Time-Domain Optimal Experimental Design in Human Seated Postural Control Testing

Short abstract

We are developing a series of systems science-based clinical tools that will assist in modeling, diagnosing, and quantifying postural control deficits in human subjects. In line with this goal, we have designed and constructed a seated balance device and associated experimental task for identification of the human seated postural control system. In this work, we present a quadratic programming (QP) technique for optimizing a time-domain experimental input signal for this device. The goal of this optimization is to maximize the information present in the experiment, and therefore its ability to produce accurate estimates of several desired seated postural control parameters. To achieve this, we formulate the problem as a nonconvex QP and attempt to locally maximize a measure (T-optimality condition) of the experiment’s Fisher information matrix (FIM) under several constraints. These constraints include limits on the input amplitude, physiological output magnitude, subject control amplitude, and input signal autocorrelation. Because the autocorrelation constraint takes the form of a quadratic constraint (QC), we replace it with a conservative linear relaxation about a nominal point, which is iteratively updated during the course of optimization. We show that this iterative descent algorithm generates a convergent suboptimal solution that guarantees monotonic nonincreasing of the cost function value while satisfying all constraints during iterations. Finally, we present successful experimental results using an optimized input sequence.

1. Introduction

In recent years, clinical researchers have expanded the study of the human seated postural control system through the application of control theoretic analysis techniques [1,2]. These studies often rely on accurate models of the underlying human postural control dynamics to make the analysis tractable. However, humans possess a number of characteristics which may be impossible to measure accurately a priori, such as moments of inertia of body segments, center of mass (COM) locations, or feedback control gains. These parameters may instead be recoverable via examination of an experimental response. In the control sciences field, the set of techniques for recovering unknown or partially unknown model parameters from an experimental response are known as “system identification” techniques.

The design and optimization of system identification experiments is both a well-studied and ongoing problem [3–7]. Recent results in experimental optimization tend to favor the technique of optimizing the spectrum of the input signal [4–7]. This technique poses a number of challenges for human experiments. Human subjects tend to fatigue quickly during motor control testing, which limits the feasible length of each trial. This issue makes frequency-domain techniques for optimal experimental design difficult to use, because the time sequence may be too short to produce accurate results at low frequency or may not maintain sufficient frequency resolution over the entire spectrum. Thus, it would be preferable to design inputs in the time-domain (for short input sequences). Additionally, it is difficult to adapt frequency-domain optimization techniques to the number and variety of constraints within which an optimal solution for human testing must remain. For example, while it is obviously crucial to never apply enough force to a subject to cause injury, it is also important to make sure that the frequency characteristics of the input do not cause the subject to switch control strategies [8] (depending on the study goals). The input must not cause the subject’s motion amplitude to grow large enough to cause injury. Finally, inputs given to human subjects must not become predictable enough for the subject to adopt a feedforward-type control strategy when only the feedback mechanisms are to be estimated, which is the case in this work.

In the time-domain, a problem which optimizes the information in an input sequence while satisfying the preceding constraints can be most readily formulated as a nonconvex quadratically constrained quadratic program (QCQP), which tend to be NP-hard (nondeterministic polynomial-time hard) for many nontrivial problems [9]. While complete solutions to nonconvex QCQP’s are not yet available, current techniques for solving or approximately solving these problems tend to exploit some combination of semidefinite relaxation, linear relaxation, or randomization [9,10].

Our contributions in this work are as follows. We formulate a time-domain QP designed to optimize an experimental input for identification of parameters in a linear time-invariant (LTI) human seated postural control model. In this approach, we maximize the trace of the experiment’s FIM, an objective known as T-optimality [11], while ensuring that the system does not violate a number of input and state constraints. Maximizing a measure of the FIM will improve the quality of the estimated parameters in general. We formulate a novel QC on the input sequence’s autocorrelation function to ensure that the input is both unpredictable to subjects and possesses the desired frequency characteristics. By computing an iterative linear relaxation of this autocorrelation constraint, we are able to formulate the problem as a tractable nonconvex QCQP which can be solved locally at each iteration. We show that this iterative algorithm generates a convergent suboptimal solution that guarantees monotonic nonincreasing of the cost function value while satisfying all constraints during iterations. Our approach is applied to optimize a human seated balance identification experiment. We show simulation results for this design using model parameters derived from a preliminary set of subject parameters, and apply the optimized input to an experimental subject using a novel backdrivable robotic seat that we have developed. The experimental results demonstrate that we are able to reduce the variances of parameters estimated using the optimized input as compared to parameters estimated using a preliminary input of similar difficulty. A preliminary version of this Technical Brief without the experimental validation was presented at the 2014 American Control Conference [12].

The rest of this brief is organized as follows: In Sec. 2, we present the dynamic model for the seated balance task. In Sec. 3, we derive the QP formulation for the experimental optimization and present the constraints under which the optimization will operate. In Sec. 4, we show results from an input optimization for one subject, and apply the optimized input to the subject. Finally, in Sec. 5, we offer some concluding remarks.

