Estimation in Robotics
slides are from different sources, e.g. D. Lee(UPenn), D. Kelin & P. Abbeel (UC Berkeley) , Wikipedia, …
What is Estimation and why is important?
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§ Let’s look at an example (RoboCup competition): § Objective: Tracking a moving target (soccer ball)
• The robots are complete autonomous,
• their onboard computers need to integrate information from the inertial and vision
sensors to perceive the world around them.
• Plan their behaviors to either attack or defend, and send motor commands for
locomotion and to manipulate the orange soccer ball in various ways.
What is Estimation and why is important?
§ Let’s look at an example (RoboCup competition:): § Objective: Tracking a moving target (soccer ball)
• In this scenario, the attacking robot needs to estimate where the ball and the goal are located in order to line up a kick. Then as the ball approaches the goal, the goalie robot needs to estimate the speed and direction of the ball in order to execute an appropriate dive to save the ball from going into the goal.
Estimation
§ By estimation, we mean to estimate some aspect of the state of the world from noisy, incomplete and uncertain data.
§ What are the sources of uncertainty in robotics?
§ (1) there are a sensor noise. That is that the sensors can provide
inaccurate information.
§ (2) there could be a lack of knowledge about the world. That is, things can be hidden from view or that the robots could not perceive.
§ (3) there might be dynamic changes in the motion and in the environment. That is, that things maybe moving over time and the robots do not know exactly where things are at the current instant.
Estimation and Uncertainty
§ There are several ways to deal with this uncertainty; among those, two different aspects of dealing with this uncertainty include:
§ (1) The first is probabilistic modeling. That is, using probability distributions to account for this uncertainty.
§ (2) And the second method is by using machine learning, to learn from previous experience to be able to predict the future uncertain world.
§ Our focus: probabilistic modeling
Problem Definition: Ball/Object Tracking
§ What we know about the problem?
§ … the attacking robot needs to estimate where the ball and the goal
are located in order to line up a kick….
§ The estimation needs to be done continuously (i.e., over time)
§ the estimation happens before observation (i.e., prediction)
§ Ball needs to be modeled (e.g., using probability distribution)
§ Ball’s motion/dynamic needs to be modeled
§ If we are measure the location and velocity of the ball, we cannot attach sensors to the ball.
§ Sensors for the measurement: Vision/camera, inertia,….
§ Measurements/observations are indirect. This means the actual parameters of
interest (location, velocity,..) are hidden from the observation § Measurements are not accurate (i.e, noisy).
Object Tracking using (KF)
§ Kalman filters are one of the main topics in the field of robotic motion planning and control and can be used in trajectory optimization.
§ One of the widely used algorithm for tracking objects/states is KF.
§The Kalman filter was invented in the 60’s and first implemented as a method of trajectory estimation for the Apollo program.
§ The Kalman filter has numerous applications in for guidance, navigation, and control of vehicles, particularly aircraft, spacecraft and dynamically positioned ships. Furthermore, the Kalman filter is a widely applied concept in time series analysis used in fields such as signal processing and econometrics.
Object Tracking using
§ Track a moving target
§ One of the widely used algorithm for tracking objects is ing.
§ Optimality of the Kalman filter assumes that the errors are Gaussian.
§ KF is optimal when the motion is linear.
§ Extensions of KF could be used when the motion is non-linear.
§ The underlying model is a hidden Markov model where the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions.
§ The algorithm works in a two-step process: prediction step, and update step.
Object Tracking using
§ Track a moving target
§ The algorithm works in a two-step process. In the prediction step, the Kalman filter produces estimates of the current state variables, along with their uncertainties. Once the outcome of the next measurement (necessarily corrupted with some amount of error, including random noise) is observed, in the update step, these estimates are updated using a weighted average, with more weight being given to estimates with higher certainty. The algorithm is recursive. It can run in real time, using only the present input measurements and the previously calculated state and its uncertainty matrix; no additional past information is required.
Object Tracking using
§ Source: Wikipedia
Probability Recap
§ Conditional probability § Product rule
§ Chain rule
§ X, Y independent if and only if:
§ X and Y are conditionally independent given Z if and only if:
Gaussian Distribution
We used Gaussian distribution in KF. Why Gaussian?
• The two parameters (mean and variance) are easy to compute and interpret.
• Good mathematical properties:
e.g., product of Gaussian distributions forms Gaussian.
• Central limit theorem:
Expectation of the mean of any random variables converges to Gaussian.
