2
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Per a previous post on Mathematica StackExchange here I'm trying to utilize 3-axis accelerometer data to determine approximate 3D position. I wanted to start with a simple case. To this end I've employed a hand drill, wood plate and HOBO Pendant G Logger. The accelerometer was mounted at one edge of the plate and rotated by the drill at approximately 20 rpm. Drill was held in a vice with the z-axis perpendicular and x- and y- axes in the plane of rotation. Sample rate was 100 Hz.

HOBO Pendant G mounted to a wood plate and rotated via hand drill. Drill was held in a vice with the z-axis nearly perpendicular and x- and y- axes in the plane of rotation.

A five-second snip of the data is presented below:

time,x_g,y_g,z_g,sum_g,x_deg,y_deg,z_deg
15.00,0.925,0.525,0.825,1.346,157.7,121.7,145.6
15.01,0.975,0.525,1.05,1.526,167.2,121.7,180
15.02,0.925,0.575,1.175,1.602,157.7,125.1,180
15.03,1.075,0.525,1.025,1.575,180,121.7,180
15.04,0.9,0.525,1.15,1.552,154.2,121.7,180
15.05,1,0.5,1,1.5,180,120,180
15.06,0.85,0.525,0.925,1.362,148.2,121.7,157.7
15.07,0.925,0.55,0.7,1.284,157.7,123.4,134.4
15.08,0.825,0.5,1,1.389,145.6,120,180
15.09,1.05,0.65,1.125,1.671,180,130.5,180
15.10,1.075,0.55,1.1,1.633,180,123.4,180
15.11,1.025,0.575,1.1,1.61,180,125.1,180
15.12,1.05,0.525,1.25,1.715,180,121.7,180
15.13,1.075,0.625,1.1,1.66,180,128.7,180
15.14,0.875,0.5,0.95,1.385,151,120,161.8
15.15,0.925,0.475,0.775,1.297,157.7,118.4,140.8
15.16,0.925,0.5,1.075,1.504,157.7,120,180
15.17,1,0.575,1.15,1.629,180,125.1,180
15.18,1,0.6,1.025,1.553,180,126.9,180
15.19,1.1,0.65,1.175,1.736,180,130.5,180
15.20,1.1,0.6,1.325,1.824,180,126.9,180
15.21,0.975,0.575,0.975,1.494,167.2,125.1,167.2
15.22,0.9,0.45,0.7,1.226,154.2,116.7,134.4
15.23,0.925,0.55,0.9,1.403,157.7,123.4,154.2
15.24,0.95,0.575,1.075,1.546,161.8,125.1,180
15.25,0.975,0.575,1.1,1.578,167.2,125.1,180
15.26,1.025,0.55,1,1.534,180,123.4,180
15.27,1.025,0.6,1.2,1.688,180,126.9,180
15.28,0.95,0.65,1.25,1.699,161.8,130.5,180
15.29,1.025,0.575,1.075,1.593,180,125.1,180
15.30,0.925,0.625,0.875,1.418,157.7,128.7,151
15.31,0.875,0.45,0.85,1.3,151,116.7,148.2
15.32,0.9,0.575,0.85,1.365,154.2,125.1,148.2
15.33,0.925,0.475,0.875,1.359,157.7,118.4,151
15.34,0.975,0.6,0.975,1.504,167.2,126.9,167.2
15.35,0.975,0.625,1.225,1.686,167.2,128.7,180
15.36,1.05,0.625,1.35,1.821,180,128.7,180
15.37,0.975,0.6,0.975,1.504,167.2,126.9,167.2
15.38,0.825,0.5,0.825,1.269,145.6,120,145.6
15.39,0.825,0.575,0.9,1.35,145.6,125.1,154.2
15.40,0.9,0.45,0.9,1.35,154.2,116.7,154.2
15.41,0.9,0.55,0.975,1.436,154.2,123.4,167.2
15.42,0.975,0.55,1.15,1.605,167.2,123.4,180
15.43,0.95,0.65,1.15,1.627,161.8,130.5,180
15.44,0.95,0.625,1.05,1.548,161.8,128.7,180
15.45,0.825,0.6,1,1.429,145.6,126.9,180
15.46,0.875,0.525,0.975,1.411,151,121.7,167.2
15.47,0.875,0.575,1.075,1.501,151,125.1,180
15.48,0.975,0.525,1,1.492,167.2,121.7,180
15.49,0.825,0.55,1.05,1.444,145.6,123.4,180
15.50,0.975,0.55,1.025,1.518,167.2,123.4,180
15.51,0.9,0.55,1.15,1.56,154.2,123.4,180
15.52,0.975,0.575,1.15,1.614,167.2,125.1,180
15.53,0.9,0.5,1.075,1.488,154.2,120,180
15.54,0.675,0.75,0.875,1.336,132.5,138.6,151
15.55,0.85,0.525,0.85,1.312,148.2,121.7,148.2
