalid_arg_or_result);
        return;
    }

    // jerk limit acceleration change
    if (is_positive(dt)) {
        Vector2f accel_delta = accel_desired - accel;
        accel_delta.limit_length(jerk_max * dt);
        accel = accel + accel_delta;
    }
}

void shape_accel_xy(const Vector3f& accel_desired, Vector3f& accel,
                    float jerk_max, float dt)
{
    const Vector2f accel_desired_2f {accel_desired.x, accel_desired.y};
    Vector2f accel_2f {accel.x, accel.y};

    shape_accel_xy(accel_desired_2f, accel_2f, jerk_max, dt);
    accel.x = accel_2f.x;
    accel.y = accel_2f.y;
}

// Shapes velocity and acceleration using jerk-limited control.
// - Computes correction acceleration needed to reach `vel_desired` from current `vel`.
// - Uses a square-root controller with max acceleration and jerk constraints.
// - Correction is combined with feedforward `accel_desired`.
// - If `limit_total_accel` is true, total acceleration is constrained to `accel_min` / `accel_max`.
// The result is applied via `shape_accel`.
void shape_vel_accel(float vel_desired, float accel_desired,
                     float vel, float& accel,
                     float accel_min, float accel_max,
                     float jerk_max, float dt, bool limit_total_accel)
{
    // sanity check accel_min, accel_max and jerk_max.
    if (!is_negative(accel_min) || !is_positive(accel_max) || !is_positive(jerk_max)) {
        INTERNAL_ERROR(AP_InternalError::error_t::invalid_arg_or_result);
        return;
    }

    // velocity error to be corrected
    float vel_error = vel_desired - vel;

    // Calculate time constants and limits to ensure stable operation
    // The direction of acceleration limit is the same as the velocity error.
    // This is because the velocity error is negative when slowing down while
    // closing a positive position error.
    float KPa;
    if (is_positive(vel_error)) {
        KPa = jerk_max / accel_max;
    } else {
        KPa = jerk_max / (-accel_min);
    }

    // acceleration to correct velocity
    float accel_target = sqrt_controller(vel_error, KPa, jerk_max, dt);

    // constrain correction acceleration from accel_min to accel_max
    accel_target = constrain_float(accel_target, accel_min, accel_max);

    // velocity correction with input velocity
    accel_target += accel_desired;

    // Constrain total acceleration if limiting is enabled
    if (limit_total_accel) {
        accel_target = constrain_float(accel_target, accel_min, accel_max);
    }

    shape_accel(accel_target, accel, jerk_max, dt);
}

// Computes a jerk-limited acceleration command in 2D to track a desired velocity input.
// - Uses a square-root controller to calculate correction acceleration based on velocity error.
// - Correction is constrained to stay within `accel_max` (total acceleration magnitude).
// - Correction is added to `accel_desired` (feedforward).
// - If `limit_total_accel` is true, total acceleration is constrained after summing.
// Ensures velocity tracking with smooth, physically constrained motion.
void shape_vel_accel_xy(const Vector2f& vel_desired, const Vector2f& accel_desired,
                        const Vector2f& vel, Vector2f& accel,
                        float accel_max, float jerk_max, float dt, bool limit_total_accel)
{
    // sanity check accel_max and jerk_max.
    if (!is_positive(accel_max) || !is_positive(jerk_max)) {
        INTERNAL_ERROR(AP_InternalError::error_t::invalid_arg_or_result);
        return;
    }

    // Calculate time constants and limits to ensure stable operation
    const float KPa = jerk_max / accel_max;

    // velocity error to be corrected
    const Vector2f vel_error = vel_desired - vel;

    // acceleration to correct velocity
    Vector2f accel_target = sqrt_controller(vel_error, KPa, jerk_max, dt);

    limit_accel_corner_xy(vel, accel_target, accel_max);

    accel_target += accel_desired;

