Human Multi-Robot Physical Interaction: a Distributed Framework

Abstract

The objective of this paper is to devise a general framework to allow a human operator to physically interact with an object manipulated by a multi-manipulator system in a distributed setting. A two layer solution is devised. In detail, at the top layer an arbitrary virtual dynamics is considered for the object with the virtual input chosen as the solution of an optimal Linear Quadratic Tracking (LQT) problem. In this formulation, both the human and robots’ intentions are taken into account, being the former online estimated by Recursive Least Squares (RLS) technique. The output of this layer is a desired trajectory of the object which is the input of the bottom layer and from which desired trajectories for the robot end effectors are computed based on the closed-chain constraints. Each robot, then, implements a time-varying gain adaptive control law so as to take into account model uncertainty and internal wrenches that inevitably raise due to synchronization errors and dynamic and kinematic uncertainties. Remarkably, the overall solution is devised in a distributed setting by resorting to a leader-follower approach and distributed observers. Simulations with three 6-DOFs serial chain manipulators mounted on mobile platforms corroborate the theoretical findings.

Appendix

To prove Theorem 1, let us first derive the closed loop dynamics or robot i. By folding (45) in (1), it holds

$$ {\boldsymbol{M}}_{\!i}\dot{\boldsymbol{s}}_{i} = -{\boldsymbol{C}}_{i}\boldsymbol{s}_{i} - \boldsymbol{K}_{s}\boldsymbol{s}_{i} - {\varDelta} \boldsymbol{u}_{i} + \boldsymbol{h}_{i} + {\boldsymbol{Y}}_{\!\!i}(\boldsymbol{x}_{i}, \dot{\boldsymbol{x}}_{i},\boldsymbol{\rho}_{i}, \dot{\boldsymbol{\rho}}_{i} )\tilde{\boldsymbol{\pi}}_{i}\!\!\!\! $$

(48)

Now, let us consider the following Lyapunov function

$$ V=\frac{1}{2}\sum\limits_{i=1}^{N}\left( \boldsymbol{s}_{i}^{\mathrm{T}}{\boldsymbol{M}}_{i}\boldsymbol{s}_{i}+\frac{1}{k_{f}}\hat{\varDelta} \boldsymbol{u}_{f,i}^{\mathrm{T}}\hat{\varDelta} \boldsymbol{u}_{f,i}+\tilde{\boldsymbol{\pi}}_{i}^{\mathrm{T}}\boldsymbol{K}_{\pi}\tilde{\boldsymbol{\pi}}_{i}\right) $$

(49)

By virtue of (48) and Property 1, the time derivative of V is

$$ \begin{aligned} {}\dot{V} \! =& \! \sum\limits_{i=1}^{N}\!\Big(\boldsymbol{s}_{i}^{\mathrm{T}}{\boldsymbol{M}}_{i}\dot{\boldsymbol{s}}_{i} + \frac{1}{2}\boldsymbol{s}_{i}^{\mathrm{T}}\dot{{\boldsymbol{M}}}_{i}\boldsymbol{s}_{i} + \hat{\varDelta} \boldsymbol{u}_{f,i}^{\mathrm{T}}\tilde{\boldsymbol{h}}_{int,i} - \tilde{\boldsymbol{\pi}}_{i}^{\mathrm{T}}\boldsymbol{K}_{\pi}{\dot{\hat{\boldsymbol{\pi}}}}_{i}\Big)\\ =&\! \sum\limits_{i=1}^{N}\Big(-\boldsymbol{s}_{i}^{\mathrm{T}}\boldsymbol{K}_{s}\boldsymbol{s}_{i}+\frac{1}{2}(\dot{\boldsymbol{M}}_{i}-2{\boldsymbol{C}}_{i})\boldsymbol{s}_{i} +\boldsymbol{s}_{i}^{\mathrm{T}}(\boldsymbol{h}_{i}-{\varDelta} \boldsymbol{u}_{i})\\ &+\hat{\varDelta} \boldsymbol{u}_{f,i}^{\mathrm{T}}\tilde{\boldsymbol{h}}_{int,i} -\tilde{\boldsymbol{\pi}}_{i}^{\mathrm{T}}(\boldsymbol{K}_{\pi}{\dot{\hat{\boldsymbol{\pi}}}}_{i}-{\boldsymbol{Y}}_{i}^{\mathrm{T}}(\boldsymbol{x}_{i}, \dot{\boldsymbol{x}}_{i}, \boldsymbol{\rho}_{i}, \dot{\boldsymbol{\rho}}_{i} )\boldsymbol{s}_{i})\Big)\\ =&\!\sum\limits_{i=1}^{N}\!\Big(\! - \boldsymbol{s}_{i}^{\mathrm{T}}{\kern-.5pt}\boldsymbol{K}_{s}\boldsymbol{s}_{i} + \hat{\varDelta} \boldsymbol{u}_{f,i}^{\mathrm{T}}\tilde{\boldsymbol{h}}_{int,i} - \boldsymbol{s}_{i}^{\mathrm{T}}{\kern-.5pt}({\kern-.5pt}\boldsymbol{e}_{int,i} + k_{f}\!\hat{\varDelta} \boldsymbol{u}_{f,i}\! +\! \frac{1}{N}{~}^{i}\tilde{\boldsymbol{h}}_{h}{\kern-.5pt})\\ & -\kappa_{i}(t)\boldsymbol{s}_{i}^{\mathrm{T}}\boldsymbol{s}_{i} -\tilde{\boldsymbol{\pi}}_{i}^{\mathrm{T}}(\boldsymbol{K}_{\pi}{\dot{\hat{\boldsymbol{\pi}}}}_{i}-{\boldsymbol{Y}}_{i}^{\mathrm{T}}(\boldsymbol{x}_{i}, \dot{\boldsymbol{x}}_{i}, \boldsymbol{\rho}_{i}, \dot{\boldsymbol{\rho}}_{i} )\boldsymbol{s}_{i}) \Big)\\ \end{aligned} $$

