Nominal trunk path feasible region of the step )ϕ L (s) ϕ R (s) [ ] [ ] [ ][ x L (s) x B (s) cosθ B (s) sinθ B (s) = + y L (s) y B (s) sinθ B (s) cosθ

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3B2 SY5/17/-177 217 SICE 1 1 Semi-autonomous Navigation of a Biped Robot Over Unlevel Terrain Based on the Combination of Discreate-Continuous Control Takanobu Yamamoto 1 and

semi-autonomous navigation system for a biped robot proposed by the authors to be applicable to locomotion on unlevel ground.

The additional functions are to automatically find a feasible landing area, adapt the foot attitude to the inclination of the landing location, and design the referential path to

The validity of the proposed system was confirmed on a dynamics simulation, where the automatic selection of the landing location and a sudden change of the desired destination

1 3B2 SY5/17/ SICE 1 1 Semi-autonomous Navigation of a Biped Robot Over Unlevel Terrain Based on the Combination of Discreate-Continuous Control Takanobu Yamamoto 1 and Tomomichi Sugihara 1 1 Department of Adaptive Machine Systems, Graduate School of Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka, Japan This paper enhances a semi-autonomous navigation system for a biped robot proposed by the authors to be applicable to locomotion on unlevel ground. It combines a discrete footstep controller and a continuous COM-foot controller, both of which are robust against perturbations. The additional functions are to automatically find a feasible landing area, adapt the foot attitude to the inclination of the landing location, and design the referential path to step over the convex portion based on the terrain profile. The control system runs at 5ms cycle due to the law computational cost. The validity of the proposed system was confirmed on a dynamics simulation, where the automatic selection of the landing location and a sudden change of the desired destination successfully worked. Key Words : Humanoid robot, Biped locomotion, Semi-autonomous navigation system, Teleoperation 1. (1) (2) (3) (4) (2) (3) (1) (3) ZMP (4) (5) takanobu.yamamoto@ams.eng.osaka-u.ac.jp, zhidao@ieee.org 第 22 回ロボティクスシンポジア (217 年 3 月 15 日 -16 日群馬 ) Morisawa (6) Kita (7) ZMP (8) ( ) (9) ( ) (1) (11) ZMP (12)

2 Nominal trunk path feasible region of the step )ϕ L (s) ϕ R (s) [ ] [ ] [ ][ x L (s) x B (s) cosθ B (s) sinθ B (s) = + y L (s) y B (s) sinθ B (s) cosθ B (s) x LC y LC (3) θ L (s)=θ B (s)+θ LC (4) ] Virtual rail (a) (b) Fig. 1: Allocator of the landing position base on virtual rails (9) (a) trajectory of the canonical stance along the nominal trunk path(virtual rail),(b) feasible region of the step 5ms 2. (8) ZMP 2 1 (9) ϕ ϕ = {x,y,θ} (1) (x,y) θ ϕ B = {x B,y B,θ B } ϕ L = {x L,y L,θ L } ϕ R = {x R,y R,θ R } d ϕ B ϕl ϕ R ϕ B d ϕ B ϕ B (s)=s d ϕ B +(1 s)ϕ B ( s 1) (2) s 1(1a) ( [ ] [ ] [ ][ x R (s) x B (s) cosθ B (s) sinθ B (s) = + y R (s) y B (s) sinθ B (s) cosθ B (s) x RC y RC ] (5) θ R (s)=θ B (s)+θ RC (6) ϕ LC = {x LC,y LC,θ LC } ϕ RC = {x RC,y RC,θ RC } ϕ LC = {,w,} ϕ RC = {, w,} 2w 1(1b) Perrin (13) d ϕ B 2 2 -ZMP (14) ẍ = ζ 2 (x x Z ) (7) ÿ = ζ 2 (y y Z ) (8) g ζ (9) z z Z (x,y,z) (x Z,y Z,z Z ) ZMP g z Z x Z y Z x y (7),(8) (7) ZMP ZMP (11)

