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

Figure 4

Sequence of configurations of the robot manipulator at times 𝑘(1/32) with 𝑘=0,1,…,32 corresponding to the optimal solution of problem 2, an initial value problem for an 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], obtained with a discretization of [𝑡_𝐼,𝑡_𝐹] into 64 intervals. The initial and final times are 𝑡_𝐼=0 and 𝑡_𝐹=1[s], respectively. The corresponding control and state variables are represented in Figure 5.

(a) 𝑡=𝑘/32, 𝑘=0,1,…,16

(b) 𝑡=𝑘/32, 𝑘=17,…,32