1. Introduction
Dynamic manipulation tasks, as found in the packaging industry, demand light-weight, stiff robotic systems, with a high payload-to-weight ratio, to achieve both energy efficiency and high positioning accuracy. Given these requirements, parallel robots are particularly well suited due to their architecture with base-fixed actuators and, hence, low moving mass [
1]. The most common parallel robots for these tasks are Delta robots [
1]. In many industrial applications, they must be operated at reduced speed, or dwell times have to be included in the motion planning, to allow frame vibrations to decay and hereby prevent unacceptably high vibrations from building up [
2]. Otherwise, the vibrations reduce the accuracy of the robot, can cause fatigue and wear in its mechanical elements, cause noise, or even affect surrounding machines [
2,
3,
4]. As a result, the full potential of Delta robots cannot be realized. The vibrations can be explained by the high speeds and accelerations required to achieve the desired short cycle times. These cause high inertia forces and moments, which act on the frame. This problem is exacerbated by the mostly slim and high structure of the frames. A typical setup of a Delta robot, including the coordinate system (COS) used in the remainder of this publication, is shown in
Figure 1. Given this geometry, they tend to vibrate laterally, especially during horizontal robot motions, typical for pick-and-place tasks.
Methods for the reduction of frame vibrations may be classified as software, hardware, and hybrid approaches. Software-based approaches comprise trajectory planning methods that aim to either minimize the excitation of natural frequencies, e.g., ref. [
5,
6], to minimize the shaking forces/moments, e.g., ref. [
7], or to suppress residual vibrations, e.g., refs. [
8,
9,
10]. A necessary condition to achieve both tool center point (TCP) control and vibration suppression is a sufficient kinematic/dynamic redundancy [
7], to control the center of mass (CoM) and TCP position independently, which is not met by Delta robots. Consequently, in the case of Delta robots, at best, trajectory planning methods achieve vibration minimization or elimination of residual vibrations after the robot motion is completed.
The goal of hardware and hybrid methods is to maintain a constant linear and angular momentum by adding additional actively or passively driven counterweights and counter rotating inertia. This is referred to as dynamic balancing. If the counterweights and inertia are directly or indirectly (e.g., by additional kinematic chains) driven by the manipulator’s actuators, it may be categorized as a hardware-based approach. In the case of an active actuation (i.e., active dynamic balancing) it may be considered a hybrid approach. Both can be used for vibration prevention. In [
2], van der Wijk et al. present a dynamically balanced Delta robot with respect to inertial forces by adding three counterweights and two additional links. As a result, the moving mass of the robot increases approximately three-fold. This is accompanied by an increase of the driving torques. The use of an active balancing mechanism has the advantage that it can be adapted in the software to a wide range of tasks and can be applied to different robot models of similar size and performance without hardware modifications. In addition, the handling object’s mass can be taken into account. A disadvantage is that an additional control system and actuators are needed, which increases the design complexity.
Despite these advantages, comparably little literature is found regarding active dynamic balancing [
11], as active research on this method is relatively young [
12] and covers mostly theoretical studies. In [
11], van der Wijk et al. presented the first active dynamic balancing unit [
13]. It was designed as an
mechanism, i.e., a five-bar linkage with an actively driven counterrotating counterweight attached to its TCP. In this representation,
refers to a parallel mechanism with two
kinematic chains, each with two revolute joints, connected to each other by a revolute joint. The underlined joints are actuated. This mechanism was used to balance a planar gantry manipulator suspended by three cables [
14]. The balancing mechanism was controlled based on the dynamics of the manipulator and the balancing mechanism [
11]. Here, only the counterrotating counterweight was taken into account; the mass and inertia of the five-bar linkage are neglected to simplify the dynamics equations. The validity of this method has been experimentally proven [
14]. An example of a passively force-balanced mechanism, in combination with one actively driven counter inertia used for moment balancing, was presented in [
15], applied to a planar
parallel robot [
2]. A similar design was presented by Wang et al. in [
16], where the authors proposed to use a planar
mechanism, the linearly actuated equivalent of the five-bar linkage, with a counterrotating counterweight attached to its TCP. In this notation,
P represents a prismatic joint. Later, Wang et al. [
12] proposed a second design consisting of three independently rotating one-link mechanisms (
), for the dynamic balancing of planar three-DoF manipulators.
The present publication presents the design of a novel planar three-DoF balancing mechanism with minimum mass and inertia, designed for the active dynamic balancing of Delta robots. The balancing mechanism presented in this publication was designed according to the engineering guideline VDI 2221 [
17]. It proposes a general process of the product design consisting of eight stages. The first step in the design of a technical product is the clarification of the problem task. The result of this step is a list of requirements, as will be presented in
Section 2. This is followed by developing and selecting design concepts. For this purpose, the required functions and their structure are defined. A function-oriented approach can help to expand the search space and support the search for alternative or innovative solution principles and also form the basis for common design methods [
17]. The functions define what the product or its components should do. They are described in a simple “object–verb form” (e.g., “constrain motion”) [
18]. The next step is to search for solution principles for the defined functions, e.g., using conventional methods such as literature and patent research as well as the analysis of natural systems, discursive methods using classification schemes, or intuitive methods like brainstorming [
18]. These functions and their solution principles are introduced in
Section 3, followed by a detailed presentation of one of the most promising balancing mechanisms. For this mechanism, the kinematics and dynamics equations are derived in
Section 4. Finally, in
Section 5, the dimensions of a prototype mechanism designed to experimentally validate the concept of active dynamic balancing are defined.