    // Constrain total acceleration if limiting is enabled 
    if (limit_total_accel) {
        accel_target.limit_length(accel_max);
    }

    shape_accel_xy(accel_target, accel, jerk_max, dt);
}

// Shapes position, velocity, and acceleration using a jerk-limited square-root command model.
// - Computes a velocity correction from position error using a square-root controller.
// - Uses sqrt_controller_accel() to bias the velocity correction based on the correction-frame closing rate.
// - Forms a velocity target by adding the correction to the feedforward velocity.
// - Computes an acceleration demand from velocity error using k_v and adds external acceleration feedforward.
// - Optionally constrains velocity and acceleration magnitudes when limit_total is true.
// - Applies jerk limiting via shape_accel() to ensure smooth acceleration transitions.
// This is the single-axis (1D) form of shape_pos_vel_accel_xy().
void shape_pos_vel_accel(postype_t pos_desired, float vel_desired, float accel_desired,
                         postype_t pos, float vel, float& accel,
                         float vel_min, float vel_max,
                         float accel_min, float accel_max,
                         float jerk_max, float dt, bool limit_total)
{
    // Sanity check limits and jerk_max.
    if (is_positive(vel_min) || is_negative(vel_max) || !is_negative(accel_min) || !is_positive(accel_max) || !is_positive(jerk_max)) {
        INTERNAL_ERROR(AP_InternalError::error_t::invalid_arg_or_result);
        return;
    }

    // Position error to be corrected.
    const float pos_error = pos_desired - pos;

    // Select sqrt_controller parameters based on error sign so the second-order limit
    // (acceleration allowance) matches the direction of motion.
    float accel_lim;
    float k_v;
    if (is_positive(pos_error)) {
        accel_lim = -accel_min;         // acceleration limit magnitude (positive)
        k_v = jerk_max / accel_lim;     // inner velocity-loop gain derived from jerk/accel limits (1/s)
    } else {
        accel_lim = accel_max;          // acceleration limit magnitude (positive)
        k_v = jerk_max / accel_lim;     // inner velocity-loop gain derived from jerk/accel limits (1/s)
    }

    // Work in the correction frame by removing the feedforward velocity.
    // vel_corr is the correction-frame velocity, so pos_error_dot = vel_desired - vel = -vel_corr.
    const float vel_corr = vel - vel_desired;

    // Velocity correction command derived from position error (second-order limited).
    float vel_corr_cmd = sqrt_controller(pos_error, k_v, accel_lim, dt);

    // Rate-of-change implied by the shaped velocity correction, using correction-frame closing rate.
    const float accel_corr_cmd = sqrt_controller_accel(pos_error, vel_corr_cmd, vel_corr, k_v, accel_lim);

    // Convert the implied rate-of-change term into an equivalent velocity correction bias.
    vel_corr_cmd += accel_corr_cmd / k_v;

    // Limit correction velocity magnitude if velocity limiting is enabled (non-zero limits).
    if (is_negative(vel_min)) {
        vel_corr_cmd = MAX(vel_corr_cmd, vel_min);
    }  
    if (is_positive(vel_max)) {
        vel_corr_cmd = MIN(vel_corr_cmd, vel_max);
    }

    // Total velocity target = feedforward + correction.
    float vel_target = vel_desired + vel_corr_cmd;

    // Constrain total velocity if limiting is enabled and velocity limits are enabled (non-zero).
    if (limit_total) {
        if (is_negative(vel_min)) {
            vel_target = MAX(vel_target, vel_min);
        }
        if (is_positive(vel_max)) {
            vel_target = MIN(vel_target, vel_max);
        }
    }

    // Acceleration demand from velocity error.
    float accel_target = (vel_target - vel) * k_v;

    // Bound acceleration command (before external feedforward is added).
    accel_target = constrain_float(accel_target, accel_min, accel_max);

    // Add external acceleration feedforward.
    accel_target += accel_desired;

    // Constrain total acceleration if limiting is enabled.
    if (limit_total) {
        accel_target = constrain_float(accel_target, accel_min, accel_max);
    }

