Balancing on Tightropes and Slacklines

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first published online 18 April 2012, doi: 10.1098/rsif.2012.007792012J. R. Soc. InterfaceP. Paoletti and L. MahadevanBalancing on tightropes and slacklines

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Balancing on tightropes and slacklines

P. Paoletti1 and L. Mahadevan1,2,3,*

1School of Engineering and Applied Sciences, 2Department of Organismic andEvolutionary Biology, and 3Wyss Institute for Bioinspired Engineering, Harvard University,

Cambridge, MA 02138, USA

Balancing on a tightrope or a slackline is an example of a neuromechanical task where thewhole body both drives and responds to the dynamics of the external environment, oftenon multiple timescales. Motivated by a range of neurophysiological observations, here we for-mulate a minimal model for this system and use optimal control theory to design a strategyfor maintaining an upright position. Our analysis of the open and closed-loop dynamics showsthe existence of an optimal rope sag where balancing requires minimal effort, consistent withqualitative observations and suggestive of strategies for optimizing balancing performancewhile standing and walking. Our consideration of the effects of nonlinearities, potential par-ameter coupling and delays on the overall performance shows that although these factorschange the results quantitatively, the existence of an optimal strategy persists.

Keywords: balance; tightrope; slackline

1. INTRODUCTION

Postural balance in humans requires active controlbecause the normal upright position is mechanicallyunstable. Indeed, a stable upright stance can only beachieved by exploiting a feedback mechanism thatcreates a corrective torque based on multiple sensoryinputs associated with body swaying [1]. In addition

to its intrinsic neuromechanical importance, under-standing the mechanisms involved with balance mayhelp in the design of biped robots [2], in the diagnosisand amelioration of motor control problems in a rangeof neurological ailments [3,4], as well as in analysinggames, such as stick-balancing [5] and even possiblyin the active stabilization of large structures, suchas buildings [6].

Most studies on postural balance are restricted tostatic or externally controlled platforms [7] where theactive feedback between body motion and the dynamicsof the environment is limited. Here, we consider theneuromechanical task of balancing on a soft dynamicsupport since this couples the internal dynamics ofthe body to the external dynamics induced by thebody in a relatively simple setting, and thus begs thequestion of how the body can sense and then optimizeits response to a dynamic environment. In addition,by considering the separate timescales associatedwith the response of the body and the controllableenvironment, we can begin to potentially probe thesensory-motor dynamics of the body as a function ofage, size, health, etc.

A variety of models have been proposed to describethe dynamics of body sway motion during balancingon a fixed or moving platform [8,9]. Most of themreduce to the dynamics of a single degree-of-freedom

system where the body is represented as an invertedpendulum; this simplification allows researchers toeasily analyse the dynamics and control issues thatarise for obtaining insight into postural balance strategiesimplemented by neural controllers [1,912]. An impor-tant aspect in all these studies requires the estimationof the bodys current spatial orientation, which is clearly

important to generate the necessary corrective action forstabilizing the mechanically unstable upright stance. Inhumans, this information about actual deviations fromthe upright position involves several proprioceptive,visual and vestibular sensory systems [13] and therelative importance of these mechanisms has led to a pro-posal of sensory channel reweighting accordingly to thestrengths of different perturbative stimuli [7,13]. Thus,vestibular feedback sensors are thought to be more rel-evant on moving platforms, whereas proprioceptive andvisual systems are the primary feedback sensors duringquiet standing [13]. Furthermore, the presence of delayin the feedback loop has a non-negligible impact on the

overall performance [8,14].When balancing on a rope rather than on a rigid

substrate, the neuromechanical dynamics of the bodyis coupled with the external dynamics of the ropewhich itself moves in response to body swaying. Toaddress how balance might be achieved in this system,we need to understand the passive coupled dyna-mics of the system, sensory modalities that allow thebody to infer its orientation and activate motorcoordination to maintain or regain balance.

In 2, we present a minimal mechanical model for thebody rope system along with a plausible sensory motorfeedback control unit based on the vestibular system

that may be used to maintain the equilibrium position.In 3, we analyse the linearized dynamics of our modeland derive an optimal feedback controller for posturecontrol in the presence of limited information about*Author for correspondence ([email protected]).

