0.6 First Trajectory Third Trajectory 0.5 Second Trajectory 0.6 Last Three Trajectories 0.5 0.4 0.4 0.3 -0.1 X (m) (f) 8 6 4 2 -2 Phase 1 Phase 2 01 2 Time (s) (h) Phase 3 Phase 4 34 External Torque Feedforward Torque Feedback Torque State Switching Point 6 4 2 -2 -4 Phase 1 Phase 2 01 2 Time (s) (i) Figure 6. (Continued) (f) and (g) Trajectories in the first and last three periods in the proposed method. (h) and (i) Joint torques and external torques in the final period in the proposed method. 4. This is consistent with the fact that reactions of human muscles are smooth and continuous and shows that the proposed method is more suitable for processing tasks in a complex environment. Experiment In this experiment, a robot arm is to follow a trajectory using a pen, which is put in a forcing environment. As shown in Figure 7, the experiment is performed in an unstable environment with four elastic ropes attached to corner rods of the wood board. A pen is fixed to the end of the Franka robot arm with a crank connecting the other ends of the elastic ropes to record trajectory in the unstable environment. The Franka robot can record contact force through the embedded force sensors as well as the joints and positions in the Cartesian space. As the forces are provided by elastic cables determined by deformations along the pulling directions, the force field in the X-Y plane is shown in the zoomed figure in Figure 7. We can see that the pen and the robot arm, except for standing the center point, will be affected by an extra force that pushes the pen to a balanced center position. If we want the robot to draw a picture on the paper, the robot has to resist the extra force to complete the drawings. Franka Robot Pen and Paper Elastic Rope Fixing Stick 0.05 Phase 3 Phase 4 34 0.1 0.2 0.3 -0.1 X (m) (g) External Torque Feedforward Torque Feedback Torque State Switching Point 0.1 -0.05 Force Field Figure 7. An experimental setup with the force field. DECEMBER 2022 * IEEE ROBOTICS & AUTOMATION MAGAZINE * 75 Torque (N.rad) Y (m) Torque (N.rad) Y (m) -0.08 -0.06 -0.04 -0.02 0.02 0.04 0.06 0.08