Object transportation using multiple mobile robots: Coupling and decoupling configuration of an object and robots

1. Introduction

In ports, warehouses, factories, and restaurants, mobile robots carrying goods or food are common. However, most of these robots are designed to handle objects with limited size, shape, and weight. Furthermore, the process of loading goods or food onto the robots requires human intervention. The common action of moving refrigerators or furniture in our daily lives is a pushing or pulling motion, and through this motion, we can move large and heavy objects with relatively small force without additional measures for transportation. This study deals with object transportation using multiple mobile robots, which makes the transport of objects in different sizes, shapes, and weights possible through pushing and pulling motions.

Many studies have been conducted in the fields of modern logistics and automated transportation on multirobot cooperation [1]–[15]. Cooperative operation by multiple robots is important when handling bulky and heavy objects because these tasks may be too arduous or dangerous for individual robots or humans. Techniques, such as pushing, grasping, and caging, have proven to be reliable methods for multirobot cooperation to address such challenges [1][2].

The movement of target objects by multiple robots directly applying pushing forces has been demonstrated [3]-[6]. The effectiveness of this method lies in the distribution of the required forces, thereby reducing the burden on individual to manipulate large objects. Grasping technology relies on the ability of robots to grasp an object securely and then move it collaboratively. This method is suitable for objects that are easily held and require precise handling. It has been shown that the grasping approach ensures operational stability and efficiency, and prevents the object from falling or overloading. The caging method uses multiple robots to form an enclosed structure and transport objects. This approach is particularly suitable for objects with irregular shapes or the ones, which are difficult to directly grasp [10]-[14].

Implementing robotic transportation in the real world provides many advantages; however, there are challenges, including precise control, environmental adaptability, inter-robot communication, and decision-making [15]. The size, shape, and weight of objects, along with environmental factors, such as terrain and space constraints, also influence transportation approaches.

This study considers the multi-robot cooperative object transportation, as shown in Figure 1, to handle objects of various sizes, shapes, and weights. Unlike pushing, grasping, and caging methods, robots use a mechanism to couple with an object, which makes transportation control easier. The mechanism is operated by applying pushing and pulling forces without an actuator, which become a free joint after coupling. The transportation operation is conducted as follows. The mobile robots first couple with the object using pushing motions. Then, they transport the object to the goal position, and finally decouple from the object using pulling motions after arriving at the destination.

Figure 1:

Procedure of object transportation using multiple mobile robots (in case of using three robots)

In this study, we focus on the coupling and decoupling operations and discuss the configurations of an object and robots for coupling and decoupling. Sections 2 and 3 present the formulation of the kinematic model and static equilibrium equation, respectively. In Section 4, it is shown that there exist configurations, in which the robots can push and pull one another for coupling and decoupling without generating a moment on the object. These configurations are singular configurations and are usually regarded as undesirable in the motion control aspect. Numerical examples are presented and discussed in Section 5 to verify the coupling and decoupling configurations.

2. Kinematic Model

Assume that the mechanism shown in Figure 2 is used for coupling and decoupling. Coupling is achieved by pushing when the gates are in contact with the axis attached to the object, and decoupling is achieved by the reverse action.

Figure 2:

Coupling mechanism

Consider an object in contact with n mobile robots, as shown in Figure 3. Let v_o ∈ ℜ² and ω_o ∈ ℜ be the linear and angular velocities of the object, respectively, r_i ∈ ℜ² (i = 1, 2, ⋯ , n) be the position vector pointing from the center of mass of the object to the contact point with the robot i, and v_ci ∈ ℜ² be the velocity of the contact point. Because v_ci = v_o + Er_iω_o, we get

Figure 3:

