Embed Size (px)
Citation preview
SENSOR-ASSISTED ADAPTIVE MOTOR CONTROLUNDER CONTINUOUSLY VARYING CONTEXT
Heiko Hoffmann, Georgios Petkos, Sebastian Bitzer, and Sethu VijayakumarInstitute of Perception, Action and Behavior, School of Informatics, University of Edinburgh, Edinburgh, UK
[email protected], [email protected], [email protected], [email protected]
Keywords: Adaptive control, context switching, Kalman filter, force sensor, robot simulation
Abstract: Adaptive motor control under continuously varying context, like the inertia parameters of a ma-nipulated object, is an active research area that lacks a satisfactory solution. Here, we present andcompare three novel strategies for learning control under varying context and show how addingtactile sensors may ease this task. The first strategy uses only dynamics information to infer theunknown inertia parameters. It is based on a probabilistic generative model of the control torques,which are linear in the inertia parameters. We demonstrate this inference in the special case of asingle continuous context variable – the mass of the manipulated object. In the second strategy,instead of torques, we use tactile forces to infer the mass in a similar way. Finally, the thirdstrategy omits this inference – which may be infeasible if the latent space is multi-dimensional –and directly maps the state, state transitions, and tactile forces onto the control torques. Theadditional tactile input implicitly contains all control-torque relevant properties of the manipu-lated object. In simulation, we demonstrate that this direct mapping can provide accurate controltorques under multiple varying context variables.
1 INTRODUCTION
In feed-forward control of a robot, an inter-nal inverse-model of the robot dynamics is usedto generate the joint torques to produce a desiredmovement. Such a model always depends on thecontext in which the robot is embedded, its envi-ronment and the objects it interacts with. Someof this context may be hidden to the robot, e.g.,properties of a manipulated object, or externalforces applied by other agents or humans.
An internal model that does not incorporateall relevant context variables needs to be re-learned to adapt to a changing context. Thisadaptation may be too slow since sufficientlymany data points need to be collected to up-date the learning parameters. An alternative is tolearn different models for different contexts andto switch between them (Narendra and Balakr-ishnan, 1997; Narendra and Xiang, 2000; Petkoset al., 2006) or combine their outputs accordingto a predicted error measure (Haruno et al., 2001;Wolpert and Kawato, 1998). However, the for-mer can handle only previously-experienced dis-crete contexts, and the latter have been testedonly with linear models.
The above studies learn robot control based
purely on the dynamics of the robot; here, wedemonstrate the benefit of including tactile forcesas additional input for learning non-linear in-verse models under continuously-varying context.Haruno et al. (Haruno et al., 2001) already usesensory information, but only for mapping a vi-sual input onto discrete context states.
Using a physics-based robot-arm simulation(Fig. 1), we present and compare three strategiesfor learning inverse models under varying context.The first two infer the unknown property (hid-den context) of an object during its manipulation(moving along a given trajectory) and immedi-ately use this estimated property for computingcontrol torques.
In the first strategy, only dynamic data areused, namely robot state, joint acceleration, andjoint torques. The unknown inertia parameters ofan object are inferred using a probabilistic gener-ative model of the joint torques.
In the second strategy, we use instead of thetorques the tactile forces exerted by the manipu-lated object. The same inference as above can becarried out given that the tactile forces are lin-ear in the object’s mass. We demonstrate bothof these steps with mass as the varying context.If more context variables are changing, estimat-
Figure 1: Simulated robot arm with gripper and forcesensors and its real counterpart, the DLR light-weightarm III. Arrows indicate the three active joints usedfor the experiments. The curve illustrates the desiredtrajectory of the ball.
ing only the mass as hidden context is insuf-ficient. Particularly, if also the mass distribu-tion changes, both the center of mass and inertiatensor vary, leading to a high-dimensional latentvariable space, in which inference may be infeasi-ble given a limited number of data points.
In our third strategy, we use a direct mappingfrom robot state, joint accelerations, and tactileforces onto control torques. This mapping allowsaccurate control even with more than one chang-ing context variable and without the need to ex-tract these variables explicitly.
The remainder of this article is organized asfollows. Section 2 briefly introduces the learn-ing of inverse models under varying context. Sec-tion 3 describes the inference of inertia parame-ters based only on dynamic data. Section 4 de-scribes the inference of mass using tactile forces.Section 5 motivates the direct mapping. Section6 describes the methods for the simulation ex-periments. Section 7 shows the results of theseexperiments; and Section 8 concludes the article.