(50)

Given the parameters update law in Eq. (47), (50) simplifies to

$$ \begin{aligned} \dot{V} =&\sum\limits_{i=1}^{N}\Big(-\boldsymbol{s}_{i}^{\mathrm{T}}\boldsymbol{K}_{s}\boldsymbol{s}_{i}+\hat{\varDelta} \boldsymbol{u}_{f,i}^{\mathrm{T}}\tilde{\boldsymbol{h}}_{int,i}-\kappa_{i}(t)\|\boldsymbol{s}_{i}\|^{2}\\ &-(\tilde{\boldsymbol{\zeta}}_{i}+\hat{\varDelta} \boldsymbol{u}_{f,i})^{\mathrm{T}}(\tilde{\boldsymbol{h}}_{int,i}+\hat{\varDelta} \boldsymbol{u}_{f,i})+\frac{1}{N}\boldsymbol{s}_{i}^{\mathrm{T}}{~}^{i}\tilde{\boldsymbol{h}}_{h} \Big)\\ \end{aligned} $$

(51)

Thus, by choosing

$$\kappa_{i}(t)>\frac{\|\tilde{\boldsymbol{\zeta}}_{i}\|}{\|\boldsymbol{s}_{i}\|^{2}} \|\tilde{\boldsymbol{h}}_{int,i}+\hat{\varDelta} \boldsymbol{u}_{f,i}\|$$

it holds

$$ \dot{V}\leq \sum\limits_{i=1}^{N}\big(-\boldsymbol{s}_{i}^{\mathrm{T}}\boldsymbol{K}_{s}\boldsymbol{s}_{i}-\hat{\varDelta} \boldsymbol{u}_{f,i}^{\mathrm{T}}\hat{\varDelta} \boldsymbol{u}_{f,i}+\frac{1}{N}\|\boldsymbol{s}_{i}^{\mathrm{T}}\|\|{}^{i}\tilde{\boldsymbol{h}}_{h}\|\big) $$

