Page - 62 - in Joint Austrian Computer Vision and Robotics Workshop 2020

and rotational velocities of the following figures are normalizedw.r.t. their physical limits. 5.1.TimeOptimalControl Fig. 4 shows the normalized motor torques and rotationalvelocitiesof timeoptimalcontrolproblem. Fortimeoptimaloptimization tasks, it isobvious that at least onemotor restriction is active. Thefinal time isgivenby tf=0.83s.Moreover, allphysical limits of the motors are well considered. Occurring joint forces and torques of the first subsystem are shown inFig. 5. M1 M2 M3 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 q˙1 q˙2 q˙3 Fz t in s 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 Figure 4: Normalized motor torques and rotational velocities of the timeoptimal control Fx Fy Fz 0 0.2 0.4 0.6 0.8 1 -40 -20 0 20 40 60 80 100 Mx Mz t in s 0 0.2 0.4 0.6 0.8 1 -2 -1 0 1 2 3 4 5 Figure5: Joint forcesandtorquesof the timeoptimal control M1 M2 M3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -1 -0.5 0 0.5 1 q˙1 q˙2 q˙3 t in s 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -1 -0.5 0 0.5 1 Figure 6: Normalized motor torques and rotational velocities of the joint loadminimization Fx Fy Fz 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -10 0 10 20 30 40 50 60 70 Mx Mz t in s 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 1 2 3 Figure 7: Joint forces and torques of the joint load minimization 5.2.MinimizingJointLoads In comparison to time optimal control, the final time tf = 2s of this optimal control is predefined due to thewalking speedof the analyzed biped. Fig. 6 and Fig. 7 shows the dynamical behavior of this task. Motor restrictions are almost inactive and the occurring joint forces and torques are reduced in comparison toFig. 5. 5.3.Maximizing theVerticalTorqueof theTorso As to the last subsection 5.2, the final time tf = 2s is also predefined in this task. Optimization re- 62