TrapezoidalProfile.md

Trapezoidal Profile (Linear Segment with Parabolic Blends)

Overview & Motivation

A point-to-point move has to reach its goal as fast as the actuator allows without saturating the drive. The trapezoidal profile — linear segment with parabolic blends (LSPB) — does this with the simplest rule that respects both a velocity ceiling and an acceleration ceiling: accelerate at the maximum rate, coast at the maximum speed, decelerate at the maximum rate. Velocity plotted against time is a trapezoid; a move too short to reach the ceiling becomes a triangle.

From the limits and the distance alone, the plan is the minimum-time one. A second mode fixes the duration instead, which multi-axis synchronization needs: all axes must finish together, so the faster ones are slowed rather than the slowest sped up.

Mathematical Theory

Problem Statement

Rest-to-rest move of length $d = |q_f - q_0|$ in direction $\sigma = \pm 1$ (positive when $q_f \ge q_0$), with limits $v_{max} > 0$ and $a_{max} > 0$. The profile is planned for the magnitude and mirrored by $\sigma$.

Minimum-Time Plan

A ramp between rest and speed $v$ takes $v/a_{max}$ and covers $v^2/(2a_{max})$, so the two ramps together consume the blend distance $v^2/a_{max}$. Comparing $d$ with the blend distance at $v_{max}$ selects the shape:

$$ \begin{cases} d \ge v_{max}^2/a_{max}: & v = v_{max}, \quad t_a = v/a_{max}, \quad t_c = (d - v^2/a_{max})/v \quad\text{(trapezoid)} \\ d < v_{max}^2/a_{max}: & v = \sqrt{d,a_{max}}, \quad t_a = v/a_{max}, \quad t_c = 0 \quad\text{(triangle)} \end{cases} $$

The total time is $T = 2t_a + t_c$. The cases agree at $d = v_{max}^2/a_{max}$, so $T$ is continuous in $d$.

Sampling

With $v$, $t_a$, $t_c$, $T$ precomputed, the unsigned state is:

Phase	Interval	Acceleration	Velocity	Position
Parabolic ramp-up	$0 \le t < t_a$	$a_{max}$	$a_{max},t$	$\tfrac12 a_{max},t^2$
Linear cruise	$t_a \le t < t_a + t_c$	$0$	$v$	$v,(t - t_a/2)$
Parabolic ramp-down	$t_a + t_c \le t < T$	$-a_{max}$	$a_{max},(T-t)$	$d - \tfrac12 a_{max},(T-t)^2$

Velocity, acceleration and the position offset are multiplied by $\sigma$. At and beyond $t = T$ the profile holds the goal at rest (velocity and acceleration zero); times before $0$ are treated as $t = 0$, holding the start with zero velocity. The reported jerk is zero in every phase.

Fixed-Duration Plan

The acceleration $a_{max}$ is kept and the cruise speed is lowered until the move takes exactly $T$. The ramps cover $v,t_a$ in total and the cruise covers $v,(T - 2t_a)$, so with $t_a = v/a_{max}$

$$ d = v\left(T - \frac{v}{a_{max}}\right) ;\Longleftrightarrow; v^2 - a_{max}T,v + a_{max},d = 0 . $$

The smaller root keeps $t_c \ge 0$. Multiplying it by its conjugate gives a form with no subtraction of nearly equal numbers:

$$ v = \frac{a_{max}T - \sqrt{a_{max}^2T^2 - 4a_{max}d}}{2} = \frac{2,a_{max},d}{a_{max}T + \sqrt{a_{max}^2T^2 - 4a_{max}d}} . $$

A real root exists iff $a_{max},T^2 \ge 4d$. Equality gives a double root $v = a_{max}T/2$ and a triangle: $T$ is then the shortest duration achievable at this acceleration. Otherwise $t_a = v/a_{max}$ and $t_c = T - 2t_a$.

Multi-Axis Synchronization

Let $T_{sync}$ be the longest axis time and $\lambda = T_{sync}/T_{axis} \ge 1$. Planning an axis again with $(v_{max}/\lambda,; a_{max}/\lambda^2)$ leaves the blend distance $v_{max}^2/a_{max}$ unchanged, so the same shape is selected, while $t_a$ and $t_c$ both scale by $\lambda$: the duration is exactly $\lambda,T_{axis}$. Velocity is divided by $\lambda$ and acceleration by $\lambda^2$.

