Biol CybernDOI 10.1007/s00422-006-0126-0

ORIGINAL PAPER

A model of cerebrocerebello-spinomuscular interactionin the sagittal control of human walking

Sungho Jo · Steve G. Massaquoi

Received: 9 May 2006 / Accepted: 5 September 2006© Springer-Verlag 2006

Abstract A computationally developed model ofhuman upright balance control (Jo and Massaquoi onBiol cybern 91:188–202, 2004) has been enhanced todescribe biped walking in the sagittal plane. The modelincorporates (a) non-linear muscle mechanics havingactivation level -dependent impedance, (b) scheduledcerebrocerebellar interaction for control of center ofmass position and trunk pitch angle, (c) rectangularpulse-like feedforward commands from a brainstem/spinal pattern generator, and (d) segmental reflex mod-ulation of muscular synergies to refine inter-joint coor-dination. The model can stand when muscles aroundthe ankle are coactivated. When trigger signals acti-vate, the model transitions from standing still to walk-ing at 1.5 m/s. Simulated natural walking displays noneof seven pathological gait features. The model can sim-ulate different walking speeds by tuning the amplitudeand frequency in spinal pattern generator. The walkingis stable against forward and backward pushes of up to70 and 75 N, respectively, and with sudden changes intrunk mass of up to 18%. The sensitivity of the model tochanges in neural parameters and the predicted behav-ioral results of simulated neural system lesions are exam-ined. The deficit gait simulations may be useful tosupport the functional and anatomical correspondencesof the model. The model demonstrates that basic human-like walking can be achieved by a hierarchical structure

S. Jo (B) · S. G. MassaquoiDepartment of Electrical Engineering and Computer Science,Computer Science and Artificial Intelligence Laboratory,Laboratory for Information and Decision Systems,Massachusetts Institute of Technology, 32 Vassar St.,Cambridge, MA 02139, USAe-mail: shjo@csail.mit.edu

of stabilized-long loop feedback and synergy-mediatedfeedforward controls. In particular, internal models ofbody dynamics are not required.

1 Introduction

Burgeoning interest in robot design and in human motorcontrol and its disorders has stimulated computationalinvestigations of natural upright balance and locomo-tion control. A range of studies have identified under-lying patterns of kinematics and muscular activation(Anderson and Pandy 2001; Ogihara and Yamazaki 2001;Perteka 2003; Jo and Massaquoi 2004; Loram et al.2004). Most have investigated the problems of balanceand locomotion separately. Human postural balance hasbeen studied and modeled chiefly with regard to pos-tural disturbances or characteristics of natural sway (Joand Massaquoi 2004; Perteka 2003; Kuo 1995; Johanssonet al. 1988). These indicate that human-like balancingcan be afforded by simple feedback circuitry. There havealso been a number of models of physiological walking.Some have focused on the role of spinal pattern gener-ators in the production of quadrupedal gait (Fukuokaet al. 2003; Kimura et al. 2001) where balancing require-ments are important, but less demanding. Kimura andFukuoka have constructed a quadruped walking robot“Tekken” whose control is based on biologicalconcepts. The actuators are viscoelastic and its neu-ral system model consists of neural pattern generators(NPGs) and vestibulospinal reflexes (Fukuoka et al.2003). The NPG uses Matsuoka’s equations to producerhythmic joint motions. This nonlinear differentialequation system models two tonically excited neuronswith a self-inhibition, which are linked reciprocally via

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inhibitory connections (Matsuoka 1987). A modifiedproportional type feedback control system representingthe vestibulospinal reflex controls body roll and pitchangles. Peripheral feedback also affects the phase tran-sitions in the NPG. Adjustable gains in the muscular con-trol based on the phase of the NPG enables the robotto adapt to different walking speeds and to irregularterrain.

Bipedal walking places particular demand on inte-grating the control of balance and gait. Two neuro-musculo-skeletal models of bipedal walking have beenparticularly successful. First, Taga’s model (Taga 1995)includes eight body segments, ground contact elements,and 20 muscles. Its neural controller defines a sequenceof global states in terms of body’s center of mass (COM)and center of pressure (COP). Inputs from neural oscil-lators [using Matsuoka’s equations (Matsuoka 1987)],located at each joint, are modulated by the neural con-troller in order to generate stable limit cycles. Low-levelfeedback control is afforded by impedance controllersthat represent muscles. This mechanism generated sta-ble gait in the sagittal plane. Secondly, the model ofOgihara and Yamazaki (2001) employs NPGs and, incomparison with Taga’s mdoel, emphasizes detail in theactuation and peripheral neural feedback. The modeldefines muscles that include Hill-type force–length–velocity relationships, reflexes from muscle spindles,tendon organs, and foot tactile receptors. A genetic algo-rithm was used to find realistic walking motion. Thesemodels show that qualitatively realistic bipedal walkingkinematics can be achieved using biomorphic compo-nents. However, these models have not demonstratedthe capacity to walk at different speeds or to balanceupright when stationary without changes in physicalparameters. Also, the sensitivity to changes in physicalcharacteristics and to disturbances has not been exam-ined. Most of all, the nature of the higher levels of centralnervous system control of bipedal gait remains essen-tially unexplored.

It has been shown that realistic control of uprightbalance can be accounted for in terms of stabilizedlong-loop (trans-cerebrococerebellar) proprioceptiveand force feedback (Jo and Massaquoi 2004). This sys-tem affords energetic efficiency because muscular co-activation and associated active body stiffness can bereduced, It also affords flexibility and automaticity ofresponse patterns because feedback gainsets can bemodified according to sensed changes in body state.The model also manages robustly the potentially de-stabilizing effects of long-loop neural signal transmis-sion delays and muscular excitation-activation phaselags that have not been heretofore included in neuro-morphic locomotor control models. At the same time,

considerable work on spinal physiology has shown thatcertain basic muscle activation patterns (synergies) maybe coded within the cord (Tresch et al. 1999; d’Avellaet al. 2003). Simulations of muscle synergies in a modelof the frog hindlimb with simple switch-like commandshave been shown to generate a range of movements con-sistent in both kinematics and muscle activations withreal behaviors (Cajigas-González 2003). Thus, given theintimate relationship between upright balance andbipedal locomotion, and the wealth of physiological datathat indicates important cerebral, cerebellar, brainstemand spinal roles in balance, patterned leg motion andlocomotion, an integrated model describing the hier-archical neural control of both balance and walkingappears valuable. In particular, it may be expected apriori that long-loop control may afford similar func-tional advantages to locomotor control as it does forstanding balance. It may also be suspected that recr-uitable muscular synergies organized at a spinal levelmay help to reduce control demands upon higher levelsystems.

Ultimately, it is also of conceptual interest to deter-mine the simplest formulation that is consistent withrecognized functional anatomy. Fundamental questionsinclude: What are the sufficient feedforward commandvariables? How complex must feedforward signals be tomanage body dynamics during walking? What types oflong-loop and segmental feedback processing are nec-essary and sufficient to support the basic feedforwardmotor command? What are possible roles of spinallyorganized muscle activation synergies and what is theirrelationship to central NPGs?

In this article, a basic model of Sagittal control ofBipedal Balance and Walking (SBBW) is proposed. Itshows that long-loop responses, if stabilized and modu-lated by cerebrocerebellar interaction, can afford suffi-cient upright balance to allow basic walking to be drivenat different speeds simply by adjusting the frequency andamplitude of two five-state rectangular pulse generatorsthat engage four, potentially five, time-invariant muscu-lar control synergies. Peripheral reflexes provide modu-lation that improves walking efficiency, and security, butare not fundamentally required for the basic gait pattern.Simulated lesions of the cerebellar and peripheral feed-back systems give rise to certain control defects that aregrossly similar to those observed clinically. This featuresupports the model’s proposed structure-function cor-respondences. On the other hand, certain features ofexperimentally observed muscle activation patterns arenot predicted by the SBBW model. This indicates thatthere exist more than one muscle control pattern thatare consistent with observed walking kinematics. Theimplications of this observation are discussed.

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2 Neurophysiological background and principalmodeling assumptions

2.1 Neural control of basic locomotor muscleactivation patterns

A large number of systems potentially influence postureand gait including cerebral cortex (King 1927; Nielsen2003; Dietz 1992), cerebellum (Dietz 1992; Morton andBastian 2004; Mortan and Bastian 2003), basal gangliaincluding subthalamic locomotor region (Zijlstra et al.1998; Shik and Orlovsky 1976; Dietz 1992), midbrainlocomotor region (Grillner 1975; Shik and Orlovsky1976; Kandel et al. 2000) and spinal cord with segmentalreflexes (Duysens et al. 2000; Knikou et al. 2005; Brookeet al. 1997; Grillner 1975; Dietz 1992; Shik and Orlovsky1976). Here a parsimonious mechanism is sought withthe intention of identifying the minimal or near-minimalneural control requirements for human-like bipedal gait.At least in primates, upright walking appears to requirethe integrity of cerebral cortical control of legs, midlinecerebellum including at least the fastigial nucleus (Moriet al. 2004) and possibly the interpositus (Amstrong andEdgley 1988), the brainstem and spinal cord. While basalganglionic dysfunction leads to a host of walking defi-cits (Zijlstra et al. 1998; Kandel et al. 2000; Shik andOrlovsky 1976) it is not clear that explicit representa-tion of basal ganglionic function is required to accountfor basic locomotion pattern. Certainly in a supporteddecerebrate cat, highly coordinated walking motions canbe elicited by stimulation of cerebellar or mesencephal-ic centers (Grillner 1975; Mori et al. 1999; Shik andOrlovsky 1976; Kandel et al. 2000). So, for the purposesof this study, basal ganglia function will be taken to besubsumed within cerebrocrebellar interaction discussedbelow.