    // Jerk-limit acceleration toward accel_target.
    shape_accel(accel_target, accel, jerk_max, dt);
}

// Shapes lateral position, velocity, and acceleration using a jerk-limited square-root command model.
// - Computes a velocity correction from position error using a square-root controller.
// - Uses sqrt_controller_accel() to bias the velocity correction based on correction-frame closing rate.
// - Forms a velocity target by adding the correction to the feedforward velocity.
// - Computes an acceleration demand from velocity error using k_v and adds external acceleration feedforward.
// - Limits acceleration magnitude with a braking-priority limiter based on the current velocity direction.
// - Optionally constrains velocity and acceleration magnitudes when limit_total is true.
// - Applies jerk limiting via shape_accel_xy() to ensure smooth acceleration transitions.
void shape_pos_vel_accel_xy(const Vector2p& pos_desired, const Vector2f& vel_desired, const Vector2f& accel_desired,
                            const Vector2p& pos, const Vector2f& vel, Vector2f& accel,
                            float vel_max, float accel_max,
                            float jerk_max, float dt, bool limit_total)
{
    // sanity check vel_max, accel_max and jerk_max.
    if (is_negative(vel_max) || !is_positive(accel_max) || !is_positive(jerk_max)) {
        INTERNAL_ERROR(AP_InternalError::error_t::invalid_arg_or_result);
        return;
    }

    // inner velocity-loop gain derived from jerk/accel limits (1/s)
    const float k_v = jerk_max / accel_max;

    // Velocity correction vector (added to vel_desired to form vel_target).
    Vector2f vel_corr_cmd;

    // Position error to be corrected (direction preserved; magnitude used for shaping).
    const Vector2f pos_error = (pos_desired - pos).tofloat();
    const float pos_error_length = pos_error.length();
    if (is_positive(pos_error_length)) {
        // Correction-frame velocity projected onto the position error direction.
        // For a moving setpoint, pos_error_dot = vel_desired - vel, so the closing rate is -(vel - vel_desired).
        float vel_corr_proj = (vel - vel_desired).dot(pos_error) / pos_error_length;

        // Velocity correction magnitude from square-root position controller.
        float vel_corr_cmd_length = sqrt_controller(pos_error_length, k_v, accel_max, dt);

        // Rate-of-change implied by the shaped velocity correction, using correction-frame closing rate.
        float accel_corr_cmd_length = sqrt_controller_accel(pos_error_length, vel_corr_cmd_length, vel_corr_proj, k_v, accel_max);

        // Convert the implied rate-of-change term into an equivalent velocity correction bias.
        vel_corr_cmd_length += accel_corr_cmd_length / k_v;

        // Limit correction velocity magnitude if velocity limiting is enabled (non-zero limits).
        if (is_positive(vel_max)) {
            vel_corr_cmd_length = constrain_float(vel_corr_cmd_length, -vel_max, vel_max);
        }

        // Map scalar correction back onto the position error direction.
        vel_corr_cmd = pos_error * (vel_corr_cmd_length / pos_error_length);
    }

    // Total velocity target is the sum of feedforward velocity and correction.
    Vector2f vel_target = vel_desired + vel_corr_cmd;
    if (limit_total && is_positive(vel_max)) {
        // Constrain total velocity magnitude if limiting is enabled.
        vel_target.limit_length(vel_max);
    }

    // Acceleration demand from velocity error.
    Vector2f accel_target = (vel_target - vel) * k_v;

    // Limit acceleration magnitude while prioritising braking along the current velocity direction.
    limit_accel_corner_xy(vel, accel_target, accel_max);

    // Add external acceleration feedforward.
    accel_target += accel_desired;

    if (limit_total) {
        // Constrain total acceleration magnitude if limiting is enabled.
        accel_target.limit_length(accel_max);
    }

    // Apply jerk limiting to smoothly approach the target acceleration.
    shape_accel_xy(accel_target, accel, jerk_max, dt);
}

