An improved inverse dynamics formulation for estimation of external and internal loads during human sagittal plane movements

Planar musculoskeletal models are common in the inverse dynamics analysis of human movements such as walking, running and jumping. The continued interest in such models is justified by their simplicity and computational efficiency. Related to a human planar model, a unified formulation for both the flying and support phases of the sagittal plane movements is developed. The actuation involves muscle forces in the lower limbs and the resultant muscle torques in the other body joints. The dynamic equations, introduced in absolute coordinates of the segments, are converted into useful compact forms using the projective technique. The solution to a determinate inverse dynamics problem allows for the explicit determination of the external reactions (presumed to vanish during the flying phases) and the resultant muscle torques in all the model joints. The indeterminate inverse dynamics problem is then focused on the assessment of muscle forces and joint reaction forces selectively in the supporting lower limb. Numerical results of the inverse dynamics simulation of sample sagittal plane movements are reported to illustrate the validity and effectiveness of the improved formulation.     

Is a single or double arm technique more advantageous in triple jumping?

Triple jumpers employ either an asymmetrical ‘single-arm’ action or symmetrical ‘double-arm’ action in the takeoff of each phase of the jump. This study investigated which technique is more beneficial in each phase using computer simulation. Kinematic data were obtained from an entire triple jump using a Vicon automatic motion capture system. A planar 13-segment torque-driven subject-specific computer simulation model was evaluated by varying torque generator activation timings using a genetic algorithm in order to match performance data. The matching produced a close agreement between simulation and performance, with differences of 3.8%, 2.7%, and 3.1% for the hop, step, and jump phases, respectively. Each phase was optimised for jump distance and an increase in jump distance beyond the matched simulations of 3.3%, 11.1%, and 8.2% was obtained for the hop, step, and jump, respectively. The optimised technique used symmetrical shoulder flexion whereas the triple jumper had used an asymmetrical arm technique. This arm action put the leg extensors into slower concentric conditions allowing greater extensor torques to be produced. The main increases in work came at the joints of the stance leg but the largest increases in angular impulse came at the shoulder joints, indicating the importance of both measures when assessing the impact of individual joint actions on changes in technique. Possible benefits of the double-arm technique include: cushioning the stance leg during impact; raising the centre of mass of the body at takeoff; facilitating an increase in kinetic energy at takeoff; allowing a re-orientation of the body during flight.     

A neuromusculoskeletal tracking method for estimating individual muscle forces in human movement

A neuromusculoskeletal tracking (NMT) method was developed to estimate muscle forces from observed motion data. The NMT method combines skeletal motion tracking and optimal neuromuscular tracking to produce forward simulations of human movement quickly and accurately. The skeletal motion tracker calculates the joint torques needed to actuate a skeletal model and track observed segment angles and ground forces in a forward simulation of the motor task. The optimal neuromuscular tracker resolves the muscle redundancy problem dynamically and finds the muscle excitations (and muscle forces) needed to produce the joint torques calculated by the skeletal motion tracker. To evaluate the accuracy of the NMT method, kinematics and ground forces obtained from an optimal control (parameter optimization) solution for maximum-height jumping were contaminated with both random and systematic noise. These data served as input observations to the NMT method as well as an inverse dynamics analysis. The NMT solution was compared to the input observations, the original optimal solution, and a simulation driven by the inverse dynamics torques. The results show that, in contrast to inverse dynamics, the NMT method is able to produce an accurate forward simulation consistent with the optimal control solution. The NMT method also requires 3 orders-of-magnitude less CPU time than parameter optimization. The speed and accuracy of the NMT method make it a promising new tool for estimating muscle forces using experimentally obtained kinematics and ground force data.     

Leg stiffness can be maintained during reactive hopping despite modified acceleration conditions

AimThe aim of the present study was to evaluate reactive hops under systematically modified acceleration conditions. It was hypothesized that a high preactivity of the leg extensors and phase-specific adjustments of the leg muscle activation would compensate the alterations caused by the various acceleration levels in order to maintain a high leg stiffness, thus enabling the jumper to perform truly reactive jumps with short ground contact times despite the unaccustomed acceleration conditions.MethodsGround reaction forces (GRF), kinematic and electromyographic data of 20 healthy subjects were recorded during reactive hopping in a special sledge jump system for seven different acceleration levels: three acceleration levels with lower than normal gravity (0.7g, 0.8g, 0.9g), one with gravitational acceleration (1g) and three with higher acceleration (1.1g, 1.2g, 1.3g).ResultsThe increase of the acceleration from 0.7g to 1.3g had no significant effect on the preactivity of the leg extensors, the leg stiffness and the rate of force development. However, it resulted in increased peak GRF (+15%), longer ground contact time (+10%) and increased angular excursion at the ankle and knee joints (+3°).DiscussionThroughout a wide acceleration range, the subjects were able to maintain a high leg stiffness and perform reactive hops by keeping the preactivity constantly high and adjusting the muscle activity in the later phases. In consequence, it can be concluded that the neuromuscular system can cope with different acceleration levels, at least in the acceleration range used in this study.     

