Thin-film piezoelectric and high-aspect ratio polymer leg mechanisms for millimeter-scale robotics

Abstract

Millimeter-scale micro-robotic leg actuators are described that integrate thin-film lead zirconate titanate (PZT) piezoelectric actuation with compliant structures formed from parylene-C polymer. Models for out-of-plane rotation using piezoelectric cantilevers and in-plane rotation using high-aspect ratio polymer beams are discussed. Opportunities are highlighted for thin polymer films to aid in load-bearing, flexibility, and resilience of piezoelectric micro-robot appendages. A simple 5 mm × 2.4 mm × 0.15 mm hexapod robot prototype incorporating multiple actuators in each leg assembly is fabricated and tested within the silicon wafer in which it was built. Completed robot leg performance shows close agreement between modeled and tested static and dynamic characteristics, with potential benefits for future walking micro-robots from low power requirements relative to payload capacity and large amplitude motion at resonance.

Appendix

Combined behavior of multiple polymer elements from in-plane rotation mechanisms is modeled by series of force balance, compliance, and geometric constraint equations for the elements previously introduced in Fig. 5. These equations are applied in up to five dimensions, neglecting torsion of beams in the parallel direction of the actuators (about the x-axis as defined in Fig. 5). Force balances about the individual components can be summarized as

$$ {\mathbf{F}}_{1} = {\mathbf{T}}_{1} {\mathbf{F}}_{0} , { }{\mathbf{F}}_{2} = {\mathbf{T}}_{2} {\mathbf{F}}_{1} , \, {\mathbf{F}}_{3} = {\mathbf{T}}_{3} {\mathbf{F}}_{1} $$

(8)

where F ₀ is the input force from the piezoelectric beam and F ₁, F ₂, and F ₃ are the force vectors at the right-hand side of the elements (tether, rigid link, and tether respectively), where T ₁, T ₂, and T ₂ are matrices computing static force balances from forces on the left-hand side of the elements in their neutral position, i.e.

$$ \begin{aligned} {\mathbf{T}}{}_{1} & = \left[ {\begin{array}{*{20}c} 1 & {\quad 0} & {\quad 0} & {\quad 0} & {\quad 0} \\ 0 & {\quad 1} & {\quad 0} & {\quad 0} & {\quad 0} \\ 0 & {\quad - L_{t} } & {\quad 1} & {\quad 0} & {\quad 0} \\ 0 & {\quad 0} & {\quad 0} & {\quad 1} & {\quad 0} \\ 0 & {\quad 0} & {\quad 0} & {\quad L_{t} } & {\quad 1} \\ \end{array} } \right], \\ {\mathbf{T}}_{2} & = \left[ {\begin{array}{*{20}c} 1 & {\quad 0} & {\quad 0} & {\quad 0} & {\quad 0} \\ 0 & {\quad 1} & {\quad w_{b} } & {\quad 0} & {\quad 0} \\ 0 & {\quad 0} & {\quad 1} & {\quad 0} & {\quad 0} \\ 0 & {\quad 0} & {\quad 0} & {\quad 1} & {\quad - w_{b} } \\ 0 & {\quad 0} & {\quad 0} & {\quad 0} & {\quad 1} \\ \end{array} } \right] \\ \end{aligned} $$

(9)

where L _t is the tether length, w _b is the width of the rigid link, and T ₃ is identical to T ₁ but using flexure dimensions.

Deformation of the elastic parylene elements is calculated as the difference between tip positions and their projected neutral location based on their initial rotation angles,

$$ \begin{aligned} {\mathbf{u}}_{3} - {\mathbf{V}}_{f} {\mathbf{u}}_{2} = {\mathbf{C}}_{f} {\mathbf{F}}_{3} \hfill \\ {\mathbf{u}}_{1} - {\mathbf{V}}_{t} {\mathbf{u}}_{0} = {\mathbf{C}}_{t} {\mathbf{F}}_{1} \hfill \\ \end{aligned} $$

