research article payload mass identification of a single

Research ArticlePayload Mass Identification of a Single-Link Flexible ArmMoving under Gravity: An Algebraic Identification Approach

Juan Carlos Cambera, Andres San-Millan, and Vicente Feliu-Batlle

Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain

Correspondence should be addressed to Vicente Feliu-Batlle; [email protected]

Received 8 December 2014; Accepted 27 May 2015

Academic Editor: Reza Jazar

Copyright © 2015 Juan Carlos Cambera et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We deal with the online identification of the payload mass carried by a single-link flexible arm that moves on a vertical plane andtherefore is affected by the gravity force. Specifically, we follow a frequency domain design methodology to develop an algebraicidentifier. This identifier is capable of achieving robust and efficient mass estimates even in the presence of sensor noise. In orderto highlight its performance, the proposed estimator is experimentally tested and compared with other classical methods in severalsituations that resemble the most typical operation of a manipulator.

1. Introduction

Flexible link robotics is a research field focused on buildingrobots with better performance than the conventional robots.The flexibility of these robots is a consequence of usinglinks with lower sectional area and lighter materials than itsrigid counterpart. Higher operational speed, lower energyconsumption, better transportability, and lower cost are onlya few advantages of the flexible link robots over the traditionalrigidmanipulators.These advantages can only be obtained byfacing very challenging problems on modelling and controlthat after four decades have not been completely solved.

The flexible link robots are systems characterised by non-linear ordinary, coupled, and partial differential equations,whose exact solution is not viable practically. This had led tolook for models with manageable complexity but still reliableand useful for the design of controllers. From the controlperspective, the same characteristics that improve the per-formance of these robots have led to vibration problems thatundermine the positioning of the end effector.The solution tothese problems can be very difficult considering the complex-ity of the model and, in the most general case, its nonmini-mum phase nature. Surveys dealing with dynamic modellingand control of flexible link robots can be found in [1–3].

In flexible link robotics, to guarantee an accurate posi-tioning in pick and place tasks is a very important problemto solve. One of the main obstacles to overcome is to designa control algorithm capable of cancelling the vibrationswhen the dynamics is affected by changes in the payloadmass. When these changes are not considered in the controldesign, the algorithm may lose accuracy and effectiveness inthe vibration suppression and, in some cases, may becomeunstable.

Several works have addressed the problem from the adap-tive control point of view. Most of them rely on the indirectmethods category, which consists of two clearly differentiatedstages (there are some early research works that use a directapproach; Siciliano et al. [4] and Yuh [5] applied the ModelReference Adaptive Control (MRAC)). In the first stage, anonline identification of the system parameters is needed. Inthe second one, the parameters identified in the first stageare used to adjust the adaptive control law, in such a way thatthe overall performance of the system is improved.This paperis devoted to the real time characterization of the parameterthat is most likely to change in a robot: the payload. In par-ticular, we want to identify the tip mass, as we assume thatthe payload polar moment of inertia is negligible. Once thisparameter has been identified, to update the dynamic model

Hindawi Publishing CorporationShock and VibrationVolume 2015, Article ID 294157, 13 pageshttp://dx.doi.org/10.1155/2015/294157

2 Shock and Vibration

of the arm is immediate, and to recompute the controllerparameters is straightforward.

A payload change affects in two ways a flexible linkrobot, it changes the vibration frequencies of the links, and itchanges themotor torques demanded for a specificmaneuver.Hence the identification algorithms can be classed, on asimilar way, into frequency based approaches and modelbased approaches.

The frequency based approaches, normally, do notdepend explicitly on the robot’s model but on the outputwhere the vibrations appear and in some cases on the input.The more classical approaches are normally based on theFFT [6]. The adaptive notch filter [7] is one of the preferredmethods because of its fast convergence rates and its low com-putational burden. Other approaches, like [8], have consid-ered adaptive observers to perform simultaneously frequencyand states estimation. A more recent work uses algebraicidentification to estimate amplitude, frequency, and phase ofa sinusoidal signal in the presence of noise andDC-offsets [9],which are two common problems not explicitly considered inthe methods aforementioned.

In the second category, which we referred to as modelbased approaches, the payload mass is identified by using thedynamicmodel of the robot and the input and output signals.Least square based techniques, like [10–12], cover most of thework carried out under this category, but there are some otheralternatives based on algebraic manipulations of the modeltransfer functions [13], Kalman filtering [14], or the alreadymentioned algebraic identification technique [15] that are alsoworth mentioning.

From the vibration control point of view, the frequencybased approaches are at a disadvantage. The frequency basedapproaches require the system to vibrate at least a cyclefraction before the identification can be carried out. Thiscondition goes against themain goal of the control algorithm,the vibration suppression, where a very fast identificationis required. In order to update the controller as soon aspossible during the trajectory execution, on the other hand,the model based approaches have proved to be highly reliablein problems concerning dynamic linear systems, but itsapplicability to nonlinear systems is not straightforward andmay imply in some cases numerical differentiation of noisysignals.

In this paper, we will focus on the problem of real timeidentification of the tip mass of a single-link flexible arm thatmoves in a vertical plane under the effects of the gravity. Thealgorithm follows a model based approach and it is basedon the algebraic identification framework, proposed in [16],and it generalizes a previous research work presented in [15].Unlike the previous research work, where onlymovements inthe horizontal plane were considered and then a linear modelwas used, in this paper we deal with a nonlinear dynamicmodel as a consequence of taking the gravity into account.This leads to a more complex problem, where most of the realtime identification techniques developed up to date cannot beapplied. Preliminary results of this identification algorithmwere presented in [17]. This paper details our new identifica-tion algorithm, proposes several improvements, and presentsa comparative analysis with other identification methods.

