Guy Bessonnet

An anthropomorphic biped robot: dynamic concepts and technological design

The authors of this study are a part of a joint project, involving four French laboratories, whose goal is the design and construction of a mechanical biped robot with anthropomorphic characteristics. In the first section of this paper, we will examine mechanical architectures of some representatives of state-of-the art biped robots by focusing on their kinematic arrangement. It is widely known that the existence of natural gaits is closely linked to the intrinsic dynamic characteristics of the mechanical structure of the biped robot. In order to further develop this idea, two studies will be presented in the second section: the first is relative to the lateral instability of the system while the second deals with the existence of passive pendular gaits during the swing phase of walking in the sagittal plane. In the last section, in correlation with the observations made, we will gain insight into main characteristics of the mechanical architecture that we have designed for the BIP project: 15 active degrees of freedom (DOF), joints actuated by special transmission system, anthropometric mass distribution.

A Parametric Optimization Approach to Walking Pattern Synthesis

Walking pattern synthesis is carried out using a spline-based parametric optimization technique. Generalized coordinates are approximated by spline functions of class C3fitted at knots uniformly distributed along the motion time. This high-order differentiability eliminates jerky variations of actuating torques. Through connecting conditions, spline polynomial coefficients are determined as a linear function of the joint coordinates at knots. These values are then dealt with as optimization parameters. An optimal control problem is formulated on the basis of a performance criterion to be minimized, representing an integral quadratic amount of driving torques. Using the above spline approximations, this primary problem is recast into a constrained non-linear optimization problem of mathematical programming, which is solved using a computing code implementing an SQP algorithm. As numerical simulations, complete gait cycles are generated for a seven-link planar biped. The only kinematic data to be accounted for are the walking speeds. Optimization of both phases of gait is carried out globally; it includes the optimization of transition configurations of the biped between successive phases of the gait cycle.

Optimal Gait Synthesis of a Seven-Link Planar Biped:

In this paper, we carry out the dynamics-based optimization of sagittal gait cycles of a planar seven-link biped using the Pontryagin maximum principle. Special attention is devoted to the double-support phase of the gait, during which the movement is subjected to severe limiting conditions. In particular, due to the fact that the biped moves as a closed kinematic chain, overactuation must be compatible with double, non-sliding unilateral contacts with the supporting ground. The closed chain is considered as open at front foot level. A full set of joint coordinates is introduced to formulate a complete Hamiltonian dynamic model of the biped. Contact forces at the front foot are considered as additional control variables of the stated optimal control problem. This is restated as a state-unconstrained optimization problem which is finally recast, using the Pontryagin maximum principle, as a two-point boundary value problem. This final problem is solved using a standard computing code. A gait sequence, comprising starting, cyclic, and stopping steps, is generated in the form of a numerical simulation.

Generating globally optimised sagittal gait cycles of a biped robot

The paper is aimed at generating optimal gait cycles in the sagittal plane of a biped, the locomotion system of which has anthropomorphic characteristics. Both single and double support phases are globally optimised, considering incompletely specified transition postural configurations from one phase to the other. An impactless heel-touch is prescribed. Full dynamic models are developed for both gait phases. They are completed by specific constraints attached to the unilaterality of contact with the supporting ground.A parametric optimisation method is implemented. The biped joint coordinates are approximated by cubic splines functions connected at uniformly distributed knots along the motion time. The finite set of unknowns consists of the joint coordinate values at knots, some gait pattern parameters at phase transitions, and the motion time of each phase. The step length is adjusted to the prescribed gait speed by the optimisation process. Numerical simulations concerning slow and fast optimal gaits are presented and discussed.

Impactless sagittal gait of a biped robot during the single support phase

The problem of generating optimal sagittal reference gaits in bipedal walking is addressed. In our study the single-support phase during which the biped reaches its highest instability is considered. The approach developed allows for a fully dynamic model of the biped, and is based on minimizing the integral of quadratic joint actuating torques. Impactless and non-sliding heel-touch is accounted for, ensuring a more stable and easier controlled walking. Optimal motion synthesis is achieved by applying the Pontryagin maximum principle. Two numerical simulations are presented. Computed optimal motions reveal anthropomorphic gait characteristics.

