2015 15th International Conference on Control, Automation and Systems (ICCAS 2015)Oct. 13–16, 2015 in BEXCO, Busan, Korea

Physical Modelling of Energy Consumption of Industrial Articulated Robots

Akeel Othman ,1 Kvetoslav Belda 2 and Pavel Burget 1∗

1 Department of Control Engineering, Faculty of Electrical Engineering, Czech Technical University in PragueKarlovo nam. 13, 121 35 Prague 2, Czech Republic (pavel.burget@fel.cvut.cz) ∗ Corresponding author

2 Department of Adaptive Systems, Institute of Information Theory and Automation of the CASPod Vodarenskou vezı 4, 182 08 Prague 8, Czech Republic (belda@utia.cas.cz)

Abstract: The paper deals with a modelling of the energy consumption of industrial articulated robots. The proposedmodelling way is based on graphically-oriented computer-aided concept that exploits CAD software Solidworks and simu-lation environment MATLAB/Simulink with SimMechanics and SimPowerSystems libraries. These software tools areused for the composition of a dynamical simulation model that represents both mechanical robot structure and robotdrives during robot motions. The paper addresses mathematical analysis and interpretation of the considered simulationmodel. Here, equations of the motion for the mechanical robot structure and appropriate dynamical equations of the robotdrives are introduced. Using these equations, the energy consumption equation is defined. The proposed way is demon-strated by simulation experiments for several different velocities of the robot motion along a selected trajectory. For the ex-periments, industrial articulated robot KUKA KR 5 arc driven by PMSM drives is considered.

Keywords: Industrial articulated robots, energy consumption, mathematical analysis, computer-aided modelling.

1. INTRODUCTION

The robots are utilized widely in industrial applica-tions due to their flexibility, accuracy and high perfor-mance. They comprise the essential parts of modern pro-duction lines and belong to the parts with the substantialenergy consumption. Nowadays, the energy cost contin-ually rises with correspondence to increasing energy de-mands. In this respect, new methods for measuring, pre-diction and optimization of a robot energy consumptionare required to be developed. There exist several methodsthat determine energy consumption. They are based ei-ther on mathematical analysis, or on direct measurementon real robot. Mathematical approach, e.g. in [1] and [2],follows from a specific expression of an energy consump-tion equation in dependence on given robot trajectory [3].On the other hand, the approach based on real measure-ments focuses on processing and analysing real data suchas in [4], [5] and [6]. It combines specific direct mea-surements of the energy consumption with its computerevaluation.

This paper deals with a description of the fullygraphically-oriented computer-aided modelling and itsmathematical analysis that is used for the determinationof the robot energy consumption. It represents specificmathematical approach. The computer-aided modellingfollows from a pure geometrical 3D CAD model. Thismodel is split by the CAD software Solidworks into par-ticular robot components that are supplemented with ap-propriate physical parameters as volumes, masses andmoments of inertia. It is shown that such componentmodel, which represents physically a mechanical robotstructure, can be converted into a simulation model oper-ated in the Matlab/Simulink environment. This model iscompleted by blocks representing the robot drives.

The significant contribution of the paper consistsin a mathematical analysis and interpretation of com-posed graphically-oriented simulation model of the robot.On the basis of the mathematical analysis, the robot en-ergy consumption function is expressed, which allowsfinding the amount of energy needed for a robot motionalong a given trajectory for an admissible range of veloci-ties. Then, the energy consumption function can illustratethe optimal speed for the given trajectory. The evaluationof the robot consumption is shown for selected point-to-point motion task applied to the robot KUKA KR 5 arcas an example of an industrial articulated robot. The ener-gy function can be used e.g. in a global optimisationof the energy consumption for a robotic cell that con-tains several robots. Such an approach has been presentedin [7] and is subject to further research.

The proposed design demonstrates a full exploita-tion of the advantages of the software tools Solidworksand MATLAB/Simulink with SimMechanics and Sim-PowerSystems libraries, as single environments for mul-tiple physical domains with customisation of physicalcomponents and extensive range of tools for variousanalyses and 3D visualisations. Such a design enablesuser to create efficient models very close to kinematicand dynamic behaviour of real robots.

The paper is organized in the following way. Sec-tion II deals with graphically-oriented computer-aidedmodelling used for the design of physical simula-tion model. Section III outlines mathematical analysisof the physical model including the model of the robotstructure and the model of the robot drives. Section IVis dedicated to the equations of electric input power androbot energy consumption. The illustrative evaluationof the robot energy consumption is presented and demon-strated in Section V by a set of simulation experiments.