2 LEARNING DYNAMICS
Complex robot structure and non-rigid bodydynamics (e.g, due to hydraulic actuation) canmake analytic solutions to the dynamics inaccu-rate and cumbersome to derive. Thus, in our ap-proach, we learn the dynamics for control frommovement data. Particularly, we learn an in-verse model, which maps the robot’s state (here,
joint angles θ and their velocities θ) and its de-
sired change (joint accelerations θ) onto the mo-tor commands (joint torques τ) that are requiredto produce this change,
τ = µ(θ, θ, θ) . (1)
Under varying context, learning (1) is insufficient.Here, the inverse model depends on a contextvariable π,
τ = µ(θ, θ, θ, π) , (2)
In Sections 3 and 4, we first infer the hiddencontext variable and then plug this variable intofunction (2) to compute the control torques. InSection 5, the context variable π is replaced bysensory input that implicitly contains the hiddencontext.
3 INFERRING CONTEXT
FROM DYNAMICS
During robot control, hidden inertia parame-ters can be inferred by observing control torquesand corresponding accelerations (Petkos and Vi-jayakumar, 2007). This inference can be carriedout efficiently because of a linear relationship inthe dynamics, as shown in the following.
3.1 Linearity in robot dynamics
The control torques τ of a manipulator are linearin the inertia parameters of its links (Sciaviccoand Siciliano, 2000). Thus, µ can be decomposedinto
τ = Φ(θ, θ, θ)π . (3)
Here, π contains the inertia parameters,π = [ m1, m1l1x, m1l1y, m1l1z , J1xx, J1xy, ..., mn,mnlnx, mnlny, mnlnz, Jnxx, ..., Jnzz], where mi isthe mass of link i, li its center-of-mass, and Ji itsinertia tensor. The dynamics of a robot holdingdifferent objects only differs in the π parametersof the combination of object and end-effector link(the robot’s link in contact with the object). Toobtain Φ, we need to know for a set of contexts cthe inertia parameters πc and the correspondingtorques τc. Given a sufficient number of τc andπc values, we can compute Φ using ordinaryleast squares. After computing Φ, the robot’sdynamics can be adjusted to different contextsby multiplying Φ with the inertia parameters π.The following two sections show how to estimateπ once Φ has been found.
3.2 Inference of inertia parameters
Given the linear equation (3) and the knowledgeof Φ, the inertia parameters π can be inferred.Assuming Gaussian noise in the torques τ , theprobability density p(τ |θ, θ, θ, π) equals
p(τ |θ, θ, θ, π) = N (Φπ, Σ) , (4)
where N is a Gaussian function with mean Φπand covariance Σ. Given p(τ |θ, θ, θ, π), the mostprobable inertia parameters π can be inferred us-ing Bayes’ rule, e.g., by assuming a constant priorprobability p(π):
p(π|τ, θ, θ, θ) ∝ p(τ |θ, θ, θ, π) . (5)
3.3 Temporal correlation of inertia
parameters
The above inference ignores the temporal correla-tion of the inertia parameters. Usually, however,context changes infrequently or only slightly.Thus, we may describe the context π at time stept + 1 as a small random variation of the contextat the previous time step:
πt+1 = πt + εt+1 , (6)
where ε is a Gaussian-distributed random vari-able with constant covariance Ω and zero mean.Thus, the transition probability p(πt+1|πt) isgiven as
p(πt+1|πt) = N (πt, Ω) . (7)
Given the two conditional probabilities (4) and(7), the hidden variable π can be described with aMarkov process (Fig. 2), and the current estimateπt can be updated using a Kalman filter (Kalman,1960). Written with probability distributions, thefilter update is
p(πt+1|τt+1, xt+1) = η p(τt+1|xt+1, πt+1)
·
∫
p(πt+1|πt)p(πt|τt, xt)dπt , (8)
where η is a normalization constant, and x standsfor θ, θ, θ to keep the equation compact. A vari-able with superscript t stands for all observations(of that variable) up to time step t.
3.4 Special case: Inference of mass
We will demonstrate the above inference in thecase of object mass m as the hidden context. This
...
τt
πt
xt+1
πt+1
τt+1
xt
...
