Trapezoidal Trajectory Calculator

What is a Trapezoidal Trajectory?

When a stage moves between two points, it typically doesn't accelerate and decelerate instantly. There is a period of constant acceleration a, a period of cruising at top speed v, and then finally a period of constant deceleration at the same rate a. The resulting velocity profile has the shape of a trapezoid.

A trapezoidal trajectory is uniquely defined by 3/4 of these parameters: total time t, distance d, acceleration a, and top speed v. Only 3 of these 4 values are required to solve for the 4th.

Each acceleration/deceleration takes time ta and covers distance da. The cruise covers the rest of the distance dc in time tc:

ta = v / ada = v² / (2a)dc = d − v² / atc = (d − v² / a) / vt = d / v + v / a

No cruise: if d ≤ v² / a, the move never reaches v. It becomes a triangular profile:

vpeak = √(a·d)t = 2√(d / a)

To solve for a top speed instead of a time, the following equation can be used if t ≥ 2√(d / a).

v = (a·t − √(a²t² − 4ad)) / 2

Here are three examples of possible trajectories that cover the same distance in the same time, using different accelerations. Each shape has the same area, which represents the distance travelled.

Introducing Thrust-Speed Data

Typically, the parameters of the trapezoidal trajectory are highly dependent on the physical device being used to perform the movement. Device specifications such as the max speed, maximum acceleration, and most importantly thrust-speed, can all limit an otherwise feasible move from point A to point B.

Thrust-speed data is used to compute an appropriate acceleration setting for the device when the distance and time (or speed) are known. This requires computing the effective moving mass of the payload plus all other components being accelerated during a move.

Effective moving mass

In addition to the payload being moved, the mass of the carriage, the inertia of the motor rotor, and the inertia of the lead screw must be included in the calculation.

meff = payload + carriage + rotor equivalent + screw equivalent

On belt-driven stages, pulleys and belt would take the place of the screw.

To convert the rotational inertia of the motor rotor and lead screw, we convert it to an equivalent mass at the carriage. This is the mass that would have the same kinetic energy as the rotating part when moving at the carriage's speed v. Here the lead L is the distance the carriage travels per motor revolution, J is the rotor's inertia, and ω is the rotor's angular speed:

ω = 2π·v / L½·J·ω² = ½·meq·v²meq = J·(2π / L)²

The speed v appears on both sides and cancels out, so the equivalent mass is the same at every speed. It depends only on the rotor's inertia and the lead, not on the trajectory.

The lead screw is treated as a solid steel cylinder, with diameter d and length ℓ taken from the product's data, a density of ρ = 7850 kg/m³, and a radius of r = d / 2. Because the screw spins with the rotor, it is converted to an equivalent mass in the same way. On actuators where the screw moves with the load instead of spinning, the screw is simply part of the moving mass.

ms = ρ·π·r²·ℓJs = ½·ms·r²meq = Js·(2π / L)²

For example, a stage with a 1.27 mm lead and a rotor inertia of 7.3 × 10⁻⁶ kg·m² (such as the LSQ075A) has an equivalent rotor mass of meq = 7.3 × 10⁻⁶ × (2π / 0.00127)² ≈ 179 kg. On stages with a fine lead, the stage's own rotor often outweighs the payload.

Rotary stages work the same way, using torque and inertia instead of force and mass. The torque T needed is the effective moving inertia Jeff multiplied by the angular acceleration α (in rad/s²). The rotor's inertia is scaled by the gear ratio N, which is 360° divided by the stage's rotation per motor revolution:

T = Jeff · αJeff = load inertia + rotor inertia × N²

You can see each term, and where its value came from, in the Effective moving mass table under the thrust–speed plot (Effective moving inertia for rotary stages). Where a product's data doesn't include a value, it is estimated.

Automatically Calculating Acceleration

Typically for a given move, you may want to either minimize the top speed or the required acceleration. Minimizing top speed (maximizing acceleration) lowers the kinetic energy and makes the movement safer in case of unexpected impacts. Minimizing acceleration (higher top speed) makes sense if your payload is sensitive to handling (e.g. liquid handling).

This calculator allows you to automatically select your desired constraint (maximize or minimize acceleration) and it will compute a feasible trajectory based on the stage's specifications. This is useful for determining if a stage is suitable for your application, or for planning particularly demanding moves with existing stages. This calculation is done numerically using the thrust-speed data and does not have a closed form expression.

Assumptions

• The acceleration and deceleration use the same rate, with no jerk limit (no S-curve smoothing).
• The stage is horizontal. Any additional friction you enter is treated as a constant force opposing the motion. The stage's internal friction is already included in the measured thrust-speed data.
• The curves are derated, and only apply at their stated test conditions (voltage, current and controller).
• Where a product has no data for a mass or inertia term, that term is estimated.
• Continuous thrust isn't checked, so long or repeated moves aren't checked for motor heating.