Multi‐input enhanced model reference adaptive control strategies and their application to space robotic manipulators

The Enhanced Model Reference Adaptive Control (EMRAC) algorithm, augmenting the MRAC strategy with adaptive integral and adaptive switching control actions, is an effective solution to impose reference dynamics to plants affected by parameter uncertainties, unmodeled dynamics and disturbances. However, the design of the EMRAC solutions has so far been limited to single‐input systems. To cover the gap, this paper presents two extensions of EMRAC to multi‐input systems. The adaptive mechanism of both solutions includes the σ$$ \sigma $$ ‐modification strategy to assure the boundedness of the adaptive gains also in presence of persistent disturbances. The closed‐loop system is analytically studied, and conditions for the asymptotic convergence of the tracking error are presented. Furthermore, when the plant is subjected to unmatched disturbances, the ultimate boundedness of the closed‐loop dynamics, which are made discontinuous by the adaptive switching control actions, is systematically proven by using Lyapunov theory for Filippov systems. The problem of trajectory tracking for space robotic arms in presence of unknown and noncooperative targets is used to test the effectiveness of the novel multi‐input EMRAC algorithms for taming uncertain systems. Four EMRAC solutions are designed for this engineering application, and tested within a high fidelity simulation framework based on the Robot Operating System. Finally, the tracking performance of the EMRAC implementations is quantitatively evaluated via a set of key performance indicators in the joint space and operational space, and compared with that of four benchmarking controllers.


INTRODUCTION
Model reference adaptive control (MRAC) is an effective control design method for imposing the dynamics of a reference model to plants with uncertain parameters. An established theoretical framework supports this control approach, which, over the past decades, has been shown to be a viable solution to control engineering plants with unknown parameters. 1,2 Nowadays, MRAC research focuses on improving the closed-loop performance of the original algorithms (e.g., by combining MRAC with other control techniques such as sliding mode control, 3 iterative learning control 4 or fuzzy algorithms, 5 just to name a few) and further extend the MRAC theory, for example, to fractional order systems, 6 switching control systems, 7,8 and piecewise affine systems. [9][10][11] To improve the closed-loop tracking performance in presence of plant parameter mismatches, unmodeled plant dynamics, rapid varying disturbances, and unknown system nonlinearities, in Reference 12 the MRAC strategy was augmented by an adaptive integral control action and an adaptive switching control action. Since then, the MRAC algorithm equipped with these additional adaptive control actions, also known in the literature as Enhanced MRAC (EMRAC), 13 has been shown to be an effective solution for steering the dynamics of engineering plants affected by disturbances and model uncertainties toward those of a reference model. Examples of applications where the EMRAC has been successfully implemented include electronic throttle valves, 12 common rail systems, 14 thermo-hygrometric control for multi-enclosed thermal zones, 15 and path tracking control for autonomous vehicles. 16 Furthermore, in Reference 17 a discrete-time version of the EMRAC algorithm was proposed, and it was shown experimentally that the adaptive integral and switching control actions are crucial for improving the tracking of the reference dynamics, compared to other robust adaptive solutions and classical MRAC techniques.
However, the EMRAC algorithms available in the literature are limited to single-input systems, which is a severe limitation of the current EMRAC theory to real systems characterized by multiple inputs. Hence, this paper aims to fill in this gap in the literature by proposing multi-input EMRAC strategies. Specifically, two multi-input EMRAC algorithms are designed and investigated. These solutions differ in the formulation of the multi-input adaptive switching control action. A first formulation for the adaptive switching control action is based on the unit vector (UV) of the closed-loop tracking error and is named EMRAC-UV, while the second one considers each component of the tracking error by using an element-wise (EW) approach, and the resulting algorithm is denoted as EMRAC-EW. Furthermore, the adaptive mechanism for the gains of the switching control action of the novel multi-input EMRAC solutions generalizes and extends that of the single-input EMRAC, while guaranteeing also the limitation of the switching adaptive gains and their first derivative.
To prevent the onset of unbounded adaptive gains in the presence of external disturbances and unmodeled dynamics, which can degrade the closed-loop tracking performance and lead to instability, 18 the adaptive laws of the proposed multi-input EMRAC algorithms are equipped with the -modification strategy to systematically limit the growth of the adaptive gains to persistent disturbances (i.e., disturbances belonging to  ∞ ).
The closed-loop system is systematically analyzed, and the closed-loop error dynamics are proven to be globally uniformly ultimately bounded also in presence of unmatched persistent disturbances, and the ultimate bound is provided. Furthermore, as the EMRAC switching control action makes the closed-loop system nonsmooth, it is not possible to use results available in the non-linear control theory for Lipschitz vector fields, 19 and thus the extensions of Lyapunov theory to Filippov systems presented in Reference 20 are used for proving the closed-loop ultimate boundedness. Moreover, conditions for guaranteeing the asymptotic tracking of the reference model are also presented.