Figure 2: Hidden Markov model for dependence oftorques τ on context π. Here, the state and statetransitions are combined to a vector x = θ, θ, θ.
restriction essentially assumes that the shape andcenter of mass of the manipulated object are in-variant. If m is the only variable context1, thedynamic equation is linear in m,
τ = g(θ, θ, θ) + m h(θ, θ, θ) . (9)
In our experiments, we first learn two mappingsτ1(θ, θ, θ) and τ2(θ, θ, θ) for two given contexts m1
and m2. Given these mappings, g and h can becomputed.
To estimate m, we plug (4) and (7) into thefilter equation (8) and use (9) instead of (3). Fur-thermore, we assume that the probability distri-bution of the mass at time t is a Gaussian withmean mt and variance Qt. The resulting updateequations for mt and Qt are
mt+1 =hT Σ−1(τ − g) + mt
Qt+Ω
1
Qt+Ω+ hT Σ−1h
, (10)
Qt+1 =
(
1
Qt + Ω+ hT Σ−1h
)
−1
. (11)
Section 7 demonstrates the result for this infer-ence of m during motor control. For feed-forwardcontrol, we plug the inferred m into (9) to com-pute the joint torques τ .
4 INFERRING CONTEXT
FROM TACTILE SENSORS
For inferring context, tactile forces measuredat the interface between hand and object may
1This assumption is not exactly true in our case. Achanging object mass also changes the center of massand inertia tensor of the combination end-effector linkplus object. Here, to keep the demonstration simple,we make a linear approximation and ignore terms ofhigher order in m – the maximum value of m wasabout one third of the mass of the end-effector link.
serve as a substitute for the control torques. Wedemonstrate this inference for the special case ofobject mass as context.
The sensory values si (the tactile forces) arelinear in the mass m held in the robot’s hand, asshown in the following. In the reference frame ofthe hand, the acceleration a of an object leads toa small displacement dx (Fig. 3).
Object
dx
dx
u h
Sensor
Figure 3: A force on the object held in the robot’shand leads to a displacement dx. This displacementshifts each sensor at position u (relative to the ob-ject’s center) by h.
This displacement pushes each sensor by theamount hi depending on the sensor’s position ui.Let ei be a vector of unit length pointing in thedirection of ui, then hi = eT
i dx. Our sensorsact like Hookean springs; thus, the resulting forceequals fi = κhiei, with κ being the spring con-stant. Since the object is held such that it cannotescape the grip, the sum of sensory forces fi mustequal the inertial force m a,
m a =
n∑
i=1
fi = κ∑
i
(eTi dx) ei . (12)
This linear equation allows the computation ofdx,
dx =m
κ(ET E)−1a , (13)
0.007
0.008
0.009
0.01
0.011
0.012
0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.018
sens
or 2
sensor 1
Figure 4: Two-dimensional projection of sensor val-ues during figure-8 movements with four differentmasses. From left to right, the mass increases as0.005, 0.01, 0.02, and 0.03.
where E is a matrix with row vectors ei. Thus,each fi is proportional to m. The total force mea-sured at a sensor equals fi plus a constant gripforce (whose sum over all sensors equals zero).Therefore, the sensory values s can be written as
s = s0 + m ϕ(θ, θ, θ) , (14)
where ϕ is a function depending on the state andacceleration of the robot arm. This linearity is il-lustrated in Fig. 4 using data from our simulatedsensors. Based on (14), the same inference as inSection 3 is possible using (10) and (11), and theestimated m can be used for control by using (9).
5 DIRECT MAPPING
The above inference of mass fails if we have ad-ditional hidden context variables, e.g., if the robothand holds a dumb-bell, which can swing. In thisgeneral case, we could still use the inference basedon dynamics. However, since for the combinationof end-effector and object, both the inertia tensorand the center of mass vary, we need to estimate10 hidden context variables (Sciavicco and Sicil-iano, 2000). Given the limited amount of trainingdata that we use in our experiments, we expectthat inference fails in such a high-dimensional la-tent space.
As alternative, we suggest using the sen-sory values as additional input for learning feed-forward torques. Thus, the robot’s state anddesired acceleration are augmented by the sen-sory values, and this augmented input is directlymapped on the control torques. This step avoidsinferring unknown hidden variables, which arenot our primary interest; our main goal is com-puting control torques under varying context.
The sum of forces measured at the tactile sen-sors equals the force that the manipulated objectexerts on the robotic hand. This force combinedwith the robot’s state and desired acceleration issufficient information for predicting the controltorques. Thus, tactile forces contain the relevanteffects of an unknown varying context. This con-text may be, e.g., a change in mass, a change inmass distribution, or a human exerting a force onthe object held by the robot.
