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Dynamics Model Abstraction Scheme Using Radial Basis Functions

Tolu, Silvia; Vanegas, Mauricio; Agis, Rodrigo; Carrillo, Richard; Cañas, Antonio

Published in:Journal of Control Science and Engineering

Link to article, DOI:10.1155/2012/761019

Publication date:2012

Document VersionPublisher's PDF, also known as Version of record

Link back to DTU Orbit

Citation (APA):Tolu, S., Vanegas, M., Agis, R., Carrillo, R., & Cañas, A. (2012). Dynamics Model Abstraction Scheme UsingRadial Basis Functions. Journal of Control Science and Engineering, [761019].https://doi.org/10.1155/2012/761019

Journal of Control Science and Engineering manuscript No.(will be inserted by the editor)

Dynamics Model Abstraction Scheme Using Radial BasisFunctions

Silvia Tolu · Mauricio Vanegas · RodrigoAgıs · Richard Carrillo · Antonio Canas

Received: date / Accepted: date

Abstract This paper presents a control model for object manipulation. Properties ofobjects and environmental conditions influence the motor control and learning. Sys-tem dynamics depend on an unobserved external context, e.g., work load of a robotmanipulator. The dynamics of a robot arm change as it manipulates objects with dif-ferent physical properties, e.g., the mass, shape or mass distribution. We address ac-tive sensing strategies to acquire object dynamical models with a radial basis functionneural network (RBF). Experiments are done using a real robot’s arm and trajectorydata are gathered during various trials manipulating different objects. Biped robots donot have high force joint servos and the control system hardly compensates all the in-ertia variation of the adjacent joints and disturbance torque on dynamic gait control.In order to achieve smoother control and lead to more reliable sensorimotor com-plexes we evaluate and compare a sparse velocity-driven vs. a dense position-drivencontrol scheme.

Keywords Dynamics Model · Sensorimotor · Radial Basis Function (RBF) ·Velocity-Driven Control Scheme

1 Introduction

Current research of biomorphic robots can highly benefit from simulations of ad-vance learning paradigms [19, 38, 43, 58, 59] and also from knowledge acquiredfrom biological systems [36, 51, 52]. The basic control of robot actuators is usuallyimplemented by adopting a classic scheme [20, 25, 44, 46, 54, 61]: a) the desired

Silvia Tolu · Rodrigo Agıs · Richard Carrillo · Antonio CanasCITIC-Dept. of Computer Architecture and Technology, ETSI Informatica y de Telecomunicacion, Uni-versity of Granada, Spain.Tel.: +34 958241000 - 31218E-mail: stolu@atc.ugr.esMauricio VanegasPSPC-group. Department of Biophysical and Electronic Engineering (DIBE), University of Genoa, Italy

