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This paper presents an overview of the literature on kinematic and calibration models of parallel mechanisms, the influence of sensors in the mechanism accuracy and parallel mechanisms used as sensors. The most relevant classifications to obtain and solve kinematic models and to identify geometric and nongeometric parameters in the calibration of parallel robots are discussed, examining the advantages and disadvantages of each method, presenting new trends and identifying unsolved problems. This overview tries to answer and show the solutions developed by the most uptodate research to some of the most frequent questions that appear in the modelling of a parallel mechanism, such as how to measure, the number of sensors and necessary configurations, the type and influence of errors or the number of necessary parameters.
In recent years a number of specialized papers presenting the most relevant methods for modelling and calibrating serial robots have been published, but these methods are not always suitable for parallel robots. With this in mind, this paper is intended to be a guide for those researchers that are familiar with designing, modelling and calibrating parallel mechanisms.
Parallel kinematic systems are closedchain mechanisms in which the moving platform is joined to the base by two or more independent kinematic chains. In recent years these mechanisms have developed extensively [
One of the first moving platforms was patented by Gwinnett in 1931 for the entertainment industry [
On the basis of these designs, in recent years new families of manipulators have been developed with improved performances, obtaining multiple mechanisms of 3, 4 and 6DOF. However, few researchers have developed 2 and 5DOF systems due to the fact that constraints have to be added for these two types of systems.
Once the design is developed, a calibration process must be carried out in order to evaluate and improve the accuracy of the moving platform. The first decision that the researcher has to make in the calibration process is the calibration method. Once the calibration model is defined, the number and type of internal or external sensors required, the data acquisition procedure, the necessary configurations and the calibration model are determined. The model will determine the nature and number of necessary parameters. Finally, the evaluation of the results and the correction model will allow the accuracy of the system to be improved.
The following sections present examples of manipulators with different number of DOFs, focusing mainly on prototypes developed in the last ten years. A summary of the most relevant methods is presented, with respect to the kinematic model and the calibration of parallel robots, and also new trends such as the methods based on matrix computation or computational intelligence.
A classification of the most wellknown parallel robots depending on their number of DOF can be found in [
There are little literature focusing on 2DOF spatial parallel mechanisms. An analysis of different alternatives of 2DOF rotational parallel systems can be found in [
In [
There are multiple examples of 3DOF parallel mechanisms, some of which are shown in
The prismatic actuators lie on a common plane and have radial direction of action. The distal end of each actuator is joined to the lower end of a constant length leg by means of a passive revolute joint. The rotation axis of the revolute joint is perpendicular to the direction of the actuator and parallel to the horizontal base plane. The other end of each leg is attached to the moving platform by a passive spherical ball joint. The design variables minimize parasitic motion. This mechanism can be used in those applications that require elevation perpendicular to the base platform and pointing of a payload. Chablat [
FourDOF manipulators are extensively used for pickand place applications. These mechanisms are divided into two important groups—mechanisms based on the Delta robot [
Few researchers have focused on 5DOF spatial parallel mechanisms. Gao [
There are numerous examples of 6DOF mechanisms in the literature. Many of these designs are based on the Stewart platform and try to improve its performance. The design proposed by Shim [
Pritschow [
Parallel mechanisms are also used in the kinematic structure of several types of sensors. A Stewart platform can be used as a wrist force sensor, where sixaxis units (multiaxis forcetorque sensor) measure the three Cartesian coordinates: X, Y and Z. Gaillet and Reboulet [
In recent years a number of specialized papers have presented new designs based on the Stewart platform. Dwarakanath [
Yao [
Sui [
Chen [
Frigola [
The
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The first known studies on parallel robot kinematics were performed by Fichter [
The initial position problem has two different phases:
Obtaining the nonlinear equations that relate the joint variables and the endeffector position and orientation, thereby arriving at the forward and inverse position kinematic model
Solving the nonlinear equation system obtained in the previous step
Depending on the approach to the problem, there are three important groups:
 Graphical methods
 Analytical methods
 Numerical methods
Graphical methods are especially used in simple mechanisms, and they can be divided into three subgroups. Dyadic decomposition methods [
An analytic approach is used in the analytical methods, although the solution procedure is usually numerical. There are three methods to obtain all the solutions: polynomial continuation methods, elimination methods and polynomial Gröbner bases.
The polynomial continuation method [
Computation methods are based on systematic algorithms that allow us to automate the analysis of the kinematics of the mechanisms, independently of the number of DOF or complexity of the mechanism. For instance, programs based on multibody systems [
The position kinematic model can be solved by the direct or inverse kinematics, depending on the input and output variables.
