Emulation of the Dynamic Effects of Human Torso During a Walking Gait B. Mohamed, F. Gravez, F.B. Ouezdou Laboratoire d’Instrumentation et Relations Individu Système, U.V.S.Q., CNRS FRE 2508, 10-12 Avenue de l’Europe, 78140 Vélizy (France) [email protected] Abstract Human being model is very complex and full reproduction of its kinematic structure is a difficult task. The main objective of this paper is to identify a minimal mechanism of human upper part able to replace it during the walk. Virtual mannequin was built and dynamic simulations are carried out. A dynamic equivalence based on approach is used to identify this mechanism. Simulation results proved that four dofs mechanism is necessary and sufficient to reproduce human upper part dynamic effects during walking gait. Simplicity of design and control led us to adopt an RPPP mechanism. Future work concerns a full design of the RPPP mechanism. Mounted on ROBIAN biped locomotion system, dynamic walk experiments will be realized.

1

Introduction

Development of walking legged robots constitutes a big challenge for robotics community. A large number of static or dynamic locomotion systems composed by two, four or six legs were built all over the world [Song & Waldron, 1989, Saranli et al., 1995, Hirose & Yoneda, 1995, Minor & Mukherjee, 2003]. Some others machines were built based on biologically considerations such as snakelike robots [Ma et al., 2002, Chirikjian & Burdick, 1995]. However, during the last decades, there have been a growing interest and continuous increase of bipedal robotics applications [Kato, 1973, Kaneko et al., 2002, Kim & Hemmami, 1998, Robinson et al., 1999, Pfeiffer et al., 2002, Espiau & Sardain, 2000, Kanehira et al., 2002]. Indeed, biped robots can operate in a human environment more efficiently than other robots such as the wheeled ones. Assistance to humans in carrying out tedious and dangerous tasks can be considered also as an important use of biped robots. Another major application of bipedal robotics research and development is the enhancement of prosthesis devices development and testing [Sellaouti et al., 2002]. To make a useful biped robot prototype allowing human being locomotion system analysis, two main approaches can be developed [Kato, 2003]. Usually, humanoid robots are built with anthropomorphic legs, arms, head and even eyes [Hirai et al., 1998, Ishida et al., 2001]. The second approach consists in ignoring the human aspects of the prototype upper part and to focus only on its dynamic effects on the lower limbs [Hemmami & Farnsworth, 1977]. Our main motivation for bipedal research concerns a significant contribution to the study of human being locomotion system. Several investigations based on dynamic simulations have been carried out [Bruneau & Ouezdou, 1999, Konno et al., 2002] . These simulations demonstrate the major contribution of the feet flexibility during walking and running gaits [Bruneau & Ouezdou, 1999]. To improve the model on which dynamic simulations are based on, experimental investigations have to be completed. For this purpose, a multi-degrees of freedom (dof) biped prototype provided with flexible feet, called ROBIAN (RObot BIpède ANthropomorphique), has been developed [Konno et al., 2002]. The major application of ROBIAN prototype is the development of a real test bed of active/passive prosthesis devices enhancing research on the human being locomotion mechanism handicap. Hence, the second approach to design the upper part (torso) of ROBIAN robot is adopted. This paper presents a work dealing with the kinematic and dynamic synthesis of ROBIAN upper part. An analysis of the human being torso dynamic effects during different walking gaits is carried out using three virtual mannequin: a normal, a fat and a small one. A minimum-dof mechanism able to reproduce these dynamic effects is then identified. 1

The reference model used for this analysis is detailed in section 2.Section 3 deals with the dynamic analysis of the human being upper part dynamic effects on the lower limbs during walking gait. A method to prove dynamic equivalence between two mechanisms based on General State Equation (GSE) formalism is proposed in section 4. Several kinematic solutions are then identified and simulated in section 5. Retained kinematic structure solution and simulation results are presented in section 6. The last section will include both conclusion and further developments of the current work.

