In modern manufacturing, tasks such as grinding, polishing, and deburring remain significant challenges for automation. Traditional manual methods are inefficient, labor-intensive, and yield inconsistent quality heavily reliant on operator skill. While industrial robots offer a path toward automation, precise control of the contact force between the tool and the workpiece is paramount for achieving high-quality surface finishing. This force control is typically implemented in two ways: direct control through the robot’s joint servos or, more effectively, through a dedicated force-controlled end effector mounted at the robot’s wrist. The latter approach offers superior control bandwidth, accuracy, and dynamic response, making the design of high-performance end effectors a critical research area.
Many existing commercial and research end effectors are limited to a single degree of freedom (DOF), usually providing axial compliance. While suitable for simple surfaces, their lack of flexibility hinders application in complex scenarios, such as finishing internal surfaces of parts with narrow cavities or conforming to contoured surfaces. Although some three-degree-of-freedom designs exist, they often prioritize rotational compliance for curvature matching, leaving a gap for solutions requiring controlled translational compliance in multiple lateral directions. This paper addresses this gap by presenting the kinematic and performance analysis of a novel 3-DOF translational parallel mechanism-based force-controlled end effector. This device is specifically designed for automated polishing within confined spaces and on large workpieces, offering independent control of contact forces along three Cartesian axes.
The core of the proposed end effector is a 3-PPaR parallel mechanism. Parallel architectures are renowned for their high stiffness, precision, and payload-to-weight ratio compared to serial counterparts. The chosen 3-PPaR topology yields a compact, symmetrical structure with three identical kinematic chains. Each chain is driven by a voice coil motor (providing the actuated prismatic joint ‘P’), chosen for its high acceleration, precise positioning, and direct drive characteristics essential for fine force control. The chain continues with a parallelogram linkage (Pa) composed of two spherical (S) and two revolute (R) joints, which effectively constrains the orientation of the output link. Finally, each chain connects to the moving platform via a revolute joint (R). This arrangement, as will be proven, grants the moving platform—and thus the attached polishing tool—with three pure translational degrees of freedom (x, y, z). The key design parameters are defined in the following table.
| Symbol | Description | Value (Example) |
|---|---|---|
| $$ R $$ | Radius of the fixed platform (distance from center O to actuator axis B_i) | 46 mm |
| $$ r $$ | Radius of the moving platform (distance from center O’ to connection point P_i) | 40 mm |
| $$ l $$ | Length of the parallelogram rod (distance between spherical joints) | 45 mm |
| $$ q_i $$ | Stroke of the i-th voice coil motor (actuated joint variable) | ±12.5 mm (25 mm total) |
| $$ \alpha_i $$ | Angular layout of the i-th limb, $$ \alpha_i = 2\pi(i-1)/3 $$ | 0°, 120°, 240° |
Mobility Analysis
The mobility of the 3-PPaR mechanism is determined using the modified Grübler-Kutzbach (G-K) formula, which accounts for the presence of the parallelogram closed-loop sub-chain. This sub-chain is first treated as a generalized kinematic pair. The motion screw system of a single limb (i=1,2,3) in the fixed coordinate frame can be expressed. After calculating the reciprocal screws (constraint wrenches) for each limb and then the secondary reciprocal screws for the entire platform, it is found that the moving platform is subject to constraints that allow only three translational motions. The modified G-K formula is given by:
$$ F = d(n – g – 1) + \sum_{i=1}^{g} f_i + v – \zeta $$
where $$ F $$ is the mobility (degrees of freedom), $$ d $$ is the order of the mechanism (d=6-λ, with λ being the number of common constraints), $$ n $$ is the number of links (including the base), $$ g $$ is the number of joints, $$ f_i $$ is the DOF of the i-th joint, $$ v $$ is the number of redundant constraints, and $$ \zeta $$ is the number of local passive DOF. For the 3-PPaR mechanism, analysis yields λ=1 (one common constraint), v=1 (one redundant constraint from symmetric constraints). Substituting the values (d=5, n=11, g=12, Σf_i=12, ζ=1):
$$ F = 5 \times (11 – 12 – 1) + 12 + 1 – 1 = 3 $$
This confirms that the mechanism possesses three translational degrees of freedom, validating its design as a pure translational end effector.
