Rigidbody and friction model parameters for the parallel robot PaLiDA
1. Introduction
The proposed chapter presents a selfcontained approach for the dynamics identification of parallel manipulators. Major feature is the consequent consideration of structural properties of such machines in order to provide an experimentally adequate identification method. Thereby, we aim to achieve accurate model parameterization for control, simulation or analysis purposes. Despite the big progress made on identification of serial manipulators, it is interesting to state the missing of systematic identification methodologies for closedloop and parallel kinematic manipulators (PKM’s). This is due to many factors that are discussed and treated systematically in this chapter.
First, the issue of modelling the dynamics of PKM in a linear form with respect to the parameters to be identified is addressed. As it is already established in the field of classic serial robotics, such step is necessary to ensure model identifiability and to apply computationally efficient linear estimation (Swevers et al., 1997, Khalil & Dombre, 2002, Abdellatif & Heimann, 2007). The case of parallel manipulators is more complicated, since a multitude of coupled and closed kinematic chains has to be considered (Khalil & Guegan, 2004, Abdellatif et al., 2005a). Beside the rigidbody dynamics, friction plays a central role in modelling, since its accurate compensation yields important improvement of control accuracy. If friction in the passive joints is regarded, the dimension of the parameter vector grows and affects the estimation in a negative way. To cope with such problem, a method for the reduction of friction parameter number is proposed, which is based on the identifiability analysis for a given manipulator structure and by considering the actual measurement noise. The calculation procedure of a dynamics model in minimal parametrized form is given in section 2.
Another important issue of PKM’s is the appropriate design of the identification experiment, in order to obtain reliable estimation results is. Two aspects are here crucial: The choice or the definition of the experiment framework at the one hand and its related experiment optimization at the other hand. Regarding the first aspect, the harmonic excitation approach proposed a couple of years ago for serial manipulators is chosen (Swevers et al., 1997). The method provides bounded motion that can be fitted in the usually highly restricted and small workspace of parallel robots. Thus, we propose an appropriate adaptation for PKM’s. The experiment optimization is carried out within a statistical frame in order to accounts for the cross correlation of measurement noise and the motion dependency of the coupled actuators (Abdellatif et al., 2005b). The Experiment design is discussed in section 3.
The typically non measurable information of the endeffector postures, velocities and accelerations are necessary to calculate the dynamics model and therefore to obtain the regression equation. Since in general only actuator measurements are available, there is a need for an adequate estimation of the executed endeffector motion during the identification experiment. However, the numerical computation of the direct and the differential kinematics yields a spectral distortion and noise amplification in the calculated data. Therefore, an appropriate and simple frequencydomain data processing method is introduced in section 4. An accurate and noisepoor regression model is then provided, which is crucial for biasfree estimation of the model parameters. Additionally, we provide useful relationships to evaluate the resulting parameter uncertainties. Here, uncertainties of single parameters as well as the uncertainties of entire parameter sets are discussed and validated.
Finally and in section 5, an important part of the chapter presents the experimental substantiation of the theoretical methods. The effectiveness of our approach is demonstrated on a six degreesoffreedom (dof’s) directly actuated parallel manipulator PaLiDA (see Fig. 1). We address the important issue of exploiting the identification results for modelbased control. The impact of accurately identified models on the improvement of control accuracy is illustrated by numerous of experimental investigations.
2. Parameterlinear formulation of the dynamics model
The objective of this section is to derive the inverse dynamics model in a linear form with respect to a set of the parameters to be identified. Such formulation allows for using linear techniques to provide the estimation of model parameters from measurement data. This kind of approach is well established for serial robots (Khalil & Dombre, 2002, Abdellatif & Heimann, 2007). Thereby, the model accounts for the rigidbody as well as for friction dynamics. We consider the case of 6dof’s parallel manipulator, that is constituted of a moving platform (endeffector platform) attached with six serial and nonredundant actuated kinematic chains to the base platform. Fig. 2 shows a general sketch of such robotic manipulator. Let
The major difference between serial and parallel manipulators is the definition of configuration variables or the configuration space. For classic serial manipulators, the actuation variables
The bodyfixed frames can be defined according to the modified DenavitHartenberg (MDH) notation (Khalil & Dombre, 2002).
The aimed dynamics model consists of the following equation:
with
2.1. Parameterlinear formulation of the rigidbody dynamics
Generally, it is recommended to use the Jourdain’s principle of virtual power to derive the dynamics in an efficient manner. In analogy to the virtual work, a balance of virtual power can be addressed:
where
The generalized rigidbody forces for a manipulator with
with the dynamic parameters of each body
with
which helps the simplification of the generalized rigidbody forces:
Equation (7) is already linear with respect to the parameter vector
can be achieved. The matrices
Then
The recursive relationship given in (9) can be used for parameter reduction. If one column or a linear combination of columns of
The rules can be directly applied to the struts or legs of the manipulator, since they are considered as serial kinematic chains. For revolute joints the 9^{th}, the 10^{th} and the sum of the 1^{st} and 4^{th} columns of
The endeffector platform closes the kinematic loops and further parameter reduction is possible. The velocities of the platform joint points
After Applying every possible parameter reduction the generalized rigidbody forces are obtained from (7) with respect to a minimal set of parameters
