M. Cody Priess

Solutions to the Inverse LQR Problem With Application to Biological Systems Analysis

In this brief, we present a set of techniques for finding a cost function to the time-invariant linear quadratic regulator (LQR) problem in both continuous- and discrete-time cases. Our methodology is based on the solution to the inverse LQR problem, which can be stated as: does a given controller K describe the solution to a time-invariant LQR problem, and if so, what weights Q and R produce K as the optimal solution? Our motivation for investigating this problem is the analysis of motion goals in biological systems. We first describe an efficient linear matrix inequality (LMI) method for determining a solution to the general case of this inverse LQR problem when both the weighting matrices Q and R are unknown. Our first LMI-based formulation provides a unique solution when it is feasible. In addition, we propose a gradient-based, least-squares minimization method that can be applied to approximate a solution in cases when the LMIs are infeasible. This new method is very useful in practice since the estimated gain matrix K from the noisy experimental data could be perturbed by the estimation error, which may result in the infeasibility of the LMIs. We also provide an LMI minimization problem to find a good initial point for the minimization using the proposed gradient descent algorithm. We then provide a set of examples to illustrate how to apply our approaches to several different types of problems. An important result is the application of the technique to human subject posture control when seated on a moving robot. Results show that we can recover a cost function which may provide a useful insight on the human motor control goal.

Reliability of assessing trunk motor control using position and force tracking and stabilization tasks

System-based methods have been applied to assess trunk motor control in people with and without back pain, although the reliability of these methods has yet to be established. Therefore, the goal of this study was to quantify within- and between-day reliability using systems-based methods involving position and force tracking and stabilization tasks. Ten healthy subjects performed six tasks, involving tracking and stabilizing of trunk angular position in the sagittal plane, and trunk flexion and extension force. Tracking tasks involved following a one-dimensional, time-varying input signal displayed on a screen by changing trunk position (position tracking) or trunk force (force tracking). Stabilization tasks involved maintaining a constant trunk position (position stabilization) or constant trunk force (force stabilization) while a sagittal plane disturbance input was applied to the pelvis using a robotic platform. Time and frequency domain assessments of error (root mean square and H2 norm, respectively) were computed for each task on two separate days. Intra-class correlation coefficients (ICC) for error and coefficients of multiple correlations (CMC) for frequency response curves were used to quantify reliability of each task. Reliability for all tasks was excellent (between-day ICC≥0.8 and CMC>0.75, within-day CMC>0.85). Therefore, position and force control tasks used to assess trunk motor control can be deemed reliable.

A thermogenic secondary sexual character in male sea lamprey

SUMMARY Secondary sexual characters in animals are exaggerated ornaments or weapons for intrasexual competition. Unexpectedly, we found that a male secondary sexual character in sea lamprey (Petromyzon marinus) is a thermogenic adipose tissue that instantly increases its heat production during sexual encounters. This secondary sexual character, developed in front of the anterior dorsal fin of mature males, is a swollen dorsal ridge known as the ‘rope’ tissue. It contains nerve bundles, multivacuolar adipocytes and interstitial cells packed with small lipid droplets and mitochondria with dense and highly organized cristae. The fatty acid composition of the rope tissue is rich in unsaturated fatty acids. The cytochrome c oxidase activity is high but the ATP concentration is very low in the mitochondria of the rope tissue compared with those of the gill and muscle tissues. The rope tissue temperature immediately rose up to 0.3°C when the male encountered a conspecific. Mature males generated more heat in the rope and muscle tissues when presented with a mature female than when presented with a male (paired t-test, P<0.05). On average, the rope generated 0.027±0.013 W cm−3 more heat than the muscle in 10 min. Transcriptome analyses revealed that genes involved in fat cell differentiation are upregulated whereas those involved in oxidative-phosphorylation-coupled ATP synthesis are downregulated in the rope tissue compared with the gill and muscle tissues. Sexually mature male sea lamprey possess the only known thermogenic secondary sexual character that shows differential heat generation toward individual conspecifics.

Trunk muscle coactivation is tuned to changes in task dynamics to improve responsiveness in a seated balance task.

When balancing, instability can occur when the object being balanced moves at a rate that is beyond the abilities of human motor control. This illustrates that responsiveness of motor control is limited and can be investigated by changing the dynamics of the task. In this study, the responsiveness of trunk motor control was investigated by changing the seat stiffness of an unstable seat. At decreasing levels of seat stiffness the probability of successfully balancing on the seat, speed of the seat, speed of the trunk relative to the seat (trunk-seat) and muscle activation of five trunk muscles were assessed. Also, across the different stiffness levels, the relation between trunk muscle activation and seat speed was determined. As hypothesized, with decreasing seat stiffness the probability of success decreased, seat speed and trunk-seat speed increased, and both agonist and antagonist activation increased. This shows that limits in the responsiveness of trunk motor control were reached during seated balancing. Furthermore, in line with our hypothesis, a positive relation was found between trunk muscle activation and seat speed. It appears that the central nervous system regulates trunk stiffness (via muscle coactivation) in relation to the dynamics of the task, possibly to maintain adequate responsiveness.