Standard notation will be used. Let $\mathbb{R},{\mathbb{R}}_{+}$, and $\mathbb{B}$ denote, respectively, the sets of real, positive real, and binary (i.e., {0, 1}) numbers. The operators of expectation and covariance matrix are denoted by $\mathbb{E}$ and Cov, respectively. A random vector x, which has a multivariate normal distribution of mean vector μ and covariance matrix Σ, is denoted by $x~\mathfrak{N}(\mu ,\Sigma )$. An identity matrix of size n × n is denoted as I_n. A vector of zeros of length n is denoted as 0_n. The Kronecker product is denoted by the operator ⊗. The vectorization of a matrix A is denoted by vec(A). Other notation will be explained as it is used.

2. Experimental Modeling

We have developed a highly backdrivable torque-control robot that we intend to use for this and future studies on human seated postural control. This robot consists of a direct-drive backdrivable electric motor (CDDR C062C; Kollmorgen, Inc.) coupled to a free-spinning seat platform (Fig. 1), displacement sensors in the motor, and a real-time electronic control unit (cRIO-9022, National Instruments, Inc.). The motor is capable of providing peak torque inputs of up to 117 Nm. Since there is no gearbox or flexible coupling between the motor and seat, we can safely control the torque applied to the seat in a feedforward manner by specifying the motor current. This highly backdrivable configuration allows us to easily generate haptic effects (virtual springs, dampers, and other force fields) in addition to torque disturbances without needing direct torque measurements for feedback. Applying these effects through a direct-drive motor means that both stability and disturbance characteristics can be fine-tuned without physically reconfiguring the system and without needing to compensate for complicated gearbox effects (stiction, backdrivability, etc.) in the control algorithm. For safety purposes, the robot has mechanical stops at ±15 deg (±0.26 rad) which prevent motions of the seat platform from exceeding this range. The combined seat and actuator, along with control hardware, we refer to as the “backdrivable robot.”

Fig. 1.

Subject on the backdrivable robot

Using this robot, we have designed a seated balance experiment based on the one performed in Ref. [1]. In the current experiment, the subject sits atop the backdrivable robotic seat which is free to pivot about an axis perpendicular to the coronal plane (Figs. 1 and 2). The angle of the lower body from vertical is α ₁ and the angle of the upper body from vertical is α ₂. Similar to the convention in Ref. [1], the portion of the subject and seat below the fourth lumbar (L4) vertebrae is lumped into a single rigid element with mass M ₁ and moment of inertia (about the COM) of J ₁. The COM is at a distance l ₁ from the pivot point of the seat. Similarly, the portion of the subject above the L4 vertebrae is lumped into a rigid element with mass M ₂ and moment of inertia J ₂ about the COM. The COM of the upper body is a distance l ₂ from the L4 vertebrae. The L4 vertebrae itself is at a distance l ₁₂ from the seat pivot. The human can apply a control torque u _h about the L4 vertebrae, and additionally possesses an intrinsic rotational stiffness k _h and intrinsic rotational damping c _h about L4. We apply (through feedback) a virtual stiffness k _r and a virtual damping c _r about the pivot point, in addition to a torque disturbance u. The sum of these torques produce the total robot torque u _r about the pivot point, i.e., $u_{\mathrm{r}}=u-k_{\mathrm{r}}{\alpha }_1-c_{\mathrm{r}}{\stackrel{·}{\alpha }}_1$. The resulting dynamics can be determined by application of Lagrange’s equation to the model in Fig. 2.

Fig. 2.

Simplified mechanical diagram of the seated balance experiment

We model the closed-loop dynamical structure of the coupled human/backdrivable robot system as shown in Fig. 3. The plant model P represents the dynamic equations of the system (shown in Ref. [13]) linearized about the upright equilibrium point. The first output $z={\left[\begin{array}{cccc}\multicolumn{1}{c}{{\alpha }_1}& \hfill {\stackrel{·}{\alpha }}_1\hfill & \hfill {\alpha }_2\hfill & \hfill {\stackrel{·}{\alpha }}_2\hfill \end{array}\right]}^{\mathrm{T}}$ contains measurements of all the states of the system in Fig. 2 and is assumed to be exactly measurable by the human (via vestibular and proprioceptive mechanisms). The second output $z_r={\left[\begin{array}{cc}\multicolumn{1}{c}{{\alpha }_1}& \hfill {\stackrel{·}{\alpha }}_1\hfill \end{array}\right]}^{\mathrm{T}}$ contains measurements of the subset of states (seat angle and rate) that are measurable by the robot via its displacement sensors.

Fig. 3.

Block diagram of the seated balance experiment

There is a feedback controller R utilizing z _r such that the robot can simulate a desired dynamical system (in this case, a spring–damper system). The purpose of this controller is to slow the unstable poles of the closed-loop system enough for the system to be stabilizable by a human subject. Other studies of unstable seated balance commonly employ similar techniques, such as adding physical springs [14,15] or having the seat balance on a hemisphere instead of a point [1]. The backdrivable robot can additionally apply a torque disturbance u to the seat that can be used as an excitation signal for system identification [16]. Both of these signals are combined and converted into a torque through the robot motor M.