2 Silvia Tolu et al.

trajectory, in joint coordinates, is obtained using the inverse kinematics [44, 56, 61]and b) efficient motor commands drive, e.g. an arm, making use of the inverse dy-namics model or using high power motors to achieve rigid movements. Differentcontrolling schemes using hybrid position/velocity and forces have been introducedmainly for industrial robots [20, 46]. In fact, industrial robots are equipped with dig-ital controllers, generally of PID type with no possibility of modifying the control al-gorithms to improve their performance. Robust control (control using PID paradigms[44, 47, 50, 56, 61–63]) is very power consuming and highly reduces autonomy ofnon-rigid robots. Traditionally, the major application of industrial robots is related totasks that require only position control of the arm. Nevertheless, there are other im-portant robotic tasks that require interaction between the robot’s end-effector and theenvironment. System dynamics depend on an unobserved external context [16, 43],e.g., work load of a robot manipulator.Biped robots do not have high force joint servos and the control system hardly com-pensates all the inertia variation of the adjacent joints and disturbance torque on dy-namic gait control [13]. Motivated by these concerns and the promising preliminaryresults [57], we evaluate a sparse velocity-driven control scheme. In this case, smoothmotion is naturally achieved, cutting down jerky movements and reducing the prop-agation of vibrations along the whole platform. Including the dynamic model intothe control scheme becomes important for an accurate manipulation, therefore it iscrucial to study strategies to acquire it. There are other bio-inspired model abstrac-tion approaches that could take advantage of this strategy [7, 32–34]. Conventionalartificial neural networks [17, 21] have been also applied to this issue [11, 14, 29]. Inbiological systems [36, 51, 52], the cerebellum [10, 23] seems to play a crucial roleon model extraction tasks during manipulation [4, 24, 55, 61]. The cerebellar cortexhas a unique, massively parallel modular architecture [3, 4, 37] that appears to beefficient in the model abstraction [24, 25].Human sensorimotor recognition [37, 52, 61] continually learns from current and pastsensory experiences. We set up an experimental methodology using a biped robot’sarm [4, 55] which has been equipped, at its last arm-limb, with three acceleration sen-sors [22] to capture the dynamics of the different movements along the three spatialdimensions. In human body, there are skin sensors specifically sensitive to accelera-tion [31, 52]. They represent haptic sensors and provide acceleration signals duringthe arm motion. In order to be able to abstract a model from object manipulation,accurate data of the movements are required. We acquire the position and the accel-eration along desired trajectories when manipulating different objects. We feed thesedata into the radial basis function network (RBF) [17, 29]. Learning is done off-linethrough the knowledge acquisition module. The RBF learns the dynamic model, i.e.it learns to react by means of the acceleration in response to specific input forcesand to reduce the effects of the uncertainty and nonlinearity of the system dynamics[26, 65].To sum up, we avoid adopting a classic regular point-to-point strategy (see Fig. 1.b)with fine PID control modules [25, 44, 56] that drive the movement along a finely de-fined trajectory (position-based). This control scheme requires high power motors inorder to achieve rigid movements (when manipulating heavy objects) and the wholerobot augments vibration artefacts that make difficult accurate control. Furthermore,

Dynamics Model Abstraction Scheme Using Radial Basis Functions 3

a

P1 Pi = PcdPPi Pf Pf

P1 …

d.1

b c

TT

…

Change of direction

Sparse velocity-driven trajectory

Smooth velocity-driven trajectory

Change of direction d.2

Fig. 1 a) Computing scheme. An FPGA processing platform embeds all the real-time control interfaces.The robot is connected to the FPGA which receives commands from the host PC. The robot is equippedwith three accelerometers in its arm. The computer captures the sensor data. A RBF is used for learning theobjects models from sensorimotor complexes. b) Dense Position-Driven scheme. A motor moves from aninitial position P1 to a final position Pf temporally targeting each intermediate point Pi along this trajectory.c) Sparse Velocity-Driven scheme. For each position Pi indicated above a target velocity is calculated fora motor with a time step T of 0.015s. Using a fixed sampling frequency for obtaining feedback position,we calculate the distance ∆P from this current position to a position Pcd corresponding to a change ofdirection. d.1) Dense position-driven trajectory. The trajectory is defined in terms of regularly sampledintermediate points (dashed line). d.2) The thiner continuous line represents the real curve executed bythe robot’s arm in the space under the sparse velocity-driven control. The black arrows indicate points inwhich changes of direction take place. The thicker continuous line corresponds to the smooth velocity-driven trajectory executed anticipating the points in which changes of direction are applied before the armreaches them.

these platform jerks induce high noise in the embodied accelerometers. Contrary tothis scheme, we define the trajectory by a reduced set of target points (Fig. 1.c) andimplement a velocity-driven control strategy that highly reduces jerks.

2 Control Scheme

The motor-driver controller and the communication interface are implemented on anFPGA (Spartan XC31500 [8] embedded on the robot) (Fig. 1.a). Robonova - I [18] isa fully articulating mechanical biped robot, 0.3 m, 1.3 kg, controlled with sixteen mo-tors [18]. We have tested two control strategies for a specific ”L” trajectory previouslydefined for the robot’s arm controlled with three motors (one in the shoulder and twoin the arm, (see Fig. 2). This whole movement along three joints makes use of threedegrees of freedom during the trajectory. The movement that involves the motor atthe last arm-limb is the dominant one, as is evidenced from the sensorimotor valuesin figure 4.b.

