Biol Cybern (2008) 99:29–41 DOI 10.1007/s00422-008-0240-2
ORIGINAL PAPER
The inverted pendulum model of bipedal standing cannot be stabilized through direct feedback of force and contractile element length and velocity at realistic series elastic element stiffness A. J. “Knoek” van Soest · Leonard A. Rozendaal
Received: 17 September 2007 / Accepted: 9 April 2008 / Published online: 27 June 2008 © Springer-Verlag 2008
Abstract Control of bipedal standing is typically analyzed in the context of a single-segment inverted pendulum model. The stiffness K SE of the series elastic element that transmits the force generated by the contractile elements of the ankle plantarflexors to the skeletal system has been reported to be smaller in magnitude than the destabilizing gravitational stiffness K g . In this study, we assess, in case K SE +K g < 0, if bipedal standing can be locally stable under direct feedback of contractile element length, contractile element velocity (both sensed by muscle spindles) and muscle force (sensed by Golgi tendon organs) to alpha-motoneuron activity. A theoretical analysis reveals that even though positive feedback of force may increase the stiffness of the muscle–tendon complex to values well over the destabilizing gravitational stiffness, dynamic instability makes it impossible to obtain locally stable standing under the conditions assumed. Keywords Human bipedal standing · Motor control · Feedback control · Muscle spindle · Golgi tendon organ
1 Introduction Standing symmetrically on two feet without falling over is not a trivial task, as the force of gravity tends to topple the body. When standing, the body behaves like an inverted multipendulum that is destabilized by the gravitational force. In order not to fall over, the forces exerted by muscle-tendon complexes must vary so as to counteract the destabilizing effect of gravity. Taking a hierarchical view of the motor A. J. “Knoek” van Soest (B) · L. A. Rozendaal Faculty of Human Movement Sciences and Research Institute MOVE, VU University, Van der Boechorststraat 9, 1081 BT Amsterdam, The Netherlands e-mail:
[email protected]
system (Bernstein 1967), the lowest-level mechanism is formed by the intrinsic properties of muscle–tendon complexes (MTC’s), endowing these with intrinsic stiffness and viscosity (Hogan et al. 1987; van Soest and Bobbert 1993). At an intermediate level, direct feedback of proprioceptive information, possibly including simple dynamic processing of the sensory signals, modulates alpha-motoneuron activity. In particular, muscle spindle afference, signaling length and velocity of the contractile element (CE), and afference from Golgi tendon organs (GTOs), signaling muscle force, are considered important (e.g. Mauritz and Dietz 1980; Dietz et al. 1992; Pratt 1995). Direct negative feedback of CE length and velocity contributes to the effective stiffness and damping of the CE. In the simplest autogenic reflexive case, the neural loop involved may be the mono-synaptical myotatic loop; in the case of heterogenic feedback, where muscle spindles from one muscle affect the alpha motoneuron activity of other muscles, a very limited number of spinal interneurons are involved. Regarding direct feedback of muscle force, it has long been known theoretically that negative force feedback reduces MTC stiffness (e.g. Nichols and Houk 1973), or, in other words, that feedback of muscle force must be positive in order to contribute to MTC stiffness. Positive GTO feedback has indeed been observed in some postural tasks (Dietz et al. 1992; Pratt 1995), and has been shown to contribute to the placing reaction (Prochazka et al. 1997a,b). It is generally assumed that this type of feedback can be implemented at the spinal level, involving a very limited number of interneurons. Finally, at the highest level in the hierarchy of the motor system, sensory integration and, based on that, model-based control are generally acknowledged to take place (Horak and Macpherson 1996; Kuo 1995; van der Kooij et al. 1999). While high-level control is generally considered to be predominant in situations where changes in control strategy are required (e.g. during platform perturbations), the necessity
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mg
ϕ
ϕ
ϕCE K C Tc E
l BC
TMTC a
Fankle
E
KSE
ϕSE
b
Fig. 1 a Schematic representation of the part of the musculoskeletal system involved in bipedal standing, and superimposed in grey a schematic representation of the model analyzed in this study. This model consists of a single rigid segment connecting the ankle joint axis with the body center of mass, subject to gravity, a muscle torque TMTC (negative as drawn), and an ankle joint reaction force. b Schematic representation of the linearized model of the muscle actuator, an SE spring in series with a CE consisting of a spring in parallel to a damper in parallel to a neurally controlled active torque Tc
for high-level control during unperturbed standing is somewhat unclear (Verdaasdonk et al. 2004; Winter et al. 1998; Loram et al. 2005b). In high-level control, several supraspinal structures are generally assumed to be involved (for review, see Horak and Macpherson 1996); in particular the cerebellum is often implicated in the context of internal models. Analysis of sagittal plane unperturbed bipedal stance generally takes place in the context of a simple model of the musculo-skeletal system that was first proposed by Gurfinkel and Osevets (1972): a single rigid segment representing the human body except the feet, rotating around the ankle joint, and acted upon by the force of gravity and the net torque produced by all muscle–tendon complexes that span the ankle joint. In the context of this model, postural sway is usually interpreted as the result of noise that acts on a system that has an equilibrium that is locally asymptotically stable (e.g. Kuo 1995; van der Kooij et al. 1999; Peterka 2002; Verdaasdonk et al. 2004). An essential requirement for local asymptotic stability in this model is that the sum of the (positive, stabilizing) effective ankle joint stiffness (whatever its source) and the (negative, destabilizing) “gravitational stiffness” must be larger than zero: K MTC,eff + K g > 0. The negative gravitational stiffness K g captures how the torque of the gravitational force relative to the ankle axis changes as a function of change in lean angle ϕ as defined in Fig. 1a. Note that we use the term “effective” MTC stiffness for the joint stiffness that results from intrinsic muscle properties and reflexive mechanisms combined.
