mechanical models for insect locomotion: stability and parameter studies

Physica D 156 (2001) 139–168

Mechanical models for insect locomotion:stability and parameter studies

John Schmitt a, Philip Holmes a,b,∗a Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA

b Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA

Received 31 May 2000; accepted 27 March 2001Communicated by C.K.R.T. Jones

Abstract

We extend the analysis of simple models for the dynamics of insect locomotion in the horizontal plane, developed in[Biol. Cybern. 83 (6) (2000) 501] and applied to cockroach running in [Biol. Cybern. 83 (6) (2000) 517]. The models consistof a rigid body with a pair of effective legs (each representing the insect’s support tripod) placed intermittently in groundcontact. The forces generated may be prescribed as functions of time, or developed by compression of a passive leg spring.We find periodic gaits in both cases, and show that prescribed (sinusoidal) forces always produce unstable gaits, unless theyare allowed to rotate with the body during stride, in which case a (small) range of physically unrealistic stable gaits doesexist. Stability is much more robust in the passive spring case, in which angular momentum transfer at touchdown/liftoff canresult in convergence to asymptotically straight motions with bounded yaw, fore-aft and lateral velocity oscillations. Using anon-dimensional formulation of the equations of motion, we also develop exact and approximate scaling relations that permitderivation of gait characteristics for a range of leg stiffnesses, lengths, touchdown angles, body masses and inertias, from asingle gait family computed at ‘standard’ parameter values. © 2001 Elsevier Science B.V. All rights reserved.

Keywords: Insect locomotion; Stability; Parameter

1. Introduction

In [1,2], we developed a class of mechanical models for the horizontal plane body-limb dynamics of insectlocomotion. Motivated by experiments of Full et al. [3–5], and by Full’s suggestion that, in rapid running, ‘detailed’neural feedback (reflexes) might be partially or wholly replaced by mechanical feedback (preflexes), we showed thata simple model with passive elastic legs could indeed produce asymptotically stable gaits, including segments ofstraight running and turning, characteristic of those observed in experiments on the cockroach Blaberus discoidalis.Since, these animals do not significantly flex their head–thorax-body, and their limbs constitute only about 6% oftotal mass, we modelled them as rigid bodies with massless legs. We focussed on models in which a pair of effective

∗ Corresponding author. Present address: Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544,USA.E-mail address: [email protected] (P. Holmes).

0167-2789/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S0 1 6 7 -2 7 89 (01 )00271 -8

140 J. Schmitt, P. Holmes / Physica D 156 (2001) 139–168

or virtual legs, each representing a tripod of front, rear and contralateral middle legs used during the stance phasesof hexapedal locomotion, are intermittently placed in ground contact at a fixed extension length and angle relativeto the body, allowed to compress freely during stance phase, and lifted off when the force developed by the axialspring in the leg returns to zero, at which point the other leg touches down. The resulting gait has a 50% duty cycle,compared with 55–60% in the animal itself. The legs are attached to the body at a moment-free pivot (‘hip’ joint),the center of pressure, which may be fixed or may move relative to the body during stance, and the foot contactpoint is also assumed moment free. We consider both fixed and moving center of pressure (COP) protocols; [1,2]focussed on the fixed COP case.

In the present paper, we extend our initial work to include analytical and perturbative studies of periodic gaitsand their stability for both prescribed forces and the passive spring model. In [1], most of the results relied onnumerical solutions of fixed point equations and numerical linearization of Poincaré maps. As pointed out in [1],the six physical parameters of the model (body mass and moment of inertia, leg stiffness, length and touchdownangle, and leg attachment point) can be reduced to four non-dimensional parameters, but this was not pursued inthat paper. Here we show that exact and approximate scaling arguments may be used to derive kinematic, dynamicand stability characteristics of families of gaits, from a single family of gaits computed at a ‘standard’ parameterset.

The paper is structured as follows. Section 2 contains brief reviews of the models; an analytical proof thatprescribed sinusoidal forces can never yield stable gaits if they are not allowed to rotate with the body, and nu-merical studies that reveal that, while prescribed forces can yield stable gaits when allowed to rotate, all stablegaits found thus far are physically unrealistic. In contrast, as found in [1], forces generated by passive elasticlegs typically yield robustly stable motions. These gaits are characterized numerically and, for the fixed COPmodel, analytically via perturbation in a small parameter, and their stability characteristics are derived. Section 3contains parameter studies in which the dependence of gaits on physical parameters are investigated. Using thenon-dimensional formulation, we show how the effects of changes in body mass, leg stiffness, and length, can beexactly or approximately computed. We also remark on the effects of inertia and touchdown angle changes. Wenote some implications of these results and draw conclusions in Section 4. Computational details are relegated toAppendix A.

2. Bipedal model: stability and perturbation studies

The equations of motion for the fixed and moving COP models were derived in [1] and are summarized here forconvenience.

The equations of motion in the inertial frame are

mr = R(θ(t))f, (1)

I θ = (rF − r) × R(θ(t))f . (2)

Here m and I are the mass and (planar) moment of inertia of a rigid body moving in the horizontal plane, rdenotes mass center (COM) position, θ body orientation, R(θ) is the rotation matrix, and rF is the foot position attouchdown, which remains fixed during each stride. A 50% duty cycle is assumed. Fig. 1 shows the model geometryand coordinate systems. The foot force f is specified in the (moving) body frame. Following Kubow and Full [6], wemay specify foot forces as functions of time derived from measurements characteristic of a summed tripod, or wemay allow force generation in a passive elastic spring. Prescribed force models — the bipedal analogs of [6] — arediscussed first.

J. Schmitt, P. Holmes / Physica D 156 (2001) 139–168 141

Fig. 1. A cockroach, viewed from below [7], and the bipedal spring model, viewed from above. F,F ′ denote the current and next foot positions,P is the leg attachment point,G is the body mass center, f1, f2 are the body-relative components of force generated in the leg, r(t) = (x(t), y(t))

is the absolute mass-center position, and θ(t) is the body orientation.

2.1. Prescribed force models

In this model, we prescribe foot forces, relative to the body frame, as

f(t) = ((−1)nA sin(ωt), B sin(2ωt))T, (3)

where the integer n counts stance phases, with n even and odd for left and right stance phases, respectively. ChoosingA = 0.0032, B = −0.004, and ω = 2π/Ts where Ts = 0.1, as in [6], closely approximates averaged and filtereddata described, e.g., in [3,4].

2.1.1. Non-rotating forcesFor most gaits in the prescribed force model, we observe that θ deviates only slightly (±5) from its value at

leg touchdown. As a result, the rotation matrix remains approximately constant through the stride. We thereforefirst study a prescribed force model with a constant rotation matrix, the values of which are determined with θ

held at its initial value at leg touchdown. In this case, the equations of motion uncouple to yield an integrablesystem. Integrating directly, and setting t = π/ω, we obtain the following single stride Poincaré map, as detailedin Appendix A.1:

xn+1 = xn + (−1)n2A cos(θn)

mω, yn+1 = yn + (−1)n2A sin(θn)

mω,

θn+1 = θn − (−1)n(2A(l cos(β) + d) − (πAB/mω2))

Iω

− (π/ω)( 12 (−B(xn cos(θn) + yn sin(θn))) + (−1)nA(xn sin(θn) − yn cos(θn)))

Iω,

θn+1 = θn + πθn

ω+ (−1)n(−π( 1

2 Bl sin(β) + A(l cos(β) + d)))

Iω2

+ (−1)n((8AB/3mω2) + (4A(−xn sin(θn) + yn cos(θn)))/ω)

Iω2, (4)


where l, β, m, I , and d denote the leg length, leg touchdown angle, body mass, body moment of inertia, and thedistance between the center of mass and the leg attachment point, respectively. A symmetric periodic stride, in thiscase, has xn+1 = −xn, yn+1 = yn, θn+1 = −θn, and θn+1 = −θn. Fixed points are determined as

xn = −(−1)nA

mω, yn = AB(3π2 − 32) + 6mω2Blπ sin(β)

6mωA(8 − π2),

θn = −4ABπ + 6Amω2(−8 + π2)(d + l cos(β)) + 3Blmπ2ω2 sin(β)

6Imω3(−8 + π2), θn = 0. (5)

Linearizing the Poincaré map about the fixed point for each stride and multiplying the Jacobians results in a Jacobianfor the full stride (see (A.14)). Stability is determined from the resulting characteristic equation. One eigenvalue is1, corresponding to overall rotation (θ) invariance, leaving a third-order characteristic equation for the remainingeigenvalues, λ. To ensure stability, we must determine that all of the remaining eigenvalues have magnitude less than1. Any eigenvalue with magnitude greater than one renders the gait unstable. One approach is to make a bilineartransformation and apply Routh’s criterion to the transformed characteristic equation. Unfortunately, our systemproves too complex for this method to be analytically tractable. However, since the characteristic equation is ofthird order, we know that at least one of the eigenvalues is real. We can thus examine the characteristic equationin λ2 = λ − 1 and apply Routh’s criterion [8] to the resulting equation. Examining the Routh array, if any oneof the coefficients of λ3

2, λ22 or λ0

2 is positive and any other coefficient of the same set is negative, then at leastone eigenvalue has real part greater than one, resulting in instability. The relevant coefficients of the characteristicequation in λ2 are

λ32 : 144A2I 2m2(−8 + π2)2ω8, (6)

λ22 : −24A(−8 + π2)(96A5(−8 + π2) + 12A3Imω4(32 − 12π2 + π4) − B2Imπ2ω4C), (7)

λ02 : −48A3B2π2(−8 + π2)C, (8)

where C = A(−32 + 3π2) + 6lmπω2 sin(β). The coefficient of λ32 is always positive, so we need to determine if

any of the remaining coefficients is negative. If A > 0, then for the coefficient of λ02 to be positive, C < 0. In this

case, the coefficient of λ22 is always negative. Conversely, if A < 0, then for the coefficient of λ0

2 to be positive,C > 0. Once again, the coefficient of λ2

2 is always negative. In each instance, a sign change occurs in the Routharray, indicating that one eigenvalue has a real part greater than 1. Thus, all periodic gaits with non-rotating forcesof the form (3), for any A, B, and ω, are unstable.

