IEEE Robotics & Automation Magazine - March 2016 - 38
Actuation
StarlETH is equipped with SEAs in all joints, as shown in Figure 1,
which enables both accurate joint position and torque tracking
using the sophisticated feedback control on the joint level.
For applications that do not push the actuators to the limits [17], the actuator dynamics can be ignored, and the actuators can be considered as perfect torque and position
sources. However, this idealization does not hold for highly
dynamic maneuvers. In fact, various saturation and frictional
effects can lead to a significant divergence of the commanded joint torque or position signals. To design a controller that
takes these effects into account and even exploits the dynamics of the actuators (the compliance of the SEAs), a model of
the actuation system is required.
A discrete-time model of the drive train of StarlETH is
shown in Figure 2, which includes the following components.
● Velocity controller: The desired motor velocity {
o )m is regulated by a proportional-integral controller with gains k vp
and k vi together with an antiwindup, which saturates the
state of the integrator at s v. The loop is updated with time
step Tv = 1/1 kHz and outputs the desired motor current
I ), which is limited to I max = 9.4 A on the motor drives.
● Current controller: A faster proportional-integral current
controller with time step Tm = 1/10 kHz, gains k Ip and k Ii ,
and antiwindup saturation s I determines the desired
motor voltage V.
● Motor electronics: Motor resistance R and inductance L
together with the back electromotive force (1/l v {o m) are
part of the motor l v electronics, which define the mapping of the applied voltage V to the motor current I. The
dynamical effects of the power electronics and power supply are neglected. This loop is evaluated at the same rate as
the current controller to minimize computational load.
● Motor mechanics: The motor current I multiplied with the
torque constant l a yields the motor torque x m, which acts
on the motor shaft together with the friction torque x f and
the load at the joint cx j . The equations of motion of the
motor shaft defining the motor velocity {o m and position
{ m can be solved with the lumped motor, gearbox, and
bearing inertia H.
● Gearbox and joint friction: The friction of the harmonic
drive gearbox can be approximated with a constant (c 0),
linear (c 1), and cubic (c 2) term of the motor velocity [18].
The frictional torque x l due to radial load fr in the bearings is also affecting the dynamics of the actuator.
● Spring mechanics: The joint torque x j is a function of the
spring deflection d, the spring stiffness c (d), the time derivative of the deflection do , and the damping d (d) . For
the detailed models of the spring characteristics, see [5].
Most parameters of the actuator model and the joint controllers are known from datasheets. The remaining parameters are identified based on different experiments [5].
The desired motor velocity {o )m is finally generated by the
joint controllers, which regulate either the desired joint
torque x )j or the desired joint position { )j and velocity {o )j , as
described in [5].
38
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IEEE ROBOTICS & AUTOMATION MAGAZINE
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march 2016
Locomotion Control
The (deterministic) locomotion control problem for periodic
gaits, which minimizes a cost function c, can be stated as:
q ), u ), x )j , m ), I ) (q )) = argmin c (q, u)
s. t. z (t) = z (t + T) 6t, , periodicity
M (q) uo - h (q, u) - S < x j - J < m = 0,
3 dynamics
qo = F (q) u, c i = J i u 6i ! I (q),
(2)
q j ! Q, u j ! U, x j ! T, , limits,
where z (t) is the periodic solution with a period time of T.
The solution fulfills all dynamic constraints and does not violate any position limits Q, velocity limits U, or actuator limits T. The periodic constraint can be replaced by the
constraint z (T ) = z ) for nonperiodic motions like jumps.
In general, this is a hard control problem to solve because
the motion planning/generation problem, which specifies
how the legs and the main body should move (q ), u )), and
the motion execution/control problem, which determines
the desired motion through the desired torques and forces
(x )j, m )), is closely coupled by the set of desired contacts
I ) (q )). Moreover, robust locomotion is only achieved if the
controller is capable to handle large disturbances due to unanticipated terrain irregularities or external pushes. As a consequence, motor actions cannot be planned fully offline and
tracked in an open-loop fashion, but have to be generated
online by a combination of a feedforward and state-feedback
controller instead.
We employ a model-based controller with a compact, yet
flexible parameter space and fine-tune these parameters by
repeating the same motion task with slight parameter variations until a desired motion is found and all constraints are
met. The proposed controller generates a desired motion
based on predefined motion primitives, which are superimposed by control actions needed for balancing. To find these
motion primitives , along with other parameters, we reformulate the problem stated in (2) to
i
)
= argmin / w k c k (q, u, m, { m, {o m),
i!P
(3)
where the optimal parameter set i ) in the admissible set P
minimizes the weighed sum of the cost terms c k, which depend on the state trajectory of the robot q, u and the actuation signals { m, {o m .
Motion Parameterization
Contact Scheduling
The desired motions of the legs need to be parameterized
in space and time. We split the motions into individual motion primitives according to the contact schedule, which
defines the set of contacts I ) (q )) and the role of the legs
between the contacts. A stance leg supports the main body,
whereas a swing leg moves its foot to a new location. A gait
pattern can be employed to predefine the contact schedule.
For periodic motions, the gait pattern is defined as nine
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