Y

X

Flexible link

Payload

Motor

g

l, �, c

Γ 𝜃m

𝜃t

Figure 1: Robotic system scheme.

The algebraic identification provides a very fast and sim-ple solution for online parameter estimation in systemswherethe parameters are piecewise constant (change from oneconstant value to another unpredictably). This methodologyis fundamentally different from other approaches in somebasics aspects: (a) it does not require any statistical knowl-edge of the noise corrupting the signals; therefore, classicalGaussian noise assumptions are not necessary; (b) it does notneed to compute iterative time derivatives of noise corruptedsignals; (c) it is not an asymptotic approach; and (d) it doesnot require persistently exciting inputs in order to make thesystem identifiable [18].

This paper is organised as follows. Section 2 presents thedynamic model of the flexible-link robot. In Section 3, thedesign of the algebraic identification algorithm is presented.Section 4 is devoted to the analysis and comparison of theexperimental results. Conclusions and future work are pre-sented in Section 5.

2. Dynamic Model

Figure 1 shows a schematic representation of the flexiblelink robot. It consists of a motor and a flexible beam thatbends on the vertical plane and therefore is affected by thegravitational force. One end of the beam is clamped to theshaft of the motor, while the other end moves freely andcarries a payload.Themodel we introduce in this section wasprepared according to the lumped mass method presented in[19] and relies on the following assumptions.

(i) The link mass is negligible in comparison to the tipmass.

(ii) The payload is considered as a point mass; therefore,its polar moment of inertia can be neglected.

(iii) The deflections are elastic and small in relation tothe link’s length, so that geometrical linearity can beassumed.

(iv) Torsion and compression effects of the link are smallin relation to the deflections.

Shock and Vibration 3

The dynamic model relates the coupling torque betweenthe motor and the link (Γ) to the angular motor position (𝜃

𝑚)

and the angular tip position (𝜃𝑡). The dynamic model of the

system is given by

Γ (𝑡) = 𝑚𝑙2 𝜃𝑡(𝑡) + ] 𝜃

𝑡(𝑡) +𝑚𝑔𝑙cos (𝜃

𝑡(𝑡)) , (1)

Γ (𝑡) = 𝑐 (𝜃𝑚(𝑡) − 𝜃

𝑡(𝑡)) , (2)

where 𝑚 is the payload mass, 𝑙 is the link’s length, ] isthe viscous friction damping coefficient, and 𝑐 is a stiffnessconstant that can be defined in function of Young’s modulus(𝐸) and the cross-sectionalmoment of inertia (𝐼) as 𝑐 = 3𝐸𝐼/𝑙.This last expression is a result of applying the lumped massmethodology developed in [19] to a single-link flexible robotthat considers the assumptions presented above.

For the algebraic estimation algorithm that we develop inthis paper, the torque measured at the base of the link, theangular motor position, and the angular tip position need tobe known.The sensory system of our robot provides the angleof the motor (𝜃

𝑚) and the coupling torque (Γ) at the base of

the link. The angular tip position (𝜃𝑡) is not measured but it

can be estimated as follows:

𝜃𝑒

𝑡(𝑡) = 𝜃

𝑚(𝑡) −

Γ (𝑡)

𝑐. (3)

Here it is important to notice that the estimation of tipposition, 𝜃𝑒

𝑡, is completely independent from the payload

mass𝑚, whose value is in principle unknown.

3. Algebraic Identification Algorithm

In this section the algebraic methodology, presented in [16],is used to design an online algorithm to identify the payloadmass of a flexible link robot moving under gravity. The basicmethodology works from the premise that the parametersare constant throughout time, but it can be extended toidentification problems where the parameters are piecewiseconstant; that is, payload changes in a pick and place task.For the sake of clarity, the designmethodology is explained inthree stages. The design procedure considers (1) and (3). Thepresence of the cosine function in (1) determines its nonlinearcharacteristic in the tip position variable (𝜃

𝑡). From the iden-

tification point of view, however, and considering that the tipposition can be estimated from (3), this nonlinear equationcan be considered linearly identifiable in its parameters𝑚 and]. Taking this into consideration the designmethodology is asfollows.

3.1. First Stage: Algebraic Reformulation of the Problem. Per-forming an online estimation of the parameters in (1) implies,for some identification techniques, numerical differentiationsof a noise corrupted signal 𝜃𝑒

𝑡. As it is well known, these

operations amplify the noise in the resulting signals and affectadversely the performance of the identification algorithms.The algebraic identification framework provides a method-ology to avoid this issue. In this section, we formulate a newmathematical relationship between the measured signals andthe parameters of themodel by applying standard operational

calculus and algebraic operations.This new expression, com-pared with the original model, is purposefully formulated tobe independent from time derivatives and initial conditions.The procedure is as follows.

First, let us apply the Laplace transform to (1):

Γ (𝑠) = 𝑚𝑙2[𝑠

2𝜃𝑡(𝑠) − 𝑠𝜃

𝑡(𝑡0) −

𝜃𝑡(𝑡0)]

+ ] [𝑠𝜃𝑡(𝑠) − 𝜃

𝑡(𝑡0)] +𝑚𝑔𝑙𝜒 (𝑠) ,

(4)

where 𝜃𝑡(𝑡0) and 𝜃

𝑡(𝑡0) represent unknown initial conditions

and 𝜒(𝑠) is the Laplace transform of the signal cos(𝜃𝑡(𝑡)).

