1 Vehicle Attitude in Euler Angle Form. Kinematics 3: Kinematic Models of Sensors and Actuators

Transcription

1 1.1Axi Convention 1 Kinematic 3: Kinematic Model of Senor and Actuator 1 Vehicle Attitude in Euler Angle Form 1.1 Axi Convention In aeropace vehicle, z point downward. Good for airplane and atellite. Here z point up, y forward, and x out the right ide. Thi ha the advantage that the projection of 3D information onto the x-y plane i more natural. The convention ued here correpond to a z-x-y Euler angle equence. It i not adviable to ue the homogeneou tranform developed here until they are verified to be correct for any enor and actuator that are ued. 1.2 Frame Aignment Some common frame are indicated in the figure below:

2 1.2Frame Aignment 2 z n x n y n y n xw z w x p z p The Navigation Frame Coordinate ytem with repect to which the vehicle poition and attitude i ultimately required. Normally, the z axi i aligned with the gravity vector; the y, or north axi i aligned with the geographic pole 1 ; and the x axi point eat to complete a right-handed ytem. 1. The geographic pole i determined by the earth pin axi, not the magnetic field. y w y p y p y w z b x b y b y b z h x h, z y h, y h y The Body Frame Poitioned at the point on the vehicle body which i mot convenient and i conidered to be fixed in attitude with repect to the vehicle body The Poitioner Frame Poitioned at the point on or near any poition etimation ytem which report it own poition. If the poitioner ytem generate attitude and attitude rate only, thi frame i not required becaue the attitude of the device will alo be that of the vehicle (rigid body). For an INS, thi i typically the center of the IMU and for GPS it i the phae center of the antenna The Senor Head Frame Poitioned at a convenient point on a enor head uch a: the interection of of rotary axe 2. The antenna may be nowhere near the GPS receiver.

3 1.3The RPY Tranform 3 center of mounting plate optical center of the hoted enor At time, a rigid enor head can and hould be defined which tilt the body axe into coincidence with thoe of the enor The Senor Frame() For video camera, it i poitioned on the optical axi at the center of projection behind the len or on the image plane. For tereo ytem, it i poitioned either between both camera or i aociated with the center of projection of the image plane of one of them. For imaging laer rangefinder, it i poitioned a the average point of convergence of the ray through each pixel The Wheel Frame Thi frame i poitioned at the center of the wheel, on the axle. 1.3 The RPY Tranform It i uually mot convenient to expre vehicle attitude in term of three pecial angle called roll, pitch, and yaw. Luckily, mot pan/tilt mechanim are kinematically formed from a yaw rotation followed by a pitch with no roll, o they are a degenerate form of the above, more general, tranform. A general homogeneou tranform, called the RPY tranform, can be formed which i imilar in principle to the DH matrix, except that it ha three rotation, and which can erve to tranform between the body frame and all other. There are ix degree of freedom involved, three tranlation and three rotation, and each can be either a parameter or a variable. Let two general frame be defined a a and b and conider the moving axi operation which tranform frame a into coincidence with frame b. In order, thee are:

4 1.3The RPY Tranform 4 tranlate along the (x,y,z) axe of frame a by (u,v,w) until it origin coincide with that of frame b rotate about the new z axi by an angle ψ called yaw rotate about the new x axi by an angle θ called pitch rotate about the new y axi by an angle φ called roll Angle are meaured counterclockwie poitive according to the right hand rule. Thee operation are indicated below for the cae of tranforming the navigation frame into the body frame. z a x a y a y a θ ψ z ab x b y b y b The forward kinematic tranform that repreent thi equence of operation i, according to our rule for forward kinematic: x b x a z a z b φ

5 1.4Invere Kinematic for the RPY Tranform 5 T b a Tran( u, v, w)rotz( ψ)rotx( θ)roty( φ) xyzθφψ T T b a 1u 1v 1w 1 cψ ψ ψ cψ 1 1 cθ θ θ cθ cφ φ 1 φ cφ to a coordinate frame. 1.4 Invere Kinematic for the RPY Tranform T b a ( cψcφ ψθφ) ψcθ ( cψφ + ψθcφ) u ( ψcφ + cψθφ) cψcθ ( ψφ cψθcφ) v cθφ θ cθcφ w Thi matrix ha the following interpretation: it rotate and tranlate point through the operation lited, in the order lited, with repect to the axe of a. it column repreent the axe and origin of frame b expreed in frame a coordinate it convert coordinate from frame b to frame a The matrix can be conidered to be the converion from a poe vector of the form The invere kinematic olution to the RPY tranform ha at leat two ue: it give the angle to which to drive a enor head, or a directional antenna given the direction coine of the goal frame. it give the attitude of the vehicle given the body frame axe, which often correpond to the local tangent plane to the terrain over which it move. Thi olution can be conidered to be the procedure for extracting a poe from a coordinate frame. There are many different way to get the olution from different element of the RPY tranform. The one ued here i ueful for modelling terrain following of a vehicle.