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2. PHYSICAL SIMULATION MODEL

This section deals with a design of the physical model.The design arises from a pure geometrical model and itsmodifications within software Solidworks (Subsec-tion 2.1). A conversion of the modified Solidworks modelinto a Simulink model is considered (Subsection 2.2),where the Simulink is taken into account as the targetenvironment for the simulation. The obtained physicalmodel represents the robot structure (part 2.2.1) to whicha suitable model of the robot drives (part 2.2.2) is con-nected. In this way, the complete simulation model iscomposed.

2.1 Physical 3D CAD Model in Solidworks

Let pure geometrical model be considered, see e.g. [8].Such models usually represent the overall robot surfaceonly, i.e. the model does not consist of individual physi-cal robot elements with appropriate physical parameters.Thus, for physical parameter inclusion, the initial modelhas to be split into individual robot elements. This op-eration is realized by Solidworks software. By specifi-cation of the robot element density, as only one neces-sary parameter, other physical parameters (masses, mo-ments of inertia and centers of gravity) are computed au-tomatically by Solidworks. To prepare proper kinematicrelation for a simulation model, the individual elementsof the physical model are connected to particular coordi-nate systems for articulated robots to Denavit and Harten-berg frames (DH-frames) [9] as it is shown in Fig. 1.

The physical robot model is synthesized usingthe kinematic constraints among individual robot ele-ments. The kinematic constraints define the motion be-tween two neighbourhood elements in correspondenceto the appropriate admissible number of degrees of free-dom (DOF). In case of articulated robots, there is alwaysone DOF for rotation in each robot joint. The individ-ual robot elements are assembled in Solidworks by def-initions of their constraints via mates as it is describedin [10]. Each mate applies a geometric relationship be-tween mate entities on each robot element (robot joint,link, flange etc.).

One DOF in rotation is obtained by applyingSolidworks-mates as follows. First, one concentric matebetween the cylindrical joint hinge of the two robot ele-ments is applied. Then, the joint DOF are reduced fromsix to two DOF – translation and rotation along the ro-tation axis. Second, an appropriate coincident mate be-tween two planes normal to the cylindrical axis is used.Thus, the joint DOF are finally reduced to only one DOF– rotation around the hinge axis of the joint [10].

Applying this procedure for all robot elements, the fi-nal physical model, consisted of just all individual robotelements, is ready for a conversion into the target simula-tion model under MATLAB/Simulink environment.

Fig. 1 CAD model of articulated robot in Solidworks.

2.2 Simulation Model in MATLAB/SimulinkThe simulation model consists of a model of the robot

structure and robot drives. The former component is gen-erated automatically from Solidworks [11] and later com-ponent (drives) is constructed directly in Simulink envi-ronment [12]. The following two subsection parts showthe arrangement of these two components. The completemodel will be shown in third part.

2.2.1 Modelling of Robot StructureThe initial Simulink model of the robot structure

is generated from Solidworks according to procedure de-scribed in [10]. The obtained model Fig. 2 consistsof standard blocks of SimMechanics library [11].

The meaning of the blocks in Fig. 2 is the follow-ing. (A) is a World frame block. It represents the globalreference frame (basic frame), therefore other framesare defined with respect to this frame. (B) is a Mech-anism configuration block of general parameters usedin the simulation e.g. gravity acceleration and sam-pling. (C) is a Solver configuration block defining sim-ulation property information. (D) is a Rigid transformblock representing a transformation matrix that allowsa follower (following mechanical robot element) to ro-tate or to shift (or both motions together) with respectto the basic frame. (E) is a Link block, which representsthe rigid body with its DH frame and appropriate infor-mation about body mass, moment of inertia in respect toits center of gravity. (F) is the Revolute joint block withone DOF. The information about its angle, angular ve-locity, angular acceleration and actuating torque are ob-tained from a built-in joint sensor. (G) is the next Linkconnected to the next revolute robot joint. (H) is a Pathgenerator. It feeds the trajectory coordinates to each jointthrough block (I) called Simulink-PS Converter that con-verts the unit-less signal inputs into physical signals withappropriate physical units. On the contrary block (J),called PS-Simulink Converter, converts physical signalsinto Simulink value signals. (K) is a Scope block thatserves for drawing the signals of kinematical quantities.