Velocity relationship when n number of robots are contact with object (coupling mechanism of each robot is in contact with the axis at the object)

where, $$ {\mathit{v}}_c=\left(\begin{array}{c}{\mathit{v}}_{c1}\\ ⋮\\ {\mathit{v}}_{cn}\end{array}\right)\in {\mathfrak{R}}^{2n}, A_O=\left(\begin{array}{cc}{I}_2& E{\mathit{r}}_1\\ ⋮& ⋮\\ I_2& E{\mathit{r}}_n\end{array}\right)\in {\mathfrak{R}}^{2n\times 3},{\mathit{w}}_O=\left(\begin{array}{c}{\mathit{v}}_O\\ {\omega }_O\end{array}\right)\in {\mathfrak{R}}^3$$I₂, is a 2 × 2 identity matrix, and E is the orthogonal rotation matrix rotating an arbitrary vector counterclockwise by 90^o in a plane and is expressed as:

The contact point velocities can also be obtained from the linear and angular velocities of each robot. Let v_Ri ∈ ℜ² and ω_Ri ∈ ℜ be the linear and angular velocities of the robot i, respectively (refer to Appendix A.1 for obtaining v_Ri and ω_Ri from the angular velocities of the wheels) and l_ci ∈ ℜ² is the position vector pointing from the center of mass of the robot to the end of the coupling mechanism, i.e., the contact point. From Figure 4 and v_ci = v_Ri + El_ciω_Ri, we get

Figure 4:

Velocity relationship between velocity of contact point and linear and angular velocities of robot

where, $$ {\mathit{w}}_R=\left(\begin{array}{c}{\mathit{v}}_{R1}\\ {\omega }_{R1}\\ ⋮\\ {\mathit{v}}_{Rn}\\ {\omega }_{Rn}\end{array}\right)\in {\mathfrak{R}}^{3n},$$

O ₂ is a 2 × 2 zero matrix, and 0 is a 2 × 1 zero vector.

From Equations (1) and (3), we can obtain the velocity constraints as:

Here, to consider only the pushing and pulling motions for coupling and decoupling, the angular velocities ω_Ri are moved to the left side of Equation (4), and the following kinematic model can be obtained.

where, $$ {\mathit{v}}_R=\left(\begin{array}{c}{\mathit{v}}_{R1}\\ ⋮\\ {\mathit{v}}_{Rn}\end{array}\right)\in {\mathfrak{R}}^{2n},{\stackrel{-}{\mathit{v}}}_O=\left(\begin{array}{c}{\mathit{v}}_O\\ {\omega }_O\\ {\omega }_{R1}\\ ⋮\\ {\omega }_{Rn}\end{array}\right)\in {\mathfrak{R}}^{3+n}$$, and $$ A=\left(\begin{array}{cccccc}{I}_2& E{\mathit{r}}_1& -E{\mathit{l}}_{c1}& 0& \cdots & 0\\ I_2& E{\mathit{r}}_2& 0& -E{\mathit{l}}_{c2}& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ I_2& E{\mathit{r}}_n& 0& 0& \cdots & -E{\mathit{l}}_{cn}\end{array}\right)\in {\mathfrak{R}}^{2n\times (3+n)}$$.

3. Static Equilibrium Equation

Let f_o ∈ ℜ² and m_o ∈ ℜ be the force and moment generated on the object, respectively, f_Ri ∈ ℜ² and m_Ri ∈ ℜ be the force and moment of the robot i, respectively (refer to Appendix A.2 for obtaining f_Ri and m_Ri from the torques of the motors attached at the wheels), and f_ci ∈ ℜ² (i = 1, 2, ⋯ , n) be the force generated at the contact point. From Figure 5, we obtain the following static equilibrium equation for the object as:

Figure 5:

Force and moment relationship (when n number of robots are in contact with the object)

where, $$ {\mathit{f}}_c=\left(\begin{array}{c}{\mathit{f}}_{c1}\\ ⋮\\ {\mathit{f}}_{cn}\end{array}\right)\in {\mathfrak{R}}^{2n}$$ and $$ {\mathit{\tau }}_O=\left(\begin{array}{c}{\mathit{f}}_O\\ m_O\end{array}\right)\in {\mathfrak{R}}^3$$. Here, the contact force f_ci can be exerted by robot i and we obtain:

This relation can be expressed for all the robots as:

where, $$ {\mathit{\tau }}_R=\left(\begin{array}{c}{\mathit{f}}_{R1}\\ m_{R1}\\ ⋮\\ {\mathit{f}}_{Rn}\\ m_{Rn}\end{array}\right)\in {\mathfrak{R}}^{3n}$$.

