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.     

Computer simulation of human body dynamics

Understanding human body dynamics is important in many situations, such as automobile and aircraft crashes, aircraft ejections, falls, and other acceleration environments. The design of automobile interiors, cockpits, and safety equipment requires knowledge of the forces and accelerations encountered during an emergency. Because of the limited information available from actual events and the various constraints in testing, computer simulations are often the only means of obtaining detailed information. The Armstrong Laboratory (AL) developed the Articulated Total Body (ATB) model to predict the human body dynamics in many of these environments. This model is a three-dimensional rigid body dynamics program in which the human body is modeled as a series of segments. Forces on the body segments are calculated based on their interaction with the surroundings including seat and cockpit surfaces. The model also calculates the internal joint resistive and constraint forces. Because of this capability to predict both internal and external forces acting on the body, the ATB model can be used in investigating injuries. It is also a valuable design tool for evaluating safety of proposed systems before prototypes are built or costly tests conducted. When testing is conducted, the model provides data that cannot be measured, such as forces within the body, and supplementing test data with parameter variation simulations. To validate the model, tests such as those conducted on the AL impact sled are simulated. Test films and instrumentation data are compared with simulation graphics and quantitative output to gain confidence in the simulation results.     

Biologically inspired kinematic synergies enable linear balance control of a humanoid robot

Despite many efforts, balance control of humanoid robots in the presence of unforeseen external or internal forces has remained an unsolved problem. The difficulty of this problem is a consequence of the high dimensionality of the action space of a humanoid robot, due to its large number of degrees of freedom (joints), and of non-linearities in its kinematic chains. Biped biological organisms face similar difficulties, but have nevertheless solved this problem. Experimental data reveal that many biological organisms reduce the high dimensionality of their action space by generating movements through linear superposition of a rather small number of stereotypical combinations of simultaneous movements of many joints, to which we refer as kinematic synergies in this paper. We show that by constructing two suitable non-linear kinematic synergies for the lower part of the body of a humanoid robot, balance control can in fact be reduced to a linear control problem, at least in the case of relatively slow movements. We demonstrate for a variety of tasks that the humanoid robot HOAP-2 acquires through this approach the capability to balance dynamically against unforeseen disturbances that may arise from external forces or from manipulating unknown loads.     

Modeling locomotion of a soft-bodied arthropod using inverse dynamics

Most bio-inspired robots have been based on animals with jointed, stiff skeletons. There is now an increasing interest in mimicking the robust performance of animals in natural environments by incorporating compliant materials into the locomotory system. However, the mechanics of moving, highly conformable structures are particularly difficult to predict. This paper proposes a planar, extensible-link model for the soft-bodied tobacco hornworm caterpillar, Manduca sexta, to provide insight for biologists and engineers studying locomotion by highly deformable animals and caterpillar-like robots. Using inverse dynamics to process experimentally acquired point-tracking data, ground reaction forces and internal forces were determined for a crawling caterpillar. Computed ground reaction forces were compared to experimental data to validate the model. The results show that a system of linked extendable joints can faithfully describe the general form and magnitude of the contact forces produced by a crawling caterpillar. Furthermore, the model can be used to compute internal forces that cannot be measured experimentally. It is predicted that between different body segments in stance phase the body is mostly kept in tension and that compression only occurs during the swing phase when the prolegs release their grip. This finding supports a recently proposed mechanism for locomotion by soft animals in which the substrate transfers compressive forces from one part of the body to another (the environmental skeleton) thereby minimizing the need for hydrostatic stiffening. The model also provides a new means to characterize and test control strategies used in caterpillar crawling and soft robot locomotion.     