2.2. Parameterlinear formulation of the friction forces
In analogy to the rigidbody dynamics, the Jourdain’s principle can be applied for friction forces. By defining an arbitrary steadystate model at jointlevel
Equation (13) means that the friction dissipation power in all joints (passive and active) has to be overcome by an equivalent counteracting actuation power. We notice that the case of classic openchain robots correspond to the special case, when the jointJacobian
For identification purpose, friction in robotics is commonly modelled as superposition of Coulomb (or dry) friction and viscous damping depending on joint velocities
Regrouping friction forces in all
Considering (13) and (15) the linear form of the resulting friction forces in the actuation space is obtained
Unlike the rigidbody dynamics, there is no uniform or standard approach for the reduction of the parameter vector dimension. In a former publication, we proposed a method that is highly adequate for identification purposes. Thereby, the expected correlation of the friction parameter estimates is analyzed for a given and statistically known measurement disturbance. Parameters whose effects are beneath the disturbance level are eliminated. Parameters with high correlation are replaced by a common parameter. The interested reader is here referred to (Abdellatif et al., 2005c) and (Abdellatif et al., 2007) for a deep insight.
3. Identification experiment design for parallel manipulators
Almost all identification methods in robotics are based on the parameterlinear form that is given by (1) in combination with (12) and (19) (Swevers et al., 1997, Khalil & Dombre, 2002, Abdellatif & Heimann, 2007). Given experimentally collected and noise corrupted
with the measurement vector
The crosscoupling is regarded by the full covariance matrix
3.1. Design of the excitation trajectory
An important step in identification is the choice of the measurement data to be collected. A classic choice consists in the socalled excitation trajectory, which ensures that the effects of all considered parameters are contained in the measurement data. A challenging issue with parallel manipulators is their restricted and highly constrained workspace. Such property reduces the possibility of highly dynamic and variable motion that is necessary for the excitation of all parameters to be identified. The appropriate choice should be a trajectory that is naturally bounded to fit into a small workspace. An attractive approach is the harmonic excitation approach originally proposed by Swevers et al. (Swevers et al., 1997) and adapted in the following for the case of parallel manipulators.
For each posture coordinate corresponding to the
providing a proper trajectory parameter vector
with
3.2. Optimization of the excitation trajectory
The next step consists in determining the values of all trajectory parameters
to provide a best possible excitation of the dynamics parameters. Such procedure is called optimal input experiment design. The design is performed by using constrained nonlinear optimization (Swevers et al., 1997, Gevers, 2005). The required constraints are expressed with respect to the actuation variables
to account for actuator limitation and therefore indirectly for workspace constraints and dynamics capabilities of the manipulator. The inverse kinematics has to be performed while the optimization, which does not introduce any significant computational cost due to its simplicity (Khalil & Guegan, 2004, Abdellatif et al., 2005a, Abdellatif & Heimann, 2007). Of course, it is possible to express the constraints
that aims increasing the volume of the asymptotic confidence ellipsoid for the parameter estimates, which is equivalent to the determinant of the inverse of the asymptotic parameter covariance matrix
4. Identification procedure: data processing, implementation and parameter uncertainties
At this stage, the dynamics of the manipulator is available in linear form (section 2). Additionally, the appropriate choice of an excitation experiment is proposed (section 3.1) with a recommended method for its optimal design (section 3.2). Therefore, the experiment can be executed and the data can be collected to achieve an estimation according to (21). Here, the next challenge for parallel manipulators is evident. The measurements are provided in the actuation space in form of actuation forces and actuator positions, whereas the information matrix
4.1. Data processing
The first step consists in calculating the direct kinematics to provide a first estimate of the posture
By taking advantage of the periodic and harmonic nature of the excitation trajectory, exact filtering in the frequencydomain can be achieved. First, it is recommended to calculate the DFTtransform of each component
Transforming back to the time domain yields the filtered signals
4.2. Parameter uncertainties
To validate the results of the identification, statements on the uncertainties of the obtained parameters are necessary. For the given linear model structure (20) and by assuming Gaussian disturbance vector
The confidence area of the estimated parameter set
where
Equations (29) and (30) are useful to evaluate the confidence of the estimate results for the complete parameter set or for the single parameters, respectively.
5. Experimental results for modelbased control
This section is dedicated to the experimental results achieved on the hexapod PaLiDA.
5.1. Description and modelling of the hexapod
The parallel robot PaLiDA (see Fig. 1) was developed by the Institute of Production Engineering and Machine Tools at the University of Hannover as a Stewart–Gough platform. It is designed with electromagnetic linear direct drives used as extensible struts for use in fast handling and light cutting machining like deburring. The actuation principle has several advantages compared to conventional ball screw drives: Fewer mechanical components, no backlash, low inertia with a minimized number of wear parts. Furthermore, higher control bandwidth and extremely high accelerations can be achieved. A commercial electromagnetic linear motor originally designed for fast lifting motions is improved for use in the struts. Each strut of the hexapod is composed of three bodies as depicted in Fig. 4. Thus, the system is modelled with 19 bodies: The movable platform (index
The dynamics model in parameterlinear form results by applying the rules discussed in section 2. The rigidbody part contains 10 base parameters (see Table 1). According to the friction modelling approach (14) the actuated joints
rigidbody  friction 



























