A hybrid impedance control scheme for underwater welding robots with a passive foundation in the controller domain

Abstract

A hybrid impedance control scheme for the force and position control of an end-effector is presented in this paper. The interaction of the end-effector is controlled using a passive foundation with compensation gain. For obtaining the steady state, a proportional–integral–derivative controller is tuned with an impedance controller. The hybrid impedance controller is implemented on a terrestrial (ground) single-arm robot manipulator. The modeling is done by creating a bond graph model and efficacy is substantiated through simulation results. Further, the hybrid impedance control scheme is applied on a two-link flexible arm underwater robot manipulator for welding applications. Underwater conditions, such as hydrodynamic forces, buoyancy forces, and other disturbances, are considered in the modeling. During interaction, the minimum distance from the virtual wall is maintained. A simulation study is carried out, which reveals some effective stability of the system.

Keywords

Underwater robot proportional–integral–derivative controller bond graph impedance control flexible link

1. Introduction

The interaction of underwater robot manipulators with objects is a very important aspect for completing autonomous missions. For a number of applications performed in underwater conditions, the robot end-effector has to make contact or keep a constant distance with the environment, as in the case of welding. The interaction of the underwater robot with such nonlinear environmental conditions is of great interest. The underwater robot manipulators should have reliable communication with the underwater environment conditions in their work space when subjected to interaction forces while performing tasks. From this point of view, control of the position and constant force between the robot end-effector and environment is necessary.

There are several interaction control schemes related to robot force control¹ in friction-less environments.² Stiffness control, direct force control, hybrid force/position control, parallel force/position control, and impedance control are some of the examples.³ By applying the designated impedance between the robot end-effector position and constant force, a controlled interaction can be achieved.⁴ Hogen^5–7 developed an approach to facilitate the application of robots involving static and dynamic interaction between the manipulator and its environment considering impedance components of the manipulator behavior using a bond graph. Krishnan et al.⁸ dealt with the issue of interaction between the hopping robot toe and the physical ground by controlling the driving point stiffness (impedance) using bond graph modeling software. Using a pneumatic actuator with passive properties of the impact and force control, the desired interaction force can be achieved for stable and dissipative contact tasks with an arbitrary environment.⁹ Wang and Xie¹⁰ investigated the force/position track control of a free flying space manipulator kinematic and dynamics using multiple impedance control (MIC). In the more practical case of contact with friction, velocity/force can be used to determine the surface normal direction.¹¹ Further, the asymptotic stability of the force regulator using a proportional–integral (PI) force plus proportional–derivative (PD) position feedback can be done in the condition where the environment is not rigid.¹² Cui et al.,¹³ through simulation, have shown the effectiveness of impedance controller Underwater Vehicle Manipulators (UVMs). Antonelli et al.^14,15 developed an external force control scheme for UVMs that does not require dynamic compensation. Farivarnejad and Moosavian¹⁶ successfully implemented the MIC and augmented object model (AOM) methods for an explicit dynamic model of a dual arm UVM showing better tracking error and good reaction in the case of a collision with the environment. Seraji and Colbaugh¹⁷ proposed on-line schemes for force tracking within the impedance-control framework for providing robustness in the presence of large uncertainties or variations in environmental parameters. According to Pathak et al.,¹⁸ an overwhelming control strategy can be extended to an impedance control strategy using a bond graph for the impedance control of space robots and the passive foundation concept as a flexible base approach can been applied. It has been applied in space exploration with a linear controller, which has a very limited capability to provide performance under any external disturbances and steady-state accuracy.

In this paper, a hybrid impedance control scheme has been modeled for a two-link flexible arm underwater robot manipulator for welding applications, which is inspired by the work done by Pathak et al.¹⁸ To achieve more effective stability and robustness during interaction, a proportional–integral–derivative (PID) controller has been embedded in the impedance controller to model a hybrid impedance controller for underwater robotic systems, which have various disturbances due to underwater environments. The PID controller has the advantage of the ability to enhance the performance of the controller and removes steady-state errors arising due to environmental disturbances.¹⁹

This concept was first implemented in a terrestrial (ground) robot manipulator, and is further extended for a two-link flexible arm underwater welding robot manipulator. For modeling and simulation of the hybrid impedance controller, the bond graph technique was used. The flexible link has been modeled as a Euler flexible beam with six equal segments and considering the mass of the welding torch as end-effector. Bond graph modeling and simulation has been carried out in SYMBOLS sonata^® software.²⁰ Simulations were carried out on the developed control system for performing the task and proved the effectiveness of the proposed control system in the underwater welding torch contact interaction.