(10)

where u ₁ and u ₃ are tip displacements of the tether and flexure, respectively (with u ₃ = 0 when anchored to the substrate). In (10), V _f and V _t are matrices projecting the neutral position of the tether and flexure tips given rotation of earlier links, and C _f and C _t are compliance matrices for the tether and flexure, respectively. Projection matrices are given by

$$\begin{aligned} {\mathbf{V}}{}_{1}& = \left[ {\begin{array}{*{20}c} 1 &\quad 0 &\quad 0 &\quad 0 &\quad 0 \\ 0 &\quad 1 &\quad {L_{t} } &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad 1 &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad 0 &\quad 1 &\quad { - L_{t} } \\ 0 &\quad 0 &\quad 0 &\quad 0 &\quad 1 \\ \end{array} } \right], \\ {\mathbf{V}}_{2} &= \left[ {\begin{array}{*{20}c} 1 &\quad 0 &\quad 0 &\quad 0 &\quad 0 \\ 0 &\quad 1 &\quad {L_{f} } &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad 1 &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad 0 &\quad 1 &\quad { - L_{f} } \\ 0 &\quad 0 &\quad 0 &\quad 0 &\quad 1 \\ \end{array} } \right]\end{aligned} $$

(11)

with L _f being the flexure length. The compliance matrix for the flexure in the tether-flexure assembly is calculated from simple Bernoulli bending theory with a simple shear stress adjustment in the y-direction,

$$ {\mathbf{C}}_{f} = \left[ {\begin{array}{*{20}c} {\tfrac{{L_{f} }}{{E_{p} A_{f} }}} &\quad 0 &\quad 0 &\quad 0 &\quad 0 \\ 0 &\quad {\tfrac{{L_{f}^{3} }}{{3E_{p} I_{f,z} }} + \tfrac{{L_{f} }}{{G_{p} A_{f} }}} &\quad {\tfrac{{L_{f}^{2} }}{{2E_{p} I_{f,z} }}} &\quad 0 &\quad 0 \\ 0 &\quad {\tfrac{{L_{f}^{2} }}{{2E_{p} I_{f,z} }} + \tfrac{1}{{G_{p} A_{f} }}} &\quad {\tfrac{{L_{f} }}{{E_{p} I_{,zf} }}} &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad 0 &\quad {\tfrac{{L_{f}^{3} }}{{3E_{p} I_{f,y,} }}} &\quad {\tfrac{{L_{f}^{3} }}{{2E_{p} I_{f,y,} }}} \\ {} &\quad {} &\quad {} &\quad {\tfrac{{L_{f}^{3} }}{{2E_{p} I_{f,y,} }}} &\quad {\tfrac{{L_{f} }}{{3E_{p} I_{f,y,} }}} \\ \end{array} } \right] $$

(12)

where A _f is the flexure’s cross-sectional area, I _f is its area moment of inertia about z or y axes, and E _p and G _p are the elastic and shear moduli of parylene-C, respectively.

Geometric constraints for the rigid element are compiled in matrix form,

$$ {\mathbf{u}}_{2} = {\mathbf{S}}_{2} {\mathbf{u}}_{1} $$

(13)

where S ₂ is a matrix projecting displacement at u ₂ from u ₁ for small angular rotations,

$$ {\mathbf{S}}_{2} = \left[ {\begin{array}{*{20}c} 1 &\quad 0 &\quad 0 &\quad 0 &\quad { - L_{act} } \\ 0 &\quad 1 &\quad {w_{b} } &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad 1 &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad 0 &\quad 1 &\quad { - w_{b} } \\ 0 &\quad 0 &\quad 0 &\quad 0 &\quad 1 \\ \end{array} } \right] $$

(14)

Equations (8)–(14) can be solved by linear algebra to either relate a specified input force to a multi-axis displacements of the flexure and tether endpoints and rigid link, or more generally to calculate the combined stiffness matrix, K, used to calculate actuator displacement numerically when accounting for nonlinear arc lengths effects introduced in (5)–(7) in Sect. 2.3.