Posture Maintenance Control of 2-Link Object By Nonprehensile Two-Cooperative-Arm Robot Without Compensating Friction

I. Introduction

IN recent years, shortage of the caregivers is becoming a serious social problem as the result of a falling birth rate and an aging population. At the caregiving workplace, transferring a care-receiver from a bed to a wheelchair frequently is quite heavy burden for caregivers. In order to relieve the burden on them, a lot of nursing care assistant equipment has been developed by different research groups [1]–[3], such as Resyone wheelchair-bed developed by Panasonic AGE-FREE Co., Ltd. and ROBOHELPER SASUKE developed by Muscle, Inc. etc. [4]. RIKEN-SRK Collaboration Center for Human-Interactive Robot Research has developed a high-functionality nursing care assistant robot ROBEAR [5] (see Fig. 1), which can realize soft movement of two 6 DoF arms to assist a care-receiver to stand up or to transfer a care-receiver from a bed.

Figure  1.  ROBEAR [5]

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From the perspective of the actual application at the caregiving workplace, safety is the most important thing. For guaranteeing it, one caregiver must stand beside the caregiving device. Also, auxiliary belt or handrail which is used to avoid the care-receiver falling from the side is usually employed simultaneously. However, auxiliary parts are rarely used for fall protection in vertical direction because of their installation inconvenience in service. In this research, the movement of robot arm in 2-dimensional space is considered to maintain the care-receiver’s posture and avoid his/her falling in vertical direction. So a nonprehensile two-cooperative-arm robot will be applied to manipulate a 2-link object with one passive joint which is regarded as a care-receiver in a 2-dimensional space. Many research works on different types of nonprehensile manipulation can be found in [6]. They have applied nonprehensile manipulation to realize many functions, such as throwing [7], [8], catching [9], batting [10], sliding [11], [12] and so on. An application on holding N-link object by using nonprehensile two-cooperative-arm robot has been considered in [13], [14]. In [13], a dynamic model for manipulating a two-rigid-link object by applying two cooperative arms is built. And holding and lifting-up motion without considering friction is realized. However, in a real-world application, the friction cannot be ignored and it will affect the arms’ performance of holding and lifting-up motion. So the general way to solve this problem is to design a friction compensator to cancel the effect of the friction. Many researchers tried to build a friction model which is near to the real friction and designed a compensator according to the friction model [15], [16]. However, the effect of this method is very dependent on the precision of the friction model. In this research, instead of compensating the friction, static friction is utilized reasonably during holding and lifting up the 2-link object.

On the other hand, considering that fast variance of movement of robot arms may lead care-receivers to feel afraid, uncomfortable, and even pained during holding and lifting up, an optimal regulator which can be modified according to care-receivers’ feeling is designed [17]. In this research, care-receivers’ feeling is also considered. In details, for reducing discomfort and pain caused by relative motion between robot arms and care-receivers, the contact points will be kept constant during holding and lifting up. To realize this purpose, a feedback control system is designed to keep the 2-link object regarded as a care-receiver not dropping down from the arms with static friction. Satisfying the stable condition under the static friction, the region of two orientation angles of the 2-link object can be found. And the most robust pair of the angles (hardest to drop down) among the obtained region can be picked up. Then the relationship between the robust pair of the angles and contact places on the arms can be constructed. According to the relationship, a controller is designed to maintain the posture of the object by moving two robot arms. By changing the target position of the 2-link object, two arms can be controlled to hold or lift up the object to the target position with the proposed method. Finally, in order to verify the effectiveness of the proposed method, experimental results on holding and lifting up the 2-link object with nonprehensile two-cooperative-arm robot are shown.

The rest of this paper is organized as follows. In Section II, model of 2-link object is built. Then, motion controller is designed to maintain the posture of 2-link object during holding and lifting-up motion without compensating friction in Section III. Experimental results are shown and analyzed in Section IV. Section V is conclusion of this paper.