15.56,0.9,0.475,0.975,1.409,154.2,118.4,167.2
15.57,0.9,0.55,1.15,1.56,154.2,123.4,180
15.58,0.925,0.525,1.15,1.566,157.7,121.7,180
15.59,0.875,0.525,0.95,1.394,151,121.7,161.8
15.60,0.75,0.575,0.95,1.34,138.6,125.1,161.8
15.61,0.9,0.6,0.975,1.456,154.2,126.9,167.2
15.62,0.875,0.475,1,1.411,151,118.4,180
15.63,0.975,0.65,1.125,1.624,167.2,130.5,180
15.64,0.975,0.5,1.325,1.719,167.2,120,180
15.65,0.925,0.55,1.175,1.593,157.7,123.4,180
15.66,0.9,0.55,1.05,1.488,154.2,123.4,180
15.67,0.8,0.45,0.85,1.251,143.1,116.7,148.2
15.68,0.8,0.55,0.8,1.258,143.1,123.4,143.1
15.69,0.875,0.525,1.025,1.446,151,121.7,180
15.70,0.95,0.575,1.075,1.546,161.8,125.1,180
15.71,0.95,0.425,1.025,1.461,161.8,115.2,180
15.72,0.875,0.55,1.05,1.473,151,123.4,180
15.73,0.875,0.425,0.95,1.36,151,115.2,161.8
15.74,0.825,0.575,0.875,1.333,145.6,125.1,151
15.75,0.775,0.45,0.975,1.324,140.8,116.7,167.2
15.76,0.925,0.525,0.85,1.362,157.7,121.7,148.2
15.77,0.9,0.475,1.05,1.462,154.2,118.4,180
15.78,0.925,0.475,1.025,1.46,157.7,118.4,180
15.79,0.9,0.475,1.1,1.499,154.2,118.4,180
15.80,0.9,0.475,1.1,1.499,154.2,118.4,180
15.81,0.875,0.575,1.075,1.501,151,125.1,180
15.82,0.95,0.5,1.05,1.502,161.8,120,180
15.83,0.925,0.6,1.075,1.54,157.7,126.9,180
15.84,0.925,0.5,1.025,1.468,157.7,120,180
15.85,0.825,0.475,0.925,1.327,145.6,118.4,157.7
15.86,0.925,0.45,0.825,1.319,157.7,116.7,145.6
15.87,0.95,0.475,1.075,1.511,161.8,118.4,180
15.88,0.975,0.6,1.25,1.695,167.2,126.9,180
15.89,0.95,0.525,1.125,1.563,161.8,121.7,180
15.90,0.95,0.575,0.95,1.461,161.8,125.1,161.8
15.91,0.925,0.575,1.125,1.566,157.7,125.1,180
15.92,0.925,0.55,1.025,1.486,157.7,123.4,180
15.93,0.975,0.475,0.825,1.363,167.2,118.4,145.6
15.94,1,0.475,0.975,1.475,180,118.4,167.2
15.95,1,0.575,1.15,1.629,180,125.1,180
15.96,1.025,0.575,1.275,1.734,180,125.1,180
15.97,0.975,0.65,1.125,1.624,167.2,130.5,180
15.98,0.875,0.475,1,1.411,151,118.4,180
15.99,0.925,0.575,0.875,1.397,157.7,125.1,151
16.00,0.9,0.475,0.825,1.31,154.2,118.4,145.6
16.01,0.925,0.45,0.9,1.367,157.7,116.7,154.2
16.02,1.05,0.575,1.1,1.626,180,125.1,180
16.03,1.025,0.525,1.225,1.681,180,121.7,180
16.04,0.95,0.65,1.1,1.592,161.8,130.5,180
16.05,0.925,0.45,0.9,1.367,157.7,116.7,154.2
16.06,0.875,0.5,0.675,1.213,151,120,132.5
16.07,0.825,0.45,0.85,1.267,145.6,116.7,148.2
16.08,0.9,0.525,1.025,1.462,154.2,121.7,180
16.09,1.075,0.625,1.075,1.644,180,128.7,180
16.10,0.95,0.5,1.25,1.648,161.8,120,180
16.11,0.925,0.625,0.975,1.482,157.7,128.7,167.2
16.12,0.975,0.625,1.2,1.668,167.2,128.7,180
16.13,1,0.6,0.975,1.52,180,126.9,167.2
16.14,0.85,0.5,0.725,1.224,148.2,120,136.5
16.15,0.875,0.55,1,1.438,151,123.4,180
16.16,1.05,0.525,1.225,1.697,180,121.7,180
16.17,1.025,0.7,1.325,1.816,180,134.4,180
16.18,1.025,0.525,1.125,1.61,180,121.7,180
16.19,0.9,0.625,0.975,1.467,154.2,128.7,167.2
16.20,0.875,0.5,1.025,1.437,151,120,180
16.21,0.9,0.575,0.8,1.334,154.2,125.1,143.1
16.22,0.85,0.525,0.95,1.379,148.2,121.7,161.8
16.23,0.95,0.65,1.15,1.627,161.8,130.5,180
16.24,0.975,0.55,1.075,1.552,167.2,123.4,180
16.25,0.925,0.575,1.075,1.53,157.7,125.1,180
16.26,0.925,0.575,1.05,1.513,157.7,125.1,180
16.27,0.875,0.625,0.975,1.452,151,128.7,167.2
16.28,0.9,0.525,0.925,1.393,154.2,121.7,157.7