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Notwithstanding a suggestion to the contrary made by Winter et al. (1998), it is generally accepted (Morasso and Sanguineti 2001; Peterka 2002) that the MTC stiffness that results from intrinsic muscle properties is too low to allow locally stable bipedal stance; thus, local stability is not achieved at the lowest hierarchical level at the muscle activation levels prevalent in normal stance. When analyzing the contribution of direct feedback of proprioceptive information (the intermediate level), it is now clear that the assumptions made regarding the properties of the series elastic element (SE) are crucial. If it is assumed that the SE is rigid, muscle spindle feedback is equivalent to feedback of joint angle and joint angular velocity. At sufficiently high feedback gain, joint angle feedback has been shown to result in an ankle joint stiffness that is higher than the destabilizing gravitational stiffness (e.g. Peterka 2002). However, in reality triceps surae SE behaves as a nonlinear spring, with a stiffness that increases with load (Proske and Morgan 1987). Triceps surae SE elasticity is considered important in tasks such as jumping (e.g. Bobbert et al. 1986) and running (e.g. Alexander and Bennett-Clark 1977). In these tasks, tendon forces are much higher than during standing. Due to the nonlinearity in its characteristic, SE stiffness during standing is bound to be substantially lower than in running and jumping. Indeed, the assumption that the SE is rigid during bipedal stance has been abandoned in recent years (e.g. Loram and Lakie 2002a; Verdaasdonk et al. 2004; van Soest et al. 2003). Verdaasdonk et al. (2004), using a model that incorporates an elastic SE, investigated the stability properties of their model (that is unstable under open loop control), using timedelayed direct feedback of muscle length, muscle velocity and muscle force. They argued among other things that, in the absence of co-contraction, the equilibrium position can be stabilized through muscle spindle feedback. This point of view is not shared by Loram et al. (2005a) who estimated the SE stiffness (in rotational terms) during standing, and concluded that this SE stiffness is lower than the destabilizing, negative, gravitational stiffness. The conclusion by Loram et al. (2005a) that K SE +K g < 0 was recently confirmed by the same authors using a different experimental approach (Loram et al. 2007). Further support for this conclusion follows from Appendix A, where we estimate the SE stiffness during standing on the basis of previously reported results of quick release experiments concerning the ankle plantarflexors (Hof 1998; de Zee and Voigt 2001). In these studies, the nonlinear elastic characteristic of the triceps surae SE was experimentally derived in rotational terms, that is, in terms of the relation between the joint angle change that corresponds to SE stretch and the SE torque that corresponds to SE force. As argued by de Zee and Voigt (2001), an advantage of this functional description in rotational terms at the joint level is that this description is directly derived from measurements at the joint level and thus
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does not depend on any assumptions on joint kinematics and muscle moment arm (and thus MTC length) as a function of joint angle. In Appendix A, we used data on individual subjects presented in these two studies to estimate K SE + K g during standing (see Appendix A for details and Table 1 for results). First, we estimated K g and the ankle joint torque required for equilibrium at a typical lean angle value for each of the subjects in these studies. Next, we determined for each subject K SE during standing from the individual SE torque-angle characteristic as reported, in essence by taking the slope of this characteristic at the ankle torque corresponding to standing. Finally, we considered K SE + K g for each of the subjects, and found that according to our estimation K SE + K g < 0 for 22 out of the 22 subjects in the Hof (1998) and de Zee and Voigt (2001) studies. This is directly seen from the range of values for K SE + K g as reported in Table 1. Loram et al. (2005a) agree with Verdaasdonk et al. (2004) that bipedal standing cannot be locally stable under open loop control when K SE + K g < 0; however, these authors disagree regarding local stability under direct muscle spindle feedback. Regarding force feedback, Loram et al. (2005a) recently stated that, in their view, under positive force feedback “... the muscle–tendon complex as a whole could produce a spring-like change in tension that stabilizes the body...”, when K SE + K g < 0. However, it has not been formally analyzed if, for K SE + K g < 0, GTO-based force feedback (possibly in combination with muscle spindle feedback) results in a dynamical system that can be locally stable in the neighborhood of an equilibrium state corresponding to bipedal standing. In this study, we present such a formal analysis of the linearized dynamics of the classical single inverted pendulum model. As will be shown, this analysis indicates that for K SE + K g < 0, the feedback-controlled system is locally unstable irrespective of the assumed intrinsic CE properties and irrespective of the values of muscle spindle and GTO feedback gains.
Joseph and Nightingale 1952; Fitzpatrick et al. 1992), and as other plantarflexors have much smaller moment arms, this “muscle” represents the triceps surae group to a good approximation. Regarding muscle behavior, the assumptions that are essential for the analysis to be presented are that (A) muscle consists of a series element SE and a contractile element CE, connected in series, and (B) that SE is a purely elastic element, i.e., SE force is a function of SE length only (Zajac 1989). No parallel elastic element is included because at the near-neutral ankle joint angles considered in this study, the parallel elastic torque is negligible. As the precise characteristics of the CE behavior will turn out to be irrelevant for the conclusion of the formal analysis presented, a simple lowdimensional phenomenological description of CE behavior is adopted in which CE force depends on CE length and CE velocity and on neural input, as is commonly assumed in Hill-type muscle models (Zajac 1989). As muscle mass is neglected, SE force equals CE force at all times, resulting in first-order contraction dynamics (Zajac 1989). Regarding feedback, muscle spindles are assumed to provide perfect and independent information on CE length and velocity, and GTOs are assumed to be perfect sensors of muscle force. Direct, proportional feedback of CE length, CE velocity and muscle force to the neural input of the CE will be investigated.
