Controlling a system with redundant degrees of freedom. I. Torque distribution in still standing stick insects

The question is investigated as to how a stick insect solves the task of distributing its body weight onto its six legs, i.e., how are the torques coordinated that are produced by the 18 joints (3 per leg). Three-dimensional force measurements of ground reaction forces have been used to calculate the torques developed by each of the 18 joints. Torques were found to change considerably although the body and the legs of the animal did not move. This result implies a tight cooperation between the 18 joint controllers. Indeed, in each individual experiment, strong correlations could be observed between specific pairs of joints. However, in spite of thorough analysis, no general correlation rules between torques could be detected. The only common attribute found for all experiments was that high absolute torques observed at the beginning of the experiment tend to converge to some minimum over time. Thus, the insects tend to decrease the torques while standing still, but do not use fixed rules. Rather they appear to exploit their extra degrees of freedom and produce time courses that can strongly vary between experiments. Possible mechanisms underlying this behaviour are discussed in a companion paper [Lévy and Cruse (2008) Controlling a system with redundant degrees of freedom: ii. solution of the force distribution problem without a body model, submitted].     

Walknet, a bio-inspired controller for hexapod walking

Walknet comprises an artificial neural network that allows for the simulation of a considerable amount of behavioral data obtained from walking and standing stick insects. It has been tested by kinematic and dynamic simulations as well as on a number of six-legged robots. Over the years, various different expansions of this network have been provided leading to different versions of Walknet. This review summarizes the most important biological findings described by Walknet and how they can be simulated. Walknet shows how a number of properties observed in insects may emerge from a decentralized architecture. Examples are the continuum of so-called “gaits,” coordination of up to 18 leg joints during stance when walking forward or backward over uneven surfaces and negotiation of curves, dealing with leg loss, as well as being able following motion trajectories without explicit precalculation. The different Walknet versions are compared to other approaches describing insect-inspired hexapod walking. Finally, we briefly address the ability of this decentralized reactive controller to form the basis for the simulation of higher-level cognitive faculties exceeding the capabilities of insects.     

A structurally optimal control model for predicting and analyzing human postural coordination

This paper proposes a closed-loop optimal control model predicting changes between in-phase and anti-phase postural coordination during standing and related supra-postural activities. The model allows the evaluation of the influence of body dynamics and balance constraints onto the adoption of postural coordination. This model minimizes the instantaneous norm of the joint torques with a controller in the head space, in contrast with classical linear optimal models used in the postural literature and defined in joint space. The balance constraint is addressed with an adaptive ankle torque saturation. Numerical simulations showed that the model was able to predict changes between in-phase and anti-phase postural coordination modes and other non-linear transient dynamics phenomena.     

Inter-joint coordination of posture on a seesaw device

Even though specific adjustments of the multi-joint control of posture have been observed when posture is challenged, multi-joint coordination on a seesaw device has never been accurately assessed. The current study was conducted in order to investigate the multi-joint coordination when subjects were standing on either a seesaw device or on a stable surface, with the eyes open or closed. Eighteen healthy active subjects were recruited. A principal component analysis and a Self-Organizing Maps analysis were performed on the joint angles in order to detect and characterize dominant coordination patterns. Intermuscular EMG coherence was analysed in order to assess the neurophysiological mechanisms associated with these coordination patterns. The results illustrated a multi-joint organization of posture on both stable ground and on the seesaw, with a higher variability among the individual postural responses observed when standing on the seesaw. These findings challenge the classical assumption of ankle mechanisms as dominating control on seesaw devices and confirm that inter-joint coordination in postural control is strongly modulated by stance conditions. When standing on the seesaw without vision, a decrease in intermuscular coherence was observed without any impact on the joint coordination patterns, likely due to an increase dependence on proprioceptive information.     

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.     