The paper is mainly divided into four sections. Section 2 explains the concept of the hybrid impedance controller and its implementation in the terrestrial robot manipulator. Section 3 covers the implementation of the hybrid impedance controller on a two-link flexible arm underwater welding robot. In this section, the modeling of the subsystems (base, link, and Jacobian) created is presented and the integrated bond graph model is explained. Further in this section, simulation and results are presented. In the last section, conclusions are drawn.

2. Hybrid impedance controller

The ratio of effort (force) to flow (velocity) of the system is known as the impedance. Because of complexity, the force/position varies persistently for robot manipulators in different conditions. This complexity arises due to the nonlinearity, uncertainties in the system, and underwater environment interaction. The high tracking precession leads to high impedance and further makes the control of interaction forces difficult because of the inefficient adjustment of wake disturbances during interaction. Suitable impedance control can balance the environment interaction, trajectory robustness, and uncertainties in the model of a two-link flexible arm underwater robot manipulator for welding applications. To cope with nonlinear conditions and to achieve more stability, the hybrid impedance controller is modeled by embedding a PID controller along with a passive foundation based on the overwhelming impedance controller. With the addition of a PID controller, that is, a third-order system, zero steady error is achieved for any constant disturbance. With an increase in proportional gain, the system becomes faster. However, if derivative gain is increased, the system becomes slower.¹⁹ With proper choices of proportional, derivative, and integral gains, the performance of the hybrid impedance controller is enhanced and a steady state is achieved.

The control system is illustrated by the conceived schematic of the one-arm robotic manipulator shown in Figure 1. On the basis of the schematic diagram shown in Figure 1, the bond graph model for impedance control is modeled as shown in Figure 2. The bond graph shown in Figure 2 is created in two domains, that is, the controller domain and the physical domain. The physical domain covers modeling of the base of the robot, the link of the robot, and interaction with the environment. The controller domain involves the modeling of the passive foundation, the impedance controller, and the PID controller. The passive foundation is designed in such a way that interaction force can be adjusted within the significant level by modulating impedance. This control concept bridges the gap between the trajectory and the force controller by making robot stiffness low during force control and high during trajectory control.

Figure 1.

One-degree-of-freedom (DOF) robot manipulator with a passive foundation.

Figure 2.

Bond graph of the hybrid impedance controller applied on single-arm robot manipulator.

The overwhelming control strategy is a robust control strategy that makes the system rigid to variations in the manipulator parameters, and ensures bounded tracking errors.^18,21 The overwhelming control strategy consists of a set of controllers coupled to the robotic manipulator through a set of high feed forward gain µ_H with an effort activated bond. Further, the modulated source of flow (MSf) as feedback with the unity transformation ratio is modeled through flow activated bonds, as shown in Figure 2. An equivalent passive foundation is mapped within the controller domain through high feed forward gain β_H, which has been modeled through the effort activated bond and modulated source of flow (MSf) as feedback with the unity transformation ratio through flow the activated bond, as shown in Figure 2. The velocity of the passive foundation is sensed and fed back to controller through the flow activated transformer, which has a transformation ratio of σ (feedback compensation). In Figure 2, M_p, R_p, and K_p are the mass, the damping resistance, and the stiffness of passive foundation, respectively. M_b is the mass of the base and m_p is the lumped mass of the arm. In Figure 2, f_vref is the reference excitation angular velocity, which is generated by the motor shown in Figure 1. The motor provides the reference angular velocity to the link.

The transfer function between robotic tip flow and environmental effort of the system is derived by following the detailed steps given by Pathak et al.¹⁸ and Mukherjee et al.²¹ The transfer function is achieved through the signal flow diagram of the bond graph model, shown in Figure 2:

\begin{array}{l} \frac{f_{tip}}{e_{env}} = \\ \frac{R (s) [1 + (1 - σ) β_{h} μ F_{pf} (s) - (1 - Ψ) μ μ_{h} F_{b} (s)]}{1 + (μ_{h} μ + G_{p} μ_{h} + μ_{h} G (s)) R (s) + (1 - σ) β_{h} μ F_{pf} (s) + (1 - Ψ) μ_{h} μ F_{b} (s)} \end{array}

(1)

Here $R (s) = \frac{1}{m_{l} s}$ is the transfer function of the robotic manipulator, $F_{pf} (s) = s / (M_{pf} s^{2} + R_{pf} s + K_{pf})$ is the transfer function of the passive foundation, $F_{b} (s) = 1 / M_{b} s$ is the transfer function for the base of the robotic manipulator, $G (s) = \frac{G_{i}}{s}$ is the integral gain related to PID control, $G_{p}$ is the proportional gain related to PID control, µ is the derivative gain related to PID control, $μ_{H}$ is the high feed forward gain, $β_{H}$ is the high gain, $σ$ is the feedback compensation from the passive foundation to the controller, and $Ψ$ is the feedback compensation from the robot base to the controller.

