A computational method for keyperformanceindicatorbased parameter identification of industrial manipulators
 Felix Jost^{1},
 Manuel Kudruss^{2}Email authorView ORCID ID profile,
 Stefan Körkel^{3} and
 Sebastian F Walter^{2}
DOI: 10.1186/s1336201700397
© The Author(s) 2017
Received: 27 April 2016
Accepted: 26 June 2017
Published: 10 July 2017
Abstract
We present a novel derivativebased parameter identification method to improve the precision at the tool center point of an industrial manipulator. The tool center point is directly considered in the optimization as part of the problem formulation as a key performance indicator. Additionally, our proposed method takes collision avoidance as special nonlinear constraints into account and is therefore suitable for industrial use. The performed numerical experiments show that the optimum experimental designs considering key performance indicators during optimization achieve a significant improvement in comparison to other methods. An improvement in terms of precision at the tool center point of 40% to 44% was achieved in experiments with three KUKA robots and 90 notional manipulator models compared to the heuristic experimental designs chosen by an experimenter as well as 10% to 19% compared to an existing stateoftheart method.
Keywords
industrial manipulators parameter identification optimum experimental design key performance indicator quantity of interest collision avoidance1 Introduction
The total worldwide stock of operational industrial manipulators at the end of 2013 is estimated between 1,332,000 and 1,600,000 units [1]. Of all the different industries, the automotive industry requires the largest share of nearly 70,000 manipulators [1]. One application area of manipulators in automotive industry are flexible measurement systems (FMS) in assembly lines. During measurements with FMS, industrial manipulators collect measurement data for quality and process control, e.g. of automotive bodies. This working process requires a high tool center point (TCP) precision to detect production errors. The TCP defines the position and orientation of the working tool, which is attached to the last link of the manipulator.
1.1 Problem description

length of a link

angle between two consecutive joint axes.
1.1.1 Parameter identification of industrial manipulators
Laser tracker measurement system
Parameter identification
1.2 Previous work on optimum experimental design