2.1.2. Rotating forcesIn this model, the forces rotate with the body, and the equations of motion are coupled and hence no longer

directly integrable. We resort to numerical simulations to determine stride characteristics and stability dependenceon parameters.

We find that behavior and stability are remarkably sensitive to the touchdown foot position. As the leg is thrownfurther out to the side, the magnitude of the moment increases. Placing the leg out further to the side, with theforces prescribed in (3), causes the force resultant to intersect the centerline of the body behind the center of massat the start of the stride. This produces large positive turning moments (on the order of those observed in [6]) in thefirst half of each left stride and negative moments in the second (and vice versa in the right stride), but opposite tothose observed in the animal. In addition, the incorrect signs for the turning moments yield yawing oscillations ofopposite sign to those observed. However, by decreasing the angle β that the leg is thrown out relative to the bodycenterline, the resultant force can be made to intersect the body centerline in front of the center of mass at the start


Fig. 2. Numerically computed eigenvalues versus β for the prescribed force model with rotating forces. Parameter values are: m = 0.0025,I = 2.04 × 10−7, l = 0.008, d = −0.0025, β = 0.175–1.025.

of the stride. While this produces qualitatively correct turning moments, the moment magnitude decreases by anorder of magnitude, and evidently precludes stabilization via yaw coupling. The stability dependence on touchdownangle, β, for fixed leg length l, is illustrated in Fig. 2 (larger β values correspond to throwing the leg out further tothe side). Note that, for β 0.43, the eigenvalue escapes the unit circle and gaits are unstable.

Thus, while non-rotating forces always yield unstable gaits, allowing forces to rotate can produce stable, albeitunrealistic motions. Evidently, the ability of force direction to respond to body motions is important for stability. Themodels to be considered next incorporate mechanical feedback of force magnitudes and directions in an energeticallyconservative framework, and, as documented in [2,9], yield robustly stable gaits reasonably similar to those observedin Blaberus.

2.2. Passive spring models

In this formulation, the foot force f is supposed to act along an ‘effective leg’, derived from the displacementof a passive elastic spring which is compressed during the stance phase, starting from and returning to its naturallength l at touchdown and liftoff, respectively (cf. Fig. 1). As in [1], to make use of conserved quantities, we findit convenient to replace r by polar coordinates (ζ, ψ) which describe the center of mass (COM) position relativeto the active stance foot. With θ as before denoting the absolute body orientation, and η denoting the varying leglength, the kinetic and potential energies and angular momentum about the stance (fixed) foot are

T = 12m(ζ 2 + ζ 2ψ2) + 1

2I θ2, V = V (η) with η =

√ζ 2 + d2 + 2ζd sin(ψ − (−1)nθ),

LF = mζ 2ψ + (−1)nI θdef=pψ + (−1)npθ . (9)


We can determine the equations of motion for both the fixed and moving COP models from a generalized modelin which d, denoting the point where the leg is pivoted on the body centerline, is a constant offset plus a functionof (approximate) leg angle relative to the body:

d = d(ψ, θ) = d0 + d1[ψ − (−1)nθ ]. (10)

Here the integer n counts stance phases, using the convention n even for left (L) and n odd for right (R). Left-rightsign changes in θ occur due to the fact that ψ increases counterclockwise about the left foot and clockwise aboutthe right, while the absolute body orientation θ takes the same sense throughout (increasing counterclockwise). Themoving COP model has d1 = 0, d0 = 0, whereas the fixed COP model has d0 = 0, d1 = 0. Introducing the aboverepresentation for d into the Lagrangian changes Lagrange’s equations as derived in [1], due to terms containing∂d/∂ψ and ∂d/∂θ . The equations become

mζ = mζψ2 − Vη

η[ζ + d sin(ψ − (−1)nθ)],

m(2ζ ζ ψ + ζ 2ψ) = −Vη

η[dζ cos(ψ − (−1)nθ) + d1d + d1ζ sin(ψ − (−1)nθ)],

I θ = (−1)nVη

η[dζ cos(ψ − (−1)nθ) + d1d + d1ζ sin(ψ − (−1)nθ)]. (11)

The models are characterized by six physical parameters: body mass, m, moment of inertia, I ; leg stiffness,k; relaxed leg length, l; pivot position (COP) relative to COM, dj ; and leg touchdown angle, β. They may benon-dimensionalized by defining a non-dimensional time t along with four other non-dimensional groups:

k = kl2

mv2, I = I

ml2, dj = dj

l, j = 0 or 1, β with t = vt

l, (12)

where v is a characteristic COM speed (e.g. at touchdown). Assuming a linear spring law, the non-dimensionalizedLagrange’s equations are

¨ζ = ζ ψ2 − k

(1 − 1

η

)[ζ + d sin(ψ − (−1)nθ)],

2ζ ˙ζ ψ + ζ 2ψ = −k

(1 − 1

η

)[ζ d cos(ψ − (−1)nθ) + dd1 + ζ d1 sin(ψ − (−1)nθ)],

I θ = (−1)nk

(1 − 1

η

)[ζ d cos(ψ − (−1)nθ) + dd1 + ζ d1 sin(ψ − (−1)nθ)], (13)

where ( ) derivatives are taken with respect to non-dimensional time t , ζ = ζ/ l, and η = η/l. These are theequations studied in this section.

2.2.1. The case d0 = d1 = 0For d0 = d1 = 0, Lagrange’s equations simplify considerably, since ζ ≡ η and the angular momenta pψ and

pθ are individually conserved during each stance phase. The system is completely integrable, and a single-stancereturn map can be computed through stride geometry and conservation of energy as in [1] (cf. Fig. 16). Written interms of COM velocity magnitude v and heading δ relative to the body and defined increasing in the leg direction,along with the body orientation θ and angular velocity θ , all at touchdown, this is

vn+1 = vn, δn+1 = δn + π − ()ψ(vn, δn) + 2β) + (−1)nωτ(vn, δn),

θn+1 = θn + ωτ(vn, δn), θn+1 = θn = ω. (14)


Fig. 3. The single-stance return map for δ for several different mass center touchdown velocities in the case of a linear spring with m = 0.0025,l = 0.01, k = 2.25 and β = 1.

Here )ψ is the angle through which the leg turns during stance and τ the stance duration. (In [1], Eq. (26), wedefined a three-dimensional map, omitting the last (trivial) component of (14). We retain θ here, since for d = 0 itplays an important role in the perturbation calculations to follow.) We once again use the convention that the left footis down for n even and the right foot down for n odd. The full stride map is obtained by iterating (14) twice. Fixedpoints of (14), corresponding to symmetric (left = right) gaits, come in two-parameter (v, θ ) families given by

v = v, θ = θ , ω(= θ ) = 0, )ψ(v, δ) = π − 2β, (15)

and the eigenvalues of the associated linearized map are easily found as 1, with multiplicity 3, and 1−∂)ψ/∂δ|(v,δ).The only non-unity eigenvalue is determined by computation of the quadrature for)ψ : the angle the leg turns throughbetween touchdown and liftoff.

The quadratic potential, V (η) = 12k(η− l)2, corresponding to a linear spring law, yields complicated expressions

involving elliptic functions for the quadratures τ(v, δ) and )ψ(v, δ) [1]. The second component of (14) yieldstypical families of single-stance return maps parameterized by v as shown in Fig. 3. When )ψ has a uniquemaximum and its slope is always less than 2 (as it is here for k > 1) then the δ return map for ω = 0 is unimodaland has at most one stable fixed point, an unstable fixed point, and no other invariant sets.

A saddle-node bifurcation occurs at a critical k, or critical speed (dependent on the other parameters), belowwhich forward motion is unsustainable and above which the domain of attraction of the stable gait rapidly increasesuntil it includes most of the interval (β − π/2, β). As v increases further and k drops below 1, the character of thereturn map changes radically, but we still observe a unique asymptotically stable fixed point. This implies that thetwo-parameter family of ‘smaller δ’ fixed points of the full four-dimensional map (14) are partially asymptoticallystable: |1 − ∂)ψ/∂δ|(v,δ) < 1.