The fixed-duration plan is the alternative. It keeps $a_{max}$ and only lowers the cruise speed: the smaller root decreases as $T$ grows, so for $T \ge T_{min}$ the cruise speed never exceeds $v_{max}$.

Complexity Analysis

Operation	Time	Space	Notes
Minimum-time plan	$O(1)$	$O(1)$	One comparison, one square root (triangle only)
Fixed-duration plan	$O(1)$	$O(1)$	One square root, one division
Sample	$O(1)$	$O(1)$	One phase branch, a few multiply-adds; no state change

Memory is a fixed handful of scalars: start, direction, distance, acceleration, peak speed and three phase times.

Step-by-Step Walkthrough

Trapezoid. $d = 5$, $v_{max} = 1$, $a_{max} = 2$. The blend distance is $1/2 = 0.5 \le 5$, so a cruise exists: $t_a = 0.5$, $t_c = (5 - 0.5)/1 = 4.5$, $T = 2(0.5) + 4.5 = 5.5$.

$t$	Phase	Position	Velocity	Acceleration
$0.25$	ramp-up	$\tfrac12\cdot 2\cdot 0.25^2 = 0.0625$	$0.5$	$2$
$3$	cruise	$1\cdot(3 - 0.25) = 2.75$	$1$	$0$
$5.25$	ramp-down	$5 - \tfrac12\cdot 2\cdot 0.25^2 = 4.9375$	$0.5$	$-2$
$\ge 5.5$	held	$5$	$0$	$0$

The ramp-up ends at position $0.25$, and the cruise formula gives the same value at $t = 0.5$. The triangle threshold for these limits is $d = 0.5$; for $d = 0.25$, $v = \sqrt{0.5} \approx 0.7071$ and $T \approx 0.7071$, with no cruise.

Fixed duration. $d = 4$, $a_{max} = 1$, $T = 5$: discriminant $25 - 16 = 9$, $v = 2\cdot 4/(5 + 3) = 1$, $t_a = 1$, $t_c = 3$; check $1\cdot(5 - 1) = 4$. For $d = 5$, $a_{max} = 2$, $T = 6$: discriminant $144 - 40 = 104$, $v = 20/(12 + \sqrt{104}) \approx 0.900980$.

Pitfalls & Edge Cases

• Non-positive limits. The minimum-time plan requires $v_{max} &gt; 0$ and $a_{max} &gt; 0$; a violation is an assertion failure, not a recoverable status. The fixed-duration plan returns no result for $a_{max} \le 0$ or $T \le 0$.
• Infeasible duration. If $a_{max}T^2 &lt; 4d$ the plan is rejected, never clamped; the shortest feasible duration is $2\sqrt{d/a_{max}}$. At the boundary, single-precision rounding can push the discriminant just below zero, so plan the slowest axis with the minimum-time plan.
• Cancellation. The textbook root $(a_{max}T - \sqrt{\cdot})/2$ loses its significant digits in single precision when $4a_{max}d \ll a_{max}^2T^2$; the conjugate form does not. The computed cruise time is clamped at zero.
• Zero distance. The minimum-time plan has $v = 0$ and $T = 0$: a zero-length profile that is at rest at the goal for every $t \ge 0$.
• Discontinuous acceleration. Position and velocity are continuous, but acceleration jumps between $0$ and $\pm a_{max}$ at the joins. The jerk impulses are not represented in the reported jerk and can excite structural modes; use the S-curve profile in that case.

Variants & Generalizations

The S-curve profile adds a jerk limit (seven segments) for continuous acceleration at the cost of a longer move. Polynomial profiles (cubic, quintic) are fixed-duration and smooth by construction but enforce no velocity or acceleration ceiling. Unequal acceleration and deceleration limits use the same construction with unequal ramp times.

Applications

• Point-to-point axis moves — servo and stepper drives, gantries, pick-and-place, gimbal slews.
• Multi-axis coordination — time-scaled or fixed-duration plans make every axis arrive together.

Connections to Other Algorithms

The S-curve profile is the jerk-limited upgrade under the same velocity and acceleration limits. The polynomial profile is fixed-time and smooth but offers no limit guarantees. The saturation rate limiter slews a changing setpoint online, whereas the trapezoid plans a whole move offline.

References & Further Reading

• Biagiotti, L. and Melchiorri, C., "Trajectory Planning for Automatic Machines and Robots," Springer, 2008, §3.2.
• Siciliano, B., Sciavicco, L., Villani, L., and Oriolo, G., "Robotics: Modelling, Planning and Control," Springer, 2009, §4.2.