In any case, the impression of most investigators isthat during locomotion, higher systems drive and mod-ulate spinal level systems that are responsible for muchof the basic patterning of muscle activity. The precisehierarchical partitioning of function has not yet beendetermined. However, several observations are rele-vant. (1) Muscular activation and leg function duringlocomotion (Ivanenko et al. 2004) indicate that theelectromyogram (EMG) pattern is consistent acrossmovement speeds, with only duration and intensitychanging (inversely). Several investigations (Ivanenkoet al. 2004, 2005, Davis and Vaughan 1993; Olree andVaughan 1995) have used factor and principalcomponent analysis, and nonnegative matrix factoriza-tion to decompose to such activity into a combinationof four or five principal waveforms, most of which haveonly modest, but not zero temporal overlap. (2) Exper-

imental observations support a spinal locus for impor-tant rhythmic locomotor EMG pattern generation inhumans (Dimitrijevic et al. 1998; Grasso et al. 2004;Calancie et al. 1994; Dietz and Harkema 2004) oftenin response to tonic electrical stimulation (Dimitrije-vic et al. 1998). (3) Experimental stimulation of eitherthe cerebellum or midbrain produces rhythmic locomo-tor movement in both intact and decerebrate cat withvigor and frequency that increase with stimulus inten-sity (Mori et al. 1998, 1999). These locomotor regionsboth strongly recruit vestibulospinal, reticulospinal andother direct spinal efferent pathways (Shik and Orlov-sky 1976). A parsimonious proposition is that the cer-ebellar and midbrain locomotor regions can each driveand potentially modulate a spinal locomotor controlsystem consistent with previous suggestions (Grillner1975; Mori et al. 1998, 1999; Kandel et al. 2000; Shikand Orlovsky 1976). (4) Tonic stimulation of frog spinalcord demonstrates synergistic patterns of muscle activ-ities (Tresch et al. 1999; Cheung et al. 2005; d’Avellaet al. 2003; d’Avella and Bizzi 2005). Such a mechanismcollapses multiple muscle control into lower degree-of-freedom control for each leg. The lower dimensionalrepresentation can still account for a wealth of frog legEMG activities and behaviors including wiping, crawl-ing and swimming (Cajigas-González 2003).

These findings suggest that a rhythmic central pat-tern generator, modulated from above, may interactwith muscle control synergies to provide a simple andeffective walking control. Numerous studies indicate thecomplexity of segmental spinal cord circuitry (Rossig-nol et al. 2006; Grillner 1975; Brooke et al. 1997), whichwill not be modeled explicitly here. Rather, a simplerepresentation of spinal control has been sought thatmay be functionally equivalent to the actual systemunder the restricted, but important condition of mod-erate velocity forward walking with modest push distur-bances. In particular, here it will be examined whetherbasic walking can be driven by four or five sequential-rectangular pulses during the gait cycle for each leg.At the level of the spinal pulse generator, locomotorfunction will be therefore viewed in terms of five hypo-thetical control epochs: “loading” (LOA), “regulation”(REG), “thrust” (THR), “retraction” (RET) and “for-ward” (FOW) (Fig. 4). The last is almost, but not pre-cisely coextensive with actual swing phase. Plausibly,these five phases may be driven at a range of frequen-cies. Sequential, multi-state transitioning appears to beconsistent with simple physiological oscillator modelsstudied by Iwasaki and Zheng (2006). Other proposedmulti-state oscillator implementations include pairs ofhalf-center oscillators (Fukuoka et al. 2003) and asymmetric ring of n coupled nonlinear oscillators

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(Collins and Stewart 1994). Although a priori it is notnecessary that the relative durations of each controlepoch remain fixed, this is the simplest assumption andwill be adopted here. If the output intensities of sucha pulse generator were scaled appropriately with fre-quency, effective muscle activation might be afforded bylinear combinations of independently recruitable mus-cle activation patterns, or time-invariant control syner-gies (e.g. Cajigas-González 2003). We note here thatthe term “synergy”, while generally used to refer totask-related groupings of multiple muscle activities, mayhave a variety of interpretations. For specificity andconsistency with a motor control perspective, the termtime-invariant control synergy is used herein to specify alinear distribution of a single control signal into multiplecommand signals that are directed to a set of muscles.This distribution will be represented by columns withinsignal distribution matrix. The total EMG will then bedetermined by the superposition of several operativecontrol synergies. Each control synergy vector is con-sidered time-invariant in the sense that any change oradaptation of its elements is assumed to occur at a timescale vastly slower than any individual movement. Allmoment-to-moment variation in command signal inten-sity is therefore attributed to variation in the controlinput signal, and/or the selection or “recruitment” ofdifferent existing distribution matrices. The combina-tion of the cyclic pulse generator and the linear distri-bution matrix will be referred to as a spinal locomotorpattern generator.

Other experiments have pointed out the role ofperipheral neural input in modulating the locomotorpattern. The most important influences appear to be:(a) detection of ground contact (Rossignol et al. 2006;Nielsen 2003; Duysens et al. 2000; Zehr and Stein 1999),(b) the monitoring of hip angle which appears to beassociated with releasing or triggering the forward steptransition (thrust to retraction, in Fig. 4) (Knikou et al.2005; Kriellaars et al. 1994; Duysens et al. 2000), (c)the peripheral position-dependent modulation of seg-mental reflexes (Baxendale and Ferrell 1981; Brookeet al. 1997; Duysens et al. 2000; Rossignol et al. 2006).It is known that presynaptic inhibition contributes toposition-dependent modulation during locomotion(Duysens et al. 2000; Brooke et al. 1997; Stein 1995).Presynaptic inhibition refers to a depression of monsy-naptic excitatory potentials that occurs in the absenceof any postsynaptic potential change or any changein motoneuronal excitability (Rudomin and Schmidt1999). This paper uses a simple model of the presyn-aptic inhibition to implement the position-dependentmodulation at spinal cord level (see Sect. 3.2.1).

2.2 Cerebrocerebellar control of COM and trunk pitch

Without the cerebrum and cerebellum, the locomotorpatterns are much simpler than normal stepping(Kandel et al. 2000). The cerebellum is critical for bal-ance and studies have shown that selective lesions ofdescending control from the motor cortex compromiseirrevocably certain fine control of especially swing legtrajectory in the cat (Drew 1993). The motor cortexalso has been shown to contribute to structure and tim-ing of step cycle during locomotion in the intact cat(Bretzner and Drew 2005). In humans, where fine inte-gration of bipedal balance and stepping must be exqui-site, disconnection of cerebral control of legs due tostroke or tumor yields devastating effects in posturalbalance and locomotion (Porter and Lemon 1993). Thus,normal bipedal function in humans appears to dependsignificantly upon activity in transcerebral pathways(Nielsen 2003; Peterson et al. 1998; Nathan 1994). More-over, it has been noted that most of cerebral activityduring locomotion appears to be generated by sensoryafferent feedback (Christensen et al. 2000; Nielsen 2003)which suggests a prominent role for long-loop feedbackmechanisms. For these reasons, the modeling here as didthat in upright balance control (Jo and Massaquoi 2004)includes at least a rudimentary trans-cerebrocerebellarfeedback loop.

The control of sagittal planar upright posture duringwalking is similar to that exerted during stationary bal-ance except that it is potentially more intermittentand/or weaker in the sense that the upper body gen-erally remains close to vertical in both situations so thattrunk posture can help to control the body’s COM withina supporting area (Gilchrist and Winter 1997). It is natu-ral therefore to consider beginning with the upright bal-ance model explored previously by the authors [FRIPIDmodel, (Jo and Massaquoi 2004)]. Still, it is possibleto walk or even run with the trunk bent forward orbackward. And during running, the trunk is maintainederect even while there is no ground contract and thelegs are not vertical. Therefore, it is plausible that reg-ulation of trunk pitch and COM position relative tothe leg configurations are managed by separate circuits.This is consistent with recent experimental work byFreitas et al. (2006) and appears to be a necessary modelextension. The FRIPID model showed that smoothswitching or scheduling cerebellar gains associatedwith proportional-integral-derivative (PID) controlcircuits as a function of sensed body state enabledthe cerebrocerebellar system to automatically modify itsresponses according to disturbance strength. The switch-ing system was argued to be easily compatible with

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known cerebellar cortical microcircuitry. Importantly,interpolation between just two gainsets was sufficient toaccount for the full range of balancing strategies. Sim-ilarly, the current model considers the possibility thata single or small set of cerebellar gains would also besufficient for walking control.

The availability of spinal synergies also enables animportant simplification of the cerebellar control in theSBBW model with respect to that proposed in the FRI-PID model (Jo and Massaquoi 2004). Under the assump-tion that certain signals ascending the spinal cord e.g. viaspinocerebellar tracts, may consist of linear combina-tions of proprioceptive information (see Jo andMassaquoi 2004; Osborn and Poppele 1992), the pres-ence of synergies in the efferent path allows multiplejoints to be both sensed and controlled by single cer-ebellar modules. Specifically, one single-input single-output (SISO) cerebellar channel can engage a synergythat spans ankle, knee and hip to control the trunkand stance. This could simplify tracking of the intendedCOM position as specified in the cerebral cortex. Anindependent SISO channel can engage muscles at thehip to control trunk pitch relative to vertical based onvestibular input and/or ankle, knee and hip angles, toge-ther with an assumed pitch for the stance surface. Suchdecoupling is similar to that used recently to controlrobotic locomotion (Hofmann 2006). Hence, in theSBBW model, cerebellar balance control is implemen-ted by 2×2 diagonal matrices instead of fully populated3 × 3 matrices as used in Jo and Massaquoi (2004) (seeSect. 5.6). On the other hand, the SBBW model is notdesigned to account for all features of human uprightbalancing such as postural strategies (Jo and Massaquoi2004) because the model focuses on walking.

2.3 Summary of principal physiological modelingassumptions

In summary, the SBBW model assumes:(PA-1) The feedforward action of brainstem and

spinal cord in locomotor muscle patterning can be sum-marized as a five-state activation/relaxation process thatdrives time-invariant muscle activation synergies. Thisprocess controls the intensity and frequency of synergyactivation.

(PA-2) The influence of peripheral input can be con-sidered to consist most importantly of (a) an indicationof which leg is in contact with the ground, (b) a modula-tion of synergies to improve step morphology by alter-ing a spinal muscle activation threshold via presynapticinhibition.

(PA-3) The control afforded by the cerebrocerebellarlong-loop circuitry is implemented by three independent

Fig. 1 (Left) Body configuration joint angle convention: θ1, θ3, θ5are respectively ankle, knee, and hip angles for right leg, and θ2,θ4, θ6 ankle, knee, and hip for left leg. θtr is the trunk pitch angle.Arrows indicate directions where angle values increase. (Right)Muscle diagram (one leg for simplicity) with muscles identified inTable 2

channels. One operates continuously to regulate trunkverticality, and two others represent independent, par-allel control of respectively left and right leg posturesat stance needed to regulate intended relative COMposition. Detection of ground contact (PA-2a) is usedto gate trans-cerebellar long-loop COM control to thesupporting leg (Duysens et al. 2000; Nielsen 2003; Zehrand Stein 1999; Morton and Bastian 2003).

3 Methods

3.1 Spinomusculoskeletal plant model

3.1.1 Skeletal system and ground contact

A seven-segment kinematic chain with pivot joints wasused to represent human walking in the sagittal plane(Fig. 1). Positive angular motion was consistent with ana-tomical flexion at the hip and knee, and dorsiflexion atthe ankle. Each leg incorporates nine muscles.

The length of each segment is represented withrespect to the total body’s height HB (Fig. 2a). Eachsegment’s mass and moment of inertia are calculatedwith respect to HB and the body’s mass MB in Table 1.The position of center of mass at feet is detailed inFig. 2b.

