Probabilistic Motion Model Chapter 7

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.

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.