MODELING OF PLANAR BIOMECHANICAL LINKAGES BY MEANS OF NATURAL COORDINATES

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MODELOWANIE INŻYNIERSKIE ISNN 1896-771X 32, s. 77-82, Gliwice 2006

MODELING OF PLANAR BIOMECHANICAL LINKAGES BY MEANS OF NATURAL COORDINATES

ADAM CZAPLICKI

Department of Biomechanics, Academy of Physical Education in Warsaw, External Faculty of Physical Education in Biała Podlaska

Abstract. A multibody biomechanical model of the lower leg of the human being is presented. The model is based on natural coordinates. It is applied to a case of raising of a leg to solve three classic biomechanics problems i.e., inverse dynamics, direct dynamics, and static optimization. The time characteristics of the resultant net torques at the basic joints of the lower leg (inverse dynamics), the time histories of natural coordinates (direct dynamics) and the time-varying muscle activation patterns (static optimization) are shown. All the achieved results are compared to the ones received through generalized coordinates.

1. INTRODUCTION

The natural coordinates for planar mechanical systems were introduced by Garcia de Jalón et al. [6]. They were described in detail for both planar and spatial systems in Garcia de Jalón and Bayo’s textbook [7]. The natural coordinates can be defined at the joints of a biomechanical model. This feature seems to be particularly useful from the biomechanical standpoint as it reduces the total number of natural coordinates required to define a multibody system and simplifies considerably the definition of the joint constraint equations. It also eliminates the preprocessing phase needed to calculate the time histories of traditional, not directly registered, generalized coordinates, relative or Cartesian ones.

Natural coordinates also provide a framework for the description of motion of multibody systems without direct using rotation coordinates. This aspect of natural coordinates can be constructive from a programming perspective.

This work is an attempt to validate the usefulness of natural coordinates in modeling planar biomechanical linkages. Two biomechanical models of the lower extremity of the human being have been established for the validation process. The first model is defined in natural coordinates, the other one in generalized ones. Both models are used to solve three classic dynamics problems i.e., inverse dynamics, direct dynamics, and static optimization. The analysis covers raising of a leg. Due to the simplicity of the movement of this motor task, a straight qualitative interpretation of the obtained results can be expected.

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2. BIOMECHANICAL MODEL

The biomechanical model of the lower extremity of the human being consists of three anatomical segments described by eight natural coordinates located at the joints (Fig. 1, left).

The model has 5 degrees-of-freedom, which means that 3 rigid body constraint equations are to be formulated. They express the constant distance conditions between two successive basic points. The muscle net torques occurring at the joints are replaced by the pairs of forces acting on the basic and appropriately calculated Pi points. The gravitational forces are distributed among the basic points. The corresponding model based on generalized coordinates is shown in Figure 3 (right). The configuration of this model is unequivocally determined by 2 coordinates of the hip joint and 3 angular coordinates ϕi. The muscle apparatus consists of nine muscles for both models: iliopsoas (ILPSO), rectus femoris (RF), vastus (VAS), gluteus (GL), hamstrings (HAMS), biceps femoris short head (BFSH), gastrocnemius (GAS), soleus (SOL) and tibialis anterior (TA). The muscle structure, for clarity reasons, has been marked on the right model only.

l1

c2

l2

c3

l3

(x2,y2)

(x4,y4)

ϕ1

(x1,y1)

−ϕ2

ϕ3

c1

τ1

τ2

τ3

u1

P1

F1

-F1

u2

P2

F2

-F₂

u3

P3

F3

-F3

(x3,y3)

(x1,y1) GL

VAS RF ILPSO

GAS

SOL

TA HAMS

BFSH H

K

A

Fig. 1. Biomechanical model in natural (left) and generalized (right) coordinates

The forces of the muscles, indispensable for solving the optimization problem, have been introduced into the equations of motion of the multibody model by means of driver actuators defined as algebraic kinematic constraints. A four-element Hill type muscle model has been used to calculate individual muscle forces. The physiological, geometrical and, elastic and damping properties of the muscles have been estimated according to data described in [5, 8, 9].

3. DATA ACQUISITION

A twenty-two-year-old male, with the height of 180 [cm] and body mass 70 [kg], raised his right leg several times. One of the trials (Figure 2) has been chosen as the most appropriate and it was used in the calculations. The data acquisition process and handling the raw kinematic

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data were similar to those described in the papers [2, 4].