Inverse and forward dynamics: models of multi-body systems

Connected multi-body systems exhibit notoriously complex behaviour when driven by external and internal forces and torques. The problem of reconstructing the internal forces and/or torques from the movements and known external forces is called the 'inverse dynamics problem', whereas calculating motion from known internal forces and/or torques and resulting reaction forces is called the 'forward dynamics problem'. When stepping forward to cross the street, people use muscle forces that generate angular accelerations of their body segments and, by virtue of reaction forces from the street, a forward acceleration of the centre of mass of their body. Inverse dynamics calculations applied to a set of motion data from such an event can teach us how temporal patterns of joint torques were responsible for the observed motion. In forward dynamics calculations we may attempt to create motion from such temporal patterns, which is extremely difficult, because of the complex mechanical linkage along the chains forming the multi-body system. To understand, predict and sometimes control multi-body systems, we may want to have mathematical expressions for them. The Newton-Euler, Lagrangian and Featherstone approaches have their advantages and disadvantages. The simulation of collisions and the inclusion of muscle forces or other internal forces are discussed. Also, the possibility to perform a mixed inverse and forward dynamics calculation are dealt with. The use and limitations of these approaches form the conclusion.     

Optimization-based prediction of asymmetric human gait

An optimization-based formulation and solution method are presented to predict asymmetric human gait for a large-scale skeletal model. Predictive dynamics approach is used in which both the joint angles and joint torques are treated as unknowns in the equations of motion. For the optimization formulation, the joint angle profiles are treated as the primary unknowns, and velocities and accelerations are calculated using them. In numerical implementation, the joint angle profiles are discretized using the B-spline interpolation. An algorithm is presented to inversely calculate the joint torques and the ground reaction forces. The sum of the joint-torques squared, called the dynamic effort, is minimized as the human performance measure. Constraints are imposed on the joint strengths (torques) and joint ranges of motion along with other physical constraints. The formulation is validated by simulating a symmetric gait and comparing the results with the experimental data. Then asymmetric gait motion is simulated, where the left and right step lengths are different. The kinematics and kinetics results from the simulation are presented and discussed. Predicted ground reaction forces are explained by using the inverted pendulum model. Predicted kinematics and kinetics have trends that are similar to those reported in the literature. Potential practical applications of the formulation and the solution approach are discussed.     

Improving joint torque calculations: optimization-based inverse dynamics to reduce the effect of motion errors

The accuracy of joint torques calculated from inverse dynamics methods is strongly dependent upon errors in body segment motion profiles, which arise from two sources of noise: the motion capture system and movement artifacts of skin-mounted markers. The current study presents a method to increase the accuracy of estimated joint torques through the optimization of the angular position data used to describe these segment motions. To compute these angular data, we formulated a constrained nonlinear optimization problem with a cost function that minimizes the difference between the known ground reaction forces (GRFs) and the GRF calculated via a top-down inverse dynamics solution. To evaluate this approach, we constructed idealized error-free reference movements (of squatting and lifting) that produced a set of known “true” motions and associated true joint torques and GRF. To simulate real-world inaccuracies in motion data, these true motions were perturbed by artificial noise. We then applied our approach to these noise-induced data to determine optimized motions and related joint torques. To evaluate the efficacy of the optimization approach compared to traditional (bottom-up or top-down) inverse dynamics approaches, we computed the root mean square error (RMSE) values of joint torques derived from each approach relative to the expected true joint torques. Compared to traditional approaches, the optimization approach reduced the RMSE by 54% to 79%. Average reduction due to our method was 65%; previous methods only achieved an overall reduction of 30%. These results suggest that significant improvement in the accuracy of joint torque calculations can be achieved using this approach.     

A pilot study investigating motor adaptations when learning to walk with a whole-body powered exoskeleton

Evidence is emerging on how whole-body powered exoskeleton (EXO) use impacts users in basic occupational work scenarios, yet our understanding of how users learn to use this complex technology is limited. We explored how novice users adapted to using an EXO during gait. Six novices and five experienced users completed the study. Novices completed an initial training/familiarization gait session, followed by three subsequent gait sessions using the EXO, while experienced users completed one gait session with the EXO. Spatiotemporal gait measures, pelvis and lower limb joint kinematics, muscle activities, EXO torques, and human-EXO interaction forces were measured. Adaptations among novices were most pronounced in spatiotemporal gait measures, followed by joint kinematics, with smaller changes evident in muscle activity and EXO joint torques. Compared to the experienced users, novices exhibited a shorter step length and walked with significantly greater anterior pelvic tilt and less hip extension. Novices also used lower joint torques from the EXO at the hip and knee, and they had greater biceps femoris activity. Overall, our results may suggest that novices exhibited clear progress in learning, but they had not yet adopted motor strategies similar to those of experienced users after the three sessions. We suggest potential future directions to enhance motor adaptations to powered EXO in terms of both training protocols and human-EXO interfaces.     