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Fig. 2 a Body segment length and b foot dimensions (Winter1990). Centers of component masses are indicated by small check-ered discs

The natural locking of the knee that prevents hyper-extension is modeled by a high impedance damped elas-ticity:

τi,lock ={

max(Kk(θi,min − θi)− Bkθi, 0) if θi < θi,min

min(Kk(θi,max − θi)− Bkθi, 0) if θi > θi,max

(1)

where Kk, Bk are, respectively, spring and dampercoefficients, θi,min is a minimum knee angle, and θi,max

is a maximum knee angle, and θi is an actual knee angle.For this research, θi,min is set to be −160 degrees, andθi,max to be zero degree.

The body model’s dynamics in response to appliedtotal muscular and ground reaction torques applied to

the joints, τM(�, �, act) and τR(Fgx, Fgy, �) , respectively,is given by

H(�)�+C(�, �)=τM(�, �, act)+τR(Fgx, Fgy, �)+G(�)

(2)

where � = [θ1 θ2 θ3 θ4 θ5 θ6

]T, � =[θ1 θ2 θ3 θ4 θ5 θ6

]T, and act = [act1 act2 · · · act9]T

represents the muscle activations defined below (Eq.11), H(�) is the symmetric configuration-dependentbody inertia matrix, C(�, �) is the matrix related tocentrifugal and Coriolis forces, G(�) is the gravitationaleffect matrix, and τR(Fgx, Fgy, �) is the torque gener-ated by horizontal and vertical reaction forces to theground at heel and toe (the details in section ). Thedynamics are executed using SimMechanics in Matlab(MathWorks Inc., Natick, MA, USA).

3.1.2 Muscle structure and activation

Muscular torque is determined by the total muscularforce (passive + active) F(l, l, act) and the moment armsof each muscle according to

τM(�, �, act) = aT F(l, l, act) (3)

AT

=⎡⎢⎣

0 0 0 0 −a5 a6 0 0 aa9

0 0 a3 −a4 0 0 ak7 −ak

8 −ak9

−a1 a2 0 0 0 0 −ah7 ah

8 0

⎤⎥⎦

(4)

where ajoi is the estimated average moment arm over

the usual range of motion of the ith muscle in Table 2.jo = a, k, or, h to distinguish ankle, knee and hip jointmoment arms, respectively, in biarticular muscles. Flexormoment arms are negative reflecting the relationshipbetween length change and direction of rotation.

The model views the anatomically redundant musclesof the trunk and legs as operating together as functionalgroups of uni- and biarticular flexors and extensors asshown in Fig. 1. Assuming that stiffness is proportionalto physiological cross-sectional area (PCA) (Brand et al.1986), the relative muscle stiffness scaling is given basedon morphmetric data in Table 2.

Passive muscular force is expressed by

Fpass=[Kpass(leq − l) − Bpass l

]+ (5)

where [x]+ ={

x,0,

x > 0x ≤ 0

, Fpass is passive tension vec-

tor, Kpass, Bpass passive muscle stiffness and viscosity

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Table 1 Body segment’smasses and moments ofinertia calculated based onWinter (1990)

Body physical parameters

Body segment, HB = 1.8 m, MB = 80 kg

Trunk Upper leg Lower leg Foot

Mass (kg) 0.678MB 0.1MB 0.047MB 0.015MB

Moment of 0.031MBH2B 6.262 × 10−4MBH2

B 2.566 × 10−4MBH2B 4.976 × 10−6MBH2

Binertia (kg m2)

matrices, leq is muscle length vector at equilibrium, l isactual muscle length vector. Active muscular force as afunction of neural input to each muscle (act ) is repre-sented by

Fact= Kact(act)[l(act) − l]+ − Bact(act)l (6)

where Fact is active tension vector, l(act) = leq+act.Kact(act), and Bact(act) are the active muscle stiffnessand viscosity matrices which are functions of muscularactivation.

Kact(act) = Ka(α[act]++βmin(γ [act]+, 1)

)Bact(act) = Ba

(α[act]++βmin(γ [act]+, 1)

) (7)

where Ka, Ba are constant matrices, and α, β, and γ

are constant coefficients. When both passive and activetensions are applied together,

F(l, l, act) =[Fpass(l, l) + Fact(l, l, act)

]+ (8)

l = leq+A(� − �eq) (9)

where � is joint angle vector at equilibrium.The positive brace means that each muscle is con-

strained to exert only contractile force. This formulationsubstantially follows that employed by Katayama and

Kawato (1993). The effective pre-set (e.g. before reflexneural activation) rotational stiffness of the ankle dur-ing standing is about 90 Nm/rad (Fujita and Sato 1998).This value was used to determine the absolute passivestiffness of each muscle given their relative scaling whichis proportional to the cross-sectional area in Table 2. Itwas assumed that Ka = 2.5Kpass and Ba = 2.5Bpass.α[act]++βmin(γ [act]+, 1) < 1 holds over simulations,therefore, active stiffness is maximally less than 250% ofpassive stiffness and damping ratio could approximatelyincrease up to 50% because the ratio is proportional toB/

√K. This was considered based on human arm mod-

eling study where stiffness and damping ratio have beenshown to increase up to 500% and 50%, respectively(Lacquaniti and Soechting 1986), with strong activation.In general, magnitude of muscle viscosity was set at one-tenth that of the muscle stiffness as has been done in armmodeling (Flash 1987).

The activation of muscle force by neural input occursaccording to low-pass dynamics that can be approxi-mated by the filter with Laplace transfer function:

EC(s) = ρ2

(s + ρ)2 , ρ = 30 rad/sec (10)

Table 2 Length, momentarm, and physiologicalcross-sectional area (PCA)parameter values of musclesdetermined from Ogihara andYamazaki (2001), and Winter(1990)

Muscle Location leq (m) ajoi (m) PCA (cm2)

Iliopsoas (IP) mono, hip flexor 0.35 0.132 17Gluteus Maximus (GM) mono, hip extensor 0.30 0.092 30.4Rectus femoris bi, hip flexor, 0.48 0.049(h), 12.5(RF) knee extensor 0.025(k)Biceps femoris long bi, knee flexor, 0.46 0.054(h), 15.8(BFL) hip extensor 0.049(k)Vastus (VA) mono, knee extensor 0.26 0.04 30Biceps femoris short (BFS) mono, knee flexor 0.29 0.049 6.8Tibialis anterior (TA) mono, ankle dorsiflexor 0.30 0.023 9.1Gastrocnemius bi, knee flexor, 0.56 0.050(k), 30(GC) ankle plantarflexor 0.040(a)Soleus (SO) mono, ankle plantarflexor 0.35 0.036 58

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(Fuglevand and Winter 1993) where s is the Laplacevariable, and then

act = EC(s) (uα) (11)

meaning that the filter is applied to uα that is the alphamotor neuronal output.

The dynamics of series elasticities, filtering actionof spindles, segmental proprioceptive and force feed-back, and spinal processing by alpha motorneuron —Renshaw cell networks were not modeled explicitly,although it is likely that these could improve the accu-racy of the simulations (Winters 1995). However, it wasnot felt that these features would bear significantly uponthe basic neuromuscular mechanisms of gait control.

3.1.3 Foot interaction with the ground

When the heel or toe contacts the ground, horizontal andvertical reaction forces are typically generated (Fig. 12).During (ygy(xi) > yi), vertical reaction force is modeledby

Figy = (

Kgy(ygy

(xi)−yi)− Bgyyi) (12)

where (xi, yi) indicates the positions of either heel ortoe with i = heel, toe. ygy (xi) represents the ground pro-file as a function of xi. Kgy and Bgy are coefficients. Fi

gyis always nonnegative. The horizontal reaction force ismodeled in two ways.

During contact, the horizontal reaction force is mod-eled by a spring and damper system as long as the hori-zontal reaction force is smaller than the maximal frictionforce (|Fi

gx| ≤ |µsFigy| where µs is the static frictional

coefficient).

Figx = (

Kgx(xi

o − xi) − Bgxxi) (13)

where xio is a location where either heel or toe touches

the ground initially and Kgy and Bgy are coefficients.However, if the horizontal reaction force is larger

than the maximal friction force (|Figx| > |µsFi

gy|), Eg. 14represents the horizontal reaction force.

Figx = −µkFi

gysgn(xi) (14)

where µk is the dynamic frictional coefficient.As as result, the total vertical and horizontal reaction

forces between foot and ground are respectively:

Fgy = Ftoegy + Fheel

gy , Fgx = Ftoegx + Fheel

gx . (15)

Fig. 3 Periodic spinal command pulse generation. Top Schematicrepresentation of the membrane potential of a hypothetical neu-ron within an oscillating circuit. Action potential pikes represent-ing output are fired when a threshold is crossed, and firing rate(intensity) is modeled by a rectangular pulse that occurs whenthreshold is traversed. Bottom The spinal pulse output commandsare summarized

3.2 Central neural control model

3.2.1 Spinal pulse generator and locomotor controlsynergies

Many models of central pattern generators have beenconsidered (Matsuoka 1987; Patla et al. 1985; Iwasakiand Zheng 2006; Collins and Stewart 1994). For thepresent purposes, it is sufficient to consider a five-statemachine whose states transition in a strict, fixedsequence. Each state is comparable with each controlepoch. For simplicity and generality, the state machinewill be considered to have scaled binary output thatis processed by a linear signal distribution matrix. Thecommand output of each state register is designateduPG,i(t). As a further simplification, the command out-put of the fifth state register, uPG,5(t), is intentionally setto zero to correspond to an almost completely passiveswing phase of each leg.

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Fig. 4 Decomposedspino-locomotor signals and amodel of the locomotorcontrol synergy network. TopProposed control epochs andspinally generated commandby muscle and phase of gait.Shaded lines indicatesuppressions by presynapticinhibition. BottomHypothetical model ofconnection between patterngenerator and muscle. Neuralsignals are subsequentlydelayed as much asTpr = [

Tpr,a Tpr,k Tpr,h]T

As in Fig. 3, for i = 1, . . . , 4, its periodic activationcan be modeled as

uPG,i = ηPG · 1[cos(2π fPGt − φi) − hi

]+ . (16)

where 1[x]+ ={

1 where x > 00 where x ≤ 0

, fPG determines the pat-

tern frequency, ηPG is an activation intensity factor, φi

is the phase shift, and hi activity discharge threshold.Specification of φi and hi determines the sequence andpotential overlap between the state activations. The val-ues used in simulation are given in Table A-1 in Appen-dix B. The values of φi and hi were chosen to yield(a) no command overlap, (b) empirically most realis-tic gait patterns subject to constraint (a). Two identicalpulse generators were used, one for each leg. A phase

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difference of 180 degrees between the generators wasenforced artificially as gaits other than simple walkingwere not entertained in this study.