Taking two times derivatives with respect to 𝑠, the initialconditions are eliminated:

𝑑2Γ (𝑠)

𝑑𝑠2= 𝑚𝑙

2[2𝜃𝑡(𝑠) + 4𝑠

𝑑𝜃𝑡(𝑠)

𝑑𝑠+ 𝑠

2 𝑑2𝜃𝑡(𝑠)

𝑑𝑠2]

+ ][2𝑑𝜃𝑡(𝑠)

𝑑𝑠+ 𝑠

𝑑2𝜃𝑡(𝑠)

𝑑𝑠2]+𝑚𝑔𝑙

𝑑2𝜒 (𝑠)

𝑑𝑠2.

(5)

In order to eliminate the derivatives in the time domain(positive power of 𝑠), we multiple both sides by 𝑠−2 to obtain

𝑠−2 𝑑

2Γ (𝑠)

𝑑𝑠2= 𝑚𝑙

2[2𝑠−2𝜃

𝑡(𝑠) + 4𝑠−1

𝑑𝜃𝑡(𝑠)

𝑑𝑠+𝑑2𝜃𝑡(𝑠)

𝑑𝑠2]

+ ][2𝑠−2𝑑𝜃𝑡(𝑠)

𝑑𝑠+ 𝑠−1 𝑑

2𝜃𝑡(𝑠)

𝑑𝑠2]

+𝑚𝑔𝑙𝑠−2 𝑑

2𝜒 (𝑠)

𝑑𝑠2.

(6)

Equation (6) can be written in the time domain byapplying the inverse Laplace transform (there are fewproperties that are specially useful for this problem; if wedenote the inverse Laplace transform by L−1, then it is wellknown that L−1𝑠(⋅) = 𝑑/𝑑𝑡(⋅), L−11/𝑠(⋅) = ∫

𝑡

0 (⋅)(𝜎)𝑑𝜎, andL−1𝑑V/𝑑𝑠V(⋅) = (−1)V𝑡V(⋅)). The resulting equation is givenby (we denote by (∫

(𝑗)

𝜙(𝑡)) the integral expression ∫𝑡

0 ∫𝜎1

0 ⋅ ⋅ ⋅

∫𝜎𝑗−1

0 𝜙(𝜎𝑗)𝑑𝜎𝑗⋅ ⋅ ⋅ 𝑑𝜎1 with the definition (∫ 𝜙(𝑡)) =

∫𝑡

0 𝜙(𝜎1)𝑑𝜎1)

∫

(2)𝑡2Γ (𝑡) = 𝑚𝑙

2[2∫(2)

𝜃𝑡(𝑡) − 4∫ 𝑡𝜃

𝑡(𝑡) + 𝑡

2𝜃𝑡(𝑡)]

+ ][−2∫(2)

𝑡𝜃𝑡(𝑡) +∫ 𝑡

2𝜃𝑡(𝑡)]

+𝑚𝑔𝑙 ∫

(2)𝑡2 cos (𝜃

𝑡(𝑡))

(7)

and in a more compact form as

𝑞 (𝑡) = 𝑚 [𝑙2𝛽 (𝑡) + 𝑔𝑙𝜉 (𝑡)] + ]𝜂 (𝑡) , (8)


where

𝑞 (𝑡) = ∫

(2)𝑡2Γ (𝑡) ,

𝛽 (𝑡) = 2∫(2)

𝜃𝑡(𝑡) − 4∫ 𝑡𝜃

𝑡(𝑡) + 𝑡

2𝜃𝑡(𝑡) ,

𝜉 (𝑡) = ∫

(2)𝑡2 cos (𝜃

𝑡(𝑡)) ,

𝜂 (𝑡) = − 2∫(2)

𝑡𝜃𝑡(𝑡) +∫ 𝑡

2𝜃𝑡(𝑡) .

(9)

Functions 𝑞(𝑡), 𝛽(𝑡), 𝜉(𝑡), and 𝜂(𝑡) can be expressed as theoutputs of the following time-varying linear and unstablefilters in perturbed Brunovsky’s canonical form:

𝑞 (𝑡) = 𝑧1,

��1 = 𝑧2,

��2 = 𝑡2Γ (𝑡) ,

𝛽 (𝑡) = 𝑧3 + 𝑡2𝜃𝑡(𝑡) ,

��3 = 𝑧4 − 4𝑡𝜃𝑡(𝑡) ,

��4 = 2𝜃t (𝑡) ,

𝜉 (𝑡) = 𝑧5,

��5 = 𝑧6,

��6 = 𝑡2 cos (𝜃

𝑡(𝑡)) ,

𝜂 (𝑡) = 𝑧7,

��7 = 𝑧8 + 𝑡2𝜃𝑡(𝑡) ,

��8 = − 2𝑡𝜃𝑡(𝑡) .

(10)

The implementation of this algorithm requires as inputsthe coupling torque measurements (Γ) and the estimationof the tip position (𝜃𝑒

𝑡), presented in (3). Notice that even

though the original expression (1) and the resulting equation(8) have the same structure, the latter does not depend ontime numerical differentiations.

3.2. Second Stage: Parameter Calculation. To identify theparameters of a system we need at least the same numberof linearly independent equations as unknown parameter.The algebraic identification exploits the advantages of theresulting expression for the algebraic reformulation of theproblem, expression (8) in our case, to define these inde-pendent equations. In this context, several approaches havebeen proposed, not only to define these equations, but alsoto enhance the performance of the identifier in the presenceof noise in the measurements. In the following paragraphs,first, we briefly describe the most relevant approaches, and

then we detail the procedure followed in this paper. Theapproaches we mention are not necessarily referring to ourspecific application, but they could be easily adapted to ourproblem.

