Rectilinear motion is the movement of a body along a single straight line, where every point on the body travels the same distance in the same direction at the same time. Industrial linear stages routinely deliver this kind of motion at speeds up to 5 m/s with positioning accuracy under 5 µm. We use it whenever a process needs predictable straight-line travel — feeding stock into a press, raising an elevator car, or driving the X-axis of a CNC router. The result is a motion path you can measure, repeat, and control with a single position variable.
Rectilinear Motion Interactive Calculator
Vary initial position, velocity, acceleration, time, and stroke length to see the carriage position and travel on a single straight guide axis.
Equation Used
This calculator applies the standard rectilinear kinematic relation for one-axis translation. The initial position is entered in mm, while velocity and acceleration use m/s and m/s^2; the calculation converts travel back to mm for the carriage position.
- Motion is constrained to one straight axis.
- Acceleration is constant over the selected time interval.
- Position is measured from the rail zero point.
- Guide stiffness, friction, backlash, and compliance are not included.
Inside the Rectilinear Motion
Rectilinear motion is the simplest case of translational motion — every particle in the moving body follows a parallel straight line. No rotation, no curve, no yaw. You drive a carriage along a guide rail, and as long as the rail is straight and the load is constrained, the carriage translates in pure linear motion. Position depends on one variable, x. Velocity is the time derivative, dx/dt. Acceleration is d²x/dt². That's the whole kinematic story — but the engineering sits in how you constrain the body so it actually behaves that way.
The constraint comes from the guideway. A profiled linear guide rail with recirculating ball bearings, a pair of round shafts with linear bushings, or a dovetail slideway — each blocks the five degrees of freedom you don't want and leaves only the one you do. If the rails are out of parallel by more than 0.02 mm/m, you'll feel binding. If preload is too tight, you'll see heat and premature ball brinelling. If preload is too loose, the carriage rocks under cutting load and your straight line becomes a wavy line. We see this constantly on retrofits — a customer reports vibration at high speed, and the cause is almost always rail parallelism or a missing preload spacer.
The actuator converts rotary input from a motor into rectilinear translation. A ball screw turns rotation into linear travel with around 90% efficiency. A belt drive does the same job with lower stiffness but higher speed — up to 5 m/s on a 25 mm wide steel-cord belt. A Linear Actuator gives you direct straight-line force without any external rotary stage. Failure modes are predictable: lost steps from undersized motors, screw whip above critical speed, belt stretch under sudden acceleration, or a guide carriage running dry because the auto-luber emptied a month ago. Get the constraint, the drive, and the lubrication right and rectilinear translation is the most reliable motion you can build.
Key Components
- Guide Rail and Carriage: The profiled rail constrains the moving body to a single straight axis. A standard 20 mm THK SSR or HIWIN HGR rail holds straightness within 13 µm per 300 mm; carriage preload is typically Z0 (light) for general work or ZA (medium) for cutting loads. Parallelism between paired rails must hold under 0.02 mm/m or the carriages will bind.
- Linear Drive (Screw, Belt, or Linear Actuator): Converts input power into straight-line force. A 16 mm × 5 mm lead ball screw delivers around 1.5 kN dynamic load at 3000 RPM input. A Linear Actuator with integrated motor and gearbox skips the conversion step entirely — pick stroke, force, and speed directly from the spec sheet.
- Position Feedback: A linear encoder or rotary encoder closes the loop on actual position. Glass-scale linear encoders resolve to 0.1 µm; rotary encoders on the screw shaft give resolution equal to lead divided by counts per rev — a 5 mm lead with a 10,000-count encoder gives 0.5 µm per count, but only if backlash and screw error are calibrated out.
- End Limits and Hard Stops: Inductive limit switches or magnetic sensors catch overtravel before mechanical hard stops engage. Hard stops are typically polyurethane bumpers rated for the carriage's full kinetic energy at rapid speed — size them for ½ × m × v² of the moving load.
- Lubrication System: Recirculating ball carriages need grease replenishment every 100 km of travel or 3 months, whichever comes first. Run dry and the balls flat-spot the raceway within hours, destroying the straight-line accuracy that justified the rail in the first place.
Where the Rectilinear Motion Is Used
Rectilinear motion shows up anywhere a process needs repeatable straight-line travel. The choice of drive — ball screw, belt, rack-and-pinion, hydraulic cylinder, or Linear Actuator — depends on stroke, speed, force, and duty cycle, but the underlying motion is identical. You'll find pure linear motion driving the cutting head of a Trumpf laser, the platen of a Husky injection moulder, and the carriage of a residential stairlift.
- Machine Tools: X, Y, and Z axes on a Haas VF-2 vertical machining centre — three orthogonal rectilinear axes driven by ball screws on profiled rails delivering ±5 µm positioning.
- Vertical Transport: Elevator car travel in a KONE MonoSpace 500 — the car is the moving body in pure rectilinear translation along guide rails, driven by a gearless traction machine.
- Material Handling: Linear shuttle carts on a Daifuku ASRS rack — straight-line travel between aisles at up to 4 m/s, positioning to a tote location within ±3 mm.
