Optimal Control Simulations Research Articles

A novel biomechanical model of the mouse forelimb predicts muscle activity in optimal control simulations of reaching movements.

Mice are key model organisms in neuroscience and motor systems physiology. Fine motor control tasks performed by mice have become widely used in assaying neural and biophysical motor system mechanisms. Although fine motor tasks provide useful insights into behaviors which require complex multi-joint motor control, there is no previously developed physiological biomechanical model of the adult mouse forelimb available for estimating kinematics nor muscle activity or kinetics during behaviors. Here, we developed a musculoskeletal model based on high-resolution imaging of the mouse forelimb that includes muscles spanning the neck, trunk, shoulder, and limbs. Physics-based optimal control simulations of the forelimb model were used to estimate in vivo muscle activity present when constrained to the tracked kinematics during reaching movements. The activity of a subset of muscles was recorded and used to assess the accuracy of the muscle patterning in simulation. We found that the synthesized muscle patterning in the forelimb model had a strong resemblance to empirical muscle patterning, suggesting that our model has utility in providing a realistic set of estimated muscle excitations over time when given a kinematic template. The strength of the similarity between empirical muscle activity and optimal control predictions increases as mice performance improves throughout learning of the reaching task. Our computational tools are available as open-source in the OpenSim physics and modeling platform. Our model can enhance research into limb control across broad research topics and can inform analyses of motor learning, muscle synergies, neural patterning, and behavioral research that would otherwise be inaccessible.

A six-compartment model for COVID-19 with transmission dynamics and public health strategies.

The global crisis of the COVID-19 pandemic has highlighted the need for mathematical models to inform public health strategies. The present study introduces a novel six-compartment epidemiological model that uniquely incorporates a higher isolation rate for unreported symptomatic cases of COVID-19 compared to reported cases, aiming to enhance prediction accuracy and address the challenge of initial underreporting. Additionally, we employ optimal control theory to assess the cost-effectiveness of interventions and adapt these strategies to specific epidemiological scenarios, such as varying transmission rates and the presence of asymptomatic carriers. By applying this model to COVID-19 data from India (30 January 2020 to 24 November 2020), chosen to capture the initial outbreak and subsequent waves, we calculate a basic reproduction number of 2.147, indicating the high transmissibility of the virus during this period in India. A sensitivity analysis reveals the critical impact of detection rates and isolation measures on disease progression, showing the robustness of our model in estimating the basic reproduction number. Through optimal control simulations, we demonstrate that increasing isolation rates for unreported cases and enhancing detection reduces the spread of COVID-19. Furthermore, our cost-effectiveness analysis establishes that a combined strategy of isolation and treatment is both more effective and economically viable. This research offers novel insights into the efficacy of non-pharmaceutical interventions, providing a tool for strategizing public health interventions and advancing our understanding of infectious disease dynamics.

Comparison of joint kinematics from optical marker-based and inertial sensor-based motion capture during change-of-direction movements

