Robust and adaptive dynamic controller for fully-actuated robots in operational space under uncertainties

Abstract

A practical control method that can perform multiple tasks in operational space is proposed for a fully actuated robot with high kinematic redundancy. Its dynamic control is often realized through force-level operational-space control framework, which computes joint torques for the required forces of prioritized multiple tasks. This approach requires an accurate dynamic model that is a major hurdle to overcome for implementation in real robots. To exempt from complex and demanding inverse dynamics computations, the proposed controller incorporates adaptive sliding-mode and online dynamics estimation schemes. The proposed operational space controller has two merits: it can obtain highly accurate control performance without calculating complex robot dynamics, and it can adapt the control parameters online to effectively compensate the uncertainties when the posture of a humanoid robot is substantially changed during operation; thus, a relatively simple, adaptive and robust control is realized for practical use for a kinematically redundant robots in operational space. The effectiveness of the proposed control method is numerically validated using a dynamic simulator. The simulated scenarios include that a fully-actuated humanoid robot undergoes severe changes in its inertia with respect to changes in posture. Experiments with a 23 degrees-of-freedom torque-controlled humanoid, CoMan, verify that the controller can perform three multiple operational-space tasks under uncertain external disturbances with high accuracy.

Appendices

Appendix A: attenuation of noise effects

In the control law (32), the operational space force reference for ith task is given by (22) and (31) as

$$\begin{aligned} {\mathbf{f}}_i = {\mathbf{f }}_{i(t - L)} + \bar{\mathbf{A}}_i({\ddot{{\mathbf {x}}}}_i^* - {\ddot{{\mathbf {x}}}}_{i(t - L)} ). \end{aligned}$$

(A.1)

If a first-order digital LPF is applied to $\mathbf{f}_i$ with the cutoff frequency $\omega$, it can be modified as follows:

$$\begin{aligned} {\mathbf{f }}_i^{f} = \frac{\omega '}{(1 + \omega ')} {\mathbf{f }}_i + \frac{1}{(1 + \omega ')} {\mathbf{f }}_{i(t - L)}^{f} , \end{aligned}$$

(A.2)

where ${\mathbf{f}}_i^{f}$ denotes the output from the filter and $\omega ' \triangleq \omega L$. Substituting (A.1) into (A.2), one can obtain the following filtered control law:

$$\begin{aligned} {\mathbf{f}}_i^{f}&= {\mathbf{f}}_{i(t - L)}^{f} + \frac{\omega '}{(1 + \omega ')} {{\bar{\mathbf{A}}}}_i\left( {\ddot{{\mathbf {x}}}}_i^* - {\ddot{{\mathbf {x}}}}_{i(t - L)} \right) \nonumber \\&= {\mathbf{f}}_{i(t - L)}^{f} + \alpha _\omega \bar{\mathbf{A}}_i\left( {\ddot{{\mathbf {x}}}}_i^* - {\ddot{{\mathbf {x}}}}_{i(t - L)} \right) , \end{aligned}$$

(A.3)

where $\alpha _\omega \triangleq \omega '(1 + \omega ')^{ - 1}$ denotes a positive number lower than 1, i.e., $0<\alpha _\omega <1$. Comparing (A.1) with (A.3), one can conclude that lowering the elements of ${{{\bar{\mathbf{A}}}}}_i$ has the same effect as using the first-order digital LPF for the operational space force references calculated in the proposed control law. With excessively high gains, the control performance will be degraded due to the noise, with too small gains the control performance will also be degraded due to the small gain. In this paper, as a practical way referring to the calculation of ${\bar{\mathbf{A}}}_i$ shown in Eq. (20), the diagonal ${\bar{\mathbf{M}}}$ is reduced with $\alpha _\omega$. The same filtering effect is thus simply applied to all tasks.

Appendix B: Proof of stability

The bounded-input-bounded-output (BIBO) stability of the overall closed-loop system with the proposed control law can be shown in the same manner of Jin et al. (2009).