4 Silvia Tolu et al.

PWM

Position acquisition modules

Motor-driver controller circuits

(PWM generation)

Motion Plan

Synchronisation

Batch Learning

Fig. 2 Diagram of the whole system architecture. The robot’s picture in the left is showing the differentjoints of the 3 DOF Robonova arm involved in the movement. The PC applied appropriate motor com-mands to each joint of the arm at all times during the movement to follow the desired trajectory. To relievethe computer from interface computation and allow real-time communication with the robot, an FPGAboard containing position acquisition modules and motor-driver controller circuits with PWM (Pulse WithModulation) was connected to the robot. The communication task between devices and the operations de-scribed above were synchronised by the computer speeding up the process. Finally, the batch learning withRBF of sensorimotor data collected during the movements was performed in the PC. The dataset was splitup into training data (75%) and test data (25%).

2.1 Control Strategies

The trajectory is defined in different ways for the two control strategies:Dense Position-Driven. The desired joint trajectory is finely defined by a set of tar-get positions (Pi) that regularly sample the desired movement (see Fig. 1.b).Sparse Velocity-Driven. In this control strategy, the joint trajectory definition is car-ried out by specifying only positions related to changes in movement direction.During all straight intervals, between a starting position P1 and the next change ofdirection Pcd, we proceed to command the arm by modulating the velocity towardsPcd (see Fig. 1.c). Velocities (Vn) are calculated taking into account the last capturedposition and the time step in which the position has been acquired. Each target ve-locity for period T is calculated with the following expression, Vn = (Pcd − Pn)/T.The control scheme applies a force proportional to the target velocity. Figure 1.d.2 il-lustrates how a smoother movement is achieved using sparse velocity-driven control.The dense position-driven strategy reveals noisy vibrations and jerks because eachmotor always tries to get the desired position in the minimum time, whereas it wouldbe better to efficiently adjust the velocity according to the whole target trajectory.The second strategy defines velocities dynamically along the trajectory as describedabove. Therefore, we need to sample the position regularly to adapt the velocity.

Dynamics Model Abstraction Scheme Using Radial Basis Functions 5

2.2 Modelling Robot Dynamics

The dynamics of a robot arm have a linear relationship to the inertial properties of themanipulator joints [5, 28]. In other words, for a specific context r they can be writtenin the form (1),

τ = Υ(q, q, q) · πr. (1)

where q, q, and q are joint angles, velocities and accelerations respectively. ThisEquation 1, based on fundamentals of robot dynamics [12], splits the dynamics intwo terms. Υ(q, q, q) is a term that depends on kinematics properties of the arm suchas direction of the axis of rotation of joints, link lengths, and so on. Term πr is ahigh dimensional vector containing all inertial parameters of all links of the arm [5].Now, let us consider that the manipulated objects are attached at the end of the arm,so we model the dynamics of the arm as the manipulated object being the last link ofthe arm. Then, manipulating different objects is equivalent to changing the physicalproperties of the last link of the arm. The first term of the equation remains constantin different models. It means that all kinematic quantities of the arm remain the samebetween different contexts. Under this assumption, we could use a set of models withknown inertial parameters to infer a predictive model (RBF) of dynamics [17, 27, 29]for any possible context. From another point of view, the previous dynamic Equation(1) could be written in the form (2),

τ = A(q)(q) + H(q, q). (2)

Each dynamic term is non linear and coupled [53] where A(q) is the matrix of inertia,symmetric and positive and H(q, q) includes Coriolis, gravitational, and centrifugalforces. A(q) could be inverted for each robot configuration. Approximating A(q) =A(q) and H(q, q) = H(q, q) we obtain (3),

τ = A(q) · u + H(q, q). (3)

Where u is a new control vector. A(q) and H(q, q) are estimations of the real robotterms. So we have the following equivalence relation (4),

u = q. (4)

Component u is influenced only by a second order term, it depends on the joint vari-ables (q, q), independently from the movement of each joint. To sum up, we replacedthe nonlinear and coupled dynamic expressions with a second order linear systemof non coupled equations. The inputs to the RBF network are the positions and ve-locities of the robot joints, while the outputs are the accelerations. The input-outputrelationship is in accordance with the Equations of motion. In this work, we proposeand show the RBF to be useful in approximating the unknown nonlinearities of thedynamical systems [1, 2, 26, 30, 45, 48, 49, 60, 65] through the control signal (4)found for the estimation method for the dynamics of the joints of a robot.

