A robust control of robot manipulators for physical interaction: stability analysis for the interaction with unknown environments

Abstract

A robust control designed for multiple degrees-of-freedom (DOF) robot manipulators performing complex tasks requiring frequent physical interaction with unknown and/or uncertain environments is analyzed to provide complete stability conditions and explain its robustness to environmental changes and disturbances. Nonlinear bang–bang impact control was introduced about two decades ago. High-velocity impact experiments using a one DOF robot and a stiff aluminum wall showed superior performance than other controllers. Moreover, it does not use robot dynamics and environmental dynamics for its design. Furthermore, intriguingly, it utilized the nonlinear joint friction, which was commonly regarded as a factor deteriorating the control performance, to subside impact energy sensibly. To date, the stability was, however, not completely proved. Thus, NBBIC was not widely adopted. In this study, thus, complete and sufficient stability conditions of NBBIC for multi-DOF robots are derived based on energy comparisons and $L_{\infty }^{n}$ space analysis. It was found that the NBBIC stability condition does not require information on the environmental dynamics and disturbances. Stability was affected by the intentional time delay, which was needed to efficiently and effectively estimate the environment and robot dynamics and the accuracy of robot inertia estimate. As was expected, larger friction was better for subsiding the impact force that is expected when impacting an environment at high velocity.

Appendices

Appendices

1.1 Appendix 1. Proof of Lemma 1

For impact force, ${{\varvec{\uptau}}}_{s} \in L_{\infty }^{n}$ because $\left\| {{{\varvec{\uptau}}}_{s} } \right\|_{\infty } = \mathop {\sup }\limits_{t > 0} \left| {\delta_{\alpha } (t)} \right| = {1 \mathord{\left/ {\vphantom {1 {2\alpha }}} \right. \kern-\nulldelimiterspace} {2\alpha }}$. δ_α(t) represents the unit pulse function defined as [4].

$$\delta_{\alpha } (t) = \left\{ {\begin{array}{*{20}l} {{1 \mathord{\left/ {\vphantom {1 {(2\alpha )}}} \right. \kern-\nulldelimiterspace} {(2\alpha )}} \, - \alpha \le t \le \alpha } \hfill \\ {0 \, t < - \alpha {\text{ or }}t > \alpha } \hfill \\ \end{array} } \right..$$

(62)

One can have stable transfer function matrices from the desired trajectory to the error (e), from external torque (τ_s) to error, and from ε to error with properly chosen control gains. The initial condition-related terms (ρ_is) of stable linear systems are bounded. $L_{\infty }^{n}$ gains β_i, γ_i, η_i (i = 1, …, 6) of the stable transfer functions are finite. Thus, the inequality, (17), holds. □

1.2 Appendix 2: $L_{\infty }^{n}$ Norm Inequality of Linear Time-Invariant Systems

The following inequality holds for stable linear time-invariant (LTI) systems.

$$ \left\| {{\tilde{\mathbf{y}}}_{{{\rm out}}} (t)} \right\|_{T\infty } \le \left\| {{\mathbf{x}}_{{{\rm in}}} (t)} \right\|_{T\infty } \int\limits_{0}^{\infty } {\left\| {{\tilde{\mathbf{P}}}(\sigma )} \right\|_{i2} {\rm d}\sigma } + \left\| {{\tilde{\mathbf{C}}}(t)} \right\|_{T\infty } , $$

(63)

where x_in(t) denotes input; y_out(t) output; P(t) is is impulse-response matrix; and C(t) represents the part of the output, y_out, which relies on the initial conditions. The inequality (63) can be obtained from the convolution integral as follows.