2 Methods
J ϕ¨ = mgl sin(ϕ) + TMTC + Tn
2.1 Modeling assumptions
For convenience of notation, the MTC dynamics is described in rotational terms; that is, CE and SE length are described in terms of angles ϕCE and ϕSE (where ϕCE + ϕSE = ϕ), and muscle action is described directly in terms of the MTC torque TMTC produced relative to the joint axis. This rotational description is fully equivalent to a translational description; for a standard Hill-type muscle model (e.g. Zajac 1989), it is given in abstract form by the following:
The model consists of a “body” modeled as a single rigid segment connecting the ankle joint to the body center of mass that can rotate around a frictionless ankle joint, the axis of which is assumed not to move. The skeletal model is actuated by a single muscle model. Strictly speaking, this muscle model represents the combined action of all active muscle–tendon complexes spanning the ankle joint. However, as co-activation (as assessed on the basis of EMG of the tibialis anterior muscle) has been reported to be minimal during unperturbed standing (Loram and Lakie 2002b;
2.2 Definition of variables and description of the plant dynamics The segment orientation ϕ as defined in Fig. 1a will be referred to as the “lean angle”. This segment with mass m and moment of inertia J (relative to the ankle) is subject to gravity, acting at the body center of mass that is located at a distance l from the ankle joint axis, to a joint reaction force at the ankle, to a torque at the ankle that represents actuation by MTCs and to a torque Tn that represents internal or external noise acting on the system. The dynamics of the skeletal system is most concisely described by the rotational equation of motion, relative to the ankle joint axis:
TMTC = TSE (ϕSE ) = TCE (ϕCE , ϕ˙CE , a)
(1)
(2)
where a represents the neural input. Note that, notwithstanding the fact that the mathematical description is in rotational
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terms, in the text we will refer to ϕCE simply as CE length, to TMTC as MTC force, etc. 2.3 Analysis of local stability in the vicinity of equilibrium Local stability will be assessed on the basis of linearization of the system dynamics in equilibrium. As the sole aim of our analysis is to establish if the system can be locally stabilized by proportional feedback of CE length, CE velocity and MTC force, it is not necessary to analyze in full the effect of feedback on the system dynamics; instead, we will formulate the Routh–Hurwitz criteria (Routh 1877; Hurwitz 1895) for the local stability of the feedback-controlled system and investigate if the feedback gains can be chosen in such a way that these criteria are met. To that aim, the Laplace transfer function linking lean angle ϕ to the noise torque Tn will first be derived. First, the skeletal dynamics (Eq. 1) is linearized around a reference equilibrium posture, indicated by index 0: J ϕ¨ ≈ d{mgldϕsin(ϕ)} ϕ + TMTC + Tn 0 (3a) = mgl cos(ϕ0 ) ϕ + TMTC + Tn = −K g ϕ + TMTC + Tn
Here, in order to simplify notation, we have defined the gravitational stiffness K g : d {mgl sin(ϕ)} Kg = − = − mgl cos(ϕ0 ) dϕ 0 Note that in the context of standing, K g has a negative value, reflecting its destabilizing effect. Transformation of Eq. 3a to the Laplace domain is straightforward. Throughout this study, variables in the Laplace domain will be distinguished from their equivalents in the time domain by a tilde superscript, yielding the following expression for the linearized skeletal dynamics: (J s 2 + K g ) ϕ˜ = T˜MTC + T˜n
(3b)
Next, the muscle dynamics (Eq. 2) is linearized, yielding: dTSE ∂ TCE TMTC ≈ ϕSE ≈ ϕCE dϕSE 0 ∂ϕCE 0 ∂ TCE ∂ TCE + ϕ˙CE + a ∂ ϕ˙ CE 0 ∂a 0 To simplify notation, the following definitions will be used: ∂ TSE ∂ TCE K SE = − , K = − CE ∂ϕSE 0 ∂ϕCE 0 ∂ TCE ∂ TCE BCE = − , K = a ∂ ϕ˙ CE 0 ∂a 0 K SE and K CE , having units Nm/rad, are the linearized intrinsic stiffness of the SE and CE in the equilibrium position,
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respectively; BCE , having units Nms/rad represents the linearized intrinsic angular damping coefficient of the CE. The minus signs in the expressions for K SE , K CE and BCE ensure that when the values of these parameters are positive, the torque changes oppose the changes in angle like in conventional springs and dampers. The gain K a (Nm) represents the linearized change of active muscle torque per unit change in neural input a (note that activation dynamics is considered to be part of Hr , see below). As explained above, ϕCE + ϕSE = ϕ and TSE = TCE ; together with the newly defined parameters this results in the following linearized equation for the muscle dynamics: TMTC = −K SE · (ϕ − ϕCE ) = −K CE ϕCE − BCE ϕ˙CE + K a a
(4a)
In the Laplace domain, this corresponds to the following: T˜MTC = −K SE · (ϕ˜ − ϕ˜CE ) = −(K CE + BCE s)ϕ˜CE + K a a˜
(4b)
In mechanical terms, Eq. 4a represents the actuator that is schematically shown in Fig. 1b, where the control torque T˜C equals K a a. ˜ Finally, the feedback law is formulated, constituting proportional feedback of CE length and velocity (originating from muscle spindles, sp’s) and MTC force (originating from Golgi tendon organs, GTO’s) to the neural input of the muscle, each with an appropriate feedback gain (K sp , Bsp and κGTO , respectively). This feedback law is formulated directly in the Laplace domain. In addition to the feedback terms, a transfer function Hr (s) is included (but not yet specified) that will be used to represent reflex loop dynamics such as muscle activation dynamics and time delays associated with the sensory transport and processing. The transfer function for the feedback law is defined as: K sp Bsp κGTO ˜ + s ϕ˜ CE − TMTC a˜ = Hr (s) − Ka Ka Ka (5) Hr (s) will be chosen so as to have low-frequency gain equal to one (i.e., Hr = 1 for s = 0). All feedback gains are scaled by K a with the intent that this gain K a (Eq. 4b) will vanish from the expressions for the controlled system, K sp , Bsp and κGTO then becoming the total loop gains for CE length, CE velocity and CE force, respectively. Based on the transfer functions for the component systems (Eqs. 3b, 4b, 5), a block diagram representing the controlled system can be constructed; this block diagram is shown in Fig. 2. Using standard methods, one can derive the transfer function that describes how the lean angle ϕ˜ depends on the noise torque T˜n from this block diagram; writing this
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transfer function as a ratio, results in the following: ϕ˜ =
3 Results and discussion
N (s) ˜ Tn D(s)
with N (s) = (K CE + BCE s) + Hr (K sp + Bsp s) + (1 + Hr κGTO )K SE D(s) = (J s 2 +K g +K SE ) (K CE +BCE s)+ Hr (K sp + Bsp s) + (J s + K g )(1 + Hr κGTO )K SE 2