5.2. Experiment design and data processing
The experiment design has been carried out according to the method given in section 3. An example of a resulting excitation trajectory with the order
The left side of Fig. 5 depicts the frequencydiscrete spectral amplitudes of the signals along with the used selection window that corresponds to an ideal lowpass filter. The respective signals in the timedomain are given on the right side of the picture. The effectiveness of the proposed filter is obvious, since the calculated signals exhibit almost no noise or disturbance corruption. Such property is a central requirement for a robust and reliable identification of parallel manipulators, because the necessary but nonmeasurable information has to be extracted from corrupted and limited measurements of the actuator displacements.
In the following three models are compared, that all result from the identification using the same trajectory but after implementing three different dataprocessing techniques. The first one results directly from rough data without any filtering. For the second, the measurements of the actuator displacements were filtered in the time domain. The third model has been identified according to the proposed frequency domain method. The validation of the models on a circular benchmark trajectory, that was not used for identification, is depicted in Fig. 7. The frequencydomain processing yields the best prediction quality corresponding to the smallest error variance
5.3. Estimation results and parameter uncertainties
The filtered data resulting from the investigated trajectory (Fig. 5) are used to compute the regressor matrix
The validation of the parameter estimation robustness can be provided, e.g. after repeating the identification experiment 100 times. The resulted parameter sets are compared to the 95% confidence intervals (see eq. (30)). Such investigation is depicted for some exemplarily chosen parameters in Fig. 8. The history of the weighted parameter estimate




