Abstract: A method and apparatus for determining routes vehicles (4, 6), the method comprising: measuring positions of a first vehicle and a second vehicle (6); providing a specification of a region (12) having a fixed position in relation to the first vehicle (4); using the measurements and the region specification, determining a first route for the first vehicle (4) and a second route for the second vehicle (6). Determining the routes comprises: constructing a graph (34) within a joint state space (X) of the vehicles (4, 6); identifying, within the graph (34), a path from a first vertex to a second vertex, the first vertex corresponding to the measured positions of the vehicles (4, 6), and the second vertex corresponding to the second vehicle (6) being at least partially located within the region (12); and, using the identified path, determining the first and second routes.

Abstract: A method and apparatus for determining routes vehicles (4, 6), the method comprising: measuring positions of a first vehicle and a second vehicle (6); providing a specification of a region (12) having a fixed position in relation to the first vehicle (4); using the measurements and the region specification, determining a first route for the first vehicle (4) and a second route for the second vehicle (6). Determining the routes comprises: constructing a graph (34) within a joint state space (X) of the vehicles (4, 6); identifying, within the graph (34), a path from a first vertex to a second vertex, the first vertex corresponding to the measured positions of the vehicles (4, 6), and the second vertex corresponding to the second vehicle (6) being at least partially located within the region (12); and, using the identified path, determining the first and second routes.

Abstract: A method and apparatus for determining a trajectory for a vehicle are disclosed, wherein the method can include identifying a starting position (p0) for the vehicle; identifying a desired terminal position (P) for the vehicle; linearly approximating dynamics of the vehicle; and using the starting position (p0), the desired terminal position (P), and the linear approximation, determining the trajectory for the vehicle. The linear approximation can be constrained by requirements that: an acceleration of the vehicle during the trajectory is less than a threshold value for the acceleration; and the threshold value for the acceleration is dependent on an infinity norm of a velocity of the vehicle. The vehicle may have a curvature limit.

Abstract: A method and apparatus for determining a trajectory for a vehicle can include: identifying a starting position (p0) and desired terminal position (P) for the vehicle; linearly approximating dynamics of the vehicle; and using the starting position (p0), terminal position (P), and linear approximation, determining the trajectory; wherein the linear approximation can be constrained by a requirement (e.g., specification) that the vehicle may not travel in a region; a center point of the region is at a distance from a predetermined point on the vehicle substantially equal to a minimum turn radius of the vehicle, in a direction substantially perpendicular to a velocity of the vehicle; and a distance from the center point of the region to a point on a perimeter of the region is greater than or substantially equal to the minimum turn radius of the vehicle.

Abstract: A method and apparatus are disclosed for determining a trajectory for a vehicle, wherein the method includes: identifying a starting position (p0) for the vehicle; identifying a desired terminal position (P) for the vehicle; linearly approximating dynamics of the vehicle; and using the starting position (p0), the desired terminal position (P), and the linear approximation, determining the trajectory for the vehicle. The linear approximation can be constrained by a requirement (e.g., specification) that a magnitude of a velocity of the vehicle is greater than or equal to a threshold value for the velocity. The vehicle may have a curvature limit.

Abstract: A method and apparatus for determining a trajectory for a vehicle are disclosed, wherein, the method includes: identifying a starting position (p0) and a desired terminal position (P) for the vehicle; linearly approximating dynamics of the vehicle; and using the starting position (p0), desired terminal position (P), and linear approximation, determining the trajectory for the vehicle. The linear approximation can be constrained by requirements (e.g., specifications) that: (i) an acceleration applied to the vehicle at a point on the trajectory is relatively large when the acceleration acts in a direction that is substantially perpendicular to the velocity of the vehicle; and (ii) an acceleration applied to the vehicle at a point on the trajectory is relatively small when the acceleration acts in a direction that is substantially parallel to the velocity of the vehicle. The vehicle may have a curvature limit.

Abstract: A method and apparatus for determining a trajectory for a vehicle can include: identifying a starting position (p0) and desired terminal position (P) for the vehicle; linearly approximating dynamics of the vehicle; and using the starting position (p0), terminal position (P), and linear approximation, determining the trajectory; wherein the linear approximation can be constrained by a requirement (e.g., specification) that the vehicle may not travel in a region; a centre point of the region is at a distance from a predetermined point on the vehicle substantially equal to a minimum turn radius of the vehicle, in a direction substantially perpendicular to a velocity of the vehicle; and a distance from the centre point of the region to a point on a perimeter of the region is greater than or substantially equal to the minimum turn radius of the vehicle.

Abstract: A method and apparatus for determining a trajectory for a vehicle are disclosed, wherein the method can include identifying a starting position (p0) for the vehicle; identifying a desired terminal position (P) for the vehicle; linearly approximating dynamics of the vehicle; and using the starting position (p0), the desired terminal position (P), and the linear approximation, determining the trajectory for the vehicle. The linear approximation can be constrained by requirements that: an acceleration of the vehicle during the trajectory is less than a threshold value for the acceleration; and the threshold value for the acceleration is dependent on an infinity norm of a velocity of the vehicle. The vehicle may have a curvature limit.

Abstract: A method and apparatus are disclosed for determining a trajectory for a vehicle, wherein the method includes: identifying a starting position (p0) for the vehicle; identifying a desired terminal position (P) for the vehicle; linearly approximating dynamics of the vehicle; and using the starting position (p0), the desired terminal position (P), and the linear approximation, determining the trajectory for the vehicle. The linear approximation can be constrained by a requirement (e.g., specification) that a magnitude of a velocity of the vehicle is greater than or equal to a threshold value for the velocity. The vehicle may have a curvature limit.

Abstract: A method and apparatus for determining a trajectory for a vehicle are disclosed, wherein, the method includes: identifying a starting position (p0) and a desired terminal position (P) for the vehicle; linearly approximating dynamics of the vehicle; and using the starting position (p0), desired terminal position (P), and linear approximation, determining the trajectory for the vehicle. The linear approximation can be constrained by requirements (e.g., specifications) that: (i) an acceleration applied to the vehicle at a point on the trajectory is relatively large when the acceleration acts in a direction that is substantially perpendicular to the velocity of the vehicle; and (ii) an acceleration applied to the vehicle at a point on the trajectory is relatively small when the acceleration acts in a direction that is substantially parallel to the velocity of the vehicle. The vehicle may have a curvature limit.