3 Normal vector of step foot leader(vertual foot) Rotation axis Referential path Rotation angle Max height Normal vector of landing surface follower (actual foot) Step foot Fig. 2: Adaptation to the terrain profile ZMP 1 t+t x Z = argmin (x Z x P ) 2 dt x Z 2 t subject to x(t + T )+ẋ(t + T ) = d x S (1) ζ T d x S x P ( t t +T ) x Z = x P 2eζ T 1 e 2ζ T (eζ T x CP d x SP ) (11) x CP x + ẋ ζ x P (12) d x SP d x S x P (13) 3. 3 (8) Shimizu (15) 2 a ψ Fig. 3: Referential path to the landing position abug which the foot is tracted by a virtual leader a = ν F ν T ν F ν T ψ = atan2 ( ν F ν T, ν T ) F ν T (14) (15) ν F ν T 3 2 (12), (leader) (follower) 3 s ( s 1) p A (s) p A (s A ) s A = u A dt Aa S d T Ad A (16) d A δ p A δs (s A) (17) u A a S =(a Sx,a Sy,a Sz ) T u A 1 t+t u A = argmin u 2 A dt u A 2 t subject to s A (t + T )=1., ṡ A (t + T )= (18)

Foot landing position determiner COM controler ZMP controler Inverse kinematcs solver Robot Path designer Referential path following controller Fig.

9s - : d ϕ B = {1,,} T = αt a S = 6 ( pa (s A ) p S T T 2ṗp ) S (2) 3 α p S =(x S,y S,z S ) 4. 4 (11) ZMP p p A (s) (2) p S p P d r S ( ) (16) 1 1ms 5ms 5. mighty (17) w =.6m T = 1.4s,α = 2/3.

4 Foot landing position determiner COM controler ZMP controler Inverse kinematcs solver Robot Path designer Referential path following controller Fig. 4: Block diagram a S = u A = 6 T ( 1 sa T 2ṡ ) A 3 (19) Fig. 5: Trajectory of COM, ZMP and the feet in xy-plane,.s s : d ϕ B = {.5,,3}, 11.9s - : d ϕ B = {1,,} T = αt a S = 6 ( pa (s A ) p S T T 2ṗp ) S (2) 3 α p S =(x S,y S,z S ) 4. 4 (11) ZMP p p A (s) (2) p S p P d r S ( ) (16) 1 1ms 5ms 5. mighty (17) w =.6m T = 1.4s,α = 2/3.3m 2,3 x 5 2,3 x 5 ϕ B = {,,} 11.9s d ϕ B = {.5,,3} 11.9s Fig. 6: Trajectory of COM, ZMP and the feet in xz-plane,.s s : d ϕ B = {.5,,3}, 11.9s - : d ϕ B = {1,,} x-position[m] Support center.3.2 COM ZMP.1 Right foot Left foot Time[s] Fig. 7: Trajectory of COM, ZMP and the feet in x-axis,.s s : d ϕ B = {.5,,3}, 11.9s - : d ϕ B = {1,,} d ϕ B = {1,,} x-y ZMP 5 x-z ZMP 6 ZMP x 7 ZMP {.4,,} 5.s(4.5s ) {,,} x-y ZMP 8 x-z ZMP

.3.2.1 -.1 -.2 -.3 -.1.1.2.3.4.5 (a). -5.s : d ϕ B = {.4,,} (b) 5.

8: Trajectory of COM, ZMP and the feet in xy-plane with a sudden chase of the

target.3.2.1 -.1 -.2 -.3 -.1.1.2.3.4.5.6.57.54.51.48.45.42.39.36.33.3.27.24.21.18.

1: Trajectory of COM, ZMP and the feet in xyplane with a sudden chase of the

5 (a). -5.s : d ϕ B = {.4,,} (b) 5.s - : d ϕ B = {,,} Fig. 8: Trajectory of COM, ZMP and the feet in xy-plane with a sudden chase of the target (a).s - 5.s : d ϕ B = {.4,,} (b) 5.s - : d ϕ B = {,,} Fig. 9: Trajectory of COM, ZMP and the feet in xz-plane with a sudden chase of the target Fig. 1: Trajectory of COM, ZMP and the feet in xyplane with a sudden chase of the target,.s-5.2s : d ϕ B = {.4,,}, 5.2s - : d ϕ B = {,,} Fig. 11: Trajectory of COM, ZMP and the feet in xzplane with a sudden chase of the target,.s-5.2s : d ϕ B = {.4,,},5.2s - : d ϕ B = {,,}