From Lemma 2, $\|{}^{i}\tilde {\boldsymbol {h}}_{h}\|$ converges to the origin after a finite time T_h; then after this time it holds

$$ \dot{V}\leq \sum\limits_{i=1}^{N}\big(-\boldsymbol{s}_{i}^{\mathrm{T}}\boldsymbol{K}_{s}\boldsymbol{s}_{i}-{~}^{i}\hat{\varDelta} \boldsymbol{u}_{f,i}^{\mathrm{T}}{~}^{i}\hat{\varDelta} \boldsymbol{u}_{f,i}\big) $$

which implies that $\dot {V}$ is semi-negative definite and, consequently, that V is bounded. By leveraging the boundedness of V and, then, of s_i, $\hat {\varDelta } \boldsymbol {u}_{f,i}$ and ${\tilde {\boldsymbol {\pi }}}_{i}$, it can be easily shown the $\ddot V$ is bounded as well. Thus, by virtue of Barbalat’s lemma, $\dot {V}$ is uniformly continuous and converges to the origin, as well as s_i and ${\hat {\varDelta } \boldsymbol {u}_{f,i}=k_{f}{\int \limits }_{t_{0}}^{t}{\tilde {\boldsymbol {h}}_{int,i}} d\tau }$ (and, therefore, by definition $\tilde {\boldsymbol {h}}_{int,i}$). The main implication of the latter is that, since because of Lemma 2, $\tilde {\boldsymbol {h}}_{int,i}$ converges to the origin in finite time (that is the internal wrenches estimated via observer converges to the real one), then also e_{int, i}, ∀i, converges to the origin.

In view of the expression of s_i in (43) and since $\hat {\varDelta } \boldsymbol {u}_{f,i}$ converges to the origin, it follows

$$ \dot{\boldsymbol{e}}_{x,i}+k_{p}\boldsymbol{e}_{x,i} =-\hat{\varDelta} \boldsymbol{u}_{f,i} $$

which represents an asymptotically stable system (in the state variable e_{x, i}) with vanishing input $-\hat {\varDelta } \boldsymbol {u}_{f,i}$. Therefore, e_{x, i} asymptotically converges to the origin. Based on the expression of e_{x, i} in (42), it asymptotically holds

$$ ({~}^{i} \hat{\boldsymbol{x}}_{v}-\boldsymbol{x}_{i})+k_{c}{\int}_{t_{0}}^{t}\sum\limits_{j\in\mathcal{N}_{i}}(\boldsymbol{x}_{j}-\boldsymbol{x}_{i})d\tau\rightarrow\boldsymbol{0}_{p} $$

(52)

Let us now introduce the object trajectory estimate error ${~}^{i}\tilde {\boldsymbol {x}}_{v} = {\boldsymbol {x}}_{v} - {~}^{i} \hat {\boldsymbol {x}}_{v}\in \Re ^{p}$ and the object trajectory tracking error e_{v, i} = x_v −x_i ∈R^p. From (52), it asymptotically holds

$$ \begin{aligned} ({\boldsymbol{x}}_{v}-\boldsymbol{x}_{i})+k_{c}{\int}_{t_{0}}^{t}\sum\limits_{j\in\mathcal{N}_{i}}(\boldsymbol{x}_{j}-{\boldsymbol{x}}_{v}-\boldsymbol{x}_{i}+{\boldsymbol{x}}_{v})d\tau&=-{~}^{i}\tilde{\boldsymbol{x}}_{v}\\ \end{aligned} $$

which can be rewritten as

$$ \begin{aligned} \boldsymbol{e}_{v,i}+k_{c}{\int}_{t_{0}}^{t}\sum\limits_{j\in\mathcal{N}_{i}}(-\boldsymbol{e}_{v,j}+\boldsymbol{e}_{v,i})d\tau&=-{~}^{i}\tilde{\boldsymbol{x}}_{v}\\ \end{aligned} $$

(53)

By denoting with $\tilde {\boldsymbol {x}}_{v}\in \Re ^{Np}$ and e_v ∈R^Np the stacked vectors of the errors ${~}^{i}\tilde {\boldsymbol {x}}_{v}$ and e_{v, i}, respectively, (6) leads to

$$ \boldsymbol{e}_{v}(t)=-k_{c}(\boldsymbol{L}\otimes \boldsymbol{I}_{p}){\int}_{t_{0}}^{t}\boldsymbol{e}_{v} d\tau-\tilde{\boldsymbol{x}}_{v} $$

(54)

in which, from (40), $\tilde {\boldsymbol {x}}_{v}$ converges to the origin in finite-time. Finally, since the communication graph is connected, the immediate consequence of (54) is that ${\int \limits }_{t_{0}}^{t}\boldsymbol {e}_{v,i}={\int \limits }_{t_{0}}^{t}\boldsymbol {e}_{v,j}$, ∀i, j which, based on (42) and (52), implies that e_{v, i} = 0_p ∀i. This completes the proof.