Static and Dynamic Analysis

Static analysis ensures to model the structure deflection with regard to static loads due to gravity or forces introduced by task. Dynamic analysis allows computing the necessary actuator torques to realize a task with the kinematic and/or force requirement. In this case, the robot structure is considered to be composed of rigid elements. To analyze the influence of flexibilities on the end-effector movement, static and dynamic models are mixed together. Thus, the proposed dynamic analysis takes into account the structure flexibility and the inertial forces.
During a task, such as machining, process forces stress the structure and cause non-negligible tool pose defects (Pritschow et al. 2002; Weck and Staimer 2002). Moreover, critical issue with PRs is their nonconstant behavior in their geometrical workspace. So, static and dynamic behavior causes different defects, depending on tool pose in the workspace (Chanal et al. 2006). Taking the PR in machining operations as an example, this section will introduce the static and dynamic analyses of PKMs (parallel kinematic machine tools).
In order to simulate those pose defects, many authors propose different modeling strategies for PKM (Chanal et al. 2006; Cobet 2002; Wang et al. 2006; Rizk et al. 2006; Deblaise and Maurine 2006; Wang and Mills 2006; Bouzgarrou et al. 2002; Katz and Li 2004; Cosson et al. 2006). Static models ensure to compute tool pose defects induced by given static forces which are exerted on PKM structure (Chanal et al. 2006). They are based on parameters which model flexibility of PKM legs and joints. Flexible dynamic models allow simulating the influence of stiffness characteristics and natural frequencies of PKM structure on machining quality (Bouzgarrou et al. 2002). They take into account component weight and inertia of PKM structure.
Static and dynamic models of PKM can be sorted into two main categories:

• Local models: representation of PKM structure around a reference pose
• Global models: representation of PKM structure and its movements

Local and global terms are defined according to kinematic description used in the model. In the case of local model, results given by model simulation are valid for a given PKM pose. For global model, kinematic description ensures to modify PKM pose without developing a new model.

Local Models

Local models impose, by definition, to define a new model for each studied PKM pose. Thus, the time for conduct static or dynamic behavior analysis of PKM structure can be significant if several PKM poses must be studied. Most local modeling methods are based on structural transfer matrix or on finite element (FE) methods.

(a) Structural Transfer Matrix

The aim of this method is to determine a transfer function which can link load parameters to displacements of PKM components. For example, relation between cutting forces f(ω) and tool tip acceleration ẍ(ω) with regard to part can be expressed as

where ω represents the spindle rotational speed.
Transfer function defined from matrix G(ω) is identified from experimental measures (Tounsi and Otho 2000). In case of PKM, the behavior can vary highly according to the tool pose; the transfer function is then different for each tool pose. The behavior modeling in all workspace leads to a large number of measures and to an extrapolation of every measured point. Thus, the use of this kind of models is not relevant for PKM (Weck and Staimer 2002).

(b) FE Methods

FE methods have been widely used for designing or simulating mechanism or structure behavior since the 1980s. These methods are generally used for designing (Bouzgarrou et al. 2002) or computing stiffness (Deblaise and Maurine 2006) or natural frequencies (Cobet 2002) of PKM. FE modeling applied to PKM encompasses two main kinds of models: models with volumic elements and models with beam elements. However, a model with a part discretized with FE can be considered as a local model or a global model.
The most accurate models seem to be those using volumic elements to represent the machine components as they rely on fewer hypotheses than the others (Wang and Mills 2006). The main problem of these models is their computation time for calculating stiffness for a given position of the machine. As an example, a complete FE model with volumic elements and nonlinear joint models needs several hours to determine the stiffness in a given direction for one pose of the mobile platform (Bouzgarrou et al. 2002). Moreover, the computation of the stiffness in another position requires remeshing the model. The development of a parameterized FE model like in Bouzgarrou’s work (Bouzgarrou et al. 2002) ensures to avoid remeshing all parts during a PKM pose modification. Moreover, PKM is designed with a large number of joints associated with nonlinear static behavior (Huang and Lee 2001). Thus, complex elements have to be used for modeling joints behavior, which will increase computation time but allow accounting joints and legs flexibility (Wang and Mills 2006).
To conclude, significant computation and modeling time of this kind of FE models make them difficult to use during the process optimization. However, it allows determining accurate PKM structure stiffness in the design stage without the need of real tests.