It is proposed that pulse generator commands uPG =[uPG,1 uPG,2 uPG,3 uPG,4

]T are distributed to themuscles according to a spinal locomotor control syn-ergy network represented by the matrix WPG (Fig. 4and Appendix A) according to five functional tasks dur-ing the gait cycle. The fifth task, active control of swingleg, is, for the moment, ignored as mentioned above.

uSP = WPGuPG (17)

The combination of spinal pulse generator and con-trol synergy distribution matrix is considered an NPG.To develop a minimal model, synergies involving themonoarticular muscles alone were formed. Thus, thespinal activations uSP,j(t) are defined for j = 1, ..., 6 inEq. 17 and Fig. 4. The signals for j = 7, 8, 9 were set tobe zero. As discussed later, simulation of simple walkingdoes not require biarticular muscle activation.

Each rhythmic spinally generated muscle commanduSP,j(t) is thus a train of rectangular pulses (Fig. 4, Top).The primary function of each pulse is approximately to:

– IP-1: prevent the upper body from falling backward.– IP-2: raise leg and trigger forward swing around hip.– GM-1: prevent the upper body from forward rota-

tion just after heel strike.– BFS-1: help swing leg avoid ground contact by flex-

ing knee.– VA-1: maintain body support on stance leg by

extending knee at mid-stance.– TA-1: help COM move forward to insure step size.– TA-2: help swing leg avoid ground contact by dors-

iflexing ankle.– SO-1: provide forward thrust using stance leg.

As detailed in the results, it was assumed that a sep-arate additional pulse could be delivered (superposedon other commands) to IP to launch walking from astanding start. The physiological basis of such a pulse isnot specified. However, as discussed below, it may be ofcerebral origin.

Finally, the SBBW model proposes that peripher-ally triggered spinal inhibition can help to modify cer-tain synergy components. One possible neural circuitinvolves presynaptic inhibition (Baxendale and Ferrell1981; Brooke et al. 1997; Rossignol et al. 2006;Duysens et al. 2000) as shown in Fig. 4 and in greaterdetail in Fig. 5. A descending signal conveys a tonic exci-tation θth,jo, jo = a, k, h(ankle, knee, hip), that inhibits

Fig. 5 Presynaptic inhibition: depolarization of an afferent ter-minal (presynaptic inhibition) is indicated by the filed triangleending on afferent pathway, inhibitory effect by the filled circle.MN: motorneuron, IN: interneuron

the proprioceptive afferent θjo until superseded. There-after, the interneuron is activated and the motor neuronactivity is suppressed. It is speculated here that sucha mechanism could truncate BFS-1 and TA-2 activityin early FOW to prevent excessive leg retraction. Thismechanism is empirically useful to improve the timingbetween knee and ankle motions to prevent ground con-tact during swing phase. The spinal reflex action is there-fore modeled as

ureflex = −WSR · 1[θpr − θth

]+ (18)

where θth = [θth,a θth,k θth,h

]T, θpr = [θa(t − Tpr,a)

θk(t − Tpr,k) θh(t − Tpr,h)]T where Tpr,jo, jo = a, k, h,

are unidrectional peripheral neural transmission delaysfrom the spinal cord to the anke, knee and hip, respec-tively and WSR is a 9 × 3 matrix that scales and distrib-utes joint-related signals from the uniarticular musclesto the other muscles via the 9 element vector ureflex.

3.2.2 Suprasegmental control

The second component of the central neural controlsystem modeled consists of two long-loop feedback path-ways that add stability. The first is concerned with track-ing the intended forward position of the body’s COMrelative to the stance foot as specified by the tonic ref-erence signal xcom,ref . The second concerns maintainingthe trunk-head segment close to vertical at all times.The pitch angle of the trunk-head segment is repre-

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Fig. 6 The RIPIDcerebro-cerebellar controlmodel applied to locomotorcontrol

sented as θtr (Fig. 1) and therefore its intended posi-tion is θtr,ref = 0. The cerebrocerebellar control model(Fig. 6) is simplified from Jo and Massaquoi (2004) inthat a high speed “catching” gainset and force feed-back are not employed. This is because the SBBW isnot designed to cope with rapid foot slippage or trans-lation of the support surface. Physiologically, it is alsoquite possible that important long-loop feedback con-trol operates via sub(cerebral)cortical levels. However,from the perspective of managing long signal transmis-sion delays the most challenging possibility is if signifi-cant processing occurs via trans(cerebral) cortical loopsas it does in arm control. For conservatism, therefore,trans(cerebral) cortical long-loop feedback is modeled.

During simulated walking xcom,ref = 25 cm and this iscompared to a linear estimate of COM position relativeto the stance foot computed as

xcom = p11θa(t − Tspr,a) + p21θk(t − Tspr,k)

+ p31θh(t − Tspr,h) = pT1 θspr (19)

where p1 = [p11 p21 p31

]T are constants, and θspr=[θa(t − Tspr,a) θk(t − Tspr,k) θh(t − Tspr,h)

]T. Tspr =[Tspr,a Tspr,k Tspr,h

]T are the afferent signal transmis-sion delays including spinal and peripheral components.The scaling factors pi1 were obtained by linearizing trigono-metric relationships and neglecting contributions of theswing leg, as detailed in Appendix A.

The estimate of θtr is presumed to depend upon asimilar linear estimate:

θtr= pT2 θspr (20)

where p2 = [p12 p22 p32

]T. The derivation of the pi2assumes a horizontal walking surface for the moment.In the case of non-horizontal support surfaces, thetrunk-head segment pitch estimate is presumably adju-

sted on the basis of or replaced by visual and/or vestib-ular input. However, this issue is not examined here.

When xcb is the input to the cerebellar system, ucb isthe output, and we take I1 = 0 (cerebellar forward inte-gral control turns out unnecessary empirically) in Fig. 6,we can then describe cerebellar control as

ucb = GCBxcb = diag(gki + gbid(·)/dt)xcb, i = 1, 2

(21)

and

xcb =(

I + I2

∫ )−1 (Ia

∫ (uref − PTθspr

)− F2PTθspr

)

= L−1{(sI+I2)−1

(Ia

(uref −PTθspr(s)

)−sF2PTθspr(s)

)}(22)

where L−1 is the inverse Laplace transform, I is the iden-tity matrix, P = [

p1 p2], uref = [

xcom,ref θtr,ref]T,

gki and gbi are, respectively, proportional and deriva-tive control gains, d(·)/dt is the differentiation operator,∫

is the integration operator. Thus, matrices Gk andGb in Fig. 6 are diagonal and GCB is a diagonal opera-tor to represent the total processing by the cerebrocer-ebellar system of joint-specific proprioceptive signals.As assumed in PA-3, the cerebrocerebellar long-loopcontrol of COM is applied to a supporting leg only.Therefore, each leg’s cerebellar long-loop circuitry isturned on and off repeatedly by ground detection. Thisis represented by the switch in Fig. 6. No precise loca-tion of the switch is specified for present purposes. Theswitch implicates that cerebellar action may be sched-uled according to sensory input. The cerebellar compu-tation is described as PID signal processing based onneuroanatomical cerebellar corticonuclear modules asdetailed in Jo and Massaquoi (2004). Diagonal matrixI2 scales integrated signal that is projected to cerebralcortex recurrently. The circuit constructs closed-loop

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Fig. 7 Hierarchical neuralcontrol of walking

transfer matrix sI (sI + I2)−1 that provides phase lead. Ia

represents scaling related to sensorimotorcortical signalintegration. Diagonal matrix F2 expresses the relativemagnitude of the afferent signal that bypasses sensori-motorcortical integration.

Finally,

udesc = WCucb(t − Tsp) (23)

where WC represents the distribution network in eithercerebral cortical area 4 or in the spinal cord (for currentpurposes the location does not matter).

3.2.3 Summary of hierarchical neural control model

Figure 7 summarizes the hierarchical neural control ofthe SBBW model. Alpha motor neuronal output uα isthen represented by a nine component vector. From Eqs.17, 18 and 23,

uα = uSP(t − Tpr) + udesc(t − Tpr) + ureflex(t − Tpr) (24)

The cerebellum presumably plays a role in the genera-tion of appropriate patterns of limb movements,

dynamic regulation of balance, and adaptation of pos-ture and locomotion through practice (Morton and Bas-tian 2004). The last function is not treated in the currentmodel, but the first two are represented. The cerebro-cerebellar system controls the COM and trunk vertical-ity, and implicitly descends neural signals affecting theparameters in the pattern generator, which implicate apresetting or adjustment of the gain of proprioceptivereflexes and a sequence of feedforward programs. Sup-raspinal control including the cerebellum also can adjustthe tonic excitation which coordinates the retraction ofa leg. In this perspective, the cerebellar system demon-strates the first two functions.

3.3 Implementational assumptions and modelevaluation

For tractability of the initial study, and to evaluate mini-mal requirements for active, moderately stable locomo-tion, the following implementational assumptions havebeen made:

− (IA-1) Movements will be driven by monoarticularmuscle pairs.

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− (IA-2) Spinal synergy activation will be driven by astrictly sequential train of on-off pulses.

− (IA-3) Inter-leg coordination is achieved by artifi-cially enforced 180 degree phase difference betweenthe pulse generators controlling each leg, ratherthan modeling interactive circuitry.

− (IA-4) For movement initiation from a standingstart, it was assumed that a single additional pulsecould be applied to hip flexors to flex the leadinghip transiently.

The implications of these assumptions are consideredin the discussion.

The SBBW model was then evaluated first in termsof its steady state walking features both kinematic andneuromuscular. Specifically, clinical investigators havedefined the determinants of normal and pathologicalgait by observing features of human locomotion patternthat minimize displacement of the body’s COM. This hasyielded the six determinants of normal gait (Saunderset al. 1953; Della Croce et al. 2001). However, all but one— knee bending at foot impact — are not observable inthe sagittal plane and therefore are not immediately use-ful for analysis of the SBBW model. As an alternative,the model was examined to determine whether it exhib-its any of nine pathological gait features (Perry 1992).Next, the robustness of its performance was assessed bydetermining its ability transition to steady state walkingat different speeds from a standing start, and to subse-quently slow to a stop. Stability robustness was tested bysubjecting the model to forward and backward impulsivedisturbances during different phases of the gait cycle,and by simulating sudden additions of mass to the trunk.Finally, the sensitivity of the model behavior to severalsimulated neural and muscular lesions as well as con-trol parameter variations was observed. During the sim-ulations, all model parameters were held fixed unlessexplicitly stated otherwise.

