Morphing of and writing with a scissor linkage mechanism

TL;DR

The paper addresses programming complex shapes with a single-degree-of-freedom scissor-linkage by deriving an intrinsic unit curvature $\kappa_o(\alpha,\phi,l)$ and developing a differentiable framework to solve inverse-design tasks. It demonstrates shape morphing by optimally choosing per-unit aspect ratios $\alpha_j$ and actuating with $\Psi$, and shows writing trajectories by matching the distal-tip curvature $\kappa_{\text{tip}}(s)$ to targets. A multi-section formulation enables tracing varied trajectories, including circular paths and scripted letters, validated through experiments with 3D-printed units and servo actuation. The approach enables automated navigation and inspection in complex environments, while highlighting challenges from sensitivity to interior geometries and the need for feedback to achieve error-free experimental deployment.

Abstract

Kinematics of mechanisms is intricately coupled to their geometry and their utility often arises out of the ability to perform reproducible motion with fewer actuating degrees of freedom. In this article, we explore the assembly of scissor-units, each made of two rigid linear members connected by a pin joint. The assembly has a single degree of freedom, where actuating any single unit results in a shape change of the entire assembly. We derive expressions for the effective curvature of the unit and the trajectory of the mechanism's tip as a function of the geometric variables which we then use as the basis to program two tasks in the mechanism: shape morphing and writing. By phrasing these tasks as optimization problems and utilizing the differentiable simulation framework, we arrive at solutions that are then tested in table-top experiments. Our results show that the geometry of scissor assemblies can be leveraged for automated navigation and inspection in complex domains, in light of the optimization framework. However, we highlight that the challenges associated with rapid programming and error-free implementation in experiments without feedback still remain.

Figures (10)

