Four Dof TORSO Dynamic Effects on Biped Walking Gait
Bachar Mohamed , Fabrice Gravez , Olivier Bruneau and Fethi Ben Ouezdou
Laboratoire d’Instrumentation et de Relations Individus Systmes, V´elizy, France Laboratoire Vision et Robotique, Bourges, France
Abstract. In this paper, a study of the upper part dynamic effects on locomotion system allows us to identify the kinematic structure of a new prototype called ROBIAN II. The biped has 16 degrees of freedom (dofs). Initially, a bio-mimetic approach is used to model a virtual manikin biped having 25 dofs based on common European male (75 kg, 1.78 m). Using, human being motion recording, foot/ground contact model, inverse kinematics and Newton-Euler equations, a 3D dynamic simulation of this virtual manikin is carried out. Scale factorization is used to get a new manikin which weight and height are those of ROBIAN II prototype. A 3D dynamic simulation of the obtained virtual manikin is carried out in order to identify the effort wrench exerted by its torso on the lower limbs. An analysis of the six components of this wrench shows the existence of two coupling relations. A study of 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. An RPPP mechanism is presented in order to replace the virtual manikin upper part. Results of 3D simulation of the 16 dofs resulting biped are then presented. A control method is used to ensure dynamic stability of this biped during walking gait.
1 Introduction Technological progress in the field of actuators, sensors and computers opens new potential applications for walking robots, mainly the biped ones. Bipedal robots can operate in human environment more efficiently than other types of robots like wheeled ones. Bipedal robots have namely a significant interest in the field of human assistance for domestic or dangerous tasks. 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 and more efficient than the static walking patterns. Therefore, research in bipedal robotics has progressed to study dynamic walking gaits (Raibert (1986), Miller et al. (1991), Vukobratovic and Timocenko (1995), Kajita et al. (2001)). Dynamic walking has been realized by some bipedal robots, most notably are Sony dream robot, the Honda P2 and P3 robots and the Wabian robot of the university of Waseda showing quite impressive performances in carrying out human like tasks (walking, dancing, turning) (Hirai et al. (1998), Ishida et al. (2001), Lim et al. (2001), Kanehiro et al. (2001)). 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 developed without any upper part. Its major objectives are increase of our understanding of the human being locomotion system and the development of a real testing bed of active/passive prosthesis. In this paper, a study of the upper part dynamic effects on locomotion system allows us to identify the kinematic structure of a new prototype. In section 2, we present the bio-mimetic ap-
proach used in order to model a virtual manikin of the human body based on a chosen kinematic structure having 25 dofs. Next, section 3 deals with the virtual manikin of ROBIAN II having 25 kg weight and 1.10 m height obtained by scale factorization. Emulation of realistic model torso of ROBIAN II without any anthropomorphic consideration is then presented. Analysis of the human torso dynamic effects during walking gait is detailed. A minimum dofs mechanism able to reproduce these effects is then identified and verified using a method based on the General State Equation (GSE) formalism. Section 4 deals with the dynamic behavior simulation and analysis of the ROBIAN II virtual manikin with a simplified mechanism constituting its upper part. In the last section, conclusions and further developments are given.
2 Bio-mimetic Approach
Figure 1. Biped manikin , Torso with 13 dofs and Example of scale factorization.
