To attain a high level of accuracy, the model must consider the most significant geometric and non-geometric parameters for the mechanism designed.

Geometric errors

Geometric errors may appear from manufacturing errors or from the deviation of the offsets of the components. Joints in the links are not perfect, so the axes cannot be perpendicular between them, and they cannot intersect in the exact center of the joint. Errors when assembling actuators can cause the axis of each actuator not to pass through the center of the joint. Other errors can appear when measuring the offset of the components at the location of the mechanism’s joint.

One of the most widely used geometric methods for modelling an open-loop or a closed-loop mechanism is the well-known Denavit-Hartenberg method [98]. This method allows us to model the joints with four parameters. One of the limitations of this method appears when it is applied to those mechanisms that present two consecutive parallel joint axes. In this case, an infinite number of common normals of the same length exist, and the location of the axis coordinate system may be made arbitrarily. In [97], Everett mentioned the most relevant publications which propose some solutions for this limitation. In [99–102], the authors developed methods to obtain a complete, equivalence and proportional model. Some research has been carried out on obtaining an accuracy model [103,104], in which manufacturing tolerances, assembly errors and offsets are studied to develop an algorithm for the identification of the kinematic parameters of the Stewart platform. In each joint–link chain three types of parameters appear: measurable variables for describing the extension of the prismatic joints, un-measurable variables describing joint angles and geometric parameters describing the dimensions of the platform.

Once the kinematic model has been determined the number of parameters will be fixed, and will depend on the selected method. Wang [103] determined that the number of geometric parameters to define a kinematic chain is 22 [103], and this number can be reduced to 7 if passive joints are considered as perfect joints. Kinematic parameters usually correspond to the position of spherical, universal or revolute joints. Therefore, each spherical ball joint has three kinematic parameters, and each prismatic joint adds a new parameter, corresponding to its elongation [105]. The method employed to calibrate the Stewart platform establishes the orientation constraint by maintaining two attitude angles of the end-effector constant.

Non-geometric errors

The non-geometric errors can appear from backlash, gear transmission, friction, gravity, temperature or compliance [106]. The non-geometric models try to predict and compensate these errors.

Some authors [107] have developed non-geometric models to achieve this. In [108], Renders has gathered the influence of non-geometric errors. The errors that have the most significant effect on accuracy are joint flexibility, link flexibility, gear transmission error, backlash in gear transmission, and temperature effect. Flexibility in joints and in links causes 8% - 10% of the position and orientation errors of the end-effector, and link flexibility is usually 5%. Joint flexibility errors can be reduced by mounting the joint encoders directly on the joint after the transmission units, instead of mounting them on the motor shaft. Gear transmission errors are mainly due to runout and orientation errors. The contribution of backlash is from 0.5%–1%. The error due to temperature effects causes 0.1% of the total error. Calibration is usually performed in an environment where the temperature is controlled. Next, a correction model that considers the working temperature is applied. Judd [109] developed a model to correct problems with robot accuracy resulting from imperfections in the main spur and encoder pinion gears, errors in the link and joint parameters and structural deformations. Hollerbach [110] introduced a calibration index that considers sensed and un-sensed joints and single and multiple loops. In [111], Gong developed an algorithm for non-geometric error identification and compensation by means of the inverse calibration of the system, analyzing the effect of geometric errors with temperature variation and compliance. These methods separated the influence of the errors due to geometric and non-geometric parameters, in order to optimize geometric parameters by means of a traditional static calibration, and to model and correct non-geometric errors by means of a dynamic calibration. However, they are not generalizable and they do not allow us to know the individual influence of each non-geometric component.

To consider all the possible errors in the same kinematic model is a laborious task that, due to the complexity of the model, does not always increase the accuracy of the result. Moreover, it can add errors in the resolution of the problem, for example in the case where the model adds discontinuous functions for backlash or gearing errors, or when parameters are a function of joint variables instead of being a function of constants.

It is not possible to know a priori what parameters must be used in the calibration process to obtain the desired accuracy, but the mechanism repeatability must be considered in order to predict the order of magnitude of the accuracy that can be reached.

