JOINT dynamic stiffness defines the dynamic relation between

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1 A Subspace Approach to the Structural Decomposition and Identification of Ankle Joint Dynamic Stiffness Kian Jalaleddini, Member, IEEE, Ehsan Sobhani Tehrani, Student Member, IEEE, Robert E. Kearney, Fellow, IEEE Abstract Objective: The purpose of this paper is to present a Structural Decomposition SubSpace (SDSS) method for decomposition of the joint torque to intrinsic, reflexive and voluntary torques and identification of joint dynamic stiffness. Methods: Firstly, it formulates a novel state-space representation for the joint dynamic stiffness modelled by a parallel-cascade structure with a concise parameter set that provides a direct link between the state-space representation matrices and the parallel-cascade parameters. Secondly, it presents a subspace method for the identification of the new state-space model that involves two steps: (i) the decomposition of the intrinsic and reflex pathways; (ii) the identification of an impulse response model of the intrinsic pathway and a Hammerstein model of the reflex pathway. Results: Extensive simulation studies demonstrate that SDSS has significant performance advantages over some other methods. Thus, SDSS was more robust under high noise conditions, converging where others failed; it was more accurate, giving estimates with lower bias and random errors. The method also worked well in practice and yielded high quality estimates of intrinsic and reflex stiffnesses when applied to experimental data at three muscle activation levels. Conclusion: The simulation and experimental results demonstrate that SDSS accurately decomposes the intrinsic and reflex torques and provides accurate estimates of physiologically meaningful parameters. Significance: SDSS will be a valuable tool for studying joint stiffness under functionally important conditions. It has important clinical implications for the diagnosis, assessment, objective quantification and monitoring of neuromuscular diseases that change the muscle tone. Index Terms Biomedical Signal Processing, Biological System Modeling, Hammerstein Structure, Muscle Tone, Nonlinear Dynamical Systems, Parallel-Cascade Structure, State-Space Methods, Subspace Identification, System Identification. I. INTRODUCTION JOINT dynamic stiffness defines the dynamic relation between the position of a joint and the torque acting about it []. It plays an important role in control of posture since it determines the resistance to external perturbations. Moreover, it is important in movement since it determines the load the central nervous system (CNS) must control. Consequently, modelling and identification of dynamic stiffness, sometimes K. Jalaleddini is with the Division of Biokinesiology and Physical Therapy, 4 Alcazar St, Los Angeles, CA, 933, USA, seyed.jalaleddini@mail.mcgill.ca. E. Sobhani Tehrani and R. E. Kearney are with the Department of Biomedical Engineering, 377 University, Montreal, QC, H3A 2B4, Canada, ehsan.sobhani@mail.mcgill.ca, robert.kearney@mcgill.ca. This work has been supported by CIHR, FRQNT and NSERC. referred to as impedance or its inverse compliance, have been the topic of active research for many years [2], [3], [4]. The torque developed at a joint in response to a position perturbation arises from both neural and non-neural mechanisms [], [6]. Torques from non-neural mechanisms, i.e. intrinsic stiffness, arise from the mechanical (visco-elasticinertial) properties of the joint, active muscles and passive tissues (tendons, ligaments, etc). Torques from neural mechanisms arise from the changes in muscle activation due to reflex mechanisms, i.e. reflex stiffness, and voluntary contributions. Accurate decomposition of the intrinsic and reflex contributions is important. Firstly, it provides the quantitative information needed to understand the role of reflexes in the control of posture and locomotion. Secondly, it has important clinical implications for the diagnosis and monitoring of neuromuscular diseases such as spinal cord injury, cerebral palsy, multiple sclerosis, stroke, parkinson s disease [7], [8], [9], [], []. The current clinical tests do not yet provide objective, reliable estimates of the intrinsic and reflex contributions [2], [3]. Consequently, the development of accurate and objective methods to decompose stiffness and quantify spasticity has been highly recommended in clinical research [4], []. The intrinsic and reflex torques change together and cannot be measured individually which makes their decomposition challenging. A number of analytical methods have been developed to solve the decomposition problem. These methods differ in the assumed model structure, e.g. static or dynamic, linear or nonlinear, and the algorithm used for their identification, e.g. frequency response identification or transfer function identification, etc. Schouten et al. assumed a linear model and used a nonlinear optimization technique to separate the intrinsic and reflex torques based on a priori information about the expected range of the parameters [6]. They applied this technique to the ankle, elbow and trunk joints [4], [7]. Our laboratory has focused on ankle stiffness and documented strong, hard nonlinearities in the reflex response which cannot be modelled using linear methods [8]. Consequently, we developed a nonlinear, parallel-cascade (PC) technique that uses little a priori information. It incorporates a nonlinear model of the reflex pathway and uses an iterative algorithm to estimate the intrinsic and reflex parameters [8]. This technique has been applied with success to the ankle and trunk joints [9], [2]. However, the method does encounter convergence problems under some operating conditions. Zhao