A 25 dofs kinematic structure presented on figure 1 was chosen for the virtual manikin of our biped robot. The human body is modeled by 16 solid primitives according to the Hanavan model Hanavan (1964) as shown on figure 1. Using a description of the 3D-bio-mechanical data a mass distribution has been associated to each solid (Seward et al. (1996), Bruneau et al. (1998)). Faithful reproduction of human movements during walking gait is of a primary importance for dynamic simulation of a virtual manikin. Therefore, a series of measurements using Vicon 1 motion analysis system was carried out to obtain positions of 16 markers placed on a human being at points where relative motion between the skin and the bones are minimal during walking gait. These markers positions are the input of the biped inverse kinematic model, which allows us to obtain the time evolution of the 25 joint variables. 1
Vicon Motion Systems is part of OMG public company
A distributed feet/ground contact model based on spring damper combination is used to include the external efforts in the dynamic model of the biped (Bruneau and Ouezdou (1999)). Under Adams2 software, the above model biped is built using parametrical construction depending on total weight and height. Figure 1 shows how useful parametrical construction is by presenting several model of the biped with different weights and heights. Finally, the 3D biped manikin is simulated using Adams (Gravez et al. (2000)). The simulation attempts to produce motion close to the recorded data. Joints are controlled using a proportional derivative controller giving joint torques according to its position ( ). The biped virtual manikin achieves 4 stages during 4.8 seconds of simulation. A positioning stage (0-1.2 sec), a launching stage (1.2-2.4 sec) and two established walking cycles of period 1.2 sec.
3 ROBIAN II Project Manikin As the objective of ROBIAN is the development of a simple testing bed of active/passive locomotion system prosthesis devices, we intend to build a small biped with 1.10 m of height and no more than 25 kg of weight. The ROBIAN mechanical structure is already finished. To this end, we proceed to a scale factorization of our humanoid according to these parameters. In a first approximation, the human upper part, which presents 13 actuators (3 per arm, 3 for neck and 4 for the trunk), is not essential to accomplish this objective. A minimal mechanism is, then, looked for, in order to replace the upper part of this model (presumed to be realistic) containing 13 dofs (figure 1). This model must reproduce the dynamic effects on the lower limbs during walking gait. 3.1 Upper part dynamic analysis 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 of the realistic biped built under Adams is isolated and embedded at the center of mass of the down-torso. Thereafter, the structure is driven 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 (forces: , , & moments: , , ). If the 6 components are independent, the equivalent system must have, at least, 6 dofs. Our objective is to determine the number of coupling relations between the wrench components in order to reduce the number of these necessary dofs. Figure 2 shows the six components of this wrench. The analysis of simulation results shows the existence of two coupling relations. The first one relates the moment component around the x axis (motion direction) to the force component in z axis (lateral direction). The second relation concerns the moment component and the force component (axes are depicted on figure 1). In a first approximation, these relations can be written as follows:
! !"
#$ #$ #$"
!"
#
#$ ')(*,+ !" 2
#%"
&
and
#$"-.(0/+ !
Adams is a trade mark of Mechanical Dynamics Incorporating
(1)
Figure 2. Components of the effort wrench at embedding point of realistic model of torso during walk.
12
103
142-5%6
1038796
where, and are two constants ( and ). Hence, the minimal quasi-equivalent mechanical system should be at least a four dofs spatial mechanism. 3.2 Proposed method for dynamic equivalence The proposed method is based on General State Equation (GSE) formulation. The objective is to identify a mechanism able to produce at its embedding point a desired wrench which components are equal to those of the reference model (13 dofs torso system). It is well known that an embedded mechanism is capable to reproduce as much components of efforts wrench at embedding point as its number of independent actuated dofs. Nevertheless, the walking gait presents coupling relations between wrench components. These relations can be satisfied by changes on the mechanism geometrical and inertial parameters. At first, a mechanism topology, number of dofs (less or equal to 4) and joints kind (R for rotational joint and P for prismatic one) is selected. Due to the two coupling relations (Equation 1), a maximum number of four efforts are chosen among the six wrench components depicted on figure 2 as inputs. The outputs are the joint variables motion laws of the candidate mechanism. Dynamic equations based on Newton-Euler formalism are written for each mechanism link at the embedding point. This leads to get the motion equations of the system which their number is equal to the number of dofs of the system. These equations can be written in the general following form:
where:
:; > = 2 A B ;: &2 F >@GIHK@J> GLHNM!O HNM!P4HNM!Q0HSR$O4HSR$PHSR$QHNTUHNT!GWV A B .. < = ... C9D < E C >@? ? F @> GIHK@J> GLHNM!O HNM!P4HNM!Q0HS.R$O4HSR$PHSR$QHNTUHNT!GWV E
X>@G is the joint variable of the ith joint. XM!O HNM!PHNM!QYHSR$O4HSR$PYHSR$Q are effort wrench components at embedding point.
