Optimal Robot Dynamics Local Identification usingGenetic-based Path Planning in Workspace Subregions

Enrico Villagrossi, Nicola Pedrocchi, Federico Vicentini and Lorenzo Molinari Tosatti

Abstract— Methods for dynamic calibrations of IndustrialRobots (IR) are increasing their importance in many applica-tions because of high performances attained by model-basedcontrol strategies. Most of known state-of-the-art methodsaim at modeling robots along the complete workspace, oftenaffecting the identified parameters with loss of physical meaning(e.g. negative inertia values) and requiring a wide explorationof the workspace both in term of joint positions and velocities(accelerations). Actually, many IR tasks require dynamic accu-racy in limited portion of the workspace and commonly displaymild dynamics. Local identification of dynamics parameters intask conditions could therefore increase the predictive capabilityof the model for that operation. This work proposes the useof a parametric-description of trajectories in Cartesian space,corresponding to the standard industrial path-description asa series of via-points in most of programming languages. Theidentification of the optimal exciting Cartesian trajectory ina local sub-region of the workspace is made by a geneticalgorithm over the template trajectory description. The useof an IR real interpolator allows to match computational andtask execution conditions.

I. INTRODUCTION

The dynamic calibration of industrial robots (IRs) isreckoned to be of utmost importance in model-based controlstrategies, since early examples [1], [2], [3], because it isgenerally accepted [4], [5] that an accurate dynamical modelparameters estimation attains high prediction capabilities formotor torques during movements. At present, almost allcalibration methods propose a similar paradigmatic proce-dure consisting in (i) deriving the robot model in symbolicform (ii) identifying the set of parameters to be (iii) excitedalong experimental trajectories computed in order to (iv)obtain a well conditioned set of data for the (v) estimationalgorithm. Within this general framework, the trajectoriesoptimization step has witnessed many different approaches,e.g. optimization of trajectory parameters for polynomial [6]or Fourier series [7], [8], all of which consistently aim atfinding excitatory patterns along the whole workspace able toarise dynamical contributions from all dynamic parametersand to set good conditioning properties for the estimationalgorithm proper, e.g. the conditioning number and singularvalues [9], [10], [11], [12] of trajectory regressors to beoptimally inverted. Specifically in scope of this work, wellconditioning properties for harmonic joint-space trajecto-ries can be encoded as fitnesses functions in evolutionary-based techniques as in [13], [14]. Regardless the excitatory

E. Villagrossi, N. Pedrocchi, F. Vicentini and L. Molinari Tosatti are withthe Institute of Industrial Technologies and Automation (ITIA) of ItalianNational Research Council (CNR), via Bassini, 15 - 20133 Milan, Italynicola.pedrocchi@itia.cnr.it

technique, general implicit calibration procedures aim atachieving good average estimation properties in large partof, even all, the workspace through wide-spanning excitationtrajectories. Nevertheless, local calibration in a sub-region ofthe workspace could be of interest when any task at handencompasses a well defined family of trajectories. As an ex-ample, automatic shoe-roughing is a task with an archetypalclass of shoe-profiled contouring trajectories, varying verymuch in the same workspace sub-region according to theparticular shoe items worked, yet belonging to a commonclass of shoe-patterns. In fact, the manipulator configurationis likely to change very little, and often the task at handrequires to be executed at almost constant low speed. Theseconditions are nominally poorly excitatory therefore the localcalibration should be able to provide an estimation model thatmaintain predictive power even in mild dynamical excitatorypattern. Despite the availability of a general good modelestimation per se, in fact, such model could lose its predictivepower in case of task patterns substantially different from theoriginally excitatory ones, mainly due to friction and inertiacontributions.The methodology proposed in this work features a task-dependent class of local exciting trajectories defined bya set of via-points in the task-space. Via-points can bein fact directly associated with a loosely defined patternrepresentative of task trajectories, e.g. reproducing a coarsepattern of a known process, like in most manufacturingtasks. This approach is closely correlated to the input modeof task-space trajectories in many industrial controllers, i.e.a set of discrete poses to be internally interpolated (seeFig. 1). The resulting trajectory is therefore generally definedby the interpolator itself on the basis of global tunableparameters (fly-by accuracy, velocity profiles, etc). Sincetrajectories are obtained from an array of points, i.e. with alimited number of constraints, the template task descriptionhas a very limited complexity and can be easily translatedinto an optimization problem, e.g. a genetic algorithms,for the identification of the optimal excitation trajectory.A remarkable aspect of the proposed methodology is thatthe interpolator of a commercial IR is embedded in theestimation algorithm (Comau-ORL library, see Section III) inorder to directly provide realistic optimized trajectories. Thepaper is structured in four Sections over the Introduction.Section II poses the problem, while suggested methodsis detailed in Section II-B. Section III reports the resultsof the procedure applied to a paradigmatic test-case anddemonstrates the efficacy of the proposed strategy. SectionIV reports final considerations.

