Workspace and Payload-Capacity of a New Reconfigurable Delta Parallel Robot

Abstract

In this paper the workspace and payload capacity of a new design of reconfigurable Delta-type parallel robot is analysed. The reconfiguration is achieved by adjusting the length of the kinematic chains of a given robot link simultaneously and symmetrically during the operation of the robot. This would produce a dynamic workspace in shape and volume. A numerical analysis of the variation of shape and volume of the workspace and payload capacity of the robot is presented. Based both on the results of this analysis and on practical requirements, a proposal for the design of a reconfiguring mechanism is presented.

Keywords

Workspace Payload Capacity Reconfigurable Delta Robot

1. Introduction

In recent years the number of studies and applications of parallel robots have increased. One of the most popular applications is in industry packaging. The above is due to their ease of construction, the lightness of their structure and the high accelerations obtained by these devices.

Unlike the serial-type robot manipulators, which only have an open-loop kinematic chain, parallel configuration allows for a distribution of payload among their two, or more closed-loop, kinematic chains. To illustrate this point consider Fig.1a, which shows a Parallix LKF-2040 parallel-architecture robot, used for object loading and unloading. Fig. 1b shows a SCARA-type serial-architecture robot. By comparing the images it is easy to appreciate the difference between the two types of architecture. In the case of the serial manipulator greater robustness is required, as each link carries not only the weight of the successive links but also the motors and payload. This creates a cantilever effect in each link and, as a result, a greater deformation overall. In contrast, in the parallel architecture the actuators are fixed to the base of the manipulator so that the weight of the motors is not supported by the kinematic chains. In addition, the payload is distributed among the kinematic chains that conform the manipulator. This results in thinner and lighter kinematic chains, which in turn results in an increased payload capacity of the manipulator, relative to its total mass.

Figure 1.

(a) Parallix LKF-2040 robot (b) A Scara-type robot

A disadvantage of parallel robots is their typically low cost effectiveness, based on complex kinematics and rather expensive control units, as well as the poor workspace to robot-dimension ratio [1]. On the other hand, the advantages of parallel robots stated before indicate that their capabilities can be optimally oriented if their specifications are task-adapted to the desired application. To facilitate flexibility and to enlarge the field of application, it is reasonable to use a reconfigurable robot design [2]. This will also help to overcome the typical challenges of parallel robots, such as high costs and undersized workspaces [3].

In order to improve the efficiency and flexibility of a robot system and to follow a demand-oriented strategy, two elementary reconfiguration concepts are possible; a static reconfiguration including a physical modification of the system and on the other hand, dynamic reconfiguration where no physical components are modified, [4]. The work in [6] studied the problem associated with the load-carrying capacity of mechanical manipulators for point-to-point motion and found that it can be formulated as a trajectory-optimization problem. The work presented in [7] evaluates the performance changes for a reconfigurable parallel robot, considering modularity properties of the mechanism. The paper in [8] describes a reconfigurable parallel robot composed of two tripods that modifies its number of DOF by detaching some links. While one of the most popular parallel robots is the Delta-type (due to its simplicity and the advantages described above), to the authors' knowledge, no reconfiguration for a Delta-type robot has been proposed nor studied.

The motivation for reconfiguring a Delta-type parallel robot is to create a more flexible device in terms of its ability to adapt to different workspace and payload capacity conditions. In this work a physical reconfiguration of a Delta-type, parallel robot is analysed, varying the length of some of their links, which results in a modified workspace and carrying capacity of the robot.

2. Delta-type parallel robot

The well-known Delta robot structure was proposed by R. Clavel in [9]. Fig. 2 shows the main components of this robot, which consists of three closed-loop kinematic chains. The fixed and the mobile platforms are marked with numbers 1 and 8, respectively. The robot has three degrees of freedom. The parallelograms (5) ensure the constant orientation between the fixed and the mobile platform, allowing only translation movements of the latter. The end-effector of the manipulator is located on the mobile platform.

Figure 2.

Clavel's Delta robot

For the work presented herein a Parallix LKF-2040 Delta-type, parallel robot, the same as the one depicted in Fig. 1a, is used as a reference (more information on this device can be found in [10]). In this device, the radius of the fixed platform is 150mm, the radius of the mobile platform is 50mm, the input link has a length of 200mm and the parallelogram has a length of 400mm.