J. R. Soc. Interface (2012) 9, 20972108

doi:10.1098/rsif.2012.0077

Published online 18 April 2012

Received31 January 2012Accepted 23 March 2012 2097 This journal is q 2012 The Royal Society

mailto:[email protected]://rsif.royalsocietypublishing.org/http://rsif.royalsocietypublishing.org/http://rsif.royalsocietypublishing.org/http://rsif.royalsocietypublishing.org/http://crossmark.crossref.org/dialog/?doi=10.1098/rsif.2012.0077&domain=pdf&date_stamp=2012-04-18mailto:[email protected]


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the state space. The performance of this control lawis tested in 4 on the full nonlinear model. Finally, in5, we conclude with a discussion of the sensitivity ofour model to additional effects such as the coupling

between parameters, the effects of delay and outlinesome future directions of study.

2. MATHEMATICAL MODEL

2.1. Mechanics

A schematic of a man on a high wire is shown infigure 1. To develop a minimal model of the systemthat allows for analysis, we make several simplifyingassumptions. Firstly, although the number of degreesof freedom of a human body is large, it is only theslow modes associated with the long limbs that are typi-

cally relevant since they are relatively easy to actuatefor the purposes of balance. Here, we consider a modelwith only one degree of freedom associated with theorientation of the body, modelled as a simple inverted

pendulum. Similarly, we model the rope as a (possibly)tensioned string that is deformed by the body, envisa-ging two extreme limits: large tension that is typicalof tightropes, and small tension typical of slacklines.In both cases, we neglect the additional rope tensioncaused by the body sway motion since it is negligiblerelative to the static tension caused by either the pre-

tension or the body weight [15]. Within this setting,the presence of the rope thus does no more than intro-duce a kinematic constraint on the motion of the cart,forcing this latter to move on a circular track associatedwith the swinging of the rope at constant extension, anexcellent approximation for slacklines. The limit of verylarge pretension, i.e. small sag, however is not so trivialas there are three possible scenarios. In the simplest set-ting, but also probably the least realistic, we assumethat the presence of the rope can be still assimilatedvia a kinematic constraint on the cart motion. Thisdoes not have to be true alwaysin particular, if onetries to recover the case of a pendulum on a fixed base,

the ratio between the pendulum and cart inertia mustvanish as R 0 to make sure the cart is not moving;this second situation is analysed in 5.1. Finally, elasticropes with large pretension can still admit oscillationsin orthogonal directions, but the horizontal and verticalmotions become uncoupled. In this case, the rope intro-duces a dynamic, rather than a kinematic constraint onthe motion of the cart and in 5.2, we introduce a thirdmodel for describing this situation.

Thus, our minimal dynamical system is an invertedpendulum (the body) placed on a cart (the rope) thatmoves on a circular track of radius R characterizingthe envelope of transverse motions of the rope, shown

in figure 1b. We note that R varies from zero near therope supports to a maximum value when the man isat midspan, i.e. R is a parameter that identifies the pos-ition of the body along the rope. The invertedpendulum is assumed to have length l and a concen-trated mass m on top, and its deviation from theradial direction is the angle a [8,16]. The cart isassumed to have mass M, with an angular positionalong the track given by the angle f. We will initiallyconsider R and M m as independent parameters byassuming, for example, that the rope pretension, andthus the sag, can be set at will, but will relax it even-tually and in 5, we discuss the effects of a possible

linear dependence between them induced by theelasticity of the rope.

The position of the cart (rope) relative to a fixed origincentred at O in figure 1b is xM; yM Rsinf;R1 cosf, while the position of the pendulum bob(body) is xm; ym xM lsina f; yM lcosa f, so that the Lagrangian of the system then reads

L 12M mR2 _f2 mRlcosa _a _f _f

12

ml2 _a _f2 M mgR1 cosf

mglcosa f;

2:1

where the first three terms represent the kinetic energyand the last two terms the gravitational potential energy.

(a)

(b)

5 0 5

5

0

5

Im(

l)

Re (l)

xO

y

R m

l M

a

f

l2 l1

l4

l3

(c)

P

P

Figure 1. (a) Schematic of a man balancing on a rope and (b)

associated mechanical model describing the transverse motionon the plane P. The linearized equations of motion forthis system are reported in (3.1) and (3.2). ( c) Spectrum ofthe linearized system defined in (3.1) and (3.2) as a func-tion of R (top). The circular markers correspond toR 0; 0:5; 1; 1:5; 2; 3. The markers for the imaginary eigen-values for R 0 is missing because jl3j jl4j 1 as R 0. (Online version in colour.)