The jump shot - a biomechanical analysis focused on lateral ankle ligaments

Handball is one of the top four athletic games with highest injury risks. The jump shot is the most accomplished goal shot technique and the lower extremities are mostly injured. As a basis for ankle sprain simulation, the aim of this study was to extend the ankle region of an existing musculoskeletal full-body model through incorporation of three prominent lateral ankle ligaments: ligamentum fibulotalare anterius (LFTA), ligamentum fibulotalare posterius (LFTP), ligamentum fibulocalcaneare (LFC). The specific objective was to calculate and visualise ligament force scenarios during the jumping and landing phases of controlled jump shots. Recorded kinematic data of performed jump shots and the corresponding ground reaction forces were used to perform inverse dynamics. The calculated peak force of the LFTA (107 N) was found at maximum plantarflexion and of the LFTP (150 N) at maximum dorsiflexion. The peak force of the LFC (190 N) was observed at maximum dorsiflexion combined with maximum eversion. Within the performed jump shots, the LFTA showed a peak force (59 N to 69 N) during maximum plantarflexion in the final moment of the lift off. During landing, the force developed by the LFTA reached its peak value (61 N to 70 N) at the first contact with the floor. After that, the LFTP developed a peak force (70 N to 118 N). This model allows the calculation of forces in lateral ankle ligaments. The information obtained in this study can serve as a basis for future research on ankle sprain and ankle sprain simulation.     

Models incorporating pin joints are suitable for simulating performance but unsuitable for simulating internal loading

Simulation models of human movement comprising pin-linked segments have a potential weakness for reproducing accurate ground reaction forces during high impact activities. While the human body contains many compliant structures such a model only has compliance in wobbling masses and in the foot-ground interface. In order to determine whether accurate GRFs can be produced by allowing additional compliance in the foot-ground interface, a subject-specific angle-driven computer simulation model of triple jumping with 13 pin-linked segments was developed, with wobbling masses included within the shank, thigh, and trunk segments. The foot-ground interface was represented by spring-dampers at three points on each foot: the toe, ball, and heel. The parameters of the spring-dampers were varied by a genetic algorithm in order to minimise the differences between simulated GRFs, and those measured from the three phases of a triple jump in three conditions: (a) foot spring compression limited to 20 mm; (b) this compression limited to 40 mm; (c) no restrictions. Differences of 47.9%, 15.7%, and 12.4% between simulation and recorded forces were obtained for the 20 mm, 40 mm, and unrestricted conditions, respectively. In the unrestricted condition maximum compressions of between 43 mm and 56 mm were obtained in the three phases and the mass centre position was within 4mm of the actual position at these times. It is concluded that the unrestricted model is appropriate for simulating performance whereas the accurate calculation of internal forces would require a model that incorporates compliance elsewhere in the link system.     

A model for insect locomotion in the horizontal plane: Feedforward activation of fast muscles, stability, and robustness

We develop a neuromechanical model for running insects that includes a simplified hexapedal leg geometry with agonist-antagonist muscle pairs actuating each leg joint. Restricting to dynamics in the horizontal plane and neglecting leg masses, we reduce the model to three degrees of freedom describing translational and yawing motions of the body. Muscles are driven by stylized action potentials characteristic of fast motoneurons, and modeled using an activation function and nonlinear length and shortening velocity dependence. Parameter values are based on measurements from depressor muscles and observations of kinematics and dynamics of the cockroach Blaberus discoidalis; in particular, motoneuronal inputs and muscle force levels are chosen to approximately achieve joint torques that are consistent with measured ground reaction forces. We show that the model has stable double-tripod gaits over the animal's speed range, that its dynamics at preferred speeds matches those observed, and that it maintains stable gaits, with low frequency yaw deviations, when subject to random perturbations in foot touchdown and lift-off timing and action potential input timing. We explain this in terms of the low-dimensional dynamics.     

