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.

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 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.by Chow and Jacobson 15 with the idea of generating an optimal gait step. In fact, the authors, considering a leg with two degrees of freedom, focused their study on the development of the necessary conditions for optimality. More recently, such a problem, stated for a planar biped, was solved numerically for optimizing the single support phase 14 of gait. Ultimately, both single-support and double-support phases of a cyclic step were optimized using the PMP. 17 The maximum principle proved to be an efficient means of optimizing the double-support of gait, during which the locomotion system moves as a closed kinematic chain subjected to constrained dynamics. This technique is generally easy to use due to the fact that it results in a two-point boundary value problem. However, global optimization of a walking cycle would result in a multi-point boundary value problem with inner boundary conditions set at unknown transition times between successive phases and steps. This adds significant intricacy to the formulation of the necessary conditions for optimality. Furthermore, solving algorithms need accurate guess solutions which could be very hard to obtain in this case.Parametric optimization techniques offer the possibility of coping with the complexity of the structure-varying aspect of walking cycles. However, most papers dealing with walking optimization using parameterization are limited to generating the swing phase. 18-21 The single-support ends with a heel-strike of the swing foot that initiates the next step. State variables, considered either in the task space or in the joint space, were approximated using polynomials defined along th...