juggling ball proyect.pdf

7/27/2019 juggling ball proyect.pdf

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Bouncing an Unconstrained Ball in Three Dimensionswith a Blind Juggling Robot

Philipp Reist and Raffaello DAndrea

Abstract We describe the design of a juggling robot thatis able to vertically bounce a completely unconstrained ballwithout any sensing. The robot consists of a linear motoractuating a machined aluminum paddle. The curvature of this paddle keeps the ball from falling off while the apexheight of the ball is stabilized by decelerating the paddleat impact. We analyze the mapping of perturbations of thenominal trajectory over a single bounce to determine the designparameters that stabilize the system. The rst robot prototypeconrms the results from the stability analysis and exhibitssubstantial robustness to perturbations in the horizontal degreeof freedoms. We then measure the performance of the robot andcharacterize the noise introduced into the system as white noise.This allows us to rene the design parameters by minimizingthe H 2 norm of an input-output representation of the system.Finally, we design an H 2 optimal controller for the apex heightusing impact time measurements as feedback and show that theclosed-loop performance is only marginally better than what isachieved with open-loop control.

I. INTRODUCTION

We present a robot that can vertically juggle a single, com-pletely unconstrained ball without any sensing and explainthe process and analysis that led us to the design of therobot. The robot consists of a linear motor that actuates amachined aluminum paddle. Therefore, the robot stabilizesa ball in three dimensions with only one degree of freedom.

The paddle has a slightly concave parabolic shape whichkeeps the ball from falling off the robot. In addition, adecelerating paddle motion at impact stabilizes the apexheight of the ball. We nd suitable values for these twodesign parameters by analyzing the stability of the systemfor a range of apex heights and ball properties.

In order to analyze stability, we perform a perturbationanalysis on the nominal trajectory of the ball over a singlebounce from apex to apex. This analysis produces a linearmapping of the perturbations over a bounce. By studying thestability regions of this mapping over a range of apex heightsand ball properties, we nd suitable design parameter valuesfor the rst robot prototype. Using these values, the robot is

able to continuously juggle a variety of balls at differentheights. The robot also exhibits substantial robustness toperturbations in the horizontal degree of freedoms.

We evaluate the performance of the robot prototype using avideo camera and an impact detector. We measure the impacttimes and impact locations during extended runs. Using themeasurement results, we can estimate the deviations of theballs post-impact velocities from the values predicted by

P. Reist and R. DAndrea are with the IMRT at ETH Zurich,Switzerland. Supporting material, including videos, may be found atwww.blindjuggler.org .

Fig. 1. The Blind Juggler. The origin of the coordinate system lies atnominal impact height and in the center of the paddle. The rotational degreesof freedom x , y are dened by the right-hand rule.

the impact models. We characterize these deviations as awhite noise input into the system. This allows us to renethe design parameters using the H 2 system norm. Roughlyspeaking, an interpretation of the H 2 norm is the gain of asystem when the input is white noise [1]. After setting up aninput-output model of the system, we are then able to ndthe optimal design parameters by minimizing the H 2 norm.

Finally, we design an H 2 optimal controller for the apexheight using the measured impact time as feedback signal.The control input to the system is the deviation fromthe nominal impact state of the paddle. We compare the

closed-loop to the open-loop H 2 norm and nd that it onlymarginally improves. Therefore, actively controlling the apexheight is not necessary.

The main contributions of this paper are extending the sta-bility analysis of an underactuated, open-loop bouncing ballsystem to three dimensions including spin and conrmingthe results using a robot prototype. In order to achieve this,we present the novel strategy of stabilizing the ball by anappropriate paddle curvature.

The paper is structured as follows: First, we give anoverview of related work. We then show the derivation of


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C. Impact

The impact function maps the pre-impact ( ) to thepost-impact (+) ball state, given the paddle state. It con-tains the impact parameters such as CR and geometricalparameters. We assume the impact to be instantaneous, i.e.T 1 = T

2 = T +2 .