For this and subsequent studies, we adopted a ‘standard’ set of parameter values characteristic of the death-headcockroach Blaberus discoidalis [3,4]: m = 0.0025 kg, I = 2.04 × 10−7 kg m2, k = 2.25–3.5 N m−1. Springconstants were chosen to match peak forces characteristic of steady running at the preferred speed of 0.2–0.25 m s−1

with reasonable leg compressions (≤ 50%), cf. [2,9]. We considered various touchdown angles and leg lengths for


Fig. 4. Periodic gaits for the standard parameter set, characteristic of the cockroach Blaberus discoidalis, fixed COP model: d0 = 0, d1 = 0,m = 0.0025, I = 2.04 × 10−7, l = 0.01, β = 1, k = 2.25. Panels show δ, θ at touchdown, maximum leg force magnitude |F | and eigenvaluemagnitude |λ| versus touchdown COM speed v.

the fixed and moving COP cases (see below); using l = 0.01 m (resp. l = 0.008 m) with β = 1 (resp. β = 1.125)gives stride lengths and cycle rates of about 10 Hz, close to those observed in the animal. From simple geometry,the stride length for this case, and the fixed COP case d0 = 0, is Ls = 4l cos(β), cf. [2]: Fig. 2.

Fig. 4 exhibits the saddle-node bifurcation described above in the family of symmetric periodic gaits withd0 = d1 = 0. While each periodic orbit could be found through the analytic relations developed in [1], here weintegrate the stance period numerically and use a Newton–Raphson algorithm to determine fixed points. For COMspeeds v below vc, no symmetric periodic gaits exist: the kinetic energy at touchdown is insufficient to overcomethe potential ‘hill’ due to leg spring compression. At vc (≈ 0.16 for the parameters of Fig. 4), the saddle-nodebifurcation produces two branches of gaits. One gait family has δ, and |F | increase with v and is always unstable.The other gait family has δ and |F | decrease with increasing v and is always stable, for this particular case. Thisspecial case illustrates neutral stability in speed, orientation and angular velocity, and stability/instability in headingdepending on the branch selected. While we will have neutral speed stability in all cases considered here due toenergy conservation, the neutral stability in angular velocity is specific to this case; perturbing either d0 or d1 fromzero will perturb a second eigenvalue from 1.

2.2.2. The case d0 = constant, d1 = 0While conservation of angular momentum about the stance foot still reduces the system to two degrees of freedom,

the system appears to be non-integrable for d0 = constant, d1 = 0. The family of periodic gaits illustrated in Fig. 4are still preserved, provided that v exceeds the critical (saddle-node) velocity, vc. Below vc, no symmetric periodicgaits exist. For v > vc, two branches of gaits emerge. One has increasing δ, |θ |, and foot force |F | for increasingv and is always unstable. The other family has decreasing δ, |θ |, and foot force |F | for increasing v and is alwaysstable, provided that d < 0.


Fig. 5. Periodic gaits for the standard parameter set, characteristic of the cockroach Blaberus discoidalis, fixed COP model: d0 = −0.0025,d1 = 0, m = 0.0025, I = 2.04 × 10−7, l = 0.01, β = 1, k = 2.25. Panels show δ, θ at touchdown, maximum leg force magnitude |F | andeigenvalue magnitude |λ| versus touchdown COM speed v.

The stability properties for each gait are inherently different from those observed in the d ≡ 0 case. For d0 < 0(leg attachment point behind the mass center), an eigenvalue corresponding to rotation moves within the unitcircle, yielding a family of gaits asymptotically stable with respect to relative COM velocity heading δ and bodyangular velocity θ at touchdown. The body asymptotically runs straight with bounded fore-aft, lateral, and yawingoscillations, but still lacks asymptotic speed stability. Conversely, for d0 > 0, the perturbed eigenvalue movesoutside of the unit circle and all gaits are unstable. It is of interest to note that the yawing behavior of this modeldiffers qualitatively from observations, due to incorrect applied torques [2]. While the yawing behavior should becosinusoidal [6], we find sinusoidal yawing in all cases. A representative family of periodic orbits for d0 = −0.0025is shown in Fig. 5.

2.2.3. The case d0 = 0, d1 = constantThe moving COP model was created to better capture the yawing behavior and the motion of the COP in the

insect. Once again, the system is not integrable, but the family of periodic gaits identified through numerics is similarto that found above. Below vc no symmetric periodic gaits exist. For v greater than vc, two gait families emerge,with increasing (resp. decreasing) δ, |θ |, |F | for increasing v. The upper branch does restabilize, as shown by theeigenvalue plot in Fig. 6 , but its gait characteristics are not physically realistic. The lower branch is always stable,provided that |I/d1| once again is greater than some critical value, and d1 < 0. Instead of having non-zero valuesfor the angular velocity θ , we have θ = 0, θ = 0 at touchdown, matching the cosinusoidal yawing oscillationsobserved in the insect. For d1 < 0 (resp. d1 > 0), we find that the COP moves aft (resp. forward) during eachstance phase. The d1 < 0 yaw oscillation patterns match experimental observations (although they are significantlysmaller in magnitude) due to the correct torque pattern generated by moving the leg pivot from ahead to behind theCOM during each stride, and gaits again display asymptotic heading and angular velocity stability, while for d1 > 0,


Fig. 6. Periodic gaits for the standard parameter set, characteristic of the cockroach Blaberus discoidalis, moving COP model: d0 = 0,d1 = −0.0035, m = 0.0025, I = 2.04 × 10−7, l = 0.008, β = 1.125, k = 3.516. Panels show δ, θ at touchdown, maximum leg force |F | andeigenvalue magnitude |λ| versus touchdown COM speed v. Note restabilization on ‘upper’ branch.

they are unstable. Fig. 6 provides a representative family of periodic orbits. Note that here, and for the fixed COPcase of Fig. 9, a speed range exists for which the ‘stability’ eigenvalues remain near their minima (0.2–0.35 and0.2–0.25 m s−1, respectively). It is interesting to note that this includes the preferred speed (0.25 m s−1) for Blaberus.

2.2.4. Analytical approximation of periodic stridesIn [1], and as summarized above in Section 2.2.2, we showed numerically that the stability of periodic gaits

depends on d: for d < 0 a family of stable gaits exists whereas for d > 0 no stable gaits exist. Here, we verify thatresult, for the fixed COP case (d = d0 = constant), using perturbation theory.

Before discussing stability, we must first find conditions for periodic gaits. Spurred by the apparent linear depen-dence of the eigenvalues on d (cf. [1], Fig. 6), we develop a first-order perturbation expansion in d. Details of thecalculations summarized here are given in Appendix A.2.1. We begin by simplifying the equations of motion, byexpanding η and retaining only terms up to and including order d, and appealing to angular momentum conservation:

¨ζ = ζ ψ2 − k(ζ − 1) − kd sin(ψ − θ), LF = ζ 2ψ + I θ , I θ = (−1)nkd(ζ − 1) cos(ψ − θ). (16)

Considering d a small parameter, we assume the following expansions:

ζ = ζ0 + d ζ1 +O(d2), ψ = ψ0 + dψ1 +O(d2), θ = dθ1 +O(d2). (17)

We substitute the above expressions into (16) and retain only terms up to and including order d. The order zeroequations are

¨ζ 0 = ζ0ψ20 − k(ζ0 − 1), (18)

ζ 20 ψ0 = LF . (19)


Since angular momentum, LF , is conserved throughout the stride, (19) can be used in (18), resulting in one nonlineardifferential equation (that is integrated by quadratures in [1]). Using the symmetries of the vector field, we assumeeven and odd expansions in time about the midpoint of the stride:

ζ0 = A + Bτ 2 + Cτ 4, (20)

ψ0 = Dτ + Eτ 3 + Fτ 5, (21)

where the start and end of the stride occur at −τ0 and τ0, respectively. In addition, contrary to standard perturbationtechnique, we expand the initial conditions in d and match orders of d in the initial conditions and expansions. Weexpand the initial conditions in order to break the symmetry of the solution for d = 0. For the order d0 case, wealso use the fact that the Schwind–Koditschek [10] approximation, employed in [1], yields a good approximationfor τ0 and )ψ , and can therefore replace the awkward exact quadratures of [1]. Using these approximations, theinitial conditions, and the equations of motion, we determine an approximation for the order zero solution, withA,B,C,D,E, and F given explicitly in terms of the non-dimensional parameters (see (A.17), (A.18) and (A.25)).

To find an approximation to the order d1 solution, we begin by analyzing the order d1 equations of motion:

¨ζ 1 = ζ1ψ20 + 2ζ0ψ0ψ1 − kζ1 − k sin(ψ0), 0 = ζ 2

0 ψ1 + 2ζ0ζ1ψ0 + I θ1, I θ1 = k cos(ψ0)(ζ0 − 1).

