1 Introduction↩︎

Robotic systems are increasingly deployed in dynamic and uncertain environments, where safety, stability, and high performance are essential. Achieving reliable operation in such settings is particularly challenging due to the nonlinear Euler–Lagrange (EL) dynamics of the robots, and due to the effects of uncertain parameters and unmodeled terms. In these contexts, control strategies must operate reliably despite incomplete system knowledge, while enforcing state constraints to ensure task feasibility and safety. The simultaneous need for uncertainty compensation and real-time constraint satisfaction defines a central challenge in robotics.

1.1 Related Works and Motivation↩︎

Artificial time delay control (also known as time-delay estimation or TDE-based control) was introduced to reduce reliance on system models by approximating unknown dynamics using delayed state/input measurements [1], [2]. Its simplicity and low computational cost have led to widespread adoption in robotics [3]–[10]. However, most TDE-based methods [3]–[10] assume an a priori known bound on the approximation error—an assumption that limits robustness to fast-varying or state-dependent uncertainties. To overcome this limitation, adaptive TDE control frameworks have been proposed by [11]–[13], wherein a structured state-dependent upper bound on the TDE error is analytically formulated and its parameters are updated online via adaptive laws. However, these methods do not incorporate mechanisms for enforcing state constraints, making them unsuitable for safety-critical applications where real-time enforcement of position and velocity limits is essential.

In many practical applications, ensuring that system states remain within prescribed bounds is as critical as achieving tracking accuracy. Robotic arms operating near humans, aerial vehicles navigating constrained spaces, or manipulators handling delicate payloads, all demand strict enforcement of position and velocity limits for safety and reliability. While Model Predictive Control (MPC) can tackle such constraints, it requires accurate model and imposes high computational overhead — making it unsuitable for many embedded robotic systems [14]–[17]. Barrier Lyapunov Functions (BLFs), by contrast, encode constraints directly into the control design, enabling real-time enforcement without online optimization [18]–[22].

Although early BLF-based control strategies required precise knowledge of system dynamics, more recent designs have relaxed this dependency by incorporating robust [23]–[25] or adaptive [26]–[31] mechanisms. However, the majority of these methods enforce position-only (i.e., partial state) constraints, which significantly reduce flexibility (cf. [18], [19], [24], [25], [27]). To address this, time-varying BLF formulations have been developed ([21], [32]–[35]), offering improved constraint scheduling. Nevertheless, many of these approaches still rely on accurate model information (cf. [21], [32], [33]) or assume a priori bounded uncertainty (cf. [34], [35]). This is a critical limitation for EL systems, where uncertainty is inherently state-dependent and cannot be bounded in advance. As shown in [36], relying on a priori bounded uncertainty in adaptive control design can even compromise stability.

1.2 Contribution↩︎

These limitations motivate the development of an adaptive-robust control framework that explicitly addresses both constraint enforcement and uncertainty adaptation in a unified manner. This framework simultaneously addresses state-dependent uncertainty and time-varying position and velocity constraints—two critical but often separately treated challenges in nonlinear control. The key contributions are:

• State-dependent uncertainty adaptation: A novel TDE formulation is derived with an explicit, structured upper bound on the approximation error. The bound’s parameters are adapted online, enabling compensation of unknown, state-dependent terms.

• Time-varying constraint enforcement: A BLF-based control strategy is developed to rigorously enforce time-varying position and velocity constraints, accommodating initial errors and transients.

• Cohesive design: The controller tightly couples adaptive uncertainty estimation and constraint satisfaction in a provably stable architecture. Stability and constraint compliance are formally established through Lyapunov analysis.

To the best of our knowledge, this is the first framework to couple Adaptive TDE with BLF-based constraint enforcement for nonlinear EL-based robotic systems. The approach is validated on a 5-degrees-of-freedom (DoF) robotic manipulator, outperforming the state of the art.

The rest of the paper is organised as follows: Section II describes the EL robotics dynamics and the control problem; Section III details the proposed control framework, while corresponding stability analysis is provided in Section IV; comparative experimental results are in Section V, while Section VI provides concluding remarks.