It is assumed that complete velocity feedback compensation to the controller from the robotic manipulator is one, that is, $(Ψ = 1)$ :

\frac{f_{tip}}{e_{env}} = \frac{R (s) [1 + (1 - σ) β_{h} μ F_{pf} (s)]}{1 + (μ_{h} μ + G_{p} μ_{h} + μ_{h} G (s)) R (s) + (1 - σ) β_{h} μ F_{pf} (s)}

(2)

In Equation (2), if σ = 1, then the robust trajectory is attained as the passive foundation effect is neglected, and if σ < 1 it is possible to adjust the interaction forces through the modulation of impedance. In this way, the relationship between tip velocity and environmental forces acting on the tip through impedance is developed. So, impedance can be used by modulating the value of σ. At the time of interaction, when the force approaches its limiting value, the motion of the tip is detained and the command motion is compensated for by the strength of the passive foundation, which leads to a low value of impedance. Due to this, the actual manipulator trajectory differs from the reference trajectory. When compensation gain $σ$ is toggled to one, the interaction is over and reinstates the high impedance state of the system. The tip further follows the reference trajectory.

To implement this concept efficiently, the heuristic expression used by Pathak et al.¹⁸ for achieving modulation is applied by the following equation:

σ (F, F_{l}, t) = 1 - swi {F (t), F_{l}} {K_{in} + K_{pg} (F (t) - F_{l}) + K_{ig} X (t)}

(3)

In Equation (3), F(t) is the interaction force received from the force sensor, F_l is the limited interaction force and K_in is a constant (a bias), and K_pg and K_ig are constant, proportional gain, and integral gain terms. There exist different techniques by which one can arrive at a proper choice of controller gains.¹⁹ In the present study, gains have been obtained by the hit and trial method. swi is the switch function in SYMBOL sonata^®, defined as follows:

swi[F(t), F_l] = 1 when F(t) ≥ F_l, that is, the impedance controller will act as the force controller;

swi[F(t), F_l] = 0 when F(t) < F_l, that is, the impedance controller will act as the trajectory controller.

X(t) is the expression that is obtained as the state of the error integrating element. With reference to this equation, SF is defined in Figure 2. X(t) used in Equation (3) is obtained by the following:

X (t) = swi {e_{s}, F_{l}} * {e_{s} - F_{l}}

(4)

where e_s is the force sensed at the bond, which gives the interaction force between the tip and the environment.

2.1 Simulation and results

Bond graph simulation is conducted to examine the efficacies of the proposed controller. To illustrate the performance, it is considered that the tip follows a half rectified sine trajectory with amplitude a, and the reference velocity given to bond f_vref may be written as follows:

f_{vref} = a * (π / 2) * \cos (t * π / 2) * swi {\cos (t * π / 2), 0}

(5)

e _env is the effort used to negate the effort given by the environment stiffness K_e, because of the unobstructed travel by the tip, and is incorporated by the following:

e_{env} = swi (Q_{tip}, z_{w}) * (K_{e} * z_{w})

(6)

where K_e is environment stiffness, z_w is the distance of the wall from the tip, and Q_tip is the tip velocity. Table 1 shows the parameters used for simulation, which was run for 10 s. The initial position of the tip is assumed to be at zero radians. In the simulation, it is assumed that tip has to keep a distance of 10 mm from the virtual wall with an amplitude (a) of 18 mm.

Table 1.

Parameters used for modeling of the two-link flexible arm underwater robot manipulator for welding applications.

Sr. no.	Parameters	Nomenclature	Value
1	Modulus of Elasticity	E	$7 \times 10^{10} N / m^{2}$
2	Link Length	L	0.5 m
3	Moment of inertia of cross-section of link	I	$2.13 \times 10^{- 7} m^{4}$
4	Density of Aluminium(alloy)	ρl	$2700 kg / m^{3}$
5	Cross section area of link	A	$1.6 \times 10^{- 3} m^{2}$
6	Surface Area of link	As	0.02
7	Mass of link	Mlink	0.5 kg
8	Volume of link	v	$8 \times 10^{- 4} m^{3}$
9	Mass of base	Mb	50 kg
10	Moment of Inertia of base	Ib	10 kg-m²
11	Distance between center of base and first joint	r	0.1 m
12	Joint Resistance	R_j	0.1 Nm/(rad/s)
13	Gain value	µ_h	10
13	Gain value	β_h	2
14	Passive foundation characterstics	Kp,	1000 Nm/rad
		Rp	100 Nm/(radians/s)
		Ip	40 kgm²
		Mp	200 kg
15	Gain modulations	Kini	0.9
		Kgi	0.4
		Kgp	0.6
16	Interaction force limit	Flimt	600 N
17	Input Reference Angular Velocity	ω	1 rad/s
18	Force error integration	K_fe	100 N/m
18	Force error integration	R_fe	80 N(m/s)
19	PID gain modulations	Gp	0.001
		Gi	0.1
		µ	7
20	Environment stiffness	Ke	4000 N/m
21	Location of underwater robot from obstruction (Wall)	y_w	0.1 m

PID: proportional–integral–derivative.

