Velocity-Tracking Controller via Quadratic Programming

• The paper demonstrates a QP-based velocity-tracking controller that minimizes deviation from reference commands while ensuring safety through convex optimization.
• It employs a cascade structure where an outer-loop QP computes ideal velocities and an inner-loop feedback law achieves rapid, exponential convergence.
• Empirical studies reveal significant improvements in obstacle avoidance, cooperative manipulation, and trajectory tracking in safety-critical robotics.

A velocity-tracking controller based on quadratic programming (QP) is a control paradigm that utilizes real-time optimization, specifically QP, to generate velocity commands that closely follow a user-supplied reference while adhering to safety constraints, such as obstacle avoidance, velocity limits, and actuator bounds. This framework has become pivotal in safety-critical robotics, cooperative multi-agent manipulation, and path-tracking scenarios, leveraging QP's convexity, real-time feasibility, and amenability to rigorous Lyapunov-based stability analysis. Key contemporary variants include cascaded QP-based controllers for nonlinear Euler-Lagrange (EL) systems under obstacle and speed constraints (Liu et al., 3 Jun 2024), cooperative object transport using multi-robot QP coordination (Wu et al., 14 Dec 2025), and output-space QP controllers for state-constrained trajectory tracking with real-time reachability certification (Gholampour et al., 16 Sep 2025).

1. Fundamental Concepts and System Models

The velocity-tracking QP controller operates in contexts where the system dynamics can be manipulated via real-valued control inputs to effect desired velocity evolution while respecting constraints. Core models include:

• Euler–Lagrange Systems: Described by

$M(q)\ddot{q} + C(q, \dot{q})\dot{q} + g(q) = \tau,$

where $q \in \mathbb{R}^n$ represents generalized positions, $\tau$ is the control torque, $M(q)$ is inertia, $C(q,\dot{q})$ is Coriolis/centrifugal, and $g(q)$ is gravity (Liu et al., 3 Jun 2024).

• Multi-Robot Cooperative Transport: Spherical robots (modeled as single-integrators) collaboratively control the motion of a double-integrator object through distributed contact forces,

$$\dot{p}_i = v_i,\quad \dot{p}_o = v_o,\quad \dot{v}_o = f(\{p_i - p_o\}),$$

with a repulsive contact model and kinematic coupling (Wu et al., 14 Dec 2025).

• Feedback-Linearized Output Path Tracking: Systems (e.g., manipulators) are feedback-linearized to double integrator form in task-space variables, with bounded disturbances and prescribed kinematic limits (Gholampour et al., 16 Sep 2025).

In all cases, the velocity-tracking objective is formalized as minimizing the error $||\dot{q} - v_c||$ or equivalent, subject to state, input, and safety constraints determined by the operational context.

2. Quadratic Programming Formulation

The central element is the reduction of a constrained velocity command generation problem to a convex QP. The general QP prototype is:

• Objective: Minimize the deviation from the reference command, frequently

$\min_{v} \| v - v_c \|^2,$

and, in multi-agent settings, also minimize total effort or contact force magnitudes.

• Constraints: Encode safety and feasibility:
  • Obstacle avoidance: Linearized or reshaped to avoid non-Lipschitzness (Liu et al., 3 Jun 2024).
  • Velocity and acceleration limits: Hard bounds, potentially regularized for robustness (Gholampour et al., 16 Sep 2025).
  • Contact/force nonnegativity: In cooperative manipulation scenarios (Wu et al., 14 Dec 2025).

A distinctive technical contribution is the use of a positive basis to linearize and reshape otherwise nonlinear (or intersecting) constraints, ensuring that the QP mapping from state and reference to solution is Lipschitz continuous even in the presence of multiple, nearly active constraints (Liu et al., 3 Jun 2024). This avoids non-differentiability-induced instability in the closed-loop system.

In cooperative object transport, robot penetration depths (proxying contact forces) are the QP decision variables, designed so that the realized object acceleration matches the ideal dynamics for error correction, subject only to nonnegativity and minimality conditions (Wu et al., 14 Dec 2025).

For output-space tracking, the QP determines the optimal virtual acceleration to minimize a one-step lookahead position-velocity error, with constraints on feasibility and disturbance tolerance embedded via one-step reachability analysis (Gholampour et al., 16 Sep 2025).

3. Cascade and Interconnection Structures

Velocity-tracking QP controllers are often realized in a two-loop (cascade) structure:

• Outer loop (QP block): Computes an idealized or "virtual" velocity (or acceleration) $v^\ast$ by solving the QP, guaranteeing constraint satisfaction and optimal proximity to the reference (Liu et al., 3 Jun 2024, Gholampour et al., 16 Sep 2025).
• Inner loop: Implements a velocity-tracking or position-tracking law to ensure the system state tracks $v^\ast$. Commonly, this is a feedback law leveraging energy shaping,

$$u(q, \dot{q}, v_c) = g(q) + C(q, \dot{q}) v^\ast - k_D (\dot{q} - v^\ast),$$

for mechanical systems. In the multi-robot case, each agent follows a position-tracking routine towards its QP-specified ideal position (Wu et al., 14 Dec 2025).