The following notations are used in the paper: \((\bullet)_{L}\) denotes that \((\bullet)(t)\) is delayed by \(L\), i.e., \((\bullet)(t-L)\); \(\lambda _{\min}(\bullet)\), \(\lambda _{\max}(\bullet)\), \(|| \bullet ||\), \(| \bullet |\) denote minimum and maximum eigenvalue, 2-norm and absolute value of \(\bullet\) respectively; \({I}_n \in\mathbb{R}^{n\times n}\) denotes Identity matrix; \(\mathop{\mathrm{diag}}\lbrace \cdot, \cdots, \cdot \rbrace\) denotes a diagonal matrix with diagonal elements \(\lbrace \cdot, \cdots, \cdot \rbrace\); \(\exp\) denotes exponential function; the saturation function \(\mathop{\mathrm{sat}}({s,\varpi})=(s/||s||)\) if \(||s||\geq \varpi\) and \(\mathop{\mathrm{sat}}({s,\varpi})=(s/\varpi)\) if \(||s|| < \varpi\) with \(\varpi\in \mathbb{R}^{+}\).

2 System Dynamics and Problem Formulation↩︎

Let us consider the following class of robots modeled via Euler-Lagrange (EL) dynamics [3]–[10]: \[\begin{align} M( q(t))\ddot{ q}(t) &+ H( q(t),\dot{ q}(t), t)= \tau(t), \label{eq:TDdynamics} \end{align}\tag{1}\] where \({q}, \dot{ q}\in\mathbb{R}^{n}\) are the system states (position and velocity), \(\tau\in\mathbb{R}^{n}\) is the generalized control input, \({M}\in\mathbb{R}^{n\times n}\) is the mass/inertia matrix and \(H \in\mathbb{R}^{n}\) represents the combined effects of dynamic forces (e.g., Coriolis, dissipative, restoring, external disturbances etc.).

The following system property holds from standard EL mechanics [37]:

Property 1: The matrix \({M}( q)\) is uniformly positive definite for all \(q\), i.e., \(\exists \psi_1, \psi_2 \in \mathbb{R}^{+}\) such that \[\psi_1 {I} \leq {M}( q) \leq \psi_2 {I} \Rightarrow (1/\psi_2) {I} \leq {M}^{-1}( q) \leq (1/\psi_1) {I}.\label{prop1}\tag{2}\]

Let us introduce the amount of uncertainty in system as follows:

Introducing a constant diagonal matrix \({\bar { M}}\), the dynamics (1 ) can be compactly written as \[\begin{align} &{\bar{ M}\ddot { q}} + { {N}}({ {q}},{{\dot{ q}},\ddot{ q}}) = { \tau },\tag{3}\\ \text{with}~~~~&{ {N}}({ {q}},{{\dot{ q}},\ddot{ q}}) = [{ {M}}({ {q}}) - {\bar { M}}]{\ddot { q}} + H( q,\dot{ q})\tag{4} \end{align}\] and the selection of \({\bar { M}}\) is discussed later (cf. Remark 3). The desired trajectories \({q}^d(t)\) and their time-derivatives \(\dot{ q}^d(t), \ddot{ q}^d(t)\) are designed to be smooth and bounded. The signals \(({ {q}}(t),{{\dot{ q}}(t),\ddot{ q}}(t))\) are available for control design. Control Objective: Let the position and velocity tracking errors \(( {e}, \dot{ {e}})\) be defined as \[\label{error} \begin{align} {e} = {q} - {q}^d ; ~~~\dot{ {e} } = \dot{ {q}} - \dot{ {q}}^d. \end{align}\tag{5}\] Let us decompose \(e=[e_1~ \;e_2 \ldots e_n]^T\), \(\dot{e}=[\dot{e}_1~ \dot{e}_2 \ldots \dot{e}_n]^T\), where \(e_i\) and \(\dot{e}_i\) the tracking errors for the \(i^{th}\) degree-of-freedom, \(i=1,2, \cdots, n,\). Since we are interested in tracking control, the problem of imposing constraints on states \({q}, \dot{ q}\) can be equivalently considered to be the problem of imposing constraints on tracking errors \({e}, \dot{ e}\) around desired trajectories (cf. (5 )). Accordingly, we define constraint functions \[\label{eq:bounds95p} \begin{align} k_{pi}(t) &= (k_{0pi} - k_{sspi}) \exp^{-\alpha_{p} t} + k_{sspi}, k_{0pi} > {e}_i (0) \\ k_{vi}(t) &= (k_{0vi} - k_{ssvi}) \exp^{-\alpha_{v} t} + k_{ssvi}, k_{0vi}> \dot{e}_i (0) \end{align}\tag{6}\] where \(\alpha_{p}, ~\alpha_{v} \in \mathbb{R}^{+}\) are scalars; \((k_{0pi}, k_{0vi})\) and \((k_{sspi}, k_{ssvi} )\) are the initial and steady-state values of the respective constraints. Our control objective is to keep the tracking errors within the constraints as \(|e_{i}(t)|< k_{pi}(t) , ~| \dot{e}_{i}(t)|< k_{vi}(t)\) for all time \(t \geq 0\). In the following, we discuss the design of the constraint functions (6 ).

