Minimum-Time Planar Paths with up to Two Constant Acceleration Inputs and $L_2$ Velocity and Acceleration Constraints

Abstract

Given starting and ending positions and velocities, $L_2$ bounds on the acceleration and velocity, and the restriction to no more than two constant control inputs, this paper provides routines to compute the minimal-time path. Closed form solutions are provided for reaching a position in minimum time with and without a velocity bound, and for stopping at the goal position. A numeric solver is used to reach a goal position and velocity with no more than two constant control inputs. If a cruising phase at the terminal velocity is needed, this requires solving a non-linear equation with a single parameter. Code is provided on GitHub at https://github.com/RoboticSwarmControl/MinTimeL2pathsConstraints.

Figures (8)

• Figure 1: Left: trajectory of a particle starting from $\textbf{p}_0$ with initial velocity $\textbf{v}_0$ and ending at $\textbf{p}_g$ with ending velocity $\textbf{v}_g$ under two constant acceleration inputs $\textbf{u}$ applied at directions $\theta_1$ and $\theta_2$ for durations $t_1$ and $t_2$; the $\bullet$ shapes show the switching points at $t_1$, $t_c$, and $t_2$ along the path, where $t_c$ is the duration the particle cruises at its maximum velocity $v_m$. Right: $L_2$ position, velocity, and acceleration profiles. $x$ in blue, $y$ in orange, $\sqrt{x^2+y^2}$ in purple. Bounds on velocity and acceleration are highlighted in pink $\blacksquare$. The control switch times are shown on the position profile. See video overview at https://youtu.be/2J-p6CDF4FE.
• Figure 2: Two examples of accelerating a particle from a starting position and velocity to a goal position as fast as possible with a bounded input. The locus of reachable positions is a circle whose center moves with $\mathbf{v}_0$. 100 isochrones (gray circles $\bigcirc\!\!\!\!\!\circ$ ) are evenly spaced in squared time: $t= \sqrt{k}$ for $k \in [0,100]$ to show these loci.
• Figure 3: Locus of positions where the particle reaches $v_m = 1$ in light blue $\blacksquare$. Top row shows four different starting velocities (blue). The velocity of the particle due to thrust $a_m=1$ in directions $\theta \in k \pi /16$, $k\in [0,31]$ is shown with pink arrows, all of length $v_m$. These arrows point in every direction. The bottom row shows $\mathbf{v}_0 = [1/2,1/2]$ for four values of $a_m$.
• Figure 4: Solution with nonzero starting and zero ending velocity. The locus of positions where the particle reaches $v_m = 1$ from the initial position are shown with a light blue set for $\mathbf{p}(t_1)$, $\blacksquare$. A similar set constructed starting from the goal position is shown in orange, $\blacksquare$. The solution trajectory is drawn in red. The positions where thrust 1 stops and thrust 2 starts are indicated by black dots. Top row shows four different initial positions with $\mathbf{v}_0 = [-0.5,-0.5]$ or $[-1,0]$ and $a_m = 1/2$. The system never exceeds terminal velocity. Bottom row shows the same positions and velocities, but with $a_m = 1$, so each requires a coasting phase.
• Figure 5: Solution with nonzero starting and ending velocity. The locus of positions where the particle reaches $v_m = 1$ from the initial position are shown with a light blue set for $\mathbf{p}(t_1)$. A similar set constructed starting from the goal position using $-\mathbf{v}_G$ is shown in orange. The solution trajectory is drawn in red. The positions where thrust 1 stops and thrust 2 starts are indicated by black dots. Top row shows four different initial positions with $\mathbf{v}_0 = [-0.5,-0.5]$, $\mathbf{v}_G = [-0.5,0]$ and $a_m = 1/2$ (the right differs to ensure $v_m$ is not reached). The system never reaches terminal velocity. Bottom row shows the same positions and velocities, but with $a_m = 1$. The system reaches terminal velocity $v_m$ in each case.