$$ \begin{aligned} \left\| {{\tilde{\mathbf{y}}}_{{{\rm out}}} {\mathbf{(}}t{\mathbf{)}}} \right\|_{T\infty } & = \left\| {\int\limits_{0}^{t} {{\mathbf{\tilde{P}(}}t - \sigma {\mathbf{)x}}_{{{\text{in}}}} {\mathbf{(}}\sigma {\mathbf{)}}} {\text{d}}\sigma + {\tilde{\mathbf{C}}}(t)} \right\|_{T\infty } \\ & \le \int\limits_{0}^{t} {\left\| {{\mathbf{\tilde{P}(}}t - \sigma {\mathbf{)x}}_{{{\text{in}}}} {\mathbf{(}}\sigma {\mathbf{)}}} \right\|_{2} {\text{d}}\sigma } + \left\| {{\tilde{\mathbf{C}}}(t)} \right\|_{T\infty } \\ & \le \int\limits_{0}^{t} {\left\| {{\mathbf{\tilde{P}(}}t - \sigma {\mathbf{)}}} \right\|_{i2} \left| {{\mathbf{x}}_{{{\text{in}}}} {\mathbf{(}}\sigma {\mathbf{)}}} \right|} {\text{d}}\sigma + \left\| {{\tilde{\mathbf{C}}}(t)} \right\|_{T\infty } \\ & \le \left\| {{\mathbf{x}}_{{{\text{in}}}} {\mathbf{(}}t{\mathbf{)}}} \right\|_{T\infty } \int\limits_{0}^{t} {\left\| {{\mathbf{\tilde{P}(}}\sigma {\mathbf{)}}} \right\|_{i2} } {\text{d}}\sigma + \left\| {{\tilde{\mathbf{C}}}(t)} \right\|_{T\infty } \\ & \le \left\| {{\mathbf{x}}_{{{\text{in}}}} {\mathbf{(}}t{\mathbf{)}}} \right\|_{T\infty } \int\limits_{0}^{\infty } {\left\| {{\mathbf{\tilde{P}(}}\sigma {\mathbf{)}}} \right\|_{i2} } {\text{d}}\sigma + \left\| {{\tilde{\mathbf{C}}}(t)} \right\|_{T\infty } \, \forall t \le T. \\ \end{aligned} $$

(64)

Therefore, (63) is valid.

1.3 Appendix 3: Proof of Lemma 2–1

M(t), Δ, and G(t) are bounded because elements of those matrices and vectors are combinations of sine/cosines of joint angles. ${\tilde{\mathbf{M}}}$ and ${\tilde{\mathbf{G}}}$ are finite in magnitude because they are the difference in M(t) and G(t) over time. ${\tilde{\mathbf{w}}}(t)$ is bounded (Appendix 5).

Substituting (3), (12) (20) (21) and (22), into (19) yields.

$$ \begin{aligned} {\mathbf{\varepsilon (}}t{\mathbf{)}} & = {\mathbf{\Delta \varepsilon (}}t - L{\mathbf{)}} + {\mathbf{\Delta G}}_{v} {\mathbf{M}}_{s}^{{{\mathbf{ - 1}}}} {\mathbf{K}}_{d} {\tilde{\mathbf{e}}}_{I} {\mathbf{(}}t{\mathbf{)}} \\ & \quad {\mathbf{ + }}\left( {{{\varvec{\Delta}}}\left( {{\mathbf{G}}_{v} {\mathbf{M}}_{s}^{ - 1} {\mathbf{B}}_{d} {\mathbf{ + K}}_{d} - {\mathbf{c}}_{r} } \right){\mathbf{ + M}}^{ - 1} (t){\mathbf{q}}_{2} {\mathbf{(}}t{\mathbf{)}}} \right){\mathbf{\tilde{e}(}}t{\mathbf{)}} \\ & \quad {\mathbf{ + }}\left( {{{\varvec{\Delta}}}\left( {{\mathbf{G}}_{v} {\mathbf{ + B}}_{d} } \right){\mathbf{ + M}}^{ - 1} (t){\mathbf{q}}_{1} (t)} \right){\mathbf{\tilde{\dot{e}}(}}t) \\ & \quad {\mathbf{ + }}\left( {{\mathbf{M}}^{ - 1} (t){\tilde{\mathbf{q}}}_{2} } \right){\mathbf{e(}}t - L{\mathbf{) + }}\left( {{\mathbf{M}}^{ - 1} (t){\tilde{\mathbf{q}}}_{1} } \right){\mathbf{\dot{e}(}}t - L) \\ & \quad - \left( {{\mathbf{M}}^{ - 1} (t){\tilde{\mathbf{M}}}(t)} \right){\mathbf{\ddot{e}(}}t - L{\mathbf{)}} + {\mathbf{\Delta c}}_{r} {\tilde{\mathbf{\theta }}}_{d} (t) - {\mathbf{\Delta G}}_{v} {\mathbf{\tilde{\dot{\theta }}}}_{d} (t) \\ & \quad + {\mathbf{M}}^{ - 1} (t)({\mathbf{\tilde{M}(}}t{\mathbf{)\ddot{\theta }}}_{d} {\mathbf{(}}t - L{\mathbf{) + \tilde{Q}}}_{d} (t){\mathbf{ + \tilde{G}}}(t){\mathbf{ + \tilde{w}}}(t)). \\ \end{aligned} $$

(65)

