Clamping force manipulation in 2D compliant gripper topology optimization under frictionless contact

Abstract

This paper exploits a clamping force-constrained topology optimization method for compliant grippers, wherein the ability of clamping force manipulation enables safe grasping actions on fragile objects. Specifically, flexibility of the compliant grippers is realized by employing the neo-Hookean material model that comprises both geometric and material non-linearity. To simulate the clamping force, the node-to-segment method is employed for contact behavior modeling. Two types of clamping force-related constraints are formulated: the P-norm aggregated peak clamping force constraint and the clamping force variance constraint. The sensitivities of these constraints are derived with the adjoint method and integrated into the topology optimization algorithm to restrict the peak force through clamping force redistribution. In the numerical implementation, two types of objective functions: the strain energy for finger-type grippers and displacement output for compliant mechanism-type grippers are utilized and all the related sensitivities are rigorously derived. Additionally, to enhance robustness of the optimized structures, the minimum length scale is strictly restricted to prevent the single-node hinge problem. The effectiveness of the proposed method is proved through several 2D numerical examples.

Appendix

1.1 Sensitivity of objective

In this study, the strain-energy objective is only considered under the contactless state. According to Eqs. (20), (41), and (46), the sensitivity regarding physical variables is derived as

$$\frac{{\partial \psi_{{\text{s}}} }}{{\partial \overline{\rho }}} = \mathop \sum \limits_{e = 1}^{{n_{{{\text{ele}}}} }} p\left( {E_{{\text{o}}} - E_{{{\text{min}}}} } \right)\overline{\rho }^{p - 1} \phi_{{\text{E}}} - p\left( {E_{{\text{o}}} - E_{{{\text{min}}}} } \right)\overline{\rho }^{p - 1} {{\varvec{\uplambda}}}_{{{\text{sg}}}}^{{\text{T}}} {\mathbf{f}}_{{{\text{intg}}}} ,$$

(59)

where ${\mathbf{f}}_{{{\text{intg}}}}$ is the internal force under contactless condition. ${{\varvec{\uplambda}}}_{{{\text{sg}}}}$ is the solution of the adjoint equation as

$${\mathbf{K}}_{{{\text{tg}}}} {{\varvec{\uplambda}}}_{{{\text{sg}}}} = \mathop \sum \limits_{e = 1}^{{n_{{{\text{ele}}}} }} \frac{{\partial \phi_{e} }}{{\partial {\mathbf{u}}_{{\text{g}}} }},$$

(60)

with

$$\frac{\partial \phi }{{\partial {\mathbf{u}}_{{\text{g}}} }} = \frac{\partial \phi }{{\partial {\mathbf{C}}}}\frac{{\partial {\mathbf{C}}}}{{\partial {\mathbf{u}}_{{\text{g}}} }},$$

(61)

where ${\mathbf{K}}_{{{\text{tg}}}}$ is the structural tangent stiffness without contact. The right Cauchy–Green deformation tensor ${\mathbf{C}}$ equals ${\mathbf{F}}^{{\text{T}}} {\mathbf{F}}$. ${\mathbf{F}}$ is the deformation gradient. Then, the derivative of ${\mathbf{C}}$ with respect to ${\mathbf{u}}_{{\text{g}}}$ is calculated as

$$\frac{{\partial {\mathbf{C}}_{ij} }}{{\partial {\mathbf{u}}_{{\text{g}}} }} = N_{mk,i} F_{mj} + F_{mi} N_{mk,j} ,$$

(62)

in which $N_{mk}$ is one component of the element shape function matrix. The symbol $mk,j$ denotes the derivative of the $mk$ component regarding the $j^{\text{th}}$ component of the spatial coordinates.