Balancing a force on the fingertip of a two-dimensional finger model without intrinsic muscles

A slightly flexed human middle finger can balance an external force on the fingertip. Internal stabilization is also possible, which means that the externally unloaded finger can be kept stiff. We want to analyse whether in these situations the intrinsic hand muscles are needed. Distances from tendons to flexion axes are taken from the literature and are substituted in the moment equilibrium equations of a two-dimensional finger model. Diagrams illustrate the statically indeterminate problem of solving tendon forces. The possibilities for equilibrium without intrinsics appear to depend mainly on four tendon-to-joint distances. These distances determine to which of two groups a finger belongs: (1) one in which intrinsics are not necessary for internal stabilization nor for balancing a force on the fingertip in any direction in the sagittal plane; (2) one in which, without intrinsics, internal stabilization is impossible and only dorso-distally directed forces on the fingertip can be balanced.     

Change in knee contact force with simulated change in body weight

The relationship between obesity, weight gain and progression of knee osteoarthritis is well supported, suggesting that excessive joint loading may be a mechanism responsible for cartilage deterioration. Examining the influence of weight gain on joint compressive forces is difficult, as both muscles and ground reaction forces can have a significant impact on the forces experienced during gait. While previous studies have examined the relationship between body weight and knee forces, these studies have used models that were not validated using experimental data. Therefore, the objective of this study was to evaluate the relationship between changes in body weight and changes in knee joint contact forces for an individual's gait pattern using musculoskeletal modeling that is validated against known internal compressive forces. Optimal weighting constants were determined for three subjects to generate valid predictions of knee contact forces (KCFs) using in vivo data collection with instrumented total knee arthroplasty. A total of five simulations per walking trial were generated for each subject, from 80% to 120% body weight in 10% increments, resulting in 50 total simulations. The change in peak KCF with respect to body weight was found to be constant and subject-specific, predominantly determined by the peak force during the baseline condition at 100% body weight. This relationship may be further altered by any change in kinematics or body mass distribution that may occur as a result of a change in body weight or exercise program.     

On the mechanics of viscous bodies and elongation in ellipsoidal cells

After deriving some auxiliary equations for the average elongation of a viscous body under the action of forces derived from a potential, the diffusion problem for an ellipsoidal cell with a constant rate of reaction is solved for the case of an infinite permeability. The equation of elongation of such a cell under the influence of diffusion forces is derived, and compared with the, approximate expression obtained by N. Rashevsky for any kind of oblong cell. The two equations are in fair agreement. Effects of constant and variable surface tension are studied.     

Elbow and wrist joint contact forces during occupational pick and place activities

A three-dimensional, mathematical model of the elbow and wrist joints, including 15 muscle units, 3 ligaments and 4 joint forces, has been developed. A new strain gauge transducer has been developed to measure functional grip forces. The device measures radial forces divided into six components and forces of up to 250N per segment can be measured with an accuracy of +/-1%. Ten normal volunteers were asked to complete four tasks representing occupational activities, during which time their grip force was monitored. Together with kinematic information from the six-camera Vicon data, the moment effect of these loads at the joints was calculated. These external moments are assumed to be balanced by the internal moments, generated by the muscles, passive soft tissue and bone contact. The effectiveness of the body's internal structures in generating joint moments was assessed by studying the geometry of a simplified model of the structures, where information about the lines of action and moment arms of muscles, tendons and ligaments is contained. The assumption of equilibrium between these external and internal joint moments allows formulation of a set of equations from which muscle and joint forces can be calculated. A two stage, linear optimisation routine minimising the overall muscle stress and the sum of the joint forces has been used to overcome the force-sharing problem. Humero-ulnar forces of up to 1600N, humero-radial forces of up to 800N and wrist joint forces of up to 2800N were found for moderate level activity. The model was validated by comparison with other studies.     

