ReM-QP Controller for Enhanced Robot Manipulability

• The paper introduces a convex QP formulation that integrates manipulability maximization and joint limit avoidance, achieving improvements such as a 36% increase in manipulability and notable failure rate reductions.
• It employs artificial redundancy via slack variables, enabling safe and dexterous motion control in both redundant and non-redundant robotic systems, even near singularities.
• The controller leverages standard robot modeling and efficient real-time QP solving to ensure robust performance in industrial, visual servoing, and human-robot interaction applications.

A Reduced and Manipulability-aware Quadratic Programming (ReM-QP) Controller is an optimization-based motion control framework designed to maximize robot manipulability while efficiently respecting task and safety constraints. It operates seamlessly across redundant and non-redundant manipulators, leveraging artificial redundancy when needed, and formalizes control synthesis as a strictly convex QP problem. This controller targets both instantaneous velocity-level control and robust real-time performance, ensuring avoidance of joint limits, singularities, and offering compatibility with standard robot modeling formats. Its design integrates manipulability maximization into the cost function and enables controlled deviation from strict end-effector paths for improved dexterity, robustness, and failure reduction.

1. Quadratic Program Formulation

The ReM-QP controller reframes resolved-rate control as a QP problem, combining minimal joint velocity cost with manipulability maximization. The classical resolved-rate scheme $J(q)\,\dot{q} = \nu$ is posed as:

$$\begin{aligned} \min_{\dot{q}} \quad & \frac{1}{2}\dot{q}^\top I_n \dot{q} \ \text{s.t.} \quad & J(q)\,\dot{q} = \nu \end{aligned}$$

To enable manipulability maximization and accommodate both redundant ($n>6$) and non-redundant ($n \leq 6$) robots, an artificial slack variable $\delta \in \mathbb{R}^6$ is introduced:

$x = [\dot{q}; \delta] \in \mathbb{R}^{n+6}$

$J(q)\,\dot{q} + \delta = \nu$

The augmented QP takes the form:

$$\begin{aligned} \min_x \quad & \frac{1}{2} x^\top \mathcal{Q} x + \mathcal{C}^\top x \ \text{s.t.} \quad & \mathcal{J}\,x = \nu \ & \mathcal{A}\,x \leq \mathcal{B} \ & x^- \leq x \leq x^+ \end{aligned}$$

Where $\mathcal{Q}$ penalizes both $\dot{q}$ and $\delta$; $\mathcal{C}$ encodes the manipulability Jacobian $J_m$ for $[\dot{q}; 0]$; and constraints $\mathcal{A}, \mathcal{B}$ enforce task/velocity boundaries, joint limits, and safety measures via inequality constraints ("velocity dampers").

The manipulability measure is $m(q) = \sqrt{\det(J(q)\,J(q)^\top)}$; its time derivative is $\dot{m} = J_m^\top \dot{q}$, and maximizing $m$ is encoded as minimizing $-\mathcal{C}^\top x$.

2. Manipulability Maximization and Joint Limit Avoidance

Manipulability maximization is achieved by optimizing the instantaneous rate of change of $m(q)$. For redundant robots, this is often done via null space projection; in ReM-QP, artificial redundancy via $\delta$ generalizes the approach to non-redundant robots by relaxing strict path constraints. The system penalizes slack ($\lambda_\delta$ in $\mathcal{Q}$), scheduling it—more tolerant away from the goal, sharply penalized near convergence.

Joint limit avoidance and singularity handling are realized by:

• Imposing velocity dampers:

$$\dot{q}_i \leq \eta ((\rho_i - \rho_s)^{-1}(\rho - \rho_s))$$

where $\rho$ quantifies joint distance to its limit, $\rho_i$ is influence distance, $\rho_s$ is the safety margin, and $\eta$ is a gain.

• Penalizing slack/deviation, allowing controlled trade-off between path accuracy and manipulability.

This approach enables the controller to recover from near-singular configurations and to circumvent joint range boundaries without sacrificing overall motion feasibility.

3. Implementation and Computational Performance

The ReM-QP controller is implemented as part of the open-source Robotics Toolbox for Python. Core implementation features include:

• Strictly convex QP solving via the "qpsolvers" Python library leveraging the Goldfarb-Idnani algorithm, with typical solve times around 2.5 ms (on a 12-core i7).
• Model compatibility: URDF, Denavit–Hartenberg, and Elementary Transform Sequence (ETS) parametric formats.
• Provision for real-time deployment with full ROS middleware integration; demonstrated on hardware such as the Franka Emika Panda.

The algorithm computes the manipulator Jacobian, its Hessian for $J_m$, and supports both simulation and real-hardware deployment. Matrix and QP computations are arranged to allow for efficient updates and scheduling, facilitating practical real-time use.

4. Comparative Evaluation

Extensive benchmarks and experimental trials compare ReM-QP (MMC variant) against conventional resolved-rate controllers and earlier manipulability-maximizing designs. Notable performance metrics include:

• On 7-DoF Panda:
  • $\sim$36% higher mean manipulability versus baseline.
  • 38% improvement at final pose.
  • Failure rate reduced from 12.4% to 8.0%.
• On 6-DoF UR5:
  • Failure rate reduced from $\sim$39.7% (baseline) to $\sim$25.6%.
  • Manipulability improvement by 20–30%.

Physical experiments verify maintenance of high manipulability, robust recovery from near-singularities, and improved trajectory execution compared to rigid path-following controllers.

5. Extension to Non-redundant and Modular Robots

The ReM-QP formalism is extensible—by virtue of the slack approach—to non-redundant robots and modular architectures. For modular robots (as in (Liu et al., 2021)), QP-based controllers stack multiple tracking constraints and incorporate linearized geometric constraints for environment and obstacle avoidance. Manipulability is maintained via soft priorities in the cost, ensuring feasible dexterous configurations while complying with hardware and collision restrictions.

Artificial redundancy via relaxation parameters or slack variables extends the utility to mobile base/arm configurations and safety-critical multi-agent networks.

6. Practical Applications

Direct applications of ReM-QP include:

• Visual servoing and reactive grasping (rapid feedback, adaptability).
• Industrial manipulation (high-precision, singularity/image avoidance).
• Human-robot interaction (fluid, safe motion in proximity).
• Mobile manipulation (whole-body coordination with both redundant and non-redundant arms).

Adoption across diverse platforms is facilitated by the modular implementation and compatibility with standard robot model descriptors.

7. Impact and Future Directions

The ReM-QP controller architecture delivers both theoretical and practical advances. By constructing a QP that integrates manipulability maximization, path deviation, and joint limit avoidance—while optimizing for computational efficiency—it provides a robust solution compatible with a wide range of robotic platforms and task requirements.

Further research may focus on integration with hierarchical MPC frameworks, recursive feasibility for safety-critical control (Parwana et al., 2021), SOS programming for state-dependent safety tuning (Lin et al., 28 Apr 2025), and manipulation-aware multi-agent architectures. The approach serves as a foundational building block for responsive, dexterous, and generalizable motion control in modern robots.