Depending on the number of sensors, the sen-sory input may be high-dimensional. However,since the sensors encode only a force vector,the intrinsic dimensionality is only three. Forlearning the mapping τ(θ, θ, θ, s), regression tech-niques exist, like locally-weighted projection re-
gression (Vijayakumar et al., 2005), that can ex-ploit this reduced intrinsic dimensionality effi-ciently. In the following, we demonstrate the va-lidity of our arguments in a robot-arm simulation.
6 ROBOT SIMULATION
This section describes the methods: the sim-ulated robot, the simulated tactile sensors, thecontrol tasks, the control architecture, and thelearning of the feed-forward controller.
6.1 Simulated robot arm
Our simulated robot arm replicates the realLight-Weight Robot III designed by the GermanAerospace Center, DLR (Fig. 1). The DLR armhas seven degrees-of-freedom; however, only threeof them were controlled in the present study; theremaining joints were stiff. As end-effector, weattached a simple gripper with four stiff fingers;its only purpose was to hold a spherical objecttightly with the help of five simulated force sen-sors. The physics was computed with the OpenDynamics Engine (http://www.ode.org).
6.2 Simulated force sensors
Our force sensors are small boxes attached todamped springs (Fig. 1). In the simulation,damped springs were realized using slider joints,whose positions were adjusted by a proportional-derivative controller. The resting position of eachspring was set such that it was always under pres-sure. As sensor reading s, we used the current po-sition of a box (relative to the resting position).
6.3 Control tasks
We used two tasks: moving a ball around an eightfigure and swinging a dumb-bell. In the first, onecontext variable was hidden, the mass of the ball.The maximum mass (m = 0.03) of a ball wasabout one seventh of the total mass of the robotarm. In the second task, two variables were hid-den: the dumb-bell mass and its orientation. Thetwo ball masses of the dumb-bell were equal.
In all tasks, the desired trajectory of the end-effector was pre-defined (Figs. 1 and 5) and itsinverse in joint angles pre-computed. The eightwas traversed only once lasting 5000 time steps,and for the dumb-bell, the end-effector swungfor two periods together lasting 5000 time steps
and followed by 1000 time steps during which theend-effector had to stay in place (here, the con-trol torques need to compensate for the swingingdumb-bell). In both tasks, a movement startedand ended with zero velocity and acceleration.The velocity profile was sinusoidal with one peak.
For each task, three trajectories were pre-computed: eight-figures of three different sizes(Fig. 1 shows the big eight; small and mediumeight are 0.9 and 0.95 of the big-eight’s size, seeFig. 11) and three lines of different heights (Fig.13). For training, data points were used from thetwo extremal trajectories, excluding the middletrajectory. For testing, all three trajectories wereused.
Figure 5: Robot swinging a dumb-bell. The blackline shows the desired trajectory.
6.4 Control architecture
We used an adaptive controller and separatedtraining and test phases. To generate trainingpatterns, a proportional-integral-derivative (PID)controller provided the joint torques. The pro-portional gain was chosen to be sufficiently highsuch that the end-effector was able to follow thedesired trajectories. The integral component wasinitialized to a value that holds the arm againstgravity (apart from that, its effect was negligible).
For testing, a composite controller providedthe joint torques (Fig. 6). The trained feed-
Motionplan
Controller
Plant+_
++
θd
.θd
..θd
θd
.θd
.θ
τ ff
fbττ
Sensormor
θ
PD
Figure 6: Composite control for the robot arm. Afeed-forward controller is put in parallel with errorfeed-back (low PD gain).
forward controller was put in parallel with a low-gain error feed-back (its PD-gain was 1% of thegain used for training in the ball case and 4%in the dumb-bell case). For those feed-forwardmappings that require an estimate of the ob-ject’s mass, this estimate was computed using aKalman filter as described above (the transitionnoise Ω of m was set to 10−9 for both dynamicsand sensor case).
6.5 Controller learning
The feed-forward controller was trained on datacollected during pure PID-control. For each masscontext, 10 000 data points were collected in theball case and 12 000 data points in the dumb-bellcase. Half of these points (every second) wereused for training and the other half for testing theregression performance. Three types of mappingswere learned. The first maps the state and accel-eration values (θ, θ, θ) onto joint torques. Thesecond maps the same input onto the five sensoryvalues, and the third maps the sensor augmentedinput (θ, θ, θ, s) onto joint torques. The first twoof these mappings were trained on two differentlabeled masses (m1 = 0.005 and m2 = 0.03 forball or m2 = 0.06 for dumb-bell). The last map-ping used the data from 12 different mass contexts(m increased from 0.005 to 0.06 in steps of 0.005);here, the contexts were unlabeled.