4 Results

4.1 Kinematic features of walking

After an initial transient, body kinematics convergedto a consistent walking pattern that was qualitativelyvery similar across a range of speeds. Fig. 8 shows atypical human speed of about 1.2 m/s that is simulatedwith fPG = 1.3 and ηPG = 1.2. Initiation is discussedbelow and initial conditions are specified in appendix. Ittakes several steps to converge to steady state walkingmotion. The steady state motion displays a number of

Fig. 8 Steady state walking at 1.21 m/s sampled every 10 ms

the kinematic features (Saunders et al. 1953; Perry 1992)of natural human walking. The ankle joint trajectory(Fig. 9 Top) includes a valley corresponding to toe pushoff. In Fig. 9 d–f, this motion at the end of ground con-tact can also be seen as a sharp transient in coordinationplots. Small bending motions at knee joint during a gaitcycle correspond to impact at the transition from swingto stance, while large bending motions correspond tothe retraction phase. During each cycle, the hip has amonophasic oscillation in the anterior-posterior direc-tion. Each joint trajectory approximates a limit cycle(Fig. 9 Top). The ankle motion is most variable, consis-tent with proximity to ground contact. During simulatedsteady walking at 1.2 m/s with the nominal sets of param-eters, each leg spends about 59% of each cycle in stanceand the rest in swing. Double support phase accountsfor about 9% of each cycle. These values mirror those of60% and 10%, respectively, measured in humans (Perry1992). The coordination of the joints is shown pairwisein Fig. 9a–c.

It has been suggested that at least 9 features can beidentified in defective human gait (Perry 1992). Sevenof these are observable in the sagittal plane:

1. Foot slap on the ground during loading response.2. Flat foot contact at transition from swing to stance.3. Knee hyperextension from loading response to mid-

stance.4. Inadequate knee flexion during swing.5. Excessive forward lean of trunk during stance.6. Backward lean of trunk during loading response.7. Asymmetrical step length.

The third pathological feature is the complement ofthe third determinant of normal gait. Model steady statewalking shows none of the above features (Fig. 10). Thesimulated motion is also qualitatively smooth.

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Fig. 9 Steady state walkingkinematics. Top: averagedtime courses of hip (red),knee (green) and ankle (blue)joints during a gait cycle,(left): experimental dataadapted from the web: CGAnormative gait database(http://guardian.cur-tin.edu.au/cga/data/index.html),(right): simulation. Middle:joint phase-plane behaviora ankle, b knee, c hip (eachtypical cycle extracted andlabeled beneath). Bottom:joint coordination plots:d ankle vs. knee, e knee vs.hip, f ankle vs. hip

4.2 Walking initiation

While a human maintains standing posture, the erectbody tends to tilt slightly forward around ankle about2–5◦ to maintain the COM within stable supporting area(see Loram et al. 2004). Therefore, an initial posturalcondition is set to be θa = 0.05, θk = θh = 0 for bothlegs. Walking can be initiated from stance by applyingpulses of muscular activity to IP (Fig. 11b) and thenstarting the NPG. The added pulse may be interpreted

as a walking trigger from CNS. Fig. 11a shows that thereis a transient initial response. The first step is with theright leg while the left leg supports body. Steady statewalking is substantially achieved by the third gait cycle.

4.3 Dynamics and muscular activation

Dynamic realism is confirmed in Fig. 12 where both com-ponents of the reaction force waveform are biphasicwith values of appropriate magnitude (Winter 1990).

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Fig. 10 Gait analysis with respect to clinical featuers of defectivegait

The horizontal reaction force is negative just after heelcontact the ground, indicating a backward horizontalfriction force. Then, the force becomes positive, indicat-ing the forward reaction as the foot to pushes backwardagainst the ground. The vertical reaction force showsa rapid rise at heel contact in excess of body weight(80 kg) to account for vertical deceleration. As the kneeflexes during midstance, the force is below body weight.At push-off, a second peak greater than body weightis caused by plantarflexors. The second peak in simula-tion is greater than the first, which is not physiologicallytypical. This is due to force at the heel that is not fullyrealistic as will be discussed.

The patterns of muscular activation during steadystate walking are shown in Fig. 13. Examining first thephysiological data from Ivanenko et al. (2004, 2006),activations of GM and VA occur the LOA control epoch.Activity of SO occurs most strongly during the THRepoch, while that of IP occurs during RET, just beforeor during the onset of forward leg swing. BFS and TAoperate during retraction and forward swing to preventthe swinging foot from touching the ground.

The activities of GM, BFS and SO are best pre-dicted by the model. Except for minor phase shifts thewaveforms are quite similar in simulation and data. IPis also fairly similar. However in the ventral muscles,the model predicts unrealistically additional activationroughly during the REG control epoch. In experimentaldata, there is little muscular activity during this periodin any muscle. GM, BFS, SO and IP are monoarticularmuscles and appear to be responsible for the bulk of the

basic leg motion during walking and apparently accountfor much of the functional realism of the SBBW model.Biarticular muscles BFL and VA responsible for flex-ing and extending the knee (not shown), and uniartic-ular muscle TA which is responsible for lifting the footto provide ground clearance, are least well predicted.The former are redundant from a control perspectiveand therefore may be inherently less predictable. TheTA in vivo appears to stay active during the FOW con-trol epoch while in simulation this was not necessary toafford ground clearance.

Principal component analysis (PCA) was applied tosimulated muscle commands act. Four principal compo-nents (PC1 – PC4, Fig. 14) captured over 98% (52.2, 36,5.6, 5.1%, respectively) of the variance in the signals.These components resemble major factor waveformsderived from published muscle activity data duringhuman walking (Ivanenko et al. 2004; Davis and Vaughan1993; Olree and Vaughan 1995). PC1 is comparable withFACTOR 1 in Ivanenko et al. 2004, and PC2 with FAC-TOR 2, PC3 with FACTOR 3, and PC4 with a com-bination of FACTOR 1 and 5 and negative FACTOR4. Olree and Vaughan (1995) also retained four majorfactors and one of them included a combination of FAC-TOR 4 and 5. According to Olree and Vaughan (1995),it was inferred that FACTOR 1 presents propulsion, andFACTOR 2 loading or weight acceptance, and suggestedthat FACTOR 1 and 3 were in fact the same as FAC-TORS 2 and 4, phase shifted by 50% of a step cycle sothat there are only three basic factors. They referred toFACTOR 5 as the coordinating factor because it main-tained the phase shift between the left and right sides.Temporarily, FACTOR 4 corresponds to retraction ofthe leg. Presumably, this closely predicts the timing ofloading of the opposite leg and thus may be involvedin inter-limb coordination. These results indicate thatdespite the SBBW model’s somewhat unrealistic activ-ity during the REG control epoch, it captures overallEMG activity fairly well. It is then also interesting thatsome of Ivanenko’s subjects displayed EMG activity inREG (red and blue traces in FACTOR 2, Fig. 14, column(b)).

4.4 Control of walking speed

Several parameters affect walking speed. Increase in thefrequency fPG and magnitude ηPG of spinal pattern gen-erator signal causes faster walking (Figs. 7, 15). Controlduring the REG and THR epochs is responsible. Selec-tive enhancement of IP, VA and TA, via muscle activa-tion components IP-1, VA-1 and TA-1 (Fig. 4) duringREG was found to be particularly potent (not shown).

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Fig. 11 a Stick figure plotand joint trajectories ofsimulated walking initiationfrom stance. In the stick figureplot the motion is sampledevery 100 ms and separatedhorizontally for clarity. Redlines indicate joint trajectoriesof initial swing leg, and bluestance. b Neural pattern forinitiation: Trigger pulses areadded to uSP,1(t) of both legsin Fig. 4 in order to initiatewalking at only first step

These help move the COM forward during stance. Speedcan also be increased within narrower limits by augment-ing GC and SO action during the thrust epoch (compo-nent SO-1 in Fig. 4). However, this eventually has theundesirable effect of lifting the body off of the ground.The SBBW model is not equipped to tolerate this occur-rence. Presumably, however, this effect could be usefulin the control of running.

Appropriately, speed is also partially controllableusing the reference signal xcom,ref . Increasing xcom,refcauses the body to accelerate forward slightly duringthe REG epoch. For example, xcom,ref = 0.18 with fPG =0.8, and ηPG = 0.9 generates steady state walking speedof 0.65 m/s, while xcom,ref = 0.25 with the same fPGand ηPG yields a speed of 0.72 m/s. The acceptabilityof increasing xcom,ref alone is limited, however. UnlessfPG and ηPG are changed concordantly, the model even-tually falls forward.

4.5 Stability to push disturbances and mass increase

The application of 200 ms duration forward and back-ward force impulses to the center of mass of the trunk-head segment were simulated at 0, 25, 50, and 75% of thegait cycle. At each gait cycle fraction, the maximal force

levels from which the model could recover steady statewalking were determined. Figure 16a,b shows the effectsof a 70 N maximal forward impulse applied at 0% phase(toe-off) and a 75 N maximal backward push applied at50% phase (mid-stance). Figure 16a demonstrates for-ward pushes cause more distance, and backward pushesless distance to be traveled within a given time. Figure16b indicates that such a push disturbs primarily anklemotion (of the stance leg) transiently. The model toler-ates forward disturbances slightly better than backwarddisturbances except at mid-stance when resistance tobackward pushes is slightly better (Fig. 16c). After aninitial deviation, the COM motion pattern is recoveredwithin 3 or 4 gait cycles modulo a fixed phase lag orgain (Fig. 16d) that demonstrates again the absence ofabsolute position control shown in Fig. 16a.

The model was also tested by changing the mass ofthe trunk-head segment, and, therefore, also its momentof inertia, without alteration of feedforward neural com-mands. Thus, this change tested the inherent viscoelasticand neural feedback mechanisms. It was found that upto a 10-kg increase (18.5% of the trunk mass) could betolerated without falling. Fig. 17b shows the effect inthe phase plane. It is evident that the walking speed islower with the increase of mass. Phase plots indicate thatdifferent limit cycles result at ankle and hip.

