High Speed Parallel Kinematic Manipulator State Estimationfrom Legs Observation

Erol Ozgur, Nicolas Andreff, Redwan Dahmouche, Philippe Martinet

Abstract— To control dynamics of a parallel robot, we shouldmeasure the state feedback accurately and fast. In this paper,we show how to estimate positions and velocities simultaneously(i.e., the state feedback) at a reasonable accuracy and speed. Wedid this using only the sequential visual contours of the legs. Asingle-iteration virtual visual servoing scheme regulates rapidlyan error of these contours. We validated this theory, a step tocontrol parallel robots at high speed by their leg kinematics,with simulations and experiments.

I. INTRODUCTION

Dynamic control of robots requires fast feedback. Motorencoders feed back the joint values fast enough to controldynamics of serial robots. However, one still wishes to usevision-based control schemes [1] for better performances ofrobots, even though it is difficult to satisfy a reasonableaccuracy and frequency for visual feedback.

In serial robots, the solution for vision-based dynamiccontrol is found in a cascade of two control loops: (i) Thefirst inner loop, and also the fast one, compensates for thedynamics. This inner loop uses an inverse dynamic modelbased on joint values provided by the motor encoders athigh frequency. (ii) The second outer loop, slower one, usesfeedback of a vision sensor. This outer loop is actually akinematic control made possible by the inner loop whichcompensates for the dynamics of the serial robot.

This approach does not work for parallel robots. Becausethe joint values do not always represent the state of a parallelrobot. The state of a parallel robot for dynamic control canbe expressed by its end-effector pose and velocity – yet it ispossible to express the state with different variables.

Therefore, we should now think how to compute poseand velocity at high speed with vision sensor. Some of theattempts to adapt vision for control schemes are as follows:Since the state of a parallel robot is expressed by its end-effector pose and velocity, these variables are computedwith classical pose estimation algorithms. Unfortunately,these algorithms cannot give directly the velocity. The posevelocity is usually computed by numerical differentiationof the estimated pose, consequently introducing additionalnoise. Besides, predictive control techniques are exploitedto adapt the visual sampling rate to the control samplingrate, but this increases the complexity [2]. Instead, from acontrol viewpoint, increasing visual feedback frequency tothe control frequency is more appropriate [3], [4]. Then to

This work is supported by the ANR-VIRAGO project.The authors are with Pascal Institute, CNRS, Clermont Universites,

Clermont-Ferrand, France. N. Andreff is also with FEMTO-ST Institute, Universite de Franche-Comte, Besancon, France.firstname.lastname@lasmea.univ-bpclermont.fr

do so, one compresses image data [5], builds fast communi-cation interfaces, or embeds image processing unit closer tothe camera [6], [7], [8], etc. These attempts complicate thesystem, too.

So here, we propose a simpler solution inspired by thefollowing two ideas: (i) Is there a visual information whichcan give the statics and velocity of a parallel robot? Inkinematic control, it was shown that the image projectionof a parallel robot’s legs is a relevant alternative to expressthe state, since they similarly encode the static pose ofthe whole mechanism. (ii) In vision-based applications, afull image is grabbed and processed. The processing stepis simply to extract meaningful features from a predictedregion. It appears that the rest of the image is untouched.Grabbing and transmitting this untouched part waste time.Now then, to increase the visual feedback frequency to thecontrol frequency, why not just grab a region of an image[9], [10] where only the meaningful features exist?

Hence, the reader may wonder: (i) How shall previoustwo ideas be integrated for fast estimation? and (ii) Whichcompatible method shall be chosen to discover the dynamicstate of the robot?

Since a small sub-image acquisition allows only for a localobservation and legs exist plentifully in a parallel robot, thisimplies a sequential grabbing strategy (one by one) to collectenough information from the whole mechanism. In these sub-images, we observe contours of the legs. Image contours areone of the simplest visual features to detect. Then, requiredinformation is the contours of the legs which are grabbedsequentially at discrete instants of the motion of the robot.

