Robotics Lab > Research > Themes > Parallel Mechanisms > Balancing

Static and dynamic balancing of parallel mechanisms

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Static balancing

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. For example:

  • 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.

Prototype du manipulateur hybride à 6 ddl équilibré statiquement Modèle DAO du manipulateur hybride à 6 ddl équilibré statiquement

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)

Video clips

The following video clip illustrate the concept of static balancing.

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Format: mpg    Length: 0:50    Size: 7.9 Mb

Dynamic balancing

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:

  1. the counter-rotations of the first leg;
  2. the counterweights of the first leg;
  3. the effector.
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.

Prototype du manipulateur plan à 3 ddl équilibré dynamiquement        Modèle DAO du manipulateur plan à 3 ddl équilibré dynamiquement

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)

Video Clips

The following video clips illustrate the concept of dynamic balancing.

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Format: mpg    Length: 1:14    Size: 14.1 Mb

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Format: mpg    Length: 0:42    Size: 10.2 Mb