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Fig. 12 Ground reactionforces. Left: simulated forceprofiles divided by totalbody’s weight. Right: typicalforce profiles adapted fromNeptune et al. (2004). Top:horizontal reaction forceprofile, and Bottom: verticalreaction force profile:components of the reactionforce are shown: blackindicates the total force, blueforce at the heel, and redforce at the toe

4.6 Sensitivity analysis and simulated system lesions

Sensitivity analysis of the model parameters associatedwith the cerebrocerebellar system helps clarify theirroles in walking simulation. Several gait cycles weresimulated with either a 30% increase and decrease inthe value of different parameters relative to their val-ues used for nominal walking simulation (Fig. 8). Stablewalking is characterized by a reproducible limit cycleas in Fig. 9. In Fig. 18, it is observed that stability islost when there is (a) an increase of cerebellar gains( gk1 or gk2), or of the gain of direct long-loop path-way to cerebellum (F2), (b) a decrease of the recur-rent integrator gain (I2) or of gk2. The model losesstability gradually with a decrease of gk2. Otherwise,walking is stable. Walking is least sensitive to varia-tion in I2 indicating that the less direct long feedbackloop is less important for walking that it is for bal-ance (see Jo and Massaquoi 2004). gk1, the propor-tional feedback control gain of the COM, is not asimmediately critical for stability in walking as is gk2, theproportional feedback control gain of the trunk pitch.Increased gain values at the moment have no obviousclinical counterpart, but indicate that there are prac-tical limits to the feedback loop strength. Decreasedgki would correspond to loss of either deep cerebellarnuclei or mossy/parallel fibers that is generally associ-ated with clinical ataxia. Reduced I2 is consistent withinterruption of cerebellar outflow signals as occurs com-monly in multiple sclerosis where it produces aviolent tremor (Massaquoi and Hallett 2002). Inappro-priate selection of F2 would correspond to malfunc-tion in either spinal or supraspinal sensory nerve.Sensitivity to the parameters proposes a rudimentaryidea about behaviors of neural lesions. Simulations withinappropriate parameter values presumably consistent

with neurophysiological malfunctions are demonstratedin Fig. 19. These do not analyze lesions rigorously, butpropose simple insights into the SBBW model withrespect to biological plausibility.

Figure 19 shows the effects of removing trans-cerebel-lar long-loop control of COM and trunk pitch, weaken-ing the recurrent integrator and eliminating theperipheral modulation of synergies. In Figs. 19a–c, thebody fails to maintain a sufficient step size causing for-ward tripping. Figure 19d demonstrates that swing legretraction is not fully achieved due to inappropriatelarge knee excursion so that toe eventually scratchesthe ground. That causes the forward trip again. In themodel, the simulated lesions in the trans-cerebellar long-loop control give rise to forward tripping due to a fail-ure of balance control of supporting leg while the lesionof the presynaptic inhibition disturbs swing leg excur-sion. To test pathological walking behaviors in evenfurther detail, cerebellar long-loop control of the swingleg should be modeled along with a possible role of thecerebellum in scaling feedforward NPG signals.

5 Discussion

The SBBW model attempts to account for the primarykinematic, dynamic and physiological features of humanlocomotor control with a formulation that is simple instructure and control principles. This approach is moti-vated by the assumption that nature may often pre-fer simple, robust solutions to motor control problems.The work continues along the lines of Taga (1995) andOgihara and Yamazaki (2001), and provides more detailregarding possible neural control mechanisms and moreextensive evaluation of the stability and performancecharacteristics that result. Specifically, it is demonstrated

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that when balance is stabilized by two specific long-loop stretch responses, stable walking with many real-istic features can be afforded by a five sequential-statespinal pulse generator that distributes pulse-like acti-vation to muscles organized in four control synergies.Model walking demonstrates natural convergence toa consistent steady-state gait pattern over a fair rangeof speeds and simultaneously displays significant resis-tance to pushes and weight changes. Walking speed canbe increased by a monotonic increase in central pulsegenerator intensity and frequency without other controlmodifications. The simulated muscle activation patternsshare important similarities to experimental observa-tions, but also show differences that remain to be rec-onciled. Successful walking is demonstrated to dependon the intactness of most components of the controlsystem suggesting that the model is of fairly minimalcomplexity. Importantly, there is no apparent require-ment for internal models of body dynamics, detailedprogramming of joint motions or explicit computation oftorque demand at any joint. Also, it appears that musclesynergies, together with composite signal feedback, canenable multi-joint feedback control to be implementedby simple Single-Input Single-Output (SISO) modules.The proposed control mechanism is argued to be broadlyconsistent with cerebrocerebellar and spinal/brainstemsystems.

5.1 Kinematic qualities of simulated gait

The SBBW model accounts for a number of kinematicfeatures of natural walking without requiring detailedprogramming of joint time-courses. The feedforwardcontrol signals consist of a steady intended forward dis-placement of the center of mass, a vertical reference forthe trunk angle and a series of non-overlapping rect-angular spinal neural pattern generator signals to causestepping. Apparently, the long-loop and segmental neu-ral feedback, muscular synergies, muscle viscoelasticityand activation dynamics are sufficient to add the coor-dination and modulation necessary to produce smooth,stable, natural appearing movements. The convergenceto a consistent steady state cadence is not surprisinggiven the regularity of the NPG. However, this was not anecessary outcome. The system dynamics include insta-bility and many significant nonlinearities and groundcontact is intermittent. Therefore, the gait could becomeunsteady, as is the case when there is reduced control atjoints (Fig. 19).

Body kinematics are well summarized by joint angle,joint coordination, phase-plane and single/double sup-port fraction data. Joint angle data shows slightly lessknee flexion and ankle plantarflexion during the RET

Fig. 13 Simulated (upper trace) and observed EMG patterns(bottom filled gray) during a gait cycle arranged anatomically. Datais adapted from Ivanenko et al. (2006)

and FOW control epochs than seen in experimentaldata. This may relate to the model’s failture to repro-duce hip drop in the swing leg, as discussed below.However, the kinematic analysis indicates that bodymotion is generally quite realistic. Consistent with this,it was found (but not shown here) that the kinematicslie naturally close to a plane in three dimensional joint-space. This feature has been identified in human subjects(Lacquaniti et al. 1999) in terms of elevation angles, theangles between body segments and a vertical reference.The transformation between joint angles as used in thisstudy, and elevation angles is affine. Therefore planarityin either system implies planarity in the other. Func-tionally, planarity corresponds to the fact that althoughthe four segments defining body and leg configurationhave in principle four degrees of freedom, during walk-ing they exhibit two. The trunk is constrained to bealways nearly vertical and the legs function primarilyas swinging pistons that exhibit rotation of the leg at thehip and flexion and extension of the leg. For the mostpart, the upper and lower leg and foot move in unison.That the trajectory is substantially a single closed loopindicates that the two underlying degrees of freedom

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Fig. 14 a Principal components of simulated EMG patterns ina gait cycle. b and c Factors summarized from human EMGdata: b Several individual subjects from Ivanenko et al. (2004).

c Comparison of principal factors from other studies (Winter 1991;Davis and Vaughan 1993; Olree and Vaughan 1995) adapted fromIvanenko et al. (2004)

are coordinated in a consistent manner at the same fre-quency. The principal partial exception to this analysis isthat the ankle motion during ground contact is slightlymore complicated. During early stance phase, θ1 (or θ2)increases somewhat while θ3 (or θ4) declines slightly asthe stance leg straightened. This is opposite to the over-all coordination pattern between knee and ankle andresults in a transient deviation from the plane. This canbe seen in human data as well (Lacquaniti et al. 1999).Planarity is recovered quickly with the rapid ankle plan-tar flexion flick at the end of ground contact. On average,the behavior remains that of a swinging piston.

As the SBBW model addresses only sagittal planeperformance and approximates the action of musclegroups without including, for example, variations inmoment arm with joint angle, full Hill-type muscledynamics (Winters and Stark 1985; Zajac 1989) and theaction of toes. Therefore, no attempt was made to matchthe kinematics of a particular human subject precisely.Nonetheless, the most characteristic aspects of natu-ral walking appear to have been captured. This is alsotrue in terms of clinically-oriented walking assessmentscales.

Fig. 15 Steady state walking speeds afforded by changing fre-quency fPG and magnitude ηPG of spinal pulse generator

5.2 Feedforward command signals

The major features of gait kinematics were producedby the empirically-developed feedforward action ofa spinal pulse generator and four muscle control

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Fig. 16 (Simulations of disturbed and normal walking. (a top,black) undisturbed; (a middle, blue) walking pushed forward by70 N at 0% phase, (a: bottom, red) walking pushed backward by75 N at 50% phase; the black stick figures under arrows indicatethe timing of the impulse applications in duration of 20 ms. b Cor-responding phase plane plots of ankle, knee, and hip (left legonly) to compare disturbed and normal walking. c Maximal forces

tolerated in each phase. d Phase plane plot for COM with dis-turbed walking: Response to 70 N forward impulse at 0% phase(blue), 75 N backward impulse at 50% phase (red). e Trajectory ofhorizontal COM (COMx) deviation between disturbed and nor-mal walking, (left) walking pushed forward by 70 N pulse at 0%phase — normal walking, and (right) walking pushed backwardby 75 N at 50% phase — normal walking

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Fig. 17 a COM patterns(forward COM position vs itsvelocity), b phase plots ofankle, knee, and hip from leftto right (left leg only);nominal walking pattern(black), increased trunk mass(red)

Fig. 18 Sensitivity to neuralparameters. Simulatedmotions with each parameterincreased by 30% anddecreased by 30% are drawnand compared with nominalwalking motion with nochange of parameters

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Fig. 19 Stick plots (red) of a no trans-cerebellar long-loop con-trol of COM, b weakened trans-cerebellar long-loop control oftrunk pitch, and c weakened recurrent integrator; d eliminatedsegmental reflex in comparison with normal walking (black). Each

motion is sampled every 100 ms, but is horizontally relocated forclarity. Thick blue line indicates the ankle trajectory, the dottedgreen the knee, and the thin red the hip

synergies that it drives. A steady (tonic) reference com-mand that specified a forward displaced COM enhancedwalking speed as discussed below, but actually was notrequired for walking as long as COM and trunk pitchwere under feedback control. Other central neural pat-tern generator schemes have used separate pattern gen-erators assigned to each joint to define independent andcontinuously varying intended trajectories. The SBBWmodel uses a simpler approach. Differently scaled butsimilarly shaped pulses delivered during five sequentialcontrol epochs: Loading, Regulation, Thrust, Retrac-tion, and Forward swing, were found to be the minimalarrangement that could account for a full range of walk-ing speeds by scaling epoch durations proportionately.This resulted in the constancy of EMG pattern acrossspeeds that has been noted experimentally (Ivanenkoet al. 2004). This is also consistent with the suggestionthat the CNS may generate only a few basic patternsof muscular activity for locomotion (Patla et al. 1985).The pulse generator command was selected to consistof rectangular pulses for simplicity. In fact, because ofthe significant low-pass filtering characteristics of mus-cle activation and impedance, net locomotor behavior isnot especially sensitive to high-frequency details of mus-cle command signal morphology. Any type of pulse-likewaveform (see also Cajigas-González 2003) can have

substantially similar effect. Thus, the spinal level is con-sidered to implement equilibrium trajectory type con-trol (Fledman 1986; Bizzi et al. 1992), the performanceof which is enhanced by more realistic activation level-dependent muscular viscoelasticity.

