A Stewart Platform is a 6-DOF parallel manipulator that uses six linear actuators connecting a fixed base to a moving top plate via spherical joints. It solves the problem of generating fully controlled motion in all six degrees of freedom — surge, sway, heave, roll, pitch, yaw - from a compact, stiff structure. Each actuator extends or retracts independently, and the combined lengths define the platform's pose. You see this in commercial flight simulators like the CAE 7000XR, Hexagon machine-tool hexapods, and high-end driving rigs.
Stewart Platform (Hexapod) Interactive Calculator
Vary joint play, target accuracy, leg count, and update rate to see platform wobble, required actuator feedback resolution, and timing.
Equation Used
This calculator follows the article accuracy example: joint play is summed across the Stewart platform legs to estimate centre wobble, and actuator length feedback should resolve to about one tenth of the desired platform precision.
- Small radial play errors are estimated as a linear stack across the active legs.
- Feedback resolution should be about one tenth of the desired platform precision.
- Controller timing is represented by update period = 1000 / update_rate.
How the Stewart Platform (hexapod) Works
The platform sits on a fixed base. Six linear actuators rise from that base and meet a moving top plate, with spherical (ball) joints at the top and universal or spherical joints at the bottom. Change any actuator length and the top plate tilts, slides, or rotates. Change all six together in a coordinated way and you get any pose inside the workspace — that is the whole point of a 6-DOF parallel manipulator. Unlike a serial robot arm where errors stack joint by joint, every actuator here works in parallel against the same load, so stiffness is high and positional error is low.
The geometry matters more than people expect. The standard octahedral hexapod arrangement places the six top joints in three close pairs, rotated 60° from three close pairs at the base. That layout gives a roughly symmetric workspace and avoids the worst singularities near the home position. If you build the platform with the joints too close to symmetric — base hexagon and top hexagon nearly the same size and aligned — you land in a Type-II singularity where the platform gains an uncontrolled DOF and can collapse under load. The fix is well known: offset the top joint pattern by 60° from the base and make the top plate roughly 60-80% the diameter of the base.
Tolerances drive the rest. Joint slop kills accuracy fast — 0.05 mm of radial play at each ball joint can show up as 0.3 mm of wobble at the platform centre once you sum across six legs. Actuator length feedback needs to resolve to roughly 1/10 of your target platform precision, so for ±0.1 mm pose accuracy you need encoders or pots resolving 10 µm on each leg. If timing between actuators drifts more than a few milliseconds during a coordinated move, you get a visible jitter — pilots in motion simulators describe it as the cabin "hunting" — and the cause is almost always a control loop that updates each leg independently rather than synchronising the trajectory.
Key Components
- Six Linear Actuators (legs): Each leg changes length to drive the top plate. Strokes typically run 100 mm to 1000 mm depending on the platform size. For a desktop simulator we run Linear Actuator units with 200 mm stroke and 750 N rated force; for a full flight sim the legs are hydraulic with 1.5 m stroke and 50 kN per leg.
- Upper Spherical Joints: Connect the top of each actuator to the moving plate. Must allow at least ±30° of swing in two axes. Radial play needs to stay under 0.02 mm — anything more and you cannot hold sub-millimetre pose accuracy because the slop multiplies across six legs.
- Lower Universal or Spherical Joints: Connect the actuator base to the fixed frame. Universal joints are cheaper but lock out one rotation; full spherical joints give cleaner kinematics. Either way, the joint centre must sit on the actuator's line of action within 0.5 mm or your inverse kinematics solver will produce pose errors that look like calibration drift.
- Moving Top Plate: Carries the payload — a cockpit, a workpiece, an antenna. Stiffness matters: any flex in the plate shows up as unmodelled compliance in the controller. We use 25 mm aluminium tooling plate or welded steel weldments with deflection under 0.1 mm at full rated load.
- Fixed Base: Anchors the lower joints. Must be at least 3-4× stiffer than the top plate or the whole platform vibrates as a single mode. Bolt it to concrete or a 50+ mm steel plate for any serious application.
- Controller and Inverse Kinematics Solver: Takes the desired 6-DOF pose (x, y, z, roll, pitch, yaw) and computes six leg lengths in real time. Update rate needs to hit at least 1 kHz for smooth motion-simulator response. Lower than 200 Hz and you feel the steps.
Industries That Rely on the Stewart Platform (hexapod)
The Stewart Platform earns its keep wherever you need precise, stiff, six-axis motion under load. The reason it dominates flight simulation and parallel-kinematic machining is the same: parallel legs share the load, so payload-to-weight ratio beats any serial arm by a factor of 5-10×. The catch is workspace — a hexapod's reachable volume is small relative to its footprint, which is why you do not see Stewart Platforms picking parts off conveyors. They live where motion fidelity, stiffness, and accuracy matter more than reach.
- Flight Simulation: CAE 7000XR full-flight simulators use a six-leg hydraulic Stewart Platform certified to FAA Level D, with ±1.2 m stroke per leg and 60° pitch/roll authority.
