The pursuit of automation in agriculture, particularly in harvesting delicate produce like apples, presents a significant challenge: achieving reliable physical manipulation without causing bruising or damage. The end effector, the component that directly interacts with the fruit, is critical to this task. Traditional end effector designs often rely on complex sensor feedback and control algorithms to regulate gripping force, which can increase cost and complexity. This work focuses on the development of a fundamentally different approach—integrating a passive, compliant constant-force mechanism into the end effector. This mechanism is designed to exhibit a near-constant reaction force over a specified range of motion, inherently limiting the maximum clamping force applied to the fruit, thereby enabling non-destructive harvesting with a simpler, more robust system.
The proposed end effector concept is based on a two-finger gripper. The key innovation lies in the connection between the driving actuator and the finger. Instead of a rigid link, a specially designed compliant buckling beam is inserted. This beam is engineered to deform in such a way that for a significant input displacement (the closing motion of the finger), the output force opposing that motion remains almost constant. This creates a natural “force-limiting” behavior.
1. Mechanical Design and Mathematical Modeling of the Compliant Beam
We consider one half of the symmetric end effector mechanism, consisting of two compliant beam segments, Beam 1 and Beam 2, connected in series. The beams have a rectangular cross-section with width \( t \) and thickness \( w \). The complete mechanism is anchored at point \( n_1 \), and point \( n_3 \) is subjected to a vertical displacement input \( y_{in} \) from the actuator, which represents the gripping motion. The goal is to determine the force \( F_y \) at \( n_3 \) as a function of \( y_{in} \) and to shape the beams so that \( F_y \) is constant over a desired range.
The undeformed shape of each beam \( i \) (where \( i=1,2 \)) is defined by a shape function \( \eta_i(u) \), which describes the angle of the beam’s neutral axis relative to the x-axis. The variable \( u \in [0,1] \) is the dimensionless arc length along the beam. We parameterize this shape using a polynomial:
$$ \eta_i(u) = c_{i0} + c_{i1}u + c_{i2}u^2 + … + c_{im}u^m $$
The coefficients \( c_{i0} … c_{im} \) define the initial curvature of the beam.
When a load is applied, the beam deforms. The deformed shape is described by another function \( \psi_i(u) \). The governing nonlinear differential equation for the large deflection of a compliant beam, based on Euler-Bernoulli beam theory and moment equilibrium, is:
$$ \frac{EI}{L_i^2} \frac{d^2\psi_i}{du^2} – [h_i \sin(\psi_i) – v_i \cos(\psi_i)] + \frac{d\eta_i}{du} = 0 $$
where \( EI \) is the flexural rigidity (with \( I = tw^3/12 \)), \( L_i \) is the beam length, and \( h_i \) and \( v_i \) are the internal reaction force components aligned with the x and y axes at the beam’s end, respectively. For our serial configuration, \( h_1 = h_2 = F_x \) and \( v_1 = v_2 = F_y \), which is the output force of interest.
The spatial coordinates of any point on the beam are given by:
$$ x_i(u) = x_i(u_0) + L_i \int_{u_0}^{u} \cos(\eta_i) \, du $$
$$ y_i(u) = y_i(u_0) + L_i \int_{u_0}^{u} \sin(\eta_i) \, du $$
The bending stress \( \sigma_i(u) \) in the beam is:
$$ \sigma_i(u) = \frac{E w}{2L_i} \left( \frac{d\psi_i}{du} – \frac{d\eta_i}{du} \right) $$
The system of equations is subject to the following boundary conditions:
- At fixed point \( n_1 \) (u=0 for Beam 1): \( x_1(0)=0, y_1(0)=0, \psi_1(0)=\eta_1(0) \).
- At connection point \( n_2 \): Continuity of position, angle, and moment, and equilibrium of forces.
$$ x_1(1) = x_2(0), \quad y_1(1) = y_2(0) $$
$$ \psi_1(1) = \psi_2(0) $$
$$ \frac{EI}{L_1} (\psi_1′(1) – \eta_1′(1)) = \frac{EI}{L_2} (\psi_2′(0) – \eta_2′(0)) $$
$$ h_1 = h_2, \quad v_1 = v_2 $$ - At input point \( n_3 \) (u=1 for Beam 2): \( x_2(1)=x_{n_3}, \quad y_2(1)=y_{in}, \quad h_2=F_x, \quad v_2=F_y \).
