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LLRR
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Figure 93: Net vehicle position changes resulting from various sequences of two left and right rotational increments of the gure only Typical values of C are much larger Note that C is a perfectly known constant value The algorithm for processing the encoder outputs must run fast enough to not miss increments and to maintain the order in which the increments occur Figure 93 illustrates four possible scenarios involving two left increments indicated by an L and four right increments indicated by an R The only di erences between the scenarios are the ordering of occurrence of the incremental wheel rotations Note that, as shown in Figure 93, a LRLR sequence of increments is distinct from a RRLL sequence While both result in the vehicle having the same yaw angle, the rst moves the vehicle slightly to the right, while the second moves the vehicle slightly to the left The algorithm must also be designed to correctly accommodate the over ow that occurs at integer multiples of C counts
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Assuming that there is no wheel slip and that the lateral vehicle velocity is zero, the vehicle kinematics are described by n cos( ) p= = u e sin( ) (93) = where is the body frame yaw rate and vb = [u, v] is the body frame velocity vector The body and tangent frame velocity vectors are related by vt = Rt vb , where b Rt = b cos( ) sin( ) sin( ) cos( )
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By the zero lateral velocity assumption, the lateral velocity v = 0 Using the kinematic relationship vw = vb + b R where the supertb script w denotes wheel and the subscript b denotes body, the body frame linear velocity of the center of each wheel is uL = u + uR = u
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L 2 L 2
(94)
92 KINEMATIC MODEL
These linear velocities are illustrated in Figure 91 The function in eqn (94) from (u, ) to (uL , uR ) can be inverted so that the speed and angular rate can be computed when the wheel velocities are known: u= =
1 2 (uL + uR ) 1 L (uL uR )
(95)
Let (sL , sR ) denote the arc length traveled by the left and right wheels, respectively Due to the no slip assumption and assuming that sL (0) = sR (0) = L (0) = R (0) = 0, we have the conditions L = 1 sL sL = RL L RL (96) sR = RR R R = 1 sR
where the angular rotation of the left and right axles ( L , R ) are de ned in eqn (91) By the de nition of arc length, we have sL = uL sR = u R (97)
Therefore, by substituting eqn (97) into the derivative of eqn (96) and solving for (uL , uR ) we obtain uL = RL L u R = RR R Finally, combining eqns (91), (93), (95) and (98) we obtain n = 1 2 (RL eL + RR eR ) cos( ) + n 2 C e = 1 2 (RL eL + RR eR ) sin( ) + e 2 C = 1 2 (RL eL RR eR ) +
(98)
(99)
The signals n , e , and represent errors due to, for example, violation of the no-slip assumption These errors are not necessarily stationary and mutually uncorrelated random signals; however, unless some form of slip detection algorithm is incorporated, n , e , and would typically be modeled as independent random processes The length of the wheel radii RL and RR vary relative to their nominal values The variation depends on the type of tire as well as various unpredictable factors The wheel radii are modeled as RL RR = R0 + RL = R0 + RR (910) (911)
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