a priori 

0.0447  0.0039  [0.0526 0.0369]  0.0074 

1.0892  0.0070  [1.0753 1.1032]  0.9439 

1.0077  0.0045  [0.9988 1.0166]  0.9458 

0.5995  0.0036  [0.5922 0.6068]  0.6201 

1.2885  0.0056  [1.2998 1.2772]  1.2295 

0.3078  0.0061  [0.3049 0.3106]  0.2878 

0.3021  0.0014  [0.2996 0.3045]  0.2878 

0.1176  0.0012  [0.1152 0.1201]  0.1217 

1.8896  0.0012  [1.8774 1.9017]  1.9012 

16.3081  0.0460  [16.2161 16.4002]  16.1920 

0.5756  0.0158  [0.5440 0.6072]   

0.9195  0.0179  [0.8837 0.9552]   

11.9772  0.2485  [11.4803 12.4742]   

4.8071  0.1861  [4.4350 5.1793]   

20.1528  0.3226  [19.5075 20.7980]   

5.1518  0.1817  [4.7884 5.5151]   

1.5857  0.2618  [1.0620 2.1094]   

5.0057  0.3519  [4.3018 5.7096]   

16.8771  0.5268  [15.8235 17.9307]   

16.7406  0.3712  [15.9981 17.4830]   

6.3408  0.5720  [5.1968 7.4848]   

23.1662  0.3799  [22.4065 23.9259]   

26.4675  0.4461  [25.5754 27.3596]   

22.8053  0.5539  [21.6974 23.9131]   
due to the variation of friction at the beginning of the measurement process until a nearly stationary state is reached. Additionally to the single parameters, the confidence of the entire parameter set can be validated. The outer bound of the 95% confidence ellipsoid
5.4. Identification and modelbased control
The impact of identification on the control and tracking accuracy of the hexapod PaLiDA is studied in the following. Hereby three control strategies are investigated. The first variation passes on any model knowledge, i.e. by implementing only linear controller for the single actuators. The second uses the inverse dynamics model to compensate for the nonlinear dynamics by considering only nominal parameter values. The third variation uses the identified model for the feedforward compensating control. All approaches are substantiated experimentally on two different trajectories: The first trajectory is a circular one and allows reaching high actuation forces, whereas the second is quadratic and is characterized with high actuator velocities.
Both trajectories were executed at different velocities
As expected, the use of standard linear control (variation 1) exhibits a significant decreasing accuracy with increasing speeds, since the impact of nonlinear and coupled dynamics increases with higher velocities and accelerations. Using modelknowledge (variation 2 and 3) improves always the tracking performance. Furthermore, the compensation of identified model (variation 3) outperforms clearly variation 2 that just uses the nominal parameter values. The latter statement can be proven at the level of actuator tracking accuracy like depicted in Fig. 11. For the same arbitrarily chosen actuator, the tracking accuracy is higher if the identified model is implemented. The same results are noticeable for the cartesian tracking accuracy
6. Conclusions
The present chapter discussed most significant aspects to achieve accurate and robust dynamics identification for parallel manipulators with 6 dof's. Hereby, the adequate consideration of structural properties of such systems has been stressed out. First, an efficient methodology to determine the inverse dynamics in a parameterlinear form has been presented, which enables the use of linear estimation techniques. Periodic excitation has been proved to be a powerful method for parallel robots, since it allows for appropriate consideration of hard workspace constraints. Due to measurement noise and cross coupling between the actuators, the achievement of the identification in a statistical framework is recommended. This includes the consideration of disturbance covariances in the experiment design, the use of GaussMarkov estimation approach as well as the frequencydomain filtering to extract non measurable information from rough data. The robustness of the results has been substantiated on a direct driven hexapod. The obtained estimates have presented high confidence in terms of single parameters, as well as in terms of the whole parameter set. Additionally, the benefits of accurate identification on the enhancement of control performance have been clearly and experimentally demonstrated.
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Notes
 The bodyfixed frames can be defined according to the modified DenavitHartenberg (MDH) notation (Khalil & Dombre, 2002).