2. Numerical Solution Strategy: Shooting Method and Optimization
Solving the above boundary value problem (BVP) directly is complex. We employ the shooting method to convert it into an initial value problem (IVP). We define an unknown initial condition for the derivative \( s = \psi_1′(0) \) and the unknown reaction forces \( F_x \) and \( F_y \). We then integrate the following state-space formulation of the ODE system from \( u=0 \) to \( u=1 \) for each beam:
For each beam segment \( i \), the state vector is \( \mathbf{S}_i = [\psi_i, \psi_i’, x_i, y_i]^T \). The derivative of this state vector is:
$$
\frac{d\mathbf{S}_i}{du} =
\begin{bmatrix}
\psi_i’ \\
\frac{L_i^2}{EI} \left[ F_x \sin(\psi_i) – F_y \cos(\psi_i) \right] + \eta_i'(u) \\
L_i \cos(\psi_i) \\
L_i \sin(\psi_i)
\end{bmatrix}
$$
We start from initial states at \( n_1 \) for Beam 1, propagate to \( n_2 \), use continuity conditions to get initial states for Beam 2, and propagate to \( n_3 \). The correctness of the guessed unknowns (\( s, F_x, F_y \)) is determined by how well the final state at \( n_3 \) matches its boundary conditions: \( \psi_2(1)=\eta_2(1) \), \( x_2(1)=x_{n_3} \), and \( y_2(1)=y_{in} \). This defines a system of three nonlinear equations. To solve it robustly and avoid local minima, we first use a Genetic Algorithm (GA) to find a good initial guess for (\( s, F_x, F_y \)), which is then refined using a standard nonlinear solver (like Newton-Raphson).
3. Shape Optimization for Constant-Force Behavior
With the solver in place, we can now optimize the design parameters of the beams to achieve the constant-force property. The design variables are the polynomial coefficients \( c_{10}, c_{11}, c_{12}, c_{20}, c_{21}, c_{22} \) and the lengths \( L_1, L_2 \) (using a quadratic shape function, m=2). The objective is to minimize the variation of the output force \( F_y \) over a target displacement range from \( y_{in} = a \) to \( y_{in} = b \).
Optimization Problem Formulation:
Minimize:
$$ f(\mathbf{c}, L_1, L_2) = \left( \frac{F(b)}{F(a)} – 1 \right)^2 $$
where \( \mathbf{c} \) is the vector of all shape coefficients, and \( F(y) \) is the force-displacement function obtained from the solver.
Subject to:
- Geometric constraints to ensure the beams fit within the end effector envelope (e.g., \( x_{min} \leq x_i(u) \leq x_{max} \), \( y_{min} \leq y_i(u) \leq y_{max} \)).
- Length constraints: \( L_{1,min} \leq L_1 \leq L_{1,max}, \quad L_{2,min} \leq L_2 \leq L_{2,max} \).
- Stress constraint: The maximum bending stress \( \sigma_{max} \) at the maximum operational displacement \( y_{in}=c \) must be less than the allowable stress \( \sigma_{allow} = \sigma_y / SF \), where \( \sigma_y \) is the material yield strength and \( SF \) is a safety factor.
We solve this nonlinear constrained optimization problem using a Sequential Quadratic Programming (SQP) algorithm. The workflow is iterative: for each set of design parameters proposed by the optimizer, the shooting method solver calculates \( F(a) \), \( F(b) \), and \( \sigma_{max} \), and the objective and constraints are evaluated.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Elastic Modulus (POM) | \( E \) | 2.6 | GPa |
| Yield Strength | \( \sigma_y \) | 76 | MPa |
| Safety Factor | \( SF \) | 1.5 | – |
| Beam Width | \( t \) | 10 | mm |
| Beam Thickness | \( w \) | 1 | mm |
| Constant-Force Start | \( a \) | 0.7 | cm |
| Constant-Force End | \( b \) | 2.0 | cm |
| Max Displacement (for stress check) | \( c \) | 2.2 | cm |
4. Results: Optimized Design and Performance Analysis
Applying the optimization framework with the parameters in Table 1, we obtained an optimized compliant mechanism. The initial and final shape functions for the two beams are:
Initial Guess:
Beam 1: \( \eta_1(u) = 2.62u – 3.57u^2, \quad L_1 = 5.0 \, \text{cm} \)
Beam 2: \( \eta_2(u) = 0.82u – 2.00u^2, \quad L_2 = 4.5 \, \text{cm} \)
Optimized Design:
Beam 1: \( \eta_1(u) = 2.46u – 6.46u^2, \quad L_1 = 4.13 \, \text{cm} \)
Beam 2: \( \eta_2(u) = 0.45u – 1.71u^2, \quad L_2 = 4.54 \, \text{cm} \)
The force-displacement characteristics of the initial and optimized designs are shown conceptually below. The optimized beam exhibits a distinct plateau in the force response between displacements \( a \) and \( b \), while the initial design shows a monotonically increasing force.
