The present project deals with the elastostatic modeling and calibration experiment of spacial industrial manipulators using optimal selection of measurements pose , for the calibration procedure, the optimal pose selection aims to the efficiency improvement of identification procedure for serial manipulators which reduces noise impact on the parameters identification precision, it is usually used for planar manipulators, our work is mainly to extend the approach for a more complicated manipulator in 3D space using a wise decomposition of the spacial manipulator into a set of serial sub-chains , the optimal pose configuration is then used in the calibration procedure using the complete and irreducible model for the 7 dof serial manipulator . The methodology is illustrated with the anthropomorphic industrial robot KUKA iiwa14 R820 for which, we performed the calibration and constructed the stiffness modeling using two different approaches namely VJM (Virtual Joint Modeling) and MSA (Matrix Structural Analysis).
2 Related Works
In contrast to previous works for calibration the proposed in  yields simple geometrical patterns that allow users to take into account the joint and workspace constraints and to find measurement configurations without tedious computations. The main theoretical results are expressed as a set of several properties and rules, which allow user to obtain optimal measurement configurations without any computation, just using superpositions and permutations of the proposed patterns. They presented an example for a 6 dof manipulator showing the efficiency.
proposes simple rules for the selection of manipulator configurations that allow the user to essentially improve calibration accuracy and reduce identification errors. The results are mainly for planar manipulators(two-, three- and four-link planar manipulators), but they can be used as a base for more complicated ones. The main contributions have been obtained for the planar case, the developed rule has been heuristically generalized for articulated robots. However, a strict theoretical proof of this approach remains unsolved for calibration of non-planar serial and parallel manipulators.
In contrast to previous works,  developed calibration technique based on the direct measurements only. To improve the identification accuracy, it is proposed to use several reference points for each manipulator configuration.The obtained theoretical results have been successfully applied to the geometric and elastostatic calibration of serial industrial robot employed in the machining work-cell for for aerospace industry.
3.1 Elastostic Modeling:
To build the extended stiffness model of the KUKA iiwa robot, we are going to use the simplifications shown in Figure 1
In the case of our manipulator, we have 6 elastic links and 7 actuated joints, the robot model approximation is shown in Figure 2
Each elastic link is represented as a rigid link and a 6 dof virtual spring, similarly, each joint is represented as an actuated joint and a 1 dof virtual spring. The final VJM model is given by Figure 3
This VJM model correpond to the following extended model equation :
The general equation between increments is as follows:
The cartesian stiffness matrix has the final expression given by equation 3
where : is the classical cartesian stiffness matrix
The basic expression for stiffness model is:
Using the cantilever beam representation of a link, one can write the expression of the stiffness matrix as follow:
We need to transform this model into a global coordinates system:
Then, the general expression of the wrenches applied in both sides of the link is:
The MSA model of the robot is shown in Figure 4
The details for each joint modeling is depicted below:
elastic support 0,1:
elastic joints with rotation about x-axis 2,3 6,7 10,11:
elastic joints with rotations about the z-axis 4,5 8,9 12,13:
Equation 14 can be written in the form:
And the final expression of is given by:
3.2 Geometric Calibration:
The basic equations for the identification using full pose measurements is
is six dimensional location vector,is the manipulator extended geometric model, is the vector of actuated coordinates and is the vector of parameters.
The goal is the identification of robot model parameters using direct measurement only without using orientation components, the basic equation for identification is:
The geometric model obtained via homogeneous transformations can be presented by the matrix product:
The cartesian coordinates of reference points corresponding to the configuration can be expressed in the following form:
Where: are unknown parameters
The procedure of identification is divided into two steps :
Step1: parameters identification of tool and base transformations
Step2: Identification of the Electrostatic and geometric parameters of the manipulator
Basic equation of identification : , where are the unknown parameters to be identified
The solution of the identification problem is given by
To perform the identification task, one need to develop a suitable geometric model, which properly describe the relation between the manipulator geometric parameters (link length and joint angles) and the end effector location (position and orientation), for this, we construct the complete and obviously irreducible model in the form of homogeneous matrices product
The base transformation is
The joint and link transformations (for revolute joint) Where is the joint axis and and are the axes orthogonal to
The tool transformation
The next step is the elimination of non identifiable and semi identifiable parameters in accordance with specific rules for different nature and structure of consecutive joints
In the case of consecutive revolute joints
if eliminate the term or that correspond to
if eliminate the term or that define the translation orthogonal to the joint axes for which k is minimum
3.4 Design of calibration experiments:
In the case of serial manipulator with revolute joints, the expression of end effector position is computed using the following formula:
are the nominal links lengths and their deviations, are nominal joints coordinates, are defined as and are the joints offsets
To take into account the impact of measurement noise, the calibration equation derived from 23 becomes:
To find the desired parameters using the noise corrupted measurements, the least square technique is applied, this approach aims at minimizing the square sum of the residuals in 24 simultaneously
Collecting the unknown parameters and into the vector and the measurements into the vector equation 23 can be rewritten as:
Where is the jacobian matrix, then one can get the unknown parameters using the least squares technique that leads to
Where the subscript k indicate the experiment number and m is the number of measurements Considering that each measurement is corrupted by an unbiased random Gaussian noise with standard deviation, the identification accuracy of the parameters can be evaluated via the covariance matrix, which is computed as follow:
With this expression, it is possible to choose the measurement configuration that yield parameters less sensitive to measurement noise, this procedure is referred to as design of calibration experiments, the optimality condition for the calibration plan proposed in  where one need to ensure that the information matrix is diagonal yields to the D-optimal plan of experiments that is satisfied when:
The above presented equations define the desired set of optimal measurements configurations.
