Design of a large stroke compliant gripping mechanism for constant-force applications

Abstract

A constant force compliant mechanism (CFCM) prevents a mechanical gripper from exerting excessive contact force through passive force control. In response to changes in mechanical stiffness induced by structural deformation, it can generate nearly constant extrinsic loads throughout the effective range of input displacement. This research aims to design an embedded CFCM end effector for manipulators capable of grasping geometrically and mechanically unknown objects without the need of sensory equipment. The proposed topology optimization method synthesizes CFCMs through discrete parameterization, loop detection, deformation analysis, objective domain enumeration, and function error evaluation, utilizing a genetic algorithm. The algorithm minimizes errors and improves solution accuracy by deducing solution feasibility through several penalty functions. Finite element analysis verifies the performance, deformation, and stress conditions of the optimal CFCM. The study includes manufacturing a CFCM prototype and assembling for end-effector, followed by experiments to validate analytical results and assessing actual grasping ability. The proposed design provides an extended range of motion and improved force stability, laying a groundwork for future research in this area.

Appendices

Appendix 1: Mathematical framework for VFIFE

In this study, we employed the vector form intrinsic finite element (VFIFE) method [36, 37] and incorporated it into the topology optimization process.

The VFIFE framework describes the motion of a continuous body by using a finite number of points (or particles) associated with the theory of vector analysis. As shown in Fig. 

Fig. 20

Description of motion and the path of a particle

20, a set of points representing the configuration of a frame structure is displaced from one position to another, where m denotes the mass of an arbitrary point in the structure and x denotes the position of the point of interest. Assuming that the interaction among points is modeled by a beam element, the properties of each element can be characterized by nodal displacements and corresponding nodal forces.

1.1 Discrete path and governing equation

Assume the particle in the structure undergoes a motion from time t_a to t whose path is called the path element and within which the deformation of the structure is infinitesimal. Then, applying infinitesimal strain and engineering stress to evaluate continuum stress and compute virtual work is feasible. The subsequent formulation involves displacement and deformation processes, calculating equivalent internal forces and external forces, and solving the equation of motion for each particle.

The VFIFE method begins by modeling the individual movement of every particle following Newton’s 2nd Law of Motion as its governing equation:

$${m\varvec{\ddot{d}}} = {\varvec{F}}_\text{ext} { - }\sum\limits_{i = 1}^{{n_{e} }} {{\varvec{f}}_{{\text{int}}}^{i} }$$

(A1)

where d is the displacement vector, including translational and rotational displacements of the particle, F_ext is the external forces including the concentrated external force vector applied at the particle and the equivalent nodal force vector from the load on the element,${\varvec{f}}_{{\text{int}}}^{i}$ is the equivalent nodal forces due to deformation in elements, n_e is the number of elements incident with the particle of interest, and m is the nodal mass whose elements contain the mass and mass moment of inertia of the particle.

1.2 Separation of rigid body motion from total displacement

To obtain the deformation in an element, the displacement of rigid body motion should be first separated from the overall displacement. Figure 

Fig. 21

Reverse rotation of a beam element

21 shows the positions of an element containing two nodes (j, k) at two consecutive time instances. If the positions of node j at two-time instances are fictitiously aligned, the angular displacement of the rigid body motion can be found as

$$\varphi = \sin^{ - 1} \left| {{\hat{\mathbf{e}}}_{x} \times {\mathbf{\hat{e}^{\prime}}}_{x} } \right|$$

(A2)

where ${\hat{\mathbf{e}}}_{x}$ is the x-axis of local coordinate system established on the element at time t_a and ${\mathbf{\hat{e}^{\prime}}}_{x}$ is the x-axis of local coordinate system established at time t_b. Meanwhile, the change of the element displacement follows

$${\mathbf{u}}_{{\mathbf{k}}} {\mathbf{ = u}}^{{\mathbf{r}}}_{{\mathbf{k}}} { + }{\mathbf{u}}^{{\mathbf{d}}}_{{\mathbf{k}}}$$

(A3)

where ${\mathbf{u}}_{k}^{d}$ and ${\mathbf{u}}_{k}^{r}$ are respectively the changes of displacement in the axial and lateral directions. Also, the angular deformation of the nodes (θ ^j, θ^k) can be written as

$$\theta^{j} = \beta^{j} - \varphi {\text{ and }}\theta^{k} = \beta^{k} - \varphi$$

(A4)

1.3 Internal force evaluation

The incremental internal forces in the element during two-time instances in a motion are raised by the pure deformation in the element. For detailed derivations of the formula, please refer to [34]. The resulting equation can be expressed as:

$$\Delta {\varvec{\hat{f}}}{ = }\frac{{{E}}}{{{{l}}_{{\text{a}}} }}\left[ {\begin{array}{*{20}c} {{A}} & 0 & 0 \\ 0 & {4{{I}}} & {2{{I}}} \\ 0 & {2{{I}}} & {4{{I}}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} \Delta \\ {{\theta }^{{{j}}} } \\ {{\theta }^{{{k}}} } \\ \end{array} } \right]$$