To confirm the effectiveness of the novel multi-input EMRAC algorithms to control the dynamics of challenging multi-input engineering plants, four EMRAC solutions are designed for the path tracking control of robotic arms in the space sector. Specifically, the EMRAC-UV and EMRAC-EW strategies have been implemented with and without feedback linearization (FL) techniques, which are commonly adopted in robotics for compensating nominal system nonlinearities.
The nascent space robotics industry has been strongly demanding on-orbit servicing. On-orbit servicing particularly focuses on using robotic arms sitting on spacecrafts to carry out manipulations and services for the potential targets in space. These targets can be identified as cooperative and non-cooperative, while the service spacecraft are usually classified as free-floating or free-flying. 21,22 The target is said to be cooperative when it can communicate and be controlled in attitude and orbit to cooperate with the service mission; on the contrary, a target is noncooperative when it has no ability to be controlled during the service mission. 23 In the case of a free-floating spacecraft, the attitude of the spacecraft platform remains uncontrolled during the manipulation, while the platform attitude is actively controlled in the case of a free-flying spacecraft. The best example is represented by the Canadarms on the International Space Station (ISS), where the attitude of the ISS is automatically managed to compensate the impact on the ISS dynamics caused by the operation of the robot arm.
The objectives of on-orbit services include the removal of malfunctioning spacecrafts, refueling, deorbiting, and in-space manufacturing. The services are usually achieved by executing three control tasks in sequence, that is, premanipulation, grasp/capture and postcapture manipulation. [24][25][26] To date, a good amount of research in space robotic manipulation has been carried out to address the approaching, premanipulation, and grasping/capturing phases. These studies cover the control and planning of space robotic systems, and address nonlinearities and nonholonomic planning problems, as well as impedance and grasping/capturing control for both cooperative and noncooperative targets. 23,27,28 For the postcapturing phase, accurate trajectory tracking is critical for mission safety and servicing performance. However, the robust manipulation and control of noncooperative targets during the postcapturing phase are still challenges that limit the operating capability and range of the potential space applications. Path tracking control for noncooperative targets poses a formidable control engineering challenge, due to the unknown inertial properties and dynamics of the targets, and the noncontrollable but still active Spacecraft Attitude and Orbit Control Systems (AOCS) of the targets, which can significantly reduce the tracking accuracy of the space robotic arm.
Although adaptive control algorithms and MRAC strategies are effective solutions for control plants with parameter mismatches, and have been successfully applied to robotic arms for terrestrial applications (for instance, see the recent survey 29 ), to the best of the authors' knowledge, in the context of space applications, only few studies tried to address the post-capturing trajectory tracking challenge for noncooperative missions by including some level of adaptation in the control strategy. For instance, to counteract the unknown mass and inertia properties of the target, an adaptive reactionless motion controller with a nonlinear regressor was proposed in Reference 30, while a control solution exploiting an RLS-based algorithm for identifying the inertia parameters of the target was suggested in Reference 31. However, the MRAC theory for the design of path tracking control solutions for noncooperative targets for space applications has not been explored yet. Hence, in this paper four multi-input EMRAC solutions are devised to tackle the trajectory tracking control problem during the postcapturing phase for unknown and noncooperative targets for single space robotic arm systems with a large attitude-controlled servicing spacecraft. Thus, for this case study, it is assumed that the target has unknown inertial properties and its AOCS autonomously counteracts the motion that the robotic arm tries to impose after the docking/connecting or grasping phase, thus generating a reaction force that acts as a disturbance on the robotic arm dynamics.
For testing the effectiveness of the four EMRAC solutions, the EMRAC controllers have been compiled as a C++ code and deployed within a real-time Robot Operating System (ROS) environment, with the simulations being carried out through the state-of-the-art Gazebo dynamic simulator. The use of the industrial-standardized ROS framework can help to efficiently and quickly implement controllers directly on real robots, and also shows the capability of the multi-input EMRAC algorithms to run in real-time, despite the two additional adaptive control actions compared to the MRAC solutions from the literature. 2 Furthermore, the ROS-based framework is also used to show that the four EMRAC solutions outperform four benchmark controllers with fixed gains.
The contributions of this paper are summarized as follows.
• To extend the EMRAC theory to multi-input systems through the design of two multi-input EMRAC strategies, along with a systematic analysis of the closed-loop system by using extensions of Lyapunov theory to Filippov systems.