6 Silvia Tolu et al.

V1

V2

V3

D1

D2

D3

I N P U T

RBF

Network

O U T P U T

S1

S2

Tr

a)

Tr

V1

V2

V3

D1

D2

D3

O U T PU T

I N P U T

Obj

S1r

S2r

b)

RBF

Network

Fig. 3 RBF inputs and outputs. a) The network is used for function approximation of the sensor responsesto specific inputs along different trajectories. b) This network includes an output label (OBJ) for objectclassification and uses as inputs the actual sensor responses (connected to the upper part of the block).

2.3 The RBF Based Modelling

The RBF function is defined as (5):

z(x) = ϕ(∥x − µ∥). (5)

Where x is the n-dimensional input vector, µ is an n-dimensional vector called centreof the RBF, ∥·∥ is the Euclidean distance, and ϕ is the RBF outline.We built our model as a linear combination of N RBF functions with N centres. TheRBF output is given by Equation (6):

y(x) =N

∑j=1

β jzj(x). (6)

Where Bj is the weight of the jth radial function, centered at µ and with an activationlevel zj.RBF has seven inputs: two per each of the three motors in the robot arm (joints) andone for the label of the trajectory which is executed (Tr). Three inputs encode the dif-ference between the current joint position and the next target position in each of thejoints (D1, D2, and D3). The other three inputs (V1, V2, and V3) encode the velocitycomputed on the basis of the previous three inputs. We aim to model the accelera-tion sensory outputs (S1 and S2) (S3 does not provide further information than S2 forthe movement trajectory defined) and we also include one output (label) to classifydifferent kinds of objects (OBJ). See the illustrative block in Fig. 3.b). Therefore,we use two RBF networks. The first one (see figure 3.a)) aims to approximate theacceleration sensor responses to the input motor commands. The second one (see fig-ure 3.b) aims to discriminate object models (Classification task) using as inputs themotor commands and also the actual acceleration estimations given by the sensors[64]. During learning, we feed the inputs and actual outputs of the training set in thenetwork [17, 29]. During the test stage, we evaluate the error obtained from the sen-sors [22] that indicates how accurate the acquired model is and therefore, how stable,

Dynamics Model Abstraction Scheme Using Radial Basis Functions 7

Fig. 4 Plots a) and c) show the motor commands (in joint coordinates) and plots b), d), and e) show thesensor responses during the movement. We compare the dense position-driven vs. the sparse velocity-driven strategies for the case related to object 0 (see the experimental results). In figure b) (Sensor 1) itcan be clearly seen how the sensor responses are directly related with the motor command changes inplot a). The sensor responses, obtained using the sparse velocity-driven strategy, are more stable. Motorcommands increasing the derivative of joint coordinates cause increments in sensor 1 response during acertain period, while motor commands decreasing the derivative of joint coordinates cause decrements insensor 1 response, plot b). The piece of trajectory monitored by these plots is mainly related to sensor1. It can be seen in the right plots d) and e) that the other sensors are not so sensitive to the motions inthis example. Nevertheless, sparse velocity-driven control still leads to much more stable sensor responses.Dense position-driven control causes large vibrations that highly affect the capability of the neural networkto approach the sensor response function.

predictable, and repeatable is the object manipulation. The error metric shown in theresult figures (see Fig. 5) is the Root Mean Square Error (RMSE) in degrees/s2 of thedifferent sensors along the whole trajectory. Finally, the classification error (relatedto the label output) indicates how accurately the network can classify a specific objectfrom the positions, velocities, and accelerometer responses along the trajectory with acorresponding target output. A lower error in the predicted sensor responses indicatesthat the motion is performed in a smoother and more predictable way. Therefore, ascan be seen in Figs. 4, 5, and 6, strategies such as sparse velocity-driven controllead to a more reliable movement control scheme. Furthermore, this allows a betterdynamic characterisation of different objects. This facilitates the use of adaptationstrategies of control gains to achieve more accurate manipulation skills (although thisissue is not specifically addressed in this work).