Global Models

In order to determine PKM behavior in the entire workspace without having to develop multiple models, some researchers propose global models which take into account PKM kinematic behavior (Paccot et al. 2009a; Nabat et al. 2008; Ramdani et al. 2008; Bonnemains et al. 2013). These models can be grouped under the term multibody models. Multibody systems are defined by Shabana as a set of bodies, rigid or flexible, linked by joints (Shabana 2005). For defining a dynamic multibody model of PKM, three formalisms can be used as follows (Dwivedy and Eberhard 2006):

• Lagrangian formulation. It describes the behavior of a dynamic system in terms of work and energy stored in the system. The Lagrange equations are used to compute dynamic equations.
• Newton–Euler formulation. It describes the behavior of a dynamic system in terms of forces and moments acting on each link.
• Hamilton principle. It ensures to link the kinetic and the potential energy.

In case of PKM, Lagrangian formulation is generally preferred due to the ability of the method to take into account different kinds of mechanical parameters such as friction or flexibility (Bonnemains et al. 2013). However, in a control context, Newton–Euler algorithm is often employed due to its efficiency for real-time application (Paccot et al. 2009a).
In case of machining simulation, the use of direct dynamic model ensures to express tool pose at each time. Indeed, this model defines active joints acceleration q̋ with regard to active joint speed q́, position q, motor torques or forces τ, and exterior actions fext:

However, most of the work about PR modeling defines inverse dynamic models (Book 1984; Bayo et al. 1988; Boyer and Coiffet 1996). Generally, the aim of inverse model is used to choose motors or to develop an adapted command strategy:

The main assumptions made on multibody models of PKM concern the compliances of the legs and the joints (Dwivedy and Eberhard 2006).

(a) Rigid Body Model

The first way to model a PKM is to consider the structure as bodies linked with joints. This hypothesis is generally used to define a new command strategy (Paccot et al. 2009a), to compute dynamic machine capacity, to optimize dynamic behavior of PKM (Nabat et al. 2008), or to dimension PKM elements with regard to the application (Ramdani et al. 2008). The rigid body model can also be used for identifying joints preloads and viscous friction, reduction ratio and mass, and moment of inertia of each solid composing PKM structure (Bonnemains et al. 2013).
For defining the dynamic model, the mechanism has to be parameterized. Indeed, the Jacobian matrices of each part of the mechanism have to be computed. The Jacobian matrix gives the relation between the end-effector operational velocity and the articular velocity:

where pv is the end-effector operational velocity, pω is the end-effector operational rotational velocity, q́ is the articular velocity, pJh is the Jacobian matrix, h is the reference point for the end-effector velocity, and p is the frame where the velocity is expressed.
The main difficulty with PKMs is the choice of the parameters in the array q́. It is often simpler to consider more parameters than only those of the motorized joints. Some constraint equations have to be written for modeling the mechanism kinematic behavior (Bouzgarrou et al. 2002).
Thus, for modeling the parallel unit of Exechon PKM, Bonnemains introduces 12 parameters with only 3 independent (Bonnemains et al. 2013; Fig. 10). Indeed, if only three parameters (the leg lengths) are kept, the equations of motion are too large for any further computation. Finally, motion equations are deduced from Lagrange’s equations. However, this method does not ensure to take into account structure flexibility.

(b) Model with Flexible Legs or Joints

Authors often choose to neglect the stiffness of the joints with regard to the stiffness of the legs or vice versa (Chanal et al. 2006; Cobet 2002; Bouzgarrou et al. 2002; Katz and Li 2004; Cosson et al. 2006). Cobet and Bouzgarrou considered in their studies that legs are rigid and flexibilities are located in joints to perform a dynamic analysis or natural frequencies computation of a PKM (Cobet 2002; Bouzgarrou et al. 2002). Chanal, Katz, and Cosson considered that joints are perfect and that only legs can warp (Chanal et al. 2006; Katz and Li 2004; Cosson et al. 2006). Legs are here modeled by Euler-Bernoulli or Timoshenko beams. To improve the accuracy of such model, flexible parts can be modeled with FE method by using volumic elements (Bouzgarrou et al. 2002).