(22)

We now assume odd and even expansions, breaking the O(d0) symmetry:

ζ1 = Mτ + Nτ 3, (23)

ψ1 = P + Qτ 2, (24)

θ1 = G + Hτ 2 + Jτ 4. (25)

The values for the coefficients M , N , P , Q, G, H , and J are determined by substituting the expansions (23)–(25)into the order d1 equations of motion (22) and matching order zero terms and order d1 initial conditions (see(A.26)–(A.29)). Solving for the variables yields an approximation to a periodic orbit, for d = 0, with error of orderd2. An example of the approximation for a single stride is given in Fig. 7; it is clearly good in this case.

2.2.5. Stability analysis of a perturbed Poincaré mapTo determine the stability of a periodic gait, we first need to derive or approximate the Poincaré map and its

relevant fixed point. To do the latter, we use the even and odd symmetry properties of the approximate solution,along with the initial conditions, to determine the starting and ending values of a periodic stride:

ζn,n+1 = 1 ± d cos(β), (26)

˙ζ n,n+1 = ∓ cos(β − δ) − d sin(β) sin(β − δ), (27)

ψn,n+1 = sin(β − δ) ∓ d sin(2β − δ). (28)

Using the above relations in conservation of angular momentum at the beginning and end of the stride yields themap:

θn+1 = θn − 2d sin(δ)

I. (29)


Fig. 7. Periodic gait approximation at O(d) (dashed) compared with numerically simulated gait (solid) for the standard parameter set withd = −0.1, v = 0.205.

Since θ is of order d , conservation of energy yields the velocity map, vn+1 = vn+O(d2). While more complicated,the heading angle map is derived from geometry, as shown in Appendix A.2.2 (cf. (14)), as

δn+1 = δn + π − )ψ − 2β + (−1)n(θn+1 − θn). (30)

Fixed points of the mapping are thus given by

˙θ = d sin δ

I, θ = θ , v = v, )ψ(v, δ) = π − 2β. (31)

The eigenvalues of the Jacobian determine stability of the fixed points. Unfortunately, the conditions presentedabove are only valid for periodic orbits. To determine the terms of the Jacobian, we need to analyze a general returnmap. We derive this from the d = 0 map of (14). For d = 0, the time of flight, the angle swept through duringthe stride, and the angular velocity all may change. Explicitly incorporating these perturbations into the return mapyields, at order d

vn+1 = vn, δn+1 = δn + π − ()ψ0 + d)ψ1 + β) + (−1)n(θn+1 − θn),

θn+1 = θn + 2θn(τ0 + d τ1) + (−1)ndK(vn, δn), θn+1 = θn + (−1)ndL(vn, δn), (32)

where L and K are integrals representing the changes in θ and θ through a stride. To determine the eigenvalues, welinearize the map, premultiply the left stride map by the right stride map, and determine the resulting characteristicequation. Two eigenvalues remain equal to 1; the remaining ‘relevant’ eigenvalues are obtained by solving the(factored) quadratic characteristic equation (cf. (A.34)–(A.37)). Stability is determined by the eigenvalue whichperturbs from λ = 1 at d = 0; the other eigenvalue λ = 1 − ∂)ψ/∂δ is less than one on the stable (smaller δ)


Fig. 8. Numerically (solid) and analytically (dashed) computed eigenvalues versus d for the standard parameter set.

branch and greater than one on the unstable (larger δ) branch for d = 0, and this behavior persists for small d. Thesecond ‘stability’ eigenvalue is found to be

λ1 = 1 − 4d τ0(∂L/∂δ)

2 − (∂)ψ/∂δ)+ O(d2), (33)

where ∂L/∂δ and ∂)ψ/∂δ may be approximated via the (complicated) expressions derived in Appendix A.2.2. Tocalculate L, we must use the Schwind–Koditschek approximation twice. First, we obtain the quadrature for L byintegrating the θ equation of motion once. Using energy conservation, we convert this integral from a function oftime to a function of η (cf. (A.43)–(A.45)). Since ψ0 appears in the integrand, we need to find ψ0 as a function ofη. To do this, we use the Schwind–Koditschek approximation for ψ0, with an arbitrary end length (A.46)–(A.48).With all integrand terms as functions of η, we use the Schwind–Koditschek approximation again to obtain ourapproximation for L (A.49)–(A.51). The derivative, ∂L/∂δ is then obtained through the quotient rule (A.52) and(A.53). In a similar manner, we obtain ∂)ψ/∂δ from the Schwind–Koditschek approximation to)ψ (A.39)–(A.43),although it may also be obtained from the quadrature developed in [1]. As shown in Fig. 8, the numerical resultsmatch the analytical approximation with an error of O(d2), as expected. Although, we have the (semi-) explicitformulae of (A.52) and (A.53), we have been unable to prove monotonicity of ∂L/∂δ, but the above semi-analyticalresults are clear.

3. Parameter studies

The fixed and moving COP models were developed with insects, cockroaches in particular, in mind. However,the model may have relevance for other sprawled posture animals, of differing sizes, through similarity relations.For geometrically similar animals, m ∝ l3, I ∝ m5/3, and k ∝ m1/3. However, stiffnesses are often taken toscale according to elastic similarity: k ∝ m2/3 [11]. Animals are most often compared at equal Froude numbers,


Fr = v/√

gl, as in the SLIP model [12]. While, we do not include gravity in our model, the leg lengths consideredare simply the horizontal projections of the leg. Since, the horizontal projection is related to its vertical projection,we propose that one can also use Froude number similarity here. Comparing animals at equal Froude number revealsthat v ∝ l1/2 ∝ m1/6. Substituting the similarity relations into our non-dimensional parameters (12) reveals thatall remain constant for geometrically similar animals. Thus, the model predicts that geometrically similar animalswould possess the same gait characteristics and stability, merely scaled in size and time (frequency).

However, animals of different species are, in general, not geometrically similar. Variations in geometry lead todifferent values for the non-dimensional parameters. In [9], we estimate ranges for such parameters, but in order todetermine the effects of changing physical parameters on gait families in general, a parameter study is necessary.

3.1. Exact scaling solutions: changes in k and m

Exact scaling requires that all non-dimensional parameters are held constant, such that all gaits are identical innon-dimensional configuration space, modulo rescaling of time, t = vt/l. Non-dimensional parameters can be heldconstant in special cases either by changes in physical parameters, or by changes in velocity. If such scaling lawscan be found, given one family of solutions (v, δ, θ, θ) for a given set of parameters, then all other periodic orbitsfor different values of a physical parameter can be found simply through scaling relations.

For the ‘standard’ parameter sets, we have the families of solutions (v, δ, θ, θ), illustrated in Figs. 5 and 6. Tosee how this picture changes for a different leg stiffness k, we use the following argument. First, we keep k constantby changing the characteristic (COM touchdown) speed:

k = k1l2

mv21

= k2l2

mv22

,v2

v1=√k2

k1≡ Rk. (34)

All other non-dimensional parameters remain constant, since they do not contain k. Thus, displacements andvelocities scale like

(D, v) → (D,Rkv). (35)

Scaling for velocity, acceleration and force quantities involves the time rescaling:

t = v1t1

l= v2t2

l⇒ ∂/∂t2

∂/∂t1= v2

v1= Rk,

∂2/∂t22

∂2/∂t21

= R2k . (36)

Thus, velocity, acceleration, and force quantities scale as follows:

(V ,A, F, v) → (RkV,R2kA,R

2kF,Rkv). (37)

Given a one-parameter family (v, δ, θ, θ , |F |), changing k therefore results in a shift in the family and corre-sponding eigenvalues according to

(v2, δ2, θ2, θ2, |F2|; |λ2|) = (Rkv1, δ1, θ1, Rkθ1, R2k |F1|; |λ1|). (38)

Changes in k simply shift the original family of curves by the scaling factorRk . Since all non-dimensional parametersremain constant, each periodic orbit for different gait families, corresponding to the same δ, traverses the same pathin (x, y) space and has the same eigenvalues.

Changes in m can be dealt with in a similar manner. In this case, both k and I are affected simultaneously. Tokeep I constant, we must also change I in conjunction with m:

I = I1

m1l2= I2

m2l2⇒ m2

m1= I2

I1. (39)


To keep k constant, as before we scale v with m:

k = kl2

m1v21

= kl2

m2v22

⇒ v2

v1=√m1

m2≡ 1

Rm

. (40)

The time rescaling follows through as before, and displacement, velocity, acceleration and force quantities scale asfollows:

(D, V,A, F, v) → (D,R−1m V,R−2

m A,F,R−1m v). (41)

Hence, given a one-parameter family (v, δ, θ, θ), changing I and m according to (39) results in the following shiftin the family of gaits and corresponding eigenvalues:

(v2, δ2, θ2, θ2, |F2|; |λ2|) = (R−1m v1, δ1, θ1, R

−1m θ1, |F1|; |λ1|). (42)

These scaling relations hold for both the fixed and moving COP models, since the non-dimensional parameters arethe same in both. Note that in both cases eigenvalues remain unchanged (at the rescaled touchdown speed).

