Rolling Biped Polynomial Motion Planning

This work discloses a kinematic control model to describe the geometry of motion of a two-wheeled biped’s limbs. Limb structure is based on a four-bar linkage useful to alleviate damping motion during self-balance. The robot self-balancing kinematics geometry combines with user-customized polynomial vector fields. The vector fields generate safe reference trajectories. Further, the robot is forced to track the reference path by a model-based time-variant recursive controller. The proposed formulation showed effectiveness and reliable performance through numerical simulations.

Invariant Algebraic Surfaces of Polynomial Vector Fields in Dimension Three

We discuss criteria for the nonexistence, existence and computation of invariant algebraic surfaces for three-dimensional complex polynomial vector fields, thus transferring a classical problem of Poincaré from dimension two to dimension three. Such surfaces are zero sets of certain polynomials which we call semi-invariants of the vector fields. The main part of the work deals with finding degree bounds for irreducible semi-invariants of a given polynomial vector field that satisfies certain properties for its stationary points at infinity. As a related topic, we investigate existence criteria and properties for algebraic Jacobi multipliers. Some results are stated and proved for polynomial vector fields in arbitrary dimension and their invariant hypersurfaces. In dimension three we obtain detailed results on possible degree bounds. Moreover by an explicit construction we show for quadratic vector fields that the conditions involving the stationary points at infinity are generic but they do not a priori preclude the existence of invariant algebraic surfaces. In an appendix we prove a result on invariant lines of homogeneous polynomial vector fields.

Polynomial Vector Fields on the Clifford Torus

First, we characterize all the polynomial vector fields in [Formula: see text] which have the Clifford torus as an invariant surface. Then we study the number of invariant meridians and parallels that such polynomial vector fields can have on the Clifford torus as a function of the degree of these vector fields.