Description

Graceful motion can be loosely defined as the one we observe in natural movements executed by animals and humans, which are characterized by being agile, efficient, and fluid. The generation of graceful robot motions is typically sought through the minimization of cost functions involving not only path length, but also aspects related to smoothness, like the time derivative of acceleration, called jerk, or that of the controls. A widely used approach to compute optimal trajectories is through direct collocation, a technique that converts the continuous-time optimal control problem into a finite-dimensional NLP problem. Collocation proceeds by discretizing the trajectory using so-called collocation points, and imposing the dynamics constraints at such points. The formulation of most collocation methods, however, assumes that the system is governed by a first order ODE, whereas robotic systems are typically described by second or higher order ODEs. As a result, the usual practice is to initially convert those ODEs into first order form via introducing new variables, and adding new equations that link these variables with their integral counterparts. An often overlooked effect of this transformation is that it generates inconsistencies between the trajectory of each variable and that of its time derivative. This is so because a collocation method only imposes the differential relationships at the collocation points, but not in between such points. A closer examination of this effect reveals that the dynamic equations, which should be satisfied in the collocation points, are actually violated in them, despite apparently having been enforced. This thesis introduces new collocation methods designed to overcome these problems. Specifically, we develop improved versions of the most popular piecewise and pseudospectral collocation schemes, including the trapezoidal and Hermite-Simpson methods, as well as the LG, LGR, and LGL methods. The new algorithms are able to treat differential equations of arbitrary order M ≥ 1 without having to convert them into first order form. In all of them, the trajectory obtained for each variable coincides exactly with the time derivative of its corresponding integral variable, and the dynamic constraints are satisfied accurately at the collocation points. These properties allow a drastic reduction of the dynamics error of the obtained trajectories in many cases, meaning that the governing equations are better respected along the continuous time horizon of the problem. Our methods also provide trajectories that are smoother than those of conventional ones, and easily treat variables such as jerk or the time derivative of the controls in the cost function. An adaptive refinement hp algorithm is also proposed to combine the benefits of our piecewise and pseudospectral methods so as to speed up convergence to the solutions.

The work is under the scope of the following projects:

• KINODYN: Kinodynamic planning of efficient and agile robot motions (web)
• KINODYN+: Synthesis of Optimally Agile and Graceful Robot Motions (web)