Gait analysis of a human walker wearing robot feet as shoes

This work takes part in the project of development of an anthropomorphic biped robot, Bip. The technological construction has been completed and the robot has performed some slow static walks. The next stage is the implementation of fast and dynamic walking gaits, in other words anthropomorphic gaits. The robot feet have been designed with a three-axes force-moment sensor for reconstructing the center of pressure-zero moment point. To study and test these feet, the idea is simple: the experimental setup consists of a human walker wearing the Bip feet as shoes, a force-plate and a motion analysis system. The purpose is double. The first one is aimed at comparing the contact pressure forces reconstructed with the Bip sensor feet, with the data recorded by the force-plate. The second one consists in determining the influence of (relatively) heavy and rigid metallic shoes on the gait of a human walker. If these feet were proving to be a handicap for efficient and elegant human walking gaits, how could it be any different for the robot Bip? Fortunately the analysis shows that they do not really handicap the walker.

Parametric-based dynamic synthesis of 3d-gait

This paper describes a dynamic synthesis method for generating optimal walking patterns of biped robots having a human-like locomotion system. The generating principle of gait is based on the minimisation of driving torques. A parametric optimisation technique is used to solve the underlying optimal control problem. Special attention is devoted to foot-ground interactions in order to ensure a steady dynamic balance of the biped. Transition states between step sub-phases are fully optimised together with step length and sub-phase lengths with respect to a given walking velocity. The data needed to generate purely cyclic steps can be reduced to the forward velocity.

Optimal Motion Synthesis – Dynamic Modelling and Numerical Solving Aspects

A general approach for generating optimal movements of actuatedmulti-jointed systems is presented. The method is based on theimplementation of the Pontryagin Maximum Principle (PMP) used as amathematical optimization tool. It applies to mechanical systems withkinematic tree-like topology such as serial robots, walking machines,and articulated biosystems. Emphasis is put on the choice of anappropriate dynamic model of the multibody system, together with thechoice of relevant performance criteria to be minimized for generatingthe optimal motion. It is shown that the Hamiltonian formalism isperfectly suitable to deal with the optimization problem using the PMP.On the other hand, prominence is given to performance criteria ensuringsoft and efficient functioning of the articulated systems. Two computingtechniques for solving the optimization problem are presented. Threenumerical simulations demonstrate the applicability of the method.

Optimal dynamics of constrained multibody systems. Application to bipedal walking synthesis

Optimal motion synthesis of closed-loop multibody systems is carried out using a penalty technique designed in order to avoid dealing with dynamic models described by differential algebraic equations. This approach allows for both implementing Pontryagins maximum principle as an optimization tool, and using the same state equation to describe the dynamics of structure-varying kinematic chains. The method presented is especially suitable to deal with unilateral mechanical constraints. We take advantage of this possibility to generate an optimal complete step of a planar biped.

GENERATING OPTIMAL WALKING CYCLES USING SPLINE-BASED STATE-PARAMETERIZATION

Optimal gait cycles are generated for a seven-link biped using a parametric optimization method. A sagittal walking pattern, including a double-support phase divided into two sub-phases, is considered. Generalized joint coordinates are approximated by three-time differentiable spline-functions. These are the concatenation of 4-order polynomials linked together up to their third derivatives at connecting points — or knots — distributed along the motion time of each phase. Optimization parameters are the values of joint coordinates at the knots, plus the joint velocities, and possibly the joint accelerations, at transitions between successive phases. An integral amount of driving torques is minimized throughout the walking cycle. During the double support, constraint forces in the kinematically closed locomotion system are dealt with as additional actuating forces. For this reason, these are also minimized. Using the above optimization parameters, this basic optimal control problem is transformed into an optimization problem of mathematical programming. The latter is efficiently solved using a Sequential Quadratic Programming algorithm. The only kinematic data required for generating a gait cycle is the walking speed. Postural configurations between successive phases, step length, and relative length of single and double supports are optimized with respect to a given walking speed.