IEEE Robotics & Automation Magazine - March 2016 - 39

parameters, which include liftoff and landing time for each
leg as well as the stride duration T that defines the period
of the cycle. We use a reduced parameterization that can
represent both symmetrical and asymmetrical gaits that allows us to define the parameter space for a specific type of
gait, which helps to optimize the time parameters for a predefined gait [13].
Terrain Adaptation
The motions of the legs need to be planned and replanned according to the encountered terrain. We introduce a control
frame C, as shown in Figure 3, to properly define the motions
on the arbitrary terrain. The z-axis of the control frame (e Cz )
is aligned with the estimated surface normal, and the x-axis
(e Cx ) is parallel to the projected x-axis of the body fixed base
frame B on the ground. The origin of the coordinate system
C is fixed to the origin of the world frame W such that only
the orientation of the frame is changing over time.
By describing the desired motion (e.g., body orientation)
in this control frame, the motion is properly defined and automatically aligned with the terrain. The same holds true for
high-level velocity commands c ) = [v x, v y, }o ] <, which are
typically given in heading direction (v x e Cx ) or lateral direction
(v y e Cy ) of the robot, as well as desired turning rate }o e Cz
around the vertical axis.
Motion of Stance Legs
Since the configurations of all support legs are defined as the
fixed foothold locations and the pose of the torso, we define
the motions of all support legs through a desired motion of
the torso. For balancing, the center of mass is simply kept
above the support region. By averaging the position of the
legs with weights depending on their role, a smooth motion
of the target of the center of mass is planned [19]. The remaining degrees of freedom (DoF) like body height above
ground and orientation are prescribed by motion primitives.
We employ polyharmonic and quintic polynomial splines
with periodic constraints to define the feedforward motion
of the main body. To reduce kinematic singularities, the default orientation of the torso is adapted to the local inclination of the terrain and superimposed by gait-specific
motions, such as pitching for a bounding gait.
Motion of Swing Legs
The desired footpoint F t of a swing leg is defined as the sum
of several position vectors, as shown in Figure 3. The basic
idea is to plan a desired foothold F td at the end of the swing
phase and then to interpolate between liftoff position F lo and
this position at touch-down as a function of the swing phase.
For the ground clearance, a trajectory in the z-direction of
the control frame C is defined using a spline with zero velocity constraints at the beginning and ending of the swing
phase to avoid step inputs to the actuators and minimize impact losses.
The desired foothold location F td is determined at every
control step based on a desired feedforward motion and a su-

perimposed balancing controller. First, a location on the
slope F s is selected with respect to the hip joint H t . For locomotion on the sloped terrain, there are two different strategies: 1) the lever mechanism and 2) the telescopic strut. The
lever mechanism projects the hip position along the slope
normal, whereas the telescopic strut strategy projects the hip
position along the gravity. We apply the latter strategy as discussed in [16]. From the selected location on the ground, the
desired foothold is defined as the desired traveling distance
given by the speed command and the timing given by the
gait pattern. In presence of a disturbance, the location of the
foothold is corrected by the balancing controller. We employ
an adapted version of Raibert's flight controller [6] for each
leg, which predicts the next foothold according to
)
k FB
p ^ v ref - v ref h

h,
g

(4)

where v )ref is the desired and v ref is the measured reference velocity between associated hip and middle of the
torso, h is the height of the hip above the ground, and g
is the gravitational acceleration. The component of the
balancing control is weighted by k FB
p and only active if
there is a velocity error.
Motion Execution
Leg Coordination
A leg coordinator decides based on the planning given by the
gait pattern and contact sensing if a leg is considered to be a
support leg and, thus, is force controlled or if it is a swing leg
and position control can be safely used. We further employ
an event detector, which identifies for each leg events such as
early and late touchdown, early and late liftoff, slipping contact, lost contact during stance, and hitting an obstacle during swing to trigger different reflex mechanisms.
No special treatments are required for late liftoff and early
touchdown. In case of slipping and lost contact during the
stance phase, the leg is lowered with respect to the ground
plane to regain contact as quickly as possible.
Execution of Swing Leg Motion
The desired swing foot positions are enforced by mapping the
Cartesian positions to the desired joint positions { )j and velocities {o )j using inverse kinematics, which are subsequently
tracked by the joint position controller.
Execution of Torso Motion
We apply a virtual model controller in combination with a
quasi-static force distribution to track the desired motion of
the main body while optimally distributing the ground reaction forces. The virtual model controller outputs a desired
net force f ) and torque t ), which should act on the torso to
execute the desired motion defined as the desired position
r )WB, velocity v )B, and acceleration a )B as well as desired rotation quaternion p )WB and angular velocity ~ )WB . The controller has the form of
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