Fig. 2. The beginning (left) and the middle phase (right) of the rising of the leg.

A significant drift of the marker of the hip joint is visible.

4. INVERSE DYNAMICS

The dynamic equations of motion for the model described in natural coordinates can be written in the generic form

λ q τ C

τ q B f q

M&&= + ^T( )( + _pas)− ^T( ) (1) where M is the global mass matrix of the system, q is the vector of natural coordinates, f is the vector of external loads containing the gravitational forces and reactions in the hip joint, B is the matrix of control distribution, T τ_pas denotes the torques exerted by the passive joint structures (ligaments), C is the Jacobian matrix of the constraints, and λ is the vector of Lagrange multipliers. The vector of Lagrange multipliers is proportional to the reaction forces associated with the kinematic constraints.

The dynamic equations of the motion for the model based on generalized coordinates are described in detail elsewhere [2].

Using kinematic data as inputs, equation (1), after a slight rearrangement, can be solved for 8 unknowns: 2 components of the reaction in the hip joint, 3 Lagrange multipliers and 3 net muscle torques at the basic joints of the lower extremity. The time characteristics of the latter parameters are presented in Figure 3 being identical for both models. The positive values of the net torque in the hip joint reflects the nature of the analysed motor task. The subject moves his thigh towards the trunk at the beginning of the trial, then prevents the leg bending rapidly down to the ground. It indicates a significant activity of iliopsoas and rectus femoris muscles.

The negative values of this torque in the middle of the trial ensure a smooth transition period between the raising and falling pointing out hamstrings and gluteus muscle activation.

-25 0 25 50

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Time [s]

Net torque [Nm] ^hip_knee

ankle

Fig. 3. Net torques at the basic joints of the lower leg

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Inverse dynamics as a pure algebraic problem puts aside the differences in computational efficiency of both models. However, having once calculated Lagrange multipliers, it is easy to obtain the internal reactions in the joints with a minimal effort, which is not the case when modeling in generalized coordinates.

5. DIRECT DYNAMICS

Equation (1) attached to the constraint equations at the level of accelerations (i.e. Cq&&=ξ) has been solved using the implicit Lagrange multipliers elimination scheme [1, 7]









 



 + +



 



=



 





=

 ⇒







 



 + +

=



 







 





= −

ξ τ τ B f 0 C

C M λ

v v q

ξ τ τ B f λ v 0 C

C M

v q

) ) (

( _pas ^T ¹ ^T _pas

T

T &

&

. (2)

The controls and external loads were those obtained from the solution of the inverse dynamics problem.

Trajectories of the hip joint, common for both models, are shown in Figure 4. Grey solid lines obtained by integration in natural coordinates cover the black ones (generalized coordinates). Dotted lines are specified curves which are to be matched. The difference between measured and calculated trajectories is about 1 [cm] at the end of the trial.

-0.08 -0.04 0 0.04 0.08

0 0.2 0.4 0.6 0.8 1 1.2

Time [s]

Hip joint coordinates [m]

x1 y1

Fig. 4. Hip joint displacements

The fifth-order Runge-Kutta method with a variable time step integration algorithm was applied in the computations. The results for natural coordinates were achieved in 0.45 [s] after 97 steps of integration, and in 0.88 [s] after 89 steps for generalized coordinates. A slight advantage of natural coordinates is noticeable, which is due to the simpler form of the RHS of equation (2). Integration in dependent coordinates is known as an unstable process [1, 7]. The fact that it was not necessary to stabilize obtained solutions for such a period of time is another observation.

6. OPTIMIZATION

Replacing vector τ in the equation (1) by the moments originating from individual muscle forces modelled as four-element Hill type actuators, the optimization problem can be stated in natural coordinates as follows

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





≤

= 1 0

) (

min ₀

i opt

a F

b Aq

a

, (3)

where F0 is the cost function, A is the 8x14 matrix, ai (i=1…9) are the activation levels of the muscles, and ^q^opt ⁼

[

^λ¹ ^λ² ^λ³ ^R¹^x ^R¹^y â¹ â² â³ â⁴ â⁵ â⁶ â⁷ â⁸ â⁹

]

^T^{. The}

cost function has been taken from the work [3]. Having realistic physiological justifications, this function is widely used in biomechanics.