2 et al. addressed this by developing a parametric model of stiffness and estimating its parameters using a non-iterative SubSpace (SS) approach [2]. This solves the convergence problem but has two main limitations. First, it assumes a simple, second-order model for intrinsic stiffness and so cannot capture more complex intrinsic dynamics that arise because of the complex musculotendon structure or dynamics of the joint fixation to the actuator [22]. Secondly, while the SS model has excellent prediction capabilities, its structure provides little physiological insight. This was because the SS method estimates an over-parameterized, discrete-time, state-space model of the reflex pathway in which each parameter depends on both the static nonlinearity and the linear dynamics. Thus, while the model can predict the reflex torque accurately, its parameters cannot be used directly to recover either the shape of the nonlinearity or properties of the linear element. Doing so requires an additional step by identifying a Hammerstein system relating the input velocity to the reflex torque predicted by the SS model. This paper describes a new identification approach that uses a new model structure to address these limitations. Intrinsic dynamics are represented by a two-sided Impulse Response Function (IRF) that can account for complex dynamics. Moreover, this model incorporates a novel state-space model for the reflex pathway that has a concise set of parameters, each directly related to only a single element of the parallel-cascade model. The paper then develops a Structural Decomposition SubSpace (SDSS) algorithm that uses orthogonal projections to decompose the intrinsic and reflex torques and then estimates the model parameters accordingly. II. MODEL FORMULATION AND IDENTIFICATION This section formulates a state-space model for dynamic ankle stiffness and develops a method for its decomposition and identification. Throughout this paper, vectors, matrices and scalars are shown in roman bold-face uppercase, uppercase and lowercase letters respectively. The continuous time argument is shown by t and the discrete time argument by k. The symbol ( ) indicates the value of ( ) contaminated by additive noise, and ( ) ˆ is its estimate. A detailed derivation of the method is available as supplementary materials. The code has been released in our NonLinear Identification (NLID) toolbox with a number of examples and demos available for download from our public GitHub repository (reklab public). A. Problem Formulation Fig. shows the parallel-cascade model of dynamic ankle joint stiffness that has proven to be an accurate model in both normal and pathological subjects [7]. The intrinsic pathway is a dynamic linear element while the reflex pathway has a Block Oriented NonLinear (BONL) structure that is cascade of a delay, differentiator, static nonlinearity and dynamic linear system. We will use this model in discrete time and transform it to a Multiple-Input Single-Output (MISO) linear system to take advantage of linear identification techniques. public Intrinsic pathway Frequency (Hz) pos(t) tq(t) Delay Differentiator Reflex pathway Gain (db) Linear dynamics Static nonlinearity v d (t) m z(t) th Gain (db) Linear dynamics Frequency (Hz) tq I (t) tq R (t) tq v (t) n(t) tq ~ ( t) Fig.. Parallel-cascade model of the ankle joint stiffness. The input is joint angular position (pos(t)) and output is the joint total torque ( tq(t)). The intrinsic pathway is modelled by a linear system (high-pass filter), and the reflex pathway modelled as a cascade of a delay operator, differentiator, static nonlinear element (threshold-slope) followed by a linear element (low-pass filter). The measured output torque ( tq(t)) is the sum of intrinsic (tq I (t)), reflex (tq R (t)) and voluntary (tq v(t)) torques and measurement noise (n(t)). The MISO, linear, state-space model for the total ankle joint stiffness is (see the Supplementary Materials and Table I for list of parameters and signals): { X R (k + ) = A R X R (k) + B T U T (k) tq(k) = C R X R (k) + D T U T (k) + tq v (k) + n(k) () where U T (k) is a new input signal constructed from current, past and future values of the position and a basis function expansion of the delayed velocity: U T (k) = [U R (k) U I (k)] U R (k) = [ g (v d (k)) g p (v d (k)) ] (2) U I (k) = [ pos(k T Max ) pos(k + T Max ) ] where g i ( ) is the i-th basis expansion of the delayed velocity v d (k) (Tchebychev polynomials are used in this study). This model represents the dynamics as a set of first-order differential equations where X R (k) is the state vector and {A R, B T, C R, D T } are the state-space matrices that contain the unknown parameters of the parallel-cascade elements. B. Identification Identification of this model structure proceeds as follows. First, use the Past Input-Multivariable Output Error StatesPace (PI-MOESP) method to estimate the order of the reflex dynamics and the matrices ÂR and ĈR using U T (k) as the input, and tq(k) as the output of this model. PI-MOESP is available in the Subspace Model Identification (SMI) toolbox and gives unbiased estimates in the presence of arbitrary colored noise [23]. Next, use the estimates  R and ĈR to define a data equation, separating the intrinsic and reflex parameter sets (see the Supplementary Materials VI-C for details): TQ = TQ I + TQ R + N (3) ) = Ψ I (U I ) Θ I + Ψ R (U R, ÂR, ĈR Θ R + N (4) where TQ, TQ I, TQ R are the vectors of the total, intrinsic and reflex torques and N is the noise vector. Θ I and Θ R are the intrinsic and reflex unknown parameter sets and Ψ I and Ψ R are their regressors created from U I (k) and {U R (k), ÂR, ĈR} respectively (see (VI-C) for details).

