resonance entrainment of tensegrity structures via cpg control€¦ · of an optimized locomotion...

Resonance Entrainment of Tensegrity

Structures via CPG Control

Thomas K. Bliss a, Tetsuya Iwasaki b, Hilary Bart-Smith a

aUniversity of Virginia, 122 Engineer’s Way, Charlottesville, VA 22903bUniversity of California, 420 Westwood Plaza, Los Angeles, CA 90095

Abstract

Neuronal circuits known as central pattern generators (CPG) are responsible forthe rhythmic motions in animal locomotion. These circuits exploit the resonantmodes of the body to produce efficient locomotion through sensory feedback. Assuch, the neuronal mechanisms are of interest in the control of autonomous roboticvehicles. The objective of this study is to establish a design framework that in-tegrates synthesized CPGs with tensegrity structures, with the goal of resonanceentrainment. Tensegrities are novel, nonlinear dynamic structures featuring highstrength to weight ratios, embedded actuation and sensing, and tunable stiffness.The dynamics of a general class of tensegrities are characterized and a two neuroncircuit known as a reciprocal inhibition oscillator is integrated. The neurons dictatethe actuation of two cables in the structure, where each neuron has control of itsrespective cable length. Sensory signals (current, stretched lengths) from the cablesare fed back to their respective neurons. The frequency and amplitude of closed-looposcillation are analyzed via harmonic balance, and a systematic method for controldesign is proposed to achieve entrainment to a prescribed resonance mode.

Key words: Neural control, Oscillation, Resonance, Harmonic balance, Tensegrity.

1 Introduction

Rhythmic motion in animals is ultimately controlled by central pattern gen-erators (CPGs), which are systems of neurons connected in such a way that

? Corresponding author T. K. Bliss. Tel. 434-982-4593. Fax 434-982-2037.Email addresses: [email protected] (Thomas K. Bliss), [email protected]

(Tetsuya Iwasaki), [email protected] (Hilary Bart-Smith).

Preprint submitted to Elsevier 10 February 2011

CONFIDENTIAL. Limited circulation. For review only

their membrane potentials autonomously oscillate with particular phase rela-tionships between individual neurons. Brown [1] first illustrated that the CPGwas responsible for this behavior and suggested that sensory feedback servedto regulate the CPGs activity. He also cited reciprocal inhibition, where theactivity of antagonistic neurons inhibit each other, as a fundamental mecha-nism in rhythm generation. Pearce and Friesen [29] examined the contributionof sensory feedback, as well as peripheral neuronal and mechanical effects,to the coordination of rhythmic activity by comparing intact swimming ofmedicinal leeches and “fictive swimming” of isolated nerve chords. Friesen [5]then pointed out that reciprocal inhibition is responsible for oscillatory mo-tion in many species. He also showed that dynamic properties like synapticfatigue and impulse adaptation play an important role in oscillatory circuits.Ultimately, the literature illustrates that CPGs, consisting of neurons withreciprocal inhibition, are basic control units for rhythmic patterns. Sensoryfeedback has the ability to fine tune these patterns, resulting in coordinatedsignals essential to animal locomotion [6, 19, 28, 45].

These studies in biology have attracted the attention of engineers seeking ef-ficient control methodologies in robotics. Taga et al. [40] employed neuronalcircuits to control double pendulums modeling legs for bipedal locomotion, andobserved robust entrainment of the CPG to the mechanical system’s resonancefrequency. Williamson [44] used a CPG to control an arm with entrainmentto a hanging mass system, as well as turning a crank and ‘playing’ with aslinky. Kimura et al. [18] experimentally verified synthetic neuronal controlof a quadruped walking on irregular terrain as well as running on flat ter-rain using reflex signals to coordinate the CPG. Lewis and Bekey [20] andNakanishi et al. [27] also applied neuronal circuits to gait control and Lewis etal. [21] demonstrated the ability to create in silico CPGs. Chen and Iwasaki[3] developed a method for synthesizing oscillators with prescribed frequency,phase, and amplitude, and how to design such oscillators for feedback controlof an optimized locomotion variable.

Engineers have also investigated the mechanisms that provide entrainmentof neuronal circuits to a resonance of dynamic systems for efficient oscilla-tions [12, 44]. Verdaasdonk et al. [41, 42] simulated the neuro-musculo-skeletalsystem of the human arm, observing the conditions necessary for a reciprocalinhibition oscillator (RIO) to entrain to a resonance. Williams and DeWeerth[43] performed a case study of a simulated single degree of freedom (DOF)mass-spring-damper system controlled by an RIO, and commented on theform of feedback necessary for the RIO to entrain to the dynamic systemwhen the RIO’s intrinsic frequency is either above or below the resonanceof the mechanical system. Iwasaki and Zheng [15] compared the entrainmentof different RIO models to a pendulum, mapping the frequency against neu-ronal parameters. Then they used the method of harmonic balance to predictentrainment. Futakata and Iwasaki [9] formalized the graphical analysis in

2


[15] and illustrated that negative integral feedback and positive rate feedbackare responsible for robust entrainment. All these studies focused on classicalstructures and models of one DOF systems like the pendulum and mass-spring-damper.

In the design of robotic locomotors, efficient propulsion can be achieved byexploiting a resonance. While CPGs appear to provide a fundamental controlframework for such applications, tensegrity structures are particularly suitablefor the mechanical part of the design due to their similarity to musculoskeletalsystems. Tensegrities are systems of bars held in compression with stabilityby a network of cables (or strings) in tension[32], where the bars and cablescan act like bones and muscles in biomechanical systems. Originally developedby Snelson [34] and Fuller [7], tensegrities have attracted interest due to theirhigh strength to weight ratios as well as their actuation capabilities [4, 38]. Thestatics [2, 24, 30] and dynamics [26, 31, 33] of tensegrities have been studiedextensively. Their use as deployable antennae has been analyzed [8, 39], aswell as controllable platforms for flight simulators [16, 36]. Despite variousengineering applications of tensegrity structures, there seem to have been veryfew studies on tensegrities in the context of morphing locomotors.

This paper develops a systematic method for designing a CPG-based feed-back controller that achieves entrainment to a resonance for a general classof tensegrity structures. We argue that coupling of neuronal control strate-gies with tensegrity structures provides a promising avenue for realization ofmorphing locomotors inspired by biology. The cables in tensegrities provide amethod of simultaneous actuation and sensing [22, 25], which is analogous tothe biological motor control mechanism of regulating muscle stiffness throughmotoneuron activation and sensing the resulting motion by stretch receptors.We will define a general class of tensegrity structures, and develop a methodto design a CPG controller to achieve entrainment to a prescribed resonancemode of oscillation. The design method is based on the multivariable harmonicbalance [13] and hence an oscillation near a resonance is to be achieved onlyapproximately, but numerical examples demonstrate effectiveness of the pro-posed method. A similar problem has been considered for general multi-DOFmechanical systems [10], but this contribution is novel in focusing on multi-DOF underactuated tensegrity systems controlled by a single oscillator. Theformalization of the concept of an antagonistic actuator-sensor pair is also anew contribution.