S

2 = S 1P 2 = P 1S +2 = S

2 , P

2 (4)

D. Free Fall until Apex

After the impact, the ball is subject to free fall for T 2 t T 3 . We can reuse (2) with different initial conditions

S T 2 = S +2 (5)S 3 = S T 3 . (6)

E. Boundary Condition and Solution

We require the boundary condition

S 3 = S 0 . (7)

With the above equations, we established a system of equations which we can solve for the nominal ball trajectoryand the unknown parameters. For the Blind Juggler, we nd

T 1 = 2H T 3 = 2 T 1 vP = vS 1 ez1 + ez , (8)where H is the nominal apex height, is the gravitational

acceleration and ez is the vertical CR . vP , vS are the paddles

and balls nominal impact velocities, respectively. Theseresults are straightforward. However, they are an importantprerequisite for the following perturbation analysis. Thegeneral derivation further allows introducing more complexmodels for the impact, problem geometry, or free fall (forexample, aerodynamic drag).

III. PERTURBATION ANALYSIS

In order to analyze the stability of the system, we derivea linear mapping which describes how perturbations addedto the initial conditions of the nominal trajectory map overa single bounce. Since the perturbations are small, we canpropagate them over the bounce using rst order approxima-tions to the system equations. The derived mapping allowsus to nd the stabilizing design parameters aP , the paddlesacceleration at impact, and c, the curvature of the paddle.

We perform the analysis in the coordinate system shown inFig. 1. The mapping is derived in two dimensions includingspin. However, we show that the analysis generalizes to threedimensions. We can omit the y and x degrees of freedomof the ball as we assume x, y and y, x to be uncorrelated.We present experimental results in Section V that conrmthis assumption. Therefore, the ball state for the analysis isS = ( x, x, y , z, z).

A. Introduce Perturbations

The perturbations s0 are added to the nominal initialconditions of the ball.

S 0 = S 0 + s0 s0 1 (9)

B. Free Fall

The dynamics of the ball during free fall are given byS = F (S ) = AS S + g (10)

AS =

0 1 0 0 00 0 0 0 00 0 0 0 00 0 0 0 10 0 0 0 0

g =

0000

. (11)

Since the perturbations are small, we can approximatethe solution to (10) given (9) to rst order using a Taylorseries expansion. We omit the higher order terms. We furtheridentify the perturbations s1 at nominal impact time T 1 as

S 1 = S T 1 , S 0 + S T 1 , S 0S 0S 0

s0 (12)

S 1 + s1 = $ $ $ $ $

S T 1 , S 0 + M 01 s0 . (13)

The Jacobian M 01 describes the linear mapping of s0 tos1 over the free fall.

C. Impact

Due to the introduced perturbations, the impact does notoccur at the nominal impact time T 1 , but at T 2 = T 1 + . Before we can apply the impact function, we have toapproximate the ball and paddle state at T 2 .

1) Impact States: We derive the motion of the ball for theimpact time deviation = t T 1 . We can reuse (10) withthe initial conditions (12), the ball state at nominal impacttime T 1 to get S (, S 1).

We introduce the paddle state P = ( x, x, y , z, z). Thepaddle motion in is the solution to

P ( = 0) = P 1P ( ) = G (P ( )) . (14)

The paddle dynamics are given by

G (P ) = P = AP P + aAP = AS

a = 0 0 0 0 aP T .

(15)

We approximate the motion of the ball and paddle to rstorder in . Note that to rst order, S (, S 1) = S 1 .

S 2 = S 1 + s1 + S 1 = S 1 + s

2

P 2 = P 1 + P 1 = P 1 + p

2 ,

(16)

where p is the perturbation of the nominal paddle state.


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Fig. 3. Paddle Shape, = ( c/ 2) 2 , and Impact CS

2) Impact Condition Function: We solve the generalimpact condition function for using the approximations(16). We approximate to rst order in S, P about S 1 , P 1 .

S 2 , P

2 = $ $ $ $ $

S 1 , P 1

+ ( S, P )

S S 1 , P 1s 2

+ ( S, P )

P S 1 , P 1 p2 (17)

0 = T S s

2 + T P p

2

= T S s1 + S 1 + T P P 1 (18)

T S , T B are the Jacobians of for the ball and paddleperturbations. We solve (18) for :

= T S S 1 + T P P 1 1

T S s1 . (19)

For the specic system analyzed, we use the followingimpact condition:

( S, P ) = iS S iP P RiS = iP = 0 0 0 1 0 ,

(20)

where R is the ball radius. We neglect the paddles shapein the impact condition. just compares the z-coordinatesof the ball and paddle. An impact occurs if = 0 . Sincethe specic impact condition function is already linear, weget T S = T P = iS .

