Abstract

Researchers have studied Stewart–Gough platforms, also known as Gough–Stewart platforms or hexapod platforms extensively for their inherent fine control characteristics. Their studies led to the potential deployment opportunities of Stewart–Gough platforms in many critical applications such as medical field, engineering machines, space research, electronic chip manufacturing, automobile manufacturing, etc. Some of these applications need micro- and nano-level movement control in 3D space for the motions to be precise, complicated, and repeatable; a Stewart–Gough platform fulfills these challenges smartly. For this, the platform must be more accurate than the specified application accuracy level and thus proper calibration for a parallel robot is crucial. Forward kinematics-based calibration for these hexapod machines becomes unnecessarily complex and inverse kinematics complete this task with much ease. To experiment with different calibration techniques, various calibration approaches were implemented by using external instruments, constraining one or more motions of the system, and using extra sensors for auto or self-calibration. This survey paid attention to those key methodologies, their outcome, and important details related to inverse kinematic-based parallel robot calibrations. It was observed during this study that the researchers focused on improving the accuracy of the platform position and orientation considering the errors contributed by one source or multiple sources. The error sources considered are mainly kinematic and structural, in some cases, environmental factors also are reviewed; however, those calibrations are done under no-load conditions. This study aims to review the present state of the art in this field and highlight the processes and errors considered for the calibration of Stewart–Gough platforms.

1 Introduction

In the world of conventional robots, there are three varieties: (i) serial robot, (ii) parallel robot, and (iii) hybrid robot [1]. Any robot, also called manipulator or mechanism consists of a base and an end-effector. These are connected by multiple links. In a serial manipulator, the links are connected in series. A parallel mechanism, sometimes called a parallel kinematic machine (PKM) or as a Stewart–Gough or Gough–Stewart Platform is made by linking a moving body or end-effector which is generally mounted on a platform or endplate, to a reference body or base through three or more links forming a closed-loop kinematic chain [2]. The base part remains fixed. A hybrid mechanism structure is formed by combination of serial links and parallel robots. Often, the parallel robot is mounted near the end-effector of the serial manipulator to provide a high-precision to the serial system. The rigidity of a parallel robot is relatively higher than a serial manipulator, but the workspace of parallel robot is lesser than serial robot. The hybrid structure increases the workspace of a parallel robot at some cost of the rigidity of the structure.

Parallel robots have received significant attention for high dynamic flexibility, structural rigidity, and high accuracy due to the closed kinematic loops, no error accumulating characteristics [3], higher load-to-weight ratio, and uniform load distribution capacity compared to serial manipulators [4]. For any parallel robots, the linking element numbers between the fixed base and movable platform vary between three and six. The link numbers together with the type of connections and the twist of the platform normally decide the degrees-of-freedom (DOF) of the machine.

One such parallel robot controlled by six links connected between the fixed base and movable platform with six degrees-of-freedom is termed a hexapod. The hexapod was first designed by an engineer Gough from the United Kingdom in 1954 for tire testing with six actuators acting as the links between the fixed base and its moving platform. The actuators are prismatic joints. This machine had the structure of an octahedral hexapod [5]. Using the Gough's platform, in 1965 another engineer from the United Kingdom, Stewart developed an articulated 6DOF flight simulator [6] for the training of pilots. These types of platforms are generally known as a Stewart–Gough platform or sometimes Gough–Stewart platform, or simply as a Stewart platform. In this document, these terms are used to mean the same machine. The combinations of motions of the six actuators give the platform high precision, high structural stiffness, and high dynamic performance [7]. Stewart–Gough platforms have been employed in many fields. The potential applications of parallel robots include mining machines, walking robots, both terrestrial and space applications including areas like high-speed manipulation, material handling, motion platforms, machine tools, medical fields, planetary exploration, satellite antennas, haptic devices, vehicle suspensions, variable-geometry trusses, cable-actuated cameras, and telescope positioning and pointing devices [8]. They are used in the development of high-precision machine tools by many companies like Giddings & Lewis, Ingersoll, Hexcel, Geodetic, and others [9,10]. The application options expanded from a simulator to automobile manufacturing, inspection, human-robot collaboration, space telescope, and medical tool control (by adding a hexapod at the end-effector point of a serial manipulator) [11]. For the precision and accuracy needed for these machines to perform at a specific level of operational characteristics, the platform movements must be precisely controlled. To get the necessary level of accuracy for the moving platform position and orientation, called pose, it is essential to understand the various errors related to the machine at the time of its operation and apply suitable compensation. Calibration of the hexapod identifies these errors and adds suitable amounts of compensations to get reliable and predictable [12] output data.

This paper surveys the calibration methods used for hexapod platforms based on Stewart–Gough platforms. Efforts were made to cover most of the key articles published after the year 2000 that are based on inverse kinematics calibration techniques. The paper has been divided into six main sections. The beginning section serves as an introduction. The second section reviews the kinematics of hexapods and the primary error factors that impact the accuracy of hexapods. Section 3 reviews the calibration techniques and the strategies used for successful calibration thereof. Major calibration methodologies and their outcomes are presented in Sec. 4. Section 5 discusses and compares the methods previously presented. Finally, Sec. 6 provides a conclusion and recommendations for future work.

2 Hexapod Kinematics and Error Factors

A general 6–6 hexapod configuration is shown in Fig. 1, with the appropriate terms defined using the nomenclature from Tsai [13] and Lee and Shim [14].

Fig. 1
Hexapod schematic with nomenclature
Fig. 1
Hexapod schematic with nomenclature
Close modal
In Fig. 1, the position of the fixed base coordinate system (System O) defined at point O compared to the moving platform coordinate system (System P) defined at point P is defined through the translation vector p, and a rotation matrix POR. The rotation matrix can be further defined as in Eq. (1) as
POR=[uxvxwxuyvywyuzvzwz]
(1)
where u, v, and w are unit vectors along the axes of the moving platform coordinate system. Subject to the orthogonal conditions defined in Eqs. (2)(7).
ux2+uy2+uz2=1
(2)
vx2+vy2+vz2=1
(3)
wx2+wy2+wz2=1
(4)
uxvx+uyvy+uzvz=0
(5)
uxwx+uywy+uzwz=0
(6)
vxwx+vywy+vzwz=0
(7)
If we define Oai = [aix, aiy, aiz]T as the vector from point O to point Ai (where i = 1, 2, …, 6) in coordinate system O, and in coordinate system P, Pbi = [bix, biy, biz]T as the vector from point P to point Bi (where i = 1, 2, …, 6) in coordinate system P, this allows us to define a vector loop equation for the ith arm of the platform as in Eq. (8).
AiBi¯=Op+PORPbiOai=Odi
(8)
Thus, the length of the ith arm controlled by the prismatic actuator i, is defined in Eq. (9) as
Odi2=Odi=[Op+PORPbiOai]T[Op+PORPbiOai]=OdiOdii=1,2,,6
(9)

Depending upon the inputs and outputs to and from the kinematic problem, the solution to the resulting system of equations is defined as either forward kinematics or inverse kinematics [15]. In forward kinematics, the position and orientation of the moving plate are calculated based on the length and orientations of the six actuators, defined by di and expressed as Op and POR.

The opposite calculation is inverse kinematics where the position and orientation of the moving plate, Op and POR, are known and the required length of each of the i actuators, defined by ||di||, is to be determined [16].

In both cases, the correctness of the hexapod parameters is dependent on a number of error factors which can be geometric or non-geometric [17,18]. These error parameters affect the values of all of the variables which define the kinematics of the hexapod. Depending on the source of the different error factors, the calibration process can be classified into three levels [12,19,20]:

  • Level-1 calibration considers only the joint errors that play a critical role in the accuracy of the robot. This is defined as “joint level calibration.”

  • Level-2 calibration, also known as “kinematic model calibration,” takes care of the error of the kinematic parameters.

  • Level-3 calibration, also called as “non-kinematic calibration” or “dynamic model calibration,” captures the errors of non-geometric or quasi-static parameters [21] such as stiffness, geometry of the robot structure, and errors caused by temperature variation [22].

In a hexapod platform, the components of the rigid structure like the base, frame, top platform, and other accessories are fabricated and normally made from metal stocks. So, the accuracy of these components has a direct influence on the accuracy of the whole system. The dimensions of the structure are dependent on the design tolerances or manufacturing deviations, clearances, joint errors [23], thermal deformations [24,25], and elastic deformations [26]. The actuators or struts are connected to the structure with movable joints; joints are impacted by the errors due to the assembly deviations in the form of joint run-out and ball screw deviations. Also, the mechanical joints are not free from friction, hysteresis, and backlash [3]. If the struts are operated by hydraulic fluids, there are chances of transmission errors and sensor errors [27].

Therefore, the error of a hexapod, EHexapod, can be expressed broadly by a function of all of the terms included in Eq. (10).
EHexapod=f(Emanufacturing,Eassembly,Etransmission,Edeformation,Esensor)
(10)

These error parameters are further elaborated in Table 1. Each of these errors can affect the position and orientation of the endplate. For instance, Emanufacturing, is the resultant error vector due to tolerances and manufacturing accuracy along the kinematic chains that define the hexapod. Similarly, assembly errors can also lead to changes in any of these parameters, although some errors may be more likely to be seen in certain parameters (i.e., wear is more likely to affect parameters specifically associated with the joint rather than the links). Similar arguments can be made for the other error components described in Table 1. There are more typical sources of errors [28] other than those mentioned here, contribute to the outcome of the system; however, their influences are dependent on the construction philosophy adopted for that particular system, and the type of operations and level of accuracy expected out of those systems.

