eess.SYApr 10, 2026

On Feedback Speed Control for a Planar Tracking

Authors: Xincheng Li, Tengyue Liu, Udit Halder

Organizations: Department of Mechanical and Aerospace Engineering, University of South Florida

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Abstract

This paper investigates a planar tracking problem between a leader and follower agent. We propose a novel feedback speed control law, paired with a constant bearing steering strategy, to maintain an abreast formation between the two agents. We prove that the proposed control yields asymptotic stability of the closed-loop system when the steering of the leader is known. For the case when the leader's steering is unavailable to the follower, we show that the system is still input-to-state stable with respect to the leader's steering viewed as an input. Furthermore, we demonstrate that if the leader's steering is periodic, the follower will asymptotically converge to a periodic orbit with the same period. We validate these results through numerical simulations and experimental implementations on mobile robots. Finally, we demonstrate the scalability of the proposed approach by extending the two-agent control law to an N-agent chain network, illustrating its implications for directional information propagation in biological and engineered flocks.

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Fig. 1: Modeling the motion and formation of two agents using their relative position and angles.
Fig. 2: Formation control of two agents. Agent two (the follower in orange) is tasked with becoming a Bertrand mate moving to the right of agent one (the leader in blue). Agent two employs the controls laws as of Prop. III.1 in (a) and of Prop. III.2 in (b). The resulting trajectories are shown in the panels (i). Panels (ii) show the polar plots of the normalized ρ\rho and αi,  i=1,2\alpha_{i},\;i=1,2 , and either convergence to the equilibrium z∗z^{*} (represented by the solid red and blue dots) in (a) or a periodic orbit in (b), with the green dot in (b)ii. showing the shape coordinate at t=12t=12 [s]. Panels (iii) show the convergence (a) or periodic oscillations (b) of α1\alpha_{1} in the top subplot ( α2\alpha_{2} converges in both cases), and the controls of the follower agent in the bottom subplot. The dashed lines represent the corresponding desired values.
Fig. 3: Formation control of two TurtleBot 3 Burger robots, implementing Prop. III.2 . (a) Trajectories of the leader (blue) and follower (orange) robot from t=0t=0 [s] to t=30t=30 [s]. Increasing opacity of the robots indicates the progression of time, with white arrows denoting the general directions of motion. (b) Top subplot displays the evolution of ρ\rho , α1\alpha_{1} , and α2\alpha_{2} . The equilibrium z∗z^{*} is represented by the dashed lines. Bottom subplot displays follower control inputs u2u_{2} and v2v_{2} (solid lines), compared to their desired values (dashed lines).
Fig. 4: Formation control of NN agents ( N=5N=5 ), connected in a chain network as described in Sec. V . The trajectories of the NN agents at different times are shown in (a), with a line in gray connecting each agent to highlight the whip-like motion taken by the formation of the agents. The speed and steering controls of each agent are plotted in (b)(i)-(ii) (with the same color coding as in (a)).
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