Defining the variables $\mu \,$ and $\delta_{i}$(i = 1, 2, …, 6) as in (24), and $\psi_{1\_G1}$ as

$$ \, \psi_{1\_G1} = \left\| {{\mathbf{\Delta c}}_{r} {\tilde{\mathbf{\theta }}}_{d} (t) - {\mathbf{\Delta G}}_{v} {\mathbf{\tilde{\dot{\theta }}}}_{d} (t)} \right.\left. { + {\mathbf{M}}^{ - 1} (t)\left[ {{\mathbf{\tilde{M}(}}t{\mathbf{)\ddot{\theta }}}_{d} {\mathbf{(}}t - L{\mathbf{) + \tilde{Q}}}_{d} (t){\mathbf{ + \tilde{G}}}(t){\mathbf{ + \tilde{w}}}(t)} \right]} \right\|_{\infty } $$

(66)

and taking the norms on (65) provide us

$$ \begin{gathered} \left\| {{{\varvec{\upvarepsilon}}}(t)} \right\|_{T\infty } \le \mu \left\| {{{\varvec{\upvarepsilon}}}(t - L)} \right\|_{T\infty } + \delta_{1} \left\| {{\tilde{\mathbf{e}}}_{I} (t)} \right\|_{T\infty } + \delta_{2} \left\| {{\tilde{\mathbf{e}}}(t)} \right\|_{T\infty } \hfill \\ \, + \delta_{3} \left\| {{\mathbf{\tilde{\dot{e}}}}(t)} \right\|_{T\infty } + \delta_{4} \left\| {{\mathbf{e}}(t - L)} \right\|_{T\infty } \hfill \\ \, + \delta_{5} \left\| {{\dot{\mathbf{e}}}(t - L)} \right\|_{T\infty } + \delta_{6} \left\| {{\mathbf{\ddot{e}}}(t - L)} \right\|_{T\infty } + \psi_{1\_G1} . \hfill \\ \end{gathered} $$

(67)

By applying Lemma 1 to (67), because $\left\| { \bullet (t - L)} \right\|_{T\infty }$ is smaller than or equal to $\left\| { \bullet (t)} \right\|_{T\infty }$ [20], the following inequality can be derived.

$$\begin{gathered} \left\| {{{\varvec{\upvarepsilon}}}(t)} \right\|_{T\infty } \le \mu \left\| {{{\varvec{\upvarepsilon}}}(t)} \right\|_{T\infty } + \delta_{1} \beta_{4} \left\| {{{\varvec{\upvarepsilon}}}(t)} \right\|_{T\infty } + \delta_{2} \beta_{5} \left\| {{{\varvec{\upvarepsilon}}}(t)} \right\|_{T\infty } \hfill \\ \, + \delta_{3} \beta_{6} \left\| {{{\varvec{\upvarepsilon}}}(t)} \right\|_{T\infty } + \delta_{4} \beta_{1} \left\| {{{\varvec{\upvarepsilon}}}(t)} \right\|_{T\infty } + \delta_{5} \beta_{2} \left\| {{{\varvec{\upvarepsilon}}}(t)} \right\|_{T\infty } \hfill \\ \, + \delta_{6} \beta_{3} \left\| {{{\varvec{\upvarepsilon}}}(t)} \right\|_{T\infty } + \psi_{G1} . \hfill \\ \end{gathered}$$

(68)

Rearrangement of (68) leads to,

$$ \left( {1{-}\mu - \delta_{1} \beta_{4} - \delta_{2} \beta_{5} - \delta_{3} \beta_{6} - \delta_{4} \beta_{1} - \delta_{5} \beta_{2} - \delta_{6} \beta_{3} } \right) \le \psi_{G1} . $$

(69)

Note that $\psi_{1\_G1}$ and $\psi_{G1}$ are finite. □

1.4 Appendix 4: Proof of Lemma 2–2

The outline of the proof of lemma 2–2 is almost identical to that of lemma 2–1. Substituting (3), (12) (20) (21) and (22), into (19) results in.

