Five-Axis CNC Machine Tool Error Compensation for High-Precision Machining

Five-axis CNC machine tools, as key equipment for machining complex curved surfaces, feature three linear axes and two rotary axes.

This configuration enables tool positioning relative to the workpiece in any direction, enhancing machining efficiency and surface quality.

However, maintaining precision in five-axis CNC machine tools poses significant challenges.

Compared to three-axis machines, the introduction of rotary axes in five-axis systems causes error elements to increase exponentially.

These errors couple, propagate, and amplify within the machine’s spatial domain, ultimately manifesting as relative positional errors between the tool and workpiece.

This leads to machined parts exceeding tolerances and becoming scrap.

In precision machining, errors originating from the machine tool itself account for a significant proportion of workpiece machining errors.

Therefore, conducting systematic research on accuracy control and error compensation for five-axis CNC machine tools is crucial for advancing the overall level of manufacturing.

Error Types in Five-Axis CNC Machines

• Geometric Errors

Geometric errors are inherent static errors originating from the shape, position, and motion issues of machine components during manufacturing and assembly.

For typical five-axis CNC machines, the primary geometric error elements are as follows.

(1) Positioning Error.

Positioning error is the deviation between the actual position and the commanded position during movement of each linear axis.

(2) Straightness Error.

Straightness error is the translational error in the other two directions during movement of each linear axis.

For example, the straightness error in the Y and Z directions during movement of the X-axis.

(3) Angular error. Angular error refers to the pitch, yaw, and roll errors of each moving axis around the three coordinate axes.

For example, the roll error, yaw error, and pitch error of the X-axis.

(4) Perpendicularity error. Perpendicularity error refers to the non-perpendicularity error between any two of the three linear axes.

• Thermal Error

During machine tool operation, friction and cutting processes in components such as motors, lead screws, guideways, and bearings generate heat, causing thermal deformation of the machine tool structure.

Thermal errors account for 40% to 70% of a machine tool’s total error.

They primarily manifest as: 

Thermal elongation of the spindle causing Z-axis tool length variation; 

Thermal deformation of the lead screw affecting linear axis positioning accuracy;

Deformation caused by heat in structural components such as the machine bed and column alters the relative positions of the axes.

Thermal errors exhibit time-varying and nonlinear characteristics, closely related to the machine tool’s operating state and ambient temperature.

• Force-induced errors

(1) Deformation caused by cutting forces.

During machining, cutting forces induce elastic deformation in the machine tool-cutting tool-workpiece system, leading to tool deflection.

(2) Gravity-induced deformation.

For large machine tools, the self-weight of moving components causes varying structural deformations at different positions.

(3) Workpiece deformation caused by clamping forces.

When servo systems track complex trajectories, factors like system gain mismatch and dynamic lag can lead to contour errors.

This error becomes particularly significant in high-speed, high-precision machining, and engineers should not overlook it.

Comprehensive Geometric Error Model Construction

• Topological Structure Description of Multi-Body Systems

The machine tool is regarded as a system composed of a series of “bodies” connected via moving or fixed joints.

Taking an AC dual-swivel-head five-axis machine tool as an example, its topological chains include the tool chain (bed → X-axis slide → Y-axis table → Z-axis ram → A-axis swivel head → C-axis rotary head → tool) and the workpiece chain (bed → workpiece table → workpiece).

• Establishing the Transformation Relationship Between Ideal and Actual Motion

In constructing kinematic models for CNC machine tools, engineers describe the motion relationship between adjacent components using coordinate transformation matrices.

Under ideal conditions, the transformation between motion axes involves only pure translation or rotation operations, disregarding any manufacturing, assembly, or operational error factors.

At this stage, the X-axis motion follows a predefined ideal trajectory, with its transformation matrix entirely determined by theoretical displacement values, exhibiting idealized rigid-body transformation.

This ideal motion transformation matrix reflects the machine tool’s geometric relationships in an error-free state, serving as the foundation for constructing error analysis models.

During actual operation of CNC machine tools, factors such as guide rail manufacturing errors, assembly tolerances, thermal deformation, and force-induced deformation cause the actual motion of the X-axis to deviate from the ideal state, exhibiting complex error behavior.

Specifically, X-axis movement introduces six geometric errors: straightness errors along the X, Y, and Z directions, and angular errors around the X, Y, and Z axes.

During modeling, engineers apply the small-angle assumption and treat angular errors as sufficiently small to allow linearization, thereby simplifying calculations.

Based on these error terms, they construct the actual motion transformation matrix for the X-axis, which comprehensively reflects its true spatial pose changes.

Similarly, engineers construct the actual motion transformation matrices for the Y-axis, Z-axis, and rotational axes (A-axis and C-axis) one by one following the same principle.

They incorporate displacement errors and angular errors in their respective directions.

• Comprehensive Geometric Error Model Construction

To achieve precise evaluation of machine tool spatial errors, engineers must construct a complete kinematic chain error model from the tool to the workpiece.

Under ideal conditions, the position of the tool tip in the tool coordinate system serves as the origin, meaning its ideal position vector is the zero vector.

Similarly, engineers locate the reference point in the workpiece coordinate system at its origin.

According to the ideal transformation relationship, CNC commands entirely determine the relative position between the tool and the workpiece.

« Introduction of Actual Motion and Geometric Errors

In practice, geometric errors exist in each motion segment, causing deviations between the actual and ideal poses.

To address this, engineers introduce the actual total transformation matrix of the tool chain.

This matrix is obtained by sequentially multiplying the actual transformation matrices between adjacent components: from the machine bed to the X-axis, X-axis to Y-axis, Y-axis to Z-axis, Z-axis to A-axis, A-axis to C-axis, and C-axis to the tool.