3 3 R^ I R R Use ˆΘ I to estimate the intrinsic and reflex torques: R I R^ R TQ R Fig. 2. 2D geometrical representation of intrinsic, reflex and total torques and their spaces used for the decomposition of the pathways. R I is the column space of the intrinsic torque and RI is its perpendicular complement. R R is the column space of the reflex torque and RR is its perpendicular complement. Note that projections of the intrinsic and total torques onto RR, the perpendicular complement of the reflex space, are equal (C) which is the key equation for decomposition. TQ C TQ I ) Decomposition: Use orthogonal projections to decompose the total torque into its intrinsic and reflex components. Fig. 2 demonstrates the decomposition geometrically, in 2D for simplicity. The total torque is the vector sum of intrinsic and reflex torques, and noise. R I (R R ) is the range, i.e. column space, of the intrinsic (reflex) torque and RI (RR ) is its perpendicular complement. Define orthogonal projection operators on R I, RI, R R, RR : { P I = I P I P R = Ψ I Ψ I = Ψ R Ψ R { P I P R = I P R where I is the identity matrix. Fig. 2 makes it evident that the projection of total torque on R R is equal to the projection of intrinsic torque on R R : Now, project C on R I : C = PR TQ I = PR TQ () P I (I P R ) TQ I = P I (I P R ) TQ (6) Projection of TQ I on R I is equal to itself (P I TQ I = TQ I ), hence (6) becomes: (I P I P R ) TQ I = P I (I P R ) TQ (7) Replace TQ I by Ψ I Θ I : (I P I P R ) Ψ I Θ I = P I (I P R ) TQ }{{} (8) F 2) Identification of the Intrinsic Pathway: Estimate the intrinsic parameters using least squares: ˆΘ I = F P I (I P R ) TQ (9) ˆTQ I = Ψ I ˆΘ I ˆTQ R = TQ ˆTQ I () This has decomposed the intrinsic and reflex torques besides estimating the intrinsic IRF. Remark : This decomposition will be unbiased provided that noise (tq v (k) + n(k)) is not correlated with the input signal. If this assumption is satisfied, then the noise vector will be perpendicular to the intrinsic and reflex subspaces and their perpendicular complements. Thus, projection of the measured torque ( TQ) in () on to RR and R I will eliminate the effects of noise. 3) Identification of the Reflex Pathway: Use the New Sub- Space (NSS) Hammerstein identification method to estimate the reflex pathway using v d (k) as input and ˆTQ R as output. This method divides the parameter set Θ R into two subsets: the coefficients of the nonlinearity Λ, and the parameters of linear system s state-space model BD = {b b m d} T. These are then estimated iteratively. C. Algorithm The following algorithm summarizes the decomposition and identification steps.. Record N samples of position input and total torque output. 2. Construct the input signal U T (k) using (2). 3. Use PI-MOESP with U T (k), as the input and noisy torque, tq(k), as the output to estimate the order of the reflex system m and the state-space matrices ÂR and ĈR. 4. Construct the regressors Ψ I and Ψ R by using the estimated ÂR and ĈR as in Supplementary Materials (9).. Estimate the intrinsic parameters ( ˆΘ I ) using (9). 6. Estimate reflex torque, TQ ˆ R, using (). 7. Use the NSS method [24] to identify the reflex Hammerstein structure from the delayed velocity (v d (k)) as the input and the estimated reflex torque ( ˆtq R (k)) as the output. A. Methods III. SIMULATION STUDIES The performance of the SDSS method was evaluated in studies designed to replicate important experimental conditions by simulating a realistic model with experimentally recorded input and noise sequences. Note that in this paper, we identify an IRF model for the intrinsic stiffness and a state-space model for the reflex linear dynamics (see model formulation and identification). However, we use the frequency response function representation for visualization purposes. ) Model: The model in Fig. was simulated in MATLAB Simulink. The intrinsic pathway was modelled using an IRF estimated previously [22]; Fig. shows the gain of its frequency response. The reflex pathway parameters were based on those from [2]; the nonlinear element was a threshold at.7 rad/s, and the linear element was the second-order, lowpass filter:

4 4 Position (rad) Noise (Nm) Total ankle torque (Nm) (A) (B) (C) Time (s) Fig. 3. A second segment of one realization of the simulated stiffness signals (SNR=dB) (A) position input; (B) output noise; (C) total torque output (sum of intrinsic, reflex and voluntary torques and noise). H = G s 2 + 2sζω + ω 2 () where G, ζ and ω were 8.( Nms rad rad ),.62 and 3.6 ( s ) respectively. 2) Input and Noise: We used realizations of input and noise from a bank of experimentally recorded signals to ensure realistic simulation for validation of the new method. These experimental signals are described in detail in the next section. Each noise realization was scaled to generate the required signal to noise ratio (SNR) defined as: N k= SNR(dB) = log tq2 (k) N k= (tq (2) v(k) + n(k)) 2 Fig. 3 A,B,C demonstrate s segments of input realization, noise realization scaled to generate an SNR of db and the total measured torque (total torque plus noise) respectively. Thus, in these experiments, the input has random amplitude whose probability is non-uniform and skewed toward larger values and the noise has a slow-varying component. 3) Identification: The performance of SDSS was evaluated and compared to that of the PC [8], and SS methods [2]. The basis functions used to describe the static nonlinearity of the reflex pathway were 2 th -order Tchebychev polynomials for all methods. Polynomials with higher orders did not improve the identification accuracy and we found this order adequate to approximate complex nonlinearities observed in experimental data. The gain of the reflex stiffness pathway can be distributed arbitrarily between its nonlinear and linear components since the intermediate signal z(k) is not measured (Fig. ). For consistency reasons, the gain of the linear element of the reflex pathway was set to so that the reflex gain was assigned to its nonlinear element only. For this normalization, the gain of the linear element was estimated as the average of the gain in the pass-band region (- Hz) of its frequency response. 4) Monte-Carlo studies: The algorithms performance was evaluated in a series of Monte-Carlo simulations. For each series, trials were simulated with the same system parameters but with different realizations of the input and noise signals, selected randomly from the input-noise libraries. Simulations were done at khz and signals were decimated to Hz for analysis. Fig. 3 shows s of the input, noise, and output+noise from a typical simulation trial. ) Errors in estimates: SDSS and PC provide direct estimates of all parallel-cascade elements. Two types of errors were used to quantify the accuracy of the estimates: { bias error = ρ E(ˆρ) random error = E (ˆρ E(ˆρ)) 2 (3) where ρ is the true system and ˆρ is its estimate. The bias error provides a measure of how well the estimates retrieve the true system on average and the random error is a measure of the trial-to-trial variance of the estimates. 