II. Model of $2 $-Link Object

In this research, a 2-link object with one passive joint is regarded as a care-receiver (see Fig. 2). The left link of the 2-link object is called Link1, the right one is called Link2. All the symbols of the system are shown in Fig. 2. $ (x,y) $ is the position of the passive joint. The marks “$ \oplus $” located at $ L_1 $ and $ L_2 $ from the passive joint show the centers of mass of two links. $ m_1 $, $ d_1 $, $ \theta_1 $ and $ m_2 $, $ d_2 $, $ \theta_2 $ are the masses, radii and orientation angles of two links, respectively. The positions of two nonprehensile arms are denoted by $ (x_{A1},y_{A1}) $ and $ (x_{A2},y_{A2}) $, respectively. $ r_1 $ and $ r_2 $ are the radii of the arms. $ l_1 $ and $ l_2 $ are the distances from the contact points to the passive joint. $ F_{N1} $, $ F_{N2} $ and $ F_{F1} $, $ F_{F2} $ denote the normal forces and friction forces which act at the contact points.

Figure  2.  Schematic diagram of the system

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For keeping the 2-link object stable on the nonprehensile arms, the dynamics model of the object can be described by applying Newton-Euler equations of motion [13], [17]. According to the actual application at the caregiving workplace, the arms could not move fast. Otherwise, the care-receiver would feel afraid, uncomfortable even painful. Considering this situation, a static model of the 2-link object is built as follows.

where

and $ s_i = \sin\theta_i $, $ c_i = \cos\theta_i(i = 1,2)$.

In order to reduce the discomfort and pain caused by relative motion between nonprehensile arms and the care-receiver, the contact points would be kept constant during holding and lifting up. So the distances $ l_1 $ and $ l_2 $ would be constant, and the friction forces $ F_{F1} $ and $ F_{F2} $ would be static friction forces. According to the geometric relationship in Fig. 2, we can obtain that

where

$ s_{21} = \sin (\theta_2-\theta_1) $, $ c_{21} = \cos (\theta_2-\theta_1) $. If (1) and (2) are satisfied, we can decide the desired positions of the nonprehensile arms with (3), where the arms are able to keep the 2-link object stable on them. Since (1) and (2) are related to the orientation angles $ \theta_1 $ and $ \theta_2 $, it is to find the relationship between $ \theta_i $ and $ l_i $ $ (i = 1,2) $ in next section such that (1) and (2) hold.

III. Holding and Lifting-Up Motion Control

As mentioned above, in order to make the care-receivers comfortable, it is better to fix the contact points during holding and lifting up. That is, $ l_1 $ and $ l_2 $ should be kept constant, and friction forces should be kept as static ones and satisfy (4) and (5).

where $ \mu_s $ is coefficient of maximum static friction. In practice, since care-receivers wear same kind of uniforms, $ \mu_s $ can be regarded same and can be measured in advance. In [13], the following solvability condition of the system defined by (1) and (2) is obtained as the function of $ (l_1,l_2) $ when $ F_{F_1} = F_{F_2} = 0 $.

1) $ a<1 $ and $ b<1 $

2) $ a>1 $ and $ b>1 $ and $ 1<a+b<2 $

where

According to the above solvability condition, we can obtain the solvable and unsolvable area for $ l_i\in [0,0.8] $ (m) (see Fig. 3). In Fig. 3, the solvable areas are enclosed by the red dashed curve and lines. Considering that it is uncomfortable when two robot arms are too close to the waist (passive joint), we assume that the desired distances $ (l_1, l_2) $ from the passive joint to the contact points where the care-receivers feel comfortable can be decided in the area which is enclosed by red rectangle shown in Fig. 3. That is, the distances $ l_i\in [0.22,0.37] $ $ (i = 1,2) $ are considered in this research. Then from (1) and (2), it is known that $ F_{Ni} $ and $ F_{Fi} $ ($ i = 1,2 $) are related to $ \theta_i $ ($ i = 1,2 $) and there are many pairs of $ \theta_i $ can make $ F_{F_i} $ belong to the range $ [-\mu_s F_{N_i},\mu_s F_{N_i}] $. In order to make the movement of arms robust, we try to find a pair of $ \theta_i $ among them which makes the stable region of the care-receiver (2-link object) maximum. This problem is equivalent to the following optimization problem

Figure  3.  Solvable and unsolvable area

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such that

where, $ \Delta_{11} = \theta_1-\underline{\theta_1} $, $ \Delta_{12} = \overline{\theta_1}-\theta_1 $, $ \Delta_{21} = \theta_2-\underline{\theta_2} $, $ \Delta_{22} = \overline{\theta_2}-\theta_2 $.