We use the following notation. The set of positive integers up to n is denotedby In. The partial derivative of a vector-valued function f(x) ∈ Rm withrespect to the vector x ∈ Rn is defined to be the m× n matrix whose (i, j)th

entry is ∂fi/∂xj. The Hessian ∂2f/∂x2, or second partial derivative, of a scalarfunction of f(x) ∈ R with respect to the vector x ∈ Rn is defined to be then× n matrix whose (i, j)th entry is given by ∂2f/(∂xi∂xj).

3


2 Tensegrity Dynamics

In this section, we consider a general class of tensegrity structures and brieflyreview their equations of motion derived in [35, 37]. The positive realness ofthe linearized dynamics are studied and the concept of an antagonistic pairis developed. The control design method proposed later will rely on theseproperties.

2.1 General Equations of Motion

We consider tensegrity structures formed by r rigid bars connected by mflexible cables. Let θ ∈ Rn be a set of generalized coordinates representingthe shape, and li be the length of the ith cable, where i ∈ Im. This length lichanges with time, and is uniquely determined by the current shape θ. Thevector l(θ) ∈ Rm, with ith entry li, denotes this dependence. Let ì be the restlength of the ith cable, and define the rest length vector ` ∈ Rm. Consideringthe inertia of the bars, and the damping and stiffness associated with thejoints and cables, dynamics of the tensegrity structure can be described bythe Euler-Lagrange equation [35, 37]:

J(θ)θ + d(θ, θ) + k(θ, `) = 0, (1)

where J(θ) is the inertia matrix, d(θ, θ) contains the damping (friction, as-sumed smooth) and centrifugal terms, and k(θ, `) is the elastic force due tocables. No external forces are considered to act on the structure, and the de-pendence of the elastic force k(θ, `) on the rest lengths ` is made explicit asthis variable will allow direct actuation of the structure.

Suppose the rest lengths of the q cables, specified as ì for i ∈ Ma ⊆ Im, arechanged by actuators. We consider the dynamical motion of (1) resulting fromsmall perturbations of the rest lengths from a static equilibrium [37], satisfyingk(θe, è) = 0 where θ(t) ≡ θe is the shape when the rest lengths of the cablesare `(t) ≡ è. Let u(t) := Ta(`(t)− è) be the actuator inputs, where a positiveinput means that the cable is lengthened, and Ta ∈ Rq×m is a matrix formedby the ith rows of the identity matrix for i ∈ Ma. Then, assuming that theinput u and the resulting perturbed motion ϑ(t) := θ(t) − θe are both small,the equation of motion in (1) can be approximated by its linearization [37]:

Jeϑ+Deϑ+Keϑ = Beu. (2)

For feedback control design, the current lengths of the p cables, li(θ) for i ∈Ms ⊆ Im, are assumed available. Let Ts ∈ Rp×m be a matrix formed by theith rows of the identity matrix for i ∈Ms so that Tsl(θ) is the available sensor

4


signal. The output of the linearized system is taken as the perturbation of thelengths Tsl(θ) from those at the nominal configuration Tsl(θe) as follows:

y = Ceϑ, Ce := Ts∂l

∂θ(θe). (3)

Let the transfer function from u to y be denoted by P (s). This model will beuseful for analyzing the closed loop system, playing an important role in thedesign of a CPG controller to entrain to a tensegrity’s resonant modes. Theoriginal nonlinear model (1) can be used to evaluate the design by simulations.

We assume that a point on the structure is fixed to the inertial frame, all thebars are tightly constrained by the cable tensions at the equilibrium, and anyadmissible motions of the bars are subject to friction. In this case, matricesJe, De, and Ke are all symmetric positive definite, and there is no rigid bodymode and the system is stable. The linear model of the tensegrity structurehas n pairs of natural frequencies and mode shapes, the ith represented byωi and ξi respectively, satisfying (Ke − Jeω2

i ) ξi = 0. The mode shapes arethe eigenvectors of J−1e Ke, and the natural frequencies are the square roots ofthe corresponding eigenvalues. When the linear system is driven by sinusoidalinputs u, the forced response of the outputs ϑ has a locally maximum ampli-tude at a resonance frequency, which is close to a natural frequency, providedthe damping in the system is sufficiently small. The resonance and naturalfrequencies of the actual (nonlinear) system would depend on the amplitudeof oscillation in general, but are close to those of the linear model when theamplitude is small.

2.2 Positive Realness

A system is easily controllable to achieve high bandwidth tracking with areasonable amount of control effort if the system is positive real [14]. Analogousto this fact, it turns out that the positive realness of the tensegrity system isimportant for achieving entrainment to the second or higher resonance mode.Recall that a transfer function G(s) is said to be positive real if G(s)+G(s)∗ ≥0 holds for all s in the closed-right half complex plane except for those thatare poles of G(s), if any. Mechanical systems with collocated force actuatorsand velocity sensors are positive real. One may also expect for a tensegritystructure that sensor/actuator collocations lead to a positive real transferfunction. Below, we show that this is not true in general, but a scaled versionof sP (s) is positive real.

The potential energy stored in all the cables, V (l), is given as the sum ofVi(li) for i ∈ Im, where Vi depends on difference between the current and restlengths of the ith cable, li := li(θ) − `i. From the Lagrange formulation, the

5


elastic force k(θ, `) in (1) is the partial derivative of V (l(θ)− `) with respectto θ, and is given by

k(θ, `) = A(θ)τ(θ, `), A(θ) :=

(∂l

∂θ(θ)

)T

, τ(θ, `) :=

(∂V

∂l(l(θ)− `)

)T

.

The linearization of the elastic force contains the control input term

Beu = −(∂k

∂`(θe, è)

)(`− è),

where the ith entry of the vector `−è is zero for the unactuated cables i 6∈Ma,and Be in (2) is thus obtained as

Be = A(θe)TT

aM, M := Ta

(∂2V

∂l2(le)

)T T

a , le := l(θe)− è.

Note from (3) that the output coefficient is given by Ce = TsA(θe)T. Hence, we

have Be = CTeM if the sensors and actuators are collocated at the same cables

(i.e. Ma = Ms and Ta = Ts). It then follows from the standard collocationargument that G(s) := sP (s)M−1 is positive real. The Hessian ∂2V/∂l2 is adiagonal matrix whose ith entry is the stiffness of the ith cable at its equilibriumlength, and M is obtained by keeping the ith diagonal entries of the Hessian fori ∈ Ma. Therefore, G(s) is the transfer function from the tension force inputto the velocity output, leading to positive realness as formally shown above.Further, when the cables are linearly elastic with uniform spring constant ε,we have M = εI and hence sP (s) is positive real without the scaling factor.

2.3 Antagonist Pairs

Tensegrity structures are analogous to musculoskeletal systems in biology inthat the cables act like muscles that actuate the body. An important propertyshared by cables and muscles is that they are only able to impart force throughtension, not compression. The property necessitates at least a pair of antag-onistic muscles to drive a skeletal joint in biological systems. This motivatesus to choose a pair of antagonistic cables to actuate the tensegrity structure.