6) Index of the Probability of Superior Performance (IPSP): The simulation gave access to the true intrinsic and reflex torques. So, the different methods were compared in terms of how well their estimates predicted the torques. This was measured in terms of identification percentage Variance Accounted For (%VAF): ( %VAF intrinsic = var( ) ˆtq I (k) tq I (k)) var(tq I (k)) ( %VAF reflex = var( ) ˆtq R (k) tq R (k)) var(tq R (k)) The %VAF distributions were not Gaussian, so the significance of differences between methods was evaluated nonparametrically. For each Monte-Carlo trial, the difference between the %VAF of the SDSS and each of the other method was computed. The percentage of times that this difference was positive was defined as the: index of the probability of superior performance (IPSP). Thus, an IPSP of indicates that the SDSS predictions are always worse than those of the other method; an IPSP of. means that predictions of the two methods are equally well; and an IPSP of would mean that the SDSS predictions are always the best. 7) Decomposition Error: A %VAF less than could arise from either over or underestimating the output of a pathway. To distinguish these, the difference between the true and predicted output powers for each pathway was examined. If a pathways is estimated correctly, this should be close to zero, values greater than zero would indicate underestimation of the pathhway and values less than zero would indicate overestimation. B. Results ) Accuracy/Precision: The first series of Monte-Carlo simulations examined the accuracy with which elements of the

5 Random error Bias error (A) (B) INTRINSIC INTRINSIC 3 PC SDSS.8 Gain (db) Gain (db) (C) - Frequency (Hz) REFLEX NONLINEARITY (D) Frequency (Hz) REFLEX NONLINEARITY Output (rad/s) Output (rad/s) Input (rad/s) 2 (E) REFLEX LINEAR DYNAMICS Input (rad/s) 2 3 LINEAR DYNAMICS (F)REFLEX Gain (db) Gain (db) Fig. 4. Estimates of the elements of the parallel-cascade model (blue) using the PC and SDSS methods from a Monte-Carlo simulation of trials with SNR=dB superimposed on the true model (red). Left column: PC results; Right column: SDSS results. Estimates of the intrinsic stiffness using: (A) PC and (B) SDSS; reflex static nonlinear element using: (C) PC ;(D) SDSS; reflex linear element using: (E) PC; (F) SDSS. parallel-cascade model were estimated by the PC and SDSS methods. Noise was scaled to generate an SNR of db. Note that the SS method does not explicitly estimate the elements of the parallel-cascade model. Consequently SS results were not included when evaluating the accuracy with which the elements of the parallel-cascade model were estimated. Fig. 4 shows the estimates produced by PC (left column) and SDSS (right column). The true values used in the simulation are shown in red. By inspection, it is evident that the estimates of all three elements produced by SDSS are closer to the true values and are less variable than those from PC. Fig. demonstrates the bias and random errors. Fig. A shows that the bias error for the SDSS estimates of the intrinsic pathway was close to zero at all frequencies. In contrast, the bias error of PC was positive at low frequencies indicating that it overestimated the low frequency gain of the intrinsic pathway (elastic gain). The PC error peaked around Hz (near the break frequency) that indicates that the pathway s break frequency was located incorrectly. Fig. B shows that the random error was substantially lower for SDSS than for PC. Fig. C,D show the estimation errors for the static nonlinearity. The bias errors of both methods were similar for negative Frequency (Hz) - Frequency (Hz) Fig.. Bias and random errors in the estimates of the parallel-cascade elements using the PC (blue) and SDSS (dashed red) methods from a MonteCarlo simulation of trials with SNR=dB. Left column: bias error; Right column: random error. Error in the intrinsic estimates: (A) bias error and (B) random error; Error in the reflex nonlinear estimates: (C) bias error and (D) random error; Error in the reflex linear estimates: (E) bias error and (F) random error. velocities where there was little reflex response. However, for positive velocities, where the reflex response was larger, PC estimates consistently underestimated the gain while the SDSS bias error was close to zero. The random errors of both estimates were near zero in the central region and increased near the ends. However, random error remained low over a much wider range of velocities for SDSS than for PC. Fig. E,F shows the estimation errors for the linear dynamic element of the reflex pathway. The bias error was close to zero at frequencies below Hz for both methods and then increased at higher frequencies where there was less input power. In contrast, the random error was much lower for SDSS than PC at frequencies up to Hz. The large bias errors of the PC estimates indicates that the decomposition of the intrinsic and reflex torques was less accurate. This interpretation is confirmed in Fig. 6 which shows a box and whisker plot of the decomposition error. Thus, it was distributed symmetrically about zero for SDSS, indicating that the torques were decomposed accurately. In contrast, it was distributed asymmetrically for PC; skewed to negative values for the intrinsic pathway and to positive values

6 6 %Normalized power estimation error SDSS INTRINSIC PC %Normalized power estimation error SDSS REFLEX Fig. 6. The decomposition error, difference between the true and predicted output powers for the (A) intrinsic; (B) reflex pathways. The black line shows the mean value; the light box shows the 2% and 7% percentiles, the dark box shows the 2.% and 97.% percentiles, and the filled circles show the residuals. Mean %VAF Mean %VAF Intrinsic 9 SDSS PC SS SNR(dB) 9 9 Reflex power/intrinsic power Mean %VAF Reflex (A) (B) * ** ** ** ** ** * ** ** ** ** ** ** oo oo oo oo oo o o oo oo oo oo oo PC SNR(dB) (C) (D) ** * * * * ** ** * ** ** ** ** ** ** o o o o o o o o o o o Mean %VAF Reflex power/intrinsic power Fig. 7. Performance of the SDSS, SS and PC as functions of SNR(A,B) and the ratio of reflex to intrinsic torque power at SNR = db (C,D): (A) Mean value of %VAF intrinsic ; (B) Mean value of %VAF reflex ; (C) Mean value of %VAF intrinsic ; (D) Mean value of %VAF reflex. The circles and stars show the range of the IPSP for comparison to the PC and SS methods respectively. One star (circle) indicates that SDSS was more accurate with probability of.8 or higher. Double stars (circles) indicates that SDSS was more accurate with probability of.9 or higher. for the reflex pathway. Thus, the PC method consistently overestimated the intrinsic stiffness and underestimated the reflex stiffness. 