Table 1

Error function illustrations

Error sourceDependencyRemarks
ManufacturingComponent tolerancesThese components act as the basic structure of the machine and any deviation impacts permanently. This structure bears all the loads generated in the static and dynamic conditions of the machine and provides rigidity to the machine.
Assembly
  • Assembly tolerances

  • System age

  • Amount of usage/Wear

Generally, assembly deviations are controllable and minimized by the replacement of old, worn-out parts with new parts.
Transmission
  • Actuation response time

  • Joint clearance and backlash

  • Platform position

  • Operation speed and lag

  • Hysteresis effects

This error depends on the robustness of the system and system configuration. Some default limitations cannot be avoided.
Deformation
(Mechanical)
  • Material properties

  • Applied loads

  • Component geometry

  • Platform position

Dependent on the structural materials used and their response property under load.
Deformation
(Thermal)
Working temperature variationsChanges in operating conditions due to temperatures may affect structural components, joint tolerances, and actuator performance.
Sensor
  • Specification tolerances/Accuracy

  • Calibration drift

Modern sensors tend to be the least inaccurate.
Error sourceDependencyRemarks
ManufacturingComponent tolerancesThese components act as the basic structure of the machine and any deviation impacts permanently. This structure bears all the loads generated in the static and dynamic conditions of the machine and provides rigidity to the machine.
Assembly
  • Assembly tolerances

  • System age

  • Amount of usage/Wear

Generally, assembly deviations are controllable and minimized by the replacement of old, worn-out parts with new parts.
Transmission
  • Actuation response time

  • Joint clearance and backlash

  • Platform position

  • Operation speed and lag

  • Hysteresis effects

This error depends on the robustness of the system and system configuration. Some default limitations cannot be avoided.
Deformation
(Mechanical)
  • Material properties

  • Applied loads

  • Component geometry

  • Platform position

Dependent on the structural materials used and their response property under load.
Deformation
(Thermal)
Working temperature variationsChanges in operating conditions due to temperatures may affect structural components, joint tolerances, and actuator performance.
Sensor
  • Specification tolerances/Accuracy

  • Calibration drift

Modern sensors tend to be the least inaccurate.

Conceptually, each degree-of-freedom in a kinematic chain is defined by four Denavit–Hartenburg (DH) parameters, representing the information necessary to transform the coordinate system across the component providing the DOF. So, on a 6DOF kinematic chain defining one limb of a Stewart platform, there are 4(6) or 24 DH parameters. With one prismatic actuator, there are 23 uncontrolled parameters and 1 controlled parameter in each limb. With six limbs, there are a total of 6 controlled DH parameters and 138 uncontrolled DH parameters in a Stewart platform. However, because these parameters are coupled into closed-loop kinematic chains, these parameters are coupled to the other parameters in the system, resulting in six independent DOF for the endplate, and the remaining 138 uncontrolled parameters are dependent upon each other, and the six independent DOF from the prismatic actuators. Errors originating from manufacturing tolerances and accuracy can have a cumulative effect that alters all of the uncontrolled DH parameters. Some of the error sources, in particular, the transmission and sensor errors, also can affect the controlled parameters associated with the actuators of each limb. Consequently, the task of calibrating a hexapod involves confirming that each of these 144 DH parameters is known throughout the operating range of the system.

3 Accuracy Improvement Strategies

The purpose of kinematic calibration of parallel robots is to improve the motion and position accuracy of the moving platform by correctly evaluating and calculating the kinematic parameters within its defined workspace. A parallel robot can be calibrated in three ways (Fig. 2): external calibration, constrained calibration, and auto- or self-calibration [29,30].

Fig. 2
Strategies for parallel robot calibration
Fig. 2
Strategies for parallel robot calibration
Close modal

External calibration is done by using one or more external instruments such as an electronic theodolite or a laser tracker [31] for the measurement of multiple poses of the end-effector. In the Constrained calibration process, motions of the mechanical elements, usually the movement of robot actuators are constrained to gather the error data. This method is comparatively simple and the least expensive [32]. Auto- or self-calibration is one of the most expensive and complex calibration techniques. In this method, the robot itself automatically monitors error parameters measured by redundant sensors with the help of the built-in algorithms installed on controllers. The error correction process can take place during normal robot operation. Several extra sensors are installed in the joints and links of the robot to gather calibration data continuously. Alternatively, additional sensors can be added to directly measure the end plate motion through extensometers or feeler arms [33,34]. Generally, the number of sensors used in a parallel robot is equal to its degrees-of-freedom. In all these strategies, sensors play an important role and they are an essential part of the calibration process; the difference occurs in how these sensors are employed.

Conventionally, the four steps shown in Fig. 3 are followed in the calibration process of a parallel robot: kinematic modeling, measurement, identification, and implementation [3537].

Fig. 3
Steps in parallel robot calibration process
Fig. 3
Steps in parallel robot calibration process
Close modal

Kinematic modeling of the platform is to build the relationship between the joint variables and the platform pose, along with the measurement device readings. The result of the modeling phase is a set of analytical equations showing relationships between these parameters as shown in Eqs. (1)(9) [17,18,38]. The measurement phase gathers the data related to the actual platform position and orientation with the help of the measurement devices [21]. The identification step will identify the optimal set of unknown parameters based on the residual between the kinematic model and measurements to fit the actual behavior of the mechanism while considering the major sources of errors for the context [7,39] and finally, through implementation, the compensation for the calculated model errors is included in the robot controller [37].

The results of a robot calibration process are expressed in terms of the pose errors for a set of positions and orientations [40]. To generate a reliable, accurate result the mechanical structure of the robot must be defined with an adequate number of parameters without repetition in the calibration model. A “good” calibration model must have three criteria: completeness, equivalence, and proportionality. Completeness refers to the fact that the model must have enough parameters to completely define the motion of the robot. Equivalence means that the derived functional model can be related to any other similar acceptable model. And the proportionality property must give the model the ability to reflect small changes in the robot geometry with small changes in the model parameters [19]. A “good” calibration model can be established by building relationships between the independent parameters that are used to define the robot system. For a multi-loop parallel robot such as the 6DOF hexapod or Stewart–Gough platform, the number of independent parameters (C) can be calculated as follows [41]:
C=3R+P+S+N+E+6L+6(F1)
(11)
where
  • L, number of independent link loops in the robot,

  • R, number of unsensed (non-instrumented) revolute joints,

  • P, number of prismatic joints,

  • S, number of unsensed (non-instrumented) spherical joints,

  • N, number of pairs of S-joints connected by a simple link without any intermediate joint,

  • E, number of measurement devices or transducers, and

  • F, number of arbitrarily located frames.

For a Stewart–Gough platform with one universal (U), one prismatic (P), and one spherical (S) joints for each actuator considering (UPS ≈ 2RP3R), the minimum total number of independent parameters necessary for the complete calibration model is as follows:
C=3*6*5+6+0+6+6*5+6(21)=138

4 Calibration Approach

To achieve a high level of kinematic accuracy, it is necessary to formulate a robust and reliable calibration method. After the introduction of hexapod by Gough, and then by Stewart, calibration became the field of interest for researchers. Calibration research is typically performed either by analytical approaches or by physical experiments. The analytical approaches are independent of the physical hexapod platform artifacts. Still, they are worth studying to get an idea of the research direction on the analytical hexapod calibration process.

4.1 Calibrations Through Analytical Approaches.

Jing et al. [42] considered that the final position and orientation of a hexapod platform are dependent on the joint radius and angles of the movable platform. In their analysis, they applied an interior-point algorithm. To validate their considerations, analytical techniques were used to reduce the errors of the six actuators. Through their analysis, they were able to reduce the average actuator length error from 0.1 mm to 2 × 10−7 mm.

In their research, Agheli and Nategh [43] took into account the lengths of the actuators as the main sources of error for the moving platform accuracy in the hexapod and accordingly designed the calibration process to minimize the cost function through an analytical calibration process. They achieved ∼50% error reduction in platform position and orientation errors at the workspace boundaries. For their analytics, Agheli and Nategh used Levenberg–Marquardt algorithm to minimize the cost function obtained through inverse kinematics calculations.

In Ref. [44], the researchers Daney et al. presented a method of calibration based on interval arithmetic and interval analysis to solve an over-constrained equation. They used Taylor expansion to obtain a linear approximation to determine the kinematic parameters. The least-square method and their developed certification algorithm provide exact kinematic parameters when no errors are considered for the measurements. With consideration of kinematic errors, the results are comparable with the classical least-square method.

For small numbers (∼10) of measurements, Daney and Emiris [45] used algebraic variable elimination and monomial linearization to calibrate the platform pose. They used the actuator lengths as the error source and compared their algorithm with classical nonlinear least-square methods. They obtained the same results. The advantage of this algorithm is that for small numbers of measurements, there is no need for any hypothesis for noise distribution and no initial estimate of solution is taken.

Wang and Ehmann [46] identified that the errors from the actuator lengths are the most dominant error factor for the overall accuracy of the 6DOF platform. They considered the errors from the actuator lengths, ball joint location, and motion errors in their analysis. Based on the sensitivity analysis of the errors, they came out with a graphical presentation of the optimal working region for the machine tool used for the experiment.

Daney [27] used the constrained optimization method and an algorithm named DETMAX for inverse kinematics applications. In this case, they studied the impact of the errors at joint positions and actuator lengths. They found that the mean error on kinematic parameters improved from 0.2705 cm to 0.0335 cm for random poses and 0.2705 cm to 0.0023 cm for selected poses.

In another study [47], Daney and Emiris carried out an analytical analysis using symbolic variable elimination with numerical optimization. These yielded superior results compared to the classical direct methods and were able to reduce initial pose error by up to 99% near the boundary configurations. With this study, the authors concluded that their algorithm is reliable irrespective of the robot configuration.

A summary of research on analytical-based calibration is shown in Table 2.