// Shapes angular position, velocity, and acceleration using a jerk-limited square-root command model.
// - Computes an angular velocity correction from angular position error using a square-root controller.
// - Uses sqrt_controller_accel() to bias the angular velocity correction based on correction-frame closing rate.
// - Forms an angular velocity target by adding the correction to the feedforward angular velocity.
// - Computes an angular acceleration demand from angular velocity error using k_v and adds external angular
//   acceleration feedforward.
// - Optionally constrains angular velocity and angular acceleration magnitudes when limit_total is true.
// - Applies jerk limiting via shape_accel() to ensure smooth angular acceleration transitions.
// This is the angular (wrapped) form of shape_pos_vel_accel().
void shape_angle_vel_accel(float angle_desired, float angle_vel_desired, float angle_accel_desired,
                           float angle, float angle_vel, float& angle_accel,
                           float angle_vel_min, float angle_vel_max, float angle_accel_max,
                           float angle_jerk_max, float dt, bool limit_total)
{
    // Wrap desired angle to the nearest equivalent setpoint relative to the current angle.
    const float angle_desired_wrapped = angle + wrap_PI(angle_desired - angle);
    shape_pos_vel_accel( angle_desired_wrapped, angle_vel_desired, angle_accel_desired, angle, angle_vel, angle_accel, angle_vel_min, angle_vel_max, -angle_accel_max, angle_accel_max, angle_jerk_max, dt, limit_total); 
}

// Limits a 2D acceleration vector to prioritize lateral (cross-track) acceleration over longitudinal (in-track) acceleration.
// - `vel_norm` is the normalised velocity (velocity divided by maximum speed) that sets the reference direction used to split the acceleration: the in-track component is parallel to it and the cross-track component is perpendicular. Normalising keys the cross-track prioritisation fade to a fraction of max speed.
// - `accel` is modified in-place to remain within `accel_max`.
// - If the full acceleration vector exceeds `accel_max`, it is reshaped to prioritize the cross-track component.
// - If `vel_norm` is zero, a simple magnitude limit is applied.
// Returns true if the acceleration vector was modified.
bool limit_accel_xy(const Vector2f& vel_norm, Vector2f& accel, float accel_max)
{
    // check accel_max is defined
    if (!is_positive(accel_max)) {
        return false;
    }
    // nothing to do unless the acceleration vector exceeds the limit
    if (accel.length_squared() <= sq(accel_max)) {
        return false;
    }

    // isotropic (direction-preserving) magnitude limit. Used directly when there
    // is no meaningful reference direction, and blended in at low reference
    // magnitude below.
    Vector2f accel_isotropic = accel;
    accel_isotropic.limit_length(accel_max);

    const float ref_mag = vel_norm.length();
    if (!is_positive(ref_mag)) {
        // We do not have a reference direction so do a simple vector length limit
        accel = accel_isotropic;
        return true;
    }

    // limit acceleration to accel_max while prioritizing cross track acceleration
    // calculate acceleration along and perpendicular to the reference direction
    const Vector2f ref_unit = vel_norm / ref_mag;
    // acceleration along the reference direction (in-track)
    float accel_dir = ref_unit * accel;
    // cross track acceleration
    Vector2f accel_cross = accel - (ref_unit * accel_dir);
    if (accel_cross.limit_length(accel_max)) {
        accel_dir = 0.0;
    } else {
        // limit_length can't absolutely guarantee this subtraction
        // won't be slightly negative, so safe_sqrt is used
        float accel_max_dir = safe_sqrt(sq(accel_max) - accel_cross.length_squared());
        accel_dir = constrain_float(accel_dir, -accel_max_dir, accel_max_dir);
    }
    const Vector2f accel_prioritised = accel_cross + ref_unit * accel_dir;

