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Edited by: Bernardo Innocenti, Université libre de Bruxelles, Belgium

Reviewed by: Nicola Francesco Lopomo, University of Brescia, Italy; Peter Betsch, Karlsruhe Institute of Technology (KIT), Germany

This article was submitted to Biomechanics, a section of the journal Frontiers in Bioengineering and Biotechnology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The final subject position is often the only evidence in the case of the fall of a human being from a given height. Foreseeing the body trajectory and the respective driving force may not be trivial due to the possibility of rotations and to an unknown initial position and momentum of the subject. This article illustrates how multibody models can be used for this aim, with specific reference to an actual case, where a worker fell into a stair well, prior to stair mounting, and he was found in an unexpected posture. The aim of the analysis was establishing if this worker was dead in that same place, if he had been pushed, and which was his initial position. A multibody model of the subject has been built (“numerical android”), given his stature and his known mass. Multiple simulations have been performed, following a design of experiments where various initial positions and velocity as well as pushing forces have been considered, while the objective function to be minimized was the deviation of the numerical android position from the actual worker position. At the end of the analysis, it was possible to point how a very limited set of conditions, all including the application of an external pushing force (or initial speed), could produce the given final posture with an error on the distance function equal to 0.39 m. The full analysis gives a demonstration of the potentiality of multibody models as a tool for the analysis of falls in forensic inquiries.

Fall from height has a significant incidence among work-related injuries, reaching over 40% for the construction industry (Dong et al.,

Numerical models to be used for dynamic analyses are made of masses, connected to one another through joints, simulating skeletal articulations. These joints may have linear or, more often, non-linear elastic behaviors which are able to produce more accurate results (Richard et al.,

Multibody models have one more advantage over anthropomorphic dummies that is the possibility of simulating voluntary movements produced by muscle activation [(Milanowicz and Kedzior,

The case here analyzed refers to the fatal fall from height of a man at work. A clinical trial followed and the judge appointed one of the authors as a prosecutor to establish if it was possible for the victim to fall and land where the cadaver was found, or if the cadaver had probably been moved from elsewhere else. Secondly, the prosecutor had to establish if such a fall needed a voluntary action (murder or suicide) or it could be simply due to a fatal accident; the authors have tried to give an answer to these questions through a numerical analysis, based on a multibody model. The model has been described in detail, especially with reference to the simulation of articular joints since joint stiffness and contact parameters have been seldom reported in a systematic way, while their knowledge is mandatory in order to be able to discuss the respective model behavior compared to other works in literature. The numerical analysis has produced a new insight into the accident kinematics, providing valuable information for the forensic dispute. The model introduced can be generalized to study different body anthropometries thanks to regression functions, allowing to calculate mass and geometries from subject weight and height (Robbins,

The authors have chosen to use a numerical multibody model (MSC Adams software v. 17, by MSC Software Corporation): the subject body is made of rigid segments, with mass and inertial moments assigned to each of them; all segments are articulated to one another through elastic joints. The initial conditions of each part belonging to the android articulated model (in terms of the respective position and speed of the center of mass) have been established through a design of experiments (DOE); the known outcome on which this analysis was based had to be the final position of the body, according to pictures and measurements taken by legal prosecutors.

In the following the multibody model is described in details, as well as input variables for DOE and their respective range of variation. Finally, the objective function, used to measure the “goodness of fit” of the supposed fall kinematics, is reported.

The articulated total body model is made of 15 ellipsoidal elements, connected to one another by means of 14 joints, as detailed in

Segments description.

1 | Head |

2 | Neck |

3 | Upper Torso |

4 | Central Torso |

5 | Lower Torso |

6 | Right Upper Arm |

7 | Right Lower Arm |

8 | Left Upper Arm |

9 | Left Lower Arm |

10 | Right Upper Leg |

11 | Right Lower Leg |

12 | Right Foot |

13 | Left Upper Leg |

14 | Left Lower Leg |

15 | Left Foot |

Ellipsoids geometry is completely defined by a center of mass coordinate system and two coordinate systems located, respectively, at the proximal and distal ends. The respective geometry is detailed in

Body segments have been assigned also a mass and inertial properties, according to anthropomorphic measurements referred to the fiftieth percentile having the input weight and height [UMTRI reports (Robbins,

Simple mechanical joints or more complex joints (generated as a combination of simple ones) have been used to reproduce natural human joints with the respective degrees of freedom (DOF). More in detail, three type of constrains have been applied: spherical (DOF: 3 rotations), revolute (DOF: 1 rotation), and primitive perpendicular (DOF: 2 rotations), as described in

Mechanical—Body joints correspondence.

