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[{"categories":null,"contents":" Tags Stewart Platforms Reference (Legnani {\\it et al.}, 2012) Author(s) Legnani, G., Fassi, I., Giberti, H., Cinquemani, S., \u0026amp; Tosi, D. Year 2012 Concepts of isotropy and decoupling for parallel manipulators isotropy: the kinetostatic properties (same applicable force, same possible velocity, same stiffness) are identical in all directions (e.g. cubic configuration for Stewart platform) decoupling: each DoF of the end effector can be controlled by a single actuator (not the case for the Stewart platform) Example of generated isotropic manipulator (not decoupled).\n\n Figure 1: Location of the leg axes using an isotropy generator\n \n Figure 2: Isotropic configuration\n Bibliography Legnani, G., Fassi, I., Giberti, H., Cinquemani, S., \u0026amp; Tosi, D., A new isotropic and decoupled 6-dof parallel manipulator, Mechanism and Machine Theory, 58(nil), 64–81 (2012). http://dx.doi.org/10.1016/j.mechmachtheory.2012.07.008 ↩\n","permalink":"/paper/legnani12_new_isotr_decoup_paral_manip/","tags":null,"title":"A new isotropic and decoupled 6-dof parallel manipulator"},{"categories":null,"contents":" Tags Position Sensors Reference (Andrew Fleming, 2013) Author(s) Fleming, A. J. Year 2013 Define concise performance metric and provide expressions for errors sources (non-linearity, drift, noise) Review current position sensor technologies and compare their performance Sensor Characteristics Calibration and nonlinearity Usually quoted as a percentage of the fill-scale range (FSR):\n\\begin{equation} \\text{mapping error (%)} = \\pm 100 \\frac{\\max{}|e_m(v)|}{\\text{FSR}} \\end{equation}\nWith \\(e_m(v)\\) is the mapping error.\n\n Figure 1: The actual position versus the output voltage of a position sensor. The calibration function \\(f_{cal}(v)\\) is an approximation of the sensor mapping function \\(f_a(v)\\) where \\(v\\) is the voltage resulting from a displacement \\(x\\). \\(e_m(v)\\) is the residual error.\n Drift and Stability If the shape of the mapping function actually varies with time, the maximum error due to drift must be evaluated by finding the worst-case mapping error.\n\n Figure 2: The worst case range of a linear mapping function \\(f_a(v)\\) for a given error in sensitivity and offset.\n Bandwidth The bandwidth of a position sensor is the frequency at which the magnitude of the transfer function \\(P(s) = v(s)/x(s)\\) drops by \\(3,dB\\).\nAlthough the bandwidth specification is useful for predicting the resolution of sensor, it reveals very little about the measurement errors caused by sensor dynamics.\nThe frequency domain position error is\n\\begin{equation} \\begin{aligned} e_{bw}(s) \u0026amp;= x(s) - v(s) \\\\\\\n\u0026amp;= x(s) (1 - P(s)) \\end{aligned} \\end{equation}\nIf the actual position is a sinewave of peak amplitude \\(A = \\text{FSR}/2\\):\n\\begin{equation} \\begin{aligned} e_{bw} \u0026amp;= \\pm \\frac{\\text{FSR}}{2} |1 - P(s)| \\\\\\\n\u0026amp;\\approx \\pm A n \\frac{f}{f_c} \\end{aligned} \\end{equation}\nwith \\(n\\) is the low pass filter order corresponding to the sensor dynamics and \\(f_c\\) is the measurement bandwidth.\nThus, the sensor bandwidth must be significantly higher than the operating frequency if dynamic errors are to be avoided.\nNoise In addition to the actual position signal, all sensors produce some additive measurement noise. In many types of sensor, the majority of noise arises from the thermal noise in resistors and the voltage and current noise in conditioning circuit transistors. These noise processes can usually be approximated by a Gaussian random process.\nA Gaussian random process is usually described by its autocorrelation function or its Power Spectral Density.\nThe autocorrelation function of a random process \\(\\mathcal{X}\\) is\n\\begin{equation} R_{\\mathcal{X}}(\\tau) = E[\\mathcal{X}(t)\\mathcal{X}(t + \\tau)] \\end{equation}\nwhere \\(E\\) is the expected value operator.\nThe variance of the process is equal to \\(R_\\mathcal{X}(0)\\) and is the expected value of the varying part squared
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