IEEE International Workshop on Robot and Human Interactive Communication
Emulation of the Human Torso Dynamic Effects During Walking Gait B. Mohamed
F. Gravez
F. B. Ouezdou
Laboartoire de Robotique de Paris 10-12 avenue de l'Europe 78140 Vélizy, France Abstract In this paper, an approach consisting on the identification of a minimal mechanical system able to emulate the dynamic effects of the human torso during walking gait is presented. A 3D dynamic simulation of a virtual manikin with 27 degrees of freedom (dofs) is carried out in order to identify the effort wrench exerted by the torso on the lower limbs. An analysis of the six components of this wrench shows that some components are dependant. A systematic study of two, three and four dofs mechanisms based on General State Equation (GSE) formalism leads us to an interesting result. Indeed, four dofs are necessary and sufficient to emulate the dynamic effects. We show also the efficiency and the generality of the proposed approach.
1
Introduction
Applications of walking robots, mainly the biped ones, have, and will probably, increase. They can operate in human environment more efficiently than other types of robots like wheeled ones. Assistance to humans in the accomplishment of domestic tasks, which are difficult or dangerous, is an important use for bipedal robots. In addition, a very useful application of research into bipedal robotics will be the enhancement of prosthetic devices development and testing. Humans walking gaits are dynamic, as they are faster and more efficient than the static walking patterns. Therefore, research in bipedal robotics has progressed to study dynamic walking gaits. [1] & [2] Dynamic walking has been realised by some bipedal robots, most notably are the Honda P2 and P3 robots and the Wabian robot of the University of Waseda shown on figure 1. [3], [4], [5] & [6]
Figure 1: humanoid robots P2, P3 of Honda, Wabian of Waseda and Sony dream robot.
Most recently, Sony dream robot, a fully dynamic humanoid robot built by Sony company (figure 1) shows quite impressive performances in carrying out human like tasks (walking, dancing ...) [www.sony.co.jp] To carry out a useful biped robot prototype allowing the human locomotion system analysis, two main approaches can be developed. Generally, an anthropomorphic upper limb with arms and head is used in humanoid robots building. The other approach consists on ignoring the human like aspects of the prototype upper part. Due to the fact that our main interest concerns significant contribution to the study of the human locomotion system, a multi-degrees of freedom (dof) biped prototype provided with flexible feet, called ROBIAN, is under development. The Robian prototype major application will be the development of a real testing bed of active/passive prosthesis devices enhancing research on the locomotion mechanism handicap. This paper presents a work dealing with the emulation of realistic model torso of Robian without any anthropomorphic consideration. An analysis of the human torso dynamic effects during walking gait is carried out. A minimum dofs mechanism able to reproduce these effects is then identified. The reference model used for this analysis is detailed on section 2. A method based on General State Equation (GSE) is developed in section 3. Kinematic solutions are then presented. In the last section conclusion and further developments of this work are given.
2
Reference model
Figure 2a shows a simplified model of the human body realised under the Adams1 software with rigid articulated links connected by one to three dofs joints. It contains 15 joints with 27 dofs: 6 per leg, 3 per arm, 3 for the head and 4 for the thorax. A passive joint are used for each foot. The human body is modelled by 16 solid primitives. [7] Using biomechanical data, a mass distribution has been associated to each solid [8]. The weight of the biped is 80 kg including 54 kg for the upper part. The height is 1.78m. 1
Adams is a product of the MDI Corporation.
Faithful reproduction of human movements during walking gait is of a primary importance for dynamic simulation of a virtual manikin. To this end, it is interesting to know 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. Position of each marker is recorded by five cameras with a 20ms recording time-step. These positions are introduced into the biped inverse geometrical model to obtain at output the time evolution of the 27 joint variables [9]. Finally, the 3D biped manikin is simulated using Adams software. The simulation attempts to produce motion closed to the recorded data. A distributed feet/ground contact model based on spring damper combination is used [10]. Joints are controlled using a proportional derived giving joint torques according to its position τ = k p (q d − q ) + k v (q& d − q& ) .