3 Proposed Adaptive Control Solution↩︎

Variable dependency will be removed subsequently for brevity whenever it is obvious. The control input \(\tau\) is designed as \[\begin{align} \tau &={\bar{ M} u + \hat{ N}}({ {q}},{{\dot{ q}},\ddot{ q}}), \tag{7}\\ \text{with} ~~ {u} &= {u}_0- \Delta u, \tag{8} \\ \hat{ N}( q,\dot{ q} , \ddot{ q}) & \cong N( q_L,\dot{ q}_L,\ddot{ q}_L)= \tau_L-\bar{ M}\ddot{ q}_L, \tag{9}\\ u_0 & = {\ddot { q}^d} - D_{D}\dot{ e}- D_{P} e, \tag{10}\\ {D}_{P}& = \mathop{\mathrm{diag}}{\bigg\{\frac{1}{k_{p1}^{2}- e^{2}_1}, \cdot \cdot \cdot, \frac{1}{k_{pn}^{2}- e^{2}_n}\bigg\}} , \tag{11} \\ {D}_{D}&=\mathop{\mathrm{diag}}{\bigg\{\frac{1}{k_{v1}^{2}- \dot{e}^{2}_1}, \cdot \cdot \cdot, \frac{1}{k_{vn}^{2}- \dot{e}^{2}_n}\bigg\}}, \tag{12} \end{align}\] and \(\Delta u\) is the adaptive control term to be designed later; \(\hat{ N}\) is the estimate of the uncertainty function \(N\) derived from the past state/input data; \(L>0\) is a small time delay introduced to collect the past data and its choice is discussed later (cf. Remark 4). This estimation process via artificial/intentional injection of delay is conventionally called time-delay estimation (TDE).

Substituting (7 ), (8 ) and (10 ) into (3 ) gives the following error dynamics: \[\begin{align} \ddot{ e}& =- D_{D}\dot{ e}- D_{P} e + \sigma- \Delta u, \label{eq:errdynamics951} \end{align}\tag{13}\] where \(\sigma={\bar{ M}}^{-1}({ N}-{\hat{ N}})\) is the estimation error stemming from TDE process (9 ) and it is termed as the TDE error. The adaptive control term \(\Delta u\) is designed based on the structure of the upper bound of TDE error \(|| \sigma||\) derived subsequently.