A first approach is to generate as many linearly indepen-dent equations as parameters have to be estimated by succes-sive differentiation of (8), which was obtained after algebraicmanipulations. This approach was developed in [20] inthe context of multiple harmonic signals identification. Notethat (8) is a single equation which involves two parameters tobe estimated. Then we need an extra equation for the iden-tification. Once the resulting system of equations is solved,and in those cases where high frequency noise is present inthe measurements, a low-pass filtering has to be carried outin order to improve the signal to noise ratio of the parameterestimates. Depending on the nature of the measurementsconsidered, the equations defined in this approach mightnot be completely linearly independent all the time. Whenthis happens, a local loss of identifiability may occur. In[21], this drawback was solved by using invariant nonlinearfiltering.

Other approaches to define this system of equations are touse the equation resulting from the algebraic reformulation ofproblem, (8) in our case, and to evaluate it at different times.Under this approach, an overdetermined system of equationsis defined by taking a large number of points in time. Thisaction allows us to achieve the two goals mentioned above atthe same time: to identify the parameters and to reduce theeffects of the noise present in the measurements. The biggerthe number of evaluation instants is, the greater the low-pass filtering effect is. To solve this overdetermined system ofequations a least squares fitting based method is normallyconsidered. In [22], the least squares method in connectionwith algebraic identification was proposed to identify theparameters for induction motors. In order to increase thecomputational efficiency of this stage, in [23], a continuousleast squares approximation was implemented in a recursiveway to identify the two main vibration modes of a flexiblestructure. In [24] the adjustment of the overdetermined sys-tem was carried out by using a recursive least squares algo-rithm. This last identifier was designed to determine theparameters of a servo model. In the present paper we usethe continuous least square approximation with a recursiveimplementation of the involved integrals. A detailed descrip-tion is presented below.

The goal of the following steps is to determine theunknown parameters 𝑚 and ] from (8). However, at the endof this section we will only center our attention on the valueof the parameter 𝑚, while the value of the parameter ]will be discarded. The parameter ], associated with theviscous friction, was introduced in themodel only to improvethe behaviour of the estimator. According to our obser-vations, an approximate model of the damping enhancesthe performance in terms of precision. Having made thisclarification, we define the following cost function:

𝜀 = ∫

𝑡

0{[𝑙

2𝛽 (𝜏) + 𝑔𝑙𝜉 (𝑡) 𝜂 (𝜏)] ⋅ [

𝑚

]]− 𝑞 (𝜏)}

2

𝑑𝜏, (11)


and its minimization leads to

[

𝑚

]] = [

[

∫

𝑡

0[𝑙2𝛽 (𝜏) + 𝑔𝑙𝜉 (𝜏)

𝜂 (𝜏)

]

⋅ [𝑙2𝛽 (𝜏) + 𝑔𝑙𝜉 (𝜏)

𝜂 (𝜏)

]

𝑇

𝑑𝜏]

]

−1

⋅ ∫

𝑡

0[𝑙2𝛽 (𝜏) + 𝑔𝑙𝜉 (𝜏)

𝜂 (𝜏)

] 𝑞 (𝜏) 𝑑𝜏.

(12)

This calculation can be efficiently implemented in a recur-sive way as follows. Let us rewrite (12) as

[

𝑚

]] = 𝐴

−1(𝑡) 𝐵 (𝑡) = [

𝑎11 (𝑡) 𝑎12 (𝑡)

𝑎21 (𝑡) 𝑎22 (𝑡)]

−1

[

𝑏1 (𝑡)

𝑏2 (𝑡)] , (13)

where

𝑎11 (𝑡) = ∫

𝑡

0(𝑙

2𝛽 (𝜏) + 𝑔𝑙𝜉 (𝜏))

2𝑑 (𝜏) ,

𝑎12 (𝑡) , 𝑎21 (𝑡) = ∫

𝑡

0(𝑙

2𝛽 (𝜏) + 𝑔𝑙𝜉 (𝜏)) (𝜂 (𝜏)) 𝑑 (𝜏) ,

𝑎22 (𝑡) = ∫

𝑡

0(𝜂 (𝜏)

2) 𝑑 (𝜏) ,

𝑏1 (𝑡) = ∫

𝑡

0(𝑙

2𝛽 (𝜏) + 𝑔𝑙𝜉 (𝜏)) 𝑞 (𝜏) 𝑑 (𝜏) ,

𝑏2 (𝑡) = ∫

𝑡

0𝜂 (𝜏) 𝑞 (𝜏) 𝑑 (𝜏) .

(14)

Assume that functions 𝛽(𝑡), 𝜉(𝑡), 𝜂(𝑡), and 𝑞(𝑡) aresampled at discrete times 𝑡 = 𝑘𝑇

𝑠, 𝑘 = 1, 2, 3, . . ., where 𝑇

𝑠

is the sampling time. If we define the function 𝜙𝑇

[𝑘] =

[(𝑙2𝛽[𝑘] + 𝑔𝑙𝜉[𝑘]) 𝜂[𝑘]], the matrices 𝐴(𝑡) and 𝐵(𝑡) can be

computed recursively in discrete time as follows:

𝐴 [𝑘] = 𝐴 [𝑘 − 1] + 𝜙 [𝑘] 𝜙𝑇

[𝑘] 𝑇𝑠,

𝐵 [𝑘] = 𝐵 [𝑘 − 1] + 𝜙 [𝑘] 𝑞 [𝑘] 𝑇𝑠,

𝐴 [0] = 0[2×2],

𝐵 [0] = 0[2×1],

(15)

where 0[2×2] and 0

[2×1] are zero matrices and their subscriptsrepresent their dimensions. The parameters 𝑚 and ] can becomputed from these matrices as

[

𝑚 [𝑘]

] [𝑘]] = 𝐴

−1[𝑘] 𝐵 [𝑘] . (16)

3.3. Third Stage: Resetting and Switching-Off Considerations.In the algorithm presented above, we considered that theuncertain parameters remain constant throughout the time.