The model of the human has a feedback loop presumed to consist of a sensory delay e ^{− τs} implemented as a fifth-order Padé approximation, and an output feedback controller K such that (if we ignore delays), the human control is u _h = Kz, where $K=\left[\begin{array}{cccc}\multicolumn{1}{c}{-K_1}& \hfill -K_2\hfill & \hfill -K_3\hfill & \hfill -K_4\hfill \end{array}\right]$. We also include an approximation of muscle dynamics using a first-order filter with time constant T_ω. This formulation of the human feedback loop is similar to that used in other studies on postural control [1] and muscle control [17].

A motion capture system using LED markers is used to capture the upper and lower body angles for external processing (Visualeyez Motion Capture System, Phoenix Technologies Inc., Burnaby, Canada). However, the angular rates $({\stackrel{·}{\alpha }}_1,{\stackrel{·}{\alpha }}_2)$ are not directly measurable, so we reduce the plant output z to $y={\left[\begin{array}{cc}\multicolumn{1}{c}{{\alpha }_1}& \hfill {\alpha }_2\hfill \end{array}\right]}^{\mathrm{T}}$ via the operator D_y, i.e.,

$$\begin{array}{cc}\multicolumn{1}{c}{y}& \hfill =\left[\begin{array}{cccc}\multicolumn{1}{c}{1}& \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]z=D_{y}z.\hfill \end{array}$$

Additive sensory white noise w in the motion capture system is also presumed to exist.

A preliminary experiment was performed on a single subject in order to determine an initial parameter vector estimate ${\stackrel{\wedge }{\theta }}_0$ that could be used in subsequent optimizations. Because it only involved a single subject, this testing was designated as nonregulated research by the MSU Institutional Review Board (IRB). For this experiment, the virtual spring k _r and damper c _r were empirically tuned so that the subject needed to apply feedback to stabilize the seat, but did not fatigue excessively while maintaining upright balance. These values are listed in Ref. [13]. Ten trials of 30 s duration were performed. During each trial, the subject was given an identical torque input u designed as a pseudorandom binary sequence (PRBS) with significant power only below 1 Hz. A PRBS sequence was attractive for initial identification because it is commonly used for system identification [16] and has spectral characteristics similar to the “reduced-power” input method [18] that has been applied successfully in human studies. The amplitude of this sequence was tuned to 6 Nm, which was the maximum amplitude that the subject could consistently stabilize for 30 s without the seat contacting the mechanical stops at ±0.26 rad. The subject was given instructions to maintain stable upright posture on the seat while the perturbations were being applied. For each trial, the resulting angles α ₁ and α ₂ were measured using the motion capture system. Successful completion of a trial was defined as the subject being able to complete the entire 30 s trial without contacting the mechanical stops.

We have determined a set of estimated model parameter values ${\stackrel{\wedge }{\theta }}_0$ for the subject through a combination of nonlinear least-squares fitting to this preliminary experiment, mean parameters fitted in a similar study [1], and tabulated data from subject height and weight [19] with $\theta :={\left[K_1\hspace{1em}{K}_2\hspace{1em}{K}_3\hspace{1em}{K}_4\hspace{1em}{J}_1\hspace{1em}{J}_2\hspace{1em}{l}_1\hspace{1em}{l}_{12}\hspace{1em}{l}_2\hspace{1em}\tau T_{\omega }\right]}^{\mathrm{T}}$. The initial estimated values ${\stackrel{\wedge }{\theta }}_0$ of these parameters are listed in Ref. [13].

3. Experimental Optimization

3.1. QP.

Because all of the subsystems are linear and rational-ordered, the closed-loop system in Fig. 3 with the true parameter vector θ ₀ can be formulated as a discrete-time LTI state-space model of the form

$$\begin{array}{c}\multicolumn{1}{c}{x_{k+1}=A({\theta }_0)x_k+B({\theta }_0)u_k{y}_k=C({\theta }_0)x_k{\tilde{y}}_k=C({\theta }_0)x_k+w_k}\end{array}$$ (1)

with $x_k\in {\mathbb{R}}^{n_x},u_k\in \mathbb{R},y_k\in {\mathbb{R}}^{n_y},w_k\underset{˜}{\mathrm{i.i.d.}}\mathfrak{N}(0,\Sigma )\in {\mathbb{R}}^{{\mathrm{n}}_{\mathrm{y}}}$, i.e., independent and identically distributed (i.i.d.) Gaussian white noise process in time, $\theta \in {\mathbb{R}}^{n_{\theta }}$, and some sampling time T. The true parameter vector θ ₀ is presumed to belong to a compact (hyper-rectangular) set Θ such that ${\theta }_0\in \Theta :={\prod }_{i=1}^{n_{\theta }}[{\theta }_i^{min},{\theta }_i^{max}]$. If the parameter vector θ ₀ is known, then the matrices A(θ ₀), B(θ ₀), and C(θ ₀) of the closed-loop model in Eq. (1) can be computed numerically using the Matlab connect command (see more details in Ref. [13]). The system is defined over the time indices $k\in \mathfrak{K}:=\{0,\dots ,N\}$ such that t_k = kT. We define the error e_k between the nominal output y_k and the noisy output ${\tilde{y}}_k$ for a given time index k and the true parameter vector θ ₀ as

$$e_k({\theta }_0):={\tilde{y}}_k-y_k={\tilde{y}}_k-C({\theta }_0)A({\theta }_0)x_{k-1}-C({\theta }_0)B({\theta }_0)u_{k-1}$$ (2)

For the remainder of this brief, we will drop the explicit notational dependence on θ in A, B, and C.