    // Fade between the isotropic limit (weak reference) and the cross-track
    // prioritised limit (reference magnitude at or above LIMIT_ACCEL_XY_MIN_REF).
    // When the reference is small its direction is ill-defined and the prioritised
    // split would re-project the saturated braking command into a lateral
    // acceleration spike (e.g. a hard stick reversal in Loiter). Fading to the
    // direction-preserving limit removes that spike. Both blend inputs have
    // magnitude <= accel_max, so the result does too.
    const float prioritise_ratio = constrain_float(ref_mag / LIMIT_ACCEL_XY_MIN_REF, 0.0f, 1.0f);
    accel = accel_isotropic * (1.0f - prioritise_ratio) + accel_prioritised * prioritise_ratio;

    return true;
}

// Limits a 2D acceleration vector with direction-dependent prioritisation.
// - Acceleration is decomposed into along-track (parallel to velocity) and cross-track components.
// - If braking is requested (negative along-track component), braking is prioritised and the
//   remaining acceleration budget is allocated to cross-track.
// - If no braking is requested (along-track acceleration or zero), cross-track acceleration
//   is prioritised and the remaining budget is allocated to along-track.
// - Ensures the final acceleration magnitude does not exceed accel_max.
// - If velocity is zero (no defined direction), a simple magnitude limit is applied.
// Returns true if the limiting logic was applied.
bool limit_accel_corner_xy(const Vector2f& vel, Vector2f& accel, float accel_max)
{
    // Check accel_max is defined.
    if (!is_positive(accel_max)) {
        return false;
    }

    if (vel.is_zero()) {
        // No along/cross decomposition possible; apply a simple magnitude limit.
        return accel.limit_length(accel_max);
    }

    // Pre-limit to keep the acceleration direction well-conditioned.
    // This allows cross-track components to appear earlier when the upstream
    // acceleration demand (often proportional to velocity error) is very large.
    accel.limit_length(2.0 * accel_max);

    // Unit velocity direction defines the along-track axis.
    const Vector2f vel_unit = vel.normalized();

    // Signed scalar projection of acceleration onto the velocity direction.
    // Negative values correspond to braking.
    float accel_dir_scalar = accel.dot(vel_unit);

    // Along-track and cross-track acceleration components.
    Vector2f accel_dir   = vel_unit * accel_dir_scalar;
    Vector2f accel_cross = accel - accel_dir;

    if (is_positive(accel_dir_scalar)) {
        // Non-braking regime
        // Prioritise cross-track acceleration and allocate the remaining budget to along-track.

        // Limit cross-track magnitude first.
        const float accel_cross_mag = MIN(accel_cross.length(), accel_max);
        const float accel_along_max = safe_sqrt(sq(accel_max) - sq(accel_cross_mag));

        accel_cross.limit_length(accel_max);
        accel_dir.limit_length(accel_along_max);

        accel = accel_cross + accel_dir;
        return true;
    }

    // Braking regime
    // Prioritise along-track deceleration and allocate the remaining budget to cross-track.

    // Limit braking magnitude.
    accel_dir_scalar = MAX(accel_dir_scalar, -accel_max);
    accel_dir = vel_unit * accel_dir_scalar;

    // Allocate remaining acceleration budget to cross-track.
    const float accel_cross_max = safe_sqrt(sq(accel_max) - sq(accel_dir_scalar));
    accel_cross.limit_length(accel_cross_max);