Optimal combination of minimum degrees of freedom to be actuated in the lower limbs to facilitate arm-free paraplegic standing

Arm-free paraplegic standing via functional electrical stimulation (FES) has drawn much attention in the biomechanical field as it might allow a paraplegic to stand and simultaneously use both arms to perform daily activities. However, current FES systems for standing require that the individual actively regulates balance using one or both arms, thus limiting the practical use of these systems. The purpose of the present study was to show that actuating only six out of 12 degrees of freedom (12-DOFs) in the lower limbs to allow paraplegics to stand freely is theoretically feasible with respect to multibody stability and physiological torque limitations of the lower limb DOF. Specifically, the goal was to determine the optimal combination of the minimum DOF that can be realistically actuated using FES while ensuring stability and able-bodied kinematics during perturbed arm-free standing. The human body was represented by a three-dimensional dynamics model with 12-DOFs in the lower limbs. Nakamura's method (Nakamura, Y., and Ghodoussi, U., 1989, "Dynamics Computation of Closed-Link Robot Mechanisms With Nonredundant and Redundant Actuators," IEEE Trans. Rob. Autom., 5(3), pp. 294-302) was applied to estimate the joint torques of the system using experimental motion data from four healthy subjects. The torques were estimated by applying our previous finding that only 6 (6-DOFs) out of 12-DOFs in the lower limbs need to be actuated to facilitate stable standing. Furthermore, it was shown that six cases of 6-DOFs exist, which facilitate stable standing. In order to characterize each of these cases in terms of the torque generation patterns and to identify a potential optimal 6-DOF combination, the joint torques during perturbations in eight different directions were estimated for all six cases of 6-DOFs. The results suggest that the actuation of both ankle flexionextension, both knee flexionextension, one hip flexionextension, and one hip abductionadduction DOF will result in the minimum torque requirements to regulate balance during perturbed standing. To facilitate unsupported FES-assisted standing, it is sufficient to actuate only 6-DOFs. An optimal combination of 6-DOFs exists, for which this system can generate able-bodied kinematics while requiring lower limb joint torques that are producible using contemporary FES technology. These findings suggest that FES-assisted arm-free standing of paraplegics is theoretically feasible, even when limited by the fact that muscles actuating specific DOFs are often denervated or difficult to access.     

Postural dynamics in the standing human

The purpose of this study was to develop a mathematical model of the linkage dynamics in upright standing, and to use this model to study output principles for postural control. The standing human was modelled in the sagittal plane as a three-segment linkage. Mechanical disturbances were simulated as forces which could be applied at various points in this linkage. An iterative approach was used to find joint torque combinations which would restore balance within 80 ms of these mechanical disturbances. The model predicted that a specific proportional relationship was necessary between the hip, knee and ankle torques in order for balance to be restored. This proportional relationship was shown to be a function of the model structure, but independent of the location, direction and amplitude of the disturbance. These predictions were tested experimentally. A disturbance apparatus was designed to apply an impulsive force to the subjects. The joint torque responses of the subjects were in quantitative agreement with the predictions of the model. The results suggest that a fixed relationship between joint torques may be required to restore balance, and this fixed relationship may make the task of postural control simpler for the nervous system.     