3.2. Approximate scaling solutions

Approximate scaling solutions differ from the exact solutions above, since we now isolate the effects of singlephysical parameter changes on families of solutions, even though all non-dimensional parameters cannot be keptconstant.

3.2.1. Changes in mass, mChanging the mass causes both k and I to change. We proceed as before, changing v to compensate for changes

in m such that k remains constant:

k = kl2

m1v21

= kl2

m2v22

⇒ v2

v1=√m1

m2≡ 1

Rm

. (43)

Time rescaling yields scaling relations for velocity and acceleration similar to the previous cases (cf. (36)):

∂/∂t2

∂/∂t1= v2

v1≡ 1

Rm

,∂2/∂t2

2

∂2/∂t21

= 1

R2m

. (44)

Since F = ma, forces scale like:

F2

F1= m2

m1

1

R2m

= 1. (45)

Here, we do not change I to keep I constant. Rather, we contend that, for small |d/I |, changes in I do not significantlyaffect translational dynamics (cf. Eq. (13) and Sections 2.2.4, 2.2.5, 3–3.3.2 and 4 and Appendix A). Therefore, δshould approximately scale like distance variables: δ2 ≈ δ1. Examining the non-dimensional equations of motion(13) for θ , however, reveals that changes in I will significantly affect the scaling of θ and θ . For small θ , which aregenerally the only physically reasonable gaits, the terms cos(ψ − θ) and sin(ψ − θ) are primarily dominated byψ . As such, one can effectively replace ψ − θ by ψ for small θ and integrate directly to obtain an approximationfor θ . In doing so, one finds that

θ ∝ kd

I. (46)


This proportionality leads to the approximate scaling relations, valid only for small values of θ :

θ2

θ1= kd I1

kd I2≈ R2

m,dθ2/dt2dθ1/dt1

≈ 1

Rm

kdI1

kd I2= Rm. (47)

Thus, given any fixed point, (v1, δ1, θ1, θ1, |F1|), changing m yields a new fixed point approximately representedby

(v2, δ2, θ2, θ2, |F2|) ≈ (R−1m v1, δ1, R

2mθ1, Rmθ1, |F1|). (48)

Here, as in the case below, we cannot easily deduce scaling for the eigenvalues |λj |, since they result from thecoupled translation–rotation dynamics (cf. Section 2.2.5).

3.2.2. Changes in leg length, lChanging the leg length causes k, I , and d to change. As before, we compensate for the changes in length by

changing the velocity to keep k constant:

k = kl21mv2

1

= kl22mv2

2

⇒ v2

v1= l2

l1≡ Rl. (49)

The time rescaling yields slightly different results for velocity and acceleration quantities, since l appears in theform for non-dimensional time:

t = v1t1

l1= v2t2

l2,

∂/∂t2

∂/∂t1= v2l1

v1l2= 1 ⇒ ∂2/∂t2

2

∂2/∂t21

= 1. (50)

Since F = ma and a ∝ l/t2, we find that forces scale like:

F2

F1= l2

t22

t21

l1= Rl. (51)

Inspection of the non-dimensional equations of motion yields the same scaling argument for θ as in the case forvarying mass:

dθ2/dt2dθ1/dt1

≈ 1 · kd2I1

kd1I2= Rl,

θ2

θ1≈ Rl. (52)

Given any set of fixed points (v1, δ1, θ1, θ1), if we change l we can therefore approximate the new set of fixed pointsthrough the scaling:

(v2, δ2, θ2, θ2, |F2|) ≈ (Rlv1, δ1, Rlθ1, Rl θ1, Rl |F1|). (53)

3.2.3. Changes in moment of inertia, IAs noted in Section 3.2.1, (modest) changes in I primarily affect yawing (θ, θ) motions, while translational

dynamics are relatively insensitive. Scaling of θ , θ is thus approximately:

(v2, δ2, θ2, θ2, |F2|) ≈ (v1, δ1, R−1I θ1, R

−1I θ1, |F1|), RI = I2

I1. (54)

However, the stability of gaits may also change if I , or more specifically |I/d|, passes out of a certain range. AsI → ∞, we essentially obtain the d = 0 case, resulting in neutral stability. In this case, it is difficult to rotatethe body, due to the large moment of inertia, resulting in ‘sluggish’ behavior. As I decreases, it becomes easier to


Fig. 9. Stability contours for variations in |I/d| and v, fixed COP case. All other parameters were held constant at values characteristic of thecockroach Blaberus discoidalis: d0 = −0.0025, d1 = 0, m = 0.0025, l = 0.01, k = 2.25, and β = 1.0.

rotate the body, eventually reaching a point where the behavior becomes too ‘lively’. This behavior is illustratedthrough the stability contours of Fig. 9. As |I/d0| increases, the largest non-unity eigenvalue increases in magnitude,eclipsing a value of |λ| = 0.70 at |I/d0| = 3.2 × 10−4. As |I/d| decreases, initially the gaits become more stable,but as |I/d0| decreases further, the largest non-unity eigenvalue increases past |λ| = 0.70 at |I/d0| = 8 × 10−6. Itis interesting to note that the region of maximum stability includes the range of values for |I/d0| that are typical ofBlaberus (modulo our estimate of d0).

3.2.4. Changes in leg touchdown angle, βChanges in β do not affect k, I , or d directly, so it is not possible to determine scaling laws for changes in β as

above. Attempts to formulate such laws will be presented in Section 3.3.2.

3.3. Verification of scaling laws

The scaling laws were verified numerically through simulations in Matlab, using Runge–Kutta integration of theequations of motion nested within Newton–Raphson routines to find periodic orbits [1].

3.3.1. Exact scaling solutionsAs expected, the exact scaling solutions presented in Section 3.1 were borne out through the numerical simulations

for both the fixed and moving COP cases. For the moving COP, the spring constant k was changed from its nominalvalue to values between 0.7 and 17.6, respectively, one-fifth and five times the nominal value. The numericallydetermined fixed point values for v and θ lay within 0.01 and 0.07%, respectively, of the predicted (scaled) values.For the fixed COP case, the numerically determined fixed point values for v and θ lay within 0.06 and 0.44%,


Fig. 10. Numerically computed periodic gaits (solid) for k = 0.7–17.6 versus the exact scaling solutions (dashed), for the moving COP case.All other parameters were held constant at values characteristic of the cockroach Blaberus discoidalis: d0 = 0, d1 = −0.0035, m = 0.0025,I = 2.04 × 10−7, l = 0.008, β = 1.125. The curves coincide, as expected.

respectively, of the predicted (scaled) values. In both cases, the critical velocity vc increases exactly like√k as k

increases, as expected. An example of exact scaling in the moving COP case for k is presented in Fig. 10. Note thatthe δ and λ curves are simply translated horizontally.

The exact scaling solutions for changing I in conjunction with m were also numerically confirmed, but resultsare not presented here.

3.3.2. Approximate scaling solutionsApproximate mass scaling. The approximate mass scaling results match the numerically computed periodic gaits

closely for both the moving and fixed COP cases, as shown in Figs. 11 and 12. In both cases, m was changed fromits nominal value to values between 0.0005 and 0.005. As argued previously, changes in I did not significantlyaffect the translational dynamics in either case, and for small θ , the approximate scaling arguments presented inSection 3.2.1 agreed well with the numerical results. For the moving (resp. fixed) COP case, the scaled v waswithin 8.8% (resp. 2.6%) of the numerically computed values, while θ (resp. θ ) was within 2.8% (resp. 1.4%) ofthe numerically computed values throughout the range considered. In both cases, vc decreases like 1/

√m as m

increases, as predicted, and we observe that eigenvalue magnitudes at rescaled speeds also decrease.Approximate length scaling. The approximate length scaling results matched the numerically computed results

well only in the fixed COP case. In the fixed (resp. moving) COP model, the length was changed from its nominalvalue to values between 0.005 and 0.02 (resp. 0.005 and 0.16). Changes in I and d did not significantly affect thetranslational dynamics in the fixed COP case, and the approximate scaling arguments for θ were borne out: thescaled values of v and θ were within 0.32 and 2.9% of the numerically computed values, as shown in Fig. 13.In addition, we find that vc increases almost exactly like l for increasing l. The same scaling arguments did notwork well for the moving COP case, as shown in Fig. 14. While we found that vc scaled approximately linearly


Fig. 11. Numerically computed periodic gaits (solid) form = 0.0005–0.005 versus the approximate scaling solutions (dashed), moving COP case.All other parameters were held constant at values characteristic of the cockroach Blaberus discoidalis: d0 = 0, d1 = −0.0035, I = 2.04×10−7,l = 0.008, β = 1.125, k = 3.516.

Fig. 12. Numerically computed periodic gaits (solid) for m = 0.0005–0.005 versus the approximate scaling solutions (dashed), fixed COP case.All other parameters were held constant at values characteristic of the cockroach Blaberus discoidalis: d0 = −0.0025, d1 = 0, I = 2.04×10−7,l = 0.01, β = 1, k = 2.25.