3.5 Geometrical patterns for measurement pose selection:
Superposition of optimal plans also gives an optimal plan for this ; this is due to the additivity of the the operations included in 29.Using this property , it is possible to generate optimal plan with a large number of measurements configurations using simple sets.
The angles can be rearranged in the optimal plan in an arbitrary way without loss of the optimality condition 29
Optimal plan for n-link manipulator can be obtained using two lower-order optimal plans for n1- and n2-link manipulators, where , This property gives an elegant technique to generate optimal plan of calibration experiments without straightforward solution of the system 29
the following present geometrical patterns for typical serial manipulators that can be used to generate optimal plans. In the frame of these patterns, all variables and and are treated as arbitrary angles:
For n = 3, m = 3, the geometrical pattern can be presented as
For n = 4, m = 4, the geometrical pattern can be presented as
3.6 Extension to the 3D case:
The simplest way to extend the proposed “rule of thumb” to the 3D case is to apply the following procedure:
Step (a): decompose the spatial manipulator into a set of planar serial sub-chains;
Step (b): apply the developed rule to each sub-chain separately,without assigning certain values to the joint coordinates that can be selected arbitrary;
Step (c): aggregate the obtained sub-chain joint coordinates in order to find configurations of the entire manipulator (where some values are still arbitrary);
Step (d): apply the developed rule to the each set of the arbitrary coordinates, i.e. ensuring that the sums of sines and cosines are equal to zero for all of them.
4 Case study: calibration of the 7 dof manipulator KUKA IIWA
4.1 Calibration experiment:
Let us consider the 7 dof. serial manipulator with seven revolute joints and six links, Kuka IIWA 14 R820 depicted in Figure 6
To complete the identification task, the irreducible model in the form of homogeneous matrix product as described before was build for Kuka iiWA:
To simulate the calibration procedure, we take ideal parameters for which we add some random noise and we compute the calibration procedure described before using a Matlab code, Figure 8 shows the results obtained for three different trajectories (trajectory obtained with and without calibration).
We can clearly see the trajectory with calibration follow better the desired trajectory in all three cases, especially in the last two pictures where the trajectories are almost colinear.
Table 1 shows the results of the parameters identification with and without calibration, the parameters obtained after calibration are more accurate and closer to the real ones.
|Parameter||Real value||Calibration||No Calibration||Improvement Factor|
4.2 Calibration using design of experiments:
In order to find the optimal pose configuration for the manipulator, the first thing to do is to decompose the spatial manipulator into a set of planar kinematic serial sub-chains, we define each joint of the manipulator as where n denotes the joint number of the robot. According to the geometrical pattern presented above, and by studying the robot structure, we naturally divide the manipulator into the two following planar sub-chains:
Note that, because the first angle doesn’t appear in the system 29 (the calibration plan is invariant with respect to the first joint), we added a virtual joint at the beginning of the second sub-chain. Using this decomposition, the property 3 described above and the results of previous works shown in Figure 5  corresponding to the 4-link manipulator with (equation 31), we built 16 configurations that reflect all the possible poses obtained when combining the two sub-chains. (table2)
Also, to build the final optimal plan, a Matlab code was developed to set the still remaining arbitrary angles that satisfy the optimality condition 29 for the entire manipulator. Taking into account the joint limits, the arbitrary angles have been set to: , , , , , , , , , , , , and the Table 3 was built which shows 16 optimal configurations for our robot. One point that one should noted is that, because the calibration plan is invariant with respect to the first joint like stated before, we can arbitrarily assign joint to make the end effector facing the direction where the laser tracker should be positioned when performing the experiment.
|Optimal Pose Measurements|
|Configuration||Joint 1||Joint 2||Joint 3||Joint 4||Joint 5||Joint 6||Joint 7|