However, this condition is not wide enough to solve thepayload identification problem of the typical pick and placetasks. In this application, the uncertain parameter 𝑚 maysuddenly change to a new constant value as the manipulatortakes or drops an object. To detect payload changes, in thiscase, we require the algorithm to be reinitiated once it hasachieved an estimate of the current parameters. But theinstant when the parameter is accurately computed is, inprinciple, unknown and amethod to determine it is thereforeneeded.

To detect when the identification algorithm has con-verged, we consider in this paper the strategy proposed in[25]. In this strategy, the convergence time is expressedin terms of the moving average and the moving standarddeviation of the parameter to identify. In our case, if wedefine 𝑚[𝑘] as the payload mass estimate at the sample 𝑘, itsmoving average (𝐸[𝑚[𝑘]]) and its moving standard deviation(𝜎[𝑚[𝑘]]) are defined as follows:

𝐸 [𝑚 [𝑘]] =1𝑀

𝑀−1∑

𝑖=0𝑚[𝑘− 𝑖] ,

𝜎 [𝑚 [𝑘]] = √1𝑀

𝑀−1∑

𝑖=0(𝑚 [𝑘 − 𝑖] − 𝐸 [𝑚 [𝑘]])

2,

(17)

where the length of thewindow (𝑀) defines the set of samplesconsidered in the calculations. The payload mass estimated𝑚[𝑘] is said to have converged when the following criteriumis fulfilled:

𝜎 [𝑚 [𝑘]]

|𝐸 [𝑚 [𝑘]]|≤ Δ. (18)

The constant Δ is the tolerance parameter and shouldbe set up according to the application’s requirements. If theapplication needs a high precision, a small Δ should beprovided. On the other hand, if the application requires a fastestimation, a greater Δ should be considered.

There are some other alternatives to the procedure pre-sented above. In [21] the convergence time was computed interms of an artificial parameter called the sentinel parameter.This parameter emulates, in terms of time of convergence, thebehaviour of the remaining system parameters, but, unlikethe others, its value converges to an arbitrary value explicitlydefined beforehand.

On a final note, wewould like to point that for those appli-cationswhere the parameters never change, and only a uniqueestimate of the parameters is necessary, it is advisable toswitch off the algorithm once it has converged. The unstablenature of the filters involved in the identificationmay producean arithmetic overflow when long time has passed. Given thefast nature of the algebraic identification, the algorithm isexpected to be reset long before any numerical problemmightarise.

4. Experiments

In this section, the algebraic identification algorithm pre-sented above is experimentally validated and comparedagainst least squares fitting based approaches.


Payload

Flexiblelink

Motor

Figure 2: Flexible single-link robot platform.

4.1. Experimental Platform. Figure 2 shows the single-linkflexible robot considered in the experiments. It is composedof a Maxon DC motor with gear reduction, a tubularduraluminium link, and amass-adjustable payload structure.The link is connected at one side to the gear shaft, whileat the same time holding the payload structure at its freeend. The motor position commands are sent from a NationalInstrument PXI real time systemand are executed by aMaxonEPOS motor driver. The control loops are executed with asampling time of 𝑇

𝑠= 2 mseconds. The sensory system

consists of strain gages placed at the base of the link and twoincremental encoders tomeasure themotor position and gearshaft position.The strainmeasurements at the base of the linkare used to estimate the coupling torque (Γ), while the outerencoder is used to measure the exact orientation of the linkat its base (𝜃

𝑚). The inner encoder measurements were not

taken into account because the backlash in the gear reductionwould produce erroneous estimates of the position of the baseof the link. However, the outer encoder has a relatively lowresolution of 1024 pulses/revolution. Under these conditions,the measured signals used by the identification algorithm,(𝜃𝑚) and (Γ), are very noisy, and their time derivatives, as

might be expected, are even noisier and should not be usedfor an identification algorithm. To illustrate this, Figure 3shows, for an arbitrary experiment, the signals relevantfor the identification algorithm: the estimation of the tipposition (𝜃

𝑒

𝑡), computed as it was shown in (3), from the

experimental records of 𝜃𝑚

and Γ; the first- and second-order time numerical differentiation of such estimation ofthe tip position, ( 𝜃

𝑒

𝑡) and ( 𝜃

𝑒

𝑡); and the coupling torque (Γ).

As a consequence of the limited resolution of the outerencoder, the estimated signals ( 𝜃

𝑒

𝑡) and ( 𝜃

𝑒

𝑡) present important

discontinuities as shown in Figures 3(b) and 3(c). These twodifferentiated signals are relevant for the other identificationalgorithms that we will use as references, in the followingsections, to evaluate our algebraic identification algorithm.

The most relevant features of the robot are describedin Table 1. The stiffness constant 𝑐 and damping coefficient

Table 1: Flexible robot parameters.

Parameter Units Symbol ValueBeam length (m) 𝑙 1.045Beam rotational stiffness (N⋅m⋅rad−1) 𝑐 2498.2Beam viscous dampingcoefficient (N⋅m⋅s⋅rad−1) ] 0.2

Payload mass 1 (Kg) 𝑚1 1.150Payload mass 2 (Kg) 𝑚2 1.690Payload mass 3 (Kg) 𝑚3 2.930

] were experimentally identified. A frequency based iden-tification, explained in [26], was considered to adjust theparameter 𝑐. The damping coefficient ] provided in this tableis given for indicative purpose only, and it is not required forthe identification algorithm. This parameter was computedby adjusting approximately the decay rate of the residualvibrations in the time domain for a specific experiment, butit might change considerably depending on the experiment.