2 2.1

ROBIAN Project Motivation

Virtual mannequins are usually used in order to simulate the behavior of the human-being while achieving manipulation or locomotion tasks [McKenna & Zelter, 1990, Seigler et al., 1982, Burderlin & Calvertr, 1989]. In our approach, a bio-mimetic reasoning was adopted in order to model a fully parametric virtual mannequin biped (See Figure 1(a)) having 27 dofs [Gravez et al., 2002] which is able to emulate the locomotion tasks. This mannequin has two parameters, total weight and total height, from which geometrical and inertial parameters of links are calculated using a bio-mimetic model. Figure (1(a)) shows a simplified model of the human body built using ADAMS software. The virtual mannequin model is based on rigid links articulated using one to three dofs joints. The kinematic structure shows that the mannequin has 15 joints with 27 dofs: 6 per leg, 3 per arm, 3 for the head and 4 for the thorax. A passive joint is used for each foot. The human body is modeled by 16 solid primitives. Using biomechanical data, a mass distribution has been associated to each solid. The weight of the virtual mannequin is 80 kg including 54 kg for the upper part and the total height is 1.78m according to a common European male. A faithful reproduction of human movements during walking gait is of a primary importance for the dynamic simulation of a virtual mannequin. It is also interesting to identify the time evolution of joint variables of a human being. Therefore, a series of measurements using VICON motion analysis system was carried out. A total number of 16 markers were laid out on a human being. These markers are placed where the relative motion between the skin and the bones are minimal during walking gait. The position of each marker is recorded by five cameras with a 20ms recording time-step. These positions which are the input of the biped inverse kinematic model allow us to get the time evolution of the 25 actuated joint variables. The simulation of the 3D biped mannequin enables us to produce motion that is close to the recorded data. A distributed feet/ground contact model based on spring damper combination is used [Bruneau & Ouezdou, 1999]. Joints torques are computed using proportional derivative controller ˙ (τ = Kp (qd − q) + Kv (q˙d − q)).

2.2

Locomotion System

Based on virtual mannequin simulation results, the ROBIAN prototype was designed and built. ROBIAN is composed from two parts : a locomotion system (lower limbs) and a torso mechanism (upper part or body). The kinematic structure of the locomotion system was derived from the virtual mannequin one. Each leg has a total of seven dofs, three actuated dofs for the hip [Sellaouti et al., 2002], one actuated dof for the knee, two actuated dofs for the ankle and one passive dof for the foot providing flexible feet system. The number of contact points on the ground varies from four to five: three points on the toe and one to two points on the heel. The total height of the locomotion system is 1.10 m and its weight is 14 kg. Since the main objective of this biped is to offer a real test bed of active/passive prosthetic devices, modular design was developed in a way that the concerned kinematic module: Hip, knee, ankle or foot can be easily replaced by the prosthetic device one needs to test. Mechanical construction of the modular locomotion system has already been done and a static walking gaits were achieved. Figure (1(b)) shows the prototype of ROBIAN biped robot.

2.3

Torso mechanism

The upper part of human being during walking attempts to stabilize the walk [Setiawan et al., 1999, Kubica et al., 2001]. Indeed, the ROBIAN upper part has to reproduce the dynamic effects of the human torso on the locomotion system. Two approaches can be considered for the design of ROBIAN upper part. The first one consists on building an anthropomorphic upper part having a kinematic structure as 2

(a)

(b)

Figure 1: Biped Virtual mannequin and ROBIAN Locomotion System prototype close as possible to the virtual mannequin one (Figure 1(a)). This approach, usually adopted in building biped robots [Hirai et al., 1998, Ishida et al., 2001], reproduce better the dynamic effects of the upper part on the locomotion system. Nevertheless, the presumed realistic upper part used for the virtual mannequin dynamic simulation has a great number of dofs (i.e. 13 actuated dofs). This implies the use of 13 actuators which will increase the design and the control complexities. The second approach, which has been adopted in our project, consists on identifying a dynamically equivalent mechanism able to produce the same dynamic effects as the realistic torso without any anthropomorphic considerations. The main advantage of this approach lies in the use of a simplified mechanism with a minimum number of actuated degrees of freedom reducing the complexity of the two aspects : design and control. A minimal mechanism is looked for in order to replace the upper part of this model (presumed to be realistic) containing 13 dofs (figure 2). This model should reproduce the dynamic effects on the lower limbs during walking gait.

