Decel-limited Constant-jerk Stopping Profile

Problem

Calculate the stopping distance of a vehicle under constant negative jerk with deceleration limits.

Given Parameters

Initial position: \(s_0\)
Initial velocity: \(v_0\)
Initial acceleration: \(a_0\)
Constant jerk: \(j\) (negative for braking)
Minimum acceleration: \(a_{min}\) (optional limit)

Solution

Kinematic Equations

With constant jerk \(j\):

Acceleration: \(a(t) = a_0 + jt\)
Velocity: \(v(t) = v_0 + a_0 t + \frac{1}{2}jt^2\)
Position: \(s(t) = s_0 + v_0 t + \frac{1}{2}a_0 t^2 + \frac{1}{6}jt^3\)

Stopping Time

Vehicle stops when \(v(t) = 0\):

\begin{equation} v_0 + a_0 t + \frac{1}{2}jt^2 = 0 \end{equation}

Using the quadratic formula:

\begin{equation} t_{stop} = \frac{-a_0 - \sqrt{a_0^2 - 2jv_0}}{j} \end{equation}

Stopping Distance

Substitute \(t_{stop}\) into position equation:

\begin{equation} d_{stop} = v_0 t + \frac{1}{2}a_0 t^2 + \frac{1}{6}jt^3 \end{equation}

With Acceleration Limit

If \(a_{min}\) is specified and reached before stopping:

Phase 1 (jerk phase) - Time to reach limit:

\begin{equation} t_1 = \frac{a_{min} - a_0}{j} \end{equation}

Phase 2 (constant acceleration) - Time at limit:

\begin{equation} t_2 = -\frac{v_1}{a_{min}} \end{equation}

where \(v_1\) is velocity at end of phase 1.

Total stopping distance: \(d_{stop} = d_1 + d_2\)

Implementation

import math
import matplotlib.pyplot as plt
import numpy as np

def stopping_distance(s0, v0, a0, j, a_min=None):
    """
    Calculate stopping distance with constant negative jerk.

    Args:
        s0: initial position
        v0: initial velocity
        a0: initial acceleration
        j: jerk (negative for braking)
        a_min: minimum acceleration limit (optional)

    Returns:
        (stopping_time, stopping_distance)
    """
    if a_min is None:
        # No acceleration limit
        discriminant = a0**2 - 2*j*v0
        t_stop = (-a0 - math.sqrt(discriminant)) / j
        d_stop = v0*t_stop + 0.5*a0*t_stop**2 + (1/6)*j*t_stop**3
        return t_stop, d_stop

    # Time to reach acceleration limit
    t1 = (a_min - a0) / j
    a1 = a0 + j*t1
    v1 = v0 + a0*t1 + 0.5*j*t1**2
    d1 = v0*t1 + 0.5*a0*t1**2 + (1/6)*j*t1**3

    if v1 <= 0:
        # Stops before reaching limit
        discriminant = a0**2 - 2*j*v0
        t_stop = (-a0 - math.sqrt(discriminant)) / j
        d_stop = v0*t_stop + 0.5*a0*t_stop**2 + (1/6)*j*t_stop**3
        return t_stop, d_stop

    # Constant acceleration phase
    t2 = -v1 / a_min
    d2 = v1*t2 + 0.5*a_min*t2**2

    return t1 + t2, d1 + d2

# Example
if __name__ == "__main__":
    s0, v0, a0, j, a_min = 0, 20, 0, -2, -4
    t, d = stopping_distance(s0, v0, a0, j, a_min)
    print(f"Stopping time: {t:.3f} s")
    print(f"Stopping distance: {d:.3f} m")

    # Plot
    t1 = (a_min - a0) / j
    v1 = v0 + a0*t1 + 0.5*j*t1**2

    if v1 > 0:
        time1 = np.linspace(0, t1, 50)
        time2 = np.linspace(t1, t, 50)
        time = np.concatenate([time1, time2])

        jerk = np.concatenate([np.full_like(time1, j), np.zeros_like(time2)])
        accel = np.concatenate([a0 + j*time1, np.full_like(time2, a_min)])

        vel1 = v0 + a0*time1 + 0.5*j*time1**2
        v1_val = v0 + a0*t1 + 0.5*j*t1**2
        vel2 = v1_val + a_min*(time2 - t1)
        vel = np.concatenate([vel1, vel2])

        pos1 = s0 + v0*time1 + 0.5*a0*time1**2 + (1/6)*j*time1**3
        d1 = v0*t1 + 0.5*a0*t1**2 + (1/6)*j*t1**3
        pos2 = d1 + v1_val*(time2 - t1) + 0.5*a_min*(time2 - t1)**2
        pos = np.concatenate([pos1, pos2])
    else:
        time = np.linspace(0, t, 100)
        jerk = np.full_like(time, j)
        accel = a0 + j*time
        vel = v0 + a0*time + 0.5*j*time**2
        pos = s0 + v0*time + 0.5*a0*time**2 + (1/6)*j*time**3

    fig, axes = plt.subplots(2, 3, figsize=(15, 8))
    axes[0,0].plot(time, pos)
    axes[0,0].set_ylabel('Position (m)')
    axes[0,1].plot(time, vel)
    axes[0,1].set_ylabel('Velocity (m/s)')
    axes[0,2].plot(pos, vel)
    axes[0,2].set_xlabel('Position (m)')
    axes[0,2].set_ylabel('Velocity (m/s)')
    axes[1,0].plot(time, accel)
    axes[1,0].axhline(a_min, color='r', linestyle='--', label=f'Limit: {a_min} m/s²')
    axes[1,0].set_ylabel('Acceleration (m/s²)')
    axes[1,0].set_xlabel('Time (s)')
    axes[1,0].legend()
    axes[1,1].plot(time, jerk)
    axes[1,1].set_ylabel('Jerk (m/s³)')
    axes[1,1].set_xlabel('Time (s)')
    axes[1,2].axis('off')
    plt.tight_layout()
    plt.savefig('stopping_profile.png', dpi=150)
    print("\nPlot saved to stopping_profile.png")

Example Results

Without Acceleration Limit

Initial conditions:

\(v_0 = 20\) m/s
\(a_0 = 0\) m/s²
\(j = -2\) m/s³

Results:

Stopping time: 4.472 s
Stopping distance: 59.628 m

With Acceleration Limit

Initial conditions:

\(v_0 = 20\) m/s
\(a_0 = 0\) m/s²
\(j = -2\) m/s³
\(a_{min} = -4\) m/s²

Results:

Stopping time: 6.000 s
Stopping distance: 69.333 m

Figure 1: constant jerk stopping profile