$$\begin{gathered} {\mathbf{\varepsilon (}}t{\mathbf{)}} \cong {\mathbf{\Delta \varepsilon (}}t - L{\mathbf{)}} \hfill \\ \, {\mathbf{ + \Delta G}}_{v} {\mathbf{M}}_{s}^{ - 1} {\mathbf{K}}_{d} {\tilde{\mathbf{e}}}_{I} {\mathbf{(}}t{\mathbf{)}} \hfill \\ \, {\mathbf{ + }}\left( {{{\varvec{\Delta}}}\left( {{\mathbf{G}}_{v} {\mathbf{M}}_{s}^{ - 1} {\mathbf{B}}_{d} {\mathbf{ + K}}_{d} - {\mathbf{c}}_{r} } \right){\mathbf{ + M}}^{ - 1} (t){\mathbf{q}}_{2} {\mathbf{(}}t{\mathbf{)}}} \right){\mathbf{\tilde{e}(}}t{\mathbf{)}} \hfill \\ \, {\mathbf{ + }}\left( {{{\varvec{\Delta}}}\left( {{\mathbf{G}}_{v} {\mathbf{ + B}}_{d} } \right){\mathbf{ + M}}^{ - 1} (t){\mathbf{q}}_{1} (t)} \right){\mathbf{\tilde{\dot{e}}(}}t) \hfill \\ \, {\mathbf{ + }}\left( {{\mathbf{M}}^{ - 1} (t){\tilde{\mathbf{q}}}_{2} (t)} \right){\mathbf{e(}}t - L{\mathbf{) + }}\left( {{\mathbf{M}}^{ - 1} (t){\tilde{\mathbf{q}}}_{1} (t)} \right){\mathbf{\dot{e}(}}t - L) \hfill \\ \, - \left( {{\mathbf{M}}^{ - 1} (t){\tilde{\mathbf{M}}}(t)} \right){\mathbf{\ddot{e}(}}t - L{\mathbf{)}} \hfill \\ \, + {\mathbf{\Delta c}}_{r} {\tilde{\mathbf{\theta }}}_{d} (t) - {\mathbf{\Delta G}}_{v} {\mathbf{\tilde{\dot{\theta }}}}_{d} (t) \hfill \\ \, + {\mathbf{M}}^{ - 1} (t)({\mathbf{\tilde{M}(}}t{\mathbf{)\ddot{\theta }}}_{d} {\mathbf{(}}t - L{\mathbf{) + \tilde{Q}}}_{d} (t){\mathbf{ + \tilde{G}}}(t){\mathbf{ + \tilde{w}}}(t)) \hfill \\ \, + {\mathbf{M}}^{ - 1} (t){\tilde{\mathbf{\tau }}}_{s} (t) + {\mathbf{\Delta G}}_{v} {\mathbf{M}}_{s}^{ - 1} \int\limits_{t - L}^{t} {{{\varvec{\uptau}}}_{s} (\sigma ){\text{d}}\sigma } . \hfill \\ \end{gathered}$$

(70)

Taking norms of both sides of (70) with defining ψ_{1_G2} as

$$ \begin{gathered} \psi_{1\_G2} = \left\| {{\mathbf{\Delta c}}_{r} {\tilde{\mathbf{\theta }}}_{d} (t) - {\mathbf{\Delta G}}_{v} {\mathbf{\tilde{\dot{\theta }}}}_{d} (t)} \right. + {\mathbf{\Delta G}}_{v} {\mathbf{M}}_{s}^{ - 1} \int\limits_{t - L}^{t} {{{\varvec{\uptau}}}_{s} (\sigma )d\sigma } \hfill \\ \left. { \, + {\mathbf{M}}^{ - 1} (t)\left[ {{\mathbf{\tilde{M}(}}t{\mathbf{)\ddot{\theta }}}_{d} {\mathbf{(}}t - L{\mathbf{) + \tilde{Q}}}_{d} (t){\mathbf{ + \tilde{G}}}(t){\mathbf{ + \tilde{w}}}(t) + {\tilde{\mathbf{\tau }}}_{s} (t)} \right]} \right\|_{\infty } , \hfill \\ \end{gathered} $$

(71)

yields

$$ \begin{gathered} \left\| {{{\varvec{\upvarepsilon}}}(t)} \right\|_{T\infty } \le \mu \left\| {{{\varvec{\upvarepsilon}}}(t - L)} \right\|_{T\infty } + \delta_{1} \left\| {{\tilde{\mathbf{e}}}_{I} (t)} \right\|_{T\infty } + \delta_{2} \left\| {{\tilde{\mathbf{e}}}(t)} \right\|_{T\infty } \hfill \\ \, + \delta_{3} \left\| {{\mathbf{\tilde{\dot{e}}}}(t)} \right\|_{T\infty } + \delta_{4} \left\| {{\mathbf{e}}(t - L)} \right\|_{T\infty } \hfill \\ \, + \delta_{5} \left\| {{\dot{\mathbf{e}}}(t - L)} \right\|_{T\infty } + \delta_{6} \left\| {{\mathbf{\ddot{e}}}(t - L)} \right\|_{T\infty } + \psi_{1\_G2} . \hfill \\ \end{gathered} $$