Kinematic Modeling
Inverse Position Kinematics
The inverse kinematics problem is straightforward: given the desired position of the moving platform center O’ = [x, y, z]^T in the fixed frame {O: x, y, z}, find the required actuator inputs $$ q_i $$. The position of the connecting point P_i on the moving platform and the point E_i on the actuator carriage are:
$$ ^{B}\mathbf{P}_i = \begin{bmatrix} x + r\sin\alpha_i \\ y – r\cos\alpha_i \\ z \end{bmatrix}, \quad ^{B}\mathbf{E}_i = \begin{bmatrix} R\sin\alpha_i \\ -R\cos\alpha_i \\ q_i \end{bmatrix} $$
The constant distance l within the parallelogram provides the constraint equation:
$$ (X_i)^2 + (Y_i)^2 + (z – q_i)^2 = l^2, \quad i = 1,2,3 $$
where $$ X_i = (x + r\sin\alpha_i – R\sin\alpha_i) = x + (r-R)\sin\alpha_i $$ and $$ Y_i = (y – r\cos\alpha_i + R\cos\alpha_i) = y + (R – r)\cos\alpha_i $$. Solving for $$ q_i $$ yields the inverse kinematic solutions:
$$ q_i = z \pm \sqrt{l^2 – X_i^2 – Y_i^2} $$
The “±” sign corresponds to two possible assembly modes for each limb. For a given configuration, a consistent sign (typically + for the working mode) is chosen.
Forward Position Kinematics
The forward kinematics problem is more complex: given the actuator inputs $$ q_1, q_2, q_3 $$, find the position [x, y, z]^T of the moving platform. By expanding and manipulating the three constraint equations from (3), a system of equations is solved. The closed-form solutions are:
$$ \begin{aligned} x &= \frac{A_1 B_1 \pm B_1\sqrt{B_1^2 – 4A_1 C_1}}{2A_1^2} \\ y &= \frac{A_2 B_2 \pm B_2\sqrt{B_2^2 – 4A_2 C_2}}{2A_2^2} \\ z &= \frac{-B_3 \pm \sqrt{B_3^2 – 4A_3 C_3}}{2A_3} \end{aligned} $$
where the coefficients A_j, B_j, C_j (j=1,2,3) are functions of $$ q_i, R, r, $$ and $$ l $$. For example:
$$ A_1 = \frac{q_2 – q_3}{\sqrt{3}(R – r)}, \quad A_2 = \frac{2q_1 – q_2 – q_3}{3(r – R)}, \quad A_3 = A_1^2 + A_2^2 + 1 $$
The other coefficients are derived similarly from algebraic elimination. These solutions indicate up to eight possible configurations for a given set of inputs, corresponding to different assembly modes.
Velocity Analysis and Jacobian Matrix
The velocity Jacobian matrix defines the linear mapping between the actuator joint velocities $$ \dot{\mathbf{q}} = [\dot{q}_1, \dot{q}_2, \dot{q}_3]^T $$ and the moving platform velocity $$ \dot{\mathbf{v}} = [\dot{x}, \dot{y}, \dot{z}]^T $$. Differentiating the loop-closure equation and projecting it along the direction of the connecting rod vector $$ \mathbf{d}_i $$ (unit vector from E_i to P_i) yields a simple relation:
$$ \mathbf{d}_i^T \dot{\mathbf{v}} = (\mathbf{d}_i^T \mathbf{c}) \dot{q}_i, \quad \text{with} \quad \mathbf{c} = [0, 0, 1]^T $$
Assembling this for all three limbs gives:
$$ \mathbf{J}_{dir} \dot{\mathbf{v}} = \mathbf{J}_{inv} \dot{\mathbf{q}} $$
where the direct and inverse Jacobian matrices are:
$$ \mathbf{J}_{dir} = \begin{bmatrix} \mathbf{d}_1^T \\ \mathbf{d}_2^T \\ \mathbf{d}_3^T \end{bmatrix}, \quad \mathbf{J}_{inv} = \operatorname{diag}(\mathbf{d}_1^T \mathbf{c}, \mathbf{d}_2^T \mathbf{c}, \mathbf{d}_3^T \mathbf{c}) = \operatorname{diag}(d_{1z}, d_{2z}, d_{3z}) $$
Here, $$ d_{iz} $$ is the z-component of the unit vector $$ \mathbf{d}_i $$. The overall Jacobian matrix $$ \mathbf{J} $$ relating $$ \dot{\mathbf{q}} = \mathbf{J} \dot{\mathbf{v}} $$ is therefore $$ \mathbf{J} = \mathbf{J}_{inv}^{-1} \mathbf{J}_{dir} $$, provided $$ \mathbf{J}_{inv} $$ is non-singular.