The development of inertial measurement units (IMUs) has opened up possibilities to test kinematics of sports movements in natural environments (Di Paolo et al., 2023). This advantage over lab-based optical motion capture (OMC) comes at the cost of inherent inaccuracies. Especially, fast multidirectional movements including de- and accelerations pose a challenge for inertial-based movement reconstruction with limited task-specific validation research available (Heuvelmans et al., 2022). Therefore, we aimed to test the validity of IMU-based kinematics in a 180° change of direction (COD) task to provide insight into the applicability for future on-field ACL injury risk research. Twenty-six sports science students were equipped with eight IMU sensors of the Noraxon Ultium Motion System and an adaption of the Plug-in-Gait markerset (53 markers) recorded by a VICON OMC system. They performed five 180° anti-clockwise CODs at their maximum speed. As an example for sagittal and frontal plane angles relevant for ACL injury risk, the knee flexion and abduction angle at the initial contact of the cutting step (right leg) were calculated using the Noraxon MyoMotion software for IMU-recorded data and OpenSim for marker-based OMC data. At the time of abstract submission, the data of five (randomly chosen) participants had been analysed. More participants and more joint angle outcomes will be included in follow-up analyses. Deviations between IMU-generated and OMC-generated joint angles are depicted in Figure 1. Within subjects, deviations below 1.5 standard deviations were observed. Between subjects, deviations ranged from 0° and 22° in knee flexion angles and 0° and 11° in knee abduction angles. A general trend towards higher joint angles measured by the IMU system is noticeable, suggesting a possible off-set error. For four out of five participants, the knee flexion deviation was below 10°, which can be considered an accuracy which is tolerable but requires consideration in the interpretation (< 2°: good accuracy, 2-5°: acceptable accuracy, 5-10°: tolerable accuracy, > 10° unbearable accuracy; Bessone et al., 2022). Deviations in knee abduction angles appear too large given the small physiological joint range and motion in this plane. This preliminary analysis of individual deviations suggests acceptable deviations between IMU and OMC measurements regarding knee flexion but problematic deviations for knee abduction. Prospectively, we aim to investigate possible off-set corrections, e.g. through a better alignment of the underlying biomechanical models, and alternative approaches for IMU-based movement reconstruction, e.g. through optimal control simulations (Nitschke et al., 2024).

Understanding HIV/AIDS dynamics: insights from CD4+T cells, antiretroviral treatment, and country-specific analysis.

In this article, we present a mathematical model for human immunodeficiency virus (HIV)/Acquired immune deficiency syndrome (AIDS), taking into account the number of CD4+T cells and antiretroviral treatment. This model is developed based on the susceptible, infected, treated, AIDS (SITA) framework, wherein the infected and treated compartments are divided based on the number of CD4+T cells. Additionally, we consider the possibility of treatment failure, which can exacerbate the condition of the treated individual. Initially, we analyze a simplified HIV/AIDS model without differentiation between the infected and treated classes. Our findings reveal that the global stability of the HIV/AIDS-free equilibrium point is contingent upon the basic reproduction number being less than one. Furthermore, a bifurcation analysis demonstrates that our simplified model consistently exhibits a transcritical bifurcation at a reproduction number equal to one. In the complete model, we elucidate how the control reproduction number determines the stability of the HIV/AIDS-free equilibrium point. To align our model with the empirical data, we estimate its parameters using prevalence data from the top four countries affected by HIV/AIDS, namely, Eswatini, Lesotho, Botswana, and South Africa. We employ numerical simulations and conduct elasticity and sensitivity analyses to examine how our model parameters influence the control reproduction number and the dynamics of each model compartment. Our findings reveal that each country displays distinct sensitivities to the model parameters, implying the need for tailored strategies depending on the target country. Autonomous simulations highlight the potential of case detection and condom use in reducing HIV/AIDS prevalence. Furthermore, we identify that the quality of condoms plays a crucial role: with higher quality condoms, a smaller proportion of infected individuals need to use them for the potential eradication of HIV/AIDS from the population. In our optimal control simulations, we assess population behavior when control interventions are treated as time-dependent variables. Our analysis demonstrates that a combination of condom use and case detection, as time-dependent variables, can significantly curtail the spread of HIV while maintaining an optimal cost of intervention. Moreover, our cost-effectiveness analysis indicates that the condom use intervention alone emerges as the most cost-effective strategy, followed by a combination of case detection and condom use, and finally, case detection as a standalone strategy.

Estimating 3D kinematics and kinetics from virtual inertial sensor data through musculoskeletal movement simulations.