As shown in Fig. 4, $ \left[\underline{\theta_i},\overline{\theta_i}\right](i = 1,2) $ is the range of $ \theta_i $ which can make $ F_{F_i} $ belong to the range $ [-\mu_s F_{N_i},\mu_s F_{N_i}] $. The solution $ (\theta_{opt1},\theta_{opt2}) $ of the above optimization problem is called robust pair of orientation angles of 2-link object when $ l_1 = l_2 = 0.28 $ (m). If the distances $ (l_1,l_2) $ vary, the stable region varies with their variation (see Fig. 5). So sweeping the distances which satisfy the above solvability condition can obtain the optimal result $ (\theta_{\rm opt1},\theta_{\rm opt2}) $ for each stable region.

Figure  4.  Stable region ($ l_1 = l_2 = 0.28 $ (m))

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Figure  5.  Stable region (Left: $ l_1 = l_2 = 0.28 $ (m); Right: $ l_1 = 0.34 $ (m), $ l_2 = 0.26 $ (m))

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In order to achieve optimal results quickly and avoid local minima under different distances, we let the optimal result of previous step be the initial values of current step, and employ MATLAB function $ fminmax $ to solve this problem. To obtain $ \Delta_{ij} $ $ (i,j = 1,2) $, we employ MATLAB function $ fmincon $ which is based on trust-region-reflective algorithm. The relationships between $ \theta_{opt1} $, $ \theta_{opt2} $ and $ (l_1,l_2) $ are obtained by solving the above optimization problem and marked by “$ \blacklozenge $” in Figs. 6 and 7.

Figure  6.  Relationship between $ \theta_{\rm opt1} $ and $ (l_1,l_2) $

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Figure  7.  Relationship between $ \theta_{\rm opt2} $ and $ (l_1,l_2) $

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By using least squares method, we fit the relationships (marked by “$ \square $” in Figs. 6 and 7) and obtain the following polynomials (7) and (8). So for some $ (l_1,l_2) $, we can obtain the corresponding robust pair of orientation angles $ (\theta_{\rm opt1},\theta_{\rm opt2}) $ based on (7) and (8).

During holding and lifting-up motion control, we set the desired position of the passive joint as $ (x_r,y_r) $ and the desired distances as $ (l_{1r},l_{2r}) $. Then according to (3), we can derive (9) because $ A_{4\times 4}\neq 0 $. If we can place the robot arms at $ (\tilde x_{A1},\tilde y_{A1}) $ and $ (\tilde x_{A2},\tilde y_{A2}) $, the posture of 2-link object will be maintained stably under the robust orientation angles $ (\theta_{\rm opt1},\theta_{\rm opt2}) $ substituted into $ A_{4\times 4}^{-1} $ and $ B_{4\times 1} $ because (1) and (2) are satisfied.

IV. Experimental Results

To verify the effectiveness of our proposed posture maintenance control method, we implement it to the nonprehensile two-cooperative-arm robot (see Fig. 8). The robot arms can be driven by 4 DC motors (RE40, Maxon motor) and drivers (EPOS2, Maxon motor) on X-Y axis. The positions of the robot arms can be calculated by using the encoders (HEDL5540, Maxon motor) which are fixed at the end of the DC motors. For measuring the orientation angles of 2-link object, we pasted FSLPs (Force-Sensing Linear Potentiometer, Interlink Electronics) on the surfaces of the robot arms and calibrated the relationship between the measured signal from FSLP and the orientation angles of the 2-link object. The related parameters of 2-link object and robot arms are given in Table I.

Figure  8.  Nonprehensile two-cooperative-arm robot

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Table  1.  Parameters

	$ i=1 $	$ i=2 $
$ m_i $ (Kg)	$ 2.71 $	$ 2.5 $
$ d_i $ (m)	$ 0.036 $	$ 0.036 $
$ r_i $ (m)	$ 0.015 $	$ 0.015 $
$ L_i $ (m)	$ 0.367 $	$ 0.399 $
$ {\mu_s} $	$ 0.161 $	$ 0.161 $

