Application and Analyses of Dynamic Reconfiguration Manipulability Shape Index into Humanoid Biped Walking

Transcription

1 Proceedings of the 216 IEEE International Conference on Robotics and Biomimetics Qingdao, China, December 3-7, 216 Application and Analses of Dnamic Reconfiguration Manipulabilit Shape Index into Humanoid Biped Walking Keli Shen 1, Xiang Li, Honghi Tian, Daiji Iawa and Mamoru Minami, Takauki Matsuno Abstract In this paper, we define a new index of dnamic manipulabilit for humanoid biped walking to measure dnamic flexibilit of changing mechanics b using residual redundanc, when primar work is being assigned, such as face and ees being targeted to certain object. Although some measurements have been arranged so far to evaluate statical or dnamical flexibilit of hand manipulator. This paper displas a new measurement of Dnamic Reconfiguration Manipulabilit Shape Index () which is the combination of dnamic manipulabilit and reconfiguration manipulabilit, and we have applied the into humanoid robot experiments to evaluate its dnamical reconfiguration abilit during walking. According to the value of in our experiments, we have verified our humanoid robot model is reliable and effective since the results have indicated that the kinematic and dnamic characteristics of our model are similar to human-being. I. INTRODUCTION In the world, man researchers made their efforts to control humanoid walking and proposed man effective methods to measure walking flexibilit. As we all known, ZMP-based walking motion is considered as the most effective and hopeful method, which has been certified to be a useful control tactic to perform stabilit of practical biped walking, because it can make sure that the humanoid can get the balance of walking and standing b keeping the ZMP within the convex hull of supporting space [1]. Nakamura [2] showed the wa of representing connection between motions and environment combining the dnamical equation of motion and put into execution on human. Since dimension number of biped walking dnamical states is variable b contacting conditions of foot with floor, we need some measurements that discuss continuall kinematic or dnamical flexibilit of humanoid according to the bipedal gaits variet. The behavior of human s walking seems to use redundanc during undertaking a prior task of biped walking or ees staring. Thus, we are willing to talk about dnamical redundanc of humanoid biped walking in this paper, proposing the concept of dnamical reconfiguration manipulabilit Shape Index (), which evaluates flexibilit and stabilit of a dnamical sstem which can possibl produce a motion in a workspace with standardied input torque, which is the combination of the dnamic manipulabilit (DM) [7] with reconfiguration manipulabilit (RM) [8]. The new index evaluates the flexibilit of the dnamical sstem of humanoid robots and possesses the abilit of shape-changing acceleration in workspace using unit torque input for all joints 1 Keli Shen is with Division of Mechanical and Sstems Engineering, Graduate school of Natural Science and Technolog Okaama Universit, Tsushima-naka, Kita-ku, Okaama 7-853, Japan pue85qr6@s.okaama-u.ac.jp DRMEs(acceleration ellipsoids) Undertaking a hand work(primar work) DRMEs keeping the height of head (primar work) Walking or standing Fig. 1. Applications of DRME for redundant manipulator and humanoid robot. DRME Work-1: keeping head-height Work-2 : lowering waist-height DRME Work-1: keeping head-height Work-2 : lowering waist-height Fig. 2. Walking on uneven floor with no-singular mechanics has a redundanc to accelerate the waist position during keeping head posture and making the foot reach the uneven floor partiall singular mechanics (from waist to head) has no abilit to keep the current head posture, during the foot reaching floor. This result can