2.4 Routh–Hurwitz conditions for stability The Routh–Hurwitz stability conditions (Routh 1877; Hurwitz 1895) are formulated in terms of the polynomial coefficients of the denominator of a transfer function that is written as the ratio of two pure polynomials in the Laplace variable s (i.e., N p (s)/D p (s)). In the following, we will investigate several variants of Hr that all have a unit numerator and a polynomial denominator: Hr (s) = 1/Dr (s). Substitution of this expression for Hr (s) in N (s) and D(s) as derived above, and subsequent multiplication of both N (s) and D(s) by Dr (s) yields a pair of numerator N p (s) and denominator D p (s) that is in the desired polynomial form for all variants of Hr that will be considered: N p (s) = Dr (s)(K CE + BCE s) + (K sp + Bsp s) + (Dr (s) + κGTO )K SE D p (s) = (J s 2 + K g + K SE )(Dr (s)(K CE + BCE s)
α0 α1 α2 α3
= (K CE + K sp ) (K SE + K g ) + (1 + κGTO ) K SE K g = (BCE + Bsp ) (K SE + K g ) (8) = J (K CE + K sp + (1 + κGTO ) K SE ) = J (BCE + Bsp )
These expressions for the polynomial coefficients have to be substituted in the Routh–Hurwitz conditions (a)–(c) as specified earlier. This yields the following inequalities that have all to be satisfied if the system is to be asymptotically stable: α0 /α3 > 0 (K CE +K sp ) (K SE + K g )+(1+κGTO ) K SE K g >0 J (BCE +Bsp ) K SE + K g α1 /α3 > 0 → >0 J K CE + K sp + (1 + κGTO ) K SE α2 /α3 > 0 → >0 BCE + Bsp 2 >0 α1 α2 −α0 α3 > 0 → J (BCE +Bsp ) (1+κGTO ) K SE →
(9) (6)
+ (K sp +Bsp s))+(J s 2 +K g )(Dr (s)+κGTO )K SE In case Hr consists of k first-order systems in series (i.e., k Hr (s) = 1/ i=1 (τi s + 1)), the order of D p is found to be 3 + k. As we will consider zero- and first-order Hr in the main text, we limit the description here to third- and fourthorder systems. In particular, when the fourth-order D p (s) is written as: D p (s) = α4 s 4 + α3 s 3 + α2 s 2 + α1 s + α0
First, the local stability of the feedback-controlled system as given by Eq. 6 and Fig. 2 will be analyzed when activation dynamics and neural time delay are neglected, i.e., when Hr is set to unity. In that case, the system is of order three; comparison of Eqs. 6 and 7 leads to the following expressions for the polynomial coefficients:
(7)
then the Routh–Hurwitz theorem states that the system is stable if and only if (a) the coefficients αi all have the same sign, (b) α1 α2 − α0 α3 > 0, and (c) α1 α2 α3 /α43 − α12 /α42 − α0 α32 /α43 > 0. In a third-order system only conditions (a) and (b) have to be satisfied. When the order of D p (s) is higher than four, the conditions (a)–(c) must still be satisfied; for each increase in order, the number of individual conditions specified by (a) increases by one, and furthermore additional conditions have to be satisfied. Even though, in Appendix B, higher-order systems are considered, for our present purpose it is not necessary to specify and analyze these additional conditions.
From the condition α1 /α3 > 0, it is immediately seen that, irrespective of the values of the other parameters, the system cannot be locally stable if K SE + K g < 0. As it was argued in the introduction and confirmed in Appendix A that, indeed, K SE + K g < 0, this concludes the intended analysis for the case Hr = 1: it is impossible to stabilize the system when the SE stiffness value does not exceed the destabilizing (negative) gravitational stiffness. Our understanding of the above conclusion can be enhanced by considering the conditions α0 /α3 > 0 and α1 α2 − α0 α3 > 0. Considering that BCE + Bsp > 0 (indicating normal damping), these conditions can be rewritten as: (K CE + K sp ) K SE + Kg > 0 K CE + K sp + (1 + κGTO ) K SE (10) (1 + κGTO ) > 0 It is straightforward to show that the bracketed term in the first of these conditions represents the effective MTC stiffness; thus, this “MTC stiffness requirement” states that the effective MTC stiffness must exceed the negative K g in magnitude, as was already argued by Gurfinkel and Osevets (1972). In the absence of all feedback (K sp = κGTO = 0), the bracketed expression for the effective MTC stiffness in Eq. 10 reduces to that for two springs in series, one having
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Fig. 2 Block diagram of the linearized model including feedback of muscle force and CE length and velocity. Laplace transfer functions are given inside the blocks. Thicker lines represent the reflexive feedback loops. See main text for definition of variables
stiffness K SE and the other having stiffness K CE . It follows directly from this well-known expression that the stiffness of the series arrangement will always be lower than that of the most compliant component. Introduction of spindle feedback (K sp = 0, κGTO = 0), replaces the intrinsic CE stiffness K CE by the effective CE stiffness K CE + K sp , but otherwise does not affect the structure of the expression for effective MTC stiffness. Thus, in case K SE + K g < 0, this “MTC stiffness requirement” can be satisfied neither under open loop control nor under CE length feedback. When force feedback is considered (K sp = 0, κGTO = 0), it is seen from the MTC stiffness requirement that the force feedback gain has to be negative-valued, implying positive force feedback, in order to increase the effective MTC stiffness. At κGTO = −1, the effective MTC stiffness (the bracketed term) equals K SE , which is not yet sufficient for local stability. Note that it can only be the case that K MTC,eff = K SE if the CE is rigid, which is indeed the case at κGTO = −1. When κGTO is made more and more negative, the effective MTC stiffness increases without bound; beyond some κGTO that is more negative than -1, the MTC stiffness requirement will be satisfied. However, such a value clearly does not satisfy the second condition of Eq. 10. The reason why κGTO < −1 is not compatible with stability can be appreciated from the relation between MTC torque and CE length, that can be derived by combining Eqs. 4b and 5: K CE + K sp BCE + Bsp + s ϕ˜ CE T˜MTC = − 1 + κGTO 1 + κGTO = − K CE,eff + BCE,eff s ϕ˜ CE From this equation it is seen that both the effective CE stiffness and the effective CE damping change sign at κGTO = −1. The negative effective CE stiffness for κGTO < −1 indicates that that for κGTO < −1, the CE shortens when the MTC is stretched, as is illustrated in Fig. 3. Thus, for κGTO < −1 the CE response amplifies the SE stretch imposed through lean angle perturbation per se; this gives rise to the increase in MTC stiffness discussed above. In passing we note that this behavior was recently termed “paradoxical movements” by Loram et al. (2005a); furthermore, these authors were
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CE
SE initial
κGTO > -1 CE eccentric energy dissipation
κGTO = -1
stretched
CE isometric energetically neutral
κGTO < -1 CE concentric energy liberation
Fig. 3 Schematic representation of the different behavioral regimes, as defined by the force feedback gain κGTO . Top panel, reference situation. Subsequent panels represent the steady state equilibrium of the muscle– tendon complex when a constant stretch is externally imposed