3

Dynamic Analysis

To carry out the kinematic synthesis process for the ROBIAN torso mechanism, a careful analysis of the dynamic effects exerted on the lower limbs has to be achieved. For this purpose, the upper part (Figure 2) of the realistic biped built under ADAMS is isolated and embedded at the center of mass of the down-torso (the lowest link of the upper part). Thereafter, the structure is animated with the time laws of the joint variables in order to extract the 6 components of the effort wrench at the point of embedding (efforts: Fx , Fy , Fz and moments: Mx , My , Mz ).This is done by using dynamic simulations for several walking gaits and different heights and weights of the biped mannequin. If the six components are independent, the equivalent system must have, at least, six dofs. The aim is to determine whether these components are independent and thereby to identify the number of coupling relations between them in order to reduce the number of the necessary dofs for the equivalent mechanism. Figure 3 shows the six components of this wrench during the several simulation stages for a walking gait at speed of 1.2 m/s. The analysis of simulation results (as shown on Figure (3)) demonstrates the existence of two coupling relations. The first one links the moment component Mx around the x axis (motion direction) to the effort component Fz in z axis (lateral direction). The second relation concerns the moment component Mz and the effort component Fx (axes are depicted on figure 2 ). These relations can be written as

3

Figure 2: Virtual mannequin upper part

Figure 3: Effort wrench components of virtual mannequin upper part.

4

follows: Mx = k1 × Fz Mz = k2 × Fx

(1) (2)

where, k1 and k2 are two constants (k1 > 0 and k2 < 0). Hence, the minimal equivalent mechanical system should have no more than four dofs. In order to generalize this analysis, three sizes of the parametric virtual mannequin were simulated for three different walking gaits as illustrated on table(1). mannequin 1 (small) 2 (standard) 3 (fat)

weight (Kg) 25 80 115

height (m) 1.10 1.78 1.80

gait 1 period (sec) 1.2 1.2 1.2

gait 2 period (sec) 1 1 1

gait 3 period (sec) 0.8 0.8 0.8

Table 1: Simulated mannequins and gaits. Simulation results are formal. Indeed, the coupling relations (eq(1) & eq(2)) are identified for all sizes and gaits. Figures (4), (5), (6) show respectively mannequin 1 for gait 3, mannequin 2 for gait 2 and mannequin 3 for gait 1.

Figure 4: Effort wrench components of mannequin 1 torso with period 0.8 sec. The identified coupling relations need to be validated. Moreover, one has to check the non-existence of other coupling relations. This reduces the number of necessary dofs for the final torso mechanism. It will be done through the proposed approach and in a progressive manner starting with a spatial two-dofs mechanism.

4

Proposed Approach

In order to identify the torso mechanism, the adopted approach has been based on looking for a mechanism which is able to reproduce the dynamic (forces, moments and accelerations) effects of the upper part of the virtual mannequin of the lower limbs. This can be stated as a kind of dynamic equivalence between 5

Figure 5: Effort wrench components of mannequin 2 torso with period 1 sec.

Figure 6: Effort wrench components of mannequin 3 torso with period 1.2 sec.

6

two mechanisms. Indeed, the dynamic equivalence is defined as follows:Two mechanisms are said to be dynamically equivalent, during the accomplishment of a given task, if the effort wrenches at embedding point of each mechanism are equal over all the task period. To illustrate the adopted approach, let us consider two mechanisms depicted on figure (7). The objective is to look for (i.e. identify) the mechanism of figure (7b) which is dynamically equivalent to the one depicted on figure (7a) (to produce the same dynamic effects) according to the above definition.The lower part of both mechanisms will stay the same and equivalence will concern only the upper part.