3.1 Upper bound structure of \(|| \sigma||\)↩︎

From (1 ) and (13 ), the following relations can be achieved: \[\begin{align} \hat{ N}&= {N}_L=[{ {M}}({ {q}_L}) - {\bar { M}}]{\ddot { q}_L} + H_L ,\tag{14} \\ \sigma&=\ddot{ q}- u. \tag{15} \end{align}\] Using (14 ), the control input \(\tau\) in (7 ) can be rewritten as \[\begin{align} \tau &= \bar{ M} u+[{ {M}}({ {q}_L}) - {\bar { M}}]{\ddot { q}_L} + H_L. \label{tau32new} \end{align}\tag{16}\] Multiplying both sides of (15 ) with \(M\) and using (1 ) and (16 ) \[\begin{align} {M} \sigma &= \tau - H- {M} {u}, \nonumber \\ & = \bar{ M} u+[{ {M}}({ {q}_L}) - {\bar { M}}]{\ddot { q}_L} + H_L - H - {M} {u}. \label{sig323} \end{align}\tag{17}\] Defining \({D} \triangleq [ {D}_P ~ {D}_D]\), \(\xi \triangleq \begin{bmatrix} { e}^T& \dot{ e}^T \end{bmatrix}^T\) and using (13 ) we have \[\begin{align} \ddot{ {q}}_L& = \ddot{ {q}}^d_L - D \xi_L + \sigma_L - \Delta u_L.\label{sig324} \end{align}\tag{18}\] Substituting (18 ) into (17 ), and after re-arrangement yields \[\begin{align} \sigma &= \underbrace{ {M}^{-1}\bar{ {M}}( \Delta u_L - \Delta u}_{ \chi_1}) + \underbrace{ {M}^{-1}( {M} \Delta u - {M}_L \Delta u_L)}_{ \chi_2} \nonumber\\ &+\underbrace{ {M}^{-1}\lbrace (\bar{ {M}} - {M} )\ddot{ {q}}^d + ( {M}_L - \bar{ {M}})\ddot{ {q}}^d_L+ H_L - H \rbrace }_{ \chi_3} \nonumber\\ &+ \underbrace{( I - {M}^{-1}\bar{ {M}}) {D} \xi}_{ \chi_4} + \underbrace{ {M}^{-1}(\bar{ {M}} - {M}_L) {D} \xi_L}_{ \chi_5} \nonumber\\ &+ \underbrace{ {M}^{-1}( {M}_L-\bar{ {M}}) \sigma_L}_{ \chi_6}. \label{sig325} \end{align}\tag{19}\]

Both \({M}\) and \({M}^{-1}\) are bounded from system property (2 ). Using the relation \((\cdot)_L= (\cdot)(t)-\int_{-L}^0 {\frac{\mathrm{d}}{\mathrm{d}\theta}(\cdot)}(t+\theta)\mathrm{d}\theta\) and the fact that integration of any continuous function over a finite interval (here \(-L\) to \(0\)) is always finite [38], the following conditions hold for unknown constants \(\delta_i\), \(i=1, \cdots,6\): \[\begin{align} &|| { \chi_1} ||=|| -{M}^{-1}\bar{ {M}} \int_{-L}^{0} \frac{\mathrm{d}}{\mathrm{d}\theta} \Delta { u}(t+\theta) \mathrm{d}\theta || \leq \delta_1 \tag{20}\\ &|| { \chi_2} ||=|| {M}^{-1} \int_{-L}^{0} \frac{\mathrm{d}}{\mathrm{d}\theta} {M}(t+\theta) \Delta { u}(t+\theta) \mathrm{d}\theta || \leq \delta_2 \tag{21}\\ &|| { \chi_3} ||=|| {M}^{-1}\lbrace (\bar{ {M}} - {M} )\ddot{ {q}}^d + ( {M}_L - \bar{ {M}})\ddot{ {q}}^d_L \nonumber\\ &\qquad \qquad \qquad -\int_{-L}^{0} \frac{\mathrm{d}}{\mathrm{d}\theta}{ H}(t+\theta)\mathrm{d}\theta \rbrace ||\leq \delta_3 \tag{22}\\ &|| { \chi_4} ||= || ( I - {M}^{-1}\bar{ {M}}) {D} \xi || \leq || {E} {D}|| || \xi|| \tag{23}\\ &|| { \chi_5} ||=|| {M}^{-1} \int_{-L}^{0} \frac{\mathrm{d}}{\mathrm{d}\theta}( \bar{ {M}} - {M}(t+\theta)) {D} \xi(t+\theta) \mathrm{d}\theta \\ & \qquad \qquad +( I - {M}^{-1}\bar{ {M}} ) {D} \xi ||\leq || E D || || \xi || + \delta_5 \tag{24}\\ &|| {\chi_6} || = || E \sigma + {M}^{-1}\int_{-L}^{0}\frac{\mathrm{d}}{\mathrm{d}\theta}\lbrace( {M}(t+\theta)-\bar{ {M}}) \sigma (t+\theta) \rbrace \mathrm{d}\theta || \nonumber \\ & \qquad \leq ||{ E}|| || \sigma || + \delta_6 . \tag{25} \end{align}\] Here \({M}\) and \(H\) are explicitly represented in time for ease of notation. The upper bound of \(|| \sigma ||\) is formulated using (20 )-(25 ) from (19 ) as \[\begin{align} \lVert \sigma \rVert &\leq \beta_{0}^{*} + \beta_{1}^{*} || {D} || \lVert \xi \rVert, \label{eq:upbound} \\ \text{where}~ & \beta_{0}^{*} =\frac{\sum_{i=1}^{6}\delta_i}{1-\lVert E \rVert}, ~ \beta_{1}^{*} = \frac{2 \lVert E \rVert }{1-\lVert E\rVert} \nonumber \end{align}\tag{26}\] under the following condition \[\begin{align} || E ||= || {I} - {M}^{-1}\bar{ M} || <1. \label{condition} \end{align}\tag{27}\]