| Displacement \( y_{in} \) (cm) | Optimized Force \( F_y \) (N) | Initial Design Force (N) |
|---|---|---|
| 0.7 (a) | 3.9116 | ~2.1 |
| 2.0 (b) | 3.9605 | ~6.8 |
| 2.2 (c) | 3.9640 | ~8.1 |
The optimized mechanism provides a nearly constant clamping force of approximately 3.95 N per side. For a two-finger end effector, this results in a total gripping force of about 7.9 N. Literature suggests that a force above 3 N is sufficient to hold a typical apple (with adequate friction), while forces exceeding 20 N may cause internal bruising. Therefore, this design falls well within the safe, non-destructive range. The constant-force behavior arises from a careful balance: Beam 2 is designed to transition from positive to negative stiffness over the displacement range, while Beam 1 provides compensating positive stiffness. Their combined effect produces a net near-zero stiffness (constant force) region.
The maximum stress was calculated to be 48.1 MPa at the maximum displacement \( c \), which is below the allowable stress of 50.7 MPa (\( \sigma_y / SF \)).
5. Validation: Finite Element Simulation and Physical Testing
The mathematical model’s accuracy was validated in two ways: nonlinear Finite Element Analysis (FEA) in Abaqus and physical tests on a 3D-printed prototype (using POM material).
FEA Validation: The optimized beam shape was modeled in Abaqus. The simulated force-displacement curve showed excellent agreement with the theoretical model. The average error in force was -0.18%, and the maximum stress at \( y_{in}=2.2 \, \text{cm} \) was 47.87 MPa, differing by only -0.48% from the calculated 48.10 MPa.
Physical Experiment: A test rig with a linear stage and a load cell was used to measure the force-displacement response of the fabricated compliant beam. The results confirmed the constant-force trend, though with a slight offset (~5% higher average force) compared to the model. This discrepancy is attributed to minor manufacturing imperfections and the simplified theoretical model which assumes perfectly thin beams, whereas the physical beam has a finite thickness that adds slight rigidity at the connections.
| Validation Method | Avg. Force Error (vs. Model) | Max Stress at \( y_{in}=c \) | Key Observation |
|---|---|---|---|
| Nonlinear FEA (Abaqus) | -0.18% | 47.87 MPa | Excellent agreement with theoretical model. |
| Physical Prototype Test | +5.06% | Not measured directly | Confirmed constant-force plateau; slight positive offset due to manufacturing. |
6. Apple Gripping Demonstration and Conclusion
Finally, a simplified gripping test was performed using two fingers, one of which incorporated the optimized compliant mechanism. An apple was placed between the fingers, and the moving finger was actuated. The measured clamping force stabilized in the range of 7.46 N to 8.42 N, with an average of 8.03 N. This aligns closely with the expected 7.9 N from the model and the physical beam test. In repeated tests on 20 apples, no fruit was dropped, and only one showed minor surface marking after 24 hours, indicating a 95% non-destructive success rate.
In conclusion, this work successfully developed a methodology for designing and optimizing a compliant constant-force mechanism for integration into a robotic end effector. The key accomplishment is the creation of a passive mechanical system that inherently limits gripping force, eliminating the need for sensitive force sensors and complex real-time control algorithms in the end effector. The mathematical model, solved via the shooting method and optimized with SQP, accurately predicts the mechanism’s behavior, as verified by FEA and physical tests. The demonstrated apple gripping tests confirm its practicality for non-destructive harvesting. This design approach for the end effector is highly adaptable; by scaling the beam thickness \( w \) or width \( t \), the magnitude of the constant-force output can be tuned, making this principle applicable to a wide variety of fruits and vegetables, paving the way for simpler, more robust, and cost-effective harvesting robots.