Abstract
This paper deals with the design of a motion and force control scheme for underwater vehicles, each of which has a manipulator. For a subsea operation that requires a contact between the manipulator’s tip (e.g., the hand) and various types of environments, it is desirable that the mechanical impedance of the manipulator is adjusted according to the contact surface. From this point of view, the paper focuses on the design of an impedance control scheme. Several impedance controllers have been developed. Most of them were designed on the assumption that the control capability of the vehicle is the same as that of the manipulator. However, it has been pointed out in the literature that for a real underwater robot, its vehicle control is more challenging than its manipulator control, because the vehicle has much larger inertia, and many more inaccurate position sensors and actuators than the manipulator. In view of this fact, we develop an impedance control scheme for the manipulator under the condition that the vehicle is independently controlled by a motion controller with poor performance. Moreover, we provide the results of simulations for comparing an existing controller with the proposed one.
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The availability of other types of vehicle-manipulator systems and the issue on consumption energy were pointed out by a reviewer for this journal. In line with the useful comment by the reviewer, we wrote the contents in this paragraph. In addition, as far as we know, an impedance controller designed in consideration of consumption energy is not reported in the literatere on underwater robotics. We think that this topic is one of our future studies, and cannot currently provide the data of simulations or experiments. Therefore, we state that "This is an important issue for future research".
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Acknowledgements
This work was supported by JSPS KAKENHI Grant Number JP21K04503.
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This work was presented in part at the joint symposium of the 28th International Symposium on Artificial Life and Robotics, the 8th International Symposium on BioComplexity, and the 6th International Symposium on Swarm Behavior and Bio-Inspired Robotics (Beppu, Oita and Online, January 25–27, 2023).
Appendices
Appendix 1: derivation of (3) with (4) to (8)
Replacing \(a_\textrm{CV}(t)\lambda (t)\) by \(J_{V}(\phi )^{\textrm{T}}f_{S}(t)\) in the vehicle’s dynamic model (15) with (16) and (17) of [33], we can acquire the vehicle’s dynamic model (3) with (4) to (8). The signal \(\lambda (t)\in R\) is a contact force perpendicular to the tangent plane of the contact surface, and \(a_{CV}(t)\in R^{6}\) is expressed as
where \(n_{C}(t)\in R^{3}\) is a normal vector of the tangent plane of the contact surface. Moreover, the derivations of the vehicle’s dynamic models of [33] and this paper are similar to that of [23]. The detailed derivation of the dynamic model of the vehicle is shown in Appendix 1 of [23]. In this paper, we provide the following equations utilized in the derivation of (3) with (4) to (8):
It should be noted that the first Eq. (54) is a standard type of dynamic model of the vehicle (e.g., the Eq. (1.52) of [1]), whereas the second and third equations of (54) are dynamic models of marine thrusters [50].
Appendix 2: derivation of (9) with (10) to (12)
Solving the model (3) for \(\ddot{x}_{V}(t)\), and then substituting it into the standard type of manipulator’s dynamic model (e.g., the Eq. (1.52) of [1])
we obtain the following dynamic model:
It is noteworthy that \(M_{V} (\phi )\) is a non-singular matrix, because it is a symmetric and positive definite matrix (as shown in the property P2 of [23]). The manipulator’s dynamic model (56) is represented in a joint space [i.e., q(t) ]. However, a control objective is generally established in a task space [i.e., \(p_{M}^{V}(t)\)], and hence we transform the dynamic model (56) from the joint space to the task space. Solving the time derivative of (11) for \(\ddot{q}(t)\) [on the assumption that \(J_M^V (q)\) is a non-singular (full rank) matrix], then substituting it into the dynamic model (56) for \(\ddot{q}(t)\), and then multiplying both its sides by \(J_{M}^{V}(q)^{-\textrm{T}}\), we can acquire the manipulator’s dynamic model (9).
Appendix 3: derivation of (17) with (18)
Substituting the constants \(\bar{c}_{6}\) and \(\bar{c}_{15}\) into 0 in the inequalities (74) of [33], and then replacing \(|\lambda (t)|\) by \(\Vert f_{S}(t) \Vert\) in the Eqs. (76) and (79) of [33], we can acquire the inequality (17) with (18). The signal \(\lambda (t)\) is a contact force perpendicular to the tangent plane of the contact surface. The detailed derivation of the inequality of \(\Vert z_{P}(t)\Vert\) with the filter \(z_\textrm{Pf}(t)\) is shown in Appendix 3 of [33].
Appendix 4: derivation of second inequality of (19)
In the derivation of the second inequality of (19), we utilize the following inequalities:
where \(\bar{c}_{1}\) to \(\bar{c}_{4}\) are positive constants. Moreover, the assumptions of the boundedness of the disturbances and their time derivatives lead to the following inequalities:
where \(\bar{c}_{D1}\) to \(\bar{c}_{D4}\) are positive constants.