Our learning task requires a non-linear re-gression technique that provides error boundariesfor each output value (required for the Kalman-filtering step). Among the possible techniques,we chose locally-weighted projection regression(LWPR)2 (Vijayakumar et al., 2005) because itis fast – LWPR is O(N), where N is the numberof data points; in contrast, the popular Gaussianprocess regression is O(N3) if making no approx-imation (Rasmussen and Williams, 2006).
7 RESULTS
The results of the robot-simulation experi-ments are separated into ball task – inference and
2LWPR uses locally-weighted linear-regression.Data points are weighted according to Gaussian re-ceptive fields. Our setup of the LWPR algorithm wasas follows. For each output dimension, a separateLWPR regressor was computed. The widths of theGaussian receptive fields were hand-tuned for each in-put dimension, and these widths were kept constantduring the function approximation. Other learningparameters were kept at default values.
control under varying mass – and dumb-bell task– inference and control under multiple varyingcontext variables.
7.1 Inference and control under
varying mass
For only one context variable, namely the mass ofthe manipulated object, both inference strategies(Sections 3 and 4) allowed to infer the unknownmass accurately (Figs. 7 to 10).
-0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0 1000 2000 3000 4000 5000m
ass
time step
true massdynamic estimate
0
0.01
0.02
0.03
nMSE
Figure 7: Inferring mass purely from dynamics. Theinference results are shown for all three trajectories.The inset shows the normalized mean square error(nMSE) of the mass estimate. The error bars on thenMSE are min and max values averaged over an entiretrajectory.
-0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0 1000 2000 3000 4000 5000
mas
s
time step
true masssensor estimate
0
0.01
0.02
0.03
nMSE
Figure 8: Inferring mass using tactile sensors. Fordetails see Fig. 7.
Both types of mappings from state and accel-eration either onto torques or onto sensors couldbe learned with low regression errors, which wereof the same order (torques with m = 0.005: nor-malized mean square error (nMSE) = 2.9 ∗ 10−4,m = 0.03: nMSE = 2.7 ∗ 10−4; sensors with m =
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0 1000 2000 3000 4000 5000
mas
s
time step
true massdynamic estimate
0
0.01
0.02
0.03
nMSE
Figure 9: Inferring mass purely from dynamics. Fordetails see Fig. 7.
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0 1000 2000 3000 4000 5000
mas
s
time step
true masssensor estimate
0
0.01
0.02
0.03
nMSE
Figure 10: Inferring mass using tactile sensors. Fordetails see Fig. 7.
0.4
0.6
0.8
1
1.2
1.4
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
z-co
ordi
nate
y-coordinate
targetlow-gain PID
sensor estimatedynamic estimate
wrong mass
Figure 11: Following-the-eight task. The figure com-pares low-gain PID control (on large eight only) withthe composite-controller that uses either the pre-dicted mass estimate (from dynamics or sensors) or awrong estimate (m = 0.03, on large eight only). Thetrue mass decreased continuously from 0.03 to 0. Theresults for composite control based on the predictedmass overlap with the target for all three test curves.
0.005: nMSE = 1.3 ∗ 10−4, m = 0.03: nMSE =2.2 ∗ 10−4). The error of the inferred mass wasabout the same for dynamics and sensor pathway.However, the variation between trials was lowerin the sensor case.
Given the inferred mass via the torque andsensory pathways (Sections 3 and 4), our con-troller could provide accurate torques (Fig. 11).The results from both pathways overlap with thedesired trajectory. As illustrated in the figure, aPID controller with a PD-gain as low as the gainof the error feed-back for the composite controllerwas insufficient for keeping the ball on the eightfigure. The figure furthermore illustrates thatwithout a correct mass estimate, tracking was im-paired. Thus, correctly estimating the mass mat-ters for this task.
7.2 Inference and control under
multiple varying context
The inference of mass based on a single hiddenvariable failed if more variables varied (Fig. 12).We demonstrate this failure with the swingingdumb-bell, whose mass increased continuouslyduring the trial. In the last part of the trial,when the dumb-bell was heavy and swung, themass inference was worst. The wrong mass es-timate further impaired the control of the robotarm (see Fig. 13).