Combining the τ_o in Equation (6) and equations related with m_Ri in Equation (8), the static equilibrium condition for pushing and pulling can be obtained as:

where, $$ {\stackrel{-}{\mathit{\tau }}}_O=\left(\begin{array}{c}{\mathit{f}}_O\\ m_O\\ m_{R1}\\ ⋮\\ m_{Rn}\end{array}\right)\in {\mathfrak{R}}^{3+n}$$.

4. Coupling and Decoupling Configuration

As shown in Figures 1 and 2, coupling and decoupling are performed using pushing and pulling motions. Thus, such actions must be achieved without generating a moment on the object; otherwise, the motion of the object becomes unstable, and the robots can detach from the object.

From Equation (5), when rank(A) < min(2n, 3 + n), there exists $$ {\overline{v}}_o\ne 0$$ (output) even if v_R = 0 (input). This is a singular configuration. On the contrary, from Equation (9), if dimKer(A^T) ≠ 0, we have

This indicates that there exists a certain contact force f_c that maintains static equilibrium without exerting force and moment on the object (τ_o = 0) and moments on the robots (m_Ri =0). Thus, it can be concluded that a configuration satisfying the following conditions is appropriate for pushing and pulling one another for coupling and decoupling.

In this case, the contact force f_c can be obtained as:

where, h is the basis of Ker(A^T) and γ is a real number. In addition, the force and moment exerted by each robot to accomplish coupling and decoupling can be obtained using Equation (8).

5. Numerical Examples

5.1 Using Two Robots

Consider Cases (a) and (b), as shown in Figure 6. Table 1 presents the values of r_i and l_ci. From Equation (5), if two robots are used, i.e., n = 2, we have:

Figure 6:

Examples of coupling and decoupling configuration using two robots

Table 1:

Parameter values of configurations (Figure 6)

$\[ A=\left(\begin{array}{cccc}{I}_2& E{\mathit{r}}_1& -E{\mathit{l}}_{c1}& \mathbf{0}\\ I_2& E{\mathit{r}}_2& \mathbf{0}& -E{\mathit{l}}_{c2}\end{array}\right)\in {\mathfrak{R}}^{4\times 5} \]$

(13)

First, in the case of Figure 6(a), rank(A) = 3 < min(4, 5) and dimKer(A^T) = 1, thus, this configuration is a singular configuration, and certain contact forces exist to push and pull each other while maintaining static equilibrium. Equation (12) gives f_c = (-0.7071, 0, 0.7071, 0)^Tγ; the robots can pull each other for decoupling if γ is a positive real number and push for coupling if γ is a negative. Equation (8) yields the forces and moments of the robots to exert f_c : τ_R = (-0.7071, 0, 0, 0.7071, 0, 0)^Tγ , where, γ is positive for decoupling (pulling) and negative for coupling (pushing).

Next, in the case of Figure 6(b), rank(A) = 3 < min(4, 5), dimKer(A^T) = 1 , and f_c = (–0.7071, 0, 0.7071, 0)^Tγ . It is known that this configuration is singular and appropriate for coupling and decoupling. For this case, the force and moment of the robots are τ_R = (-0.7071, 0, 0, 0.7071, 0, 0)^Tγ. However, if r₂ is changed to (0.4, 0)^T as shown in Figure 7, rank(A) = 4 = min(4, 5) and dimKer(A^T) = 0. Hence, the configuration is not singular, and there are no contact forces for static equilibrium. Figure 7 shows that the contact forces exert a moment on the object in this configuration.