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Fig. 2 Block diagram of robot structure in Simulink.

Fig. 3 Block diagram of robot drives in Simulink.

The simulation model in Fig. 2 is done generally.However, it requires to specify proper physical param-eters of a robot motion. They are inserted to blocks B,F and H. While blocks B and F need only modificationof parameters (gravity vector etc.), block H requires a de-tailed user specification defining the robot motion, whichcan be done in different ways that depend on the order ofthe acceleration polynomial. In our case we choose theacceleration polynomial of zero order i.e. constant ac-celeration and deceleration leading to the fastest motion[13], which is suitable for the purpose of this paper.

2.2.2 Modelling of Robot DrivesLet the Permanent Magnet Synchronous Motor

(PMSM) drives be considered as widely used robot ac-tuators. An appropriate PMSM model block from Sim-PowerSystems library of MATLAB/Simulink is shownin Fig. 3, whereas the PMSM block consists of six mainblocks as shown in Fig. 4. These blocks form togetherone PMSM drive system, which has three inputs (onethree-ABC-phase power source, load torque and speedset-point) and four outputs (stator current, rotor speed,electromagnetic torque and DC Bus voltage). For a de-termination of the energy consumption of PMSM motor,the electromagnetic torque is crucial. It is in the propor-tion with the input energy needed for the robot motionalong a desired trajectory.

2.2.3 Complete Simulation ModelAs mentioned, the complete simulation model of the

robot consists of the model of a robot structure and themodel of the appropriate robot drives. It determines nec-essary torques for the required robot motion. The modelcan be used for the determination of the energy con-sumption. The consumption is computed by additionalSimulink block as indicated in Fig. 5.

Fig. 4 Conceptual control diagram of PMSM drive.

Fig. 5 Complete simulation model of the robot.

3. MATHEMATICAL ANALYSISOF SIMULATION MODEL

In this section, the mathematical analysis of the simu-lation model Fig. 5 is described with considering 6-DOFrobot structure in Fig. 1. Since it consists of both mechan-ical robot structure and drives, the mathematical descrip-tion has to address pure mechanical equations of motionand also dynamical electro-mechanic equations.

3.1 Mathematical Model of Robot StructureTo obtain mathematical model of the mechanical

structure, the kinematic and dynamic relations of givenrobot have to be considered. The following subsec-tion parts make appropriate overview and show specifickey-points of the mathematical description leading to theequations of motion.

3.1.1 Description of the Robot KinematicsLet two main descriptive coordinate systems be con-

sidered. One, usually Cartesian coordinate system, de-scribes the motion of robot end-effector in natural Carte-sian coordinates. The second system is connected withthe description of the robot motion from the drive pointof view. It uses joint coordinates. In our case, for pur-pose of the evaluation of energy consumption, the initialcoordinates are the joint coordinates that are also the co-ordinates of equations of motion. The transformationsamong individual joint coordinates system (DH framesas in Fig. 1) are necessary for construction of these equa-tions. The transformations describing kinematic relationsare defined as a specific matrix according to DH approach

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as follows

T ii−1=

cosϑi −cosαi sinϑi sinαi sinϑi ai cosϑisinϑi cosαi cosϑi −sinαi cosϑi ai sinϑi0 sinαi cosαi bi0 0 0 1

(1)

where ai, di, αi and ϑi are DH parameters and this trans-formation matrix represents transformation between twoneighborhood DH frames (between two separate rigidbodies). For the whole coordinate transformation frombasic frame (i = 0) to end-effector (i = n, where n isa number of rigid bodies), the individual coordinates areconnected by the following chain equation

Tf = T n0 = T 1

0 T 21 · · · T n

n−1 =n∏

i=1

T ii−1 (2)

Individual transformations in Eq. (2) and their meaning-ful combinations are involved in the composition of equa-tions of motion. This DH approach represents a suitableway, which allows simple and systematic compositionof the mentioned equation.