2. Design Requirements
In [
19], a detailed analysis of the shaking forces and moments caused by Delta robots and the interaction between these forces/moments and the dynamics of typical industrial robot frames was performed. The results of this analysis form the basis for the mechanism design and are therefore briefly summarized below.
Due to the mostly slender and high structure of the frames, the first three natural frequencies belong to eigenmodes describing a horizontal translation in the
-plane and a rotation around the
z-axis, as shown in
Figure 2.
The motion of the first two mode shapes coincides with the direction of the dominant shaking forces, which occur in the
-plane and coincide with the main direction of motion of the robots. As a result, simulations and experiments conducted in [
19] to analyze the frame vibrations indicate a dominance of vibrations in the horizontal plane. Against this background, the balancing mechanism is designed to eliminate the vibration forces in the
-plane. If the lines of action of the balancing forces do not cross the center of the frame, they lead to additional shaking moments about the
z-axis. These moments would excite the third, rotational eigenmode of the frame. To compensate for these parasitic moments, moments about the
z-axis are also compensated.
In addition to the requirements derived from the analysis of the vibration characteristics of Delta robot frames, we will now examine design requirements that originate from the active dynamic balancing process itself. These requirements pertain to the following:nolistsep
Parasitic shaking forces and moments: If the shaking forces are not collinear with the balancing forces, force balancing will result in additional parasitic shaking moments. This is shown schematically in
Figure 3a. The motion of the counterweight’s and the robot’s CoM are parallel but not colinear. Even if the sum of the inertia forces is zero, they create a force couple around the attachment point, which leads to a parasitic shaking moment.
High ratio of counterweight’s mass to stationary mass: The mass of the balancing mechanism placed on top of the robot’s frame inevitably leads to a reduction in the frame’s natural frequencies, which could make the system more susceptible to frame vibrations. It also places higher demands on the structural rigidity of the frame to prevent unacceptable static deflections due to gravitational loads. Therefore, the ratio of the counterweight’s mass and all moving masses that contribute to the balancing, to the stationary mass of the balancing mechanism, which does not participate in balancing (e.g., base mass), should be as high as possible.
Kinematics and Dynamics Complexity: Most control designs require the solution of kinematics and dynamics equations for each time step. Thus, complex, nonlinear dynamics that can only be solved numerically place high demands on the control hardware. To allow the use of low-performance, low-cost control hardware, it is desirable to design a balancing mechanism with linear kinematics and dynamics equations.
Stiffness of the connection to the robot: If the balancing mechanism is mounted far from the robot mounting point, the balancing forces/moments must be transmitted through the frame. The compliance of the frame can cause non-negligible dynamic effects in the transmission of forces/moments to the robot base. It is therefore desirable to have a high rigidity in the connection between the robot and the balancing mechanism.
Size: To avoid collisions and a high adaptability to different frame designs, it is desirable to keep the balancing mechanism as small as possible. At the same time, if the counterweight is also used as a counter rotating inertia, a large diameter of the counterweight is desirable to keep the mass low while achieving a high inertia [
20].
Enhancement Effects: When a rotating actuator is used to drive a counterrotating inertia, the direction of rotation of the actuator and the counterrotating inertia should coincide. In this case, the actuator inertia participates in the moment balancing [
11]. This and similar enhancements should be used when applicable.
Costs: Costs include both operating and acquisition costs. The sum of the two must be offset by the increased productivity provided by shorter cycle times.
The power requirement of the balancing unit depends on the mass and moment of inertia of the counterweight. For simplicity and without loss of generality, this shall be shown on the example of a one-dimensional balancing task. The acceleration,
, of the counterweight,
, can be calculated by the equilibrium of forces:
Here,
denotes the total mass and
the CoM acceleration of the mechanism that is to be balanced. Consequently,
represents the shaking force caused by the manipulator. Dividing by the counterweight’s mass,
, and substitution with
yields
The mechanical power requirement of the balancing unit is defined as
The term
is fully defined by the mass of the mechanism being balanced and its CoM trajectory. The only variable that can be influenced by the mass of the counterweight is
. Since
is a hyperbolic function of the counterweight’s mass (cf.
Figure 3b), so is the power requirement (
3).
Regarding the target of low operation costs, a high counterweight mass and inertia are favorable. At the same time, a higher mass inevitably leads to a reduction of the frame’s natural frequencies and increased demands on its structural stiffness. Therefore, a trade-off between these two goals has to be made.
