In this paper, 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. 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. 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.

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.

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 of assessing trunk motor control 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 for assessing trunk motor control can be deemed reliable.

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 0.72-0.82. 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.

SUMMARYSecondary 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. Supplementary material available online at