2) Robustness: Fig. 7A-B summarizes the results of the robustness of the three methods as a function of the noise power. The SDSS estimates of the intrinsic pathway were the most accurate, they had the largest mean %VAF at SNRs greater than and the IPSP analysis demonstrated that the probability was larger than.9 at SNRs above db. Results for the reflex pathway was similar.the SDSS estimates were always the most accurate. They had the highest mean %VAFs at all SNRs. The IPSP analysis showed that the probability was greater than.9 for all SNRs larger than db. 3) Relative contribution of intrinsic and reflex pathways: Monte-Carlo simulations were also used to evaluate sensitivity to changes in the relative magnitudes of the intrinsic and reflex torques. Simulations were run using the nominal model but with reflex gains spanning the range reported for normal and spastic subjects (-7 Nms rad ) [9], [7]. The noise level was adjusted to maintain a constant SNR of db. Fig. 7C shows the results for the intrinsic pathway identification. The mean %VAF for the SDSS was greater than other methods at all reflex gains. IPSP values were large for low reflex gains and decreased as the reflex power increased. Thus, SDSS was the most accurate method for low reflex contributions while all three methods performed well for larger reflex contributions. Fig. 7D shows the results for the reflex pathway; SDSS had the largest mean %VAF. IPSP values were large indicating that SDSS was significantly the most accurate method for the reflex pathway identification. All three methods performed poorly for the lowest reflex gain but SDSS degraded the least. IV. EXPERIMENTAL STUDIES The practical application of SDSS was demonstrated by estimating the intrinsic and reflex stiffnesses at the ankle during tonic isometric contractions. Three levels of contraction were examined to explore the performance of the intrinsic and reflex estimators at different total stiffnesses and relative contributions of the pathways. A. Methods ) Apparatus: The experimental apparatus was similar to that described in [24]. The subjects left foot was attached to a hydraulic actuator which delivered position perturbations. By convention, the neutral (zero) position was defined to be 9 degrees angle between the subjects foot and shank. Positions dorsiflexed from the neutral and torques tending to dorsiflex the ankle were taken as positive. EMG signals were recorded from the Soleus and Tibialis Anterior muscles using single differential surface electrodes supplied with the Bagnoli systems. They were amplified ( times), bandpass filtered (2-2Hz), and full-wave rectified. The reference electrode was a DermaSport placed on the knee which was immobilized throughout the experiment. All signals were low-pass filtered at 486.3Hz and digitized (24bits) using a set of NI-4472 A/D cards. 2) Subjects: Five subjects (three females) aged between 2 and 29 with no history of neuromuscular disorders were examined on three different days. Subjects gave informed consent to the experimental procedures that had been approved by McGill University Institutional Review Board. Subjects were instructed to ignore the ongoing position perturbations and generate a constant muscle activation level only with the aid of the visual feedback on an overhead monitor. The visual feedback contained the low-pass filtered ankle torque to remove the effect of position perturbations. 3) Trials: Experimental Input: The properties of the input signal will strongly influence the identification performance. Consequently, we built a library of realistic inputs by recording the position perturbations resulting when different realizations of a piecewise constant, Pseudo Random Arbitrary Level Distributed Signal (PRALDS) were applied as input to a hydraulic actuator while subjects maintained an isotonic contraction. The PRALDS switching

7 7 time varied randomly over [-2] (ms) and its peak amplitude was scaled to.4 (rad). Trials were acquired with contraction levels at,, and % of the subjects Maximum Voluntary Contraction (MVC) in both plantarflexing and dorsiflexing directions and the joint angles placed at -.4, and.22 (rad). Experimental Noise: Output noise is often treated as white and Gaussian when evaluating the performance of identification algorithms. However, the noise associated with the torque in stiffness identification experiments is neither white nor Gaussian [26]. Rather, it comprises (i) a white Gaussian component due to electronics and signal conditioning circuitry and (ii) a slow-varying, non-gaussian component due to variations in the voluntary torque. Consequently, to ensure that our simulations represented noise realistically, we built a library of noise signals by recording the torque generated while subjects attempted to maintain a constant muscle contraction with no position perturbation. Trials were acquired at the same experimental conditions as experimental inputs. The noise library comprised 2 records of voluntary torque, each 6s long. Stiffness Identification: The mean joint position was placed at.22rad. The joint was perturbed using realizations of PRALDS inputs similar to those described in the experimental input section. Subjects were instructed to generate constant voluntary torques proportional to their MVC: % of plantarflexion MVC (PF trial), no muscle activation (REST trial), and % of dorsiflexion MVC (DF trial). One trial, lasting 6 seconds was recorded at each contraction level. 4) Analysis: The torque records were inspected for nonstationarity using the R locits package by examining their Haar wavelet coefficients across time [27]. The test demonstrated that the coefficients were not large enough to reject the null hypothesis of non-stationarity. Consequently, the data were weak-sense stationary. We used the SDSS and PC methods to identify the parallelcascade structure. We also compared SDSS to the first stage of a set of analytical methods that use a two-stage approach to separate intrinsic and reflex torques. These methods first use linear methods to estimate a frequency response function describing the overall dynamics relating position and torque. Then, a constrained optimization technique is used to separate the intrinsic and reflex responses based on a priori knowledge of their properties [4], [6], [7], [28]. The common parameter of the parallel-cascade and the linear model structures was the elastic gain. Thus, we compared those estimated from the SDSS, PC and linear methods. The elastic parameter was parameterized as the DC gain of the frequency response function of the intrinsic pathway from the PC and SDSS methods. We also took the DC gain of the identified frequency response function as a measure of the elastic parameter from the linear method. The reflex pathway nonlinearities were parameterized by fitting a model consisting of a threshold and slope using the trust-region-reflective method from MATLAB s optimization toolbox to the Tchebychev estimates. We used the NSS Hammerstein identification method as described in [24] to model and predict the EMG signals. In this Position(rad) Torque(Nm) Torque(Nm) Torque(Nm) EMG(v).2.2. (C) (D).2 (A) (B) (E).4 Position Total torque Intrinsic torque Reflex torque Soleus EMG Measured torque Predicted torque Measured EMG Predicted EMG Time(s) Fig. 8. Experimental data and SDSS predictions for a typical subject during a PF trial (% of plantarflexion MVC): (A) measured position; (B) measured torque along with the predicted total torque; (C) predicted intrinsic torque; (D) predicted reflex torque; (E) measured EMG