Table 2

Summary of analytical work on hexapod calibration

ReferenceErrors consideredMethodsOutcomePerformanceImportant points
Jing et al. [42]Moving platform joint radius and anglesInterior-point algorithmAverage actuator length error reduced from 0.1 mm to 2 × 10−7 mmUse of matlab optimization toolbox could find local optimal solutionThough other optimization methods may work better, the authors preferred this to try out.
Agheli and Nategh [43]Actuator lengthA least-square method based on Levenberg–Marquardt algorithmReduction in platform pose error by ∼50% at the boundaryThe position and orientation errors reduced 500 and 15,000 times, respectively.Maximum kinematics parameter errors were observed at the boundary of the workspace.
Daney et al. [44]Actuator lengthInterval arithmetic and interval analysisThe new measurement method shows comparable results to the classical least-square methodThe interval analysis provided numerically certified result to kinematic calibration problemThe classical least-square method may not provide realistic solutions for all cases.
Daney and Emiris [45]Actuator lengthAlgebraic variable elimination and monomial linearizationClassical nonlinear least-squares method and this method generates exactly same resultsA superior method for small numbers (=10) of measurements.Need no hypothesis for noise distribution and no initial estimate of the solution.
Wang and Ehmann [46]Actuator length, location, and motion errors of ball jointsAn automated error analysis model of first- and second-order inverse kinematicsGraphically sensitivity analysis results to select optimal working region1 mm actuator length (z-axis) error causes platform deviation in x-axis from −140 μm to −180 μm, y-axis from 520 μm to 650 μm, and z-axis from −150 μm to −350 μmThe length error (z-direction) of the actuators influences the accuracy of the machine much larger than any other errors.
Daney [27]Joint position and actuator lengthConstrained optimization method and DETMAX algorithmMeasurement noise in kinematic parameters reduced by 10 to 15 timesMean error on kinematic parameters improved from 0.2705 cm to 0.0335 cm for random poses and to 0.0023 cm for selected poses.The error value decreases steadily with an increase in the number of randomly chosen poses and remains usually constant for carefully chosen configurations.
Daney and Emiris [47]Platform poseSymbolic variable elimination with numerical optimizationMore reliable algorithm irrespective of configurations compared to the standard direct methodsInitial pose error was reduced by 99%.
  • An efficient technique to enhance the robustness of the measurement process.

  • Possibility of this method for the self-calibration process.

ReferenceErrors consideredMethodsOutcomePerformanceImportant points
Jing et al. [42]Moving platform joint radius and anglesInterior-point algorithmAverage actuator length error reduced from 0.1 mm to 2 × 10−7 mmUse of matlab optimization toolbox could find local optimal solutionThough other optimization methods may work better, the authors preferred this to try out.
Agheli and Nategh [43]Actuator lengthA least-square method based on Levenberg–Marquardt algorithmReduction in platform pose error by ∼50% at the boundaryThe position and orientation errors reduced 500 and 15,000 times, respectively.Maximum kinematics parameter errors were observed at the boundary of the workspace.
Daney et al. [44]Actuator lengthInterval arithmetic and interval analysisThe new measurement method shows comparable results to the classical least-square methodThe interval analysis provided numerically certified result to kinematic calibration problemThe classical least-square method may not provide realistic solutions for all cases.
Daney and Emiris [45]Actuator lengthAlgebraic variable elimination and monomial linearizationClassical nonlinear least-squares method and this method generates exactly same resultsA superior method for small numbers (=10) of measurements.Need no hypothesis for noise distribution and no initial estimate of the solution.
Wang and Ehmann [46]Actuator length, location, and motion errors of ball jointsAn automated error analysis model of first- and second-order inverse kinematicsGraphically sensitivity analysis results to select optimal working region1 mm actuator length (z-axis) error causes platform deviation in x-axis from −140 μm to −180 μm, y-axis from 520 μm to 650 μm, and z-axis from −150 μm to −350 μmThe length error (z-direction) of the actuators influences the accuracy of the machine much larger than any other errors.
Daney [27]Joint position and actuator lengthConstrained optimization method and DETMAX algorithmMeasurement noise in kinematic parameters reduced by 10 to 15 timesMean error on kinematic parameters improved from 0.2705 cm to 0.0335 cm for random poses and to 0.0023 cm for selected poses.The error value decreases steadily with an increase in the number of randomly chosen poses and remains usually constant for carefully chosen configurations.
Daney and Emiris [47]Platform poseSymbolic variable elimination with numerical optimizationMore reliable algorithm irrespective of configurations compared to the standard direct methodsInitial pose error was reduced by 99%.
  • An efficient technique to enhance the robustness of the measurement process.

  • Possibility of this method for the self-calibration process.

4.2 External Approach.

External approaches are widely used as a method for hexapod calibration. In this approach, the calibration is done through experiments on the hexapods using additional measuring instruments. Instruments like a double ball bar (DBB), laser interferometer, laser tracker, digital cameras, etc., are used [30]. Highlights of external approach-based hexapod calibration have been summarized in Tables 3 and 4. The system name used for the experiment, instruments used, type of error considered, and kinematics are presented in Table 3; whereas the methods, performance, and key findings of the study are presented in Table 4 for the same experiments mentioned in Table 3.

Table 3

Summary of external approaches (part 1—left side)

ReferenceSystem nameInstruments usedError considered/measuredKinematics
Song et al. [48]FARO measuring armJoint errorsInverse
Mahmoodi et al. [49]Six rotary sensors on six actuatorsActuator lengthForward/Inverse
Jáuregui et al. [17]Secondary mirror of a radio-telescopeLaser interferometerActuator lengthInverse
Ren et al. [35]XJ-HEXABiaxial inclinometer and laser trackerActuator lengthInverse
Nategh and Agheli [50]Hexapod tableDigital cameraPlatform poseForward/Inverse
Großmann et al. [51]FELIXDouble ball barPlatform poseInverse
Liu et al. [52]3D laser trackerActuator lengthsInverse
Ting et al. [53]Micro-positioning platformDMT22 dual sensitivity systems with C5 probeHysteresis of piezoelectric actuatorsInverse
Daney et al. [16]DeltaLab robotCharge-coupled device (CCD) cameraJoint imperfection and backlash of each actuator.Inverse
Dallej et al. [54]Omnidirectional cameraPosition and orientation of the actuatorsInverse
Daney et al. [55]DeltaLab “Table of Stewart”Sony digital video cameraPlatform poseInverse
Gao et al. [56]FFCM of FASTLaser tracker LTD500Platform poseInverse
Renaud et al. [57]A CCD camera and 1D laser interferometryPlatform poseInverse
Week and Staimer [58]Ingersoll HOH600Double ball bar as seventh actuator and a CMMActuator lengthInverse
Ihara et al. [59]Telescoping magnetic ball bar (DBB)Actuator length and joint positionsInverse
ReferenceSystem nameInstruments usedError considered/measuredKinematics
Song et al. [48]FARO measuring armJoint errorsInverse
Mahmoodi et al. [49]Six rotary sensors on six actuatorsActuator lengthForward/Inverse
Jáuregui et al. [17]Secondary mirror of a radio-telescopeLaser interferometerActuator lengthInverse
Ren et al. [35]XJ-HEXABiaxial inclinometer and laser trackerActuator lengthInverse
Nategh and Agheli [50]Hexapod tableDigital cameraPlatform poseForward/Inverse
Großmann et al. [51]FELIXDouble ball barPlatform poseInverse
Liu et al. [52]3D laser trackerActuator lengthsInverse
Ting et al. [53]Micro-positioning platformDMT22 dual sensitivity systems with C5 probeHysteresis of piezoelectric actuatorsInverse
Daney et al. [16]DeltaLab robotCharge-coupled device (CCD) cameraJoint imperfection and backlash of each actuator.Inverse
Dallej et al. [54]Omnidirectional cameraPosition and orientation of the actuatorsInverse
Daney et al. [55]DeltaLab “Table of Stewart”Sony digital video cameraPlatform poseInverse
Gao et al. [56]FFCM of FASTLaser tracker LTD500Platform poseInverse
Renaud et al. [57]A CCD camera and 1D laser interferometryPlatform poseInverse
Week and Staimer [58]Ingersoll HOH600Double ball bar as seventh actuator and a CMMActuator lengthInverse
Ihara et al. [59]Telescoping magnetic ball bar (DBB)Actuator length and joint positionsInverse
Table 4

Summary of external approaches (part 2—right side)

ReferenceMethodsPerformanceImportant points
Song et al. [48]ANN-based nonlinear functionsMean pose error reduction from 0.642 mm and 0.184 deg to:
Coupled network: 0.076 mm and 0.024 deg respectively;
Decouple network: 0.052 mm and 0.018 deg respectively
  • The coupled and decoupled networks show a similar results pattern though the optimal numbers of hidden nodes for coupled network is 13 and decoupled network is ∼6.

Mahmoodi et al. [49]A new method with six rotary sensorsPositional and orientation variances improved by 0.16 m2 and 0.16 rad2 respectively.
  • The new method is less accurate in orientation measurement.

  • This method is better than conventional method for position measurement.

Jáuregui et al. [17]Simplified method (same error amount considered for all actuators) and comprehensive method (each actuator error is not equal)Majority of pose deviations fall within 10 μm.
  • The simplified method creates linear relationships and is easy to solve.

  • The comprehensive method is complex, nonlinear, but more accurate.

Ren et al. [35]Keeping any two attitude angles of the end-effector constant.
  • Position accuracy = 0.1 mm

  • Orientation accuracy = 0.011 deg

  • Exempting the need for precise pose measurement and mechanical fixtures.

  • Independent of inclinometer range and accuracy.

Nategh et al. [50]A least-square approach based on Levenberg–Marquardt algorithm with singular value decomposition.The position and orientation errors as per analytical methods were 0.1 mm and 0.01 deg respectively and were 1.45 mm and 0.27 deg as per experiment.Employed observability index to find the most visible and optimum number of measurement configurations.
Großmann et al. [51]Genetic algorithm based trajectory optimization.The deviation of platform pose reduced from 0.7 mm to 0.17 mm.Genetic algorithms are slow to get the most accurate solution, also rarely improve the solution.
Liu et al. [52]Genetic algorithmAfter 5000 generations, the platform position and orientation errors improved 1.4 and 2.4 times respectively without measurement noise filter.The genetic algorithm showed good calculation stability, though it is not sensitive to measurement noises.
Ting et al. [53]Preisach modelPlatform accuracy level achieved 1 μm in position and 10 μ deg in orientation.The convergence of errors for fixed points can happen after several iterations.
Daney et al. [16]Interval arithmetic and analysis methodsYielded intervals for the position and orientation that include noise and robot repeatability error.Finds ranges of parameters that satisfy the calibration model.
Dallej et al. [54]Linear regressionExperimental validation of the method yielded 0.8 cm median error with respect to the CAD geometry.
  • An omnidirectional camera overcomes the self-occlusion problem arising in the single perspective camera.