- Driving and Motorsport Simulators: Dynisma DMG-1 and Cruden Hexatech rigs use electric Stewart Platforms running 1 kHz update rates to reproduce road-surface micro-vibration for F1 driver-in-the-loop testing.
- Machine Tool Hexapods: Hexagon Metrology and the historic Geodetic Variax milling machine use the Stewart geometry to mount a spindle, achieving 5-axis machining without rotary tables.
- Astronomy and Optics: Large telescope secondary-mirror positioners — including units on the Gran Telescopio Canarias — use compact hexapods to align mirrors to micron precision.
- Surgical and Medical Robotics: Stryker Mako and similar orthopaedic surgery platforms use small-stroke hexapods to position cutting guides within 0.1 mm during knee and hip arthroplasty.
- Vibration Test Stands: Team Corporation Tensor 18 kN multi-axis shakers use a Stewart geometry to apply controlled 6-DOF vibration to satellites and avionics during qualification testing.
The Formula Behind the Stewart Platform (hexapod)
The core math of a Stewart Platform is the inverse kinematics — given the desired pose of the top plate, what length must each leg be? The forward problem (legs to pose) is nasty and has up to 40 solutions; the inverse problem is clean and closed-form, which is why every controller runs inverse kinematics in real time. At the centre of the workspace, leg lengths sit near nominal and the math behaves well. Push the platform toward the edges of its travel — high pitch, high heave, high yaw combined — and leg lengths diverge fast, with one leg hitting full extension while the opposite leg compresses to its hard stop. The sweet spot for most simulators sits around 30-50% of maximum stroke and ±15° of rotation, where actuator velocity demands stay low and you stay clear of singularities.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| Li | Length of leg i (i = 1 to 6) | m | in |
| T | Translation vector of top plate centre relative to base | m | in |
| R | 3×3 rotation matrix for top-plate orientation (roll, pitch, yaw) | dimensionless | dimensionless |
| bi | Position of upper joint i in the top-plate frame | m | in |
| ai | Position of lower joint i in the base frame | m | in |
Worked Example: Stewart Platform (hexapod) in a 2-seat home racing simulator hexapod
You are building a 2-seat home racing simulator on a Stewart Platform. The base hexagon has a 600 mm radius to the lower joint pairs, the top plate has a 400 mm radius, and the home-position vertical gap between plates is 500 mm. You want to compute leg 1's length when the cockpit pitches forward 10° (nominal driving input), and check what happens at low input (2°) and at the limit you plan to allow (25°).
Given
- rbase = 0.600 m
- rtop = 0.400 m
- h0 = 0.500 m
- θpitch,nom = 10 °
- Leg 1 base joint a1 = (0.600, 0, 0) m
- Leg 1 top joint b1 (in plate frame) = (0.400, 0, 0) m
Solution
Step 1 — at nominal 10° pitch, build the rotation matrix R for rotation about the y-axis and apply it to b1:
Step 2 — add the home translation T = (0, 0, 0.500) m to get the top joint's position in base coordinates, then subtract a1:
Step 3 — leg length is the magnitude of Δ:
Step 4 — at the low end of the typical operating range, 2° pitch, the same calculation gives L1,low ≈ 0.4485 m. The leg has shortened by only 3 mm from the home length of about 0.4516 m. That is a barely perceptible motion — the driver feels it as a faint cue, the kind of micro-pitch you want for road-camber feedback at speed.
Step 5 — at the high end, 25° pitch:
That is 64 mm of stroke change from home — a strong, clearly felt brake-dive cue. Above 25°, the opposite leg (leg 4) starts running out of extension and the platform approaches a workspace boundary; push past 30° and you risk a Type-II singularity where the platform briefly loses control authority in heave.
Result
Leg 1 needs to measure 477. 2 mm at the nominal 10° pitch pose. At 2° pitch the leg sits at 448.5 mm — a 3 mm change you would barely see but the driver feels as subtle road texture. At 25° pitch it drops to 413.5 mm, a clear 64 mm stroke that delivers a hard brake-dive cue, and that is roughly the limit before the opposite leg bottoms out and the workspace closes in. If your measured leg length differs from 477.2 mm by more than 1 mm, the usual suspects are: (1) joint-centre offset error — your CAD model placed b1 at the bolt centre but the actual ball-joint pivot sits 2-3 mm above that, which shows up as systematic pose error; (2) base-frame deflection under load, which moves a1 away from its theoretical position when the cockpit is occupied; or (3) actuator zero-point drift after a power cycle if you are using relative encoders without a homing routine.