Solving the first filter of (6), and then taking its norm, we obtain
Taking the norm of the first filter of (6), and then applying (59) to it, we have
Similarly, solving the second filter of (6), then taking its norm, and then applying the first inequality of (58) to it, we obtain
Taking the norm of the second filter of (6), and then applying (61) and the first inequality of (58) to it, we have
Representing the second Eq. (4) in the form of a norm, and then applying (59), (61) and the first inequality of (58) to it, we obtain
where \(\bar{c}_{5}\in R^{+}\) is provided by the following equation:
Differentiating the second Eq. (4) with respect to time, then taking its norm, then applying (60), (62) and the third inequality of (58) to it, and then substituting (64) into it, we have
Taking the norm of the third Eq. (12), and then applying (63), the second and fourth inequalities of (57), and the second inequality of (58) to it, we obtain
where \(\bar{c}_{6}\in R^{+}\) is written as
Differentiating the third Eq. (12) with respect to time, then taking its norm, and then applying (63), (65), the first to fourth inequalities of (57) and the fourth inequality of (58) to it, we have
where \(\bar{c}_{7}\in R^{+}\), \(\bar{c}_{8}\in R^{+}\) and \(\bar{c}_{9}\in R^{+}\) are, respectively, provided by the following equations:
Differentiating the third Eq. (10) with respect to time, then taking its norm, and then applying (66), (68), the first and second inequalities of (15) and the inequality \(\Vert \dot{q}(t)\Vert \le \Vert \dot{\phi }(t)\Vert\) to it, we obtain
The inequality (70) leads to the second inequality of (19).
Appendix 5: derivations of inequalities of (20)
The condition of all the revolute joints of the manipulator leads to the first inequality of (20). Solving (11) for \(\dot{q}(t)\) [on the assumption that \(J_{M}^{V}(q)\) is a non-singular (full rank) matrix], then taking its norm, and then applying the first inequality of (15) to it, we obtain the second inequality of (20). Taking the norm of the kinematic equation (e.g., the Eq. (2.17) of [47])
and then applying the eighth inequality of (13) and the first inequality of (20) to it, we acquire the third inequality of (20). We can rewrite (5) as
where the meaning of the skew-symmetric matrix \(S(\cdot )\) is described in Table 1. Representing (72) in the form of a norm, then utilizing the property \(\Vert S(\bar{x})\Vert = 6\Vert \bar{x}\Vert\) for any vector \(\bar{x}\in R^{3}\), and then applying the fifth and eighth inequalities of (13) and the first inequality of (20) to it, we acquire the fourth inequality of (20).
Appendix 6: derivations of equations of (29)
Replacing \(p_{M}(t)\) and \(p_{M}^{V}(t)\) by \(p_\textrm{MR}(t)\) and \(p_\textrm{MR}^{V}(t)\) in (71) of Appendix 5, respectively, and then solving it for \(p_\textrm{MR}^{V}(t)\), we acquire the first equation of (29). In this derivation, we utilize the property \(R(\phi _{V})^{-1} = R(\phi _{V})^{\textrm{T}}\) (e.g., the Eq. (2.7) of [47]). Substituting (14) into (72) of Appendix 5 for \(J_{R}(\phi _{V})\dot{\phi }_{V}(t)\), we have
Applying the skew-symmetric property \(S(\bar{x})\bar{z}=-S(\bar{z})\bar{x}\) for any vectors \(\bar{x}\in R^{3}\) and \(\bar{z}\in R^{3}\) (e.g., the Eq. (A.32) of [48]) to (73), then replacing \(p_{M}^{V}(t)\), \(\dot{p}_{M}^{V}(t)\) and \(\dot{p}_{M}(t)\) by \(p_\textrm{MR}^{V}(t)\), \(\dot{p}_\textrm{MR}^{V}(t)\) and \(\dot{p}_\textrm{MR}(t)\), respectively, and then solving it for \(\dot{p}_\textrm{MR}^{V}(t)\), we obtain the second Eq. (29).