Substituting the control force input (22) into (18) yields the closed-loop error dynamics shown in Eq. (27). If $\mathbf{J}_{(i|P)}$ is assumed to be full rank, Eq. (27) can be expressed as

$$\begin{aligned}&({\ddot{{\mathbf {x}}}}_i^* - {\ddot{{\mathbf {x}}}}_i ) = \varvec{\epsilon }_i, \quad \text { or } \quad \end{aligned}$$

(B.1)

$$\begin{aligned}&{\ddot{\mathbf{e}}}_i + 2 \lambda {\dot{\mathbf{e}}}_i + \lambda ^2 \mathbf{e}_i = \varvec{\epsilon }_i, \end{aligned}$$

(B.2)

where $\varvec{\epsilon }_i \triangleq \bar{\mathbf{A}}_i^{-1} ({\bar{{\varvec{\eta }}}}_i - {\breve{{\varvec{\eta }}}}_i)$ denotes the remaining error of the TDE for the ith operational-space task. Multiplying ${{\bar{\mathbf{A}}}}_i$ to (B.1) and combining (8),(9), (12), (22), and (31) give the following:

$$\begin{aligned} {\mathbf{A}}_i \varvec{\epsilon }_i&= \left( \mathbf{A}_i - {{\bar{\mathbf{A}}}}_i\right) {\ddot{{\mathbf {x}}}}_i^* + \left( {\bar{{\varvec{\eta }}}}_i - {\breve{{\varvec{\eta }}}}_i\right) ,\nonumber \\&= \left( \mathbf{A}_i - {{\bar{\mathbf{A}}}}_i\right) {\ddot{{\mathbf {x}}}}_i^* + {\bar{{\varvec{\eta }}}}_i - [ \mathbf{f}_{i(t-L)} - {{\bar{\mathbf{A}}}}_i {\ddot{{\mathbf {x}}}}_{i(t-L)} ],\nonumber \\&= \left( \mathbf{A}_i - {{\bar{\mathbf{A}}}}_i\right) {\ddot{{\mathbf {x}}}}_i^* + \left[ \bar{\mathbf{A}}_i - {\mathbf{A}}_{i(t-L)}\right] {\ddot{{\mathbf {x}}}}_{i(t-L)}\nonumber \\&\quad +\, \left[ {{{\varvec{\eta }}}}_i - {{{\varvec{\eta }}}}_{i(t-L)} \right] . \end{aligned}$$

(B.3)

Then, substituting ${{\varvec{\epsilon }}}_{i(t-L)}$ from (B.1) into (B.3) yields

$$\begin{aligned} {{\varvec{\epsilon }}}_{i} = \left( \mathbf{I} - \mathbf{A}_i^{-1} \bar{\mathbf{A}}_i\right) {{\varvec{\epsilon }}}_{i(t-L)} + \varvec{\kappa }_{1,i} + \varvec{\kappa }_{2,i}, \end{aligned}$$

(B.4)

where $\varvec{\kappa }_{1,i} \triangleq (\mathbf{I} - \mathbf{A}_i^{-1} {{\bar{\mathbf{A}}}}_i) [{\ddot{{\mathbf {x}}}}_i^* - {\ddot{{\mathbf {x}}}}_{i(t-L)}^*]$ and $\varvec{\kappa }_{2,i} \triangleq \mathbf{A}_i^{-1} (\mathbf{A}_i - \mathbf{A}_{i(t-L)}) {\ddot{{\mathbf {x}}}}_{i(t-L)} + [{{{\varvec{\eta }}}}_i - {{{\varvec{\eta }}}}_{i(t-L)} ]$ which are bounded for a sufficiently small L. Refer to Jin et al. (2009); Jung et al. (2004); Su et al. (2000) for details on the boundedness of $\varvec{\kappa }_{1,i}$ and $\varvec{\kappa }_{2,i}$.

In the discrete-time domain, equation (B.4) for kth sampling can be represented as

$$\begin{aligned} {{\varvec{\epsilon }}}_{i(k)} = \left( \mathbf{I} - \mathbf{A}_{i(k)}^{-1} {\bar{\mathbf{A}}}_{i(k)}\right) {{\varvec{\epsilon }}}_{i(k-1)} + \varvec{\kappa }_{1,i(k)} + \varvec{\kappa }_{2,i(k)}. \end{aligned}$$

(B.5)

From the viewpoint of ${{\varvec{\epsilon }}}_{i(k)}$, $\varvec{\kappa }_{1,i(k)}$, and $\varvec{\kappa }_{2,i(k)}$ are forcing functions.