(
)
The biped achieves 3 stages during five seconds of simulation. A positioning stage (0-1.2sec), a launching stage (1.2-2.4sec) and two established walking cycles of period 1.2sec.
Mx, My, Mz). If the 6 components are independent, the equivalent system must have, at least, 6 dofs. The interest here is to determine the number of coupling relations between the wrench components in order to reduce the number of these necessary dofs. Figure 3 shows the six components of this wrench. 200
100 M z .sim Fx .s im
100
50
(N )
(N .m )
0
0
-1 0 0
-5 0
-2 0 0
-1 0 0 0
1
2 T em p s (sec)
3
4
5
575 Fy .sim
550 (N )
525
500 0
1
2
T em p s (se c )
3
4
75
5
150
M x .sim Fz .s im
50
100 50
25 (N .m )
(N )
0
0
-2 5
-5 0
-5 0
-1 0 0 -1 5 0
-7 5 0
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2 T em p s (sec)
3
4
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20 M y.s im
10 (N .m ) 0
-1 0
y z Figure 2a: Biped manikin
-2 0
x
Figure 2b: Torso with 13 dofs
In this paper a minimal mechanism is looked for in order to replace upper part of this model (presumed to be realistic) containing 13 dofs (figure 2b). This model should reproduce the dynamic effects on the lower limbs. The proposed approach is based on making equivalence in term of efforts between the selected model and the realistic one built under Adams. Initially, the upper part (figure 2b) of the realistic biped built under Adams is isolated and embedded at the centre of mass of the down-torso. Thereafter, the structure is animated with the time laws of the joint variables in order to extract, using a dynamic simulation, the 6 components of the effort wrench at point of embedding (efforts: Fx, Fy, Fz & moments:
0
1
2
T e m p s (se c)
3
4
5
Figure 3: Components of the effort wrench at embedding point of realistic model of torso during walking gait. The analysis of simulation results shows the existence of two coupling relations. The first one relates 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 2b). These relations can be written as follows: M x = k 1 ⋅ Fz (1) M z = k 2 ⋅ Fx (2) Where, k1 and k2 are two constants (k1> 0 and k2< 0). Hence, the minimal equivalent mechanical system should be a four dofs spatial mechanism.
3
r Y is the output vector. f1 = x2 , f 2 = g1 ,K , f 2 n −1 = x2 n , f 2 n = g n .
Proposed method – Dynamic analysis
The proposed method is based on General State Equation (GSE) formulation. The objective is to identify a mechanism able to produce at its embedding point a required wrench which components are equal to those of the reference model (13 dofs torso system). It is well known that a mechanism is capable to reproduce as much entries as its number of dofs. Nevertheless, the walking gait presents coupling relations between wrench components. These relations can be validated by changes on the mechanism geometrical and inertial parameters. At first, a mechanism topology, for instance, number of dofs (less or equal to 4) and joints kind (R for rotational joint and P for prismatic one) is selected. The inputs of this mechanism are chosen among the components depicted on figure 3 according to the kinematic structure. The outputs are the joint variables motion laws of the candidate mechanism. Dynamic equations based on Newton-Euler formalism are written for each mechanism link at the embedding point. This leads to get the motion equations of this system. The number of motion equations is equal to the number of dofs of the system. These equations can be written in the general following form: q&&1 g 1 q i , q& i , Fx , F y , Fz , M x , M y , M z , Pg , Pi M M = q&&n g n q i , q& i , Fx , F y , Fz , M x , M y , M z , Pg , Pi where: qi is the joint variable of the ith joint. Fx , Fy , Fz , M x , M y , M z are effort wrench components