Everett [112] analyzed models based on forward kinematics and explained that there are a maximum number of parameters that must be identified, and that the model accuracy cannot be improved by adding extra parameters. The author explained that unless all joints are moved, not all parameters can be identified, because if one or more joints do not move, some unknowns can be decomposed, shifted and absorbed into others. He determined that four parameters must be considered for each revolute joint, two of which must be orientational. And for a prismatic joint two orientational parameters are necessary, applied about the non-colinear axes before and perpendicular to the translational joint axes. Thus, it is more frequent to add parameters to compensate non-geometric errors in the geometric model [113].

Regardless of the method selected, calibration can be solved by means of inverse or forward kinematics. The calibration problem can be formulated in terms of residual measurements. For example differences between the joint variable measurements and the values obtained by the inverse kinematic model. This model offers significant advantages compared to the one based on forward kinematics, since calculations in the latter case are more complex and require more time to be solved. Besides, the solution in the inverse model is unique, unlike the forward one in which several possible solutions can appear. The inverse model allows us to decouple the calibration problem for every kinematic chain, and the constraints can be expressed by analytical equations. This method gives numerical efficiency but that measurement of the positions has to be very precise.

Another possibility is to perform partial measurements of the position, posing the problem in terms of errors between measured values and computed values via forward kinematics. In [114], the kinematic model was solved using this method.

In [103], Wang presented a method for the calibration of a 6-UPS robot that uses parallel kinematics according to the lengths of kinematic chains and positioning parameters of the platform. Zhuang [115] developed a model in terms of residual measurements of the difference between the measurement length of the kinematic chain and the one obtained from the model without using forward kinematics. The author solved the problem by means of minimizing the sum of squares of the constraint equations. Parameter errors are mainly due to errors in the assembly of spherical and universal joints, and solutions obtained for some parameters are out of the range provided by the method, which means that some constraint equations are not satisfied. In [116], Daney developed an algorithm to perform the calibration by means of partial measurements of the position, thanks to the elimination of the rotation parameters. This algorithm endeavours to obtain the advantages of the two models, inverse and forward, combining the elimination of symbolic variables through numerical optimization, which allows us to obtain numerical stability. The method is applied for three different cases in which the number of restrictions and equations to perform the calibration varies. The conclusions are that methods that obtain better results do not match those which have a higher algebraic computational cost, and that the elimination of the rotation parameters improves the accuracy of initial estimations. In [117], the kinematic problem was solved by equalizing constraint equations to a value, ɛ_i, to obtain a solution.

Classical robot calibration methods use additional sensors to measure the position and orientation of the end-effector and the joint variables of the ball joints, where the calibration process optimizes the error between the measured and computed variables.

The type of sensors used in a parallel mechanism affects not only the design process but also the calibration procedure. Sensors can be used to measure the variables of the mechanism, usually the active ones, in order to obtain the necessary data to solve the kinematic problem. In the calibration procedure, mechanism internal sensors are used to obtain information of the system. These data will be the input to the mathematical model. The output of the forward model will be the calculated position and orientation of the end-effector. In the calibration procedure, the nominal and the calculated position and orientation of the end-effector are compared and the mechanism geometric parameters are obtained. On the other side, external devices having measurement systems allow us to measure the nominal position and orientation of the end-effector. It is important to note that every measurement error of the measurement device will be propagated to the calibration results. Therefore, measurement devices must be more accurate than the desired accuracy of the mechanism that is going to be calibrated.

In parallel mechanisms, the most used internal sensors are lineal optical sensors (for measuring the elongation of the actuator), rotary optical sensors (for measuring the motor rotation of the actuator), linear variable differential transformer (LVDT) and force-torque sensors (for the dynamic calibration).

