Probabilistic Motion Model Chapter 7

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Questions and Answers

What is the purpose of the atan2 function?

To calculate the tangent of an angle.

To calculate the cosine of an angle.

To calculate the sine of an angle.

To calculate the inverse tangent of an angle, taking into account the signs of x and y. (correct)

What is the purpose of the parameters α1 to α4 in the noise model for odometry?

To model the direction of the robot.

To model the speed of the robot.

To model the environment the robot is operating in.

To model the accuracy of the robot. (correct)

Why does odometry error affect translation error?

Because the robot may rotate slightly while translating. (correct)

Because the robot's wheels may slip.

Because the robot's sensors are not perfectly accurate.

Because the robot's environment may be uneven.

Which of the following are examples of signals that can be modeled by a stochastic process?

Speech, music, image, time-varying channels, noise Signup and view all the answers

Which of the following is NOT a commonly controlled drive train using a velocity motion model?

Articulated Drive Signup and view all the answers

What is the purpose of the function `prob(a, b)` as used in the calculation of the posterior probability?

To calculate the likelihood of a given state given a measurement. Signup and view all the answers

What is the primary principle behind "Dead Reckoning"?

Calculating a vehicle's current position based on its velocities and elapsed time. Signup and view all the answers

How is the motion model applied in practice?

By applying it repeatedly for short movements. Signup and view all the answers

What is the main source of odometry information in robotic systems?

Wheel encoders Signup and view all the answers

What is the purpose of sampling from a normal distribution in the context of probabilistic motion models?

To approximate the probability distribution of the robot's state. Signup and view all the answers

Which of the following is a cause of motion errors in wheeled robots?

Uneven wheel diameters. Signup and view all the answers

In the context of odometry, what does 'trans' represent?

The robot's translational velocity. Signup and view all the answers

What is the significance of the "Odometry information u  rot1 , rot 2 , trans" in the text?

It describes the input data used for calculating the robot's motion. Signup and view all the answers

How is the "Odometry Model" related to the concept of "Dead Reckoning"?

Odometry Model provides a more advanced and precise way to implement Dead Reckoning using wheel encoders. Signup and view all the answers

What are the primary advantages of using odometry for robot motion estimation?

It provides a real-time estimation of the robot's position based on integrated wheel encoder feedback. Signup and view all the answers

What is the primary challenge in modeling robot motion?

Inherent uncertainty in robot motion Signup and view all the answers

Which of the following statements about odometry-based models is true?

They use wheel encoders to track rotation. Signup and view all the answers

In a probabilistic motion model, what does the term specifying a posterior probability represent?

The action's effect on the robot's future state Signup and view all the answers

What is the primary advantage of using a velocity-based model?

It does not require wheel encoders. Signup and view all the answers

Which coordinate parameter does not represent orientation in a robot's configuration?

Y-axis position Signup and view all the answers

Study Notes

Chapter 7: Probabilistic Motion Model

Robot motion is inherently uncertain.
Uncertainty needs to be modeled.

Robot Motion

Robot motion is inherently uncertain.
How can this uncertainty be modeled?
Presented images show a robot navigating a maze, with the paths representing different possible trajectories.

Dynamic Bayesian Network

A dynamic Bayesian network is used for controls, states, and sensations.
A network diagram illustrates the relationships between control inputs (u), robot states (X), and sensor measurements (Z) over time.
The motion model (Gaussian distribution) returns potential outputs or end points of the robot given its previous state and control input.

Motion Model

The probability of a state (Xt) given control input (Ut) and previous state (Xt-1) is denoted as p(xt | Ut, Xt-1).
This model details the probability of a robot's state or location change given a control input

Probabilistic Motion Filter Models

Implementing the Bayes filter requires the transition model, p(xt | Xt-1, Ut).
This model calculates the probability of a new state post control input, conditional on the prior state and input, used within a bayesian filter
This section discusses how p(xt| Xt-1, Ut) can be modeled based on motion equations and uncertain outcomes of movements

Coordinate Systems

A wheeled robot's configuration in 3D space is described by six parameters: three Cartesian coordinates and three Euler angles (representing roll, pitch, and yaw).
For simplicity, this chapter focuses on robots operating in a 2D plane. Thus, the robot pose is represented by the coordinates (x, y, θ).

