The Configuration Space Method for Kinematic Design of Mechanisms (MIT Press)

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First, one has the following formula:. According to Equation 1 , and for the 2- to 6-DOF of parallel manipulators, we can list a table that illustrates all the possible solutions. Here it is assumed that the number of limbs equals the DOF of the manipulator. Let us consider a case study: say one wants to invent a 3-DOF pure translational parallel manipulator. This involves the following steps:. In reference [ 30 ], the theory of groups of displacements is used to develop some new architecture of 4-DOF 3T1R parallel mechanisms by resorting to the parallelograms.

The general function Gf set theory was put forward recently to synthesize design parallel mechanisms [ 32 ], and our new proposed manipulator is based on this method. According to the general function set theory, it has two classes. For the first class: first of all, we need to determine how many linkages, how many active linkages, how many passive linkages and how many actuators on the i th active linkage should be employed by using the following Equations 2 to 6 based on the end-effector characteristics:.

Finally, the particular ideal parallel mechanism can be synthesized through gathering the kinematic limbs.

In fact there are multiple 3-DOF parallel mechanisms that can be derived based on the general function set theory [ 33 ], but some of them are not useful at all. In reference [ 34 ], a hybrid robotic manipulator was proposed to be used as the neck of a mine rescue robot. This robot was inspired by a bio-structure, and the distinct attribute of that manipulator is that a passive leg which is in the form of prismatic-universal structure was put in the middle inside the system so that it can constrain the whole structure to be 3-DOF, i.

Inspired by the design in reference [ 34 ], the middle passive limb was switched from the original type to the prismatic-universal structure pattern, this passive limb equipped with a universal joint it is fixed at the moving platform center and a prismatic joint it is connected to the base , by this way the mechanism has the three desired DOF, i. If one wants to design manipulator systems with more than three degrees of freedom on the basis of the hybrid mechanism we propose, one can remove the middle passive leg, then the system will come to be a 6-DOF mechanism.

How the 6-DOF manipulator can be controlled will be addressed in future work. When mechanisms and parallel manipulators move, the position of the center of mass is changing and the angular momentum is also changing, therefore vibration is produced inside the system. The purpose of dynamic balancing is to make the CoM fixed and the angular momentum unchanging. Normally, the existing shaking force and shaking moment that the whole system produces can be dynamically balanced by adding supplementary components.

However, the potential issue that more weight and inertia will be included inside the system, which can make the system heavy and produce higher inertia effects, appears. Here it is proposed to achieve dynamic balance through a reconfiguration method, and the mass relation index is proposed as a basis to further derive new reactionless mechanisms. Furthermore, after a single dynamic balanced limb is created, those limbs can be assembled together to construct the entire parallel structure.

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Here the authors suggest that one can accomplish dynamic balancing conditions based on employing naturally dynamically balanced mechanisms rather than resorting to the old counterweights approaches. For instance, one can accomplish reactionless conditions based on the reconfiguration concept.

Here a new approach is put forward, i. For a SteadiCam system, as shown in Figure 2 , counter-weights are employed in order to make the system force balanced, and by altering the mass relations, make the system dynamically balanced. This is a mass relations concept. There are two linkages at the bottom of the system, and those two linkages can function as the counter-mass, so the system is force balanced.

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If one whirls the SteadiCam system, the system is dynamic balanced as well. If the link 2 is relocated somewhere towards the up direction, as illustrated in Figure 2 , the system is still force balanced, but the dynamic balance condition will be lost. Therefore the problem one faces is how to reposition or reconfigure the layout of the system in order to recover the reactionless condition. Let us consider the situation where one shifts link 2 to a different position, i.

It is clear that if the reactionless condition needs to be brought back, one has to shift the camera in a counter-clockwise direction, as shown in the Figure 2 , and the same goes for mass 1. Therefore one has the same scenario, the only difference is that there are two weights in the top instead of one and one weight in the bottom instead of two.

Put it in another way, if the linkage 2 is shifted counter-clockwise, one also has to shift the camera counter-clockwise, and the same goes for mass 1. In this way, reactionless conditions can be regained.


The whole process concerns the mass relations, so as long as one has the required mass relationships, one can have reactionless conditions. What dominates here is the correlation of those three masses. Inspired by the above design, here dynamic balancing based on the reconfiguration notion is put forward.

Through this approach, one does not employ a counter-mass but via reconfiguring the system by shifting the linkage, which does not make the system heavier. The following Figure 3 illustrates the notion of balancing based on reconfiguration. The origin of the coordinate frame x, y coincides with the revolute joint, the x axis horizontally points towards right, and the y axis vertically points up. The linear momentum of the linkage is therefore:.

For the aim of getting force balancing conditions, the linear momentum has to be set constant. By observing the above equation, and since the mass cannot be set to zero, the only way to make it a constant is to set d 1 to zero, which means the CoM of the linkage is set to the revolute joint:. Employing this concept and applying it to two-bar three-bar, four-bar and other mechanisms, new balanced two-bar three-bar, four-bar and other mechanisms can be derived. The objective of employing counter-weights is to shift the center of mass to a fixed spot, so the concern one faces is whether it is possible to not employ counter-weights to accomplish the same aim.

The link can be reconfigured so that center of mass is shifted to a fixed spot. Now for the two link scenario, we have the following result, as shown in Figure 4. For the aim of getting the force balancing conditions, the linear momentum has to be constant. From observation of the above equation, in order to satisfy the above condition, the following force balancing conditions are therefore obtained:.