16.29,0.875,0.575,0.9,1.381,151,125.1,154.2
16.30,0.925,0.525,0.95,1.426,157.7,121.7,161.8
16.31,0.925,0.525,1.125,1.548,157.7,121.7,180
16.32,0.925,0.6,1.1,1.557,157.7,126.9,180
16.33,0.925,0.65,1.125,1.595,157.7,130.5,180
16.34,0.9,0.6,1.025,1.49,154.2,126.9,180
16.35,0.825,0.575,0.825,1.301,145.6,125.1,145.6
16.36,0.875,0.525,1.025,1.446,151,121.7,180
16.37,0.875,0.625,1.1,1.538,151,128.7,180
16.38,0.925,0.5,1.075,1.504,157.7,120,180
16.39,0.95,0.575,1.025,1.511,161.8,125.1,180
16.40,0.875,0.5,1.1,1.492,151,120,180
16.41,0.9,0.625,1.075,1.535,154.2,128.7,180
16.42,0.9,0.5,1.2,1.581,154.2,120,180
16.43,0.875,0.6,0.95,1.424,151,126.9,161.8
16.44,0.875,0.525,1,1.429,151,121.7,180
16.45,0.9,0.575,1.05,1.498,154.2,125.1,180
16.46,0.925,0.525,1.1,1.53,157.7,121.7,180
16.47,0.875,0.525,1.1,1.5,151,121.7,180
16.48,0.875,0.525,1,1.429,151,121.7,180
16.49,0.75,0.475,0.9,1.264,138.6,118.4,154.2
16.50,0.85,0.625,0.95,1.42,148.2,128.7,161.8
16.51,0.9,0.5,1.025,1.453,154.2,120,180
16.52,0.95,0.525,1.025,1.493,161.8,121.7,180
16.53,0.875,0.5,1.175,1.548,151,120,180
16.54,0.95,0.475,1.1,1.529,161.8,118.4,180
16.55,0.9,0.525,0.975,1.427,154.2,121.7,167.2
16.56,0.875,0.475,1.05,1.447,151,118.4,180
16.57,0.85,0.65,1.1,1.535,148.2,130.5,180
16.58,0.725,0.525,1.075,1.399,136.5,121.7,180
16.59,0.875,0.475,0.85,1.309,151,118.4,148.2
16.60,0.9,0.5,1.05,1.471,154.2,120,180
16.61,0.9,0.525,1.2,1.589,154.2,121.7,180
16.62,0.9,0.525,1.125,1.533,154.2,121.7,180
16.63,0.875,0.525,0.95,1.394,151,121.7,161.8
16.64,0.85,0.5,0.975,1.387,148.2,120,167.2
16.65,0.925,0.525,0.975,1.443,157.7,121.7,167.2
16.66,0.925,0.525,1,1.46,157.7,121.7,180
16.67,0.925,0.525,1.025,1.477,157.7,121.7,180
16.68,0.9,0.5,1.05,1.471,154.2,120,180
16.69,0.9,0.475,0.95,1.392,154.2,118.4,161.8
16.70,0.875,0.525,0.925,1.377,151,121.7,157.7
16.71,0.85,0.475,1.15,1.507,148.2,118.4,180
16.72,0.925,0.5,1.05,1.486,157.7,120,180
16.73,0.95,0.5,0.775,1.324,161.8,120,140.8
16.74,0.875,0.475,0.925,1.359,151,118.4,157.7
16.75,0.9,0.475,1.075,1.48,154.2,118.4,180
16.76,0.975,0.5,0.975,1.467,167.2,120,167.2
16.77,0.95,0.5,1.075,1.519,161.8,120,180
16.78,1,0.525,1.1,1.577,180,121.7,180
16.79,0.9,0.525,1.1,1.515,154.2,121.7,180
16.80,0.975,0.575,0.95,1.478,167.2,125.1,161.8
16.81,0.875,0.475,0.975,1.394,151,118.4,167.2
16.82,0.95,0.575,0.95,1.461,161.8,125.1,161.8
16.83,0.975,0.525,1,1.492,167.2,121.7,180
16.84,1.05,0.525,1.025,1.558,180,121.7,180
16.85,1.025,0.525,1.3,1.737,180,121.7,180
16.86,1.025,0.55,1.2,1.671,180,123.4,180
16.87,0.925,0.5,0.975,1.434,157.7,120,167.2
16.88,0.85,0.55,1,1.423,148.2,123.4,180
16.89,0.975,0.5,1.075,1.535,167.2,120,180
16.90,1,0.625,1.05,1.579,180,128.7,180
16.91,1,0.6,1.2,1.673,180,126.9,180
16.92,1.075,0.625,1.1,1.66,180,128.7,180
16.93,0.925,0.55,1.025,1.486,157.7,123.4,180
16.94,0.975,0.525,0.9,1.427,167.2,121.7,154.2
16.95,0.875,0.5,1.025,1.437,151,120,180
16.96,0.925,0.525,0.875,1.377,157.7,121.7,151
16.97,0.825,0.5,0.9,1.319,145.6,120,154.2
16.98,1,0.6,0.975,1.52,180,126.9,167.2
16.99,1.025,0.55,1.275,1.726,180,123.4,180
17.00,1.025,0.65,1.1,1.638,180,130.5,180
17.01,0.925,0.55,1.075,1.521,157.7,123.4,180
17.02,0.925,0.6,1.15,1.593,157.7,126.9,180
17.03,0.95,0.55,0.975,1.468,161.8,123.4,167.2