correct in suggesting that these paradoxical movements can result from positive force feedback. However, the negative effective CE damping for κGTO < −1 implies that the dissipative nature of the contractile element has been annihilated; at κGTO < −1 the velocity-dependent contribution to CE force is an energy source instead of an energy sink, which is incompatible with stability. The previous analysis assumed Hr = 1, or, in other words, neglected activation dynamics and the time delay in the feedback loop. It is well known (e.g. Rack 1981) that the maximal feedback gain for which a system controlled through negative feedback is stable is reduced when dynamic components are introduced that add phase lag to the feedback loop. As time delays as well as activation dynamics introduce additional phase lag, one would expect that the results obtained for Hr = 1 describe the best-case performance for the type of feedback control investigated. However, (Prochazka et al. 1997a,b), in analyzing positive feedback of muscle force, reported that inclusion of activation dynamics and feedback time delay increased the range of positive force feedback gains for which their system was stable. Prochazka argued that the classical point of view outlined above is related to negative feedback control and does not necessarily apply
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to positive feedback. The model investigated here differs in important respects from that analyzed by Prochazka; the model considered here is open loop unstable and has elastic series elements. Nevertheless, as we have shown above that GTO feedback must be positive in order to improve stiffness, we consider it necessary to assess the implications of additional dynamics. To that aim, we first substitute Dr = τa s +1 in Eq. 6, that is, we include a first-order system that may be thought to represent activation dynamics. Rearranging the resulting transfer function to a ratio of polynomials results in the following coefficients of D p (s) (see Eq. 7): α0 α1 α2 α3 α4
= K CE + K sp (K SE + K g ) + K SE K g (1 + κGTO ) = BCE + Bsp (K SE + K g )+ K CE K SE + K g + K SE K g τa = τa BCE (K SE + K g )+ J K sp + K CE + J K SE (1+κGTO ) = J (BCE + Bsp ) + J (K CE + K SE ) τa = J BCE τa
We have found that in order to show that stability is impossible with the added loop dynamics, it suffices to consider the conditions stating that all coefficients should have identical sign, and more in particular to consider the following two of these conditions: (BCE +Bsp )(K SE +K g )+(K CE (K SE +K g )+K SE K g )τa >0 J BCE τa (11) (BCE + Bsp ) + (K CE + K SE ) τa α3 /α4 > 0 → >0 BCE τa α1 /α4 > 0 →
As noted previously, J, τa and BCE are always positive; thus, the divisions by J BCE τa and BCE τa , respectively can be ignored. As before, we are interested in the case K SE + K g < 0; division by this factor changes the sign of the inequality. This yields the following: Kg τa −(BCE + Bsp ) > K CE + K SE K SE + K g −(BCE + Bsp ) < (K CE + K SE ) τa from which it follows that, if the system is to be stable, we must have the following: Kg (K CE + K SE ) > K CE + K SE K SE + K g In the conditions of interest here, K g /(K SE + K g ) is larger than one, in which case the latter inequality cannot be satisfied. Therefore, introduction of a first-order lag in the feedback loop does not affect our conclusion that force feedback cannot stabilize the system. In Appendix B, the case is considered where Hr is the product of an arbitrary number of first-order lags. This type of transfer function is relevant in particular because a time delay can be approximated to any degree of accuracy by such a transfer function (This can be seen as follows: the transfer function of a puretime delay n of τd seconds is e−τd ·s = 1/eτd ·s = 1/ lim τnd s + 1 ; n→∞ the latter transfer function represents a cascade of identical
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first-order systems, each having a time constant τd /n). The analysis presented in Appendix B leads to the conclusion that additional phase lags in the feedback loop do not locally stabilize the single-segment stance model. Thus, the earlier conclusions can be generalized to models incorporating additional dynamics, including time delays: when SE stiffness is lower than the destabilizing gravitational stiffness (K SE + K g < 0), direct feedback of CE length, CE velocity and muscle force cannot result in locally stable bipedal standing under the assumptions adopted in this study.
4 General discussion Given the modeling framework adopted in this study, the conclusion that direct force feedback is insufficient to obtain local stability if K SE + K g < 0, follows in a straightforward way from the analysis, and thus requires no further discussion. Here, we will discuss to what extent this conclusion depends on the assumptions underlying the analysis, and which directions for future research are considered promising. Sensitivity for parameter values. Given the model structure (single inverted pendulum; SE and CE in series), our conclusion that, if K SE + K g < 0, the investigated feedback scheme cannot result in local stability has the nice property that it is independent of any parameter values. The only quantitative question that may be asked is how strongly our support for the claim in literature that K SE + K g < 0, as derived from re-analysis of the quick-release experimental data as reported by Hof (1998) and de Zee and Voigt (2001) (see Appendix A), depends on the assumed average lean angle. At a higher lean angle, K g will be slightly less negative, the required ankle joint torque will be higher, and, due to the nonlinear characteristic of the SE, the linearized SE stiffness will also be higher; together these changes might result in K SE + K g > 0. To investigate this, we recalculated Table 1 for an assumed average lean angle value that is 50% higher than that reported by Winter et al. (2001). At this (in our view unrealistically high) lean angle, K SE + K g is negative for 19 out of the 22 subjects and ranges between −477 and +47 Nm/rad; our conclusion regarding the sign of K SE + K g on the basis of re-analysis of the quick-release experimental data is not very sensitive to the assumed lean angle. The assumed absence of co-activation of antagonists. Activity of antagonists (most importantly m. tibialis anterior) would result in dorsiflexor SE stiffness, and would entail stronger contraction of the plantarflexors, resulting in higher SE stiffness of the plantarflexors. Thus, substantial co-activation of m. tibialis anterior would contribute to the SE ankle stiffness. However, reports on EMG activity
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indicate that activity in this muscle is hardly detectable during unperturbed standing (Loram and Lakie 2002b; Joseph and Nightingale 1952; Fitzpatrick et al. 1992). Regarding activity of muscles other than triceps surae and tibialis anterior, to our knowledge no information is available. However, these muscles have substantially smaller moment arms, resulting in a smaller contribution to stiffness. All in all it is unlikely that equating triceps surae SE stiffness with total SE ankle stiffness (as was done in Appendix A) results in a gross underestimation of the linearized SE ankle stiffness during standing.
