B-spline based filters for multi-point trajectories planning

Abstract

In this paper, the relation between B-splines and FIR (Finite Impulse Response) filters is demonstrated and exploited to design a digital filter for trajectory planning, combining the very simple structure and computational efficiency of FIR filters with the flexibility of splines. In particular, the trajectory generator consists of two main elements. The former is devoted to the solution of an optimization problem that, given a set of points to be interpolated (or approximated), provides the control points defining the spline. The latter, a cascade of moving average filters, gives the trajectory profile at each sampling time on the basis of such points. The proposed method has been applied to several robotic and industrial applications, and in this paper two case studies are reported as examples: an industrial robot performing a welding operation and a mobile robot moving in an environment with obstacles. With respect to these tasks, the main features of the trajectory generator are shown: the possibility of planning trajectories with high degree of smoothness (continuity of the derivatives), the possibility of easily changing the duration of the trajectory (and therefore the velocity, acceleration, jerk, etc. of the trajectory) maintaining the same geometric path, the possibility of locally modifying the pre-planned path.

Improving the Accuracy of Industrial Robots via Iterative Reference Trajectory Modification

“…is a B-spline basis function of degree d, the control points p i must be computed by imposing interpolation conditions on the given data points q ⋆ i , see [6]. Note that, as shown in Fig.…”

“…A similar idea has L. Biagiotti is with the Department of Engineering "Enzo Ferrari", University of Modena been already proposed in the continuous-time domain in [5], where a two-degrees-of-freedom local control, and a plug-in RC is used to update the reference trajectory. The novelty of this paper lies in the assumption that the reference trajectories are defined by spline functions, which are de-facto the standard tool used in the industrial field for planning complex motions interpolating a set of given via-points [6]. Thanks to the possibility of generating B-spline trajectories by means of dynamic filters [7], in the seminal works [8] and [9] the trajectory planner has been inserted inside a control loop that modifies in real-time the control points of the B-spline curve so that the tracking error at the desired via-points converges to zero.…”

Automatic elliptical trajectory planning algorithm for pick and place operation

“…Hence, in order to obtain the expect trajectory, not only the constraints of the controller, but also the specification of the task are all need to be taken into consideration. One trajectory with trapezoidal velocity profile was proposed in [8].…”

Repetitive Control Meets Continuous Zero Phase Error Tracking Controller for Precise Tracking of B-Spline Trajectories

“…B-spline trajectory generation, ZPETC and repetitive control, that are well established not only in the literature but also in the industrial practice, have merged together for the first time. As a matter of fact, the possibility to modify online the control points that defines a B-spline with an Repetitive Control mechanism in a unique discrete-time scheme has been made possible only by the particular way of generating the B-spline curve that we proposed in the paper [16], that is by means of a chain of dynamic filters. On the other hand, the B-spline generator that, along with the reference trajectory profile, generates online its time derivatives allows the implementation of the ZPETC, which is usually not causal, in its continuous time version (CZPETC).…”