In the field of robot technology, compliance control plays a pivotal role in enabling safe and efficient physical human-robot collaboration. However, uncertainties in the environment, such as sudden contact loss between the robot and the workpiece, can lead to physical impacts that compromise system stability and safety. These events may result in damage to the robot or task components, and in human-robot interaction scenarios, they pose risks to operators. Traditional impedance control methods often focus on maintaining continuous contact but fail to address the energy and power surges that occur during contact loss. To mitigate these issues, we propose an energy- and power-restricted Cartesian impedance control framework that limits the total energy and transmitted power of the robot system. Our approach incorporates an energy tank to ensure passivity and introduces a power scaling factor to regulate energy exchange rates, thereby enhancing robustness and safety in robot technology applications. This method is rigorously analyzed using Lyapunov theory and validated through simulations, demonstrating its effectiveness in real-world scenarios.
The advancement of robot technology has expanded its applications to dynamic interactions with unstructured environments, where energy exchange is fundamental. For instance, tasks like grinding, polishing, and assembly require the robot to exert controlled forces, while collaborative operations involve force feedback and compliant motion. Impedance control, which emulates a mass-damper-spring behavior, is widely used due to its adaptability. However, standard impedance controllers do not inherently restrict the energy and power injected into the system, leading to potential instability during contact loss. In robot technology, ensuring that the robot’s energy and power remain within safe thresholds is crucial for preventing abrupt accelerations and impacts. Our work addresses this by dynamically scaling stiffness and damping parameters and integrating an energy tank with a power regulator. This not only maintains passivity but also controls the rate of energy release, offering a novel solution in robot technology for safe interaction under uncertain conditions.
To formalize the problem, consider a robot interacting with an environment under impedance control. The robot’s dynamics in Cartesian space are given by:
$$ V(x)\ddot{x} + U(x, \dot{x})\dot{x} + F_g(x) = F_{\text{cart}} + F_{\text{ext}} $$
where \( x, \dot{x}, \ddot{x} \in \mathbb{R}^{6 \times 1} \) denote the end-effector position, velocity, and acceleration, respectively; \( V(x) \in \mathbb{R}^{6 \times 6} \) is the inertia matrix; \( U(x, \dot{x}) \in \mathbb{R}^{6 \times 6} \) represents the Coriolis and centrifugal matrix; \( F_g(x) \in \mathbb{R}^{6 \times 1} \) is the gravity vector; \( F_{\text{cart}} \in \mathbb{R}^{6 \times 1} \) is the Cartesian control force; and \( F_{\text{ext}} \in \mathbb{R}^{6 \times 1} \) is the external force. The desired impedance behavior is expressed as:
$$ F_{\text{ext}} = V_d \ddot{\tilde{x}} + D_d \dot{\tilde{x}} + K_d \tilde{x} $$
with \( \tilde{x} = x – x_d \), \( \dot{\tilde{x}} = \dot{x} – \dot{x}_d \), and \( \ddot{\tilde{x}} = \ddot{x} – \ddot{x}_d \), where \( x_d, \dot{x}_d, \ddot{x}_d \) are the desired trajectories. The matrices \( V_d, D_d, K_d \in \mathbb{R}^{6 \times 6} \) define the desired inertia, damping, and stiffness. By setting \( V_d = V(x) \) and incorporating the dynamics, the control force simplifies to:
$$ F_{\text{cart}} = F_g(x) + V(x)\ddot{x}_d + U(x, \dot{x})\dot{x}_d – D_d \dot{\tilde{x}} – K_d \tilde{x} $$
Mapping to joint space via the Jacobian \( J \), the control torque becomes:
$$ \tau_{\text{cart}} = G(q) + J^T \left( V(x)\ddot{x}_d + U(x, \dot{x})\dot{x}_d – D_d \dot{\tilde{x}} – K_d \tilde{x} \right) $$
where \( G(q) \) is the joint-space gravity vector. The closed-loop dynamics ensure the robot exhibits the desired impedance特性. However, during contact loss, the uncontrolled energy and power can cause instability, necessitating restrictions on these quantities.
