In almost every aspect of life, there is a concept of state (for example, where are you right now?) and a concept of uncertainty (for example, how soon can you reach home?). And whenever there is uncertainty, there is a tool that can help you get a better estimate of that state. It is called a Kalman Filter, named after an American Mathematician, Rudolf Kalman.

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What can Software
learn from Robots
Naman Kumar
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● Examples
○ Autonomous Driving
○ Robot Localization
○ Project Management
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● Autonomous Driving
3https://www.economist.com/the-economist-explains/2015/05/12/how-does-a-self-driving-car-work

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● Robot Localization
Source: Probabilistic Robotics, Sebastian
Thrun et al. 4

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● Project Management
○ Input:
■ Jira Tasks completed
■ Unit tests passed
■ Integration testing status
■ Code coverage
○ Output: Precise estimate of the project
○ Also, an ability to predict accurately if and when you will
meet the milestone
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● TartanSense (Agriculture Robotics)
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● TartanSense
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● Bayes Filter
○ Terminology
■ X: State
■ Z: Measurement
■ u: Control/ Action
■ t: Timestep
………………..
………………..
X0
X1
’ X1
Xt
’ Xt
u1
Z1
ut
Zt
Source: Probabilistic Robotics, Sebastian
Thrun et al.
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● Bayes Filter
○ Estimates a probability density function recursively over time using measurements and some
maths
○
○ Algorithm bayes_filter(bel(xt-1
), ut,
zt
):
■ For all xt
do
● bel’(xt
) = ∫p(xt
| ut
,xt-1
) bel(xt-1
) dx
● bel(xt
) = η p(zt
|xt
) bel’(xt
)
■ Endfor
■ Return bel(xt
) 9

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● But what the heck is a Kalman Filter?
○ It is just a Bayes filter with following conditions:
■ Normally distributed variables
■ Linear transitions
○ To reiterate, it has got two main steps:
■ PREDICT: It generates estimates of the state variables with some
associated uncertainty
■ UPDATE: The new measurements (or data) are incorporated and the
estimates are updated using a weighted average with more weight
given to more certain estimates
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● KALMAN FILTER I
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● Kalman Filter equations:
○ Next State probability: xt
= At
*xt-1
+ Bt
*ut
+ εt
○ Measurement probability: zt
= Ct
*xt
+ δt
● Kalman Filter Algorithm
○ kalman_filter(μt-1
,Σt-1
,ut
, zt
):
■ μ’t
= At
μt-1
+ Bt
ut
■ Σ’t
= At
Σt-1
AT
t
+ Rt
■ Kt
= Σ’t
CT
t
(Ct
Σ’t
CT
t
+ Qt
)-1
■ μt
= μ’t
+ Kt
(zt
- Ct
μ’t
)
■ Σt
= (I - Kt
Ct
)Σ’t
○ return μt
, Σt
● *K = Error in Prediction /
(Error in Prediction + Error in Measurement)
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● Shortcomings of a vanilla Kalman Filter
○ Works only for linear systems
○ Works only for Gaussian noise
○ It can compute beliefs only for continuous states
● Real life Example: Driving home
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● Extended Kalman Filter (EKF)
○ Overcomes the linearity assumption. State and measurement probabilities are computed
using nonlinear functions g and h:
■ xt
= g(ut
,xt-1
) + εt
■ zt
= h(xt
) + δt
○ Comparing EKF and KF:
○ What will be the repercussions of this?
■ The belief is no longer gaussian
■ EKF can only calculate an approximate true belief unlike exact belief calculated by KF
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● Extended Kalman Filter
○ The key here is the linearization. EKF uses First order Taylor Expansion
○ Algorithm:
Source: Probabilistic Robotics, Sebastian
Thrun et al.
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● Where will EKF fail?
○ Lost Robot (Use Sum of Gaussians -> MHEKF)
Source: Probabilistic Robotics, Sebastian
Thrun et al. 16

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● Where will EKF fail?
○ Degree of Nonlinearity
○ Degree of Uncertainty
● Is there a better way to linearize?
○ Unscented Kalman Filter
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● Unscented Kalman Filter
○ How can we linearize better?
○ In KF, we have one point (the mean) and we linearize the function around it
○ In UKF, we have a bunch of points and we linearize around them
● When to use what?
○ If you have a linear system, use Kalman Filter
○ If you have a small nonlinear system, use Extended Kalman Filter
○ If you have a highly nonlinear system, use Unscented Kalman Filter
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● Alternative to Gaussian Filters: Non-parametric filters
○ Histogram Filters
○ Particle Filters
Source: Probabilistic Robotics, Sebastian
Thrun et al. 19

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Questions/ Thoughts?
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