3. Preliminary modelling

Fig. 3 shows the schematic top view of a Delta-type parallel robot. F₀:=(O₀, X₀, Y₀, Z₀) is defined as the inertial reference frame of the robot, as shown in the figure. αi are the angles associated with each robot leg relative to the X0 axis and have the following values:

α_{i} = [\begin{matrix} \frac{π}{6} & \frac{5 π}{6} & \frac{3 π}{2} \end{matrix}]

(1)

The subscript i=(1,2,3) refers to each of the three legs of the robot.

Figure 3.

Schematic top view of a Delta-type robot

In Fig. 4, a side and front view of a robot's leg are shown. In this figure, R_i=R is the radius of the fixed platform, r_i=r is the radius of the mobile platform, L1i=L1 is the length of the actuated link, L2i=L2 is the length of the parallelogram, θ_i1 is the actuated angle (associated to the active joint A_i), while θ_i2 and θ_i3 are the passive angles (associated to the passive joint B_i). Using the previous notation, it is clear from Fig. 4 that

R_{i} + L 1_{i} + L 2_{i} + r_{i} = p,

(2)

p - R_{i} - L 1_{i} - r_{i} = L 2_{i},

(3)

where p is the vector going from the centre of the fixed platform (O₀) to the centre of the mobile platform (point P), R_i is vector $\bar{O_{0} A_{i}}$ ,L1_i is vector $\bar{A_{i} B_{i}}$ , L2_i is vector $\bar{B_{i} C_{i}}$ and r_i is vector $\bar{C_{i} P}$ . For the joints' angles to become apparent, we can write (3) in frame F_Ai:=(A_i, X_i, Y_i, Z_i) i.e.

​_{Ai} T^{0} {(α_{i})}_{0} p -_{Ai} R_{i} -_{Ai} T^{Bi} {(θ_{i 1})}_{Bi} L 1_{i} -_{Ai} r_{i} =_{Ai} T^{Bi} {(θ_{i 2}, θ_{i 3})}_{Bi} L 2_{i}

(4)

Figure 4.

Side and front views of a Delta-robot leg

where kT^l is the rotation matrix allowing a change in the expression of a vector from frame l to frame k and jp is the expression of vector p in frame Fj.

In order to eliminate the passive angles (θ_i2, θ_i3) we can premultiply each side of (4) by its transpose, hence

​_{A i} v^{T} ​_{A i} v = {| | L 2_{i} | |}^{2}

(5)

where AiV=_AiT⁰(α_i)_0p-Ai R_i-Ai T_Bi(θ_i1)_BiL1_i-Airi.

Developing (5) in components and rearranging terms

{(x_{p} - x_{i})}^{2} + {(y_{p} - y_{i})}^{2} + {(z_{p} - z_{i})}^{2} = L 2^{2}

(6)

where x_p, y_p and z_p are the coordinates of vector 0p and:

\begin{matrix} x_{i} = (R + L 1 \cos θ_{i 1} - r) \cos α_{i} \\ y_{i} = (R + L 1 \cos θ_{i 1} - r) \sin α_{i} \\ z_{i} = - L 1 \sin θ_{i 1} \end{matrix}

Equation (6) corresponds to the three spheres of radius L2.

4. Robot modelling

In this section the well-known kinematic model of the Delta robot is briefly recalled.

4.1 Direct kinematic model

The direct kinematics of the Delta robot are defined by (6), where the unknowns are the position of point P= [x_p y_p z_p] for a given set of angles θ_i1, for i = 1, 2, 3.

The solution to this system of equations can be represented by a point defined by the intersection of the three spheres mentioned above. In general, there are two possible solutions, which means that for a set of actuated angles, the mobile platform may have two possible configurations with respect to the fixed platform [1].

4.2 Inverse kinematic model

The inverse kinematics are defined by solving equation (6), for the unknowns θ_i1 given a position P= [X_p Y_p Z_p]:

\tan \frac{θ_{i 1}}{2} = \frac{- 2 z_{p i} \pm \sqrt[2]{4 z_{p i} ​^{2} + 4 R_{1} ​^{2} - S_{i} ​^{2} + Q_{i} ​^{2} (1 - \frac{R_{1} ​^{2}}{L 1^{2}}) + Q_{i} (\frac{- 2 R_{1} S_{i}}{L 1} - 4 R_{1})}}{- 2 R_{1} - Q_{i} (\frac{R_{1}}{L 1} - 1) - S_{i}}