2098 Balancing on tightropes and slacklines P. Paoletti and L. Mahadevan

J. R. Soc. Interface (2012)

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Maintaining balance implies staying in a restrictedregion around the upright position corresponding tof 0, a 0 so that we start with a focus on a linear-ized model of the dynamics (although we will laterconsider the complete dynamics). This linearizedmodel around the unstable equilibrium with at mostquadratic terms in the Lagrangian (2.1) then reads

~L 12M mR2 _f2 mRl _f2 _a _f mRla2

12

ml2 _a _f2 M mgRf2

mgl1 a f2:

2:2

The equations of motion for the system are thengiven by

d

dt

@~L

@q @

~L

@q Fq; 2:3

where Fq

Ff; Fa

represents the generalized force

acting in the direction of the generalized coordinate q.Humans use their arms or a pole to maintain balance,this corresponds to a torque Fa on the inverted pendu-lum. Moreover, as the angle of the cart is not controlled,we have Ff; 0. We have ignored the role of dissipationand damping in our model given its equivalence to theaction of the torque Fa. It is convenient to expressthe dynamics in dimensionless form by definingR R=l, m m=M, t

ffiffiffiffiffiffiffig=l

pt, F

a Fa=mgl. Here,the scaled sag parameter R determines if one is in thetightrope or slackline limit, while the scaled body massm is initially considered constant. Later, we assume

that the cart mass is the sum of the mass of the feet(about 2% of the body mass [16]) and that ofthe rope; we initially set m 30, but the overallbehaviour of the system is relatively insensitive to theexact value of this parameter, as discussed in 5.1.For notational simplification, from now on we willalways refer to dimensionless quantities without usingthe overbars.

2.2. Neurobiology

To complement the physical model with the neurobiolo-gical controller requires the specification of a suitable

sensory cue that can be implemented via a physiologicalsystem and a concomitant control action implementedvia an internal torque.

Complete knowledge of the full state of the systemrequires separate measurements of f, _f, a and _a,which is both unreasonable and unlikely. The threemain sources of orientation measurements in humansare proprioceptive, visual and vestibular systems [13].Precise information about absolute orientation withrespect to the environment can be obtained by thevisual system, but this introduces delays of the orderof 100200 ms [13], and are thus unlikely to be usefulhere given the fast timescales induced by the movable

support, as we will see later. Proprioceptive sensors canmeasure the relative position and orientation betweentwo joints of the body, but during balancing thisinformation is inaccurate owing to the lack of a fixed

reference for the cart (feet). However, fast and reliablemeasurements of body rotational and translationalvelocities are provided by the vestibular system locatedin the inner ear. The relative importance of these threesources varies accordingly to the motor task [13].Experimental studies on quiet standing atop asway-referenced platforms, i.e. the closest existing

experimental set-up to our problem we are aware of,show that proprioceptive information quality isdegraded and that the vestibular system starts to playa prominent role [2,7]. Since velocity measurementsare more reliable than position and acceleration [17],we will assume that only the measurement of absoluteangular velocity _f _a is available for feedback control.This is motivated by the fact that vestibular system vel-ocity estimate is the most accurate among availablebody sway measures and as we will prove, this limitedinformation is enough to correctly estimate the fullstate of the system. This is also a very conservativemodel since it corresponds to the worst case scenario

where we completely neglect visual, proprioceptiveand absolute inclination measurements. However, it isworth noting that our results are robust to differentchoices of sensory input; for example, the same qualitat-ive behaviour is observed if we assume that _a and _faremeasured separately by the proprioceptive system, oreven in the ideal case when the whole state can bemeasured. For the control action, we assume thatbalancing around the upright position is achievableusing the torque Fa [8] that is assumed to be bounded,i.e. jFatj Famax (see appendix C for a simplederivation of an upper bound on the scaled torqueFamax 0:3). Once again, our results are quite insensi-

tive to the particular choice of Fmax; we can vary thisquantity by a factor of 10 (for example, by consideringpotential torso and leg movements) without affectingthe overall qualitative behaviour of the system.

For control strategies, we have a choice of conti-nuous or discontinuous control strategies. The latterhas been proposed for stabilization of quiet uprightstance [18 21], based on the assumption that smalldeviations from upright position are not detected bythe control unit and that active corrective torques aregenerated only when the position [19] or the velocity[20] exceeds a given threshold, i.e. when jatj . acr orj _aj . _acr. In the case under study, the coupling with therope dynamics amplifies the sway dynamics and there-fore the time spent inside the uncontrolled regionjatj acr, j _atj _acr is small, so that a continuousstrategy might be practically equivalent to an intermit-tent control. Intermittent control strategies have alsobeen proposed to cope with the intrinsic delay that ispresent in the feedback loop during quiet standing,exploiting the intrinsic stable manifold of the pendulumdynamics to passively drive the system towards theequilibrium [22]. Although such an approach canexplain several experimental observations about quietstanding, its relevance for the problem at hand is notobvious. In fact, replacing the fixed base with a rope

couples the dynamics of the rope to that of thehuman on it, so that the argument used for designingswitching regions is invalid because the state spacehas now four-dimensional instead of two-dimensional.

