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When a robot equipped with compliant joints driven by elastic actuators contacts an object and its joints are deformed, multi-modal information, including the magnitude and direction of the applied force and the deformation of the joint, is used to enhance the performance of the robot such as dexterous manipulation. In conventional approaches, some types of sensors used to obtain the multi-modal information are attached to the point of contact where the force is applied and at the joint. However, this approach is not sustainable for daily use in robots,

A compliant joint is an important mechanism that provides increased dexterity and dynamic behavior when compared with the performance of a joint driven by an electric servo motor with a higher reduction gear. A compliant joint can enable a robot hand to passively and adaptively grasp objects of many different shapes [

Some previous researches have adopted approaches in which an end effector force is estimated or calculated without force sensor [

The authors have adopted McKibben pneumatic actuators [

In this work, we employ a two-degrees-of-freedom (2-DoF) joint mechanism driven by four pneumatic actuators. For 2-DoF joint mechanism, the joints (at least one joint) are rotated passively even though the direction of the force is parallel to the top link. We derive the relationship between the contact information, including the magnitude of the force perpendicular and parallel to the link and the angles of the joint, and pressures of the four pneumatic actuators. In this paper, conditions relative to 2-DoF joint mechanism driven by pneumatic actuators, such as geometrical constraints, equilibrium condition of the applied force and the restoring force of the actuator, typical properties of a McKibben pneumatic actuator, and Boyle’s law are adopted. Subsequently, we explain how we utilize such conditions to derive the relationship between the contact information and the actuator pressures. For the evaluation of the derived equations, a physical 2-DoF joint mechanism is developed, and the accuracy of the derived relationship is verified by comparing the calculated and observed forces and angles. We also demonstrate an application in which various kinds of materials are categorized based on their stiffness upon bringing the materials in contact with the joint mechanism from arbitrary direction.

McKibben pneumatic actuator. (

There are some nonlinear models concerned with a relationship between the tension force and the extended length of the McKibben pneumatic actuator ([

As explained above, the McKibben pneumatic actuator is extended by the tension force. The actuator has an interesting characteristic that the inner pressure changes without more air being supplied to or removed from it when an external force is applied.

We adopt the 2-DoF joint model driven by four antagonistic actuators in our study. _{x} and F_{y} and the joint angles θ_{1}_{2}

2-DoF joint model before and after the force is applied. (

As shown in the left panel in _{ij0}_{1}_{2}_{3}_{si}_{1} + _{i}_{10} = _{si}_{1} + _{ni}_{1} + Δ_{i}_{10} = _{1}_{si}_{2} + _{i}_{20} = _{si}_{2} + _{ni}_{2} + Δ_{i}_{20} = _{1} + _{2}
_{sij}_{nij}_{ij0}_{sij}_{nij}_{ij0}_{1}_{1} + h_{2}_{ij}_{1} + Δ_{R}_{1})^{2} = (_{1} + _{1} sinθ_{1})^{2} + (_{1} − _{1} cosθ_{1})^{2}
_{1} + Δ_{L}_{1})^{2} = (_{1} − _{1} sinθ_{1})^{2} + (_{1} − _{1} cosθ_{1})^{2}
_{1} + Δ_{R}_{θ1})^{2} = (_{1} + _{2} sinθ_{1})^{2} + (_{2} − _{2} cosθ_{1})^{2}
_{1} + Δ_{L}_{θ1})^{2} = (_{1} − _{2} sinθ_{1})^{2} + (_{2} − _{2} cosθ_{1})^{2}
_{2} + (Δ_{R}_{2} − Δ_{R}_{θ1}))^{2} = (_{2} + _{2} sinθ_{2})^{2} + (_{2} − _{2} cosθ_{2})^{2}
_{2} + (Δ_{L}_{2} − Δ_{L}_{θ1}))^{2} = (_{2} − _{2} sinθ_{2})^{2} + (_{2} − _{2} cosθ_{2})^{2}
_{1}_{2}_{ij}_{L}_{θ}_{1}_{R}_{θ}_{1}_{1}

Natural and extended length of the actuator.

