A Minimum-Energy Control Approach for Redundant Mobile Manipulators in Physical Human-Robot Interaction Applications

1. Problem Statement

The paper addresses the challenge of controlling redundant mobile manipulators (a robotic arm mounted on a mobile base) during Physical Human-Robot Interaction (pHRI).

• Context: In pHRI tasks (e.g., co-transportation, peg-in-hole), a human operator interacts with the robot's end-effector via a virtual admittance model. The robot must translate the human's applied forces into desired end-effector velocities.
• The Core Issue: Mobile manipulators possess more degrees of freedom (DoF) than required for a specific task (redundancy). Solving the Inverse Differential Kinematics (IDK) to map end-effector velocities to joint velocities (arm + base) is non-trivial.
• Limitations of Existing Methods:
  • Standard pseudoinverse methods often do not optimize for energy or safety.
  • Prioritized control schemes (e.g., switching between "manipulation" and "locomotion" modes) rely on user intervention or complex heuristics, leading to cognitive load, execution delays, and suboptimal energy usage.
  • Moving the heavy mobile base unnecessarily increases the system's kinetic energy, posing safety risks to the human operator and reducing interaction smoothness.

2. Methodology

The authors propose a control strategy that formulates the IDK problem as a constrained optimization problem to minimize the total kinetic energy of the whole-body system.

A. System Architecture

• Interaction Model: The human interacts with the robot through a virtual admittance model ($M \dot{v}_d + B v_d = f_h$), generating a desired twist $v_d$ in operational space.
• Kinematic Relation: The goal is to find the joint velocity vector $\dot{q}_d$ (comprising arm joints $\dot{q}_a$ and base virtual joints $\dot{q}_b$) such that $v_d = J(q)\dot{q}_d$.

B. The Proposed Optimization Approach

Instead of using a standard pseudoinverse or a switching logic, the authors define an optimization problem:

• Objective Function: Minimize the total kinetic energy ($E_K$) of the system:
  $$E_K = \frac{1}{2}\dot{q}_a^T M_a(q_a) \dot{q}_a + \frac{1}{2}\dot{q}_b^T M_b \dot{q}_b$$
  Where $M_a$ and $M_b$ are the inertia matrices of the arm and the mobile base, respectively.
• Constraint: The kinematic constraint $J(q)\dot{q}_d = v_d$ must be satisfied.
• Solution: The optimal solution is derived analytically using the dynamically consistent Jacobian pseudoinverse ($J_M^\dagger$):
  $$\dot{q}^*_d = J_M^\dagger(q) v_d$$
  Where $J_M^\dagger(q) = M(q)^{-1}J(q)^T (J(q)M(q)^{-1}J(q)^T)^{-1}$.
• Secondary Task: To prevent the robot from drifting from a preferred configuration, a null-space projector is applied to add a secondary task (minimizing distance to a desired configuration $q_{des}$) without affecting the primary energy-minimization objective.

3. Key Contributions

1. Energy-Optimal IDK Formulation: The paper introduces a closed-form solution for the IDK problem specifically designed to minimize the kinetic energy of the entire mobile manipulator system, rather than just minimizing joint velocities.
2. Automatic Base Coordination: Unlike "switch" methods that require manual mode changes, this approach automatically determines the optimal trade-off between moving the arm and moving the base. It inherently minimizes the movement of the heaviest component (the base) to reduce energy and improve safety.
3. Experimental Validation: The method is rigorously tested on a real-world peg-in-hole task involving 27 human participants, comparing the proposed method against two benchmarks: a pure "locomotion" mode and a "switch" mode.

4. Experimental Results

The study utilized a UR10e cobot on a Robotnik RB-KAIROS+ omnidirectional base. Key performance metrics included average kinetic energy ($\bar{E}$), average human force ($\bar{F}_h$), execution time ($T_f$), and final displacement error ($x_f$).

• Kinetic Energy Reduction:
  • The proposed Minimum-Energy approach reduced total stored kinetic energy by 66% compared to the locomotion approach and achieved similar energy levels to the switch approach (69% reduction vs. locomotion).
  • Crucially, the energy profile was smoother with lower peaks compared to the switch approach, which exhibited high energy spikes during locomotion phases.
• Human Effort:
  • The average human force required was reduced by 22% compared to the locomotion approach and 41% compared to the switch approach.
  • The locomotion approach required the highest effort due to excessive base movement.
• Execution Time:
  • The proposed method achieved the fastest execution time (median 28.33s), which was 24% faster than the switch approach (37.49s) and 11% faster than the locomotion approach (31.77s).
  • The switch approach suffered from delays caused by manual mode switching.
• Accuracy:
  • The final displacement error ($x_f$) for the minimum-energy approach was comparable to the locomotion approach (0.08m vs 0.09m) and significantly better than the switch approach (0.28m), which suffered from suboptimal configurations during user switching.
• User Experience:
  • Velocity profiles showed the minimum-energy approach provided the most stable motion with fewer abrupt fluctuations, enhancing user comfort and safety.

5. Significance

This work demonstrates that minimizing kinetic energy is a highly effective criterion for controlling redundant mobile manipulators in collaborative settings.

• Safety: By naturally minimizing the movement of the heavy mobile base, the approach reduces the momentum and potential impact forces during interaction, directly enhancing human safety.
• Efficiency: It eliminates the cognitive load and time delays associated with manual mode switching, resulting in faster and smoother task completion.
• Scalability: The closed-form solution ensures real-time computability, making it suitable for dynamic pHRI scenarios where reaction times are critical.

In conclusion, the proposed minimum-energy control approach outperforms existing benchmark methods across all evaluated metrics, offering a robust solution for safe and efficient human-robot collaboration.