Balancing on tightropes and slacklines P. Paoletti and L. Mahadevan 2099




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Moreover, since the maximum admissible delay in oursystem is of the order of 4070 ms (see 5.3), andthus shorter than the typical 100200 ms, the perform-ance gap between intermittent and continuousstrategies is minimal, and therefore we focus here onlyon continuous feedback control.

3. ANALYSIS AND CONTROL

Before we discuss the controller, we first discuss the lin-earized dynamics of the system about the uprightposition.

3.1. Linearized dynamic analysis

The state of the system can be represented asx f; _f; a; _a with the input torque u Fa.Then the linearized non-dimensional dynamics (seeappendix A for the full nonlinear dynamics) is given by

x

t

Ax

t

Bu

t;

3:1

and

A0 1 0 0

1=R 0 m=R 00 0 0 1

1=R 0 1 m m=R 0

2664

3775

B 0

mR 1R20

1 m m1 2RR2

2664

3775:

9>>>>>>>>>>=>>>>>>>>>>;

3:2

The linear stability of the upright position is completely

characterized by the spectrum of the matrix A and isshown in figure 1c as a function of R (see alsoappendix B). We note that as R increases the magni-tude of the real eigenvalues also increases, so that thesystem instability becomes severe, making controlmore difficult for large R. However, simultaneously,the imaginary eigenvalues move towards the origin, sothat the natural frequencies of the system decreaseand make control easier due to the absence of high fre-quencies in the system response. On the other hand, asR decreases, the system instability becomes less severebut the presence of high-frequency oscillations makesthe model less likely to be accurate, because the

dynamics of the rope can no longer be neglected andthe control task becomes harder because the controllerhas to damp out such fast oscillations. The competitionbetween these two effects suggests the existence of anoptimum sag R, and thus a location along the tight-rope/slackline where it is easiest to balance for a givenmass ratio m.

In any case, for all R, the presence of one unstablemode requires the presence of a controller to stabilizethe upright position. To ensure the existence of a feed-back controller capable of such a task requires that thesystem is controllable, i.e. that there exists an input Fa

capable of steering the system to the desired position. In

a linear setting, this condition states that the system iscontrollable if and only if the controllability matrixJ B AB . . . An1B has full rank [23]. In our model,J is always full-rank so that a suitable linear controller

always exists (appendix B). However, as R 0 orR 1 the matrix has a weak rank deficiency;in figure 2b, the condition number k of the matrix Jis plotted as a function ofR and shows that a minimumoccurs at R 1.

As discussed in 2, partial measurements of the fullstate f; _f; a; _a are provided by _f _a via the vestibu-lar system. To check that the system is observable, i.e.that the reconstruction of the full state x from _f _ais possible, we first write the measurement equationyt _ft _at as

yt Cxt 3:3with C [0, 1, 0, 1]. In the linear setting, observabilityreduces to testing that the associated observabilitymatrix V has full rank [23], where

VC

CA

..

.

CAn1

26664

37775: 3:4

In our model, Vhas full-rank for each value ofR and M(appendix B), but as R 0 and R 1, V showsagain a weak rank deficiency as reported in figure 2that also shows that the minimum condition numberoccurs when R 1.

3.2. Optimal control strategy

Given the complete controllability and observabilityproperties of our linearized model, it is also possible tostabilize the pendulum dynamics via a feedback control-

ler, but the control strategy is not unique. Here, we focuson an optimal output feedback controller that takes theform of an LQG (linear quadratic Gaussian) controller,consisting of two parts: an optimal controller (linearquadratic or simply LQ regulator) given full statemeasurements and an optimal observer (Kalman filter)to estimate the full state given partial measurements,with the overall control scheme shown in figure 2 (seeappendix C for a comparison of the case with completemeasurements and that with limited information).

Since we want to minimize deviations from the uprightposition f a 0, we choose a performance index

Ju 1

0qff2 qaa2 Fa2 dt; 3:5

wherewe have assumed an infinite time horizon forsimpli-city and we have set the weights qf qa 10 in order toobtain stabilization for a wide range ofR1, while enlargingthe basin of stabilization by limiting large variations in fthat would bring the system out of the range where linear-ization is valid. We do not weight the velocities _fand _a sothat we do not affect the controlled dynamics and deterio-rate the performance. Recently, other authors [24] haveanalysed quiet standing using a two-link inverted

1The magnitude of these weights is quite different from Qu et al. [9]

where the authors study the balancing problem on the sagittalplane with the feet on a fixed platform. However, what matters inLQ design is the relative magnitude among the various weights andin this perspective our choice is quite close to the one reported inthe study of Qu et al. [9].