Let us consider an experiment with an input sequence defined as u := [u ₀ ⋯ u_{N −1}]^T. Note that we can determine the system output y_k at an arbitrary time index k ≥ 1 when the input sequence [u ₀, u ₁,$\hspace{1em}\dots \hspace{1em}$,u_{k −1}]^T and initial state condition x ₀ are known. The complete solution to the discrete-time state–space system given in Eq. (1) is

$$y_k={\mathit{CA}}^k{x}_0+C\sum _{i=0}^{k-1}{A}^{k-i-1}{\mathit{Bu}}_i$$

Note that we can reconfigure this solution as a matrix operation

$$y=\left[\begin{array}{c}\multicolumn{1}{c}{y_1}\\ \multicolumn{1}{c}{y_2}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{y_N}\end{array}\right]=\left[\begin{array}{cccc}\multicolumn{1}{c}{\mathit{CA}}& \hfill \mathit{CB}\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{{\mathit{CA}}^2}& \hfill \mathit{CAB}\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{⋮}& \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \multicolumn{1}{c}{{\mathit{CA}}^N}& \hfill {\mathit{CA}}^{N-1}B\hfill & \hfill \dots \hfill & \hfill \mathit{CB}\hfill \end{array}\right]\left[\begin{array}{c}\multicolumn{1}{c}{x_0}\\ \multicolumn{1}{c}{u_0}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{u_{N-1}}\end{array}\right]=:\mathit{GU}$$

We now have a nonrecursive solution $y\in {\mathbb{R}}^{{\mathit{Nn}}_y}$ for all time k ≥ 1 given $U\in {\mathbb{R}}^{N+n_x}$. Note that the first element in U is x ₀. We can now define a vector form of the error $e=\tilde{y}-y={\left[\begin{array}{cccc}\multicolumn{1}{c}{e_1^{\mathrm{T}}}& \hfill e_2^{\mathrm{T}}\hfill & \hfill \dots \hfill & \hfill e_N^{\mathrm{T}}\hfill \end{array}\right]}^{\mathrm{T}}\in {\mathbb{R}}^{{\mathit{Nn}}_y}$.

The log likelihood function for a data set $\tilde{y}:={\left[\begin{array}{ccc}\multicolumn{1}{c}{{\tilde{y}}_1^{\mathrm{T}}}& \hfill \dots \hfill & \hfill {\tilde{y}}_N^{\mathrm{T}}\hfill \end{array}\right]}^{\mathrm{T}}$ given the true parametrization θ ₀ is

$$\begin{array}{c}\multicolumn{1}{c}{\ln p\left(\tilde{y}\right|{\theta }_0)=\sum _{k=1}^N\ln p\left({\tilde{y}}_k\right|{\theta }_0)=-\frac{N}{2}\ln 2\pi -\frac{N}{2}\ln \left|\Sigma \right|\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}-\frac{1}{2}\sum _{k=1}^N{e}_k^{\mathrm{T}}({\theta }_0){\Sigma }^{-1}{e}_k({\theta }_0)}\end{array}$$

The maximum likelihood estimator for θ ₀ is then given by

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{\wedge }{\theta }}_N}& \hfill =\mathit{arg}\underset{\theta \in \Theta }{min}(\frac{1}{N}\sum _{k=1}^N{e}_k^{\mathrm{T}}(\theta ){\Sigma }^{-1}{e}_k(\theta ))=\mathit{arg}\underset{\theta \in \Theta }{min}{J}_N(\theta )\hfill \end{array}$$

Under mild conditions [16,20], it can be shown that ${lim}_{N\to \infty }{\stackrel{\wedge }{\theta }}_N={\theta }_0=\mathit{arg}\underset{\theta \in \Theta }{min}{lim}_{N\to \infty }\mathbb{E}\{J_N(\theta )\}\mathrm{w}.\mathrm{p}.1,$ and that the prediction error converges in distribution to a normally distributed random variable [16,20,21] $\sqrt{N}({\stackrel{\wedge }{\theta }}_N-{\theta }_0)\stackrel{d}{\to }\mathfrak{N}(0,{\mathbb{I}}^{-1}(u;{\theta }_0))$ where $\mathbb{I}(u;{\theta }_0)$ is the FIM.