The coordination of force oscillations and of leg movement in a walking insect (Carausius morosus)

As in the preceding paper stick insects walk on a treadwheel and different legs are put on platforms fixed relative to the insect's body. The movement of the walking legs is recorded in addition to the force oscillations of the standing legs. The coordination between the different legs depends upon the number and arrangement of the walking legs and the legs standing on platforms. In most experimental situations one finds a coordination which is different from that of a normal walking animal.Supported by DFG (Cr 58/1)     

Virtual slope control of a forward dynamic bipedal walker

Active joint torques are the primary source of power and control in dynamic walking motion. However the amplitude, rate, timing and phasic behavior of the joint torques necessary to achieve a natural and stable performance are difficult to establish. The goal of this study was to demonstrate the feasibility and stable behavior of an actively controlled bipedal walking simulation wherein the natural system dynamics were preserved by an active, nonlinear, state-feedback controller patterned after passive downhill walking. A two degree-of-freedom, forward-dynamic simulation was implemented with active joint torques applied at the hip joints and stance leg ankle. Kinematic trajectories produced by the active walker were similar to passive dynamic walking with active joint torques influenced by prescribed walking velocity. The control resulted in stable steady-state gait patterns, i.e. eigenvalue magnitudes of the stride function were less than one. The controller coefficient analogous to the virtual slope was modified to successfully control average walking velocity. Furture developments are necessary to expand the range of walking velocities.     

Ankle intrinsic stiffness changes with postural sway

In standing, the human body is inherently unstable and its stabilization requires constant regulation of ankle torque, generated by a combination of ankle intrinsic properties, peripheral reflexes, and central contributions. Ankle intrinsic stiffness, which quantifies the joint intrinsic properties, has been usually assumed constant in standing; however, there is strong evidence that it is highly dependent on the joint torque, which changes significantly with sway in stance. In this study, we examined how ankle intrinsic stiffness changes with postural sway during standing. Ten subjects stood on a standing apparatus, while subjected to pulse perturbations of ankle position. The mean torque of a short period before the start of each pulse was used as a measure of background torque. Responses with similar background torques were grouped together and used to estimate the parameters of an intrinsic stiffness model. Stiffness estimates were normalized to the critical stiffness and the background torque was transformed to the center of pressure location. We found that in most subjects, the normalized stiffness increased linearly with the movement of center of pressure towards the toes, with an average slope of 2.11 ± 0.80 1/m·rad. This modulation of ankle intrinsic stiffness seems functionally appropriate, since the intrinsic stiffness increases quickly, as the center of pressure moves toward the toes and the limits of stability. These large changes of ankle intrinsic stiffness with postural sway must be incorporated in any model of stance control.     

Crank inertial load has little effect on steady-state pedaling coordination

Inertial load can affect the control of a dynamic system whenever parts of the system are accelerated ordeclerated. During steady-state pedating, because within-cycle variations in crank angular acceleration still exist, the amount of crank inertia present (which varies widely with road-riding gear ratio) may affect the within-cycle coordination of muscles. However, the effect of inertial load on steady-state pedaling coordination is almos always assumed to be negligible, since the net mechanical energy per cycle developed by muscles only depends on the constant cadence and workload. This study tests the hypothesis that under steady-state conditions, the net joint torques produced by muscles at the hip, knee, and ankle are unaffected by crank inertial load. To perform the investigation, we constructed a pedaling apparatus which could emulate the low inertial load of a standard ergometer or the high inertial load of a road bicycle in high gear. Crank angle and bilateral pedal force and angle data were collected from ten subjects instructed to pedal steadily (i.e. constant speed across cycles) and smoothly (i.e. constant speed within a cycle) against both inertias at a constant workload. Virtually no statistically significant changes were found in the net hip and knee muscle joint torques calculated from an inverse dynamics analysis. Though the net ankle muscle joint torque, as well as the one- and two-legged crank torque, showed statistically significant increases at the higher inertia, the changes were small. In contrast, large statistically significant reductions were found in crank kinematic variability both within a cycle and between cycles (i.e. cadence), primarily because a larger inertial load means a slower crank dynamic response. Nonetheless, the reduction in cadence variability was somewhat attenuated by a large statistically significant increase in one-legged crank torque variability. We suggest, therefore, that muscle coordination during steady-state pedaling is largely unaffected, though less well regulated, when crank inertial load is increased.     

