Linköping University Electronic Press
Report
Modeling and Experiment Design for Identification of Wear in a
Robot Joint under Load and Temperature Uncertainties based on
Constant-speed Friction Data
André Carvalho Bittencourt and Patrik Axelsson
Series: LiTH-ISY-R, ISSN 1400-3902, No. 2058
ISRN: LiTH-ISY-R-3058
Available at: Linköping University Electronic Press
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-90023
Modeling and Experiment Design for
Identification of Wear in a Robot Joint
under Load and Temperature
Uncertainties based on Constant-speed
Friction Data
André Carvalho Bittencourt, PAtrik Acelsson
Division of Automatic Control
E-mail: andrecb@isy.liu.se, axelsson@isy.liu.se
15th March 2013
Report no.: LiTH-ISY-R-3058
Submitted to IEEE-ASME Trans. on Mechatronics
Address:
Department of Electrical Engineering Linköpings universitet
SE-581 83 Linköping, Sweden
WWW: http://www.control.isy.liu.se
AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET
Technical reports from the Automatic Control group in Linköping are available from http://www.control.isy.liu.se/publications.
Abstract
The eects of wear to friction are studied based on constant-speed friction data collected from dedicated experiments during accelerated wear tests. It is shown how the eects of temperature and load uncertainties produce larger changes to friction than those caused by wear, motivating the con-sideration of these eects. Based on empirical observations, an extended friction model is proposed to describe the eects of speed, load, temper-ature and wear. Assuming availability of such model and constant-speed friction data, a maximum likelihood wear estimator is proposed. A crite-rion for experiment design is proposed which selects speed points to collect constant-speed friction data which improves the achievable performance bound for any unbiased wear estimator. Practical issues related to exper-iment length are also considered. The performance of the wear estimator under load and temperature uncertainties is found by means of simulations and veri ed under three case studies based on real data.
Keywords: industrial robotics, wear, friction, identi cation, condition mon-itoring
Modeling and Experiment Design for Identification
of Wear in a Robot Joint under Load and
Temperature Uncertainties based on Constant-speed
Friction Data
Andr´e Carvalho Bittencourt and Patrik Axelsson
Abstract—The wear effects to friction are studied based on
constant-speed friction data collected from dedicated experiments during accelerated wear tests. It is shown how the effects of temperature and load uncertainties produce larger changes to friction than those caused by wear, motivating the consideration of these effects. Based on empirical observations, an extended friction model is proposed to describe the effects of speed, load, temperature and wear. Assuming availability of such model and constant-speed friction data, a maximum likelihood wear estimator is proposed. A criterion for experiment design is proposed which selects speed points to collect constant-speed friction data which improves the achievable performance bound for any unbiased wear estimator. Practical issues related to experiment length are also considered. The performance of the wear estimator under load and temperature uncertainties is found by means of simulations and verified under three case studies based on real data.
Index Terms—industrial robotics, wear, friction, identification,
condition monitoring
I. INTRODUCTION
F
RICTION can be defined as the tangential reaction force between two surfaces in contact. It is not a fundamental force but the result of complex interactions between contacting surfaces down to a nanoscale perspective. Friction always opposes motion, dissipating kinetic energy. A part of the work produced by friction appears as heat transfer, vibrations and acoustic emissions. Other outcomes of friction are due to mechanical action between the surfaces, such as plastic deformation, adhesion and fracture. The later are related towear, which can be defined as “the progressive loss of material
from the operating surface of a body occurring as a result of relative motion at its surface” (quoted from [1]). The need for relative motion between surfaces implies that the wear mechanisms are related to mechanical action between surfaces. This is an important distinction to other processes with a similar outcome and very different nature, e.g. corrosion (see [2] for basics on wear related phenomena).
Excessive wear may lead to a deterioration of the system’s performance and to an eventual failure. The potential damages caused by excessive wear can however be prevented if good maintenance practices are performed. In order to support
The authors are with the Department of Electrical Engineering, Link¨opings University, Link¨oping, Sweden.{andrecb, axelsson}@isy.liu.se
This work was supported by ABB and the Vinnova Industry Excellence Center LINK-SIC at Link¨oping University.
maintenance actions, the use of methods to determine the condition of the equipment is desirable, allowing for condition-based maintenance (CBM). The wear processes inside a robot joint cause an eventual increase of wear debris in the lubricant. Monitoring the iron content of lubricant samples taken from the robot joint can thus be used as an indication of the joint condition. The study of wear debris particles is known as ferrography and was first introduced in [3]. Since then, the science has evolved and helped to understand wear related phenomena (see [4] for a historical review on ferrography). These techniques are however intrusive and costly, requiring laboratory analyses and alternatives which do not rely on additional sensory information are preferred.
The accumulated wear in a tribosystem may lead to varia-tions in friction (see [5] for a review on the relation between wear and friction). As shown by experimental studies in this paper, such relation is also present for a robot joint. Alternatives for wear monitoring are thus possible provided it is possible to observe friction and the relation between friction
and wear is known. Monitoring friction to infer about wear
is however challenging since friction is significantly affected by other factors than wear such as temperature and load (see Fig. 3). The effects of temperature are specially difficult since temperature is not measured in typical robot applications. These co-effects should nevertheless be considered when ver-ifying the reliability of a solution.
In the literature, little can be found about wear estimation
for industrial robots. This may be attributed to the lack of wear models available and the high costs and time required to perform wear experiments. There are related approaches used for fault detection, where the objective is to decide whether a change from nominal is present. In the related literature, faults are typically consider as actuator malfunctions, modeled as changes in the output torque signals or in parameters of a robot model, including the case of friction changes, which is important since they can relate to wear.
The estimation of friction parameters in a robot model from measured data is a natural approach for fault detection because of their physical interpretation. A shortcoming is the need for excitation in the data, which might be restricting in some appli-cations. In [6], a least-squares method is used to estimate robot parameters over a moving window. Estimates of the Coulomb and viscous friction parameters are compared to confidence
values of their nominal behavior for fault detection. In the paper, an experimental study is shown where the estimated friction parameters could indicate some of the faults but could not readily distinguish between them, e.g. the increase of joint temperature had a similar effect as a fault in the drive-chain.
Monitoring estimates of the Coulomb and viscous param-eters is also considered in [7]. The paramparam-eters are estimated from constant-speed friction data collected from experiments in an off-line manner, using a procedure similar to what is used in this paper (see Sec. II-A). These data are collected for different speed levels, forming a so-called friction curve (see e.g. Fig. 3 for examples of friction curves). In [7], the area under the friction curve is also monitored, which is related to the energy needed to overcome friction, i.e. a “friction curve energy”. Experimental work suggested that monitoring the Coulomb and/or viscous friction parameters are not robust solutions to detect wear changes even under rather constant load and temperature conditions. The use of the friction curve energy improved the robustness, but as it was shown (compare Fig. 6.3(d) with Fig. A.1 in [7]) it still cannot distinguish from the effects of temperature. Its behavior was also found to be dependent on properties of the lubricant used. As illustrated here, the effects of wear and temperature affect friction in a similar manner and the simple friction model used in [6], [7] did not consider these effects. As it will be shown, the use of a more detailed friction model allows for a reliable estimation of the effects of wear even under temperature and load uncertainties.