Figure 3(a) shows the comparison of the reference tip displacement and the actual tip displacement. It is observed that the tip follows the reference trajectory in the environment until 8 mm from the initial tip position. As the limit force is reached, the trajectory follows straight path with a small dip. This dip is observed when the reference trajectory starts moving back from the amplitude. The tip again follows the reference trajectory when force reduces the limited force. Figure 3(b) shows the relation of contact force between the tip and environment versus time. This force is observed by a detector e_s, shown in Figure 2. It is observed that force rise until it reaches the limited force. Due to flexibility between the two inertias, considerable vibrations are observed at the limited force and force starts decreasing almost after 0.5 s in the first cycle. As the reference trajectory reaches its amplitude, force almost becomes zero until the next cycle starts. From Figures 3(b) and (c), it is observed that the compensation gain and interaction force are interrelated. The compensation gain modulates as the interaction force reaches its limited value and keeps the interaction force almost at 800 N, that is, the limit force. Figure 3(d) shows the base displacement with respect to time. The interaction force is passed to the base of the robot manipulator, leading to movement from its starting position, as shown in Figure 3(d).

Figure 3.

Plots for the results attained for the one-arm robotic manipulator: (a) plot of tip displacement versus time; (b) plot of force versus time; (c) plot of compensation gain versus time; (d) plot of base displacement versus time.

3. Hybrid impedance control for the two-link flexible arm underwater welding robot

Implementation of hybrid impedance control for the two-link flexible arm underwater robot manipulator is represented by a block diagram, as shown in Figure 4. The modeling incorporates a model of the base, a Euler flexible beam with six equal segments, the attached mass at the end of second link as a welding torch, underwater nonlinear conditions (buoyancy force, the gravity force matrix, and hydrostatic force), and the proposed hybrid impedance control scheme. The models are created using subsystems (capsules) of the hybrid impedance controller (impedance controller with PID), the base of the underwater robot, flexible links (LK₁ and LK₂) and a Jacobian. Further, these subsystems are connected to each other according to the block diagram, as shown in Figure 4.

Figure 4.

Block diagram for implementation of the hybrid impedance controller to a two-link flexible arm underwater robot manipulator for welding applications. HIC: Hybrid-impedance-controller.

Figure 5 shows the subsystem of a hybrid impedance controller, which is applied to the two-link flexible arm underwater robot manipulator for welding applications. The modeling of the hybrid impedance controller is based on the discussion in the previous section. The modeling of the subsystem base and subsystem link is based on the concept of an open kinematic chain configuration and the physical system shown in Figure 6. The two links are considered as a Euler flexible beam, divided into six equal segments. The joints considered in this configuration are revolute joints. To study the kinematics of the physical system, different frames have been considered, as shown in Figure 6. {A} represents the absolute frame, {R} represents the robot frame, and {1} represents the frame being located at the joint of the base with the first link. Frame {2} is at second joint, that is, at the joint of the first link with the second link, and frame {t} locates the tip of the second link. Figure 7 shows the subsystem (capsule) of the base for the two-link flexible arm underwater robot manipulator for welding applications, which is created through bond graphs. In Figure 7, M_b and I_b represent the mass of the base and the inertia of base. Zcm and Ycm are the coordinates of the base with respect to the absolute frame {A}, as shown in Figure 6.

Figure 5.

Bond graph subsystem (capsule) model of the hybrid impedance controller.

Figure 6.

Physical model of the two-link flexible arm underwater robot manipulator for welding applications.

Figure 7.

Bond graph subsystem (capsule) model of the flexible link for the underwater robot.

In the bond graph model of the flexible link, the length of each link is represented by L, flexural rigidity as EI, density as ρ, and cross-sectional area as A. The change in angular displacement between the base and the first link (the first joint angle) is θ₁ and the change in angular displacement between the first and the second link is θ₂ (the joint angle of the second link). As per underwater conditions, in the downwards direction (the Z direction), buoyant force, hydrostatic force, and gravity act on the mid-point of each segment.