Figure 7:

Example of coupling or decoupling operation (normal configuration, which does not satisfy the condition in Equation (11); the robots cannot push or pull each other because the contact forces exert a moment on the object)

5.2 Using Three Robots

The four configurations and parameter values are given in Figure 8 and Table 2, respectively. If three robots are used (n = 3), we obtain:

Figure 8:

Examples of coupling and decoupling configuration using three robots

Table 2:

Parameter values of configurations (Figure 8)

$\[ A=\left(\begin{array}{ccccc}{I}_2& E{\mathit{r}}_1& -E{\mathit{l}}_{c1}& \mathbf{0}& \mathbf{0}\\ I_2& E{\mathit{r}}_2& \mathbf{0}& -E{\mathit{l}}_{c2}& \mathbf{0}\\ I_2& E{\mathit{r}}_3& \mathbf{0}& \mathbf{0}& -E{\mathit{l}}_{c3}\end{array}\right)\in {\mathfrak{R}}^{6\times 6} \]$

(14)

Checking the case of Figure 8(a), rank(A) = 5 < min (6, 6), dimKer(A^T) = 1 , f_c = (–0.5, –0.2887, 0.5, –0.2887, 0, 0.5774)^Tγ , and τ_R = (–0.5, –0.2887, 0, 0.5, –0.2887, 0, 0, 0.5774, 0)^Tγ, where, γ is positive for decoupling and negative for coupling. The configuration is singular, and robots can push and pull one another for coupling and decoupling. In this case, the contact forces meet at the center of mass of the object. Unlike this, Figure 8(b) is the case when the contact forces meet at another place. This configuration is proper for coupling and decoupling; we can confirm this by rank(A) = 5 < min (6, 6) , dimKer(A^T) = 1, f_c = (–0.3536, –0.3536, 0.3536, –0.3536, 0, 0.7071)^Tγ, and τ_R = (–0.3536, –0.3536, 0, 0.3536, –0.3536, 0, 0, 0.7071, 0)^Tγ.

In case of Figure 8(c), rank(A) = 5 < min (6, 6) , dimKer(A^T) = 1 , f_c = (0.7071, 0, –0.7071, 0, 0, 0)^Tγ , and τ_R = (0.7071, 0, 0, –0.7071, 0, 0, 0, 0, 0)^Tγ, where, γ is positive for coupling and negative for decoupling. This configuration has a local singular configuration and is appropriate for the coupling and decoupling of Robots 1 and 2. However, Robot 3 cannot exert a contact force for coupling and decoupling.

In addition, if a robot must be decoupled from the object, the configuration shown in Figure 8(d) can be used. In this case, rank(A) = 5 < min (6, 6) , dimKer(A^T) = 1 , f_c = (0.3536, –0.3536, –0.3536, –0.3536, 0, 0.7071)^Tγ , and τ_R = (0.3536, –0.3536, 0, –0.3536, –0.3536, 0, 0, 0.7071, 0)^Tγ . When γ is a positive, Robot 3 can decouple from the object by pulling motion, while the other robots push the object.

6. Conclusion

In this study, we discussed the coupling and decoupling configuration of an object and robots for cooperative object transport. The following conclusion were drawn:

• (1) When two or three robots are used, a singular configuration is appropriate for coupling and decoupling, because forces exist that push and pull one another without generating a moment on the object.
• (2) When more than three robots are used, coupling and decoupling operations can be achieved regardless of the formation of a singular configuration.
• (3) A specific robot or some robots out of all the robots can couple/decouple with/from the object using at a local singular configuration.

Our future work will involve conducting experimental verification and developing a path-following control method for multirobot cooperative object transportation.

Author Contributions

Conceptualization, J. -K. Choi; Methodology, J. -K. Choi; Software, J. -K. Choi and H. Tian; Formal Analysis, J. -K. Choi, H. Tian and Y. -S. Ha; Investigation, J. -K. Choi and H. Tian; Resources, J. -K. Choi and H. Tian; Data Curation J. -K. Choi and H. Tian; Writing-Original Draft Preparation, J. -K. Choi and H. Tian; Writing-Review & Editing, J. -K. Choi, H. Tian and Y. -S. Ha; Visualization, J. -K. Choi and H. Tian; Supervision, J. -K. Choi.

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