6 1.4Invere Kinematic for the RPY Tranform 6 Proceeding a for a mechanim, the element of the tranform are aumed to be known: T b a r 11 r 12 r 13 p x r 21 r 22 r 23 p y r 31 r 32 r 33 p z The premultiplication et of equation will be ued. The firt equation i: T b a r 11 r 12 r 13 p x r 21 r 22 r 23 p y r 31 r 32 r 33 p z Tran( u, v, w)rotz( ψ)rotx( θ)roty( φ) ( cψcφ ψθφ) ψcθ ( cψφ + ψθcφ) u ( ψcφ + cψθφ) cψcθ ( ψφ cψθcφ) v cθφ θ cθcφ w The tranlational element are trivial. From the (1,2) and (2,2) element: ψ atan2( r 22, r ) 12 Thi implie that yaw can be determined from a vector which i known to be aligned with the body y axi. The econd equation i: r 11 r 12 r 13 r 21 r 22 r 23 r 31 r 32 r 33 [ Tran( u, v, w) ] 1Ta Rotz ( ψ )Rotx( θ)roty( φ) b ( cψcφ ψθφ) ψcθ ( cψφ + ψθcφ) ( ψcφ + cψθφ) cψcθ ( ψφ cψθcφ) cθφ θ cθcφ which generate nothing new. The next equation i: [ Rotz( ψ) ] 1 [ Tran( u, v, w) ] 1Ta Rotx ( θ )Roty( φ) b ( r 11 cψ + r 21 ψ) ( r 12 cψ + r 22 ψ) ( r 13 cψ + r 23 ψ) ( r 11 ψ + r 21 cψ) ( r 12 ψ + r 22 cψ) ( r 13 ψ + r 23 cψ) r 31 r 32 r 33 cφ φ θφ cθ θcφ cθφ θ cθcφ From the (2,2) and (3,2) element:

7 1.4Invere Kinematic for the RPY Tranform 7 θ atan2( r 32, r 12 ψ + r 22 cψ) Which implie that pitch can alo be determined from a vector known to be aligned with the body y axi. A good olution for φ i available from the (1,1) and (1,3) element. However, for reaon of convenience, the olution will be delayed until the next equation. The next equation i: [ Rotx( θ) ] 1 [ Rotz( ψ) ] 1 [ Tran( u, v, w) ] 1Ta Roty ( φ ) b ( r 11 cψ + r 21 ψ) ( r 12 cψ + r 22 ψ) cθ[ r 11 ψ + r 21 cψ] + r 31 θ cθ[ r 12 ψ + r 22 cψ] + r 32 θ θ[ r 11 ψ + r 21 cψ] + r 31 cθ θ[ r 12 ψ + r 22 cψ] + r 32 cθ... ( r 13 cψ + r 23 ψ) cθ[ r 13 ψ + r 23 cψ] + r 33 θ θ[ r 13 ψ + r 23 cψ] + r 33 cθ 1 cφ φ 1 φ cφ From the (1,1) and (3,1) element: φ atan2( θ[ r 11 ψ + r 21 cψ] r 31 cθ, ( r 11 cψ + r 21 ψ) ) Thi implie that roll can be derived from a vector known to be aligned with the body x axi.