Accuracy of linear and rotary optical sensors is highly dependent on the method used to couple the encoder to a shaft. This value can commonly reach ±0.5 μm and ±1 arcsecond, respectively, and resolution 1 nm and 0.02 arcseconds, respectively. LVDTs present a very high reliability. Accuracy and resolution are limited only by the signal conditioning electronics and the analog-to-digital converters. Resolution can reach the nanometer range. These types of sensors are used to measure relative motion between objects whose surfaces only move a little bit with respect to each other. Besides, their measurement range is low (about 0.5 m). On the contrary, linear optical sensor measurement range is up to 30 m and rotary sensors offer not rotation limit for incremental encoders and several turns for absolute encoders. Force-torque sensors are commonly used to measure the applied forces on the mechanism links. These devices frequently present a force sensor accuracy of 6 mN and a torque sensor accuracy of 30 mN·mm

External devices typically used in the calibration procedure to improve the mechanism accuracy are cameras, laser trackers, coordinate measuring machines (CMM) or autocollimators. Cameras and autocollimators are non-contact measurement instruments. These devices are therefore more suitable when the influence of measurement forces can affect the results. Cameras and 3D imaging sensors present compactness, robustness and flexibility [118]. The rapid development of these devices in the last decades has significantly improved their accuracy. Another advantage of these sensors is their portability. Moreover, the recent development in this technology allows us to perform a massive data acquisition. CMMs offer high resolution and accuracy (about a few micrometers). The laser tracker volumetric accuracy is about tens of micrometers. A typical measurement range is about 900 mm × 1,200 mm × 700 mm in a CMM and up to 40 m radius range in a laser tracker. Therefore, these last devices are suitable for large parallel mechanisms. Although autocollimator resolution (0.1 arcsecond) and accuracy (0.2 arcseconds) is very high too, the measurement range is very low (from arcseconds to a few degrees). By contrast, measurements with a CMM take a lot of time. In order to obtain a high accuracy and efficiency, optical and contact sensors are used in combination. A visual sensor provides global information of the object surface. On the contrary, force and tactile sensors obtain local information. In recent years, several studies have focused on the combination of visual and force/tactile sensors to obtain a high knowledge of the mechanism behavior [119].

Everett [120] designed a sensor for measurements in the calibration of mechanisms. This sensor did not apply external constraint forces on the mechanism. The sensor used three LED/phototransistor pairs as optical switches. Each switch used a LED that shone through an optical fiber. The light left the fiber and passed through a gap. On the other side, another fiber received the beam, and a phototransistor received the light. This work described the sensor construction, the sensor performance and calibration. The sensor was installed in the gripper of a robot as a tool. Precision spheres, having a diameter of 12.7 mm, were mounted in the workspace of the robot. The spheres were located within an area of 1,600 cm². Firstly, the relative positions of the spheres were measured with a coordinate measuring machine. Secondly, the sensor was positioned over these spheres automatically. Three light beams defined position with respect to the sensor. Two beams were separated 10 mm and the third beam was mounted 30° away from them. The robot was programmed to find the trip point of the sensor. This point defined the origin of the sensor coordinate frame. Then, the sensor input was examined to determine which light beams were broken. The test obtained 100 trip points and the measurement error was calculated as the standard deviation of the position errors. The measurement error was 0.06 mm and the repeatability was between 0.06 and 0.08 mm. To calibrate the mechanism by means of the developed sensor, the phases are the following:

- To develop a model that relates measurable joint positions to mechanism pose

A widespread classification in the kinematic calibration of parallel robots is the one presented by Merlet in [13]. Calibration models are classified into three types: external calibration, constrained calibration and auto-calibration or self-calibration.

Although the simplest way of obtaining the necessary data is by using internal sensors, their assembly is difficult in most of the systems. In external measurement systems, it is usually necessary to establish, in an approximate way, the relation between the measurement system and the reference system of the end-effector. And this procedure has the problems described above.

In self-calibration methods, additional sensors are added to passive joints and each pose of the mechanism can be used as a calibration pose. These methods require that the number of internal sensors is greater than the number of DOF of the mechanism. Self-calibration methods are usually low-cost and can be performed on-line. They can be divided into two groups: (a) the mechanism has more internal sensors than necessary; (b) a passive chain is added to the mechanism.