3) Impact Location: Due to the perturbations, the balldoes not hit the paddle in its center at x = = 0 .The curvature of the paddle and the x-coordinate at impactdetermine the angle at which the impact coordinate system(CS ) C is in respect to the inertial CS I , see Fig. 3.

The impact function is applied in the impact CS C ; wedene the rotation matrix ACI that rotates a vector from I to C .

4) Impact Model: For the vertical impact velocities, weuse Newtons impact laws assuming the ratio of the ballsmass to the paddles mass to be small; the ratio for the robotprototype is 3 10 4 .

z+S = ez z

S + (1 + ez )zP , (21)

where zS and zP are the vertical velocities of the ball andpaddle, respectively. For the horizontal directions and spin,we use the impact model derived in [21]. The horizontal CRex is dened as:

ex = x+ R +yx R y

. (22)

R is the ball radius. ex relates the velocity of the contactpoint of the ball over the impact. The values of ex rangefrom -1 to 1.

Combined with conservation of angular momentum and(21), we obtain the following impact function:

(S, P ) = S + = C S S + C P P

. (23)

C S , C P are the mappings of the pre-impact ball and paddlestate to the post-impact ball state, respectively. Note that (23)is dened in the impact coordinate system.

C S =

1 0 0 0 00 1 k kR 0 00 1 R 0 00 0 0 1 00 0 0 0 ez

C P =

0 0 0 0 00 k 0 0 00 0 0 00 0 0 0 00 0 0 0 ez + 1

=ex + 11.4R

, k =ex + 1

3.5, = k 1 ez

(24)

5) Impact Function: (16) and (19) dene the ball andpaddle state before impact. We apply the impact function(23) on the rotated vectors and rotate the result back into theinertial frame. We dene

I S

2 , P

2 := AT CI ACI S

2 , ACI P

2 . (25)

I represents the impact function acting directly in I . Weapproximate I to rst order to obtain the post-impact ballstate.

S +2 = I S

2 , P

2

+ I (S, P )

S S 2 , P 2s 2

+ I (S, P )

P S 2 , P 2 p2

= S +2 + s+2 (26)

D. Free Fall until Apex

After the impact, the ball is in free fall until the nominalapex time T 3 . The free fall is analogous to Section III-B. Wereuse M 01 from (13) to map the perturbations at T 1 + toT 3 + . We get

S T 3 + = S 3 + M 01 s+2 . (27)

In order to obtain the ball state at the nominal apex timeT 3 , we compensate for analogous to (16):

S 3 = S 3 + M 01 s+2 S 3 . (28)

We combine (13), (16), (19), (26), (28) and obtain thematrix M 03 which maps the initial perturbations s0 over asingle bounce.

s3 = M 03 s0 (29)


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Fig. 4. maxe x

(Ax ) for various Apex Heights H , ez = 0 .8

For the specic system, the mapping M 03 is a block-diagonal matrix with blocks corresponding to x, x, y andz, z .

M 03 =Ax 00 Az

, (30)

with

Ax =

2c T 21e z +1 + 1 T 1

2c T 21e z +1 k + 2 kR T 1

2c T 1e z +1

2c T 21e z +1 k + 1 kR

2c T 1e z +1 2c T 21

e z +1 1 R

Az =a P (e z +1) 2 + (e 2z +1 )

2 ( (e z 1) 2 +( e z +1) 2 a P )T 1

2 (e z +1) 2 ( + a P )

2 T 1a P (e z +1) 2 + (e 2z +1 )

2

.

IV. DESIGN PARAMETERS

With the mapping M 03 , we analyze the local stability of the system. The goal is to nd stabilizing values for the un-

known design parameters curvature c and paddle accelerationaP for a range of apex heights and ball properties.We analyze the spectral radius of M 03 to determine the

stability of the system; if is smaller than one, the system isstable. The spectral radius of a matrix is dened as (A) :=max

i| i (A)| , where i is the i-th eigenvalue of A.