A nominal configuration of practical importance is one of symmetry, and thenatural oscillations about this configuration are symmetric as well. If two actu-ators are placed at symmetric locations, then they would act as an antagonisticpair. To elaborate on this point, consider a symmetric structure driven by twosymmetrically placed actuators with collocated sensors. Let P (s) be the 2× 2transfer matrix with its (i, j)th entry denoted by Pij(s) with i, j = 1, 2. Sincethe effect of u1 on y2 is the same as that of u2 on y1 by symmetry, we have

6


P12(s) = P21(s). By a similar argument, we arrive at P11(s) = P22(s). Hence,antagonistic actuations u1 = −u2 leads to y1

y2

=

P11 P12

P12 P11

u1

−u1

= (P11 − P12)

u1

−u1

,indicating that the resulting outputs y1 and y2 are also antagonistic; y1 = −y2.

For a general tensegrity structure, it is not obvious how to choose an antag-onistic pair of cables for actuation and sensing. Furthermore, there may wellbe no such choice in general. The objective of this section is to provide a def-inition and a verifiable condition for an extended notion of antagonicity thatbroadens applicability of controlled tensegrity structures. To this end, considerthe following definition:

Definition 1 A two-input, two-output system P (s) is said to have antago-nistic pairs of actuators and sensors if there exists a nonzero scalar transferfunction ρ(s), nonzero η ∈ R, and diagonal matrices H,G ∈ R2×2 such that

HP (jω)Ge = ηρ(jω)e, ∀ ω ∈ R, where e :=

1

−1

. (4)

Condition (4) means that if a pair of anti-phase inputs u = e sin(ωt) is ap-plied to the augmented system HP (s)G, then its outputs oscillate also in ananti-phase manner in the steady state. The parameter η is redundant in thisdefinition but will become useful later as a design freedom. The gain matri-ces H and G provide conditioning of the inputs and outputs (i.e. scaling oftheir magnitudes) to the original system P (s). The following result gives acomputationally verifiable condition for antagonicity.

Lemma 1 Let a stable 2×2 transfer function P (s) be given. The system P (s)has an antagonistic pair of actuators and sensors if and only if there exists anonzero vector f ∈ R4 such that

fTCXCT = 0, f1f4 = f2f3, (5)

where X is the unique symmetric solution of the Lyapunov equation

AX +XAT +BBT = 0,

and (A,B,C) is a minimal realization of the transfer function

q(s) :=[P11(s) P12(s) P21(s) P22(s)

]T.

7


Proof. For each ω ∈ R, there exists ρ(jω) satisfying (4) if and only if

e⊥HP (jω)Ge = 0 ⇔ hTP (jω)g = 0,

⇔ tr(FP (jω)) = 0,

⇔ fTq(jω) = 0,

where

e⊥ :=[1 1

], h := (e⊥H)T, g := Ge, F := ghT, f :=

[F11 F21 F12 F22

]T,

with Fij and Pij(s) being the (i, j) entries of F and P (s), respectively. Furthernote that

fTq(jω) = 0, ∀ ω ∈ R ⇔ fTQ = 0, Q :=1

2π

∫ ∞−∞

q(jω)q(jω)∗dω.

The matrix Q is given by Q = CXCT with the controllability grammian X.Thus, existence of f such that fTQ = 0 is necessary. Finally, for such f , thegain matrices G and H, such that F = ghT, can be constructed if and only ifdet(F ) = 0, or f1f4 = f2f3 holds. 2

The definition and the associated verifiable condition for antagonicity can beapplied to any stable 2× 2 transfer function and are not limited to tensegritysystems. When the condition in Lemma 1 is satisfied, the parameters in (4)can be obtained from a vector f ∈ R4 satisfying (5) as follows:

G := gGo, H := hHo, η := gh, ρ(s) := eTHoP (s)Goe/2, (6)

where g, h ∈ R are scalar design parameters to be specified later, Go :=diag(a1,−a2) and Ho := diag(b1, b2) are defined by any full rank factor a, b ∈R2 of F such that F = abT, and F ∈ R2×2 is defined by f = [ F11 F21 F12 F22 ]T.The freedom in the sign of f can be used is to introduce normalizationρ(0) > 0.

3 Neuronal Control

This section proposes a control design method for the tensegrity structureP (s) described by (2) and (3). We assume that the structure has an antago-nistic pair of actuators with collocated sensors, and satisfies (4) for (6) withpositive real ρ(s). The control architecture based on a CPG is first introduced,the closed-loop system is analyzed to examine its oscillation properties, andthen, reversing the analysis, a control design method is proposed to achieveentrainment to a prescribed resonance mode.

8


3.1 Framework for CPG Control

The most basic CPG unit is the reciprocal inhibition oscillator. It is a systemof two neurons with mutually inhibitory synaptic connections. Each neuroncan be modeled as a mapping from input w to output v as shown by [9]:

vi = ϕ(qi), qi = f(s)wi, f(s) :=2ωos

(s+ ωo)2,

for i = 1, 2. The transfer function f(s) is a bandpass filter centered at thefrequency ωo, capturing the lag and adaptation characteristics of neuronaldynamics. The nonlinearity ϕ(x) captures the threshold property of neurons,and is assumed to be an odd, bounded, increasing function that is strictlyconcave on x > 0 and satisfies ϕ′(0) = ϕ(∞) = 1. A suitable choice for ϕ(x)is the hyperbolic tangent function, tanh(x), which is used in our numericalstudy. To build an RIO, inhibitory connections link two neurons as shown inthe upper half of Fig. 1, where µ > 0 is the neuronal coupling strength. TheRIO is regarded as a system with input r and output v, and is described by

v = Φ(q), q = f(s)(Mv + r), M :=

0 −µ

−µ 0

, (7)

where Φ := ϕI acts on q elementwise.

The idea from biology is to place the RIO in a feedback loop with the me-chanical system to be controlled. The outputs vi are thus used to control thetensegrity through actuation gains gi, and sensory feedback will return as rithrough gain parameters hi, as shown in Fig. 1. The resulting closed loopsystem is described byJes2 +Des+Ke −BeGΦ

f(s)HCe f(s)MΦ− I

ϑq

= 0 (8)

where the gain matrices G and H are 2× 2 diagonal matrices with entries giand hi, respectively, and are chosen as in (6) with design parameters g, h ∈ R.

When µ is sufficiently large, the RIO exhibits anti-phase oscillations of q1and q2 without any input r, where the frequency is roughly equal to ωo [9].With sensory feedback from the system, the RIO could autonomously modifyits oscillation frequency to conform with the dynamics it is facing (e.g., toachieve entrainment to a mechanical resonance). In particular, when hi aresmall in magnitude, tensegrity oscillations will be induced near the intrinsicfrequency of the RIO through the almost open-loop control. When hi are larger

9


than a threshold value, however, the oscillation frequency may become closeto a mechanical resonance under a certain condition. The task here is to derivesuch condition on the controller parameters gi, hi, µ, and ωo, and find a designmethod to determine the parameter values satisfying the condition.