(c) Model with Flexible Legs and Joints

Recent studies have shown that both joints and legs compliances have an impact on tool pose defects (Shabana 2005; Dwivedy and Eberhard 2006; Majou et al. 2007). Majou proposed a model considering legs and joints compliances (Majou et al. 2007). Virtual joints are added with a given stiffness in order to represent strains of the joints or elongation of the legs.
Bonnemains proposed on his work a generic model of PKM by computing strain energy ED of a PKM structure consisting both joint and link compliances (Bonnemains et al. 2013):

where qm is leg m length and Nm, Mfxm, Mfym, and Mtm are, respectively, the compression force, bending moments around ym and zm, and torsional moment in leg m. fp is the effort transmitted by joint p and Kp is the stiffness matrix of joint p. Sm, IGxm, IGym, and I0m are, respectively, the leg section, the moments of inertia about xm and ym axes, and the polar moment of inertia of leg m. E and G are Young’s and shear moduli of the material. The most time-consuming step in the computing of the ED is the determination of the efforts Nm, Mfym, Mfzm, Mtm, and fp. Their expressions rely on the kinematic model of the given architecture. The displacements of the mobile platform for a given effort Fe applied on it at Oe can then be computed using Castigliano’s theorem:

where δe is the displacement in the direction of the applied effort Fe.
To determine the displacement in a different direction than the applied effort, a fiction effort can be applied to the platform in the desired direction. The strain energy is then differentiated with respect to this fictitious effort, and this effort is finally set to zero in order to obtain the displacement in the desired direction.

Modeling Method

Many authors propose models of PKM in order to determine their stiffness in a design point of view (Bouzgarrou et al. 2002) (Pashkevich et al. 2009). Their goal is only to optimize the architecture design. However, according to the simulation goal and studied machining operation, it can be relevant to develop a model of PKM adaptable to different given architectures and compact enough to determine quickly (less than 1 h) the tool displacements for a given machining operation (efforts known along a given tool path). The interest of this model arises after the design stage; this can be an element of machining simulation like the detection of collision before the real machining of the part.
So, FE models with beam elements or multibody models can be good answers to this problem. Indeed, they are easily adaptable to different architectures, and they can be computed in a lower computation time than FE models with volumic elements. Nevertheless, it seems necessary to simplify again legs’ geometry by discretizing them. Using material parameter description in a multibody model does not require longer computation times or give more accurate results (Bonnemains et al. 2013).

Calibration

The description of PKM mechanical behavior implies the definition of geometrical, static, and dynamic models. Thus, geometrical, static, and dynamic parameters have to be identified. In the following paragraph, calibration methods are introduced for these tree kinds of parameters.

Geometrical Calibration

The geometrical calibration or identification of a PKM consists of determining the best geometrical model which describes the mechanism. Three principal steps have to be verified during PKM calibration (Renaud et al. 2006):

• Geometrical modeling of the mechanism
• Parameter identification method based on the optimization of a dedicated cost function
• Choice of the experimentation approach and associated measurement method

(i) PKM Modeling and Definition of Identified Parameters

The geometrical model dedicated to calibration is generally the inverse kinematics model (Weck and Staimer 2002; Fan et al. 2003). For PKM, the inverse kinematics model (IKM) expresses motor displacements with regard to end-effector posture and geometrical dimensions of joints and links (Bi and Jin 2011; Fig. 11).

(ii) Cost Function Definition

The geometrical parameters are computed from a comparison of experimental measurements with theoretical estimation of the same machine pose (Renaud et al. 2006). In the general case, the geometrical parameters are identified by minimizing a cost function which compares these two estimations. This cost function depends on the means used to obtain measurement redundancy.
In the case of external measurements, Renaud states that the machine should be compared to the machine coordinates system in order to decrease the influence of measurement noise (Renaud et al. 2006). Indeed, the cost function can be generally expressed analytically.
In the case of PKM, IKM computes the motor positions V in the machine coordinates system as a function of tool position and orientation X in the Cartesian coordinates system and the identified geometrical parameters ξ:

V = IKM(X,ξ)

To compare machine poses in the machine coordinates system, the theoretical motor positions Vi = IKM(Xdi, ξd) have to be compared with the real position Vmi = IKM(Xmi, ξ) computed from the tool pose measurements (Fig. 12). The theoretical motor positions Vi are computed for given nominal geometrical parameter values ξd. The identification of the geometrical parameters ξ is successful if Vmi is equal to Vi.