This separation facilitates modular safety analysis: the outer loop enforces constraint satisfaction, while the inner loop ensures convergence to the QP output.

4. Constraint Handling, Lipschitz Properties, and Robustness

QP-based velocity-tracking is distinguished by precise handling of system, obstacle, and actuation constraints:

• Obstacle Constraints: In EL systems, obstacle avoidance is ensured by projecting nonlinear safety conditions onto linearized, positive-basis-generated barriers, with smoothing functions $\varphi$ and class $\mathcal{K}^e$ penalty functions $\alpha_c$. This method reshapes constraints to maintain QP feasibility and solution uniqueness, provided a mild ball-separation condition is satisfied (Liu et al., 3 Jun 2024).
• Lipschitz Continuity: Non-smoothness in generic QP mappings can compromise closed-loop stability. The reshaping via a positive basis and smoothing ensures the QP solution $v^\ast$ (or $s^\ast$ in multi-agent cooperation) is locally Lipschitz in state and reference, a necessary condition for robust Lyapunov-based analysis and ISS properties (Liu et al., 3 Jun 2024, Wu et al., 14 Dec 2025).
• Reachability and Safety Margins: Output-space tracking incorporates one-step reachability analysis both offline (to certify trajectory feasibility) and online (to adapt penalty weighting and ensure robust feasibility under bounded disturbances). When axial reachability is lost, a regularization penalty blends position and velocity performance in the QP, and a KKT-inspired correction further improves tracking (Gholampour et al., 16 Sep 2025).

5. Stability, Feasibility, and Theoretical Guarantees

Rigorous analysis underpins the QP-based velocity-tracking approach:

• Existence and Uniqueness: With convex costs and constraint sets shaped under the positive-basis or similar constructions, the QP has a unique minimizer at each time step. For multi-agent systems, strict convexity is mandated by $\epsilon$-regularization (Wu et al., 14 Dec 2025).
• Stability: Lyapunov (or ISS) theory is used to show exponential convergence of tracking error up to bounded residuals set by the time variations in the velocity reference and permissible disturbances. For cascaded controllers, composite nonsmooth Lyapunov-like functions encapsulate both constraint satisfaction and convergence properties (Liu et al., 3 Jun 2024).
• Small-Gain Theorem Application: In interconnected multi-agent systems, stability analysis employs a nonlinear small-gain theorem to certify that cascade gains are chosen so that the overall closed loop remains input-to-state stable as long as the command derivatives are bounded (Wu et al., 14 Dec 2025).
• Satisfiability Domains: Feasibility of the QP formulation is reduced to geometric conditions, e.g., minimum ball-separation in EL systems or non-interpenetration in robot collectives, and parameter choices for robustness margins (Liu et al., 3 Jun 2024, Wu et al., 14 Dec 2025).

6. Empirical Validation and Performance

Extensive simulation and experimental studies support the practical efficacy of QP-based velocity-tracking controllers:

• For EL systems, safety and performance are validated both in simulation, with thousands of spherical obstacle constraints, and in real-robot experiments on 2-link and 6-DOF manipulators. The outer loop produces velocity references that strictly obey joint-space ball constraints and velocity limits, while the inner loop achieves rapid exponential convergence with no collisions (Liu et al., 3 Jun 2024).
• In cooperative object transport, simulation with three robots pushing a planar object demonstrates that with sufficiently high position feedback gain, object and formation tracking errors decay exponentially to small tubes, and effort (penetrations) is minimized across agents. Improper gain selection (too low position gain) visibly degrades performance, verifying small-gain predictions (Wu et al., 14 Dec 2025).
• In path tracking under reachability-constrained QP, performance on randomized spline tests shows a 95.5% reduction in position RMSE and 81.7% reduction in velocity RMSE compared to pure pursuit methods, with real-time handling of safety stops and "freeze-resume" scenarios without replanning (Gholampour et al., 16 Sep 2025).

7. Extensions and Practical Considerations

Velocity-tracking controllers leveraging QP are highly generalizable. They support:

• Plug-in of more complex linear/convex constraints, e.g., friction cones, actuation bounds, and prioritized soft constraints.
• Localized QP formulations that consider only "active" constraints to greatly reduce computational load while retaining provable safety (Liu et al., 3 Jun 2024).
• Smooth integration with higher-level planning, as in pre-run reachability checks and per-iteration adaptation of penalty weights for blending position and velocity objectives (Gholampour et al., 16 Sep 2025).
• Modularization suitable for distributed and multi-agent systems with explicit stability and performance guarantees (Wu et al., 14 Dec 2025).

A plausible implication is that such controllers will underpin future safety-critical robotic applications—especially where constraint enforcement, real-time optimization, and robustness to disturbances must be explicitly guaranteed.

---

Key references:

• "Safety-Critical Control of Euler-Lagrange Systems Subject to Multiple Obstacles and Velocity Constraints" (Liu et al., 3 Jun 2024)
• "Quadratic-Programming-based Control of Multi-Robot Systems for Cooperative Object Transport" (Wu et al., 14 Dec 2025)
• "Trajectory Tracking with Reachability-Guided Quadratic Programming and Freeze-Resume" (Gholampour et al., 16 Sep 2025)