• To propose four multi-input EMRAC solutions to tackle the trajectory tracking control problem for space manipulators operating in a free-flying mode for a noncooperative scenario during the postcapturing phase, and to test them in a ROS based simulation environment. Due to the high risk of manipulating an unknown and noncooperative target in space, the space industry and the blooming in-orbit servicing market only focus on the in-orbit servicing of communicable and healthy satellites. This study discusses the potential robust trajectory tracking solution for inertia-changed and noncooperative targets. This can greatly increase the servicing quality and expand the number of servable spacecrafts.
• To quantitatively compare the performance of the four proposed EMRAC solutions with four benchmarking feedback controllers, including a proportional integral derivative (PID) controller embedded in ROS and three full-state FL-based strategies, augmented either with a proportional integral (PI) controller, a proportional derivative (PD) controller, or a robust strategy. The comparison is carried out via a set of key performance indicators (KPIs) in the joint space and operational space.
In the remainder,  n denotes the identity matrix in R n×n , while  n,m is the zero matrix in R n×m . Given a symmetric matrix  ∈ R n×n , then min () and max () denote the minimum and maximum eigenvalues of , respectively. Moreover, for a sequence of matrices  j ∈ R n j ×n j , with j = 1, … , m, Δ( 1 ,  2 , … ,  m ) denotes the block diagonal matrix in R (n 1 +···+n m )×(n 1 +···+n m ) with the jth diagonal block being  j .
The paper is organized as follows. Section 2 presents the novel multi-input EMRAC algorithms, that is, the EMRAC-UV and EMRAC-EW strategies, while the main theoretical results established therein are proved in Section 3. Section 4 presents EMRAC solutions for trajectory tracking control for space manipulators. Simulation results and the quantitative comparison of the closed-loop tracking performance are presented in Section 5. Finally, conclusions are drawn in Section 6, while Appendix A and Appendix B, respectively, provide the mathematical background on Filippov systems, used to prove the ultimate boundedness of the closed-loop tracking error, and details of the ROS simulation framework for the simulation analysis.

EMRAC STRATEGIES FOR MULTI-INPUT SYSTEMS
Consider a multi-input plant of the forṁx where x ∈ R n x is the state vector of the plant, u ∈ R n u is the plant input vector, n x and n u are the dimensions of the state space and control input, respectively, and t 0 ∈ R is the initial time instant. System (1) is subjected to two types of disturbances, that is, the measurable disturbance d ∈ R n d , with n d being the dimension of the space of the measurable disturbance, and the nonmeasurable disturbance  ∈ R n x . Both disturbances are assumed to be bounded (i.e., there exist two constants  ∞ > 0 and d ∞ > 0, such that ||(t)|| ≤  ∞ and ||d(t)|| ≤ d ∞ , ∀ t ≥ t 0 ). Moreover, A ∈ R n x ×n x , B ∈ R n x ×n u , and E ∈ R n x ×n d are the dynamic matrix, the input matrix and the matrix of the measurable disturbance, respectively, which are assumed constant with unknown entries. The control objective for the EMRAC is to steer the dynamics of system (1) toward those of a linear reference system while guaranteeing the boundedness of all closed-loop signals. The reference model dynamics are given by an asymptotically stable LTI system of the forṁx where x m ∈ R n x is the reference model state, r ∈ R n u is the reference input assumed to be bounded, while A m ∈ R n x ×n x , B m ∈ R n x ×n u , and E m ∈ R n x ×n d are the dynamics matrix, the input matrix, and the disturbance matrix of the reference model, respectively, with A m being a Hurwitz matrix. It is assumed that there exist some constant matricesΦ R ∈ R n u ×n u ,Φ X ∈ R n u ×n x ,Φ D ∈ R n u ×n d , and an invertible matrix S ∈ R n u ×n u such that following matching conditions are satisfied The ideal gainsΦ R ,Φ X , andΦ D can be collected in the matrixΦ ∈ R n u ×n w , with n w = 2n x + n u + n d , and in the vector ∈ R n u n w defined asΦ whereΦ I =  n u ,n x ,̂j, j = 1, 2, … , n w , is the jth column ofΦ and M is a known upper bound of where both ∈ R n u ×n u and are assumed to be bounded, that is, there exist two constants ∞ > 0 and ∞ > 0, such that || (t)|| ≤ ∞ and ||(t)|| ≤ ∞ , ∀ t ≥ t 0 . The control action provided by the EMRAC algorithm for the multi-input system (1) is where where x I ∈ R n x is the integral of the tracking error whose dynamics are computed aṡ where x e is the state tracking