Structural and kinematic modeling of a quadruped biomechanism.

Geonea, Ionut Daniel ; Ungureanu, Cezar Alin ; Micu, Cristina 等

1. INTRODUCTION

In case of four legged mammals, the structure of anterior and posterior legs is very similarly by the structure of most majorities of actual four legs quadrupeds (Buzea, 2005). To some quadrupeds, the anterior legs are short that those posterior. To remark, that at quadrupeds, the anterior legs have the degree of mobility larger than the posterior legs.

By physical modeling of a dog (fig. 1) we obtain a biomechanism (mobile biorobot), in which the legs are realized like plane articulated kinematic chains (Antonescu&Buzea, 2005).

[FIGURE 1 OMITTED]

2. KINEMATIC SCHEME AND THE MOBILITY OF THE BIOMECHANISM

The kinematic scheme of the quadruped biomechanism is realized in vertical longitudinal plane (fig. 2), in which are represented the plane articulated mechanisms of those two legs, from rear (fig. 2a) and front (fig. 2b). The booth mechanisms are articulated in the upper side to a horizontal link, which represent the body of the physically modeled dog.

The joints [A.sub.0] and [B.sub.0] of each mechanism to the upper mobile platform (fig. 2) are considered as basis joints, by this reason this platform has been noted with 0.

Each from those two mechanisms (rear and front) has a first kinematic chain, the four bar mechanism [A.sub.0]AB[B.sub.0], which is formed from the kinematics chains 0, 1, 2 and 4. The second kinematic chain of each mechanism is the four bar articulated mechanism ACED, with the kinematic elements 1, 2, 4 and 5 (fig. 2a) or BCED, from the elements 2, 3, 4 and 5 (fig. 2b). The mechanism of the front leg contain another kinematic chain DGHF (fig. 2b), which is formed from the kinematic elements 2, 5, 6 and 7.

The mobility [M.sub.b] of each from those two plane mechanisms is calculated with the Dobrovolski formula:

[M.sub.bf] = (6 - f)n - [5.summation over (k=f+1)] (k - f)[C.sub.k] (1)

which for f - 3 (plane mechanisms) become the Grubler-Cebasev formula:

[M.sub.b3] = 3n - [5.summation over (k=4)](k - 3)[C.sub.k] = 3n - 2[C.sub.5] - [C.sub.4] (2)

where the class of the kinematic joint distinguish the imposed restrictions (k=5, k=4).

To calculate the mobility of those two mechanisms (fig. 2a, 2b) we use the formulas (1) and (2):

a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[FIGURE 2 OMITTED]

3. KINEMATIC MODELLING OF THE ANTERIOR LEG

The mechanism has three independent contours (fig. 2b) 1) - [A.sub.0][B.sub.0]BA[A.sub.0] (03210), 2) - BCEDB (34523) and 3) - GDFHG (25762). We choose a coordinate system with the origin in the fixed joint [A.sub.0] (fig. 3), having the axis [A.sub.0]x and [A.sub.0]y orientated from right to left, respectively from upper to bottom.

The closing vectorial equation of the first contour (03210) is writhed explicitly (fig. 3):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

[FIGURE 3 OMITTED]

We arrange the terms of equations (3), that in the left part to be the vectors which contain the unknown (the angles [[phi].sub.2] and [[phi].sub.3]), and in the right side to be the vectors known as size and direction (the angle [[phi].sub.1] is the independent parameter, being imposed in certain give interval):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3')

We introduce the notations: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], else the vectorial equation (3') is writhed under a convenient form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Projecting the vectorial contour on the coordinate's axis [A.sub.0]x and [A.sub.0]y (fig. 4) we obtain two scalar equations equivalent to the vectorial equation (5):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The nonlinear system of equations (5) can be resolved by eliminating one of two unknown [[phi].sub.2] and [[phi].sub.3].

For that the system is writhed more compactly, under the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5')

where we have used the notations:

[b.sub.1] ([[phi].sub.1]) = [l.sub.0] cos [[phi].sub.0] + [l.sub.1] cos [[phi].sub.1];

[b.sub.2] ([[phi].sub.1]) = [l.sub.0] sin [[phi].sub.0] + [l.sub.1] sin [[phi].sub.1].

To calculate the angle [[phi].sub.2] we isolate the terms which contain the other unknown [[phi].sub.3]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

After square up of those two equations (6) and summing, it results:

[l.sup.2.sub.3] = [l.sup.2.sub.2] + [b.sup.2.sub.1] + [b.sup.2.sub.2] - 2[l.sub.2][b.sub.1] cos [[phi].sub.2] - 2[l.sub.2][b.sub.2] sin [[phi].sub.2] (7)

The obtained expression (8) is a trigonometrically equations with variable coefficients, under the form:

[A.sub.1]([[phi].sub.1])sin [[phi].sub.2] + [B.sub.1]([[phi].sub.1])cos [[phi].sub.2] + [C.sub.1]([[phi].sub.1]) = 0 (8)

where the variable coefficients have the expressions:

[A.sub.1]([[phi].sub.1]) = 2[l.sub.2][b.sub.2]([[phi].sub.1]); [B.sub.1]([[phi].sub.1]) = 2[l.sub.2][b.sub.1]([[phi].sub.1]); [C.sub.1]([[phi].sub.1]) = [l.sup.2.sub.3] - [l.sup.2.sub.2] - [b.sup.2.sub.1]([[phi].sub.1]) - [b.sup.2.sub.2]([[phi].sub.1]) (9)

With the help of formulas sin [phi] = 2tg 1/2 [phi]/1+t[g.sup.2] 1/2 [phi] cos [phi] = 1 - t[g.sup.2] 1/2 [phi]/1 + t[g.sup.2] 1/2 [phi]

the solutions of the equations (7) are deducted under the form:

[[phi].sub.2] = 2arctg ([A.sub.1] [+ or -] [square root of [A.sup.2.sub.1] + [B.sup.2.sub.1] [C.sup.2.sub.1]]/[B.sub.1] - [C.sub.1]) (10)

4. DIAGRAMS OF ANGULAR DISPLACEMENTS VARIATIONS

We consider the uniform movement of the motor element 1 (fig. 3). With the help of the MSC.ADAMS software we simulate the movement of the anterior leg for the angle [[phi].sub.1] = 52[degrees].

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

5. REFERENCES

Antonescu, P. (2003). Mechanisms, Printech Publishing House, Bucharest

Antonescu, O.; Antonescu, P. (2006). Mechanisms and Manipulators, Printech Publishing House, Bucharest

Buzea, E. M. (2005). The pattern of movement on jumping of frogs (rana type), SYROM 2005, Vol. III, pp. 611-616

*** (2009) www.smart-toys.ro, Accesed on: 2009-08-08

Micu C.; Geonea I. D. (2010). Kinematic modeling and simulation of quadruped biomechanism, International Conference of Mechanical Engineering ICOME 2010--27th-30th of April 2010, Craiova--Romania