These geometrical parameters ξ are generally identified by using a least square function (Khalil and Dombre 2004):

where Xmi is the measured tool pose in the Cartesian coordinates system, Xdi is the theoretical tool pose, and n is the number of measured poses. The identified geometrical parameters ξ are the parameters which minimize the cost function.

(iii) External Measurement Methods

Several experimental methods exist to identify PKM geometrical parameters. Some of them use a one-off measurement performed by measuring a machined part by using external measuring equipment (Pritschow et al. 2002; Weck and Staimer 2002; Terrier et al. 2004; Song et al. 1999). Others are carried out with integrated measuring equipment inside the PKM structure (Pritschow et al. 2002; Patel and Ehmann 2000). These later methods ensure to control geometric machine defect during machining in all the workspace. However, this kind of methods seems to be less accurate if measured joints are far away from the effector and can be difficult to realize according to the joint nature (Renaud et al. 2006; Merlet 2002). In the following, the external measurement methods are discussed.
A first external measurement method consists of adding mechanical constraints to the tool movement in order to measure tool pose errors. For example, Weck (2002) designs a redundant leg consisting of a linear guideway to connect two ball bearings which are attached to the rotary table and an HSK interface to the spindle. This mechanism is like the solution introduced by Patel, Chen, and Martinez (Patel and Ehmann 2000; Chen and Hsu 2004; Martinez and Collado 2004; Fig. 13). Huang and Neugebauer have developed a specific calibration method for PKM based on the measure of parallelism, straightness, and orthogonality of virtual axis of machining coordinate system (Huang et al. 2005b; Neugebauer et al. 1999). A probe is put on the machine spindle in order to “scan” standard parts. To limit the spindle movement of the machine tool, Bleicher has developed a serial mechanical system which ensures to measure tool pose defect in all the workspace (Bleicher and Gunther 2004).

A second method consists of using exteroceptive sensors. Song uses a 5D laser interferometer system which simultaneously provides five measurements (three linear displacements and two rotational displacements) (Song et al. 1999). These methods measure only some of the tool pose defects. However, the relation between the tool pose defects and the machined surface quality is not direct (Chanal et al. 2006), and the quality of the machined part is therefore not guaranteed, even if calibration is well performed.
Thus, the measurement of a machined part is relevant. Pritschow designed a specific part which is machined on the PKM to be calibrated (Pritschow et al. 2002). The part is milled with a ball end mill with changing spindle orientation (Fig. 14). Measurement of the ball cups provided the positions of the ball end mill centers. However, ball cups do not represent usual machined surfaces. Indeed during machining, several continuous surfaces are generally created by the motion of a tool along a trajectory. So it is relevant to use a machined part which allows the real tool path to be compared with the programmed one as in Chanal’s work (Chanal et al. 2007; Fig. 15).

Standard identification methods ensure that tool pose defects are minimized, which then may enable the part to be machined with the required quality. However, usual methods need a certain level of mastering to implement and can take a long time. In a machining context, a method which identifies the geometrical parameters in a part of the workspace for a given part shape may increase the accuracy and decrease the cost of identification by minimizing experimentation time.

Static Calibration

Static calibration allows identifying flexibility parameters of PKM structure. Thus, the experimental procedure has to ensure the measurement of the structure deformation with regard to an effort applied on the PKM end effector.
To identify the static behavior of a Tricept, Robin loads the structure with known masses and measures induced tool pose movement with a comparator (Robin et al. 2007). However, with this method, the structure can only be loaded in one direction; thus, Bonnemains uses a stroke combined with a LVDT and force sensors (Bonnemains et al. 2013; Fig. 16).

In the case of PKM, the observed stiffness depends on the tool pose in the workspace and can highly vary. For example, Bonnemains observes an increase of the stiffness with a factor of 3 between the center and edge of workspace. A nonlinear stiffness behavior can appear due to the influence of bearing stiffness with regard to the legs (Fig. 17).
To conclude, static calibration should be achieved with regard to the task load in order to ensure a local linearization of the stiffness model. Thus, the numerical minimization can be expressed as a linear problem.