(A5)

where $\Delta \hat{\varvec{f}} = [\Delta \hat{{f}}_{{{j,x}}} {,}\;\Delta {m}_{{{j,z}}} {,}\;\Delta {m}_{{{k,z}}} ]^{{\text{T}}}$, Δ is the magnitude of ${\mathbf{u}}_{k}^{d}$, and E, I, A, and l_a are the Young’s modulus, second moment of area, area of cross section, and element length of the object, respectively. In addition, three unknown force components in each node can be obtained by the static equilibrium of the element:

$$\begin{array}{*{20}c} {\sum {F_{{\hat{x}}} } = 0} & : & {\Delta \hat{f}_{1x} = - \Delta \hat{f}_{2x} } \\ {\sum {F_{{\hat{y}}} } = 0} & : & {\Delta \hat{f}_{1y} = - \Delta \hat{f}_{2y} } \\ {\sum {M_{1a} } = 0} & : & {\Delta \hat{f}_{2y} = - \left( {\Delta m_{1z} + \Delta m_{2z} } \right)/l_{a} } \\ \end{array}$$

(A6)

Once the incremental internal forces for each node are known, the new nodal forces, ${\hat{\varvec{f}}}$, can be obtained as:

$${\hat{\varvec{f}}}= {\hat{\varvec{f}}}_{{a}} { + }\Delta \hat{f}$$

(A7)

Note that the hat indicates the internal forces are calculated in the local coordinate system at time t_a. This force must be transformed to the global coordinate system first and then to the time state t_b by the following equation:

$${\varvec{f}}_\text{int} = {\mathbf{R\Omega} \varvec{\hat{f}}}$$

(A8)

where Ω is the transformation matrix from local to global coordinate system and R is the rotation matrix from coordinate at time t_a to coordinate at time t_b.

1.4 Time integration

To ensure a rapid attainment of a stable static solution, we incorporate a DRM [38] with kinetic damping to minimize the system’s potential energy efficiently. Additionally, we can extract a steady reaction force corresponding to the input node’s displacement. Time integration techniques are required to solve the equation of motion for particles within a path element as described in Eq. (A1). In this work, we apply the central difference explicit scheme. Alternatively, if we opt for DRM, we need to derive the kinetic energy of the particle during the time segment. In this approach, the static equilibrium in a conservative system aligns with the condition where the potential energy reaches a minimal value. Let the displacement equation without damping be expressed as:

$$d_{n + 1} = \frac{{(\Delta t)^{2} }}{m}(F_{n}^{{{\text{ext}}}} - F_{n}^{{\text{int}}} ) + 2d_{n} - d_{n - 1}$$

(A9)

The nodal velocities for the next step can be represented by the half-interval method as

$$\dot{d}_{n - 1/2} = (d_{n} - d_{n - 1} )/\Delta t$$

(A10)

$$\dot{d}_{n + 1/2} = (d_{n + 1} - d_{n} )/\Delta t$$

(A11)

Then, the minimum potential energy can be detected by checking if the inequality below holds at each instance:

$$\sum {\frac{1}{2}(m_{i} \dot{d}_{n + 1/2}^{2} )} < \sum {\frac{1}{2}(m_{i} \dot{d}_{n - 1/2}^{2} )}$$

(A12)

If it holds, then set $\dot{d}_{n - 1/2}$ to obtain $d_{n} = d_{n - 1}$. This yields a new displacement:

$$d_{n + 1} = \frac{{(\Delta t)^{2} }}{m}(F_{n}^{{{\text{ext}}}} - F_{n}^{{\text{int}}} ) + d_{n}$$

(A13)

By iteratively applying Eq. (A9) and Eq. (A13) based on the condition of kinetic energy, we can attain the static solution for a dynamic system. Fig.

Fig. 22

Flowchart of VFIFE algorithm for quasi-static nonlinear analysis

22 depicts the algorithm of VFIFE coupled with kinetic DRM for quasi-static nonlinear structural analysis.

In the explicit quasi-static nonlinear analysis, the time step is set to Δt = 10⁻⁵, and the convergence threshold is defined as the kinetic energy E_k < 10⁻¹⁵. The relationship between the applied displacement and time is given as:

$$d_{{t,{\text{control}}}} = \left\{ {\begin{array}{*{20}c} {\frac{t}{{t_{{{\text{set}}}} }}d_{{{\text{control}}}} { , }t < t_{{{\text{set}}}} } \\ {d_{{{\text{control}}}} { , }t \ge t_{{{\text{set}}}} } \\ \end{array} } \right.$$

(A14)

where d_t,control is the displacement at time t, t_set is the time at which the control displacement is reached, and d_control is the goal of displacement control value.

Appendix 2: Efficiency of the optimization

The GA toolbox in MATLAB was used in this study. The optimization was initiated with 80 individuals and terminated at a maximum generation limit of 70 iterations. The algorithm will stop if tolerance between two steps is less than 10⁻⁵. In the finite element analysis performed by VFIFE, the number of iterations for each displacement control point (1–30 mm) to reach a steady state for three types of structures, are shown in Fig.

Fig. 23

Number of iterations for three types of synthesized structure

23. Calculating one individual of the data took approximately 25–30 s of CPU time, resulting in about 45 h to achieve one final successful result.