However, although simultaneous multiple sub-imagesgrabbing strategy (non-sequential) provides all the informa-tion at once, it is not preferable because of addressing prob-lem and because it is slower for estimation. To compute thefull state at each control sample time, the following methodsare proposed: An extended Kalman filter predicts relativestate of the robot by fusing a set of image feature points be-longing to an object with some redundant measurements [6].A CMOS Rolling Shutter camera captures a single image rowby row. This causes image artifacts for the moving objects.Then, exploiting these visual artifacts, the pose and velocityof a moving object are simultaneously estimated with a non-linear least squares method [11]. A virtual visual servoingscheme [12] estimates the state variables by sequentiallygrabbing blobs of a rigid pattern at high speed [10], [13].

Our work also exploits the virtual visual servoing schemeassociated with sequential grabbing strategy as in [13]. Andimproves it to track an articulated set of legs of a parallel

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Preprint submitted to 2013 IEEE/RSJ International Conferenceon Intelligent Robots and Systems. Received March 22, 2013.

robot at high speed rather than to track a rigid patternfastened to the end-effector at high speed. Yet, since we aregood without any pattern, in a roundabout way, this gets ridof calibration between the pattern, the camera and the end-effector. The objectives of this work are as follows:

• to estimate the posture and velocity (without derivation)of a parallel robot through a copy virtual robot imitatingits motion.

• to progress towards achieving dynamic control of aparallel robot based on leg kinematics [14].

The rest of this paper goes on as follows: Section IIprovides background information about the kinematics oflegs. Section III explains the estimation of the state variablesthrough a single-iteration virtual visual servoing scheme.Section IV shows the simulation and experimental results.Section V concludes the paper and offers some perspectives.

II. EDGE KINEMATICS OF A CYLINDRICAL LEG

Fig. 1. The Quattro parallel robot with a base-mounted camera.

A. Notation

• B∈ℜ3×1 is one of the tip points lying on the revolutionaxis of the cylindrical leg (see Fig. 1).

• x ∈ ℜ3×1 is the orientation unit vector and also corre-sponds to the revolution axis direction of the leg.

• s ∈ {L , R} is the literal representation for the (L)eft orthe (R) ight side of the leg seen from the camera.

• ps ∈ ℜ3×1 is a projection contour point lying on theimage plane and located inside a visual edge of thecylindrical leg, ps = [x,y,1]T (see Fig. 2).

• ns ∈ ℜ3×1 is a unit vector orthogonal to the planedefined by the projection center O of the camera andby a side visual contour of the leg. Hence, it stands fora mathematical representation of such a visual contour.

• X ∈ ℜn×1 is the end-effector pose of a parallel robot.• Unless specified in a left supper-script, all the variables

are expressed in the camera frame.

B. Edge Kinematics

Let Mx ∈ℜ3×n and LB ∈ℜ3×n be the differential kinematicmodels that relate the velocity of the end-effector pose to thevelocity of the orientation unit vector x and to the velocityof the tip point B of the cylindrical leg in a kinematic chain:

x = Mx X , B = LB X (1)

Fig. 2. View of the geometry of a cylindrical leg from its 3D orientationdirection x (perpendicular to the paper plane).

The geometry of a cylindrical leg (see Fig. 2) imposes thefollowing constraints [15]:

BT ns =−r , xT ns = 0 , pTs ns = 0 (2)

where r is radius of the leg. Since we would like to exploitcontours of a leg, the third constraint will be used in thevirtual visual servoing scheme. This requires completion ofa differential model defined between an edge ns and theend-effector pose X. Exploiting the second constraint andknowing that nT

s ns = 0, the image velocity of the contourns is expressed as follows:

ns = α x + β (x × ns ) (3)

where α and β are two unknown scalars. The α comes outby differentiating the second constraint in (2) and replacing(3) into the differentiated second constraint, which yields:

xT ns + xT (α x + β (x × ns )) = 0 (4)

that gives α as below:

α = Mα

[Bx

]with Mα =

[01×3 −nT

s]

(5)

Afterwards, the β is computed by differentiating the firstconstraint in (2) and replacing (3) and (5) into the differen-tiated first constraint. This yields:

BT ns + BT (α x + β (x × ns )) = 0 (6)

which allows to calculate β as follows:

β = Mβ

[Bx

]with Mβ =

[−nT

sBT (x×ns )

BT x nTs

BT (x×ns )

](7)

Thus, the differential model between an edge velocity andthe end-effector pose velocity appears when (5), (7) and (1)are inserted into (3). This gives:

ns = Ms X with Ms = (x Mα + (x × ns ) Mβ )

[LBMx

](8)

Finally, the complete differential model for both of the leftand right edges of a cylindrical leg is defined as follows:

n = Mn X (9)

where n ∈ ℜ6×1 and Mn ∈ ℜ6×n are as below:

n =

[n

Ln

R

], Mn =

[ML

MR

](10)

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Preprint submitted to 2013 IEEE/RSJ International Conferenceon Intelligent Robots and Systems. Received March 22, 2013.