Dynamic Calibration

The accuracy of a dynamic behavior simulation is linked on the accuracy of the parameters that describe the dynamic model (Khalil and Dombre 2004). The dynamic parameters of a PKM are generally inertial parameters, friction parameters, and preloads (Bonnemains et al. 2013). The identification procedure requires the joint torque measurement during the execution of a tool path. The followed tool paths should be sufficiently exciting in order to obtain a unit conditioning of the minimization problem (Khalil and Dombre 2004). In fact, tool paths used for dynamic identification are composed of (Bonnemains et al. 2013):

• Several acceleration and speed. Low acceleration and speed are used in order to identify friction and preload parameters. Various high acceleration and speed ensure to identify inertial parameters.
• Movements throughout the task workspace in order to deal with the nonconstant behavior of PKM.

Figure 18 illustrates a tool path used for dynamic calibration. In this case, joint torque measurements are applied when the PKM follows the tool path in low speed and high speed respectively compared to the speed required for the machining operation.

Control

The control of a machine is the last stage before realizing the task. In fact, the aim of the control is to ensure the best accuracy with regard to the task requirement in terms of static and dynamic errors. Moreover, the control strategy should ensure perturbations reject. Therefore, lots of control schemes have been developed from simple linear control to complex robust and adaptive one. The following sections deal with control strategies used for parallel kinematic machines (PKM).

Linear Control

Most of the industrial machine motion controllers are based on linear single-axis control. There are two kinds of controls strategies: classical speed control and torque control.

(a) Joint Speed Control

The joint speed linear control is the most developed control strategy in an industrial context because of a simple architecture implementation and wellknown tuning (Franck et al. 2004; Kelly et al. 2005; Khalil and Dombre 2002; Spong et al. 2006; Sun et al. 2007; Tournier 2010). The control scheme is based on the proportional-integral (PI) joint speed controller and proportional (P) position loop (see Fig. 19). Hence, the speed control allows compensating load variations, and the position loop deals with the dynamic answer (i.e., settling time). However, such a control strategy is not clearly relevant when dynamic accuracy is required especially when the dynamic solicitations are great. Indeed, the speed controller imposes the joint behavior during transitional period with relatively slow time response. Moreover, the classical tuning rules do not take into account the dynamic behavior of the mechanical structure under task solicitations.

(b) Joint Torque Control

An improvement of the dynamic accuracy during transitional period comes from using a linear proportional-derivative (PD) torque control, sometimes called linear computed torque control (Franck et al. 2004; Kelly et al. 2005; Khalil and Dombre 2002; Spong et al. 2006). The control scheme is based on a PD position loop imposing each actuator torque (see Fig. 20). The tuning of the PD controller is linked to the simplified dynamic of the mechanical structure: 

The above assumption is verified when dynamic solicitations allow neglecting nonlinearity of the dynamic behavior, such as dynamic coupling between legs or Coriolis and centrifugal forces. The static accuracy is thus assured in the joint space by the control loop and in the task space by geometrical identification. Dynamic accuracy is assured by controller tuning and dynamic identification of J and f. In this case such a control scheme could be relevant in an industrial context because of simple implementation, simple tuning, and relevant accuracy.
However, the disturbance on torque input such as gravity, dry friction, and task solicitations (e.g., such as cutting forces) can degrade the accuracy. One may find some solutions in gravity and friction compensation (Franck et al. 2004) or identification under task solicitations (Bonnemains et al. 2009). To improve accuracy performance, specific tuning can be employed such as fuzzy logic or neural network (Sun et al. 2007; Bing€ul and Karahan 2011; Carmona Rodriguez et al. 2012; D’Emilia et al. 2007).
Nevertheless, due to the nonlinearity of the dynamic behavior in the PKM case, linear control is not always sufficient to ensure a constant accuracy in the whole workspace (A° stro¨m and Ha¨gglund 2001; Paccot et al. 2009b). To solve this problem, nonlinear control can be envisaged.

Nonlinear Control

In the machine tool or robot case, nonlinear control consists in employing a suitable model of the machine structure behavior in the control loop. The motion controller could be based on linear PID controller coupled with predictive, adaptive, or intelligent control strategies.

