Robotics 2 Target Tracking. Kai Arras, Cyrill Stachniss, Maren Bennewitz, Wolfram Burgard
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1 Robotics 2 Target Tracking Kai Arras, Cyrill Stachniss, Maren Bennewitz, Wolfram Burgard Slides by Kai Arras, Gian Diego Tipaldi, v.1.1, Jan 2012
2 Chapter Contents Target Tracking Overview Applications Types of Problems and Errors Algorithms Linear Discrete Systems (LDS) Kalman Filter (KF) Overview Error Propagation Esimation Cycle Limitations Extended Kalman Filter (EKF) Overview First-Order Error Propagation Tracking Example
3 Target Tracking Overview Tracking is the estimation of the state of a moving object based on remote measurements. [Bar-Shalom] Detection is knowing the presence of an object (possibly with some attribute information) Tracking is maintaining the state and identity of an object over time despite detection errors (false negatives, false alarms), occlusions, and the presence of other objects
4 Target Tracking Applications Fleet Management Surveillance Air Traffic Control Motion Capture Military Applications Robotics, HRI
5 Tracking: Problem Types Track stage Track formation (initialization) Track maintenance (continuation) Number of sensors Single sensor Multiple sensors Sensor characteristics Detection probability (PD) False alarm rate (PF) Target behavior Nonmaneuvering Maneuvering Number of targets Single target Multiple targets Target size Point-like target Extended target
6 Tracking: Error Types 1. Uncertainty in the values of measurements: Called noise Solution: Filtering (State estimation theory) 2. Uncertainty in the origin of measurements: measurement might originate from sources different from the target of interest. Reasons : False alarms Decoys and countermeasures Multiple targets Solution: Data Association (Statistical decision theory)
7 Tracking: Problem Statement Given Model of the system dynamics (process or plant model). Decribes the evolution of the state Model of the sensor with which the target is observed Probabilistic models of the random factors (noise sources) and the prior information Wanted System state estimate such as kinematic (e.g. position), feature (e.g. target class) or parameters components In a way that... Accuracy and/or reliability is higher than the raw measurements Contains information not available in the measurements
8 Tracking Algorithms Single non-maneuvering target, no origin uncert. Kalman filter (KF)/Extended Kalman filter (EKF) Single maneuvering target, no origin uncertainty KF/EKF with variable process noise Muliple model approaches (MM) Single non-maneuvering target, origin uncertainty KF/EKF with Nearest/Strongest Neighbor Data Association Probabilistic Data Association filter (PDAF) Single maneuvering target, origin uncertainty Multiple model-pdaf
9 Tracking Algorithms Multiple non-maneuvering targets Joint Probabilistic Data Association filter (JPDAF) Multiple Hypothesis Tracker (MHT) Multiple maneuvering targets MM-variants of MHT (e.g. IMMMHT) Other Bayesian filtering schemes such as Particle filters have also been sucessfully applied to the tracking problem
10 Linear Dynamic System (LDS) A continuous-time Linear Dynamic System (LDS) can be described by a state equation of the form Called process or plant model The system can be observed remotely through Called observation or measurement model This is the State-Space Representation, omnipresent in physics, control or estimation theory Provides the mathematical formulation for our estimation task
11 Linear Dynamic System (LDS) Stochastic process governed by is the state vector is the input vector is the process noise is the system matrix is the input gain The system can be observed through is the measurement vector is the measurement noise is the measurement matrix
12 Discrete-Time LDS Continuous model are difficult to realize Algorithms work at discrete time steps Measurements are acquired with certain rates In practice, discrete models are employed Discrete-time LDS are governed by is the state transition matrix is the discrete-time input gain Same observation function of continuous models
13 Discrete-Time LDS Continuous model are difficult to realize Algorithms work at discrete time steps Measurements are acquired with certain rates In practice, discrete models are employed Discrete-time LDS are governed by In target tracking, the input is unknown! is the state transition matrix is the discrete-time input gain Same observation function of continuous models