17.04,0.95,0.6,1.075,1.555,161.8,126.9,180
17.05,0.975,0.6,1.175,1.641,167.2,126.9,180
17.06,0.975,0.575,1.275,1.705,167.2,125.1,180
17.07,0.975,0.575,1.025,1.527,167.2,125.1,180
17.08,0.9,0.5,0.925,1.384,154.2,120,157.7
17.09,0.9,0.55,1.075,1.506,154.2,123.4,180
17.10,0.925,0.575,1.1,1.548,157.7,125.1,180
17.11,0.925,0.625,0.975,1.482,157.7,128.7,167.2
17.12,0.975,0.675,1.175,1.669,167.2,132.5,180
17.13,0.975,0.65,1.175,1.659,167.2,130.5,180
17.14,0.9,0.575,1,1.463,154.2,125.1,180
17.15,0.875,0.475,0.925,1.359,151,118.4,157.7
17.16,0.875,0.525,0.9,1.361,151,121.7,154.2
17.17,0.875,0.575,1,1.448,151,125.1,180
17.18,0.875,0.525,1.025,1.446,151,121.7,180
17.19,0.9,0.575,0.95,1.429,154.2,125.1,161.8
17.20,0.875,0.5,1.075,1.474,151,120,180
17.21,0.925,0.525,1.1,1.53,157.7,121.7,180
17.22,0.875,0.55,1.025,1.456,151,123.4,180
17.23,0.9,0.525,1.025,1.462,154.2,121.7,180
17.24,0.825,0.575,1,1.418,145.6,125.1,180
17.25,0.825,0.7,1.025,1.49,145.6,134.4,180
17.26,0.925,0.475,1.15,1.55,157.7,118.4,180
17.27,0.825,0.6,1.025,1.446,145.6,126.9,180
17.28,0.875,0.525,1.05,1.464,151,121.7,180
17.29,0.875,0.55,1.025,1.456,151,123.4,180
17.30,0.9,0.5,0.975,1.418,154.2,120,167.2
17.31,0.9,0.55,1.125,1.542,154.2,123.4,180
17.32,0.8,0.625,1.075,1.479,143.1,128.7,180
17.33,0.8,0.5,0.925,1.321,143.1,120,157.7
17.34,0.9,0.5,1.05,1.471,154.2,120,180
17.35,0.825,0.575,1.025,1.436,145.6,125.1,180
17.36,0.825,0.525,1,1.399,145.6,121.7,180
17.37,0.925,0.525,0.975,1.443,157.7,121.7,167.2
17.38,0.825,0.425,0.975,1.346,145.6,115.2,167.2
17.39,0.85,0.575,1,1.433,148.2,125.1,180
17.40,0.875,0.475,1.075,1.465,151,118.4,180
17.41,0.775,0.55,0.925,1.326,140.8,123.4,157.7
17.42,0.875,0.5,0.975,1.402,151,120,167.2
17.43,0.85,0.65,1.025,1.482,148.2,130.5,180
17.44,0.925,0.475,1.1,1.514,157.7,118.4,180
17.45,0.975,0.55,1.1,1.569,167.2,123.4,180
17.46,0.9,0.45,1.075,1.472,154.2,116.7,180
17.47,0.85,0.525,0.925,1.362,148.2,121.7,157.7
17.48,0.8,0.525,0.925,1.331,143.1,121.7,157.7
17.49,0.85,0.425,0.825,1.258,148.2,115.2,145.6
17.50,0.85,0.525,0.825,1.296,148.2,121.7,145.6
17.51,0.875,0.4,1,1.388,151,113.6,180
17.52,0.975,0.525,1.05,1.526,167.2,121.7,180
17.53,0.925,0.5,1.05,1.486,157.7,120,180
17.54,0.9,0.525,1.075,1.497,154.2,121.7,180
17.55,0.875,0.6,1.175,1.583,151,126.9,180
17.56,0.875,0.525,1.05,1.464,151,121.7,180
17.57,1,0.525,0.775,1.37,180,121.7,140.8
17.58,0.775,0.475,1.1,1.427,140.8,118.4,180
17.59,0.975,0.425,0.975,1.443,167.2,115.2,167.2
17.60,0.9,0.5,0.975,1.418,154.2,120,167.2
17.61,0.925,0.475,1.025,1.46,157.7,118.4,180
17.62,0.85,0.5,1,1.404,148.2,120,180
17.63,0.95,0.55,0.85,1.388,161.8,123.4,148.2
17.64,0.875,0.475,0.975,1.394,151,118.4,167.2
17.65,0.875,0.475,0.825,1.293,151,118.4,145.6
17.66,0.875,0.425,0.925,1.342,151,115.2,157.7
17.67,0.975,0.625,1.05,1.563,167.2,128.7,180
17.68,1.025,0.525,1.25,1.7,180,121.7,180
17.69,1.075,0.65,1.275,1.79,180,130.5,180
17.70,1.025,0.575,1.15,1.644,180,125.1,180
17.71,0.85,0.475,0.8,1.26,148.2,118.4,143.1
17.72,0.825,0.45,0.875,1.284,145.6,116.7,151
17.73,0.85,0.475,0.85,1.293,148.2,118.4,148.2
17.74,0.825,0.475,1,1.381,145.6,118.4,180
17.75,1,0.6,1.1,1.603,180,126.9,180
17.76,0.975,0.525,1.275,1.689,167.2,121.7,180
17.77,1.125,0.625,1.175,1.743,180,128.7,180