The entrainment property has been fully analyzed and confirmed in a differentsetting (single input, single output one-DOF system) [9, 15]. Unlike thesestudies, two RIO output signals are not combined to form one output, nor arethe feedback signals derived from a single state of the controlled structure;stated more clearly, the dynamic system controlled by the RIO is MIMO, notSISO as used in earlier studies.

P11(s) P12(s)

P21(s) P22(s)

φ

μ

-

-

w2

RIO

Tensegrity

q2v2

r2

v1q1w1

r1

u2

u1g1

h2

h1

g2

μ

φ

f(s)

f(s)

y2

y1

Fig. 1. RIO - Tensegrity system

3.2 Oscillation Analysis

In this section, we apply the method of multivariable harmonic balance (MHB)[13] to the closed-loop system and derive a condition under which the MHBequation predicts existence of oscillations. The frequency and amplitude ofoscillation are also characterized.

To this end, recall that the Fourier series approximation of a generic periodicsignal x(t) is given by

x(t) =∞∑k=0

αk sin(kωt+ φk) ∼= α1 sin(ωt+ φ1),

where the approximation is accurate when the bias and higher order harmonicsare all small. The phasor of x(t) is defined by the complex number x = α1e

jφ1 .The describing function [17] of the nonlinearity ϕ(·) is an amplitude dependentgain κ such that ϕ(x) ∼= κ(a)x for x(t) := a sinωt, where the approximationis based on the Fourier series as described above. When ϕ is an odd sigmoidfunction as assumed earlier, κ(a) is real valued and monotonically decreasingfrom one to zero on a > 0.

10


Under the nominal operating condition, the variables q1 and q2 in the RIOcontroller should oscillate 180o out of phase to each other. Hence, we chooseto enforce this anti-phase property as a constraint in the control design. Tocharacterize the condition for such oscillations to occur, let us consider thesituation where the closed-loop system has a periodic solution such that q1 =−q2. In this case, the phasor of q is given by q = ae for some a ∈ R. With theuse of Fourier series approximations and the describing function, (8) becomes

Je(jω)2 +De(jω) +Ke −κaBeG

fωHCe κafωM − I

ϑae

= 0, (9)

which is the MHB equation, where fω := f(jω), κa := κ(a), and κ is thedescribing function of the nonlinearity in the RIO. Recalling that f(s) actsas a band pass filter centered at the intrinsic frequency ωo, it is reasonableto assume that higher order harmonics will be sufficiently attenuated and thebias component will be blocked, making the MHB equation a reliable tool foroscillation analysis. Under the antagonicity assumption (4) with the choice ofG and H in (6), the MHB equation (9) holds if and only if

ϑ = aκa(−Jeω2 +De(jω) +Ke)−1BeGe, (10a)

1 = ηκafωgω, g(s) := µ/η + ρ(s) (10b)

where gω := g(jω).

The closed loop system is expected to oscillate at a frequency ω, amplitude a,and mode shape ϑ that satisfy (10). Since κ(a) is real positive, from (10b) weobtain the phase angle condition

∠f(jω) + ∠g(jω) =

0, (η > 0),

−π, (η < 0).(11)

Solutions ω to (11) represent possible closed-loop, steady-state oscillation fre-quencies. These candidates, however, may not admit any solution a to (10b).In particular, the range of the describing function κ is the interval (0, 1), andhence there exists a satisfying (10b) if and only if

1 < ηf(jω)g(jω), (12)

If this condition is violated for a particular ω, it means that oscillations maynot exist for the frequency. On the other hand, if (12) holds, the oscillationamplitude a of qi can be estimated from

a = κ−1(1/ηf(jω)g(jω)). (13)

11


The process described above would predict multiple oscillations for the closed-loop system in general, and some of them may be unstable and are not observedin simulations.

Stability of the closed-loop oscillations can be predicted from the associatedcharacteristic equation obtained by replacing the nonlinearity with its describ-ing function [11]:

1 = ηκ(a)f(s)g(s).

As shown by Iwasaki and Zheng [15], the predicted oscillation with amplitudea is expected to be stable if the following conditions are satisfied:

λ(a) = 0, λ′(a) < 0, (14)

where λ(a) is the maximum value of the real parts of the characteristic roots,and prime denotes the derivative, provided such maximal root is unique. Thefirst condition implies that all other roots lie in the open left half plane, and aretherefore stable. The root on the imaginary axis is the solution that causes theoscillatory behavior. To examine stability of the oscillation, let us assume thatthe amplitude of oscillation is perturbed, causing a∗ > a. When (14) holds,λ(a∗) < λ(a) = 0, stabilizing the system, causing the amplitude to decreaseback to a. If the amplitude is perturbed such that a∗ < a, then λ increases,destabilizing the system and increasing the amplitude to a. In conclusion, givenan RIO controller and a tensegrity structure, stable entrainment is predictedat a frequency ω and amplitude a such that (11)–(14) are satisfied.

Graphically, candidate oscillation frequencies are characterized by intersec-tions of the Nyquist plot of ηf(jω)g(jω) with the real axis greater thanone. The coordinate of each intersection d specifies the amplitude througha = κ−1(d). Since the critical point 1/κ(a) moves to the right/left when theamplitude a is increased/decreased, the stability condition (14) is satisfiedwhen the intersection is the rightmost one. Summarizing the argument, wehave the following result.

Proposition 2 The MHB equation predicts existence of a stable limit cyclefor the closed-loop system if and only if the Nyquist plot of ηf(jω)g(jω) crossesthe real axis at a value greater than one. The frequency ω and amplitude a ofqi oscillation are estimated by (11) and (13), respectively, where ω is the onethat gives the maximum real value of ηf(jω)g(jω).

The phase balance condition =[fωgω] = 0 gives estimates for the oscillationfrequencies. The Nyquist criterion indicates that the stable oscillation wouldbe the one having the largest gain value ηfωgω. Thus, the harmonic balance

12


equation predicts that a stable limit cycle exists when γ > 1 where

γ := maxω

ηfωgω subject to =[fωgω] = 0. (15)

The maximizer gives the estimated frequency of oscillation, and the amplitudea is estimated from κ(a) = 1/γ. The goal of entrainment to resonance is for ωto be near a desired resonance frequency ωk. Properly selecting the controllerparameters ωo, η, and µ is the designer’s challenge.

3.3 Resonance Entrainment Mechanism

To gain insights into the entrainment mechanisms, let us consider the limitingcase where µ/η and the damping ratios are small, say of ε-order. Since ρ(s) ispositive real, it is given in the form

ρ(s) =z1 · · · znp1 · · · pn

,

zj := s2 + 2ζjωjs+ ω2j ,

pi := s2 + 2ζiωis+ ω2i ,

where i ∈ In, j ∈ In−1, zn is a (real) high frequency gain, ζi and ζj are thedamping ratios, and ωi and ωj are the resonance and anti-resonance frequenciessatisfying the interlacing property (see e.g., [23])

ω1 < ω1 < ω2 < ω2 < · · · < ωn.

The gain plot of ρ(s) has alternative peaks and valleys at which the phasecurve goes down and up, respectively, between 0o and −180o as shown inFig. 2. The band pass filter f(s) has a maximum gain at ωo and its phasedrops around ωo from 90o to −90o. Since µ/η is small, g(s) is close to ρ(s),and the oscillation would occur near an intersection of the phase curves forρ(s) and −ηf(s) due to (11).