 | Show Table

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Based on the proposed posture maintenance control method, the control scheme of the system is structured (see Fig. 9). Since $ l_1 $ and $ l_2 $ are not considered as controllable variables, they would be changed by external force. So we set their initial values as desired distance $ (l_{1r},l_{2r}) $ and use current values which are calculated by (3) after the start. The initial state of the 2-link object for holding motion control and lifting-up motion control is stable. That is, after putting 2-link object onto the robot arms and it can stay there stably, the control system is started. As shown in Fig. 9, the positions of the robot arms and orientation angles are obtained by encoders and FSLPs, respectively. According to the current values of $ l_1 $ and $ l_2 $, the robust orientation angles $ (\theta_{\rm opt1},\theta_{\rm opt2}) $ can be calculated by (7) and (8). Then the desired positions of the robot arms are decided by (9) and sent to motor driver EPOS2s which are used to drive DC motor to move to the desired positions.

Figure  9.  Block diagram of the structured control scheme

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A Holding Motion Control

In this experiment, we controlled nonprehensile cooperative arms to hold 2-link object at the constant position ($ 0.00025 $ (m),$ -0.24435 $ (m)) under the robust orientation angles. The results are shown in Figs. 10–12.

Figure  10.  $ l_1 $ and $ l_2 $

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Figure  12.  Real and desired orientation angles of 2-link object

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In Fig. 10, we can see $ l_1 $ and $ l_2 $ tend to constants after the start. After adding the disturbance by hand (Pull & Push Link 1) during the holding motion, they can also tend to other constants. From Fig. 11, we know that the real passive joint position of 2-link object can be held at the desired position approximately. The real and desired orientation angles of 2-link object are shown in Fig. 12. $ \theta_1 $ and $ \theta_2 $ can track the desired robust orientation angles $ \theta_{\rm opt1} $ and $ \theta_{\rm opt2} $, respectively. From the above results, the effectiveness of our proposed method can be verified in holding motion.

Figure  11.  Real and desired passive joint positions of 2-link object

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B Lifting-Up Motion Control

In this experiment, we controlled robot arms to lift up 2-link object along the trajectory $ (x_r(t), y_r(t)) $. The trajectory was planned by using 5th-order polynomial as follows

with the following boundary conditions.

So we can obtain the coefficients of (10) and (11) as follows.

The steps of this experiment are

1) Hold 2-link object at 0 (m),$ -0.2411 $ (m))

2) Lift up 2-link object to (0 (m),$ -0.1411 $ (m))

3) Hold 2-link object at (0 (m),$ -0.1411 $ (m))

4) Lift up 2-link object to ($ -0.05 $ (m),$ -0.1411 $ (m))

5) Hold 2-link object at ($ -0.05 $ (m),$ -0.1411 $ (m))

6) Lift up 2-link object to ($ -0.05 $ (m),$ -0.2411 $ (m))

7) Hold 2-link object at ($ -0.05 $ (m),$ -0.2411 $ (m)).

and each step costs 5 s. The trajectories of each lifting-up motion are planned by using (10) and (11). We added disturbance by hand (Pull & Push Link 1), and got the results (see Figs. 13–15).

Figure  13.  $ l_1 $ and $ l_2 $

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Figure  15.  Real and desired orientation angles of 2-link object

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In Fig. 13, the time of adding disturbance is shown. During holding motion and lifting-up motion, $ l_1 $ and $ l_2 $ are almost constant. That means, relative motion between robot arms and 2-link object is small, and this method is able to be applied to reduce pain caused by the relative motion between robot arms and care-receivers. The real and desired passive joint positions of 2-link object are shown in Fig. 14. We can see the robot arms can stably hold and lift up 2-link object to track the planned trajectory under the robust orientation angles which are shown in Fig. 15. From the above results, the effectiveness of our proposed method can be also verified in lifting-up motion.

Figure  14.  Real and desired passive joint positions of 2-link object

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V. Conclusion

In this paper, stable regions for posture maintenance control were obtained based on the built static model of 2-link object while it was kept stable on the nonprehensile robot arms within static friction force. Among the obtained regions, the robust pairs of orientation angles corresponding to the distances from the contact points to the passive joint of 2-link object were found. Under the obtained robust orientation angles, a feedback control system scheme was designed to control the robot arms to hold and lift up 2-link object. Experimental results showed the effectiveness of the proposed method. In the future work, another link which is regarded as the care-receiver's shank will be added. The new model of 3-link object will be built and the stable region and robust orientation angles will be recalculated.