Spherical | 3 Rotations | Upper Neck |

Spherical | 3 Rotations | Lower Neck |

Spherical with Perpendicular | 2 Rotations (rotation along the long axis segment is removed) | Right/Left Shoulder |

Revolute | 1 Rotation in the sagittal plane | Right/Left Elbow |

Spherical | 3 Rotations | Lumbar Spine |

Spherical | 3 Rotations | Thoracic Spine |

Spherical with Perpendicular | 2 Rotations (rotation along the long axis segment is removed) | Right/Left Hip |

Revolute | 1 Rotation in the sagittal plane | Right/Left Knee |

Revolute | 1 Rotation in the sagittal plane | Right/Left Ankle |

Axial rotations around the long bone's axis and abduction/adduction movements of elbows, knees and ankles have not been taken into consideration in the following simulations, in order to simplify the model, in relation of its purpose. Indeed, preliminary tests have demonstrated that these movements did not take place or were very limited for this case study.

The passive resistance of all joints has been defined. This job has represented a major burden in the modeling process, due to the high number of degree of freedoms involved, and to many different analytical laws having been implemented in literature, sometimes with peculiar reference systems. After a wide literature survey (Engin,

Limiting the joint range of motion

Joint stabilization, preventing segments collapsing under their own weight.

Passive resistive moments characteristics.

^{°}] |
|||||
---|---|---|---|---|---|

Upper/Lower Neck (Haug et al., |
Flexion | 0°-30° | 1.4 | 0.0678 | |

Extension | 0°-35° | 2.5 | |||

Lateral Bending | 0°-45° | 2.2 | |||

Twist | 0°-50° | 0.5 | |||

Shoulder (Engin, |
Flexion/Extension | −50°−180° | 0.0678 | ||

Abduction/Adduction | −50°-160° | ||||

Abduction in Frontal Plane | 0°-160° | ||||

Thoracic (Bergmark, |
Flexion | 0°-10° | 3 | 0.0565 | |

Extension | 0°-5° | 3.4 | |||

Lateral Bending | 0°-20° | 2 | |||

Twist | 0°-30° | 2.5 | |||

Lumbar (Kapandji, |
Flexion | 0°-45° | 1.8 | 0.0565 | |

Extension | 0°-10° | 2.5 | |||

Lateral Bending | 0°-20° | 1.3 | |||

Twist | 0°-5° | 0.9 | |||

Elbow (Engin and Chen, |
Flexion | 0°-150° | 0.0339 | ||

Hip (Riener and Edrich, |
Flexion/Extension | −30°-150° [−30°−50°] | 0.0339 | ||

Abduction in the Frontal Plane | 0°−80° | 1.2 | |||

Adduction in the Frontal Plane | 0°−30° | 0.8 | |||

Knee (Riener and Edrich, |
Flexion | 0°-150° | 0.0339 | ||

Ankle (Haug et al., |
Plantar flexion | 0°-50° | 0.3 | 0.0339 | |

Dorsiflexion | 0°-30° | 0.5 |

_{S}: shoulder flexion/extension angle

_{s}: shoulder abduction angle

_{E}: elbow flexion angle

_{H}: hip flexion/extension angle

_{K}: knee flexion/extension (in the application of this formula for the flexion/extension resistance of the hip, this angle has been set equal to zero)

Whenever the passive joint resistance has been modeled with a non-linear behavior, the force/displacement function was analytically described through a spline, whose trend is similar to the one reported in

Passive resistive moment for shoulder flexion/extension: the general trend including a “Hard stop”

Joint resistive properties have been completed with constant viscous damping coefficients (

The numerical model has been validated for one specific anthropometry, comparing its results with experimental results obtained by Hajiaghamemar et al. (

These same configurations have been simulated with the developed model (

Scenario 1, Scenario 2, Scenario 3, Scenario 4, Scenario 5:

Head impact force.

Experimental (Dummy) | 22.8 ± 2.1 | 14.9 ± 4.6 | 20.3 ± 3.7 | 21.6 ± 6.1 | 17.1 ± 2.2 |

Simulation (Model) | 22.9 | 14.83 | 21.46 | 24 | 18.6 |

Analytical deviation | Δ = 0.1 [ |
Δ = −0.07 [ |
Δ = 1.16 [ |
Δ = 2.4 [ |
Δ = 1.5 [ |

These five scenarios have been realized applying suitable motion laws to joints for the first few instants, and the only gravity action was simulated from that point on.

The choice of input parameters to be varied, according to the design of experiments, has not been trivial, since it was necessary to list all unknown variables, and to select a limited set of those variables which were likely to play a significant influence on the final victim position. According to first trials, the authors have chosen to consider five variables, defining the body position on the upper floor, its orientation, and the initial speed of the central torso (which simulates an impulsive action due to a shove); the respective representation is reported in

Initial parameters definition and actual scenario representation.

Input variables of DOE.

