The demand for automating complex, contact-intensive tasks such as deburring, chamfering, and precision polishing on large-scale components is rapidly growing across aerospace, energy, and automotive sectors. While industrial robots offer the necessary workspace and flexibility, their inherent lack of fine force-control capability often renders them unsuitable for these advanced applications, where consistent contact force is paramount for quality and surface integrity. To bridge this technological gap, we propose a novel add-on force-controlled end effector, designed to be mounted on the flange of a standard industrial robot. This device aims to decouple the robot’s coarse, large-scale positional movements from the fine, rapid force adjustments required during continuous contact operations. We introduce a robust adaptive admittance control strategy to achieve fast and precise tracking of desired contact forces in the operational space, specifically along the surface normal direction. This approach enhances the robot’s capability to perform high-quality grinding and polishing on complex geometries, including lateral surfaces, internal holes, and narrow structural features.
The core of our system is a 3-degree-of-freedom (3-DOF) translational parallel mechanism of the 3-P(UU)₂ type. This architecture constrains all rotational motions, providing pure translational movement of the end-effector’s tool platform along the X, Y, and Z axes. To achieve high performance, we employ a unique hybrid actuation system within each of the three identical kinematic chains. The active prismatic joint in each chain is driven by a custom-designed pneumoelectric actuator. This actuator synergistically combines a voice coil motor (VCM) for high-bandwidth, high-precision force control with an integrated nitrogen gas spring. The gas spring provides several critical benefits: a high force-to-mass density, inherent compliance and vibration damping, and a near-constant force output over its stroke, which simplifies control. Mechanical springs are added to balance the gas spring’s preload, ensuring the actuator returns to a neutral position. A multi-axis force sensor is mounted on the moving platform to measure the output contact force vector, while linear optical encoders track the displacement of each actuator. The integration of these elements results in a high-performance force-controlled end effector.
To effectively control this system, we first establish its kinematic and dynamic models. Using a closed-loop vector approach, we define the inverse and forward kinematics. Let $\mathbf{p} = [p_x, p_y, p_z]^T$ be the position vector of the moving platform’s center in the base frame. The inverse kinematics, calculating the required actuator displacements $q_i$ from the platform position, is given by:
$$
\begin{aligned}
q_1 &= p_z – \sqrt{l^2 – (p_x – R + r)^2 – p_y^2} \\
q_2 &= p_z – \sqrt{l^2 – (p_x + \frac{R – r}{2})^2 – (p_y – \frac{\sqrt{3}}{2}(R – r))^2} \\
q_3 &= p_z – \sqrt{l^2 – (p_x + \frac{R – r}{2})^2 – (p_y + \frac{\sqrt{3}}{2}(R – r))^2}
\end{aligned}
$$
where $l$ is the constant length of the parallelogram links, $R$ is the radius of the base platform, and $r$ is the radius of the moving platform. The forward kinematics solution is more complex but can be derived numerically or through analytical elimination.
The dynamics of the hybrid pneumoelectric actuator are crucial for controller design. Modeling the nitrogen gas spring involves applying thermodynamic principles under an adiabatic assumption. The force output from the combined actuator, considering the VCM force ($F_m = K_m i$), gas pressures ($P_a$, $P_b$), and mechanical dynamics, can be linearized and described in the Laplace domain. The transfer function from the actuator’s input current to its output displacement $q_o$ approximates to:
$$
G_q(s) = \frac{q_o(s)}{i(s)} \approx \frac{K_{q1}}{K_{q2} + s(B_{eq} + K_c) + s^2 M}
$$
Here, $K_{q1}$ and $K_{q2}$ are steady-state gains, $B_{eq}$ is the equivalent viscous damping (enhanced by the gas spring’s damping orifice), $K_c$ is a stiffness coefficient, and $M$ is the moving mass. This model highlights the gas spring’s role in augmenting system damping for improved stability.
The contact force $F_o$ between the end effector’s tool and the workpiece environment is modeled as a spring: $F_o = K_e (x_o – y)$ for $x_o \geq y$, where $K_e$ is the environmental stiffness, $x_o$ is the end-effector position, and $y$ is the environment position. The overall plant transfer function from actuator command to output force becomes:
$$
G_F(s) = \frac{F_o(s)}{q_i(s)} \approx \frac{K_{qx} K_{q1} K_e}{K_{q2} + K_{qx} K_{F\tau} K_e + s(B_{eq} + K_c) + s^2 M}
$$
where $K_{qx}$ and $K_{F\tau}$ are kinematic Jacobian-related transformation coefficients from joint to operational space.