Portable measurement systems using inertial sensors enable motion capture outside the lab, facilitating longitudinal and large-scale studies in natural environments. However, estimating 3D kinematics and kinetics from inertial data for a comprehensive biomechanical movement analysis is still challenging. Machine learning models or stepwise approaches performing Kalman filtering, inverse kinematics, and inverse dynamics can lead to inconsistencies between kinematics and kinetics. We investigated the reconstruction of 3D kinematics and kinetics of arbitrary running motions from inertial sensor data using optimal control simulations of full-body musculoskeletal models. To evaluate the feasibility of the proposed method, we used marker tracking simulations created from optical motion capture data as a reference and for computing virtual inertial data such that the desired solution was known exactly. We generated the inertial tracking simulations by formulating optimal control problems that tracked virtual acceleration and angular velocity while minimizing effort without requiring a task constraint or an initial state. To evaluate the proposed approach, we reconstructed three trials each of straight running, curved running, and a v-cut of 10 participants. We compared the estimated inertial signals and biomechanical variables of the marker and inertial tracking simulations. The inertial data was tracked closely, resulting in low mean root mean squared deviations for pelvis translation (≤20.2mm), angles (≤1.8 deg), ground reaction forces (≤1.1 BW%), joint moments (≤0.1 BWBH%), and muscle forces (≤5.4 BW%) and high mean coefficients of multiple correlation for all biomechanical variables . Accordingly, our results showed that optimal control simulations tracking 3D inertial data could reconstruct the kinematics and kinetics of individual trials of all running motions. The simulations led to mutually and dynamically consistent kinematics and kinetics, which allows researching causal chains, for example, to analyze anterior cruciate ligament injury prevention. Our work proved the feasibility of the approach using virtual inertial data. When using the approach in the future with measured data, the sensor location and alignment on the segment must be estimated, and soft-tissue artifacts are potential error sources. Nevertheless, we demonstrated that optimal control simulation tracking inertial data is highly promising for estimating 3D kinematics and kinetics for a comprehensive biomechanical analysis.

Transfemoral limb loss modestly increases the metabolic cost of optimal control simulations of walking.

In transtibial limb loss, computer simulations suggest that the maintenance of muscle strength between pre- and post-limb loss can maintain the pre-limb loss metabolic cost. These results are consistent with comparable costs found experimentally in select cases of high functioning military service members with transtibial limb loss. It is unlikely that similar results would be found with transfemoral limb loss, although the theoretical limits are not known. Here we performed optimal control simulations of walking with and without an above-knee prosthesis to determine if transfemoral limb loss per se increases the metabolic cost of walking. OpenSim Moco was used to generate optimal control simulations of walking in 15 virtual "subjects" that minimized the weighted sum of (i) deviations from average able-bodied gait mechanics and (ii) the gross metabolic cost of walking, pre-limb loss in models with two intact biological limbs, and post-limb loss with one of the limbs replaced by a prosthetic knee and foot. No other changes were made to the model. Metabolic cost was compared between pre- and post-limb loss simulations in paired t-tests. Metabolic cost post-limb loss increased by 0.7-9.3% (p < 0.01) depending on whether cost was scaled by total body mass or biological body mass and on whether the prosthetic knee was passive or non-passive. Given that the post-limb loss model had numerous features that predisposed it to low metabolic cost, these results suggest transfemoral limb loss per se increases the metabolic cost of walking. However, the large differences above able-bodied peers of ∼20-45% in most gait analysis experiments may be avoidable, even when minimizing deviations from able-bodied gait mechanics. Portions of this text were previously published as part of a preprint (https://www.biorxiv.org/content/10.1101/2023.06.26.546515v2.full.pdf).

Gender differences in cycling motions: On objective functions for urban cycling

AbstractSocietal challenges such as climate change and the energy crisis are putting bicycles at the centre of many people's mobility considerations ‐ but not all. In addition to the need for improved infrastructure, stability and comfort when cycling also play a major role. Simulations offer the possibility to investigate the influence of biometric differences on comfort and stability more cost‐effectively than the measurement methods currently used for bike fitting. Bicycle dynamics and multibody simulation of cycling motions have been the subject of research for a long time. Often data‐driven models are used that follow pre‐established measurement data. Moreover, optimal control simulations for competitive sports are available, where for example, the travel distance during a given time is maximised. However, such models are usually based on the biometric data of an average 18–25 year old male, while the influence of gender differences on cycling motions are rarely explored. Yielding towards closing this gender data gap, we use a discrete mechanics and optimal control framework (DMOCC), which benefits from its structure preserving formulation and has been successfully used for biomechanical applications before. The implemented multibody model of a leg performing a cycling motion can be adapted to individual 3D scans via the geometry parameters and the bounds on the joint angles and torques, providing the possibility to investigate the influence of biometric diversity on the resulting motions. In this first approach, we discuss several possibilities to formulate appropriate objective functions for cycling. The final aim of this study is to supplement a given bike frame by software chosen adaptations so that it optimally fits to individual biometric conditions, thus increasing comfort, the sense of safety and performance, which ultimately enables greater participation in the mobility transition for women, children, and seniors.