Whole body inverse dynamics over a complete gait cycle based only on measured kinematics

This paper presents a three-dimensional (3D) whole body multi-segment model for inverse dynamics analysis over a complete gait cycle, based only on measured kinematic data. The sequence of inverse dynamics calculations differs significantly from the conventional application of inverse dynamics using force plate data. A new validated "Smooth Transition Assumption" was used to solve the indeterminacy problem in the double support phase. Kinematic data is required for all major body segments and, hence, a whole body gait measurement protocol is presented. Finally, sensitivity analyses were conducted to evaluate the effects of digital filtering and body segment parameters on the accuracy of the prediction results. The model gave reasonably good estimates of sagittal plane ground forces and moment; however, the estimates in the other planes were less good, which we believe is largely due to their small magnitudes in comparison to the sagittal forces and moment. The errors observed are most likely caused by errors in the kinematic data resulting from skin movement artefact and by errors in the estimated body segment parameters. A digital filtering cut-off frequency of 4.5Hz was found to produce the best results. It was also shown that errors in the mass properties of body segments can play a crucial role, with changes in properties sometimes having a disproportionate effect on the calculated ground reactions. The implication of these results is that, even when force plate data is available, the estimated joint forces are likely to suffer from similar errors.     

Estimation of trunk muscle forces using the finite element method and in vivo loads measured by telemeterized internal spinal fixation devices.

Direct quantitative measurement of muscle forces is not possible. Forces in the trunk muscles were estimated for standing and flexion of the upper body using three-dimensional, nonlinear finite element models of the lumbar spine with and without an internal spinal fixation device. Muscle forces assumed were two pairs dorsally and one ventrally, each representing several muscles. Muscle forces in the model with internal fixators were varied in discrete steps until the implant loads calculated closely corresponded to those measured in a patient with an instrumented implant. The calculated angles between adjacent lumbar vertebrae were compared with corresponding values measured on X-ray films of a patient as well as with literature values and served as a second criterion for predicting muscle forces. For the model without an implant, the muscle forces of the first model were slightly varied until the lumbar spine shape and the intradiscal pressure were physiological. The abdomen was shown to have a considerable supporting function for flexion.     

Takeoff and landing forces of leaping strepsirhine primates.

Knowledge of the forces animals generate and are exposed to during locomotion is an important prerequisite for understanding the musculoskeletal correlates of locomotor modes. We recorded takeoff and landing forces for 14 animals representing seven species of strepsirhine primates with a compliant force pole. Our sample included both specialized vertical clingers and leapers and more generalized species. Takeoff forces are higher than landing forces. Peak forces during acceleration for takeoff ranged from 6 to 12 times body weight, and the peak impact forces at landing are between 5 and 9 times body weight. There is a size-related trend in peak force magnitudes. Both takeoff and landing forces decrease with increasing body size in our sample of animals from 1 kg to over 5 kg. Peak forces increase with distance leapt. The distance effect is less clear, probably due to the narrow range of distances represented in our sample. A comparison of subadult and adult animals of two species of sifakas reveals a tendency for the young animals to exert relatively higher peak forces in comparison to their adult conspecifics. Finally, Lemur catta and Eulemur rubriventer, the "generalists" in our sample, tend to generate higher forces for equal tasks than the specialized vertical clingers and leapers (i.e., the indriids and Hapalemur).A broad-scale comparison of peak leaping forces and peak forces for quadrupedal and bipedal walking and running shows that leaping at small body size is associated with exceptionally high forces. Whereas relative forces (i.e., forces divided by body weight) decrease with increasing body mass for leaping, forces for walking and running do not change much with size. Leaping forces in our sample scale to (mass)(-1/3), which is consistent with expectations derived from geometric similarity models. Forces associated with other locomotor activities do not appear to follow this pattern. The very high forces found in strepsirhine leapers do not seem to be matched by bone robusticity beyond that documented for quadrupedal species.     