Z ₁₁–Z₁₆ are the junctions for the first link and Z₂₁–Z₂₆ are the junctions for the second link. The velocity of the flexible link at junction 11 (middle of the first segment of the first link) in the Z and Y directions with respect to frame {1} can be written as follows:

{\overset{\cdot}{Z}}_{11} = - (r \sin β) \overset{\cdot}{β} - {\overset{\cdot}{Y}}_{11} \sin (θ_{1} + β)

(7a)

{\overset{\cdot}{Y}}_{11} = (r \cos β) β + {\overset{\cdot}{Y}}_{11} \cos (θ_{1} + β)

(7b)

In the above equations, $β$ represents the rotation of the base while θ₁ represents the joint angle and r is the distance between the center of the base and the first joint. The velocities in the Z and Y directions at each segment depend upon the velocity at previous junctions and can be derived with respect to the absolute frame by using segment angles $(γ_{11}, γ_{12}, γ_{13}, γ_{14}, γ_{15}, γ_{16})$ . This relation in the bond graph model is taken care of by the transformers. The absolute velocity at the tip of the first link with respect to frame {1} may be expressed as follows:

{\overset{\cdot}{Z}}_{tip 1} = {\overset{\cdot}{Z}}_{16} - (\frac{L}{12} \sin γ_{16}) {\overset{\cdot}{γ}}_{16}

(8a)

{\overset{\cdot}{Y}}_{tip 1} = {\overset{\cdot}{Y}}_{16} + (\frac{L}{12} \cos γ_{16}) {\overset{\cdot}{γ}}_{16}

(8b)

In Equations (8a) and (8b) ${\overset{\cdot}{Z}}_{16}$ and ${\overset{\cdot}{Y}}_{16}$ are the velocities in the Z and Y directions, respectively, at the midpoint of the sixth segment of the first link. The second expression in Equations (8a) and (8b) gives the velocity due to the distance between the tip and the center of the sixth segment of the first link, which is half the length of the segment, that is, L/12. The absolute velocity at the first segment of the second link with respect to frame {1} may be derived as follows:

{\overset{\cdot}{Z}}_{21} = {\overset{\cdot}{Z}}_{tip 1} - {\overset{\cdot}{Y}}_{21} \sin (γ_{21})

(9a)

{\overset{\cdot}{Y}}_{21} = {\overset{\cdot}{Y}}_{tip 1} + {\overset{\cdot}{Y}}_{21} \cos (γ_{21})

(9b)

In Equations (9a) and (9b), ${\overset{\cdot}{Y}}_{21}$ is the velocity of the second link at junction 21 in the perpendicular direction of the second link. The segment angles of the second link $(γ_{21}, γ_{21}, γ_{23}, γ_{24}, γ_{25}, γ_{26})$ transform their velocity to their next segment and finally to the tip of the second link. The velocities in the Z and Y directions at the tip of second link with respect to frame {1} can be evaluated as follows:

{\overset{\cdot}{Z}}_{tip 2} = {\overset{\cdot}{Z}}_{26} - (\frac{L}{12} \sin γ_{26}) {\overset{\cdot}{γ}}_{26}

(10a)

{\overset{\cdot}{Y}}_{tip 2} = {\overset{\cdot}{Y}}_{26} + (\frac{L}{12} \cos γ_{26}) {\overset{\cdot}{γ}}_{26}

(10b)

${\overset{\cdot}{Z}}_{26}$ and ${\overset{\cdot}{Y}}_{26}$ are the velocities in the Z and Y directions at the midpoint of the sixth segment of the second link. The kinematic of the flexible link provides the modulus of transformation functions used in the subsystem (capsule) of the link created through the bond graph, as shown in Figure 8. When the motion of the body is to be simulated in an underwater environment, a number of additional forces must be considered, which are governed by the various forces. In the subsystem (capsule) of the link shown in Figure 8, the underwater condition has been taken care of by the source of effort SE (F_uw) attached to each segment. F_uw represents the result of hydrostatic force, gravity force, and buoyancy force acting on each segment. In this work, it is assumed that the welding is performed at a depth of 10 m so the effect of surface waves is ignored. Further, it is assumed that fluid is irrotational, unbounded, and still. The effect of drag and fluid acceleration is not considered. The forces applied are based on the following equation^22,23:

(P, τ) = (m) (\overset{\cdot}{V}, \overset{\cdot}{ω}) + (m_{C}) (V, ω) + F_{uw}

(11a)

Figure 8.

Bond graph subsystem (capsule) model of the base of the underwater robot.

In Equation (11a), $(P, τ)$ is the external input forces and torques, $(m)$ is the mass and added mass-inertia matrices, and $(m_{C})$ is the Coriolis and centripetal matrices and underwater forces given as follows:

F_{uw} = B + G + H

(11b)

Here, $B$ is the buoyancy force matrix, G is the gravity force matrix, and H is the hydrostatic force. The buoyancy force and hydrostatic force can be easily expressed in the earth fixed frame in the Z direction as follows:

B = - ρ gv and H = ρ g A_{s}

(11c)

where ρ is the fluid density, υ is the link volume, g is the gravity acceleration, and A_s is the surface of the link on which water pressure is acting.