Since Ax is independent of aP and Az is independent of c,we can determine the two design parameters independently.

A. Paddle Curvature

Stability of the balls horizontal perturbations can beachieved by choosing an appropriate paddle curvature assketched in Fig. 3. We evaluate the spectral radius of Axover a range of apex heights and ball properties. We nd that is sensitive to the apex height. The CR ez appears in Ax ,but has negligible inuence on stability.

In Fig. 4 we show the maximum spectral radius of Ax overthe range ex [ 0.5, 0.5] as a function of c for various apexheights H .

We set the design specication for the rst robot proto-type to juggle at heights up to 1.2 m. Therefore, choosingc = 0 .36 m 1 satises the stability requirement. We furtherpredict that the ball should fall off the paddle for heightslarger 1.4 m.

Fig. 5. Fitted Gaussians for Apex Height Deviations with StandardDeviations H

B. Paddle Acceleration

Stability of the apex height is achieved by the properpaddle acceleration. It has been shown in previous work thata negative acceleration is stabilizing [8], [11], [12].

The spectral radius of Az is independent of the apexheight. For a range of aP [ 1, 10] ms 2 , the spectralradius is constant and smaller than one for any given ez [0.7, 0.9]. We choose aP = / 2 which lies in the center of the stable region.

V. ROBOT PROTOTYPE

We built a robot prototype (Fig. 1) with the determineddesign parameters. It consists of a linear motor actuating asolid aluminum paddle. The paddle was CNC-milled to thedesired shape and has a diameter of 0.3 m. We chose themotor to be able to continuously juggle balls at heights upto at least 1.2 m and ez larger 0.7 (tennis ball).

The quadratic trajectory of the paddle is driven by a servocontroller. A high level state machine running on a D-Spacereal time control system [22] provides the prole parametersto the controller and is timing the motion.

For given parameters ez , a P and H , the paddles trajectoryis fully dened except the duration the paddle is deceleratingaround the nominal impact time. We determine this durationby assuming an upper bound for a static error in ez .

A. Impact of Ball Properties on Performance

After some experiments with different balls, we observedthat the main source of noise in the measured impacttimes and locations are stochastic deviations of the impactparameters, i.e. ball roundness and CR . High precision balls,which are specically manufactured to have very preciseroundness for valve applications, introduce the least amountof noise. Only these high precision balls allow the robot to juggle at apex heights larger than 0.6 m. Less precise balls(Superball, table tennis ball, etc.) generate too much noise inthe horizontal degrees of freedom and eventually fall off atlarger apex heights. We show a ball comparison in the videosubmitted with this paper.


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Fig. 6. Impact Location Data for Paddle Curvature c = 0 .36 m 1 . Top:Accumulated Impact Locations (x) and Paddle Center (+) for H = 1 .05 m .Bottom: Fitted Impact Coordinate Distributions.

B. Measurements

A piezo lm strain gage attached to the paddle acting asa contact microphone lets us measure the impact times. Anencoder on the linear motor provides information about thepaddle position. We measure the impact locations using avideo camera mounted above the paddle. We collected dataat three different nominal apex heights: 0.5, 0.8 and 1.05 m.In the three experiments, we recorded a total of 5301, 4549and 5789 consecutive impacts, respectively.

1) Apex Height: From the paddle positions of two con-secutive impacts and the impact time difference, we estimate

the apex height of the ball. In Fig. 5, we show the Gaussiandistributions tted to the data.

2) Impact Locations: In Fig. 6, we show the results of theimpact location measurements. The distributions are rathernarrow. However, this performance can only be achieved witha precision ball. For the measurements presented, we used asolid PA-6.6 ball with a diameter of 0.012 m.

The measurements conrm the assumption that x, y andy, x of the ball are uncorrelated. The covariance matrix of n = 5789 measured impact coordinates X = ( x, y ) withx, y R n 1 for H = 1 .05 m is

Fig. 7. Noise Histograms and t Gaussians for H = 1 .05 m .