The
The
The differential kinematic model is usually used to determine singular configurations or to control the mechanism.
The
The
The inverse kinematics of closedchain mechanisms can be solved through geometric [
To solve the direct kinematic problem, the use of analytical methods is complex, given that the chains share the same unknown factors; therefore, the most suitable resolution methods tend to be numerical [
Many studies have developed methods to obtain a mathematical model which allow us to solve the direct kinematic of parallel mechanisms. In [
Merlet and Bonev [
Bonev solved the direct kinematic problem of parallel manipulators by adding three linear extra sensors. Linear Variable Differential Transformers (LVDTs) and Cable Extension Transducers (CETs) are linear extra sensors commonly used in parallel mechanisms. The LVDT sensor has a low measurement range (0.5 m), it requires support electronics, its installation requires universal joints and its price is high. The cable end of a CET is joined to the moving object and the CET is fixed. The cable extends or retracts when the object moves. The CET produces electrical signals proportional to the movement of their extension cables. And the linear displacement is converted to angular displacement with the cable being wound onto a cylindrical spool. A rotary sensor measures the spool rotation. The range provided by these sensors is from 0.04 to 40 m, the accuracy is about 0.02% of full scale for potentiometers and 0.02% for shaft encoders and the repeatability is 0.02% of full scale. The sensors selected to this design were the CETs, and they connect the planar base and the planar moving platform at distinct points. The linear extra sensors were implemented considering sensor misalignment range, link sensor interference and singularity of a matrix which depends on the geometry of the moving platform and the arrangement of the sensor base attachment points and base joints. To solve the model, three coordinates are obtained directly from the extra sensory data. In [
Subsequently new contributions appear, trying to search for all the direct kinematic solutions, based on a new concept, the multibody system [
In recent years computational intelligence, such as artificial network, genetic algorithms or fuzzy logic, is becoming important in solving mechanisms.
Artificial intelligence is concerned with intelligent behaviour in machines, and it involves perception, learning, reasoning, communicating and acting in different environments.
An artificial neural network is a mathematical model that tries to simulate brain hardware structure and reproduce its low level capabilities, such as pattern recognition or data classification, through a learning process. It consists of an interconnected group of artificial neurons that process information. This method presents the capability to derive meaning from imprecise data or complicated systems, so it is used in complex mechanisms where there is not enough data or it is very difficult to obtain the kinematic equations by means of classical methods. Other advantages of these artificial systems are [




Fuzzy systems try to reproduce high level capabilities of the brain, such as approximate reasoning, because the capabilities are usually non precise or fuzzy in the real world [
Different systems that apply these methods can be found in literature. In [
In [
In this section it has been shown that neural networks and fuzzy systems have important advantages, and they have been applied in different areas such as robotics. These methods can be suitable for those systems where it is very complex or impossible to obtain the model by means of classical methods, for example, in very complex mechanisms, non welldefined problems, when the environment is unknown [
Robot calibration consists of identifying the geometric parameters in order to improve the model accuracy. In parallel mechanisms, the objective is to reduce the endeffector position error by means of an accuracy identification of the kinematic parameters. This procedure allows us to obtain correction models to establish corrections in the measurement results. Moreover, the calibration procedure quantifies the effects of the influence variables in the final measurement. The steps to achieve this goal can be divided in five phases: determination of the kinematic model by means of nonlinear equations, data acquisition, optimization or geometric parameter identification, model evaluation and, finally, identification of the error sources and implementation of correction models.
The first step, determination of the kinematic model, consists of obtaining the nonlinear equations that relate the joint variables with the position and orientation of the endeffector and the initial values of nominal geometric parameters.
The second step is data acquisition. The home position is a position, within the robot working range, where all joint angles have a predefined value. The displacements of the endeffector are usually measured with respect to this defined position.
The following step, optimization or geometric parameter identification, is usually carried out by means of approximation procedures based on leastsquare fitting.
Once the optimization is applied, an evaluation of the model in different positions than those used in the identification process must be carried out to test the model obtained.
Finally, an identification of the error sources and a modelling and implementation of the correction models can be performed.