Figure 7: Definition of the dynamic equivalence between two mechanism The approach consists in isolating the reference upper part and determining, through the use of dynamic simulation, the effort wrench exerted by the reference upper part on the common lower part at embedding point O (figure 7a). Then, the equivalent upper part looked for should reproduce all effort wrench components at its embedding point O
(figure 7b). Dynamic equations according to the Newton-Euler formalism are written for all links of the studied system at its embedding point in order to compute the motion equations of this mechanism. The number of the needed motion equations is equal to the number of degrees of freedom of the mechanism. Figure (8) gives a simplified diagram of the approach steps. Indeed, both the studied mechanism and the reference one are built and loaded in the same ADAMS session. During simulation, effort wrench components of reference model are extracted and used in real-time as input into the General State Equations (GSE). Then, the GSE computes the joints variables which are applied, also in real-time, to the corresponding joints of the studied system. Finally, based on multiple dynamic simulations, time evolutions of wrenches components are compared to decide whether the studied mechanism is (or isn’t) equivalent to the reference one. To prevent homogeneities problems when comparing two wrenches [Duffy, 1990], the metric used here to compare the two models is based on the following norm: M=


i=x,y,z



Fimod − Firef

2 +

Firef


i=x,y,z



Mimod − Miref Miref

2 (3)

where: Fimod (Mimod ) and Firef (Miref ) are the force (moment) components of the studied and the reference mechanisms respectively. Geometrical and inertial parameters of the studied model are considered as design variables. These parameters values has to be changed in an iterative way to achieve the equivalence objective. A detailed optimization procedure will be used for the adopted mechanism in order to determine its parameters. Dynamic equivalence is achieved when all six components of reference and studied mechanisms effort wrench are identical. To carry out the GSE formulation, the motion equations of a general n dofs mechanism are first

7

Figure 8: Dynamic equivalence approach written in the following form:

⎡

⎤ ⎡ ⎤ q¨1 gi (qi , q˙i , WO
, PG , PI ) ⎣ ... ⎦ = ⎣ ⎦ .........
gn (qi , q˙i , WO , PG , PI ) q¨n

(4)

where: • g1 , g2 , . . . , gn are non linear functions of qi , q˙i , WO
, PG and PI . • qi , q˙i , q¨i are, respectively, generalized position, velocity and acceleration of ith joint. • WO
is the effort wrench exerted by the mechanism at embedding point and given by: WO
= [Fx , Fy , Fz , Mx , My , Mz ] where Fx , Fy , Fz are force components in the (O
, x, y, z) frame and Mx , My , Mz are moments components in the same frame. • PG , PI are the geometrical and inertial parameters of the studied mechanism. Due to the fact that all intermediate unknown joint forces are eliminated, the system of equations given by (4) forms a system of second order differential equations which can be transformed into a first order differential system given as follows: ⎤ ⎡ r1 (x1 , . . . x2n , u1 , . . . , um , PG , PI ) ⎦ ......... (5) X˙ = R = ⎣ r2n (x1 , . . . x2n , u1 , . . . , um , PG , PI ) ⎡ ⎤ h1 (x1 , . . . x2n , u1 , . . . , um ) ⎦ ......... Y =⎣ (6) hn (x1 , . . . x2n , u1 , . . . , um ) ⎤ ⎡ u1 (7) U = ⎣ ... ⎦ um where: X is the state vector which components are x1 = q1 , , x2 = q˙1 , . . . , x2n−1 = qn , x2n = q˙n . U is the input vector which has m components chosen among the six effort wrench components (Fx , Fy , Fz , Mx , My ,Mz ) of the reference model. Y is the output vector which components are generally taken as joints variables qi . Vector R components are given as follows : r1 = x2 , r2 = g1 , . . . , r2n−1 = x2n , r2n = gn 8