3.2 Adaptive law design↩︎

The error dynamics (13 ) can be re-written in form of \[\begin{align} \dot{ \xi}& = ( A_0 + {\bar{A}}) \xi+ {B} ( \sigma- \Delta u), \label{eq:errdynamics952} \end{align}\tag{28}\] where \({B}=\begin{bmatrix} {0}\\ {I}_{n} \end{bmatrix}\), \(A_0 = \begin{bmatrix} - \lambda_P & 0 \\ 0 & - \lambda_D \end{bmatrix}\), \({\bar{A}} = \begin{bmatrix} \lambda_P & {I}_n \\ - {D}_P & - {D}_{D}+ \lambda_D \end{bmatrix}\) and \(\lambda_P= \mathop{\mathrm{diag}}\lbrace \lambda_{p1}, \cdots, \lambda_{pn} \rbrace, \lambda_D = \mathop{\mathrm{diag}}\lbrace \lambda_{d1}, \cdots, \lambda_{dn} \rbrace\) are user-defined positive definite diagonal matrices with \(\lambda_{pi}, \lambda_{di} \in \mathbb{R}^{+}\) for each \(i=1,2, \cdots,n\).

The term \(\Delta u\) is designed as \[\begin{align} \label{eq:auxinput952} \Delta { {u}}(t) &= \rho \frac{ s}{|| s||} , \end{align}\tag{29}\] where \({s=B}^T {D_\xi} \xi\) and \(D_{\xi =}\) diag\(\{ {D}_{P}, {D}_{D} \}\). The adaptive gain \(\rho\) in (29 ) is formulated based on the upper bound structure of \(|| \sigma ||\) from (26 ) as \[\begin{align} \rho=\hat{\beta}_0+\hat{\beta}_1 || {D}|| || \xi || + \zeta, \label{sw32gain} \end{align}\tag{30}\] where \(\hat{\beta}_0, \hat{\beta}_1\) are the estimates of \(\beta_0, \beta_1 \in \mathbb{R}^{+}\), respectively and adapted via the following laws: \[\tag{31} \begin{align} &\dot{\hat{\beta}}_{0} = || s || - \nu_0\hat{\beta}_{0},~\hat{\beta}_{0} (0) > 0, \tag{32} \\ &\dot{\hat{\beta}}_{1} = || s || || D|| || \xi|| - \nu_1 \hat{\beta}_{1},~\hat{\beta}_{1} (0) > 0, \tag{33} \\ &\dot{\zeta} = -\left( 1 + || D_{\xi}||||\bar{ A}|||| \xi||^2 \right. \tag{34} \\ &\left. \quad - \sum_{i=1}^{n} \frac{\dot{k}_{pi}}{k_{pi}} \left[ \frac{e_{i}^{2}}{k_{pi}^{2}-e_{i}^{2}} \right] - \sum_{i=1}^{n} \frac{\dot{k}_{vi}}{k_{vi}} \left[ \frac{\dot{e}_{i}^{2}}{k_{vi}^{2}- \dot{e}_{i}^{2}} \right] \right) \zeta + \epsilon, \nonumber \end{align}\] where \(\nu_i, \epsilon \in\mathbb{R}^{+}\) are user-defined scalars. Finally, combining (7 ) and (29 ) becomes \[\label{theproposedcontrollaw} \begin{align} { {\tau }} &= \underbrace {{{ {\tau }}_L} - {\bar{ M}}{{{\ddot { q}}}_L}}_{{\text{TDE~part}}}+\underbrace {{\bar{ M}}({{{\ddot { q}}}^d} + {{ {D}}_D}{\dot{ e}} + {{ {D}}_P}{ {e}})}_{{\text{Desired~error~constraint~part}}} +\underbrace { \rho { s}/{|| s||} }_{{\text{Adaptive~part}}}. \end{align}\tag{35}\]

4 Closed-Loop System Stability↩︎

5 Experimental Validation and Results↩︎

The experimental platform consisted of UFactory 5-DoF xArm-5 robotic manipulator (cf. Fig. 2), with an NVIDIA Jetson AGX Xavier serving as the compute unit. A custom-designed end-effector was attached, incorporating a Dynamixel XM430-W210-T servo motor to enable rapid switching between two tools: a marker and an eraser. This configuration allows for seamless tool transitions during task execution. Real-time joint state data were acquired via the manipulator’s internal feedback interface, supporting synchronized, high-frequency closed-loop control.