4.2. Definition of the Experiments. To validate the alge-braic identifier, three types of experiments were defined toreproduce the typical operations of a manipulator. Theseexperiments consider three payload masses (𝑚1, 𝑚2, 𝑚3),whose values are given in Table 1. The motor position wascontrolled using the algebraic controller explained in [26],while no tip vibration control algorithm was considered. Itis noteworthy that the algebraic identifier presented does notdepend on the control algorithm used for themotor position.In fact, it only considers the dynamics of the flexible link.

4.2.1. Experiment Type 1. The motor position (𝜃𝑚) tracks a

fourth-order trajectory between 90 degrees, the vertical pose,and 45 degrees in 2 seconds.This experiment replicates a pickand place operation, where the robot lifts, moves, and placesan object between two points following a defined trajectory.

4.2.2. Experiment Type 2. The motor position (𝜃𝑚) tracks

a 4-second sinusoidal trajectory between 90 degrees and 45degrees. This experiment is used to prove the stability of thealgorithm when a persistent input is applied. Identificationstability is not evident taking into account that the algorithmis based on unstable filters.

4.2.3. Experiment Type 3. The motor is blocked at 0 degrees(𝜃𝑚

= 0), and then the link is gently hit in its tip. Thisexperiment reproduces those cases where the controlledrobot is not capable of completely cancelling the vibrationsat the tip position or the robot at rest picks or releases anunknown payload.

Before ending this section, we present the motor position(𝜃𝑚) and torque (Γ)measurements for each experiment type.

Due to space constrains, we only display the measurementsfor the payload mass𝑚2. The plots for the payload masses𝑚1and 𝑚3 are similar and do not add value to the discussion.Figures 4, 5, and 6, show the sensor measurements used


0 1 2 3 4 50.6

0.8

1

1.2

1.4

1.6

Time (s)

𝜃e t

(rad

)

(a)

0 1 2 3 4 5

0

0.5

1

Time (s)

−0.5

−1

𝜃e t

(rad

/s)

.

(b)

0 1 2 3 4 5

0

500

Time (s)

−500

𝜃e t

(rad

/s2)

..

(c)

0 1 2 3 4 5

0

20

40

Time (s)

Γ(N

m)

(d)

Figure 3: Example of the signals involved in the identification algorithm to discuss (a) estimation of the tip position (𝜃𝑒

𝑡), (b) first-order

numerical time differentiation of the estimation of the tip position ( 𝜃𝑒

𝑡), (c) second-order numerical time differentiation of the estimation of

the tip position ( 𝜃𝑒

𝑡), and (d) coupling torque (Γ).

Table 2: Algebraic identification: convergence time for the different experiments and payloads.

Errors Experiment type 1 Experiment type 2 Experiment type 310% 5% 2% 10% 5% 2% 10% 5% 2%

Payload𝑚1 0.57 s 0.77 s 1.08 s 0.68 s 0.84 s 1.24 s 0.022 s 0.028 s 0.030 sPayload𝑚2 0.33 s 0.61 s 1.27 s 0.40 s 0.65 s 0.88 s 0.036 s 0.132 s 0.238 sPayload𝑚3 0.29 s 0.39 s 0.58 s 0.58 s 0.73 s 1.16 s 0.048 s 0.050 s 0.196 s

for the identification in the experiment types 1, 2, and 3,respectively.

4.3. Algebraic Identification Results: Convergence Times andPrecision. In this section we present the experimental resultsof applying the first two stages of the algebraic identificationalgorithm that we detailed in Section 3.The third stage of thisalgorithm will be addressed in the next section.

The first two stages of the identification method can besummarised in three steps: first, to use the sensor measure-ments, 𝜃

𝑚and Γ, to estimate the tip position 𝜃

𝑒

𝑡by means

of (3); second, to compute the functions 𝑞(𝑡), 𝛽(𝑡), 𝜉(𝑡),and 𝜂(𝑡) from (10); and, third, to sample these functions to

obtain the matrices 𝐴[𝑘] and 𝐵[𝑘], as shown in (15), anduse them to finally compute the payload mass by applying(16). Table 2 summarizes the performance of the algebraicestimation algorithm in terms of speed of convergence andprecision. In this table, we present the convergence times ofthe algorithm for the three types of experiments presentedabove, when the estimates reach and stay in the band of±10%,±5%,±2%of the real payloadmass value.The followingparagraphs are devoted to the discussion of these results inthe context of pursuing the implementation of an adaptivecontrol.

The convergence rates, shown in Table 2, demonstratethat algebraic identification produces accurate and fast


0 1 1.5 2 3 3.5 4 4.5 5

0.81

1.21.41.6

Time (s)2.5

𝜃m

(rad

)

05

Time (s)

Γ(N

m)

0.5

0 0.5 1 1.5 2 3 3.5 4 4.5 52.5−10−505101520

Figure 4: Experiment type 1: motor position and torque measure-ment for mass𝑚2.

1 2 3 4 5 6 7 8 9 10Time (s)

0

0.81

1.21.41.6

𝜃m

(rad

)

1 2 3 4 5 6 7 8 9 10Time (s)

0−5

05

101520

Γ(N

m)


121416182022

𝜃m

(rad

)Γ

(Nm

)

0 0.5 1 1.5 2 3 3.5 4 4.5 5Time (s)

2.5

0 0.5 1 1.5 2 3 3.5 4 4.5 5Time (s)

2.5

0

0.5

1

−0.5

−1


payload mass estimates in all the cases studied. This is in theexperiments that considered large trajectories (experimenttypes 1 and 2), persistent inputs (experiment 2), and alsoin those that considered quasistatic situations (experimenttype 3). Specifically speaking, this algorithm achieves pay-load estimates with errors lower than 5% in less than 0.84seconds.This means that, in the specific case of the trajectoryconsidered in the experiment type 1, once the parameters inthe adaptive controller have been updated, the controller hasmore than one half of the trajectory time interval to improvethe tracking and the vibration suppression.