5

Kinematic solutions

5.1

Two dofs mechanism

The studied two-dof model is presented in figure (9). It is an RR mechanism (R for rotational joint) with two perpendicular joints. The above proposed approach is applied to this model by choosing input vector t U = [My , Mz ]

Figure 9: A two-dof model and its parameters The motion equations can be written as function of the parameters:   q¨1 r1 (q1 , q2 , q˙1 , q˙2 , My , Mz , a, b, mi , Iij ) = q¨2 r2 (q1 , q2 , q˙1 , q˙2 , My , Mz , a, b, mi , Iij )

(8)

where: a is the distance between embedding point (O) and second joint center point (A). b is the distance between second joint center and the second link center of mass (G). mi is the mass of ith link. (i ∈ 1, 2, 3) Iij is the ith link moment of inertia around jth axis. ( j ∈ x2, y2, z2 ) The built GSE is given by:

t X = q1 , q˙1 , q2 , q˙2 (9)

t Y = q1 , q2 (10)

t U = My , Mz (11)

t R = r1 , r2 , r3 , r4 with r1 = x2 and r2 = x4 (12) The model is built under the ADAMS software using cylindrical primitives for the 3 links while respecting the geometrical and the inertial properties of the realistic torso. The 6 components of the effort wrench at point O are extracted and compared with those of the realistic torso. Several simulations show that, not only components My and Mz (input vector components) are respected, but also the Fx component. In this model, the coupling relation between components Mz and Fx is verified. Figure (10) shows simulation results obtained with the final geometrical and inertial parameters given by table (2). a (m) 0.1

b (m) 0.25

m1 + m2 (Kg) 24

m3 (Kg) 30

I2y (Kg.m2) 1

I3x (Kg.m2) 3

I3y (Kg.m2) 2

I3z (Kg.m2) 3

Table 2: RR mechanism geometrical and inertial parameters.

5.2

Three dofs mechanism

The studied two-dof mechanism allowed us to validate one of the two coupling relations identified during the reference model analysis, but simulation results showed that more dofs should be considered in order to have dynamic equivalence. In this section two models will be studied. The first one is a 3D inverted pendulum mechanism, and the second is an RPP (P for prismatic one) mechanism. 9

Figure 10: Two dofs model simulation results 5.2.1

3D Inverted pendulum

Figure (11) shows the kinematic structure of an inverted pendulum which is composed of two links and one spherical joint and related frames.

Figure 11: Inverted pendulum model and its parameters The motion equations are: ⎤ ⎤ ⎡ ⎡ r2 (q1 , q2 , q3 , q˙1 , q˙2 , q˙3 , Fx , Fy , My , a, b, mi , I2j ) q¨1 ⎣ q¨2 ⎦ = ⎣ r4 (q1 , q2 , q3 , q˙1 , q˙2 , q˙3 , Fx , Fy , My , a, b, mi , I2j ) ⎦ r6 (q1 , q2 , q3 , q˙1 , q˙2 , q˙3 , Fx , Fy , My , a, b, mi , I2j ) q¨3

(13)

where: a is the distance between embedding point (O) and the spherical joint center (A). b is the distance between spherical joint center and the second link center of mass (G). mi is the mass of ith link. (i ∈ {1, 2}) I2j is the second link moment of inertia around jth axis. ( j ∈ {x3, y3, z3} ) 10

The built GSE is given by:



t q1 , q˙1 , q2 , q˙2 , q3 , q˙3 t

Y = q1 , q2 , q3 t

U = Fx , Fz , My t

with r1 = x2 , r3 = x4 and r5 = x6 R = r1 , r2 , r3 , r4 , r5 , r6 X=

(14) (15) (16) (17)

The inverted pendulum mechanism is built under the ADAMS software by taking cylindrical primitives for both links. The geometrical and the inertial properties of the realistic torso are respected. The 6 components of the effort wrench at point (O) are extracted and compared with those of the realistic torso. Several simulations show that, not only components Fx , Fz and My (input vector components), but also components Mx and Mz are respected. In this model, both coupling relations already found in the analysis of the human torso are checked. But the Fy component is still not respected. Figure (12) shows simulation results obtained with the final geometrical and inertial parameters which are given by table (3).