5.1 Experimental Scenario↩︎

The robot was commanded to perform a constrained drawing and erasing task on a surface: drawing of \(3\) layers of concentric semicircles and then erase the middle semicircle without erasing/ disturbing the inner and outer semicircles. Involving physical contact and tight position and velocity limits in the task was designed to evaluate tracking performance and constraint handling under state-dependent and external disturbances arising from unknown surface contact forces and frictional variations at the end-effector. The specific sequences of the task are as follows:

• Starting from the initial position, draw an innermost semicircle with a radius of \(20\) cm; lift the pen and come back to the initial position.

• Draw an outermost semicircle with a radius of \(24\) cm; lift the pen and come back to the initial position.

• Draw a middle semicircle with a radius of \(22\) cm; lift the pen and come back to the initial position.

• Rotate the tool (from drawing to erasing) and erase the middle (\(22\) cm) semicircle using the eraser starting from the initial position.

Both the drawing and erasing of the middle semicircle occur in a confined region between the inner and outer paths, thereby emphasizing the critical role of real-time constraint enforcement to ensure safe and feasible motion. The desired trajectory \(({ q}^d, \dot{ q}^d)\) for the manipulator is pre-computed using the open-source package named ruckig [40]. The \(4\) cm radial distance between the inner and outer semicircles was selected to accommodate the \(2\) cm wide eraser. It is important to note that the initial condition of the robot is always set at zero (as provided by the manufacturer). Therefore, the initial error always starts from zero (cf. the experimental error plots later).

To evaluate the significance of the proposed control strategy, its performance was compared against two representative baselines: (i) an adaptive BLF-based controller [32] (referred as ABLF) and (ii) an adaptive TDE-based controller [11] (referred ATDC). The parameters used in the experiments for the proposed controller are listed in Table 1 along with parameters to design the constraints. For both baselines, the controller gains were carefully tuned according to the respective design guidance to achieve their best possible performance under identical task conditions.

5.2 Results and Analysis↩︎

The performance of the controllers is illustrated in Figs. 3–5. Figure 3 clearly illustrates that the semicircles drawn by the proposed controller closely match the intended trajectories, while those produced by the baseline controllers exhibit noticeable distortions. The results clearly demonstrate that the proposed controller outperforms the baselines in both tracking accuracy and constraint satisfaction. As shown in the tracking error plots 4–5, it consistently keeps the tracking errors within the prescribed position and velocity bounds—a direct result of its integrated adaptive time-delay estimation and BLF-based constraint handling. These qualitative trends are further substantiated by the quantitative results in Table 2, where the proposed controller achieves the lowest Root Mean Square (RMS) errors across all joints. ABLF, equipped with constraint handling capability, could honor the constraints; however, it cannot negotiate state-dependent and unmodelled uncertainties, leading to distorted drawings. Although ATDC can tackle state-dependent uncertainty, it lacks the ability to handle constraint; consequently, ATDC partially erases the inner and outer semicircles, which is unintended. These observations highlight that the capability to handle either the state-dependent uncertainty or the state constraints alone is not sufficient; a control framework, like the proposed one, must combine both these features to achieve the desired result.

6 Conclusions↩︎

This paper presented a unified adaptive-robust control framework that integrates time-delay estimation with BLF-based constraint enforcement for Euler–Lagrange robotic systems under state-dependent uncertainties. The proposed approach achieves real-time tracking while rigorously satisfying time-varying position and velocity constraints—without requiring prior model knowledge. Experimental validation on a 5-DoF robotic manipulator demonstrated the superior performance of the proposed controller over the state-of-the-art in contact-rich tasks, achieving lower tracking errors and strict constraint compliance. A future work would be to extend the proposed framework for underactuated robotic systems.

References↩︎