For a MIMO (multi-input multi-output) system, the FIM is an extension of the SISO (single-input single-output) case given in Refs. [22] and [23]

$$\begin{array}{c}\multicolumn{1}{c}{\mathbb{I}(u;{\theta }_0)={\mathbb{E}}_{\tilde{y}|{\theta }_0}[\left(\frac{\partial \ln p(\tilde{y}|\theta )}{\partial \theta }{|}_{\theta ={\theta }_0}\right){\left(\frac{\partial \ln p(\tilde{y}|\theta )}{\partial \theta }{|}_{\theta ={\theta }_0}\right)}^{\mathrm{T}}]=\left[\sum _{k=1}^N{\left(\frac{\partial e_k}{\partial \theta }{|}_{\theta ={\theta }_0}\right)}^{\mathrm{T}}{\Sigma }^{-1}\left(\frac{\partial e_k}{\partial \theta }{|}_{\theta ={\theta }_0}\right)\right]}\end{array}$$

Taking the partial of e_k with respect to the ith element of θ yields $(\partial e_k/\partial {\theta }_i)=-(\partial y_k/\partial {\theta }_i),i=\{1,\dots ,n_{\theta }\}$. Then, we have ${(\partial e/\partial {\theta }_i)|}_{\theta ={\theta }_0}=-{(\partial y/\partial {\theta }_i)|}_{\theta ={\theta }_0}=-{(\partial G/\partial {\theta }_i)|}_{\theta ={\theta }_0}U=:-H_{i}U$. We can combine these matrices H_i for each θ_i to form

$$\mathfrak{H}=\left[\begin{array}{cccc}\multicolumn{1}{c}{H_1}& \hfill H_2\hfill & \hfill \dots \hfill & \hfill H_{n_{\theta }}\hfill \end{array}\right]\in {\mathbb{R}}^{{\mathit{Nn}}_y\times n_{\theta }(n_x+N)}$$

Additionally, we form $\mathfrak{U}=I_{n_{\theta }}\otimes U\in {\mathbb{R}}^{n_{\theta }(N+n_x)\times n_{\theta }}$. We can then form the FIM for the system in Eq. (1) as

$$\mathbb{I}(u;{\theta }_0)={\left(\mathfrak{H}\mathfrak{U}\right)}^{\mathrm{T}}(I_N\otimes {\Sigma }^{-1})\left(\mathfrak{H}\mathfrak{U}\right)\in {\mathbb{R}}^{n_{\theta }\times n_{\theta }}$$ (3)

where all elements in $\mathfrak{H}$ are assumed to be bounded, i.e., ${\ell }_{h1}\le {\mathfrak{H}}_{\mathit{ij}}\le {\ell }_{h2}.$ Note that the FIM is defined using the true parameter vector θ ₀ [16]. However, in reality an optimization can only be performed based on the current best-estimate ${\stackrel{\wedge }{\theta }}_0$ [16]. Therefore, we will proceed from this point using ${\stackrel{\wedge }{\theta }}_0$ in place of θ ₀.

Amongst a number of different optimality conditions [11], we choose the T-optimality condition, which will maximize the trace of the FIM [3,24], and in turn provides an objective that is quadratic in u. Because of the potentially large number of free variables in u (for time-domain experimental design), choosing a cost function that is purely quadratic in u will allow us to efficiently solve the problem using a QP algorithm later. We therefore use a cost $J(u;{\stackrel{\wedge }{\theta }}_0)$ defined by

$$J(u;{\stackrel{\wedge }{\theta }}_0)=-\mathrm{trace}(\mathbb{I}(u;{\stackrel{\wedge }{\theta }}_0))$$ (4)

Note that both the FIM and $J(u;{\stackrel{\wedge }{\theta }}_0)$ are functions of the input sequence u, the initial condition x ₀, and estimated parameters ${\stackrel{\wedge }{\theta }}_0$ only. While the cost function $J(u;{\stackrel{\wedge }{\theta }}_0)$ is nonconvex in u [3], a general QP solver can be used to perform the unconstrained local minimization

$$u^{\star }=\mathit{arg}\underset{u}{min}J(u;{\stackrel{\wedge }{\theta }}_0)$$ (5)

3.2. Design Constraints.

In this brief, the quadratic optimization in Eqs. (4) and (5) is subject to the following constraints.

3.2.1. Input Limits.

Since the direct-drive motor should be restricted to only apply a safe amount of torque, we apply a constraint such that $-u_m\le u\le u_m,u_m\in {\mathbb{R}}_{+}$.

3.2.2. Output Constraints.

There is a finite angular range over which both the robot seat platform and the human torso can move. We therefore apply the constraint $-1^N\otimes y_m\le \mathit{GU}\le 1^N\otimes y_m$, where 1^N is a vector of ones of length N, and $y_m\in {\mathbb{R}}_{+}^p$ defines the maximum amplitude of each output individually. Additionally, the angular difference $\tilde{\alpha }={\alpha }_2-{\alpha }_1$ is limited by both the structure and flexibility of the subject’s lower back. By reformulating the closed-loop system in Fig. 3, we can form a structure G_δ similar to G where u is the input and $\tilde{\alpha }$ is the output. If ${\stackrel{\wedge }{\theta }}_0$ is known, then this reformulation can be performed numerically in Matlab using connect [13]. We then apply the constraint $-{\delta }_{\alpha }\le G_{\delta }U\le {\delta }_{\alpha },{\delta }_{\alpha }\in {\mathbb{R}}_{+}$.