Suppose that the desired trajectory and initial error are bounded, then the following condition assures the boundedness of ${{\varvec{\epsilon }}}_{i}$:

$$\begin{aligned} ||\mathbf{I} - \mathbf{A}_{i}^{-1} {{\bar{\mathbf{A}}}}_i ||< 1. \end{aligned}$$

(B.6)

For the adaptive gain ${{\bar{\mathbf{A}}}}_{i}$, one can prevent the drift using a saturation technique with $\gamma ^-$ and $\gamma ^+$ of ${\bar{m}}_j$ as shown in (23) and (25); this can limit the search space of the gain ${{\bar{\mathbf{A}}}}_{i}$, which is widely accepted for adaptive control. Therefore, ${{\varvec{\epsilon }}}_{i}$ in the closed-loop error dynamics of ith task, shown in Eq. (B.1) and (B.2), is bounded, and the overall system is BIBO stable.

Appendix C: Control of floating base system

The dynamics of a floating base system such as a humanoid robot can be expressed as two coupled dynamic equations: floating base dynamics of six under-actuated DoFs and the rigid body dynamics of n-DoFs actuated joints. Its equation of motions can be represented as follows (Righetti et al. 2011; Henze et al. 2016):

(C.1)

where denote the inertia and the Coriolis matrices, respectively, $\ddot{\mathbf{x}}_b, \dot{\mathbf{x}}_b \in {{\mathbb {R}}}^6$ denote acceleration and velocity of the floating base, respectively, denotes a gravity vector, denotes nonlinear effects, and denotes a vector of the generalized external forces.

The dynamic equation can be expressed as the following partitioned form (Featherstone 2014):

(C.2)

where $\mathbf{M}_b^c \in {{\mathbb {R}}} ^{ 6 \times 6 }$ denotes the composite rigid body inertia of the robot, ${\mathbf{F}}_s \in {{\mathbb {R}}}^{ 6 \times n }$ denotes a matrix which includes the spacial forces required at the floating-based system, and $\mathbf{h}_b^c$ is the spatial bias force for the composite rigid body containing the full floating-based system while $\mathbf{h}$ is the corresponding force vector of Centrifugal, gravity, friction, and external torques of the actuated joints.

Multiplying $\mathbf{F}_s^T (\mathbf{M}_b^c)^{-1}$ to the first row of (C.2) and subtracting from the second row yield

(C.3)

From (C.3), one can obtain the dynamic equation of fully-actuated joints as follows:

$$\begin{aligned} {{\varvec{\tau }} } = \left[ \mathbf{M}(\mathbf{q}) - \mathbf{F}_s^T \left( \mathbf{M}_b^c\right) ^{-1}{} \mathbf{F}_s\right] \ddot{\mathbf{q}} + \mathbf{h} - \mathbf{F}_s^T \left( \mathbf{M}_b^c\right) ^{-1} \mathbf{h}_b^c, \end{aligned}$$

(C.4)

which provides a direct relation between ${{\varvec{\tau }} }$ and $\ddot{\mathbf{q}}$ influenced by the dynamics of the floating base (Nakanishi et al. 2007). Since the coefficients of (C.4) can be regarded as those of the standard fully-actuated joint dynamics, one can rearrange (C.4) in the same form of (1) as follows:

$$\begin{aligned} {{\varvec{\tau }} } = \mathbf{M}' \ddot{\mathbf{q}} + \mathbf{c} + \mathbf{g} + {{\varvec{\nu }}} + {{\varvec{\tau }}}_d' \end{aligned}$$

(C.5)

with

$$\begin{aligned} \left\{ \begin{array}{l} \mathbf{M}' \triangleq \mathbf{M}(\mathbf{q}) - \mathbf{F}_s^T (\mathbf{M}_b^c)^{-1}{} \mathbf{F}_s \\ {{\varvec{\tau }}}_d' \triangleq {{\varvec{\tau }}}_d - \mathbf{F}_s^T (\mathbf{M}_b^c)^{-1} \mathbf{h}_b^c. \end{array} \right. \end{aligned}$$

Therefore, the proposed controller (32) can be applied to the actuated joints in the same manner derived in Sect. 2.2. The dynamic effects due to the floating base are regarded as additional disturbances in the composite dynamic term $\varvec{\zeta }(\mathbf{q}, \dot{\mathbf{q}}, \ddot{\mathbf{q}})$ in (17), which are effectively tackled by the adaptive gain and TDE component in the controller.