In [121], Yang used built-in sensors in the passive joints and the parameter errors were identified by the least-square algorithm. In [122], Hesselbach developed algorithms to determine the resolution of the required sensors to reach the desired accuracy. These algorithms can be easily adapted to any 6-DOF parallel mechanisms that consist of kinematic chains with 6-DOF. Chains must include at least one ball joint. The author designed an absolute angle measuring micro sensor system. This sensor is added into passive joints of parallel mechanisms. One of the objectives is therefore to obtain a robust compact sensor. The use of passive joint sensors usually simplifies the optimization procedure. In self-calibration, the measured passive joint angles and the calculated joint angles, by means of the model, can be compared. Hollerbach [123] used nine precision potentiometers to calibrate a 6-DOF parallel mechanism. The joint angle offsets and the joint angle gains were the identified kinematic parameters. These parameters related the raw analog input data from the potentiometers to the joint angles. The mechanism was placed into a number of poses, and the joint angles were read. The obtained potentiometer readings were converted to predicted joint angles and to predicted poses by solving the forward kinematics. In the following step, these values were compared with the theoretical joint angles obtained by means of the inverse kinematics.

Developments based on constraining the mobility do not require extra sensors [124,125]. In [124], Ryu analyzed a design which consisted of a link of fixed length with spherical ball joints in its ends. The measurement data is obtained by the internal measurement system of six actuators which measure the 5-DOF moving platform motion. This information is used in the calibration procedure without using any extra sensing device. The results show that the position and orientation error and the measurement noise and the link inaccuracy have the same order. Therefore, the calibration accuracy depends on the sensor accuracy and on the constraint link.

Constrained calibration methods decrease the number of DOF of the mechanism restricting the movement of the end-effector or the mobility of any joint. In these methods, the mechanism mobility is constrained during the calibration, thus some geometric parameters will remain constant in this process.

In [126], the mechanism was constrained in such a way that only platform rotations around a fixed point were allowed, and constraint equations were solved by means of the forward model. Chiu [127] limited the movement of a 6-UPS mechanism by adding a seventh leg, connected to the base through a universal joint linked to the end-effector. However, this design considerably restricts the working range and causes interferences between links. Besides, some parameter errors related to immobilize joints can occur unseen. In [128], Ren kept two attitude angles of the end-effector constant at different measurement configurations using a biaxial inclinometer.

These methods have lower costs than external calibration, but on the other hand they are more complex than self-calibration. Another problem is that not all the workspace is available due to constraints, and it is usually a less accurate method than external and self-calibration.

However, in practice it is not easy to add extra redundant sensors or restrictions, so the most frequently used method is external calibration in which the necessary information is obtained by means of external devices such as theodolite [129], inclinometers [130,131], vision devices [117,132], the laser tracker [133,134] or the coordinate measuring machine [135–137].

Whitney [129] developed a forward calibration method. The method defined link lengths and joint sensor offsets as parameters. The theodolite measurements of tool position and the readings of the robot joint sensors were introduced in the model to perform the calibration procedure. The results show that the calibration model predicts theodolite readings with an error of 0.13 mm. Daney [117] developed a vision-based measurement method. Although the measurement data, for example the measured poses of the mechanism, are given by the sensor, it is necessary to consider the noise of this device. The mechanism poses were measured and sensors measured the six leg lengths for every pose.

Besnard [130] developed a method for the kinematic calibration of a 6-DOF parallel mechanism where the calibration model considered the error on the angle between the inclinometer axes. The two inclinometers were fixed to the platform. These sensors were used to measure the prismatic joint variables and the platform orientation. Each inclinometer measured its orientation from terrestrial horizontal, and both inclinometers had a null value when the platform was horizontal. The prismatic joint values and the inclinometer values were used to calibrate the geometric parameters for a number of configurations. The calibration procedure minimized the residual between the inclinometer measured values and the calculated values. The results show that inclinometers with a precision of 1e-3° and motorized joint sensors with a precision of 0.02 mm are necessary to obtain a position accuracy of about 0.4 mm. Rauf [131] developed a calibration method of parallel mechanism with partial pose measurements, by measuring the rotation of the end-effector along with its position. The device consists of a linear variable differential transformer and a biaxial inclinometer, to measure the position, and an optical encoder to measure the rotation. In [105], inclinometers were not used to measure precise values, rather to indicate if the values measured in one configuration were equal to the ones measured in a different configuration, thereby making the calibration method independent of the range of the measurements carried out and the positioning accuracy of the inclinometer. Although is not easy to obtain high accuracy and great workspace with conventional inclinometers, results obtained in calibration are usually satisfactory. The inclinometers were installed on the end-effector. The calibration was performed by keeping two attitude angles of the end-effector constant. The results show that the position and orientation accuracy, after calibration, can be 0.1 mm and 0.01°, when the inclinometer repeatability is 0.001°. And the precision of the leg length measurements is 0.002 mm. The results show that, before calibration, the errors were from 4 to 8 mm, and after calibration they were from −2 to 2 mm.

