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Ortiz et al. Intell Robot 2021;1(2):131-50 I http://dx.doi.org/10.20517/ir.2021.09 Page 137

Proof 1 Consider the Lyapunov function as

−1 (24)
= ( ) P ( )

where P +2 is the prior covariance matrix in Equation (21), and P +2 > 0. From Equation (22),

+1 = ( + 1) P −1 ( + 1)
+1
−1
= ( ) ( − K ) P +1 ( − K ) ( ) (25)

+2( + )P −1 ( − K ) ( )

+1

+( + )P −1 ( + )
+1
Because k k ≤ ¯ , k k ≤ ¯, the last term on the right side of Equation (25) is
2
2

( + )P −1 ( + ) ≤ max P −1 ( ¯ + ¯ ) (26)
+1 +1

where max P −1 is the maximum eigenvalue of P −1 .
+1 +1
The second term of Equation (25) is

2( + )P −1 ( − K ) ( )
+1
= 2( +)P −1 ( − K ) ( ) (27)
+1
+ P −1 ( − K ) ( )

+1
where K is the gain of EKF in Equation (5). In view of the matrix inequality

−1
+ ≤ Λ + Λ (28)
× ×
which is valid for any , ∈ < and for any positive definite matrix 0 < Λ = Λ ∈ < , the first term of
Equation (27) is

2 P −1 ( − K ) ( )
+1
−1
≤ Λ( +) + ( ) P −1 ( − K ) Λ ( )

+1
(29)
−1 −1
2

≤ ¯ max [Λ] +
P +1 ( − K ) Λ
k ( )k

≤ ¯ max [Λ] + ( ) P −1 ( − K ) Λ −1 ( )

+1
We apply the sliding mode compensation in Equation (20) to the second term of Equation (27):

× [ ( )] Υ ( )

(30)
Í Í
= − | ( )| + ([ ( )] ( ))
=1 =1, ≠

where are the elements of the matrix Υ, Υ = P −1 ( − K ) . When the the orientation is not big,
+1
sin ≈ 0, cos ≈ 0,

Õ
| | , > 0, = 1, . . . , (31)
=1, ≠
Thus, the second term of Equation (27) is negative.
The first term on the right side of Equation (25) has the following properties:

P +2 ≥ ( − K )P +2 ( − K )