By subtracting Equation (3) from Equation (2), we obtain the joint angle θ_{1}_{1}^{2} + 2_{1}Δ_{R}_{1} + Δ_{1}^{2} − (_{1}^{2} − 2_{1}Δ_{L}_{1} + Δ_{L}_{1}^{2}) = 4_{1}_{1} sinθ_{1}

Because _{i1}^{2}_{i1}

By subtracting Equation (5) from Equation (4), θ_{1}_{R}_{θ}_{1}_{L}_{θ}_{1}

From Equations (9) and (10),
_{1}(Δ_{R}_{θ}_{1} + Δ_{L}_{θ}_{1}) = _{2}(Δ_{R}_{1} + Δ_{L}_{1})

By subtracting Equation (7) from Equation (6), the joint angle θ_{2}

considering that (_{ij}-Δl_{i}_{θ}_{j}^{2} is negligible compared with (Δ_{ij}-Δl_{i}_{θ}_{j}_{2}

From the principle of virtual work, the external force ^{−1})^{T} τ
_{x},F_{y}^{T}, and _{1},τ_{2}^{T} as shown in ^{T} _{2}_{1}−_{1}+_{2}_{1} + h_{2}_{1}+_{2}^{T}, the Jacobian matrix is given by:

As shown in _{1}, is driven by four actuators. The top joint, whose angle is denoted as θ_{2}, is driven by two actuators. In order to derive the joint torques _{1}_{2}_{L2}_{R2}_{1}_{L1}_{L2}_{1_L1} = _{L}_{1}_{1} cosθ_{1}
_{1_L2} = _{L}_{2}_{2} cosθ_{1} − _{L}_{2}_{2} cosθ_{2}
_{2_L1} = _{L}_{2}_{2} cosθ_{2}

Joint torques by the single forces _{L1}_{L2}

Note that the second term on right side of Equation (16) is a reaction force of τ_{2_L2}_{1_R1}_{1_R2}_{2_R2}_{R1}_{R2}_{R1}_{R2}_{L1}_{L2}_{1} =_{1_L1} +_{1_L2} +_{1_R1} +_{2_R2}_{2} =_{2_L2} +_{2_R2}

From Equation (1), the restoring forces of the actuators are expressed as:
_{ij}_{ij}_{ij}_{ij}_{ij}_{0})
_{ij}_{Rj}_{Rj}_{0}Δ_{Rj}_{0} = _{Lj}_{Lj}_{0}Δ_{Lj}_{0}

Using Equations (14), (18) and (20), the force is expressed as:

In order to derive the extended length _{ij}_{ij0}_{ij}_{ij}_{0}_{ij}_{0}_{ij}_{0} = _{ij}_{ij}_{ij}_{0} + Δ_{ij}

The cross-sectional areas of the actuators before and after the force is applied (_{ij0}_{ij}

As shown in _{ij}

By substituting Equations (24–27) into Equation (23), it is rewritten as:
_{ij}_{0}(_{ij}_{0}_{ij}^{2} − _{ij}_{0}^{3}) = _{ij}_{ij}_{0}_{ij}^{2} + _{ij}^{2}Δ_{ij}_{ij}_{0}^{3} + 3_{ij}_{0}^{2}Δ_{ij}_{j}_{0}Δ_{ij}^{2} + Δ_{ij}^{3}))

Because _{ij}^{2}_{ij}^{3}_{ij}_{ij}

Note that β_{ij}

Fiber of the nylon sleeve.

Using Equation (29), the joint angles θ_{1}_{2}_{x}_{y}_{1}

From Equations (13) and (29), the equilibrium angle θ_{2}

These equations indicate that the angles can be expressed in terms of the ratio of the pressures _{ij0}_{ij}_{θij}

Using Equations (21), (22) and (29), the force can be expressed by the pressures as:

Therefore, forces _{x}_{y}

and
_{F1} = _{1}_{L}_{1}(β_{1} − Δ_{L}_{10}), β_{F2} = −_{1}_{R}_{1}(β_{1} − Δ_{R}_{10}), β_{F3} = _{2}_{L}_{2}(β_{2} − Δ_{L}_{20})_{F4} = −_{2}_{R}_{2}(β_{2} − Δ_{R}_{20}), β_{F5} = _{2}_{L}_{2}(β_{2} − Δ_{L}_{20}), β_{F6} = −_{2}_{R}_{2}(β_{2} − Δ_{R}_{20})

These equations indicate that the magnitude of the applied force _{ij0}-P_{ij}_{F1}-_{F6}_{θ}_{ij}_{ij}

For the evaluation of the relationship between the contact information including the magnitudes of the force components _{x}_{y}_{1}_{2}_{1}_{2}_{1}_{2}_{3}

Developed 2-DoF joint mechanism and experimental setup. (

The forces and the angles can be calculated using Equations (30–33). However, as explained in the previous section, it is difficult to obtain the coefficients β_{θ}_{ij}_{F1–}_{F6}_{ij} in Equations (32) and (33) are different from each other as mentioned in

The procedure for obtaining the contact information and the corresponding pressure data is as follows:

Supplying compressed air. Compressed air is supplied to the actuators. The pressures _{ij0}

Applying an external force. The force

Measuring contact information and pressure; After a certain interval of time subsequent to applying a force, the joint mechanism adopts an equilibrium posture. The angles θ_{1}_{2}_{x}_{y}_{ij}

In order to obtain the coefficients, 90 sets of contact information including θ_{1},_{2}, F_{x}_{y}_{ij0}_{ij}_{1}_{2}_{x}_{y}_{θ1L1} and β_{θ1R1} for θ_{1}_{θ2L1}, β_{θ2R1}, β_{θ2L2}, and β_{θ2R2} for θ_{2}_{Fx1}-_{Fx6}_{x}_{Fy1}-_{Fy6}_{y}

After estimating the coefficients, another 18 sets of observed data are recorded to evaluate the accuracy of the obtained multiple regression equations.