Renaud [132] developed a monocular high precision measuring device based on a vision sensor. A calibration target was placed in different positions and some images were taken with the vision sensor. The mathematical model obtained the pose of the target with respect to the vision sensor. The measuring device, based on a vision sensor, presents an accuracy on the order of 10 μm in position and 5e-4° in orientation; and, in [138], the author performed a calibration for a mechanism having linear actuators on the base. The forward model offers more stability, although it cannot converge if there are noise problems, and it is possible to perform the calibration by means of partial measurements of the end-effector pose.

In [133], Koseki used a laser tracking coordinate measuring machine. This device consisted of 4 laser stations and a wide-angle retro-reflector. When the laser beam was incident upon the retro-reflector, the reflected beams returned parallel to the incident beam. The retro-reflector had a lens whose focus coincided with its spherical surface and the surface mirrors incident beam. The interferometer measured the change in distance from the intersection of two axes of pan and tilt to the retro-reflector incrementally. Some advantages of this device are non-contact measurement, wide measuring range and ability to measure a high speed object. The results show that the accuracy obtained as the distance between measured position and calculated position presents an average error of 1.63 mm, before calibration, and 0.30 mm after calibration.

In [135] three methods were compared: a method using external measurement, a method using additional redundant sensors and a method using both. The author affirms that by using the implicit calibration, the basic system of equations may be obtained by using only the sensor information. The implicit calibration considers the basic system of equations obtained by the closed loop nature of parallel robots, and the system is specified by a function of the available data. The kinematic parameters can be identified by solving this system. The results show that the kinematic parameters were well identified as a function of the dimension of the redundant information on the state of the robot, and that the accuracy was higher for the method that used both: external measurements and additional redundant sensors.

In [139], Last classified the calibration techniques following the criteria of degree of automation and data-acquisition method. Thus, calibration techniques can be divided in four groups:

- Type 1: Non-autonomous methods by additional sensors from data acquisition, such as calibration by means of a laser tracker, camera systems or an extensible ball bar

To assure that the number of equations is not smaller than the number of unknowns the minimum number of measurements, m, is given by Equation 3: (3) m ≥ ρ + ζ

Defining η as the coefficients that relate the transducer signal (corresponding to the joint) to the real displacement of the joint and a as the coefficients in the kinematic model, ρ is the number of elements in the vector η and ζ is the number of elements in the vector a [94]. Besides, the positions chosen for data acquisition of the optimization process should guarantee the influence of all the parameters that are going to be identified, guaranteeing the generality of the parameters obtained for all the workspace [110]. In [140], Driels analyzed the optimum positions for data acquisition, and concluded that all the possible variation range of the joints in the mechanism should be covered. Nahvi [141] concluded that for performing a calibration, the number of joint sensors must be higher than the mobility of the mechanism. The author defined the noise amplification index and demonstrated that this index is an indicator of the amplification of the sensor noise and unmodeled errors. Moreover, the results show that the effects of sensor noise and unmodeled sources of error dominate the effect of length and other kinematic parameter variations of the mechanism. Merlet [13] explained that the number of constraint equations must be greater or equal than the number of unknowns. In practice the number of equations is usually greater, reducing the sensitivity of calibration to the uncertainty associated with measurements. This uncertainty is usually caused by measurement device noise. Thus, it is usual to develop an over-constrained system [142]. Huang [143] identified the parameter errors with an endpoint sensor and a dial indicator by measuring the flatness of a fictitious plane, the straightness and squareness of two orthogonal axes and the orientation error of the end-effector. These methods are usually based on data acquisition for fixed positions similar to work positions. Besides, it should be possible to generalize the parameter identification to positions different from those used for the identification process.

