IEEE Robotics & Automation Magazine - June 2018 - 33

L

L
LIP

HIA
Mechanism
(Gantry)

x3

HIA

B
x2
x1

LIP
Mechanism
(Tripteron)

A

Figure 5. A macro-mini robotic system for pHRI. The mini LIP
mechanism is mounted on the end effector of the macro HIA
mechanism.

Figure 6. A schematic representation of a 1 DoF passive-active
mechanism. (Figure courtesy of P.D. Labrecque et al. [13].)

design, they can be modeled as a 1-DoF manipulator; therefore, the analysis of a 1-DoF system is presented. The limits of
the experimental prototype of the macro-mini mechanism
are then specified and used to determine the capabilities of
the macro-mini robot.
In macro-mini architecture, the operational limits are dictated by the size of the LIP mechanism's workspace and by the
HIA mechanism's velocity and acceleration limits. To achieve
intuitive behavior using the passive mechanism, the end effector of the LIP mechanism must be kept within its workspace,
i.e., the HIA mechanism must respond quickly enough to
always reposition the LIP joints in, or close to, their neutral
configuration. If the HIA mechanism is not fast enough, then
the user will reach the limits of the LIP mechanism's workspace and the rendering of a low-impedance interaction to
the user will be limited and unintuitive.
Figure 6 represents a simple, 1-DoF passive-active (macro-
mini) mechanism. In this model, the rail connection between
A and the base represents the macro HIA mechanism, while
the rail connection between A and B represents the mini LIP
mechanism, whose range of motion is 2 L. The position of
body A and body B with respect to the fixed frame are x 1
and x 3, respectively, while x 2 denotes the position of platform B with respect to platform A. It is therefore preferable
to establish the capability of this macro-mini robot to follow
the motion produced by the operator (x 3) given the limited
range of motion of the LIP mechanism (- L # x 2 # L) and
the limited velocity and acceleration of the HIA mechanism.
Figure 6 shows
x3 = x1 + x2 .

(5)

To evaluate the capabilities of the system, it is assumed that
x 3 undergoes a harmonic motion, e.g.,
x 3 = R sin (~t),

(6)

where t is the time, and ~ and R are the frequency and amplitude, respectively, of the input motion. Assuming that the HIA
mechanism is controlled to keep body B at the neutral

(midrange) configuration of the LIP mechanism, the HIA
mechanism driving body A reacts proportionally and in the
same direction as body B. Its motion is therefore expressed as a
harmonic motion of identical frequency and phase, e.g.,
x 1 = D sin (~t),

(7)

where D is the amplitude of motion of the macro HIA mechanism. If the amplitude of motion of the end effector (body B)
is larger than the range of motion of the LIP mechanism, then
the HIA mechanism must contribute accordingly to a displacement whose amplitude is given by
D = (R - L) .

(8)

Given the velocity and acceleration constraints of the actuator of the HIA mechanism, noted here as xo 1, max and xp 1, max, it
is possible to determine the maximum amplitude of the HIA
mechanism as a function of the frequency of the motion
using the first two time derivatives of (7), yielding
D vmax =

xo 1, max
~

,

(9)

,

(10)

and
D amax =

xp 1, max
~

2

where D vmax and D amax, respectively, represent the maximum
amplitudes of x 1 given the velocity and acceleration constraints. Substituting (9) into (8), the maximum amplitude of
input motion x 3 for a given frequency ~ and the passive
joint's motion range L is expressed as
R vmax =

xo 1, max
~

+ L,

(11)

where R vmax is the maximum range of input motion that is
feasible due to the velocity constraint. Similarly, the relationship for R amax, the maximum range of input motion that is
feasible due to the acceleration constraint, is obtained by
substituting (10) into (8), yielding
JUNE 2018

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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33
IEEE Robotics & Automation Magazine - June 2018

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