3.1.2 Description of the Robot Dynamics

Since the articulated type of robots is considered,a suitable way for the composition of equations of mo-tion is the Lagrange approach based on Lagrange equa-tions of second type, defined as

d

dt

(∂L∂q

)T

−(∂L∂q

)T

= ξ (3)

whereL is Lagrangian, q is generalized coordinates and ξis generalised force effects. Lagrangian represents differ-ence of kinematic Ek and potential energy Ep as follows

L(q, q) = Ek(q, q)− Ep(q) (4)

Considering Eq. (4), Lagrange equations can be ex-pressed in the following modified form

d

dt

(∂Ek

∂q

)T

−(∂Ek

∂q

)T

+

(∂Ep

∂q

)T

= ξ (5)

By application of Eq. (5) and appropriate physical param-eters of given articulated robot, equations of motion canbe composed. For given articulated robots, generalizedcoordinates are identical with joint coordinates q = ϑand generalized force effects correspond to input torquesof appropriate joints ξ = τ , then the equations of motionare

B(ϑ) ϑ+ C(ϑ, ϑ) ϑ+ g(ϑ) = τ (6)

where B(ϑ) is an inertia matrix, C(ϑ, ϑ) is a coefficientmatrix of Coriolis and centrifugal force effects, g(ϑ) isa vector of gravitational effects, ϑ = [ϑ1 ϑ2 · · · ϑn ]Tis a vector of joint angles and τ = [ τ1 τ2 · · · τn ]T isa vector of torques acting on appropriate joints [9].

3.2 Mathematical Model of Robot Drives

In this paper, the PMSM drives are considered in re-spect to given robot used for energy consumption eval-uation. The description of i-th motor arises from thevoltage distribution in the individual phases of three ACphase system and from equilibrium equation. Consid-ering Clarke and Park transformation, the initial set ofequations of voltage distribution defined in d-q rotatingfield coordinate system (rotating reference frame) can beexpressed as follows

uSdi=RSi

iSdi+ Ldi

d

dtiSdi− Lqi

ωeiiSqi

(7)

uSqi=RSi

iSqi+ Lqi

d

dtiSqi

+ Ldiωei

iSdi+ ψMi

ωei(8)

where RSi, Ldi

, Lqiand ψMi

are motor parameters,uSdi

, uSqiare d-q voltages (system inputs), iSdi

, iSqiare d-q currents, ωei

is the electrical rotor speed.Then, the equilibrium equation among electromag-

netic torque and (mechanical) load torques is defined as

τei = τi +Bi ωmi+ Ji

d

dtωmi

(9)

where Ji and Bi are other motor parameters; ωmiis

mechanical rotor speed related to electrical speed asωmi

= ωei/p considering p for a number of pole pairs;

τi is a one load torque component of the torque vector τfrom Eq. (6); and τei is electromagnetic torque definedas follows

τei =3

2p (ψMi

iSqi+ (Ldi

− Lqi) iSdi

iSqi) (10)

A dynamic model of i-th PMSM, representing appro-priate simulation block, can be formulated by the follow-ing first-order differential equations:

d

dtiSdi

=1

Ldi

(uSdi−RSi

iSdi+ Lqi

p ωmiiSqi

) (11)

d

dtiSqi

=1

Lqi

(uSqi−RSi

iSqi− Ldi

p ωmiiSdi

−ψMip ωmi

) (12)d

dtωmi

=1

Ji(τei − τi −B ωmi

) (13)

Considering appropriate components of torques τifrom the equations of motion in Eq. (6) the modelin Eqs. (11)∼ (13), formally identical for all robot drivesbut different appropriate physical parameters, representscomplete dynamical description of the robot i.e. mathe-matical interpretation of the complete simulation modelin Simulink environment shown in Fig. 5.

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4. ROBOT ENERGY CONSUMPTIONThe energy consumption is an important parameter for

each electrical appliance like robots, because it showstheir efficiency in respect to all energy inputs. For robots,its determination arises from the input power of the in-dividual drives that are loaded by the mechanical robotstructure and possibly other further outside loads. In thispaper, the unloaded case is considered so the drive loadis represented only by mechanical robot structure.

4.1 Equation of Electric Input PowerThe electric input power for a set of n 3-phase PMSM

drives as a balanced system in considered d-q rotatingreference frame is defined as

P =3

2

n∑i=1

(uSdiiSdi

+ uSqiiSqi

) (14)

Considering Eqs. (7), (8) and (10), the electric inputpower is expressed as

P =3

2

n∑i=1

[RS (i2di + i2qi) + idi Ldd

dtidi + iqi Lq

d

dtiqi ]

+n∑

i=1

ωriτei (15)

The equation of input power can be simplified by the as-sumption iSd = 0 due to no need of field-weakening op-tion and by omitting negligible term d/dt(iSq)

P =3

2

n∑i=1

Ri2qi +

n∑i=1

ωmiτei (16)

Considering Eq. (9), the complete equation for the inputpower is

P =3

2

n∑i=1

Ri2qi+n∑

i=1

ωmi(τi+Biωmi

+Jid

dtωmi

) (17)

Now it is possible to specify energy consumption.