The arm of the robot manipulator has been modeled as a flexible link using the Euler–Bernoulli beam theory. The lumped inertia of the beam is taken into account through rotary inertia, whereas shear deformation is neglected. The 1-junctions along the upper line of the ladder structure represent the velocities of the mass centers of the segments in the Z and Y directions to which the corresponding inertia elements (m_is) are attached. The subscript is represents the segment number and the corresponding mass of each segment is ρAL/6. The 1-junctions along the lower line represent segment interface rotations. The C elements at 0-junctions along the lower line model represent the flexural stiffness of the segments represented by the K_fs symbol in Figure 8, and stiffness is given by 4EI/L. The underwater force F_uw is attached to the 1-junction corresponding to the segment velocities. In the bond graph model bond SP is soft pad attached for computational simplicity, that is, to avoid the differential casualty, attached at the upper part of bond graph shown in Figure 8. The symbol used for the sub-system of the link is LK.

3.1 Jacobian

The velocity from the base of the two-link flexible arm underwater robot manipulator has been mapped with the torch of the robot using the Jacobian. The torch velocity has been sensed and sent to the PID-based impedance controller.

The kinematic relations for the tip motion in the Z and Y directions can be written as follows:

Z_{tip 2} = Z_{cm} + r \cos β + L (\cos (β_{1}) + \cos (β_{12}))

(12a)

Y_{tip 2} = Y_{cm} + r \sin β + L (\sin (β_{1}) + \sin (β_{12}))

(12b)

Here $β_{1} = β + θ_{1}$ and $β_{12} = β + θ_{1} + θ_{2}$ . The Jacobian is calculated in the bond graph with the help of transformers on the bases of the above equations. This is achieved by derivations of these equations and then neglecting small values, given in matrix form as follows:

[\begin{matrix} {\overset{\cdot}{Z}}_{tip 2} \\ {\overset{\cdot}{Y}}_{tip 2} \end{matrix}] = L [\begin{matrix} - \sin (β_{1}) - \sin (β_{12}) & - \sin (β_{12}) \\ \cos (β_{1}) + \cos (β_{12}) & \cos (β_{12}) \end{matrix}] [\begin{matrix} {\overset{\cdot}{θ}}_{1} \\ {\overset{\cdot}{θ}}_{2} \end{matrix}]

(13)

[J] = L [\begin{matrix} - (\sin (β_{1}) + \sin (β_{12}) & - \sin (β_{12}) \\ \cos (β_{1}) + \cos (β_{12}) & \cos (β_{12}) \end{matrix}]

(14)

The transformer moduli used in the bond graph capsule of a Jacobian, shown in Figure 9, as per the above equation are expressed as follows:

μ_{1} = - L (\sin (β_{1}) - \sin (β_{12})), μ_{2} = - L (\cos (β_{1}) + \cos (β_{12})), μ_{3} = - L \sin (β_{12}), and μ_{4} = - L \cos (β_{12})

(15)

Figure 9.

Bond graph subsystem (capsule) model of the Jacobian.

3.2 Integrated bond graph

The integrated bond graph of the two-link arm underwater robot manipulator with a hybrid impedance controller is shown in Figure 10. It comprises subsystems, that is, one base (B), two links (LK1 and LK2), one Jacobian (J), and two hybrid impedance controllers (ImPid) in the Y and Z directions. These subsystems are attached to each other according to the strategy shown in Figure 4. The reference velocity is given to the system in both directions through the respective hybrid impedance controller, as shown in lower part of Figure 10. Both of the hybrid impedance subsystems are attached to the flexible foundation through the 1-junction and are also connected to the base and Jacobian through 0-junctions in respective directions. The base is connected to the Jacobian and the first link, which are further connected to the second link.

Figure 10.

Integrated bond graph model of the two-link flexible arm underwater robot manipulator for welding applications.

To consider the mass (m_wt) effect of the welding torch, an I-element is attached in both directions and in order to remove the differential casualty, C_pad and R_pad are attached to the 0-junction, as shown in Figure 10. In the present model, the environment restrictions are considered in only the Y direction, according to which the force detector has been place on the Y direction bond. Bond SE is attached to the 1-junction, representing the forces (F_c) required for contradicting the effort of environmental stiffness K_e. The modulation of hybrid impedance is done by compensation gain (σ) in the Y direction only. When the gain (σ) value is one, impedance is changed, which leads to control of the trajectory. In another situation, when the compensation gain (σ) value is more than one, force at the end-effector remains controlled. The flow activated C elements provide trajectory in the Y and Z directions and this flow is also given to the hybrid impedance controller for comparison with the reference velocities.