Z[\]N[!^
are geometrical and inertial parameters of the mechanism.
These equations form a system of second order differential equations that can be grouped into a first order system called ordinary differential equations system or General State Equation GSE. It’s obvious that analytical solutions for a more than 2 dofs mechanism is impratical. Hence numerical solving of this system of equations is carried out using Adams software. The GSE is built as follows:
_Y` _0acbedfgih0j&kmlnj ]@o@o@op] lrqNs I] t j ]@o@o@op]Itvu']N[\Y]N[!^Ww x y b|dfgr}~j x y .. .. z ]I{ . . z h,qNs8kmlnj ]@o@o@o@] lrqNs ]It j ]@o@o@o@]Itvui]N[\]N[!^Ww }@s ` b b b
b b where: Z bln j ]@o@o@o@] lrqNs is the state vector with lKj }~j , lrq }~j , o@o@o , lrqNsKj }@s , lrqNs }@s . Z t ]@o@o@o@]Itvu is the input vector whose components are chosen among ] ! ] ! ] ] j ] of the b9 vector.realisticb torso. b9 Z${ isb the output Z h0j lrq ] h,q j ]@o@o@o@] h,qNsKj lrqNs ] h,qNs s .
3.3 Equivalent four dofs RPPP model A recent work showed that no more than two coupling relations exist between wrench components which means that a four dofs mechanism is necessary and sufficient to ensure dynamic equivalence with human upper part during walking gait (Mohamed et al. (2001)).
Figure 3. RPPP Model and Simulation of RPPP Model.
Several models with four dofs are possible. Nevertheless it is necessary to have a thought about the realization of the mechanism. Indeed, it is very difficult to build then to control some
kinds of joints (e.g. spherical joints). A decoupled RPPP mechanism (figure 3) was chosen because it fulfills the requirements of simple design and controllability. For the RPPP model, dynamic equations at point O can be reduced to:
$p~ $ ~ p, ¡ £~¢ ¤ ¢ ~¥£¦ § ~ p,¨ ¡ £~¢ ¤ ¨ ¢ ¨ ¥ ©!ª' ¬« ¦ ,¢ ¥Kp®Y¯@° ~p± ¦9° ~ ¡ £~¢ &¢ ± ¯I² ³° ~p±@´ µ§ «! ¨ ¦ ¨ ¢ ¥ ¯I² ³° ~p± ° ¨ ~ ¡ ¶~¢ !¢ ¨ ± p®Y¯@° ~@± ´ ©!i ° · $ $¬$ ¨ $§±Y¸$ ¨ @ ©!¹8 ¦ «! £¦ ,¢ ¥ ¯I² ³° ~p± ° ~ ¡ ¶~¢ !¢ ± p®Y¯@° ~@± ´ µ§ «! ¨ ¦ ¨ ¢ ¥ p®Y¯@° ~@±K ° ¨ ~ ¡ £~¢ !¢ ¨ ± ¯I² ³° ~p± ´ ¾½À¿rÁ ) and p is the sum of inertia of the 5 links around the where, º is the link » º mass (¼ vertical axis (figure 3). The four motion equations are obtained as follows: Âà ~ Å Æ Âà ¸ ° ~~È@É@É@É@ÈN ¨ È~¢ ~È@É@É@É@ÈK¢ ¨ È ©!ª È ©! È ©!¹ È $ ÈNÊËÈNÊ!ºW± Å Æ .. Ç Ä ... Ç Ä ¨ ¸ ¨ ° ~~È@É@É@É@ÈN ¨ È~¢ ~È@É@É@É@ÈK¢ ¨ .È ©!ª È ©! È ©!¹ È $ ÈNÊËÈNÊ!ºW± where:
Ì@º is the joint variable of the ith joint. Ì ©!ª È ©! È ©!¹ È $ are chosen wrench components at embedding point. ÌÊËÈNÊ!º are geometrical and inertial parameters of the RPPP mechanism. The related GSE is given as follows:
Í Î ~~ÈK~¢ ~ÈN È¢ ÈN@ÈK@¢ ÈN ¨ È¢ ¨@ÏÐ , Ñ Î $ È ©!ª È ©! È ©!¹ ÏÐ , ÒÓÎ ~~ÈN ÈN@ÈN ¨pÏÐ , Ô Í Î Ö ~È Ö È Ö ,È Ö ¨ È Ö §È Ö,× È Ö,Ø È Ö,Ù ÏmÐ Ö ÛÚ , Ö ¸ ,Ô0Ö Õ ÛÚ ¨ , Ö ¨ ¸ , Ö § 9Úr× , Ö,×µ ¸0 , Ö,صÛÚrÙ , Ö,Ùµ ¸ ¨ . with Table 1. Parameters of the RPPP model.