3 Results

Studies have shown high correlations between the inertia tensor and various hapticproperties like length, height, orientation of the object, and position of the hand grasp[15, 41, 64]. The inertia tensor has the central role to be the perceptual basis for mak-

8 Silvia Tolu et al.

ing haptic judgments of object properties.In order to describe the approach and the pursued movements (several repetitions)and to test different dynamic models of the whole system (robot arm with object)[6, 42], the robot hand manipulated objects with different weights, shapes and posi-tion of grasp. We acquired all the data from the position and accelerator sensors andwe used them for the RBF off-line training. We considered the following cases ofstudy:0. NO object;1. Scissors (0.05 kg, 0.14 m);2. Monkey wrench manipulated fixing the centre (0.07 kg, 0.16 m);3. Monkey wrench manipulated fixing an extreme (0.07 kg, 0.16 m);4. Two Allen keys (0.1 kg, 0.10 m).We repeated manipulation experiments ten times for each case of study collectingsixty thousand data. We applied a cross-validation method defining different datasetsfor the training and test stages. We used the data of seven experiments out of ten fortraining and three for the test stage. We repeated this approach three times shufflingthe experiments of the datasets for training and test. The neural network is built us-ing the MBC model toolbox of MATLAB [35]. We chose the following algorithms:TrialWidths [40] for the width selection and StepItRols for the Lambda and centresselection [9]. We measured the performance calculating the RMSE in degrees/s2 inthe acceleration sensory outputs (S1 and S2).We addressed a comparative study evaluating how accurately the neural networkcan abstract the dynamic model using a dense position-driven and a sparse velocity-driven schemes. Plots in Fig. 4 illustrate how the motor commands are highly relatedwith specific sensor responses (mainly in sensor 1 along this piece of trajectory).Figs. 4.a and 4.c illustrate a piece of movement. Figs. 4.b, 4.d, and 4.e show thatsparse velocity-driven control leads to much more stable sensor responses.

3.1 Abstracting the Dynamic Model during Manipulation

The goal of the RBF network is to model the sensory outputs given the motor com-mands along well defined trajectories (manipulating several objects). The purpose isto study how accurate models can be acquired if the neural network is trained withindividual objects or with groups of ”sets of objects” (as a single class). When weselect the target ”models” to abstract in the training stage with the neural network,we can define several target cases: 5 classes for abstracting single object models (Fig.5.a), 10 classes for abstracting models for pairs of objects (Fig. 5.b), 10 classes forabstracting models of sets of ”three objects” (Fig. 5.c), 5 classes for models of setsof ”four objects” (Fig. 5.d) and finally, a single global model for the set of five ob-jects (Fig. 5.e). The neural network, trained with data of two objects, presents theminimum of RMSE using 25 neurons in the hidden layer. For three objects, we used66-80 neurons depending on the specific case, and for four objects, 80 neurons. Buteven with these larger numbers of neurons, the network was able to accurately clas-sify the objects in the case of sets of two of them. During manipulation, if the modelprediction significantly deviates from the actual sensory outputs, the system can as-

Dynamics Model Abstraction Scheme Using Radial Basis Functions 9

0

0 .2

0 .6

S1 S2

4

S 1 S 2

3

S1 S2

2

S1 S2

1

S1 S2

O B J 0

0 .4

0 .8

1 .2

1 .0

RM

SE

(d

egre

es/s

2)

S1 S2 S1 S2 S1 S2 S1 S2 S1 S2 S1 S2 S1 S2 S1 S2 S1 S2 S1 S2

A B C D E F G H I J Objs

0

0.2

0.4

1.2

1

0.6

0.8

RM

SE

(d

egre

es/s

2)

a) b)