Our primary control objective is to make the normal component of the contact force $F_o$ track a desired reference force $F_r$, even in the presence of unknown environmental stiffness $K_e$, position $y$, and other disturbances. A standard admittance control law is given by:
$$
M_d (\ddot{x}_r – \ddot{x}_n) + B_d (\dot{x}_r – \dot{x}_n) + K_d (x_r – x_n) = F_r – F_o
$$
where $x_r$ is the reference position, $x_n$ is the commanded position in the force-controlled direction, and $M_d, B_d, K_d$ are the desired inertia, damping, and stiffness parameters. However, this approach leads to a steady-state force error if $F_r$ is not chosen correctly relative to the unknown environment. To achieve perfect tracking, we propose a Robust Adaptive Admittance Control scheme. The control law is designed as:
$$
F_r = k_0(e, F_i)F_i + k_1(e, F_o)F_o + k_2(e, \dot{e})\dot{e} + d_c(e)
$$
where $e = F_{mr} – F_o$ is the force tracking error ($F_{mr}$ is the output of a stable reference model), and $F_i$ is the input command. The terms $k_0, k_1, k_2$ are adaptive gains updated by laws such as $\dot{k}_0 = \lambda_0 e F_i$, and $d_c(e)$ is a robust term like $d_c = \delta \cdot sat(e/\phi)$ to counteract bounded disturbances and unmodeled dynamics. This composite controller ensures asymptotic stability, as proven via Lyapunov analysis, enabling the force-controlled end effector to adapt to varying contact conditions and reject disturbances.
We constructed a prototype and a dedicated testbed to validate the performance of our force-controlled end effector and control algorithm. The experimental setup includes the 3-DOF end effector, a multi-axis force sensor, a mock-up workpiece with flat and curved surfaces, a dSPACE real-time controller, and a host PC running Control Desk. Key experiments were conducted to evaluate step response, impact rejection, and force tracking during motion.
Step Response Performance: The end effector was commanded to track a step change in desired force from 0 N to 20 N while in contact. The results, comparing standard admittance control and our robust adaptive admittance control, are summarized below.
| Control Method | Rise Time (ms) | Steady-State Error (N) | Overshoot (%) |
|---|---|---|---|
| Standard Admittance | 9.27 | ~1.0 | – |
| Robust Adaptive Admittance | 19.27 | -4.5×10⁻⁴ | 38.8 |
The robust adaptive controller effectively eliminates the steady-state error at the cost of a slightly longer rise time, demonstrating precise force tracking capability. Tests with different tool materials (diamond grit, alloy steel, sandpaper) showed consistent performance, confirming the adaptability of the controller.
Impact Rejection and Robustness: To evaluate the disturbance rejection and compliance of the system, a sudden 1mm step displacement was applied to the environment while the controller tried to maintain a constant 20 N force. The system’s recovery was measured.
| Control Method | Settling Time (ms) | Max Overshoot (%) | Steady-State Error (N) |
|---|---|---|---|
| Standard Admittance | 89.0 | 50.5 | ~1.0 |
| Robust Adaptive Admittance | 116.0 | 57.5 | ~0 |
The gas spring’s damping effect contributes to a well-damped response. While the robust adaptive control has a marginally longer settling time, it successfully drives the force error back to zero, showcasing excellent impact absorption and robustness.
Force-Motion Hybrid Control: Finally, we tested the system’s ability to maintain a constant normal force while the end effector followed prescribed trajectories over flat and cylindrical surfaces. This simulates real grinding/polishing tasks. The force error was quantified using the Root-Mean-Square Error (RMSE): $$F_{RMSE} = \sqrt{\frac{1}{N-1} \sum_{j=1}^{N} (F_j – F_{rv})^2}$$
| Trajectory / Control Method | Force RMSE (N) | Avg. Force Peak (N) | Steady-State Error (N) |
|---|---|---|---|
| Flat Surface (Standard Adm.) | ~1.5 (est.) | High | N/A |
| Flat Surface (Robust Adaptive) | 0.143 | 0.694 | -4.5×10⁻⁴ |
| Cylindrical Surface (Robust Adaptive) | 0.125 | 0.471 | -1.08×10⁻³ |
The standard admittance control failed to maintain a stable force due to environmental position errors. In contrast, our robust adaptive admittance control successfully maintained a consistent force with very low error and minimal force spikes during motion reversals, proving its effectiveness for complex, continuous contact operations.
In conclusion, we have successfully developed and validated a 3-DOF translational force-controlled end effector based on a parallel mechanism with hybrid pneumoelectric actuation. The integration of nitrogen gas springs provides inherent compliance and damping, while the voice coil motors enable high-fidelity actuation. More importantly, the proposed robust adaptive admittance control algorithm allows this end effector to achieve rapid, precise, and stable force tracking. It effectively compensates for unknown and varying environmental stiffness, rejects disturbances from impacts or motion errors, and maintains excellent force regulation during complex trajectories. This system effectively augments the capability of standard industrial robots, enabling them to perform high-precision continuous contact tasks like grinding and polishing on complex workpieces. The presented design and control framework offers a significant step towards more flexible and intelligent robotic manufacturing systems.