  • No mechanical modification of the robot is necessary.

Daney et al. [55]Constrained optimization method with Tabu searchImprovements in accuracy were not as per expectation due to the biasness error of 1.29 mm on the z-axis.
  • The workspace boundary has a concentration of optimal poses.

  • By maximizing the observability index, the robustness of calibration increases with respect to measurement noise.

Gao et al. [56]Least-square methodAccuracy improved to 0.2 mm
  • Method is effective even in lack of sufficient measurements.

  • Some false parameters may occur for fewer measurement configurations.

Renaud et al. [57]Error function minimizationPrecision obtained in camera measurement is in the order of 1 μm in translation and 1 × 10−3 deg in rotations for an axial displacement of 400 mm
  • Low cost and easy to use compared to the measurements by interferometer.

  • Precision level depends on the camera resolution.

Week et al. [58]Gravity compensation
  • Roundness accuracy improved by 3.7×

  • Squareness accuracy improved by 7×.

The redundant actuator can be used to measure and compensate for the deflections due to gravity and thermal error.
Ihara et al. [59]Fourier transformationMachine's motion error decreased to ¼.The measurement is easy and can take care of circularity, absolute radial error, and circle center position error.
ReferenceMethodsPerformanceImportant points
Song et al. [48]ANN-based nonlinear functionsMean pose error reduction from 0.642 mm and 0.184 deg to:
Coupled network: 0.076 mm and 0.024 deg respectively;
Decouple network: 0.052 mm and 0.018 deg respectively
  • The coupled and decoupled networks show a similar results pattern though the optimal numbers of hidden nodes for coupled network is 13 and decoupled network is ∼6.

Mahmoodi et al. [49]A new method with six rotary sensorsPositional and orientation variances improved by 0.16 m2 and 0.16 rad2 respectively.
  • The new method is less accurate in orientation measurement.

  • This method is better than conventional method for position measurement.

Jáuregui et al. [17]Simplified method (same error amount considered for all actuators) and comprehensive method (each actuator error is not equal)Majority of pose deviations fall within 10 μm.
  • The simplified method creates linear relationships and is easy to solve.

  • The comprehensive method is complex, nonlinear, but more accurate.

Ren et al. [35]Keeping any two attitude angles of the end-effector constant.
  • Position accuracy = 0.1 mm

  • Orientation accuracy = 0.011 deg

  • Exempting the need for precise pose measurement and mechanical fixtures.

  • Independent of inclinometer range and accuracy.

Nategh et al. [50]A least-square approach based on Levenberg–Marquardt algorithm with singular value decomposition.The position and orientation errors as per analytical methods were 0.1 mm and 0.01 deg respectively and were 1.45 mm and 0.27 deg as per experiment.Employed observability index to find the most visible and optimum number of measurement configurations.
Großmann et al. [51]Genetic algorithm based trajectory optimization.The deviation of platform pose reduced from 0.7 mm to 0.17 mm.Genetic algorithms are slow to get the most accurate solution, also rarely improve the solution.
Liu et al. [52]Genetic algorithmAfter 5000 generations, the platform position and orientation errors improved 1.4 and 2.4 times respectively without measurement noise filter.The genetic algorithm showed good calculation stability, though it is not sensitive to measurement noises.
Ting et al. [53]Preisach modelPlatform accuracy level achieved 1 μm in position and 10 μ deg in orientation.The convergence of errors for fixed points can happen after several iterations.
Daney et al. [16]Interval arithmetic and analysis methodsYielded intervals for the position and orientation that include noise and robot repeatability error.Finds ranges of parameters that satisfy the calibration model.
Dallej et al. [54]Linear regressionExperimental validation of the method yielded 0.8 cm median error with respect to the CAD geometry.
  • An omnidirectional camera overcomes the self-occlusion problem arising in the single perspective camera.

  • No mechanical modification of the robot is necessary.

Daney et al. [55]Constrained optimization method with Tabu searchImprovements in accuracy were not as per expectation due to the biasness error of 1.29 mm on the z-axis.
  • The workspace boundary has a concentration of optimal poses.

  • By maximizing the observability index, the robustness of calibration increases with respect to measurement noise.

Gao et al. [56]Least-square methodAccuracy improved to 0.2 mm
  • Method is effective even in lack of sufficient measurements.

  • Some false parameters may occur for fewer measurement configurations.

Renaud et al. [57]Error function minimizationPrecision obtained in camera measurement is in the order of 1 μm in translation and 1 × 10−3 deg in rotations for an axial displacement of 400 mm
  • Low cost and easy to use compared to the measurements by interferometer.

  • Precision level depends on the camera resolution.

Week et al. [58]Gravity compensation
  • Roundness accuracy improved by 3.7×

  • Squareness accuracy improved by 7×.

The redundant actuator can be used to measure and compensate for the deflections due to gravity and thermal error.
Ihara et al. [59]Fourier transformationMachine's motion error decreased to ¼.The measurement is easy and can take care of circularity, absolute radial error, and circle center position error.

The artificial neural network (ANN) was used by Song et al. [48] in the calibration process for their Stewart–Gough platform. They corrected the joint variables by embedding the compensations in their numerical control system for online real-time error compensation. They experimented with their hexapod and demonstrated that the proposed ANN-based robust compensator can substantially enhance static pose accuracy for both coupled and decoupled networks. The ANN approach was implemented with inverse kinematics. The results obtained show that mean pose error reduced from 0.642 mm and 0.184 deg to 0.076 mm and 0.024 deg respectively for coupled network and to 0.052 mm and 0.018 deg respectively for decoupled network though the optimal numbers of hidden nodes for couple network is 13 and decoupled network is ∼6.

Mahmoodi et al. [49] proposed a new method of calibration for the Stewart–Gough platform-based parallel robot. They used rotary sensors in place of the linear sensors in actuators. The method is not too sensitive to the orientation measurement but showed better results in position measurement for the platform. In this study, six rotary sensors were used on six actuators to correct the pose of the platform. They used a mix of forward and inverse kinematics for their platform and observed the positional and orientation variances improved over the conventional methods to 0.16 m2 and 0.16 rad2 respectively for both small and moderate movements.

The studies by Jáuregui et al. [17] used a laser interferometer as the measuring instrument to calibrate the hexapod. They used inverse kinematics and considered the error related to the actuator length. Their experiment consisted of two methods. In the first method, they considered the error from all actuators to be the same, applying linear relationships. This was simple and easy to solve. In the second method, labeled as the comprehensive method, each actuator error was measured separately and modeled in a nonlinear relationship. As expected, the second method resulted in complex calculations but yielded greater accuracy.

Ren et al. [35] did their experiment with their hexapod named XJ-HEXA using a biaxial inclinometer with a length precision of 0.002 mm and repeatability for angles of 0.001 deg. They reached a position and orientation accuracy up to 0.1 mm and 0.01 deg respectively after calibration of 80 configurations. They kept two of the three orientations defining angles of the moving platform constant during the measurements.

Nategh and Agheli [50] studied their “hexapod table” with the use of an image-capturing system. In this research, the results obtained by the analytical approaches and experiments matched very closely. The platform position and orientation errors as per the analytical methods were 0.1 mm and 0.01 deg respectively, whereas those from the experimental methods were 1.45 mm and 0.27 deg, respectively. A least-squares approach based on the Levenberg–Marquardt algorithm was employed in this calibration process with singular value decomposition.

Another study by Großmann et al. [51] used a DBB to identify and collect the kinematic parameters by moving the platform on a specific trajectory in the 3D workspace. They used a genetic algorithm and simulated measurements to finalize the parameters. Their hexapod named FELIX was designed and manufactured with a focus on simplicity and capacity for compensating the motion errors generated due to the thermal and elastic deformations. Thermal and elastic deformations were considered in the algorithm by incorporating fixed factors. By this method, they measured the error along a trajectory and were able to reduce the initial deviation from 0.7 mm to 0.17 mm after optimizing the trajectory orientation through kinematic calibration.

Liu et al. [52] also used a genetic algorithm for calibrating their hexapod using inverse kinematics. They measured the errors coming from the actuator lengths and used a 3D laser tracker for the measurements. The genetic algorithm converged initially very fast but gradually became slow little by little. For their experiments, though they obtained an improvement in the error of the platform for the position by 1.4 times and for the orientation by 2.4 times, they found that the genetic algorithm is not sensitive to measurement noises.

The applications of hexapod did not remain restricted to the dimension level of millimeters or inches, it has attracted attention for micro-level applications too. Ting et al. [53] did their experiment with a 6DOF micro-positioning platform to evaluate the platform error due to the hysteresis of the piezoelectric actuators. By using inverse kinematics with the Preisach model, they achieved a platform accuracy at the level of 1 μm in position and 10 μ deg in orientation after several iterations. For their experiment, they used Lion Precision DMT22 dual sensitivity systems with C5 probe for measurement.

A popular method of hexapod calibration includes vision-based data collection. Daney et al. [16] employed calibration processes using a 1024 × 768 CCD camera. In their experiments, they used plates with dot marks as visual targets and obtained their images for measurement analysis. The final errors measured by them were relatively large with respect to the length of the actuators and with that they concluded that the kinematic model used was not robust.

Dallej et al. [54] used an omnidirectional camera. The measurement of the actuators had been investigated by using images from external cameras. They used an omnidirectional camera which overcomes the self-occlusion problem arising in the single perspective camera. By using the omnidirectional camera, Dallej et al. got a median error of approximately 1 cm when compared to the computer-aided design (CAD) geometry and data obtained from the camera.

Researchers Daney et al. did several experiments with hexapod and other parallel robots. They used this work [55] on the machine DeltaLab's “Table of Stewart” which involved Sony digital video camera (1024 × 768) with a 4.2 mm focal length for measuring the joint positions and actuator lengths. In this case, they used the constrained optimization method by combining it with the Tabu search, but the results obtained were not satisfying due to the error resulting from the bias of the system along the z-axis in the range of 1.29 mm. They selected 18 and 64 random poses for analyzing the pre- and post-calibration error values.