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pendulum model to show that balance can also beachieved by minimizing a linear combination of angulardeviations and velocities of the two links instead ofusing the classical centre-of-mass displacement minimiz-

ation. We argue that the presence of a swaying ropemakes our strategy similar to that proposed in the studyof Kiemel et al. [24] because deviations from the verticalare automatically reflected in rope swaying motion andcentre-of mass displacement. Defining three matricesQ diagqf; 0; qa; 0, H 1, N 0, the performanceindex (3.5) can be written as

Ju 1

0

xTQx uTHu xTNudt: 3:6

Given the values ofQ, H and N, the control input thatminimizes the performance index J(u) can be expressed as

ut Kxt; 3:7where the optimal feedback gain K is determined by theunique symmetric positive definite solution P of an

algebraic matrix Riccati equation [25]

K H1BTP N 3:8

and

ATP PA PB NH1PB NT Q 0:3:9

In order to compare the influence of the system parametersR and m on the controller performance, we fix the cost(3.6) and allow the feedback gain K to change accordingto (3.8) and (3.9).2

In the presenceof noise in thesensor measurementsandinformation processing, and because we only have partialinformation, the LQ regulator (3.7) is not sufficient andmust be supplemented by an estimate x of the true state

1 2 3 4 50

5000

10000

absolute velocity sensor

LQ regulator

f(t)+a(t)

LQG controller

Kalman filteru(t)x(t) x(t)

R

50

500

1000(i) (ii)

(iii) (iv)

50

0.5

50

0.5

1.0

R

S

50

0.05

0.10

R

U

F amax

(a)

(b)

(c)

ts

max(Fa)

k(X)

k(W)

Figure 2. (a) Schematics of the balancing control loop. (b) Condition number kof the controllability matrix J(solid) and obser-vability matrix V (dashed) as a function of normalized radius R. (c) Influence of the normalized radius R on time to steady statets (i), maximum control torque maxFa (ii), stabilization S(iii) and energy spent for control U(iv). Data obtained by simulatingthe full nonlinear model (A 3) and (A 4) with m 30 and LQG controller based on measurement of _f _a (see 3.2). (Onlineversion in colour.)

2The opposite strategy, i.e. the comparison of controller performancewith fixed feedback gain K is not viable because the feedback gaindesigned for a given value of the parameters (and thus for a given Aand B) can be even non-stabilizing for different values of parameters.





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variable x, so that the optimal control input now reads

ut Kxt: 3:10:To estimate x given noisy and limited measurement

yt _ft _at, while preserving the optimality ofthe control design, we design a Kalman filter as an opti-mal observer that gives a minimum-variance estimate of

the system state variables. In the presence of noise onthe internal dynamics and in the measurement process,(3.1) and (3.3) read

xt Axt But Gwt 3:11and

yt Cxt vt: 3:12where w(t), v(t) are stochastic variables, G is a gainvector linking the scalar noise w(t) to the vectorialstate variable xt, and the stochastic variables havezero mean and variance S and Q, respectively. Thenthe Kalman filter dynamics can be written as

_xt Axt But Lyt yt 3:13and

yt Cxt; 3:14where the matrix gain L is derived from the positivedefinite solution S of an algebraic Riccati equation [25]

L SCT Q1 3:15and

AS SAT GSGT SCT Q1CS 0: 3:16We note that the Kalman filter (3.12) and (3.13) can beviewed as a copy of the original system (3.1)(3.3) withan additional input yt Cxt that allows for correc-tions to the estimates based on actual measurements.

In absence of noise, i.e. ifS Q 0, equations (3.15)and (3.16) can be satisfied by taking S 0 and arbitraryL. However, we use a virtual noise to allow the systemto be controlled in a weakly nonlinear regime. We chooseS Q 106 and G [100, 100, 10, 10]T so that thenoise amplitude on a is in agreement with the resultreported in thework of Boulet etal. [26] andthecoefficientsof variation ofGw(t) with respect to a(0) and f(0) read,respectively, 0.5 and 5. The noise in f is assumed tobe larger than the noise in a because deviations fromf 0 quickly invalidate the linearized model and thusthe controller design. Finally, we note that, although wedo not have any experimental measurement of such noiseparameters, the qualitative behaviour of the system isrobust to changes in these parameter values, the more sen-sitive being the ratio between the noise on a and the noisein f, and all the results presented below still hold.