10 3 0.267 0.003 0.003 0.273 m2 . (31)

We measured small non-zero means in the impact locationdistributions (Fig. 6). This is due to a slightly misleveledpaddle and would be zero for a perfectly leveled paddle.

C. Parameter Identication

From the impact times, paddle positions and velocities at

impact, we identify ez from the data. The parameter slightlydecreases with increasing heights and lies in the interval ez [0.79, 0.82].

We estimate the horizontal impact parameter ex from thedata by minimizing the deviation of the predicted post-impacthorizontal velocities to the measured velocities. As we haveno measure of the balls spin, this proves to be difcult; weobtain three different values for the three impact heights. ForH = 1 .05 m, we even obtain ex = 1 .2, which lies outsidethe range of possible values for ex . Without measuring thespin of the ball, we cannot properly estimate ex .

D. Noise Identication

We identify the noise as the deviations from the measuredto the predicted post-impact velocities. For the prediction, weuse the linear model derived in Section III with the identiedparameters of ez . For the unknown parameter ex , we choosea xed value of ex = 0 .2, assuming a gripping ball lesselastic than a Superball [21].

S +2,meas = S +2,pred + , (32)

with = (0 , x , 0, 0, z )T . We also measure the noise

in y-direction. This is straightforward, since this direction


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is independent of the others (31). We can extend the linearmodel to include the y-direction.

Since we cannot measure the balls spin, we assume zerospin at all times and capture the inuence of spin on thesystem in the noise. In Fig. 7 we show tted Gaussiandistributions to the measurement histograms of x , z . Thehistogram of z appears less Gaussian since the actual paddledeceleration is not perfectly linear.

We analyze the auto-correlation sequences of the noisedata and learn that a white noise model is a good approxi-mation to the measured noise.

VI. REFINING THE DESIGN PARAMETERS

Now that we characterized the noise introduced into thesystem as white noise, we can use the H 2 norm to renethe design parameters. An interpretation of the H 2 norm is,roughly speaking, the gain of a system when the input iswhite noise [1]. By minimizing the H 2 norm, we can ndthe design parameters that maximally reject the noise andimprove the performance of the system.

Analogous to the stability analysis in Section IV, we canoptimize the design parameters c and aP independently asM 03 is block diagonal and the noise introduced in the verticaland horizontal directions only impacts the respective degreeof freedoms.

A. Optimized Paddle Curvature

We want to nd the paddle curvature parameter c thatminimizes the H 2 norm over a range of ball properties. Inorder to apply the norm to the system, we have to derive theinput-output representation of the system for the horizontaldirections. The x and y directions are independent, hencethe paddle curvature minimizing the two dimensional systemalso minimizes the three dimensional one. The state is sx =(x, x, y ). The system output wx is equivalent to the impactlocation on the paddle, which is what we want to keep small.The input is dx = x , the post-impact velocity deviation.

sx (k + 1) = Ax sx (k) + Bx dx (k)wx (k) = C x sx (k)

(33)

We call this LTI system Gx . It is discrete with a sampletime of 2T 1 . The matrices are

Bx = T 1 , 1, 0T C x = 1, T 1 , 0 . (34)

We nd c = 0 .25 from

minc

maxe x

Gx (c, ex ) 2

J (c )(35)

for H = 1 .05 m and ez = 0 .8. Since we could not identifythe parameter ex from the measurements, we minimize aworst case value of the H 2 norm over the range ex [ 0.5, 0.2]. This range captures a slipping to gripping impactof the ball which is less elastic than a Superball, see [21].

TABLE I

IMPACT LOCATION STANDARD D EVIATIONS

H c = 0 .36 m 1 c = 0 .24 m 1 Change0.5 m 0.0120 m 0 .0130 m 9.1%0.8 m 0.0134 m 0 .0141 m 4.8%

1.05 m 0.0163 m 0 .0147 m -10.1%

1) Results: We manufactured a paddle with a slightlysmaller c = 0 .24 m 1 , which is acceptable as J (0.24)and J (0.25) are both equal to 1.72 for unscaled input. Wemeasured the impact locations using the new paddle andshow a comparison of the measured standard deviations of the impact x-coordinate in Table I.