In [
As is known, the calibration procedures present three well differentiated levels, level one calibration or joint level calibration, level two or kinematic calibration and level three or dynamic calibration [
The joint calibration consists of determining the relations between the signal produced by the joint displacement transducer and the actual joint displacement. By means of this modelling, two more parameters are added for each joint in the mathematical model of the mechanism. For prismatic joints, d_{0i} (joint displacement in the model initial pose, with respect to the sensor reference mark) and k_{i}, (function curve of the sensor output). In case of rotary joints, these parameters are θ_{0i} (joint rotation in the model initial pose, with respect to the sensor reference mark) and k_{i}. At this level, data acquisition is performed by means of some external measurement devices to determine the actual joint angle accurately, or by moving the joint to any known configuration. Usually, easily measureable configurations are chosen. In openloop mechanisms, configurations in which several elements are aligned are frequently used. In closedloop mechanisms, due to their geometry, those configurations with a known joint angle are more suitable. In this case, the forward model is required, and in closedloop mechanisms this problem presents more complexity. Another possibility is to place the endeffector in a known position and orientation belonging to the workspace, in order to solve the problem by means of the inverse kinematics. The equations developed in the modelling phase allow us to carry out the parameter identification process. The correction phase ensures that the parameter values are precise, using a controller to convert the signal that comes from the joint transducer into a representative value of the actual joint angle. In [
The kinematic calibration consists of determining the kinematic geometry of the mechanism and the correct joint angle relationship.
(a) Kinematic model
Just like in the joint calibration, in this level the first phase is to determine the kinematic model. This model allows us to obtain equations which relate the joint variables of the mechanism with the position and orientation of the endeffector.
It is not an easy task to obtain a suitable model that ensures the optimal accuracy of the system, and one of the unsolved problems in this working area is that it is very difficult to obtain a generalizable model – it is specific to the design under study.
In a parallel mechanism, the endeffector position is limited by certain restrictions. Calibration will obtain this position for several poses of the endeffector or configurations. Constraint equations are a function of m geometric parameters of the robot, of the measurements obtained at the position and of the n pose parameters of the process of calibration. For N calibration poses, there are (N × c) constraint equations with (m + N × n) unknowns, and the number of constraint equations must be greater or equal to the number of unknowns. The solution to this nonlinear system is usually obtained by means of numerical methods, such as minimizing the sum of squares of the constraint equations.
Everett divided kinematic calibration models into two categories [
The second category includes those models in which it is considered that some of the joints can contain higher pairs. In these models, besides revolute and prismatic joints, some additional movements can appear and must be expressed as a function of the variable joints. These models add an offset to each joint, adding three new parameters to each one. This fact originates a multitude of possible functions to model the joint, and the concepts of equivalent and complete model are not applied to this category.
To attain a high level of accuracy, the model must consider the most significant geometric and nongeometric 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 openloop or a closedloop mechanism is the wellknown DenavitHartenberg method [
Once the kinematic model has been determined the number of parameters will be fixed, and will depend on the selected method. Wang [
Nongeometric errors
The nongeometric errors can appear from backlash, gear transmission, friction, gravity, temperature or compliance [
Some authors [
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 [
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 [
In [
This is probably one of the steps with more unresolved questions, mainly due to the difficulty in finding a general methodology. Therefore researchers try to find the best procedure, usually applied to a specific mechanism. This subsection shows how different authors have dealt with these unsolved problems.
Any measurement error of the external instrument is propagated to the results of the identified parameters. Therefore, it is recommended to use an instrument for data acquisition that is, at least, one magnitude order more accurate than the mechanism whose parameters are going to be identified.
The ability to measure the global reference system of the mechanism which is going to be calibrated usually determines the different options of data acquisition and the sensors that are going to be used at this step.
If the global reference system can be measured by means of an external measurement instrument (for example a laser tracker or a coordinate measuring machine), a direct geometric transformation can be established. This transformation obtains the coordinates of the measured points in the global reference system of the mechanism. In this case, direct comparisons in the objective function, between measured data (or their geometric composition) and mechanism model nominal data, can be made providing both are expressed in the same reference system.
Unfortunately this relation is not usually easy to obtain through a direct measure. In these cases, leastsquare methods can be used with a finite set of data. These methods allow us to obtain an approximation of this transformation, which depends on the mechanism error in the points and configurations used in data acquisition. Moreover, this approximation is absorbed by the objective function. For that reason, it has direct influence on the value of the identified parameters. This method is therefore not suitable for parameter identification procedures in which positioning accuracy is mandatory or when it is necessary to generalize the positioning accuracy obtained in the identification process to other areas of the workspace.
For all these reasons, the geometric relation between the reference system of the measurement instrument and the mechanism global reference system must be established accurately. Otherwise, the objective function should be obtained starting from a reference position and evaluating Euclidean distances between datasets.
Classical robot calibration methods use additional sensors to measure the position and orientation of the endeffector 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 endeffector. In the calibration procedure, the nominal and the calculated position and orientation of the endeffector 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 endeffector. 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 forcetorque 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 analogtodigital 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. Forcetorque 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 noncontact 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 [
Everett [
 To develop a model that relates measurable joint positions to mechanism pose
 To measure a sufficient set of joints positions and their corresponding mechanism poses
 To identify the parameters of the model
 To determine the spheres centre relative to the fixture datum (for example by the coordinate measuring machine)
 To collect calibration data by the sensor.