    accel = accel_cross + accel_dir;
    return true;
}

// Piecewise square-root + linear controller that limits second-order response (acceleration).
// - Behaves like a P controller near the setpoint.
// - Switches to sqrt(2·a·Δx) shaping beyond a threshold to limit acceleration.
// - `second_ord_lim` sets the max acceleration allowed.
// - Returns the constrained correction rate for a given error and gain.
float sqrt_controller(float error, float p, float second_ord_lim, float dt)
{
    float correction_rate;
    if (is_negative(second_ord_lim) || is_zero(second_ord_lim)) {
        // No second-order limit: use pure linear controller
        correction_rate = error * p;
    } else if (is_zero(p)) {
        // No P gain, but with acceleration limit — use sqrt-shaped response only
        if (is_positive(error)) {
            correction_rate = safe_sqrt(2.0 * second_ord_lim * (error));
        } else if (is_negative(error)) {
            correction_rate = -safe_sqrt(2.0 * second_ord_lim * (-error));
        } else {
            correction_rate = 0.0;
        }
    } else {
        // Both P and second-order limits defined — use hybrid model
        const float linear_dist = second_ord_lim / sq(p);
        if (error > linear_dist) {
            // Positive error beyond linear region — use sqrt branch
            correction_rate = safe_sqrt(2.0 * second_ord_lim * (error - (linear_dist / 2.0)));
        } else if (error < -linear_dist) {
            // Negative error beyond linear region — use sqrt branch
            correction_rate = -safe_sqrt(2.0 * second_ord_lim * (-error - (linear_dist / 2.0)));
        } else {
            // Inside linear region
            correction_rate = error * p;
        }
    }
    if (is_positive(dt)) {
        // Clamp to ensure we do not overshoot the error in the last time step
        return constrain_float(correction_rate, -fabsf(error) / dt, fabsf(error) / dt);
    } else {
        return correction_rate;
    }
}

// Vector form of `sqrt_controller()`, applied along the direction of the input error vector.
// - Returns a correction vector with magnitude shaped using `sqrt_controller()`.
// - Direction is preserved from the input error.
// - Used in 2D position or velocity control with second-order constraints.
Vector2f sqrt_controller(const Vector2f& error, float p, float second_ord_lim, float dt)
{
    const float error_length = error.length();
    if (!is_positive(error_length)) {
        return Vector2f{};
    }

    const float correction_length = sqrt_controller(error_length, p, second_ord_lim, dt);
    return error * (correction_length / error_length);
}

// Inverts the output of `sqrt_controller()` to recover the input error that would produce a given output.
// - Useful for calculating required error to produce a desired rate.
// - Handles both linear and square-root regions of the controller response.
float inv_sqrt_controller(float output, float p, float D_max)
{
    // Degenerate case: second-order limit (D_max) is positive, but P gain is zero
    if (is_positive(D_max) && is_zero(p)) {
        return (output * output) / (2.0 * D_max);
    }

    // Degenerate case: no D_max, but P gain is non-zero → use linear model
    if ((is_negative(D_max) || is_zero(D_max)) && !is_zero(p)) {
        return output / p;
    }

    // Degenerate case: both gains are zero — no useful model
    if ((is_negative(D_max) || is_zero(D_max)) && is_zero(p)) {
        return 0.0;
    }

    // Compute transition threshold between linear and sqrt regions
    const float linear_velocity = D_max / p;

    if (fabsf(output) < linear_velocity) {
        // Linear region: below transition threshold
        return output / p;
    }

    // Square-root region: above transition threshold
    const float linear_dist = D_max / sq(p);
    const float stopping_dist = (linear_dist * 0.5f) + sq(output) / (2.0 * D_max);
    return is_positive(output) ? stopping_dist : -stopping_dist;
}

// Computes the rate-of-change implied by sqrt_controller() for the commanded correction rate.
// - Uses the chain rule to estimate rate_cmd_dot = d(rate_cmd)/dt based on the actual closing rate.
// - For a fixed target: error = target - state, and error_dot = -rate_state (since state_dot = rate_state).
// - In the linear region of sqrt_controller(): d(rate_cmd)/d(error) = p
// - In the sqrt region of sqrt_controller():   d(rate_cmd)/d(error) = second_ord_lim / |rate_cmd|
// - Therefore:
//     linear region: rate_cmd_dot = -p * rate_state
//     sqrt region:   rate_cmd_dot = -(second_ord_lim / |rate_cmd|) * rate_state
// Notes:
// - If second_ord_lim <= 0, the controller is linear everywhere.
// - If p == 0, the controller is pure sqrt everywhere.
// - rate_cmd must be the output of sqrt_controller() for the same error.
float sqrt_controller_accel(float error, float rate_cmd, float rate_state, float p, float second_ord_lim)
{
    // If we are moving away from the target return zero.
    if (!is_positive(rate_cmd * rate_state)) {
        return 0.0;
    }