14 LDS Example Throwing ball We want to throw a ball and compute its trajectory This can be easily done with a LDS No uncertainties, no tracking, just physics The ball s state shall be represented as We ignore winds but consider the gravity force g No floor constraints We observe the ball with a noise-free position sensor
15 LDS Example Throwing ball Throwing a ball from s with initial velocity v Consider only the gravity force, g, of the ball y o s v x State vector Process matrices Initial state Input vector (scalar) Measurement vector Measurement matrix
16 LDS Example Throwing ball Initial State No noise
17 LDS Example Throwing ball System evolution Observations
18 LDS Example Throwing ball System evolution Observations
19 LDS Example Throwing ball System evolution Observations
20 LDS Example Throwing ball System evolution Observations
21 LDS Example Throwing ball System evolution Observations
22 LDS Example Throwing ball System evolution Observations
23 LDS Example Throwing ball System evolution Observations
24 LDS Example Throwing ball System evolution Observations
25 LDS Example Throwing ball System evolution Observations
26 LDS Example Throwing ball System evolution Observations
27 LDS Example Throwing ball System evolution Observations
28 LDS Example Throwing ball System evolution Observations
29 LDS Example Throwing ball Initial State It s windy and our sensor is imperfect: let s add Gaussian process and observation noise
30 LDS Example Throwing ball System evolution Observations
31 LDS Example Throwing ball System evolution Observations
32 LDS Example Throwing ball System evolution Observations
33 LDS Example Throwing ball System evolution Observations
34 LDS Example Throwing ball System evolution Observations
35 LDS Example Throwing ball System evolution Observations
36 LDS Example Throwing ball System evolution Observations
37 LDS Example Throwing ball System evolution Observations
38 LDS Example Throwing ball System evolution Observations
39 LDS Example Throwing ball System evolution Observations
40 LDS Example Throwing ball System evolution Observations
41 LDS Example Throwing ball System evolution Observations
42 Kalman Filter (KF) The Kalman filter is the workhorse of tracking Under linear Gaussian assumptions, the KF is the optimal minimum mean squared error (MMSE) estimator. It is still the optimal linear MMSE estimator if these conditions are not met An optimal estimator is an algorithm that processes observations to yield a state estimate that minimizes a certain error criterion (e.g. RMS, MSE) It is basically a (recursive) weighted sum of the prediction and observation. The weights are given by the process and the measurement covariances See literature for detailed tutorials
43 Kalman Filter (KF) Consider a discrete time LDS with dynamic model where is the process noise (no input assumed) The measurement model is where is the measurement noise The initial state is generally unknown and modeled as a Gaussian random variable State estimate Covariance estimate
44 Kalman Filter State Prediction Measurement Prediction Update
45 EKF: Error Propagation Error Propagation is everywhere in Kalman filtering From the uncertain previous state to the next state over the system dynamics From the uncertain inputs to the state over the input gain relationship From the uncertain predicted state to the predicted measurements over the measurement model
46 Error Propagation Law Given A linear system Y = F X X X, Y assumed to be Gaussian Input covariance matrix C X System matrix F X the Error Propagation Law computes the output covariance matrix C Y 46
47 Error Propagation Law Derivation in Matrix Notation Blackboard... 47
48 Error Propagation Law Derivation in Matrix Notation 48
49 Kalman Filter Cycle
50 Kalman Filter Cycle motion model predicted state posterior state innovation from matched landmarks observation model predicted measurements in sensor coordinates targets raw sensory data
51 KF Cycle 1/4: State prediction State prediction In target tracking, no a priori knowledge of the dynamic equation is generally available Instead, different motion models (MM) are used Brownian MM Constant velocity MM Constant acceleration MM Constant turn MM Specialized models (problem-related, e.g. ship models)
52 Motion Models: Brownian No-motion assumption Useful to describe stop-and-go motion behavior State representation Ball example Initial state Transition matrix
