Formulation of the Problem:

Case Study: Overstepping

Using optimal control to investigate potential improvements in whole-body walking generation for the Humanoid Robot HRP-2

Henning Koch, Katja Mombaur (ORB-IWR::University of Heidelberg), Philippe Souères (LAAS-CNRS::Gepetto)

OPTIMIZATION IN ROBOTICS AND BIOMECHANICS

O BR Katja.Mombaur@iwr.uni-heidelberg.de

Henning.Koch@iwr.uni-heidelberg.dePhilippe.Soueres@laas.fr

min Joint Torque

max postural stability

min Joint Velocity

max Efficiency

max Forward Velocity

Objectives used for case study

min Joint Torque

max postural stability

min Joint Velocity

max Efficiency

max Forward Velocity

References:

[1] S. Kajita, F. Kanehiro, K. Kaneko, K. Fujiwara, K. Harada, K. Yokoi, and H. Hirukawa. Biped walking pattern generation by using preview control of zero-moment point. In Proceedings of the 2003 IEEE International Conference on Robotics & Automation, 2003.[2] M. Morisawa, S. Kajita, K. Kaneko, K. Harada, F. Kanehiro, K. Fujiwara, and H. Hirukawa. Pattern generation of biped walking constrained on parametric surface. In Proceedings of the 2005 IEEE International Conference on Robotics and Automation, Barcelona, Spain, 2005.[3] T. Takenaka, T. Matsumoto, and T. Yoshiike. Real time motion generation and control for biped robot - 1st report: Walking gait pattern generation. In Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems, 2009.[4] L.Saab, O. Ramos, N.Mansard, P.Souères, and J.-Y. Fourquet. Generic dynamic motion generation with multiple unilateral constraints. In IEEE International Conference on Intelligent Robots and Systems, 2011.[5] T. Buschmann, S. Lohmeier, H. Ulbrich, and F. Pfeiffer. Optimization based gait pattern generation for a biped robot. In Proceedings of 5th IEEE-RAS International Conference on Humanoid Robots, 2005.[6] K. H. Koch, K. Mombaur, and P. Soueres. Optimization-based walking generation for humanoid robot. In SYROCO, Dubrovnic, Kroatia, 2012.[7] K. D. Mombaur, H. G. Bock, J. P. Schlöder, and R. W. Longman. Human-like actuated walking that is asymptotically stable without feedback. In Proceedings of IEEE International Conference on Robotics and Automation, pages 4128 – 4133, Seoul, Korea, May 2001.[8] K. Kaneko, F. Kanehiro, S. Kajita, H. Hirukawaa, T. Kawasaki, M. Hirata, K. Akachi, and T. Isozumi. Humanoid robot HRP-2. In Proceedings of the IEEE International Conference on Robotics & Automation, 2004.

[9] S.Nakaoka, S.Hattori, F.Kanehiro, S.Kajita, and H.Hirukawa. Constraint-based dynamics simulator for humanoid robots with shock absorbing mechnisms. In Proceedings of the 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2007.[10] S.Kajita, M.Morisawa, K.Miura, S.Nakaoka, K.Harada, K.Kaneko, F.Kanehiro, and K.Yokoi. Biped walking stabilization based on linear inverted pendulum tracking. In International Conference on IntelligentRobots and Systems, 2010.[11] B.Verrelst, K.Yokoi, O.Stasse, H.Arisumi, and B.Vanderborght. Mobility of humanoid robots: Stepping over large obstacles dynamically. In Proceedings of the 2006 IEEE International Conference on Mechatronics and Automation, 2006.[12] M. Vukobratovic and J. Stephanenko. On the stability of anthromomorphic systems. Mathematical Biosciences, 15:1–37, 1972.[13] H. Bock and K. Plitt. A multiple shooting algorithm for direct solution of optimal control problems. pages 243–247. Pergamon Press, 1984.[14[ D. B. Leineweber. The theory of MUSCOD in a nutshell. Master’s thesis, IWR University of Heidelberg, 1995.[15] P.-B.Wieber, F.Billet, L.Boissieux, and R.Pissard-Gibollet. The humans toolbox, a homogenous framework for motion capture, analysis and simulation. In International Symposium on the 3D Analysis of Human Movement, 2006.[16] R. Featherstone. Rigid Body Dynamics Algorithms. Springer-Verlag New York, Inc., Secaucus, NJ, USA, 2007.[17] Y.Guan, K.Yokoi, and K.Tanie. Feasibility: Can humanoid robots overcome given obstacles. In Proceedings of the 2005 IEEE International Conference on Robotics and Automation, 2005.