Energy-conserving impact algorithm for the heel-strike phase of gait

Significant ground reaction forces exceeding body weight occur during the heel-strike phase of gait. The standard methods of analytical dynamics used to solve the impact problem do not accommodate well the heel-strike collision due to the persistent contact at the front foot and presence of contact at the back foot. These methods can cause a non-physical energy gain on the order of the total kinetic energy of the system at impact. Additionally, these standard techniques do not quantify the contact force, but the impulse over the impact. We present an energy-conserving impact algorithm based on the penalty method to solve for the ground reaction forces during gait. The rigid body assumptions are relaxed and the bodies are allowed to penetrate one another to a small degree. Associated with the deformation is a potential, from which the contact forces are derived. The empirical coefficient-of-restitution used in the standard approaches is replaced by two parameters to characterize the stiffness and the damping of the materials. We solve two simple heel-strike models to illustrate the shortcomings of a standard approach and the suitability of the proposed method for use with gait.     

Consequences of forced silking

The forced silking of a spider to obtain major ampullate (MA) silk for experiments is a standard practice; however, this method may have profound effects on the resulting silk's properties. Experiments were performed to determine the magnitude of the difference in the forces required to draw silk from the MA gland between unrestrained spiders descending on their draglines and restrained spiders from which MA silk was drawn with a motor. The results show that freely falling spiders can spool silk with as little as 0.1 body weights of force, which generates a stress that is about 2% of the silk's tensile strength. In contrast, forcibly silked spiders apply as much as 4 body weights of force with an internal braking mechanism, and this force creates silk stresses in excess of 50% of the silk's tensile strength. The large forces observed in forced silking should strongly affect the draw alignment of the polymer network in the newly spun fibers, and this may account for the differences in material properties observed between naturally spun and forcibly spun MA silks. In addition, the heat produced by the internal friction brake during forced silking may set the upper limit of forced silking speed.     

Effect of load position on muscle forces, internal loads and stability of the human spine in upright postures

A novel kinematics-based approach coupled with a non-linear finite element model was used to investigate the effect of changes in the load position and posture on muscle activity, internal loads and stability margin of the human spine in upright standing postures. In addition to 397 N gravity, external loads of 195 and 380 N were considered at different lever arms and heights. Muscle forces, internal loads and stability margin substantially increased as loads displaced anteriorly away from the body. Under same load magnitude and location, adopting a kyphotic posture as compared with a lordotic one increased muscle forces, internal loads and stability margin. An increase in the height of a load held at a fixed lever arm substantially diminished system stability thus requiring additional muscle activations to maintain the same margin of stability. Results suggest the importance of the load position and lumbar posture in spinal biomechanics during various manual material handling operations.     

Joint contact loading in forefoot and rearfoot strike patterns during running

Research concerning forefoot strike pattern (FFS) versus rearfoot strike pattern (RFS) running has focused on the ground reaction force even though internal joint contact forces are a more direct measure of the loads responsible for injury. The main purpose of this study was to determine the internal loading of the joints for each strike pattern. A secondary purpose was to determine if converted FFS and RFS runners can adequately represent habitual runners with regards to the internal joint loading. Using inverse dynamics to calculate the net joint moments and reaction forces and optimization techniques to estimate muscle forces, we determined the axial compressive loading at the ankle, knee, and hip. Subjects consisted of 15 habitual FFS and 15 habitual RFS competitive runners. Each subject ran at a preferred running velocity with their habitual strike pattern and then converted to the opposite strike pattern. Plantar flexor muscle forces and net ankle joint moments were greater in the FFS running compared to the RFS running during the first half of the stance phase. The average contact forces during this period increased by 41.7% at the ankle and 14.4% at the knee joint during FFS running. Peak ankle joint contact force was 1.5 body weights greater during FFS running (p<0.05). There was no evidence to support a difference between habitual and converted running for joint contact forces. The increased loading at the ankle joint for FFS is an area of concern for individuals considering altering their foot strike pattern.     

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.     

