4Sensors for Mobile Robots Contact sensors: BumpersInternal sensorsAccelerometers (spring-mounted masses)Gyroscopes (spinning mass, laser light)Compasses, inclinometers (earth magnetic field, gravity)Proximity sensorsSonar (time of flight)Radar (phase and frequency)Laser range-finders (triangulation, tof, phase)Infrared (intensity)Visual sensors: CamerasSatellite-based sensors: GPS
5Proximity SensorsThe 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.
6Noise IssuesSensors are imperfect!Specular Reflection
7Beam-based Sensor Model Scan z consists of K measurements.Individual measurements are independent given the robot position and the map of environment.A cyclic array of k ultrasound sensors
9Typical Measurement Errors of an Range Measurements Beams reflected by obstaclesBeams reflected by persons / caused by crosstalkRandom measurementsMaximum range measurements
10Proximity Measurement Measurement can be caused by …a known obstacle.cross-talk.an unexpected obstacle (people, furniture, …).missing all obstacles (total reflection, glass, …).Noise is due to uncertainty …in measuring distance to known obstacle.in position of known obstacles.in position of additional obstacles.whether obstacle is missed.
11Beam-based Proximity Model Measurement noiseUnexpected obstacleszexpzmaxzexpzmaxLikelihood of sensing unexpected objects decreased with rangeGaussian DistributionExponential Distribution
12Beam-based Proximity Model Random measurementMax range/FailureszexpzmaxzexpzmaxIf fail to detect obstacle, report maximum rangeUniform distributionPoint-mass distribution
13Resulting Mixture Density Weighted average, andHow can we determine the model parameters?System identification method: maximum likelihood estimator (ML estimator)
14Measured distances for expected distance of 300 cm. Raw Sensor DataMeasured distances for expected distance of 300 cm.SonarLaser
15Approximation The likelihood of the data Z is given by Our goal is to identify intrinsic parameters that maximize this log likelihoodMathematic derivation can be found pp163~167Psudo-code can be found in pp160An iterative procedure for estimating parametersDeterministically compute the n-th parameter to satisfy normalization constraint.
16Approximation Results Laser is more accurate than sonar, narrower GaussiansLaserThe smaller range, the more accurate the measurementSonar400cm300cm
17ExamplezP(z|x,m)Laser scan, projected into a previously acquired mapThe likelihood, evaluated for all positions and projected into the map. The darker a position, the largerP(z|x,m)
18Summary Beam-based Model Assumes independence between beams.Justification?Overconfident! Suboptimal resultModels physical causes for measurements.Mixture of densities for these causes.Assumes independence between causes. Problem?ImplementationLearn parameters based on real data.Different models should 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.
19Likelihood Fields Model Beam-based model is …not smooth for small obstacles and at edges (small changes of a robot’s pose can have a tremendous impact on the correct range of sensor beam, heading direction is particularly affected).not very efficient (computationally expensive in ray casting)Idea: Instead of following along the beam, just check the end point.
20Likelihood Fields Model Project the end points of a sensor scan Zt into the global coordinate space of the mapProbability is a mixture of …a Gaussian distribution with mean at distance to closest obstacle,a uniform distribution for random measurements, anda small uniform distribution for max range measurements.Again, independence between different components is assumed.
21Likelihood Fields Model Distance to the nearest obstacles. Max range reading ignored
22Example Example environment Likelihood field The darker a location, the less likely it is to perceive an obstacleP(z|x,m)Sensor probabilityOi : Nearest point to obstaclesO1O2O3Zmax
23San Jose Tech MuseumOccupancy grid mapLikelihood field
24Scan MatchingExtract likelihood field from scan and use it to match different scan.Sensor scan, 180 dotsThe darker a region, the smaller likelihood for sensing an object there
25Properties of Likelihood Fields Model Highly efficient, uses 2D tables only.Smooth w.r.t. to small changes in robot position.Allows gradient descent, scan matching.Ignores physical properties of beams.Will it work for ultrasound sensors?
26Additional Models of Proximity Sensors Map matching (sonar,laser): generate small, local maps from sensor data and match local maps against global model. (e.g., transform scans into occupancy maps)Scan matching (laser): map is represented by scan endpoints, match scan into this map. (e.g. ICP algorithm to stitch scans together)Features (sonar, laser, vision): Extract features such as doors, hallways from sensor data.
27Features/Landmarks Active beacons (e.g., radio, GPS) Passive (e.g., visual, retro-reflective)Standard approach is triangulationSensor providesdistance, orbearing, ordistance and bearing.
29Feature-Based Measurement Model Feature vector is abstracted from the measurement:Sensors that measure range, bearing, & a signature (a numerical value, e.g., an average color)Conditional independence between featuresFeature-Based map: withi.e., a location coordinate in global coordinates & a signatureRobot pose:Measurement model:Zero-mean Gaussian error variables with standard deviations
30Sensor Model with Known Correspondence A key problem for range/bearing sensors: data association, (arise when the landmarks cannot be uniquely identified)Introduce a correspondence variable , between feature and landmark If , then the i-th feature observed at time t corresponds to the j-th landmark in the mapAlgorithm for calculating the probability of a landmark measurement: inputs: feature , with know correspondence , the robot pose and the map, Its output is the numerical probability
31Summary of Sensor Models Explicitly modeling uncertainty in sensing is key to robustness.In many cases, good models can be found by the following approach:Determine parametric model of noise free measurement.Analyze sources of noise.Add adequate noise to parameters (eventually mix in densities for noise).Learn (and verify) parameters by fitting model to data.Likelihood of measurement is given by “probabilistically comparing” the actual with the expected measurement.This holds for motion models as well.It is extremely important to be aware of the underlying assumptions!