IEEE Robotics & Automation Magazine - December 2015 - 83

x eb

= k d Dn.

Human
Locomotion

Motor
Generator

Induced
ac Voltage

Diodes
Rectification

10
5

0
1,
00
0
1,
10
0
1,
20
0
1,
30
0

0
Motor Speed (r/min)
(a)

(6)

As the duty cycle D determines the relationship between
the torque and the speed, it can be regarded as the damping
coefficient. k d is the proportionality coefficient in the unit of
Nm/(r/min).
To verify the effectiveness of (6), the braking torque was
evaluated under two experimental conditions.
● D is constant (50%), and n varies from 400 to 1,250 r/min.

where C T refers to the motor torque constant.
According to (5), the braking torque x b is proportional to
the motor speed n. As the motor armature resistance R a is
usually very small ( 0.98 X for our motor), the resulting braking torque x b will be quite large once the motor rotates.
If we switch the motor-winding-short on or off with a
pulsewidth modulation (PWM) signal, the braking torque
during the switch-on period will be very large and the ankle
joint will only be able rotate at a low speed, while the braking
torque during the switch-off period will be small and the joint
will be able to rotate quickly. With an appropriate on/off frequency, the braking torque will be positively correlated with
the duty cycle ^Dh of the PWM signal, and the resulting
equivalent braking torque (x eb) can be approximated as

(5)

CT CE z2
n,
Ra

= C T zI a =

where R a refers to the motor armature resistance.
I a will produce a braking torque x b that prevents the
motor from rotating

(4)

Ea
,
Ra

Ia =

where C E refers to the electromotive constant, z refers to the
magnetic field intensity, and n refers to the motor speed.
If the stator windings of the brushless motor (or rotor
windings of the brushed motor) are shorted, the induced voltage will generate a current I a, and its instantaneous value can
be estimated by

(3)

Braking
Torque (mNm)

E a = C E zn,

n is constant (600, 800, and 1,000 r/min, respectively), and
D varies from 20 to 80%.
The evaluated motor, which acted as a generator during
the experiment, was driven by another motor whose output
shaft was connected in series with that of the evaluated motor.
The braking torque was
estimated by the average
armature current, with
A prosthesis in the maximal
the torque constant of
33.5 mNm/A provided by
damping mode without
the motor data sheet.
The evaluation results
control can only move
of the proposed method
are shown in Figure 2.
within a limited angle
The braking torque is
proportional to the motor
range (less than 8°).
speed at a constant PWM
duty cycle D, as shown in
Figure 2(a). The braking torque is proportional to the duty
cycle D at a constant speed, as shown in Figure 2(a). These
results verified that the braking torque resulted from the
motor winding short can be effectively estimated by (6).
The implementation process of the proposed damping
control onto the prosthesis prototype is shown in Figure 3.
The human bipedal locomotion generates a driving torque,
●

Braking
Torque (mNm)

results in the rotation torque x m during the CF phase. x m
drives the ankle joint to plantar flex during the early stance
and dorsiflex during the middle stance, resulting in the motor
rotation. A motor can behave as a generator when the motor
rotates, and the rotation will generate an induced voltage E a,

15
10

600 r/min
800 r/min
1,000 r/min

5
0
20

50
40
60
Duty Cycle (%)
(b)

Figure 2. An evaluation of the proposed method for braking
torque control under two different conditions: (a) the motor
speed and (b) the duty cycle.

Rectified
dc Voltage

PWM
Resistance

Controllable
dc Current

Motor

Controllable
Braking Torque

Prosthesis, Adaptor
Figure 3. The realization process of the proposed braking torque control on the prosthesis prototype.

DECEMBER 2015

IEEE ROBOTICS & AUTOMATION MAGAZINE

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - December 2015