- Medical Equipment: Patient-table travel on a Siemens MAGNETOM Vida MRI scanner — rectilinear motion into the bore with sub-millimetre repeatability over a 2 m stroke.
- Furniture and Ergonomics: Height adjustment on a sit-stand desk frame using a paired Linear Actuator on each leg — synchronised rectilinear translation lifting up to 100 kg of desktop and monitor load.
- Packaging Machinery: Bag-feed stroke on a Bosch SVE 2520 vertical form-fill-seal machine — the film carriage indexes downward in a precise linear pull controlled by a servo-driven belt drive.
The Formula Behind the Rectilinear Motion
The core kinematic equation for rectilinear motion under constant acceleration tells you where the body will be at any time given its starting conditions. At the low end of typical motion-control duty — slow, smooth indexing under 0.1 m/s — the velocity term dominates and acceleration barely matters. At the nominal range most CNC and packaging axes operate in, both terms contribute and you size the drive around the peak ½at² contribution during ramp-up. Push to the high end — rapid traverses above 2 m/s on a long stroke — and acceleration becomes the limiting factor because motor torque, screw critical speed, or belt stretch caps how fast you can change velocity. The sweet spot for most servo-driven linear stages sits around 1 m/s peak velocity with 5–10 m/s² acceleration.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| x(t) | Position at time t | m | in |
| x0 | Initial position | m | in |
| v0 | Initial velocity | m/s | in/s |
| a | Constant acceleration | m/s2 | in/s2 |
| t | Elapsed time | s | s |
Worked Example: Rectilinear Motion in a semiconductor wafer-handling gantry
You are sizing the rectilinear X-axis on a wafer-handler gantry for a Tokyo Electron CLEAN TRACK ACT12 coater line. The carriage carries a 4 kg end-effector across a 600 mm stroke between the load port and the spin chuck. The motion profile starts at rest, accelerates at 8 m/s², cruises at 1.2 m/s, then decelerates symmetrically. You need to know how far the carriage travels during the acceleration phase and how that changes if you scale acceleration up or down across the typical 4–16 m/s² range that the linear stage supplier rates.
Given
- x0 = 0 m
- v0 = 0 m/s
- anom = 8 m/s²
- vcruise = 1.2 m/s
- Stroke = 0.600 m
Solution
Step 1 — find the time to reach cruise velocity at the nominal 8 m/s² acceleration:
Step 2 — compute the distance covered during that acceleration ramp using the rectilinear motion equation with x0 = 0 and v0 = 0:
So at nominal tuning the carriage burns 90 mm of its 600 mm stroke just getting up to speed, leaves another 90 mm for the deceleration ramp, and only 420 mm of the stroke runs at full cruise. That's a comfortable working zone — the stage spends most of its travel at the cruise setpoint where the encoder loop is well-behaved.
Step 3 — at the low end of the supplier's rated range, 4 m/s²:
That eats 360 mm of the 600 mm stroke between accel and decel ramps, leaving only 240 mm of cruise. The handler still works but the cycle time stretches and the carriage barely settles into steady-state before it needs to slow again.
Step 4 — at the high end, 16 m/s²:
The ramps shrink to 45 mm each, leaving 510 mm of cruise — best cycle time on paper. But at 16 m/s² the inertial reaction force on the 4 kg payload is 64 N, which puts a 1.6× higher torque demand on the servo and risks lost steps if the drive is undersized. It also tightens the demand on belt stiffness or screw whip margin.
Result
At the nominal 8 m/s² setting the carriage covers 90 mm during the acceleration phase, which leaves a usable 420 mm cruise window inside the 600 mm stroke. That feels right — the wafer handler hits cruise velocity early, runs steady through the middle of travel, and decelerates cleanly into the chuck position. At 4 m/s² the ramps consume 180 mm each and cruise drops to 240 mm; at 16 m/s² ramps shrink to 45 mm with 510 mm of cruise, but you pay for it in servo current and mechanical stress. If you measure 110 mm of ramp travel instead of the predicted 90 mm, the most common causes are: (1) the servo current limit clamping below the commanded 8 m/s², so actual acceleration is closer to 6.5 m/s² — check the drive's torque-saturation flag, (2) coupling backlash between motor and ball screw stretching the effective ramp, typical on jaw couplers worn past 0.05 mm play, or (3) belt drives running below the manufacturer's tension spec, which adds compliance and lengthens the ramp by 15–20% before the carriage actually moves with the motor.
