Sensors for Mobile Robots CSE-571 • Contact sensors: Bumpers • Internal sensors Robotics • Accelerometers (spring-mounted masses) • Gyroscopes (spinning mass, laser light) • Compasses, inclinometers (earth magnetic field, gravity) Probabilistic Sensor Models • Proximity sensors • Sonar (time of flight) • Radar (phase and frequency) Beam-based • Laser range-finders (triangulation, tof, phase) • Infrared (intensity) Scan-based • Visual sensors: Cameras, depth cameras • Satellite-based sensors: GPS Landmarks 4/15/20 CSE-571 - Robotics 2 SA-1 1 2 Proximity Sensors Beam-based Sensor Model • Scan z consists of K measurements. z = { z , z ,..., z } 1 2 K • The central task is to determine P(z|x) , i.e. the probability of a measurement z given that the robot is at position x . • Question : Where do the probabilities come from? • Approach : Let ’ s try to explain a measurement. 4/15/20 CSE-571 - Robotics 3 4/15/20 CSE-571 - Robotics 4 3 4 1
Beam-based Sensor Model Beam-based Sensor Model • Scan z consists of K measurements. z = { z , z ,..., z } 1 2 K • Individual measurements are independent given the robot position. K Õ = P ( z | x , m ) P ( z | x , m ) K Õ k = P ( z | x , m ) P ( z | x , m ) = k 1 k = k 1 4/15/20 CSE-571 - Robotics 5 4/15/20 CSE-571 - Robotics 6 5 6 Proximity Measurement Beam-based Proximity Model • Measurement can be caused by … Measurement noise Unexpected obstacles • a known obstacle. • cross-talk. • an unexpected obstacle (people, furniture, …). • missing all obstacles (total reflection, glass, …). • Noise is due to uncertainty … 0 z exp z max 0 z exp z max • in measuring distance to known obstacle. • in position of known obstacles. ( z − z exp ) 2 − 1 1 = h l - l z • in position of additional obstacles. P ( z | x , m ) e hit ( z | x , m ) = η σ 2 2 P 2 πσ 2 e unexp • whether obstacle is missed. 4/15/20 CSE-571 - Robotics 7 4/15/20 CSE-571 - Robotics 8 7 8 2
Beam-based Proximity Model Mixture Density Random measurement Max range T a æ ö æ P ( z | x , m ) ö ç hit ÷ ç hit ÷ a ç ÷ ç P ( z | x , m ) ÷ = unexp × unexp P ( z | x , m ) ç ÷ ç ÷ a P ( z | x , m ) ç ÷ ç ÷ max max ç ÷ ç ÷ a P ( z | x , m ) è ø è ø rand rand 0 z exp z max 0 z exp z max 1 How can we determine the model parameters? 1 = h P ( z | x , m ) = h P ( z | x , m ) max rand z z small max 4/15/20 CSE-571 - Robotics 9 4/15/20 CSE-571 - Robotics 10 9 10 Approximation Raw Sensor Data • Maximize log likelihood of the data z: Measured distances for expected distance of 300 cm. P ( z | z ) exp • Search parameter space. • EM to find mixture parameters • Assign measurements to densities. Sonar Laser • Estimate densities using assignments. • Reassign measurements. 4/15/20 CSE-571 - Robotics 11 4/15/20 CSE-571 - Robotics 12 11 12 3
Approximation Results Example Laser Sonar z P(z|x,m) 300cm 400cm 4/15/20 CSE-571 - Robotics 13 4/15/20 CSE-571 - Robotics 14 13 14 Summary Beam-based Model Scan-based Model • Assumes independence between beams. • Beam-based model is … • Justification? • not smooth for small obstacles and at edges. • Overconfident! • Models physical causes for measurements. • not very efficient. • Mixture of densities for these causes. • Idea: Instead of following along the • Implementation beam, just check the end point. • Learn parameters based on real data. • Different models can be learned for different angles at which the sensor beam hits the obstacle. • Determine expected distances by ray-tracing. • Expected distances can be pre-processed. 4/15/20 CSE-571 - Robotics 15 4/15/20 CSE-571 - Robotics 16 15 16 4
Scan-based Model Example • Probability is a mixture of … • a Gaussian distribution with mean at distance to closest obstacle, • a uniform distribution for random Likelihood field measurements, and • a small uniform distribution for max range measurements. Map m • Again, independence between different components is assumed. P(z|x,m) 4/15/20 CSE-571 - Robotics 17 4/15/20 CSE-571 - Robotics 18 17 18 San Jose Tech Museum Scan Matching • Extract likelihood field from scan and use it to match different scan. Occupancy grid map Likelihood field 4/15/20 CSE-571 - Robotics 19 4/15/20 CSE-571 - Robotics 20 19 20 5
Scan Matching Properties of Scan-based Model • Extract likelihood field from first scan • Highly efficient, uses 2D tables only. and use it to match second scan. • Smooth w.r.t. to small changes in robot position. • Allows gradient descent, scan matching. • Ignores physical properties of beams. • Works for sonars? ~0.01 sec 4/15/20 CSE-571 - Robotics 21 4/15/20 CSE-571 - Robotics 22 21 22 Landmarks Additional Models of Proximity Sensors • Map matching (sonar,laser): generate • Active beacons ( e.g . radio, GPS) small, local maps from sensor data and • Passive ( e.g . visual, retro-reflective) match local maps against global model. • Standard approach is triangulation • Scan matching (laser): map is represented by scan endpoints, match scan into this • Sensor provides map using ICP, correlation. • distance, or • Features (sonar, laser, vision): Extract • bearing, or features such as doors, hallways from • distance and bearing. sensor data. 4/15/20 CSE-571 - Robotics 23 4/15/20 CSE-571 - Robotics 24 23 24 6
Distance and Bearing Probabilistic Model 1. Algorithm landmark_detection_model (z,x,m): = a = q z i , d , , x x , y , 2. ˆ = - 2 + - 2 d ( m ( i ) x ) ( m ( i ) y ) x y 3. α = atan2( m y ( i ) − y , m x ( i ) − x ) − θ ˆ ˆ = - e × a ˆ - a e p prob( d d , ) prob( , ) 4. det d a + 5. Return z p z P ( z | x , m ) det det fp uniform 4/15/20 CSE-571 - Robotics 25 4/15/20 CSE-571 - Robotics 26 25 26 Summary of Parametric Motion Distributions and Sensor Models for P(z|x) • Explicitly modeling uncertainty in motion and sensing is key to robustness. • In many cases, good models can be found by the following approach: 1. Determine parametric model of noise free motion or measurement. 2. Analyze sources of noise. 3. Add adequate noise to parameters (eventually mix in densities for noise). 4. Learn (and verify) parameters by fitting model to data. 5. Likelihood of measurement is given by “ probabilistically comparing ” the actual with the expected measurement. • It is extremely important to be aware of the underlying assumptions! 4/15/20 CSE-571 - Robotics 27 4/15/20 CSE-571 - Robotics 28 27 28 7
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