There are important differences in the convergence timebetween those experiments that involved large trajectories(experiment types 1 and 2) and the one that considered aquasistatic situation (experiment type 3). In the experimentsof the first group, the payload estimates were achieved in anaverage time of 0.48, 0.67, and 1.03 secondswith errors of 10%,5%, and 2%, respectively, and with little difference betweenthem. In the quasistatic situation, experiment type 3, thealgorithm needed 0.035, 0.07, and 0.15 seconds to guaranteethe same margin errors. This means that in the experimenttype 3 the payload estimation was achieved about 8 timesfaster than in the experiment types 1 and 2. These differencesshould be taken into account by the adaptive controller.

4.4. Algebraic Identification Results: Resetting Algorithm. Thepurpose of the convergence criterium (18) is to automaticallydetect when the algebraic identification algorithm has con-verged in order to immediately reset the identification algo-rithm if necessary. In this sectionwe present the results of thisconvergence criterium when it was applied to experimentstype 1.

At this point, we assumed that the sensor measurementsin the experiments type 1, 𝜃

𝑚and Γ, were used to estimate

the tip position 𝜃𝑒

𝑡, as shown in (3), and, subsequently, to

estimate the payload mass by applying (10), (15), and (16).The convergence criterium used the payload mass estimatesprovided by the previous algorithm and it was configuredin terms of the tolerance parameter (Δ) and the windowssize (𝑀). In the context of experiments type 1, we assumedthat it was desirable to have an identification algorithmcapable of performing payload estimates in a period of timeshorter than one half of the trajectory, that is, in less than1 s. With this condition in mind, we defined the toleranceparameter (Δ) of 0.015 and a windows size (𝑀) of 150 samples(0.3 s). Table 3 summarizes the performance of the resettingalgorithm evaluated for these experiments. As can be seenin this table, convergence times stay less than or equal to 0.9seconds for all the payload masses considered. It means thatthe payloadmass is computed in the 45% of the time requiredfor the trajectory. In the same table, it can be also noticedthat the estimation errors are below the 5% margin of error,which is an acceptable error for a potential adaptive controlapplication. To count with a resetting algorithm is especiallyimportant because the convergence time varies depending onthe type experiment trajectory performed. Taking estimatesof the parameter based on a fixed time implies not exploitingthe full potential of the identifier in terms of updating speedand, what it worse, it could make us take incorrect estimates


Table 3: Algebraic identifier resetting algorithm: evaluation forexperiment type 1.

Experiment type 1Convergence

time% of the trajectory

duration Error

Payload𝑚1 0.90 s 45.0% 3.99%Payload𝑚2 0.63 s 31.7% 4.61%Payload𝑚3 0.65 s 32.4% 1.06%

of the parameter if the algorithm has not already converged.This is apparent in our experiment where, as it was pointedin the last section, the convergence time for the experimenttype 3 was achieved in only a fraction of time required for theexperiment types 1 and 2.

4.5. Comparison with Least Square Fitting. In this sectionwe compare the performance of the linear least squaresfitting approach against the algebraic identification approach.For the purpose of a fair comparison between these, weconsidered two versions of the linear least squares fitting.

In the first case, we obtained the tip payload mass (𝑚)and viscous friction coefficient (]) by applying the regularlinear least square fitting to (1) that is linear in parameters.Here we considered the motor position (𝜃

𝑚) and coupling

torque (Γ) measurements to compute an estimation of the tipposition (𝜃𝑒

𝑡) by taking into account expression (3). Taking

these equations into account, the procedure is reduced toapply a linear least squares fitting to the following equation:

Γ (𝑡) = 𝑚 (𝑙2 𝜃𝑒

𝑡(𝑡) + 𝑔𝑙cos (𝜃𝑒

𝑡(𝑡))) + ] ( 𝜃

𝑒

𝑡(𝑡)) , (19)

where the first- and second-order time derivatives of theestimation of the tip position, ( 𝜃

𝑒

𝑡) and ( 𝜃

𝑒

𝑡), respectively, were

computed by numerical differentiation.In the second case, we filtered the terms of (19) through a

low-pass second-order Butterworth filter, before applying theregular linear least square fitting indicated in the first case.This is shown in

𝐹 {Γ (𝑡)} = 𝑚𝐹 {𝑙2 𝜃𝑒

𝑡(𝑡) + 𝑔𝑙cos (𝜃𝑒

𝑡(𝑡))}

+ ]𝐹 { 𝜃𝑒

𝑡(𝑡)} ,

(20)

where the letter 𝐹 indicates the low-pass filtering operation.This filtering operation was performed to soften the effectsof the noise present in the input signals of the linear squaresfitting. The noise effects were shown in Figure 3.

Looking at the convergence rates of the regular leastsquares fitting, shown inTable 4, it is easy to evidence that thismethod is far behind in performance in comparison to thealgebraic identifier. If we consider all the experiment types,this method requires 3.27 seconds to reach the 10% marginerror while the algebraic identification algorithm achievesthis in only 0.68 seconds. This means that the algebraicidentifier is about 5 times faster than the regular least squaresfitting.

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

3

Tim

e of c

onve

rgen

ce fo

r esti

mat

ions

w

ith an

erro

r belo

w 1

0% (s

)

Algebraic estimatorLeast square fitting without filterLeast square fitting with filter

Cut-off frequency of the low-pass filter (Hz)

Figure 7: Comparison of the convergence times respect cut-off frequency for the identification of the payload mass 𝑚3 inexperiment type 1.