Rectilinear Motion vs Alternatives
Rectilinear motion is one option among several when you need controlled travel between two points. The honest comparison is against curvilinear translation (where the body follows a curved path) and rotary motion converted to oscillating linear travel via a crank-slider. Each has a place — pick on stroke, accuracy, and how the downstream process consumes the motion.
| Property | Rectilinear Motion (linear stage) | Curvilinear Motion (arc / cam follower) | Crank-Slider Reciprocating Motion |
|---|---|---|---|
| Peak speed | Up to 5 m/s on belt drive, 2 m/s on ball screw | Limited by follower contact, typically under 3 m/s tip speed | Up to 1500 strokes/min on a high-speed press |
| Positioning accuracy | ±5 µm with linear encoder feedback | ±50 µm typical, cam profile tolerance dependent | Stroke endpoints fixed by geometry, no mid-stroke control |
| Stroke length | Practical up to 6 m, longer with rack-and-pinion | Limited to follower arc, typically under 200 mm | Fixed by crank radius, typically under 500 mm |
| Capital cost (1 m stroke, 5 µm class) | $3,000–$8,000 for stage + drive | $1,500–$4,000 for cam + follower assembly | $800–$2,000 for crank-slider with motor |
| Application fit | CNC, semiconductor, metrology, packaging | Indexing, rotary cam-driven assembly | Presses, pumps, fixed-stroke feeders |
| Complexity / control overhead | High — needs servo, encoder, tuning | Low — purely mechanical once cam is cut | Low — single rotary input, no feedback needed |
Frequently Asked Questions About Rectilinear Motion
That gap is reversal error, and it almost always lives in the drivetrain rather than the rail. Ball screws have axial backlash from the nut preload setting — a Z0 preload class on a typical 16 mm screw gives 5–15 µm reversal, while a P-class preloaded nut drops it under 2 µm. Couplings add their own slop: a worn jaw coupler can contribute another 10–20 µm.
Quick diagnostic — command a 1 mm move forward, then 1 mm reverse, and read the linear encoder. The difference is your reversal error. Fix it by upgrading to a preloaded nut, switching to a bellows coupling, or compensating in software with a backlash table.
Above roughly 1.5 m stroke the ball screw runs into critical speed limits — a 16 mm screw whips at around 2500 RPM with fixed-supported ends, which caps your top speed near 0.2 m/s on a 5 mm lead. A belt drive doesn't care about length and happily runs at 3–5 m/s on the same stroke.
Pick the screw if you need 5 µm class accuracy or high thrust at low speed. Pick the belt if you need speed, long stroke, and you can live with 50–100 µm positioning. For a 2 m wafer-handler gantry, belt almost always wins. For a 2 m grinder Z-axis, screw wins.
You've found a structural resonance. The carriage, rails, and supporting frame form a mass-spring system with a natural frequency, and at 1.8 m/s the ball passing frequency of the recirculating bearings or a belt tooth-meshing frequency lines up with that mode.
Calculate ball pass frequency as v / ball-pitch — a typical 20 mm carriage with 3.175 mm balls passes a ball every 3.175 mm, giving 567 Hz at 1.8 m/s. If your frame's first bending mode is near there, you'll feel it. Fixes: stiffen the frame, add a tuned mass damper, or notch out that velocity in your motion profile.
End-of-stroke stalls usually come from one of two causes. First, mechanical end stops or sealed end caps add resistance the motor wasn't sized for — Acme-screw actuators in particular show a 30–50% torque spike in the last 5 mm. Second, voltage sag under sustained current pulls the motor below its rated speed-torque curve, especially on 12 V systems running long supply leads.
Check input voltage at the actuator terminals during the stall, not at the power supply. If you see more than 10% drop, the wire gauge is undersized. If voltage holds steady, the issue is mechanical end-of-stroke binding and you need to back off your software limit by 2–3 mm.
For most positioning work, no — straightness error of 5–15 µm over a 500 mm stroke is below what your process cares about. For metrology, semiconductor lithography, or coordinate measuring, yes — you map it. A profiled rail's straightness comes from the rail mounting surface, not the rail itself, so a rail bolted to a poorly machined base can show 50 µm of waviness even though the rail spec is 5 µm.
The rule of thumb: if your application tolerance is more than 10× the rail's stated straightness, ignore it. If it's less than 3×, map it with a laser interferometer or straightedge and apply software compensation.
Linearly, until you hit the motor's torque limit. The drive sees inertia from the payload, the carriage, the screw or belt pulley, and the motor rotor — and most undersized axes have the motor rotor itself contributing 30–50% of total reflected inertia. Doubling payload from 4 kg to 8 kg might only drop max acceleration from 16 m/s² to 12 m/s², not to 8 m/s², because the motor inertia doesn't change.
Always compute the inertia mismatch ratio: load inertia divided by motor inertia. Stay under 10:1 for crisp servo response. Above 20:1 you'll see overshoot and ringing no matter how well you tune.
That's belt stretch under acceleration combined with encoder placement. If your encoder is on the motor shaft rather than directly on the carriage, the controller measures motor position and assumes the belt is rigid. On a long fast move, the belt elastically stretches by 0.2–0.5 mm during acceleration and recovers during deceleration — but recovery isn't instant, so the carriage settles short.
Two fixes. Add a linear encoder on the carriage and close the loop on actual position, which costs more but eliminates the issue. Or pre-tension the belt to manufacturer spec — most stretch-related errors disappear when belt tension is brought from 60% of nominal up to 100%, measured with a sonic tension meter.
References & Further Reading
- Wikipedia contributors. Linear motion. Wikipedia
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