3.2.3. Human Control Constraint.

The human subject is only capable of generating a finite amount of torque u _h. We can again reformulate the closed-loop system in Fig. 3 to form a structure G_u similar to G where u is the input and u _h is the output. Then, we apply the constraint $-u_{\mathrm{h}m}\le G_{\mathrm{u}}U\le u_{\mathrm{h}m},u_{\mathrm{h}m}\in {\mathbb{R}}_{+}$.

3.2.4. Autocorrelation Constraint.

In addition to the preceding linear constraints, it was desired to constrain the autocorrelation of the input sequence so as to reduce predictability of the signal while maintaining desirable spectral characteristics. The autocorrelation of a discrete real-time sequence u_k at lag j can be computed as $R_{\mathit{uu}}(u;j)=\sum _k{u}_k{u}_{k-j}$. We can reformulate this as the quadratic matrix multiplication

$$R_{\mathit{uu}}(u;j)=u^{\mathrm{T}}Q(j)u$$ (6)

where $Q(j)\in {\mathbb{B}}^{N\times N}$ is a Toeplitz matrix containing ones on its jth upper off-diagonal and zeros everywhere else, e.g.

$$R_{\mathit{uu}}(u;1)=u^{\mathrm{T}}\left[\begin{array}{ccccc}\multicolumn{1}{c}{0}& \hfill 1\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 1\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{⋮}& \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill & \hfill ⋮\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 1\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \end{array}\right]u$$

We consider the term R_uu(u) (with j omitted) to be the autocorrelation vector for all lags $j=\{0,\dots ,(N/2)-1\}$.

We desired the normalized autocorrelation of the first N/2 lags of the optimal input sequence autocorrelation to be within some region of our preliminary experiment’s PRBS signal autocorrelation $R_{\mathit{uu}}^{\star }$, i.e.

$$R_{\mathit{uu}}^{\star }-\beta \le \frac{R_{\mathit{uu}}(u)}{R_{\mathit{uu}}(u;0)}\le R_{\mathit{uu}}^{\star }+\beta$$ (7)

where β > 0 is a scalar constant. The constraint in Eq. (7) is quadratic in u based on the definition of R_uu(u; j) in Eq. (6).

Unfortunately, the minimization of $J(u;{\stackrel{\wedge }{\theta }}_0)$ subject to the constraints listed above is a nonconvex QCQP, the solution of which is still an open research question. Therefore, we propose an iterative linearization technique to find a good solution to Eq. (5).

3.3. Proposed Iterative Descent Algorithm.

Since we can not directly apply a QC such as the one in Eq. (7) to the QP, we propose to compute a linear relaxation of the autocorrelation about a nominal vector $\stackrel{\wedge }{u}$. This relaxation takes the form of a linearization based on a Taylor series expansion about $\stackrel{\wedge }{u}$, i.e.,

$${\stackrel{\wedge }{R}}_{\mathit{uu}}(\stackrel{\wedge }{u};u;j)={\stackrel{\wedge }{u}}^{\mathrm{T}}Q(j)\stackrel{\wedge }{u}+{\stackrel{\wedge }{u}}^{\mathrm{T}}(Q(j)+Q^{\mathrm{T}}(j))(u-\stackrel{\wedge }{u})$$

This constraint is made slightly more conservative than the true QC in Eq. (7) by shrinking the constraint boundary, i.e.,

$$R_{\mathit{uu}}^{\star }-\beta +\gamma \le \frac{{\stackrel{\wedge }{R}}_{\mathit{uu}}(\stackrel{\wedge }{u};u)}{{\stackrel{\wedge }{R}}_{\mathit{uu}}^0}\le R_{\mathit{uu}}^{\star }+\beta -\gamma$$ (8)

where γ s.t. 0 < γ < β is a small constant. Note that we normalize ${\stackrel{\wedge }{R}}_{\mathit{uu}}(\stackrel{\wedge }{u};u)$ by ${\stackrel{\wedge }{R}}_{\mathit{uu}}^0$, which we define as ${\stackrel{\wedge }{R}}_{\mathit{uu}}^0:=R_{\mathit{uu}}(\stackrel{\wedge }{u};0)$. Now, by ensuring that $\tilde{u}=u-\stackrel{\wedge }{u}$ is constrained to be small, a local solution can be found that satisfies the linear constraint in Eq. (8) but does not violate the quadratic autocorrelation constraint Eq. (7).