along with the predicted EMG response. experimental condition, we did not find any strong correlation between the joint perturbation and the EMG response of the Tibialis Anterior muscle which has been demonstrated previously [29]. Thus, we only present results for the Soleus muscle. Finally, we obtained parameter values from the SDSS method across the PF, REST and DF conditions and normalized them to the REST condition for comparison purposes. We used a sign test with % significance level to test the significance of differences between the different conditions. B. Results Fig. 8 shows a segment of the PF trial with the least VAF. Fig. 8A shows the recorded perturbations which demonstrates the random switching nature of the perturbations. Fig. 8B shows the measured total ankle torque (blue) along with that predicted from the model identified by SDSS (red). It is evident that the two curves were very similar; indeed the identification %VAF was 87%. Fig. 8C,D show the time history of the predicted intrinsic and reflex torques, illustrating how the two mechanisms contributed to the overall torque. Fig. 8E shows the rectified EMG signal together with the predicted one. See Supplementary Materials VI-D for the identified models for this subject in the REST and DF conditions as well. We compared the SDSS estimates of stiffness with those obtained using linear frequency response techniques. The SDSS models were much more accurate than the linear models;

8 8.8 (A) %VAF total 6 4 (B) Intrinsic Gain (elastic stiffness) Coherence SDSS Linear Frequency(Hz) Fig. 9. The coherences between the measured torque and that predicted by the linear model (red) and the SDSS model (blue) demonstrates that SDSS accounted for more output power below Hz. the average torque identification VAF of SDSS models was 32% larger than that of the linear fits for all subjects and trials. Fig. 9 shows the coherence between the observed torque and the SDSS (blue) and linear (red) model predictions. It is evident that coherence of the linear model was significantly lower than that of the SDSS in the 2-Hz bands. On average, the elastic parameter from SDSS was 6.8% (with 9% range of [38.2,92.]%) larger than that from the linear method and 6.4% (with 9% range of [-8.2,3.]%) lower than that from the PC method. We also quantified the accuracy of the PC and SDSS methods in predicting the torque responses in terms of identification VAF% (see methods). Fig. summarizes the individual VAFs. Both methods predicted the measured total torque equally well for all trials (Fig. A); the %VAF total was 9.4% ± 2.% for the SDSS method and 9.2% ± 2.% for the PC method. The identification %VAFs for the Soleus EMG were 8.2% ± 4.%, 42.% ± 27.% and.8% ± 28.% for the PF, REST and DF conditions respectively. The two algorithms, however, assigned the pathway gains differently. Thus, Fig. B shows that the intrinsic gain (elastic stiffness) estimated using the PC method was always larger than or equal to that obtained from SDSS. In contrast, for the reflex pathways, the thresholds and gains (the slope of the nonlinearity) estimated with PC were generally lower than those estimated by SDSS (Fig. C-D). We then used SDSS to estimate the parallel-cascade parameters for all subjects. Fig. shows the parameters changed consistently with activation direction: (A) the elastic parameter of intrinsic stiffness was smallest in the REST and increased to PF and DF conditions; (B) the threshold of the static nonlinearity of the reflex stiffness was smallest in the PF and increased to DF and REST trials; (C) the slope of static nonlinearity was largest in the PF and decreased to DF and rest trials. Since both threshold and slope of the static nonlinearity contribute to the power of the reflex torque, we quantified the relative contribution of the reflex pathway to the total torque by computing %VAF reflex in (D) across all conditions which demonstrates that it decreased significantly from PF to REST and DF conditions; (E) similar to the reflex stiffness results, the threshold of the static nonlinearity of the EMG reflex PC %VAF threshold(rad/s) PC 9 9 PF REST DF SDSS VAF total.. (C) Nonlinearity s threshold.. threshold(rad/s) SDSS K(Nm/rad) PC slope(nms/rad) PC K(Nm/rad) SDSS 2 2 (D) Nonlinearity s slope 2 2 slope(nms/rad) SDSS Fig.. The distribution of pathway gains between SDSS and PC methods for all subjects at all contraction levels: PF, REST, DF: (A) identification %VAF using SDSS vs. PC; (B) estimates of the intrinsic pathway gain using SDSS vs. PC; (C) estimates of the reflex pathway threshold using SDSS vs. PC; (D) estimates of the reflex pathway slope using SDSS vs. PC. response was the smallest in the PF and increased to the DF and REST trials which was consistent with those identified from the stiffness estimates; (F) similar to the reflex stiffness results, the slope of static nonlinearity of the EMG was largest in the PF and decreased in the REST and DF trials. V. DISCUSSION This paper develops a state-space representation for parallelcascade model of ankle joint stiffness whose parameters are directly related to the underlying dynamics of the system. It presents a novel Structural Decomposition SubSpace (SDSS) method that decomposes the torque and identifies all elements using a subspace based method. Extensive simulations show SDSS decomposes the intrinsic and reflex torques and accurately estimates the key parameters. Pilot studies with experimental data also confirm that the new method works well in practice. A. Comparison to Previous Works We have presented a novel approach to decompose the intrinsic and reflex pathways only based on measurement of the position and torque. In this section, we compare and contrast it to other approaches to this problem. ) Empirical Methods: An experimental approach to the decomposition problem is to perform two trials: one with reflex activity and one where it is eliminated by electrical stimulation, ischemia, or vibration of muscle tendons [3], [3]. This early approach provided useful insights about the mechanisms of intrinsic and reflex stiffnesses. However, it suffers from two problems. First, the methods used to suppress

