Rigid Body Attitude Estimation- An Overview and Comparative Stud.pdf

Combining the two observer systems of 4130 and 4131

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Combining the two observer systems of (4.130) and (4.131), the full observer system is obtained as ˙ ˆ R = ˆ RS ( ω + σ ) , σ = k m S ( b m ) ˆ R T r m + k a S ( b a ) ˆ R T ( Ab a + k A ( v - ˆ v )) , ˙ ˆ v = k v ( v - ˆ v ) + ge 3 + Ab a , ˙ A = AS ( ω ) + k A ( v - ˆ v ) b T a . (4.133) Unlike the rotation-matrix-based estimation approach in [Hua, 2010], the authors in [Roberts and Tayebi, 2011b] take a quaternion-based approach with a much simpler ob- server stability proof. Their observer is given by ˙ ˆ Q = 1 2 ˆ Q 0 ω + σ , σ = k m S ( b m ) ˆ R T r m + k a k v S ( b a ) ˆ R T ( v - ˆ v ) , ˙ ˆ v = k v ( v - ˆ v ) + ge 3 + ˆ Rb a + 1 k v ˆ RS ( σ ) b a , (4.134)
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C hapter 4. D ynamic A ttitude F iltering and E stimation 94 where almost global exponential convergence is proven for all initial states except for quaternion error ˜ Q characterized by ˜ q 0 = 0. The Lyapunov function considered for the observer’s stability proof is V = γ 2 ˜ r T ˜ r + γ q (1 - ˜ q 2 0 ) , (4.135) with ˜ r = k v ˜ v - ( I - ˜ R ) r a . (4.136) The definition of the signal ˜ r helped the authors to fuse the accelerometer readings with the linear velocity obtained from GPS. In fact, the update law for ˆ v in (4.134) approaches the equation describing the relationship between accelerometer output and linear acceleration when velocity estimate error ˜ v goes to zero. It is reported that the observer performed well under assumptions of a relatively large linear acceleration of rigid body. In applications where the GPS data is not available, such as indoor, or in environ- ments such as forests where GPS data is weak and discontinuous, air pressure measure- ments can replace the velocity and position measurements in the observer design since they give a sense of vehicle’s linear velocity through the surrounding air. The authors in [Mahony et al., 2011] used this idea to estimate the attitude of a fixed-wing UAV in accel- erated mode without GPS measurements. The filter, which in nature is quite similar to an explicit complementary filter, is a SO (3) observer of the form ˙ ˆ Q = 1 2 ˆ Q 0 ω meas + δ , (4.137) where the correction term δ is a PI filter of an error vector e given by δ = k P e + k I Z t 0 ed τ, e = ¯ υ × ˆ υ, (4.138) with k P and k I being the proportional and integral gains, respectively. In their notation, ¯ υ : = ˆ g / k ˆ g k is the low-frequency normalized estimate of the gravitational direction and ˆ υ = ˆ R T e 3 is the expected gravitational direction in the body-fixed frame. Vector ˆ g is the
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C hapter 4. D ynamic A ttitude F iltering and E stimation 95 estimate of the gravitational direction given by ˆ g = - ( y a - ˆ b a ) , (4.139) where y a is the accelerometer measurement in the body frame and ˆ b a is the estimate of the rigid body’s acceleration with respect to the inertial frame expressed in body frame. Acceleration of the rigid body can be estimated as ˆ b a = ω × V air , (4.140) where the body frame expressed airspeed V air can be formulated as a function of the chang- ing angle of attack α and its magnitude | V air | is measured from the calibrated dynamic pressure measurements. The angle of attack α
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