manipulator specifications,

measurement system specifications and

collision avoidance.
In [10] and [11] the quality of a given set of measurement configurations is defined by five socalled observability indices (OIs) which were first proposed by Borm and Menq in [12] and [13]. They are based on the sensitivities w.r.t. geometric parameters and configurations \({F}_{1}:=\frac{\mathrm{d} {f}_{1}}{\mathrm{d} \boldsymbol {p}}( \boldsymbol {p},\boldsymbol {q}^{\mathrm{meas}})\) from the parameter estimation problem (6). In [10] the five quality criteria for parameter identification of industrial manipulators are related to those of OED problems [14]. Each of the quality criteria can be used as a cost function in the optimization problem (7a)(7b).
The approach of nonlinear experimental design for explicit key performance indicators (KPIs) applied to a chemical process has been discussed in [23].
1.3 Contribution of the article
All presented formulations for the objective function in (7a)(7b) only consider the statistical uncertainties of the geometric parameters of the kinematic model. We present a new optimization problem formulation in which additionally the TCP position or other KPIs are considered in the objective. Thereby, the optimization problem (7a)(7b) is extended with the statistical error of TCP positions given by a number of predefined working configurations. The OED problem is solved by a sequential quadratic programming (SQP) method. The avoidance of collisions has to be considered in the optimization method to be applicable in industrial practice. This is achieved by introducing additional nonlinear constraints to the optimization problem. The segments of the manipulator are approximated by capsules. The avoidance of collisions between two manipulator segments is described by a single nonlinear constraint in the optimization problem.
1.4 Organization of the article
In Section 2, we explain the methodology of OED for parameter identification of industrial manipulators in detail. First, we introduce the general formulation of KPIs. Second, we formulate the OED problem for KPIs. Afterwards, we introduce a collision avoidance strategy to generate collisionfree measurement configurations. In Section 3, we present numerical results from two simulation studies. First, the KPI approach is presented for a simulation of realworld flexible measurement systems (FMS) example with three different KUKA robots (KR15, KR300 and KR500) considering collision avoidance. Second, for a set of 90 different manipulator geometries the proposed methodology is statistically compared against a heuristic and a stateoftheart method in simulation. Afterwards, the results are discussed. We finish with a conclusion and give an outlook to future work.
2 Parameter identification of manipulators with key performance indicators
In the following, we will present a mathematical concept in which the TCP precision as the main quantity of interest is explicitly considered. In the remainder of this article we will denote this main quantity of interest as key performance indicator (KPI). Primarily, we are interested in the TCP precision and, secondarily, in the precision of the geometric parameters.
2.1 Formulation of key performance indicators
In the following, we derive how the error propagation from the measurements to parameters as described by Equation (11) can be extended to specific quantities of interest, the socalled key performance indicators (KPIs). For this, the error propagation from the data η to the KPIs s is quantified by \(\phi(C( \boldsymbol {s}))\).
Analogously to (6), the solution \(\hat{ \boldsymbol {v}} = (\hat{ \boldsymbol {p}}, \hat{ \boldsymbol {s}})^{T}\) of the constrained nonlinear leastsquares problem (13a)(13b) is subject to statistical uncertainty due to the stochasticity of the measurement data η. The propagated uncertainty for both, the parameter and the KPI estimates \(\hat{ \boldsymbol {p}}\), \(\hat{ \boldsymbol {s}}\), can be considered for the minimization in OED.
Lemma
Proof
See [24]. □
Proposition
Proof
See [24]. □
2.2 Collision avoidance
In the following, we present a collision avoidance strategy required for the practical realization of OED for KPIs in industrial applications. The optimized measurement configurations \(\boldsymbol {q}^{\mathrm{meas}}\) are required to be free of collisions between the manipulator and its environment. This is achieved by introducing a collision avoidance strategy to the formulation of the OED problem.
We give an overview of existing collision avoidance approaches. Afterwards, our approach is presented in a general manner for the fact that the collision avoidance formulation is not restricted to flexible measurement systems. In Section 3.2 the collision avoidance technique is then applied on a specific experimental setting.
Collision avoidance is a major research field in which different approaches have already been successfully applied. In [25] the manipulator segments and obstacles are approximated by the union of convex polyhedra and the collision avoidance criterion between two polyhedra is based on Farkas’ Lemma. In [26] the body of a humanoid manipulator is described by ‘strictly convex hulls’. These hulls result from slightly blowing up the usual convex hulls through patches of spheres and tori. As a consequence, the gradient of the proximity distance function becomes continuous, which is useful in a derivativebased optimization framework.
Not all segments can be described by just one suited capsule. If the first or last joint is described by k capsules, then additionally \((k1)\cdot(n2) \) constraints have to be added to (22). If one of the interior segments is described by k capsules this results in \((k1)\cdot(n3)\) additional constraints. The total number of constraints can be decreased by further investigation of the geometry of the manipulator and motion to find segments which will never collide and therefore do not have to be considered in the collision avoidance strategy. A study in [29] has shown, that the blow up from inlying to enclosing capsules only has a slightly negative influence on the optimization performance. As the description of an industrial manipulator by capsules is just an approximation of reality, the use of enclosing capsules guarantees by higher probability that the optimized measurement configurations are collisionfree in a practical application without losing much performance.
2.3 Optimum experimental design for key performance indicators
3 Numerical results
DenavitHartenberg and Hayati zero offset values with \(\pmb{\lambda\in\{ 650,700,\ldots, 1\text{,}050 \} }\) and \(\pmb{\mu\in\{ 600, 650,\ldots , 1\text{,}050 \} }\)
Joint i  \(\boldsymbol{\theta_{i}}\) ( \(\boldsymbol{{}^{\circ}}\) )  \(\boldsymbol{\alpha_{i}}\) ( \(\boldsymbol{{}^{\circ}}\) )  \(\boldsymbol{a_{i}}\) (mm)  \(\boldsymbol{d_{i}}\) (mm)  \(\boldsymbol{\beta_{i}}\) ( \(\boldsymbol{{}^{\circ}}\) ) 

1  0.00  \( \underline{90}\underline{.}\underline{00} \)  \( \underline{350}\underline{.}\underline{00} \)  0.00   
2  \( \underline{0}\underline{.}\underline{00} \)  \( \underline{0}\underline{.}\underline{00} \)  \( \underline{0}\underline{.}\underline{00} \)    λ 
3  \( \underline{90}\underline{.}\underline{00} \)  \( \underline{90}\underline{.}\underline{00} \)  \( \underline{145}\underline{.}\underline{00} \)  \( \underline{0}\underline{.}\underline{00} \)   
4  \( \underline{0}\underline{.}\underline{00} \)  \( \underline{90}\underline{.}\underline{00} \)  \( \underline{0}\underline{.}\underline{00} \)  μ   
5  \( \underline{0}\underline{.}\underline{00} \)  \( \underline{90}\underline{.}\underline{00} \)  \( \underline{0}\underline{.}\underline{00} \)  \( \underline{0}\underline{.}\underline{00} \)   
6  180.00  180.00  0.00  −170.00   
Position and orientation of laser tracker and reflector
Laser tracker  Reflector ( \(\boldsymbol{{^{k}} \boldsymbol {c}_{j}}\) )  