Computational performance of musculoskeletal simulation in OpenSim Moco using parallel computing.

Optimal control musculoskeletal simulation is a valuable approach for studying fundamental and clinical aspects of human movement. However, the high computational demand has long presented a substantial challenge, creating a need to improve simulation performance. The OpenSim Moco software package permits musculoskeletal simulation problems to be solved in parallel on multicore processors using the CasADi optimal control library, potentially reducing the computational demand. However, the computational performance of this framework has not been thoroughly examined. Thus, we aimed to investigate the computational speed-up obtained via multicore parallel computing relative to solving problems serially (i.e., using a single core) in optimal control simulations of human movement in OpenSim Moco. Simulations were solved using up to 18 cores with a variety of temporal mesh interval densities and using two different initial guess strategies. We examined a range of musculoskeletal models and movements that included two- and three-dimensional models, tracking and predictive simulations, and walking and reaching tasks. The maximum overall parallel speed-up was problem specific and ranged from 1.7 to 7.7 times faster than serial, with most of the speed-up achieved by about 6 processor cores. Parallel speed-up was generally greater on finer temporal meshes, while the initial guess strategy had minimal impact on speed-up. Considerable speed-up can be achieved for some optimal control simulation problems in OpenSim Moco by leveraging the multicore processors often available in modern computers. However, since improvements are problem specific, achieving optimal computational performance will require some degree of exploration by the end user.

Malware propagation model of fractional order, optimal control strategy and simulations

In this paper, an improved SEIR model of fractional order is investigated to describe the behavior of malware propagation in the wireless sensor network. Firstly, the malware propagation model of fractional order is established based on the classical SEIR epidemic theory, the basic reproductive number is obtained by the next-generation method and the stability condition of the model is also analyzed. Then, the inverse approach for the uncertainty propagation based on the discrete element method and least square algorithm is presented to determine the unknown parameters of the propagation process. Finally, the optimal control strategy is also discussed based on the adaptive model. Simulation results show the proposed model works better than the propagation model of integer order. The error of proposed model is smaller than integer order models.

Change the direction: 3D optimal control simulation by directly tracking marker and ground reaction force data.

Optimal control simulations of musculoskeletal models can be used to reconstruct motions measured with optical motion capture to estimate joint and muscle kinematics and kinetics. These simulations are mutually and dynamically consistent, in contrast to traditional inverse methods. Commonly, optimal control simulations are generated by tracking generalized coordinates in combination with ground reaction forces. The generalized coordinates are estimated from marker positions using, for example, inverse kinematics. Hence, inaccuracies in the estimated coordinates are tracked in the simulation. We developed an approach to reconstruct arbitrary motions, such as change of direction motions, using optimal control simulations of 3D full-body musculoskeletal models by directly tracking marker and ground reaction force data. For evaluation, we recorded three trials each of straight running, curved running, and a v-cut for 10 participants. We reconstructed the recordings with marker tracking simulations, coordinate tracking simulations, and inverse kinematics and dynamics. First, we analyzed the convergence of the simulations and found that the wall time increased three to four times when using marker tracking compared to coordinate tracking. Then, we compared the marker trajectories, ground reaction forces, pelvis translations, joint angles, and joint moments between the three reconstruction methods. Root mean squared deviations between measured and estimated marker positions were smallest for inverse kinematics (e.g., 7.6 ± 5.1 mm for v-cut). However, measurement noise and soft tissue artifacts are likely also tracked in inverse kinematics, meaning that this approach does not reflect a gold standard. Marker tracking simulations resulted in slightly higher root mean squared marker deviations (e.g., 9.5 ± 6.2 mm for v-cut) than inverse kinematics. In contrast, coordinate tracking resulted in deviations that were nearly twice as high (e.g., 16.8 ± 10.5 mm for v-cut). Joint angles from coordinate tracking followed the estimated joint angles from inverse kinematics more closely than marker tracking (e.g., root mean squared deviation of 1.4 ± 1.8 deg vs. 3.5 ± 4.0 deg for v-cut). However, we did not have a gold standard measurement of the joint angles, so it is unknown if this larger deviation means the solution is less accurate. In conclusion, we showed that optimal control simulations of change of direction running motions can be created by tracking marker and ground reaction force data. Marker tracking considerably improved marker accuracy compared to coordinate tracking. Therefore, we recommend reconstructing movements by directly tracking marker data in the optimal control simulation when precise marker tracking is required.

