A multi-axis stage combines several single-axis stages to move the carriage. This article answers one question: how do you combine multiple axes, and what is a hexapod?
The answer: serial multi-axis stages combine single-axis platforms in sequence (XY/XYZ/XYθ), with the lower carriage carrying the upper stage. Errors must be assessed at the actual workpoint. A hexapod connects two plates through six parallel actuators to coordinate three translations and three rotations. Selection depends on DOF, travel, load and workspace.
This article covers multi-axis orthogonal combination, error accumulation, the hexapod principle, and how to choose between the two.
Figure 1 | Top: the X carriage carries the Y stage, whose carriage carries the workpiece. Bottom: six variable-length actuators join the plates through end joints; a translucent top plate exposes the connections. All six legs cooperate on X/Y/Z and θx/θy/θz; leg numbers are not Cartesian axes. This conceptual projection is not commercial CAD or a physical angle/dimension specification[1]. Click for full size.
1. Multi-Axis Orthogonal (XY/XYZ/XYθ)
A multi-axis stage combines single-axis stages orthogonally:
XY: two perpendicular axes (X and Y), two-dimensional planar motion
XYZ: three perpendicular axes, three-dimensional motion
XYθ: two translations plus one rotation (θ is rotation about the vertical axis)
Each axis can use a single-axis stage (screw, linear motor, piezo or voice coil). The advantage is modularity: travel and drive type can be selected separately. In Figure 1, the lower X carriage carries the entire Y stage, so its payload includes that upper stage and the workpiece; both axes’ geometric errors affect the same workpoint.
Multi-axis stages are the workhorse of semiconductor inspection, PCB fabrication, and optical alignment — these applications need large travel and precise in-plane positioning, where an XY or XYZ stage is the standard configuration.
2. Error Accumulation and Squareness
A multi-axis stage’s accuracy is not the simple sum of single-axis accuracy; it is the result of the axes’ errors combining:
Combined errors: XY positioning includes axis-position, straightness, angular and squareness errors. Two “1 μm single-axis specifications” do not determine whole-stage planar accuracy; check component/vector definitions and calibration or compensation conditions.
Squareness: the X and Y axes must be perpendicular (90°). If they are not (e.g., 90°±0.01°), the carriage produces positioning deviation as it moves — the larger the travel, the larger the deviation from squareness error.
Figure 2’s geometric example uses δ = 0.01° between actual and ideal Y directions and L = 100 mm travel along the actual axis. The transverse deviation is ΔX = L sin δ ≈ Lδ. Converting δ to 1.745 × 10⁻⁴ radians gives ΔX ≈ 17.5 μm. The drawn angle is exaggerated; this calculation excludes other errors and is not an accuracy guarantee.
- Abbe error: a workpoint offset h perpendicular to the measuring line turns relevant angular motion φ into linear workpoint error. Figure 2’s planar example gives Δx = h sin φ ≈ hφ, with radians in the approximation. The sensed reference and actual machining/alignment point must be evaluated within the same error model[5].
Figure 2 | Top view: ideal versus actual Y. X–Z side view: measuring reference versus an offset workpoint. Squareness is an inter-axis angle error; Abbe error involves workpoint offset. Angles and deviations are exaggerated. These are planar geometric/small-angle examples, not measurements[5]. Click for full size.
So a multi-axis stage’s acceptance reads the whole-machine plane accuracy and squareness specs, not the single-axis accuracy.
3. Hexapod (Stewart Platform)
The hexapod (Stewart platform) here is a parallel manipulator: six variable-length actuators connect a fixed base to a moving plate. Coordinated leg lengths control six degrees of freedom within the reachable workspace: X/Y/Z translation and θx/θy/θz rotation. A single Cartesian motion generally requires multiple legs; one leg does not control one Cartesian axis[1].
Figure 1 uses six distinct joints on each plate, arranged in three pairs. Inclined, alternating legs control position and orientation; the end joints allow their angles to change. Joint placement varies by design and is not universally “three points per plate, offset by 60°”.
Kinematics has two directions:
Inverse kinematics (specified pose → leg lengths): for straight telescoping legs with known joint geometry, transform the top joints into base coordinates and calculate the six endpoint distances. Then check stroke, joint angles and interference to establish reachability[3].
Forward kinematics (leg lengths → platform pose): solve coupled nonlinear/polynomial equations and select a pose consistent with assembly and workspace constraints. Multiple mathematical solutions can occur. Numerical methods are common; it is too broad to claim that every geometry lacks a closed-form solution[4].
Applications include optical/fiber alignment, metrology and precision positioning. Precision hexapods can use electromechanical screw or piezo drives; a linearly extending leg does not necessarily contain a linear motor[1][2].
A hexapod’s workspace depends on leg lengths, joint angles and interference, and changes with plate orientation. Individual X/Y/Z or rotation maxima cannot generally be combined into a simultaneously reachable rectangular range. Check the required pose, path, load and rotation center[2][3].
4. Selection: Multi-Axis vs Hexapod
| Multi-axis orthogonal | Hexapod | |
|---|---|---|
| DOF | Depends on assembly; XY/XYZ/XYθ can add rotation stages | 6 (three linear + three rotational) |
| Travel | Selected separately for each stage | Translation and rotation workspace are coupled |
| Structure | Modular, intuitive | Compact, six legs in parallel |
| Kinematics | Serial order and coordinate transforms | Coordinated legs; forward solution needs branch selection |
| Stiffness | Depends on guides, stacking and load | Parallel paths support stiffness; pose/load still matter |
| Suits | Modular choice of individual travel ranges | Compact position and orientation control |
Multi-axis suits applications needing individual travel-range choices; a hexapod suits compact position and orientation control. A serial system can also reach six DOF by adding rotary axes, so axis count alone does not decide the choice.
A hexapod’s multiple load paths support compact, stiff designs. Actual stiffness and dynamics still depend on model, pose, payload and center of mass; every hexapod is not automatically stiffer or faster than every serial system[2].
5. Applicable Range and Limits
Serial stages require workpoint evaluation of combined errors, squareness and Abbe offset under whole-machine acceptance conditions. A hexapod requires pose-dependent workspace, joint-geometry and calibration checks; six-leg control does not automatically eliminate geometric, thermal or load errors.
Common Misconceptions
“Multi-axis accuracy equals single-axis accuracy”: single-axis numbers cannot be transferred directly. Workpoint, error definitions, geometry, load and compensation conditions matter.
“All six axis maxima are simultaneously available”: individual maxima do not determine the reachable combinations; check workspace for the complete pose and path.
“A hexapod is a typical six-joint serial arm”: the structures differ. Six legs join a hexapod’s two plates, while a serial arm connects joints in sequence. “Six-axis” alone only describes controlled axis count.
Self-Check for the Reader
Why does single-axis accuracy not directly determine accuracy at the complete machine’s workpoint?
What are a hexapod’s six degrees of freedom?
How do you obtain the six leg lengths from a specified plate pose? Which constraints can make it unreachable?
Back to the opening question: serial stages carry the upper stages in sequence; a hexapod uses six parallel legs to control one plate’s position and orientation. The drawings explain connections and error sources. Selection and acceptance must use the actual workpoint, path, pose, payload and calibration conditions.