III. VIRTUAL VISUAL SERVOING

The non-sequential (simultaneous) acquisition approachgives the complete static pose of a robot straightaway. On theother hand, the sequential acquisition approach takes several(k) successive sub-images to collect the same information.Since the sequential acquisition approach gives the same in-formation with k successive sub-images, one needs previousk−1 sub-images to be stored. Thus, the static pose can becomputed at each sub-image grabbing instant with previouslystored k − 1 sub-images. Given the sampling time Tcamera,the beauty of this scenario is that it provides simultaneouslyboth the static pose and velocity. Furthermore, sequentialapproach is a lot faster than the non-sequential one. TableI compares requirements of the tasks of the sequential andnon-sequential approaches for the computation of the staticpose and velocity. To keep the comparison simple, a tasktime is assessed in either a sub-image processing time Tsubor a full image processing time Tf ull , where Tsub ≪ Tf ull .

TABLE ICOMPARISON OF SEQUENTIAL AND NON-SEQUENTIAL APPROACHES.

Task Sequential Non−SequentialImage acquisition 1Tsub 1Tf ullFeature extraction 1Tsub 1Tf ullPose computation k Tsub k TsubPose and velocity computation k Tsub 2k Tsub

For example: If a 48×48 pixels sub-image is grabbedrather than a full 1024×1024 pixels image, then in sequentialapproach: (i) Image acquisition is reduced to about 455times; (ii) Feature extraction is reduced to about 455 times;(iii) Pose computation time stays same; (iv) Pose and velocitycomputation time is reduced to about 2 times.

The estimation of the dynamic state of the robot is fasterwhen the information grabbed is smaller. However, whensmaller pieces of information are grabbed, less informationis kept for the present time and less accuracy is expected.Consequently, an optimum should be determined regardingthis tradeoff within theoretical and physical limits.

A. Notation

• t ∈ { tc, tc } denotes the time, where tc is an acquisitioninstant of the camera and tc is an estimation instant forthe state variables of the virtual robot.

• T ∈ {Tc , T c } are time periods of sub-image acquisitionof the camera and of the update of the virtual robotstate variables, respectively. The virtual time period T cshould be equal to or greater than the acquisition timeperiod Tc of the camera (T c > Tc) so that the virtualrobot can catch the motion of the real robot.

• j (t) ∈ {1, 2, . . .} is a function of time instants thatindicates which cylindrical leg is observed at time t.

B. Sequential Postures Error

A posture error is formed with a pair of reference projec-tion contour points (in metric units), {p∗

jL i, p∗

jR i}, extracted

from the sub-image of a cylindrical leg of the real robot, and

Fig. 3. A full image of lower-legs with their 48× 48 pixel2 sub-imagesfrom the base-mounted camera of the Quattro robot. These sub-images aregrabbed consecutively at discrete time instants of the motion of the Quattro.

their associated edges (feedback signal) computed from thevirtual robot’s cylindrical leg:

e ji =

p∗TjL i

njL

p∗TjR i

njR

(11)

where i = 1, . . . ,m is the index of a detected contour point.Then, the error vector e j ∈ℜ2m×1 of the j th leg of the virtualrobot is noted for all the contour points as follows:

e j =C∗j n j (12)

where n is as in (10) and C∗j ∈ ℜ2m×6 is a constant reference

contour matrix:

C∗j =

[P∗T

jL 00 P∗T

jR

](13)

with {P∗jL , P∗

jR } detected left and right side contours of thej th cylindrical leg of the real robot at an instant of time:

P∗jL =

[p∗

jL1· · · p∗

jLm

]∈ ℜ3×m (14)

P∗jR =

[p∗

jR1· · · p∗

jRm

]∈ ℜ3×m (15)

Finally, having the sets of contour matrices from the realrobot {C∗} and their corresponding feedback edge pairs {n}from the virtual robot which are saved at k sequential discreteinstants, the complete error vector e ∈ ℜ2k m×1 is formed bystacking the last k posture errors of the legs:

e =

C∗

j ( tc)n j ( tc)

...C∗

j ( tc −(k−1)Tc)n

j ( tc −(k−1)T c)

(16)

where j (·) circular-wise enumerates the cylindrical legs atconsecutive instants of time. Figure 3 shows an example forthe enumeration of the lower-legs of the Quattro robot.

