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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This paper presents a unified control framework for robust trajectory tracking and moving obstacle avoidance applicable to a broad class of mobile robots. By formulating a generalized kinematic transformation, we convert diverse vehicle dynamics into a strict feedback form, facilitating the design of a Sliding Mode Control (SMC) strategy for precise and robust reference tracking. To ensure operational safety in dynamic environments, the tracking controller is integrated with a Collision Cone Control Barrier Function (C3BF) based safety filter. The proposed architecture guarantees asymptotic tracking in the presence of external disturbances while strictly enforcing collision avoidance constraints. The novelty of this work lies in designing a sliding mode controller for ground robots like the Ackermann drive, which has not been done before. The efficacy and versatility of the approach are validated through numerical simulations and extensive real-world experiments on three distinct platforms: an Ackermann-steered vehicle, a differential drive robot, and a quadrotor drone. Video of the experiments are available at https://youtu.be/dWcxwum96vk

Abstract

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