Quantitative measures of sagittal plane head-neck control: A test-retest reliability study

Determining the reliability of measurements used to quantify head-neck motor control is necessary before they can be used to study the effects of injury or treatment interventions. Thus, the purpose of this study was to determine the within- and between-day reliability of position tracking, position stabilization and force tracking tasks to quantify head-neck motor control. Ten asymptomatic subjects performed these tasks on two separate days. Position and force tracking tasks required subjects to track a pseudorandom square wave input signal by controlling their head-neck angular position (position tracking) or the magnitude of isometric force generated against a force sensor by the neck musculature (force tracking) in the sagittal plane. Position stabilization required subjects to maintain an upright head position while pseudorandom perturbations were applied to the upper body using a robotic platform. Within-day and between-day reliability of the frequency response curves were assessed using coefficients of multiple correlations (CMC). Root mean square error (RMSE) and mean bandpass signal energy, were computed for each task and between-day reliability was calculated using intra-class correlation coefficients (ICC). Within- and between-day CMCs for the position and force tracking tasks were all ≥0.96, while CMCs for position stabilization ranged from 0.72 to 0.82. ICCs for the position and force tracking tasks were all ≥0.93. For position stabilization, ICCs for RMSE and mean bandpass signal energy were 0.66 and 0.72, respectively. Measures of sagittal plane head-neck motor control using position tracking, position stabilization and force tracking tasks were demonstrated to be reliable.

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

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 experiments 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.

The Inverse Problem of Continuous-Time Linear Quadratic Gaussian Control With Application to Biological Systems Analysis

In this paper, we demonstrate two methods for solving the inverse problem of continuous-time LQG control. This problem can be defined as: given a known LTI system with feedback controller K and Kalman gain L, can we find the weighting matrices Q, R (for state and input, respectively) and estimated noise intensities W, V (for process and measurement noise, respectively) such that the LQG control synthesis problem using these weights generates K and L? We formulate a regularized version of this problem as a minimization problem subject to a set of Linear Matrix Inequalities (LMIs). If feasible, a unique exact solution to the inverse LQR problem exists. If the LMIs are infeasible, we show a gradient descent algorithm that will find Q, R, W, and V to minimize the error in the recovered gain matrices K and L. We demonstrate these techniques through several numerical examples and formulate a human postural control case study to which we intend to apply our proposed techniques.Copyright

Determination of body segment masses and centers of mass using a force plate method

We would like to call the readers’ attention to a serious algeraic error we identified in the technical note co-authored by amavandi, Farahpour, and Allard, entitled “Determination of body egment masses and centers of mass using a force plate method n individuals of different morphology,” and published in Medical ngineering and Physics [1]. Subject-specific body segment masses nd center of mass (COM) locations are used in biomechanical analses aiming to assess accurately joint forces and moments arising uring any activity. There is no direct way to measure these varibles in a living individual, and therefore, various indirect methods ave been used in the past for this purpose [1]. Damavandi et al. laim to have developed a simple force plate method that can be sed to determine subject-specific segment masses and COM locaions [1]. Unfortunately, the authors defined a single variable xs ith two different, incompatible geometric definitions and then sed that incorrectly defined variable to solve erroneously for their nal result. To date, this technical note has been cited eight times Google Scholar) [2–9] and this incorrect technique was applied in hree other biomechanical studies [2–4], indicating an urgent need o expose the error. The two definitions occur in two different sections of the paper eferencing Figures 1 and 2, respectively. The variable xs, indicating egment COM location, is first defined using Eqs. (1)–(5) and Figure , as the distance between the shoulder joint and the center of mass f the arm. In Figure 1, the distance xs is shown as xs and is solved or in Eq. (5) as a function of segment and body masses ms and mb s well as change in center of pressure COP. The second, and diferent, definition of variable xs is shown explicitly in Figure 2 as he initial distance from the arm COM to the reaction board pivot. hese two distances are clearly different, yet the same variable xs s used to represent both of them. The two different xs definitions rroneously yield algebraically independent equations, Eqs. (7) and 8), that can be solved for ms and xs to yield the final results. Clearly, hese final results are in error because the two geometrical definiions for the variable xs have different values and the same value annot be applied simultaneously to these two different distances. he same error in the derivation also yields an erroneous result for egment mass ms. It is impossible to obtain a unique solution for s and xs from such force plate experiments using correct notation, nless one of the unknown variables is assumed to be known. In general, rotating a segment from vertical to horizontal in ny body configuration, vertical or horizontal, always produces he same moment change equal to the product of the segment eight and the distance from the segment’s joint to COM. Because his change is always the product of the same two unknowns, hey simply cannot be separated by algebraic manipulation. Merely [

Time-domain optimal experimental design in human postural control testing

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 an experimental device and associated experimental task for identification of the human 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 postural control parameters. To achieve this, we formulate the problem as a non-convex QP and attempt to maximize a measure (T-optimality condition) of the experiments 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 non-increasing of the cost function while satisfying all constraints during iterations. Finally, we present example experimental results using an optimized input sequence.

Determining human control intent using inverse LQR solutions

In this paper, we have developed a method for determining the control intention in human subjects during a prescribed motion task. Our method is based on the solution to the inverse LQR problem, which can be stated as: does a given controller K describe the solution to a time-invariant LQR problem, and if so, what weights Q and R produce K as the optimal solution? We describe an efficient Linear Matrix Inequality (LMI) method for determining a solution to the general case of this inverse LQR problem when both the weighting matrices Q and R are unknown. Additionally, we propose a gradient-based, least-squares minimization method that can be applied to approximate a solution in cases when the LMIs are infeasible. We develop a model for an upright seated-balance task which will be suitable for identification of human control intent once experimental data is available.Copyright