be analed b the DRME, indicating the width in -axis is nearl ero and ellipse shape is flat. during undertaking prior work. We have applied to a humanoid robot, whose primar work is assigned to keep head position or orientation close to desired one as much as possible. The concept is shown in Fig. 1 and the Dnamic Reconfiguration Manipulabilit Ellipsoid (DRME) of floorfixed four link robot is shown in Fig. 1, and we think that ellipsoid also includes ellipse and line segment in this paper. Simulations showed the value of DRME was relative to the change of the human s waist height, indicating that index can be useful such as the case that the whole shapechange control can be improved through DRME based on request of secondar work different from walking. We can also conclude that our new humanoid robot model is valid and has the similar motion characteristics like human-being. Walking on the uneven floor is shown in Fig. 2. Considering condition of work-1 (keeping the height of head) and work-2 (lowering the height of waist) is set. These work give an assumption that humanoid s face and ees should be targeted to certain object during walking on the uneven floor. There is space for -acceleration of waist in, which enable robot to undertake work-1 and work-2 simultaneousl at the /16/$ IEEE 1436

2 TABLE I PHYSICAL PARAMETERS. Fig. 3. Ü Definition of humanoid s link, joint and angle number. Link l i m i d i Head Upper bod Middle bod Lower bod Upper arm Lower arm Hand Waist Upper leg Lower leg Foot Total same time. However, there is little space for -acceleration of waist in, work-1 can be undertaken but work-2 cannot be undertaken because head will go down when the foot approach floor. Therefore, can be considered as an index to evaluate flexibilit of plural work of biped walking. From our group simulation experiments, We find that humanoid robot similar to human-being is suitable to stand, walk or run on the even floor than on the uneven floor, which is indicated b the change of the value. We will tr to use to find out how to keep the highest flexibilit of motion on the road with the uncertain and complexed condition such as stairs with different ladder heights or challenging terrain in the future. II. DYNAMICAL BIPED WALKING MODEL We discuss a humanoid robot whose definition is described in Fig. 3. Table I shows length l i [m], mass m i [kg] of links and joints coefficient of viscous friction d i [N m s/rad], which are decided according to [3]. Our model figures rigid whole bod feet including toe, torso, arms and bod having 17 degree-of-freedom. Specific explanation of this model is ignored, which is mentioned in [4]. III. VISUAL LIFTING METHOD This section proposes a vision-feedback control to strengthen biped standing/walking stabilit as shown in Fig. 4. We appl a model-based matching method to evaluate posture of a static target object described b ψ(t) representing the robot s head based on Σ H. The relativel desired posture of Σ R (coordinate of reference target object) and Σ H is predefined b Homogeneous Transformation as H T R. The difference of the desired head posture Σ Hd and the current posture Σ H is defined as H T Hd, it can be described b: H T Hd ( d(t), (t)) = H T R ( (t)) Hd T R 1 ( d(t)) (1) where H T R is calculated b ψ(t) that can be measured b on-line visual posture evaluate approach, which is proposed b [5], [6]. However, in this paper, we assume that this parameter is given directl. Here, the force which is exerted é (t) x x ÜH ÜHd fv Fig. 4. Ü R x fv isforcethatliftupheadto prevent from falling down Static object Jh(q) Supporting-leg Concept of Visual Lifting Stabiliation. on the head to minimie δψ(t)(= ψ d (t) ψ(t)) calculated from H T Hd is considered to be directl proportional to δψ(t). The deviation of the robot s head posture is caused b gravit force and the