_{i} [m] |
0.00 : 1.00 |

_{i} [m] |
−0.25 : 0.15 |

_{i} [°] |
−90 : 90 |

_{i} [m/s] |
−10.00 : −0.10 |

_{i} [°/s] |
−10 : 10 |

The objective function to be minimized was the distance between the actual victim position (“A” configuration in the following) and the position of the multibody android at the end of the simulation (“M” configuration in the following). Seven different functions have been tested in order to choose the best formulation that is the simplest one, leading to the same results as the most complex one. It can be so defined:

Where:

_{Ai}, _{Ai} are the coordinate of the center of mass of “i” body segment (

_{Mi}, _{Mi} are the coordinate of the center of mass of “i” segment belonging to the multibody android model (

Values to be assigned to “i” are detailed in

Body segments considered by each objective function.

OBJ 1 | x | |||||||||||||||

OBJ 2 | x | x | x | |||||||||||||

OBJ 3 | x | x | x | x | x | |||||||||||

OBJ 4 | x | x | x | x | x | x | x | |||||||||

OBJ 5 | x | x | x | x | ||||||||||||

OBJ 6 | x | x | x | x | x | |||||||||||

OBJ 7 | x | x | x | x | x | x | x |

The validation of the model has been performed comparing numerical model results with experimental results obtained by Hajiaghamemar et al. (_{1} to instant t_{5}, as extracted from both models. As can be seen, trends of these curves are very similar as well as rotations' values.

The model has proved to be able to simulate both body kinematics and the respective impact forces with a maximum peak error equal to 11% (

Optimization process workflow.

As specified in the above section, the first design of experiments was performed considering five input variables (

Preliminary results.

_{i} |
_{i} [m/s] |
_{ι}^{°}/s] |
_{i} [m] |
_{i} [m] |
|||
---|---|---|---|---|---|---|---|

OBJ1 | 156 | 0.0 | −0.10 | 10 | 0.0 | 0.15 | 0.14 |

OBJ2 | 137 | 0.0 | −0.10 | −10 | 0.0 | −0.05 | 0.63 |

OBJ3 | 137 | 0.0 | −0.10 | −10 | 0.0 | −0.05 | 0.92 |

OBJ4 | 137 | 0.0 | −0.10 | −10 | 0.0 | −0.05 | 1.7 |

OBJ5 | 127 | 0.0 | −5.05 | 10 | 0.0 | −0.25 | 0.80 |

OBJ6 | 127 | 0.0 | −5.05 | 10 | 0.0 | −0.25 | 0.82 |

OBJ7 | 119 | 0.0 | −5.05 | 0.0 | 0.0 | −0.05 | 1.33 |

According to preliminary results, the following statements can be made:

◦ OBJ2, OBJ3, and OBJ4 reach their minimum value for the same set of input parameters (Trial 137); therefore, considering also the center of mass of upper arms or of lower legs is not relevant.

◦ OBJ5 and OBJ6 reach their minimum for the same combinations of parameters. Therefore, the addition of the lower torso center of mass to the objective function is not relevant for the analysis.

◦ All objective functions reach their minimum for ϑ_{i} equal to zero (the android position is on the back, with respect to the aperture).

◦ OBJ1 can reach its minimum value also for incorrect final body positions such as supine or with feet-to-head vector pointing to the door, that is opposite to the x-axis direction (see the reference system in

Taking into account all these observations, the next analysis has been focused on three objective functions, that are OBJ2, OBJ5, and OBJ7. A new analysis has been performed considering these three objective functions; however ϑ_{i} range of variation has been set equal to −5° and 15°, since the previous analysis had demonstrated that its optimized value was zero, and this result was confirmed by optimization analyses which always produced values close to zero. The new design of experiments has produced results reported in

Second DOE results.

_{i} [^{°}] |
_{i} [m/s] |
_{i}^{°}/s] |
_{i} [m] |
_{i} [m] |
|||
---|---|---|---|---|---|---|---|

OBJ2 | 201 | 15 | −5.05 | 0.0 | 0.0 | 0.15 | 0.39 |

OBJ5 | 201 | 15 | −5.05 | 0.0 | 0.0 | 0.15 | 0.42 |

OBJ7 | 201 | 15 | −5.05 | 0.0 | 0.0 | 0.15 | 0.71 |

The results of this second factorial analysis can be so summarized:

◦ OBJ2, OBJ5, and OBJ7 reach their minimum value for the same set of input parameters (Trial 201); therefore, considering the only center of mass of the head and upper legs allows to reach accurate results;

◦ Even when ω_{i} varies between its extreme values, the respective objective functions variation is below 2%; as such, input variable ω_{i} has been removed from the analysis since it did not play a significant influence (in relation to the hypothesized range of variation).