In general, there are multiple intersections, i.e., solutions ω to the phase bal-ance equation. However, every solution is close to one of the following; ωo, ωi,and ωi, in the sense that the distance measured by

$i :=1

2

(ω

ωi− ωiω

), $i :=

1

2

(ω

ωi− ωiω

)

13


10−1

100

101

102

103

104

105

10−6

10−4

10−2

100

102

Gai

n

10−1

100

101

102

103

104

105

−300

−200

−100

0

100

Phas

e [d

eg]

Frequency [rad/sec]

Fig. 2. Entrainment condition from the Bode perspective

is small. In particular we have $i = O(ε) for some i ∈ In ∪ {o} or $j = O(ε)for some j ∈ In−1. To see this, note that

=[fωgω] = 0 ⇔ =[j(2ωoω)

po

(µ

η+z1 · · · znp1 · · · pn

)]= 0

⇔ R

[(µ

η· p1 · · · pn + z1 · · · zn

)p∗op∗1 · · · p∗n

]= 0

⇒ $o$1 · · ·$n · $1 · · · $n−1 = O(ε)

⇒

$i = O(ε) for some i ∈ In ∪ {o}, or

$j = O(ε) for some j ∈ In−1.

A typical situation is visualized in Fig. 2. When the damping ratios are small,the phase curve of ρ(s) abruptly changes by 180o near resonance and anti-resonance frequencies. This approximates the phase curve of g(s) when therelative gain µ/η is small, and forces the phase balance to occur near ωi, ωi, orωo. Entrainment to the kth resonance mode would occur if the phase balanceoccurs at ω ∼= ωk where the gain |fωgω| is the largest among all frequencies atwhich phase balance occurs.

14


Consider the positive feedback case

η > 0, ωk < ωo < ωk.

In this case, the phase balance occurs near resonance and anti-resonance modeslower than ωo (see blue curves in Fig. 2). Among these candidates for oscilla-tions, the one with the largest gain |fωgω| is expected to be stable. Since thegain at ωi is smaller than the gain at ωi+1, oscillations at anti-resonance fre-quencies would be unstable. The gain |f(jωi)| increases with i, while |g(jωi)|decreases with i. Hence, the two effects tend to cancel each other and the sta-bility property depends on the magnitudes of the relative effects. Therefore,the mode to which stable entrainment occurs can be different for differentplant transfer function g(s). It turns out that the CPG tends to entrain to thefirst mode for the tensegrity structure studied later.

Next consider the negative feedback case

η < 0, ωk−1 < ωo < ωk. (16)

The phase balance for this case occurs near resonance and anti-resonancemodes higher than ωo (see red curves in Fig. 2). The gain |f(jωi)| is smallerfor higher frequency ωi. Hence, if the peak gain g(jωi) has the same decreasingproperty, so does the overall gain |f(jωi)g(jωi)|. Thus, the stable oscillationwould occur at the lowest resonance mode ωk among those higher than ωo.Approximate entrainment to the kth resonance mode would thus be expectedif (16) holds, provided ζi, ζi, and |µ/η| are small, and the damping ratios ofthe modes higher than ωk are comparable to or smaller than ζk.

In the other cases where

η > 0, ωk < ωo < ωk+1 or η < 0, ωk < ωo < ωk,

the phase balance occurs near ωo. Whether the oscillation at ω ∼= ωo is stableor unstable depends on the actual amount of damping and the location ofωo within the interval between a resonance and an anti-resonance. The rangeof the parameters for resonance entrainment can be more precisely estimatedfrom the mode partition diagram (MPD) explained in the next section. Thecontrol design then simply reduces to picking an appropriate point on theMPD.

3.4 Design by Mode Partition Diagrams

In the previous analysis, we have shown that the frequency of closed-looposcillation, if any, is close to a resonance frequency or the intrinsic frequency

15


of the RIO. Below, we will provide an analytical formula for the entrainmenterror in the frequency.

When the tensegrity structure is lightly damped and has collocated actuatorsand sensors, the transfer function ρ(s) has a frequency response illustrated inFig. 2, and can be approximated by a second order transfer function near aresonance mode. Let ρ(s) be an approximation of ρ(s) in the neighborhood ofthe kth resonance mode s ∼= jωk:

ρ(s) ∼= ρ(s) :=δkω

2k

s2 + 2ζkωks+ ω2k

.

The following result can be proved through a slight modification of results in[9, 10]. The proof is omitted for brevity.

Proposition 3 Suppose the harmonic balance condition

1 = ηκafω(µ/η + ρω)

holds, and the associated quasi-linear system is marginally stable. Then wehave ∣∣∣∣ω − ωkωk

∣∣∣∣ ≤∣∣∣∣∣ζk$

∣∣∣∣∣ if (ωo − ωk)(η/µ− 2$ζk/δk) > 0,

∣∣∣∣ω − ωoωo

∣∣∣∣ ≤∣∣∣∣∣ζk$

∣∣∣∣∣ if (ωo − ωk)η < 0,

where

$ :=1

2

(ωoωk− ωkωo

).

The error bound |ζk/$| is independent of the value of δk, and is determinedby the damping ζk and the distance $ between ωk and ωo. Hence, the boundis valid even if the approximation parameter value δk is inaccurate as longas the effects of the other poles and zeros in ρ(s) can be captured by a realconstant δk. In reality, the imaginary part of δk is close to but not exactlyequal to zero, and this could potentially make the bound violated. However,we find the quantitative bounds useful in constructing MPDs.

The mode partition diagrams illustrate the expected entrainment characteris-tics of the closed loop system by separating the (ωo, η/µ) parameter space intoregions where entrainment is expected close to the kth resonance frequency,the intrinsic RIO frequency, or neither. Let these sets be denoted Ok, Oo, andN respectively. For every x := (ωo, η/µ) pair, the frequency candidate sat-isfying (15) is determined and denoted ω(x). If such ω(x) exists, define the

16


distance to ωi by

ei :=

∣∣∣∣∣ω(x)− ωiωi

∣∣∣∣∣ ,for i ∈ In∪{o}. The mode ωk to which ω(x) is closest is determined such that

ek < ei, ∀ i ∈ In\{k}.

The sets Ok and Oo are then defined by

x ∈

Ok if ek < eo and ek < |ζk/$|Oo if eo < ek and eo < |ζk/$|

The set N is defined by x for which there is no feasible solution to (15) or theestimated closed loop frequency ω(x) is outside of the error bound |ζi/$| forall i.

The analyses above, and the resulting mode partition diagrams, can now beused to design an RIO controller for a given tensegrity morphing structureusing the linearized dynamics with the goal of entrainment to the kth resonantfrequency.

(1) Select actuation gain g to attain a desired amplitude of oscillation atresonance, where the amplitude is estimated from the phasor

ϑ := Θ(jωk)Ge, Θ(s) :=(Jes

2 +Des+Ke

)−1Be.