To ensure that the linearization in Eq. (8) is both always valid and more conservative than the true QC Eq. (7), we constrain the difference $\tilde{u}=u-\stackrel{\wedge }{u}$ such that

$$-{\delta }_u\le \tilde{u}\le {\delta }_u,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{\delta }_u\in {\mathbb{R}}^{+}$$ (9)

Therefore, when we allow only a small change in u, we may solve the following optimization:

$$u^{\star }=\mathit{arg}\underset{u}{min}J(u;{\theta }_0)$$ (10)

$$\begin{array}{cc}\multicolumn{1}{c}{\mathrm{subject}\mathrm{\hspace{1em}}\mathrm{to}:}& \hfill -u_m\le u\le u_m\hfill \\ \multicolumn{1}{c}{}& \hfill -1^N\otimes y_m\le \mathit{GU}\le 1^N\otimes y_m\hfill \\ \multicolumn{1}{c}{}& \hfill -{\delta }_{\alpha }\le G_{\delta }U\le {\delta }_{\alpha }\hfill \\ \multicolumn{1}{c}{}& \hfill -u_{\mathrm{h}m}\le G_{u}U\le u_{\mathrm{h}m}\hfill \\ \multicolumn{1}{c}{}& \hfill R_{\mathit{uu}}^{\star }-\beta +\gamma \le \frac{{\stackrel{\wedge }{R}}_{\mathit{uu}}(\stackrel{\wedge }{u};u)}{{\stackrel{\wedge }{R}}_{\mathit{uu}}^0}\le R_{\mathit{uu}}^{\star }+\beta -\gamma \hfill \\ \multicolumn{1}{c}{}& \hfill -{\delta }_u\le \tilde{u}\le {\delta }_u\hfill \end{array}$$ (11)

An overall solution is found by computing a series of successive solutions u ^{⋆ i} to the problem of Eq. (10) subject to the constraints in Eq. (11). For each iteration i, we perform a local linearization Eq. (8) of the quadratic autocorrelation constraint in Eq. (7) about $\stackrel{\wedge }{u}=u^{\star (i-1)}$ and solve for u ^{⋆ i}. Each solution u ^{⋆ i} becomes $\stackrel{\wedge }{u}$ in the next iteration of the solution. This is done so as to allow u to traverse a wide range while not violating the input linearization constraint in Eq. (9) at any point during the optimization. Each solution u ^{⋆ i} is found using Matlab’s quadprog general QP solver in combination with the YALMIP modeling toolbox.

Note that we are computing the optimization based on the estimate ${\stackrel{\wedge }{\theta }}_0$, instead of the true parameter vector θ ₀. This is a common problem in system identification, and can be dealt with via a number of methods, such as iterative system identification techniques [25].

3.4. Convergence Analysis.

In this section, we discuss the convergence properties of the proposed iterative descent algorithm proposed in Sec. 3.3.

First note that $J(u^{\star i};{\stackrel{\wedge }{\theta }}_0)\ge J(u^{\star i+1};{\stackrel{\wedge }{\theta }}_0)$ by the construction. Next, we show that J has a lower bound. This can be shown by the fact that the FIM in Eq. (3) has an upper bound with an assumption that all elements in $\mathfrak{H}$ are bounded, i.e., ${\ell }_{h1}\le {\mathfrak{H}}_{\mathit{ij}}\le {\ell }_{h2}.$ This follows from the fact that

$$\mathrm{trace}\left(\mathbb{I}(u;{\stackrel{\wedge }{\theta }}_0)\right)=\mathrm{trace}(I_N\otimes {\Sigma }^{-1}{\mathfrak{H}\mathfrak{U}\mathfrak{U}}^T{\mathfrak{H}}^T)=\mathrm{vec}{(I_N\otimes {\Sigma }^{-1})}^T\mathrm{vec}({\mathfrak{H}\mathfrak{U}\mathfrak{U}}^T{\mathfrak{H}}^T)\le {\ell }_T$$ (12)

since all elements in $\mathfrak{U}$ are also bounded due to the input constraints in the constrained optimization in Eq. (10).

Since the value J has a lower bound which is −ℓ_T from Eq. (12) and is monotonically nonincreasing during the iterations, it will converge to some value as iterations proceed.

Therefore, this iterative descent algorithm generates a convergent suboptimal solution that guarantees monotonic nonincreasing of the cost function while satisfying all constraints during iterations.

4. Case Study

We have performed a case study on a single subject to demonstrate our experimental optimization. The goal of the optimization is to determine an experimental input sequence that will minimize a measure of the covariance for the estimated parameters. This is achieved via a maximization of the experiment’s FIM trace subject to constraints as described in Eqs. (10) and (11). Using parameters ${\stackrel{\wedge }{\theta }}_0$ from [13], G, G_u, and G_δ from Sec. 3 were computed numerically using Matlab’s connect function. The limits applied to the optimization are listed, along with their sources, in Ref. [13]. We let $x_0={\left[\begin{array}{cc}\multicolumn{1}{c}{0.01}& \hfill 0_9^{\mathrm{T}}\hfill \end{array}\right]}^{\mathrm{T}}$, and since the sensor noise for both elements of y_k were approximately equal and uncorrelated, we let Σ = I. The initial input $\stackrel{\wedge }{u}$ was the same PRBS signal given to the subject in the preliminary experiment. Note that, in the preliminary experiment, the initial $\stackrel{\wedge }{u}$ was challenging enough that the subject required considerable practice to complete the trials successfully.