4. SIMULATIONS

To verify the performance of our linear control design, we

now test it on the full nonlinear model. This is requiredas the same linearized model used for controller designcould potentially hide performance degradation due tolarge deviations from the equilibrium position that

dramatically changes the local response of the model toactuation. Such large deviations are indeed the maincause for instability in the closed loop system, as shownbelow, but they will be correctly compensated in the lin-earized model, although with large controller efforts. Wesimulate the system with initial conditions a0 18,f0 18 and _a0 _f0 0 until tf 1000 whichin dimensional terms corresponds to a few minutes.To obtain a quantitative measure of the perform-ance, we need to define one or more performanceindexes. Two natural choices are the time ts needed toreach the steady state, defined as ats 0:1a0 andfts 0:1f0, and the maximum torque maxFaapplied by the controller. To detect situations wherestabilization is achieved by using a maximum torquemaxFa . Famax, we define an index of success S as

S 1 maxfFatg Famax and ts tf

0:5 maxfFatg . Famax and ts tf0 ts . tf or system unstable;

8>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

5:3

where now xT xM; _xM; a; _aT and khas been normal-ized by Mg=l. We note that as R 0 for the case whenthe cart is placed on a circular track the natural fre-

quency of the system scales asffiffiffiffiffiffiffiffiffi

1=Rp

, whereas in thecurrent model the natural frequency scales as

ffiffiffik

pso

that large values of k correspond to small sag, asdesired. Assuming as before that only the total angular

velocity is measured, then

CT 0 0 0 1: 5:4

Finally, in order to obtain a fair comparison ofthe controller performance, we chose the same weightsQ, H and N for the LQ controller design and thesame parameter values S, Q and G of the Kalmanfilter as in 3.2.

In figure 4b, we report the total energy spent forstabilization as a function of the spring stiffness k. Wenote that the energy converges to a minimum ask 1 that corresponds to balancing on a rigid sup-port. As the stiffness decreases, the control becomesmore difficult because the cart displacement tends tobe larger and thus the system can be driven out of therange of validity of the linearized model.

5.3. Influence of delay

Measurement of the body orientation show a naturaldelay due to processing time of the information by thenervous system. Moreover, once the measurement isobtained, additional delays are introduced by theneural feedback controller that has to calculate the cor-rect amount of torque necessary for stabilization and bythe motor control system due to the neural signal travel-ling time and muscle activation delay. A delay in thefeedback loop can severely reduce controller perform-ance during balancing [8]. There is no agreement onthe magnitude of the delay t mainly because humansensors/controller tend to adapt it accordingly to differ-ent tasks and different accurate measurements available

[13]. Common estimates from experimental data rangefrom 25 ms to 200 ms depending on the experimentalconditions, with the delay decreasing proportionallyto the utilization of the vestibular system [9,13,16] cor-responding to values of scaled delay t t ffiffiffiffiffiffiffig=lp between0.07 and 0.63.

In the presence of delay in the control loop, thedynamics reads

xt Axt But t; 5:5i.e. it is represented by a delay differential equation,where we have lumped all the delays into a single

term, using the linearity of our model to advantage.Since the magnitude of the delay is smaller than thetypical timescale associated with the pendulumdynamics, i.e. t, 1, we approximate the delay variableut t with its Taylor expansion up to the secondorder [9]

ut t ut t_ut t2

2ut 5:6

thus obtaining an extended system

_~xt Axt But 5:7and

~yt Cxt; 5:8

0.02 0.04 0.06 0.08 0.100

0.05

0.10

xM

kM

m

k1

U

(a)

(b)

a

Figure 4. (a) Mechanical model for the motion on a tightropewith large pretension. The linearized equation of motion arereported in (5.2) and (5.3). (b) Total energy spent for stabil-ization as a function of spring stiffness k. Data obtained bysimulating the full nonlinear model with LQG controllerbased on measurement of _a. (Online version in colour.)





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where

x x

u

_u

24

35 and ~u u; 5:9

and

A A B tB0 0 10 0 0

2

4

3

5; B

t2=2B01

2

4

3

5and

C C 0 0:5:10

This allows us to design an LQG controller for theextended system that gives ~u as a function of x, andthen we can integrate twice ~u to obtain the input ufor the original system, as illustrated by the blockdiagram in figure 5a.