Estimates of the viscous friction parameter were also con-sidered in [8] to monitor the lubricant health in a mechanical transmission. The parameter estimates are achieved based on the energy balance for the system over a time window. As it is shown in experiments, changes in the lubricant viscosity can be monitored by comparison of the estimated viscous parameter with that generated by a model. Because viscous friction is highly dependent on the lubricant temperature, a temperature-dependent model is used. The lubricant tempera-ture is estimated based on a Kalman filter using environment temperature measurements and a heat transfer model. A similar approach but based on an observer of the viscous friction torque is also presented in [9] with simulation studies for a robot joint.
Considering the nonlinear nature of a manipulator, the use of nonlinear observers is another common approach for fault detection, see e.g. [10], [11]. Different design approaches are used and the observer stability is typically guaranteed by analyses of the decay rate of a candidate Lyapunov function. Due to uncertainties in the modeling assumptions, approaches have been suggested to improve robustness. In [12]–[14], nonlinear observers are used together with adaptive schemes while in [15] support vector machines are trained to model the uncertainties. In [16], a nonlinear fault observer is suggested based on a neural network model for the abnormal robot behavior and defines a robust adaptation rule based on known uncertainties’ bounds. In [13], [17], the residuals of a nonlinear observer of generalized momenta are used for detection of actuator faults, the derivation of such observer is rather simple and compared to [18], where the residuals of a torque observer
are used, it does not require inversion of the inertia matrix; an efficient implementation for the former is also discussed in [19].
The use of unknown input (fault) observers, as presented in [11], [16], is important to support, e.g., control law re-configuration and diagnosis. In [20], an extended Kalman-Bucy filter is used to estimate friction torques in a rotating machine; the presence of a friction change is detected based on a multiple hypotheses test in a Bayesian framework where each hypothesis is associated to a known friction model. A friction observer for control is also suggested in [21] using joint torque measurements; the observer is inspired by the generalized momenta observer presented in [13], [17], its structure is of a linear low-pass filter of the actual friction torque.
While observer design is mainly dependent on a robot model, observers can also be achieved by direct processing of input-output data, where a black-box model is identified to provide estimates of measured signals. The use of neural
network and neuro-fuzzy models are common, see e.g. [22],
[23], but also support vector machines have been consid-ered [24]. Disadvantages of these methods are the need for training data and the difficulties in guaranteeing their gener-alization capacities, with a compromise between adaptation and detection performance. Similar approaches can be used for fault classification, requiring faulty data for the training stage, which is difficult in practice; see [25] for a review of industrial use of neural networks.
Less studied for nonlinear systems are parity-space ap-proaches, where a model of the system is used to find a projection of the input-output data to a space dependent on faults and errors but not on the states. In [26], the authors propose a projection which is robust to modeling errors and noise. Practical difficulties with such approaches for nonlinear systems is the need for higher order derivatives of the input and output data and high computational complexity. Note that while parity spaces and observers are equivalent for linear systems (as shown in [27]), this is not the case for nonlinear systems (see the discussion in [28]).
In [29], the passivity property of Lagrangian systems is used to define energy balance equations which are monitored for fault detection and isolation; the framework is illustrated with a simulation study of a robot manipulator with faults in dissipative components (e.g. friction changes) and energy-storing components (e.g. load changes). As advantages to classical methods, the author mentions the simplicity of the models used in an energy model compared to, e.g., models used for nonlinear observers, and the low complexity for on-line implementation. Because the energy balance is also affected by disturbances, knowledge of these effects to the system’s energy can be used to achieve robustness; some approaches are discussed in [29]. In [30], the energy balance is computed over a moving window to perform fault diagnosis in a robotic system. A known robot model is considered available and thresholds are determined based on available bounds on the model uncertainties, allowing for fault detection. Fault isolation and identification are achieved based on investigation of the correlation of the residuals to multiple fault energy patterns. A simulation study illustrates the framework, where
actuator faults are successfully diagnosed; the case of con-comitant faults is also illustrated.
The vibration patterns generated from a robot joint also contain valuable information about its condition. In [31], neural networks are used to learn the vibration patterns of a robot based on accelerometers’ measurements. Similarly, the acoustic emissions of the robot joints may change under a fault. In [32], features of sound measurements, i.e. peaks of a wavelet transformation, are monitored and determination of a fault is done based on labeled data using a nearest neighbor classifier. Besides the extra sensors needed, these approaches require data from a pre-defined trajectory which makes them unsuitable for on-line monitoring.
In this paper, a wear estimator is proposed based on a known
friction model and constant-speed friction data which are achieved through dedicated experiments, in an off-line manner. A solution based on a dedicated experiment will decrease the robot availability which is undesired from the perspective of a robot user. The trade-off between experiment length and the estimator accuracy is therefore important and is studied in detail. The main contributions leading to the proposed solution are listed
• first, the effects of wear to friction are modeled based on
empirical observations;
• an extended friction model is proposed and identified
that describes the effects of speed, temperature, load and wear;
• experiment design is considered based on the achievable
performance for any unbiased wear estimator;
• with a known friction model, a maximum likelihood wear
estimator is proposed;
• the estimator is validated through simulations and case
studies based on real data.
These results are presented through Secs. III to V. Sec. II reviews earlier results presented in [33] which are used in this paper; namely, an experiment routine used to provide constant-speed friction data and the friction model to be extended. The conclusions and proposals for further research are presented in Sec. VI.
A preliminary version of this work was presented in [34]
where the wear model was first presented and a prediction-error wear estimator was suggested and verified. This paper suggests wear estimators based on a statistical framework, with a more in-depth study of experiment design, achievable performance and verification studies.
II. FRICTION IN A ROBOT JOINT
Due to the complex nature of friction, it is difficult to describe it from physical principles. In a robot joint, with several components interacting such as gears, bearings and shafts, which are rotating/sliding at different velocities and under different lubrication levels, it is difficult to estimate and model friction at a component level. A typical approach is to consider these effects collectively, as a joint friction and to study friction based on the input-output behavior of the joint. Friction is a dynamic phenomenon; at a contact level, the surfaces’ asperities can be compared to (very stiff) bristles
in a brush, each of which can be seen as a body with its own dynamics connected by the same bulk (see e.g. [35], [36] for details on asperities friction models). Because the internal friction states are not measurable, it is common to study friction in steady-state, when friction presents a static behavior. Experimental data show that under constant speed, the friction in a robot joint is static (see e.g. [33]).
The simplified behavior of steady-state friction1 makes it
easier to be modeled and to identify the sources of changes, e.g. caused by wear or temperature. However, steady-state friction data are typically not available from a robot’s normal operation but can be achieved based on dedicated experiments. In Sec. II-A, a simple experimental procedure, described first in [33], is presented which is used to provide estimates of friction at a given speed level. Friction data collected using such procedure simplifies the wear estimation problem since the experiment is performed in a controlled manner, reducing the effects of external disturbances (found, e.g., in contact applications) and it does not rely on a robot model, which may contain uncertainties. These type of data will be used as input to the wear estimators described here.