The determination of the optimal number of configurations to the data acquisition, in order to perform a successful calibration, is still one of the unsolved problems, and in the specialized literature different criteria and opinions can be found. Therefore decisions are made without a specific methodology to obtain the configurations for a calibration process. According to Zhuang [144], the number of necessary configurations is given by n+1, n being the number of DOF of the mechanism. In [145], Borm defined the index of observability based on the non-zero singular values of the Jacobian matrix, and represented the data scatter. By maximizing this index, the errors of the parameters can be better observed. In [146], Sun related five observability indexes and analyzed how to minimize the variance of the parameters and minimize the uncertainty of the end-effector position. Agheli [147] showed that the boundaries of the workspace should be examined for the maximum observability errors. In [121], Yang illustrated the effect of the measurement noise and robot repeatability on the calibration results. If the measurement noise exists, more measured end-effector poses must be considered. The author simulated 100 end-effector poses (50 poses to calibrate the robot and 50 to verify the results), and he concluded that the quantified orientation and position deviations and the calibrated initial poses of the modules frames become stable when the number of poses used for calibration is greater than 20. The results show that the quantified orientation deviation becomes stable in 0.004 radians, and the quantified position deviation becomes stable in 0.09 mm. Bai [148] recommended that the calibration should consider more than 10 measured poses to improve the calibration accuracy. Moreover, in [105] Ren concluded that selecting an optimal set of configurations is more efficient in decreasing the influence of measurement noise. Also, increasing the number of measurement configurations will decrease the pose error but only in a limited way, as when the number of measurement configurations is increased over a certain amount the improvement is not clear but the runtime is increased considerably. The number of configurations needs to be adjusted to reach a balance between accuracy and efficiency. On the contrary, Horne [149] studied the effectiveness of five pose selection criteria: the geometric mean of the singular values, the inverse condition number, the minimum singular value, the noise amplification index and the inverse of the sum of the reciprocals of all of the singular values, and the results show that the pose selection criteria did not significantly improve the calibration process for the 4-DOF parallel mechanism studied, and moreover some of the results using the criteria are worse than those results where no criteria had been used.

(b) Optimization procedure

The calibration process can be solved by two ways [93]: (a) by multiplying the constraints by Lagrange multipliers and using a modified objective; (b) by solving the constraint equations and substituting into the objective functions.

In [93,114] authors applied Lagrange multipliers to perform the calibration. The model can be solved by means of Newton method.

The second method presents non-linear equations and the inverse matrix is not easy to obtain, however nowadays there are powerful computers that allow us to perform this type of procedure.

The objective of the parameter identification or optimization is to search for the optimum values of all parameters included in the model that minimize the position error of the platform.

The objective function to minimize can be formulated in terms of a linear least-square problem. This function is usually defined as the quadratic difference of the error (obtained between the measured value of the end-effector position and the value computed by the kinematic model). The increment established for parameters must be defined for each iteration, and its value will depend on the optimization method chosen. In most of the cases numerical optimization techniques are used to minimize the end-effector error.

The equation that relates joint variables with the final position of the end-effector by means of the forward method is given by the Equation 4: (4) y = f ( θ i , p )where y is the vector that expresses the position and orientation of the end-effector according to the Euler angles, y=[x, y, z, α, β, γ], θ_i, are the joint variables, with i from 1 to the number of DOF of the mechanism, and p=[p₁, p₂,…, p_j]^T is the vector of the model parameters. The number of parameters j will depend on the model chosen. Thus, it is possible to identify the geometric parameters from vector p by iterative optimization methods. These methods minimize the difference between the coordinates obtained by the model and the nominal values measured in the same position. These differences are denominated residues (see Equation 5): (5) ϕ = ∑ i = 1 n ( y i − f ( θ i , p ) ) T ( y i − f ( θ i , p ) )