x (mm)  2,730.88  −89.58 
y (mm)  4,554.68  −2.84 
z (mm)  1,397.67  327.03 
α (^{∘})  −95.63   
β (^{∘})  −95.63   
γ (^{∘})  0.23   
3.1 Software
The optimization problem is solved with the software package VPLAN [30], developed in the research group of the authors. The evaluation of derivatives is performed by ADIFOR [31]. ADIFOR is able to evaluate the derivatives of the vectorvalued objective function (6). The OED problem (24a)(24e) is formulated as a nonlinear optimization problem and solved by the sparse SQP solver SNOPT7 [32].
3.2 Experimental setup
Range of motion of joints and admitted incident angle between sensor surface and laser beam
Joint i  Range of motion  Incident angle  

\(\boldsymbol{q^{\mathrm{lo}}}\) ( \(\boldsymbol{{}^{\circ}}\) )  \(\boldsymbol{q^{\mathrm{up}}}\) ( \(\boldsymbol{{}^{\circ}}\) )  
1  −82  82  β ≥ 30^{∘} 
2  −77  31  
3  −13  154  
4  −347  347  
5  −87  87  
6  −347  347 
Values of the constraints to avoid collisions between manipulator bodies, laser beam and object bodies
Manipulator bodies (mm)  Laser beam (mm)  

x direction  ≥450.0   
y direction     
z direction  ≥0.0   
Manipulator bodies  ≥0.0  ≥0.0 
Object  ≥0.0   
Car part 1  ≥0.0  ≥0.0 
Car part 2  ≥0.0  ≥0.0 
The radii of the capsules are chosen in a way, that the capsules nearly surround their appropriate manipulator segments. The surrounding capsules are used as a conservative and robust approach to guarantee that the optimized measurement configurations are applicable in reality. Each simulated parameter identification consists of 60 measurement configurations from a heuristic approach and from the optimization method introduced in Section 2.3. The 60 heuristic measurement configurations are randomly and uniformly chosen from the motion range of the industrial manipulator. In the experiment two sets of measurement configurations are computed differing by the choice of variancecovariance matrix projections in the cost function of optimization.
3.3 Computation of optimum experimental designs
3.3.1 KUKA robots (KR15, KR300, KR500)
Key performance indicator (KPI) average variances (mm ^{ 2 } ) of three KUKA robots with heuristic and optimized experimental designs for parameter identification
Cost function  Average variance of KPI (mm ^{ 2 } ): \(\boldsymbol{\frac{1}{3}\operatorname{tr}( {C}_{s})}\)  

Heuristic  Optimized with collision avoidance  
\(\boldsymbol{\phi_{A}( {C}_{p})}\)  \(\boldsymbol{\phi_{A}( {C}_{s})}\)  
KR 15  0.59  0.40  0.33 
KR 300  0.58  0.39  0.35 
KR 500  0.61  0.40  0.35 
Heuristic  Standard method  KPI method 
3.3.2 Variety of 90 industrial manipulators
Minimum and maximum value of the averaged three dimensional variances of the tool center point (mm ^{ 2 } ) from the 90 experiments with heuristic and optimized experimental design for parameter identification
Cost function  Range of KPI average variance (mm ^{ 2 } )  

Min  Max  Min  Max  
Heuristic      0.55  0.71 
\(\phi_{A}( {C}_{p})\)  0.39  0.39  0.42  0.45 
\(\phi_{A}( {C}_{s})\)  0.33  0.34  0.34  0.37 
Collision avoidance  No  Yes 
Key performance indicator (KPI) average variances (mm ^{ 2 } ) of 90 experiments with heuristic and optimized experimental design for parameter identification
Cost function  Average variance of KPI (mm ^{ 2 } ): \(\boldsymbol{\frac{1}{3}\operatorname {tr}( {C}_{s})}\)  

