## 1 Introduction  
Among small-scale transducers, piezoelectric ceramics can achieve exceptionally high work and power densities [1]. They can operate with comparatively simple geometry, and consume lower currents and reach higher bandwidths than shape memory alloy or electrothermal transducers. Their combination of high force generation, fast response time, and modest power consumption has made piezoelectric actuators promising for small-scaled robotics. For example, piezoelectric actuation has been used in miniature walking robots [2; 3; 4; 5] and in anthropomorphic robotic hands [6]. However, when it becomes necessary to generate large displacement amplitudes, piezoelectric actuators are rarely ideal due to low intrinsic strain. While operation at resonance can amplify motion substantially, it is also difficult to design piezoelectric ceramics to produce resonance in low frequency ranges, especially at small size scales [7]. This can result in a mismatch between the highest-amplitude operating frequencies of a piezoelectric actuator and those of the natural dynamics of a small robotic system, when seeking to produce naturalistic walking or swimming gaits. To address the above shortcomings, many researchers have used amplification mechanisms to increase piezoelectric actuator stroke. For example, a compact out-of-plane architecture used a half-scissor mechanism to convert the small in-plane elongation of a piezoceramic into a much larger out-of-plane displacement [8]. Flextensional microactuators consist of a bulk piezoelectric material bonded to endcaps that flexes in response to piezoelectric contraction [9]. Thus, the small piezoelectric contraction is amplified through buckling of the flextensional structure. Significant amplification can also be achieved via a combination of a chevron bridge and lever arms [10], though requiring a much-increased total footprint. Drawbacks of these schemes are that substantial actuator work must be expended to deform the elastic actuation structures, and dynamic amplification tends to remain most effective at high frequency ranges for electromechanical impedance matching between the transducer and mechanism. Alternatively, piezoelectric actuation has been used to produce large or even continuous output displacements through contact interactions [11]. Piezoelectric wiggle motors are among the world's smallest linear motors, using ultrasonic standing wave vibrations to directly rotate a screw [12]. Various other piezoelectric motors designs, based variously on impact events, traveling waves, or stick-slip behavior, have been developed to drive a free rotor or body [13; 14; 15; 16]. This permits large displacement or even continuous-displacement outputs, but tends to require complex fabrication and provide small load-bearing capacity. Impact dynamics have also been used in energy harvesting to couple piezoelectric transducers to vibration sources in widely disparate frequency domains [17; 18]. Here, we explore a concept for dynamic amplification of a piezoelectric transducer via impact interaction with a compliant structure. Conceptually, the opportunity is to build kinematic energy within the piezoelectric ceramic and transfer it to deformation of the elastic output mechanism at impact. This is similar to the operating principle of linear impact-based actuators [19; 20], sometimes driven piezoelectrically. However, in the proposed scheme, there is a permanent elastic connection between elements to permit simple assembly, and frequency differences are used to let the active element return to its full resonant amplitude between impacts. This contrasts with the use of individual accelerations of the transducer to induce each impact. With proper frequency selection, in the proposed scheme, impact periods can settle into quasi-periodic patterns, including examples with down-conversion in frequency by a factor of 10 or more.