SUMMARY 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.

A computerized method based on optimal dynamic synthesis was developed for generating the flight phase of somersaults. A virtual gymnast is modeled as a planar seven-segment multibody system with six internal degrees of freedom. The aerial movement is generated using a parametric optimization technique. The performance criterion to be minimized is the integral quadratic norm of the torque generators. The method produces realistic movements showing that somersaults perfectly piked or tucked appear spontaneously according to the value of the rotation potential of the initial movement. It provides accurate knowledge of the evolution of joint actuating torques controlling the somersault, and makes it possible to investigate precisely the configurational changes induced by modifications of the rotation potential. Four simulations are presented: one with a reference value for the rotation potential, two with reduced values, and the last with a different hip flexion limit. They give an insight into the coordination strategies which make the movement feasible when the rotation potential is decreased. The method gives accurate assessments of the energetic performance required, together with precise evaluations of the mechanical efforts to be produced for generating the acrobatic movement.

A simplified method for generating three-dimensional bipedal gait is presented. The approach consists in combining a lateral sway motion of the biped with an optimal sagittal movement. A method for computing the latter was presented in an earlier paper. The lateral movement is generated on the base of a simplified kinematic model and using a stochastic search method minimizing a dynamic performance criterion. Sagittal and frontal movements are successfully associated to produce well-balanced 3D-gait.

Dynamic synthesis of human movements raises the question of the selection of a suitable performance criterion able to generate proper dynamic behaviors. Two quite different criteria are likely to be appropriate candidates: the minimum effort cost (or sthenic criterion) and the minimum energy cost. The paper is aimed at clarifying the dynamic effects of these two fundamental criteria when considering movements executed with liveliness as they are in gymnastic. It is well known that the former cost generates movements with smooth dynamics. A special attention is devoted to the latter. The optimal control theory shows that minimizing the energy consumption results in actuating inputs of bang-off-bang type producing momentum impulses. When achieving dynamic synthesis, this criterion makes necessary to account for bounds set on driving torques. Moreover, when dealing with one-sided contacts, as in floor handstands, the unilaterality of contact forces must be explicitly accounted for since it tends to be infringed by impulsive efforts.Numerical simulations of these formal properties are carried out using a parametric optimization technique, and considering the raising phase of floor handstands. It is shown that the energetic criterion tends to generate movements which exhibit similarities with their real counterpart performed by an expert gymnast. Conversely, the sthenic criterion produces movements quite different. But, a salient fact is that these ones proved to be easier to perform by young beginners. As a result, they could help to coach novice gymnasts.

Walking pattern synthesis is carried out using a spline-based parametric optimization technique. Generalized coordinates are approximated by spline functions of class C 3 fitted 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 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.

In this work we address the control problem of biped robots by using the task function approach. A problem arrives when one of the feet is in contact with the ground, which presents imperfections. There is then the possibility that the biped robot undergoes a fall. It is difficult to track any trajectory due to the presence of unevenness on the ground. What we propose is to use the task function approach combined with the application of the total rate of angular momentum to obtain a control law for the ankle. By this technique, the tracking becomes more smooth and the balance is assured. The control law proposed allows the upper part of the robot to be controlled independently since only the ankle actuators are concerned. We enounce the formal problem and present some simulations with real parameters of a 21 degrees of freedom biped robot.

The problem of modelling dynamics of open-loop multibody systems is addressed in the present Part-1 of the paper with optimization of motion in mind. The optimization technique implemented in the complementary Part-2 is the Pontryagin Maximum Principle (PMP) which requires deriving equations of motion in state space form, and makes necessary to carry out higher order differentiation in order to formulate some optimality conditions. So as to fulfil these requirements, an algebraic differentiation technique is developed in the present paper, which results in formulating, in the same global computational scheme, Lagrangian and Hamiltonian equations of motion together with the Jacobian matrix of phase-velocities involved in the conditions for optimality stated by the PMP. All formulations required are formally exact and essentially non-redundant which will ensure safer and faster numerical processing. The structure of the final algorithm was used to develop a computerized symbolic formulation of the entire optimization problem. The file which results can be used directly by the numerical solver. Such a symbolic computation code has proved to be an essential tool to cope with the huge complexity of formulations involved in the statement of the optimization problem dealt with in part 2 of the paper.

A general method for generating optimal motions of actuated multibody systems is presented. A dynamics-based optimization technique is developed using the Pontryagin Maximum Principle. For the reasons stated in Part 1 of the paper, this powerful mathematical tool is associated with a Hamiltonian dynamic model for the mechanical system. In the present Part 2 of the paper, a unified approach is designed for dealing with unconstrained as well as constrained mechanical systems. In the latter case, kinematic loops are cut at appropriate joints or mechanical contacts in order to deal with open tree-like kinematic structures. Pontryagin's Maximum Principle makes it possible to account for related closure constraints together with interaction efforts at cut joints in a quite general and efficient way in stating and solving the dynamic optimization problem. The salient features of the method lies in considering interaction forces at cut joints as additional control variables, and dealing with closure (holonomic) constraints using a penalty method. A new solving strategy was developed for the final problem, making easier the search for guess solutions. Three numerical simulations demonstrate the effectiveness of the method.