In the final analysis, the DOE retained four factors and assigned five levels to each of them, for a total number of trials equal to 625, and the objective function OBJ2 was calculated. According to results, the best input variables set is the one reported in

Final results.

_{I} [^{°}] |
_{i} [m/s] |
_{i} [m] |
_{i} [m] |
||
---|---|---|---|---|---|

OBJ2 | 15 | −5.05 | 0 | 0.15 | 0.39 |

Initial and final configuration for the best parameters' combination: the wireframe model represents the actual victim position, the solid model represents the numerical android position at the end of the simulation.

Even if the final value of the objective function (0.39 m) may not seem so low, it should be reminded that it is a sum of three distances: 0.05 m for the head, 0.15 m for the left upper leg and 0.18 m for the right upper leg.

With the reported “optimal” combination of parameters, the maximum segment distance was obtained for lower arms segments (

The computational effort required for all the performed simulations was in general very low (PC with i7-8700 CPU and 32 GB RAM). The longest time was required by DOE simulations, being strictly related to the number of trials which have been tested: the final analysis with 625 experiments has taken about 1 h. All other simulations have been performed in few seconds.

The methodology used by the authors to establish the initial conditions has been a sort of “trial and error” where a wide spectrum of possibilities has been inquired. As such, the procedure is heavily biased by the choice of input variables to be varied with the respective range. A promising alternative approach could be based on evolutionary algorithms where the system is able to “auto-tune” itself to individuate the best solution (Dasgupta and Michalewicz,

The first part of this work concerns the creation of an articulated multibody model suitable for the main purpose that is the analysis of a fall from a given height, knowing only the final actual position. First of all, segments, representing body parts, and connection joints between them have been created. Inertial and geometrical properties of segments were based on anthropometric data calculated from the victim height and weight, through regression equations. However, it is quite obvious that two variables are very few to fully determine body segments geometry and inertial properties; more accurate results could be obtained through a deeper examination of the victim anthropometry, for example by means of laser scanning (Pandis and Bull,

Connection articular joints have been modeled with classical mechanical joints or with a combination of these; some joint's degrees of freedom have been neglected since they demonstrated to undergo null or very limited movements. This simplification could not hold when analyzing other cases of fall/accidents.

The effective operation of joints has been guaranteed by the implementation of passive resistive properties retrieved from literature. Many joints have been modeled with a non-linear elastic behavior (

Validation of numerical models it's a key point for their application. Human multibody model validation is not so trivial, mainly due to the problems in raising appropriate experimental data or to the possibility of performing necessary tests (Griffin,

In this work the model has been validated reproducing different fall scenarios and comparing them with results obtained by Hajiaghamemar et al. (

For the model here presented, the performed validation should be deemed sufficient, having taken into account that all inertial and geometrical properties were obtained from well-known regression laws as well as resistive joints properties were the results of a comparison between many experimental results performed over last 40 years.

The model here introduced is not able to simulate trauma and injuries, and the corresponding energy absorption, therefore it behaves more elastically compared to the actual body response. Nonetheless, the likelihood of injuries can be established, on the basis of injury criteria (King,

The final objective function does not take into account appendicular skeleton movements (lower arms, lower limbs and feet): this result agrees with findings from other researcher who demonstrated the respective negligible influence (Milanowicz and Kedzior,

All the procedure has been here developed and tuned for the case of a fall from a height. Despite the application of the developed model to a single case study, is the authors' opinion that the model can be generalized to study different forensic backgrounds. Indeed, for this kind of applications, where the input parameters are final configuration's evidences, the whole procedure is the same and so the model application is quite straightforward once inertial and geometrical properties have been tuned to the person specific characteristics.

This article illustrates a well-established approach where a validated multibody numerical model is used to simulate the dynamics of a human body, given its initial conditions. Special care has been paid to the accurate simulation of the passive properties of articular joints, reporting the respective elastic behavior in detail. In the specific case here analyzed, the dynamic analysis has allowed establishing the position of the victim prior to the fall and, more important, that a voluntary action had to be included in the model (in the form of an initial velocity at the central torso) in order to justify the final position of the victim. The result of the analysis was somehow unexpected since at a first glance the victim position seemed quite odd and unlikely, leaving the suspect that it had been moved after death. On the whole, a demonstration has been given of how biomechanics can give a contribution to the forensic analysis of a fall from height, together with legal medicine, suggesting that the best approach should be multidisciplinary.

The datasets generated for this study are available on request to the corresponding author.

GP and DC have set up the numerical model. EZ has discussed model details with GP and DC, and she has supervised the whole work with FC and PC. PC has analyzed and organized all experimental data.

DC was employed by the company MSC Software. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The Supplementary Material for this article can be found online at:

This video shows the simulation of a fall from a height of a human subject. The sequence of falling here represented is referred to the best parameters combination which best match to the actual final configuration (wireframe model in this video).