Machine Vision and Applications (2006) 17(2): 133–138 DOI 10.1007/s00138-006-0019-1 ORIGINAL PAPER Joaqu´ ın Salas · Wendy Avalos · Rafael Casta ˜ neda · Mario Maya A machine-vision system to measure the parameters describing the performance of a Foucault pendulum Received: 20 September 2005 / Accepted: 22 January 2006 / Published online: 24 March 2006 C  Springer-Verlag 2006 Abstract We describe a machine-vision system that makes real-time measurements of the kinematics of a Foucault pen- dulum. Images are taken from a downward-facing camera placed close to the pendulum suspension point. The bob is detected via background subtraction and located by fitting circles to the resulting contour segments. The bob trajec- tory is then modelled by fitting ellipses to recent positions. Parameters are improved through Kalman filtering. Exper- imental results are shown. Our implementation is a robust and accurate tool for visualization of the pendulum kine- matics as well as troubleshooting and maintenance of the mechanical elements. Keywords Visual tracking · Object recognition · Target detection · Vision system · Visual inspection 1 Introduction The Foucault pendulum has long fascinated mankind. With his simple yet effective experiment, Jean Bernard Leon Fou- cault [1] showed in 1851 that the Earth rotates without re- lying on extraterrestrial reference points. With skilled hand- iwork [2], Foucault managed to construct a pendulum with negligible torque force. Once the pendulum was set swing- ing, the oscillation plane veered slowly but perceptibly. Since there is no force that makes the pendulum rotate with respect to the floor, Foucault concluded that it must be the floor that rotates with respect to the pendulum. J. Salas (B ) CICATA Unidad Quer´ etaro-IPN, Jos´ e Siurob 10, Col. Alameda, Quer´ etaro, Qro. 76040, M´ exico E-mail: salas@ieee.org W. Avalos Instituto Tecnol´ ogico de Morelia, Av. Tecnologico 1500, Colonia Lomas de Santiaguito, C.P. 58120, Morelia, Mexico R. Casta ˜ neda · M. Maya Instituto Tecnol´ ogico de Zitacuaro, Ex-Hacienda de Manzanillos s/n Carr. Toluca-MoreliaApdo., Postal No. 112, Zitcuaro C.P. 61500, Mexico Currently, there are fairly large number of Foucault pen- dulums around the world that are being used to show that the Earth spins [3]. Briefly, at the north/south pole, a Fou- cault pendulum’s oscillation plane would seem to veer 360 ◦ every 24 (sidereal) hours. On the other hand, on the equa- tor, a pendulum would not veer at all because the building where it is would be travelling eastward around the Earth’s axis. For places at different latitude some veering and some travelling takes place. The degree of veer in t seconds de- pends on the latitude φ by n = 360 ◦ sin φ t /86164.1, where 86164.1 is the number of seconds in a sidereal day. In this paper, we describe a machine-vision system to measure, in real-time, some parameters related to the kine- matics of a Foucault pendulum, such as the oscillation plane’s orientation θ(t ) and ellipticity (t ). The images are taken by a downward-looking camera located close to the suspension mechanism at the top of the dome, close to the gripping system, and aligned with the vertical direction us- ing the Zhang extrinsic parameters calibration procedure [4]. Physically, realizing a Foucault pendulum is rather difficult. Due to mechanically anisotropy, a pendulum’s period of os- cillation is not the same irrespective of direction, θ(t ); rather the pendulum is characterized by two orthogonal ‘principal planes’ with slightly different periods. Over time, a phase difference is built up between oscillations [2], opening a tra- jectory, known as a Lissajous curve [5], that instantaneously approximates an ellipse whose major axes are α(t ) and β(t ). The Lissajous curve veers and backs intrinsically as well as because of the synchronicity of the pendulum (its swing pe- riod depends slightly on amplitude), so a Foucault pendu- lum rarely veers at the expected terrestrial rate. During os- cillation, the pendulum will be damped by factors such as [6] air resistance and internal friction in the suspension wire and gripping system. For this reason, Foucault pendulums in museums often incorporate a mechanical or electromagnetic drive to maintain the amplitude of the oscillation. Over the last decades, there has been some controversy about certain phenomena observed in mechanical oscilla- tors. Back in 1959, Maurice Allais [7] reported that dur- ing eclipses the oscillation plane direction deviated with