Friction in a robot joint is not only affected by wear; the effects of temperature and load are also significant. In [33], the behavior of steady-state friction is studied in detail and a static nonlinear model is suggested to describe the effects of speed, temperature and load. This model is reviewed in Sec. II-B and is extended in Sec. III to include the effects of wear to friction.
A. A Procedure to Estimate Friction at a fixed Speed Level
A manipulator is a multivariable, nonlinear system that can be described in a general manner through the rigid multi body dynamic model
M(ϕ) ¨ϕ+ C(ϕ, ˙ϕ) + τg(ϕ) + τf = u (1)
where M(ϕ) is the inertia matrix, C(ϕ, ˙ϕ) relates to speed
dependent terms (e.g. Coriolis and centrifugal),τg(ϕ) are the
gravity-induced joint torques andτf contains the joint friction
components. The system is controlled by the input torque,u,
applied by the joint motor (in the experiments the torque reference from the servo was measured2).
When only one joint is moved (C(ϕ, ˙ϕ) = 0 at that joint)
under constant speed ( ¨ϕ≈ 0), Eq. (1) simplifies to
τg(ϕ) + τf = u. (2)
The resulting applied torqueu drives only friction and
gravity-induced torques. The required torques to drive a joint in forward,u+, and reverse,u−
, directions at the constant speed level ¯ϕ and at a joint angle value˙ ϕ (so that τ¯ g( ¯ϕ) is equal in
both directions), are
τf++ τg( ¯ϕ) = u+ (3a)
τ−
f + τg( ¯ϕ) = u −
. (3b)
1In this paper, the term steady-state friction is used as a synonym of the friction observed in constant-speed conditions.
2It is known that using the torque reference from the servo as a measure of the joint torque might not always hold because of the temperature dependence of the torque constant of the motors. The deviations are however considered to be small and are neglected in this paper.
0 1 2 3 4 5 6 7 8 9 −100 −50 0 50 100 t(s) ϕ (r ad ), ˙ϕ (r ad /s ) −1 −0.5 0 0.5 1 u u ϕ ˙ ϕ ¯ ϕ
Fig. 1. Example of excitation signals used for the constant-speed friction estimation at ¯ϕ= 42 rad/s and ¯˙ ϕ= 0.
In case an estimate ofτg( ¯ϕ) is available, it is possible to isolate
the friction component in each direction using (3). If such estimate is not possible (e.g. not all masses are completely known),τf can still be achieved in case friction is independent
of the rotation direction. Subtracting the equations in (3) yields
τf+− τ− f = u
+− u−
and if τf+ = −τ−
f = τf, the resulting direction independent friction evaluated at the constant speed ¯ϕ is:˙
τf=
u+− u−
2 . (4)
To generate suitable data, single joint movements are per-formed with the desired speed ¯ϕ in forward and backward˙
directions around a joint angle ϕ. An example of the joint¯
angle-, speed- and torque3 data generated from such
experi-ment in joint 2 of an ABB IRB 6620 is shown in Fig. 1. The signals were sampled at2 kHz4for ¯ϕ˙= 42 rad/s around ¯ϕ= 0.
The constant-speed data is segmented aroundϕ and the steady-¯
state friction levels can be achieved using (3) or (4).
The procedure can be repeated for several ¯ϕ’s and a friction˙ curve can be drawn, which contains steady-state friction values
plotted against speed. A friction curve can be seen in Fig. 2, when computations have been made using (3) and (4). As seen in the figure, there is only a small direction dependency of friction for the investigated joint. Therefore, in this paper, friction levels are computed using (4), which is not influenced by deviations in the gravity model of the robot.
The average time required to execute a trajectory to estimate friction at one speed was optimized down to2.5 s. As it will be
shown, the choice of which and how many speed levels where friction data is collected is an important design parameter, affecting the quality of the wear estimates and the length of the experiments. Considering a 6 axes robot, restricting the maximum experiment time to a minute allows the choice of a
maximum of 4 speeds levels per axis.
B. A Friction Model describing Speed, Temperature and Load
In [33], steady-state friction data were studied to derive a friction model that can describe the effects of speed,
temper-3Throughout the paper all torques are normalized to the maximum ma-nipulation torque at low speed and are therefore displayed as dimensionless quantities.
4Similar results have been experienced with sampling rates down to220 Hz.
0 50 100 150 200 250 0.05 0.1 0.15 | ˙ϕ| (rad/s) |τf |
Fig. 2. Friction curve. Crosses indicate friction levels achieved using (4), with the assumption that friction is direction independent. Dotted/dashed lines indicate friction levels achieved using (3a) and (3b) respectively.
ature and load. The model takes the form:
τf( ˙ϕ, τl, T) = {Fc,0+ Fc,τlτl} + Fs,τlτle − ˙ ϕ ˙ ϕs,τl α + (5a) + {Fs,0+ Fs,TT}e − ˙ ϕ { ˙ϕs,0+ ˙ϕs,T T } α + (5b) + {Fv,0+ Fv,Te −T TVo} ˙ϕ, (5c)
whereτlis the absolute value of the manipulated load torque
and T is the joint temperature. The remaining variables are
parameters used to model the friction behavior. In the same paper, the parameters for the model were identified for a robot joint equipped with same type of gearboxes as the ones studied in this paper; the parameter values are given in Table I.
To illustrate the model behavior, Fig. 3(a) presents observed and model-based predictions of friction curves for high and low values of τl and T . Notice the effects of τl, which
give an offset increase of the whole curve together with an exponential-like increase at speeds below25 rad/s. The effects
of T can be seen as an exponential increase at speeds below 80 rad/s and a decrease of the curve slope at higher speeds.
Notice further that for such temperature and load values, there is a speed range where the effects are less pronounced, in this case around 80 rad/s.
As shown in [33], this model can be used to predict the normal behavior of steady-state friction under broad operation conditions. The mean and standard deviation of the prediction error for the model (5), nominated here as ε, were estimated
based on more than 5800 steady-state friction data points col-lected under different speed, temperature and load conditions as [µε, σε] = [−9.2410−4,4.2310−3].
III. MODELING WEAR EFFECTS TO FRICTION Monitoring a robot until a failure takes place is a costly and time consuming task and it is thus difficult to fully comprehend the effects of wear in a robot joint. To study these effects, accelerated wear tests were performed with a robot joint. Friction curves were estimated periodically during the tests until failure. Even during accelerated tests, a wear related fault may take several months or years to appear.
The resulting friction curves from such experiment are shown in Fig. 3(b), which were obtained under the same
load-TABLE I
IDENTIFIED PARAMETERS FOR THE MODEL(5),VALUES TAKEN FROM[33].