3. Structural Design of the Balancing Mechanism
The balancing task was divided into two sub-functions describing the balancing of shaking forces and the balancing of shaking moments. These two functions were further subdivided into a principle mechanism or solution to provide a multi- or unidirectional force/moment, a method to combine these principle mechanisms and to attach them to the robot frame. Following [
18], the proposed solutions and their combinations were summarized in a morphological matrix (see
Figure A1,
Figure A2 and
Figure A3). In this step, candidate solutions with up to six DoFs were considered. These concepts include novel solutions as well as solutions or adaptations of known solutions from the literature. Examples of the latter include the use of a five-bar mechanism to provide counterbalancing forces in two planar DoF [
11], the use of counterrotating eccentric masses as used in the Lanchester balancer [
21] and the active three-DoF balancer proposed in [
12], or the movement of a fluid or steel balls similar to the solution proposed in [
22] to change the mass of the counterweight in a mechanism for the passive balancing of variable payloads. Examples of novel solutions in the context of dynamic balancing are the planar
,
, and
mechanism (see
Figure A2) or a cable-driven parallel robot. Finally, the system concepts were combined into solutions that provide at least three DoF to allow the balancing of forces in the
plane and moments around the
z-axis. These were then evaluated using the criteria presented in the previous section.
A detailed discussion of the characteristics of all the solutions and their combinations is beyond the scope of this publication, so only the most promising concept, which has been further developed, is presented below. The chosen balancing mechanism consists of three identical
kinematic chains connecting the counterweight to the base, as shown in
Figure 4. These kinematic chains consist of two perpendicular guide rails crossing at the center of an actively driven carriage. One of the guide rails is fixed to the base and the other to the counterweight by means of a revolute joint. The design of this mechanism is inspired by THK Co., Ltd. (Minato City, Tokyo).
CMX series precision alignment stages [
23]. Unlike the balancing mechanism presented here, the
CMX alignment stages have four
kinematic chains, only three of which are actuated. The actuators are arranged perpendicular to each other in a square. A variation of this mechanism with
kinematic chains has been patented by Eidelberg [
24]. The triangular arrangement of the balancing mechanism’s kinematic chains is similar to the tilt table presented by Kimura in [
25].
Using Gruebler’s Equation (
4), it can be shown that the balancing mechanism has two planar translational and one rotational DoF. Since the mechanism performs planar motions only, Gruebler’s equation is given by [
26]
with
being the number of links including the base,
the number of all joints, and
the sum of all joint DoF.
As shown in
Figure 4, the triangular arrangement of the kinematic chains results in a small footprint, making the best use of the space at the top of the Delta robot. It also has the same triangular shape as the base of many Delta robot models, which allows it to be mounted directly on the robot, resulting in a direct transmission of the balancing forces to the robot base. In this way, they are compensated at their origin before they can act on the frame. The balancing of shaking forces in the
-plane and the moments around the
z-axis are performed with the same counterweight. This is desirable to minimize the number of additional elements and thus the weight. The radius of the counterweight has been maximized and cutouts have been made to achieve a high moment of inertia with a low mass. Due to the parallel kinematic design, the mechanism has a low height. This can be seen in
Figure 5, which shows the balancing mechanism on a low-cost Delta robot called Suisui Bot, designed by the authors of this publication for experimental validation of vibration reduction methods. Please refer to [
27] for more details on the Suisui Bot.
The counterweight is only 29
away from the top of the frame. This low profile allows minimization of the parasitical shaking moments in the
-plane. Furthermore, the balancing mechanism has, in good approximation, linear kinematics and dynamics, which minimizes the computational effort on the control system (see
Section 4.3). Finally, all motors are stationary, minimizing the risk of cable damage. This mechanism was selected due to the combination of the above-mentioned properties, which make it a promising solution for active dynamic balancing. The dimensions of the balancing mechanism are defined in
Section 5 after a brief derivation of the inverse dynamics of the mechanism necessary for this step.
6. Summary and Discussion
This publication presented the design of a novel three-DoF planar parallel mechanism for the dynamic balancing of Delta robots. First, the main design requirements for a balancing mechanism were defined, and possible mechanisms were presented based on these requirements. Finally, a 3- mechanism was identified as one of the most promising solutions given the following:
its small footprint, which allows it to be mounted directly on the robot, resulting in direct force/moment transmission;
its low weight, due to the use of a counterrotating counterweight;
its low height, which makes it possible to minimize parasitic shaking moments;
its, to a good approximation, linear kinematics and dynamics.
To evaluate the performance of the mechanism, a small, low-cost prototype was built and tested on the low-cost Delta robot Suisui Bot. A description of the control system and an experimental validation of the overall system is beyond the scope of this publication, but can be found in [
19,
30]. In these publications, it has been shown that, in a scenario with 75 quasi-randomized pick-and-place positions and cycle times as described in
Section 5, the mechanism provides a significant vibration reduction of up to 86% [
19]. In future work, concepts for combining this mechanism with solutions for balancing the shaking moments about the
x- and
y-axes will be investigated, as these represent the highest shaking moments. A combination with gyroscopes seems promising to achieve high moments with minimal additional mass and inertia.