influence of walking dnamics. The joint torque τ h (t) lifting the robot s head is given as the following equation: τ h (t) = J h (q) T K p δψ(t) (2) where J h (q) in Fig. 4 is Jacobian matrix of the head posture against joint angles including q 1,q 2,q 3,q 4,q 8,q 9,q 1,q 17, and K p is proportional gain like impedance control. We appl this input to prohibit the behavior of falling down caused b gravit or dangerous slipping gaits happened unpredictabl during walking progress. What we want to point out is that the input torque for non-holonomic joint such as joint-1, τ h1 in τ h (t) in (2) is ero because of its free joint. δ ψ(t) can show the deviation of the humanoid s position and orientation, however, onl position was discussed in this stud. IV. DYNAMIC RECONFIGURATION MANIPULABILITY A. Dnamic Manipulabilit In short, motion equation of serial link manipulators can be written as M(q) q + h(q, q) + g(q) + D q = τ (3) 1437

3 Undertaking prior hand work No prior work Fig. 5. Dnamic manipulabilit ellipsoids (DMEs)(no prior work) and dnamic reconfiguration manipulabilit ellipsoids (DRMEs) (undertaking prior work). where M(q) R n n represents inertia matrix, h(q, q) R n and g(q) R n represent vectors of Coriolis force, centrifugal force and gravit, D = diag[d 1,d 2,,d n ] is matrix representing coefficients of joints viscous friction and τ R n represents joint torque. The kinematic equation of manipulators, the relation between the j-th link s velocit ṙ j R m and the angular velocit q R n is described b ṙ j = J j q ( j = 1,2,,n). (4) Then, J j R m n can be described as Jacobian matrix including ero block matrix, J j = [ J j, ]. Through differentiating calculation of (4), we can achieve (5) r j = J j (q) q + J j (q) q (5) where J j (q) q represents the acceleration rooted in nonlinearrelation of two-coordinates-space from q j to r j. Then, we can obtain (6) from (3) and (5) r j J j q = J j M 1 [τ h(q, q) g(q) D q]. (6) Here, two variables are defined as follows: τ = τ h(q, q) g(q) D q (7) r j = r j J j q = J j q. (8) Then, (6) can be rewritten as r j = J j M 1 τ ( j = 1,2,,n). (9) Considering serial desired accelerations r j of all links being producible b serial joint torques τ j satisfing an Euclidean norm condition ( τ j = ( τ τ τ2 j )1/2 1), and then the each front end acceleration makes an ellipsoid in range room of J j. These ellipsoids of each link have been known as Dnamic Manipulabilit Ellipsoid (DME) from [7] (Fig. 5) which are denoted in (1). r T j (J j (M T M) 1 J T j ) + r j 1, and r j R(J j ) (1) where, R(J j ) represents range room of J j. Fig. 6. Configuration change of intermediate links during manipulator executing work. B. Dnamic Reconfiguration Manipulabilit Here, the desired end-effector s acceleration r nd is considered as prior work. Putting j = n into (9), the desired r j is described b r nd, r nd = J n M 1 τ. (11) From (11), τ is obtained in (12) τ = (J n M 1 ) + r nd +[I n (J n M 1 ) + (J n M 1 )] 1 l (12) where, 1 l represents an arbitrar vector which satisfies 1 l R n. The left superscript 1 of 1 l represents the first dnamic configuration change subtask. In the right side of (12), the first part describes the solution of minimiing τ in the null room of J n M 1 during executing r nd. The second part describes the elements of torques at each joint, which can change the configuration of manipulator despite of the influence of arbitrar r nd as end-effector acceleration for trailing the desired path. In this research, 1 r nd is commanded b a higher-level dnamic reconfiguration control sstem and 1 r jd can be used for ordinar dnamic configuration change subtask. The relation between 1 r jd and r nd is described in (13) b substituting (12) into 1 r jd = J j M 1 τ 1 r jd = J j M 1 (J n M 1 ) + r nd +J j M 1 [I n (J n M 1 ) + (J n M 1 )] 1 l. (13) Here, two new variables are defined in the following equations 1 r jd = 1 r jd J j M 1 (J n M 1 ) + r nd (14) 1 Λ j = J j M 1 [I n (J n M 1 ) + (J n M 1 )]. (15) In (14), 1 r jd is named after the first dnamic configuration change acceleration. In (15), 1 Λ j is a m n matrix named after the first dnamic configuration change matrix. And then, (13) can be simplified as 1 r jd = 1 Λ j 1 l. (16) The relation between 1 r jd and 1 r jd is shown in Fig. 6. However, the ke is how to find out 1 l to generate 1 r jd. Solving 1 l in (16) as 1438