Gao et al. [56] carried out their study and calibration of a 500 m aperture spherical telescope (FAST) using inverse kinematics. They used a laser tracker for measurements and controlled the position and orientation of the platform with a Stewart–Gough platform-based fine feed cabin model (FFCM). In their study, they were able to achieve the desired accuracy level of 0.2 mm for the FAST. Here they had not considered any specific error factor except the final pose of the telescope.

In their research, Renaud et al. [57] used a CCD camera and an LCD monitor to calibrate a 6DOF parallel machine tool by using inverse kinematics. They compared the results by comparing the measurement data obtained for the same poses by a laser interferometer. The platform pose of the machine had been measured for comparison. The final comparison showed that the precision level obtained from the optical measurement is in the order of 1 μm in translation and 1 × 10−3 deg in rotations for an axial displacement of 400 mm.

The findings by Week and Staimer [58] considered the errors on the actuator length. They used a DBB as a redundant actuator in their Ingersoll HOH600 robot. The accuracy of roundness and squareness in their machining tests was improved by factors of 3.7 and 7.0, respectively. The system of setup they developed could be used to measure the errors due to the thermal and gravity load deflections at the end-effector pose.

An investigation by Ihara et al. [59] using a telescoping magnetic ball bar resulted in a reduction in motion error of the platform by 25% after calibration. They used Fourier transformation and included the length error of struts, and position errors of base and platform joints of the platform for optimizing the overall error of the system.

4.3 Constraint Approach.

The constraint calibration approach has the limitation of practical applications, for this reason, this calibration approach is less popular among researchers. Constraint approach for calibration is implemented by using some mechanical constraints to restrict the motion of one or more joints in the parallel robots [30,31]. Normally no external measuring instrument is used. The already existing sensors in the system act as the measuring sensor. The applied motion constraint causes a reduction in the degrees-of-freedom for the end-effector and reduction of workspace of the platform. The kinematics parameters also get reduced. The force generated due to constraining the movement may distort the structure and impact the accuracy of the calibration. As the system loses one or more degrees-of-freedom and the number of sensors becomes more than the active degrees-of-freedom, the calibration process may be considered as self- or auto-calibration process of a system with the reduced degrees-of-freedom.

Ryu and Rauf [31] used this approach to calibrate a hexapod “hexa slide mechanism (HSM)” by constraining one actuator at a time and repeating the process for all six actuators. Ryu and Rauf constrained the motion of the platform by restricting the motion of one actuator of the system and the system worked as a 5DOF system instead of the regular 6DOF system. There was no extra sensor used, the in-built existing actuator sensors were used for taking the needful measurements. By constraining one actuator, the initial error values of the platform position 8.0 × 10−3 m and orientation 1. × 10−2 rad converge to 3.8 × 10−16 m and 1.7 × 10−15 rad, respectively.

Rauf and Ryu [60] constrained three actuators and experimented with 3DOF in the same hexapod HSM mentioned in previous paragraph. Here also the final correction values are near zero. For 3DOF measurements, initial error values of position 1.0 × 10−2 m and orientation 1.7 × 10−2 rad changed to 2.4 × 10−10 m and 1.8 × 10−9 radians, respectively. The final correction values change by very small amounts depending on the value of the measurement noises. The advantage of these methods is that the locking device can be universal and need not be specific for a particular system.

4.4 Auto- or Self-Calibration Approach.

Auto- or self-calibration is one of the ways to calibrate 6DOF hexapod platforms. This method requires adding one or more redundant sensors to the passive joints or an additional redundant passive limb [30]. The addition of extra sensors increases the complexity of the design and manufacturing processes of platforms. Moreover, the addition of redundant sensors sometimes makes system development more expensive. The auto- or self-calibration method also limits the workspace for calibration. Some research studies have been done with this method.

Mura [33] used a set of wire extensometers to directly measure the position of the moving end plate relative to the fixed plate during testing of flexible automotive components. The testing motions used ensured that the extensometers remained in tension throughout the test and provided accurate positional data while not imparting a significant amount of additional stiffness into the system.

Similarly, Guo et al. [34] used a smaller set of four extensometers to directly measure the position of the end plate relative to the fixed plate during multimodal loading tests. The data were used in real-time to connect to an adams model of the system, and to generate force feedback information as part of the system control loop. This use of continuous calibration demonstrated that forces could be very precisely controlled during testing.

In their study, Chiu and Perng [61] used a cylindrical gauge block and a commercial trigger probe to do the auto-calibration of the hexapod platform. They made use of the nonlinear least-square method in their algorithm. The advantage of their method is that the instruments are standard and commercially developed, which makes them easily available and less expensive. Multiple robot configurations were used to validate their method and the results showed that different levels of accuracy were achieved.

Similarly, in another auto-calibration process, Zhuang et al. [62] used a coordinate measuring machine (CMM) with their robot FAU Stewart–Gough Platform. Here also the position and orientation of the platform were used to calibrate system error. They used Levenberg–Marquart algorithm for optimization and got error reduction by 50%. The highlights from their research are that some extra sensors needed to be installed in some of the joints to gather calibration data. Another conclusion they drew is that if the end-effector is a separate attachment on the hexapod platform, then the end-effector should be calibrated separately.

The research done by Patel and Ehmann [9], observed that for their calibration algorithm increased error of the platform poses in some cases and for around 90% of observations, they got accuracy improvement from 50% to 100%. They used the least-square method for their calibration algorithm. The advantage of their method is that the extra sensor is easily mountable and un-mountable; also, the extra sensor consisting of ball bar can remain with the system during the actual machine operation to allow online calibration.

The details are presented in Table 5. It should be noted that for each case, platform pose error was considered.

Table 5

Assessment of auto or self-calibration approaches

ReferenceRobot nameInstruments usedMethodsPerformanceImportant points
Mura [33]Wire extensometer (6)Direct kinematicsUsed to measure flexible automobile components in dynamic loading conditions such as fatigue situations.Extensometers continuously measure the platform position and represent a level of stiffness that is negligible compared to the measurement item.
Guo et al. [34]MLMTMWire extensometer (4)Direct kinematics and adams simulationsUsed to measure flexible specimens in multi-axial loading situations. Coupled with adams simulations to provide Proportional–Integral–Derivative (PID) control information for force control.Integrated into the control loop to provide deflection information to an adams model to provide data to a PID force control loop. The extensometers represent a negligible stiffness in comparison to the test specimens.
Chiu and Perng [61]A cylindrical gauge block and a commercial trigger probeNonlinear least squaresMultiple platform configurations were used to validate the calibration process and different accuracy levels have been obtained.The instruments used here are standardized and commercialized.
The method is comparatively compact and economical.
Zhuang et al. [62]FAU Stewart–Gough platformCMMLevenberg–Marquart algorithmThe average error was reduced by more than 50%.The end-effector required separate calibration since it was not part of the closed-loop kinematic chains.
It requires redundant sensors that need to be installed at some of the joints of the machine tool.
Patel and Ehmann [9]Ball barLeast-square minimizationAnalysis suggests that the measuring device accuracy needs to be five to ten times more than the desired calibration accuracy.The extra actuators can be mounted or unmounted easily. When left with the machine in certain situations, it enables online calibration.
ReferenceRobot nameInstruments usedMethodsPerformanceImportant points
Mura [33]Wire extensometer (6)Direct kinematicsUsed to measure flexible automobile components in dynamic loading conditions such as fatigue situations.Extensometers continuously measure the platform position and represent a level of stiffness that is negligible compared to the measurement item.
Guo et al. [34]MLMTMWire extensometer (4)Direct kinematics and adams simulationsUsed to measure flexible specimens in multi-axial loading situations. Coupled with adams simulations to provide Proportional–Integral–Derivative (PID) control information for force control.Integrated into the control loop to provide deflection information to an adams model to provide data to a PID force control loop. The extensometers represent a negligible stiffness in comparison to the test specimens.
Chiu and Perng [61]A cylindrical gauge block and a commercial trigger probeNonlinear least squaresMultiple platform configurations were used to validate the calibration process and different accuracy levels have been obtained.The instruments used here are standardized and commercialized.
The method is comparatively compact and economical.
Zhuang et al. [62]FAU Stewart–Gough platformCMMLevenberg–Marquart algorithmThe average error was reduced by more than 50%.The end-effector required separate calibration since it was not part of the closed-loop kinematic chains.
It requires redundant sensors that need to be installed at some of the joints of the machine tool.
Patel and Ehmann [9]Ball barLeast-square minimizationAnalysis suggests that the measuring device accuracy needs to be five to ten times more than the desired calibration accuracy.The extra actuators can be mounted or unmounted easily. When left with the machine in certain situations, it enables online calibration.

5 Discussion and Comparison

The goal of each calibration method is to make the hexapod machines more accurate, improve precision, and obtain correct results at the platform pose during their operations. Based on the literature survey done, it can be inferred that researchers predominantly opt for conducting experiments on their hexapod platforms to validate their calibration procedures. Among these experiments, the majority employed external measurement equipment. The key focus in these studies is the improvement of pose accuracy, primarily targeting platform pose errors, joint errors, and actuator length errors. The extent of error being addressed varies among these studies. In Table 6, the crucial details from each research endeavor were summarized, aiding readers in identifying literature of interest based on the chosen approach, error considerations, methods employed, and the level of error addressed.

