1. AE-8803 (Fall 2015)
PROJECT PRESENTATION
Stochastic Estimation using Second Order
Approximations
Marcus Pereira
December 9, 2015

2. • The EKF (Extended Kalman Filter) is the most commonly used algorithm for
state estimation of non-linear dynamical systems in presence of stochastic
noise.
• It is based on the 1st order Taylor series approximation of the system dynamical
equations and measurement equations, expanded about the current state
estimate.
• These 1st order approximations don’t always hold for,
i. highly non-linear trajectories (such aggressive maneuvers of a quadcopter)
ii. when there is high uncertainty in the dynamics or external disturbances such as wind
gusts
iii. when measurement noise is high
Motivation

3. Problem Definition
• Problem Definition - To investigate the use of 2nd order Extended Kalman Filter for state estimation
of non-linear dynamical systems.
• System implemented for this purpose is a quadcopter model with non-linear dynamics and
measurement equations of the form:
• Tested for highly non-linear trajectories, in presence of high uncertainty in dynamics (simulated by
high process noise) and high measurement noise.
• Process noise w(t) and measurement noise are assumed Gaussian white noise with zero mean and
high sigma (standard deviation).
• Controller – 1st order Differential Dynamic Programming algorithm.

4. Equations of the 1st order Extended Kalman Filter (EKF)

5. 3-D visualization of the
trajectory simulated in
MATLAB
Motion of the Center of
mass of the Quadcopter
Initial
position
Final
position

6. Simulation
• These are the input vectors to the DDP algorithm which generates the
optimal trajectory seen on the previous slide.
• Simulation time – 4 seconds
• 1st Simulation – DDP + 1st order EKF with high process and
measurement noise. Because of presence of stochastic noise, using
the following control law:

11. Equations of the 2nd order Extended Kalman Filter (EKF)

12. • 2nd Simulation:
i. Generate optimal trajectory using 1st order DDP in absence of stochastic
noise (same aggressive trajectory as shown in the earlier slide)
ii. No control modification/correction based on state estimate i.e. use
optimal trajectory without optimal feedback gains to allow noise to
dominate the trajectory of the quadcopter.
iii. Compare estimates of 1st order EKF vs. 2nd order EKF vs true state.
Observation:
Attitude states were highly non-linear and had noise concentration. Huge spikes
observed in estimates of 1st order filter. For most of the states, the 2nd order
filter estimates stay close to the respective true states.

22. • 3rd Simulation:
• Using control with correction based on state estimate:
• Compare performance of (1st order EKF + DDP) vs. (2nd order EKF + DDP)
• Observation:
Due to better accuracy of state estimates, DDP control with 2nd order EKF
is close to optimal trajectory. Huge spikes observed in 1st order EKF + DDP
control scheme and trajectories are far from optimal.

29. Future Direction
• Quadcopter model considered is with state additive noise and the G
matrix is independent of state.
• The equation for state covariance prediction will change in presence
of state multiplicative noise and if the G matrix is state dependent.
• Also, equations of DDP will have to be re-derived .