The system performance with c = 0 .24 m 1 decreasesfor lower heights; we can accept this as increased impactlocation deviations are more likely to cause the ball to falloff at larger heights.

We achieve larger apex heights with the new paddle; thisis predicted in Fig. 4. For c = 0 .24 m 1 , the robot jugglesat heights up to 2.1 m. We show this in the video submittedwith the paper.

B. Optimized Paddle Acceleration

Analogous to the paddle shape optimization, we canoptimize the paddle acceleration aP . We build the input-output system for the vertical directions. The state vectorbecomes sz = ( z, z). The input is dz = z , the post-impactvelocity deviation. We seek to minimize the output wz , whichis equivalent to the z-coordinate of s1 , the perturbation inheight at nominal impact time.

sz (k + 1) = Az sz (k) + B z dz (k)wz (k) = C z sz (k)

(36)

We call this discrete LTI system with sample time 2T 1Gz . The matrices are

B z = T 1 , 1T

C z = 1, T 1 . (37)

We obtain aP = 4.95 ms 2 from

mina P

Gz (aP ) 2 (38)

with H = 1 .05 and ez = 0 .8. This value is almostidentical to the stabilizing value we chose in Section IV of 4.9 ms 2 .

VII. H 2 OPTIMAL CONTROLLER FOR APEXSince the horizontal degree of freedoms are decoupled

from the vertical ones, we cannot actively control the ballshorizontal impact location. However, we can use feedback control to increase the vertical performance. We design afeedback controller K that minimizes the H 2 norm of theclosed-loop system.

We use , the impact time deviation as feedback; thecontact microphone we attached to the paddle provides thismeasurement. As control action, we use the deviation fromnominal impact height and velocity of the paddle, thus


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u = ( uz , u z )T . In order to design the controller for Gz ,

we extend the system (36) to include and u:

q (k + 1) = Aq (k) + B1dz (k) + B2u(k)wz (k) = C 1q (k) + D11 dz (k) + D12 u(k)

(k) = C 2q (k) + D21 dz (k) + D22 u(k).

(39)

For determining u(k), we only have the previous impacttime. We capture this measurement delay in the systemdynamics by extending the state to q = sT z ,

T . This allows

us to calculate u = K .

A = Az 0C 0B1 =

Bz0 B2 =

BuD

C 1 = C z 0 C 2 = 0 0 1D11 = D12 = 0 D21 = D22 = 0

(40)

We already derived C in (19). We can derive D in theperturbation analysis by adding u to p2 in (16).

With the system established, we can nd the controllermatrix K that minimizes the closed-loop H 2 norm frominput dz to wz . For H = 1 .05 m, ez = 0 .8, a P = / 2and scaling the input with the measured z = 0 .015,we achieve a closed-loop norm of 0.014. Given the sameparameters, the open-loop norm is 0.017. We can only reducethe norm by 14% with feedback control. Considering that themeasured standard deviations of the apex height distributionsare already small (Fig. 5), the improvement is marginal.

Note that by choosing D12 = 0 , we do not constrain thecontrol action. Therefore, the improvement is an upper boundthat may not be physically possible; for example, the strokeof the linear motor is limited.

VIII. CONCLUDING REMARKS

We showed that feedback control is not necessary for theapex height. However, feedback control in an adaptive settingas in [23] can relax the a priori knowledge about the CR andcompensate for the decrease of the CR at increasing height.This reduces the required stroke of the linear motor as themean impact time deviation is eliminated.

We have recently identied that the paddle shape optimiza-

tion can be improved by considering the fact that the noiseamplitude we determined in Section V-D is also a functionof the unknown parameter ex . We are currently working onincorporating this knowledge into the optimization processand plan to manufacture a new paddle in the near future.We will also look into nding a way to reliably estimate theimpact parameter ex .

Lastly, we believe that determining the optimal paddleacceleration experimentally, i.e. by minimizing the measuredapex deviations, and comparing the results with [12] wouldyield interesting insights.

IX. ACKNOWLEDGMENTS

We would like to thank Matthew Donovan for his greathelp with the mechanical design of the robot. We fur-ther thank Angela Sch ollig and Sebastian Trimpe for theirsuggestions that helped improve the controller design andperturbation analysis.

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