1、Observability Properties and Optimal Trajectories for On-lineOdometry Self-CalibrationAgostino Martinelli and Roland SiegwartAbstract In this paper we derive theoretical results for theproblem of on-line odometry self-calibration for a mobile robot.A first series of results regards the problem of un
2、derstandingif a given system (consisting of a robot with several sensors)contains the necessary information to perform the on-lineself calibration of the odometry. We consider several casescorresponding to different odometry systems and different typesof robot sensors. Finally, we also consider the
3、problem ofmaximizing the calibration accuracy and we formulate thisproblem as an optimal control problem. For the special case of aholonomic vehicle, we derive second order differential equationscharacterizing the solution, i.e. defining the best trajectorywhich maximizes the calibration accuracy.I.
4、 INTRODUCTIONCalibration is the problem of estimating the parameterscharacterizing the systematic error of a sensor. In mobilerobotics, performing this process on-line is not only a desirewhich automatizes a work which would have to be performedby hand, but it is in many cases a real need for applic
5、ation-like scenarios. This is especially true for the odometry.Indeed, the pressure of tires can change over time and theeffective wheelbase depends on the terrain where the robotis moving. Having a system able to adapt continuously todifferent floor types and changing wheels attributes (i.e.differe
6、nt tire pressure, deterioration, etc.) is a key advantage.Several strategies have been developed to perform on-line self-calibration. In order to do this, the mobile robotis equipped with at least one other sensor. In many cases,an Extended Kalman Filter has been introduced to simul-taneously estima
7、te the robot configuration and the param-eters characterizing the systematic error of the odometrysensor (i.e., to solve the Simultaneous Localization and AutoCalibration (SLAC) problem). Regarding the odometry, theSLAC problem bas been investigated in 3, 4, 7, 8,12 and 13 both for indoor and outdoo
8、r environments.Very recently, the same idea was adopted to self calibratea vision sensor 9.Although in most cases the strategies proposed performwell, the following two questions remain open: Given a system consisting of a mobile robot withseveral sensors, does the system contain the necessaryThis w
9、ork has been supported by the Swiss National Science Foundationand by the European project BACS (Bayesian Approach to CognitiveSystems). The authors would like also to thank Antonio Bicchi for helpfuldiscussions.A.Martinelli is with the E-motion group at the Inria Rhone Alpes, 655av. de lEurope, Mon
10、tbonnot, 38334 St Ismier Cedex, France. He is alsopermanent guest at the Autonomous System Lab at the ETHZ, Zurich,Switzerland. agostino.martinelliieee.orgR. Siegwart leads the Autonomous System Lab at the ETHZ, Zurich,Switzerland. r.siegwartieee.orginformation to perform the calibration of the odom
11、etrysensor and/or to perform SLAC? What is the best robot trajectory which maximizes thecalibration accuracy?An answer to the second question for a mobile robot withdifferential drive could generalize the off-line method usuallyadopted to calibrate the odometry, the UMBmark 1. Thismethod requires th
12、at the robot moves along a square path inboth clockwise and counterclockwise direction.Very recently, we answered the first question for non-holonomic vehicles equipped with several types of extero-ceptive sensors 10. Furthermore, in 10 we answered thesecond question for holonomic vehicles in the ca
13、se whenthe final robot configuration is not assigned. In this paperwe give an answer to the previous two questions for othersystems. In section II we define these systems. In section IIIwe answer the first question both for holonomic and non-holonomic vehicles. In section IV, we answer the secondque
14、stion for the case of holonomic vehicles. With respectto 10 we take into account the additional constraint dueto an assigned final robot configuration. In particular, wederive second order differential equations characterizing thesolution. Finally, conclusions are presented in section V.II. THE SYST
15、EMSWe consider two types of mobile robot. The former isa non-holonomic vehicle equipped with a differential drivesystem, the latter is a holonomic vehicle. We also considerseveral systems, corresponding to the previous two typesof robot equipped with different exteroceptive sensors. Weintroduce the
16、systems separately for each type of robot.A. Non-holonomic VehicleThe configuration of the robot in a global reference framecan be characterized through the vector X =xR,yR,RTwhere xRand yRare the cartesian robot coordinates andRis the robot orientation. The dynamics of this vector aredescribed by t
17、he following non-linear differential equations:X = f (X, u)=xR= v cosRyR= v sinRR= (1)where v and are the linear and the rotational robot speed,respectively. The link between these velocities and the robotcontrols u depends on the considered robot system drive.We will consider the case of a differen
18、tial drive. In orderProceedings of the 45th IEEE Conference on Decision evaluation of the kinematicand odometric approach, Proceedings of the 1999 IEEE Internationalconference on Control Applications, volume: 2 , 22-27 Aug. 1999,Pages:102l-1026.8 A. Martinelli, N. Tomatis, A. Tapus and R. Siegwart,
19、Simultaneous Lo-calization and Odometry Calibration for Mobile Robot, InternationalConference on Intelligent Robots and Systems, Las Vegas, Nevada,October 2003.9 A. Martinelli, D. Scaramuzza and R. Siegwart, Automatic Self-Calibration of a Vision System during Robot Motion, InternationalConference o
20、n Robotics and Automation, Orlando, Florida, April2006.10 A. Martinelli, J. Weingarten and R. Siegwart, Theoretical Results onOn-line Sensor Self-Calibration, International Conference on Intelli-gent Robots and Systems, Benjing, China, October 2006.11 P.S. Maybeck, Stochastic Models, Estimation and
21、Control, AcademicPress, 1979, vol 141-112 Roy N., and Thrun S., Online Self-calibration for Mobile Robots,proceedings of the 1999 IEEE International Conference on Roboticsand Automation, 19 May 1999 Detroit, Michigan, pp. 2292-2297.13 H.J. von der Hardt, R. Husson, D. Wolf, An Automatic CalibrationM
22、ethod for a Multisensor System: Application to a Mobile RobotLocalization System, Interbational Conference on Robotics and Au-tomation, Leuven, Belgium, May 1998.APPENDIXWe want to show that the gradients dL0h, dL11h, dL12h,dL211h and dL3111h are independent. The system is definedin (6). The observa
23、tion h is the distance D. We start ourproof by computing the five Lie derivatives L0h, L11h, L12h,L211h and L3111h.Byusingh = D and from the expressionof f1and f2given in (7) we get:L0h = DL11h = Acos L12h = AB cosL211h = AC sin +A2sin2DL3111h = 3A3D2sin2 cos AC2cos +3A2CDsin cosTo compute the gradi
24、ents we simply need to differentiateall the previous functions with respect to D, , A, B andC. We have for instance dL0h =1, 0, 0, 0, 0. To prove thatthe previous gradients are independent we stack them in amatrix and we compute its determinant. We obtain det =AcosbracketleftBigA3C sin cos D+2A2C2cosbracketrightBigwhich is differentfrom 0 except when =2+2n and/or C = A sin 2D.45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006 ThB02.53070