On the other hand, the performance of the filteredversion of the linear least square algorithm did not showany significant and consistent improvement in comparisonwith its nonfiltered version or the algebraic identificationalgorithm. In this performance study, cut-off frequenciesbetween 0 and 40Hz were considered in filter 𝐹 of (20) todeal with the noise present in themeasurements. Discretizingthe filter using the bilinear transform and a sampling timeof 𝑇𝑠= 2ms, the poles 𝑃

𝑖of the filter must fulfill Haykin’s

condition |𝑃𝑖𝑇𝑠| < 0.5 [27]. This implies that the maximum

cut-off frequency of the filter is 0.5/2𝜋𝑇𝑠

= 39.7Hz ≈

40Hz. To illustrate the performance of this identificationmethod, Figure 7 shows its convergence time with respect todifferent cut-off frequencies.This graphic presents the resultsof identification of the payload mass 𝑚3 in the experimenttype 1. However, similar results are obtained from the otherexperiment types and masses. For ease of comparison withthe other methods, the convergence times for the nonfilteredlinear least squares algorithm and the algebraic algorithm arealso indicated in Figure 7.

Figures 8, 9, and 10 show comparisons of the identifica-tion algorithms here discussed. A cut-off frequency of 10Hzwas considered for the filtered version of the linear leastsquares algorithm.

5. Conclusions

In this paper, an online algebraic identification algorithm hasbeen presented for the identification of the payload massof a single-link flexible robot moving under gravity. Thisalgorithm was designed for a specific nonlinear dynamicstructure that can be arranged to be linear in parameters.The algorithm considers as inputs the motor position andthe torque measured at the base of the link. Its performancehas been experimentally evaluated in situations that involvelarge trajectories (experiment types 1 and 2), trajectories that


Table 4: Linear least squares fitting: convergence time for the different experiments and payloads.

Errors Experiment type 1 Experiment type 2 Experiment type 310% 5% 2% 10% 5% 2% 10% 5% 2%

Payload𝑚1 2.30 s >5 s >5 s 1.30 s 1.45 s 2.94 s 1.92 s 4.27 s >5 sPayload𝑚2 2.15 s >5 s >5 s 1.30 s 1.46 s >5 s 3.27 s >5 s >5 sPayload𝑚3 2.34 s >5 s >5 s 1.40 s 1.42 s 2.95 s 0.49 s 3.70 s >5 s

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

Estim

ated

pay

load

(kg)

Payload actual values

Time (s)

Estimation of the 1.15 kg payloadEstimation of the 1.69kg payloadEstimation of the 2.93kg payload

(a)

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4


Time (s)


Estim

ated

pay

load

(kg)

(b)

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4


Time (s)


Estim

ated

pay

load

(kg)

(c)

Figure 8: Experiment type 1: (a) algebraic identifier, (b) least square fitting, and (c) least square fitting with filter.


0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4Es

timat

ed p

aylo

ad (k

g)

Time (s)



(a)


0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

Estim

ated

pay

load

(kg)

Time (s)


(b)


0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

Estim

ated

pay

load

(kg)

Time (s)


(c)


persistently excited the link (experiment type 2), and in aquasistatic situation (experiment type 3), where the flexiblelink, initially at rest, is externally perturbed.The performanceanalysis of this algorithm shows that it is capable of achievingestimates in less than 0.84 seconds with an error lower than5%. For large trajectories, similar to those we considered inthe experiments (45 degrees in 2 seconds), this might bean acceptable convergence rate if it has to be included inan indirect adaptive control implementation. An adaptivecontroller that considers this identification algorithm mightcount with more than one half of the trajectory lasting

time to improve the trajectory tracking and the vibrationsuppression.

To highlight the performance of our identification algo-rithm, we presented a comparison with a linear leastsquare fitting. Other identification algorithms described inthe scientific literature and referenced in Introduction havenot been included in this comparison because they arenot suited to deal with nonlinear dynamics. In particu-lar, two least squares fitting based algorithms were con-sidered: the regular least square fitting and a prefilteredversion. For estimations with errors below 10%, the results of



0.5

1

1.5

2

2.5

3

3.5

4

Estim

ated

pay

load

(kg)

Time (s)


00 1 2 3 4 5

(a)


00

1 2 3 4 5

0.5

1

1.5

2

2.5

3

3.5

4

Estim

ated

pay

load

(kg)

Time (s)


(b)


00

1 2 3 4 5

0.5

1

1.5

2

2.5

3

3.5

4

Estim

ated

pay

load

(kg)

Time (s)


(c)


the comparison show that the algebraic identification algo-rithm is about 5 times faster than the regular least squaresfitting. The filtered variant, for its part, did not show anyimprovement in terms of time of convergence in comparisonwith its regular version. The performance superiority of ouralgorithmover the linear least square fitting can be attributed,in good part, to the fact that the algebraic identifier doesnot require any time derivative of the measurements. Thistrait avoids the noise amplification problems that the other

methods present and, moreover, allows using sensors of thelink position with relatively low resolution, as we showed inour experiments.

In this work we implemented a resetting algorithm todetect when the estimation convergence is achieved. Thisalgorithm was tested in some experiments that reproducethe typical trajectory tracking performed in a pick and placetask (experiment type 1). Its performance demonstrates itsfeasibility for a potential adaptive control application.


Our further work will be devoted to the study of adaptivecontrol laws based on the estimation algorithm here devel-oped.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This paper was sponsored by the Spanish FPU12/00984 Pro-gram (Ministerio de Educacion, Cultura y Deporte). It wasalso sponsored by the Spanish Government Research Programwith the Project DPI2012-37062-CO2-01 (Ministerio de Eco-nomia y Competitividad) and by the European Social Fund.

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Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


research article payload mass identification of a single

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