Page - 61 - in Joint Austrian Computer Vision and Robotics Workshop 2020

3.1.TimeOptimalControl Theoptimizationproblemfor timeoptimalcontrol isdefinedas min tf,d ∫ tf 0 1dt (7) s.t. q(0,d)=q0 (8) q(tf,d)=qtf (9) q˙(0,d)=0 (10) q˙(tf,d)=0 (11) qmin≤q(d)≤qmax (12) − q˙max≤ q˙(d)≤ q˙max (13) −Qmax≤Q(d)≤Qmax (14) Q(q(d))=M(q(d))q¨(d)+g(q(d),q˙(d)). (15) In this case, the final time tf and the set of control points d to parameterize the B-splines are regarded as optimization variables. Eq. (15) represents the dynamical behaviorof the robotic systeminminimal representation. M ∈ R3,3 is the global mass ma- trix, g∈R3 includes non-linear terms andQ∈R3 is the global vector of generalized forces. The re- strictions in Eq. (8)–(12) are associated to process requirements and those inEq. (13)–(14) are defined bychosenmotors. This equality and inequality con- straints were used for all optimization tasks in the following. 3.2.MinimizingJointLoads Aimof this optimization task is tominimize dy- namic joint forces and torquesbetweenground/torso andarmof thehumanoidwalkingmachine. Thefinal time tf for themotion ispredefined in this task. The cost function isgivenby min d 1Qc ⊤ 1Qc. (16) The set of control pointsd are regarded asoptimiza- tion variables. As mentioned above, the optimiza- tion constraints are givenbyEq. (8)–(14). Note, the occurring joint forces and torques can be calculated withEq. (5). 3.3.Maximizing theVerticalTorqueof theTorso During gait, arms are used to counterbalance the torque around the vertical axis. Amomentum con- trol approach with this quantity is presented in [6]. Hence, another optimization strategy is to find a proper setofcontrol pointsd such that thecost func- tion max d 1Q c⊤ 6 1Q c 6 (17) ismaximized. Once again, optimization constraints are given by Eq. (8)–(14). The quantity 1Qc6 is the sixth entry of 1Qc and describes the joint torque of the first subsystem in the opposite direction of the gravity vector. 4.OptimizationMethod TheSequentialQuadraticProgramming(SQP)al- gorithmwaschosen tosolveallconsideredoptimiza- tion problems. This approach is also used in [3] for trajectory planning. The SQP method requires an valid initial guess for trajectories. In this case, the initial trajectories are defined as B-splines. Prop- erties of B-splines can be found in [8]. An initial guess for thearmangelsqare foundby interpolating the initial andfinal position aswell as some support pointswith aB-spline of degree 4. Furthermore, ve- locities andaccelerations at the initial andfinalposi- tion are set to zero. Twentycontrol pointswerecho- sen to initialize eachof the threepolynomials. Fig. 3 showsexemplary an initial guess trajectory. t in s 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 Figure3: Exampleofan initial guess trajectory 5.SimulationResults In this section, relevant results of the optimiza- tion tasks are presented. The typical arm mo- tion during a step of the biped is defined by the start configuration q0 = (−pi4 00)⊤rad and the final configuration qtf = ( pi 4 0 pi 4 )⊤ rad of the minimal coordinates. Moreover, limits regarding joint angles are defined by qmin = (−pi2 00)⊤rad and qmax = ( pipi 3pi4 )⊤ rad. Maximal mo- tor rotational velocities and torques are given by q˙max = (12.66.312.6) ⊤rads−1 and Qmax = (415480165)⊤Nm. Note, that all motor torques 61