2013 IEEE/ASME International Conference onAdvanced Intelligent Mechatronics (AIM)Wollongong, Australia, July 9-12, 2013

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Fig. 1: Trajectories in IRs controllers are defined as a set of K via-points and interpolating rules. The identification of theoptimum exiting trajectory grounds on the idea that moving the via-points in bounded-regions improves the identificationof dynamic parameters. The envelop of the K bounded regions correspond to the maximum task sub-workspace.

II. MOTIVATION AND METHODOLOGY

In most of dynamical parameters identifications, excitationtrajectories are fairly different from those commonly usedin industrial procedures, which are often characterized byclasses of trajectories defined as sets of joined simple geo-metric entities (e.g. lines and circles), performed at constantregime velocity along the path. Alternatively, high accuracyin torques prediction could be obtained by Iterative LearningControl (ILC) techniques [15] resulting in implicit dynamicscalibration, i.e. without the need of any model. Nevertheless,ILC algorithms prevent the possibility to extend (extrap-olate) results for a given trajectory to similar ones, andare rarely used in tasks where robots have to interact withthe environment [16]. Therefore, a sound algorithm for thelocal dynamics estimation requires both a proper dynamicsmodel and a design method for excitatory trajectories thattakes into account the constraints imposed by ranging withina small sub-region of the whole workspace in proceduralmild-dynamics conditions. In addition, once designed andoptimally computed, several trajectories representative of duetasks/sub-regions can be stored and loaded for re-calibration.

Notation

qs ≡[q1s , . . . , q

dofs

]TJoint positions at s-th sample time.

q̇s, q̈s, τs Joint velocities, accelerations, andtorques at s-th sample time.

Q ≡ {q1, . . . ,qS} Joint position time series of S dif-ferent time samples.

{Q, Q̇, Q̈ } Trajectory.

(̃·), (̂·), (·)∗ Measured, estimated value and op-timum estimation respectively.

A. Dynamics modeling and estimation

Making use of standard rigid multi-bodies dynamics [17],the model representation of robot dynamics can be reduced[18], [19] to the linear system:

φ0 (q̈, q̇,q)π0 = τ , (1)

where π0 is a base set of dynamical parameters, and matrixfunction φ0 should be considered as generalized accelera-tions. The base set π0 includes only those parameters thatgive contribution to joint torques and are observable alongany excitatory trajectory1 that generates φ0. The minimalsize Nπ of the base set π0 is demonstrated [11] to be40 for a 6-dof anthropomorphic manipulator, in addition ofwhich other Nf coefficients of the friction model yield thecompound parameters set π.The selected friction model [20] provides the j-th jointfriction torque in the form:

τ jf = f j0 sign(q̇j) + f j1 q̇j + f j2 sign(q̇j)

(q̇i)2, (2)

requiring Nf = 3 × dof additional dynamics parame-ters2. For trajectory {Q, Q̇, Q̈ } composed by S-samples themodel (1) is expressed as:

T ≡

τ1...τS

=

φ1 (q̈1, q̇1,q1)...