Fc,0 Fc,τl Fs,0 Fs,τl Fs,T Fv,0 Fv,T ϕ˙s,0 ϕ˙s,τl ϕ˙s,T TVo α 3.1110−2 2.3410−2 −2.5010−2 1.2610−1 1.6010−3 1.3010−4 1.3210−3 −24.81 9.22 0.98 20.71 1.36 0 50 100 150 200 250 300 0 0.05 0.1 0.15 0.2 ˙ ϕ (rad/s) τf T= 33◦C, τl= 0.70 T= 80◦C, τl= 0.70 T= 33◦C, τl= 0.01 T= 80◦C, τl= 0.01 offset: 0.038
(a) Observed friction curves (circles) and model-based predic-tions (lines) for low and high values ofT and τl.
96.7742 96.7742 96.7742 93.54839 93.54839 93.54839 94.62366 94.62366 94.62366 97.84946 97.84946 97.84946 98.92473 98.92473 98.92473 100 100 100 ˙ ϕ (rad/s) τf 0 50 100 150 200 250 0 0.02 0.04 0.06 0.08 0.1 0.12 0 10 20 30 40 50 60 70 80 90 100 offset: 0.017
(b)Wear effects from accelerated tests under constant load- and temperature conditions. The colormap is related to the length of the tests. The dashed line relates to a wear level critical for CBM.
Fig. 3. Friction dependencies in a robot joint based on experimental studies. The offset values were removed for a comparison, its values are shown in the dotted lines. The data were collected for similar gearboxes and are presented in directly comparable scales. Notice the larger amplitude of effects caused by temperature and load compared to those caused by wear but the different speed dependence.
and temperature levels. The colormap relates to a normalized
time-index k with values between [0, 100], indicating the
length of the accelerated wear tests. Notice the time behavior of the friction curves, which remain fairly constant untilk≈ 90
and increase quickly thereafter. According to gear experts, it would be interesting to detect a gear wear level corresponding to the dashed line in the figure, which occurs atk≈ 97.
Notice that the effects of load/temperature are larger in amplitude than those caused by wear, but with a different speed dependency. The effects of load/temperature are concentrated in the low and high speed regions, whereas the effects of wear appear, at first, in the low to intermediate speed regions. By the end of the accelerated wear tests, the friction curves are also affected at higher speed levels, changing the viscous behavior of friction. The different speed dependencies of the effects are therefore important for the choice of speed levels in order to obtain an accurate identification of wear.
A. Wear Modeling
Resolving for coupled effects between wear, temperature, load and other parameters would require costly long term experiments. In order to make it possible to examine and model the effects of wear, a simplifying assumption is taken that considers the effects of load/temperature to be independent of those caused by wear. Under this assumption, the effects of wear in the friction curves of Fig. 3(b) can be isolated since temperature/load conditions are the same for these data.
A wear profile quantity, τ˜f, is defined by subtracting
fric-tion curve data observed before the accelerated wear tests started,τ0
f, from the ones obtained from the same robot with
accelerated tests, i.e.,
˜ τf = τf− τf0. (6) 0 50 100 150 200 250 300 20 40 60 80 100 0 0.02 0.04 0.06 0.08 k ˙ ϕ(rad/s) ˜τf τf0
Fig. 4. Friction wear profileτ˜f computed from the data in Fig. 3(b)
according to (6). The dashed line indicates an assumed wear level that should be detected. The dotted lines relate to the friction curve
τf0 before the wear tests started.
The resulting wear profile from the accelerated wear tests in Fig. 3(b) can be seen in Fig. 4, where friction is presented alongk andϕ. In the figure, the dashed line relates to a wear˙
level considered important to detect, as in Fig. 3(b). The dotted line relates toτ0
f, the friction curve before the experiment wear
tests started.
As can be noticed, the effects of wear appear as an
in-crease of the exponential-like behavior of the friction curves
up to 150 rad/s and small (linear) decrease of the velocity slope dependency at higher speeds. Introducing w as a wear
parameter, the observations support the choice of a model structure for the wear profile as
˜ τf( ˙ϕ, w) = Fs,wwe − ˙ ϕ ˙ ϕs,w w α + Fv,ww ˙ϕ. (7)
The model represents wear effects with an exponential- and a velocity dependent terms, with4 parameters. In the model,
the exponential term describes the effects in a speed range and has a similar structure as for the temperature dependence
TABLE II
PARAMETERS FOR THE MODEL(7)AND ONE STANDARD DEVIATION IDENTIFIED USING THE WEAR PROFILE DATA
ATk= 96.77WITHw = 35. Fs,w[10−4] Fv,w[10−7] ϕ˙s,w 9.02 ± 0.19 −5.15 ± 1.00 2.19 ± 0.15 0 50 100 150 200 250 0 0.01 0.02 0.03 0.04 ˙ ϕ(rad/s) ˜τf ˆ w∗= 22.39, k = 94.62 ˆ w∗= 30.35, k = 95.70 ˆ w∗= 35.00, k = 96.77 ˆ w∗= 46.73, k = 97.85
Fig. 5. Measured wear profile (circles) and model-based predic-tions (lines).
found in (5b). Similarly, a velocity dependent term is also found for the temperature effects in (5c).
The model parameters cannot be directly identified since the wear quantity w is not measurable. To overcome this, w is defined with values between [0, 100], relative to a failure
state. The value w = 35 is chosen as a reference for the wear effects associated with the dashed line in Fig. 4, which occurs atk= 96.77 and is related to a level that is important to detect.
With this convention, the parameters for (7) are identified using the wear profile dataτ˜f for the curve at k= 96.77. The
parameterα is fixed to1.36 for consistency with model (5) and
the identification method described in [33] is used to identify the remaining parameters. The values obtained are shown in Table II.
B. Validation
Considering the identified parameters for the model (7), all available wear profile data at k is used to achieve an estimate
of the wear level for the accelerated wear tests of Fig. 4 at each k. The identification method described in [33] is used
and the achieved wear levels are assigned as wˆ∗. This wear
estimate is considered the best possible given the available information since all friction data available is used at each k
and no disturbances are present. Using the identified wear values, the wear profile given by model predictions from (7) and observations are presented for the interval k= [94, 98]
in Fig. 5. As can be noticed, the model can predict well the behavior of τ˜f. The mean and standard deviation for the
prediction error of the wear model (7), nominated here as ε,˜
are [µε˜, σ˜ε] = [9.7210−4,3.8210−3]. C. Steady-state Friction Model
Under the assumption that the effects of load/temperature are independent of those caused by wear, it is possible to extend the model given in (5) to include the effects of wear as
τf( ˙ϕ, τl, T, w) = τf( ˙ϕ, τl, T) + ˜τf( ˙ϕ, w), (8) 10 10 10 20 20 20 30 30 30 40 40 40 50 50 50 60 60 60 70 70 70 80 80 80 90 90 90 100 100 100 35 35 35 35 ˙ ϕ (rad/s) τf 0 50 100 150 200 250 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0 10 20 30 40 50 60 70 80 90 100
Fig. 6. Increase of wear levels given by the model (8) with colormap indicating w. The dashed line relates to the wear level at which an alarm should be generated.
where τf( ˙ϕ, τl, T) is given by (5) and ˜τf(w) is described
in (7). Fig. 6 presents the friction predictions given by the proposed model atT= 40◦
C and τl= 0.10 for wear values in
the range w = [0, 100] when the parameters given in Tables I and II are used. Notice that the effects are concentrated to the speed range of[0, 150] rad/s. As previously, the dashed line
in Fig. 6 indicates an alarm level for the wear with w = 35, it has a friction increase of 0.017 at 50 rad/s relative to w = 0
which is consistent to the increase found in Fig. 3(b) for the same speed value.
