2 Used \sigma for variance instead of \sigma^2. Also grammar error.
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You are reading it too narrowly.

  • You don't "need" odometery. SLAM is simply a way to fuse any sensor estimates into a consistent estimate of the robot's state.

  • "Feature-based" doesn't necessarily mean you need to have identifiable features everywhere in the environment.

  • First principal of sensor fusion: Two estimates are better than one!

Example

I haven't read the book "for dummies" but if they don't do the following numeric example, I'd set the book on fire and get a better one. And if they do have this example, then I wonder why you didn't mention it!

( you can follow along in the math here )

A robot is at position $x=0$, and is moving to the right (increasing $x$). In this perfect world, the dynamics and sensor modes are linear. (otherwise use EKF, PF, or some variant).

  • There's a wall at exactly $x=10$ which the robot can measure distance to.
  • The robot has a laser scanner to get distance with sensor variance $\sigma_l=.1$$\sigma_l^2=.1$
  • The robot can measure it's distance travelled with odometers using sensor variance $\sigma_o = .5$$\sigma_o^2 = .5$. Clearly the laser is more accurate thatthan the odos.

Here's how the robot handles SLAM in this simple environment. (note this is actually localization since we aren't updating the position of the wall).

  • The robot tries to move one unit to the right.
  • Odometry measures $x=.9$
  • Laser scanner says you are $8.8$ units from the wall. (implying you are at 1.2)

Question: Where are you?

  • Do you choose the best sensor? In this case, the laser is the best right? So obviously I'm at $x=1.2$.

  • Do you choose the one "closest" to what you expect? Well in this case I think we should use odometry, since $.9$ is closer to what I intended, (moving one unit).

  • Maybe you could average the two? Well, that's better, but it is susceptible to outliers.

  • The shiny principals of sensor fusion tell you how to answer the question as follows:

Your minimum mean-squared estimate of the robot position is given by:

$$x_{mmse}=\frac{\sigma_l}{\sigma_o+\sigma_l}(.9) + \frac{\sigma_o}{\sigma_o+\sigma_l}(1.2)$$$$x_{mmse}=\frac{\sigma_l^2}{\sigma_o^2+\sigma_l^2}(.9) + \frac{\sigma_o^2}{\sigma_o^2+\sigma_l^2}(1.2)$$ $$x_{mmse}=\frac{.1}{.6}(.9) + \frac{.5}{.6}(1.2)$$ $$x_{mmse}=1.15$$

... unless I screwed up the algebra somewhere. People localize airplanes using math not much more complicated than that.

You are reading it too narrowly.

  • You don't "need" odometery. SLAM is simply a way to fuse any sensor estimates into a consistent estimate of the robot's state.

  • "Feature-based" doesn't necessarily mean you need to have identifiable features everywhere in the environment.

  • First principal of sensor fusion: Two estimates are better than one!

Example

I haven't read the book "for dummies" but if they don't do the following numeric example, I'd set the book on fire and get a better one. And if they do have this example, then I wonder why you didn't mention it!

( you can follow along in the math here )

A robot is at position $x=0$, and is moving to the right (increasing $x$). In this perfect world, the dynamics and sensor modes are linear. (otherwise use EKF, PF, or some variant).

  • There's a wall at exactly $x=10$ which the robot can measure distance to.
  • The robot has a laser scanner to get distance with sensor variance $\sigma_l=.1$
  • The robot can measure it's distance travelled with odometers using sensor variance $\sigma_o = .5$. Clearly the laser is more accurate that the odos.

Here's how the robot handles SLAM in this simple environment. (note this is actually localization since we aren't updating the position of the wall).

  • The robot tries to move one unit to the right.
  • Odometry measures $x=.9$
  • Laser scanner says you are $8.8$ units from the wall. (implying you are at 1.2)

Question: Where are you?

  • Do you choose the best sensor? In this case, the laser is the best right? So obviously I'm at $x=1.2$.

  • Do you choose the one "closest" to what you expect? Well in this case I think we should use odometry, since $.9$ is closer to what I intended, (moving one unit).

  • Maybe you could average the two? Well, that's better, but it is susceptible to outliers.

  • The shiny principals of sensor fusion tell you how to answer the question as follows:

Your minimum mean-squared estimate of the robot position is given by:

$$x_{mmse}=\frac{\sigma_l}{\sigma_o+\sigma_l}(.9) + \frac{\sigma_o}{\sigma_o+\sigma_l}(1.2)$$ $$x_{mmse}=\frac{.1}{.6}(.9) + \frac{.5}{.6}(1.2)$$ $$x_{mmse}=1.15$$

... unless I screwed up the algebra somewhere. People localize airplanes using math not much more complicated than that.