φS (q̈S , q̇S ,qS)

π = Φπ, (3)

where Φ is the observation matrix. Experimental samplingT̃, Φ̃ includes also measurements noise ν:

T̃ = Φ̃π̂ + ν, ν ∼ N (0, σν) . (4)

Several techniques are known [21], [22], [23] for the directsolution of the system in (4). Among other, ordinary least-squares technique and weighted least-squares technique [24]appear to be the most common, i.e. denoting as D the weightmatrix, the system is solved as π̂ = (Φ̃TDΦ̃)−1Φ̃TDT̃.

As previously introduced, the core problem in model iden-tification is the generation of an optimal excitation trajectory{Q?, Q̇?, Q̈? } able to provide the best regression conditionsfor the system in (4) and contribute in finding the optimum

1Dynamic calibration does not aim at estimating the full set of link figures(masses, inertias, friction coefficients) but to minimize the prediction errorof the model w.r.t. the real behavior of the robot.

2For small joints movements friction has a remarkable relevance, over-whelming the contribution by other parameters, therefore a standard linearfriction, including the static and the viscous terms, appears to be restrictivefor the dynamical representation of the phenomenon.

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Fig. 2: The model estimation procedure (mid-layer) computes the parameters using an optimized excitatory, which is pre-computed in through GA (top-layer). The obtained model is then evaluated along random trajectories (bottom-layer).

estimate π? : ν → 0 out of the (weighed) pseudo-inversionof Φ? = Φ(Q?, Q̇?, Q̈?).In the present work, the optimization problem formulationis derived from the requirements and constraints of in-dustrial setups, i.e. finding a (sub)optimal set of K via-points {P ?1 , . . . , P ?K} along which a complete trajectory iscalculated by standard IR motion interpolators (see Fig. 1).The workspace sub-region is in fact partitioned into anarbitrary number of K points through which the interpolatedtrajectory passes and each of which is responsible of pin-pointing the passing trajectory in a given pose.The use of real IR interpolator, during optimization process,allows to avoid passing through singularity points.

B. Optimal Excitation Trajectory

Optimal via-points are obtained from a Genetic Algorithmsearch over sets of candidate individuals {Pn1 , . . . , PnK}whose interpolated trajectory and derived regressors Φn areevolved over generations (see top layer in Fig. 2).The 6D coordinates of each of K via-points are concatenatedalong a single individual genome of 6K genes ranging in[0, 1] ∈ R. Genotypes are linearly mapped into physicaltrajectory parameters according to particular ranges definedfor each position/orientation coordinate: the mapping rangesare set in order to constrain a looped trajectory and converg-ing orientations for the robot end-effector. The populationcontains N = 50 genotypes (individuals), each carrying fullinformation to code a trajectory. Generations following thefirst one are produced by a combination of selection withelitism, recombination and mutation [25]. The selection ofindividuals is made on the basis of a fitness function thatmaximizes the determinant of a quadratic form associated

with the regressor Φn of each n-th individual trajectory asin [13], i.e.

fn (Φn) = log10

det[HT (Φn) H (Φn)

]M

, (5)

where H (Φ) is a torque-normalized regressor (see below(6)) and M is the unpredictable number of intermediatepoints generated by the interpolator3. Quadratic forms ofregressors are M -scaled in order to let individuals be com-parable in each generation. Torque normalization H, instead,introduces significant variations w.r.t. previously formulatedfitnesses, especially D-optimal or A-optimal fitnesses.The purpose of torque normalization is to remove dynamicalbiases due to the different contribution to the overall powerby each link in real robots. Considering, in fact, the regressorφ as a generalized acceleration, each row represents thedynamical contribution to a single-joint torque within theoverall dynamics of the system. Same absolute generalizedaccelerations in φ could give comparable torques despitethe relative exploitation of the maximum torque of different-sized actuators. The observation matrix is scaled with axis-wise ratios of nominal-over-maximum τ = maxi(τ

inom) in

order to normalize contributions of different actuators. Hencedenoting as d = diag(τ1nom/τ , . . . , τ

6nom/τ), the observation

matrix scaled results:

H (Φ) =

d . . . 0...

