IEEE Robotics & Automation Magazine - June 2020 - 95

with final time t f . The MPC solves the following constrained
optimization problem:
min
u ( t)

1
2

#0

tf

u (t) 2 dt,

d 3 x (t)
= u.
dt 3
[x (0), xo (0), xp (0)] = [x 0, xo 0, xp 0],
[x (t f ), xo (t f ), xp (t f )] = [x f , xo f , xp f ],
o
p
[x min, x min, x min] # [x, xo , xp ,] # [x max, xo max, xp max],
subject to

(5)

with initial condition [x 0, xo 0, xp 0] and terminal condition
[x f , xo f , xp f ] . The upper and lower inequality constraints of the
optimization problem prevent the MJMPC scheme from outputting trajectories the subsequent whole-body controller cannot track. We constrain the maximum vertical displacement,
velocity, and acceleration of the CoM as well as the maximum
jerk the point mass can be exposed to. For the final state, two
quantities need to be determined: the final time t f at which the
system should reach state x f and the final CoM state itself,
which is provided by a nonlinear step optimizer [13] and thus
evokes stepping behavior if the push gets too large.

Determining Final Time via Data Fitting
Although the final time t f at which the robot should come to a
rest could also be optimized, this would introduce nonlinearity
to the optimization problem. Thus, we treat the final time t f as
an open parameter: too short and it violates the physical capabilities; too long and it may get unstable or use too much energy.
We determine an appropriate value for t f via least-square-error
fitting between collected DRL data and the MJMPC. Figure 11,
shows that both a piecewise linear and quadratic approximation
are able to fit the data.
Nonlinear Step Optimization
In an outer loop, a nonlinear step optimizer [13] provides the MJMPC scheme with a reference point. The
position x f in the final state x f = [x f , 0, 0] is determined by the nonlinear step-location optimizer, and
the velocity xo f and acceleration xp f are set to zero. The
step optimizer is able find a step location and timing
within 3 ms due to the specific formulation of the
optimization problem. Consequently, we are able to
run the optimizer at real time at 100 Hz. Furthermore,
the nonlinear optimizer considers the kinematic constraints of the robot to determine the maximal step

Ankle
Hip
Toe
Stepping

Figure 7. T-SNE, with every dot indicating the activation of all 175 neurons during one time step. The classification of every dot is
determined by whether dots are nonzero (exceed a threshold) for foot velocity (step), angular momentum (hip), or vertical CoM
velocity (toe) and is an ankle strategy otherwise.

JUNE 2020

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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95



IEEE Robotics & Automation Magazine - June 2020

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