You are reading it too narrowly.

  • You don't "need" odometery. SLAM is simply a way to fuse any sensor estimates into a consistent estimate of the robot's state.

  • "Feature-based" doesn't necessarily mean you need to have identifiable features everywhere in the environment.

  • First principal of sensor fusion: Two estimates are better than one!

Example

I haven't read the book "for dummies" but if they don't do the following numeric example, I'd set the book on fire and get a better one. And if they do have this example, then I wonder why you didn't mention it!

( you can follow along in the math here )

A robot is at position $x=0$, and is moving to the right (increasing $x$). In this perfect world, the dynamics and sensor modes are linear. (otherwise use EKF, PF, or some variant).

  • There's a wall at exactly $x=10$ which the robot can measure distance to.
  • The robot has a laser scanner to get distance with sensor variance $\sigma_l^2=.1$
  • The robot can measure it's distance travelled with odometers using sensor variance $\sigma_o^2 = .5$. Clearly the laser is more accurate than the odos.

Here's how the robot handles SLAM in this simple environment. (note this is actually localization since we aren't updating the position of the wall).

  • The robot tries to move one unit to the right.
  • Odometry measures $x=.9$
  • Laser scanner says you are $8.8$ units from the wall. (implying you are at 1.2)

Question: Where are you?

  • Do you choose the best sensor? In this case, the laser is the best right? So obviously I'm at $x=1.2$.

  • Do you choose the one "closest" to what you expect? Well in this case I think we should use odometry, since $.9$ is closer to what I intended, (moving one unit).

  • Maybe you could average the two? Well, that's better, but it is susceptible to outliers.

  • The shiny principals of sensor fusion tell you how to answer the question as follows:

Your minimum mean-squared estimate of the robot position is given by:

$$x_{mmse}=\frac{\sigma_l^2}{\sigma_o^2+\sigma_l^2}(.9) + \frac{\sigma_o^2}{\sigma_o^2+\sigma_l^2}(1.2)$$ $$x_{mmse}=\frac{.1}{.6}(.9) + \frac{.5}{.6}(1.2)$$ $$x_{mmse}=1.15$$

... unless I screwed up the algebra somewhere. People localize airplanes using math not much more complicated than that.

1
source | link

You are reading it too narrowly.

  • You don't "need" odometery. SLAM is simply a way to fuse any sensor estimates into a consistent estimate of the robot's state.

  • "Feature-based" doesn't necessarily mean you need to have identifiable features everywhere in the environment.

  • First principal of sensor fusion: Two estimates are better than one!

Example

I haven't read the book "for dummies" but if they don't do the following numeric example, I'd set the book on fire and get a better one. And if they do have this example, then I wonder why you didn't mention it!

( you can follow along in the math here )

A robot is at position $x=0$, and is moving to the right (increasing $x$). In this perfect world, the dynamics and sensor modes are linear. (otherwise use EKF, PF, or some variant).

  • There's a wall at exactly $x=10$ which the robot can measure distance to.
  • The robot has a laser scanner to get distance with sensor variance $\sigma_l=.1$
  • The robot can measure it's distance travelled with odometers using sensor variance $\sigma_o = .5$. Clearly the laser is more accurate that the odos.

Here's how the robot handles SLAM in this simple environment. (note this is actually localization since we aren't updating the position of the wall).

  • The robot tries to move one unit to the right.
  • Odometry measures $x=.9$
  • Laser scanner says you are $8.8$ units from the wall. (implying you are at 1.2)

Question: Where are you?

  • Do you choose the best sensor? In this case, the laser is the best right? So obviously I'm at $x=1.2$.

  • Do you choose the one "closest" to what you expect? Well in this case I think we should use odometry, since $.9$ is closer to what I intended, (moving one unit).

  • Maybe you could average the two? Well, that's better, but it is susceptible to outliers.

  • The shiny principals of sensor fusion tell you how to answer the question as follows:

Your minimum mean-squared estimate of the robot position is given by:

$$x_{mmse}=\frac{\sigma_l}{\sigma_o+\sigma_l}(.9) + \frac{\sigma_o}{\sigma_o+\sigma_l}(1.2)$$ $$x_{mmse}=\frac{.1}{.6}(.9) + \frac{.5}{.6}(1.2)$$ $$x_{mmse}=1.15$$

... unless I screwed up the algebra somewhere. People localize airplanes using math not much more complicated than that.