Objs

S1 S2 S1 S2 S1 S2 S1 S2 S1 S2 S1 S2 S1 S2 S1 S2 S1 S2 S1 S2

A B C D E F G H I J

1.2

1

0.8

0

0.2

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0.6

RM

SE

(d

egre

es/s

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(d

egre

es/s

2)

S1 S2 S1 S2 S1 S2 S1 S2 S1 S2

A B C D E

0

0.2

0.4

0.6

0.8

1

1.2

Objs

c) d)

S1 S2

RM

SE

(deg

rees

/s2

)

0

0.2

0.4

0.6

1.2

1

0.8

e)

Fig. 5 RMSE values for predicted data of sensors S1 and S2 for sparse velocity-driven control scheme.The darker colour in the top of the columns indicates the increment of RMSE in the test stage. a) Using adifferent class (model) for each single object. b) Using a different class (model) for each set of two objects.Cases (objects): A (0 and 1), B (0 and 2), C (0 and 3), D (0 and 4), E (1 and 2), F (1 and 3), G (1 and 4),H (2 and 3), I (2 and 4), and J (3 and 4). c) Using a different class (model) for each set of three objects.Cases (objects): A (0, 1, and 2), B (0, 1, and 3), C (0, 1, and 4), D (0, 2, and 3), E (0, 2, and 4), F (0, 3, and4), G (1, 2, and 3), H (objects 1, 2, and 4), I (1, 3, and 4), and J (2, 3, and 4). d) Using a different class foreach set of four objects. Cases (objects): A (0, 1, 2, and 4), B (0, 1, 2, and 3), C (0, 1, 3, and 4), D (0, 2, 3,and 4), E (1, 2, 3, and 4). e) Using a single class for the only set of five objects. In this case the incrementof RMSE in the test stage is neglectable. Cases (objects): (0, 1, 2, 3, and 4).

sume that the model being applied is incorrect and should proceed to ”switch” toanother model or to ”learn modifications” on the current model. In this task, we eval-uate the performance of the ”abstraction process” calculating the RMSE between themodel sensor response and the actual sensor response along the movement. Fig. 6compares the performance achieved between the two control schemes under studyfor the cases of a non-fixed (Fig. 6.a) and a fixed robot (Fig. 6.b). Results indicatethat the velocity driven strategy reduces the error to very low values in both cases.Results in Fig. 5 are obtained using a sparse velocity-driven control scheme. InFig. 5.a, they show how the RBF achieves different performance when trying to ab-stract models of different objects. For object 1 and 3, both sensors lead to similarRMSE values as their dynamic models are similar. Objects 2 and 4 lead to higherRMSE values since they have more inertia than others. As we can see (comparingthe results of Fig. 5.a and Fig. 5.b, if the network has to learn the dynamic model ofmore objects in the same class (shared model learning) at the same time, the errorrate increases slightly.

10 Silvia Tolu et al.

Fig. 6 Comparison between dense position-driven control scheme (column A) and the sparse velocity-driven control scheme (column B) for the case related to object 0. RMSE (degrees/s2) of predicted valuesof the sensors S1 and S2 using a sparse velocity-driven scheme (B) are always lower than the ones obtainedusing dense position-driven control (A). The sensor that suffers more significantly from jerks when usingdense position-driven scheme is S2. The darker colour in the top of the columns indicates the incrementof RMSE in the test stage. The test results are always slightly higher. The variance value obtained with thecross-validation method for the test results is also included. Figs. 6.a and 6.b are related to the cases of anon-fixed and a fixed robot respectively.

