Static and dynamic balancing of parallel mechanisms
Static balancing involves ensuring that the motors do not contribute towards
supporting the mechanism's weight, for any of the possible configurations. A
mechanism can therefore remain stable in any position without the help of motors or brakes.
This result can be
obtained by using counterweights or springs. The most common application of this
concept at the present time is that of desk lamps (Figure 1). The balancing of these lamps
is possible through the use of springs, but also as a result of friction, which we will not
rely on in robotic mechanisms. Static balancing is mainly useful in applications
involving heavy loads (for example, flight simulators).
Fig. 1: Statically balanced desk lamps.
A statically balanced mechanism can be obtained by achieving a potential energy which
remains constant irrespective of the position and orientation of the mechanism. The potential energy
expression must thus be found, its derivative equalled to zero and the system of equations solved.
Several parallel mechanisms have been studied in the laboratory, including 3-DOF planar mechanims,
2- and 3-DOF spherical mechanisms as well as 4-, 5- and 6-DOF spatial mechanisms.
From the theoretical results obtained, a large number of designs were proposed for statically balanced
parallel mechanisms and several wooden prototypes were developped.
- A 1-DOF spring-balanced mechanism (Figure 2). This is the simplest example which leads to
a clear understanding of the relationships between the masses, the connecting/joint positions, the stiffness of
the spring and the sense of gravity.
- A 3-DOF planar mechanism, statically balanced using counterweights and springs (Figure 3). This
mechanism is of the type 3-RRR planar. The springs are placed on the base
of the mechanism and are connected by a system of pulleys.
- A 3-DOF planar mechanism, statically balanced using springs (Figure 4). A parallepiped was added
to the proximal members to achieve the balancing without the use of counterweights.
Fig. 2: Planar 1-DOF mechanism balanced using springs.
Fig. 3: Planar 3-DOF mechanism balanced using counterweights and springs.
Fig. 4: Planar 3-DOF mechanism balanced using springs.
A robotic prototype was also designed for a spatial 6-DOF mechanism statically balanced using springs (Figure 5).
The mobile platform is connected to the base by 3 legs each consisting of a 5-bar mechanism.
In the event that this architecture finds interesting applications
as a base structure for motion simulators, for example, and more specifically for
flight simulators, the use of springs was preferred over the use of counterweights,
since the latter substantially increase the total inertia of the structure when
large accelerations of the platform are required.
Fig. 5: Spatial 6-DOF mechanism balanced using springs.
A poster was prepared in 2002 on a 6-DOF spatial statically balanced
mechanism. The poster can be downloaded with the following PDF file.
Statically-Balanced 6-DOF Hybrid Parallel Manipulator (12 Mb)
The following video clip illustrate the concept of static balancing.
Format: mpg Length: 0:50 Size: 7.9 Mb
Dynamic balancing goes one step further than static balancing (a dynamically
balanced mechanism is also statically balanced) and is essentially aimed at reducing
the reaction forces and moments on the base, for all trajectories of the mechanism.
Thus, one says that a mechanism is dynamically balanced when the sum of all forces
and moments acting on the base are always nil. Thus, the mechanism will not transmit
any vibration to its environment when it is operated. This result is usually obtained
through the use of counter-rotations but it has been shown, in theoretical studies
carried out in our laboratory, that it is possible to achieve the same result without counter-rotations.
The most common application for dynamic balancing is in combustion motors. Several parts
are added so as to counter balance the movement of various parts, such as
vilbrequins (Figure 6). The vibrations of the motor are thus greatly reduced.
Dynamic balancing is essential in applications where one must reduce the
efforts and moments created at the base of the mechanism. This is the case for mechanisms
used for the active correction in telescopes (the movements of the correction mechanism
must not influence the other instruments) and for space applications.
Fig. 6: Combustion Motors.
Several parallel mechanisms have been studied in the laboratory, including
a range of 4-bar and 5-bar mechanisms, with and without counterweights
and counter-rotations. These mechanisms have then been used to analyze
planar and spatial mechanisms with several DOFs.
Two prototypes of the 4-bar mechanisms which have been developped in the laboratory are of particular
interest. The first is a mechanism which is suspended dans le vide (Figure 7).
The first is a 4-bar mechanism, without counter-rotations, whose central member
is diagonally positioned (Figure 8). The mechanism obtained in this way is dynamically balanced.
Fig. 7: Frank's Mechanism.
Fig. 8: Gabriel's Mechanism.
A robotic prototype was also built of a 3-DOF planar mechanism (Figure 9).
Five-bar mechanisms in the shape of parallelogram were chosen to build
the legs since they allow the balancing conditions to be simplified.
Figure 9 shows the following:
The counterweights serve to maintain a constant position for the centre of mass while the
counter-rotations are used to maintain a constant angular moment. The counterweights,
the counter-rotations and the effector are made of steel. The mass of the effector is 0.1 kg
and that of the mobile parts is about 4 kg.
- the counter-rotations of the first leg;
- the counterweights of the first leg;
- the effector.
Fig. 9: Planar 3-DOF dynamically balanced mechanism.
Several posters were prepared in 2002 on dynamic balancing.
The posters can be downloaded with the following PDF files.
Planar 3-DOF Reactionless Parallel Mechanism (4.9 Mb)
Synthesis of a Reactionless 6-DOF Parallel Mechanism (4.7 Mb)
The following video clips illustrate the concept of dynamic balancing.
Format: mpg Length: 1:14 Size: 14.1 Mb
Format: mpg Length: 0:42 Size: 10.2 Mb