Singularity Analysis
Singularities are configurations where the end effector loses or gains uncontrollable degrees of freedom, severely impacting its stiffness and controllability. For parallel mechanisms, singularities are classified based on the determinants of the Jacobian matrices.
| Singularity Type | Condition | Physical Interpretation & Consequence |
|---|---|---|
| Inverse Singularity | $$ \det(\mathbf{J}_{inv})=0 $$, $$ \det(\mathbf{J}_{dir}) \neq 0 $$ | Occurs when $$ d_{iz}=0 $$ for one or more limbs, i.e., the connecting rod $$ \mathbf{d}_i $$ is horizontal. The platform loses one or more DOFs; actuators can be locked while the platform moves under external wrench. |
| Forward Singularity | $$ \det(\mathbf{J}_{dir})=0 $$, $$ \det(\mathbf{J}_{inv}) \neq 0 $$ | Occurs when the three vectors $$ \mathbf{d}_1, \mathbf{d}_2, \mathbf{d}_3 $$ become linearly dependent. The platform gains uncontrollable motion; it cannot resist certain external forces even with all actuators locked. Specific cases include: all rods vertical, two rods vertical, or one rod lying in the plane of the other two. |
| Combined Singularity | $$ \det(\mathbf{J}_{dir})=0 $$ and $$ \det(\mathbf{J}_{inv})=0 $$ | Occurs under conditions satisfying both previous types. Both motion and force transmission become problematic. |
A critical observation is that by imposing a prudent design rule where the fixed platform radius is larger than the moving platform radius (R > r) and considering the physical limits of the spherical joints, all singular configurations can be excluded from the reachable workspace. This makes the 3-PPaR end effector particularly robust for practical force-control applications.
Workspace Analysis
The reachable workspace is the set of all positions that the center of the moving platform can achieve. Its size and shape are crucial for assessing the end effector‘s utility in polishing tasks. The workspace is determined numerically by considering the following constraints:
- Actuator Stroke: $$ q_{i_{min}} \le q_i \le q_{i_{max}} $$.
- Spherical Joint Angle Limit: To prevent mechanical interference or damage, the angle $$ \gamma_i $$ between the connecting rod and the vertical z-axis must be less than a maximum value (e.g., 45°). This is the most restrictive constraint.
$$ 0 < \gamma_i = \arccos\left(\frac{z – q_i}{l}\right) < 45^\circ $$ - Link Interference: Avoids collisions between adjacent limbs, which is generally mitigated by the symmetric 120° separation.
A numerical search algorithm (e.g., a discretization and boundary-tracking method) is employed to find all points [x, y, z]^T satisfying the inverse kinematics and the above constraints. For the example parameters (R=46mm, r=40mm, l=45mm, stroke=25mm, $$ \gamma_{max}=45^\circ $$), the resulting workspace is a compact, regular volume.
| Characteristic | Description | Value / Shape |
|---|---|---|
| Overall Shape | 3D volume composed of an upper quasi-conical region and a lower quasi-cylindrical region. | Regular, symmetric about z-axis. |
| Translational Range (x, y) | Maximum lateral motion of the platform center. | Approximately ±25 mm. |
| Translational Range (z) | Maximum vertical motion of the platform center. | Approximately 0 to 30 mm. |
| Cross-section (constant z) | Shape of horizontal slices through the workspace. | Approximate hexagon with 120° rotational symmetry; no internal holes. |
The lateral range is primarily governed by the actuator stroke and the link length l, while the vertical range is more directly influenced by the stroke itself. The absence of singularities and voids within this workspace confirms the mechanism’s suitability for continuous-path polishing operations.
Conclusion
This paper presented a detailed kinematic and performance analysis of a novel 3-PPaR parallel mechanism designed as a high-performance force-controlled end effector. The mechanism provides three pure translational degrees of freedom, enabling independent control of contact forces along the X, Y, and Z axes. This capability is essential for automated polishing in confined spaces and on complex surfaces where lateral force adjustment is necessary. Key advantages of this end effector design include:
- Compact and Symmetric Architecture: The 3-PPaR topology with star-configured limbs leads to a balanced and space-efficient structure.
- Simple and Decoupled Kinematics: Both inverse and forward kinematics solutions are available in closed form, facilitating real-time control.
- Singularity-Free Workspace: With the prudent design condition R > r and considering joint limits, the operational workspace is devoid of singularities, ensuring stable and full-rank force/torque transmission.
- Substantial and Regular Workspace: The reachable workspace is sufficiently large for polishing tasks and exhibits a regular shape without internal holes, ensuring uninterrupted tool paths.
The use of voice coil actuators further enhances the potential for high-bandwidth, precise force control. The analytical foundations established here—covering mobility, position analysis, Jacobian formulation, singularity conditions, and workspace characterization—provide a solid theoretical basis for the optimal dimensional design, dynamic modeling, and prototype development of this advanced force-controlled robotic end effector. Future work will focus on stiffness modeling, dynamic analysis, force control strategy implementation, and experimental validation of the polishing performance.