to consider the question if important dissipative elements outside the CE have been neglected. In particular, the SE, generally modelled to be purely elastic, shows some hysteresis, indicating dissipation. However, measurements of SE hysteresis indicate that SE is only lightly damped (Bahler 1967). Similarly, the frictional forces in the ankle joint dissipate energy. To our knowledge, no data are available from which the amount of joint dissipation can be estimated. While it seems unlikely that the dissipation in joints will be sufficient to stabilize the system, this is an area for further research.
The assumption of a pure “ankle strategy”. Our analysis leads to the conclusion that through GTO-mediated positive feedback of muscle force, the effective MTC stiffness at the ankle joint can be boosted to an appropriate value, at the expense of damping at the ankle joint. From a theoretical point of view, it is interesting to consider the question if, in a multi-segment model (e.g. allowing joint rotation in ankle and hip) energy dissipation at the hip may compensate the energy production at the ankle that results from a sufficiently high positive force feedback gain, so that the dissipation requirement can be satisfied after all. The analysis of Barin (1989), who showed that in a two-segment model of bipedal stance it is not necessary for stability to have dissipation at each of the joints, points in the direction that this might indeed be possible. In fact, consideration of multi-joint models of unperturbed bipedal stance is interesting from another perspective as well: it is now becoming clear that the smallest local stiffness at the ankle joint that is compatible with stability depends on modeling assumptions. First of all, it has been shown that in a multi-joint model with local stiffness only, the minimally required ankle joint stiffness is higher than in the corresponding single-joint model (Rozendaal and van Soest 2005; see also Edwards 2007). More interestingly, we have recently found (Rozendaal and van Soest 2007) that when the stiffness matrix for a multi-joint model is allowed to be non-symmetrical, the minimally required ankle joint stiffness can be very low. At least in the cat, it is well established that the heterogenic reflexes that can result in such a non-symmetrical stiffness matrix, make a substantial contribution to the mechanical behavior (Nichols 1989). Furthermore, according to recent experimental studies on human bipedal standing (Pinter et al. 2006; Hsu et al. 2007), there is little empirical ground for the assumption that postural sway is limited to the ankle joint. Therefore it is in our view urgent to start considering multi-joint models of bipedal standing. Future research will have to clarify if heterogenic reflexes can stabilize a multi-joint model of bipedal standing at a realistic value of K SE .
The assumed sensor properties. In this study, it was assumed that muscle spindles provide perfect, noiseless information on CE length and velocity, and similarly, that Golgi tendon organs provide perfect, noiseless information on muscle force. These assumptions, as well as the somewhat unrealistic assumption that the muscle spindles provide independent information about CE length and velocity and that, thus, effective loop gains for CE length and velocity can be chosen independently, all facilitate direct feedback control. Thus, it is clear that more realistic assumptions regarding sensor characteristics are unlikely to change the conclusions of this study. In this context, it is worthwhile to revisit the recent study by Verdaasdonk et al. (2004), who, using a model similar to the one used in this study, arrived at the conclusion that the system can be stabilized on the basis of muscle spindle feedback alone. The discrepancy between their conclusion and the conclusion reached in this study is due to a difference in the assumed variables about which muscle spindles inform. In the study by Verdaasdonk et al. (2004), SE nonrigidity was taken into account, but, at the same time, muscle spindles were assumed to provide direct information on joint angle and angular velocity, instead of CE length and velocity. Under this assumption, effective joint stiffness and viscosity are directly affected by spindle feedback, whereas in this study, it is only CE stiffness and viscosity that is directly affected. In our view, the assumption by Verdaasdonk et al. (2004) that muscle spindles sense joint angle and joint angular velocity is incompatible with the functional anatomy of muscle spindles.
The assumed absence of SE damping and joint friction. In the light of our conclusion that dissipativity becomes critical when spindle+GTO feedback is considered we need
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Is locally stable standing at all possible on the basis of intermediate-level direct feedback control? Stepping back, it is surprising that proportional feedback of three variables (CE length and velocity, as well as muscle force) fails to provide full control over the dynamics of a linearized third-order system. This state of affairs must indicate that, together, the three feedback variables do not contain information on all relevant variables. Indeed, it is easily verified that neither of the three feedback variables depends directly on ankle joint angular velocity. If locally stable standing in a single segment model is to be achieved on the basis of direct feedback, feedback of a sensory signal that is instantaneously affected by
Biol Cybern (2008) 99:29–41
ankle joint angular velocity must be added, and the question becomes which sensors may provide this information. The current consensus is that joint receptors are not accurate joint angle sensors (Horak and Macpherson 1996); consequently, it is unlikely that joint receptors provide joint angular velocity information to be used in direct feedback. This is supported by the finding that reduction of somatosensory information from feet and ankle joints has no effect on unperturbed stance (Diener et al. 1984). However, the Golgi tendon organ might play a role in lowlevel reconstruction of joint angular velocity. As SE length can be directly derived from muscle force (as sensed by GTO’s) it is clear that, together, GTO’s and muscle spindles inform on the length of the muscle–tendon complex, and thus on joint angle. Taking the time derivative of this relation, it is seen that the combination of CE velocity (sensed by muscle spindles), force (sensed by GTO’s) and time derivative of force informs directly on joint angular velocity. The question then becomes if information on the time derivative of force is available at the intermediate level. In our view there are some indications that this is indeed the case. First of all, it is well documented that GTO’s themselves are sensitive for the rate of change of force (Anderson 1974; Houk and Henneman 1967; Crago et al. 1982). Furthermore, the spinal interneurons that are part of the GTO feedback loop receive descending input and input from both Ia and Ib afferents from several muscles (Jankowska 1992), which at least suggests the possibility of dynamic processing of the Ib afferent signals at the spinal level. It is thus conceivable that the modulation of alpha-motoneuron activity by GTO afferent information depends not only on force but also on its rate of change, which, in combination with muscle spindle feedback, might effectively amount to feedback of joint angular velocity. It may thus be of interest to investigate if GTO-based feedback of force and rate of change of force, together with muscle spindle feedback, can potentially stabilize the system. Alternatively, direct feedback of body angular velocity as sensed by the visual and/or vestibular systems may be considered (for review, see Horak and Macpherson 1996). However, the observation that vestibular loss patients are able to stand with their eyes closed (e.g. Peterka 2002) suggests that visual and vestibular information and the associated supraspinal processing are not essential for postural control. Nonlinear effects. In this study, we considered linear proportional feedback of force, CE length and CE velocity. It is well established that responses to postural perturbations are not linearly related to these variables. For example, the dynamic muscle spindle response has been reported to be nonlinearly history-dependent: when a series of stretches is applied, the muscle spindle response amplitude is reduced after the first stretch (Haftel et al. 2004). Furthermore, when human wrist muscles are slowly stretched, an initial burst is observed in