Based on the previous analysis, one can see that force balancing based on the reconfiguration notion will not include an extra counter-mass, and on the other hand, if balancing is accomplished by resorting to adding an additional counter-mass, the entire system be heavier.

After applying this concept to the crank-slider mechanism, the crank-slider system which is balanced based on the reconfiguration notion is used as a Scott-Russell mechanism in place of the conventional version type [ 18 ] and we join the crank-slider system to every single limb of the 3-RPR parallel mechanism, as illustrated in Figure 5. In order to achieve complete dynamic balancing, three counter-rotations have been added within the system, as illustrated in Figure 5 , where CR stands for counter-rotation, the red solid circle represents a counter-mass, and the black solid one stands for the center of mass.

The moving platform mass can be replaced by three point masses placed at three attachment points of the moving platform and the three legs. The three point masses are represented by m pm 1 , m pm 2 , and m pm 3. If one satisfies the following, then the above condition can be obtained:. This replacement of the moving platform allows one to analyze the shaking force balancing and shaking moment balancing of each limb of the robotic system.

Through making the linear and angular momentum equal to 0, the shaking force and shaking moment are able to be balanced provided:. The mass and axial moment of inertia of link i are denoted as m li and I li , respectively. It can be seen that by employing the reactionless via reconfiguration to a crank-slider system to function as a Scott-Russell system, no extra counter-mass is included inside the system.

However, if one continues to use the conventional version type, two extra counter-masses must be included in the Scott-Russell system, which will increase the overall mass and inertia. By employing the reconfiguration approach as an alternative to adding counter-masses, it is also possible to make the 4-bar linkage equipped with an Assur group [ 13 ] achieve reactionless condition, and employ those reconfigured based reactionless 4-bar linkage equipped with an Assur group to synthesize the entire parallel robotic system, put it another way, one can decompose the system first, and after that integrate the system.

The above shows the concept of dynamic balancing based on the reconfiguration approach, as an alternative to adding counter-masses, the objective of which is to shift the center of mass, so one is able to resort to the reconfiguration technique to accomplish the same aim. Testing was conducted using Simulink and dSpace. Here, a 2-DOF link manipulator is set up and built as an illustration.

The robot is suspended by wires in the air so any unbalancing phenomena can be easily observed. The Simulink model is shown in Figure 6. For the unbalanced 2-DOF link manipulator, when the manipulator moves from one position to another, the system will swing and vibrate, which can be observed in real time, as illustrated in Figure 7. For the balanced 2-DOF link case, when the manipulator moves from one position to another, the system will remain steady, which can also be observed in real time, as shown in Figure 8.

Control of a serial mechanism can be categorized as joint control and end-effector operational control. Most robotic industries use a PID controller to control each joint of robotic manipulators.

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The problem is that it cannot make up for any load changes. Here a two-DOF link manipulator as shown in Figure 9 will be used as an example. After applying different payload masses, joint 1 motion output is illustrated in Figure 10 a while joint 2 motion output is shown in Figure 10 b. For joint 1, when the payload is 0, the motion is quite steady, but when the payload increases to 5 and 15, one can see that joint 1 motion is no longer the same, as shown in Figure 10 a, and the simulation also shows that the joint output increases and decreases.

The same applies to joint 2, as seen in Figure 10 b. This has led to the use of adaptive control, especially model reference adaptive control MRAC. Dubowsky [ 26 ] was the first to apply the MRAC concept to a robotic manipulator. The author employed a linear time-invariant differential formulation to be the reference model for each joint of the robot.

The robotic system was maneuvered through altering the feedback gains to follow the model. A steepest-descent approach was employed to update the feedback gains, after which Horowitz applied the hyper-stability method and developed an adaptive algorithm [ 34 ] for a serial fashion robot arm to which a nonlinear term in its dynamic equation was compensated and the dynamic interaction among the joints was decoupled.

In reference [ 27 ], Horowitz applied the Gibbs-Appell formulation for dynamic modelling of robotic manipulators to meet the requirement that inertia matrix and non-linear provision in the system dynamic equation are constant. An improved version of the method was later proposed in Sadegh [ 35 ]. The assumption that the inertia matrix and nonlinear provision are constant in the process of adaptation can be removed by altering the control rule and parameter adaptation rule.

It was proved that, through altering the control rule i.

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The error then is going to be processed inside the adaptation section and the result will be provided as one of the input parts to the plant. This error is processed by the integration action and the result of which will subtract the position and velocity that are processed by the Kp and Kd elements. This procedure is comparable to the PID control. The ideal system is independent from the plant, so that the plant feedback values are not being utilized to influence the reference model input.

The ideal system is not influenced with regards to the plant. In reference [ 27 ], the M and N matrices are treated as constant in the adaptation process. Under the PID control, we need to employ the dynamic model which is derived based on the Lagrange approach, however, under the MRAC, one needs to employ the dynamic model which is derived based on the Gibbs-Appell dynamic formulation. In reference [ 35 ], the author developed an improved MRAC system that is able to get rid of the situation where the M and N matrices being unchanging, in order to make the Lagrange based dynamic equation useable.

For the 2-DOF robotic manipulator, through employing the Lagrange technique, the dynamic formulation is illustrated as follows:. Through re-parametrization of the above equation,. As shown in the previous in Figure 10 , after applying different payload masses and moving the robot from one position to another, when the payload is 0, joint 1 motion is quite steady, but when the payload increases to 5 and 15, joint 1 motion is no longer the same, and the joint output also increases and decreases.