9 9 Normalized elastic Normalized threshold Normalized slope Normalized VAF Normalized threshold Normalized slope 2. (B) (C) (A) (D) (E) (F) Intrinsic Stiffness Elastic Reflex Stiffness Threshold Reflex Stiffness Slope Reflex Stiffness VAF Reflex EMG Threshold Reflex EMG Slope PF REST DF Subject Subject 2 Subject 3 Subject 4 Subject Fig.. SDSS estimates of parallel-cascade parameters: (A) intrinsic elastic stiffness in the PF (active ankle extensor muscles) and DF (active ankle flexor muscles) were larger than those of REST (no activation level) trials; (B) reflex nonlinearity s thresholds of PF were smaller than those of REST and DF; (C) VAF of reflex torque nonlinearity s slopes of PF were larger than those of REST and DF; (D) reflex torque VAF of PF were larger than those of REST and DF;(E) EMG: nonlinearity s thresholds of PF were smaller than those of REST and DF; (F) EMG: reflex nonlinearity s slopes of PF were larger than those of REST and DF. the reflex response will also alter the dynamics [32]. Second, it is difficult to match the muscle activation levels between the two trials and thus the biases in the estimates are unavoidable. The neuroflexor technique provides an overall value for the intrinsic and reflex contributions [33]. The wrist joint is perturbed twice with slow and fast velocities. It assumes that the joint response is purely mechanical to slow perturbations and the combination of mechanical and reflexive to fast perturbations. The main limitation is the assumption that the intrinsic stiffness is similar in the two trials which has been shown to change as a function of joint velocity [34]. These empirical methods do not quantify the mechanics in the dynamic phase of the response but give an overall number. The results cannot predict the torque response to the perturbations or to novel position trajectories. Consequently, their applications are less suitable for control purposes, e.g, the design of prosthesis. 2) Quantitative Methods: Some methods used EMG to guide the decomposition since it contains information about the muscle activation which includes the voluntary and reflexive drives; but is not explicitly related to the intrinsic torque [3]. These studies either identified the relation between EMG and torque in a separate isometric contraction trial [28] or assumed simple, a priori, linear models [36]. The relationship between EMG and torque may be complex and nonlinear particularly when conditions are not isometric [9]. Also, surface EMG measurement contains only a subset of motor unit action potentials and may also contain cross-talk from synergistic muscles or antagonistic muscles. This might deteriorate the confidence of such estimates. Nevertheless, the EMG-based approaches provide useful means to supplement and validate stiffness results. Indeed, in this paper, we also identified the reflex EMG response to supplement our stiffness estimates. The reflex EMG response followed the same pattern as the reflex stiffness identified with our method. This support the validity of our approach which we believe avoids potential problems with EMG based approaches. Another approach has been to assume a linear model structure for joint admittance [6], [4], [28], [7]. A linear frequency response function is fit to the position-torque signals and then parametrized using a nonlinear optimization. This effectively decomposes the intrinsic and reflex contributions by constraining the ranges of the parameter values based on a priori knowledge. While this approach is appropriate at joints with linear reflex responses, it is not not for those, such as the ankle, with strong nonlinear behaviour. We showed that the linear model could not predict the torque, as accurately as SDSS. The nonlinear model also encompasses linear structures. An added advantage of SDSS is that its convergence is guaranteed in contrast to nonlinear optimization where different sets of parameters might equally well predict the response. We did not compare the parameter values (except for the elastic parameter) resulting from SDSS to those from other quantitative methods that use linear frequency response function. This was because the assumed underlying models are structurally different. Thus, making direct comparison of parameters is difficult, if not impossible. However, it remains of interest to identify measures that are quantifiable across all different methods for a more comprehensive and objective comparison. B. Algorithmic Issues SDSS is a MOESP based subspace method [23] that is fast, and can deal with arbitrary coloured output noise, as is the case for joint stiffness identification. It has the potential for closed-loop identification [37] which is necessary when the joint interacts with a compliant load [38]. MOESP estimates the order of the reflex linear system before identifying the state-space matrices. This reduces the a priori information and is important for stiffness identification because previous experimental studies showed that the order of the reflex stiffness is variable [7]. SDSS uses a short, two-sided IRF to model the intrinsic dynamics whereas the SS uses a linear mass-spring-damper model. The use of IRF has two advantages. First, the intrinsic model may be more complex than a pure viscoelastic-inertial

10 structure due to the muscle tendon complex structure and/or fixation dynamics [22], [39]. Second, it avoids the need to numerically differentiate the position signal to estimate velocity and acceleration which is prone to numerical errors and noise amplification. The decomposition works under two assumptions: (i) that the intrinsic and reflex regressors (Ψ I, Ψ R in (3)) and their union (Ψ in (7)) are of full rank. That is when the intrinsic and reflex spaces are not parallel to each other in Fig. 2. Fortunately, this can be easily achieved by designing a persistently exciting input like PRALDS or PRBS; (ii) that the noise is not correlated with the input signal. The noise is the sum of voluntary torque and measurement noise. The measurement noise is presumably independent of the joint position. The voluntary torque is also unlikely to change significantly with the input position perturbations as they were very fast and small in amplitude. Furthermore, we instructed subjects to ignore the perturbations and generate a constant background activity. SDSS estimates the reflex elements using our NSS method which is an iterative method based on normalized alternative convex search (NACS) approach whose convergence has been recently established for any square integrable nonlinearity [4], [4]. Fortunately, the convergence of the iteration does not depend on the choice of initial condition (step 7 of the algorithm). Thus, SDSS is guaranteed to converge. We also directly used the NSS method to identify the Soleus EMG reflex response which shows that this method is capable of decoupling the reflex component of the signal from the measured total EMG. We used Tchebychev polynomials to represent the static nonlinearity of the reflex pathway mainly because they are linear in parameters (coefficients). Thus, their identification can be achieved using simple linear regression techniques. A potential limitation of the method is that the reflex delay must be known a priori while for the PC method it is estimated implicitly as part of the IRF description of the reflex linear element. Normally, this should not be a problem since the reflex delay can be estimated from an analysis of the EMG responses. It could be a problem if the reflex delay were to change dramatically from condition to condition or if more than one pathway with different delays were involved. SDSS identifies the short-range stiffness, i.e. under stationary conditions when displacement of the joint is small [42], [43]. Stiffness will change dramatically during large movements and therefore SDSS cannot be used to track stiffness. However, SDSS does show promise as the basis for new methods to identify time-varying, parallel-cascade models. We are taking steps to extend SDSS using linear parameter varying techniques to provide