(7)

where,

\begin{matrix} R_{1} = R - r \\ Q_{i} = 2 X_{p i} \cos α_{i} + 2 Y_{p i} \sin α_{i} \\ S_{i} = \frac{1}{L 1} (- X_{p i} ​^{2} - Y_{p i} ​^{2} - Z_{p i} ​^{2} + L 2^{2} - L 1^{2} - R_{1} ​^{2}) \end{matrix}

solution exists if and only if:

4 Z_{p i} ​^{2} + 4 R_{1} ​^{2} - S_{i} ​^{2} + Q_{i} ​^{2} (1 - \frac{R_{1} ​^{2}}{L 1^{2}}) + Q_{i} (\frac{- 2 R_{1} S_{i}}{L 1} - 4 R_{1}) \geq 0

(8)

4.3 Workspace

The inverse kinematics of the Delta robot were used to calculate the shape and volume of the workspace. In order to do this, the proposed methodology consists of selecting uniformly distributed points in a volume large enough to ensure that the robot workspace is contained within that volume. If for every point in question there is a valid solution of the inverse kinematic model, this point is considered part of the robot workspace. See [1] for other workspace calculation possibilities.

Fig. 5 shows the workspace of the Parallix LKF-2040 Delta robot. This workspace is bounded by two hemispherical-like surfaces (one inside the other), where all the points located between these two hemispheres represent reachable positions for the end-effector of the robot.

Figure 5.

Workspace of the Parallix LKF-2040 robot

5. Reconfiguration

In this work, a dynamic geometric reconfiguration of a given Delta robot is studied. The proposed reconfiguration is performed by simultaneously and symmetrically adjusting the length of a given link in the three kinematic chains while the robot is operating. This would provide the robot with a varying workspace (in shape and volume), increasing its flexibility and enabling it to adapt to a wider range of applications. Even though non-symmetrical reconfigurations may seem more promising than symmetrical ones, they also require more actuators (symmetrical reconfigurations can be performed with only one actuator). The choice of the reconfiguration is made with the aim of keeping the reconfiguring mechanism as simple as possible, facilitating its practical implementation.

Fig. 6 shows three possible reconfigurations for the Delta robot that meet the desired initial conditions. The first sketch (left to right) shows an increase or decrease in the radius of the fixed platform (R). The central sketch shows a reconfiguration generated by modifying the length of the first link (L1). Finally, the right sketch shows the reconfiguration generated by modifying the length of the parallelogram (L2). The dimensions of the mobile platform were not considered for reconfiguration since this is equivalent to a reconfiguration of the fixed platform (as can be deduced from eq. (7) where R₁ = R-r is the sole term containing the dimensions of both platforms in the form of their algebraic difference).

Figure 6.

Reconfiguration options. Left, R reconfiguration; centre, L1 reconfiguration; right, L2 reconfiguration

It is important to mention that each reconfiguration will only vary the length of the respective link, maintaining the length of the remaining links fixed to its original value. Finally, the results obtained from the three reconfigurations will be compared, taking into account the change in the size and shape of the workspace, as well as the change achieved in the maximum displacement of the end-effector, along the X, Y and Z axes.

Code programmed in Matlab©performs the calculation of the inverse kinematics for each point contained within the initial volume. The coordinates of the points, for which there is valid solution to the inverse kinematics, are stored. Similarly, data about workspace volume and the maximum displacement along the X, Y and Z axes is stored. A second program uses the information generated for each reconfiguration, related to the coordinates of the points that make up the workspace, creating a cloud of points in three-dimensional space.

As a result, a series of images representing the change in shape of the workspace, with respect to the change in length of the corresponding link, is obtained. In order to have a picture clearly showing the boundaries of the workspace, a third program is used in order to eliminate all intermediate points between the boundaries.

6.1 Workspace variation

For the reconfiguration of R (i.e. the dimension of the fixed platform), its length was varied from 0 to 580mm, at intervals of 10mm, to obtain a total of 59 images. The change of the shape of the workspace according to variations in the dimension of the fixed platform is shown. Figs. 7a-7c show the resulting workspace for the reconfiguration of R, corresponding to values of 50, 250 and 450mm.

Figure 7.