4 1 l = 1 Λ + j 1 r jd + (I n 1 Λ + i 1 Λ j ) 2 l. (17) In (17), 2 l is an arbitrar vector which satisfies 2 l R n. 1 l is assumed to consist with restricted condition 1 l 1, and then next equation is obtained, 1 r jd T ( 1 Λ + j )T 1 Λ + j 1 r jd 1. (18) If rank( 1 Λ i ) = m, (18) denotes an m-dimensional room ellipsoid, satisfing 1 r jd = 1 Λ j 1 Λ + j 1 r jd, 1 r jd R m, (19) which means that 1 r jd can be arbitraril obtained in m- dimensional room and (18) alwas has the value 1 l satisfing all 1 r jd R m. Besides, if rank( 1 Λ j ) = p < m, r jd is not obtained arbitraril in R m. Thus, we name reduced r jd after 1 r jd. Then (18) can be denoted as the following equation, which is proposed b [9] ( 1 r jd )T ( 1 Λ + j )T 1 Λ + j 1 r jd 1 ( 1 r jd = 1 Λ 1 j Λ + j 1 r jd ). (2) Equation (2) denotes an r-dimensional room ellipsoid. These ellipsoid described in (18) and (2) are shown in Fig.5. Therefore, if the matrix Λ conducted b the singular value dissolution, we can get 1 Λ j = 1 U 1 j Σ 1 jv T j (21) r n r 1 σ j,1 1. Σ i = r.. 1 σ j,r. (22) m r In (21) and (22), 1 U R m m, 1 V R n n denote orthogonal matrixes, and r is the number of non-ero singular values of 1 Λ j (σ j,1 σ j,r > ). Besides, r m since rank( 1 Λ j ) m. Thus, when the end-effector of manipulator executes some work, dnamic configuration change flexibilit of j-th link can be denoted as 1 w j = 1 σ j,1 1 σ j,2 1 σ j,r. (23) In this paper, we name the value of w j in (23) after dnamic reconfiguration manipulabilit measure (DRMM), indicating the degree of that configuration change acceleration of j- th link can be obtained from different direction. And then, volume of dnamic configuration change ellipsoid at the j-th link is defined as 1 V DR j = c m 1w j (24) { 2(2π) c m = (m 1)/2 /[1 3 (m 2)m] (m : odd) (2π) m/2 /[2 4 (m 2)m] (m : even). (25) For taking dnamic reconfiguration measure of the whole manipulator-links into consideration, we give an indicatrix called dnamic reconfiguration manipulabilit shape index (): 1 n 1 W DR = j=1 a j 1 V DR j. (26) In this paper, singular-values are enlarged a hundredfold to obtain formal value of ellipsoid, comparing with ellipse or line segment. For robots in sagital plane like Fig. 5, a j is denoted as a 1 = a n 1 = 1[m 1 ], a 2,3,,(n 2) = 1[m 2 ]. (27) V. APPLICATION TO HUMANOID BIPED ROBOT BASED ON This section is aimed at using the measurement tool to anale the flexibilit of biped walking. Firstl, keep the height of the head and assume the constraint condition 1 l 1, DRMEs are plotted in Fig. 7. There are 8 ellipses generating on the tip of the links. The volume of ellipse is formed b the manipulabilit to judge the mobilit performance of biped walking links. Here, DRMEs are determined b 1 Λ j, which depends on both the jacobian matrix and the inertial matrix in the (15). Furthermore, two variables mentioned above are actuall the functions about humanoid joint angles from the supporting foot to the head. Therefore, DRMEs are actuall dominated b joint angles. In other words, DRMEs are determined b the configuration of biped walking robot. Combining the shape of ellipse in Fig. 7 and the equation (22), (23), (24), we can find that ellipse area is composed of the lines representing the all the accelerations producing potentiall in the different directions of whole or part workspace. In (22), 1 σ j,1, 1 σ j,2 correspond to the macro axis and minor axis, representing the biggest acceleration and the smallest acceleration. In general, the micro axis denotes the desired acceleration. In Fig. 7, 4 ellipses in the neck, shoulders and waist locate in the X-Y plane, there is the desired acceleration in the direction Y axis, which means robot will go along the direction of Y axis potentiall. This is due to humanoid leg can onl rotate in the Y-Z plane, robot has the inertial tendenc of forwarding the direction of Y axis. On the other hand, in Fig. 7, 4 ellipses in the waist, hip and supporting leg locate in the Y-Z plane, there is also the desired acceleration in the direction Y axis. 8 ellipses of biped humanoid have the same potential forwarding performance, indicating the harmon of the biped walking. And then, we calculate all the joint reconfiguration flexibilit with (26) from the supporting foot to the head (except the head ellipsoid since keeping the height of the head is prior task). The value of is the total of the calculated joint acceleration while obtaining the optimal walking configuration b adjusting the joint angles. And then, biped walking is simulated on the even ground. In this section, we use the visual-lifting method mentioned in section 3. Here, we change the K p to control the joint torque τ h (t) in (2) from the supporting foot to the head to overcome the gravit and the influence of walking dnamics, assuming there is an upward force keeping 1439

5 X Y DRMEs DRMEs Z Y Fig. 7. The DRMEs of biped walking humanoid (c) (d) Kp = diag[2; 29; 8] (e) Kp = diag[2; 29; 12] Fig. 9. Screen-shot of biped walking with DRMEs about different gain set : K p = diag[2,29,9], K p = diag[2,29,95], (c) K p = diag[2,29,1], (d) K p = diag[2,29,15], (e) K p = diag[2, 29, 11]. (c)kp = diag[2; 29; 13] Fig. 8. Screen-shot of unstabl biped walking with DRMEs about different gain sets the height of the head directl. To begin with, we should determine the range of lifting-proportional-gain to realie the biped walking general safet and stabilit on the even ground. And then, we performed some simulations to obtain the suitable range, finding that under the conditions of K p = diag[2,29,8], K p = diag[2,29,12], and (c)k p = diag[2,29,13], humanoid robot can not walking stabl shown in Fig. 8. The red marks on the foots indicate gaits have collision with the floor since the float foot can not leave ground resulting in constraints out of work. So we set five sorts of lifting-proportional-gain, such as K p = diag[2,29,9], K p = diag[2,29,95], (c)k p = diag[2,29,1], (d)k p = diag[2,29,15] and (e)k p = diag[2,29,11], to anale that how the different heights of waist influence the biped walking flexibilit and stabilit according to comparing the values corresponding to different walking gaits. Thus, in this paper,we choose the states that float foot was going to point-contact the surface of floor and become supporting-foot, which was eas to obtain and compare shown in Fig. 9-4, -4, (c)-4, (d)-4, (e)-4. In Fig. 1, the lowest point means the minimum value obtained while the gaits is like -4, -4, (c)-4, (d)-4, (e)-4. Comparing the minimum value, we get a conclusion that as the lifting gain become larger under the stable walking, is obtained with the maximum corresponding to the same gait marked with the black circle in Fig. 1, which means there is an optimal configuration to keep better walking flexibilit. These group of experiments indicated our humanoid robot model was correct and it had Kp = diag[2;29;9] Kp = diag[2;29;1] Kp = diag[2;29;95] (c) (d) (e) Kp = diag[2;29;15] Kp = diag[2;29;11] Fig. 1. until t = 8.