Quadratic-Programming-based Control of Multi-Robot Systems for Cooperative Object Transport

This paper investigates the control problem of steering a group of spherical mobile robots to cooperatively transport a spherical object. By controlling the movements of the robots to exert appropriate contact (pushing) forces, it is desired that the object follows a velocity command. To solve the problem, we first treat the robots’ positions as virtual control inputs of the object, and propose a velocity-tracking controller based on quadratic programming (QP), enabling the robots to cooperatively generate desired contact forces while minimizing the sum of the contact-force magnitudes. Then, we design position-tracking controllers for the robots. By appropriately designing the objective function and the constraints for the QP, it is guaranteed that the QP admits a unique solution and the QP-based velocity-tracking controller is Lipschitz continuous. Finally, we consider the closed-loop system as an interconnection of two subsystems, corresponding to the velocity-tracking error of the object and the position-tracking error of the robots, and employ nonlinear small-gain techniques for stability analysis. The effectiveness of the proposed design is demonstrated through numerical simulations.

💡 Research Summary

This paper addresses the control problem of coordinating a team of spherical mobile robots to cooperatively transport a spherical object via pushing contact forces. The objective is to steer the object’s velocity to track a desired velocity command in real-time. The system models the object’s dynamics as a second-order integrator influenced by the net contact force from the robots, while each robot is modeled as a first-order integrator with velocity as its control input.

The core contribution is a novel hierarchical control design based on Quadratic Programming (QP). The design decouples the problem into two layers. In the first layer, the robots’ positions are treated as virtual control inputs for the object. A QP-based virtual controller is formulated to compute the ideal robot positions. The QP’s objective function has a dual goal: minimizing the error in generating the net contact force required for velocity tracking (which takes the form of a feedback linearizing term -k_v(v_o - v_c) + ˙v_c) and minimizing the sum of squared individual contact force magnitudes (weighted by a small parameter ε). The constraint forces the contact force variables to be non-negative, ensuring pushing-only interaction. The authors carefully design the QP, selecting constant unit vectors l_i that positively span the space and ensure any n of them are linearly independent. This guarantees the QP is always feasible, yields a unique solution, and results in a Lipschitz continuous virtual control law, which is crucial for subsequent stability analysis.

In the second layer, actual tracking controllers for the robots are designed. Using the ideal positions from the QP as references, simple proportional controllers (v_i = -k_p(p_i - p_i*) + v_o) are employed to drive each robot to its assigned position, thereby realizing the contact forces planned by the upper layer.

The main theoretical result (Theorem 1) establishes the practical stability of the overall closed-loop system. Under assumptions of bounded and sufficiently slow-changing velocity commands, and with appropriate initial conditions where robots are positioned around the object according to the vectors l_i, the controller gains k_v and k_p can be chosen to ensure all signals remain bounded and the object’s velocity tracking error ultimately converges to a small neighborhood of zero. The proof leverages a small-gain analysis for interconnected nonlinear systems. The closed-loop system is viewed as an interconnection between two subsystems: the object’s velocity tracking error dynamics and the collective robot position tracking error dynamics. Using Lyapunov functions and their Dini derivatives, input-to-state stability properties for each subsystem are derived. The nonlinear small-gain theorem is then applied to show that if the controller gains are tuned to satisfy a small-gain condition, the interconnected system is stable.

The proposed method provides a systematic and theoretically rigorous framework for cooperative transport that simultaneously handles optimal force distribution (minimizing contact forces) and guarantees closed-loop stability. The effectiveness of the control strategy is validated through numerical simulations, demonstrating successful object transport with velocity tracking while the robots maintain a caging formation around the object.