17.78,0.925,0.525,1.35,1.719,157.7,121.7,180
17.79,1,0.625,1.025,1.562,180,128.7,180
17.80,0.95,0.55,0.95,1.452,161.8,123.4,161.8
17.81,0.975,0.575,1.025,1.527,167.2,125.1,180
17.82,0.95,0.575,1.15,1.599,161.8,125.1,180
17.83,1,0.525,1,1.509,180,121.7,180
17.84,0.975,0.525,0.975,1.475,167.2,121.7,167.2
17.85,0.9,0.525,0.975,1.427,154.2,121.7,167.2
17.86,0.925,0.5,0.95,1.417,157.7,120,161.8
17.87,0.925,0.55,0.85,1.371,157.7,123.4,148.2
17.88,0.9,0.6,1.1,1.543,154.2,126.9,180
17.89,1,0.6,1.1,1.603,180,126.9,180
17.90,1.025,0.55,1.125,1.618,180,123.4,180
17.91,0.975,0.575,1.15,1.614,167.2,125.1,180
17.92,0.9,0.525,0.975,1.427,154.2,121.7,167.2
17.93,0.85,0.55,0.9,1.355,148.2,123.4,154.2
17.94,0.9,0.6,1.15,1.579,154.2,126.9,180
17.95,0.975,0.625,1.175,1.65,167.2,128.7,180
17.96,1.025,0.7,1.225,1.744,180,134.4,180
17.97,1.025,0.625,1.3,1.77,180,128.7,180
17.98,0.975,0.575,1.075,1.561,167.2,125.1,180
17.99,0.925,0.525,1.05,1.495,157.7,121.7,180
18.00,0.9,0.625,1.075,1.535,154.2,128.7,180
18.01,0.9,0.575,1,1.463,154.2,125.1,180
18.02,0.875,0.6,1.025,1.475,151,126.9,180
18.03,0.95,0.525,1,1.476,161.8,121.7,180
18.04,0.925,0.575,1.175,1.602,157.7,125.1,180
18.05,0.975,0.525,1.175,1.615,167.2,121.7,180
18.06,0.875,0.6,1,1.458,151,126.9,180
18.07,0.875,0.525,0.975,1.411,151,121.7,167.2
18.08,0.8,0.6,1.025,1.432,143.1,126.9,180
18.09,0.95,0.575,0.975,1.478,161.8,125.1,167.2
18.10,0.925,0.575,1.15,1.584,157.7,125.1,180
18.11,0.925,0.575,1.125,1.566,157.7,125.1,180
18.12,0.875,0.575,1.025,1.465,151,125.1,180
18.13,0.95,0.5,0.975,1.45,161.8,120,167.2
18.14,0.825,0.525,0.975,1.381,145.6,121.7,167.2
18.15,0.875,0.575,1.025,1.465,151,125.1,180
18.16,0.825,0.525,0.975,1.381,145.6,121.7,167.2
18.17,0.875,0.575,0.95,1.414,151,125.1,161.8
18.18,0.925,0.575,1.025,1.496,157.7,125.1,180
18.19,0.875,0.525,1.025,1.446,151,121.7,180
18.20,0.875,0.525,1.075,1.482,151,121.7,180
18.21,0.9,0.525,1.1,1.515,154.2,121.7,180
18.22,0.875,0.525,0.975,1.411,151,121.7,167.2
18.23,0.875,0.625,1.05,1.503,151,128.7,180
18.24,0.875,0.525,1.1,1.5,151,121.7,180
18.25,0.875,0.55,0.975,1.421,151,123.4,167.2
18.26,0.95,0.5,1.05,1.502,161.8,120,180
18.27,0.875,0.475,1.125,1.502,151,118.4,180
18.28,0.925,0.525,1.05,1.495,157.7,121.7,180
18.29,0.925,0.5,0.925,1.4,157.7,120,157.7
18.30,0.75,0.625,0.85,1.294,138.6,128.7,148.2
18.31,0.85,0.5,1.05,1.44,148.2,120,180
18.32,0.975,0.625,1.05,1.563,167.2,128.7,180
18.33,0.875,0.5,1.225,1.586,151,120,180
18.34,0.925,0.6,1.15,1.593,157.7,126.9,180
18.35,0.925,0.5,1,1.451,157.7,120,180
18.36,0.875,0.55,0.95,1.404,151,123.4,161.8
18.37,0.925,0.5,1.1,1.522,157.7,120,180
18.38,0.95,0.55,1.1,1.554,161.8,123.4,180
18.39,0.875,0.525,1.025,1.446,151,121.7,180
18.40,0.925,0.475,0.925,1.392,157.7,118.4,157.7
18.41,0.875,0.55,0.9,1.37,151,123.4,154.2
18.42,0.9,0.475,0.95,1.392,154.2,118.4,161.8
18.43,0.95,0.525,0.8,1.348,161.8,121.7,143.1
18.44,0.95,0.5,0.95,1.434,161.8,120,161.8
18.45,0.975,0.6,1.2,1.659,167.2,126.9,180
18.46,0.95,0.6,1.2,1.644,161.8,126.9,180
18.47,1,0.55,0.95,1.485,180,123.4,161.8
18.48,0.9,0.525,1,1.444,154.2,121.7,180
18.49,0.925,0.55,0.875,1.387,157.7,123.4,151
18.50,0.875,0.425,0.85,1.292,151,115.2,148.2
18.51,0.975,0.525,0.85,1.396,167.2,121.7,148.2