Choosing the Stewart Platform (hexapod): Pros and Cons
The Stewart Platform is not the only way to move something in six axes. The real question is whether you need stiffness and precision over a small workspace, or reach over a large one. Pick the wrong topology and you spend twice the budget for half the result.
| Property | Stewart Platform (Hexapod) | 6-Axis Serial Robot Arm | Gough-Variant 3-RPS Tripod |
|---|---|---|---|
| Degrees of freedom | 6 (full) | 6 (full) | 3 (heave, roll, pitch) |
| Payload-to-weight ratio | 5:1 to 10:1 | 0.1:1 to 0.3:1 | 3:1 to 5:1 |
| Positional accuracy | ±0.01 to ±0.1 mm | ±0.05 to ±0.3 mm | ±0.05 mm |
| Workspace volume vs footprint | Small (~10-20% of footprint) | Large (full sphere of arm reach) | Very small |
| Stiffness | Very high — parallel legs | Low — cantilevered links | High |
| Inverse kinematics | Closed-form, easy | Closed-form for most arms | Closed-form, easy |
| Forward kinematics | Hard — up to 40 solutions | Easy | Moderate |
| Typical cost (industrial) | $15k-$500k | $25k-$150k | $8k-$40k |
| Best application fit | Motion sims, machining, vibration test | Pick-place, welding, assembly | Antenna pointing, mirror tip-tilt |
Frequently Asked Questions About Stewart Platform (hexapod)
You are approaching a singularity. As the top plate rotates, the legs start aligning with each other in projection — at certain poses two or more legs become geometrically parallel or coplanar, and the platform gains an uncontrolled DOF in that direction. The Jacobian determinant drops toward zero and the controller can no longer resist load along that axis.
Map your singularity surfaces in CAD before you build, and software-limit the controller to stay at least 5° of rotation away from any singular pose. The classic fix is to redesign the joint pattern — rotating the top hexagon by 60° relative to the base buys you significantly more usable workspace before singularity.
It comes down to bandwidth and payload. Electric Linear Actuator legs give you 1-3 kW per leg, clean control, no fluid, and bandwidth up to about 20 Hz — fine for driving sims, machining, and most industrial work. Hydraulic legs deliver 50+ kN per leg with bandwidth above 50 Hz, which is why every commercial flight simulator certified to FAA Level D still runs hydraulics.
Rule of thumb: under 5000 kg payload and under 30 Hz bandwidth, go electric. Above either threshold, hydraulic still wins on power density. Hybrid designs exist but the plumbing and control complexity rarely pays off.
This is almost always a control loop issue, not a mechanical one. The six legs are running independent PID loops that each fight to hold their commanded length, but small differences in loop tuning or feedback noise cause the legs to chase each other — leg 1 overshoots, leg 4 corrects, leg 2 reacts, and the platform develops a low-amplitude limit cycle.
Check that all six leg controllers share the same gains and the same sample clock. Better still, run the inverse kinematics inside a single coordinated trajectory generator that updates all six setpoints synchronously at 1 kHz. The jitter usually disappears the moment the legs stop arguing with each other.
You can, but you usually shouldn't. The whole reason a Stewart Platform exists is high stiffness in a small workspace. Bolting a long serial axis on top throws away the stiffness advantage — the new long axis becomes the compliance bottleneck, and you've just built an expensive way to do what a SCARA or delta robot does for a quarter the price.
The exception is when you need 6-DOF orientation control during the pick — for instance, mating an aerospace fastener at a compound angle. Then the hexapod stays useful, but you size it for the orientation task, not for reach.
Edge error in a calibrated hexapod is usually geometric parameter error, not actuator error. The inverse kinematics depends on knowing all 12 joint locations (six base, six top) within tenths of a millimetre. If your CAD numbers do not match the as-built joint pivot centres — and they rarely do, because ball joints have offsets between the bolt hole and the rotation centre — the error is small at home and grows linearly toward the workspace boundary.
The fix is geometric self-calibration: command 30-50 known poses, measure the actual top-plate pose with a laser tracker or photogrammetry, and solve for the 12 joint positions plus actuator offsets as a least-squares fit. Most commercial hexapods ship with this routine built in. Do it once and edge error typically drops below 0.1 mm.
Start from the worst-case combined pose, not from individual axes. A platform that needs ±300 mm heave, ±20° roll, and ±20° pitch separately needs much more stroke than one that needs those simultaneously, because in the combined pose one leg extends to maximum while another compresses to minimum.
Run your inverse kinematics over the corner cases of the desired workspace and read off the maximum and minimum leg lengths. Add 10-15% margin on each end so you never operate at the hard stops, where actuator velocity must go to zero and singularity surfaces cluster. For a typical motion-simulator with ±300 mm heave and ±20° rotations, expect leg strokes around 600-800 mm.
At large pitch angles the legs are no longer pushing efficiently against the load. The vertical force component on the platform is the actuator force times the cosine of the leg angle from vertical, so as legs splay outward to support a tilted plate, more of their force goes sideways and gets cancelled by the opposing leg rather than supporting the payload.
The Jacobian condition number gets worse, the legs need higher velocities to produce the same platform velocity, and you hit velocity limits before force limits. If the platform feels sluggish before it feels weak, you are velocity-limited at that pose — either oversize the actuators on speed, or shrink the operational rotation envelope.
References & Further Reading
- Wikipedia contributors. Stewart platform. Wikipedia
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