In the area of biped robot research, much progress has been made in the past few years. However, some difficulties remain to be dealt with, particularly about the implementation of fast and dynamic walking gaits, in other words anthropomorphic gaits, especially on uneven terrain. In this perspective, both concepts of center of pressure (CoP) and zero moment point (ZMP) are obviously useful. In this paper, the two concepts are strictly defined, the CoP with respect to ground-feet contact forces, the ZMP with respect to gravity plus inertia forces. Then, the coincidence of CoP and ZMP is proven, and related control aspects are examined. Finally, a virtual CoP-ZMP is defined, allowing us to extend the concept when walking on uneven terrain. This paper is a theoretical study. Experimental results are presented in a companion paper, analyzing the evolution of the ground contact forces obtained from a human walker wearing robot feet as shoes.

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.

The anthropomorphic biped robot Bip is equipped with sensors for measuring the ground/feet forces in order to localize the center of pressure (CoP) and zero moment point (ZMP). This paper focuses on experimental results regarding the evolution of the ground contact forces, obtained from a human walker wearing the robot feet as shoes. First, one determines the influence of heavy and rigid metallic shoes on the gait of the subject, during experiments carried out on flat ground. Second, the evolution of the contact forces is studied while walking on parallel planes with different elevations (stairs), then one addresses the case where the feet are supported by two nonparallel planes (uneven terrain). The corresponding analysis is founded on the concepts of virtual supporting surface and pseudo-CoP-ZMP introduced in a companion paper, discussing the theoretical aspects of CoP and ZMP (Sardain and Bessonnet, 2004). Beyond the academic contribution, the data analyzed in this paper could constitute an incitement to design truly anthropomorphic feet, for Bip as well as for other biped robots, because all the current robot feet are functionally rather poor.

In this paper, a method for generating optimal cyclic sagittal gait of a bipedal robot is presented. Impactless movements are considered. Dynamics models of the two main phases of gait are extensively developed. A variety of gait constraints defining a valid walk are taken into account. Transition configurations from one phase to the other are not specified, but they are submitted to a minimum set of constraints defining the gait pattern. Joint trajectories are approximated using cubic splines functions connected at uniformly distributed knots. Joint coordinates at knots, motion times of gait phases, and independent joint coordinates and velocities at phase transitions are the unknowns to be determined. A Sequential Quadratic Programming method is used to solve the algebraic optimization problem that results. Numerical simulations concerning slow and fast optimal gaits are presented and discussed.

A general method for generating optimal motions of over-actuated mechanisms having closed kinematic topology is presented. The approach applies to controlled mechanical systems such as parallel robots, cooperating arms and walking machines. An optimal control problem is formulated and dealt with using Pontryagin's Maximum Principle. A special attention is devoted to dynamic modeling. Any closed loop is cut at an appropriate joint, and a Hamiltonian dynamic model is formulated using a full set of joint coordinates. Then, the main features of the method developed consist in considering Lagrange's multipliers as additional control variables, while closure conditions of open loops are accounted for using a penalty function integrated into the performance criterion to be minimized. The optimization problem is solved using a standard computing code.

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.

Dynamics-based optimisation of sagittal gait of a planar seven-link biped is carried out using a parametric optimisation technique. Biped joint trajectories are approximated by four-order polynomial functions defined on subintervals uniformly distributed along the motion time. These spline functions are linked together at connecting times, or knots, up to their third derivatives in order to eliminate jerks at connecting points. Connecting conditions make it possible to compute polynomial coefficients as functions of joint trajectory values at knots, these values becoming the optimisation variables. Attention is focused on dealing with the double support phase of gait during which the biped is over-actuated.

The work presented is aimed at generating optimal steps of a biped robot in its sagittal plane. The objective is to provide the robot controller, at the local level, with well organized reference gait trajectories it will be able to perform efficiently. The movements generated are the result of the minimization of a performance criterion defined as the integral of the quadratic actuating torques on the overall motion time. Special attention is devoted to the double support phase of gait during which the motion is submitted to severe limiting conditions. Especially, due to the fact that the biped works as a closed kinematic chain, over-actuation must be compatible with double, non-sliding unilateral contacts. A full dynamic model of the robot is formulated using Hamiltonian phase variables in order to apply efficiently Pontryagin's Maximum Principle. The final problem is solved using a standard computing code.