(2) Choose a value for neuronal coupling strength µ, where µ > 1 will ensureintrinsic oscillations of the RIO (by Lemma 1 of [9]).

(3) Construct the mode partition diagram, and choose a pair (ωo, η/µ) suchthat entrainment is predicted at mode k.

(4) Calculate the sensory gain h := η/g, given g, η/µ, and µ above.

Step 1 allows the designer to choose the approximate oscillation amplitude ofthe structure by using the linearized resonance peak. Note that the larger theamplitude, the higher the likelihood of cables becoming slack during part ofthe oscillation cycle. In regards to step 3, note that the further ωo is from thetargeted mode, the smaller the error bound |ζk/$| becomes, thus the designerwill want to maximize |ωo−ωk|, while remaining in the Ok region of the MPD.

4 Three Cell Beam Application

The design method outlined in the previous section is applied to the threecell tensegrity beam presented below. The nonlinear tensegrity dynamics are

17


compared to the linearized system to illustrate the robustness of controllerentrainment.

4.1 Three cell beam

For the purpose of demonstrating the methods above, as well as the incor-poration of central pattern generator control (Section 3), a three cell, classtwo tensegrity beam is modeled (Fig. 3). Each bar has equivalent mass m,moment of inertial J , and length b. The base nodes are separated by distancew. Nodes one through four have mass ma and nodes five and six have massmb. A symmetric configuration will be assumed for the static equilibrium suchthat

−θ1 = θ2 = −θ3 = θ4 = −θ5 = θ6, (17)

as shown in Fig. 3(a). In Appendix A, analytical solutions to the static equilib-rium condition k(θe, `e) = 0 are developed, providing expressions for `e givenθe. Appendix B summarizes the parameter values used for this application.This structure has six pairs of natural frequencies and mode shapes, illus-

b1

b2

b3

b4

b5

b6

l2

l1

l4

l3

l6

l5

l8l7 l9w

(a)

θ1

θ2θ3

θ4θ5

θ6

(b)

Fig. 3. Three cell, class two tensegrity beam. Bars are shown in black, cables in red.

trated in Fig. 4. Cable pairs (l1, l2), (l3, l4), and (l5, l6) are antagonistic andthere are two possible actuation and feedback gain configurations for eachpair. The first configuration produces diagonal G and H matrices of the formG = gI and H = hI, where the cables are actuated out of phase and modes1, 2, and 5 can be excited. The second configuration produces in phase actu-ation of the cables, capable of exciting modes 3, 4 and 6, and requires G andH matrices of the form G = gE and H = hE, where E := diag(e). In eithercase, the essential tensegrity response ρ(s) is given by (6), and the transferfunction sρ(s) is positive real, resulting in −180◦ < ∠ρ(jω) < 0◦.

The physical limitation of cables going slack can be included in the definitionof k(θ, `) in (1). In particular, we impose the following nonlinearity to the

18


0 0.2 0.4−0.1

0

0.1ω1 = 12.57

y [m

]

0 0.2 0.4−0.1

0

0.1ω2 = 77.63

0 0.2 0.4−0.1

0

0.1ω3 = 93.81

y [m

]

0 0.2 0.4−0.1

0

0.1ω4 = 175.74

0 0.2 0.4−0.1

0

0.1ω5 = 204.40

y [m

]

x [m]0 0.2 0.4

−0.1

0

0.1ω6 = 218.78

x [m]

Fig. 4. Linearized natural modes, frequencies in rad/sec. Strings omitted for clarity

tension in the ith cable:

τi(θ, `) :=

∂V /∂li if li ≥ 0

0 if li < 0

The slack condition is ignored for the linearized dynamics used in the controldesign, but is included in numerical simulations of the nonlinear dynamics fordesign evaluation.

4.2 Controller Design for Entrainment to a Resonance

Following the steps outlined above, we will first select (l1, l2) as the antagonistpair with which to control the structure. The dynamic properties for the threecell beam are presented in Table 1. Let us choose to target the first mode (k =1) with the first bar oscillating at an amplitude of 30◦, giving g = 2.32e − 3,and set µ = 1.1. Figure 5 presents the mode partition diagram, from which wehave chosen ωo = 5, η/µ = −1.75, giving h = −8.30e2. Numerical simulationsof the nonlinear system (1) and the RIO (7), where `i(t) = `ei + givi(t) andri(t) = hi(li(θ(t)) − la) for i = 1, 2, result in the steady state oscillationshown in Fig. 6. The Hilbert transforms of discretized signals `1, `2, l1 and l2are used to extract the average frequency of oscillation over complete cyclesin the steady state, resulting in ωs = 11.74 rad/sec. This is outside of thepredicted error bound (|ζi/$| = 2.0%, e1 = 6.6%), but we do note that thestructure is roughly achieving its predicted oscillation amplitude. We also notethat during the oscillation cycle, cables 1 and 2 are going slack, which causesa drop in stiffness, and perhaps a drop in the resonance frequency.

For this reason, we study the nonlinear frequency response, presented in Fig. 7.For a given actuation amplitude g, the structure is driven by anti-phase sinu-

19


Table 1Properties for (l1, l2)

Mode ωi ζi δi

1 12.5671 0.0212 0.7288

2 77.6487 0.0850 0.1709

3 93.8449 0.0702 0.0902

4 175.8348 0.0274 0.0393

5 204.3473 0.0314 0.0210

6 218.5771 0.0086 0.0012

η/μ

ωo

50 100 150 200 250 300−2

−1

0

1

2

Fig. 5. Mode partition diagram for cables {l1, l2}.

soidal signals: `1(t) = `a+g sin(ωt), `2(t) = `a−g sin(ωt). For each frequency ω,the Hilbert transforms of discretized signals `1, `2, l1 and l2 are used to extractthe average amplitude and phase over complete cycles, and compared to thelinearized system, ρ(s). As shown in Fig. 7, as the oscillation amplitude is in-creased, the nonlinear system begins to diverge from the linear approximation.For example, the nonlinear resonance frequency occurs at ω1∗ = 11.76 rad/secwhen g = 2.32e−3, resulting in an entrainment error of |ωs−ω1∗|/ω1∗ = 0.2%.This illustrates the robustness of the CPG controller, as it is able to entrainto the nonlinear system, even when designed for entrainment to the linearizeddynamics.

4.3 Mode switching

The mode partition diagram in Fig. 5 suggests the possibility of switchingbetween modes by modulating either the feedback η, or the RIO intrinsicfrequency ωo. For this example, we consider the latter and target an oscillationamplitude of 8◦ for θ1 for both the first and second modes. The simulationresults are presented in Fig. 8, where the controller parameters and steady

20


89.5 90 90.5 91−30

−20

−10

0

10

20

30

Time [s]

[d

eg]

(a)

0 0.2 0.4

−0.2

0

0.2

x [m]

y [m

]

(b)

89.5 90 90.5 91−0.02

0

0.02

0.04

0.06

Time [s]

[m

]

(c)

Fig. 6. Simulation of entrainment to the first resonance mode: Fig. 6a, angulardisplacement of θ1 (blue) and θ2 (red); Fig. 6b, displacement profile of the structurebetween times to and t1; Fig. 6c, l1 (blue) and l2 (red), where we note that a cableis slack when its value drops below zero.