3.2 Classifying Objects during Manipulation

In Fig. 7, the discrimination power of the RBF based on the dynamic model itself isshown. In this case, we evaluate if the RBF is able to accurately distinguish amongdifferent objects assigning the correct labels (0, 1, 2, 3, and 4) by using only theinputs of the network (i.e. sensorimotor vectors). Evaluation of the classification per-formance is not straightforward and is based on the difference between the ”discrimi-nation threshold” (half of the distance between the closest target label values) and theRMSE value of the label output for a specific set of objects. We have chosen a singledimension distribution of the classes, therefore, distances between the different classlabels are diverse. For instance, the ”distance” between labels one and two is one,while the distance between labels one and four is three and the threshold is the halfof that distance. For any set of two or five objects (i.e. A (0 and 1), B (1 and 3), andC (0, 1, 2, 3, and 4)) only one threshold value exists (Th (A) = 0.5, Th (B) = 1, andTh (C) = [0.5, 0.5, 0.5, 0.5]), while sets of three or four objects (i.e., D (1, 2, and 3),E (0, 2, and 3), and F (0, 1, 2, and 4), and G(1, 2, 3, and 4)) may have two thresholdvalues (Th(D) = [0.5, 0.5], Th(E) = [1, 0.5], Th(F) = [0.5, 0.5, 1], and Th(G) = [0.5,0.5, 0.5]). We see that the RBF is able to correctly classify the objects when usingonly two of them (values above zero) but the system becomes unreliable (negativevalues) when using three objects or more in the case of a standing robot (Fig. 7.a),

Dynamics Model Abstraction Scheme Using Radial Basis Functions 11

0 3

0 1 2 4b 0 2 3 4a 0 1 3 4a 1 2 3 4 0 2 3 4b 0 1 3 4b

1 4 1 30 22 3

0 1 2 4a

1 20 12 4 3 4 0 3 4a 0 3 4b 1 2 3 1 2 4a 1 2 4b 1 3 4a 1 3 4b 0 1 2 0 1 3a 0 1 3b 0 1 4a 0 1 4b 0 2 3a 0 2 3b 0 2 4 2 3 4 0 1 2 3

0 4

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-0.5

-1.5

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ld –

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BJ)

2 OBJ 3 OBJ 5 OBJ 4 OBJ

Thr

esho

ld –

RM

SE

(O

BJ

0

0.5

1.0

1.5

2.5

2.0

a)

b)

Fig. 7 Bars represent the object discrimination power of the network (RBF) for all combinations of two,three, four, or five objects respectively (see paragraph 3 for object classification). If the RMSE (OBJ)value of the label output for a specific set of objects lies below the discrimination threshold, an accurateclassification of objects can be achieved. Graphs show the result obtained with a non-fixed robot (Fig. 7.a)and with a fixed robot (Fig. 7.b), and each bar is associated to a combination of objects listed in the legendfrom left to right respectively. A combination with two thresholds (i.e., 0 3 4) has two bars and two labels(0 3 4a and 0 3 4b) and bars are joined together in the graph.

RM

SE

5 OBJ 4 OBJ 3 OBJ 2 OBJ

1.6

1

0

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0.4

0.6

0.8

1.2

1.4

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Fig. 8 Mean of RMSE for objects (OBJ) using the sparse velocity- driven control scheme.

while the RBF always correctly classifies the objects when the robot is fixed to theground (Fig. 7.b).

3.3 Accurate Manipulation Strategy: Classification with Custom Control

We have abstracted dynamic models from objects and can use them for accurate con-trol (Fig. 9).a. Direct control. We control different objects directly. During manipulation, the sen-sor responses are compared with a ”shared model” to evaluate the performance. Wemeasure the RMSE comparing the global model output (model of the sensor outputsof two different manipulated objects) and the actual sensor outputs.b. Classification with control. Since we are able to accurately distinguish betweentwo objects (Fig. 7.a), we select the corresponding individual model from the RBFnetwork (see 3.a) to obtain the acceleration values of the object being manipulatedinstead of using a shared model with the data of the two objects. In fact, the indi-

12 Silvia Tolu et al.

RM

SE (

degr

ee/s

2 ) R

MSE

(de

gree

/s2 )

Fig. 9 Comparison between the individual and the shared models related to predicted sensory values S1and S2. The columns of a lighter colour indicate RMSE for the direct control while the columns of a darkercolour represent RMSE for the controller of a single object once it is accurately classified. Cases (objects):A (0 and 1), B (0 and 2), C (0 and 3), D (0 and 4), E (1 and 2), F (1 and 3), G (1 and 4), H (2 and 3), I (2and 4), and J (3 and 4).