Initiation of movement required a small additionaltrigger pulse applied to hip flexors during the first step.This component essentially provides the initial swing legelevation and forward momentum that would ordinarilybe contributed by substantially passive dynamics in themiddle of the gait cycle. That participation of a sepa-rate system, such as the basal ganglia, may be requiredto begin walking is a conjecture. However, it is consis-tent with the particular deficit in Parkinson’s Diseasewherein the initiation of walking may be disturbed pref-erentially. However, more analysis will be required tocharacterize such a system.

5.3 Dynamics and muscle activation

During double support, a biped is a closed kinematiclinkage (Pandy and Berme 1988). As such, there existsno unique solution for the torques at each joint. More-over, in general, correct kinematics implies only correctnet torques. Given the agonist-antagonist organization

Biol Cybern

of the muscles and the redundancy of actuation, theforces applied by individual muscle groups are not deter-mined by the body motion. Therefore, it was importantto verify that model forces and muscle activations arerealistic. This can be done explicitly for ground con-tact for which there is considerable experimental data.The biphasic force profile with larger forces at heel-strike and toe-off is consistent with significant controlat the ankle rather than the hip. This is quite realis-tic. The somewhat greater toe-off pulse in the simula-tion appears to result from the inability to reproducein simulation the full stiffness of a typical floor. As aresult, there is a mild bounce forward from rear foot toforefoot.

For approximation of joint torques, where less exper-imentally measured force data is available, the EMGpattern can serve as a crude surrogate if it is assumed,as is here, that the muscle model is reasonably accurateunder the conditions studied. Overall, the control systemappeared to generate muscle activations that were simi-lar to those observed in vivo. In particular, the finding offour principal independent waveforms in muscle activa-tion corresponds closely with the four or five factors thathave been found in human data. These appear to corre-spond approximately to the five principal control epochsthat have been proposed here. During each epoch, themuscle activation pattern tends to be quite differentfrom the others which yields their independence, andnear orthogonality. The epoch-specificity of the EMG isrelated to the significant differences in locomotor taskduring each period. Accordingly, analysis of phase-spe-cific locomotor requirements may shed some light onthe two features of predicted EMG that were poten-tially least realistic. These were: (a) the inappropriatelypredicted more widespread inappropriately predictedmuscular activity during the REG control epoch and (b)the model’s failure to predict TA activation throughoutswing.

As pointed out above, EMG records from humansoccasionally show activity during the REG controlepoch. However, others clearly do not. Two effects maybe important. Activation of VA and RF during REG wasfound to be important to keep the knee from bucklingduring stance. In principle, however, this is not a prob-lem if the knee joint is substantially vertical or evenlocked in extension. This stabilizing requirement is verysensitive to small changes in knee angle. It is thereforequite possible that a model with more realistic muscleviscoelastic properties, and/or enhanced segmental orlong-loop reflex action during LOA could obviate orreduce muscular action during REG.

In addition, during REG and THR that the ankle hasits greatest efficacy in body control, especially during

slow walking. This is because it is during late REG andearly THR that the COM lies above the base of sup-port. During LOA the landing leg is extended forward.Because of the geometry of foot placement, musculardorsiflexion at the ankle during LOA acts to stiffen heelcontact and can thereby contribute to COM decelera-tion. But active plantar flexion can contribute little toforward COM acceleration. Conversely, late in THR,the COM is forward, beyond the foot. Active plantarflexion can contribute significantly to accelerating theCOM forward (and up). But dorsiflexion has little effecton COM deceleration. During late REG and early THR(midstance), however, the ankle can accelerate or decel-erate the body. In particular, to accelerate the body, theventral muscles, IP and VA, are activated as well asTA, as shown in Fig. 4. Importantly, throughout stancephase, the thigh adductors, especially the adductor mag-nus (well developed in sprinters), can contribute to hipextension and thus to COM acceleration. These havethe advantage over not lifting the body. These musclesare often not recorded experimentally, and were notincluded in the current simulations.

The failure of the SBBW model to anticipate pro-longed activation of TA may relate to the fact that themodel only describes motions within the midline sagittalplane. Natural walking includes a small amount (about6 degrees) of shifting pelvic tilt in the frontal plane thatresults in about 5 cm of drop of the swing leg hip (Inmanet al. 1981). While subtle, this potentially interferes withfoot ground clearance. Extended activation of the TA invivo may be required to compensate for this tilt. Thus,the absence of direct swing leg control during the FOWepoch may account for the finding of four rather thanfive principal components in the muscle activation sig-nals. Arguably, the ‘missing synergy’ includes activity ofBFS, BFL and TA during swing that helps to maintainthe leg in retraction. This would presumably correspondto the fifth FACTOR identified in experimental EMGdata. In any case, it appears that EMG control duringearly to mid-stance and early to mid swing in the oppo-site leg is likely to be important, and sensitive to manyfactors. This suggests that more data will be requiredto determine the variety of possible muscle activationpatterns that can be effective.

5.4 Stability

Steady state walking with limit-cycle characteristics wasshown to follow from a standing start using only a stepchange in forward commands and a launching pulseapplied to IP. This is not necessarily a realistic commandprocess but was employed to underscore the model’sinherent stability and to emphasize potential simplicity

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of forward control. Steady walking resisted modestimpulsive disturbances and mass increases withoutchanges in parameters or changes in feedforward con-trol signals. These alterations stressed the limit cycle-likebehavior described above and suggest that the stabiliz-ing characteristics of the mechanics and control schemeare fundamentally sound. The SBBW model’s feedbackcontrol system is particularly simple and correspondswell with the two channel control of posture positedon the basis of human studies (Freitas et al. 2006). Tosome extent, this is quite expected based on first princi-ples. Minimal necessary conditions for upright walkingare (a) maintenance of the COM near the base of sup-port, and (b) maintenance of trunk verticality. Duringsingle leg support, this means that the COM should beroughly over the foot, and during double support phase,the COM should be roughly within the convex hull ofthe two feet. These conditions are evidently fairly easyto achieve using long-loop feedback, especially whencontrol is applied grossly to whole limbs. This is facili-tated by the use of multiarticular synergies. As discussedfurther below, this approach has potentially importantimplications for cerebellar functional architecture andto our knowledge has not been used previously in sim-ulations of natural locomotion or in robot locomotioncontrol.

Limited exploration of the sensitivity of walking tosimulated system lesions indicates that it is likely thatall system components must be reasonably intact forsuccessful walking. This suggests that the model has agenerally parsimonious structure. In principle, there area limited number of failure modes. The COM can fallforward, backward or directly downward. For the latterto occur, the knees must buckle. Given adequate acti-vation of the knee extensors VA and RF, or extensionof the knee by other means during the REG epoch,this is unlikely. If the body falls forward or backward,either the legs spread in a “split”, or the swing legencounters the ground causing a trip. In all simulatedsystem lesions, the body tripped forward. This is mostlikely when there is appreciable forward momentum andtypical in cerebellar disease and severe sensory neurop-athy when vision is removed. A propensity to trip is seenin toddlers and indicates that strong management of toeclearance is critical to improved locomotor control. Thispresumably would be the primary responsibility of acti-vations of BFS, BFL and TA during swing.

5.5 Limitations in performance and stability

The model displays the ability to begin walking froma standstill, and some capacity to adjust its walkingspeed in response to simple changes in the feedforward

commands. However, control of deceleration and walk-ing speed was limited. In particular, slowing and stop-ping to a standing balance was not achieved by theSBBW model. Slow walking requires prolonged timein single leg stance phase. This places particular demandon the balance system. While it is expected that a morecomplete balance mechanism such as that possessed bythe FRIPID model (Jo and Massaquoi 2004) would bestable under these conditions, the more rudimentarySBBW model has difficulty. This finding seems grosslyconsistent with the difficulties that toddlers often havein stopping without falling down. Quickly slowing to astand apparently requires more precise control to cancelforward linear and angular momentum simultaneouslyusing the legs, and/or strong standing balance mecha-nisms to dissipate residual energy without moving thefeet. Thus, it appears that some type of smooth ‘clutch’ isneeded to properly coordinate the walking and standingbalance systems during slow walking and stopping. Assuggested above, another important factor is likely tobe control of the swing leg. This could include long-loopfeedback as well as feedforward spinal input. Moreover,the current model balances upright only with enhancedankle stiffness. While this level of stiffness is within phys-iological range, more complete feedback as used in theFRIPID balance model has the potential to afford supe-rior balance control with more modest ankle stiffness.

5.6 Implications for neural architecture

It has been shown experimentally that electrical stim-ulation of the posterior structures of the lumbar spinalcord can induce patterned, locomotor-like activity(Dimitrijevic et al. 1998). Importantly, focal stimulationelicits synchronous rhythmic EMG activities in musclesat different joints in the lower limb. This suggests thatneural oscillators are not joint specific and is highlyconsistent with the synergies used here. A synergy isa fixed relationship of activities within a set of mus-cles that may be modulated and/or time-shifted as awhole, but is otherwise kept intact. Synergies have gen-erally been defined in terms of EMG, both time-depen-dent and time-independent formulations (d’Avella andBizzi 2005; Cheung et al. 2005), and in terms of function(Cajigas-González 2003). These views are compatiblewith different emphases. The latter could be shown toaccount for the swinging piston-like behavior of a frog’sleg during walking, swimming and wiping behaviors.In fact, a significant range of frog leg behaviors couldbe accounted for by the four kinematic control syner-gies. Similarly with the kinematic control synergies, thedriving temporal waveforms in the SBBW model aretaken to be simple pulses that have scalable intensity

Biol Cybern

and duration. While a rich variety of time-varying acti-vations can be afforded, the control is ultimately con-strained significantly by these assumptions. In any case,the SBBW model supports the possibility of time-invari-ant control synergies mediated by effectively fixed, lin-ear connections between spinal pulse generator circuitryand motor neuron pools, thereby decoupling and sim-plifying the temporal and distributional components oflimb control.

Recent work in the frog indicates that as previouslysuspected, peripheral sensory feedback may modulatethe expression of motor synergies (Cheung et al. 2005).However, in general, this is found to be relatively minoreffect in that basic leg behaviors are retained followingperipheral deafferentation. On the other hand, uprightbipedal locomotion places particularly strong demandson the control of leg trajectory. The SBBW model findsit extremely useful to have some modulation of espe-cially the retraction control phase according to sensedleg position. Presynaptic inhibition in spinal cord asassumed by the model has been identified (Rudominand Schmidt 1999; Baxendale and Ferrell 1981; Duysenset al. 2000; Rossignol et al. 2006). Absence of this mech-anism results in simulated walking with excessive legretraction reminiscent of the high-stepping gait of tabesdorsalis, a syphilitic condition involving compromisedsensory input to the spinal cord (Ropper and Brown2005). This suggests the basic plausibility of the mecha-nism. However, more extensive analysis will be requiredfor validation.