3.3 Simulation and results

The proposed hybrid impedance control scheme is implemented on a two-link flexible arm underwater robot manipulator for welding applications and a simulation study was carried out for 2 s. It is assumed that the welding torch follows an elliptical path with major radius a = 0.4 m and minor radius b = 0.2 m. Accordingly, torch coordinates are given by the following equations:

Y = - a * \cos (ω t) + y_{0}

(16a)

Z = b * \sin (ω t) + z_{0}

(16b)

Here y₀ and z₀ are coordinates of the center of the ellipse. The corresponding reference velocity obtained can be written as follows:

\overset{\cdot}{Y} = V_{yref} = a ω * \sin (ω t)

(17a)

\overset{\cdot}{Z} = V_{zref} = b ω * \cos (ω t)

(17b)

It is assumed that torch motion is restricted in the Y direction by Y_w distance from the center of the ellipse. F_c force is applied in the Y direction to specify the restriction by the environment in the bond graph model, that is, to contradict the effort of the environmental stiffness effort element. This force is represented as follows:

F_{c} = swi (Q_{ytip}, y_{w}) * K_{e} * Y_{w}

(18)

Here $Q_{ytip}$ is the velocity of the welding torch in the Y direction.

The initial position of the welding torch is considered by taking zero as the value of both joint angles. It is also considered that the absolute frame {A} and the center of the base frame {R} are overlapping. The initial position of the welding torch with respect to frame {A} is as follows:

{[Y_{tip 0} Z_{tip 0}]}^{t} = {[1.10]}^{t}

(19)

and the center of the ellipse at time t = 0 is as follows:

{[Y_{0} Z_{0}]}^{t} = {[0.70]}^{t}

(20)

The parameters used in the simulation are shown in Table 1.

Figure 11(a) shows the comparison of reference tip displacement and actual tip displacement in the Y direction. It is observed that in the underwater condition at the start time, that is, 0 s, the position in the Y direction of the reference trajectory is 1.1 m. However, at this position, as per Equation (18), the force limit restricts the actual trajectory due to which it starts from 1 m. The difference in the start point of the reference trajectory and the actual trajectory is Y_w. The tip follows the reference trajectory until it reaches 0.1 m distance from the wall and at the same time force is reduced as compared to the limited force, as shown in Figure 11(e). Figure 11(b) shows the tip displacement in the Z direction. The plot shows continuous drift in the Z direction due to free floating of the base. The same trend can be seen in Figure 11(c), showing the trajectory followed by the torch. Figure 11(d) shows the graph of compensation gain in the Y direction with time and Figure 11(e) shows the force detected (e_s) versus time graph. It is observed that force reaches limited force, that is, 600 N, when the distance of the trajectory remains at 0.1 m from wall, as shown in Figure 11(e). From Figures 11(d) and (e), it is observed that compensation gain is modulated as the interaction force reaches its limited value and the interaction force remains at less than 600 N (limit force). Figure 11(f) shows the base rotation versus time and Figure 11(g) shows the displacement of CM. From Figures 11(f) and (g) it is seen that, due to the motion of the base, after some time the underwater robot will not remain in the workspace of the robot.

Figure 11.

(a) Comparison of reference and actual tip displacement in the Y direction versus time. (b) Trajectory followed in the Z direction. (c) Y displacement versus Z displacement. (d) Compensation gain versus time. (e) Interaction force versus time. (f) Base rotation versus time. (g) Displacement of CM of the base.

4. Conclusions

The hybrid impedance controller has been modeled using the bond graph technique for interaction points and is exemplified. A passive foundation has been used to achieve impedance control. This controller is able to restrict the interaction force to its limited value through compensation gain and achieve a steady state through PID control. The scheme was first implemented in an on-ground single-arm robot manipulator and its efficacy was substantiated through simulation results. Furthermore, the hybrid impedance controller was applied on a two-link flexible arm underwater robot manipulator for welding applications. The interaction of the end-effector as a welding torch with a wall was simulated. The results show that the proposed scheme is suitable for implementation in an underwater robot manipulator for welding applications and is able to control both position and force through compensation gain µ_H. The constraints of the underwater condition, such as hydrostatic force, buoyancy force, and disturbances, are taken into account. In both cases interaction force and position are limited within the commanded value. In future work, some force tracking capability of the impedance controller will be attempted.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Author biographies

Sunil Kumar is an Assistant Professor in the Department of Mechanical Engineering, Sant Longowal Institute of Engineering and Technology, Sangrur, India, and is pursuing his PhD in the field of railway dynamics. His research interest includes system dynamics and robotics.

