Vert Systems: Mountain Biking Tracking Simon Fowler Jonathan Black, Jacqueline Christmas, Mikhail Krastanov, Jakub Nowotarski, Jan Sieber, Robert Szalai, Jakub Tomczyk, Ellen Webborn Bristol 91st ESGI Mountain Biking Tracking

Problem Outline Sensor located on the helmet of the rider contains p ( t ) ∈ R 3 GPS : (1 Hz ) a ( t ) ∈ R 3 Accelerometer : ˆ (100 Hz ) ˆ Ω( t ) ∈ R 3 x 3 Gyroscope : (100 Hz ) m ( t ) ∈ R 3 Magnetometer : ˆ (100 Hz ) Aim: Obtain trajectory of rider at higher accuracy and time resolution than the GPS data. Bristol 91st ESGI Mountain Biking Tracking

Wavelet smoothing Analysed data including pre-processing raw data using wavelet transform. 20 20 Raw signal (acceleration) S 3 10 10 0 0 −10 −10 −20 −20 −30 −30 0 200 400 600 800 1000 0 200 400 600 800 1000 20 20 S 4 S 5 10 10 0 0 −10 −10 −20 −20 −30 −30 0 200 400 600 800 1000 0 200 400 600 800 1000 Figure: Example of wavelet smoother performed on linear acceleration of the sensor with respect to x axis. Bristol 91st ESGI Mountain Biking Tracking

Determining orientation matrix To evolve the rotation matrix we use the gyroscope data R ( t ) = ˆ ˙ Ω( t ) R ( t ) to evolve in discrete time steps we use � � ˆ R n +1 = exp Ω n h n R n The angular velocity matrix is defined as 0 ˆ − ˆ ω 3 ω 2 ˆ Ω = − ˆ 0 ˆ , ω 3 ω 1 ˆ − ˆ 0 ω 2 ω 1 where ˆ ω 1 , ˆ ω 2 , ˆ ω 3 are the gyroscope measurements. Bristol 91st ESGI Mountain Biking Tracking

Model ˙ X(t) = V(t) , ˙ V(t) = R(t) · ˆ a(t) + gravity , R(t) = ˆ ˙ Ω(t) · R(t) , R(t) · ˆ m(t) = magnetic north , X(t) = p(t) , R(t) T R(t) = I Where Ω is the matrix of gyro measurements, and R is the rotation matrix between the two frames. Bristol 91st ESGI Mountain Biking Tracking

Method-Least squares Two approaches were considered - the first solves the model’s equations using least squares. The system is large over-determined nearly linear solved iteratively with Gauss-Newton iteration considers a segment of trajectory at once Bristol 91st ESGI Mountain Biking Tracking

Method-Kalman Filter The second method solves the dynamical system with hidden states x n x n-1 x n and observations y n . x n = Θx n − 1 + e n , y n y n = Wx n + ǫ n where p n v n p n v n − 1 a n ˆ x n = y n = , R n ˆ g n R n − 1 m n ˆ gravity Θ updates location assuming constant velocity between time-steps. W is the transformation from states to observations. Bristol 91st ESGI Mountain Biking Tracking

Demos Bristol 91st ESGI Mountain Biking Tracking

Real time processing a a

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