Intercellular separation forces generated by intracellular pressure

Turgor pressure tends to force plant cells towards a spherical form, thus separating them at the angles from adjacent cells. In cooked vegetables containing starch, the swelling pressure of starch gelatinization generates analogous cell separation forces. A theoretical analysis of the relationship between internal pressure and cell separation forces is presented. Apart from the effect of internal pressure, cell separation forces increase with the diameter of the cell and decrease with the number of cell sides. Cell separation forces are reduced by the introduction of intercellular spaces and decrease further as these expand. The relationship between intracellular pressure and cell separation forces provides a basis upon which the strength of intercellular adhesion can be measured by experiment.     

Force transmission via axial tendons in undulating fish: a dynamic analysis

Sonomicrometrics of in vivo axial strain of muscle has shown that the swimming fish body bends like a homogenous, continuous beam in all species except tuna. This simple beam-like behavior is surprising because the underlying tendon structure, muscle structure and behavior are complex. Given this incongruence, our goal was to understand the mechanical role of various myoseptal tendons. We modeled a pumpkinseed sunfish, Lepomis gibbosus, using experimentally-derived physical and mechanical attributes, swimming from rest with steady muscle activity. Axially oriented muscle-tendons, transverse and axial myoseptal tendons, as suggested by current morphological knowledge, interacted to replicate the force and moment distribution. Dynamic stiffness and damping associated with muscle activation, realistic muscle force generation, and force distribution following tendon geometry were incorporated. The vertebral column consisted of 11 rigid vertebrae connected by joints that restricted bending to the lateral plane and endowed the body with its passive viscoelasticity. In reaction to the acceleration of the body in an inviscid fluid and its internal transmission of moment via the vertebral column, the model predicted the kinematic response. Varying only tendon geometry and stiffness, four different simulations were run. Simulations with only intrasegmental tendons produced unstable axial and lateral tail forces and body motions. Only the simulation that included both intra- and intersegmental tendons, muscle-enhanced segment stiffness, and a stiffened caudal joint produced stable and large lateral and axial forces at the tail. Thus this model predicts that axial tendons function within a myomere to (1) convert axial force to moment (moment transduction), (2) transmit axial forces between adjacent myosepta (segment coupling), and, intersegmentally, to (3) distribute axial forces (force entrainment), and (4) stiffen joints in bending (flexural stiffening). The fact that all four functions are needed to produce the most realistic swimming motions suggests that axial tendons are essential to the simple beam-like behavior of fish.     

A contact model to simulate human–artifact interaction based on force optimization: implementation and application to the analysis of a training machine

Musculoskeletal multibody models are increasingly used to analyze and optimize physical interactions between humans and technical artifacts. Since interaction is conveyed by contact between the human body and the artifact, a computationally robust modeling approach for frictional contact forces is a crucial aspect. In this contribution, we propose a parametric contact model and formulate an associated force optimization problem to simultaneously estimate unknown muscle and contact forces in an inverse dynamic manner from a prescribed motion trajectory. Unlike existing work, we consider both the static and the kinetic regime of Coulomb’s friction law. The approach is applied to the analysis of a leg extension training machine with the objective to reduce the stress on the tibiofemoral joint. The uncertainty of the simulation results due to a tunable parameter of the contact model is of particular interest.     

Effect of load position on muscle forces,internal loads and stability of the human spine in upright postures

A novel kinematics-based approach coupled with a non-linear finite element model was used to investigate the effect of changes in the load position and posture on muscle activity, internal loads and stability margin of the human spine in upright standing postures. In addition to 397 N gravity, external loads of 195 and 380 N were considered at different lever arms and heights. Muscle forces, internal loads and stability margin substantially increased as loads displaced anteriorly away from the body. Under same load magnitude and location, adopting a kyphotic posture as compared with a lordotic one increased muscle forces, internal loads and stability margin. An increase in the height of a load held at a fixed lever arm substantially diminished system stability thus requiring additional muscle activations to maintain the same margin of stability. Results suggest the importance of the load position and lumbar posture in spinal biomechanics during various manual material handling operations.