Biomedical Engineering Reference
In-Depth Information
computational complexity has not increased and
τ
d
can be efficiently computed
with the RNEA.
6.4.4 Validation and Experiment
We tested the TSID against the Unifying Framework (UF) (Peters et al.
2007
) and
the Whole-Body Control Framework (WBCF) (Sentis and Khatib
2005
)ona
customized version of the Compliant huManoid (Coman) simulator (Dallali
et al.
2013
). The robot has 23 DoFs: 4 in each arm, 3 in the torso and 6 in each
leg. We adapted the simulator to make the robot rigid and fully-actuated (we fixed
the robot base and we removed the joint passive compliance). Direct and inverse
dynamics, both in simulation and control, were efficiently computed using C
language functions, generated with the Robotran (
2012
) symbolic engine. Contact
forces were simulated using linear spring-damper models [stiffness 2
10
5
N/m
and damping 10
3
Ns/m, as proposed in Dallali et al. (
2013
)] with realistic friction.
To integrate the equations of motion, we used the Simulink variable step integrator
ode23t
, with relative and absolute tolerance of 10
3
and 10
6
, respectively. The tests
were executed on a computer with a 2.83 GHz CPU and 4 GB of RAM.
6.4.4.1 Trajectory Generation
To generate reference position-velocity-acceleration trajectories, we used the
approach presented in Pattacini et al. (
2010
) (see also Sect.
6.5
, which provides
approximately minimum-jerk trajectories). The trajectory generator is a third order
dynamical system that takes as input the desired trajectory
x
d
(
t
) and outputs the
three position-velocity-acceleration reference trajectories
x
r
(
t
),
. The
reference position trajectory follows the desired position trajectory with a velocity
that depends on the parameter “trajectory time” (always set to 1.0 s in our tests). We
set all proportional gains
K
p
ﾼ
x
r
t
ðÞ
,
x
r
t
ðÞ
5s
1
. The
pseudoinverse calculations are all performed using the “damped pseudoinverse”
technique to guarantee stability near singularities.
10 s
2
and all derivative gains
K
d
ﾼ
6.4.4.2 Test 1: Feasible Task Hierarchy
In this test the robot performs four tasks (see also Fig.
6.9
):
• F: 3 DoFs, apply a normal force of 20 N on a wall with the right hand
• T2: 3 DoFs, track a circular trajectory with the left hand
• T1: 1 DoF (
x
coordinate), track a sinusoidal reference with the neck base
• T0: 23 DoFs, maintain the initial joint posture
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