8 2 Angular Velocity The roll, pitch, and yaw angle are, a we have defined them, meaured about moving axe. Therefore, they are a equence of Euler angle, pecifically, the z-x-y equence 1. The Euler angle definition of vehicle attitude ha the diadvantage that the roll, pitch, and yaw angle are not the quantitie that are actually indicated by trapped-down vehiclemounted enor uch a gyro. The relationhip between the rate of the Euler angle and the angular velocity vector i nonlinear. The angle are meaured neither about the body axe nor about the navigation frame axe. The total angular velocity i the um of three component, each meaured about one of the intermediate axe in the chain of rotation which bring the navigation frame into coincidence with the body frame. 2 Angular Velocity 8 1.4Invere Kinematic for the RPY Tranform ω b ω b φ θ + rot( y, φ) + ω x cφθ φcθψ ω y φ + θψ ω z φθ + cφcθψ rot y, φ ( )rot( x, θ) cφ φcθ 1 θ φ cφcθ Thi reult give the vehicle angular velocity expreed in the body frame in term of the Euler angle rate. Notice that when the vehicle i level ( θ φ ), the x and y component are zero and the z component i jut the yaw rate a expected. Thi relationhip i alo very ueful in it inverted form. One can verify by ubtitution that: θ φ ψ ω x cφ+ ω z φ ω y tθω [ z cφ ω x φ] [ ω z cφ ω x φ] cθ ψ θ φ ψ cφ φ tθφ 1 tθcφ φ cφ cθ cθ ω x ω y ω z 1. The equence depend on the convention for aigning the direction of the linear axe. becaue ω z cφ ω x φ cθψ

9 3 Actuator Kinematic For wheeled vehicle, the tranformation from the angle of the teerable wheel and their velocitie onto path curvature, or equivalently angular velocitie, can be very complicated. There can be more degree of freedom of teer than are neceary. In thi cae, the equation which relate curvature to teer angle are overdetermined. 3.1 The Bicycle Model of Ackerman Steer Vehicle In one particular cae, however, the teering mechanim i deigned uch that thi will not be the cae. Thi mechanim i ued on mot conventional automobile and i called Ackerman teering. It i ueful to approximate the kinematic of the Ackerman teering mechanim by auming that the two front wheel turn lightly differentially o that the intantaneou center of rotation of the vehicle can be determined purely by kinematic mean. 3 Actuator Kinematic 9 3.1The Bicycle Model of Ackerman Steer Vehicle Let the angular velocity vector directed along the body z axi be called β. Uing the bicycle model approximation, the path curvature κ, radiu of curvature R, and teer angle α are related by the wheelbae L. α 1 tanα dβ R L d 3.2 YawRate α Rotation rate i obtained from the peed a: dβd Vtα β κv d dt L Thu, the teer angle α i an indirect meaurement of the ratio 1 of β to velocity through the function: 1. Curvature i dβ d. Dividing top and bottom by dt, it clear that curvature alway meaure the ratio of linear and angular velocitie. R L V

10 3 Actuator Kinematic 1 3.2YawRate α Lβ atan V atan( κl)

11 4 Kinematic of Video Camera Many enor ued on robot vehicle are of the imaging variety. For thi cla of enor, the proce of image formation mut be modelled. Typically, thee tranformation are not linear, and hence they cannot all be modelled by homogeneou tranform. Thi ection provide the homogeneou tranform and nonlinear equation neceary for modelling uch enor. 4.1 Perpective Projection In the cae of paive imaging ytem, a ytem of lene form an image on an array of enitive element called a CCD. Thee ytem include traditional video camera and infra red camera. The tranformation from the enor frame to the image plane row and column coordinate i the tandard perpective projection. Thi type of tranform i unique in two way: 4 Kinematic of Video Camera Perpective Projection it reduce the dimenion of the input vector by one and hence it dicard information it require a pot normalization tep where the output i divided by the cale factor in order to re-etablih a unity cale factor Thi tranformation can be derived by imiliar triangle.

12 4 Kinematic of Video Camera Perpective Projection z x i x f y + f x y f y z i z f y + f y i z y f f ( x i, z i ) ( x, y, z ) x x i y i z i w i f x y z 1 P i f Note in particular that all projection are not invertible. Here, the econd row i all zero, o the matrix i ingular. Thi, of coure, i the ultimate ource of the difficulty of meauring cene geometry with a ingle camera. For thi reaon, thi tranform i identified by the pecial capital letter P.