Similarly, the sensitivity of the displacement-output objective is computed as

$$\frac{{\partial \psi_{{\text{d}}} }}{{\partial \overline{\rho }}} = - p\left( {E_{{\text{o}}} - E_{{{\text{min}}}} } \right)\overline{\rho }^{p - 1} \left( {{{\varvec{\uplambda}}}_{{{\text{dg}}}}^{{\text{T}}} {\mathbf{f}}_{{{\text{intg}}}} + {{\varvec{\uplambda}}}_{{{\text{dc}}}}^{{\text{T}}} {\mathbf{f}}_{{{\text{intc}}}} } \right),$$

(63)

where ${\mathbf{f}}_{{{\text{intg}}}}$ and ${\mathbf{f}}_{{{\text{intc}}}}$ are the internal forces under contactless and contact conditions, respectively. ${{\varvec{\uplambda}}}_{{{\text{dg}}}}$ and ${{\varvec{\uplambda}}}_{{{\text{dc}}}}$ are the solutions of the adjoint equations of Eqs. (64) and (65), respectively.

$${\mathbf{K}}_{{{\text{tg}}}} {{\varvec{\uplambda}}}_{{{\text{dg}}}} = 2\omega \left( {{\mathbf{l}}^{{\text{T}}} {\mathbf{u}}_{{\text{g}}} - u_{{\text{g}}}^{\# } } \right){\mathbf{l}},$$

(64)

$${\mathbf{K}}_{{{\text{tc}}}} {{\varvec{\uplambda}}}_{{{\text{dc}}}} = 2\left( {{\mathbf{l}}^{{\text{T}}} {\mathbf{u}}_{{\text{c}}} - u_{{\text{c}}}^{\# } } \right){\mathbf{l}}.$$

(65)

Herein, ${\mathbf{K}}_{{{\text{tc}}}}$ is the structural tangent stiffness under contact condition.

The sensitivity analysis is typically performed with respect to the physical density variable $\overline{\rho }$, however, to update the design variables, it is necessary to calculate the sensitivity with respect to $\rho .$ This can be achieved using the chain rules according to the PDE filter in Eq. (43) and the Heaviside projection in Eq. (21) as

(66)

1.2 Sensitivity of clamping force constraint

Before deriving the sensitivity of , we first calculate the derivative of $f_{{\text{n}}}$ with respect to ${\mathbf{u}}_{{\text{c}}}$. As given in Fig. 2, the contact possibly refers 6 conditions. The inside-left, inside-right, out-of-first, and out-of-last normal contact force have an identical expression as given in Eq. (2). Thus, the partial derivative is written as

$$\frac{{\partial f_{{\text{n}}} }}{{\partial {\mathbf{u}}_{{\text{c}}} }} = \epsilon {\mathbf{n}}^{{\text{T}}} \frac{{\partial {\mathbf{g}}}}{{\partial {\mathbf{u}}_{{\text{c}}} }} + \epsilon {\mathbf{g}}^{{\text{T}}} \frac{{\partial {\mathbf{n}}}}{{\partial {\mathbf{u}}_{{\text{c}}} }}.$$

(67)

In this research, we only explore the compliant gripper against relatively stiffer objects and thus, the normal direction regards as a constant. The vector ${\mathbf{g}}$ can be calculated with

$${\mathbf{g}} = {\mathbf{G}}\left( {{\mathbf{c}} + {\mathbf{u}}_{{\text{c}}} } \right),$$

(68)

where c is the undeforming state coordinate of structure. The matrix ${\mathbf{G}}$ can extract the vector ${\mathbf{g}}$ from vector ${\mathbf{c}} + {\mathbf{u}}_{{\text{c}}}$. Hence,

$$\frac{{\partial {\mathbf{g}}}}{{\partial {\mathbf{u}}_{{\text{c}}} }} = {\mathbf{G}}.$$

(69)

Therefore, $\frac{{\partial f_{{\text{n}}} }}{{\partial {\mathbf{u}}_{{\text{c}}} }}$ defines as

$$\frac{{\partial f_{{\text{n}}} }}{{\partial {\mathbf{u}}_{{\text{c}}} }} = \epsilon {\mathbf{n}}^{{\text{T}}} {\mathbf{G}}.$$

(70)

For the out-of-both and in-of both, the normal contact force is formulated as

$$f_{{\text{n}}} = w_{{\text{r}}} \epsilon g_{{{\text{nr}}}} + w_{{\text{l}}} \epsilon g_{{{\text{nl}}}} .$$

(71)

According to Eqs. (4) and (5), the derivatives of $g_{{{\text{nr}}}}$ and $g_{{{\text{nl}}}}$ with respect to ${\mathbf{u}}_{{\text{c}}}$ read

$$\frac{{\partial g_{{{\text{nr}}}} }}{{\partial {\mathbf{u}}_{{\text{c}}} }} = {\mathbf{n}}_{{\text{r}}}^{{\text{T}}} {\mathbf{G}}_{{\mathbf{r}}} ,$$