a model whose parameters change as a function of joint kinematics and/or muscle activation [44]. C. Simulation Methods Identification methods are often developed and validated in simulation studies with ideal inputs and noise. However, the performance might degrade considerably when applied to experimental data. For example, nonlinear actuator dynamics often cause the perturbations actually delivered to the joint to differ greatly from the ideally designed ones. Furthermore, in practice, noise is rarely white or Gaussian [26]. It is known that such non-ideal behaviours will heavily affect identification accuracy. Thus, we accounted for these differences and designed simulation scenarios by mimicking real experiments using a realistic model and input and noise sequences observed experimentally. We believe that by doing so we ensured that our simulation results will be more relevant to experimental conditions. D. Simulation Results ) Decomposition Accuracy (Bias Error): The PC and SS methods had difficulty decomposing the pathways whereas the SDSS method gave unbiased and accurate decompositions. It was more accurate than SS presumably because it uses more realistic dynamics for the intrinsic pathway. It was more accurate than PC presumably because of the effectiveness of the orthogonal projections in extracting each pathway s contribution from an output contaminated with noise and contribution of the other pathway whereas PC iteration does not guarantee convergence of the decomposition. Thus, PC overestimated the intrinsic and underestimated the reflex pathway (Fig.6). This was likely because of overestimating the low frequency gain of the intrinsic pathway (Fig.A) and underestimating the slope of the static nonlinearity (Fig.C). Consequently, the PC method assigned the pathway gains differently. 2) Precision (Random Error): The SDSS intrinsic and reflex estimators provided the most precise models. Statistical tests also revealed that SDSS was more robust than PC and SS at all conditions. SDSS was more robust than SS likely because its state-space structure for the reflex pathway had fewer parameters. SDSS was more robust than PC presumably because (i) the iterations used in PC are not guaranteed to converge and (ii) the SDSS considers a structure with fewer parameters. E. Experimental Results The experimental recordings were performed at a fixed dorsiflexed position in a torque matching task when subjects provided tonic contraction levels of (REST) and % of their MVC in both plantar (PF) and dorsiflexion (DF) directions. These levels were tested because it was known from previous experimental studies that they result in a wide range of the ratio of reflex to intrinsic powers. Thus, both intrinsic and reflex gains were large in PF; the intrinsic was the lowest and reflex was medium at REST; the intrinsic was large and reflex was the lowest in DF (Fig. ). SDSS successfully extracted stiffness models and accurately predicted the measured torque. The decomposed intrinsic and reflex torques are visually consistent with what would be expected from physiology. Thus, the reflex torque is smoothed version of the EMG response, demonstrating the low-pass filtering property of muscle activation/contraction dynamic. The intrinsic torque has a higher bandwidth reflecting the sensitivity to velocity and acceleration [4], [46].

11 The SDSS and PC methods both predicted the total torque equally well; they had similar %VAF total on average. However, they assigned the pathway gains differently (Fig. ). So, the PC method s intrinsic gain estimates were consistently larger than those of SDSS. In contrast, the PC estimates of reflex stiffness gain (nonlinearity s slope) were smaller than those of SDSS. As this was also observed in the simulation studies, we believe that the SDSS results likely reflect the actual physiological behaviour. F. Clinical Implications SDSS would be a valuable tool to objectively quantify neuromuscular disorders that change the muscle tone [47]. Altered muscle tone can be due to changes in visco-elastic properties [9], resulting in altered intrinsic torque; velocity or position thresholds of the reflexes [48], [49]; gain of the reflexes [46] or a combination of thereof. 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Kian Jalaleddini Dr. Jalaleddini received his PhD in Biomedical Engineering from McGill University, Montreal, Canada (2), the MASc in Electrical and Computer Engineering from Concordia University, Montreal, Canada (29), and the B.Sc in Electrical Engineering from University of Tehran, Tehran, Iran (27). He is currently a postdoctoral scholar in the Division of Biokinesiology and Physical Therapy at University of Southern California. He is a member of IEEE Engineering in Medicine and Biology and Control System Societies. He has served as the chair of the IEEE Engineering in Medicine and Biology Chapter, Montreal Section (2-2), and secretary of the IEEE Montreal Section (2-23). His research interests include analysis of biomedical signals and systems, development and application of system identification tools, exploring biomechanics of human joints and spinal reflexes. Ehsan Sobhani Tehrani Mr. Sobhani received his M.A.Sc. in Electrical and Computer Engineering from Concordia University, Montreal, Canada (28) and B.Sc. in Electrical Engineering from Amirkabir University of Technology, Tehran, Iran (23). Since 2, he is a Ph.D. candidate at the Department of Biomedical Engineering at McGill University, Montreal, Canada. Also, Mr. Sobhani is a Senior Systems Engineer and presently the Chief Technology Officer (CTO) of GlobVision Inc., Saint- Laurent, Quebec, Canada. He is a member of IEEE Control Systems Society (CSS), Engineering in Medicine and Biology Society (EMBS), and Aerospace and Electronic Systems Society (AESS). His areas of expertise are in human biomechanics and motor control; physiological signal processing; estimation, filtering and prediction; system identification; fault diagnosis; computational intelligence and machine learning; and modeling and simulation of dynamical systems (physiological and engineering). Robert E. Kearney Dr. Kearney received his undergraduate (968), Masters (97) and PhD (97) degrees in Mechanical Engineering from McGill University. He is currently Professor and Chair of the Department of Biomedical Engineering in the faculty of Medicine at McGill University. He maintains an active research program that focuses on using quantitative engineering techniques to address important biomedical problems. Specific areas of research include: The development of algorithms and tools for biomedical system identification; the application of system identification to understand the role played by peripheral mechanisms in the control of posture and movement; and the development of signal processing and machine learning methods for respiratory monitoring. His research is supported by NSERC, CIHR, and the Qatar National Research Fund. Dr. Kearney is a professional engineering, a Fellow of the IEEE, the Engineering Institute of Canada, the American Institute of Medical and Biological Engineering.