3.16.2. Odometry We are using encoders in our wheels as instruments of measuring our robot’s movements to determine its location in the course. In order to do this, we will need to use an odometer model to convert the information read from the encoders to real world coordinates. The model we intend to use is detailed below. Begin with the following equations relating the change in position of the wheels to the circular arc of the turning axis as seen in Figure 3.21. Eq(1): = Rα Δsl Eq(2): = (R+2L)α Δsr Eq(3): = (R+L)α Δs Figure 3.21: Odometry p1- Example showing the relations of variables (Permission Requested from Princeton) Use Eq(1) and Eq(2) 2 to solve for the distance from the right and left wheels to the axis of the turn. Eq(4): Rα = Δsl Eq(5): Lα = (Δsr - Rα)/2 = /2 /2 Δsr – Δsl Substituting Eq(4) and Eq(5) into Eq(3) we represent as a function of and Δs Δsl Δsr. = (R+L)α = R + L= + /2 /2 = /2 + /2 Δs α Δsl Δsr – Δsl Δsl Δsr = (Δsl + /2 Δs Δsr) To calculate the change of the angle of the robot’s orientation is equal to the Δθ, circular arc’s center point as seen in Figure 3.22. = Δθ α Figure 3.22: Odometry p2- Example relating circular to the orientation α (Permission Requested from Princeton) Then solve for by equating from Eq(1) and Eq(2). α α = (Δsr - ) /2L Δθ 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. = cos(θ + Δx Δd Δθ/2) = sin(θ + Δy Δd Δθ/2)
3.16.2. Odometry We are using encoders in our wheels as instruments of measuring our robot’s movements to determine its location in the course. In order to do this, we will need to use an odometer model to convert the information read from the encoders to real world coordinates. The model we intend to use is detailed below. Begin with the following equations relating the change in position of the wheels to the circular arc of the turning axis as seen in Figure 3.21. Eq(1): Δsl = Rα Eq(2): Δsr = (R+2L)α Eq(3): Δs = (R+L)α Figure 3.21: Odometry p1- Example showing the relations of variables (Permission Requested from Princeton) Use Eq(1) and Eq(2) 2 to solve for the distance from the right and left wheels to the axis of the turn. 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 3.22: 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)
Initial%20Project%20Report.docx