Consequently if both robots are very close to each

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Consequently, if both robots are very close to each other or the camera is perpendicularly oriented to the floor, the real distance may be roughly estimated by the average of the perspective label lengths: ). ' ' 2 ' ' ( ' ' ' ' ' 2 1 1 0 1 0 1 0 0 y y y y d y d y d y d (5.5) In real world operations, (5.5) will result in an intolerable error when the leader is moving further away, because the fast decreasing label length of the leader will be dominant, thus yielding a large positive error. Therefore, the ratio v R denoted as 0 / y d can be estimated by its upper and lower limits as given in the following proposition. Proposition 1 : With a sufficiently small tilting angle, the ratio v R is found between an upper limit, which is the average perspective projection image ratio (PPIR), and a lower limit as given in the following inequality: ), ' ' 2 ' ' ( ' ) ' ' 2 ' ' ( ' 1 0 1 0 1 0 1 0 y y y y d R y y y y d v y (5.6) where y = ' / ' 0 1 y y . Proof: To prove the relationship (5.6), one can translate the perspective image by using Fig. 5.1 as:
5. Relative Distance Estimation for Robots Control Using Monocular Digital Camera 106 0 0 ' ky y , and 0 1 ' y y , (5.7) where λ k ≤ 1. So , with 1 0 y y and a small angle θ 0 , due to similarity, one can use the following trapezoidal formula: , ' ' 2 ' ' 2 ) ( ' 1 0 1 0 0 1 0 0 0 y y y y d y y y d y ky R d v (5.8) where v R denotes the ratio 0 / y d . Now by substituting , 2 ) ( ' ' 0 k k R y d v (5.9) and , 2 ) ( ' ' 1 k R y d v (5.10) into (5.5) and noting k k 4 2 , one has , 4 ' ' ' ' 2 1 0 2 1 0 y d k k R y d y d v (5.11) This verifies the upper limit in (5.6). The lower limit can also be derived by noting λ k as: , 4 4 0 2 2 0 0 2 y d k k R ky y k k R v v (5.12)
5. Relative Distance Estimation for Robots Control Using Monocular Digital Camera 107 So the new estimated distance range can be rewritten as in (5.6). The above proposition suggests the use of a coefficient v which is greater than y but less than 1 in order to estimate the real relative distance ratio in longitudinal direction as: v v y d y d R y d ) ' ' ' ' ( 2 1 1 0 0 , 1 v y (5.13) To approximate v in the proposed algorithm, it notes the convergent series: 1 1 2 1 i i , (5.14) which leads to the choice . ' 2 ' ' ) 1 2 ( ) 1 ( 2 1 0 1 0 1 y y y n n n i y i y v (5.15) According to the experiments, by using several generic digital cameras (normal lens camera), the reasonable value for v is obtained when n is between 3 and 4 for an expected tolerance less than 5% of the real distance. Remark 1: By using (5.15) the PPIR algorithm can avoid the calculation of trigonometric functions in (5.3), and hence such an algorithm as the CORDIC (Deschamps et al ., 2006) is not used in this design. Remark 2: It can be seen that coefficient v can be adjusted automatically without requiring any additional optical installation or information of the camera. Particularly, in
5. Relative Distance Estimation for Robots Control Using Monocular Digital Camera 108 the case when both robots are found closely to each other or the camera has a zero tilt angle, the leader’s label length ' 1 y will converge to the label length of the follower, ' 0 y

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