velocity motion model
TRANSCRIPT
![Page 1: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/1.jpg)
Velocity Motion Model
Çetin Meriçli
25.09.2007
![Page 2: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/2.jpg)
Outline
• Preliminaries
• Kinematic Configuration
• Probabilistic Kinematics
• Velocity Motion Model
• Derivation
• Exact Motion
• Real Motion
• Algorithms
![Page 3: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/3.jpg)
3
Kinematic Configuration
• In general the configuration of a robot can be described by six parameters.
• Three-dimensional cartesian coordinates plus three Euler angles pitch, roll, and tilt.
• Throughout this section, we consider robots operating on a planar surface.
• The state space of such systems is three-dimensional (x,y,θ).
![Page 4: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/4.jpg)
4
Dynamic Bayesian Network for Controls, States, and Sensations
![Page 5: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/5.jpg)
5
Robot Motion
• Robot motion is inherently uncertain.
• How can we model this uncertainty?
![Page 6: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/6.jpg)
6
Reasons for Motion Errors
bump
ideal case different wheeldiameters
carpetand many more …
![Page 7: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/7.jpg)
7
Probabilistic Motion Models
•To implement the Bayes Filter, we need the transition model p(x | x’, u).
•The term p(x | x’, u) specifies a posterior probability, that action u carries the robot from x’ to x.
• In this section we will specify, how p(x | x’, u) can be modeled based on the motion equations.
![Page 8: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/8.jpg)
8
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.
• 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.
![Page 9: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/9.jpg)
9
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.
![Page 10: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/10.jpg)
10
Velocity-Based Model
![Page 11: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/11.jpg)
11
Mathematical Derivation
On Board
![Page 12: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/12.jpg)
12
Equation for the Velocity Model
Center of circle:
with
![Page 13: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/13.jpg)
13
Posterior Probability for Velocity Model
correction: |v| => v^2, |w| => w^2
![Page 14: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/14.jpg)
14
The atan2 Function
• Extends the inverse tangent and correctly copes with the signs of x and y.
![Page 15: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/15.jpg)
15
Calculating the Probability (zero-centered)
• For a normal distribution
• For a triangular distribution
• Algorithm prob_normal_distribution(a,b):
• return
• Algorithm prob_triangular_distribution(a,b):
• return
![Page 16: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/16.jpg)
16
Typical Distributions for Probabilistic Motion Models
εσ2 x =
1
2πσ 2e
−12x2
σ2 εσ2 x ={
0 if∣x∣6σ2
6σ2−∣x∣
6σ2
Normal distribution Triangular distribution
![Page 17: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/17.jpg)
17
Sampling from Velocity Model
![Page 18: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/18.jpg)
18
How to Sample from Normal or Triangular Distributions?
• Sampling from a normal distribution
• Sampling from a triangular distribution
• Algorithm sample_normal_distribution(b):
• return
• Algorithm sample_triangular_distribution(b):
• return
![Page 19: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/19.jpg)
19
Normally Distributed Samples
106 samples
![Page 20: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/20.jpg)
20
For Triangular Distribution
103 samples 104 samples
106 samples105 samples
![Page 21: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/21.jpg)
21
Examples (velocity based)
![Page 22: Velocity Motion Model](https://reader033.vdocuments.us/reader033/viewer/2022061104/542d865c219acd4e4b8b55e0/html5/thumbnails/22.jpg)