37
muscle spindle afferents, and the peak firing rate during this initial burst codes for the starting length more strongly than the steady-state firing rate (Cordo et al. 2002). It is clear that these and other nonlinearities affect the perturbation response of the system. However, in our view nonlinearities in the feedback pathways (or, for that matter, in the musculoskeletal dynamics) cannot affect our conclusion regarding the local stability in the neighborhood of equilibrium, because local stability is a function of first-order (linear) terms only. A view of unperturbed standing that is altogether different from the classical view that we adopted in this study is that nonlinear dynamics ensure that the postural sway is bounded even though no locally stable equilibrium state exists. Such a point of view has been adopted by Loram and coworkers (e.g. Loram et al. 2005b) who have suggested that control actions are undertaken intermittently. Clearly, the methods of analysis used in this study are not suitable to analyze this type of control. High- and/or intermediate-level control of standing? The analysis in this study was limited to intermediate-level direct feedback control that might be implemented at the spinal level. We showed that, when it is assumed that K SE +K g < 0, local stability is not easily achieved on the basis of direct intermediate-level reflexive control. However, it is in our view too early to conclude that high-level control, based on extensive sensory integration, and thus requiring substantial involvement of supraspinal structures is a prerequisite for locally stable standing. In particular, stability may arise from direct feedback of other signals, and/or from dynamic processing of such signals; feedback of the rate of change of force is in our view a particularly interesting candidate, as discussed earlier. Even if high-level control, involving extensive sensory integration is important in postural control, this does not have to imply full state estimation necessarily. For example, it was recently proposed by Lockhart and Ting (2007) that time-delayed feedback of body center of mass acceleration (presumably derived on the basis of sensory integration) suffices to predict the initial response (in terms of muscle activity) to platform perturbations in the standing cat. In the human, similar initial muscle activity during platform perturbations was previously interpreted to be primarily the result of feedforward control (e.g. Diener et al. 1988). While the latter interpretation would strongly suggest model-based control implemented at the supraspinal level, this is not necessarily the case for the mechanism suggested by Lockhart and Ting (2007). Nevertheless, in a functional sense we would classify both mechanisms to be part of “high-level” control, as they both depend on sensory integration. In sum, it is too early to conclude that high-level modelbased control that effectively implements state estimation is the only mechanism that can result in locally stable bipedal standing. At the lower end of the hierarchy, this study
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indicates that local stability of the classical 1-DOF inverted pendulum model, actuated by a muscle–tendon complex for which K SE + K g < 0, is not easily achieved by direct feedback of proprioceptive signals arising from muscle spindles and Golgi tendon organs.
Appendix A: Estimation of K g and K SE during standing from previously published results of quick-release experiments Aim. In this Appendix we estimate the destabilizing gravitational stiffness K g as well as the rotational SE stiffness K SE of the plantarflexors during standing for the subjects involved in two independent experimental studies (Hof 1998; de Zee and Voigt 2001) in which SE characteristics of the plantarflexors were directly measured. Outline of the experimental paradigm. In both studies, subjects produced a constant plantarflexion torque against a motor driven ergometer. Subsequently a quick release took place during which both joint angle and muscle torque were recorded. EMG data (in both studies) as well as results from stimulation experiments (in de Zee and Voigt 2001) indicated that co-activation of tibialis anterior was negligible. Compensating for the estimated CE length change during the release, the length change of the SE (in rotational terms) was estimated from the data. SE stiffness of the plantarflexors at the respective muscle torque level was obtained as the slope of the corrected SE torque-angle relationship. Measurements were carried out for a wide range of muscle torques, around a neutral ankle joint angle. The results were summarized in terms of a nonlinear functional description of the plantarflexion SE torque as a function of SE angle. As an example we consider the study of de Zee and Voigt (2001). Here, the functional description has the form of the well known square root relation between SE torque and SE angle (Prilutsky et al. 1996), where in contrast to this study both variables are positive by definition and where TSE,Zee refers to the SE torque for one leg: ϕSE,Zee = a · TSE,Zee + b + c (A1) with a, b, c constants that were fitted and reported for each of the subjects. In both studies this functional description fitted the data closely. Individualized estimation of K g . The gravitational stiffness K g depends on m, l and ϕ0 (see Eq. 3a in main text). Body mass m was reported for individual subjects in both studies mentioned. l, the distance from ankle joint to the center of mass of the body except feet) was estimated from individually reported body length, assuming that the distance from ankle to body center of mass is 51% of body length. ϕ0 , the nominal
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value of the lean angle, was taken to be 3.67◦ (Winter et al. 2001). The range of resulting values for K g is reported in Table 1. Individualized estimation of K SE during standing. First, using the derivative d ϕSE,Zee /dTSE,Zee that is readily obtained from Eq. A1, we derived how K SE,Zee , the linearized SE stiffness (for one leg) of the plantarflexors, depends on TSE,Zee : dTSE,Zee dϕSE,Zee −1 K SE,Zee = = dϕSE,Zee dTSE,Zee
−1 2 a · TSE,Zee + b a = = a 2 a · TSE,Zee + b (A2) Next, the nominal value of the plantarflexion torque TSE,Zee during standing was estimated for each subject in the de Zee and Voigt (2001) study, by assuming left-right symmetry in loading (i.e., TSE,Zee = 0.5TMTC ) and assuming torque equilibrium at the lean angle mentioned before, using subjectspecific values for m and l. Finally, subject-specific values for a, b and TSE,Zee were entered in Eq. A2, to obtain the corresponding K SE,Zee for one leg. In Table 1, the estimated total TMTC and K SE (for two legs) during standing are reported. The same approach was used to analyze the experimental results of Hof (1998) (see Table 1). Interpretation. From the range of values of K SE + K g (Table 1) it is seen that K SE + K g is negative for each of the 22 subjects of the de Zee and Voigt (2001) and Hof (1998) studies. Put differently, the lean angle at which K SE + K g would equal zero (ϕ* in Table 1) is larger than the lean angle that is typically observed; in fact, ϕ* was found to be 7◦ for the de Zee and Voigt (2001) study, which would place the body center of mass approximately 0.1 m anterior to the ankle joint axis. This lean angle is about twice (and more than 10 standard deviations from) the average lean angle. For the Hof (1998) study, the lean angle at which K SE + K g = 0 places the body center of mass in front of the toes in most subjects. We conclude that available data from quick release experiments strongly suggest that K SE during bipedal standing is lower than K g .