However, observability analysis with A and C as in(5.10) shows that the system is no more completelyobservable, because the internal variables u; _u donot directly influence the output. However, we caneasily overcome this problem by considering

yt _at _ft

ut_ut

24

35 C C 0 00 1 0

0 0 1

24

35; 5:11

i.e. by adding measurements of the controller internalvariable [9]. This assumption is not restrictive sincethe controller generates these signals by itself, thus iteven does not need to measure them.

For the design of the extended system LQ regulator,we define a performance index as

~Ju 1

0

10f2 10a2 Fa2

_F

a2

0:01 F

a2

dt;

5:12

where we choose to add a weight on _F

ato obtain better

performance. In fact, when delay is present it can beconvenient to slow down the dynamics of the controlledsystem so that the delay remains small compared withthe typical closed loop timescale and the controllercan easily compensate for it.

In figure 5b, we show the stability region as afunction of the feedback delay t and the normalizedsag R. We note that increasing the value of the delaymakes the performance worse and can lead the systemto instability, as expected [8,27]. Since the naturalfrequency of the system decreases as ffiffiffiffiffiffiffiffiffi1=Rp as R )

1,

we expect the critical value of t to increase asR becomes larger. The counterintuitive behaviourreported in figure 5b can be explained by noticingthat the controllability matrix condition number

(a)

0

0.1

0.2

246

8

105

0

0.1

0.2

0.1

0.2

0.3

0.4

R

1 2 3 4 50

0.2

0.4

0.6

0.01

0.02

0.03

(b)

(c)

(d)

unstable

unstable

f(t) +a(t)

t

t

t

u(tt)u(t) x(t)(t)

absolute velocity sensor

delayLQ regulator

LQG controllerKalman filter

x(t)

Figure 5. (a) LQG control system structure in presence of feedback delay t. (b) Stability region and (c) controllability matrixcondition number as a function of R and t. Data obtained by simulating the full nonlinear model (A 3) and (A 4) with m30 and LQG controller based on measurement of _f _a. The white region corresponds to unstable dynamics, whereas the greylevels in the stability region code for the total amount of energy U spent for stabilizing the system. (d) Stability region as afunction of R and R

gm with g

1 as in 5.1.





11/13

increases as t increases as shown in figure 5c, thus thisfact makes the control task actually harder. The sameplot shows that for each value of t the conditionnumber reaches its minimum in a neighbourhood ofR 1 and therefore the critical value of t reaches itsmaximum in that region. Finally, in figure 5d, weshow the region of stability as a function of the feedback

delay when the sag R is proportional to the mass mas in 5.1. We note that in this case, the stable regionbecomes larger and, in particular, it encloses theorigin because as R 0 and t 0 the problem reducesto the trivial stabilization of an inverted pendulum on afixed base. However, for every value of the sag R therestill exists a critical delay above which stabilizationcannot be achieved.

5.4. Conclusions

A minimal mechanical model for the combined systemof a human on a wirean inverted pendulum mountedon a cart that is moving on a circular trackallows usto pose the problem of postural balancing on a tightropeor a slackline as an optimal control problem. We showthat the measurement of the absolute angular positionalone is enough to generate a feedback correctivetorque that stabilizes the upright position. Our analysisalso shows that there exists an optimal range of the ropesag where the balancing task is accomplished withminimal energy effort, in agreement with empiricalexperimental evidence [28].

More generally, tightrope walking and slackliningare just two examples of situations where the couplingof (active) internal body dynamics to (passive) externalsubstrate dynamics both constrains and poses challengesfor the neurophysiological performance of motor systems.Recent experimental studies have shown that posturalsway analysis can be used to quantify changes in bodyor neural control due to pathologies [3], traumas [4] orageing [29]. Complementing such data with mathemat-ical models like the one we have proposed can help todevise tests with better discrimination while possiblyuncovering the reason for performance degradationwith age.

We thank M. Venkadesan for fruitful discussions.

APPENDIX A. FULL NONLINEAR MODEL

The EulerLagrange equations (2.3) for f and a read,respectively,

ml2 mlRcosaa 2mRlsina _a _f M mR2 2mlRcosa ml2 f mRlsina _a2

mglsina f MgRsinf FfA 1

and

ml2a ml2 mlRcosa f mlRsina _f2 mglsina f Fa:

A 2

By normalizing the variables as described in 2.1, wecan write the full nonlinear equations of motions as

f msinaD

_a2 msinaRcosa 1D

_f2 2 msina

D_a _f

mcosasina fD

1 msinfD

mRcosa 1RD

Fa mRD

Ff A 3

and

a msina1 RcosaD

_a2

R2sina1 m msina1 2Rcosa

D_f

2

1 mRsina f sinf Rsinf a2D

sinf msin2a f

2D 2msina1 Rcosa

D

_a _f

R21 m m1 2Rcosa

RDFa

m1 RcosaDR

Ff; A 4

where D R1 m mcos2a.