vidual models of the five objects were previously learnt. The RMSE is always lowerwhen single object models are used (Fig. 9) (after an object classification stage). Di-rect control using a global (shared) model achieves a lower performance because theapplied model does not take into account the singularities of each of the individualmodels.The classification with control scheme is feasible when dealing with objects easy tobe discriminated and the actual trajectory of the movement follows more reliablythe single object model than a ”shared model”. In Fig. 9, the RMSE is plotted whencomparing sensor model response and the actual sensor responses. Fig. 8 shows theaccuracy of the classification task when training the network with different numbersof objects (classification errors along the trajectory). Nevertheless, we have shown(Fig. 7) that with the objects used in our study, the network can only reliably classifythe object (along the whole trajectory) when we focus on distinguishing between twoobjects.

4 Conclusions

The first conclusion of the presented work is that the applied control scheme is crit-ical to facilitate reliable dynamic model abstraction. In fact, the dynamic models orcomputed torques are used for high performance and compliant robot control in termsof precision and energy-efficiency [39, 43].We have compared a dense position-driven vs. a sparse velocity-driven control strat-egy. The second scheme leads to smoother movements when manipulating different

Dynamics Model Abstraction Scheme Using Radial Basis Functions 13

objects. This produces more stable sensor responses which allow to model the dy-namics of the movement. As model abstraction engine, we have used a well knownRBF network (widely used for function approximation tasks) and we compared itsperformance for abstracting dynamics models of the different objects using a sparsevelocity-driven scheme (Fig. 5.a).In a second part of the work, we have evaluated if the object classification is feasible.More concretely, we have checked out if we can perform it using sensorimotor com-plexes (motor commands, velocities and local accelerations) obtained during objectmanipulation (exploration task) as inputs. Figure 7.a shows that only the discrim-ination between any pair of two objects was possible. Robots with low gains willproduce different actual movements when manipulating different objects, becausethese objects affect significantly their dynamic model. If the robot is controlled withhigh gains, the differences in the dynamic model of the robot are compensated by thehigh control gains. If the robot is tightly fixed but is controlled with low gains theactual movement (sensorimotor representation of an object manipulation) is differentfor each object (Fig. 7.b). If the robot is slightly fixed, the control commands producevibrations that perturb the dynamic model and therefore, the different sensorimotorrepresentations are highly affected by these noisy artefacts (motor command drivenvibrations) (see figure 7.a).Finally, by assuming that we are able to classify objects during manipulation and toswitch to the best dynamic model, we have evaluated the accuracy when comparingthe movement’s actual dynamics (actual sensor responses) with the abstracted ones.Furthermore, we have evaluated the gain in performance if we compare it with a spe-cific dynamic model instead of a ”global” dynamics model. We have evaluated theimpact of this two-step control strategy obtaining an improvement of 30% (Fig. 9) inpredicting the sensor outputs along the trajectory. This abstracted model can be usedin predicted control strategies or for stabilisation purposes.Summarising, our results prove that the presented robot and artificial neural networkcan abstract dynamic models of objects within the stream sensorimotor primitivesduring manipulation tasks. One of the most important results of the work shown inFigs. 6.a (non-fixed robot) and 6.b (fixed robot) indicates that robot platforms canhighly benefit from non-rigid sparse velocity-driven control scheme. The high errorobtained with the dense position-driven control scheme (Fig. 6.a) is caused by vibra-tions as the slight fixation of the robot to the ground is only supported by its ownweight. Furthermore, when we try to abstract a movement model, the performanceachieved by the neural network (RBF) also represents a measurement of the stabil-ity of the movements (the repeatability of such trajectories when applying differentcontrol schemes).

Acknowledgements This work has been supported in part by the EU grant SENSOPAC (FP6-IST-028056)and by the Spanish National Grants DPI-2004-07032 and TEC2010-15396.

14 Silvia Tolu et al.

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