. . ....

0 . . . d

Φ. (6)

3Depending on location of via-points, the interpolator generates differentset of discrete points, figuring out a Φ6M×Np built with M concatenatedregressors φ6×Np - where Np = Nπ +Nf .

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Fig. 3: Comau-NS16.

(a) Fitness function in (5). (b) Optimum set of viapoints{P ?1 , . . . , P

?6

}.

Fig. 4: Genetic algorithm optimization results. Given 6D via-points the trajectoriesare automatically calculated through the Comau-ORL library replicating the standardComau motion interpolator.

III. RESULTS

Method implementation and experiments are shown usinga 6DOF industrial COMAU NS16 manipulator (see Fig. 3).Such platform is suitably used because of the availability ofthe ORL library (Open Realistic Robot Library) by COMAURobotics. The ORL library is a collection of routines repro-ducing the functionalities of COMAU motion interpolator.The usage of the ORL allows to seamlessly match simulatedand real conditions and remove from the methodology manyof the porting distortions and errors.

A. Identified Excitation Trajectory

The identification of the excitatory trajectory resulted froma plain evolutionary process where the first generations show(Fig. 4a) a fast convergence of the population to a family oftrajectories yielding well-conditioned regressors. Along thesaturation phase of the evolution, the best individuals and thepopulation in general steadily improve their properties, whileretaining consistency. This is mainly due to the selectivepower of a fitness function that includes the scaling of thegeneralized acceleration term (the regressor) on the basis ofnominal axes torques, avoiding major disruptions in terms ofhomogeneity in the interpolated trajectory.

B. Dynamic Parameters Estimation

Optimal via-points {P ?1 , . . . , P ?K} have been loaded intoNS16 interpolator using a FLY type mode with Cartesianspeed equal to 1 m/s, obtaining the excitation trajectory{Q?, Q̇?, Q̈? } of Fig. 4b. τ̃ , q̃ and ˜̇q have been sampledwith a rate fs =500 Hz and filtered with a 8th-orderButterworth filter with fc =30 Hz. ¨̃q has been numericallycomputed. Table I reports the results of the inversion of themeasured system in (3), [24].

C. Model Validation

Validation of the evolved trajectory and consequently esti-mated parameters π̂? is performed by moving robot along 30trajectories, each defined by 6 via-points randomly calculatedwithin the workspace sub-region as criteria described insection I and shown in Fig. 1. Cartesian velocity has been setto 0.1 m/s. Sampled torques measured during each single

TABLE I: COMAU-NS16 robot.

(a) Gear ratios and nominal motor torques.

Joint 1 2 3 4 5 6

Gear ratio −1

182

1

159

1

162

1

73

1

79−

1

50

Nominal torque [N/m] 3 6 3 0.7 0.7 0.7

(b) Inertia parameters (Antonelli [11])

index value index value

1 -4.664 21 0.1472 -30.103 22 -1.5783 148.069 23 2606.7424 -98.005 24 -59.8295 -90.828 25 -220.1996 18.420 26 60.4247 5.033 27 96.2718 18.264 28 0.9849 -13.666 29 -29.219

10 -14.379 30 -184.50411 0.222 31 2371.11112 2.062 32 4.24813 1.796 33 0.61714 -2.534 34 -9.59015 1.653 35 10.29816 0.375 36 0.53417 -0.007 37 -2.69518 -0.874 38 6.64019 0.072 39 2.42220 -0.048 40 -0.452