18.52,0.95,0.475,1.05,1.494,161.8,118.4,180
18.53,1.075,0.625,0.975,1.58,180,128.7,167.2
18.54,0.975,0.55,1.175,1.623,167.2,123.4,180
18.55,1.025,0.55,1.175,1.653,180,123.4,180
18.56,1.025,0.625,1.1,1.628,180,128.7,180
18.57,0.95,0.45,0.925,1.4,161.8,116.7,157.7
18.58,0.925,0.6,0.9,1.423,157.7,126.9,154.2
18.59,0.925,0.4,0.975,1.402,157.7,113.6,167.2
18.60,0.9,0.675,0.975,1.489,154.2,132.5,167.2
18.61,1,0.475,1.125,1.578,180,118.4,180
18.62,1.15,0.7,1.175,1.787,180,134.4,180
18.63,1.025,0.525,1.45,1.852,180,121.7,180
18.64,1,0.6,0.925,1.488,180,126.9,157.7
18.65,0.975,0.45,0.825,1.354,167.2,116.7,145.6
18.66,0.95,0.525,0.925,1.426,161.8,121.7,157.7
18.67,0.875,0.525,0.825,1.312,151,121.7,145.6
18.68,0.9,0.525,1.15,1.552,154.2,121.7,180
18.69,1.025,0.6,1.15,1.653,180,126.9,180
18.70,1,0.6,1.075,1.586,180,126.9,180
18.71,0.975,0.65,1.075,1.59,167.2,130.5,180
18.72,1.025,0.55,1.175,1.653,180,123.4,180
18.73,1.025,0.575,0.95,1.511,180,125.1,161.8
18.74,0.95,0.5,1.025,1.484,161.8,120,180
18.75,1,0.6,1,1.536,180,126.9,180
18.76,0.95,0.6,1.2,1.644,161.8,126.9,180
18.77,0.925,0.65,1.225,1.667,157.7,130.5,180
18.78,0.925,0.575,1.05,1.513,157.7,125.1,180
18.79,0.925,0.575,1.025,1.496,157.7,125.1,180
18.80,0.85,0.475,0.775,1.244,148.2,118.4,140.8
18.81,0.85,0.475,0.725,1.214,148.2,118.4,136.5
18.82,0.925,0.525,1.025,1.477,157.7,121.7,180
18.83,0.925,0.625,1.025,1.516,157.7,128.7,180
18.84,0.925,0.65,1.025,1.526,157.7,130.5,180
18.85,1.025,0.65,1.15,1.672,180,130.5,180
18.86,0.95,0.675,1.125,1.62,161.8,132.5,180
18.87,0.95,0.55,0.975,1.468,161.8,123.4,167.2
18.88,0.9,0.55,1.05,1.488,154.2,123.4,180
18.89,0.925,0.475,0.975,1.425,157.7,118.4,167.2
18.90,0.825,0.575,0.85,1.317,145.6,125.1,148.2
18.91,0.875,0.5,0.975,1.402,151,120,167.2
18.92,0.85,0.675,0.95,1.442,148.2,132.5,161.8
18.93,0.95,0.5,1.05,1.502,161.8,120,180
18.94,0.95,0.55,1.2,1.626,161.8,123.4,180
18.95,0.975,0.575,1,1.51,167.2,125.1,180
18.96,0.9,0.525,0.975,1.427,154.2,121.7,167.2
18.97,0.85,0.575,1.025,1.45,148.2,125.1,180
18.98,0.85,0.65,0.975,1.448,148.2,130.5,167.2
18.99,0.875,0.525,0.925,1.377,151,121.7,157.7
19.00,0.9,0.55,0.925,1.403,154.2,123.4,157.7
19.01,0.825,0.6,1.1,1.5,145.6,126.9,180
19.02,0.925,0.5,1.125,1.54,157.7,120,180
19.03,0.95,0.575,1.025,1.511,161.8,125.1,180
19.04,0.9,0.475,1.1,1.499,154.2,118.4,180
19.05,0.9,0.65,1.075,1.545,154.2,130.5,180
19.06,0.925,0.625,1,1.499,157.7,128.7,180
19.07,0.9,0.525,1.025,1.462,154.2,121.7,180
19.08,0.875,0.575,0.975,1.431,151,125.1,167.2
19.09,0.875,0.55,1,1.438,151,123.4,180
19.10,0.9,0.55,1,1.453,154.2,123.4,180
19.11,0.925,0.45,1.1,1.506,157.7,116.7,180
19.12,0.9,0.575,1.05,1.498,154.2,125.1,180
19.13,0.9,0.55,1.05,1.488,154.2,123.4,180
19.14,0.9,0.625,1.05,1.518,154.2,128.7,180
19.15,0.925,0.525,1.1,1.53,157.7,121.7,180
19.16,0.9,0.525,0.975,1.427,154.2,121.7,167.2
19.17,0.9,0.5,0.95,1.401,154.2,120,161.8
19.18,0.875,0.475,1,1.411,151,118.4,180
19.19,0.9,0.55,1,1.453,154.2,123.4,180
19.20,0.925,0.575,1,1.479,157.7,125.1,180
19.21,0.9,0.55,0.95,1.42,154.2,123.4,161.8
19.22,0.975,0.525,1,1.492,167.2,121.7,180
19.23,0.95,0.575,1.05,1.528,161.8,125.1,180
19.24,0.95,0.475,1.1,1.529,161.8,118.4,180
19.25,0.975,0.55,1.15,1.605,167.2,123.4,180