Mathematical model of COVID-19 transmission dynamics incorporating booster vaccine program and environmental contamination

COVID-19 is one of the greatest human global health challenges that causes economic meltdown of many nations. In this study, we develop an SIR-type model which captures both human-to-human and environment-to-human-to-environment transmissions that allows the recruitment of corona viruses in the environment in the midst of booster vaccine program. Theoretically, we prove some basic properties of the full model as well as investigate the existence of SARS-CoV-2-free and endemic equilibria. The SARS-CoV-2-free equilibrium for the special case, where the constant inflow of corona virus into the environment by any other means, Ω is suspended (Ω=0) is globally asymptotically stable when the effective reproduction number R0c<1 and unstable if otherwise. Whereas in the presence of free-living Corona viruses in the environment (Ω>0), the endemic equilibrium using the centre manifold theory is shown to be stable globally whenever R0c>1. The model is extended into optimal control system and analyzed analytically using Pontryagin's Maximum Principle. Results from the optimal control simulations show that strategy E for implementing the public health advocacy, booster vaccine program, treatment of isolated people and disinfecting or fumigating of surfaces and dead bodies before burial is the most effective control intervention for mitigating the spread of Corona virus. Importantly, based on the available data used, the study also revealed that if at least 70% of the constituents followed the aforementioned public health policies, then herd immunity could be achieved for COVID-19 pandemic in the community.

Patterns of asymmetry and energy cost generated from predictive simulations of hemiparetic gait

Hemiparesis, defined as unilateral muscle weakness, often occurs in people post-stroke or people with cerebral palsy, however it is difficult to understand how this hemiparesis affects movement patterns as it often presents alongside a variety of other neuromuscular impairments. Predictive musculoskeletal modeling presents an opportunity to investigate how impairments affect gait performance assuming a particular cost function. Here, we use predictive simulation to quantify the spatiotemporal asymmetries and changes to metabolic cost that emerge when muscle strength is unilaterally reduced and how reducing spatiotemporal symmetry affects metabolic cost. We modified a 2-D musculoskeletal model by uniformly reducing the peak isometric muscle force unilaterally. We then solved optimal control simulations of walking across a range of speeds by minimizing the sum of the cubed muscle excitations. Lastly, we ran additional optimizations to test if reducing spatiotemporal asymmetry would result in an increase in metabolic cost. Our results showed that the magnitude and direction of effort-optimal spatiotemporal asymmetries depends on both the gait speed and level of weakness. Also, the optimal speed was 1.25 m/s for the symmetrical and 20% weakness models but slower (1.00 m/s) for the 40% and 60% weakness models, suggesting that hemiparesis can account for a portion of the slower gait speed seen in people with hemiparesis. Modifying the cost function to minimize spatiotemporal asymmetry resulted in small increases (~4%) in metabolic cost. Overall, our results indicate that spatiotemporal asymmetry may be optimal for people with hemiparesis. Additionally, the effect of speed and the level of weakness on spatiotemporal asymmetry may help explain the well-known heterogenous distribution of spatiotemporal asymmetries observed in the clinic. Future work could extend our results by testing the effects of other neuromuscular impairments on optimal gait strategies, and therefore build a more comprehensive understanding of the gait patterns observed in clinical populations.

