Energy-Optimal Trajectory Planning for Planar Underactuated RR Robot Manipulators in the Absence of Gravity

Figure 9

Control and state variables of the optimal solution of problem 4, an initial value problem for an underactuated 𝑅𝑅 planar robot manipulator with boundary conditions 𝜃₁(𝑡_𝐼)=0[rad], 𝜃₂(𝑡_𝐼)=0[rad], 𝜃̇₁(𝑡_𝐼)=5[rad/s], 𝜃̇₂(𝑡_𝐼)=0[rad/s], 𝜃₁(𝑡_𝐹)=free, 𝜃₂(𝑡_𝐹)=𝜋[rad], 𝜃̇₁(𝑡_𝐹)=free, and 𝜃̇₂(𝑡_𝐹)=0[rad/s]. The initial and final times are 𝑡_𝐼=0 and 𝑡_𝐹=1[s], respectively. The corresponding sequence of configurations of the robot manipulator at times 𝑘(1/32) with 𝑘=0,1,…,32 is represented in Figure 8.

(a) 𝜃₁

(b) 𝜃₂

(c) 𝜃̇₁

(d) 𝜃̇₂

(e) 𝑢₂