[s]. the same kinematic and dnamic performance as humanbeing. Furthermore, In Fig. 11, the trajector is designed to estimate the Dnamical Reconfiguration Manipulabilit of humanoid walking on the uneven ground. Besides, ladder height δh was set as.2[m],.3[m],.4[m] to consider the influence of the change of it on. The gain K p = diag[2,29,1], which enables humanoid robot to walk stabl, were chosen to evaluate the influence of the change of ladder height on Dnamical Reconfiguration Manipulabilit. The value change of was shown from Fig. 12 to Fig?14. The red mark was the value of when humanoid robot walk down the first step. In the case that humanoid robot walked on the uneven ground with lifting-gain K p = diag[2,29,1], if the ladder height increased, the value of when humanoid robot walked down the first step became smaller, which was shown from Fig. 12 to Fig. 14. What we can conclude from above is that if the ladder height become larger, the Dnamical Reconfiguration Manipulabilit of humanoid walking down the stairs will become smaller. In other words, 144

6 éh éh was similar to human-being on the kinematic and dnamic characteristics such as suitable walking posture. On the other hand, we demonstrated that our new humanoid robot model was correct and effective Fig. 11. = 1813:92 The uneven ground Fig. 12. Case: K p = diag[2,29,1], δh =.2[m] = 1754: Fig. 13. Case: K p = diag[2,29,1], δh =.3[m] = 16937: REFERENCES [1] M. Vukobratovic, A. Frank and D. Juricic, On the Stabilit of Biped Locomotion, IEEE Transactions on Biomedical Engineering, Vol. 17, No. 1, 197. [2] Y. Nakamura and K. Yamane, Dnamics of Kinematic Chains with Discontinuous Changes of Constraints Application to Human Figures that Move in Contact with the Environments, Journal of RSJ, Vol. 18, No. 3, pp , 2 (in Japanese). [3] M. Kouchi, M. Mochimaru, H. Iwasawa and S. Mitani, Anthropometric database for Japanese Population , Japanese Industrial Standards Center (AIST, MITI), 2. [4] T. Maeba, M. Minami, A. Yanou, J. Nishiguchi, Dnamical Analses of Humanoid s Walking b Visual Lifting Stabiliation Based in Eventdriven State Transition Cooperative manipulations based on Genetic Algorithms using contact information, 212 IEEE/ASME Int. Conf. on Advanced Intelligent Mechatronics Proc., pp. 7 14, 212. [5] W. Song, M. Minami, F. Yu, Y. Zhang and A. Yanou, 3-D Hand & Ee-Vergence Approaching Visual Servoing with Lapunov-Stable Pose Tracking, Proceedings of IEEE International Conference on Robotics and Automation, pp , 211. [6] F. Yu, W. Song and M. Minami, Visual Servoing with Quick Ee-Vergence to Enhance Trackabilit and Stabilit, Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Sstems, pp , 21. [7] T. Yoshikawa, Dnamic Manipulabilit of Robot Manipulators, Proceedings of the IEEE International Conference on Robotics and Automation, Vol. 2, No. 1, pp , [8] M. Minami, Y. Naitoh and T. Asakura, Avoidance Manipulabilit for Redundant Manipulators, Journal of the Robotics Societ of Japan, Vol. 17, No. 6, pp , 1999 (in Japanese). [9] M. Minami, X. Li, T. Matsuno and A. Yanou, Dnamic Reconfiguration Manipulabilit for Redundant Manipulators, Journal of the Mechanisms Robotics, Vol. 8 / , 216. [1] Y. Kobaashi, M. Minami, A. Yanou and T. Maeba Dnamic Reconfiguration Manipulabilit Analses of Humanoid Bipedal Walking, Proceedings of the IEEE International Conference on Robotics and Automation (ICRA),, pp , 213. Fig. 14. Case: K p = diag[2,29,1], δh =.4[m] humanoid robot can walk more stabl and faster on even ground than on the uneven ground. Thus, this characteristic is also like our human-being. VI. CONCLUSION In this paper, we defined a new indicatrix called Dnamic Reconfiguration Manipulabilit Shape Index() for the optimiation of dnamic flexibilit using redundanc for reconfiguration. And was applied to our simulation experiments of humanoid biped walking. According to changing the waist height and ladder height, we found that walking flexibilit and stabilit were different b the value of, which indicated that our robot model 1441