(c) Friction parameters as in (2)

index ax. value

f0 41 1 -62.10842 2 -0.41243 3 89.93544 4 0.93545 5 45.28146 6 1.286

f1 47 1 2.00348 2 0.73249 3 10.50350 4 0.19651 5 -3.42952 6 -0.128

f2 53 1 -0.00454 2 0.00755 3 -0.00356 4 -0.02457 5 -0.00158 6 0.0002

trajectory test T̃ are then compared to estimated torques T̂figured out of the locally calibrated model π̂? (see bottomlayer in Fig. 2) obtained from the optimal trajectory. TheRMSE between estimated and sampled torques correspondsto the best estimator for the noise distribution in (4) and to ametric of the repeatability of the estimation over random testtrajectories. Due to the inherently different contribution ofeach robot axis to the generated torque, so to the estimationnoise, and the consideration of scaling discussed for theselected fitness function, the RMSE is expressed as:

σ̂i,jν =‖T̃i,j − Φi,j(˜̈q, ˜̇q, q̃)π̂?‖2

S, (7)

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(a) Cartesian Path (position). (b) Distribution of normalized prediction error σ̂ν for each motors as (8).

Fig. 5: Validation: 30 randomly generated trajectories.

where S is the number of samples in each i = 1, . . . , 30trajectory and j ∈ (1, . . . , 6) is the axis number. Thenσ̂i,jν has been normalized out of each corresponding motornominal torque transformed from motor side to joint side(NS16 gearmotors data are reported in Table I):

σ̂i,jν|norm =σ̂i,jντ inom

, (8)

as in Fig. 5b. Such estimator is used also in Fig. 6 to computethe upper and lower bounds for the predicted T̂ from T̂?.

D. Discussion

The use of heuristics in the optimization of the excitatorytrajectory has proven to be useful in the industrial-likecalibration scenario presented in the paper. The selectionof a limited number of via-points, as in common robottask programming, allows a straightforward representationof the task at hand through its major template features,i.e. pose to visit and task space constraints in task execu-tion. While the accuracy is given by the over-determinedlinear system solution in terms of conditioning propertiesof regressors, such figures would eventually work on theoptimization process for the excitatory trajectory in termsof analytical motion planning. Contrarily, an heuristic searchallows mapping an easy task representation directly to theconditioning properties of the trajectory regressor, increasingthe versatility of a task template made of via-points.Furthermore, it is worth noting that the method attains smalltorques prediction error variance, that is, it demonstratesthat modelling the robot with the fully-identifiable dynamicparameters allows a good accuracy. Nevertheless, variabilityof prediction error is higher for last three axes (see Fig. 5b),and this problem should be due to the fact that selectedmovements affect static friction predominating on otherparameters. A different model of static friction should makethe methods more accurate, although a higher-model ordershould make more complex the convergence of the geneticalgorithm, and non linear models are not easy to be integratedin the method suggested. A solution should consists on relaxthe task dependent constraints extending the movements forthese axes, effecting on higher dynamic excitation.

Finally, the method suggested provides the time consum-ing process is performed offline, whereas the parameters

identification consist on a fast procedure, basically, a repeti-tion of a limited movements for about a hundred of cycles.From a practical point of view, unpredicted phenomena,usually representing a major source of model drifting such asfrictions variations due to temperature, number of workingcycle, bearing wear, etc, may be also reconsidered without afull dynamic re-calibration.

IV. CONCLUSIONS

A procedure for local dynamic calibration has been in-troduced. Robot dynamic parameters are identified inside oftask-dependent workspace sub-region. Experimental resultsconfirm high accuracy in estimation of torques at joints,moreover prediction errors display good repeatability inparameters identification. The procedure is also easy toperform, allowing to frequently reload the calibration, inorder to compensate parameters variations due to externalphenomena. Future works will compare the accuracy reachedthrough a local dynamic calibration and calibration definedfor the complete workspace.

ACKNOWLEDGMENTS

This work has been partially funded by ROBOFOOTEuropean project (FP7-2010-NMP-ICT-FoF).

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