Velocity Estimation from noisy Measurements Sensor fusion using modified Kalman filter www.controltrix.com

Objective Consider a vehicle moving Desired to measure the velocity accurately Velocity is directly measured but is noisy Acceleration also measured using onboard accelerometers Integrating acceleration data gives velocity Offset errors in acc./random walk cause drift in velocity Standard solution Kalman filter with optimal gain K for sensor data fusion Estimate by combining velocity and acc. measurement

Problem specifics Acceleration and velocity are measured using noisy sensor Direct velocity measurement is noisy (sv = 10m/s) Acceleration is measured with sa = 0.1 m/s2 offset = 0.2 m/s2 (DRIFT) Superposed sine wave drive Amplitude A = 3 m/s2, frequency f = 0.05 Hz Sample time Ts = 0.1 s Simulated time = 200s - 400s

Measured velocity noisy data (True velocity is smooth sine wave of amp 10, period 20 s)

Advantages No matrix calculations Easier computation, can be easily scaled Equivalent to Kalman filter structure (easily proven) No drift (the error converges to 0) Estimate accelerometer drift in the system by default Drift est. for calib. and real time comp. of accelerometers

Advantages. Can be modified easily to make tradeoff between drift performance (convergence) and noise reduction Systematic technique for parameter calculations No trial and error

Comparison Sl No metric Kalman Filter Modified Filter 1. Drift Drift is a major problem (depends inversely on K) Needs considerable characterization.(Offset, temperature calibration etc). Guaranteed automatic convergence. No prior measurement of offset and characterization required. Not sensitive to temperature induced variable drift etc. 2. Convergence Non-Zero measurement and process noise covariance required else leads to singularity Always converges No assumptions on variances required Never leads to a singular solution 3. Method Two distinct phases: Predict and update. Can be implemented in a few single difference equation or even in continuum.

Comparison. Sl No metric Kalman Filter Modified Filter 4. Computation Need separate state variables for position, velocity, etc which adds more computation. Highly optimized computation. Only single state variable required 5. Gain value /performance In one dimension, K = process noise / measurement noise. dt ‘termed as optimal’ Gains based on systematic design choices. The gains are good though suboptimal (based on tradeoff) 6. Processor req. Needs 32 Bit floating point computation for accuracy and plenty of MIPS/ computation Easily implementable in 16 bit fixed point processor 40 MIPS/computation is sufficient Note: The right column filter is a super set of a standard Kalman filter

Sim results std Kalman filter velocity estimation error (v^ - v) vs time

Sim results of proposed solution error = v^ – v vs time

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