Case Study: Walking

Multiple studies of HRP-2 overstepping large obstacles have been previously made: - Static study [17]: 24.2 cm - Dynamic study [11] in simulation: 25 cm - Dynamic study [11] in reality: 15 cm

Based on optimal control optimisation we could extend the overstepping performance by using the whole-body dynamics: - Obstacle center point: 39.1 cm - Obstacle tip height: 44.6 cm - Obstacle diameter: 11 cm

- Step length: 0.584 m - Step width: 0.157 m ➟ 0.177 m - Step time: 4.762 s

ZMP X over timeZMP Z over time

The robot HRP-2 [8] is equipped with: - position-controlled joints - powered with brushed DC-Motors - the cantilever joint structure in the legs - ankle joint with rubber bushings [9] for shock absorption and ground contact constraint reaction control - stabilizer [10] (against external perturbations), to perform adjustments to the joint angle trajectories, in real-time to maintain dynamic stability [12] - As [11] states the stabiliser mostly compensates the effect of the elasticity in the ankle joint

➟ The elasticity in the ankle joint will not be considered:

Kinematic Structure of the Roboti - 30 degrees of freedom - 6 degrees of freedom - global free-flyer (pelvis)

The Humanoid Robot HRP-2

Motion Generation for HRP-2The research field of motion generation is far spread. Typically one would distinguish between, methods that are mainly based on heuristics (e.g. pattern generators [1,2,3]) and those that are optimisation based (optimal control [4]). Pattern generators are based on highly simplified models (linear inverted pendulum [1], lumped mass models [5]), operate sufficiantly quickly for realtime applications and are easy parametrisable. Optimal Control methods usally depend on more complex models, are therefore computationally much more expensive [6,7]. Even they rarely perform quickly enough to permit real-time applications, these methods give a much more intuitive access to complex motion characteristics.

In the following we propose an optimal control optimisation approach that depends solely on the model dynamics of the system and the governing physical events of the motion at hand.

Dynamic Modeling - DYNAMOD

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Benchmark Forward Dynamics - branched model

LibrariesRBDL

DYNAMOD

DAE-System of dynamic model with contact constraints

Inelastic impact constraint - velocity discontinuity equation

The rigid multi-body model is established in the opensource software package - DYNAMOD. Inspired by the idea of optimising the symbolic model equations and transforming them to C-Code (HuMAnS[15]), DYNAMOD is based on 6D spatial algebra [16] to establish the necessary components of the dynamic model equation including flexibly definable 6D contact constraints.

As the pure symbolic model equations are handled directly any parametrisation of the model is easily applicable. Further more the model logic is accessible to symbolic or automatic differentiation.

Currently the software package is build on top of the commercial algebra package MapleTM.

Direct Multiple Shooting Algorithm

The problem formulation is then solved by the optimal control optimization framework MUSCOD II [13,14]. The framework is based on a direct multiple shooting approach: - System controls discretized on an equidistant time grid - System states parametrized: transform boundary value problem ➟ initial value problems - Integration of the system trajectory and computation of sensitivities by internal numerical differentiation- This results into a large but highly structured NLP that is solved by a specifically taylored SPQ method.

Path Constraints

- Unilateral contact condition - Contact feasibility - Static contact friction condition - Self-Collision avoidance - ZMP-stability constraint [12]