The activation levels of the muscles are shown in Figure 5. Being similar in shape for both models these characteristics differ quantitatively. The gluteus and vastus (very weak) activation has been predicted in natural coordinates’ environment only.

Iliopsoas

0.0 0.2 0.4 0.6 0.8 1.0

0 0.2 0.4 0.6 0.8 1 1.2

Time [s]

Activation level

n. coordi nates g. coordi nates

Rectus fem oris

0.0 0.2 0.4 0.6 0.8 1.0

0 0.2 0.4 0.6 0.8 1 1.2

Time [s]

Activation level

n. coordinates g. coordinates

Vastus

0.0 0.2 0.4 0.6 0.8 1.0

0 0.2 0.4 0.6 0.8 1 1.2

Time [s]

Activation level

Gluteus

0.0 0.2 0.4 0.6 0.8 1.0

0 0.2 0.4 0.6 0.8 1 1.2

Time [s]

Activation level

Ham strings

0.0 0.2 0.4 0.6 0.8 1.0

0 0.2 0.4 0.6 0.8 1 1.2

Time [s]

Activation level

Biceps fem oris s.h.

0.0 0.2 0.4 0.6 0.8 1.0

0 0.2 0.4 0.6 0.8 1 1.2

Time [s]

Activation level

Gastrocnem ius

0.0 0.2 0.4 0.6 0.8 1.0

0 0.2 0.4 0.6 0.8 1 1.2

Time [s]

Activation level

Tibialis anterior

0.0 0.2 0.4 0.6 0.8 1.0

0 0.2 0.4 0.6 0.8 1 1.2

Time [s]

Activation level

Fig. 4. Activation levels of selected muscles during the trial

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A sequential programming method (SQP) has been used to solve the optimization problem.

Benefiting mostly from the simpler form of the vector b the computations in natural coordinates were again slightly faster (7.2 [s]) comparing to generalized coordinates (7.8 [s]), where the matrix A is of size 5x11.

7. CONCLUSIONS

The natural coordinates provide a useful framework for modeling planar biomechanical linkages. The equations of motion in natural coordinates are not minimal in the number, but they have a simple structure. This feature makes them easy to implement for computations.

Three classic biomechanical problems under analysis have been solved in natural coordinates more efficient then in generalized ones.

REFERENCES

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2. Blajer W., Czaplicki A.: Modeling and inverse simulation of somersaults on the trampoline.

Journal of Biomechanics 2001, Vol. 34, p. 1619-1629.

3. Crowninshield R. D., Brand R. A.: A physiologically based criterion of muscle force prediction in locomotion. Journal of Biomechanics 1981, Vol. 14 , p. 793-801.

4. Czaplicki A., Silva M.T., Ambrósio J.C.: Biomechanical modelling for whole body motion using natural coordinates. Journal of Theoretical and Applied Mechanics 2004, Vol. 42, p. 927-944.

5. Davy D.T., Audu M.L.: A dynamic optimization technique for prediction muscle forces in the swing phase of gait. Journal of Biomechanics 1987, Vol. 20, p. 187-201.

6. Garcia de Jalón J., Serna M.A., Avilés R.: A computer method for kinematic analysis of lower-pair mechanism, Part 1: Velocities and accelerations and Part 2: Position problems.

Mechanism and Machine Theory 1981, Vol. 16, p. 543-566.

7. Garcia de Jalón J., Bayo, E.: Kinematic and dynamic simulation of multibody systems: The real-time challenge. New York: Springer-Verlag, 1993.

8. Hatze H.: The complete optimization of the human motion. Mathematical Biosciences 1976, Vol. 28, p. 99-135.

9. Yamaguchi G.T.: Dynamic modeling of musculoskeletal motion. Boston: Kluwer Academic Publishers, 2001.

MODELOWANIE PŁASKICH ŁAŃCUCHÓW BIOMECHANICZNYCH ZA POMOCĄ WSPÓŁRZEDNYCH NATURALNYCH

Streszczenie. W pracy zaprezentowano modelowanie płaskich układów biomechanicznych we współrzędnych naturalnych. Rozwiązano zagadnienie odwrotne, proste oraz problem optymalizacyjny dla wymachu kończyny dolnej.

Uzyskane wyniki porównano z odpowiadającymi im rezultatami otrzymanymi we współrzędnych uogólnionych.

This work was supported by Academy of Physical Education in Warsaw, grant BW III 11.