53 Motion Models: Brownian No-motion assumption Useful to describe stop-and-go motion behavior State representation Ball example Initial state Transition matrix State prediction: Uncertainty grows
54 Motion Models: Brownian No-motion assumption Useful to describe stop-and-go motion behavior State representation Ball example Initial state Transition matrix State prediction: Uncertainty grows
55 Motion Models: Brownian No-motion assumption Useful to describe stop-and-go motion behavior State representation Ball example Initial state Transition matrix State prediction: Uncertainty grows
56 Motion Models: Brownian No-motion assumption Useful to describe stop-and-go motion behavior State representation Ball example Initial state Transition matrix State prediction: Uncertainty grows
57 MMs: Constant Velocity Constant target velocity assumption Useful to model smooth target motion State representation Ball example Initial state Transition matrix
58 MMs: Constant Velocity Constant target velocity assumption Useful to model smooth target motion State representation Ball example Initial state Transition matrix State prediction: Linear target motion Uncertainty grows
59 MMs: Constant Velocity Constant target velocity assumption Useful to model smooth target motion State representation Ball example Initial state Transition matrix State prediction: Linear target motion Uncertainty grows
60 MMs: Constant Velocity Constant target velocity assumption Useful to model smooth target motion State representation Ball example Initial state Transition matrix State prediction: Linear target motion Uncertainty grows
61 MMs: Constant Velocity Constant target velocity assumption Useful to model smooth target motion State representation Ball example Initial state Transition matrix State prediction: Linear target motion Uncertainty grows
62 MMs: Constant Acceleration Constant target acceleration assumed Useful to model target motion that is smooth in position and velocity changes State representation Ball example Initial state Transition matrix
63 MMs: Constant Acceleration Constant target acceleration assumed Useful to model target motion that is smooth in position and velocity changes State representation Ball example Initial state Transition matrix State prediction: Non-linear motion Uncertainty grows
64 MMs: Constant Acceleration Constant target acceleration assumed Useful to model target motion that is smooth in position and velocity changes State representation Ball example Initial state Transition matrix State prediction: Non-linear motion Uncertainty grows
65 MMs: Constant Acceleration Constant target acceleration assumed Useful to model target motion that is smooth in position and velocity changes State representation Ball example Initial state Transition matrix State prediction: Non-linear motion Uncertainty grows
66 MMs: Constant Acceleration Constant target acceleration assumed Useful to model target motion that is smooth in position and velocity changes State representation Ball example Initial state Transition matrix State prediction: Non-linear motion Uncertainty grows
67 MMs: Constant Acceleration Constant target acceleration assumed Useful to model target motion that is smooth in position and velocity changes State representation Ball example Initial state Transition matrix
68 MMs: Constant Acceleration Constant target acceleration assumed Useful to model target motion that is smooth in position and velocity changes State representation Ball example Initial state Transition matrix State prediction: Non-linear motion Uncertainty grows
69 MMs: Constant Acceleration Constant target acceleration assumed Useful to model target motion that is smooth in position and velocity changes State representation Ball example Initial state Transition matrix State prediction: Non-linear motion Uncertainty grows
70 MMs: Constant Acceleration Constant target acceleration assumed Useful to model target motion that is smooth in position and velocity changes State representation Ball example Initial state Transition matrix State prediction: Non-linear motion Uncertainty grows
71 MMs: Constant Acceleration Constant target acceleration assumed Useful to model target motion that is smooth in position and velocity changes State representation Ball example Initial state Transition matrix State prediction: Non-linear motion Uncertainty grows
72 KF Cycle 2/4: Meas. Prediction Measurement prediction Observation Typically, only the target position is observed. The measurement matrix is then Note: One can also observe Velocity (Doppler radar) Acceleration (accelerometers)