An approximate stochastic optimal control framework to simulate nonlinear neuro-musculoskeletal models in the presence of noise.

Optimal control simulations have shown that both musculoskeletal dynamics and physiological noise are important determinants of movement. However, due to the limited efficiency of available computational tools, deterministic simulations of movement focus on accurately modelling the musculoskeletal system while neglecting physiological noise, and stochastic simulations account for noise while simplifying the dynamics. We took advantage of recent approaches where stochastic optimal control problems are approximated using deterministic optimal control problems, which can be solved efficiently using direct collocation. We were thus able to extend predictions of stochastic optimal control as a theory of motor coordination to include muscle coordination and movement patterns emerging from non-linear musculoskeletal dynamics. In stochastic optimal control simulations of human standing balance, we demonstrated that the inclusion of muscle dynamics can predict muscle co-contraction as minimal effort strategy that complements sensorimotor feedback control in the presence of sensory noise. In simulations of reaching, we demonstrated that nonlinear multi-segment musculoskeletal dynamics enables complex perturbed and unperturbed reach trajectories under a variety of task conditions to be predicted. In both behaviors, we demonstrated how interactions between task constraint, sensory noise, and the intrinsic properties of muscle influence optimal muscle coordination patterns, including muscle co-contraction, and the resulting movement trajectories. Our approach enables a true minimum effort solution to be identified as task constraints, such as movement accuracy, can be explicitly imposed, rather than being approximated using penalty terms in the cost function. Our approximate stochastic optimal control framework predicts complex features, not captured by previous simulation approaches, providing a generalizable and valuable tool to study how musculoskeletal dynamics and physiological noise may alter neural control of movement in both healthy and pathological movements.

Optimal control simulations of two-finger grasps

Grasping is a complex human activity performed with readiness through a complicated mechanical system as an end effector, i.e. the human hand. Here, we apply a direct transcription method of discrete mechanics and optimal control with constraints (DMOCC) to reproduce human-level grasping of an object with a three-dimensional model of the hand, actuated through joint control torques. The equations of motions describing the hand dynamics are derived from a discrete variational principle based on a discrete action functional, which gives the time integrator structure-preserving properties. The grasping action is achieved through a series of constraints, which generate a hybrid dynamical system with a given switching sequence and unknown switching times. To determine a favourable trajectory for grasping action, we solve an optimal control problem (OCP) with different physiological objectives subject to discrete Euler–Lagrange equations, boundary conditions and path constraints.

Optimal Control of Crystal Size and Shape in Batch Crystallization Using a Bivariate Population Balance Modeling

Abstract An optimal control framework was employed to obtain optimal supersaturation/temperature policies for controlling the crystal mass, size, and shape that meet target product specifications. It uses a bivariate population balance model that includes crystal nucleation, growth, dissolution, and disappearance. The optimal control scheme, solving a dynamic optimization problem, was applied to the batch cooling crystallization of potassium dihydrogen phosphate. The population balance model was evaluated in open-loop experiments, showing good prediction for the mean characteristic lengths and number of particles, both for the supersaturation and undersaturation zones. The deterministic optimal control simulations demonstrated the application of the control action policies to produce crystals of desired mass and average shape for different control targets.