(a) Workspace corresponding to R=50 mm (b) Workspace corresponding to R=250 mm (c) Workspace corresponding to R=450 mm

As can be seen, increasing the value of R, reduces the working space. The maximum workspace is obtained when R=50mm (R=r). Although the latter reconfiguration generates the maximum workspace, a gap in the centre is detected; this set of points is not part of the workspace.

However, in the reconfiguration corresponding to R=450mm, the gap disappears and the set of points that was previously unattainable now becomes part of the workspace. This provides evidence that actual limitations for a configuration of R can be overcome by reconfiguring the manipulator.

For the reconfiguration of L1 (or actuated link), its size was varied from 100 to 300 mm, at intervals of 10mm. With these parameters, a total of 21 images were obtained, which show the change in the shape of the workspace to variations in the length of link L1. Figs. 8a-8c show the reconfiguration of L1 to values of 100, 200 and 300 mm.

Figure 8.

(a) Workspace corresponding to L1=100 mm (b) Workspace corresponding to L1=200 mm (c) Workspace corresponding to L1=300 mm

Increasing the value of L1 also increases the working space. For the configuration corresponding to L1 = 100mm, a gap in the centre can be detected, which decreases as the value of L1 is increased. For a higher value of L1 there are fewer constraints in terms of points that can be accessed by the manipulator. However, it can be seen that the maximum displacement along the X and Y axes hardly changes since the growth of the workspace is downward (along the Z axis).

For the reconfiguration of L2 (or parallelogram) the length was varied from 200 to 800 mm, at intervals of 20mm to obtain a total of 31 images, which show the change in the shape of the workspace when varying the length of the link. Figs. 9a-9c show the workspace corresponding to the reconfiguration of L2 when the size is 400, 600 and 700 mm.

Figure 9.

(a) Workspace corresponding to L2=400 mm (b) Workspace corresponding to L2=600 mm (c) Workspace corresponding to L2=700 mm

As can be seen, increasing the value of L2 increases the workspace in nearly the same proportion. In the configuration of L2=400mm a gap can be detected, which increases as the value of L2 is increased.

Although there is an increase in both the workspace and the maximum reach along the X, Y and Z axes, there exists a set of points that the manipulator will never be able to access.

Figs. 10a-10d show the graphs relating to the change in volume of the working space, as well as those relating to the change in the maximum displacement along the X, Y and Z axes respectively. In them it can be seen that the reconfiguration of the L1 parameter produces a significant change in the volume of the workspace, although it does not produce any change in the maximum reach along the X and Y axes. The reconfiguration of the L2 parameter yields the largest change in the workspace volume and the maximum reach along the X, Y, Z axes; the shape of the workspace remains practically the same and the points near the centre of the fixed platform remain unreachable.

Figure 10.

(a) Variation in workspace volume (b) Variation in the maximum reach along the X-axis (c) Variation in the maximum reach along the Y-axis (d) Variation in the maximum reach along the Z-axis

The design of a reconfiguring mechanism that modifies the L1 and L2 parameter in a dynamic way, by means of just one actuator, seems rather difficult, or even practically impossible. On the other hand, the reconfiguration of the R parameter causes a significant change in the shape and volume of the workspace, as well as in the maximum reach along the X, Y, Z axes, with a bound value in all cases. The change in the shape of the workspace allows points near the reference frame origin to be reached. The design of a reconfiguring mechanism based on the variation of this parameter by means of just one actuator seems the simplest option in terms of practical feasibility, while having a reasonable variation of the considered indices. Based on the former analysis, the reconfiguration of R parameter is selected as the most viable option for the work presented herein.

7. Payload analysis

With the proposed reconfiguration, a variation in the robot's payload capacity can be anticipated (in general, different configurations of the robot result in different carrying capacities for the same point in the workspace). Therefore, it will become necessary to study the payload capacity of the robot while reconfiguring itself. This is done for the reconfiguration of the parameter R. Payload variation may be a useful property; for example it increases the flexibility of the robot allowing it to adapt to different weight-carrying needs. It also increases the number of trajectories available to perform a given task (in particular, it offers a lowest energy cost trajectory to perform the task).

7.1 Payload calculation

Since this manipulator is general purpose, there is no prescribed trajectory at this point for a given task and no dynamic carrying capacity can be determined. Thus, in this study, only the static carrying capacity of the robot is considered.

The carried weight is applied at point P (end-effector point) with direction –Z_i i.e.