IV. MODEL-BASED WEAR ESTIMATION Consider that the experiment described in Sec. II-A is repeatedN times independently at speed levels
˙Φ=[ ˙ϕ1,· · · , ˙ϕi,· · · , ˙ϕN]T
generating the steady-state friction data points
τf= [τf1,· · · , τfi,· · · , τfN]T.
A model for each steady-state friction datum τfi can be achieved by including an additive uncertainty term to model (8). Assuming that the prediction errors for models (5) and (7) follow independent Gaussian distribu-tions, ε∼ N (µε, σ2ε) and ˜ε∼ N (µε˜, σ2ε˜), the resulting data generation model is τfi = τf( ˙ϕi, τl, T) + ε + ˜τf( ˙ϕi, w) + ˜ε (9a) = τf( ˙ϕi, τl, T, w) + ¯ε (9b) ¯ ε∼ N (µ¯ε, σ2¯ε), µε¯= µε+ µε˜, σε2¯= σ 2 ˜ ε+ σ 2 ε. (9c)
Using the values for the mean and standard deviation of ε
and ε found in Secs. II-B and III-B respectively, it is found˜
that µε¯= 4.8010−5≈ 0 and σε¯= 5.7010−3. The joint density
for this model is
p(τf|τl, T, w) = N τf; µ( ˙Φ, τl, T, w), Σ . (10a) WhereΣ = Iσ2 ¯ ε and µ( ˙Φ, τl, T, w) = [τf( ˙ϕ1, τl, T, w), · · · , (10b) τf( ˙ϕi, τl, T, w), · · · , τf( ˙ϕN, τl, T, w)]T (10c) (10d)
where τf(·) is the nonlinear function given by (8).
An unbiased estimate ofτlis considered available, achieved,
e.g., using a robot model, with distribution N (µτl, σ
2 τl). The
information from this estimate is included in the model by considering the marginal density function
¯ p(τf|T, w) = Z ∞ −∞ p(τf|τl, T, w)N (τl; µτl, σ 2 τl) dτl (11)
which forp(τf|τl, T, w) given in (10) can be found explicitly
since the dependence of µ(·) on τlis linear. It is given by (see
e.g. [37] p. 93) ¯ p(τf|T, w) = N τf; ¯µ( ˙Φ, T, w), ¯Σ( ˙Φ) (12a) where ¯ µ( ˙Φ, T, w) = µ( ˙Φ, µτl, T, w) (12b) ¯ Σ( ˙Φ) = Σ + M ( ˙Φ)M ( ˙Φ)Tσ2 τl (12c) M( ˙Φ), [m( ˙ϕ1), · · · , m( ˙ϕi), · · · , m( ˙ϕN)]T (12d) m( ˙ϕ), Fc,τl+ Fs,τle − ˙ ϕ ˙ ϕs,τl α . (12e) In this setting, the vector of unknowns isθ= [T, w]T which
has log-likelihood function given by
log L(θ) = log Nτf; ¯µ( ˙Φ, θ), ¯Σ( ˙Φ)
. (13)
It is further considered that the model parameters are known. The parameters for (5), e.g. given in Table I, can be identified for a new robot using joint temperature measurements and an estimate of the joint load torques, which can be achieved from a robot model (see e.g. [33]). The parameters for (7), describing the wear behavior, are more difficult because failure data is required. For CBM, wear estimates are needed before a failure of the system, in which case the parameters for (7) cannot be known in advance. This can be overcome with the use of historical failure data. The studies that follow in this section illustrate the case where these models are known, focusing on the effects of temperature and load uncertainties. In Sec. V, the effects of uncertainties in the wear model are studied based on real data.
The wear level w, with values larger or equal to zero, is the unknown quantity of interest. Joint temperature measurements are considered unavailable, which is typical in industrial applications, but with known lower and upper limits T , T .
For a robot operating in a controlled indoor environment, T
would be minimum room temperature while T is given by
the maximum room temperature and self heating of the joint because of actuator losses.
Under this setup, the objective is to estimate the wear level w present given data τf. The estimate is, of course,
dependent on τf and thus on the choice of ˙Φ. The problem
of experiment design is to choose ˙Φ such that the estimated
wear level is as accurate as possible.
A. Experiment Design
An estimate ˆθ of θ is dependent on the data set, i.e. τf and
the associated ˙Φ, and on the estimator used. The mean square
˙ ϕ(rad/s) |f ′|T 50 100 150 200 250 20 40 60 80 1 2 3 4 5 6 x 10−3 (a) Contour of f′ T . ˙ ϕ(rad/s) |f ′ w| 50 100 150 200 250 0 20 40 60 80 100 2 4 6 8 x 10−4 (b) Contour of|f′ w|.
Fig. 7. Information content of T and w contained in the model as
a function ofϕ. The dashed line relates to the value where they are˙
0; values to the right of the line are negative and otherwise positive. Notice that the scale used forf′
T is a factor of 10 larger than forfw′
and their almost complementary speed dependence.
error of an estimate can be used as a criterion to assess how the choice of ˙Φ affects the performance. Let the bias of an
estimate ˆθ be denoted b(θ),E[ˆθ] − θ then, from the
Cram´er-Rao lower bound (see e.g. exercise 2.4.17 in [38]), it follows
MSE(ˆθ) = Eh(ˆθ− θ)2i= Var(ˆθ) + b(θ)bT(θ) (14a)
≥ b(θ)b(θ)T + [I + ∇ θb(θ)] F (θ) −1[I + ∇ θb(θ)]T (14b) where F(θ) = E∇θlog L(θ)(∇θlog L(θ))T (15) is the Fisher information matrix. Neglecting the bias term, which is a function of the estimator used, the lower bound on the MSE can be minimized by affecting the Fisher information matrix, e.g. through ˙Φ, improving the achievable performance
for any unbiased estimator.
For the log-likelihood function in (13), the Fisher informa-tion matrix is given by (see [39] for a proof)
F( ˙Φ, θ) = [∇θµ( ˙¯ Φ, θ)] ¯Σ( ˙Φ)−1[∇θµ( ˙¯ Φ, θ)]T (16)
where the dependence on ˙Φ is highlighted. For θ = [T, w]T,
the information matrix is dependent on products of the terms
f′ T , ∂τf( ˙ϕ, θ) ∂T , f ′ w, ∂τf( ˙ϕ, θ) ∂w (17)
which relate to the information aboutT and w contained in the model. Because of the structure of the model, these derivatives are only function of ϕ and of the differentiation variable.˙
Fig. 7 shows contour plots of the amplitude of these functions when the parameters for the model are given by the values in Tables I and II.