• Figure 1: Geometry of scissor linkage mechanism. (A) A single scissor-unit is made of two rigid members of length $l$ connected by a pin joint. Each unit is described by two geometric variables: angle between members $\phi$; the aspect ratio $\alpha$ defined as relative distance of the pin joint from the end to the total length, $l$. (B) Vectors $\hat{\mathbf{t}}^{(1)}$ and $\hat{\mathbf{t}}^{(2)}$ capture the orientation of the members $ac$, $bd$. Along the edges $ac$, $bd$ we define two normal vectors $\hat{\mathbf{N}}_e$ and $\hat{\mathbf{N}}_w$. The tangent vectors, $\hat{\mathbf{T}}_e$ and $\hat{\mathbf{T}}_w$, are defined orthogonal to these normal vectors. $\Delta_o$ is the width of the segment passing through the pin joint and intersecting faces $ad, bc$. (C) Contours of $\kappa_o(\alpha, \phi)$, the effective curvature of each scissor-unit defined in Eq. \ref{['eq:kappaFinal']}. $\kappa_o$ diverges for $\phi \to 0$ and vanishes at the symmetric midpoint $\alpha=0.5$. (D) Comparison between theoretical prediction of the critical actuation angle $\Psi^*$ (solid line) required to achieve self-intersection ($N\phi^*=2\pi$) and experimentally measured angles (green markers). Error bars are smaller than the size of the markers. (E) Scissor-units are assembled by connecting to neighboring units via pin joints resulting in a single degree-of-freedom for the mechanism. The location of center pin joint of $j$-th unit is denoted by $\mathbf{r}_j$ and the radius of curvature by $\kappa_j^{-1}$. Actuation angle $\Psi$ is the angle between the members of the first unit, $\Psi = \phi_1$ and is used to control the motion of the assembly. $\phi^*$ is defined as the angle between the lines connecting the sides $ad$ and $bc$ in Fig. \ref{['fig:schmUnit']}(B)). (F) Sequence of configurations of a scissor mechanism from experiments whose aspect ratios are given by $\{\alpha_j\} = 0.45$ for $0<j<6$ and $\{\alpha_j\} = 0.55$ for $6 \leq j < 12$. Decreasing the actuation angle $\Psi$ via Arduino controlled motor drive (see Sec. \ref{['Model']} for details) steers the mechanism from a near-straight configuration (left) to a curved shape (right), illustrating large-scale shape change enabled by the intrinsic geometry.
• Figure 2: Shape morphing of scissor mechanism. (A) 3D printed scissor-units made with an adjustable slider allowing continuous variation in the aspect ratio $\alpha$ along with a screw that can be locked into position. These units are connected to neighboring units via pin joints. (B) In the shape morphing task, the mechanism is designed to take a target shape by choosing appropriate aspect ratios $\{\alpha_j\}$ and actuating the mechanism to angle $\Psi^*$. These values are arrived at by minimizing the shape error, $\mathcal{L}$ between the current configuration, $\mathbf{r}_j$ and the target shape described by a curvature $\kappa^t$ (see SI Sec. \ref{['StaticDesignDetails']} for details). (C) Examples showing the scissor mechanism deployed to target shapes (curves in blue) in experiments and simulation (see SI Video 2 for the deployment sequence). The target shapes have $(i)$ a monotonically increasing curvature (left); $(ii)$ a sinusoidal curvature profile (middle); $(iii)$ a three-petaled flower (right). (D) Evolution of the mechanism shape in experiments upon actuation by decreasing $\Psi$. Inset shows the solution aspect ratios $\alpha_j$ obtained via the optimization procedure detailed in Sec. \ref{['Static Inverse Design']}.
• Figure 3: Writing with a scissor mechanism. (A) Contour in the $(\alpha,\Psi)$ plane showing the curvature of the distal end of the mechanism $\kappa_{\textrm{tip}}(\alpha,\Psi;N)$ made of $N$ scissor-units with constant $\alpha$. Three different contours corresponding to $N=5$ (orange), $N=10$ (green), and $N=20$ (blue) are shown and iso-contour labels correspond to magnitude of $\kappa_{\textrm{tip}}$. Insets illustrate tip trajectories $\mathbf r_{\textrm{tip}}(\Psi)$ (for $N=5$) obtained by sweeping $\Psi$ from $\pi$ to $0$ at the corresponding parameter locations. As $\alpha$ departs from $0.5$ the trajectory rapidly develops tight loops (large $\kappa_{\textrm{tip}}$), whereas $\alpha\to 0.5$ yields an almost straight path with $\kappa_{\textrm{tip}}\approx 0$. We see that the mechanism is capable of tracing trajectories with rapidly changing tip curvature, however, they vary only monotonically. (B) Representative tip trajectories generated using the forward kinematics formulation for segmented mechanisms in Eq. \ref{['eq:seg_update']}. Trajectories are for $J=3$ sections with distinct aspect ratios ($\alpha_j = \{0.41, 0.17, 0.55 \}$ for solid line, $\alpha_j = \{ 0.44,0.43,0.23\}$ for short dashed line, $\alpha_j = \{0.71, 0.32, 0.51 \}$ for long dashed line, $\alpha_j = \{ 0.41, 0.62, 0.89 \}$ for dot-dashed line). The mechanism is capable of tracing complex paths with varying curvature, including spirals and self-intersecting loops. (C) The mechanism is parameterized by constant aspect ratios $\{\alpha_j\}$ (function in green), which determine the tip trajectory $\mathbf{r}_{\textrm{tip}}(\Psi)$ (shown in red) through the forward kinematic relations in Eq. \ref{['eq:seg_update']}. The resulting trajectory is compared against a prescribed target curve (blue) in the tip-trajectory error $\mathcal{L}_{\textrm{tip}}$ (see Eq. \ref{['eq:curv_loss']}), which quantifies the curvature mismatch between $\kappa_{\textrm{tip}}(s;\{\alpha_j,N_j\},\Psi)$ and the target curvature $\kappa^{t}(s)$. The differentiable simulation provides exact derivatives (such as $\partial \mathbf r_{\textrm{tip}}/\partial \alpha_j$) via automatic differentiation, enabling efficient gradient-based updates of $\{\alpha_j\}$ (for fixed $\{N_j\}$) and the actuation input $\Psi$ range. (D) Snapshots of the scissor mechanism's deployment as the tip traces a circular trajectory. The red colored stroke indicates the tip trajectory, and the inset shows the aspect-ratio, $\{\alpha_j\}$ obtained using the optimization procedure. (E) Comparison between target curves (blue) and tip trajectories (red) when the mechanism traces the characters $j$ and $D$, as a tribute to the automaton artist Jaques-Droz. The color gradient from light to dark indicates tracing direction as $\Psi$ decreases from $\Psi_{\max}$ to $\Psi_{\min}$. Insets show the optimized aspect-ratio profiles $\{\alpha_j\}$, where $j$ indexes the scissor-units. See SI Video 3 for a visualization of the mechanism performing the writing task.
• Figure S1: (A) Three scissor-units with the same aspect ratio are assembled by connecting neighboring units. Their center nodes are marked by the yellow points $\mathbf{r}_{j-1}$, $\mathbf{r}_{j}$, and $\mathbf{r}_{j+1}$. Using the turning-angle definition, the curvature is given by the angle $\vartheta_j$ between the vectors joining adjacent center nodes, namely $\mathbf{v}_{j-1}$ and $\mathbf{v}_{j}$ (visualized in green). The associated length scale is defined as $|\mathbf{v}_{j-1}|+|\mathbf{v}_{j}|$. (B) For a single scissor-unit, the green circle is the osculating circle of radius $R_{\mathrm{osc}}$; the member orientation vectors $\hat{\mathbf{t}}^{(1)}$ and $\hat{\mathbf{t}}^{(2)}$ are tangent to this circle. (C) A two-segment scissor mechanism for an arbitrary choice of the base center node position $\mathbf{r}_0$ and the initial orientation $\hat{\mathbf{p}}_1$. For a given actuation angle $\Psi$, each section $j$ is modeled as a circular arc with center of curvature $\tilde{\mathbf{q}}_j$ and radius $\kappa_j^{-1}$. At the interface between sections (node $\mathbf{r}_1$), the change in aspect ratio from $\alpha_1$ to $\alpha_{2}$ requires a shift $\zeta_j^{j+1}$ between successive centers of curvature to maintain geometric continuity (from $\tilde{\mathbf{q}}_1$ to $\tilde{\mathbf{q}}_2$ ). The final tip position $\mathbf{r}_{\text{tip}}$ is obtained by iterating these section-wise rotations and translations.
• Figure S2: (A) Scissor mechanism corresponding to $N=30$ with $\alpha = 0.52$ and $\Psi = \pi/4$. (B) Comparison between shape of scissor mechanism with $N=30$ and $\Psi= \pi/4$ constructed from full non-linear simulation (shown in blue) for $\alpha_j = \alpha_0 + \epsilon j$ and perturbative solution in Eq. \ref{['eq:phi_star_linear']} (shown in gray). We see a good match between the perturbative solution and the full solution. We have chosen $\alpha_0 = 0.52, \epsilon = 0.001$.