101

102

10−2

100

102

Gai

n

101

102

−200

−150

−100

−50

0

Phas

e [d

eg]

Frequency [rad/sec]

Gai

n

Frequency [rad/sec]

101.05

101.14

101

101.8

10−0.3

100

Gai

n

Linearg = 2.32e-3

g=6.19e-4g=1.05e-2

Fig. 7. Frequency response characteristics of the linearized and nonlinear dynamicsfor modes 1 and 2. We note that the models diverge as the oscillation amplitudeincreases (ω1∗ = 11.76, ω2∗ = 71.25).

state frequencies (ωs) are presented in Table 2, and η/µ and µ are held constantat −1.75 and 1.1 respectively, while ωo, g, and h are ramped linearly betweentimes ts1 and ts2. Note the nonlinear dynamic frequency response for thesegains g are also presented in Fig. 7. We see that, for t < ts1, entrainment to thefirst mode is achieved, as the linearized tensegrity dynamics accurately modelthe structure. For t > ts2, the steady state system oscillates at a frequency

21


Table 2Mode switching parameters and results

Mode ωo g h ωs

1 (t < ts1) 3 6.19e-4 -3.11e3 12.57

2 (t > ts2) 30 1.05e-2 -1.83e3 75.45

21 22 23 24 25 26−40

−35

−30

−25

−20

−15

Time [s]

θ 1 [deg

]

(a)

0 0.2 0.4−0.2

−0.1

0

0.1

0.2

x [m]

y [m

]

(b)

0 0.2 0.4−0.2

−0.1

0

0.1

0.2

x [m]

y [m

]

(c)

Fig. 8. Switching between modes 1 and 2 by varying ωo. Fig. 8a shows the timecourse of θ1, where ωo changes from 3 to 30 at ts. Figs. 8b and 8c illustrate thedisplacement of the tensegrity between times (t1, t2) and (t3, t4) respectively.

slightly higher than ω2∗, but well within the predicted error bound (|ωs −ω2∗|/ω2∗ = 5.75%, |ζ2/$| = 7.71%, e2 = 2.83%). This example highlights therobustness of entrainment, the capabilities of the design method developed inSection 3.4, and the potential of CPG controllers to entrain to multiple modesor quite possibly gaits.

5 Conclusion

This paper has presented a systematic method for designing a CPG-basedfeedback controller that achieves entrainment to a desired resonance of a gen-eral class of tensegrity structures. This method is based on the linearizedtensegrity dynamics and utilizes the method of multivariable harmonic bal-ance. These tools provide approximate results, and therefore do not guaranteeexact entrainment. Numerical examples, however, illustrate their effectiveness,

22


and also demonstrate the robustness of the controller in its ability to entrainto nonlinear resonance. Additional numerical simulations illustrate the capa-bility of a CPG-controller to achieve entrainment to multiple resonances bymodulating the controller’s intrinsic frequency. This property could be usefulin the control of a robot with multiple natural gaits (e.g. walking and running).

Our examples all use negative feedback (η < 0), but entrainment via positivefeedback is possible and has been verified numerically (results not presented).It is important to point out that, relative to negative feedback, the robust-ness against perturbations in η/µ or ωo is small for the higher order modes asevidenced in the mode partition diagram (Fig. 5) and agrees with the devel-opments in [10]. Thus, as an additional design guideline, the use of negativefeedback is encouraged but not necessary.

The design method has also taken advantage of the concept of antagonisticpairs, a novel contribution whose application is not limited to tensegrities. Itallows a designer to determine if a pair of collocated actuators/sensors arecapable of exciting oscillation of a structure about its nominal configurationthrough out of phase actuation.

Acknowledgements

The authors would like to acknowledge funding from the Office of Naval Re-search through the MURI program on Biologically-Inspired Autonomous SeaVehicles (Contract No. N00014-08-1-0642), the David and Lucille PackardFoundation (Contract No. 2003-25897), and the National Science Foundation(Contract Nos. CMS-0384884 and NSF-0654070). Thomas K. Bliss also thanksthe Achievement Rewards for College Scientists Metro Washington Chapterand the Virginia Space Grant Consortium.

References

[1] T.G. Brown. The Intrinsic Factors in the Act of Progression in the Mam-mal. Proceedings of the Royal Society of London. Series B, ContainingPapers of a Biological Character, 84(572):308–319, 1911.

[2] C. R. Calladine. Buckminster fuller’s “Tensegrity” structures and clerkmaxwell’s rules for the construction of stiff frames. International Journalof Solids and Structures, 14(2):161–172, 1978.

[3] Z. Chen and T. Iwasaki. Circulant synthesis of central pattern generatorswith application to control of rectifier systems. Automatic Control, IEEETransactions on, 53(1):273–286, Feb. 2008.

23


[4] B. de Jager, M. Masic, and R.E. Skelton. Optimal topology and geometryfor controllable tensegrity systems. Matrix, 6:98, 2002.

[5] W. O. Friesen. Reciprocal inhibition: A mechanism underlying oscillatoryanimal movements. Neuroscience and Biobehavioral Reviews, 18(4):547–553, 1994.

[6] W. O. Friesen and J. Cang. Sensory and central mechanisms control inter-segmental coordination. Current Opinion in Neurobiology, 11(6):678–683,2001.

[7] R. B. Fuller. Tensile-integrity structures, U.S. Pat. 3063521, 1962.[8] H. Furuya. Concept of deployable tensegrity structures in space applica-

tion. International Journal of Space Structures, 7:143–151, 1992.[9] Y. Futakata and T. Iwasaki. Formal analysis of resonance entrainment by

central pattern generator. Journal of Mathematical Biology, 57(2):183–207, 2008.

[10] Y. Futakata and T. Iwasaki. Entrainment to natural oscillations viaentrainment to natural oscillations via uncoupled central pattern gener-ators. IEEE Transactions of Automatic Control, (99):1, 2010.

[11] T. Glad and L. Ljung. Control Theory: Multivariable and Nonlinear Meth-ods. Taylor & Francis, New York, 2000.

[12] N.G. Hatsopoulos. Coupling the Neural and Physical Dynamics in Rhyth-mic Movements. Neural Computation, 8(3):567–581, 1996.

[13] T. Iwasaki. Multivariable harmonic balance for central pattern genera-tors. Automatica, 44(12):3061–3069, 2008.

[14] T. Iwasaki, S. Hara, and H. Yamauchi. Dynamical system design froma control perspective: Finite frequency positive-realness approach. IEEETransactions on Automatic Control, 48(8):1337, 2003.

[15] T. Iwasaki and M. Zheng. Sensory feedback mechanism underlying en-trainment of central pattern generator to mechanical resonance. BiologicalCybernetics, 94(4):245–261, 2006.

[16] N. Kanchanasaratool and D. Williamson. Motion control of a tensegrityplatform. Communications in Information and Systems, 2(3):299–324,2002.