In this equation, y_i is given by the vector of the nominal position and orientation values for the n configurations utilized in the parameter identification. In each configuration, the position and orientation of the end-effector will be obtained by means of the mechanism model given by f, for the joint variables θ_i, corresponding to this configuration. This equation represents the objective function to minimize, whose value will be obtained as the sum of squares of the n poses, used in the parameter identification of the mechanism. A common way to express this equation is the one shown in Equation 6: (6) ϕ = ∑ i = 1 n [ ( x m i − x p i ) 2 + ( y m i − y p i ) 2 + ( z m ii − z p i ) 2 + ( α m i − α p i ) 2 + ( β m i − β p i ) 2 + ( γ m i − γ p i ) 2 ]where the values with m_i sub-index are the external measured values and the p_i sub-index values are the computed values by means of the mathematical model, for the n identification poses. The optimum values will be given by the minimum of the objective function ϕ.

Traditionally, this equation has been widely used for open chain mechanisms [150], since in these systems is easier to obtain the forward kinematics, but in recent years it has also been widely used in parallel mechanisms such as in [93,95,104,105,136,143,147,148], where the position and orientation of the end-effector is measured and compared with the value given by applying the forward kinematic model, in function of model parameters and joint variables.

Another method to perform the calibration is by comparing the joint variables, which are given by the inverse kinematic model [115,117,123,138,141,151,152], by means of Equation 7: (7) ϕ = ∑ i = 1 n ( q i − g ( X i , p ) ) T ( q i − g ( X i , p ) )where q_i are the joint variables, which are externally measured and g(X_i, p) are the joint variables obtained by means of the inverse model, in function of the position and orientation of the platform and the model parameters.

Another widely used equation for obtaining the error is the representation as differences between the measured and the computed distances, d_mi and d_i respectively, obtained by the kinematic model, between two points, as it is shown in Equation 8: (8) ϕ = ∑ i = 1 n ( d m i − d i ) 2where the sub-index mi indicates the values externally measured, and the values with sub-index i correspond to the values obtained by the kinematic model.

Once the most suitable calibration model has been selected for the mechanism, and the objective function is defined, the next step will be to solve the system. These systems are non-linear, thus it is not possible to obtain an analytical solution to the parameter identification problem. Non-linear optimization iterative techniques are usually used to obtain the optimum parameters that minimize the error in the identification poses. For these systems, the most suitable resolution techniques are those based on least squares, specially used to adjust a parametric model to a set of data. A usual approach to the optimization problem consists of linearizing the equations of the model in an environment of the parameter to identify by means of a development in Taylor series. A suitable formulation to the optimization problem and a good approach to the function are achieved in a small interval in relation to its current value. In parallel kinematics it is usual to use developments in Taylor series for every parameter pi. This approach can be first or second order. The optimization problem can be solved through several methods:

(1) Gradient optimization methods

The simplest ones are those methods based on the gradient, also known as line search. These methods are usually used when the objective function to be minimized is approximated through a first-order Taylor series development. The following step will be to consider a stop criterion based on the convergence of the method or on small increments of parameters between iterations to obtain the set of parameters that minimizes the function.

(2) Least-square optimization methods

In the second group we can find those methods that consider second-order approaches in the Taylor series development. In this second term the Hessian matrix appears, with components that are the second derivatives of the objective function with respect to the vector of parameters. This matrix should be invertible. Problems derived from the singularity of this matrix in numerical optimization procedures must be solved by choosing a suitable mathematical model and a set of data for the optimization, or by employing optimization methods that avoid this singularity.

Optimization methods based on least-square are different from the previous ones when obtaining the direction of search and when defining the increment that parameters should have in this direction. One of the most frequently used least-square methods that take into account second order terms is the Gauss-Newton method. In this method the Hessian matrix should be positive or semi-defined positive, that is all its own values are positive or positives and zero, which will not always be produced for any mathematical model and for any set of values of the objective function. Moreover, convergence is not always guaranteed.