    // If no second-order limit, controller is linear everywhere (rate_cmd ~ p*error).
    if (!is_positive(second_ord_lim)) {
        return -p * rate_state;
    }

    // If no P gain but second-order limit exists, controller is pure sqrt everywhere.
    if (!is_positive(p)) {
        if (is_zero(rate_cmd)) {
            return 0.0;
        }
        return -(second_ord_lim / fabsf(rate_cmd)) * rate_state;
    }

    // Both P and second-order limit defined — match sqrt_controller() region selection.
    const float linear_dist = second_ord_lim / sq(p);

    if (fabsf(error) <= linear_dist) {
        // Inside linear region.
        return -p * rate_state;
    }

    // Outside linear region (sqrt branch). Guard divide-by-zero on rate_cmd.
    if (is_zero(rate_cmd)) {
        return 0.0f;
    }
    return -(second_ord_lim / fabsf(rate_cmd)) * rate_state;
}

// Calculates stopping distance required to reduce a velocity to zero using a square-root controller.
// - Uses the inverse of the `sqrt_controller()` response curve.
// - Inputs: velocity, P gain, and max deceleration (`accel_max`)
// - Output: stopping distance required to decelerate cleanly.
float stopping_distance(float velocity, float p, float accel_max)
{
    // Use inverse of sqrt_controller to compute stopping distance from current velocity
    return inv_sqrt_controller(velocity, p, accel_max);
}

// Return the largest M >= 0 that can scale a 3D direction without exceeding
// independent axis limits:
//
//   M * |unit.xy| <= max_xy
//   -max_z_neg <= M * unit.z <= max_z_pos
//
// where unit = normalize(direction). The magnitude of direction is ignored.
//
// max_z_pos limits travel in the +Z direction.
// max_z_neg limits travel in the -Z direction.
// All limits must be positive.
//
// Typical use: limit velocity or acceleration magnitude along a desired
// direction without changing that direction.
//
// Returns 0 if the direction is zero or any limit is zero.
float kinematic_limit(Vector3f direction, float max_xy, float max_z_neg, float max_z_pos)
{
    // Decompose into horizontal magnitude and vertical component
    const float dir_xy = direction.xy().length();

    return kinematic_limit(dir_xy, direction.z, max_xy, max_z_neg, max_z_pos);
}

// Return the largest M >= 0 along a direction defined by horizontal and
// vertical components, constrained by:
//
//   M * |unit.xy| <= max_xy
//   -max_z_neg <= M * unit.z <= max_z_pos
//
// dir_xy (>= 0) and dir_z define a direction; only their ratio matters
// (normalized internally).
//
// max_z_pos limits travel in the +Z direction.
// max_z_neg limits travel in the -Z direction.
// All limits must be positive.
//
// Returns 0 if any limit is zero or the direction is zero.
float kinematic_limit(float dir_xy, float dir_z, float max_xy, float max_z_neg, float max_z_pos)
{
    // Reject invalid limits
    if (is_negative(dir_xy) || !is_positive(max_xy) || !is_positive(max_z_pos) || !is_positive(max_z_neg)) {
        return 0.0;
    }

    // Check for zero length direction vector
    const float dir_length = safe_sqrt(sq(dir_xy) + sq(dir_z));
    if (!is_positive(dir_length)) {
        return 0.0;
    }

    if (is_zero(dir_xy)) {
        // Pure vertical - constrained only by vertical limits
        return is_positive(dir_z) ? max_z_pos : max_z_neg;
    }

    if (is_zero(dir_z)) {
        // Pure horizontal - constrained only by horizontal limits
        return max_xy;
    }

    // Normalize the direction vector (only ratio matters)
    dir_xy /= dir_length;
    dir_z /= dir_length;