Fig. 13. Numerically computed periodic gaits (solid) for l = 0.005–0.02 versus the approximate scaling solutions (dashed), fixed COP case.All other parameters were held constant at values characteristic of the cockroach Blaberus discoidalis: d0 = −0.0025, d1 = 0, m = 0.0025,I = 2.04 × 10−7, β = 1, k = 2.25.

Fig. 14. Numerically computed periodic gaits (solid) for l = 0.005–0.016, moving COP case. All other parameters were held constant at valuescharacteristic of the cockroach Blaberus discoidalis: d0 = 0, d1 = −0.0035,m = 0.0025, I = 2.04×10−7, β = 1.125, k = 3.5156. Predictionsfor vc shown dashed.


Fig. 15. Numerically computed periodic gaits (solid) for β = 0.6–1.3 versus the approximate scaling solutions (dashed), fixed COP case. Allother parameters were held constant at values characteristic of the cockroach Blaberus discoidalis: d0 = −0.0025, d1 = 0, m = 0.0025,I = 2.04 × 10−7, l = 0.01, k = 2.25.

with increasing l, the slope did not agree with scaling predictions. We attribute the error in scaling to the fact thatthe moving COP model is more sensitive to changes in d than the fixed COP model. In this study, d increased toalmost 50% of the leg length, which can no longer be considered a small perturbation. In this case too, eigenvaluemagnitudes at rescaled speeds decrease (here approximately linearly) with increasing l.

Approximate β scaling. In both the fixed and moving COP models, β was allowed to vary between 0.6 and 1.3 rad.Only results for the fixed COP model are presented here, in Fig. 15. However, it was apparent in both cases that vc

decreased almost linearly with increasing β. This intuitively makes sense, since as the leg is flung further out to theside, the kinetic energy required to move the body over the potential hill decreases. Ultimately, as β → π/2, onewould expect that vc would tend to zero, since any velocity would be able to overcome the (vanishing) resistance ofthe spring in such a glancing touchdown. Scaling vc in such a manner produced results that matched the numericalcomputations results rather well. The corresponding timescale change (cf. (36)) allows one to predict displacement(δ), velocity (θ), force and eigenvalue scalings much as the previous cases (see Fig. 15). Here, we find that eigenvaluemagnitudes increase with β at rescaled (reduced) speeds.

4. Conclusions and implications

In this paper, which follows [1,2] and provides the analytical background used in [9], we study single-degree-of-freedom rigid body models for the horizontal-plane dynamics of rapidly running insects. The models have a pair ofmassless effective legs, freely pivoted on the body and the ground at the stance foot, and the forces generated at thefoot may be either prescribed as functions of time, or allowed to develop due to compression of a passive, linearlyelastic leg spring. All such models can display periodic gaits. Here we focus on stability properties and on how gaitcharacteristics vary with non-dimensional and physical parameters.


We find that if force magnitudes and directions are prescribed to match those due to the tripod of support legs usedby animals in each stance phase, and do not rotate to follow body orientation during stride (being reset only at legtouchdown), then all gaits are unstable. Here, the translation and rotation equations of motion decouple, and a closedform solution may be found. Allowing force vectors to follow body orientation, the coupled equations, studied vianumerical methods, show that while ‘most’ gaits remain unstable, there is a range of leg lengths and touchdownangles which yield stable, albeit physically unrealistic, gaits. This appears to be due to the small correcting momentsdue to collapse of a support tripod to a single leg, since the simulations of [6] showed that tripod foot forces couldproduce realistic gaits capable of resisting perturbation. We conjecture that the difference is due to the significantlylarger moments that tripod forces can produce, without increase in net lateral and fore-aft forces, cf. [9]. Nonetheless,we feel that the inability of prescribed forces to adjust magnitudes in response to perturbations, is unrealistic.

In contrast, the passive spring models robustly generate corrective forces which, with appropriate leg attachmentgeometry, lead to asymptotically stable gaits. Specifically, if legs are attached behind the COM in the fixed COPcase, or move rearward during stance in the moving COP case, then a branch of stable gaits exists over a range offorward speeds. The stability is relative: since the models are energetically conservative and there is overall rotationalinvariance, speed and orientation stability are neutral (two unit eigenvalues are inevitable), but the body motionis asymptotically straight with bounded yaw, fore-aft, and lateral oscillations corresponding to relative stability ofheading and angular velocity. Using orbit symmetries and a chain of approximations to quadrature integrals, we areable to derive semi-explicit expressions for in-stride gait variations and the relevant (stability) eigenvalues of thestride to stride Poincaré map.

In [2], we studied particular models with physical parameters appropriate for the cockroach Blaberus discoidalis. Itis desirable to extend results to other species, perhaps even beyond insects. Biologists often employ simple geometricscaling, e.g. to estimate speeds and power consumption variations across species [11]. Here, we find that thenon-dimensional stiffness, inertia, pivot offset, and touchdown angle parameters defined in (12) all remain constantunder geometric scaling, assuming elastic similarity for ‘muscle’ (leg) stiffness. Thus gaits of all geometricallysimilar animals may be derived from a single gait family computed for a standard parameter set: they are identical,up to length and time rescaling.

We also developed scaling relations to allow prediction of gait families from a standard set in cases where onlya single physical parameter, such as leg stiffness or body mass, changes. These allow predictions to be made fordepartures from geometric scaling. We use the trick of adjusting a characteristic velocity used in the definitionof non-dimensional stiffness k (12) to maintain constant non-dimensional parameters, thus leaving solutions ofthe non-dimensional equations (13) unchanged. In the case of stiffness, the resulting scaling is exact: one simplyhas to translate gait curves suitably. For mass, length and inertia changes, we show that approximate scaling lawsmay still be found, based on the assumption that non-dimensional inertia I and pivot offset d do not significantlyaffect translational dynamics, and simply affect yaw angles via the ratio I /d. This is reasonable for relatively small|d/I |, as appears true for the cockroach data. We also derive an empirical law to describe the effects of changesin touchdown angle β. Roughly speaking, we find that increases in stiffness k and leg length l cause increases incharacteristic velocities and force magnitudes, while increases in mass m and touchdown angle β lead to decreasesin the same quantities. Specifically, the critical COM speed at which the saddle-node bifurcation occurs, and belowwhich periodic gaits do not exist, scales as follows:

vc ∼√k,∼ 1√

m,∼ l,∼ −β, (55)

and it is insensitive to changes in I or d .As noted in [1,2], the ability of a mechanical system with intermittent contacts to produce asymptotically stable

motions: specifically, straight running without directional feedback, appears relevant to the understanding of rapid


running in insects and other legged animals. The analyses of Section 2 imply that coupling between rotationaland translational effects is necessary for this, and that stability is significantly more robust if the legs act as passivesprings, rather than producing prescribed forces. This principle has recently been used in the design and constructionof a nimble hexapedal robot: RHex, built by M. Buehler of McGill University, Montreal; see [13]. In general, webelieve that simple models of the type described here can serve as useful templates for the understanding of complexlocomotory systems in nature [14,15] and technology.

The parameter and scaling studies of Section 3 are used in quantitative comparisons of predictions with data fromcockroach trials of [9], but more generally they yield predictions of kinematics and forces for animals of differentsizes and body types. We are currently gathering and reducing data for such cross-species comparisons.

Acknowledgements

This work was supported by DoE: DE-FG02-95ER25238 and DARPA/ONR: N00014-98-1-0747. John Schmittwas partially supported by a DoD Graduate Fellowship and a Wu Fellowship of the School of Engineering andApplied Science, Princeton University. We thank Bob Full, Mariano Garcia and Dan Koditschek for numeroussuggestions.

Appendix A. Detailed calculations

The following appendices present the calculations omitted in the text.

A.1. Non-rotating prescribed force calculations

The equations of translation (2) in component form are

x = (−1)nA cos(θn) sin(ωt)

m− B sin(θn) sin(2ωt)

m, (A.1)

y = (−1)nA sin(θn) sin(ωt)

m+ B cos(θn) sin(2ωt)

m. (A.2)

Integrating yields:

x = −(−1)nA cos(θn) cos(ωt)

mω+ B sin(θn) cos(2ωt)

2mω+ c1, (A.3)

y = −(−1)nA sin(θn) cos(ωt)

mω− B cos(θn) cos(2ωt)

2mω+ c2. (A.4)

At t = 0, x = xn, and y = yn. Solving for the constants, x and y are expressed as

x = xn + (−1)nA cos(θn)(1 − cos(ωt))

mω− B sin(θn)(1 − cos(2ωt))

2mω, (A.5)

y = yn + (−1)nA sin(θn)(1 − cos(ωt))

mω+ B cos(θn)(1 − cos(2ωt))

2mω. (A.6)

To obtain the map for a single stride, we set t = π/ω in (A.6), resulting in the first two equations of (4). To integratethe θ equation of motion, we also need expressions for x(t) and y(t). Integrating (A.6) yields:


x = xn + xnt + (−1)nA cos(θn)(ωt − sin(ωt))

mω2− B sin(θn)(2ωt − sin(2ωt))

4mω2, (A.7)

y = yn + ynt + (−1)nA sin(θn)(ωt − sin(ωt))

mω2+ B cos(θn)(2ωt − sin(2ωt))

4mω2. (A.8)

From the equations of motion, the θ equation is

I θ = (rf (tn) − r) × R(θ(tn))f(t), (A.9)

where rf is given by

rf = r(tn) + R(θ(tn))(−(−1)nl sin(β), d + l cos(β))T. (A.10)

Evaluating the cross product, integrating the result once (using Mathematica) and setting t = 0 yields:

θn = c5 + (−1)nA(d + l cos(β))

Iω+ (−1)nBl sin(β)

2Iω. (A.11)

The constant c5 is determined from (A.11). Evaluating the integrated expression at t = π/ω yields the θ mappresented in (4). Integrating (A.9) twice and evaluating the expression at t = 0 yields:

θn = c6 − 13(−1)nAB

12Imω4+ 2(−1)nA(xn sin(θn) − yn cos(θn))

Iω3+ B(xn cos(θn) + yn sin(θn))

4Iω3. (A.12)

The above equation yields an expression for c6. Evaluating the integrated expression at t = π/ω yields the θ mappresented in (4).