(
4
Studied kinematic solutions
In the following, a list of models obtained by increasing the number of dofs from 2 to 4 is analysed. At first the two coupling relations (Eq. 1&2) results of the preceding analysis have to be validated. Then, we need to check the possible existence of more coupling relations. The number of theses relations will determine the number of dofs necessary for the model of the equivalent torso. 4.1 Two dofs RR model The studied 2 dofs model is presented on figure 4. The above GSE method is applied to this model. y
)
at embedding point. Pg , Pi are geometrical and inertial parameters of the mechanism. These equations form a system of second order differential equations which can be grouped into a first order system called ordinary differential equations system or General State Equation GSE. Analytical resolution is impossible for the complex systems with more than 2 dofs. The numerical resolution of this type of equations is carried out using Adams software. The GSE is build as follows: f 1 x 1 ,K , x 2 n , u 1 , u m , Pg , Pi q1 r& r X = M ,Y = M f 2 n x 1 ,K , x 2 n , u 1 , u m , Pg , Pi q n where: r T X = [x 1 ,K , x 2 n ] is the state vector with x 1 = q 1 , x 2 = q& 1 ,K , x 2 n −1 = q n , x 2 n = q& n . r T U = [u 1 ,K , u m ] is the input vector whose components are chosen among Fx , Fy , Fz , M x , M y , and M z of the realistic torso.
(
)
(
)
q1
y, y1
x
G A
b
z1
z
)
(
x1
y2
y2
a O
x2 q2 x1
z1, z2
x
Figure 4: RR model. The motion equations are: q&&1 = f 2 q 1 , q 2 , q& 1 , q& 2 , M y , M z , a , b , mi , I ij q&&2 = f 4
( (q
1
, q 2 , q& 1 , q& 2 , M y , M z , a , b , mi , I ij
) )
Where: a is the distance between embedding point (O) and second joint centre (A). b is the distance between second joint centre and the second link centre 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 ={x,y,z} ) The built GSE is given as follows:
[
]
r r r T T T X = [q1 , q&1 , q2 , q&2 ] ,U = M y , M z ,Y = [q1 , q2 ] r& T X = [ f1 , f 2 , f 3 , f 4 ] with f1 = x2 , f 3 = x4
The model is build under the Adams software by taking cylinders 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 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 we identified the coupling relation between components Mz and Fx (Eq.1). Figure 5 shows simulation results obtained with the geometrical and inertial parameters given on table 1.
a
0.1 m
b
0.25 m
m1+m2
24 kg
m3
30 kg
I2y
1 kg.m2
I3x
3 kg.m2
I3y
2 kg.m2
I3z
3 kg.m2
y,y1 q2
x1
x1
G
y, y1
q3 z1
q1
a
z1
z
Table 1: Parameters of the RR model. 20
O z
My.mod My.sim
x
x
Figure 6: RPP model.
10 (N.m)
In this model the coupling relation between components Mx and Fz (Eq. 2) is identified.
0
575
-10
Fy.mod Fy.sim
-20 0
1
2
Temps (sec)
3
4
5 550
100 Mz.mod Mz.sim
(N)
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500
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Fz.mod Fz.sim
200 40
Fx.mod Fx.sim
(N) 0
150
-40
100 (N)
-80
50
-120
0 0
1
2
Temps (sec)
3
75
-50
4.2 Three dofs RPP model The studied RPP mechanism is given on figure 6. It consists of one rotational joint and two prismatic ones. Dynamic equations at point O can be reduced to: Fy = (m1 + m2 + m3 + m4 )g + m3q&&2
{(
)
Fz = m4 q&&3 − q q& cos q1 − (q3q&&1 + 2q&1q&3 ) sin q1
)
}
(N)
0
1
2
Temps (sec)
3
My.mod My.sim 10
(
M y = I y q&&1 − m4 q&& q + 2q&1q3q&3 2 1 3
5
25
Figure 5: Results of the RR model.
