Eq(4): = Rα Δsl = - = /2 /2 Eq(5): Lα (Δsr Rα)/2 Δsr – Δsl Substituting Eq(4) and Eq(5) into Eq(3) we represent as a function of and Δs Δsl Δsr. = = + L= + /2 = /2 + /2 Δs (R+L)α R α Δsl Δsr /2 – Δsl Δsl Δsr = + /2 Δs (Δsl Δsr) To calculate the change of the angle of the orientation is equal to the robot’s Δθ, circular center point as seen in Figure 3.22. arc’s = Δθ α Figure 3.27: Odometry p2- Example relating to the orientation circular α (Permission Requested from Princeton) Then solve for by equating from Eq(1) and Eq(2). α α = - ) /2L Δθ (Δsr Δsl With and we are able to calculate the position change in real world Δθ Δs coordinates. Using trigonometry to model the and as functions of Δx Δy Δd. = + Δx Δd cos(θ Δθ/2) = + Δy Δd sin(θ Δθ/2) As seen in Figure 3.23 the paths Δd ≈ Δs are slightly different. The measurements will be made fast enough to approximate that Δd ≈ Δs. = + Δx Δs cos(θ Δθ/2) = + Δy Δs sin(θ Δθ/2) Figure 3.28: Odometry p3- Example denoting difference of and Δd Δs (Permission Requested from Princeton) Using the above model, we will be able to approximate our robots location on the field. The coordinates will not be exact because of the assumptions that were made. Also there will be noise from the encoders that will impact the calculations. Other causes of inaccuracy to the model can arise from the wheels slipping, hitting a bump, or driving on carpet. 3.16.3. Wireless Data Transfer
Eq(4): Rα = Δsl Eq(5): Lα = (Δsr - Rα)/2 = Δsr /2 – Δsl /2 Substituting Eq(4) and Eq(5) into Eq(3) we represent Δs as a function of Δsl and Δsr. Δs = (R+L)α = R α + L= Δsl + Δsr /2 – Δsl /2 = Δsl /2 + Δsr /2 Δs = (Δsl + Δsr) /2 To calculate the change of the angle of the robot’s orientation Δθ, is equal to the circular arc’s center point as seen in Figure 3.22. Δθ = α Figure 3.27: Odometry p2- Example relating circular α to the orientation (Permission Requested from Princeton) Then solve for α by equating α from Eq(1) and Eq(2). Δθ = (Δsr - Δsl ) /2L With Δθ and Δs we are able to calculate the position change in real world coordinates. Using trigonometry to model the Δx and Δy as functions of Δd. Δx = Δd cos(θ + Δθ/2) Δy = Δd sin(θ + Δθ/2) As seen in Figure 3.23 the paths Δd ≈ Δs are slightly different. The measurements will be made fast enough to approximate that Δd ≈ Δs. Δx = Δs cos(θ + Δθ/2) Δy = Δs sin(θ + Δθ/2) Figure 3.28: Odometry p3- Example denoting difference of Δd and Δs (Permission Requested from Princeton) Using the above model, we will be able to approximate our robots location on the field. The coordinates will not be exact because of the assumptions that were made. Also there will be noise from the encoders that will impact the calculations. Other causes of inaccuracy to the model can arise from the wheels slipping, hitting a bump, or driving on carpet. 3.16.3. Wireless Data Transfer
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