Gaussian Distribution : Example
Ball color distribution Color Image
“ Ye l l o w ” ?
Gaussian Distribution : Example
Ball color distribution Color Image
Gaussian Distribution (1D)
𝑥 Variable
𝜇 Mean 𝜎2 Variance
𝜎 Standard deviation
Gaussian Distribution (1D)
𝑝 𝑥 = 1 𝑒−𝑥2 2𝜋 2
(Standard Normal Distribution)
Gaussian Distribution (1D)
1 −(𝑥+1)2 2𝜋𝑒 2
Gaussian Distribution (1D)
1 −(𝑥−1)2 2𝜋𝑒 2
Gaussian Distribution (1D)
𝑝 𝑥 = 1 𝑒−𝑥2 2𝜋4
Gaussian Distribution (1D)
𝑝𝑥= 1 𝑒−𝑥2 1.4𝜋 1.4
Gaussian Distribution (1D)
Gaussian Distribution : Example
Ball color distribution Color Image
Object Tracking using
• Now that we reviewed the probability and Gaussian distribution, let’s go back to our problem of ball tracking.
• Note: we just learnt that we could model the ball as a gaussian distribution.
• Track a moving target • Soccer ball
Object Tracking using
§ Source: Wikipedia
Intuition behind KF
• Physical model means we needs to discuss the
ddynamics of the Ball’s motion • Motion: linear, non-linear
• Constant speed, Friction, Gravity…
Intuition behind KF
• Multiple measurements: ●, ●, ●, … • Each measurement is noisy:
• What is the true state of the object?
State and Measurement
• State (x): any quantity of interest
• Measurement (z): what we observe
(Hidden Variable) z
• Example: “What characterizes the state of a ball?” – Position,Velocity,Acceleration
– Rotation – Color
– Temperature – Elasticity -…
Measurement
• Example: What do we observe or measure? – Distance
– Inertiachange – Color
Measurement
Bayesian ing
• Modeling motion and noise
• Mathematical underpinnings of Kalman filters • Position tracking example
Linear Modeling
• Discrete Linear dynamical system of motion • xt+1=Axt +But zt =Cxt
• Simple state vector, x, is position and velocity •xt+1:=[v dv/dt]
• Description of Dynamics
• A = [ 1 dt ] 01
KF: Bayesian filtering
2 𝑥t−1 𝑥t 𝑥t+1 𝑥
KF: Bayesian filtering
2 𝑥t−1 𝑥t 𝑥t+1 𝑥
• Know xt-1
Bayesian modeling
• Prediction using state dynamics model • Inference from noisy measurements
• ModelxtwithaGaussian(meanandcovariance)
Bayesian filtering
Bayesian filtering
Bayesian ing
• Apply the MAP to Bayes’ Rule • Solve the maximization
• Establish update method • MAP: Maximum A Posteriori
In Bayesian statistics, a maximum a posteriori probability (MAP) estimate is an estimate of an unknown quantity, that equals the mode of the posterior distribution. The MAP can be used to obtain a point estimate of an unobserved quantity on the basis of empirical data. It is closely related to the method of maximum likelihood (ML) estimation, but employs an augmented optimization objective which incorporates a prior distribution (that quantifies the additional information available through prior knowledge of a related event) over the quantity one wants to estimate. MAP estimation can therefore be seen as a regularization of maximum likelihood estimation.
Bayesian filtering
Bayesian filtering
Bayesian filtering
• Posterior distribution is another Gaussian • MAP Estimates “optimal” xt value
• Use MAP estimates to form a new mean and variance for the state
Bayesian filtering
Bayesian filtering
Bayesian filtering
Bayesian filtering
Bayesian filtering
Ball Tracking
• Model the ball using a Gaussian function
• Track the ball using
1D Visualization
• The position of x is moving forward
• Uncertain motion model increases the spread
• We observe a noisy position estimate, zt
• The corrected position has less spread than both
the observation and motion adjusted state
p(xt|xt-1)
1D Visualization
• The position of x is moving forward
• Uncertain motion model increases the spread
• We observe a noisy position estimate, zt
• The corrected position has less spread than both
the observation and motion adjusted state
p(xt|xt-1)
1D Visualization
• The position of x is moving forward
• Uncertain motion model increases the spread
• We observe a noisy position estimate, zt
• The corrected position has less spread than both
the observation and motion adjusted state
p(xt|xt-1)
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