Appendix S1: Relation between variables of the cost function
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Appendix S1: Relation between variables of the cost function
(S1.4). Details of these matrices can be found in [30]. Identical relationships can be derived for the motion of the. 439 flywheel segment. 440. Center of Pressure.
434
435 436
Appendix S1: Relation between variables of the cost function Variables represented with capital letters stand for column vectors of values over the prediction horizon, e.g. ck+1 Ck+1 = ... (S1.1) ck+N
437
438
Time derivatives The position, speed and acceleration of the CoM over the whole prediction horizon can be related to the T CoM state of the system at a time tk , cˆk = ck c˙k c¨k , and the piecewise constant third derivative ... ... ... T C k = [ c k , . . . , c k+N −1 ] through constant matrices: ... Ck+1 = Sp cˆk + Up C k , (S1.2) ... ˙ Ck+1 = Sv cˆk + Uv C k , (S1.3) ... ¨ Ck+1 = Sa cˆk + Ua C k . (S1.4)
440
Details of these matrices can be found in [30]. Identical relationships can be derived for the motion of the flywheel segment.
441
Center of Pressure
442
The linear dynamics (1) can be reversed to compute the position of the CoP as:
439
zx = cx − 443 444
h j ¨ c¨x − θ g mg
(S1.5)
The position of the CoP over whole prediction horizon Zk+1 can then be related to the piece-wise ... the ... constant third derivatives C k and Θk : ... cˆk C Zk+1 = Sz ˆ + Uz ...k , (S1.6) θk Θk with h i j Sa , Sz = Sp − hg Sa − mg h i j Ua . Uz = Up − hg Ua − mg
445
446 447 448 449
(S1.8)
Foot position The position of the support foot over the whole prediction horizon Fk+1 can be related to the current support foot position fk , which is fixed on the ground, and the positions F¯k+1 of the future steps, which is an optimization variable. If the step durations are already known, this can be done easily with matrices Vk+1 and V¯k+1 filled with 0s and 1s simply indicating which sampling times ti fall within which steps: Fk+1 = Vk+1 fk + V¯k+1 F¯k+1
450
(S1.7)
(S1.9)
451
452 453 454 455 456
457 458
Swing foot acceleration The motion of the swing foot is interpolated in the forward/backward direction with 5th degree polynomials between its current position, velocity and acceleration and the future positions of the foot on the ground, with zero velocity and acceleration (no impact). This trajectory is further discretized. Acceleration of the swing foot f¨j0 at each of these discretization instants tj can then be related to the step landing position via a linear relation: f¨j0 = aj F¯k+1 + bj (S1.10) where aj and bj only depend on the discretization instant tj and the current state of the swing foot. This relation can be extended over the whole prediction horizon: 0 F¨k+1 = AF¯k+1 + B
459 460 461 462
(S1.11)
where A is a matrix whose rows correspond to discretization times tj and columns to steps, and B is a column vectors. Note that the foot is assumed to be sufficiently removed from the ground during its motion and its effects on the system dynamics are neglected (LIP assumption).