The total energy of the robot system comprises kinetic and potential components:
$$ E_{\text{tot}} = T_{\text{tot}} + U_{\text{tot}} $$
where
$$ T_{\text{tot}} = \frac{1}{2} \dot{\tilde{x}}^T V(x) \dot{\tilde{x}}, \quad U_{\text{tot}} = \frac{1}{2} \tilde{x}^T K_d \tilde{x} $$
To limit the total energy, we introduce a stiffness scaling factor \( \lambda(t) \):
$$ \lambda(t) = \begin{cases}
1 & \text{if } E_{\text{tot}} \leq E_{\text{max}} \\
\frac{E_{\text{max}} – T_{\text{tot}}}{U_{\text{tot}}} & \text{if } E_{\text{tot}} > E_{\text{max}}
\end{cases} $$
This adjusts the stiffness matrix to \( K_d^\dagger = \lambda(t) K_d \), ensuring \( E_{\text{tot}} \leq E_{\text{max}} \). Similarly, the power due to motion is given by:
$$ P_{\text{motion}} = -\tilde{x}^T K_d^\dagger \dot{x} – \dot{x}^T D_d \dot{x} $$
where the first term represents task-related power and the second dissipation. A damping scaling factor \( \beta(t) \) limits the power:
$$ \beta(t) = \begin{cases}
1 & \text{if } P_{\text{motion}} \leq P_{\text{max}} \\
\frac{-\tilde{x}^T K_d^\dagger \dot{x} – P_{\text{max}}}{\dot{\tilde{x}}^T D_d \dot{x}} & \text{if } P_{\text{motion}} > P_{\text{max}}
\end{cases} $$
yielding a modified damping matrix \( D_d^\dagger = \beta(t) D_d \). The revised control law is:
$$ \tau_{\text{cart}} = G(q) + J^T \left( V(x)\ddot{x}_d + U(x, \dot{x})\dot{x}_d – \beta(t) D_d \dot{\tilde{x}} – \lambda(t) K_d \tilde{x} \right) $$
This energy- and power-restricted controller enhances safety in robot technology by preventing excessive energy buildup and rapid power transitions.
However, varying impedance parameters can violate passivity, a key stability criterion in robot technology. To address this, we employ an energy tank that stores and releases energy to compensate for the time-varying controller. The tank energy \( E_{\text{tank}} \) evolves as:
$$ \dot{E}_{\text{tank}} = \begin{cases}
\alpha P_{\text{task}} & \text{if } P_{\text{task}}^\dagger \leq 0 \\
\gamma P_{\text{task}} & \text{if } P_{\text{task}}^\dagger > 0
\end{cases} $$
where \( P_{\text{task}}^\dagger = \lambda(t) \dot{x}^T K_d \tilde{x} \) is the scaled task power, and \( \alpha, \gamma \) are tuning parameters defined as:
$$ \alpha = \begin{cases}
0 & \text{if } P_{\text{task}}^\dagger \leq 0 \wedge E_{\text{tank}} < E_{\text{tank,min}} \\
\frac{1}{2} \left[ 1 – \cos\left( \frac{E_{\text{tank}} – E_{\text{tank,min}}}{\delta E} \pi \right) \right] & \text{if } E_{\text{tank,min}} \leq E_{\text{tank}} < E_{\text{tank,min}} + \delta E \\
1 & \text{otherwise}
\end{cases} $$
and
$$ \gamma = \begin{cases}
0 & \text{if } P_{\text{task}}^\dagger > 0 \wedge E_{\text{tank}} > E_{\text{tank,max}} \\
1 & \text{otherwise}
\end{cases} $$
Here, \( E_{\text{tank,min}}, E_{\text{tank,max}} \), and \( \delta E \) are thresholds for safe operation. To further regulate the energy exchange rate, a power scaling factor \( \eta \) is introduced:
$$ \eta = \begin{cases}
\frac{P_{\text{low}}}{P_{\text{task}}^\dagger} & \text{if } P_{\text{task}}^\dagger \leq P_{\text{low}} \leq 0 \\
1 & \text{otherwise}
\end{cases} $$
where \( P_{\text{low}} \) is the maximum allowable extraction power. The tank dynamics are updated to \( \dot{E}_{\text{tank}}^\dagger = \eta \alpha P_{\text{task}}^\dagger \) for \( P_{\text{task}}^\dagger \leq 0 \) and \( \dot{E}_{\text{tank}}^\dagger = \gamma P_{\text{task}}^\dagger \) otherwise. The stiffness scaling factor is adjusted accordingly:
$$ \lambda(t) = \begin{cases}
1 & \text{if } E_{\text{tot}} \leq E_{\text{max}} \wedge \alpha \neq 0 \\