Evaluation of the roles of passive and active control of balance using a balance control model

At present there is a lack of consensus regarding the relative roles of passive and active control of quiet upright stance. In the current work, this issue was investigated using two simulation models based on contemporary theories. Specifically, the two models, both of which assumed active control torques to be generated from an optimal neural controller, differed with respect to whether or not passive control torques (stiffness and damping) were included. Model parameters were specified using experimental center-of-pressure (COP) time series obtained during upright stance, and comparisons then made between simulated and actual COP-based measures. Including both active and passive joint torques in the control model did not appear to lead to any improvement in the ability to simulate COP compared with only including active joint torque. Further, simulated passive control torques were typically less than 10% of the active control torques, though some exceptions were found. These results, along with existing empirical evidence, suggest that active control torque is dominant in maintaining balance during upright stance.     

The need for muscle co-contraction prior to a landing

In landings from a flight phase the mass centre of an athlete experiences rapid decelerations. This study investigated the extent to which co-contraction is beneficial or necessary in drop landings, using both experimental data and computer simulations. High speed video and force recordings were made of an elite martial artist performing drop landings onto a force plate from heights of 1.2, 1.5 and 1.8 m. Matching simulations of these landings were produced using a planar 8-segment torque-driven subject-specific computer simulation model. It was found that there was substantial co-activation of joint flexor and extensor torques at touchdown in all three landings. Optimisations were carried out to determine whether landings could be effected without any co-contraction at touchdown. The model was not capable of landing from higher than 1.05 m with no initial flexor or extensor activations. Due to the force–velocity properties of muscle, co-contraction with net zero joint torque at touchdown leads to increased extensor torque and decreased flexor torque as joint flexion velocity increases. The same considerations apply in any activity where rapid changes in net joint torque are required, as for example in jumps from a running approach.     

Hands-on parameter search for neural simulations by a MIDI-controller

Computational neuroscientists frequently encounter the challenge of parameter fitting--exploring a usually high dimensional variable space to find a parameter set that reproduces an experimental data set. One common approach is using automated search algorithms such as gradient descent or genetic algorithms. However, these approaches suffer several shortcomings related to their lack of understanding the underlying question, such as defining a suitable error function or getting stuck in local minima. Another widespread approach is manual parameter fitting using a keyboard or a mouse, evaluating different parameter sets following the users intuition. However, this process is often cumbersome and time-intensive. Here, we present a new method for manual parameter fitting. A MIDI controller provides input to the simulation software, where model parameters are then tuned according to the knob and slider positions on the device. The model is immediately updated on every parameter change, continuously plotting the latest results. Given reasonably short simulation times of less than one second, we find this method to be highly efficient in quickly determining good parameter sets. Our approach bears a close resemblance to tuning the sound of an analog synthesizer, giving the user a very good intuition of the problem at hand, such as immediate feedback if and how results are affected by specific parameter changes. In addition to be used in research, our approach should be an ideal teaching tool, allowing students to interactively explore complex models such as Hodgkin-Huxley or dynamical systems.     

Predictive Simulation Generates Human Adaptations during Loaded and Inclined Walking

Predictive simulation is a powerful approach for analyzing human locomotion. Unlike techniques that track experimental data, predictive simulations synthesize gaits by minimizing a high-level objective such as metabolic energy expenditure while satisfying task requirements like achieving a target velocity. The fidelity of predictive gait simulations has only been systematically evaluated for locomotion data on flat ground. In this study, we construct a predictive simulation framework based on energy minimization and use it to generate normal walking, along with walking with a range of carried loads and up a range of inclines. The simulation is muscle-driven and includes controllers based on muscle force and stretch reflexes and contact state of the legs. We demonstrate how human-like locomotor strategies emerge from adapting the model to a range of environmental changes. Our simulation dynamics not only show good agreement with experimental data for normal walking on flat ground (92% of joint angle trajectories and 78% of joint torque trajectories lie within 1 standard deviation of experimental data), but also reproduce many of the salient changes in joint angles, joint moments, muscle coordination, and metabolic energy expenditure observed in experimental studies of loaded and inclined walking.