Ü
ÝßÞvàáÝãâ
Ý ä
Ý å
Ýãæ
ç¤è
Parameter Value 0.25 m 9.77 kg 3.1 kg 1.82 kg 2.76 kg 0.295 kg.m
â
During simulation (figure 3), realistic torso wrench components are given -on line- to the GSE solving procedure. The GSE procedure computes the joint variables motion laws that are applied on line to ROBIAN II biped manikin upper part (RPPP model). Dynamic behavior simulation result using parameters depicted on table 1 shows that all 6 components of the realistic torso effort wrench are reproduced by the RPPP model (figure 4). This confirms dynamic equivalence between the two mechanisms. Resulting motions of the mechanism parts (figure 5) satisfy dimensions of the prototype to be built. These movements are periodic having 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 in the actuators sizing.
Figure 4. Results of the RPPP model forces and moments.
Figure 5. Movements produced by the RPPP model.
Figure 6.
é-ê
and
é-ë
coordinates.
4 Dynamic Behavior Simulation Analysis and Stability Control In the previous section, we identified a four dofs mechanism dynamically equivalent to the human shape torso having 13 dofs. Dynamic simulations of new virtual biped show that open loop walk using RPPP mechanism motions obtained in the last section presents instability during launching stage. Nevertheless, we use a control method to stabilize the biped robot. The control method is composed of the following stages. Firstly we compute online the total wrench exerted by the ground on the feet. Next, x and z coordinates ( ) of the intersection point denoted A between the wrench axis and the ground is calculated:
ì í)î
ïð-ñµò ôõ ó$ò ù ò$ó ü ø û ý with : öãí þ ø þ ñµî ó í)öv÷ øúù þ ø þpÿ ñµò and ñµó are x and z coordinates of point A, ø í ò ó is resultant force where : of contact wrench, ý í ò $ó is resultant moment of contact wrench. In order to avoid the fall of the robot (minimization of the roll and pitch motion), the required virtual forces to be applied on the down torso are computed as follows:
ò í.ö òÀ ò ù òÀ ò ó í.ö óÀó ù óÀ ó ò ó ò ó ò where : ö and ö are stiffness coefficients, and are damping coefficients, are the difference between x and z coordinates of point A and biped mass center. At last, the required joints torques for stabilization are obtained as follows: ïð ò ôõ í î ó where : is the jacobian matrix of each leg. ò Àtransformation ó
and
Àó
Figure 6 shows and time evolution in a frame related to the projection of the mass center on the ground. Simulation of stable walk of ROBIAN II manikin with control is shown on figure 7.
5 Conclusion The aim of this work is to carry out 3D fully dynamic simulations of the virtual manikin of ROBIAN II. This biped has 16 dofs, 25 kg of weight and 1.10 m height. Based on bio-mimetic approach (motion recording, foot/ground contact model, inverse kinematics of a standard human being), Newton-Euler equations were used to build a virtual manikin having 25 dofs and to carry out the 3D dynamic behavior simulations of the system. First of all, scale factorization was used to reach ROBIAN II weight and height. Then, dynamic analysis of human like upper part (having 13 dofs) was carried out in order to identify effort wrench components exerted by the upper part on the locomotion system.