13 5 Kinematic of Laer Rangefinder Kinematic of reflecting a laer beam from a mirror are central to the operation of the current generation of laer rangefinder. There are at leat two way to go to model them: The reflection operator The mechanim forward kinematic where we conider the operation on the laer beam to be the mechanim. Alway be careful to account for mirror gain. Thi i the amount by which the rate of rotation of the laer beam i different from the rate of rotation of the mirror. 5.1 The Reflection Operator Thi can be ued when the tranlation of the laer beam can be ignored. From Snell law for reflection of a ray: The incident ray, the normal to the urface, and the reflected ray, all lie in the ame plane. 5 Kinematic of Laer Rangefinder The Reflection Operator The angle of incidence equal the angle of reflection. From thee two rule, a very ueful matrix operator can be formulated to reflect a vector off of any urface, given the unit normal to the urface. Conider a vector v i, not necearily a unit vector, which impinge on a reflecting urface at a point where the unit normal to the urface i nˆ. Unle they are parallel, the incident and normal vector define a plane, which will be called the reflection plane.

14 Drawing both vector in thi plane, it i clear that Snell law can be written in many form: v r ( v i nˆ )nˆ v r v i 2( v i nˆ )nˆ v r v i 2v i coθnˆ nˆ v r v i 2( n ˆ nˆ )v i θ θ v r v r Ref( nˆ )vi v i Ref( nˆ ) I 2( nˆ nˆ ) 1 2n x n x 2n x n y 2n x n z 2n y n x 1 2n y n y 2n y n z 2n z n x 2n z n y 1 2n z n z Where the outer product ( ) of the normal with itelf wa ued in forming the matrix equivalent. The reult i expreed in the ame coordinate in which both the normal and the incident ray were expreed. 5 Kinematic of Laer Rangefinder Kinematic of the Azimuth Scanner Notice that a reflection i equivalent to a rotation of twice the angle of incidence about the normal to the reflection plane. A imilar matrix refraction operator can be defined. In order to model rangefinder, the laer beam will be modelled by a unit vector ince the length of the beam i immaterial. The unit vector i operated upon by the reflection operator - one reflection for each mirror. The ultimate reult of all reflection will be expreed in the original coordinate ytem. The reult of uch an analyi give the orientation of the laer beam a a function of the actuated mirror angle, but it ay nothing about where the beam i poitioned in pace. The precie poition of the beam i not difficult to calculate and i important in the izing of mirror. From the point of view of computing kinematic, beam poition can often be ignored. 5.2 Kinematic of the Azimuth Scanner The azimuth canner i a generic name for a cla of laer rangefinder with equivalent

15 kinematic. In thi canner, the laer beam undergoe the azimuth rotation/reflection firt and the elevation rotation/reflection econd. Example are the ERIM and Perceptron. Both canner are 2D canning laer rangefinder employing a polygonal azimuth mirror and a flat nodding elevation mirror. The mirror move a hown below: x nodding mirror polygon mirror y 5 Kinematic of Laer Rangefinder Kinematic of the Azimuth Scanner Forward Kinematic A coordinate ytem called the ytem i fixed to the enor with y pointing out the front of the enor and x pointing out the right ide. The beam enter along the x axi. It reflect off the polygonal mirror which rotate about the y axi. It then reflect off the nodding mirror, to leave the houing roughly aligned with the y axi. Firt, the beam i reflected from the laer diode about the normal to the polygonal mirror. Computation of the output of the polygonal mirror can be done by inpection - noting that the beam rotate by twice the angle of the mirror becaue it i a reflection operation. The z-x plane contain both the incident and normal vector. The datum poition of the mirror hould correpond to a perfectly vertical beam, o the datum for the mirror rotation angle i choen appropriately. Conider an input beam vˆ m along the x axi and reflect it about the mirror by inpection:

16 5 Kinematic of Laer Rangefinder Kinematic of the Azimuth Scanner x z vˆ p ψ nˆ p vˆ m ψ 2 vˆ m vˆ p vˆ p 1 T Ref( nˆ p)vˆ m ψ cψ T z vˆn y vˆ p α 2 θ 2 nˆ n vˆ n Ref( nˆ n )vˆp vˆp 2( vˆp nˆ n )nˆ n vˆp put ψ cψ T α -- 2 π θ nˆ n α T -- 2 cα-- 2 Notice that thi vector i contained within the x -z plane. Now thi reult mut be reflected about the nodding mirror. Notice that, at thi point, vˆ p cannot be imply rotated around the x axi ince the axi of rotation which i equivalent to a reflection i normal to both vˆ p and nˆ n. Since vˆ p i not alway in the y -z plane, the x axi i not alway the axi of rotation. vˆn ψ 2cψc α -- α -- ψ cψα cψ c α -- cψcα 2 vˆn ψ cψ π -- 2 θ cψc π -- 2 θ T [ ψ] [ cψcθ] [ cψθ] Thi reult i ummarized in the following figure:

17 5 Kinematic of Laer Rangefinder Kinematic of the Azimuth Scanner x z R ψ θ y Thu, the kinematic of the azimuth canner are equivalent to a rotation around the x axi followed by a rotation around the new z axi. Thi i alo equivalent to two rotation in the oppoite order about fixed axe Forward Imaging Jacobian The imaging Jacobian provide the relationhip between the differential quantitie in the enor frame and the aociated poition change in the image. The Jacobian i: v x y z Rψ Rcθcψ Rθcψ J i v i v ψ y R x θ z x x x R ψ θ v y v i y y R ψ θ z R z ψ z θ Invere Kinematic Rψ Rcθcψ Rθcψ ψ Rxψ cθcψ Rcθψ Rθcψ θcψ Rθψ Rcθcψ The forward tranform i eaily inverted. R ψ θ x2 + y2 + z2 atan( x y2 + z2 ) atan( z y ) hx (, y, z )

18 5.2.4 Invere Imaging Jacobian The imaging Jacobian provide the relationhip between the differential quantitie in the enor frame and the aociated poition change in the image. The Jacobian i: v i x2 R + y2 + z2 x ψ atan( x ( y2 + z2 )) v y θ atan( z y ) z R R R x y z i v i v ψ ψ ψ x y z θ x θ y θ z y ---- R z --- R x ---- R y2 + z y x z x R R y2 + z2 R y2 + z2 z y2 + z2 y y2 + z2 5 Kinematic of Laer Rangefinder Kinematic of the Azimuth Scanner Analytic Range Image of Flat Terrain Given the baic kinematic tranform, many analye can be performed. The firt i to compute an analytic expreion for the range image of a perfectly flat piece of terrain. h β z g z y y g (x g,y g,z g ) Let the enor fixed coordinate ytem be mounted at a height h and tilted forward by a tilt angle of β. Then, the tranform from enor coordinate to global coordinate i: x g x y g y cβ + z β y β + z cβ + h z g R

19 If the kinematic are ubtituted into thi, the tranform from the polar enor coordinate to global coordinate i obtained: x g y g Rψ ( Rcθcψ)cβ ( Rθcψ)β Rcθβcψ 5 Kinematic of Laer Rangefinder Kinematic of the Azimuth Scanner contour of contant range of 2 meter, 4 meter, 6 meter, etc. z g ( Rcθcψ)β ( Rθcψ)cβ + h h Rθβcψ Now by etting z g and olving for R, the expreion for R a a function of the beam angle ψ and θ for flat terrain i obtained. Thi i an analytic expreion for the range image of flat terrain under the azimuth tranform. R h ( cψθβ) Notice that when R i large θβ h R. A a check on the range image formula, the reulting range image i hown below for h 2.5, β 16.5, a horizontal field of view of 14, a vertical field of view of 3, and an IFOV of 5 mrad. It ha 49 column and 15 row. The edge correpond to The curvature of the contour of range i intrinic to the enor kinematic and i independent of the enor tilt. Subtituting thi back into the coordinate tranform, the coordinate where each ray interect the groundplane are: x g y g htψ θβ h tθβ z g Notice that the y coordinate i independent of ψ and hence, line of contant elevation in the image are traight line along the y-axi on flat terrain. From the previou reult, it can be verified by ubtitution and ome algebra that:

20 5 Kinematic of Laer Rangefinder 2 5.2Kinematic of the Azimuth Scanner x g tψ 2 y2 g h 2 Thu line of contant azimuth are hyperbola on the groundplane Reolution The Jacobian of the groundplane tranform ha many ue. Mot important of all, it provide a meaure of enor reolution on the ground plane. Differentiating the previou reult: dx g dy g h( ecψ) θβ htψcθβ ( θβ) 2 h ( θβ) 2 dψ dθ The determinant of the Jacobian relate differential area: ( hecψ) dx g dy 2 g ( θβ) 3 dψdθ Notice that when R i large and ψ, the Jacobian norm can be approximated by: ( hecψ) 2 ( θβ) 3 h 2 h R -- 3 R 2 h R -- Thu, the pixel denity on the ground i proportional to the cube of the range Azimuth Scanning Pattern The canning pattern i hown in the following figure with a 1 m grid uperimpoed for reference purpoe. Only every fifth pixel i hown in azimuth to avoid clutter. Y Coordinate in Meter X Coordinate in Meter

