机器人如何实时确定他们的状态，并从带有噪声的传感器测量量获得周围环境的信息？在这个模块中，你将学习怎样让机器人把不确定性融入估计，并向动态和变化的世界进行学习。特殊专题包括用于定位和绘图的概率生成模型和贝叶斯滤波器。

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From the course by 宾夕法尼亚大学

机器人学：估计和学习

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机器人如何实时确定他们的状态，并从带有噪声的传感器测量量获得周围环境的信息？在这个模块中，你将学习怎样让机器人把不确定性融入估计，并向动态和变化的世界进行学习。特殊专题包括用于定位和绘图的概率生成模型和贝叶斯滤波器。

From the lesson

Bayesian Estimation - Target Tracking

We will learn about the Gaussian distribution for tracking a dynamical system. We will start by discussing the dynamical systems and their impact on probability distributions. This linear Kalman filter system will be described in detail, and, in addition, non-linear filtering systems will be explored.

- Daniel LeeProfessor of Electrical and Systems Engineering

School of Engineering and Applied Science

In this lecture,

we will discuss non-linear approaches to the Kalman filter.

A standard linear model has some limitations that require different

methods to model motion uncertainty.

Non-linear motion updates break the Gaussian properties of the state

distribution.

We will now discuss the extended common filter and the unscented common filter.

With a non-linear dynamical system, the state transition can be a function

of the state, and thus it's harder to predict where the model should be going.

A differentiable function helps by allowing for

approximation through linearization at the current time step.

The Jacobian matrix represents the differentiation of this matrix

motion function, but it does not capture higher order dynamics.

If there is not a large deviation in the time step,

then this can be a valid approach to track the state distribution through time.

The linear form, shown above, is rewritten to project the state

using the non-linear function instead of the state transition matrix.

Effectively, the Jacobian replaces the state transition matrix

in the calculation of the predicted covariance.

Similarly, the Kalman gain is rewritten

with the Jacobian of the observation function.

The Jacobian is evaluated at the point x of t.

The overall update to track the state over time is very similar to the linear system.

The extended Kalman filter is a straightforward method

to retain the gassing concepts given a differentiable motion and

observation model.

The next approach to dealing with non-linearities utilizes a small set

of sample points.

This filter is called the unscented Kalman filter or UKF.

The UKF continually re-estimates the distribution statistics of the mean and

covariance, by transforming characteristic

points through the non-linear dynamical system.

First, we model the distribution based on a set of sigma points.

These sigma points typically characterize the covariance of the matrix

around about a standard deviation away from the mean and

including the mean as one of the sigma points.

The statistics of the sigma points will have the same mean and

covariance of the underlying Gaussian distribution.

The new distribution after the motion model is applied will not be Gaussian,

however, we can recalculate the mean and

covariance of the transformed characteristic signal points

in order to approximate the new distribution as Gaussian.

With the idea of tracking statistics over time

we first look at tracking the expected state over time.

The expected state is the average of the transformed sigma points.

Similarly the predicted covariance is the covariance of the sigma points

having been run through the dynamical system.

Each sigma point will have an associated expectation for the observation.

We can model the distribution of the expected observation then

by calculating the statistics again.

The Kalman gain is slightly modified from the linear system.

To calculate the Kalman gain, we disregard the observation model and

utilize the covariances of the observation points where signal points

run through the motion model.

The final update is just as the linear filter.

However, the covariance update will be slightly different.

Please look at the attached notes for good resources on further details.

Finally, in a preview to week four, we can take the unscented Kalman

filter to the limit and use many points to characterize the distribution.

This modified distribution of points then

will not limit the underlying model to be Gaussian.

We will explore this particle filter in week four.

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