Probabilistic Motion Model PDF

Chapter 7 Probabilistic Motion Model  Robot Motion  Robot motion is inherently uncertain.  How can we model this uncertainty? Dynamic Bayesian Network for Controls, States, and Sensations Motion model is treated as a function - distribution (Gaussian)- that returns potential outputs or end poi...

Chapter 7 Probabilistic Motion Model  Robot Motion  Robot motion is inherently uncertain.  How can we model this uncertainty? Dynamic Bayesian Network for Controls, States, and Sensations Motion model is treated as a function - distribution (Gaussian)- that returns potential outputs or end points of the robot given it was in Xt-1 before executing the odometry Ut Probabilistic Motion Models  To implement the Bayes Filter, we need the transition model.  The term specifies a posterior probability, that action ut carries the robot from xt-1 to xt.  In this section we will discuss, how can be modeled based on the motion equations and the uncertain outcome of the movements.  Coordinate Systems  The configuration of a typical wheeled robot in 3D can be described by six parameters.  These are the three-dimensional Cartesian coordinates plus the three Euler angles for roll pitch and yaw. For simplicity, throughout this section we consider robots operating on a planar surface.  The state space of such systems is three- dimensional vector (x,y,).  Cont. ▶ The orientation of a robot is often called bearing, or heading direction. ▶ we postulate that a robot with orientation θ = 0 points into the direction of its x-axis. ▶ A robot with orientation θ=.5π points into the direction of its y-axis. ▶ Pose without orientation will be called location.  Typical Motion Models  In practice, one often finds two types of motion models:  Odometry-based  Velocity-based (Dead Reckoning)  Odometry-based models are used when systems are equipped with wheel encoders.  Wheel Encoders located directly behind each motor is a wheel encoder. Each wheel encoder is used to count the number of times the motor (left or right) has rotated.  Velocity-based models have to be applied when no wheel encoders are given. They calculate the new pose based on the velocities and the time elapsed. Example Wheel Encoders These modules provide +5V output when they "see" white, and a 0V output when they "see" black. These disks are manufactured out of high quality laminated color plastic to offer a very crisp black to white transition. This enables a wheel encoder sensor to easily see the transitions. Source: http://www.active-robots.com/  Velocity motion model ▶ The velocity motion model assumes that we can control a robot through two velocities, a rotational and a translational velocity. ▶ Many commercial robots offer control interfaces where the programmer specifies velocities. ▶ Drive trains that are commonly controlled in this way include the differential drive, the Ackerman drive, the synchro drive, and some holonomic drives.  Dead Reckoning  Derived from “deduced reckoning.”  Mathematical procedure for determining the present location of a vehicle.  Achieved by calculating the current pose of the vehicle based on its velocities and the time elapsed.  Historically used to log the position of ships. Reasons for Motion Errors of Wheeled Robots different wheel ideal case diameters Bump - carpet and many more …  Odometry Model Alternatively, one might Robot moves from x , y , to x ' , y ' , ' want to use the. Odometry information u  rot1 , rot 2 , trans. odometry measurements as the basis for c alculating the robot’s motion over time. Odometry is c ommonly Two angles plus a translation obtained by integrating wheel encoders information;most c ommerc ial robots make  rot 2 suc h integrated pose estimation available in x ' , y ' , periodic time intervals  trans ' (e.g., every tenth x, y,   rot 1 of a sec ond).  The atan2 Function  Extends the inverse tangent and correctly copes with the signs of x and y.  Noise Model for Odometry  The measured motion is given by the true motion corrupted with noise. The parameters α1 to α4 (with αi ≥ 0 for i = 1,... , 4) are robot-specific error parameters. They model the accuracy of the robot. Rotating a robot on a spot for a while does not affect the initial rotation only but also drifts it to a side, this is why odometry error affect translation error Typical Distributions for Probabilistic Motion Models Normal distribution Triangular distribution Normal distributions are commonly used to model noise in continuous stochastic processes. Examples of signals that can be modelled by a stochastic process are speech, music, image, time-varying channels, noise, and any information bearing function of time. Calculating the Probability Density (zero-centered) Calculating the Posterior Given x, x’, and Odometry the func tion prob(a, b) implements an error distribution over a with zero mean and variance b  ASSIGNMENTS 2 1. Calculate the following question: 2. Then write a python code to find the probability Application  Repeated application of the motion model for short movements.  Typical banana-shaped distributions obtained for the 2d- projection of the 3d posterior. sample from this this is the return value distribution of motion model as shown next slide x x u u  21 Sample-Based Density Representation How to Sample from a Normal Distribution?  Sampling from a normal distribution 1. Algorithm sample_normal_distribution(b): 2. return This kind of sampling is not uniform as those random number generation algorithms found in programming languages such as Python, C++ or Matlab  23 Normally Distributed Samples 106 samples How to Sample from Normal or Triangular Distributions? 23  For Triangular Distribution 25 103 samples 104 samples 105 samples Introduction to Mobile Robotics 106 samples Prof. A MESLEH  26 Sample Odometry Motion Model Examples (Odometry-Based) High rotational error than translation error especially in the translation and rotation error initial rotational are both relatively equal translation error has error bigger variance than the rotation error  29 Velocity-Based Model Noise Model for the Velocity-Based Model  The measured motion is given by the true motion corrupted with noise. vˆ v  |v | || 1 2 ̂     |v | || 3 4  Discussion: What is the disadvantage of this noise model? Noise Model for the Velocity-Based Model  The ( vˆ,ˆ)-circle constrains the final orientation (2D manifold in a 3D space)  Better approach: Term to account for the final rotation Posterior Probability for Velocity Model 32 Mathematical derivation : section 5.3.3  33 Motion Term to account for the final rotation Sampling from Velocity Model 34 Examples (Velocity-Based) 35