Appendix 7: derivations of inequalities of (30)
Taking the norms of (14) and its time derivatives, and then applying the fifth to seventh inequalities of (13) to them, we have
where \(\bar{c}_{10}\in R^{+}\) is provided by the following equation:
Representing the first Eq. (29) in the form of a norm, and then applying the eighth inequality of (13), the first inequality of (26) and the boundedness of \(p_{V}(t)\) (in the assumption A3) to it, we acquire the first inequality of (30). Taking the norm of the second Eq. (29), then utilizing the property \(\Vert S(\bar{x})\Vert = 6\Vert \bar{x}\Vert\) for any vector \(\bar{x}\in R^{3}\), and then applying the eighth inequality of (13), the second inequality of (26), the first inequality of (30), the first inequality of (74) and the boundedness of \(\dot{p}_{V}(t)\) and \(\dot{\phi }_{V}(t)\) (in the assumption A3) to it, we obtain the second inequality of (30). We consider the following differential kinematic Eq. (e.g., the Eq. (6.10) of [47]):
Representing (76) in the form of a norm, then utilizing the property \(\Vert S(\bar{x})\Vert = 6\Vert \bar{x}\Vert\) for any vector \(\bar{x}\in R^{3}\), and then applying the eighth inequality of (13), the third inequality of (26), the first and second inequalities of (30), the first and second inequalities of (74) and the boundedness of \(\ddot{p}_{V}(t)\), \(\dot{\phi }_{V}(t)\) and \(\ddot{\phi }_{V}(t)\) (in the assumption A3) to it, we attain the third inequality of (30). Differentiating (76) with respect to time, and then using the property \(\dot{R}(\phi _{V},\dot{\phi }_{V})=S(\omega _{V})R(\phi _{V})\) (e.g., the Eq. (5.36) of [47]), we obtain
Taking the norm of (77), then utilizing the property \(\Vert S(\bar{x})\Vert = 6\Vert \bar{x}\Vert\) for any vector \(\bar{x}\in R^{3}\), and then applying the eighth inequality of (13), the fourth inequality of (26), the first to third inequalities of (30), the first to third inequalities of (74) and the boundedness of \(p_{V}^{(3)}(t)\), \(\dot{\phi }_{V}(t)\), \(\ddot{\phi }_{V}(t)\) and \(\phi _{V}^{(3)}\) (in the assumption A3) to it, we attain the fourth inequality of (30).
Appendix 8: derivation of (35)
Taking the norm of (33), then applying (17), the third and eighth inequalities of (13) and the first inequality of (16) to it, and then substituting (34) into it, we have
Applying the eighth inequalities of (13) to (78), and then rearranging it, we obtain
The inequality (79) leads to the inequality (35).
Appendix 9: proof of Theorem 1
We choose the following positive definite function:
Differentiating (80) with respect to time, then substituting the closed-loop error model (36) into it, and then applying the third inequality of (16), the first inequality of (19), the third inequality of (30) and the inequality \(\bar{a}\bar{b}\le \bar{c}\bar{a}^{2}+\bar{b}^{2}/(4\bar{c})\) for any scalars \(\bar{a}\in R\), \(\bar{b}\in R\) and \(\bar{c}\in R^{+}\) to it, we have
where \(\bar{c}_{11}\in R^{+}\) is provided by the following equation:
In the application of the inequality \(\bar{a}\bar{b}\le \bar{c}\bar{a}^{2}+\bar{b}^{2}/(4\bar{c})\) to \(\dot{V}_{1}(t)\), we choose \(\bar{a}\), \(\bar{b}\) and \(\bar{c}\) as \(\Vert e_{I}(t)\Vert\), \(\bar{c}_{11}\Vert M_{T}\Vert\) and \(\alpha /2\), respectively. Substituting (80) into (81), we obtain
Moreover, applying Lemma 3.2.4 of [51] to (83), we have
where \(c_{V1}=2V_{1}(0)\) and \(c_{V2}=\bar{c}_{11}^2\). Utilizing (80) and (84), we acquire the inequality (38), which means the ultimate boundedness of \(e_{I}(t)\).
We apply the following signals to the differential Eq. (24):
In this case, the differential Eq. (24) is valid, and furthermore \(u_{I}(t)\) and \(w_{I}(t)\) are bounded, because of the boundedness of \(e_{I}(t)\) and the assumption A10. Therefore, it follows from Lemma 1 that \(z_{I}(t)\) is bounded. This is equivalent to the boundedness of \(e_{P}^{V}(t)\) and \(\int _{0}^{t} e_{P}^{V}(\bar{\tau })d\bar{\tau }\).