While many studies show the existence of musclesynergy organization at the spinal cord level, and spi-nalized cats display roughly normal locomotor patternswhen suspended and placed on treadmills (Lam andPearson 2001; Hiebert and Pearson 1999; Kandel et al.2000), it is also clear that supraspinal control is extremelyimportant to fully normal locomotor function (Mortonand Bastian 2004; Dietz 1992; Shik and Orlovsky 1976;Brooke et al. 1997; Nielsen 2003). Therefore, it wasimportant to model supraspinal mechanisms. In partic-ular, it was of interest to determine whether feedbackcontrol of walking could be implemented stably by trans-cerebellar long-loop mechanisms using low-order lineardynamics, and that control could be tuned in terms ofsimple linear scalings and thresholds. As argued else-where (Jo and Massaquoi 2004; Massaquoi 1999) thecerebellum is very likely to provide important, but sim-ple scaling and dynamic processing for many centerswithin the central nervous system. While not explicitlyattributed to cerebellar circuits, the scheduled scalingof especially the amplitude of NPG output with move-ment speed (Fig. 15) could also be implemented or atleast refined by cerebellar circuits. More detailed study

of control derangements in cerebellar disease will beneeded to determine the validity of this conjecture.

In contrast to the FRIPID cerebrocerebellar balancecontrol model, the SBBW model indicates that the cer-ebellar contribution to body control may be representedby a diagonal matrix with elements of form (gk +gbd(·)/dt). This is potentially an important simplificationin comparison with the full 3×3 matrices of the FRIPIDmodel (Jo and Massaquoi 2004) that represent explicitcoordination of multiple joints. In the SBBW modek,cerebellar modules process signals independently whilethe coordination of different muscles is managed byextracerebellar synergy circuits represented by thecolumns of WC. While the two views are not mutu-ally exclusive, the latter is particularly compatible withrecent anatomical studies (Kelly and Strick 2003) thatemphasize very narrow point-to-point modular process-ing by cerebellum, rather than switchbroad-like fan-in and fan-out that was usually assumed. Moreover,the only gainscheduling (see Jo and Massaquoi 2004)needed by the SBBW model was that based on detectionof foot-ground contact. COM and trunk pitch feedbackgains were applied to signals obtained from the stanceleg only.

5.7 Implications for robotic locomotor control

The SBBW model demonstrates that explicit compu-tation of body dynamics is not required for control ofnatural walking that is robust to modest disturbances.Three features appear to be primarily responsible. First,its feedback systems provide basic upright stability viaindependent control of COM position and trunk pitch.Second, its feedforward pattern generator activates par-ticular synergies each of which performs a separatedynamic function that is required within every gait cycle.Thus, in the central feedforward and feedback circuits,there is substantial decoupling of the various controlcircuits, while control of muscles become coupled. Bothdesign features decrease complexity, but at the expenseof reduced behavioral flexibility. The architecture allowsconsiderable independence in neural tuning, but only ofspecific degrees of freedom that are relevant for bipedallocomotion. Control of the body into arbitrary con-figurations is not possible using this system. Presum-ably, body control for other behaviors must be providedby other neural components. Finally, the viscoelastic-ity of the muscles smoothes the effects of commandinputs and enables stable contact with the ground with-out explicit computation of joint torques or foot forces(Hogan 1985). Many of these features are relevant forthe control of artificial humanoid robots if driven byseries elastic actuators (Blaya and Herr 2004) that

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behave much more as muscles than do typical torquemotors.

The control and mechanical properties of the SBBWmodel lie between those of different artificial bipedalwalkers. Passive walkers (McGeer 1993) may exhibitvery natural appearing gait patterns, but characteristi-cally only for a narrow speed range that is dictated bythe walker’s geometry and mass distribution, and by theslope of the decline. Gently actuated, near passive walk-ers can traverse horizontal ground by injecting energyvia foot plantarflexion (Collins and Ruina 2001) andrearward rotation at the hip (van der Linde 1999). How-ever, because of the lack of active stabilizing feedback,these walkers also exhibit a fairly narrow speed rangeand also do not resist external disturbances well. Atthe other end of the spectrum, fully actuated human-oid robots such as ASIMO developed by Honda motorcorporation (Hirai et al. 1998) demonstrate highly flex-ible position control of all body segments. However,in general, they are comparatively heavy, energy ineffi-cient and have control that relies on much more com-plete specification of joint angle trajectory together withjoint torque computation based on the relative relationbetween zero moment point (ZMP) (see Vukobratovicet al. 1990) and COM.

Although not investigated extensively here, the ener-getics of the SBBW model are likely to be favorable.First, all synergies are organized to minimize muscularcoactivation. Each synergy activates only muscles thatdo not directly compete with each other at any joint, andeach central command source drives only one synergyat a time. The current implementation uses substan-tially passive leg swing that likely contributes to ener-getic efficiency. Preliminary analysis shows that Cmt, anindex used to measure mechanical energetic efficiency,is defined (Collins and Ruina 2005) as

Cmt = mechanical energy used(weight)(distance traveled)

(25)

For the SBBW model, Cmt is about 0.09 without anyattempt to optimize this value. Passive and near pas-sive walkers have generally shown Cmt less than 0.1, andhumans Cmt of about 0.05, while ASIMO has a Cmt of1.6 (Collins and Ruina 2005). This supports the impres-sion that SBBW model already provides a fairly realis-tic description of human locomotion and suggests thathuman-type actuation and control may afford consider-able locomotor performance and stability without incur-ring great loss of energetic efficiency. However, furtherinvestigation including estimation of metabolic costs ofmuscle activation will be required for a full energeticcharacterization.

Acknowledgements The authors thank Profs. Rodney Brooksand Hugh Herr for valuable comments and an anonymous reviewerfor useful suggestions. This work is partially supported by NSFKDI grant IBN-9873478 and by Fernando J Corbato fellowshipand Harold E Edgerton fellowship from MIT EECS department.

Appendix A

– Neural transmission delaysClosed-loop transmission delays are conservativelytaken to be 60, 70, and 80 ms for long-loop responseto and from the hip, knee, ankle, respectively basedon 50 m/s neural conduction velocity, and five synap-tic delays of less than 1ms.

– Tactile receptor on the foot.The interaction between foot and ground is detectedby tactile receptors on the foot. The signal from thereceptor is expressed by Rt.

Rt = 1[Ft

gy + Fhgy − δF

]+

where δF is an offset (a small amount of force). Forsimplicity, detection is based on the total reactionforce on the foot, which is a sum of reaction forceson the toe and heel. Rt is 1 when foot receive reactionforce and 0 otherwise. Therefore, the tactile recep-tors inform of whether each leg is at either swing orstance phase.

– Estimate of COM

xcom ≈ m1

(m1 + m2 + m3)r1 sin θa(t − Tspr,a)

+ m2

(m1 + m2 + m3)(l1 sin θa(t − Tspr,a)

+r2 sin(θa(t − Tspr,a)

+θk(t − Tspr,k))) + m3

(m1 + m2 + m3)

×(l1 sin θa(t − Tspr,a)

+l2 sin(θa(t − Tspr,a) + θk(t − Tspr,k))

+r3 sin(θa(t − Tspr,a) + θk(t − Tspr,k)

+θh(t − Tspr,h)))

≈ (m1r1 + m2(l1 + r2) + m3(l1 + l2 + r3))

(m1 + m2 + m3)θa

×(t − Tspr,a)

+ (m2r2 + m3l2)(m1 + m2 + m3)

θk(t − Tspr,k)

+ m3r3

(m1 + m2 + m3)θh(t − Tspr,h)

Biol Cybern

where θa, θk, and θh are ankle, knee, and hip jointangles of supporting leg, mi is the mass of each seg-ment, li is the length of each segment, and ri is thelength from the distal end to the COM of each seg-ment (i=1: lower leg, i=2: upper leg, i=3: trunk).

Appendix B

Parameter values for simulation

– Muscle parameterα, β, and γ are set to be 0.11, 0.4, and 0.6 respectively.

A =⎡⎣ 0 0 0 0 0.023 −0.036 0 0 −0.040

0 0 −0.040 0.049 0 0 −0.025 0.049 0.0500.132 −0.092 0 0 0 0 0.049 −0.054 0

⎤⎦

T

– Foot interaction to the groundKgy = 30000, Bgy = 500, Kgx = 10000, Bgx = 1000;µk = 0.6, µs = 1.2.ygy (x) = 0, which describes the flat ground.

– Spinal pattern generator

Table 3 Parameters for periodic pattern generation

uPG,1 uPG,2 uPG,3 uPG,4

φi 0.38 1.2 0.705 0.5275hi cos(0.16π) cos(0.2π) cos(0.23π) cos(0.125π)

WPG =

⎡⎢⎢⎣

0.3 0 0 0.8 0.76 0 0 0 00 0.38 0 0 0 0 0 0 0

0.4 0 0.64 0 0 0 0 0 00 0 0 0.8 0.9 0.4 0 0 0

⎤⎥⎥⎦

T

– Tactile receptor on the foot δF = 20.– Cerebrocerebellar system

Ia =[

0.2 00 1

]; F2 =

[0.6 00 0.3

];

I2 =[

100 00 100

]; I1 = 0; gb1 = 0, gk1 = 3;

gb2 = 0, gk2 = 30.

WC =[

0 0 2 −5 6 −1 3 −1 −34 −2.8 0 0 0 0 1.5 −1.6 0

]T

– Estimates of xcom, and θtr

p11 = 0.97, p21 = 0.53, p31 = 0.14; p12 = 1,p22 = −1, p32 = 1.

– Spinal segmental inhibition

θth,a = 0.35; θth,k = −0.35; θth,h = 0.55;

WSR = ρ

⎡⎣ 0 0 0 0 1 0 0 0 0

0 0 1 0 0 0 0 0 01 0 0 0 0 0 0 0 0

⎤⎦

T

where ρ is a sufficient large number (ρ > ηPG).– Initial conditions

positionsθ1 = 0.2, θ3 = 0, θ5 = −0.2 for right leg;θ2 = 0, θ4 = −0.1, θ6 = 0.4 for left leg.

velocitiesθ1 = (fPG+1)/2, θ3 = −(fPG+1)/2 , θ5 = (fPG+1)/2for right leg;θ2 = −(fPG + 1)/2, θ4 = (fPG + 1)/2, θ6 = −(fPG +1)/2 for left leg.

– Reference signalsxcom,ref = 0.25; θtr,def = 0.

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