The descent algorithm in Algorithm 3.3 was applied using the initial parameter vector ${\stackrel{\wedge }{\theta }}_0$ from Ref. [13] and the initial PRBS input $\stackrel{\wedge }{u}$. For an input sequence with length N = 300 and a sampling time of T = 0.1 s, we were able to converge to a local suboptimal input sequence (E _stop = 1 × 10⁻³) in approximately 3.5 h on a 2.2 GHz Xeon server.

4.1. Optimization Results.

The optimal input u ^⋆ along with the change in the objective function with increasing i are shown in Fig. 4. We simulate the system in Fig. 3 with u(t) = u ^⋆ to produce the corresponding outputs y and differential angle $\tilde{\alpha }$ (Fig. 5). The final signal autocorrelation $R_{\mathit{uu}}^{\star }$ and its constraints are also shown in Fig. 5. None of the other constraints for the system were active. The solution u ^⋆ produces an approximately 1.6 times improvement relative to the initial $\stackrel{\wedge }{u}$ in the value of the objective function without violating any of the listed constraints.

Fig. 4.

The upper plot shows the optimal input sequence u ^⋆. The lower plot shows the change in the objective function $J(u;{\stackrel{\wedge }{\theta }}_0)$ with increasing iteration i.

Fig. 5.

Simulated results using the optimal input u ^⋆. The upper plot shows the simulated angles α ₁ and α ₂ versus time, along with their bounds. The center plot shows the differential angle $\tilde{\alpha }$ versus time along with its bounds. The bottom plot shows the optimal input signal autocorrelation $R_{\mathit{uu}}^{\star }$ along with its bounds, and the original signal autocorrelation R_uu for comparison. The constraints on u _h were not active during simulation.

4.2. Experimental Application.

To compare the variances of the parameters estimated using the optimal experiment, we performed an experiment using the same subject tested in Sec. 2. This experiment was again designated as nonregulated research by the MSU IRB. Ten trials of the 30 s length using the optimal input u ^⋆ were performed using an experimental setup otherwise identical to that in Sec. 2. The subject was able to successfully complete the ten trials of the experiment, although the subjective difficulty of the task was very high. The resulting mean best-fit parameters ${\stackrel{\wedge }{\theta }}_N$ are shown in Ref. [13].

In Table 1, we compare the variances across ten trials of the parameters estimated in the preliminary experiment done in Sec. 2 with the parameters estimated from the optimal experiment. For almost all parameters (see Table 1), the optimal experiment reduced the variances of the estimated parameters as compared to the initial PRBS input while the mean values (i.e., bias part) from the two estimators are similar (see Ref. [13]), which shows the advantage of our approach.

Table 1.

Variance of the parameters in ${\stackrel{\wedge }{\theta }}_0$ versus the variance of the parameters in ${\stackrel{\wedge }{\theta }}_N$

Parameter	${\stackrel{\wedge }{\theta }}_0$ Variance	${\stackrel{\wedge }{\theta }}_N$ Variance
K 1	2727	2238
K 2	2374	496.6
K 3	5.306 × 104	5840
K 4	1.621 × 104	1238
J 1	1.031	0.2304
J 2	0.228	0.298
l 1	0.0007639	0.0005456
l 12	0.0008781	0.001225
l 2	0.004951	0.0009266
τ	0.0004225	0.000375
Tω	0.005771	0.0004575

Because the sequence u ^⋆ is only optimal for a parameter vector ${\stackrel{\wedge }{\theta }}_0$, this technique could be employed as part of a broader iterative procedure [25]. After a u ^⋆ is found, a subject can be tested using u ^⋆ as the input and the resulting experimental response fitted to find ${\stackrel{\wedge }{\theta }}_N$. The parameters ${\stackrel{\wedge }{\theta }}_N$ can then be fed back as ${\stackrel{\wedge }{\theta }}_0$ in the next iteration of the input optimization and the process repeated until a desired level of convergence is achieved [25]. While this iterative scheme is widely used in practice, there is, however, no guarantee that it converges to the minimum of the cost function [20].

5. Conclusions

In this work, we have demonstrated a QP technique for generating an optimal experimental input for a human seated postural control identification experiment. To this end, we have formulated a quadratic objective function based on a measure of the FIM that will maximize the information present in the experiment for the proposed testing to improve the quality of the estimated parameters. We have formulated a set of output, input, and control constraints, in addition to a unique linearized autocorrelation constraint, such that the resulting input signal will be feasible for the proposed testing. The resulting solution u ^⋆ converged to a local solution without violating any of the prescribed constraints. We have additionally demonstrated an experimental application of this input signal in conjunction with our backdrivable robot, and shown that the resulting estimated parameters from the subject have lower variances than those recovered from a preliminary experiment, which is consistent with the goal of our optimization.

In future work, we intend to apply this technique in quantitative clinical testing, and extend our formulation to produce input sequences that guarantee a minimum level of performance across a subject population.