Table 6

Comparison of experimental calibrations mentioned in this review

ReferenceCalibration approachInstruments usedError consideredMethodsPose error before calibrationPose error after calibration
Song et al. [48]ExternalFARO measuring armJoint errorsANN-based nonlinear functions0.1 mm and 0.1 deg0.1 mm and 0.1 deg
Mahmoodi et al. [49]ExternalRotary sensorsActuator lengthA new method0.1 m and 0.1 radReduced by 8 to 16 times
Jáuregui et al. [17]ExternalLaser interferometerActuator lengthTwo different methods based on error propagation calculation0–60 µm0–10 μm
Ren et al. [35]ExternalBiaxial inclinometer and laser trackerActuator lengthA new orientation constraint method9 mm and 1 deg0.1 mm and 0.01 deg
Nategh and Agheli [50]ExternalDigital cameraPlatform poseLeast-square approach based on Levenberg–Marquardt algorithm2.67 mm and 4.6 deg1.45 mm and 0.27 deg
Großmann et al. [51]ExternalDouble ball barPlatform poseGenetic algorithm based trajectory optimization.0.7 mm0.17 mm
Liu et al. [52]External3D laser trackerActuator lengthsGenetic algorithm0.7237 mm and 0.1346 deg0.1555 mm and 0.0172 deg
Ting et al. [53]ExternalDMT22 dual sensitivity systems with C5 probeActuator lengthsPreisach model1 µm and 10 µdeg
Daney et al. [16]ExternalCCD cameraJoint errorsInterval arithmetic and analysis methods1.32 mm and 0.34 deg1.10 mm and 0.26 deg
Dallej et al. [54]ExternalOmnidirectional cameraJoint errorsLinear regression1.2 cm∼1 cm
Daney et al. [55]ExternalSony digital video cameraPlatform poseConstrained optimization method with Tabu search1.45 mm and 0.27 deg
Gao et al. [56]ExternalLaser tracker LTD500Platform poseLeast-square method±2 mm±1 mm
Renaud et al. [57]ExternalA CCD camera and 1D laser interferometryPlatform poseError function minimization1 µm and 0.001 deg
Week et al. [58]ExternalDouble ball bar and CMMActuator lengthGravity compensationImprovement (µm):
Roundness 3.7×
Squareness 7×
Ihara et al. [59]ExternalTelescoping magnetic ball bar (DBB)Actuator length and joint positionsFourier transformationCircle center error
(−68, 164) µm
Circle center error
(−7, −9) µm
Chiu and Perng [61]Self-calibrationCylindrical gauge block and trigger probePlatform jointsNonlinear least squares±1 µm±0.001 µm
Zhuang et al. [62]Self-calibrationCMMPlatform jointsLevenberg–Marquart algorithm1 µin.0.1 µin.
Patel and Ehmann [9]Self-calibrationBall barPlatform poseLeast-square minimization±0.1 mm3–100%
Mura [33]Self-calibrationWire extensometersPlatform poseDirect kinematics±0.1 mm±0.005 mm
Guo et al. [34]Self-calibrationWire extensometersPlatform poseDirect kinematics±10 N, ±0.1 Nm±0.1 N, ±0.005 Nm
ReferenceCalibration approachInstruments usedError consideredMethodsPose error before calibrationPose error after calibration
Song et al. [48]ExternalFARO measuring armJoint errorsANN-based nonlinear functions0.1 mm and 0.1 deg0.1 mm and 0.1 deg
Mahmoodi et al. [49]ExternalRotary sensorsActuator lengthA new method0.1 m and 0.1 radReduced by 8 to 16 times
Jáuregui et al. [17]ExternalLaser interferometerActuator lengthTwo different methods based on error propagation calculation0–60 µm0–10 μm
Ren et al. [35]ExternalBiaxial inclinometer and laser trackerActuator lengthA new orientation constraint method9 mm and 1 deg0.1 mm and 0.01 deg
Nategh and Agheli [50]ExternalDigital cameraPlatform poseLeast-square approach based on Levenberg–Marquardt algorithm2.67 mm and 4.6 deg1.45 mm and 0.27 deg
Großmann et al. [51]ExternalDouble ball barPlatform poseGenetic algorithm based trajectory optimization.0.7 mm0.17 mm
Liu et al. [52]External3D laser trackerActuator lengthsGenetic algorithm0.7237 mm and 0.1346 deg0.1555 mm and 0.0172 deg
Ting et al. [53]ExternalDMT22 dual sensitivity systems with C5 probeActuator lengthsPreisach model1 µm and 10 µdeg
Daney et al. [16]ExternalCCD cameraJoint errorsInterval arithmetic and analysis methods1.32 mm and 0.34 deg1.10 mm and 0.26 deg
Dallej et al. [54]ExternalOmnidirectional cameraJoint errorsLinear regression1.2 cm∼1 cm
Daney et al. [55]ExternalSony digital video cameraPlatform poseConstrained optimization method with Tabu search1.45 mm and 0.27 deg
Gao et al. [56]ExternalLaser tracker LTD500Platform poseLeast-square method±2 mm±1 mm
Renaud et al. [57]ExternalA CCD camera and 1D laser interferometryPlatform poseError function minimization1 µm and 0.001 deg
Week et al. [58]ExternalDouble ball bar and CMMActuator lengthGravity compensationImprovement (µm):
Roundness 3.7×
Squareness 7×
Ihara et al. [59]ExternalTelescoping magnetic ball bar (DBB)Actuator length and joint positionsFourier transformationCircle center error
(−68, 164) µm
Circle center error
(−7, −9) µm
Chiu and Perng [61]Self-calibrationCylindrical gauge block and trigger probePlatform jointsNonlinear least squares±1 µm±0.001 µm
Zhuang et al. [62]Self-calibrationCMMPlatform jointsLevenberg–Marquart algorithm1 µin.0.1 µin.
Patel and Ehmann [9]Self-calibrationBall barPlatform poseLeast-square minimization±0.1 mm3–100%
Mura [33]Self-calibrationWire extensometersPlatform poseDirect kinematics±0.1 mm±0.005 mm
Guo et al. [34]Self-calibrationWire extensometersPlatform poseDirect kinematics±10 N, ±0.1 Nm±0.1 N, ±0.005 Nm

It's worth noting that the units used to measure errors differ across various studies, and some literature sources do not specify pose errors before the calibration process; they only report final values after calibration. Additionally, in certain instances, pose error is defined solely by position errors, with no consideration given to orientation errors. Remarkably, none of the experiments focus exclusively on improving accuracy through orientation-related factors.

As mentioned earlier, the pose accuracy is dependent on several mechanical and surrounding factors like temperature and load being experienced by the system. The ideal calibration would consider all these factors for any working condition of the machine but achieving that is not only expensive and time-consuming but also potentially unnecessary depending on the application conditions. In experiments where only the actuator lengths have been considered as the sources of error for the accuracy of the platform pose, it should be noted that the motion of the platform is dependent on all the joints which are moving to generate the motion. So, while calibration may include only the actuator length error, it also indirectly includes the error contributed by the joints. Even if all joints are not equipped with individual sensors, their error factors are indirectly accounted for in the calibration process. In general, when the platform pose accuracy is of primary interest, the calibration of it indirectly considers all the error factors existing in the hexapod system but adding appropriate compensation for each error factor becomes difficult unless they are correctly identified and accounted for in the calculations.

5.1 Error Identification.

From the tables above, it is seen that the level of improvements obtained is varied, with a few cases showing an increase in the error values. The error factors considered are also different in each study. In most cases, the platform pose error due to the actuator length errors remained common and was considered as the primary error contributor in the whole system. Also, considering the error of each actuator separately and equally impacts the overall accuracy of the platform. In all these calibration processes, use of external instruments is the most common practice. External instruments such as DBB, laser interferometer, biaxial inclinometer, laser tracker, telescopic magnetic ball bar, and wire extensometers were used. A separate study may be useful to ascertain the effectiveness of each of these instruments for the calibration of a hexapod by inverse kinematics.

Optical calibration methods have also gained importance in recent times. CCD camera, omnidirectional camera, digital video camera, and other image-capturing devices were used to achieve an accuracy level in the range of 1.0 cm to 0.1 mm. The advantage of using optical methods is that the parallel machine does not need modifications to accommodate the measurement equipment for calibration, also it is non-invasive and automatically records the event for future reference. In these cases, the quality of the optical systems and the associated analysis system configuration has a major contribution to the final accuracy and precision attained.

5.2 Algorithms.

Among the several different algorithms used for the calibration purpose, the least-square method based on the Levenberg–Marquardt algorithm was used most for both optical and non-optical calibration. The genetic algorithm had been used for a couple of studies, but they appeared to have high running time to reach an optimized level and were not sensitive to measuring noises. Constrained optimization method and Tabu search, algebraic Elimination, nonlinear least-square methods were used and all of them resulted in various levels of calibration accuracy. Different algorithms also yielded different results on the same system for the calibrations done by Daney and Ehmann [45]. Therefore, the selection of calibration methods and algorithms plays an important role in obtaining the desired accuracy level for the parallel robot system and application. In all cases, the final platform pose is the guiding parameter to evaluate the calibration outcome.

5.3 External Factors.

There are some inherent dimensional errors in the hexapod structure due to the dimensional tolerances of each component used in the fabrication. The cumulative effects of all these tolerances play a significant role in platform accuracy. Apart from these errors, other factors, hysteresis for instance, vary with the change of the operational characteristics. Normally, the effect of temperature is expected to be minimal for the parallel robot system unless the system experiences large temperature variations during its operation. From a practical point of view, such robot systems operate in a controlled environment unless they are deployed in special applications like large field telescope mounting. In these applications where exposure to fluctuating weather conditions is unavoidable, proper calibration factors for thermal deviation must be included.

Likewise, the elastic deformation error factor is not dominant in all cases. If the hexapod platform is subjected to high loads relative to its structure, a factor for elastic deflection is essential. Hexapod platforms reviewed in this paper can carry loads up to 2000 kg though none of them were subjected to such load during the calibration process. As such, the structure can undergo a substantial amount of elastic stress and that may lead to a notable amount of deflection to impact the accuracy of the robot. In these types of cases, the factor of elastic load cannot be ignored. There are potential research opportunities for evaluating the impact of load conditions on the parallel robots and adding suitable compensation factors for further improving the platform pose accuracy.

6 Conclusions

Hexapods are used for precise, complex, and repeatable operations in a variety of applications. Small errors in the system can lead to a serious impact on pose accuracy. Therefore, calibration is a critical activity for any hexapod machine to be reliable in each application. There are several sources of errors that negatively impact the accuracy of the hexapod. Selecting an appropriate calibration approach is dependent upon the purpose and constraints of the hexapod system. In some cases, external sensor calibration may be the best solution, while in other cases, constrained or auto-calibration may be a better approach. This paper attempts to provide an overview of those methods and draw an outline of the current state of the art in this field and help other researchers to take appropriate notes for the desired methods.