The objective of experiment design is to choose ˙Φ that
minimizes the bound on MSE(ˆw), i.e. ˙Φ∗ = arg min ˙ Φ [F ( ˙Φ, θ) −1] 2,2.
Dropping the argument forF( ˙Φ, θ), the analytical expression
for [F−1] 2,2 is given by [F−1] 2,2= [F ]1,1 [F ]1,1[F ]2,2− [F ]21,2 . (18)
For a positive definite ¯Σ( ˙Φ), the problem is well-posed only
if∇θµ( ˙Φ, θ) has rank equal to the number of unknowns. This
can only be achieved if N≥ 2 and if there are at least two
different speed values are chosen. To ensure the later, addi-tional constraints are added to keep a minimum separation,δϕ˙,
between each speed level in ˙Φ. Furthermore, the search is
limited to the minimum ϕ and maximum˙ ϕ speed levels for˙
which the experiment of Sec. II-A can be performed.
˙Φ∗ = arg min ˙ Φ [F ( ˙Φ, θ)−1] 2,2 (19a) s.t. ϕ˙i− ˙ϕj≤ −δϕ˙, (i < j) (19b) ˙ ϕ≤ ˙ϕi ≤ ˙ϕ (19c)
The problem (19) is a constrained nonlinear minimization which is solved here using fmincon in Matlab. To avoid local minima, the initial values are chosen from a coarse grid search.
1) The case whereN= 1: can be considered by
marginaliz-ing the effects ofT from the likelihood function. Considering
that T can occur with equal probability over its domain, the
marginalized likelihood function is,
¯ p(τf|w) = 1 T− T Z T T ¯ p(τf|T, w) dT (20)
Since there is no analytical solution for (20), Monte Carlo Integration (MCI) is used to approximate it in a symbolic expression as ˆ¯ p(τf|w) = 1 NT NT X i=1 ¯ p(τf|, T(i), w) (21)
forNT randomly generated samples,T(i), uniformly sampled
in its domain of integration. Using this approximation the Fisher information is ¯ F( ˙Φ, w) = E " ∂ log ˆ¯p(τf|w) ∂w 2# . (22)
The differentiation of ˆp(·) is performed symbolically and the¯
expectation is computed using MCI with Nτf samples taken
from p(τ¯ f|w), leading to the estimate ˆ¯F( ˙Φ, w) of ¯F( ˙Φ, w).
The associated optimization problem is thus
˙Φ∗ = arg min ˙ Φ ˆ ¯ F( ˙Φ, w)−1 (23a) s.t. ϕ˙ ≤ ˙ϕi≤ ˙ϕ (23b)
which is also a constrained nonlinear minimization and is solved in the same manner as (19).
2) Analyses: The problems (23) and (19) are solved for N = 1 and N = [2, 3, 4] respectively when [w, T ] = [35, 40].
The parameters for the friction model are taken from Tables I and II. The remaining parameters for the data generation model and optimization are given below
µε¯= 0, σε¯= 5.7010−3, (24a) µτl= 0.5, στl= 0.1, (24b) T = 30, T = 50, (24c) ˙ ϕ= 1, ϕ˙ = 280, (24d) δϕ˙ = 5. [NT, Nτf] = [100, 200] (24e)
The optimal speed values found are shown in Table III, which have values in a region around [30 − 50]rad/s. In this speed
TABLE III
CHOICE OF OPTIMAL SPEED VALUES FOR DIFFERENT VALUES OF FRICTION OBSERVATIONSN . “COST”IS THE VALUE OF THE
OBJECTIVE FUNCTION IN(23) (N= 1)OR(19) (N ≥ 2) COMPUTED ATΦ˙∗. N Cost Φ˙∗ 1 45.91 33.78 2 26.01 [35.84, 40.84]T 3 19.65 [33.68, 38.68, 43.68]T 4 16.50 [31.65, 36.65, 41.65, 46.65]T 0 50 100 150 200 250 0 0.5 1 1.5 2 2.5 x 10−3 ˙ ϕ(rad/s) |f′ T| |fw′| (a)|f′ T| and |fw′|. 10 20 30 40 50 60 70 80 30 35 40 45 50 ˙ ϕ(rad/s) T (C ◦) |f′ w| > 2|fT′| (b) Speed region where|f′ w| > 2|fT′|.
Fig. 8. (a) Behavior off′ wandf
′
T with speed evaluated at [T, w] = [40, 35]. (b) The speed regions which give |f′
w| > 2|f ′
T| when w = 35
andT varies in the band[30 − 50]C◦.
region|f′
w| is larger than |f ′
T| by a factor of 2. The terms |f ′ w|
and|f′
T| evaluated at [T, w]=[40, 35] are shown in Fig. 8(a).
The choice of ˙Φ∗
is also dependent on the operating points for w and T . To illustrate these effects, the shaded region in Fig. 8(b) displays the speed region where |f′
w| > 2|f ′ T|
when w is fixed at 35 and T varies in the range [30 − 50]C◦
. Notice that this speed region is not optimal in the sense of (19) or (23), but relates to a region where the information for w is considerably larger than for T . As it can be seen, only a
narrow band of speed values contain useful information for the estimation of w. The speed band also varies with temperature, with no overlap over all temperature values considered.
It is important to stress that the objective of the experiment design problem proposed here is to achieve as accurate as possible estimate w, irrespective of the performance for ˆˆ T .
Different criteria could be used, e.g. to minimize the trace of
F( ˙Φ, θ)−1, which would choose ˙Φ with different relevance to
the effects ofT .
B. Maximum Likelihood Estimation
Given the number of measurements allowed,N , the maxi-mum likelihood estimate ofθ given the data vector τf is the
value for which the log-likelihood function, given in (13), has a maximum, i.e.
ˆ
θ= arg max
θ log L(θ).
The terms dependent onθ in the log-likelihood function have
the form log L(θ) ∝ −hτf− ¯µ( ˙Φ, θ) iT ¯ Σ( ˙Φ)−1h τf− ¯µ( ˙Φ, θ) i .
and the problem is therefore a weighted nonlinear least-squares, where T and w are estimated jointly,
[ ˆT ,w] = arg minˆ T ,w h τf− ¯µ( ˙Φ, T, w) iT ¯ Σ( ˙Φ)−1 (25a) h τf− ¯µ( ˙Φ, T, w) i s.t. 0 ≤w (25b) T ≤T ≤ T , (25c)
where the constrains were added according to the available knowledge of the unknowns, restricting the search space. The problem can be solved using standard least-squares solvers. Here, lsqnonlin available in Matlab is used with initial values found from a coarse grid search.