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C. Approximated Edge Evolution Model

In order to regulate this time-space error to zero, adifferential model should be defined between an edge n oftime t +∆t and the effector pose X of reference time t. Tofind this model, we first write small displacement of an edge:

n t+∆ t = n t + δn t (17)

where ∆ t tells how far in time the displaced edge is, andwhere δn t is the displacement in the edge values withrespect to reference time instant t. One can approximate thedisplacement δn t through (9):

δn t ≈ Mnt δX t (18)

The displacement in the end-effector pose δX t can beapproximated with a constant acceleration model as follows:

δX t ≈ ∆t X t +12

∆t2 X t (19)

If the representation of the end-effector pose X t is notan element of a linear vector space, (19) is valid only ifthe rotational axis of the motion during ∆ t time remainsconstant. Otherwise, it will be still acceptable for a small ∆t.Finally, the approximated differential model can be expressedusing (17), (18) and (19) as below:

n t+∆t ≈n t+∆ t − n t

∆t=

δn t

∆t(20)

n t+∆t ≈ H( t, ∆t)

[X tX t

](21)

where H( t, ∆t) ∈ ℜ6×2n is the constant acceleration evolutionmodel of the edge pair of a leg:

H( t, ∆t) =[

Mn t12 ∆t Mn t

](22)

Mn t is defined in (10). This approximation allows to solvethe inverse kinematic problem of a parallel robot only onceinstead of k times, where k is equal to or greater than thenumber of observed legs.

D. Visual Servoing Control Law

The control law which regulates the error is computed byfirst differentiating (12) with respect to time, which gives:

e j = C∗j n j + C∗

j n j (23)

then replacing n j by (21) and imposing e j = −λ e j for anexponential convergence, (23) becomes:

−λ e j = Le j

[X tX t

]+ C∗

j n j (24)

where Le j ∈ ℜ2m×2n is so-called interaction matrix whichrelates the end-effector pose velocity and its derivative tothe error function of a leg:

Le j =C∗j H j( t,∆t) (25)

Then, to converge to the state of the real robot, the controllaw u =

[XT

u , XTu]T is computed to update the pose and

velocity of the virtual robot as follows:[XuXu

]= −λ L†

e (e − e) , λ > 0 (26)

where Le ∈ ℜ2km×2n and e ∈ ℜ2km×1 are as below:

Le =

Le j ( tc)

...Le j ( tc −(k−1) Tc)

, e =

C∗

j ( tc)n j ( tc)...

C∗j ( tc −(k−1)Tc)

n j ( tc −(k−1)T c)

(27)

The term e in (26) is difficult to approximate, since we donot use any correspondence between successively detectedreference contour points. Therefore, it is considered as adisturbance. Normally, (26) should not be directly calculatedwith an ordinary pseudo-inverse least squares regression.This can be unstable and slow. Here, (26) is just a represen-tation of the numerical solution of (24). The linear system in(24) can be solved with damped total least squares and QRdecomposition. This yields robust and fast solutions. Thus,one can write (24) as follows:

Au = b (28)

where A ∈ ℜ(2km+2n)×2n and b(2km+2n)×1 are the augmentedcoefficient matrix and the augmented error vector, respec-tively:

A =

[Leµ I

], b =

[−λ e

0

](29)

with µ damping parameter, I (2n by 2n) identity matrix,and 0 (2n by 1) zero vector. Linear system (28) can be nowsolved for u with QR decomposition.