Researchers have explored a variety of calibration methods, each exhibiting varying degrees of accuracy and complexity in implementation. The focus of this review is to identify the methods used to improve the positional accuracy of the hexapod platform or end-effector and accuracy improvement obtained after calibration. Instruments and techniques utilized to compensate pose errors stemming from various inaccuracies were summarized. Individual researchers searching for an appropriate calibration approach should be cognizant of the level of calibration needed for a particular application and the resulting impact upon the complexity of implementation. Three types of calibration strategies are identified for physical systems: external, constraint, and auto- or self-calibration. Double ball bars, laser trackers, laser interferometers, and optical devices are some of the most used instruments for collecting positional data. The types of errors typically considered in these studies revolve around platform pose, joint variables, and actuator length. In addition to the studies on physical systems, analytical-based studies were also discussed, and their specifics were summarized. In general, the authors agree with the sentiment of Merlet [30], who suggested that the emergence of image-based systems offers a great deal of potential improvement. Since this statement was published in 2006, we have only observed improvements in the hardware and software supporting image-based calibration and we also note that biological systems generally use manipulator-image-coordination (i.e., hand-eye-coordination) approaches with great success.

The literature reviewed in this paper predominantly covers research conducted since the year 2000 focusing hexapod platforms and on calibration of platform pose under no-load condition; however, in actual applications, these hexapods may be experiencing heavy working loads. The effect of working load propagates directly to the hexapod structure. Based on structural rigidity, the load causes elastic deformation which can affect the operational accuracy. Therefore, the response behavior and platform accuracy of a hexapod under the influence of working loads remain a subject for further studies.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