73 KF Cycle 3/4: Data Association Once measurements are predicted and observed, we have to associate them with each other This is resolving the origin uncertainty of observations Data association is typically done in the sensor reference frame Data association can be a hard problem and many advanced techniques exist More on this later in this course
74 KF Cycle 3/4: Data Association Step 1: Compute the pairing difference and its associated uncertainty The difference between predicted measurement and observation is called innovation The associated covariance estimate is called the innovation covariance The prediction-observation pair is often called pairing
75 KF Cycle 3/4: Data Association Step 2: Check if the pairing is statistically compatible Compute the Mahalanobis distance Compare it against the proper threshold from an cumulative ( chi square ) distribution Significance level Degrees of freedom Compatibility on level α is finally given if this is true
76 KF Cycle 3/4: Data Association Constant velocity model Process noise Measurement noise No false alarm No problem
77 KF Cycle 3/4: Data Association Constant velocity model Process noise Measurement noise Uniform false alarm False alarm rate = 3 Ambiguity: several observations in the validation gate
78 KF Cycle 3/4: Data Association Constant velocity model Process noise Measurement noise Uniform false alarm False alarm rate = 3 Wrong association as closest observation is false alarm
79 KF Cycle 4/4: Update Computation of the Kalman gain State and state covariance update
80 KF Cycle 4/4: Update prediction measurement correction It's a weighted mean! 80
81 Kalman Filter Cycle motion model predicted state posterior state innovation from matched landmarks observation model predicted measurements in sensor coordinates targets raw sensory data
82 Kalman Filter: Limitations Non-linear motion models and/or non-linear measurement models Extended Kalman filter Unknown inputs into the dynamic process model (values and modes) Enlarged process noise (simple but there are implications) Multiple model approaches (accounts for mode changes) Uncertain origin of measurements Data Association How many targets are there? Track formation and deletion techniques Multiple Hypothesis Tracker (MHT)
83 Extended Kalman Filter The Extended Kalman filter deals with non-linear process and non-linear measurement models Consider a discrete time LDS with dynamic model where is the process noise (no input assumed) The measurement model is where is the measurement noise The same KF-assumptions for the initial state
84 Kalman Filter State Prediction Measurement Prediction Update
85 Extended Kalman Filter State Prediction Jacobian Measurement Prediction Jacobian Update
86 First-Order Error Propagation Given A non-linear system Y = f(x) X,Y assumed to be Gaussian Input covariance matrix C X Jacobian matrix F X the Error Propagation Law computes the output covariance matrix C Y 86
87 First-Order Error Propagation Approximating f(x) by a first-order Taylor series expansion about the point X = µ X 87
88 Jacobian Matrix It s a non-square matrix in general Suppose you have a vector-valued function Let the gradient operator be the vector of (first-order) partial derivatives Then, the Jacobian matrix is defined as 88
89 Jacobian Matrix It s the orientation of the tangent plane to the vectorvalued function at a given point Generalizes the gradient of a scalar valued function 89
90 Track Management Formation When to create a new track? What is the initial state? Occlusion/deletion When to delete a track? Is it just occluded? Heuristics: Greedy initialization Every observation not associated is a new track Initialize only position Lazy initialization Accumulate several unassociated observations Initialize position & velocity Heuristics: Greedy deletion Delete if no observation can be associated No occlusion handling Lazy deletion Delete if no observation can be associated for several time steps Implicit occlusion handling
91 Example: Tracking the Ball Unlike the previous experiment in which we had a model of the ball s trajectory and just observed it, we now want to track the ball Comparison: small versus large process noise Q and the effect of the three different motion models For simplicity, we perform no gaiting (i.e. no Mahalanobis test) but accept the pairing every time
92 Ball Tracking: Brownian Small process noise Large process noise Ground truth Observations State estimate