E. Virtual Robot Motion Evolution

Pseudo-update rule for the dynamic state of the virtualrobot using the control law (26) can be written as follows:

X tc = X( tc−T c)+ T c Xu , X tc = X( tc−T c)

+ T c Xu (30)

This update rule is valid as long as representation of theend-effector pose forms a vector space. In order to assemblethe feedback edges n, the virtual robot’s pose is displacedback in time with a constant velocity motion model, sincethe updated dynamic state is up to velocity. Using the newupdated state variables and differential kinematic models forthe time tc, the feedback edge set is calculated as follows:

n tc+∆t ≈ n tc + ∆t MntcX tc (31)

where ∆t = iT c is the virtual time displacement with anindex i = 0,−1, . . . ,−(k−1) counting backwards.

F. Predicting Future Sub-Image Location

The next dynamic state estimation needs a future sub-image to be grabbed at the next future sampling time tc+Tc.The position of this sub-image on the image plane must becorrectly predicted from the current dynamic state {X, X}computed for the time instant tc. Otherwise there will not beany useful signal in the grabbed sub-image and the tracking

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will fail. Achieving a correct prediction is, itself, a proof ofthe correct performance of the proposed method. In order topredict any of the corresponding sub-image positions of thelegs, we first find the likely future pose of the real robot:

X( tc +Tc) = X tc + T c X tc (32)

Once the likely future end-effector pose X( tc +Tc) is found,it is solved for the tip point B j and for the orientationunit vector x j of a leg through the inverse kinematic model(IKM). Subsequently, the next location of the correspondingsub-image center is calculated as follows:

z j

[ imw j1

]= K ( B j − d j x j ) (33)

where imw j ∈ ℜ2×1 is the next predicted location of a sub-image center in pixel units, d j is a distance that indicateshow far along the cylindrical leg to the leg’s tip point theobserved region is, z j is the projective scale factor, and K isthe camera intrinsic matrix.

IV. RESULTS

We conducted simulation and experimentation on theQuattro robot (see Fig. 1). The Quattro robot encodes itspose in the contours of its 4 lower-leg rods. Thus, weestimated the dynamic state of the Quattro robot by usingat least 4 sub-images grabbed from each of the lower-legs atconsecutive discrete instants of the motion. For example, thefirst estimation used sub-images of the lower-legs {1,2,3,4},the second used {2,3,4,1} and so on. We set the camerasub-image acquisition frequency to 500Hz. Each sub-imagewas a 48× 48 pixel2. Figure 3 shows these 4 sub-imageson the lower-legs. Furthermore, while even the lower-legrod being observed is partially out of the field of viewof the camera, we can still measure visual contours bykeeping the sub-image location in the visible region viaparameter d j and/or by observing the other rod if the legis a parallelogram type. Figure 4 shows the simple blockrepresentation of the estimation algorithm. We did eachestimation with a single-iteration virtual visual servoing. We

Fig. 4. Single-iteration virtual visual servoing for fast estimation.

noticed that the computation of the velocity control law Xuwas very ill-conditioned. Therefore, we assigned the poseupdate control law Xu to the pose velocity. We evaluated theperformance of the estimated states with root-mean-square ofresiduals (RMSE). The end-effector of the Quattro robot has3 translational (xyz) and 1 rotational (θ ) degrees of freedom.Thus, we examined accuracies in these two parts.

Simulation: We validated the proposed estimation algo-rithm in Matlab. The test path was a 0.2m diameter circlemotion with 2m/s maximum velocity and 4G maximumacceleration. The path spans XY, XZ and YZ planes. Inorder to test the robustness, we generated the followingnoises: (i) we deflected camera orientation with 1◦ degreearound an arbitrary axis; (ii) we displaced camera position0.005m away along an arbitrary direction; (iii) we perturbedcontours [-1,+1] pixel orthogonally and uniformly. Figures 5and 6 show the errors between reference and estimated posesand velocities. Table II lists accuracies of the position andorientation estimations.

0

0.005

0.01

posi

tiona

l er

ror

(m)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.05

0.1

orie

ntat

iona

l er

ror

(rad

)

time (s)

xyz

θ

Fig. 5. Cartesian pose errors versus time (for the results of Table II).

0

0.2

0.4

posi

tiona

l vel

ocity

erro

r (m

/s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

1

2

3

orie

ntat

iona

l vel

ocity

erro

r (r

ad/s

)

time (s)

Fig. 6. Cartesian velocity errors versus time (for the results of Table II).