(72)

$$\frac{{\partial g_{{{\text{nl}}}} }}{{\partial {\mathbf{u}}_{{\text{c}}} }} = {\mathbf{n}}_{{\text{l}}}^{{\text{T}}} {\mathbf{G}}_{{\text{l}}} ,$$

(73)

in which ${\mathbf{G}}_{{\text{r}}}$ and ${\mathbf{G}}_{{\text{l}}}$ can extract the vector ${\mathbf{g}}_{{\text{r}}}$ and ${\mathbf{g}}_{{\text{l}}}$ from ${\mathbf{c}} + {\mathbf{u}}_{{\text{c}}}$. The weights have different expressions in out-of-both and in-of-both conditions. For out-of-both, according to Eqs. (6), (7), (8), and (9), the derivatives of $w_{{\text{r}}}$ and $w_{{\text{l}}}$ with respect to ${\mathbf{u}}_{{\text{c}}}$ are expressed as

$$\frac{{\partial w_{{\text{r}}} }}{{\partial {\mathbf{u}}_{{\text{c}}} }} = - \frac{{p_{{\text{r}}} {\mathbf{t}}_{{\text{l}}}^{{\text{T}}} + p_{{\text{l}}} {\mathbf{t}}_{{\text{r}}}^{{\text{T}}} }}{{\left( {p_{{\text{r}}} + p_{{\text{l}}} } \right)^{2} }}{\mathbf{G}}_{{\text{l}}} ,$$

(74)

$$\frac{{\partial w_{{\text{l}}} }}{{\partial {\mathbf{u}}_{{\text{c}}} }} = \frac{{p_{{\text{r}}} {\mathbf{t}}_{{\text{l}}}^{{\text{T}}} + p_{{\text{l}}} {\mathbf{t}}_{{\text{r}}}^{{\text{T}}} }}{{\left( {p_{{\text{r}}} + p_{{\text{l}}} } \right)^{2} }}{\mathbf{G}}_{{\text{l}}} .$$

(75)

The derivatives of $w_{{\text{r}}}$ and $w_{{\text{l}}}$ with respect to ${\mathbf{u}}_{{\text{c}}}$ under in-of-both condition have the same expression as Eq. (74) and Eq. (75). Considering Eq. (71), Eq. (72), Eq. (73), Eq. (74), and Eq. (75), $\frac{{\partial f_{{\text{n}}} }}{{\partial {\mathbf{u}}_{{\text{c}}} }}$ is formulated as

$$\frac{{\partial f_{{\text{n}}} }}{{\partial {\mathbf{u}}_{{\text{c}}} }} = \epsilon w_{{\text{r}}} {\mathbf{n}}_{{\text{r}}}^{{\text{T}}} {\mathbf{G}}_{{\text{r}}} + \epsilon w_{{\text{l}}} {\mathbf{n}}_{{\text{l}}}^{{\text{T}}} {\mathbf{G}}_{{\text{l}}} + \left( {\epsilon g_{{{\text{nl}}}} - \epsilon g_{{{\text{nr}}}} } \right)\frac{{p_{{\text{r}}} {\mathbf{t}}_{{\text{l}}}^{{\text{T}}} + p_{{\text{l}}} {\mathbf{t}}_{{\text{r}}}^{{\text{T}}} }}{{\left( {p_{{\text{r}}} + p_{{\text{l}}} } \right)^{2} }}{\mathbf{G}}_{{\text{l}}} .$$

(76)

According to the P-norm aggregation of Eq. (50), the $\frac{{\partial F_{{{\text{pn}}}} }}{{\partial {\mathbf{u}}_{{\text{c}}} }}$ is expressed as

$$\frac{{\partial F_{{{\text{pn}}}} }}{{\partial {\mathbf{u}}_{{\text{c}}} }} = \frac{{c_{{\text{p}}} }}{{f_{{\text{n}}}^{\# } }}\left( {\mathop \sum \limits_{{m \in {\mathcal{V}}}} \left( {\frac{{f_{{{\text{n}}.m}} }}{{f_{{\text{n}}}^{\# } }}} \right)^{{p_{{\text{n}}} }} } \right)^{{\frac{1}{{p_{{\text{n}}} }} - 1}} \left( {\mathop \sum \limits_{{m \in {\mathcal{V}}}} \left( {\frac{{f_{{{\text{n}}.m}} }}{{f_{{\text{n}}}^{\# } }}} \right)^{{p_{{\text{n}}} - 1}} \frac{{\partial f_{{{\text{n}}.m}} }}{{\partial {\mathbf{u}}_{{\text{c}}} }}} \right).$$

