Kalman Filter in Mobile Robotics 2015
Explore the fundamentals of the Kalman Filter in mobile robotics during the spring of 2015, covering topics such as actions, observations, data estimation, elapse time, and more. Dive into examples, initial estimates, measurement acquisition, and new belief updates to understand the basis of this linear system approach. Discover the work of Rudolf E. Kalman and gain insights into how to apply Kalman Filters in real-world scenarios through intuitive derivations.
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Presentation Transcript
Kalman Kalman Filter Example Filter Example Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
Rudolf E. Kalman Rudolf E. Kalman b. 1930 Hungary Kalman Filter NASA Ames 1960 National Medal of Science (2009) Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
Actions and Observations Actions and Observations Through Time Through Time Belief(xt) (using all evidence to date) Get Measurement Elapse Time Belief''(xt) (without latest evidence) Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
Kalman Data: Kalman Data: Our estimate: x1..t= x1, x2, x3...xt Our measurement: z1..t= z1, z2, z3...zt Our action: u1..t= u1, u2, u3...ut Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
Kalman Example Kalman Example "Understanding the Basis of the Kalman Filter Via a Simple and Intuitive Derivation" Ramsey Faragher https://www.cl.cam.ac.uk/~rmf25/papers/Understanding%20the%20 Basis%20of%20the%20Kalman%20Filter.pdf Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
Initial Estimate Initial Estimate Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
Elapse Time Elapse Time Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
Get Measurement Get Measurement Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
New Belief New Belief Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
Kalman: Assume a Linear System Kalman: Assume a Linear System xt= Ftxt-1+ Btut+ wt xt = state (estimate) ut= action Ft= state transition matrix Bt= control input matrix wt= noise wt~ N(0,Qt) Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
Kalman: Measurements Kalman: Measurements zt= Htxt+ vt zt= measurements Ht = Transformation matrix vt= observational noise vt~ N(0, Rt) Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
Kalman: Belief Kalman: Belief Gaussian Distribution x0= initial mean P0= Variance N(x0, P0) Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
Kalman Example Kalman Example Action: ft/m State: xt vt Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
Kalman Update Kalman Update State: xt Action: ft/m vt xt= xt-1+ vt-1 t + 0.5 (ft/m) * t2 vt= vt-1+ (ft/m) t 1 = t 0 t2/2 t x't v't xt-1 ft/m + vt-1 1 Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
Kalman: Elapse Time Update Kalman: Elapse Time Update x't= Ftxt-1+ Btut P't= FtPt-1FTt + Qt Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
Kalman: Measurement Update Kalman: Measurement Update Kalman Gain (how much to correct estimate) Kt= P'tHTt(HtP'tHTt+ Rt)-1 New Belief: xt= x't+ Kt(zt-Htx't) Pt= P't- KtHtP't Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
Simple Kalman Filter Example Simple Kalman Filter Example xt = just position (meters) ut= 1 m/s wt~ N(0,0.1) vt~ N(0,1.0) x0= 4.5 P2= 2.0 x't= xt-1+ ut+ wt zt= xt + vt (F=1, B=1) (H=1) Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
t=0 t=0 x0=4.5 P2=2 Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
t=1 t=1 x1= x0+ 1 = 5.5 P'1= F0P0FT0 + Q0 = 1 * 2 * 1 + 0.1 = 2.1 Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
t=1 t=1 z z1 1=5.673 =5.673 K1= P'tHTt(HtP'tHTt+ Rt)-1 = (2.1 * 1)/(1*2.1*1+1.0) = .677 x1= x'1+ K1(z1-H1x'1) = 5.5 + .677 (5.673-1*5.5) = 5.617121 Pt= P't- KtHtP't = 2.1 - .677 * 1 * 2.1 = 0.6783 Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
Kalman Filters Kalman Filters Must be linear system Stored as a gaussian Mobile Robotics Mobile Robotics Spring 2015 Spring 2015