The estimator above is valid for N ≥ 2 since at least
two equations are needed to solve for the two unknowns. ForN= 1, the marginalized likelihood function approximation
given in (21) can be used, leading to the problem
ˆ w = arg min w − 1 NT NT X i=1 ¯ p(τf|, T(i), w) (26) s.t. 0 ≤w. (27)
This is a nonlinear constrained problem which is solved here using fminconfrom Matlab with initial values taken from a coarse grid search.
1) Simulation Results.: With the optimal speed values
found in Table III, the bias and variance properties of the proposed estimators are evaluated based on Monte Carlo simulations. The same setup used for experiment design in Sec. IV-A is considered, with model parameters values given in Tables I and II and simulation/optimization parameters given in (24). The true wear level is fixed at w0= 35 and
temperature is varied in the range T0 = [30, 50]. The data
generation model (9) is used to generate τf at the different
operation points which is input to (27) or (25) for N = 1
and N = [2, 3, 4] respectively, with speed values given in
Table III. The estimation is repeated a total of NMC= 1103
per operating point to assess the estimators’ performance. Fig. 9 shows the simulation results for the bias and variance of the estimators as a function of the true temperature levelT0.
As it can be seen, the bias and variance are reduced with N .
The reduction in the variance is specially sensitive for N= 2
compared toN= 1, which is related to marginalization effects
ofT . While the variance presents a constant behavior with T0,
the bias has a nonlinear behavior.
V. STUDIES BASED ON REAL DATA
Gathering enough informative data related to wear from the field would have been inviable since wear faults take a long time to develop and are infrequent. Even in accelerated wear tests, a fault may take several months or years before it becomes significant. Another difficulty with such tests are the high costs of running several robots to obtain reliable statistics. Moreover, temperature studies are challenging since the thermal constant of a large robot is several hours.
The only viable alternative in the research project was to combine wear-free friction data collected from a robot under
30 32 34 36 38 40 42 44 46 48 50 −4 −2 0 2 4 T0(C◦) b ias N= 1 N= 2 N= 3 N= 4 (a) BiasE [ ˆw − w0]. 30 32 34 36 38 40 42 44 46 48 50 10 15 20 25 30 35 40 45 50 T0(C◦) var ian ce NN= 1= 2 N= 3 N= 4 (b) VarianceE(ˆw − E [ ˆw])2.
Fig. 9. Monte-Carlo based estimates of bias (a) and variance (b) of the estimators (27) and (25) (N= 1 and N = [2, 3, 4] respectively)
evaluated with w0= 35 and T0 in the band[30 − 50]C◦.
variations ofT and τlwith wear profile data collected from a
different robot of the same type during accelerated wear tests under constant load/temperature conditions. The first data set is assigned as Tf and describes the normal behavior of the
robot. Each element in the dataset has an associated speed, load and temperature value which is written as[Tf]( ˙ϕ, τl, T).
The wear profile data, achieved using in (6), is assigned ˜Tf,
where each datum is associated with a time-indexk and speed
value, i.e.[ ˜Tf](k, ˙ϕ). Neglecting any possible combined effects
of load/temperature and wear, these datasets are combined as a function of k according to
τf(k, ˙ϕ) = [Tf]( ˙ϕ, τl(k), T (k)) + [ ˜Tf](k, ˙ϕ) (28)
which has the same structure as the friction model used in the wear estimators, see (9). The behavior of wear with k is
determined by the wear profile data used and the effects of disturbances can be examined by choosing T(k) and τl(k).
Notice that these data are not analytically generated, but actually based on constant-speed friction data, collected with the experiment described in Sec. II-A.
A. Description of Scenarios
Three different wear profile datasets are considered, they are assigned as ˜T0
f, ˜Tf1 and ˜Tf2. The dataset ˜Tf0 was used
for the wear modeling presented in Sec. III, and is shown in Fig. 4. The other two are shown in Fig. 10. Some relevant characteristics of these data are listed below.
˜ T0
f small random variations, remaining around 0 up
to k = 90 followed by an exponential increase
thereafter.
˜ T1
f medium random variations, remaining around 0 up
to k = 70 followed by large increases. It has a
maximum amplitude which is 56% of that found
in ˜T0 f.
˜ T2
f small random variations, remaining stationary up
tok= 30 followed by small increases up to k = 97
from where it increases steeply. It has a maximum amplitude which is106% of that found in ˜T0
f.
Only one dataset is used to describe the normal behavior of friction and is assigned asT0
f. Three scenarios are considered
using the dataset pairs(T0
f, ˜Tf0), (Tf0, ˜Tf1) and (Tf0, ˜Tf2). The
scenarios are called 0, 1 or 2 according to the wear profile used.
To simplify the presentation of the results, the behavior of T(k) and τl(k) are the same for the three scenarios. The
TABLE IV
CHOICE OF OPTIMAL SPEED VALUES FOR DIFFERENT VALUES OF FRICTION OBSERVATIONSN . Possible values:[2.1, 8.7, 15.3, 21.9, 28.5, 35.1, 41.7, 82.2, 133.5, 184.7, 236.2, 287.1] N Cost Φ˙∗ 1 46.58 35.1 2 26.20 [35.1, 41.7]T 3 22.60 [28.5, 41.7, 82.2]T 4 18.00 [2.1, 28.5, 35.1, 41.7]T
associated behavior of wear-free friction for the scenarios and the values of T and τl are shown in Fig. 11. Notice that the
amplitude of the friction changes due to temperature and load are considerably larger than of those caused by wear for any of the scenarios. The maximum change value found for the wear-free friction behavior is 157% relative to the maximum
change found in ˜T0 f.
The same parameters for the friction model and estimators are used in all scenarios. The parameters for the estimators are given in (24). The model parameters used to describe the normal behavior of friction, i.e. related to τf( ˙ϕ, τl, T) in (8),
are given in Table I and were identified using the dataset T0 f.
The wear parameters used, i.e. related to τ˜f( ˙ϕ, w) in (8), are
given in Table II and were identified using the dataset ˜T0 f.
In this setting, the use of a dataset with superscript 0 in a scenario, means that the related model parameters are considered correct otherwise they are uncertain. That is, the correct parameters for τf( ˙ϕ, τl, T) are considered available
in all scenarios while the wear parameters are uncertain for Scenarios 1 and 2.
B. Results and Discussion
The datasets considered contain friction data collected un-der 12 different speed levels. The choice of speed values for experiment design is thus limited to these speed levels. The problems (19) and (23) are solved by considering every possible combination of speed level for N= [1, 2, 3, 4]. The
resulting optimal values are given in Table IV and relate well to those found in Table III. Notice that the optimal values depend on the wear model parameters used which in this case is only consistent for Scenario 0.
The resulting wear estimates for the different scenarios are shown in Figs. 12(a) to 12(c). The same axes are used in the figures so they are directly comparable. The shaded area highlights a region which should be easily distinguishable from the rest in order to allow for a simple detection of excessive wear.
The wear estimates become smoother for larger N , which
is in line with the simulation study of Sec. IV-B. For all scenarios, the larger wear estimates for k >90 allows for
a distinction of the critical (shaded) regions. Noticeably, the wear estimates are consistent to the wear profile data used in all scenarios, even for Scenarios 1 and 2 when the wear model is uncertain.