    // Compare the direction slope (|dir_z/dir_xy|) to the limit slope
    // (max_z/max_xy) to determine which axis constraint is hit first.
    const float slope = dir_z / dir_xy;

    if (is_positive(slope)) {
        // Positive-Z: constrained by max_z_pos
        if (slope < max_z_pos / max_xy) {
            // Shallow direction: horizontal limit reached first
            return max_xy / dir_xy;
        }
        // Steep direction: vertical limit reached first
        return max_z_pos / dir_z;
    }

    // Negative-Z: constrained by max_z_neg
    if (-slope < max_z_neg / max_xy) {
        // Shallow direction: horizontal limit reached first
        return max_xy / dir_xy;
    }
    // Steep direction: vertical limit reached first
    return -max_z_neg / dir_z;
}

// Applies an exponential curve to a normalized input in the range [-1, 1].
// - `expo` shapes the curve (0 = linear, closer to 1 = more curvature).
// - Typically used for pilot stick input response shaping.
// - Clipped to `expo < 0.95` to avoid divide-by-zero or extreme scaling.
float input_expo(float input, float expo)
{
    // Clamp input to normalized stick range
    input = constrain_float(input, -1.0, 1.0);
    if (expo < 0.95) {
        // Expo shaping: increases control around center stick
        return (1 - expo) * input / (1 - expo * fabsf(input));
    }
    // If expo is too close to 1, return input unchanged
    return input;
}

// Converts a lean angle (radians) to horizontal acceleration in m/s².
float angle_rad_to_accel_mss(float angle_rad)
{
    // Convert lean angle to horizontal acceleration
    return GRAVITY_MSS * tanf(angle_rad);
}

// Converts a lean angle (degrees) to horizontal acceleration in m/s².
float angle_deg_to_accel_mss(float angle_deg)
{
    // Convert degrees to radians, then to acceleration
    return angle_rad_to_accel_mss(radians(angle_deg));
}

// Converts a horizontal acceleration (m/s²) to lean angle in radians.
// Assumes: angle = atan(a / g)
float accel_mss_to_angle_rad(float accel_mss)
{
    // Inverse of angle_rad_to_accel_mss
    return atanf(accel_mss/GRAVITY_MSS);
}

// Converts a horizontal acceleration (m/s²) to lean angle in degrees.
float accel_mss_to_angle_deg(float accel_mss)
{
    // Convert result of radian-based conversion to degrees
    return degrees(accel_mss_to_angle_rad(accel_mss));
}

// Converts pilot’s normalized roll/pitch input into target roll and pitch angles (radians).
// - `roll_in_norm` and `pitch_in_norm`: stick inputs in range [-1, 1]
// - `angle_max_rad`: maximum allowed lean angle
// - `angle_limit_rad`: secondary limit to constrain output while preserving full stick range
// Outputs are Euler angles in radians: `roll_out_rad`, `pitch_out_rad`
void rc_input_to_roll_pitch_rad(float roll_in_norm, float pitch_in_norm, float angle_max_rad, float angle_limit_rad, float &roll_out_rad, float &pitch_out_rad)
{
    // Constrain angle_max to 85 deg to avoid unstable behavior
    angle_max_rad = MIN(angle_max_rad, radians(85.0));

    // Convert normalized pitch and roll stick input into horizontal thrust components
    Vector2f thrust;
    thrust.x = - tanf(angle_max_rad * pitch_in_norm);
    thrust.y = tanf(angle_max_rad * roll_in_norm);

    // Calculate the horizontal thrust limit based on angle limit
    angle_limit_rad = constrain_float(angle_limit_rad, radians(10.0), angle_max_rad);
    float thrust_limit = tanf(angle_limit_rad);

    // Apply limit to the horizontal thrust vector (preserves stick direction)
    thrust.limit_length(thrust_limit);

    // Convert thrust vector back to pitch and roll Euler angles
    pitch_out_rad = - atanf(thrust.x);
    roll_out_rad = atanf(cosf(pitch_out_rad) * thrust.y);
}