[17] H. K. Khalil. Nonlinear Systems. Prentice Hall, New Jersey, 2002.[18] H. Kimura, S. Akiyama, and K. Sakurama. Realization of Dynamic Walk-

ing and Running of the Quadruped Using Neural Oscillator. AutonomousRobots, 7(3):247–258, 1999.

[19] W. B. Kristan Jr., R. L. Calabrese, and W. O. Friesen. Neuronal controlof leech behavior. Progress in Neurobiology, 76(5):279–327, 2005/8.

[20] M.A. Lewis and G.A. Bekey. Gait Adaptation in a Quadruped Robot.Autonomous Robots, 12(3):301–312, 2002.

[21] M.A. Lewis, R. Etienne-Cummings, M.J. Hartmann, Z.R. Xu, and A.H.Cohen. An in silico central pattern generator: silicon oscillator, cou-pling, entrainment, and physical computation. Biological Cybernetics,88(2):137–151, 2003.

[22] F. Li and R. E. Skelton. Sensor/actuator selection for tensegrity struc-

24


tures. In 45th IEEE Conf. on Decision and Control, San Diego CA,December 2006. IEEE.

[23] J.L. Lin, K.C. Chan, J.J. Sheen, and S.J. Chen. Interlacing propertiesfor Mass-Dashpot-Spring Systems with proportional damping. Journalof dynamic systems, measurement, and control, 126(2):426–430, 2004.

[24] M. Masic and R. E. Skelton. Selection of prestress for optimal dynam-ic/control performance of tensegrity structures. International Journal ofSolids and Structures, 43:2110–2125, 2005.

[25] K.W. Moored and H. Bart-Smith. The Analysis of Tensegrity Structuresfor the Design of a Morphing Wing. Journal of Applied Mechanics, 74:668,2007.

[26] H. Murakami. Static and dynamic analyses of tensegrity structures. part1. nonlinear equations of motion. International Journal of Solids andStructures, 38:3599 – 3613, 2001.

[27] J. Nakanishi, J. Morimoto, G. Endo, G. Cheng, S. Schaal, and M. Kawato.Learning from demonstration and adaptation of biped locomotion.Robotics and Autonomous Systems, 47(2-3):79–91, 2004.

[28] G.N. Orlovskiı, TG Deliagina, and S. Grillner. Neuronal Control of Lo-comotion: From Mollusc to Man. Oxford University Press, 1999.

[29] R. A. Pearce and W. O. Friesen. Intersegmental coordination of leechswimming: comparison of in situ and isolated nerve cord activity withbody wall movement. Brain Research, 299(2):363–366, 1984.

[30] S. Pellegrino and C. R. Calladine. Matrix analysis of statically and kine-matically indeterminate frameworks. International Journal of Solids andStructures, 22(4):409–428, 1986.

[31] R. E. Skelton. Dynamics of tensegrity systems: Compact forms. In 45thIEEE Conf. on Decision and Control, San Diego CA, December 2006.IEEE.

[32] R. E. Skelton and M. C. de Oliveira. Tensegrity Systems. Springer, 1edition, June 2009.

[33] R. E. Skelton, J. P. Pinaud, and D. L. Mingori. Dynamics of the shell classof tensegrity structures. Journal of the Franklin Institute, 338:255–320,2001.

[34] KD Snelson. Continuous tension, discontinuous compression structures.US Patent, 1965.

[35] C. Sultan. Modeling, design and control of tensegrity structures withapplications. PhD thesis, Purdue University, 1999.

[36] C. Sultan, M. Corless, and R. E. Skelton. A tensegrity flight simulator.Journal of Guidance, Control, and Dynamics, 23(6):1055–1064, 2000.

[37] C. Sultan, M. Corless, and R. E. Skelton. Linear dynamics of tensegritystructures. Engineering Structures, 24(6):671–685, 2002.

[38] C. Sultan and R. Skelton. Deployment of tensegrity structures. Interna-tional Journal of Solids and Structures, 40(18):4637–4657, 2003.

[39] C. Sultan and R.E. Skelton. Tendon control deployment of tensegritystructures. In Proceedings of the SPIE 5th Symposium on Smart Struc-

25


tures and Materials. SPIE, 1998.[40] G. Taga, Y. Yamaguchi, and H. Shimizu. Self-organized control of bipedal

locomotion by neural oscillators in unpredictable environment. BiologicalCybernetics, 65(3):147–159, 1991.

[41] BW Verdaasdonk, H. Koopman, and F.C.T.V.D. Helm. Energy efficientand robust rhythmic limb movement by central pattern generators. Neu-ral Networks, 19(4):388–400, 2006.

[42] B.W. Verdaasdonk, H.F.J.M. Koopman, and F.C.T. Van der Helm. Reso-nance tuning in a neuro-musculo-skeletal model of the forearm. BiologicalCybernetics, 96(2):165–180, 2007.

[43] C.A. Williams and S.P. DeWeerth. A comparison of resonance tuningwith positive versus negative sensory feedback. Biological Cybernetics,96(6):603–614, 2007.

[44] M. M. Williamson. Neural control of rhythmic arm movements. NeuralNetworks, 11:1379–1394, 1998.

[45] X. Yu, B. Nguyen, and W. O. Friesen. Sensory feedback can coordinatethe swimming activity of the leech. Journal of Neuroscience, 19(11):4634–4643, 1999.

A Symmetric Prestressable Configuration

Consider the tensegrity shown in Fig. 3a. The symmetric configuration (17)is considered, and let θo := θ6. All cables are assumed linearly elastic withuniform spring constant ε. Solving k(θe, è) = 0 for è with these geometricconstraints produces a solution where è1 = è2, è3 = è4, è5 = è6, withè7, è8 and è9 as functions of è1, è3, and è5. To reduce the number of freeparameters, assume è1 = . . . = è6 =: à. This allows è to be expressed interms of one free parameter, à:

è7 =1

l2a

(2wàla + 2b2w cos(2θo) + wb(b2 + 4w2) sin(θo)− 5b2w − 3w3

),

è8 = 3w − 2wàla− 2b sin(θo) +

4bà sin(θo)

la,

è9 = 2b sin (θo)− 2w +wàla,

where la :=√b2 + w2 − 2bw sin (θo). The following condition enforces all ca-

bles to be in tension,

0 < à < la, 0 < {è7, è9} < 2b sin θo, 0 < è8 < w,

26


which reduces to

max

(3w − 4b sin(θo)) la2 (w − 2b sin(θo))

(3w − 2b sin(θo))la2w

2(w − b sin(θo))law

< à < la.

This result leads to a stiffness matrix with two free parameters, the springconstant ε of the cables and à. This allows the designer to select appropriatematerials for a physical model, then adjust the pretension using à to targetspecific dynamic properties like natural frequency.

B Tensegrity Beam Parameters

Parameter Value Units

J 5.0630e-05 kg·m2

m 2.6159e-2 kg

ma 7.0431e-2 kg

mb 4.5726e-2 kg

ε 1.3485e+03 N/m

d 0.0100 kg·m2/s

w 7.6200e-2 m

à 1.2041e-1 m

27


resonance entrainment of tensegrity structures via cpg control€¦ · of an optimized locomotion...

Documents