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 1000 2000 3000 4000 5000 6000
mas
s
time step
true masssensor estimate
dynamic estimate
Figure 12: Inference of mass in the dumb-bell task.
During the last part of the movement, track-ing was better with the direct mapping from(θ, θ, θ, s) onto torques. For this mapping, theresults still show some deviation from the target.However, we expect this error to reduce with moretraining data.
0.52
0.53
0.54
0.55
0.56
0 1000 2000 3000 4000 5000 6000
z-co
ordi
nate
time step
targetlow-gain PID
dynamic estimatedirect mapping
Figure 13: Swinging-the-dumb-bell task. The figurecompares low-gain PID control (shown only on themiddle trajectory) with the composite-controller thatuses either the predicted mass estimate from dynam-ics or a direct mapping from (θ, θ, θ, s) onto torques.For the estimate from sensors, the results were similarto the dynamics case and are omitted in the graph toimprove legibility. The true mass increased continu-ously from 0 to 0.06.
8 CONCLUSIONS
We presented three strategies for adaptive mo-tor control under continuously varying context.First, hidden inertia parameters of a manipulatedobject can be inferred from the dynamics and inturn used to predict control torques. Second, thehidden mass of an object can be inferred fromthe tactile forces exerted by the object. Third,correct control torques can be predicted by aug-menting the input (θ, θ, θ) with tactile forces andby directly mapping this input onto the torques.
We demonstrated the first two strategies withobject mass as the varying context. For more con-text variables, inferring the mass failed, and thus,the control torques were inaccurate. In principle,all inertia parameters could be inferred from thedynamics, but this inference requires modeling a10-dimensional latent-variable space, which is un-feasible without extensive training data.
On the other hand, the direct mapping ontotorques with sensors as additional input couldpredict accurate control torques under two vary-ing context variables and in principle could copewith arbitrary changes of the manipulated object(including external forces). Further advantagesof this strategy are its simplicity (it only requiresfunction approximation), and for training, no la-beled contexts are required.
In future work, we try to replicate these find-ings on a real robot arm with real tactile sensors.Real sensors might be more noisy compared to
our simulated sensors; particularly, the interfacebetween sensor and object is less well controlled.
ACKNOWLEDGMENTS
The authors are grateful to the German Aerospace
Center (DLR) for providing the data of the Light-
Weight Robot III and to Marc Toussaint and Djordje
Mitrovic for their contribution to the robot-arm sim-
ulator. This work was funded under the SENSOPAC
project. SENSOPAC is supported by the European
Commission through the Sixth Framework Program
for Research and Development up to 6 500 000 EUR
(out of a total budget of 8 195 953.50 EUR); the SEN-
SOPAC project addresses the “Information Society
Technologies” thematic priority. G. P. was funded by
the Greek State Scholarships Foundation.
REFERENCES
Haruno, M., Wolpert, D. M., and Kawato, M. (2001).Mosaic model for sensorimotor learning and con-trol. Neural Computation, 13:2201–2220.
Kalman, R. E. (1960). A new approach to linear filter-ing and prediction problems. Transactions of theASME - Journal of Basic Engineering, 82:35–45.
Narendra, K. S. and Balakrishnan, J. (1997). Adap-tive control using multiple models. IEEE Trans-actions on Automatic Control, 42(2):171–187.
Narendra, K. S. and Xiang, C. (2000). Adaptive con-trol of discrete-time systems using multiple mod-els. IEEE Transactions on Automatic Control,45(9):1669–1686.
Petkos, G., Toussaint, M., and Vijayakumar, S.(2006). Learning multiple models of non-lineardynamics for control under varying contexts. InProceedings of the International Conference onArtificial Neural Networks. Springer.
Petkos, G. and Vijayakumar, S. (2007). Contextestimation and learning control through latentvariable extraction: From discrete to continu-ous contexts. In Proceedings of the InternationalConference on Robotics and Automation. IEEE.
Rasmussen, C. E. and Williams, C. K. I. (2006).Gaussian Processes for Machine Learning. MITPress.
Sciavicco, L. and Siciliano, B. (2000). Modelling andControl of Robot Manipulators. Springer.
Vijayakumar, S., D’Souza, A., and Schaal, S. (2005).Incremental online learning in high dimensions.Neural Computation, 17:2602–2634.
Wolpert, D. M. and Kawato, M. (1998). Multiplepaired forward and inverse models for motor con-trol. Neural Networks, 11:1317–1329.