A widespread classification in the kinematic calibration of parallel robots is the one presented by Merlet in [
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 endeffector. And this procedure has the problems described above.
In selfcalibration 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. Selfcalibration methods are usually lowcost and can be performed online. 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 [
Developments based on constraining the mobility do not require extra sensors [
Constrained calibration methods decrease the number of DOF of the mechanism restricting the movement of the endeffector 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 [
These methods have lower costs than external calibration, but on the other hand they are more complex than selfcalibration. Another problem is that not all the workspace is available due to constraints, and it is usually a less accurate method than external and selfcalibration.
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 [
Whitney [
Besnard [
Renaud [
In [
In [
In [
 Type 1: Nonautonomous methods by additional sensors from data acquisition, such as calibration by means of a laser tracker, camera systems or an extensible ball bar
 Type 2: Nonautonomous methods by kinematic constraints from data acquisition, such as calibration by contour tracking or by passive joint clamping
 Type 3: Autonomous methods by additional sensors from data acquisition, such as calibration by passive joint sensors or with actuation redundancy
 Type 4: Autonomous methods by kinematic constraints from data acquisition
To assure that the number of equations is not smaller than the number of unknowns the minimum number of measurements,
Defining
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 [
(b) Optimization procedure
The calibration process can be solved by two ways [
In [
The second method presents nonlinear 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 leastsquare problem. This function is usually defined as the quadratic difference of the error (obtained between the measured value of the endeffector 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 endeffector error.
The equation that relates joint variables with the final position of the endeffector by means of the forward method is given by the
In this equation,
Traditionally, this equation has been widely used for open chain mechanisms [
Another method to perform the calibration is by comparing the joint variables, which are given by the inverse kinematic model [
Another widely used equation for obtaining the error is the representation as differences between the measured and the computed distances,
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 nonlinear, thus it is not possible to obtain an analytical solution to the parameter identification problem. Nonlinear 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:
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 firstorder 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.
In the second group we can find those methods that consider secondorder 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 leastsquare 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 leastsquare methods that take into account second order terms is the GaussNewton method. In this method the Hessian matrix should be positive or semidefined 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.
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 [
In the specialized literature we can find other alternatives to the GaussNewton method problem. For example, the singular value decomposition (SVD) [
Another alternative is the use of the decomposition QR for parameter identification [
Techniques based on leastsquare 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 leastsquare 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 [
(d) Correction model
To end the calibration process, a correction model can be obtained to improve the accuracy of the mechanism. Huang [
This paper presents an overview of the solutions developed on kinematic and calibration models of parallel mechanisms and the influence of sensors in the mechanism accuracy in recent years. The most relevant classifications to obtain and solve kinematic models and to identify geometric and nongeometric parameters in the calibration of parallel robots are presented. And the advantages and disadvantages of these methods, applications of parallel mechanisms as sensors, new trends and the identification of unsolved problems are discussed. This overview is intended to be a guide for researchers working on parallel mechanisms, in the design, modelling and calibration of these systems. In the document, some common questions are answered and the most uptodate research carried out is summarized. The document describes the different phases required to perform a calibration process, putting special emphasis on the fact that the first decision made, the calibration method, will determine the number and type of necessary sensors, internal or external, from data acquisition, the required configurations and the calibration model. The model will determine the nature and number of necessary parameters. There are different methods to perform the calibration process, and the choice of one or the other must consider the characteristics of the mechanism. Methods based on computational intelligence are able to scan a vast solution set and are not as sensitive to bad initial values as classical methods. These methods can be suitable for those systems where it is very complex or impossible to obtain the model by means of classical methods. On the contrary, in those devices in which a careful kinematic analysis is necessary to obtain complete performance knowledge, classical methods are more suitable. Depending on the type and number of selected sensors, the cost function will be formulated. This function is established in terms of position and orientation of the moving platform (DKM) or in terms of distances (IKM). And, finally, the evaluation of the results and the correction model will allow us to improve the accuracy of the system.
TwoDOF parallel robots:
ThreeDOF parallel robots:
FourDOF mechanisms:
FiveDOF mechanisms:
SixDOF mechanisms:
Known hexapod systems (reprinted from [
Scheme of different methods to modeling and solving parallel kinematic mechanisms.
Joint symbols.
Universal joint  
Prismatic joint  
Revolute joint  
Spherical joint  
Parallelogram joint 