Bearing or Orientation

Orientation, also known as bearing or heading direction, describes the robot's direction in space.
A bearing of 0° aligns with the robot's X-axis.
A bearing of 90° (π/2) aligns with the robot's Y-axis.
Pose without orientation is called location.

Typical Motion Models

Two types of motion models are prevalent in practice (odometry-based and velocity-based).
Odometry-based models are used when the robot is equipped with wheel encoders.
Velocity-based (Dead Reckoning) models are used when wheel encoders are not available; it calculates the new pose based on velocities and time elapsed.

Example Wheel Encoders

These modules provide +5V when they "see" white, and 0V when they "see" black.
Quality laminated color plastic ensures crisp black-to-white transitions, aiding in accurate wheel encoder sensing.

Velocity Motion Model

Assumes control over robot motion is achievable via rotational and translational velocities.
Many commercial robots possess control interfaces allowing programmers to specify these velocities.

Dead Reckoning

A mathematical procedure for determining the current location of a vehicle.
Calculates the current pose from vehicle velocities and elapsed time.
Historically used for logging ship positions.

Reasons for Motion Errors

Motion errors arise from various factors such as different wheel diameters and surface irregularities (bumps, carpets).

Odometry Model

Explains how a robot moves from one point to another, using a robot encoder.
Involves two angles (drot1, drot2) and translation (dtrans) to model the interval-based motion.
Odometry-based motion calculation often uses integrated wheel encoder readings to determine estimated movement.

The atan2 Function

Extends the inverse tangent function to incorporate the signs of x and y.

Noise Model for Odometry

Measured motion is the true motion corrupted by noise

Typical Distributions for Probabilistic Models

Normal distributions are commonly used to model process noise of continuous stochastic processes.
Examples of signals that can be modeled by stochastic processes include speech, music, image, time-varying channels, noise.

Calculating the Probability Density

Algorithms for calculating probability density using normal and triangular distributions are described.

Calculating the Posterior

An algorithm (motion_model_odometry) is presented to calculate the posterior given the state (x,x') and odometry data, using Gaussian distributions
Using parameters (a1-a4) to model the accuracy or error of the robot, allows for a probabilistic model.

Application

Repeated application of the motion model, for short movements produces typically banana-shaped distributions.

Sample-Based Density Representation

Sample data of the distribution can be viewed graphically; this demonstrates a visual representation of the normal probability density.

Sampling from a Normal Distribution

An algorithm for sampling from a normal distribution is given.

Normally Distributed Samples

Graphically showcasing normally distributed samples.
A graph of normal distributed samples (with 10⁶ samples).

How to Sample

Algorithms for sampling from normal and triangular distributions are included.

Distribution Graphs

Graph examples of sampling distributions (10³, 10⁴, 10⁵, and 10⁶ samples) visually comparing normal and triangular distributions.

Sample Odometry Motion Model

This is an algorithm to sample from an odometry-based model using the input values that produce an output of new positions (xt).

Examples (Odometry-Based)

Visual examples illustrating various odometry-based motion scenarios, showcasing differences in translation and rotational errors.

Velocity-Based Model

Represents robot motion as (v, ω), which represents translational and rotational velocities.

Noise Model for the Velocity-Based Model

Describes how measured motion differs from the true motion due to noise, using example equations.
Discusses the potential drawbacks of this noise model.

Noise Model for the Velocity-Based Model (advanced)

Shows how noise can be accounted for using equations, such as (v, ω).
Suggests using equations to account for rotational noise for more accurate results.

Posterior Probability for Velocity Model

Details the calculation of posterior probability using velocity-based motion model parameters, along with the equations involved.

Motion (kinematic equations)

Displays kinematic equations crucial for understanding translational and rotational movement in the context of a robot.

Sampling from Velocity Model

Presents the algorithms that enable sampling from a velocity-based motion model.

Examples (Velocity-Based)

Demonstrates visual examples of velocity-based motion scenarios.

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Description

Explore the intricacies of probabilistic motion models in robotics through this quiz. Learn how uncertainty affects robot navigation and how dynamic Bayesian networks can be used to model state changes over time. Assess your understanding of Gaussian distributions and the probabilistic motion filter models.