(a) Computed Torque Control

When the dynamic behavior of the mechanical structure cannot be considered as linear under the task dynamic solicitations, a classical solution is the well-known computed torque control (CTC) (Luh et al. 1980). The control schemes include the inverse dynamic model of the machine to compute the joint torque reference (see Fig. 21). This allows linearizing the dynamic behavior of the structure; hence, the control signal uctrl is

Therefore, the controlled system (IDM and joints) can be seen as a linear double integrator. Moreover, the tuning of the PD controller is quite simple:

Nevertheless, this assumption can be verified when dynamic modeling and sensing errors are minimized with a relevant dynamic identification process (Khalil and Dombre 2002; Lammerts 1993; Paccot et al. 2007). Furthermore, the external perturbing forces (such as load variation or cutting forces) could be taken into account to increase the control accuracy and stability.

(b) Predictive Control

When the classical CTC cannot deal with modeling errors and perturbing forces, strategies likes model predictive control (MPC) or generalized predictive control (GPC) can be employed (Belda and Bohm 2006; Belda et al. 2003; Clarke et al. 1987; Company et al. 2003; Cuvillon et al. 2005; Ginhoux et al. 2005; Richalet 1993; Vivas and Poignet 2005). These methods are based on a minimization of the predicted error with regard to the desired path on a finished horizon. The definition of the cost function is generally made from the dynamic modeling under classical statistic errors and an observation of the real robot behavior. The minimization leads to a modification of the control signal (see Fig. 22). Such a control strategy allows for reducing external perturbation impact on accuracy and transitional period by computing an adapted control signal. However, the modeling of the mechanical structure is fixed and still subjected to modeling errors. Therefore, the robustness to this kind of perturbations is not clearly established.

(c) Adaptive Control

Another way to improve the CTC robustness is the adaptive control (AC) associated with a robust control strategy (Lammerts 1993; Fernandez 2004; Honegger et al. 2000). The general control scheme is based on a modification of the model parameters (or sometimes controller tuning) (see Fig. 23). The model parameters come from the minimization of the error between modeled and real behavior. Therefore, the dynamic model of the structure can change thus improving the control robustness to modeling errors.

(d) Intelligent Control

Last but not least, intelligent control such as fuzzy logic or neural networks can also be employed to robustify the CTC behavior (Dorato et al. 1992; Oh et al. 2004; Song et al. 2005). The controller is based on the dynamic modeling of the structure and trained on several trajectories. The control action leads to a modification of the control signal or model parameters.

Joint Space Control Versus Cartesian Space Control

In the PKM case, the concern in the space control is primordial (Paccot et al. 2009b; Beji et al. 1998; Bruyninckx 1999; Kim et al. 2005). Hence, because of the duality between serial and parallel kinematic machine, there is an ambiguity on using joint space control. Indeed, contrary to SKM, ensuring the tracking in the joint space does not ensure the tracking in the Cartesian space. Moreover, by controlling each joint separately, the disparate tracking errors can lead to internal torques, especially in an overconstrained structure case. However, most of the works on PKM control in the literature reuse the joint space control theory leading to inefficient use of these structures (Merlet 2002; Paccot et al. 2008). To improve the structure behavior control, Cartesian space control associated with Cartesian models should be employed (Paccot et al. 2009b; Beji et al. 1998; Paccot et al. 2008; Khalil and Ibrahim 2007).
Nevertheless, by using, a Cartesian space control, one should resolve the end-effector pose perception. A first classical solution is the use of the forward kinematic model (FKM) in the feedback loop (Fig. 24). Indeed, the FKM is not trivial to compute and can degrade the expected performance of a Cartesian space control (Lou et al. 2004). Therefore, a direct measure of the end-effector pose with an exteroceptive sensor is a relevant solution (Fig. 25). The use of 3D laser tracker, computer vision, high-speed vision, or passive structure could be envisaged (Paccot et al. 2008; Dallej et al. 2006a, b; Corbel et al. 2008). In these cases, the perception of the end-effector motion under the task solicitations is accurately captured and thus allows for the improvement of the task quality.

In this section, the focus is on the main control strategies used for PKM. To improve the task achievement, several points should be taken into account: influence of dynamic solicitation of the task on the mechanical structure, dynamic behavior of the machine, and specificities of the PKM. The future direction could come from a control strategy which presents an adaptation of the control architecture, the controller settings, and the used models with regard to the structure behavior under the task solicitations. This online adaptation requires a pertinent sensing method. The fusion of fast proprioceptive sensors (active and passive joint variables) and accurate exteroceptive ones could be an efficient solution.