{W = - W a}_{zi}

(9)

where W is the vector associated to the weight, W=‖ W| is the weight and a_z is a unit vector along the Z direction. Now, because of the structure of the robot (universal joints in points Bi and C_i), the forces applied through each robot's leg have the directions of vectors -L2_i. i.e.

F_{i} {=F}_{i} a_{i}

(10)

with a_i=-L2_i/L2 and F_i =‖ F_i ‖ (cf. Fig. 11). The static carrying capacity must satisfy the following conditions:

Σ \bar{F} = 0

(11)

Σ \bar{M} = 0

(12)

Eq. (11) is the sum of forces acting in a static structure and eq. (12) is the sum of torques about a point. By developing (11) and using (9–10) we can obtain F_1a1+ F_2a2 + F_3a3= −W_az which can be rewritten in matrix form as:

N F =- W

(13)

Figure 11.

Schematic diagram of the Delta robot.

where N=[agi] (g={x, y, z} and i={1, 2, 3}), F=[F₁ F₂ F₃]^T.

The solution of (13) is given by F= -N⁻¹W. This solution is in terms of the unknown weight parameter W. In order to determine this parameter, we use eq. (12) about each point Ai:

M_{i} + τ_{i} = 0

(14)

with M=r_i x F_i (where r_i is the vector $\bar{A_{i} C_{i}}$ ), the moment of force F_i and τ_i the motor torque. Expressing eq. (14) in terms of frame F_i, the corresponding second scalar equation is

(r_{z i} a_{x i} - r_{x i} a_{z i}) F_{i} (W_{i}) = τ

(15)

where τ_i is the magnitude of the motor torque. Letting τ_i be the maximum torque available, the maximum weight Wi that can be lifted (considering the capacity of each of the three motors) for position P is obtained. The least Wi is then the maximum weight that the robot can lift at point P.

The payload distribution in the workspace leads to the notion of payload space. In this space information on the payload capacity at each point of the workspace is incorporated. In order to obtain a representation of this space, a graphical representation is desirable. With this purpose, “slicing” the workspace into evenly spaced directions and then assigning a colour to the points of the resulting sections, according to their payload capacity is proposed. Given the symmetry of the robot architecture around the Z₀ axis, three cuts are proposed at 0°, 30° y 90°, as shown in Fig. 12, knowing that the same results apply to all three kinematic chains.

Figure 12.

Cuts of robot's workspace used to generate graphical representations of the payload space (0°, 30° and 90°).

7.2 Payload Variation

Figs. 13 –15 show the static payload capacity for the proposed reconfiguration (varying R parameter). Cuts in the workspace at 0° with respect to X₀ axis are shown for different values of R. Cuts at 30° and 90° basically show the same payload distribution and are not shown here. In these figures the horizontal axis represents the displacement of the robot along the cutting direction, while the vertical axis represents the displacement of the robot along the vertical axis (Z) (both in mm). To the right of the figure, a colour scale was added, it ranges from 0kg (dark blue) to 8kg (red). It is worth mentioning that, in general, the static payload capacity of the robot is different from its dynamic payload capacity. Fig. 16 illustrates the minimum and maximum static payload capacities for all of the robots' workspace for different values of parameter R.

Figure 13.

Payload space for R=50 mm, cuts at 0°

Figure 14.

Payload space for R=150 mm, cuts at 0°.

Figure 15.

Payload space for R=250 mm, cuts at 0°.

Figure 16.

Static payload capacity: red=maximum, blue=minimum.

From Figs. 13 –16, the following can be emphasized:

The maximum payload capacity is located at the bottom of the workspace, as the L1 and L2 links are aligned and the load does not produce momentum. On the other hand, the minimum payload capacity is obtained at the farthest points on the workspace where the L1 and L2 links achieve a near horizontal position and the motor torque is required to preserve the equilibrium position.

For all reconfigurations, the minimum carrying capacity tends to zero.

For a given reconfiguration, the area resulting from a cutting directions lightly varies with respect to other cutting directions.

For a given reconfiguration, intermediate load points corresponding to a cutting direction are not in the same position as for all other cutting directions.

As the value of R increases, the intermediate load points move away from the centre of the payload space.

7.3 Limited Payload Space

It is desirable that, for any reconfiguration, the manipulator can manipulate a maximum weight along the entire working space. A solution for this is to limit the working space so that the areas with zero payload capacity are restricted from the workspace.