Solving (31) for \(\dot{e}_{P}^{V}(t)\) [in view of the positive definiteness of \(M_{T}\) (i.e., the non-singularity of \(M_{T}\))], then taking its norm, and then applying the assumption A10 and the boundedness of \(e_{I}(t)\), \(e_{P}^{V}(t)\) and \(\int _{0}^{t} e_{P}^{V}(\bar{\tau })d\bar{\tau }\) to it, we can show that \(\dot{e}_{P}^{V}(t)\) is bounded. Differentiating the first Eq. (22) with respect to time, then representing its time derivative \(\dot{p}_{M}^{V}(t)\) in the form of a norm, and then applying the second inequality of (30) and the boundedness of \(\dot{e}_{P}^{V}(t)\) to it, we can indicate the boundedness of \(\dot{p}_{M}^{V}(t)\). Utilizing the second inequality of (20) and the boundedness of \(\dot{p}_{M}^{V}(t)\), we can show that \(\dot{q}(t)\) is bounded. The assumption A3 and the boundness of \(\dot{q}(t)\) lead to the boundedness of \(\dot{\phi }(t)\) and \(\dot{\eta }(t)\), whose meanings are described in Table 1. Utilizing the third inequality of (20) and the assumption A3, we can show that \(p_{M}(t)\) is bounded. Similarly, using the fourth inequality of (20), the boundedness of \(\dot{p}_{M}^{V}(t)\) and the assumption A3, we can indicate the boundedness of \(\dot{p}_{M}(t)\). Therefore, the assumption A2 and the boundedness of \(p_{M}(t)\) and \(\dot{p}_{M}(t)\) lead to the boundedness of \(f_{S}(t)\). Taking the norm of the second equation of (22), and then applying (27) and the boundedness of \(f_{S}(t)\) to it, we can indicate the boundedness of \(e_{F}(t)\). In view of the fact that the filter (18) is a stable one, it follows from the assumption A3 and the boundedness of \(\dot{\eta }(t)\) and \(f_{S}(t)\) that \(z_\textrm{Pf}(t)\) is bounded. Applying the boundedness of \(\dot{\eta }(t)\), \(f_{S}(t)\), \(z_\textrm{Pf}(t)\), \(e_{P}^{V}(t)\), \(\dot{e}_{P}^{V}(t)\), \(e_{F}(t)\) and \(e_{I}(t)\) to (35), we can show that \(\tau _{M}(t)\) is bounded. We can prove that all the closed-loop signals are bounded. This completes the proof.
Appendix 10: proof of theorem 2
We choose the following positive definite function:
Differentiating (86) with respect to time, then substituting the closed-loop error model (37) into it, and then applying the second and third inequalities of (16), the first and second inequalities of (19), the fourth inequality of (30) to it, we have
In view of the stability properties of Theorem 1 and the assumption A3 [i.e., the boundedness of \(\dot{\phi }(t)\)], the constant \(c_{\phi }\in R^{+}\) is defined as
where \(\sup _{t\ge 0}\{\Vert \bar{z}(t)\Vert \}\) means the supremum of \(\Vert \bar{z}(t)\Vert\) for any vector signal \(\bar{z}(t)\). Applying the inequality \(\Vert \dot{\phi }(t)\Vert \le c_{\phi }\) to (87), and then rearranging it, we obtain
where \(\bar{c}_{12}\in R^{+}\) and \(\bar{c}_{13}\in R^{+}\) are, respectively, provided by the following equations:
Applying the inequality \(\bar{a}\bar{b}\le \bar{c}\bar{a}^{2}+\bar{b}^{2}/(4\bar{c})\) for any scalars \(\bar{a}\in R\), \(\bar{b}\in R\) and \(\bar{c}\in R^{+}\) to the second and third terms of the right-hand side of (89), and then substituting (86) into it, we have
In the application of the inequality \(\bar{a}\bar{b}\le \bar{c}\bar{a}^{2}+\bar{b}^{2}/(4\bar{c})\) to the second term of the right-hand side of (89), we choose \(\bar{a}\), \(\bar{b}\) and \(\bar{c}\) as \(\Vert \dot{e}_{I}(t)\Vert\), \(\bar{c}_{12}\Vert M_{T}\Vert\) and \(\alpha /4\), respectively. Similarly, in the application of its third term, we select \(\bar{a}\), \(\bar{b}\) and \(\bar{c}\) as \(\Vert \dot{e}_{I}(t)\Vert\), \(\bar{c}_{13}\rho \Vert M_{T}\Vert\) and \(\alpha /4\), respectively. Moreover, applying Lemma 3.2.4 of [51] to (91), we obtain
where \(c_{V3}=2V_{2}(0)\), \(c_{V4}=2\bar{c}_{12}^2\) and \(c_{V5}=2\bar{c}_{13}^2\). Utilizing (86) and (92), and then multiplying both its sides by 2, we have
Differentiating (31) with respect to time, and then substituting it into (93) for \(\dot{e}_{I}(t)\), we acquire the inequality (39). This completes the proof.
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Taira, Y., Sagara, S. & Oya, M. Impedance control based on error feedback for the manipulator of an underwater vehicle-manipulator system. Artif Life Robotics 28, 830–849 (2023). https://doi.org/10.1007/s10015-023-00896-6
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DOI: https://doi.org/10.1007/s10015-023-00896-6