2 3 1
4
50
5
0
4
-25
3
-50
Temps (sec)
-75
2
20
1
5
(N)
The related GSE is given as follows:
[
]
r r r T T T X = [q1 , q&1 ,q2 , q& 2 ,q3 , q&3 ] ,U = M y , Fy , Fz ,Y = [q1 , q2 , q3 ] r& T X = [ f 1 , f 2 , f 3 , f 4 , f 5 , f6 ] with f 1 = x2 , f 3 = x4 , f 5 = x6
-20
-10
Where: mi is the mass of ith link, Iy is the sum of inertia of the 4 bodies around the vertical axis and g is gravitational acceleration.
0
0
4 Mx.mod Mx.sim
-100
0
1
2
Temps (sec)
3
4
Figure 7: Results of the RPP model.
5
[
Figure 7 shows simulation results obtained with geometrical and inertial parameters given on table 2.
a
0.35 m
m1+m2
35 kg
m3
7 kg
r r T X = [q 1 , q& 1 , q 2 , q& 2 , q 3 , q& 3 , q 4 , q& 4 ] ,U = M y , Fx , F y , Fz r r T & T y = [q 1 , q 2 , q 3 , q 4 ] X = [ f 1 , f 2 , f 3 , f 4 , f 5 , f 6 , f 7 , f 8 ] with f 1 = x 2 , f 3 = x 4 , f 5 = x 6 , f 7 = x 8
m4
5 kg
Iy
3 kg.m2
Table 2: Parameters of the RPP model.
We notice that through the two studied models, no more than two coupling relations exist between effort wrench components of realistic torso during walking gait. Thus, a four dofs model is necessary and sufficient to reproduce dynamic effects at the waist link. 4.3 Four dofs RPPP model Several models with four dofs are possible. Nevertheless it is necessary to have a thought as for the realisation of this model. Indeed, it is very difficult to build then to control certain types of joints (e.g. spherical joints). The model to be considered should fulfil the requirements of the simple design and controllability. Thus, an RPPP mechanism (figure 8) is chosen. y,y1 q3 q4
x1
x1
G q2
Figures 9 and 10 show simulation results obtained with geometrical and inertial parameters given on table 3. Model different parts resulting motions respect dimensions of the prototype to be built.
a
0.35 m
m1+m2
35 kg
m3
7 kg
m4
5 kg
m5
7 kg
Iy
3 kg.m2
Table 3:Parameters of the RPPP model.
Velocities and accelerations of these links are also taken into account due to their importance in the actuators choice. All 6 components of the realistic torso effort wrench are reproduced by the RPPP model (figure 9&10). The movements produced by the RPPP model are presented on figure 11. These movements are periodic having the same period as the walking gait which is 1.2sec. 200
q1
y, y1
Fx.mod Fx.sim
x 100 (N)
a
z1
z1
z O
0
-100
x
z
0
1
2
Temps (sec)
3
4
5
575 Fy.mod Fy.sim
Figure 8: RPPP model. 550
For the RPPP model, dynamic equations at point O can be reduced to:
{(
)
}
Fx = m3 q&&2 − q 2 q& 12 cos q1 − (q 2 q&&1 + 2q& 1 q& 2 ) sin q1 +
{(
)
m5 q&&4 − q4 q& 12 sin q1 + (q 4 q&&1 + 2q& 1 q& 4 ) cos q1 Fy = (m1 + m2 + m3 + m4 + m5 )g + m4 q&&3
{(
)
}
}
{(
)
m5 q&&4 − q4 q& cos q1 − (q4 q&&1 + 2q& 1 q& 4 ) sin q1
(
)
(
}
M y = I y q&&1 + m3 q&& q + 2q& 1 q 2 q& 2 − m5 q&& q + 2q& 1 q4 q& 4 2 1 2
2 1 4
525
0
1
2
Temps (sec)
3
4
120
5 Fz.mod Fz.sim
60 (N)
)
where, mi is the ith link mass and Iy is the sum of inertia of the 5 links around the vertical axis. The related GSE is given as follows:
(N)
500
Fz = −m3 q&&2 − q 2 q& 12 sin q1 + (q 2 q&&1 + 2q& 1 q& 2 ) cos q1 + 2 1
]
0
-60
-120 0
1
2
Temps (sec)
3
4
Figure 9: Results of the RPPP model-efforts.