In this work the workspace is bounded by a cylinder volume that is the standard working-space limitation for commercial manipulators of this kind, see Fig. 17.

Figure 17.

Limited payload space.

Using this limited workspace the same calculations as for the preceding subsection are performed to obtain the limited workspace payload capacity. The results of these are shown in Fig. 18.

Figure 18.

Limited payload space, cut at 0°, corresponding to (a) R=50 mm, (b) R=150 mm and (c) R=250 mm.

From Fig. 18 it can be seen that:

The maximum payload capacity is a function of the reconfiguration of R and is no longer located at the bottom of the workspace but at the top.

Given a point in the workspace, the payload capacity changes depending on the reconfiguration of R.

The minimum payload capacity (non-zero load) remains at the most distant points in the workspace.

For a given reconfiguration, distribution of the payload capacity for different cutting directions is alike.

As the value of R increases, the intermediate load points move away from the centre of the payload space.

Fig. 19 illustrates the minimum and maximum static payload capacities for the limited workspace for different values of parameter R For the proposed limited workspace the maximum payload ranges between 0.6 and 0.9 Kg, depending on the reconfiguration.

Figure 19.

Limited payload space carrying capacity: red=maximum, blue=minimum.

8. Reconfiguring mechanism

In Figs. 20 –22 a proposal for a reconfiguring mechanism for parameter R is illustrated. Fig. 20 shows a subassembly of the reconfiguring mechanism. It consists of a typical ball screw mechanism where the rotary motion is applied by means of a bevel gear. The linear bearing case is also the point of support for one leg of the robot. Fig. 21 shows a general view of the complete reconfiguring mechanism. The three ball screw subassemblies integrated with the delta robot are shown. The top servomotor is the actuator for the reconfiguring mechanism. It simultaneously transmits motion to the three ball-screw subassemblies by using a bevel gear as shown in Fig. 22. This design meets the initial requirements as it allows the dynamic change of parameter R for all kinematic chains in a simultaneous and symmetrical fashion.

Figure 20.

Subassembly of the reconfiguring mechanism.

Figure 21.

General view of the reconfiguring mechanism.

Figure 22.

Bottom view of the reconfiguring mechanism.

9. Other performance criteria

9.1 Stiffness

The stiffness of a parallel manipulator can be characterized by its stiffness matrix. This matrix relates the forces and torques applied at the gripper link in Cartesian space to the corresponding Cartesian displacements. The stiffness matrix K can be obtained from the kinematics equations and is given by

K = k J^{T} J

(16)

where k is a scalar representing the stiffness of each actuator (modelled as a linear spring), while J is the Jacobian matrix. The stiffness matrix is a positive semidefinite matrix; see [11][12][1][13].

9.2 Dynamics errors

For high-speed manipulation, errors caused by dynamic sources can strongly affect accuracy and repeatability.

Among such sources we can consider elastic deformations, natural vibrations and drive errors (see [1] and the references therein). In particular, regarding vibrations and their undesirable effects on accuracy, some efforts have been made to model these effects [14][15] and to minimize them via control [15][16].

9.3 Manipulability indices

Several authors [17][18][19] have proposed kinematic performance indices in order to assess the level of manipulability of the robot. For instance, the square Jacobian matrix J presented in [20] can be used to determine Yoshikawa's manipulability index), as follows,

j = | det J |

(17)

10. Conclusions and future work

In this work, a new design of a reconfigurable Delta-type parallel robot was presented. The reconfiguration is achieved by symmetrically adjusting the length of a given link of all kinematic chains using only one actuator. Reconfiguration of the links R, L1 and L2 were considered. The variation of the length of R causes a significant change in the shape and volume of the workspace and a simple design of a reconfiguring mechanism for this parameter is feasible. Thus, the reconfiguration of R is selected as the best choice for practical purposes.

The proposed reconfiguration allows for an adjustment in the shape and volume of the robot's workspace and also produces changes in its payload capacity. In particular, for a given point in the workspace, a robot can handle different loads. This characteristic is an open research issue and can be studied in several directions; for example, finding the minimum energy path for a given task. Research on this issue is considered as future work at our research group. The manufacture and control of the proposed reconfiguring mechanism is in progress.

Footnotes

11. Acknowledgements

This work was partially funded by the National Council for Science and Technology of Mexico (CONACYT) and by the Fund for Research Support (FAI) of UASLP.

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