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Abstract: Robot-assisted surgery has advantages compared to conventional laparoscopic procedures, e.g., precise movement of the surgical instruments, improved dexterity, and high-resolution visualization of the surgical field. However, mechanical tissue properties may provide additional information, e.g., on the location of lesions or vessels. While elastographic imaging has been proposed, it is not readily available as an online modality during robot-assisted surgery. We propose modifying a da~Vinci surgical instrument to realize optical coherence elastography (OCE) for quantitative elasticity estimation. The modified da~Vinci instrument is equipped with piezoelectric elements for shear wave excitation and we employ fast optical coherence tomography (OCT) imaging to track propagating wave fields, which are directly related to biomechanical tissue properties. All high-voltage components are mounted at the proximal end outside the patient. We demonstrate that external excitation at the instrument shaft can effectively stimulate shear waves, even when considering damping. Comparing conventional and deep learning-based signal processing, resulting in mean absolute errors of 19.27 kPa and 6.29 kPa, respectively. These results illustrate that precise quantitative elasticity estimates can be obtained. We also demonstrate quantitative elasticity estimation on ex-vivo tissue samples of heart, liver and stomach, and show that the measurements can be used to distinguish soft and stiff tissue types.