(77)

Thus, according to Eqs. (50), (70), and (76), $\frac{{\partial F_{{{\text{pn}}}} }}{{\partial {\mathbf{u}}_{{\text{c}}} }}$ is calculated through

$$\frac{{\partial F_{{{\text{pn}}}} }}{{\partial {\mathbf{u}}_{{\text{c}}} }} = \frac{{c_{{\text{p}}} }}{{f_{{\text{n}}}^{\# } }}\left( {\mathop \sum \limits_{{m \in {\mathcal{V}}}} \left( {\frac{{f_{{{\text{n}}.m}} }}{{f_{{\text{n}}}^{\# } }}} \right)^{{p_{{\text{n}}} }} } \right)^{{\frac{1}{{p_{{\text{n}}} }} - 1}} \left( {\mathop \sum \limits_{{m \in {\mathcal{V}}^{1} }} \left( {\frac{{f_{{{\text{n}}.m}} }}{{f_{{\text{n}}}^{\# } }}} \right)^{{p_{{\text{n}}} - 1}} \epsilon {\mathbf{n}}^{{\text{T}}} {\mathbf{G}} + \mathop \sum \limits_{{m \in {\mathcal{V}}^{2} }} \left( {\frac{{f_{{{\text{n}}.m}} }}{{f_{{\text{n}}}^{\# } }}} \right)^{{p_{{\text{n}}} - 1}} \left( {\epsilon w_{{\text{r}}} {\mathbf{n}}_{{\text{r}}}^{{\text{T}}} {\mathbf{G}}_{{\text{r}}} + \epsilon w_{{\text{l}}} {\mathbf{n}}_{{\text{l}}}^{{\text{T}}} {\mathbf{G}}_{{\text{l}}} + \left( {\epsilon g_{{{\text{nl}}}} - \epsilon g_{{{\text{nr}}}} } \right)\frac{{p_{{\text{r}}} {\mathbf{t}}_{{\text{l}}}^{{\text{T}}} + p_{{\text{l}}} {\mathbf{t}}_{{\text{r}}}^{{\text{T}}} }}{{\left( {p_{{\text{r}}} + p_{{\text{l}}} } \right)^{2} }}{\mathbf{G}}_{{\text{l}}} } \right)} \right),$$

(78)

where ${\mathcal{V}}^{1}$ is linked to the inside-left, inside-right, out-of-first, and out-of-last conditions. And ${\mathcal{V}}^{2}$ refers to the out-of-both and in-of-both conditions.

Regarding the variance constraint, Eq. (53) is reorganized into a concise expression, as

$$F_{{{\text{var}}}} = \frac{1}{{F_{{{\text{var}}}}^{\# } }}\frac{1}{{n_{{\text{c}}} }}\mathop \sum \limits_{{m \in {\mathcal{V}}}} \left( {f_{{{\text{n}}.m}} } \right)^{2} - \frac{1}{{F_{{{\text{var}}}}^{\# } }}\left( {\mathop \sum \limits_{{m \in {\mathcal{V}}}} \frac{{f_{{{\text{n}}.m}} }}{{n_{{\text{c}}} }}} \right)^{2} .$$

(79)

