Generation of dynamic motions under continuous constraints: Efficient computation using B-Splines and Taylor polynomials

Abstract

This paper proposes a new computation method to solve semi-infinite optimization problems for motion planning of robotic systems. Usually, this problem is solved by means of time-grid discretization of the continuous constraints. Unfortunately, discretization may lead to unsafe motions since there is no guarantee of constraint satisfaction between time samples. First, we show that constraints such as joint position and velocity do not need time-discretization to be checked. Then, we present the computation method based on Taylor polynomials to evaluate more complex constraints over timeintervals. This method also applies to continuous equality constraints, to continuous maximum derivative constraint, and to compute the cost function.

Emulating human leg impairments and disabilities on humanoid robots walking

“…The Taylor expansions also apply to the states variables of the robots (joint and Cartesian position, velocity and acceleration, contact forces and joint torques) in various algebraic operations [1], and we use additional coefficients in order to compute the contact forces [2]. Note that the coefficients of the time-polynomials (approximating the constraints and the cost function) are function of the optimization variables (the control points of the BSpline).…”

“…In this paper, we use our on-going work on optimal dynamic multi-contact motions generation [1], [2] to investigate whether adding simple constraints that translate -to some extent-a given human leg impairment, results in emulating a similar walking behavior by a humanoid robot, namely the HRP-2.…”

Generation of dynamic multi-contact motions: 2D case studies

“…As discussed in [15], it is not easy to set a time-grid discretization which guarantees in any circumstances that the constraints holds in-between a pair of sample time. Recently, we presented in [4], the time-interval discretization based on Taylor polynomial approximation of the constraints, that make possible to take into account:…”

“…In Table I, N is the number of parameters, N ctr the number of constraints and iter the number of iterations of the optimization process. We try to find a solution for several values N eq : the order of the continuous equality constraint as defined in [4] (for a function f (t) = ∑ a i t i , it ensures that ∀i ∈ {1, · · · , N eq } a i = 0).…”

“…It is based on our recent work [4] which applies semi-infinite optimization to dynamic motions with one contact such as a kicking motion. We extend it to generate smooth, dynamically stable and full-body multicontact motions and transitions, from a given sequence of contact points as input.…”