93 Ball Tracking: Brownian Ground truth Observations State estimate
94 Ball Tracking: Brownian Ground truth Observations State estimate
95 Ball Tracking: Brownian Ground truth Observations State estimate
96 Ball Tracking: Brownian Ground truth Observations State estimate
97 Ball Tracking: Brownian Ground truth Observations State estimate
98 Ball Tracking: Brownian Ground truth Observations State estimate
99 Ball Tracking: Brownian Ground truth Observations State estimate
100 Ball Tracking: Brownian Ground truth Observations State estimate
101 Ball Tracking: Brownian Ground truth Observations State estimate
102 Ball Tracking: Brownian Ground truth Observations State estimate
103 Ball Tracking: Brownian Ground truth Observations State estimate
104 Ball Tracking: Brownian Ground truth Observations State estimate
105 Ball Tracking: Brownian Ground truth Observations State estimate
106 Ball Tracking: Brownian large difference! Ground truth Observations State estimate
107 Ball Tracking: Constant Velocity Small process noise Large process noise Ground truth Observations State estimate
108 Ball Tracking: Constant Velocity Ground truth Observations State estimate
109 Ball Tracking: Constant Velocity Ground truth Observations State estimate
110 Ball Tracking: Constant Velocity Ground truth Observations State estimate
111 Ball Tracking: Constant Velocity Ground truth Observations State estimate
112 Ball Tracking: Constant Velocity Ground truth Observations State estimate
113 Ball Tracking: Constant Velocity Ground truth Observations State estimate
114 Ball Tracking: Constant Velocity Ground truth Observations State estimate
115 Ball Tracking: Constant Velocity Ground truth Observations State estimate
116 Ball Tracking: Constant Velocity Ground truth Observations State estimate
117 Ball Tracking: Constant Velocity Ground truth Observations State estimate
118 Ball Tracking: Constant Velocity Ground truth Observations State estimate
119 Ball Tracking: Constant Velocity Ground truth Observations State estimate
120 Ball Tracking: Constant Velocity Ground truth Observations State estimate
121 Ball Tracking: Constant Velocity smaller difference Ground truth Observations State estimate
122 Ball Tracking: Const. Acceleration Small process noise Large process noise Ground truth Observations State estimate
123 Ball Tracking: Const. Acceleration Ground truth Observations State estimate
124 Ball Tracking: Const. Acceleration Ground truth Observations State estimate
125 Ball Tracking: Const. Acceleration Ground truth Observations State estimate
126 Ball Tracking: Const. Acceleration Ground truth Observations State estimate
127 Ball Tracking: Const. Acceleration Ground truth Observations State estimate
128 Ball Tracking: Const. Acceleration Ground truth Observations State estimate
129 Ball Tracking: Const. Acceleration Ground truth Observations State estimate
130 Ball Tracking: Const. Acceleration Ground truth Observations State estimate
131 Ball Tracking: Const. Acceleration Ground truth Observations State estimate
132 Ball Tracking: Const. Acceleration Ground truth Observations State estimate
133 Ball Tracking: Const. Acceleration Ground truth Observations State estimate
134 Ball Tracking: Const. Acceleration Ground truth Observations State estimate
135 Ball Tracking: Const. Acceleration Ground truth Observations State estimate
136 Ball Tracking: Const. Acceleration very small difference! Ground truth Observations State estimate
137 Summary Tracking is maintaining the state and identity of a moving object over time despite detection errors (false negatives, false alarms), occlusions, and the presence of other objects Linear Dynamic Systems (a.k.a. the state-space representation) provide the mathematical framework for estimation For tracking, there is no control input u in the process model. Therefore good motion models are key The Kalman filter is a recurse Bayes filter that follows the typical predict-update cycle
138 Summary The Extended Kalman filter (EKF) is for cases of non-linear process or measurement models. It computes the Jacobians, first-order linearizations of the models, and has the same expressions than the KF A large process noise covariance can partly compensate a poor motion model for maneuvering targets But: large process noise covariances cause the validation gates to be large which in turn increases the level of ambiguity for data association. This is potentially problematic in case of multiple targets
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