Bone contact forces on the distal tibia during the stance phase of running

Although the tibia is a common site of stress fractures in runners, the loading of the tibia during running is not well understood. An integrated experimental and modeling approach was therefore used to estimate the bone contact forces acting on the distal end of the tibia during the stance phase of running, and the contributions of external and internal sources to these forces. Motion capture and force plate data were recorded for 10 male runners as they ran at 3.5-4 m/s. From these data, the joint reaction force (JRF), muscle forces, and bone contact force on the tibia were computed at the ankle using inverse dynamics and optimization methods. The distal end of the tibia was compressed and sheared posteriorly throughout most of stance, with respective peak forces of 9.00+/-1.13 and 0.57+/-0.18 body weights occurring during mid stance. Internal muscle forces were the primary source of tibial compression, whereas the JRF was the primary source of tibial shear due to the forward inclination of the leg relative to the external ground reaction force. The muscle forces and JRF both acted to compress the tibia, but induced tibial shear forces in opposing directions during stance, magnifying tibial compression and reducing tibial shear. The superposition of the peak compressive and posterior shear forces at mid stance may contribute to stress fractures in the posterior face of the tibia. The implications are that changes in running technique could potentially reduce stress fracture risk.     

Explanation of the bilateral deficit in human vertical squat jumping.

In the literature, it has been reported that the mechanical output per leg is less in two-leg jumps than in one-leg jumps. This so-called bilateral deficit has been attributed to a reduced neural drive to muscles in two-leg jumps. The purpose of the present study was to investigate the possible contribution of nonneural factors to the bilateral deficit in jumping. We collected kinematics, ground reaction forces, and electromyograms of eight human subjects performing two-leg and one-leg (right leg) squat jumps and calculated mechanical output per leg. We also used a model of the human musculoskeletal system to simulate two-leg and one-leg jumps, starting from the initial position observed in the subjects. The model had muscle stimulation as input, which was optimized using jump height as performance criterion. The model did not incorporate a reduced maximal neural drive in the two-leg jump. Both in the subjects and in the model, the work of the right leg was more than 20% less in the two-leg jump than in the one-leg jump. Peak electromyogram levels in the two-leg jump were reduced on average by 5%, but the reduction was only statistically significant in m. rectus femoris. In the model, approximately 75% of the bilateral deficit in work per leg was explained by higher shortening velocities in the two-leg jump, and the remainder was explained by lower active state of muscles. It was concluded that the bilateral deficit in jumping is primarily caused by the force-velocity relationship rather than by a reduction of neural drive.     

Correlation between ground reaction force and tibial acceleration in vertical jumping

Modern electronics allow for the unobtrusive measurement of accelerations outside the laboratory using wireless sensor nodes. The ability to accurately measure joint accelerations under unrestricted conditions, and to correlate them with jump height and landing force, could provide important data to better understand joint mechanics subject to real-life conditions. This study investigates the correlation between peak vertical ground reaction forces, as measured by a force plate, and tibial axial accelerations during free vertical jumping. The jump heights calculated from force-plate data and accelerometer measurements are also compared. For six male subjects participating in this study, the average coefficient of determination between peak ground reaction force and peak tibial axial acceleration is found to be 0.81. The coefficient of determination between jump height calculated using force plate and accelerometer data is 0.88. Data show that the landing forces could be as high as 8 body weights of the jumper. The measured peak tibial accelerations ranged up to 42 g. Jump heights calculated from force plate and accelerometer sensors data differed by less than 2.5 cm. It is found that both impact accelerations and landing forces are only weakly correlated with jump height (the average coefficient of determination is 0.12). This study shows that unobtrusive accelerometers can be used to determine the ground reaction forces experienced in a jump landing. Whereas the device also permitted an accurate determination of jump height, there was no correlation between peak ground reaction force and jump height.     

Modelling, simulation and optimisation of a human vertical jump.

This paper describes an efficient biomechanical model of the human lower limb with the aim of simulating a real human jump movement consisting of an upword propulsion, a flying and a landing phase. A multiphase optimal control technique is used to solve the muscle force sharing problem. To understand how intermuscular control coordinates limb muscle excitations, the human body is reduced to a single lower limb consisting of three rigid bodies. The biomechanical system is activated by nine muscle-tendon actuators representing the basic properties of muscles during force generation. For the calculation of the minimal muscle excitations of the jump movement, the trajectory of the hip joint is given as a rheonomic constraint and the contact forces (ground reaction forces) are determined by force plates. Based on the designed musculoskeletal model and on the differential equations of the multibody system, muscle excitations and muscle forces necessary for a vertical jump movement are calculated. The validity of the system is assessed comparing the calculated muscle excitations with the registered surface electromyogramm (EMG) of the muscles. The achieved results indicate a close relationship between the predicted and the measured parameters.     