(3) Levenberg-Marquardt optimization method

The methods studied can present problems when processing the objective function, and are therefore not always suitable for the parameter identification process. The gradient method ensures that a local minimum of the function is found and usually requires more iterations to find it. It also needs the objective function to be continuous in every parameter, which not is always the case.

Numerical problems that appear in the two methods studied are solved by the algorithm developed by Levenberg and Marquardt [153,154]. The method adds a positive value λ to the Hessian main diagonal elements, obtaining a non-singular and therefore invertible matrix. The main problem of the Newton Gauss-Newton method with the processing of the second order components is therefore solved. To choose λ, a compromise between the speed of convergence of the method and the invertibility of the matrix must be reached in every iteration. This is why the Levenberg-Marquardt algorithm results in a combination of the two methods presented above. When parameters are far from the optimum solution, the value of λ increases and the algorithm behaves similarly to the gradient method; when the parameters approach their optimum value, the value of λ decreases, behaving in a similar way to the Gauss-Newton method in both search direction and parameter increment. This method is widely used in the calibration of parallel mechanisms by authors such as in [105,110,115, 123–126,130,132,135,143,151,155–158].

In the specialized literature we can find other alternatives to the Gauss-Newton method problem. For example, the singular value decomposition (SVD) [114,116,147,148,156,159].

Another alternative is the use of the decomposition QR for parameter identification [125]. In [160] an algorithm allows us to introduce range limits of the joints in a configuration selection process and avoids the problem of local minimum, although it is a high-cost computational method. In [161,162] the problem is solved by means of genetic algorithms. Yu applies the inverse kinematics model and improves the accuracy in the parallel robot position by means of an artificial neural network. In [163], Stan performs the optimization of a 2-DOF parallel robot using genetic algorithms. The author considers transmission quality index, manipulability, stiffness and workspace. In [164] Liu obtains residual measures through inverse kinematics and develops a calibration method using genetic algorithms.

Techniques based on least-square usually present lower computational cost, providing we consider as an initial value a solution close to the optimum solution for the set of parameters. However, in methods based on genetic and neural networks algorithms this premise is not usually so significant. These algorithms are usually used for parameter optimization and identification when it is not known whether the initial values are close to the optimum solution. Furthermore, the combinatorial nature of these methods is purely stochastic, which avoids problems in the definition of the search direction in traditional least-square methods.

(c) Evaluation of the identified parameters

The evaluation of the identified parameters consists of evaluating the mechanism behaviour with the set of the identified parameters obtained in the previous step. This procedure is performed in configurations different from those utilized in the optimization process. This phase must evaluate the degree of compliance of the error values obtained in other positions of the workspace. In [133], Koseki utilized a laser tracking coordinate measuring system to evaluate the accuracy of a parallel mechanism. Cheng [165] analyzed the relationship between original errors and position-stance error of a moving platform by means of the complete differential-coefficient theory and evaluates the error model. The conclusion is that improving manufacturing and assembly techniques allows us to reduce the moving platform error, and that a small change in position-stance error in different kinematic positions proves that the error-compensation of software can considerably improve the precision of parallel mechanisms.

(d) Correction model

To end the calibration process, a correction model can be obtained to improve the accuracy of the mechanism. Huang [166] performed the calibration of a parallel mechanism and compensated geometrical and position errors in x and y coordinates. Gong [111] identifies non-geometric errors and developed a method to compensate these errors with a laser tracker by means of the inverse calibration model. An extensive guide of error compensation methods can be found in [167]. In this paper, Oiwa explained in detail how to compensate joints errors, link length, forces in a measurement loop and frame deformation using a coordinate measuring machine. This method considers thermal effect and external forces. The results show that the deflection of measured Z-coordinates is not completely eliminated with the compensation system, but compensation using displacement sensors built in spherical joints improves moving accuracy of parallel kinematic mechanisms when the mechanism moves in a large working space. The thermal and elastic deformations of the limb can be compensated by connecting the scale unit with the joints through low expansion material rods. Bringing the rod end into contact with the ball of the spherical joint enables the scale unit to measure the joint error and the limb deformations. And, the measured position and the orientation of the base platform compensate the thermal and elastic deformations of the frame.