The term $\frac{{\partial F_{{{\text{var}}}} }}{{\partial {\mathbf{u}}_{{\text{c}}} }}$ is shown below

$$\frac{{\partial F_{{{\text{var}}}} }}{{\partial {\mathbf{u}}_{{\text{c}}} }} = \frac{1}{{F_{{{\text{var}}}}^{\# } }}\frac{2}{{n_{{\text{c}}} }}\mathop \sum \limits_{{m \in {\mathcal{V}}}} \left( {f_{{{\text{n}}.m}} \frac{{\partial f_{{{\text{n}}.m}} }}{{\partial {\mathbf{u}}_{{\text{c}}} }}} \right) - \frac{2}{{F_{{{\text{var}}}}^{\# } }}\left( {\mathop \sum \limits_{{m \in {\mathcal{V}}}} \frac{{f_{{{\text{n}}.m}} }}{{n_{{\text{c}}} }}} \right)\mathop \sum \limits_{{m \in {\mathcal{V}}}} \left( {\frac{1}{{n_{{\text{c}}} }}\frac{{f_{{{\text{n}}.m}} }}{{\partial {\mathbf{u}}_{{\text{c}}} }}} \right).$$

(80)

According to Eqs. (70) and (76), $\frac{{\partial F_{{{\text{var}}}} }}{{\partial {\mathbf{u}}_{{\text{c}}} }}$ yields

$$\begin{gathered} \frac{{\partial F_{{{\text{var}}}} }}{{\partial {\mathbf{u}}_{{\text{c}}} }} = \frac{1}{{F_{{{\text{var}}}}^{\# } }}\frac{2}{{n_{{\text{c}}} }}\left( {\mathop \sum \limits_{{m \in {\mathcal{V}}^{1} }} \left( {f_{{{\text{n}}.m}} \frac{\epsilon }{{n_{{\text{c}}} }}{\mathbf{n}}^{{\text{T}}} {\mathbf{G}}} \right) + \mathop \sum \limits_{{m \in {\mathcal{V}}^{2} }} f_{{{\text{n}}.m}} \left( {\epsilon w_{{\text{r}}} {\mathbf{n}}_{{\text{r}}}^{{\text{T}}} {\mathbf{G}}_{{\text{r}}} + \epsilon w_{{\text{l}}} {\mathbf{n}}_{{\text{l}}}^{{\text{T}}} {\mathbf{G}}_{{\text{l}}} + \left( {\epsilon g_{{{\text{nl}}}} - \epsilon g_{{{\text{nr}}}} } \right)\frac{{p_{{\text{r}}} {\mathbf{t}}_{{\text{l}}}^{{\text{T}}} + p_{{\text{l}}} {\mathbf{t}}_{{\text{r}}}^{{\text{T}}} }}{{\left( {p_{{\text{r}}} + p_{{\text{l}}} } \right)^{2} }}{\mathbf{G}}_{{\text{l}}} } \right)} \right) \hfill \\ - \frac{2}{{F_{{{\text{var}}}}^{\# } }}\left( {\mathop \sum \limits_{{m \in {\mathcal{V}}}} \frac{{f_{{{\text{n}}.m}} }}{{n_{{\text{c}}} }}} \right)\left( {\mathop \sum \limits_{{m \in {\mathcal{V}}^{1} }} \frac{1}{{n_{{\text{c}}} }}\epsilon {\mathbf{n}}^{{\text{T}}} {\mathbf{G}} + \mathop \sum \limits_{{m \in {\mathcal{V}}^{2} }} \frac{1}{{n_{{\text{c}}} }}\left( {\epsilon w_{{\text{r}}} {\mathbf{n}}_{{\text{r}}}^{{\text{T}}} {\mathbf{G}}_{{\text{r}}} + \epsilon w_{{\text{l}}} {\mathbf{n}}_{{\text{l}}}^{{\text{T}}} {\mathbf{G}}_{{\text{l}}} + \left( {\epsilon g_{{{\text{nl}}}} - \epsilon g_{{{\text{nr}}}} } \right)\frac{{p_{{\text{r}}} {\mathbf{t}}_{{\text{l}}}^{{\text{T}}} + p_{{\text{l}}} {\mathbf{t}}_{{\text{r}}}^{{\text{T}}} }}{{\left( {p_{{\text{r}}} + p_{{\text{l}}} } \right)^{2} }}{\mathbf{G}}_{{\text{l}}} } \right)} \right). \hfill \\ \end{gathered}$$

(81)

The sensitivity of clamping force constraint is computed through

(82)

with the adjoint equation of

(83)

where is given in Eqs. (78) and (81). Sensitivity of the constraints can be similarly obtained using the chain rule, according to Eqs. (43) and (45). The constraint sensitivity with respect to ρ is expressed as

(84)