5
T
,
75
Three mechanical structures (RR, RPP and RPPP) were studied in order to validate the existing relations and check the possibility of more coupling relations. An interesting result was found. A four dofs mechanism is necessary and sufficient to emulate the dynamic effects of a manikin torso on the lower limbs. A complete design of the identified RPPP mechanism in order to be manufactured will be one of the further developments of this work. A dynamic simulation of the Robian robot with its new upper part is being also carried out.
Mx.mod Mx.sim
50 25 (N.m) 0 -25 -50 -75 0
1
2
Temps (sec)
3
4
5
20 My.mod My.sim 10
References
(N.m) 0
[1]
-10
-20 0
1
2
Temps (sec)
3
4
5
80 Mz.mod Mz.sim 40 (N.m) 0
-40
-80 0
1
2
Temps (sec)
3
4
5
Figure 10: Results of the RPPP model-moments. 0,3
0,3
DEPz DEPy DEPx ROTy
0,15
0,2 (rad)
(m)
0
0,1
0
-0,15
-0,3 0
1
2 Temps(sec)
-0,1 3
4
5
Figure 11: Movements produced by the RPPP model.
5
Conclusion
The aim of this work was to find a minimal mechanism emulating the dynamic effects of the upper limbs on the locomotion system. This mechanism will be the upper part of the Robian prototype which is under development. A generic method based on the GSE concept was carried out in order to identify a minimal mechanism able to generate a required effort wrench. The analysis of the six effort wrench components shows the existence of two coupling relations between them.
E. Nicolls, Bipedal Dynamic Walking in Robotics, Honours Thesis , University of Western Australia, 1998. [2] W. T. Miller III, P. W. Latham, S. M. Scalera, Bipedal Gait Adaptation For Walking With Dynamic Balance, Proceedings of the 1991 American Controls Conference, Boston, Mass., June 1991, pp.1603-1608. [3] S. Hashimoto, Humanoid Robots in Waseda Univeristy-Hadaly-2 and Wabian, IARP First International Workshop on Humanoid and Human Friendly. 1998. [4] J. Yamagushi, A. Takanishi. Development of a biped walking Robot having Antagonistic Driven Joints using Nonlinear Spring Damper Mechanism. In 1997 IEEE Proc. of Int. Conf. on Rob. & Aut. [5] K. Hirai, Current and future perspective of Honda humanoid robot, Proceedings of the IEEE International Workshop on Intelligent Robots and Systems IROS 97, pp. 500-508. [6] K. Hirai. M. Hirose, Y. Haikawa. The Development of Honda Humanoid Robot. In proceeding of the 1998 IEEE, Inter. Conf. On Robotics and Automation, Leuven, Belgium. [7] F. Gravez, O. Bruneau, F.B. Ouezdou. Three Dimensional Simulation of Walk of Anthropomorphic Biped, CISM-IFToMM RoManSy, 2000, Zakopane, Pologne. [8] D. W. Sewar, A. Bradshow, F. Margrave. The Anatomy of a Humanoid Robot. In Robotica, Vol. 14, Cambridge University Press, 1996. [9] F. Gravez, O. Bruneau, F.B. Ouezdou. Capture de Mouvement pour la Simulation Dynamique de Mannequin Virtuel. XV Congrès Français de Mécanique, Nancy, Sept. 2001. [10] O. Bruneau,F .B. Ouezdou. Distributed Ground / Foot Robot Interactions. In Robotica, Vol. 17, Cambridge University Press, 1999.