TABLE IIDYNAMIC STATE ESTIMATION ERRORS.

Pose Errors Velocity Errorsxyz (m) θ (rad) ˙xyz (m/s) θ (rad/s)

RMSE 0.005 0.036 0.161 0.747

Experimentation: We coded the estimation algorithm inC++ with nT2 matrix library [16]. The test path was a 0.08mby 0.08m square motion. The path crosses XY, XZ andYZ planes. The motion was without rotation. The maximumvelocity and acceleration of the motion were 0.25m/s and1m/s2, respectively. We calibrated the camera such thata non-iterative model compensates for distortions rapidly.Figures 7 and 8 show the reference and estimated Cartesianposes and velocities. Table III lists accuracies of the positionand orientation estimations. Table IV tabulates approximate

TABLE IIIDYNAMIC STATE ESTIMATION ERRORS.

Pose Errors Velocity Errorsxyz (m) θ (rad) ˙xyz (m/s) θ (rad/s)

RMSE 0.004 0.027 0.098 0.31

time consumptions of the processes required to estimate a

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−0.020

0.020.04

0.06

0.08

0.54

0.56

0.58

0.6

0.62

X (m) Y (m)

Z (

m)

EstimatedReference

Fig. 7. Reference (blue dotted line) and estimated (red solid line) Cartesianspace curves in the camera frame (for the results of Table III).

−0.5

0

0.5

Xve

l (m

/s)

−0.5

0

0.5

Yve

l (m

/s)

−0.5

0

0.5

Zve

l (m

/s)

0 1 2 3 4 5−2

0

2

time (s)

θ vel (

rad/

s)

ReferenceEstimated

Fig. 8. Superimposed reference (red dotted line) and estimated (black solidline) Cartesian velocities in the camera frame (for the results of Table III).

dynamic state of the Quattro robot using sequential 48×48pixel2 sub-images. An estimation costed about 1400 µs. Thismeans that the posture and velocity of the Quattro can becomputed faster than 500Hz.

TABLE IVTIME CONSUMPTIONS FOR AN ESTIMATION WITH 48×48 SUB-IMAGES.

ROI exposure ROI transfer Edge detection Estimation500 µs 100 µs 200 µs 600 µs

Discussion: Simulation showed that the subjectednoises are pronounced on the estimations approximately atsame order of magnitude. This means that a good calibratedsystem will yield good results. Experimentation also val-idated this conclusion. However, experimentation revealedthat depth (i.e., along optical axis of the camera) and con-sequently orientation estimations are very sensitive to noise.These errors in depth and orientation might appear becauseof a cylindrical leg whose radius is relatively smaller thanits observational distance from the camera. This makes esti-mations, especially the depth, more sensitive to small noises.We can eliminate this sensitivity to noise by placing multiplecameras (e.g., a camera per leg) in different viewpoints.This also allows us to explore the entire robot workspace.Yet, we can improve these estimations in a dynamic controlscenario of a robot by exploiting the computed-torque controllaw as a feed-forward acceleration term for better updateof state estimations [13]. Generally, the source of errorscomes from: (i) the use of approximated theoretical models;(ii) the calibration of camera extrinsic parameters; (iii) theimage noise while detecting visual contours. The obtained

results validate the proposed theory and motivate for furtherresearch.

V. CONCLUSIONSThis paper has shown that we can estimate the postures

and velocities of all parts of a parallel robot rapidly withreasonable accuracy through sequential observation of theleg contours. This method does not need any artificial patternsince the legs are observed, and is applicable for any slimprism-shaped legs. It is feasible by an edge detection [17] in asmall and structured sub-image. The sub-image contains onlya portion of the slim leg. Some future perspectives to increasethe accuracy of estimation are as follows: (i) to investigatethe grabbing strategy with different number, size and orderof the sub-images; (ii) to explore the number of rods (4 or8) and the number of sub-images on each rod.

Finally, we conclude that this work will lead to controldynamics of parallel robots simply and robustly from theirleg observations.

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CONFIDENTIAL. Limited circulation. For review only.

Preprint submitted to 2013 IEEE/RSJ International Conferenceon Intelligent Robots and Systems. Received March 22, 2013.