Figure 12: 3D inverted pendulum simulation results

11

a (m) 0.1

b (m) 0.25

m1 (Kg) 14

m2 (kg) 40

I2x (Kg.m2) 3

I2y (Kg.m2) 3

I2z (Kg.m2) 3

Table 3: 3D Inverted pendulum geometrical and inertial parameters.

Figure 13: RPP mechanism and its parameters 5.2.2

RPP mechanism

Figure (13) shows the studied RPP mechanism. It is composed of one rotational joint and two decoupled prismatic ones. The motion equations are: ⎤ ⎤ ⎡ ⎡ r2 (q1 , q2 , q3 , q˙1 , q˙2 , q˙3 , Fz , Fy , My , a, b, mi , Iy ) q¨1 ⎣ q¨2 ⎦ = ⎣ r4 (q1 , q2 , q3 , q˙1 , q˙2 , q˙3 , Fz , Fy , My , a, b, mi , Iy ) ⎦ (18) r6 (q1 , q2 , q3 , q˙1 , q˙2 , q˙3 , Fz , Fy , My , a, b, mi , Iy ) q¨3 where: a is the distance between spherical joint center and the second link center of mass (G). mi mass of ith link. (i ∈ {1, 2, 3, 4} ) Iy is the sum of all links inertias around y axis. The built GSE is given by: t

X = q1 , q˙1 , q2 , q˙2 , q3 , q˙3

t Y = q1 , q2 , q3

t U = Fy , Fz , My

t with r1 = x2 , r3 = x4 and r5 = x6 R = r1 , r2 , r3 , r4 , r5 , r6

is the

(19) (20) (21) (22)

As for the inverted pendulum mechanism, the RPP mechanism is build under the ADAMS software using basic solid primitives (cylinders, spheres, ... ). The 6 components of the effort wrench at point O are extracted and compared with those of the realistic torso. Several simulations trials show that, not only components Fy , Fz and My (input vector components), but also component Mx are respected. In this model only the first coupling relation found in the analysis of the human torso is validated. The Fx and Mz components are not respected. Figure (14) shows simulation results obtained with the final geometrical and inertial parameters given by table (4). a (m) 0.35

m1 + m2 (Kg) 42

m3 (Kg) 7

m4 (Kg) 5

Iy (Kg.m2) 3

Table 4: RPP mechanism geometrical and inertial parameters.

12

Figure 14: RPP mechanism simulation results

5.3

Conclusion

The proposed approach was applied to two and three-dofs mechanisms. Simulations analysis showed that no more coupling relations exist between effort wrench components of rigid body model of the realistic torso during several walking gaits. Thus, a four dofs model is necessary and sufficient to reproduce dynamic effects at the waist link.

6

RPPP mechanism

The next step concerns the identification of an appropriate four dofs mechanism among the four possible kinematic structures (RRRR, RRRP, RRPP and RPPP). The criterion considered in our choice deals with the ability of the mechanism to fulfill simple design and control requirements. After several trials, the adopted solution is an RPPP mechanism. This mechanism has one rotational joint followed by three decoupled prismatic ones. Figure (15) describes the kinematic structure of this mechanism and the related frames.