(72)

From lemma 1 and (72), we can obtain

$$ \begin{gathered} \left\| {{{\varvec{\upvarepsilon}}}(t)} \right\|_{T\infty } \le \mu \left\| {{{\varvec{\upvarepsilon}}}(t)} \right\|_{T\infty } + \delta_{1} \beta_{4} \left\| {{{\varvec{\upvarepsilon}}}(t)} \right\|_{T\infty } + \delta_{2} \beta_{5} \left\| {{{\varvec{\upvarepsilon}}}(t)} \right\|_{T\infty } \hfill \\ \, + \delta_{3} \beta_{6} \left\| {{{\varvec{\upvarepsilon}}}(t)} \right\|_{T\infty } + \delta_{4} \beta_{1} \left\| {{{\varvec{\upvarepsilon}}}(t)} \right\|_{T\infty } + \delta_{5} \beta_{2} \left\| {{{\varvec{\upvarepsilon}}}(t)} \right\|_{T\infty } \hfill \\ \, + \delta_{6} \beta_{3} \left\| {{{\varvec{\upvarepsilon}}}(t)} \right\|_{T\infty } + \psi_{G2} . \hfill \\ \end{gathered} $$

(73)

Rearranging (73) yields,

$$ \left( {1 - \mu - \delta_{1} \beta_{4} - \delta_{2} \beta_{5} - \delta_{3} \beta_{6} - \delta_{4} \beta_{1} - \delta_{5} \beta_{2} - \delta_{6} \beta_{3} } \right)\left\| {{\varvec{\upvarepsilon}}} \right\|_{T\infty } \le \psi_{G2} . $$

(74)

Note that $\psi_{1\_G2}$ and $\psi_{G2}$ are bounded. □

1.5 Appendix 5: Boundedness of ${\tilde{\mathbf{w}}}(t)$

w(t) contains external disturbances and frictions, and the time difference of w(t), ${\tilde{\mathbf{w}}}(t)$, and is bounded. w(t) contains external disturbances and the Coulomb friction, but not the viscous friction. The viscous friction of i^th joint can be written as

$$ b\dot{\theta }_{i} (t) = b\dot{\theta }_{d\_i} (t) + b\dot{e}_{i} (t), $$

(75)

where b denotes viscous friction coefficient; $\dot{\theta }_{i}$ ith is joint angular velocity; $\dot{\theta }_{d\_i}$ ith is joint desired angular velocity; $\dot{e}_{i}$ ith is joint velocity error. Comparing Eq. (21) and Eq. (75) reveals that viscous friction is not included in w(t).

The difference in Coulomb friction over time is in $L_{\infty }^{n}$.

$$\begin{gathered} f_{c\_i} sgn\left( {\dot{\theta }_{i} (t)} \right) - f_{c\_i} sgn\left( {\dot{\theta }_{i} (t - L)} \right) = \hfill \\ \left\{ {\begin{array}{*{20}l} 0 \hfill & \begin{gathered} if \, \dot{\theta }_{i} (t)\dot{\theta }_{i} (t - L) > 0{\text{ or }} \hfill \\ \, \dot{\theta }_{i} (t) = \dot{\theta }_{i} (t - L) = 0 \, \hfill \\ \end{gathered} \hfill \\ {f_{c\_i} sgn\left( {\dot{\theta }_{i} (t)} \right)} \hfill & \begin{gathered} if \, \dot{\theta }_{i} (t - L) = 0 \, and \hfill \\ \, \dot{\theta }_{i} (t) \ne 0 \hfill \\ \end{gathered} \hfill \\ { - f_{c\_i} sgn\left( {\dot{\theta }_{i} (t - L)} \right)} \hfill & \begin{gathered} if \, \dot{\theta }_{i} (t) = 0 \, and \, \hfill \\ \, \dot{\theta }_{i} (t - L) \ne 0 \hfill \\ \end{gathered} \hfill \\ {2f_{c\_i} sgn\left( {\dot{\theta }_{i} (t)} \right)} \hfill & {if \, \dot{\theta }_{i} (t)\dot{\theta }_{i} (t - L) < 0} \hfill \\ \end{array} } \right., \hfill \\ \end{gathered}$$

(76)

where f_{c_i} denotes Coulomb friction. Equation (7) confirms that the difference in Coulomb friction over time is finite, and therefore in $L_{\infty }^{n}$. Therefore, if external disturbance, d(t), is in $L_{\infty }^{n}$, ${\tilde{\mathbf{w}}}(t)$ is in $L_{\infty }^{n}$.