Navigation in agricultural fields imposes various constraints on manoeuvrability, which can be tackled by using four-wheel steering (4WS) vehicles which are capable of switching between multiple steering mechanisms with distinct kinematic properties. For example, parallel positive steering (PPS) with four wheels in parallel to each other can maintain the vehicle’s heading when moving along a curve. Symmetric negative steering (SNS) with two wheels on each side sharing the same steering angle can turn with a small radius. This paper presents a controller capable of selecting and switching between the two aforementioned modes autonomously for better trajectory tracking performance with special heading requirements for agricultural applications. The controller is implemented as a Model Predictive Control (MPC) controller formulated as a mixed-integer quadratic programming (MIQP) problem for the 4WS vehicle. Practical constraints, such as limits on wheel velocities, steering angles and their rate-of-changes are taken into account. A Python implementation confirms the real-time execution capability of the controller and simulation results highlight its effectiveness.

To deal with the problem of optimal path planning in 2D space, this paper introduces a new toolbox named “Navigation with Polytopes” and explains the algorithms behind it. The toolbox allows one to create a polytopic map from a standard grid map, search for an optimal corridor, and plan a safe B-spline reference path used for mobile robot navigation. Specifically, the B-spline path is converted into its equivalent Bézier representation via a novel calculation method in order to reduce the conservativeness of the constrained path planning problem. The conversion can handle the differences between the curve intervals and allows for efficient computation. Furthermore, two different constraint formulations used for enforcing a B-spline path to stay within the sequence of connected polytopes are proposed, one with a guaranteed solution. The toolbox was extensively validated through simulations and experiments.

This paper firstly presents our optimal path planning algorithm within a $2\mathrm{D}$ non-convex, polytopic region defined as a sequence of connected convex polytopes. The path is a B-spline curve but being parametrized with its equivalent Bézier representation. By doing this, the local convexity bound of each curve's interval is significantly tighter. Thus, it allows many more possibilities for constraining the entire curve to remain inside the region by using only linear constraints on the control points of the curve. We further guarantee the existence of the valid path by pointing out an algebraic solution. We integrate the algorithm, together with our previously published results, into the Navigation with polytopes toolbox which can be used as a global path planner, compatible with ROS navigation tools. It provides a framework for constructing a polytope map from a standard occupancy gridmap, searching for an appropriate sequence of connected polytopes and finally, planning a minimal-length path with different options on B-spline or Bézier parametrizations. The validation and comparison with existing methods are done using gridmaps collected under Gazebo simulations and real experiments.

In this paper, we present a new approach to calculate the B-spline-to-Bézler conversion matrix which converts the control points of a uniform B-spline curve into the control points of an equivalent Bézier curve. It takes into account the order of the curve intervals and hence can provide the conversion of the whole curve at once. The algorithm is implemented so that the computation time only increases proportionally with the number of control points until a constant value before getting saturated. Applications include but are not limited to the efficient usage in different optimal path planning algorithms for navigation in 2D non-convex polytopic region as being presented.