Vikas Rastogi is a Professor in the Department of Mechanical, Production, Industrial and Automobile Engineering, Delhi Technological University, Delhi, India. He obtained his PhD in Mechanical Engineering (Lagrangian-Hamiltonian Mechanics) from the Indian Institute of Technology-Kharagpur. His research interests include system dynamics, modeling and simulation, structural dynamics, vehicle dynamics, and control and rotor dynamics.

Pardeep Gupta is working as Professor in the Department of Mechanical Engineering, Sant Longowal Institute of Engineering and Technology, Sangrur, India. He obtained his PhD from NIT, Kurukshetra, in 2004. His areas of interest include quality and reliability engineering, total productive maintenance, conventional and non-conventional metal machining, and optimization techniques.

References

Siciliano

Villani

. Robot force control. Boston, MA: Kluwer, 1999.

Canudas

DWC

Brogliato

. Direct adaptive impedance control including transition phases. Automatica1997; 33: 643–649.

Antonelli

Sarkar

Stefano

. External force control for underwater vehicle-manipulator system. In: Proceedings of 38th conference on decision & control, Crowne Plaza Hotel and Resort, Phoenix, AZ, 7–10 December, 1999, pp.2975–2980. New York: IEEE.

Hogan

. Impedance control: an approach to manipulation. J Dynam Syst Measur Contr1985; 107: 1–24.

Hogan

. Impedance control: an approach to manipulation: Part I-theory. J Dynam Syst Measur Contr1985; 107: 1–7.

Hogan

. Impedance control: an approach to manipulation: Part II-theory. J Dynam Syst Measur Contr1985; 107: 8–16.

Hogan

. Impedance control: an approach to manipulation: Part III-theory. J Dynam Syst Measur Contr1985; 107: 17–24.

Krishnan

Pathak

Jain

. Force control in monopod hopping robot while landing. Intell Contr Autom2010; 1: 96–104.

Zhu

Barth

. Passivity-based impact and force control of a pneumatic actuator. J Dynam Syst Measur Contr2008; 30: 1–7.

10.

Wang

Xie

. Adaptive Jacobian position/force tracking control of free-flying manipulators. J Robot Auton Syst2009; 57: 173–181.

11.

Yoshikawa

Sudou

. Dynamic hybrid position/force control of robot manipulators on-line estimation of unknown constraint. IEEE Trans Robot Autom1993; 9: 220–226.

12.

Chiaverini

Siciliano

Villani

. Force/position regulation of compliant robot manipulators. IEEE Trans Autom Contr1994; 39: 647–652.

13.

Cui

Podder

Sarkar

. Impedance control of underwater vehicle-manipulator systems (UVMS). In: Proceeding IEEE/RSJ international conference intelligent robots and systems, 7–11 September, 1997, pp.148–153. New York: IEEE.

14.

Antonelli

Chiaverini

Sarkar

. An external force control scheme for underwater vehicle-manipulator system. In: International conference on intelligent robots and systems, Kyongji, Korea, 7–21 October, 1999. Gyeonggi-do, Korea: Kyunghee printing Co. Ltd.

15.

Antonelli

Chiaverini

Sarkar

. External force control for underwater vehicle-manipulator systems with adaptive motion control. In: IEEE Hong-Kong symposium on robotics and control, Hong-Kong, 2–3 July, 1999, pp.361–366.

16.

Farivarnejad

Moosavian

ASA

. Multiple impedance control for object manipulation by a dual arm underwater vehicle-manipulator system. Ocean Eng2014; 89: 82–89.

17.

Seraji

Colbaugh

. Force tracking in impedance control. Int J Robot Res1997; 16: 97–117.

18.

Pathak

Mukherjee

Dasgupta

. Impedance control of space robots using passive degrees of freedom in controller domain. Trans ASME2005; 127: 564–578.

19.

Ghosal

. Robotics fundamental concepts and analysis. Oxford: Oxford University Press, 2006.

20.

Mukherjee

. User manual of SYMBOLS sonata^®, High Tech Consultants, STEP, Indian Institute of Techonology, Kharagpur, 2009.

21.

Mukherjee

Karmakar

Samantaray

. Bond graph in modeling simulation and fault identification. New Delhi: I.K. International Publishing House Pvt Ltd, 2006.

22.

Wang

Wan

. Modeling and motion control strategy for autonomous underwater vehicles. In: Proceedings of international conference on mechatronics and automation, Changchun, China, 9–12 August 2009, pp.4851–4856. New York: IEEE

23.

Sharkar

Podder

. Coordinated motion planning and control of autonomous underwater vehicle-manipulator systems subject to drag optimization. IEEE J Ocean Eng2001; 26: 228–239.