Appendix B: Local stability conditions for model including higher-order reflex loop dynamics and pure time delay In the main text, the analysis was restricted to systems with first-order dynamics in the feedback path (Hr = 1/(τa s + 1)). For reasons outlined in the main text, it needs to be investigated if inclusion of a time delay in the feedback path
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39
Table 1 Mean and range of mechanical variables related to stance de Zee and Voigt (2001)
Hof (1998)
Mean
Range
Mean
Range
Body length (m)
1.76
1.62–1.98
1.82
1.69–1.95
Body mass (kg)
74
53–102
76
59–89
−TSE (Nm)
41
26–55
44
32–54
K SE (Nm/rad)
467
313–664
203
91–403
−K g (Nm/rad)
636
413–864
678
492–847
K SE + K g (Nm/rad)
−169
(−278)–(−18)
−475
(−585)–(−323)
7
4–10
21
13–34
ϕ∗
(◦ )
Values are calculated from the data reported in de Zee and Voigt (2001) and Hof (1998), using a lean angle value of 3.67 degrees (Winter et al. 2001). TSE and K g are negative according to the definition used in this study. Note that |TSE | = 2 · TSE,Zee and K SE is summed over two legs. ϕ* is the lean angle at which K SE + K g = 0
may affect the conclusion of this study. In this appendix it will be shown that inclusion in the feedback path of a time delay leads to the same conclusion as reached in the main text: if K SE + K g < 0 then direct feedback of muscle force and CE length and velocity cannot stabilize the system. In this appendix we will consider Hr of the following form: Hr{k} (s) =
k i=1
1 τi s + 1
(B1)
This transfer function represents a cascade of {k} first-order lags. As argued in the main text, this class of transfer functions can be used to approximate a time delay to any desired precision. As in the main text, the stability of the feedbackcontrolled system will be analyzed based on the denominator polynomial of the transfer function, which equals (see Eq. 6 in the main text for the definition of D p (s)): {k} D {k} p (s) ≡ D(s)Dr (s) = D(s) {k}
{k}
≡ αk+3 s k+3 + αk+2 s
k
i=1 k+2
(τi s + 1)
{k}
{k}
→ −(BCE + Bsp ) > νlower
{k}
{k}
→ −(BCE + Bsp ) < νupper
α1 /αk+3 > 0 α3 /αk+3 > 0
{k}
{k}
(B3)
As both these conditions have to be satisfied if the system is to be locally stable, it is clear that the system cannot be locally {k} {k} {k} stable if νupper − νlower < 0. The expressions for νupper and {k} νlower are: {k}
νlower =
K CE K SE + K SE K g + K g K CE (k, 1) K SE + K g
{k}
νupper = (K CE + K SE ) (k, 1) BCE K SE + K g (k, 2) + J K CE K SE + K SE K g + K g K CE + (k, 3) J (B4)
{k}
{k}
+ · · · + α1 s + α0
(B2) As in the main text (see Eq. 11), we have found that only two of the Routh–Hurwitz stability requirements need to be analyzed in order to show that local stability is impossible, under
In these expressions, (k, m) stands for the sum of all possible products of m elements from the set {τ1 , . . . , τk } in case 1 m k and (k, m) = 0 otherwise; e.g., (3, 1) = τ1 + τ2 + τ3 ; (3, 2) = τ1 τ2 + τ1 τ3 + τ2 τ3 ; (3, 3) = {k} {k} τ1 τ2 τ3 ; (3, 4) = 0. The difference νupper − νlower can be written as:
{k}
{k} νupper − νlower
=
2 J K SE K SE +K g (k, 1) +
BCE K SE + K g (k, 2) + K CE K SE + K SE K g + K g K CE (k, 3) J
the same assumptions as before (J > 0; τi > 0; BCE > 0; K SE + K g < 0). These requirements concerns the signs of the denominator coefficients of s and s 3 :
(B5)
The denominator of this expression is positive; hence if all {k} {k} numerator terms are negative we have νupper − νlower < 0, which is incompatible with local stability because in that case
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the conditions in Eq. B3 cannot be satisfied simultaneously. Under the assumptions used throughout this study (J > 0; τi > 0; BCE > 0; K SE + K g < 0) the terms containing (k, 1) and (k, 2) are clearly negative. The term containing (k, 3) is negative if K CE > −K SE K g /(K SE + K g ). In words, K CE must be above some negative value which is larger in magnitude than K SE . For realistic system parameters this is undoubtedly the case, and therefore also the numerator term containing (k, 3) is negative-valued. We conclude that inclusion of an arbitrary number of firstorder lags in series in the feedback loop (and thus inclusion of a pure time delay) leads to the same conclusion as reached for the simpler model: for K SE + K g < 0 the system cannot be locally stabilized by proportional feedback of muscle force, in combination with feedback of CE length and velocity.
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