19.26,1,0.475,1.125,1.578,180,118.4,180
19.27,0.95,0.575,1.1,1.563,161.8,125.1,180
19.28,0.975,0.6,1.2,1.659,167.2,126.9,180
19.29,0.925,0.6,1.2,1.63,157.7,126.9,180
19.30,0.95,0.575,1.05,1.528,161.8,125.1,180
19.31,0.975,0.55,1.025,1.518,167.2,123.4,180
19.32,0.95,0.575,1,1.494,161.8,125.1,180
19.33,1.025,0.5,1.025,1.533,180,120,180
19.34,0.9,0.5,1.05,1.471,154.2,120,180
19.35,1.025,0.5,1,1.517,180,120,180
19.36,0.9,0.55,0.925,1.403,154.2,123.4,157.7
19.37,1,0.575,0.925,1.479,180,125.1,157.7
19.38,0.975,0.625,1.2,1.668,167.2,128.7,180
19.39,1.025,0.675,1.25,1.752,180,132.5,180
19.40,1,0.525,1.15,1.612,180,121.7,180
19.41,1.05,0.55,0.95,1.519,180,123.4,161.8
19.42,0.975,0.5,1.05,1.518,167.2,120,180
19.43,0.975,0.6,0.975,1.504,167.2,126.9,167.2
19.44,1.025,0.475,0.9,1.444,180,118.4,154.2
19.45,0.925,0.575,0.925,1.429,157.7,125.1,157.7
19.46,0.975,0.575,1.075,1.561,167.2,125.1,180
19.47,1.075,0.675,1.275,1.799,180,132.5,180
19.48,1.225,0.675,1.375,1.961,180,132.5,180
19.49,1.125,0.625,1.325,1.847,180,128.7,180
19.50,1.05,0.7,1.2,1.741,180,134.4,180
19.51,1,0.65,1.2,1.692,180,130.5,180
19.52,1.025,0.575,0.85,1.45,180,125.1,148.2
19.53,0.975,0.5,0.775,1.342,167.2,120,140.8
19.54,0.925,0.55,0.875,1.387,157.7,123.4,151
19.55,1,0.525,1.125,1.594,180,121.7,180
19.56,1.05,0.65,1.175,1.705,180,130.5,180
19.57,1.025,0.6,1.25,1.724,180,126.9,180
19.58,1.05,0.675,1.175,1.714,180,132.5,180
19.59,1.025,0.625,1.15,1.662,180,128.7,180
19.60,0.975,0.65,1.125,1.624,167.2,130.5,180
19.61,0.975,0.625,0.975,1.514,167.2,128.7,167.2
19.62,1.025,0.575,0.95,1.511,180,125.1,161.8
19.63,0.975,0.6,1,1.52,167.2,126.9,180
19.64,1,0.5,1.025,1.517,180,120,180
19.65,0.975,0.7,0.925,1.515,167.2,134.4,157.7
19.66,1,0.525,0.95,1.476,180,121.7,161.8
19.67,0.975,0.65,1.075,1.59,167.2,130.5,180
19.68,1.075,0.575,1.275,1.764,180,125.1,180
19.69,1.025,0.65,1.325,1.797,180,130.5,180
19.70,1,0.625,0.975,1.53,180,128.7,167.2
19.71,0.875,0.525,0.925,1.377,151,121.7,157.7
19.72,0.9,0.65,0.975,1.478,154.2,130.5,167.2
19.73,0.975,0.525,1.05,1.526,167.2,121.7,180
19.74,0.95,0.625,0.975,1.498,161.8,128.7,167.2
19.75,0.925,0.55,1.025,1.486,157.7,123.4,180
19.76,0.925,0.625,1,1.499,157.7,128.7,180
19.77,0.925,0.575,1,1.479,157.7,125.1,180
19.78,1,0.65,1.075,1.606,180,130.5,180
19.79,0.95,0.575,1.2,1.635,161.8,125.1,180
19.80,0.9,0.7,1.05,1.55,154.2,134.4,180
19.81,0.9,0.575,1,1.463,154.2,125.1,180
19.82,0.85,0.575,0.925,1.382,148.2,125.1,157.7
19.83,0.875,0.575,0.925,1.397,151,125.1,157.7
19.84,0.925,0.525,1,1.46,157.7,121.7,180
19.85,0.925,0.575,1.025,1.496,157.7,125.1,180
19.86,0.9,0.525,1.075,1.497,154.2,121.7,180
19.87,0.925,0.6,1.1,1.557,157.7,126.9,180
19.88,0.925,0.625,1.125,1.585,157.7,128.7,180
19.89,0.95,0.725,1.175,1.676,161.8,136.5,180
19.90,0.975,0.575,1.2,1.65,167.2,125.1,180
19.91,0.925,0.625,1.05,1.533,157.7,128.7,180
19.92,0.925,0.475,0.975,1.425,157.7,118.4,167.2
19.93,0.875,0.575,0.95,1.414,151,125.1,161.8
19.94,0.925,0.525,0.95,1.426,157.7,121.7,161.8
19.95,0.875,0.6,1.025,1.475,151,126.9,180
19.96,0.9,0.525,1.05,1.479,154.2,121.7,180
19.97,0.975,0.575,0.95,1.478,167.2,125.1,161.8
19.98,0.875,0.575,1.025,1.465,151,125.1,180
19.99,0.95,0.5,1.1,1.537,161.8,120,180
20.00,0.95,0.6,1.075,1.555,161.8,126.9,180