6 5.2s(4.3s ) {,,} x-y ZMP 1 x-z ZMP (B)( :#15H2762) ( :# ) (1) H. Arai. Motion Planning of Discretetime Nonholonomic Systems with DifferenceEquation Constraints. The Journal of Robotics Society of Japan, Vol. 18, No. 6, pp , 2. (2) J. Kuffner, S. Kagami, K. Nishiwaki, M. Inaba, and H. Inoue. Online Footstep Planning for Humanoid Robots. In Proceedings of the 23 IEEE International Conference on Robotics and Automation, pp , 23. (3) O. Kanoun, E. Yoshida, and J.-P. Laumond. An optimization formulation for footsteps planning. In Proceedings of 29 IEEE-RAS International Conference on Humanoid Robots, pp , 29. (4) S. Kajita et al. Biped Walking Pattern Generation by using Preview Control of Zero-Moment Point. In Proceedings of the 23 IEEE International Conference on Robotics & Automation, pp , 23. (5) T. Sugihara and Y. Nakamura. Boundary Condition Relaxation Method for Stepwise Pedipulation Planning of Biped Robots. IEEE Transaction on Robotics, Vol. 25, No. 3, pp , 29. (6) M. Morisawa, N. Kita, S. Nakaoka, K. Kaneko, S. Kajita, and F. Kanehiro. biped locomotion control for uneven terrain with narrow support region. In System Integration (SII), 214 IEEE/SICE International Symposium on, pp , Dec 214. (7) N. Kita, M. Morisawa, and F. Kanehiro. Foot landing state estimation from point cloud at a landing place. In th IEEE-RAS International Conference on Humanoid Robots (Humanoids), pp , Oct 213. (8) T. Yamamoto and T. Sugihara. Robust navigation system of a biped robot on the combination of responsive discretecontinuous control. In the Proceedings of 17th Annual Conference of The Society of Instrument and Control Engineers, System Integration Division, 216. (9) H. Kobayashi and T. Sugihara. Self-consistent Automatic Navigation of COM and Feet for Realtime Humanoid Robot Steering. In Proceedings of 29 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp , 29. (1) T. Sugihara. Foot-guided Control of a Biped Robot That Guarantees Standing Stability at Landing. In Proceedings of 216 Annual Conference of Japan Society of Mechanical Engineers, Robotics and Mechatronics Division, 2A1-12b5, 216. (11) T. Sugihara. Standing Stabilizability and Stepping Maneuver in Planar Bipedalism based on the Best COM- ZMP Regulator. In Proceedings of the 29 IEEE International Conference on Robotics & Automation, pp , 29. (12) T. Sugihara. Robust path following controller with a virtual leader-follower model. In Proceedings of 33th Annual Conference of The Robotics Society of Japan, 215. (13) N. Perrin et al. On the Problem of Online Footsteps Correction for Humanoid Robots. In Proceedings of 27th Annual Conference of The Robotics Society of Japan,3O1-4, 29. (14) K. Mitobe, G. Capi, and Y. Nasu. Control of walking robots based on manipulation of the zero moment point. Robotica, Vol. 18, pp , 2. (15) Y. Shimizu and T. Sugihara. Efficient Path Planning of Humanoid Robots with Automatic Conformation of Body Representation to the Complexity of Environments. In Proceedings of 212 IEEE-RAS International Conference on Humanoid Robots, pp , 212. (16) T. Sugihara. Solvability-Unconcerned Inverse Kinematics by the Levenberg-Marquardt Method. IEEE Transaction on Robotics, Vol. 27, No. 5, pp , 211. (17) T. Sugihara, K. Yamamoto, and Y. Nakamura. Hardware design of high performance miniature anthropomorphic robots. Robotics and Autonomous System, Vol. 56, pp ,

5D1 SY0004/14/ SICE 1, 2 Dynamically Consistent Motion Design of Humanoid Robots even at the Limit of Kinematics Kenya TANAKA 1 and Tomo

5D1 SY0004/14/ SICE 1, 2 Dynamically Consistent Motion Design of Humanoid Robots even at the Limit of Kinematics Kenya TANAKA 1 and Tomo 5D1 SY4/14/-485 214 SICE 1, 2 Dynamically Consistent Motion Design of Humanoid Robots even at the Limit of Kinematics Kenya TANAKA 1 and Tomomichi SUGIHARA 2 1 School of Engineering, Osaka University 2-1