4.2 Equation of Robot Energy ConsumptionThe equation of the robot energy consumption is com-

puted as a time integration of total input power along theconsidered motion trajectory within corresponding timeinterval as it is indicated by the following equation:

E =

∫ Γ

0

[3

2

n∑i=1

Ri2qi

+n∑

i=1

ωmi(τi +Bi ωmi

+ Jid

dtωmi

)

]dt

(18)

By evaluating energy consumption according to Eq. (18)for different admissible kinematic parameters but alongonly one motion trajectory, it is possible to construct en-ergy consumption curve that can be use for the selectionof the optimal kinematic parameters for the consideredtrajectory. It will be shown in the next section.

Fig. 6 Typical kinematical profiles of the robot motion.

5. RATE OF ENERGY CONSUMPTIONIn this section, the evaluation of the energy consump-

tion is demonstrated on the KUKA KR 5 arc robot (see[14] and Fig. 1). The energy consumption for this robotis evaluated for point-to-point motion task, i.e. only ini-tial and final robot position has to be kept. The trajectoryfor this task is simply prepared by considering usual kine-matic relations between individual quantities, i.e. angularposition, speed and acceleration – see Fig. 6. Accordingto this figure, the used representative trajectory is basedon zero order profile of acceleration.

Shown profiles represent pattern for all six joint anglesof the considered robot for the selected trajectory timeduration. Individual joints have modified profiles accord-ing to selected time duration to be synchronized to eachother. The details on trajectory planing are in [13].

Fig. 7 Example of time behaviours of currents iSqi.

Fig. 8 Example of time behaviours of torques τei .

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Fig. 9 Energy consumption relative to motion duration.

For the specified trajectory, individual time behaviourof current iSqi

and electromagnetic torques τei aredemonstrated in figures 7 and 8, respectively. These fig-ures show examples of specified quantities of one testingtrajectory realization for time duration equal to 2 s. Thisduration is selected from considered set of reasonable du-rations = [1.3s, 1.4s, 1.5s, 1.6s, 1.7s, 2s]. Consideringthese parameters for all duration realizations, the result-ing individual energy consumptions of the robot motionare shown in Fig. 9.

The curve in this figure is an approximation of the val-ues of energy consumptions corresponding to the men-tioned set of reasonable duration times. The appropriatevalues are listed in Table 1. The curve illustrates the en-ergy consumption profile that can serve for the selectionof optimal time for the given trajectory. In the case of ourexample, the optimal time duration is 1.5s. More detailsare provided in [7].

The result shows that the shortest time requires for therobot acceleration and braking extra energy for copingwith inertia force effects and on the opposite side for longtime durations, the energy consumption increases due todemands on more or less static support of the mechani-cal robot structure during the motion. Between these twocases, there is short interval, which represents balancedoptimum. The optimum is characterized by suitable ac-celeration and deceleration profiles not to excite increaseof inertia effects too much and at the same time, the du-ration time is not too long to require support force inputsof robot structure.

Table 1 Energy Consumption for Trajectory Realizations

Point Trajectory Robot velocity ConsumptionNo. duration [s] [m s−1] [J]1 1.3 0.11430 37512 1.4 0.09648 35643 1.5 0.09434 30524 1.6 0.09045 34295 1.7 0.08890 41436 2.0 0.06300 7167

6. CONCLUSIONThe paper presents the application of specific

graphically-oriented computer-aided modelling approachto model the energy consumption of industrial articu-lated robots. The construction of the complete simulationmodel has been described. Simultaneously, the compo-sition of the corresponding mathematical equations de-scribing the simulation model has been introduced. Ontheir basis, the equation of the energy consumption hasbeen derived. The proposed approach of the determina-tion of the energy consumption has been illustrated bythe set of simulation experiments that demonstrate evalu-ation of the energy consumption for one selected motiontrajectory for several different time durations. The exper-iments have been realized with real physical parametersof industrial robot KUKA KR 5 arc as the representativeof industrial articulated robots.

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