By using the method of least squares, the coefficients are estimated as: β_{θ1L1} _{θ1R1} _{θ2L1} _{θ2R1} _{θ2L2} _{θ2R2} _{Fx1} =_{Fx2} =_{Fx3} =_{Fx4} =_{Fx5} =_{Fx6} =_{Fy1} =_{Fy2} =_{Fy3} =_{Fy4} =_{Fy5} =_{Fy6} =

Accuracy of contact information. (_{1}; (_{2}; (_{x}; (_{y}.

As an application to demonstrate the accuracy of the estimated contact information, this paper shows that the compliance of some types of materials can be categorized by pushing the object as shown in _{1}(_{2}+h_{3}^{T} , where _{1} to O_{2}(_{o}_{o}^{T}, and the direction of the force owing to the spring corresponds with the vector from O_{2} to the contact point C(_{c}_{c}^{T}. Consequently, the stiffness of the object _{o}_{x2}+F_{y2}^{−1/2} , and _{2} can be expressed as:

Position of the object and the joint mechanism before/after the object is pushed. (

The coordinates of the contact point C are given by

Note that the coordinates of C can be estimated using pressures _{L2}_{R2}_{L1},_{R1}

Let the vector _{1}+_{2}_{1}+_{2}^{T}, and therefore, the angle θ_{F}

The angle θ_{F}_{x}_{y}

From Equations (40) and (41), the direction _{2} can be calculated using Equation (40). Therefore, the distance of deformation of the object

By using Equations (38) and (42), and the estimated force _{2}, be known. The stiffness is subsequently calculated from Equations (38) and (42). From the inner pressures of the actuator before and after the object has been pushed, the stiffness of the object is obtained. Each of the selected objects is pushed, the stiffness in each case is calculated several times, and the variation in the calculated stiffness is observed.

Acquired softness of the sample objects.

In this study, in order to acquire varied contact information by using a single type of sensor for a durable joint mechanism used in robots, we focused on the passivity of a pneumatic actuator wherein the pressure of the actuator changes passively owing to the applied force. We employed a 2-DoF joint mechanism driven by McKibben pneumatic actuators, and we aimed at obtaining contact information including the magnitude and direction of the applied force and the joint angles by using the pressures in the actuators. Because the sensors used to measure the pressures need not to be set at the contact point and the joint, it is possible to obtain sensor durability, thereby avoiding sensor damage due to excessive magnitude of force or iterative contact, which is the case in conventional joint mechanisms. Because only one type of sensor as pressure sensor is required to obtain the contact information, such multi-modal information can be obtained by a simple device system. In order to obtain the relationship between the contact information and the pressures, we constructed a 2-DoF joint model and imposed certain conditions. Using the model and conditions, we found that the joint angles can be expressed via multiple regression equations involving the ratio of the pressures before and after the force is applied. We also found that the applied forces can be expressed via multiple regression equations involving the product of the triangular function derived from the Jacobian matrix, which was determined from the difference between the directions of the force and each moment arm, and difference in the pressure before and after the force is applied.

In order to evaluate the regression models, we performed experiments using a physical 2-DoF joint mechanism. Before conducing the experiments, we estimated the coefficients in the regression equations mentioned above by using the least squares method, because it is very difficult to obtain certain physical values represented by the coefficients. By using the estimated coefficients, we found that the angles of two joints and the forces perpendicular and parallel to the link could be calculated accurately. In terms of application, three types of materials were categorized by pushing objects made of these materials with the 2-DoF joint mechanism through a certain distance in an arbitrary direction.

There are some applications on this joint mechanism. One of the remarkable characteristics of the McKibben pneumatic actuator is that the viscoelasticity can be changed by tuning the inner pressure. The joint mechanism in this paper is driven by antagonistic pneumatic actuators, and then the stiffness of the joint can be changed by tuning the inner pressure of the actuators. For example, in [

Takashi Takuma wrote this paper and gave technical advices for the experiments. Ken Takamine mainly worked on experiments and constructed the basic model of the 2-DoF joint mechanism. Tatsuya Masuda supervised succession of research process.

The authors declare no conflicts of interest.