Figure 7. Dynamic Simulation.
This analysis showed the existence of two coupling relations between wrench components, which led us to an interesting result. Indeed, a four dofs mechanism is able to emulate dynamic effects of the 13 dofs upper part system. A generic method based on General State Equation formalism was developed in order to check dynamic equivalence between two mechanical systems. An RPPP mechanism dynamically equivalent to human upper part was presented in order to replace ROBIAN II upper part. Simplicity of manufacturing and control of this mechanism were the main reasons that led us to choose such a kinematic structure. Different steps of the control method were exposed. The control insure a reduction of instability problems. Finally, results of 3D stable walking gait dynamic simulation of the resulting biped manikin having 16 dofs and flexible feet were presented. A complete design of the presented RPPP model in order to be manufactured will be one of the further developments of this work. This mechanism combined with the locomotion mechanism, which is already manufactured, will lead to the global prototype of the ROBIAN II biped.
References Bruneau, O., and Ouezdou, F. (1999). Distributed ground/walking robot interactions. Robotica 17. Bruneau, O., Ouezdou, F., and Guinot, J. (1998). Dynamic simulation tool for biped robots. In CISM IFToMM - Symposium on Theory & Practice of Robots & Manipulators (RoManSy). Gravez, F., Bruneau, O., and Ouezdou, F. B. (2000). Three-dimensional simulation of walk of anthropomorphic biped. In CISM - IFToMM - Symposium on Theory & Practice of Robots & Manipulators (RoManSy). Hanavan, E. P. (1964). A mathematical model of the human body. Amrl-tr-64-102, ad-608-463, Aerospace Medical Research Laboratories, Wright-Patterson Air Force Base, Ohio. Hirai, K., Hirose, M., Haikawa, Y., and Takenaka, T. (1998). The development of honda humanoid robot. In IEEE - International Conference on Robotics & Automation (ICRA). Ishida, K., Kuroki, Y., Yamagushi, J., Fujita, M., and Doi, T. (2001). Motion entertainment by a small humanoid robot based on open-r. In IEEE - International Workshop on Intelligent Robots & Systems (IROS). Robot humano¨ıde Sony. Kajita, S., Kanehiro, F., Kaneko, K., Yokoi, K., and Hirukawa, H. (2001). The 3d linear inverted pendulum mode : A simple modeling for a biped walking pattern generation. In IEEE - International Workshop on Intelligent Robots & Systems (IROS). Kanehiro, F., N.Miyata, Kajita, S., Fujiwara, K., Hirukawa, H., Nakamura, Y., Yamane, K., Kohara, I., and Kawamura, Y. (2001). Virtual humanoid robot platform to develop controllers of real humanoid robots without porting. In IEEE - International Workshop on Intelligent Robots & Systems (IROS). Lim, H., Setiawan, S., and Takanishi, A. (2001). Balance and impedance control for biped humanoid robot locomotion. In IEEE - International Workshop on Intelligent Robots & Systems (IROS). WABIAN de l’universit´e de WASEDA. Miller, W. T., Latham, P. W., and Scalera, S. M. (1991). Bipedal gait adaptation for walking with dynamic balance. In American Controls Conference, 1603–1608. Mohamed, B., Gravez, F., and Ouezdou, F. (2001). Emulation of the human torso dynamic effects during walking gait. In IEEE - International Workshop on Robot & Human Communication, 562–567. Raibert, M. H. (1986). Legged robots that balance. MIT Press. Seward, D. W., Bradshow, A., and Margrave, F. (1996). The anatomy of a humanoid robot. Robotica 14:437–443. Vukobratovic, M., and Timocenko, O. (1995). Stability analysis of certain class of bipedal walking robots with hybridisation of classical and fuzzy control. In ECPD - International Conference on Advanced Robotics & Intelligent Automation.