Linearizing the map about the fixed point (5) yields the single stride Jacobian:

Φss =

1 0 0 0

0 1 0(−1)n2A

mωπB

2Iω2

(−1)nπA

Iω21

−2πAxn + πByn

2Iω2

0(−1)n4A

Iω3

π

ω1 − 4Axn

Iω3

(A.13)

with n even and odd for left and right strides, respectively, and xn and yn represented by the relations given in (5).The full stride Jacobian (relevant terms only) is obtained by premultiplying the left stride Jacobian by the rightstride Jacobian to obtain

Φfs =

1 0 0 0

0 1 − 8A2

Imω4

−2Aπ

mω2

8A2xn

Imω4

· · · 2πA(−2Axn + Byn)

I 2ω51 + π2(Byn − 2Axn)

2Iω3Φfs3,4

· · · π2A

Iω3− 16A2xn

I 2ω6

2π

ω− 4Aπxn

Iω4Φfs4,4

,

Φfs3,4 = −2πA2

Imω3+ π(Byn − 2Axn)

Iω2+ 2πAxn(2Axn − Byn)

I 2ω5,

Φfs4,4 = 1 + −8A2

Imω4+ π2(Byn − 2Axn)

2Iω3− 8Axn

Iω3+ 16A2x2

n

I 2ω6. (A.14)


One eigenvalue is clearly λ = 1. The characteristic equation for the remaining three eigenvalues is determined fromthe lower right 3 × 3 block of (A.14) and the relevant terms are presented in (6)–(8).

A.2. Passive spring perturbation solution

A.2.1. Analytical approximation of periodic stridesThe non-dimensional equations of motion for left stance phase, including angular momentum conservation, and

with a linear spring are

¨ζ = ζ ψ2 − k

(1 − 1

η

)(ζ + d sin(ψ − θ)), LF = ζ 2ψ + I θ ,

I θ = (−1)nk

(1 − 1

η

)ζ d cos(ψ − θ), (A.15)

where η is the non-dimensionalized spring length. We expand η and retain only terms up to and including d:

1

η= 1√

ζ 2 + d2 + 2d ζ sin(ψ − θ)

≈ ζ − d sin(ψ − θ). (A.16)

Using the above expression for 1/η in (A.15) results in the equations of motion presented in (16).Substituting the expansions in time (20) and (21) into the order d0 equations of motion and collecting order τ 0

terms yields

2B = L2F

A3+ k(1 − A), (A.17)

D = LF

A2. (A.18)

At this point, we expand the initial conditions in d:

ζ (−τ0) =√

1 + d2 − 2d cos(π − β) ≈ 1 + d cos(β),

˙ζ (−τ0) = − cos(α − δ) ≈ − cos(β − δ − d sin(β)) ≈ − cos(β − δ) − d sin(β) sin(β − δ),

ψ(−τ0) = − 12)ψ − d sin(β),

ψ(−τ0) = sin(α − δ)

ζ (−τ0)≈ sin(β − δ − d sin(β))

1 + d cos(β)≈ sin(β − δ) − d sin(2β − δ), (A.19)

where α is the angle formed between the centerline of the body and ζ (see Fig. 16). Matching the order d0 initialconditions with the order d0 expansions in time yields:

1 = A + Bτ 20 + Cτ 4

0 , − cos(β − δ) = −2Bτ0 − 4Cτ 30 ,

− 12)ψ = −Dτ0 − Eτ 3

0 − F τ 50 , sin(β − δ) = D + 3Eτ 2

0 + 5F τ 40 . (A.20)

The expressions for τ0 and )ψ are obtained through the Schwind–Koditschek approximation to the quadratures in[1] as

τ0 = η(1 − ηb)√(1 − k(η − 1)2)η2 − L2

F

, (A.21)


Fig. 16. Stance geometry for left stride.

)ψ = 2LF (1 − ηb)

η

√(1 − k(η − 1)2)η2 − L2

F

, (A.22)

η = 14 (3ηb + 1), (A.23)

where ηb < 1 is the smallest positive root of following quartic equation:

(1 − k(1 − ηb)2)η2

b = L2F . (A.24)

Given τ0 and )ψ , we obtain equations which can be solved for the remaining parameters:

1 = A + (L2F + kA3(1 − A))τ 2

0

4A3+ cos(β − δ)τ0

4, C = cos(β − δ) − 2Bτ0

4τ 30

,

F = sin(β − δ) − D − 3Eτ 20

5τ 40

, E = 5)ψ − 8Dτ0 − 2 sin(β − δ)τ0

4τ 30

. (A.25)

Proceeding to the order d1 equations of motion (22), we begin by analyzing the equation for θ1. Substituting thesolutions obtained for ζ0 and ψ0 into the equation of motion, expanding cos(ψ0) in a Taylor series expansion, andmatching terms of order τ 0 yields:

H = k(A − 1)

2I. (A.26)

Matching the initial conditions yields:

J = −((θn/d) + 2Hτ0)

4τ 30

, (A.27)

G = θn

d− Hτ 2

0 − I τ 40 . (A.28)


We do the same for the other two equations of motion, but we find that the order τ 0 equations are already satisfied.Thus, we simply match initial conditions at order d1 to obtain the values:

M = − sin(β) sin(β − δ) − 3Nτ 20 , N = cos(β) − sin(β) sin(β − δ)τ0

2τ 30

,

P = − sin(β) − Qτ 20 , Q = sin(2β − δ)

2τ0. (A.29)

Using these values for A, . . . ,Q in (17), (20), (21) and (23)–(25) yields the approximations of Fig. 7.

A.2.2. Fixed point and Poincaré map detailsThe heading angle map is derived from the stance geometry, as shown in Fig. 16, as

δn+1 = π − )ψ − α − θn + θn+1 − δ1. (A.30)

We find δ1 from the ending values of the stride:

tan(δ1)= ζn+1ψn+1

˙ζ n+1

= (1 − d cos(β))(sin(β − δn) + d sin(2β − δn))

cos(β − δn) − d sin(β) sin(β − δn)= tan(β − δn) + d sin(β)

1 − d sin(β) tan(β − δn)

≈ tan(β − δn)

(1 + d sin(β)

sin(β − δn) cos(β − δn)

)def= tan(β − δn − dγ ). (A.31)

Expanding tan(β − δn − dγ ) and retaining only terms up to and including d yields

tan(β − δn − dγ ) ≈ tan(β − δn) − dγ sec2(β − δ) ⇒ γ = − sin(β). (A.32)

Thus, δ1 can be expressed as

δ1 = β − δn + d sin(β), (A.33)

which leads to the mapping for δn+1 presented in (30).Linearizing the general return map (32) results in the following Jacobian for a single stride (relevant entries only):

Φ =

1 0 0 0

· · · B + d(B1 − ΩB2) 0 (−1)n(C + dC1)

· · · · · · 1 · · ·· · · (−1)ndD 0 1

, (A.34)

where θn = dΩ and

B = 1 − ∂)ψ

∂δ, B1 = ∂(J (vn, δn) − )ψ1)

∂δ, B2 = 2(−1)n∂(τ0 + d τ1)

∂δ,

C = 2τ0, C1 = 2τ1, D = ∂L(vn, δn)

∂δ. (A.35)

Coefficients shown as · · · do not enter the eigenvalue calculations. Premultiplying the left stride Jacobian by the


right stride Jacobian determines the full stride Jacobian (relevant entries only):

Φ =

1 0 0 0

· · · B2 + d(2BB1 − CD) · · · C(B − 1) + d(C1(B − 1) + B1C − B2ΩC)

· · · · · · 1 · · ·· · · dD(1 − B) 0 1 − dDC

. (A.36)

From (A.36), the characteristic equation (with the two unity eigenvalues factored out) is determined:

λ2 − (1 + B2 + 2d(BB1 − CD))λ + B2 + 2Bd(B1 − CD) = 0. (A.37)

Solving the quadratic equation results in the eigenvalue specified in (33), along with the second eigenvalue:

λ2 =(

1 − ∂)ψ

∂δ

)[1 − ∂)ψ

∂δ+ 2d

(∂K

∂δ− ∂)ψ1

∂δ− 2τ0(∂L/∂δ)

2 − (∂)ψ/∂δ)

)]+O(d2). (A.38)

From (33), it is clear that we need to calculate ∂L/∂δ and ∂)ψ/∂δ in order to determine the ‘relevant’ eigenvalue.We begin by calculating ∂)ψ/∂δ. The Schwind–Koditschek approximation to )ψ can be rewritten as

)ψ = 128 sin(β − δ)(1 − ηb)

(3ηb + 1)√(16 − 9k(ηb − 1)2)(1 + 3ηb)2 − 256 sin2(β − δ)

. (A.39)

Since )ψ is a function of ηb, in evaluating ∂)ψ/∂δ, we need ∂ηb/∂δ, which can be determined implicitly fromthe equation for ηb (A.24) as

∂ηb

∂δ= sin(2(β − δ))

2ηb(−1 + k(1 − ηb)(1 − 2ηb)). (A.40)

Given ∂ηb/∂δ, ∂)ψ/∂δ is determined through the quotient rule as

∂)ψ

∂δ= g ˙f − f ˙g

g2, (A.41)

where ( ) denotes derivatives with respect to δ and

g = (3ηb + 1)√(16 − 9k(ηb − 1)2)(1 + 3ηb)2 − 256 sin2(β − δ), f = 128 sin(β − δ)(1 − ηb),

˙g = 3∂ηb

∂δ

√(16 − 9k(ηb − 1)2)(1 + 3ηb)2 − 256 sin2(β − δ)

+ (3ηb + 1)[−12(1 + 3ηb)(−8 + 3k(−1 + ηb)(−1 + 3ηb))(∂ηb/∂δ) + 512 sin(β − δ) cos(β − δ)]

2√(16 − 9k(ηb − 1)2)(1 + 3ηb)2 − 256 sin2(β − δ)

,

˙f = −128

(cos(β − δ)(1 − ηb) + sin(β − δ)

∂ηb

∂δ

). (A.42)

Determining ∂L/∂δ is much more complicated. To begin, we need to determine a relation forL from the quadratureobtained by integrating the third component of (22):

L = 2kd

I

∫ τ

0(ζ0 − 1) cos(ψ0) ds. (A.43)


From conservation of energy (order d0), and noting that ζ0 ≡ η, we find

ds = dη√(1 − k(η − 1)2) − (L2

F /η2)

. (A.44)

Thus, L can be expressed as

L = 2kd

I

∫ 1

ηb

(η − 1)η cos(ψ0) dη√(1 − k(η − 1)2)η2 − L2

F

. (A.45)

In a similar manner, we determine ψ0 as a function of η through conservation of angular momentum (order d0) andthe expression for ds (A.44) as

dψ0 = LF dη

η

√(1 − k(η − 1)2)η2 − L2

F

. (A.46)

Thus, for some arbitrary time between 0 and τ0:

ψ0 = ψ0(−τ0) +∫ 1

ηb

LF dη

η

√(1 − k(η − 1)2)η2 − L2

F

+∫ η

ηb

LF dη

η

√(1 − k(η − 1)2)η2 − L2

F

. (A.47)

The expression for ψ0 above is approximated using the Schwind–Koditschek approximation as

ψ0 = −π

2+ β + (1 − ηb)LF

η

√(1 − k(η − 1)2)η2 − L2

F

+ (η − ηb)LF

η2

√(1 − k(η2 − 1)2)η2

2 − L2F

,

η = 14 (3ηb + 1), η2 = 1

4 (3ηb + η). (A.48)

Since ψ0 is determined as a function of η, we use the Schwind–Koditschek approximation again to evaluate (A.45):

L = 2kd(η − 1)η(1 − ηb) cos(Z)

I

√(1 − k(η − 1)2)η2 − L2

F

,

Z = −π

2+ β + (1 − ηb)LF

η

√(1 − k(η − 1)2)η2 − L2

F

+ (η − ηb)LF

η3

√(1 − k(η3 − 1)2)η2

3 − L2F

,

η3 = 15ηb + 1

16. (A.49)

Expanding all the terms and simplifying yields

L = −6kd(−1 + ηb)2(1 + 3ηb) cos( 1

2 (−π + 2β + )ψ) + Y )

I

√(16 − 9k(ηb − 1)2)(1 + 3ηb)2 − 256 sin2(β − δ)

, (A.50)

Y = 1024(1 − ηb)LF

(15ηb + 1)√(256 − 225k(ηb − 1)2)(1 + 15ηb)2 − 65536 sin2(β − δ)

def= o

p. (A.51)

Calculating ∂L/∂δ is a simple application of the quotient rule, but quite complex:

∂L

∂δ= n ˙m − m ˙n

n2, (A.52)


where

m = −6kd(ηb − 1)2(1 + 3ηb) cos( 12 (−π + 2β + )ψ) + Y )

I,

n =√(16 − 9k(ηb − 1)2)(1 + 3ηb)2 − 256 sin2(β − δ),

˙m = −6kd[(ηb − 1)(9ηb − 1)(∂ηb/∂δ) cos( 12 (−π + 2β + )ψ) + Y ) + m1]

I,

m1 = 3(ηb − 1)2(1 + 3ηb) sin

(−π + 2β + )ψ

2+ Y

)(1

2

∂)ψ

∂δ+ ∂Y

∂δ

),

˙n = −12(1 + 3ηb)(−8 + 3k(ηb − 1)(3ηb − 1))(∂ηb/∂δ) + 512 sin(β − δ) cos(β − δ)

2√(16 − 9k(ηb − 1)2)(1 + 3ηb)2 − 256 sin2(β − δ)

,

∂Y

∂δ= p ˙o − o ˙p

p2, ˙o = −1024

(∂ηb

∂δsin(β − δ) + (1 − ηb) cos(β − δ)

),

˙p = 15∂ηb

∂δ

√(256 − 225k(ηb − 1)2)(1 + 15ηb)2 − 65536 sin2(β − δ)

+ −60(15ηb + 1)2(−128 + 15k(ηb − 1)(15ηb − 7))(∂ηb/∂δ)

2√(256 − 225k(ηb − 1)2)(1 + 15ηb)2 − 65536 sin2(β − δ)

+ 131072(15ηb + 1) sin(β − δ) cos(β − δ)

2√(256 − 225k(ηb − 1)2)(1 + 15ηb)2 − 65536 sin2(β − δ)

. (A.53)

Numerical evaluation of these functions, using the approximate root of ηb of (A.24), yields the eigenvalue plot ofFig. 8.

References

[1] J. Schmitt, P. Holmes, Mechanical models for insect locomotion: dynamics and stability in the horizontal plane — theory, Biol. Cybern.83 (6) (2000) 501–515.

[2] J. Schmitt, P. Holmes, Mechanical models for insect locomotion: dynamics and stability in the horizontal plane — application, Biol. Cybern.83 (6) (2000) 517–527.

[3] R.J. Full, M.S. Tu, Mechanics of six-legged runners, J. Exp. Biol. 148 (1990) 129–146.[4] R.J. Full, M.S. Tu, Mechanics of a rapid running insect: two-, four- and six-legged locomotion, J. Exp. Biol. 156 (1991) 215–231.[5] L.H. Ting, R. Blickhan, R.J. Full, Dynamic and static stability in hexapedal runners, J. Exp. Biol. 197 (1994) 251–269 .[6] T.M. Kubow, R.J. Full, The role of the mechanical system in control: a hypothesis of self-stabilization in hexapedal runners, Phil. Trans.

Roy Soc. Lond. B 354 (1999) 849–861.[7] R. Kram, B. Wong, R.J. Full, Three-dimensional kinematics and limb kinetic energy of running cockroaches, J. Exp. Biol. 200 (1997)

1919–1929.[8] K. Ogata, Modern Control Engineering, Prentice-Hall, Englewood Cliffs, NJ, 1990.[9] J. Schmitt, M. Garcia, R. Razo, P. Holmes, R.J. Full, Dynamics and stability of legged locomotion in the horizontal plane: a test case using

insects, 2001, submitted for publication.[10] W.J. Schwind, D.E. Koditschek, Approximating the stance map of a 2 d.o.f. monoped runner, J. Nonlinear Sci. 10 (5) (2000) 533–568.[11] T.A. McMahon, J. Bonner, On Size and Life, Scientific American Books/Freeman, New York, 1983.[12] R. Blickhan, The spring-mass model for running and hopping, J. Biomech. 11/12 (1989) 1217–1227.[13] G. Taubes, Biologists and engineers create a new generation of robots that imitate life, Science 288 (2000) 80–83.[14] R.J. Full, D.E. Koditschek, Templates and anchors: neuromechanical hypotheses of legged locomotion on land, J. Exp. Biol. 202 (1999)

3325–3332.[15] M.H. Dickinson, C.T. Farley, R.J. Full, M.A.R. Koehl, R. Kram, S. Lehman, How animals move: an integrative view, Science 288 (2000)

100–106.

mechanical models for insect locomotion: stability and parameter studies

Documents