\lambda(t-1) & \text{if } \alpha = 0 \wedge P_{\text{task}}^\dagger \neq 0 \\
\frac{E_{\text{max}} – T_{\text{tot}}}{U_{\text{tot}}} & \text{if } E_{\text{tot}} > E_{\text{max}}
\end{cases} $$
This ensures that when the tank energy is critically low, the controller reverts to a standard impedance mode while maintaining energy and power limits. The passivity of the closed-loop system is proven using Lyapunov theory. Define the storage function for the augmented system as \( W = S_{\text{cart}}^\dagger + E_{\text{tank}} \), where \( S_{\text{cart}}^\dagger = T_{\text{tot}} + \lambda(t) U_{\text{tot}} \). Its derivative satisfies:
$$ \dot{W} \leq \dot{\tilde{x}}^T F_{\text{ext}} $$
which confirms passivity, as the system’s energy change is bounded by the external energy supply.
To validate our approach in robot technology, we conducted simulations using MATLAB/Simscape. The robot model is a 7-DOF manipulator, and the control parameters are summarized in the following tables:
| Parameter | Value |
|---|---|
| Cartesian Stiffness \( K_d \) | [900, 900, 900, 150, 150, 150] |
| Cartesian Damping \( D_d \) | [3, 3, 3, 3, 3, 3] |
| Parameter | Value |
|---|---|
| Cartesian Stiffness \( K_d \) | [900, 900, 900, 150, 150, 150] |
| Cartesian Damping \( D_d \) | [3, 3, 3, 3, 3, 3] |
| Maximum Allowable Power \( P_{\text{motion,max}} \) | 0.5 W |
| Maximum Allowable Energy \( E_{\text{tot,max}} \) | 0.3 J |
| Parameter | Value |
|---|---|
| Maximum Allowable Extraction Power \( P_{\text{tank,low}} \) | -0.25 W |
| Maximum Tank Energy \( E_{\text{tank,max}} \) | 5 J |
| Minimum Tank Energy \( E_{\text{tank,min}} \) | 0.2 J |
| Transition Threshold \( \delta E \) | 0.05 J |
| Initial Tank Energy \( E_{\text{tank,init}} \) | 3 J |
| Environment Stiffness \( K_b \) | 1000 N/m |
| External Disturbance Force \( F_{\text{ext}} \) | 50 sin(π(t-10)/2) N |
The simulation involves the robot moving from an initial position to a target while interacting with a planar surface. Contact loss occurs when the end-effector detaches from the surface, and an external force is applied to simulate human-robot interaction. Comparisons between standard impedance control and our method show that the proposed controller effectively limits energy and power. For instance, the total energy remains below \( E_{\text{max}} \) due to \( \lambda(t) \), and the power is constrained by \( \beta(t) \). The energy tank ensures passivity, and the power scaling factor \( \eta \) prevents rapid energy release, resulting in smoother motion transitions. The real-time performance on a Linux system demonstrates that the computational overhead is minimal, with an average execution time of 259 μs, meeting the 1 kHz requirement for robot technology applications.
In conclusion, our energy- and power-restricted impedance control method enhances safety in robot technology by addressing contact loss scenarios. By dynamically scaling stiffness and damping, and integrating an energy tank with power regulation, we ensure that the robot’s energy and power remain within safe bounds while maintaining system passivity. This approach is particularly beneficial in physical human-robot collaboration, where unpredictable interactions occur. Future work will involve implementing this controller on hardware platforms and extending it to joint-level force control for broader applications in robot technology. The integration of such advanced control strategies underscores the evolving capabilities of robot technology in achieving robust and safe autonomous operations.