An experimental setup for the measurement of forces on a human cadaveric foot during inversion

An experimental setup was developed for statically measuring seven vertical and three horizontal reaction forces on the foot. In the setup, the leg can be simultaneously loaded (1) by a vertical force, (2) by an externally applied axial moment, and (3) by simulated muscle forces. The foot is free to invert under influence of the external loads. Statical analysis and test experiments were used for evaluation. The setup can be used in combination with Roentgen photogrammetry to measure bone positions simultaneously with forces.     

Side-to-side differences in bone strength in master jumpers and sprinters

Introduction: This study evaluated side-to-side difference in tibial bone structure, calf muscle cross-sectional area (CSA) and hopping force in master athletes as a result of training for sports with different magnitudes of inter-leg loading difference. Methods: Tibial bone parameters (at 4%, 14%, 38% and 66% tibial length proximal to distal end), muscle CSA (at 66% tibial length) and hopping forces of both legs of 51 master athletes (conditioned jumpers, conditioned triple jumpers, unconditioned jumpers, hurdlers and sprinters) were examined using pQCT. In epiphyseal 4% slice bone CSA (Ar.tot), total BMC (vBMC.tot), trabecular BMC (vBMC.tb) cortical BMC (vBMC.ct), and trabecular BMD (vBMD.tb) were measured. In diaphyseal slices, Ar.tot, vBMC.ct, cortical density (vBMD.ct), cross-sectional moment of inertia (CSMI) and calf muscle CSA (MuscA) were examined. Results: In conditioned jumpers, side-to-side differences in favour of take-off leg were found in 4% slice in vBMC.tb (+4.1%) (P<0.05). A side-to-side difference was found in 66% slice vBMC.ct and CSMI (both P<0.05), with conditioned jumper (+2.8% and 6.6%) and triple jumper (+2.7% and 7.2%) values higher than other groups. Conclusion: The results suggest that regular training in high-impact sports with uneven lower limb loading results in side-to-side differences in skeletal adaptation independent of age and gender, suggesting that high-impact exercise is effective in maintaining bone strength throughout human lifespan.     

Estimation of muscle forces in gait using a simulation of the electromyographic activity and numerical optimization

Clinical gait analysis provides great contributions to the understanding of gait patterns. However, a complete distribution of muscle forces throughout the gait cycle is a current challenge for many researchers. Two techniques are often used to estimate muscle forces: inverse dynamics with static optimization and computer muscle control that uses forward dynamics to minimize tracking. The first method often involves limitations due to changing muscle dynamics and possible signal artefacts that depend on day-to-day variation in the position of electromyographic (EMG) electrodes. Nevertheless, in clinical gait analysis, the method of inverse dynamics is a fundamental and commonly used computational procedure to calculate the force and torque reactions at various body joints. Our aim was to develop a generic musculoskeletal model that could be able to be applied in the clinical setting. The musculoskeletal model of the lower limb presents a simulation for the EMG data to address the common limitations of these techniques. This model presents a new point of view from the inverse dynamics used on clinical gait analysis, including the EMG information, and shows a similar performance to another model available in the OpenSim software. The main problem of these methods to achieve a correct muscle coordination is the lack of complete EMG data for all muscles modelled. We present a technique that simulates the EMG activity and presents a good correlation with the muscle forces throughout the gait cycle. Also, this method showed great similarities whit the real EMG data recorded from the subjects doing the same movement.     

