Feedback stabilizer-based trajectory planning of mobile robots with kinematic constraints

Introduction

Contribution

They propose a generic approach which first uses an (arbitrary) feedback stabilizer to generate the ‘path’ and then rebuilt the corresponding ‘trajectory’ along this ‘path’ to meet various kinematic constraints. A general framework is established to transform feedback stabilizers into a feasible and highly efficient trajectory planner by using path generation and optimal velocity planning techniques, considering both kinematic and differential constraints.

文章中反复提到的 “feedback stabilizer” 是什么意思。
貌似就是把用于posture stabilization的feedback control叫做feedback stabilizer.

The main contribution of the paper is :

1. A bridge is built between the feedback stabilization control and highly efficient local trajectory planning, which extends the function of existing stabilizers to act as new types of trajectory planners.
2. The established framework is stabilizer-free, which means that the stabilizer can be chosen arbitrarily from existing methods.

针对的问题

Few stabilization controllers are ultimately applied to the practice or integrated on the commercial mobile robots. In contrast, it is motion planning and trajectory tracking control are more focused. So, what are the key obstacles in the way of existing stabilization controllers to real applications?

• One reason is that the velocity and acceleration limits cannot be guaranteed to be satisfied theoretically for most existing stabilization controllers.
• Another reason is that stabilization controllers are not appropriate to be utilized in the scenario with large pose errors.

Essential Idea

One of the essential idea in this paper is to regard the feedback stabilization system as a “path generator” rather than a “trajectory generator”.

Another idea is to convert the implicit discrete path into explicit analytical expression.

General Framework

There are three basic procedures in the general framework:

1. Feedback stabilizer-based path generation by numerical solvers. The kinematic/dynamic model of mobile robots, together with the stabilizers, constitutes a closed-loop system represented by differential equations, which can be solved using numerical methods such as the Euler method or Runge-Kutta methods. Subsequently, appropriate discrete trajectory points in the configuration space are carefully selected to be utilized in the next procedure to generate explicit paths.
2. Explicit analytical path derivation. we discard the time label of the discrete trajectory points to yield a series of discrete path points. To deal with the possible cusp points in the trajectory, the discrete path is segmented into several path segments, and the derivation of the analytical form of every path segment becomes a constrained quadratic programming problem considering the boundary conditions.
3. Optimal velocity allocation along the analytical path, wherein the linear and angular velocity/acceleration limits are guaranteed to be satisfied. Therefore, a practically feasible and efficient trajectory is obtained. Optimal velocity allocation along the analytical path, wherein the linear and angular velocity/acceleration limits are guaranteed to be satisfied. Therefore, a practically feasible and efficient trajectory is obtained.

Problem Statement

consider a unicycle robot kinematics model:
$$\dot{x}=vcos(\theta )$$ $$\dot{y}=vsin(\theta )$$ $$\dot{\theta }=w$$
the control input is: $\mathbf{u}=[v,w{]}^T$

Feedback Stabilization-based Path Generation

choose polar coordinate-based stabilizer

The output of this module is:

• a sequence of selected path points and the corresponding path length
• the number of the cusp points, their index among the selected points; the corresponding robot orientation at these cusp points are recorded

Path Segmentation

In this section, we will introduce a path segmentation mechanism to segment the discrete path into several smooth segments.

Explicit Path Derivation

For every path segment $L_k:(x_s(i),y_s(i),s_s(i))$ $(i=N_k,N_k+1,...,N_{k+1})$, we can approximate it with polynomials to yield an analytical expression.

Optimal Velocity Planning Along the Pre-computed Explicit Path

Along with the explicit analytical paths, many velocity scheduling algorithms, usually termed as “path-constrained trajectory planning” have been proposed.

Simulation Results