References

1.
Harib
,
K. H.
,
Sharif Ullah
,
A. M. M.
, and
Hammami
,
A.
,
2007
, “
A Hexapod-Based Machine Tool With Hybrid Structure: Kinematic Analysis and Trajectory Planning
,”
Int. J. Mach. Tools Manuf.
,
47
(
9
), pp.
1426
1432
.
2.
Shi
,
H.
,
She
,
Y.
, and
Duan
,
X.
,
2015
, “
Modeling and Measurement Algorithm of Hexapod Platform Sensor Using Inverse Kinematics
,”
2015 5th Australian Control Conference (AUCC)
,
Griffith University Gold Coast Campus in Gold Coast, Australia
,
Nov. 5–6
, pp.
331
335
.
3.
Pandilov
,
Z.
, and
Dukovski
,
V.
,
2012
, “
Parallel Kinematics Machine Tools : Overview—From History to the Future
,”
International Journal of Engineering
,
2
(
1
), pp.
111
124
.
4.
Jouini
,
M.
,
Sassi
,
M.
,
Sellami
,
A.
, and
Amara
,
N.
,
2013
, “
Modeling and Control for a 6-DOF Platform Manipulator
,”
2013 International Conference on Electrical Engineering and Software Applications
,
Hammamet, Tunisia
,
Mar. 21–23
.
5.
Gough
,
V. E.
, and
Whitehall
,
S G
,
1962
, “
Universal Tyre Testing Machine
,”
Proceedings Automation Division Institution of Mechanical Engineers
,
IMechE 1, London, UK
.
6.
Stewart
,
D.
,
1965
, “
A Platform With Six Degrees of Freedom
,”
Proc. Inst. Mech. Eng.
,
180
(
1
), pp.
371
386
.
7.
Wu
,
J. F.
,
Zhang
,
R.
,
Wang
,
R. H.
, and
Yao
,
Y. X.
,
2014
, “
A Systematic Optimization Approach for the Calibration of Parallel Kinematics Machine Tools by a Laser Tracker
,”
Int. J. Mach. Tools Manuf.
,
86
, pp.
1
11
.
8.
Mazare
,
M.
,
Taghizadeh
,
M.
, and
Rasool Najafi
,
M.
,
2017
, “
Kinematic Analysis and Design of a 3-DOF Translational Parallel Robot
,”
Int. J. Autom. Comput.
,
14
(
4
), pp.
432
441
.
9.
Patel
,
A. J.
, and
Ehmann
,
K. F.
,
2000
, “
Calibration of a Hexapod Machine Tool Using a Redundant Leg
,”
Int. J. Mach. Tools Manuf.
,
40
(
4
), pp.
489
512
.
10.
Patel
,
Y. D.
, and
George
,
P. M.
,
2012
, “
Parallel Manipulators Applications—A Survey
,”
Mod. Mech. Eng.
,
02
(
03
), pp.
57
64
.
11.
Physik Instrumente (PI) GmbH & Co KG.
,
2018
,
Hexapod Parallel Robots Automate Highly Precise Production Processes
.
12.
Majarena
,
A. C.
,
Santolaria
,
J.
,
Samper
,
D.
, and
Aguilar
,
J. J.
,
2010
, “
An Overview of Kinematic and Calibration Models Using Internal/External Sensors or Constraints to Improve the Behavior of Spatial Parallel Mechanisms
,”
Sensors (Switzerland)
,
10
(
11
), pp.
10256
10297
.
13.
Tsai
,
L.
,
1999
,
Robot Analysis: The Mechanics of Serial and Parallel Manipulators
,
John Wiley and Sons, Inc
,
Hoboken, NJ
.
14.
Lee
,
T.-Y.
, and
Shim
,
J.-K.
,
2001
, “
Forward Kinematics of the General 6-6 Stewart Platform Using Algebraic Elimination
,”
Mech. Mach. Theory
,
26
(
9
), pp.
1073
1085
.
15.
Merlet
,
J. P.
,
2002
, “
Still a Long Way to Go on the Road for Parallel Mechanisms
,”
Proceedings of the ASME International Mechanical Engineering Congress and Exhibition
,
Montreal, Canada
,
Sept. 29–Oct. 2
.
16.
Daney
,
D.
,
Andreff
,
N.
,
Chabert
,
G.
, and
Papegay
,
Y.
,
2006
, “
Interval Method for Calibration of Parallel Robots: Vision-Based Experiments
,”
Mech. Mach. Theory
,
41
(
8
), pp.
929
944
.
17.
Jáuregui
,
J. C.
,
Hernández
,
E. E.
,
Ceccarelli
,
M.
,
López-Cajún
,
C.
, and
García
,
A.
,
2013
, “
Kinematic Calibration of Precise 6-DOF Stewart Platform-Type Positioning Systems for Radio Telescope Applications
,”
Front. Mech. Eng.
,
8
(
3
), pp.
252
260
.
18.
Ziegert
,
J. C.
,
Jokiel
,
B.
, and
Huang
,
C.-C.
,
1999
, “Calibration and Self-Calibration of Hexapod Machine Tools,”
Parallel Kinematic Mach
,
Springer-Verlag Ltd.
,
London
, pp.
205
216
.
19.
Everett
,
L. J.
,
Driels
,
M.
, and
Mooring
,
B. W.
,
1987
, “
Kinematic Modelling for Robot Calibration
,”
Proceedings. 1987 IEEE International Conference on Robotics and Automation
,
Raleigh, NC
,
Mar. 31–Apr. 3
, IEEE, pp.
183
189
.
20.
Elatta
,
A. Y.
,
Gen
,
L. P.
,
Zhi
,
F. L.
,
Daoyuan
,
Y.
, and
Fei
,
L.
,
2004
, “
An Overview of Robot Calibration
,”
Inf. Technol. J.
,
3
(
1
), pp.
74
78
.
21.
Meng
,
Y.
, and
Zhuang
,
H.
,
2007
, “
Autonomous Robot Calibration Using Vision Technology
,”
Rob. Comput. Integr. Manuf.
,
23
(
4
), pp.
436
446
.
22.
Nubiola
,
A.
, and
Bonev
,
I. A.
,
2014
, “
Absolute Robot Calibration With a Single Telescoping Ballbar
,”
Precis. Eng.
,
38
(
3
), pp.
472
480
.
23.
Szep
,
C.
,
Stan
,
S. D.
,
Csibi
,
V.
,
Manic
,
M.
, and
Bǎlan
,
R.
,
2009
, “
Kinematics, Workspace, Design and Accuracy Analysis of RPRPR Medical Parallel Robot
,”
Proceedings of 2009 2nd Conference on Human System Interactions (HSI ’09)
,
Catania, Italy
,
May 21–23
, pp.
75
80
.
24.
Bleicher
,
F.
,
Puschitz
,
F.
, and
Theiner
,
A.
,
2006
, “
Laser Based Measurement System for Calibrating Machine Tools in 6 DOF
,”
Annals of DAAAM and Proceedings of the 17th International DAAAM Symposium
,
Cluj-Napoca, Romania
,
Mar. 24–25
, pp.
39
40
.
25.
Ramesh
,
R.
,
Mannan
,
M.
, and
Poo
,
A.
,
2000
, “
Error Compensation in Machine Tools—A Review. Part I
,”
Int. J. Mach. Tools Manuf.
,
40
(
9
), pp.
1257
1284
.
26.
Szatmári
,
S.
,
1999
, “
Geometrical Errors of Parallel Robots
,”
Period. Polytech. Mech. Eng.
,
43
(
2
), pp.
155
162
.
27.
Daney
,
D.
,
2002
, “
Optimal Measurement Configurations for Gough Platform Calibration
,”
Proceedings of 2002 IEEE International Conference on Robotics and Automation
,
Washington, DC
,
May 11–15
, Vol. 1, pp.
147
152
.
28.
Soons
,
J. A.
,
1997
, “
Error Analysis of a Hexapod Machine Tool
,”
Trans. Eng. Sci.
,
16
, pp.
347
358
.
29.
Traslosheros
,
A.
,
Sebastián
,
J. M.
,
Torrijos
,
J.
,
Carelli
,
R.
, and
Castillo
,
E.
,
2013
, “
An Inexpensive Method for Kinematic Calibration of a Parallel Robot by Using One Hand-Held Camera as Main Sensor
,”
Sensors (Switzerland)
,
13
(
8
), pp.
9941
9965
.
30.
Merlet
,
J. P.
,
2006
,
Parallel Robotics
, 2nd ed.,
Springer
,
Dordrecht, The Netherlands
.
31.
Ryu
,
J.
, and
Rauf
,
A.
,
2001
, “
A New Method for Fully Autonomous Calibration of Parallel Manipulators Using a Constraint Link
,”
2001 IEEE/ASME International Conference on Advanced Intelligent Mechatronics. Proceedings (Cat. No. 01TH8556)
,
Como, Italy
,
July 8–12
, Vol. 1, pp.
141
146
.
32.
Traslosheros
,
A.
,
Sebastián
,
J. M.
,
Castillo
,
E.
,
Roberti
,
F.
, and
Carelli
,
R.
,
2011
, “
A Method for Kinematic Calibration of a Parallel Robot by Using One Camera in Hand and a Spherical Object
,”
IEEE 15th International Conference on Advanced Robotics New Boundaries Robot (ICAR 2011)
,
Tallinn, Estonia
,
June 20– 23
, pp.
75
81
.
33.
Mura
,
A.
,
2011
, “
Six D.O.F. Displacement Measuring Device Based on a Modified Stewart Platform
,”
Mechatronics
,
21
(
8
), pp.
1309
1316
.
34.
Guo
,
J.
,
Wang
,
D.
,
Fan
,
R.
,
Chen
,
W.
, and
Zhao
,
G.
,
2016
, “
Development of a Material Testing Machine With Multi-dimensional Loading Capability
,”
J. Adv. Mech. Des. Syst. Manuf.
,
10
(
2
).
35.
Ren
,
X. D.
,
Feng
,
Z. R.
, and
Su
,
C. P.
,
2009
, “
A New Calibration Method for Parallel Kinematics Machine Tools Using Orientation Constraint
,”
Int. J. Mach. Tools Manuf.
,
49
(
9
), pp.
708
721
.
36.
Huang
,
T.
,
Wang
,
J.
,
Chetwynd
,
D. G.
, and
Whitehouse
,
D. J.
,
2003
, “
Identifiability of Geometric Parameters of 6-DOF PKM Systems Using a Minimum Set of Pose Error Data
,”
Proc. IEEE Int. Conf. Robot. Autom.
,
2
(
1
), pp.
1863
1868
.
37.
Zou
,
H.
, and
Notash
,
L.
,
2001
, “
Discussions on the Camera-Aided Calibration of Parallel Manipulators
,”
Proceedings of 2001 CCToMM Symposium on Mechanisms, Machines, and Mechatronics
,
Saint-Hubert, Canada
,
May 31–June 1
, pp.
3
4
.
38.
Olarra
,
A.
,
Axinte
,
D.
, and
Kortaberria
,
G.
,
2018
, “
Geometrical Calibration and Uncertainty Estimation Methodology for a Novel Self-Propelled Miniature Robotic Machine Tool
,”
Robot. Comput. Integr. Manuf.
,
49
(
Jan.
), pp.
204
214
.
39.
Zhuang
,
H.
,
Yan
,
J.
, and
Masory
,
O.
,
1998
, “
Calibration of Stewart Platforms and Other Parallel Manipulators by Minimizing Inverse Kinematic Residuals
,”
J. Robot. Syst.
,
15
(
7
), pp.
395
405
.
40.
Santolaria
,
J.
, and
Ginés
,
M.
,
2013
, “
Uncertainty Estimation in Robot Kinematic Calibration
,”
Robot. Comput. Integr. Manuf.
,
29
(
2
), pp.
370
384
.
41.
Vischer
,
P.
, and
Clavel
,
R.
,
1998
, “
Kinematic Calibration of the Parallel Delta Robot
,”
Robotica
,
16
(
2
), pp.
207
218
.
42.
Jing
,
X.
,
Fang
,
Y.
, and
Wang
,
Z.
,
2020
, “
A Calibration Method for 6-UPS Stewart Platform
,”
Proceedings of 2019 Chinese Intelligent systems Conference
,
Fuzhou, China
,
Oct. 16–17
, Vol. 593,pp. 513–519.
43.
Agheli
,
M.
, and
Nategh
,
M.
,
2009
, “
Identifying the Kinematic Parameters of Hexapod Machine Tool
,”
World Academy of Science, Engineering and Technology, Int. J. Mech. Aeros. Ind., Mech. Manuf. Eng.
,
3
, pp.
392
397
.
44.
Daney
,
D.
,
Papegay
,
Y.
, and
Neumaier
,
A.
,
2004
, “
Interval Methods for Certification of the Kinematic Calibration of Parallel Robots
,”
Proc. IEEE Int. Conf. Robot. Autom.
,
2004
(
2
), pp.
1913
1918
.
45.
Daney
,
D.
, and
Emiris
,
I. Z.
,
2004
, “
Algebraic Elimination for Parallel Robot Calibration
,”
Proceedings of 11 World Congress of International Federation for the Theory of Machines and Mechanisms (IFToMM)
,
Tianjin, China
,
April
.
46.
Wang
,
S. M.
, and
Ehmann
,
K. F.
,
2002
, “
Error Model and Accuracy Analysis of a Six-DOF Stewart Platform
,”
ASME J. Manuf. Sci. Eng.
,
124
(
2
), pp.
286
295
.
47.
Daney
,
D.
, and
Emiris
,
I.
,
2001
, “
Variable Elimination for Reliable Parallel Robot Calibration
,”
2nd Workshop on Computational Kinematics (CK2001)
,
Seoul, South Korea
,
May 20–22
, p.
13
.
48.
Song
,
Y.
,
Tian
,
W.
,
Tian
,
Y.
, and
Liu
,
X.
,
2022
, “
Calibration of a Stewart Platform by Designing a Robust Joint Compensator With Artificial Neural Networks
,”
Precis. Eng.
,
77
(
June
), pp.
375
384
.
49.
Mahmoodi
,
A.
,
Sayadi
,
A.
, and
Menhaj
,
M. B.
,
2014
, “
Solution of Forward Kinematics in Stewart Platform Using Six Rotary Sensors on Joints of Three Legs
,”
Adv. Rob.
,
28
(
1
), pp.
27
37
.
50.
Nategh
,
M. J.
, and
Agheli
,
M. M.
,
2009
, “
A Total Solution to Kinematic Calibration of Hexapod Machine Tools With a Minimum Number of Measurement Configurations and Superior Accuracies
,”
Int. J. Mach. Tools Manuf.
,
49
(
15
), pp.
1155
1164
.
51.
Großmann
,
K.
,
Kauschinger
,
B.
, and
Szatmári
,
S.
,
2008
, “
Kinematic Calibration of a Hexapod of Simple Design
,”
Prod. Eng.
,
2
(
3
), pp.
317
325
.
52.
Liu
,
Y.
,
Liang
,
B.
,
Li
,
C.
,
Xue
,
L.
,
Hu
,
S.
, and
Jiang
,
Y.
,
2007
, “
Calibration of a Steward Parallel Robot Using Genetic Algorithm
,”
Proceedings of 2007 International Conference on Mechatronics and Automation (ICMA 2007)
,
Harbin, Heilongjiang, China
,
Aug. 5–9
, pp.
2495
2500
.
53.
Ting
,
Y.
,
Jar
,
H. C.
, and
Li
,
C. C.
,
2007
, “
Measurement and Calibration for Stewart Micromanipulation System
,”
Precis. Eng.
,
31
(
3
), pp.
226
233
.
54.
Dallej
,
T.
,
Hadj-Abdelkader
,
H.
,
Andreff
,
N.
, and
Martinet
,
P.
,
2006
, “
Kinematic Calibration of a Gough-Stewart Platform Using an Omnidirectional Camera
,”
2006 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)
,
Beijing, China
,
Oct. 9–15
, pp.
4666
4671
.
55.
Daney
,
D.
,
Papegay
,
Y.
, and
Madeline
,
B.
,
2005
, “
Choosing Measurement Poses for Robot Calibration With the Local Convergence Method and Tabu Search
,”
Int. J. Rob. Res.
,
24
(
6
), pp.
501
518
.
56.
Gao
,
M.
,
Li
,
T.
, and
Yin
,
W.
,
2003
, “
Calibration Method and Experiment of Stewart Platform Using a Laser Tracker
,”
Proc. IEEE Int. Conf. Syst. Man Cybern.
,
3
, pp.
2797
2802
.
57.
Renaud
,
P.
,
Andreff
,
N.
,
Dhome
,
M.
, and
Martinet
,
P.
,
2002
, “
Experimental Evaluation of a Vision-Based Measuring Device for Parallel Machine-Tool Calibration
,”
IEEE Int. Conf. Intell. Robot. Syst.
,
2
, pp.
1868
1873
.
58.
Week
,
M.
, and
Staimer
,
D.
,
2002
, “
Accuracy Issues of Parallel Kinematic Machine Tools
,”
Proc. Inst. Mech. Eng. Part K J. Multi-body Dyn.
,
216
(
1
), pp.
51
57
.
59.
Ihara
,
Y.
,
Ishida
,
T.
,
Kakino
,
Y.
,
Li
,
Z.
,
Matsushita
,
T.
, and
Nakagawa
,
M.
,
2000
, “
Kinematic Calibration of a Hexapod Machine Tool by Using Circular Test
,”
Proceedings of 2000 Japan US Symposium on Flexible Automation Conference
,
Ann Arbor, MI
,
July 23–26
, pp.
1
4
.
60.
Rauf
,
A.
, and
Ryu
,
J.
,
2001
, “
Fully Autonomous Calibration of Parallel Manipulators by Imposing Position Constraint
,”
Proc. IEEE Int. Conf. Robot. Autom.
,
3
, pp.
2389
2394
.
61.
Chiu
,
Y. J.
, and
Perng
,
M. H.
,
2003
, “
Self-Calibration of a General Hexapod Manipulator Using Cylinder Constraints
,”
Int. J. Mach. Tools Manuf.
,
43
(
10
), pp.
1051
1066
.
62.
Zhuang
,
H.
,
Liu
,
L.
, and
Masory
,
O.
,
2000
, “
Autonomous Calibration of Hexapod Machine Tools
,”
ASME J. Manuf. Sci. Eng.
,
122
(
1
), pp.
140
148
.