Each segment employs an error-accounting transformation matrix, cumulatively propagating errors throughout the tool motion chain.

« Construction of the Workpiece Kinematic Chain

Similarly, engineers construct the actual total transformation matrix for the workpiece chain based on its motion path.

Using these two total transformation matrices, they calculate the true position of the tool tip in actual space relative to the workpiece coordinate system.

Based on this result, engineers obtain the tool’s actual position through coordinate transformation and matrix operations, with all position and transformation parameters expressed numerically.

At the same time, they extract the actual direction vector of the tool axis in space from the rotational component of the total transformation matrix to analyze directional error.

They define the comprehensive geometric error as the difference between the tool’s actual position and its ideal position.

This error vector comprises spatial deviation components in the X, Y, and Z directions.

The constructed comprehensive geometric error model systematically reflects how errors in each machine tool axis affect the final tool position and orientation.

Given a set of motion commands for each machine tool axis and inputting the corresponding 21 geometric error parameters (including positioning errors, straightness errors, and angular errors for each axis), this model can precisely calculate the three-dimensional spatial error at the commanded position.

This model provides a theoretical basis for machine tool accuracy analysis, error compensation, and precision enhancement, holding significant engineering application value.

Five-Axis CNC Machine Tool Error Compensation Technology

• Offline Detection and Compensation

(1) Error Detection. Precise measurements of 21 geometric errors on the machine tool are performed using high-precision instruments.

Measurements are typically conducted after machine tool warm-up to isolate thermal errors.

(2) Error Modeling. Measured error data is fitted into polynomial or tabular forms and incorporated into a comprehensive error model.

(3) Compensation Table Generation.

The model predicts errors at each point within the machine’s entire working space and generates inverse compensation values.

For the CNC system, these compensation values are typically input as a “pitch error compensation table.”

(4) Compensation Execution. During program execution, the CNC system reads the current compensation values corresponding to each axis position in real time.

Engineers add these values to the commanded position, driving the servo motor to make fine adjustments that cancel out the errors.

• Online Detection and Real-Time Compensation

Online detection and real-time compensation represent the future development trend, aimed at compensating for time-varying errors (thermal errors, force-induced errors).

A network of temperature, displacement, and force sensors is installed on the machine tool to monitor its status in real time.

Algorithms such as neural networks, fuzzy logic, and support vector machines are employed to construct real-time mapping models from sensor data to machine tool errors.

These models predict current errors based on real-time sensor data and correct motion commands in real time through open CNC systems or external controllers.

While online compensation technology is complex and costly, it enables more comprehensive precision assurance.

Simulation Experiment and Results Analysis

• Simulation Experiment Design

(1) Machine Tool Parameters. X-axis working stroke: 600 mm; Y-axis working stroke: 500 mm; Z-axis working stroke: 400 mm; A-axis stroke: 120°; C-axis stroke: 360°.

(2) Error Data. Simulated 21 geometric error parameters for the machine tool, all modeled as position-dependent polynomial functions.

All error magnitudes remain within the machine’s permissible accuracy limits.

(3) Test Trajectory. Within the machine tool’s working space, a typical spatial diagonal trajectory was selected, moving from point P1 (0,0,0) to point P2 (600,500,400).

Concurrently, the A-axis and C-axis moved in tandem from 0° to 90° and 180°, respectively.

The researchers uniformly selected fifty points along this trajectory for calculation.

(4) Simulation Process.

① For each point on the trajectory, retrieve the corresponding 21 error values from the simulated error database based on the commanded positions of each axis.

② Substitute these error values into the comprehensive geometric error model to calculate the three-dimensional spatial error and its vector for that point.

③ Invert the calculated error values to obtain compensation quantities, assuming perfect compensation.

④ Compare the error results before and after compensation.

• Analysis of Simulation Experiment Results

The error comparison before and after compensation for selected trajectory points along the spatial diagonal is shown in Table 1.

As indicated in Table 1, without compensation, the machine tool’s spatial error increases significantly with the distance traveled along each axis.

At the trajectory endpoint, the total error reached its maximum value of 63.2 μm.

Among these, the Z-axis error was the largest, primarily caused by the positioning error of the Z-axis coupled with the error coupling from the X-axis and Y-axis movements to the Z-direction.

This phenomenon fully demonstrates the coupling and cumulative nature of errors in five-axis CNC machine tools.

Following model-based error compensation, the method significantly suppressed spatial errors across the entire trajectory.

The compensated total error was controlled below 10.0 μm, with the maximum total error decreasing from 63.2 μm to 8.7 μm—representing an approximately 86% improvement in accuracy.

This validates the effectiveness of the comprehensive geometric error model constructed and the offline compensation method adopted in this study.

Micrometer-level residual errors persist post-compensation, attributable to model idealization:

• the model assumes small angles and omits higher-order error terms;
• residuals exist when fitting actual measurement data with polynomials;
• and the experiment compensated only geometric errors without accounting for thermal errors, force-induced errors, etc.

Conclusion

This paper presents a systematic investigation into precision control issues in five-axis CNC machining.

It comprehensively analyzes various error sources in five-axis CNC machines, clearly identifying geometric errors as the fundamental and critical factor affecting precision.

Based on multibody dynamics theory and homogeneous coordinate transformation, a comprehensive error model for five-axis CNC machines was successfully developed.

This model fully reflects the coupled effects of 21 geometric error elements.

This model is universally applicable to five-axis CNC machines of different structural types.

Simulation experiments using simulated real machine error data validated the effectiveness of the model and compensation method.

Experimental results demonstrate that offline geometric error compensation can reduce the machine’s spatial positioning error to below 10.0 μm, achieving significant compensation effects.