APPENDIX B. LINEAR SYSTEM ANALYSIS

The spectrum of the uncontrolled dynamics (3.1) reads

lA

1ffiffiffiffiffiffi

2Rp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 4R1 m

q

r

1 ffiffiffiffiffiffi2RpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL


q

r

1ffiffiffiffiffiffi

2Rp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL


q

r

1 ffiffiffiffiffiffi2RpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL


q

r

26666666666664

37777777777775

B 1

and

L 1 mR R m: B 2Note that for each value of R and M there arealways one real positive eigenvalue, one real negativeeigenvalue and two purely imaginary eigenvalues, con-sistently with the Hamiltonian (conservative) natureof the system.

Finally, the controllability matrix J reads

J

0 x2 0 x2R

mR

x1

x2 0 x2R

mR

x1 0

0 x1 0x2

R 1

Rx2

x1

x1 0x2R

1 Rx2x1 0

2666666664

3777777775

B 3





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and

x1 1 m m1 2RR2

and

x2 mR 1

R2;

B 4

whereas the observability matrix V can be written as

VC

CA

..

.

CAn1

26664

37775

0 1 0 10 0 v 00 0 0 v

v=R 0 v v m=R 0

2664

3775 B 5

and

v 1 m: B 6

APPENDIX C. CONTROL WITH FULLSTATE INFORMATION

In order to check that the behaviour observed in the 4is not due to the lack of information about the true stateof the system, we consider an ideal case where we haveexact measurements of the all state variables, so that wecan directly exploit the LQ regulator without any

Kalman filter. The LQ design parameters are as in 4so that we are not changing anything except themeasurement equation (3.3) that becomes y(t) x(t),i.e. C I.

In figure 6, a comparison between the two differentcontrol schemes is reported. We note that the lack ofcomplete measurements on the full state does not mas-sively degrade the system performances. The maindifference is in the small radius limit, where a full infor-mation about the system variables allows the controllerto stabilize the system with smaller values of R withrespect to the LQG case, although with large correctivetorque. Therefore, we can conclude that in the limit

when R(1, a strong source of instability is due to thepoor state estimate provided by the Kalman filter.This result is in agreement to the observation that forsmall R the linearization validity basin is very narrow

and high-frequency oscillations are present in thesystem, so that the linear observer can easily fail.

This analysis also shows that the difficulties thatarise in the limits R 0 and R 1 are intrinsic inthe system dynamics and are not due to the particularchoice of control scheme, in agreement with the discus-sion of 3.1 about the controllability/observability

properties and the validity of linear design.

APPENDIX D. TORQUE UPPERBOUNDESTIMATE

The main limitation in human balancing ability comesfrom the maximum value that the torque Fa canassume. We can estimate an upperbound for this quan-tity by considering the case in which the human uses apole of mass mp and length lp supported by the twohands exerting forces F1 and F2 at a distance a from

the pole centre of mass.In order to support the weight mpg of the pole

and generate a pure torque T, the forces F1 and F2must satisfy F1 F2 mpg, F1a F2a T. ThenT 2Fmax mpga. For the purpose of upperboundestimate, we can neglect the pole weight mpg withrespect to the maximum force Fmax. Reasonablevalues for the parameters are Fmax 250 N anda 0:5m so that Tmax 250 Nm or, in dimension-less form, considering a body mass of aboutm M 80 kg and leg length l 1m,

TT

mgl 250

80 9:81 1 0:3:

D 1

The applied torque Fa is the negative of this quantity.Including potential torso and lateral leg movements inFmax would increase this value, but these movementsare likely to be only emergency manoeuvres used onlywhen the deviation from the upright position reachesextreme values. In any case, the qualitative behaviourof the system is quite insensitive to the exact valuechosen for Fmax and we can indeed increase it by anorder of magnitude without severely affecting theobserved behaviour, in accordance with the fact thatthe main difficulties for balancing on a rope are dueto the structural property of the dynamics, such as con-trollability and observability, and not to the saturationon the control action. Here, we neglect potential wind-up problems, i.e. situations when the arms or the polestart twirling, by assuming that the angular displace-ment of the pole from the horizontal position isalways limited to a few degrees.

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0

500

1000

0

0.5

50

0.5

1.0

R50

0.05

0.1

R

max(Fa)

S U

(a) (b)

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Balancing on Tightropes and Slacklines

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