To analyze the data I've employed the following code.

aa = Import[StringJoin[NotebookDirectory[], "LT3.csv"], "Data"];
aa[[1]]
Show[{ListLinePlot[Transpose[{aa[[All, 1]], aa[[All, 2]]}], 
   PlotRange -> {{15, 20}, {0, 1.5}}, ImageSize -> 650], 
  ListLinePlot[Transpose[{aa[[All, 1]], aa[[All, 3]]}], 
   PlotStyle -> Darker[Green]], 
  ListLinePlot[Transpose[{aa[[All, 1]], aa[[All, 4]]}], 
   PlotStyle -> Orange]}]

Acceleration [xg] vs. time [s] for x (blue), y (green) and z (orange) axes.

Next I considered the FFT.

ListLinePlot[Abs[Fourier[Rest@aa[[1000 ;; 2500, 2]]]], 
 PlotRange -> {{0, 30}, {0, 35}}, ImageSize -> 650]

FFT of acceleration in x-plane

Then I applied a moving average filter and graphed the result.

bb = Rest@aa;
adat = MovingAverage[bb, 100]
{ax, ay, az} = Interpolation /@ (adat[[All, {1, #}]] & /@ {2, 3, 4});
Plot[{ax[t], ay[t], az[t]}, {t, First@adat[[All, 1]], 
  Last@adat[[All, 1]]}, PlotLabel -> "Acceleration", Frame -> True, 
 FrameLabel -> {"Time", "Acceleration"}, 
 PlotLegends -> {"x", "y", "z"}, ImageSize -> 650]

Acceleration vs. Time

Again, per here we have velocity vs. time

{vxt, vyt, 
   vzt} = (x /. 
      Quiet@First@
        NDSolve[{x[0] == 0, x'[t] == #[t]}, 
         x, {t, First@adat[[All, 1]], Last@adat[[All, 1]]}] & /@ {ax, 
     ay, az});
Plot[{vxt[t], vyt[t], vzt[t]}, {t, First@adat[[All, 1]], 
  Last@adat[[All, 1]]}, PlotLabel -> "Velocity", Frame -> True, 
 FrameLabel -> {"Time", "Velocity"}, PlotLegends -> {"x", "y", "z"}, 
 ImageSize -> 650]

Velocity vs. time

And finally, position.

{xt, yt, zt} = (x /. Quiet@First@NDSolve[{x[0] == 0, x'[t] == #[t]}, x, {t, First@adat[[All, 1]], Last@adat[[All, 1]]}] & /@ {vxt, vyt, vzt});
Plot[{xt[t], yt[t], zt[t]}, {t, First@adat[[All, 1]], Last@adat[[All, 1]]}, PlotLabel -> "Position", Frame -> True, FrameLabel -> {"Time", "Position"}, PlotLegends -> {"x", "y", "z"}, ImageSize -> 650]

Position vs. time

ParametricPlot3D[{xt[t], yt[t], zt[t]}, {t, First@adat[[All, 1]], Last@adat[[All, 1]]}, AxesLabel -> {"X", "Y", "Z"}]

Displacement in x, y and z planes

Sorry for the long post but I wanted to be thorough. I've also tried alternative numerical integration methods but all lead to the same result. My guess is the integration error is accumulating with time. I realize the moving average filter is a halfhearted try but I have yet to identify an alternative filtering technique that works reference. If anyone would like all of the original data or notebook I'd be happy to share.

Any advice is greatly appreciated.

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  • $\begingroup$ Are you claiming that the numerical result disagrees with measured resuts? If so, I did not find the relevant part of the post, so you might want to make more clear what exactly is the troublesome aspect of the computations. $\endgroup$ – Daniel Lichtblau Dec 30 '14 at 15:40
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    $\begingroup$ As @DanielLichtblau pointed out, it's not clear what problem you've identified. That said, what you're trying to do is tricky, at best. Since you're doubly integrating accelerometer data, biases become a major error source (the bias is the value that the sensor indicates when it's not accelerating). Furthermore, biases drift, so correcting for them is challenging. Add to that noise and other error sources, and you've got a tough problem. Finally, you need to correct for the rotational motion of the earth... $\endgroup$ – Cassini Dec 30 '14 at 15:53
  • $\begingroup$ Sorry. Since the accelerometer was rotated in the x-y plane I would expect the final result to be a circle. $\endgroup$ – Grasliner Dec 30 '14 at 15:54
  • $\begingroup$ You data has consistently large z_g, larger than the other components usually, which are all positive(!). Have you correctly accounted for zero-point offset/bias? Perhaps you need to use the degree information to account for the spinning. I'm not that familiar with accelerometers, but the data or your interpretation of the data represented by your code looks wrong. Any integration error seems secondary at this point. $\endgroup$ – Michael E2 Dec 30 '14 at 18:20
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    $\begingroup$ This question appears to be off-topic because the problem is one of physics (or electronics?), not Mathematica. $\endgroup$ – Oleksandr R. Dec 31 '14 at 1:29