21 5.3 Simple Scanner Kinematic The implet poible 2D canner can be contructed by mounting a 1D canner on a pan table. Although it imple mechanically, it a tough math model relative to other. We will model thi one with our fundamental operator and rule for forward kinematic modelling. l a x i x z i yi x r z y z r yr Forward Kinematic The homogeneou coordinate of a range pixel in the mirror (m) frame are: l e ψ, azimuth l r R x m z m y m θ (elevation) 5 Kinematic of Laer Rangefinder Simple Scanner Kinematic v m R 1 T To convert the coordinate of thi pixel to the intermediate (i) frame, we mut generate the relevant tranform a follow. The operation which bring frame i into coincidence with frame m are: Tranlate along y i y m a ditance l e. Rotate around y i y m an angle θ. i Hence, the matrix T m which convert coordinate of a point from frame m to frame i i: T m i T m i Tran(, l e, )Roty( θ) T m i cθ θ 1 l m e T i θ cθ 1 1l e 1 1 cθ θ 1 l e θ cθ cθ θ 1 θ cθ

22 5 Kinematic of Laer Rangefinder Simple Scanner Kinematic Therefore, the coordinate of a range pixel in the i frame are: T i Tran( l,, a )Rotz( ψ) v i x y z w i v i T m i vm cθ θ 1 l e θ cθ R 1 Rθ, l e, Rcθ, 1 To convert the coordinate of thi pixel to the enor () frame, we mut generate the relevant tranform a follow. The operation which bring frame into coincidence with frame i are: Tranlate along z z i a ditance l a. Rotate around z z i an angle ψ. Hence, the matrix T i which convert coordinate of a point from frame i to frame i: T T i T i 1 1 1l a 1 cψ ψ ψ cψ 1 cψ ψ ψ cψ 1 l a i T cψ ψ ψ cψ 1 l a Therefore, the coordinate of a range pixel in the frame are: v T i vi x y z w v cψ ψ ψ cψ 1 l a Rcψθ l e ψ Rψθ + l e cψ Rcθ + l a 1 Rθ l e Rcθ 1

23 The lat expreion i the complete enor forward kinematic olution which convert coordinate from range, azimuth, elevation to a carteian (x,y,z) poition with repect to the enor frame Invere Kinematic The goal of the invere kinematic i to compute the range, and the azimuth and elevation angle which correpond to a given carteian (x,y,z) poition expreed with repect to the enor frame. The forward kinematic relationhip can be rewritten a follow: v i v v T m i vm T i vi T i Tm i vm The invere kinematic olution can be obtained by auming that v i known and olving for the range, azimuth, and elevation i of the point. Premultiplying the above by T : 5 Kinematic of Laer Rangefinder Simple Scanner Kinematic cψ ψ ψ cψ 1 l a i i T v T mvm ( v i ) x y z w cθ θ 1 l e θ cθ Thi generate the following three equation: 1 ) cψx + ψy Rθ 2 ) ψx + cψy l e 3 ) z l a Rcθ R 1 Firt, we olve 2) directly for the azimuth angle. Then, we olve 1) and 3) for the elevation angle. Finally, 3) can be olved for the range when the elevation angle i known. The reult i:

24 5 Kinematic of Laer Rangefinder Simple Scanner Kinematic ψ atan2( y, x ) atan2 l e, ± x + y l e θ atan2( cψx ψy, l a z ) R ( l a z ) ( cθ) which i the invere kinematic olution.

25 6 Summary Simple Scanner Kinematic 6 Summary The RPY matrix i yet another compound orthogonal operator matrix. Unlike the DH matrix, it ha 6 dof o it i completely general. Angular velocity i related in a complicated manner to the rate of roll, pitch, and yaw angle. Video camera are modelled by a perpective projection. Laer rangefinder model are nonlinear and cannot be repreented by a contant homogeneou tranform like a camera can. However, our mechanim modelling rule apply perfectly and one can alo ue a reflection operator to model them.

26 6 Summary Simple Scanner Kinematic