The fact that the wear estimates do not differ much withN
might lead to the conclusion thatN= 1 should be used, but this
is only true if the optimal speed values are chosen. To illustrate this, wear estimates were achieved for Scenario 2 using one measurement forϕ˙= 82.2 and ˙ϕ= 133.5. As it can be noticed,
the wear estimates are considerably affected by changes in temperature when these speed values are used. However, when these two measurements are used together, the estimate becomes less sensitive. The inclusion of measurements around the optimal speed values will also increase robustness to uncertainties in the wear model. This is because the optimal speed values depend on the wear model parameters used, which are typically unknown before a fault appears.
VI. CONCLUSIONS
From the simulation and experimental results, it is possible to conclude that the simultaneous estimation of temperature and wear, when N ≥ 2, provides more reliable estimates
compared toN= 1 when the marginalized likelihood function
in (27) is used. In general, it can be expected that additional measurements will increase the estimates’ accuracy, with the tradeoff of an increased experiment time.
A natural extension to this work is to consider on-line estimation, without the need for data collected from experi-ments. This could perhaps be achieved by considering data from a friction observer, e.g. as presented in [20], [21]. The sensitivity of such approach to unmodeled phenomena, e.g. due to dynamic friction, and external disturbances should be considered carefully based on experiments performed on a real robot in different applications.
The wear estimates could possibly be improved further in case estimates/measurements of the joint temperatures were available. Joint temperature sensors are typically unavailable in applications, more common is perhaps the availability of environment temperature sensors. In the later case, estimates of the joint temperature are possible based on a known heat transfer model, e.g. using the approach presented in [8]. The temperature in the joints can also be possibly inferred based on estimates of the motor temperature, which can be achieved based on a known relation between temperature and the motor constant, see e.g. [40].
The studies presented here are restricted to one type of gearbox. It would be interesting to study the behavior of friction- and wear effects in other gear types. Also interesting is to consider other types of variations and how these affect the models and framework presented. For example, a change of lubricant may require the re-estimation of all or some of the friction parameters used.
For CBM, it is important to provide an accurate decision on when to perform maintenance. Based on the laboratory test studies presented, this should be done carefully. For example, while a threshold set at 35 could be used for Scenarios 0
and 1, the same threshold would give a too early detection for Scenario 2. A too early detection is understood as less critical than a total failure of the system but may lead to unnecessary maintenance actions. More careful analyses of the wear estimates may therefore be needed in order to give an accurate support for a maintenance decisions. Perhaps the influence on the wear rate should also be considered in
0 50 100 150 200 250 300 0 20 40 60 80 100 −0.01 0 0.01 0.02 0.03 0.04 0.05 k ˙ ϕ(rad/s) ˜τf τ0 f
(a) Wear profile data ˜T1 f. 0 50 100 150 200 250 300 0 20 40 60 80 100 −0.02 0 0.02 0.04 0.06 0.08 0.1 k ˙ ϕ(rad/s) ˜τf τ0 f
(b) Wear profile data ˜T2 f.
Fig. 10. Friction wear profile data used in Scenarios 1 (a) and 2 (b). The dashed line indicates a wear level to be found. The dotted lines relate to the friction curveτf0 before the wear tests started.
0 50 100 150 200 250 300 0 20 40 60 80 100 0 0.05 0.1 0.12 k ˙ ϕ(rad/s) τf offset: 0.0376
(a) Nominal friction behavior.
0 10 20 30 40 50 60 70 80 90 100 30 35 40 45 50 k T (C ◦) 0 10 20 30 40 50 60 70 80 90 1000 0.2 0.4 0.6 0.8 τl τl T (b) Associated T and τl.
Fig. 11. Behavior of normal (wear-free) friction as a function ofϕ and k for the scenarios considered (a); an offset value corresponding to˙
the smallest friction value in the dataset was removed for a comparison to the wear effects. The associated temperature and load values are shown in (b). 0 10 20 30 40 50 60 70 80 90 100 0 20 40 60 80 100 k ˆw N= 1 N= 2 N= 3 N= 4 (a) Scenario 0. 0 10 20 30 40 50 60 70 80 90 100 0 20 40 60 80 100 k ˆw N= 1 N= 2 N= 3 N= 4 (b) Scenario 1. 0 10 20 30 40 50 60 70 80 90 100 0 20 40 60 80 100 k ˆw N= 1 N= 2 N= 3 N= 4 (c) Scenario 2. 0 10 20 30 40 50 60 70 80 90 100 0 20 40 60 80 100 k ˆw ˙Φ=82.2 ˙Φ=133.5 ˙Φ=[82.2, 133.5]T
(d) Scenario 2 with non-optimal speed values.
Fig. 12. Wear estimates for the different scenarios investigated. Figs. (a) to (c) present the estimates forN= [1, 2, 3, 4] using the optimal
speed values. Fig. (d) illustrates Scenario 2 when non-optimal speed values are used forN= [1, 2]. The shaded area in the figures relates
the decision rule. The determination of the experimentation frequency and remaining lifetime are also important. The study of lifetime models is therefore important.
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Andr´e Carvalho Bittencourt graduated in Automatic Control Engineer with
honors from the Federal University of Santa Catarina, Florian´opolis, Brazil. He received a Licentiate degree in January 2012 from Link¨oping University, Sweden, where he is currently a Ph.D. student. His main research interests are industrial robotics, diagnosis and condition monitoring.
Patrik Axelsson received the M.Sc. degree in applied physics and electrical
engineering in January 2009 and the Licentiate degree in automatic control in December 2011, both from Link¨oping University, Sweden. His research interests are sensor fusion and control for industrial manipulators.
Division, Department
Division of Automatic Control Department of Electrical Engineering
Date 2013-03-15 Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport
URL för elektronisk version http://www.control.isy.liu.se
ISBN ISRN
Serietitel och serienummer
Title of series, numbering ISSN1400-3902
LiTH-ISY-R-3058
Titel
Title Modeling and Experiment Design for Identi cation of Wear in a Robot Joint under Load andTemperature Uncertainties based on Constant-speed Friction Data
Författare
Author André Carvalho Bittencourt, PAtrik Acelsson Sammanfattning
Abstract
The eects of wear to friction are studied based on constant-speed friction data collected from dedicated experiments during accelerated wear tests. It is shown how the eects of temperature and load uncertainties produce larger changes to friction than those caused by wear, motivating the consideration of these eects. Based on empirical observations, an ex-tended friction model is proposed to describe the eects of speed, load, temperature and wear. Assuming availability of such model and constant-speed friction data, a maximum likelihood wear estimator is proposed. A criterion for experiment design is proposed which selects speed points to collect constant-speed friction data which improves the achievable performance bound for any unbiased wear estimator. Practical issues related to experiment length are also considered. The performance of the wear estimator under load and temper-ature uncertainties is found by means of simulations and veri ed under three case studies based on real data.
Nyckelord
