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

Figure 3

Control and state variables of the optimal solution of problem 1, a boundary value problem for the planar 𝑅𝑅 robot manipulator with boundary conditions 𝜃₁(𝑡_𝐼)=0[rad], 𝜃₂(𝑡_𝐼)=−𝜋/2[rad], 𝜃̇₁(𝑡_𝐼)=0[rad/s], 𝜃̇₂(𝑡_𝐼)=0[rad/s], 𝜃₁(𝑡_𝐹)=𝜋/2[rad], 𝜃₂(𝑡_𝐹)=−𝜋/2[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 2.

(a)  𝜃₁

(b) 𝜃₂

(c) 𝜃̇₁

(d) 𝜃̇₂

(e) 𝑢₁