Elevated gastrocnemius forces compensate for decreased hamstrings forces during the weight-acceptance phase of single-leg jump landing: implications for anterior cruciate ligament injury risk

Approximately 320,000 anterior cruciate ligament (ACL) injuries in the United States each year are non-contact injuries, with many occurring during a single-leg jump landing. To reduce ACL injury risk, one option is to improve muscle strength and/or the activation of muscles crossing the knee under elevated external loading. This study?s purpose was to characterize the relative force production of the muscles supporting the knee during the weight-acceptance (WA) phase of single-leg jump landing and investigate the gastrocnemii forces compared to the hamstrings forces. Amateur male Western Australian Rules Football players completed a single-leg jump landing protocol and six participants were randomly chosen for further modeling and simulation. A three-dimensional, 14-segment, 37 degree-of-freedom, 92 muscle-tendon actuated model was created for each participant in OpenSim. Computed muscle control was used to generate 12 muscle-driven simulations, 2 trials per participant, of the WA phase of single-leg jump landing. A one-way ANOVA and Tukey post-hoc analysis showed both the quadriceps and gastrocnemii muscle force estimates were significantly greater than the hamstrings (p<0.001). Elevated gastrocnemii forces corresponded with increased joint compression and lower ACL forces. The elevated quadriceps and gastrocnemii forces during landing may represent a generalized muscle strategy to increase knee joint stiffness, protecting the knee and ACL from external knee loading and injury risk. These results contribute to our understanding of how muscle?s function during single-leg jump landing and should serve as the foundation for novel muscle-targeted training intervention programs aimed to reduce ACL injuries in sport.     

Influence of body segment parameter estimation on calculated ground reaction forces in highly dynamic movements

The effect of body segment parameter estimation (BSP) on the inverse dynamics modelling results has not yet been demonstrated in specific groups during athletic movements with high segment accelerations. Therefore, the purpose of this study was to analyse this effect in ski-jumpers as representatives of a specific group (i.e. low body mass index) by comparing calculated and measured ground reaction forces during ski-jumping imitation jumps. Full body kinematics and vertical ground reaction forces were recorded of 9 ski-jumpers performing three imitation jumps each. BSP were estimated using three previously published, one individually optimized and one ski-jumper group specific model. Vertical ground reaction forces were calculated using the vertical acceleration of the segments as well as the BSP of the single models in a top-down approach. Statistical analysis revealed a main model effect concerning the root mean square error between the calculated and the measured ground reaction force with deviations between the models of 53%. Individual optimization and the application of the ski-jumper group specific model increased the accuracy of the calculated ground reaction forces by 11 and 7%, respectively, compared to the best performing published model. The results of inverse dynamics modelling are very sensitive to the BSP estimation for specific groups like ski-jumpers during movements incorporating high segment accelerations. This emphasizes the importance of selecting adequate BSP estimation models or methods when analysing specific groups in highly dynamic movements in order to increase the accuracy of the inverse dynamics analyses results.     

Are natural coordinates a useful tool in modeling planar biomechanical linkages?

The paper analyzes the use of natural coordinates in modeling and simulation of musculoskeletal models of the human body. A biomechanical model of the lower extremity of the human body was constructed for the analysis. It consists of three anatomical segments described by eight natural coordinates located at the joints. The developed model was applied to solve three classic dynamics problems of human motion: inverse dynamics, direct dynamics, and static optimization. The analysis covers the raising of a leg together with; the time characteristics of the resultant net torques at the basic joints of the leg (inverse dynamics), the time histories of natural coordinates and their velocities (direct dynamics) as well as the time-varying muscle force patterns (static optimization). In order to check the numerical efficiency of modeling in the natural coordinates' environment the results were compared with the ones received through generalized coordinates. Some conclusions drawn from this comparison and final remarks referring to the biomechanical identification of the analyzed motor task were included in the paper.     

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.     

Influence of leg stiffness and its effect on myodynamic jumping performance.

The purposes of this study are: a) to examine the possibility of influencing the leg stiffness through instructions given to the subjects and b) to determine the effect of the leg stiffness on the mechanical power and take-off velocity during the drop jumps. A total of 15 athletes performed a series of drop jumps from heights of 20, 40 and 60 cm. The instructions given to the subjects were a) "jump as high as you can" and b) "jump high a little faster than your previous jump". The jumps were performed at each height until the athlete could not achieve a shorter ground contact time. The ground reaction forces were measured using a "Kistler" force plate (1000 Hz). The athletes body positions were recorded using a high speed (250 Hz) video camera. EMG was used to measure muscle activity in five leg muscles. The data was divided into 5 groups where group 1 was made up of the longest ground contact times of each athlete and group 5 the shortest. The leg and ankle stiffness values were higher when the contact times were shorter. This means that by influencing contact time through verbal instructions it is possible to control leg stiffness. Maximum center of mass take-off velocity the can be achieved with different levels of leg stiffness. The mechanical power acting on the human body during the positive phase of the drop jumps had the highest values in group 3. This means that there is an optimum stiffness value for the lower extremities to maximize mechanical power.