13

Figure 15: RPPP mechanism and its parameters

6.1

GSE formulation for the RPPP mechanism

Considering the kinematic structure of this mechanism, the choice of the input vector components among the effort wrench ones at point (O) can be written as follows : U = [Fx , Fy , Fz , My ]

t

Dynamic equations according to the Newton-Euler formulation which involve these effort wrench components are the following:

Fx = m3 (¨ q2 − q2 q˙12 ) cos(q1 ) − (q2 q¨1 + 2q˙1 q˙2 ) sin(q1 )

+m5 (¨ (23) q4 − q4 q˙12 ) sin(q1 ) − (q4 q¨1 + 2q˙1 q˙4 ) cos(q1 ) (24) Fy = (m1 + m2 + m3 + m4 + m5 )g + m4 q¨3

2 Fz = −m3 (¨ q2 − q2 q˙1 ) sin(q1 ) + (q2 q¨1 + 2q˙1 q˙2 ) cos(q1 )

+m5 (¨ (25) q4 − q4 q˙12 ) cos(q1 ) − (q4 q¨1 + 2q˙1 q˙4 ) sin(q1 ) My

=

Iy q¨1 + m3 (¨ q1 q22 + 2q˙1 q˙2 q2 ) − m5 (¨ q1 q42 + 2q˙1 q˙4 q4 )

(26)

where: mi is the mass of the ith link (i ∈ {1, 2, 3, 4, 5} ) and Iy the total moment of inertia defined as:
Iy = Iyi 2≤i≤5

where Iyi is the ith link moment of inertia around y axis. The corresponding motion equations are then obtained : r2 = q¨1 r4 = q¨2

r6 = q¨3 r8 = q¨4

My − 2q˙1 (m3 q2 q˙2 − m5 q4 q˙4 ) Iy + m3 q22 − m5 q42 Fx cos(q1 ) − Fz sin(q1 ) = + q2 q˙12 m3

 m5 q4 My + 2q˙1 q˙4 (Iy + m3 q22 ) − 2m3 q˙1 q˙2 q2 q4 − m3 (Iy + m3 q22 − m5 q42 ) Fy − (m1 + m2 + m3 + m4 + m5 )g = m4 Fx sin(q1 ) + Fz cos(q1 ) = + q4 q˙12 m5

 m3 q2 My + 2q˙1 q˙2 (Iy + m3 q22 ) − 2m5 q˙1 q˙4 q2 q4 + m5 (Iy + m3 q22 − m5 q42 ) =

14

(27)

(28) (29)

(30)

The GSE formulation for the RPPP mechanism is given by: X=



q1 , q˙1 , q2 , q˙2 , q3 , q˙3 , q4 , q˙4

t

t Y = q1 , q2 , q3 , q4

t U = Fx , Fy , Fz , My t

withr1 = x2 , r3 = x4 , r5 = x6 and r6 = x8 R = r1 , r2 , r3 , r4 , r5 , r6 , r7 , r8

6.2

(31) (32) (33) (34)

RPPP mechanism parameters optimization

A simple representation of the RPPP mechanism is composed by three spheres able to move through x,y and z axis and two boxes having a relative rotational movement around the y axis. This model and the reference one (realistic human torso with 13 dofs) are built in the same ADAMS session as shown on figure (16).

Figure 16: RPPP Model for dynamic simulation An iterative optimization procedure has to be developed in order to determine the geometrical and inertial parameters values. The design vector XD is the following : XD = [a, m12 = m1 + m2 , m3 , m4 , m5 , Iy ]t

(35)

In all previous studied mechanisms simulations, geometrical and inertial parameters were initially taken as close as possible to the reference upper part ones. Then, they were changed in order to have reasonable motions amplitude and to insure dynamic equivalence according to the given definition. Once the torso mechanism is chosen, optimizing these parameters becomes primordial. To do so, criteria have to be stated. First one comes immediately from the metric defined in eq(3). Since the GSE formulation gives exact solution for a given entry vector, the components of this vector are consequently fully reproduced whatever involved parameters (in the occurrence masses and vertical inertia) values are. These parameters affect essentially motions amplitudes. Hence, the second criterion C2 to optimize these parameters was taken as follows: 2  2  qiu qil −1 + −1 C2 = (36) max(qi ) min(qi )

