Minimum Time Trajectories for a Car-Like Mobile Robot Moving with Rigid Wheels Under Non-Sliding Constraints

Joseph Ben-Asher; Elon Rimon; Leeor Ravina

cs.ROSep 21, 2026

Minimum Time Trajectories for a Car-Like Mobile Robot Moving with Rigid Wheels Under Non-Sliding Constraints

Authors: Joseph Ben-Asher, Elon Rimon, Leeor Ravina

Organizations: Dept. of AE, Technion, Israel. · Dept. of ME, Technion, Israel.

Abstract

This paper studies the minimum time trajectoriesvof a car-like mobile robot navigating in an obstacle free environment. The robot, with forward and backward speeds, is controlled by bounded front-wheels acceleration and limited front-wheels steering rate. The paper extends previous results which solved this problem for the kinematic car-like robot. However, the kinematic model assumes pure rolling at the wheels ground contacts. This assumption requires non-sliding constraints for the front and rear wheels that can only be handled by the robot dynamics. This paper formulates the non-sliding constraints based on the robot dynamics then augments the kinematic model time-optimal path primitives with three new path primitives associated with the non-sliding constraints. The three non-sliding path primitives together with the kinematic model twelve path primitives form the car-like robot time optimal trajectories. Approximate analytic solutions for the non-sliding path primitives are also provided. Examples study the time-optimal path primitives along representative maneuvers, illustrating how the non-sliding constraints influence the time optimal trajectories of the car-like robot.

Figures & tables

Figure 1: The car-like mobile robot state variables are $(\mbox{\scriptsize$X$},\mbox{\scriptsize$Y$},\mbox{\footnotesize$\theta$})$ , front wheels steering angle $\phi$ and front wheels midpoint speed $\nu$ . — Figure 1: The car-like mobile robot state variables are $(\mbox{\scriptsize$ X $},\mbox{\scriptsize$ Y $},\mbox{\footnotesize$ \theta $})$ , front wheels steering angle $\phi$ and front wheels midpoint speed $\nu$ .

Figure 2: The simpler bicycle-like mobile robot has the same state variables: $(\mbox{\scriptsize$X$},\mbox{\scriptsize$Y$},\mbox{\footnotesize$\theta$})$ , front wheel steering angle $\phi$ and front wheel speed $\nu$ . — Figure 2: The simpler bicycle-like mobile robot has the same state variables: $(\mbox{\scriptsize$ X $},\mbox{\scriptsize$ Y $},\mbox{\footnotesize$ \theta $})$ , front wheel steering angle $\phi$ and front wheel speed $\nu$ .

Figure 3: Side view of the bicycle-like mobile robot showing the normal loads $N_{1}$ and $N_{2}$ at the ground contacts.

	Front Wheels Linear	Steering Front Wheels
	Acceleration $u_{1}(t)$	Rotation Rate $u_{2}(t)$
${\cal L}^{+}$	${a_{max}}$	$b_{max}$
${{\cal L}^{-}}$	$-{a_{max}}$	$b_{max}$
${{\cal L}^{0}}$	$0$	${b_{max}}$
${{\cal R}^{+}}$	${a_{max}}$	$-b_{max}$
${{\cal R}^{-}}$	$-{a_{max}}$	$-b_{max}$

Table 1: The twelve time optimal path primitives of the car-like robot moving under inactive non-sliding constraints.

	Front Wheels	Rear Wheels
	Ground Force	Ground Force
${\cal FW}$	$\left\\|F_{1}\right\\|^{2}=\mu^{2}(N_{1})^{2}$	—
${\cal RW}$	—	$\left\\|F_{2}\right\\|^{2}=\mu^{2}(N_{2})^{2}$
${\cal BW}$	$\left\\|F_{1}\right\\|^{2}=\mu^{2}(N_{1})^{2}$	$\left\\|F_{2}\right\\|^{2}=\mu^{2}(N_{2})^{2}$

Table 2: The time optimal path primitives of the car-like robot moving under active non-sliding constraints. Contact forces and normal loads are state dependent and linear in

u_{1}

and

u_{2}

	$\mathcal{FW}$	$\mathcal{RW}$
$u_{1}=-a_{max}$	$(-a_{max},u^{-}_{2,I}(t))$ $(-a_{max},u^{-}_{2,II}(t))$	$(-a_{max},u^{-}_{2,I}(t))$ $(-a_{max},u^{-}_{2,II}(t))$
$-a_{max}$ left of $C_{1}$ or $C_{2}$	$(u^{-}_{1}(t),u^{-}_{2}(t))$	$(u^{-}_{1}(t),u^{-}_{2}(t))$
$u_{1}=+a_{max}$	$(+a_{max},u^{+}_{2,I}(t))$ $(+a_{max},u^{+}_{2,II}(t))$	$(+a_{max},u^{+}_{2,I}(t))$ $(+a_{max},u^{+}_{2,II}(t))$
$a_{max}$ right of $C_{1}$ or $C_{2}$	$(u^{+}_{1}(t),u^{+}_{2}(t))$	$(u^{+}_{1}(t),u^{+}_{2}(t))$
$u_{2}=-b_{max}$	$(u^{-}_{1,I}(t),-b_{max})$ $(u^{-}_{1,II}(t),-b_{max})$	$(u^{-}_{1,I}(t),-b_{max})$ $(u^{-}_{1,II}(t),-b_{max})$
$-b_{max}$ below $C_{1}$ or $C_{2}$	$(u^{-}_{1}(t),u^{-}_{2}(t))$	$(u^{-}_{1}(t),u^{-}_{2}(t))$

Table 3: Approximate solutions for

\mathcal{FW}

and

\mathcal{RW}

path primitives valid when

|u^{\pm}_{1,I}(t)|,|u^{\pm}_{1,II}(t)|\leq a_{max}

and

|u^{\pm}_{2,I}(t)|,|u^{\pm}_{2,II}(t)|\leq b_{max}

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