Performance seeking control of minimum infrared characteristic on double bypass variable cycle engine

In order to study the performance seeking control of double bypass variable cycle engine, a Feasible Sequential Quadratic Programming (FSQP) algorithm with global and local superlinear convergence characteristics is adopted. The engine model established a backward infrared radiation intensity prediction method for exhaust system, which could calculate the infrared radiation intensity of the high temperature components and nozzle stream in the system. The infrared radiation absorption effect of the nozzle stream on the high temperature components is considered in the method. According to the above model, a minimum infrared characteristic performance seeking control based on infrared prediction model is proposed. The optimal control simulations of the maximum thrust, minimum specific fuel consumption and minimum infrared characteristic mode on the double bypass variable cycle engine are carried out. Finally, the simulation results show that the algorithm could obtain the optimal operating conditions of double bypass variable cycle engine in different working conditions, which achieves the steady-state performance seeking control of the engine.

Optimal control problem arises from illegal poaching of southern white rhino mathematical model

In this paper, a novel dynamical population model of a southern white rhino with legal and illegal poaching activity is introduced. The model constructed is based on a predator–prey model with southern white rhinos as prey and humans (hunters) as predators. We divide the southern white rhino population into three classes based on their horn condition. We investigate the existence and the stability of the equilibrium points, which depend on some threshold functions. From an analytical result, it is trivial that arresting as many hunters as possible helps conserve white rhinos, but it comes at a high cost. Therefore, an optimal strategy is needed. The optimal control is then constructed using Pontryagin’s minimum principle and solved numerically with an iterative forward–backward method. Optimal control simulations are given to provide additional insight into the dynamics of the model. Analysis of the cost function effectiveness is conducted using the ACER (Average Cost–Effectiveness Ratio) and ICER (Incremental Cost–Effectiveness Ratio) indicator method. The results show that the hunter population can be more easily controlled with a time-dependent hunter arrest rate rather than by treating it as a constant.

Optimal control of coupled quantum systems based on the first-order Magnus expansion: Application to multiple dipole-dipole-coupled molecular rotors

We develop a methodology for performing approximate optimal control simulations for quantum systems with multiple interacting degrees of freedom. The quantum dynamics are modeled using the first-order Magnus approximation in the interaction picture, where the interactions between different degrees of freedom are treated as the perturbation. We present a numerical procedure for implementing this approximation, which leverages the separability of the zeroth-order time evolution operator and the pairwise nature of common interactions for a reduced computational cost. This formulation of the first-order Magnus approximation is suitable to be combined with gradient-free methods for control field optimization; to this end, we adopt a Stochastic Hill Climbing algorithm. The associated computational costs are analyzed and compared with those of the exact simulation in the large N limit. For numerical illustrations, we perform approximate optimal control simulations for systems of two and three dipole-dipole coupled molecular rotors under the influence of a global control field. For the two rotor system, we optimize fields for both orientation and entanglement control objectives. For the three rotor system, we optimize fields for orienting rotors in the same direction as well as in opposite directions.

Comparison of different optimal control formulations for generating dynamically consistent crutch walking simulations using a torque-driven model

Walking impairment due to spinal cord injury can be improved using active orthoses, together with some type of external support such as crutches for balance. This study explores how optimal control problem formulation affects the ability to generate dynamically consistent crutch walking simulations with active orthosis assistance that closely reproduce experimental data. The investigation compares eight optimal tracking problem formulations using a torque-driven full-body skeletal model of a healthy subject walking with active knee-ankle-foot orthoses and forearm crutches. We have found that small adjustments to ground reactions are required to achieve convergence, tracking of joint instead of marker coordinates significantly improves convergence, and minimising joint jerks instead of joint accelerations has only a minor effect on the problem convergence. These results provide guidance for future work that seeks to develop predictive optimal control simulations that can identify orthosis control parameters that maximise improvement in walking function on an individual patient basis.

Decisions in Designing an Australian Macroeconomic Model

This paper discusses the decisions made in designing a new Australian macroeconometric model to be used for both policy analysis and forecasting. Serving these dual functions requires a reasonable level of consistency with both macroeconomic theory (emphasised in New Keynesian dynamic stochastic general equilibrium models) and macroeconomic data (emphasised in vector autoregression models). The key decisions made in modelling household, business, government and foreign behaviour are explained. The design decisions taken are reflected in the new model, which is the latest in a series developed by the author. The use of the latest model is illustrated in optimal control simulations.