15

where : qiu and qil are the desired upper and lower bound limits of the ith joint variable qi . max(qi ) and min(qi ) are the maximum and minimum ith joint variable value during all the gait period. The following constraints corresponding to normal size of human torso were also included in the optimization process : 5


mi = 17.5 (kg)

(37)

−8 (deg) ≤ q1 ≤ 8 (deg)

(38)

−0.1 (m) ≤ q2 ≤ 0.1 (m) −0.02 (m) ≤ q3 ≤ 0.02 (m)

(39) (40)

−0.1 (m) ≤ q4 ≤ 0.1 (m)

(41) (42)

i

The last component of the design vector is the distance a. It affects the reproducing of the two remaining wrench components Mx and Mz . Hence, the first criterion C1 based on the metric definition can be written as follows:  C1 =

Mxmod Mxref

2 −1

 +

Mzmod Mzref

2 −1

(43)

This optimization procedure was applied to the three virtual mannequins described in section 3 for the normal walking gait which period is 1.2 sec. This led us to the set of optimal values given by table (5). mannequin 1 (small) 2 (standard) 3 (fat)

a (m) 0.21 0.33 0.34

m1 + m2 (Kg) 10 36 38

m3 (Kg) 3 6.8 8.1

m4 (Kg) 1.9 4.6 5.8

m5 (Kg) 2.6 6.7 8.5

Iy (Kg.m2) 0.285 2.2 2.9

Table 5: RPPP mechanism geometrical and inertial parameters. Dynamic behavior simulation results, using parameters depicted on table (5) for the reference mannequin 2 (the standard one), show that all the 6 components of the realistic torso effort wrench are reproduced by the RPPP model (figure 17). This confirms the dynamic equivalence between the two mechanisms according to the adopted definition. Resulting motions of the RPPP mechanism links, given in figure (18), satisfy dimensions of the prototype to be built. These movements are periodic and they have the same period as the walking gait, which is 1.2 sec. Velocities and accelerations of these links are also taken into account due to their influence on the actuators sizing.

7

Conclusion

Two kinds of approaches are used in bipedal design. The first is the bio-mimetic approach whereby the human being aspects (lower limbs, arms, head and thorax) are reproduced while the second approach ignores human aspects and focuses on the dynamic effects of the upper limbs on the lower ones. The main objective of the ROBIAN project is to build a real test bed of human being locomotion system prosthetic devices that enhances research into the handicap of different joints of the locomotion system (foot, knee, ankle, or hip). This explains the reason why the second approach is adopted while building our biped. The aim of the present paper was to identify a minimal dofs mechanism emulating the dynamic effects of the upper limbs on the lower ones during walking gait. This mechanism will constitute the upper part of the ROBIAN biped prototype. Its locomotion system construction has already been achieved. A generic method based on the GSE concept was used in order to identify a minimal mechanism able to generate a required effort wrench at a given point. The analysis of the six effort wrench components has shown the existence of two coupling relations between them. Four mechanical structures (RR, RPP,

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Figure 17: RPPP mechanism simulation results

Figure 18: RPPP mechanism joint motions RRR and RPPP) were studied in order to validate these coupling relations and check the non-existence of other coupling relations. This reduces the number of dofs needed. The study has pointed out an interesting result, namely that a four-dof mechanism is necessary and sufficient to emulate the dynamic effects of a the mannequin torso on the lower limbs during walking gait.

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Reducing complexity of design and control led us to chose an RPPP mechanism. Dynamic equivalence between this mechanism and the the virtual mannequin upper part was validated through 3 sizes of mannequin (small, standard and fat one) and for 3 different walking gaits. Finally, parameters of the RPPP mechanism were optimized using two criteria. The first one is based on the metric used to compare the wrench of mannequin upper part and the RPPP mechanism one. The second one concerns bounding box of the RPPP mechanism limiting its motions amplitudes. A further development of this work will address the complete design of the identified RPPP mechanism which will be built afterward. Experiments with whole prototypes: ROBIAN and the RPPP torso will be further conducted.

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