Heuristic  Optimized  Optimized  
\(\phi_{A}( {C}_{p})\)  0.61  0.39  0.43 
\(\phi_{A}( {C}_{s})\)  0.61  0.33  0.35 
Collision avoidance    No  Yes 
4 Discussion
4.1 KUKA robots (KR15, KR300, KR500)
In the simulated parameter identification of the three different KUKA robots, the optimized experimental designs \(\boldsymbol {q}_{p}^{\mathrm{meas}}\) and \(\boldsymbol {q}_{s}^{\mathrm{meas}}\) provide higher TCP accuracies in contrast to the heuristic experimental design (improvement of 32% to 44%). The largest improvement of the precision of the TCP is achieved by only using the covariance matrix \({C}_{s}\) of the KPI in the cost function for optimization (KR 15: 44%, KR 300: 40%, KR 500: 43%). A comparison between the stateoftheart optimization approach and our new approach implies that the new problem formulation (24a)(24e), which takes the TCP as a KPI into account, yields a higher TCP precision (KR 15: 17%, KR 300: 10%, KR 500: 12%). Moreover, our formulation of optimizing measurement configurations can be used for industrial manipulators with a low payload to heavy duty models.
4.2 Variety of 90 industrial manipulators
In order to show that our parameter identification method is not limited to some specific industrial manipulators, we performed a case study with 90 notional industrial manipulators. By analyzing Table 8 and Figure 8 one recognizes that the two optimum experimental designs reduce the statistical uncertainty of the 90 manipulators’ TCP accuracies by 36% and more on average in comparison to the heuristic set of configurations. The collision avoidance method, which is essential for the computation of practicable measurement configurations, has an increasing effect on the optimization performance by 6% to 9%. Table 8 indicates that the largest improvement (43%) of the TCP precision is achieved by only using the statistical uncertainty of the TCP position as cost function. Moreover, the case study stresses that the new approach using KPIs is superior to the stateoftheart optimization approach providing a 19% higher TCP precision on average. The case study underlines the benefit of the TCP precision achieved through our problem formulation and shows that the improvement is not limited on specific KUKA models but on a range of small to large industrial manipulators.
5 Conclusion
This article introduces a new problem formulation for the computation of optimal and collisionfree measurement configurations for parameter identification of industrial manipulators. The novelty lies in the fact, that the precision of the tool center point (TCP) is directly considered in the optimization problem as a key performance indicator (KPI). The approach is verified by the simulated parameter identification of three different KUKA robots and also by the quantitative results with 90 notional manipulator geometries. In the experiments an improvement of 40% to 44% of the precision of the TCP is achieved in contrast to the heuristic approach and 10% to 19% improvement compared to an existing stateoftheart method. For the computation of collisionfree configurations required in practice a collision avoidance method is introduced, which provides a minimal number of nonlinear constraints in the optimization problem. In the experiments a laser tracker system is used for parameter identification but the approach is also applicable to other measurement systems. Furthermore, the approach is not limited to a specific observability index (OI) as cost function. As our approach yields a higher TCP precision with the same number of measurement configurations it is also possible to reduce the needed number of configurations to achieve a certain TCP precision in a shorter reidentification time interval compared to the heuristic approach.
6 Outlook
Despite the fact that our problem formulation with key performance indicators (KPIs) improves the tool center point (TCP) precision by 40%, the parameter identification procedure is not adaptive at the moment and cannot incorporate an online estimation of the TCP precision. The next step to be investigated would be an online approach of parameter estimation and optimum experimental design (OED) for parameter identification of manipulators, which monitors the current precision and sequentially identifies itself using optimal measurement configurations calculated onthefly when necessary. We assume that with this approach a fast reidentification of parameters with a guaranteed TCP precision during the work cycles of the manipulator can be realized leading to more flexibility in reidentification and a higher throughput. Additionally, the online parameter identification has the advantage that the parameter identification can be stopped when a certain level of TCP precision is reached which results in shorter parameter identification intervals.
At the moment, the improvements are achieved by considering a kinematic model of the manipulator only. However, the improvement could be increased by incorporating a higher level of detail of the manipulator into the model. The higher level of detail is achieved by introducing further parameters, e.g. nongeometric errors like joint mobilities or elasticities. This would require the computation of forces acting on the links and joints, which can be achieved by using a dynamic model of the manipulator described by differential equations. As shown in [30], the same approach for differential equation models can be used when the required derivative information is available. However, this approach is yet to be investigated for the dynamics of multibody systems acting as dynamic model for the parameter identification procedure.
Another possible extension is the definition and consideration of further KPIs in the optimization problem. The article demonstrates that the use of TCP precision as KPI achieves a significant precision improvement after parameter identification over the heuristic parameter identification approach as well as the existing stateoftheart problem formulations not using KPIs. Future work will investigate the possibilities of not only taking one TCP pose into account but additional poses of important working configurations. Furthermore, the definition of KPIs is not limited to TCPs, such that working path trajectories of welding manipulators could be defined as KPIs as well. This way, the range of possible application of the proposed approach will be increased.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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