error, e ∈ R n x ×n x is a positive diagonal matrix and I (‖x I ‖) is the -modification strategy to prevent the drift of the integral of the tracking error (9) defined as where I andM I are strictly positive constants. The adaptive gains in (8) are computed as where X , X , I , I ∈ R n x ×n x , R , R ∈ R n u ×n u , and D , D ∈ R n d ×n d are strictly positive diagonal matrices and F X , F I ∈ R n u ×n x , F R ∈ R n u ×n u and F D ∈ R n u ×n d are the locking strategies for preventing the unbounded evolution of the gains (11) in the presence of disturbances and unmodeled dynamics. Moreover, y e ∈ R n u is computed as y e = B T m P e x e , with P e being the solution of P e A m + A T m P e = −Q, where Q ∈ R n x ×n x is a strictly positive matrix. The integral parts of the adaptive gains in (11) can be collected in the matrix Φ ∈ R n u ×n w and the vector ∈ R n u n w defined as where j , j = 1, 2, … , n w is the jth column of the Φ-matrix. Moreover, the leakage terms in (11) are computed as where X , I ∈ R n x ×n x , R ∈ R n u ×n u and D ∈ R n d ×n d are strictly positive diagonal matrices and (|| ||) is the -modification strategy for the adaptive gains of the smooth control actions computed as where andM are strictly positive constants such that where ⊗ is the Kronecker product and the strictly positive matrices Γ , Γ ∈ R n w ×n w are defined as The control action u N (t) can be set either as u (uv) [ sgn(y e1 ) sgn(y e2 ) · · · sgn(y en u ) where the dynamics of the adaptive gains Φ N0 ∈ R and where ‖y e ‖ Ω = y T e Ωy e with Ω ∈ R n u ×n u being a strictly positive matrix, Nj , Nj , j = 0, 1, … , n u , are strictly positive constants, and the h-functions are defined as with j , j , j , j = 0, 1, … , n u , are strictly positive constants. The -modification functions Nj ( ||Φ Nj || ) , j = 0, 1, … , n u , in (21) and (22) are defined as where Nj andM Nj , j = 0, 1, … n u , are strictly positive constants and wherec j > 0 is the j-entry on the diagonal of the matrix SP −1 S T and j∞ > 0, j = 1, … , n u , are constants such that | j | ≤ j∞ .
In the rest of the paper, when the EMRAC control action (7) is equipped u N = u (uv) N in (19), then the resulting control strategy is named EMRAC-UV (i.e., EMRAC with the u N -term based on the unit vector in the direction of y e ), while when u N is set as (20), the resulting strategy is referred to as EMRAC-EW (i.e., EMRAC with the u N -term based on each entry of the vector y e ).
Let us now define the following vectors and matrices and the positive constant and functions Notice that when T B m > 0, then ||(t)|| < ||(t)||, thus 2 ∞ <  2 ∞ and ( ∞ ) < ( ∞ ). This condition is satisfied for instance when the reference inputs for system (2) are decoupled, thus, after a permutation of the columns of B m , the reference input matrix can be expressed as Indeed, under this condition, the entries of the -vector in (6) can be set as Theorem 1 (EMRAC-UV). Consider system (1) and the reference model (2). Let the adaptive control action be given by (7), where u N is set as u (uv) N in (19) and the adaptive gains computed as in (11) and (21). Then, all the closed-loop signals are bounded. Moreover, a (a) if ≠ 0, then the tracking error dynamics are globally uniformly ultimately bounded, and there exists a time  (dependent onx e (t 0 )) such that whereP and are computed as in (27) and (28), respectively; b (b) if = 0, the closed-loop tacking error x e converges to zero as the time goes to infinite, that is, Theorem 2. (EMRAC-EW) Consider system (1) and the reference model (2). Let the adaptive control action be given by (7) where u N is set as u (ew) N in (20) and the adaptive gains are computed as in (11) and (22). If the matrix SP −1 S T is diagonal, then results (a) and (b) of Theorem 1 hold also for the EMRAC-EW algorithm.
As for the Minimal Control Synthesis (MCS) adaptive strategy, 32 Theorems 1 and 2 hold also for linear time-varying systems when the variation of the system matrices is slower than the adaptation rate of the integral part of the adaptive gains in (11). Hence, the following corollary extends the results obtained for the MCS algorithm to the multi-input EMRAC algorithm.

Corollary 1. Assume that the matrices of the plant (1) are time-varying with a rate of variation such that
then Theorems 1 and 2 still hold. Remarks: • For system (1), the measurable disturbance d models nonmanipulated plant inputs, while  represents unmodeled dynamics and unmeasurable external disturbances.
• The multi-input EMRAC strategy enhances MRAC solutions with compensation of measurable disturbances 2 by equipping the control strategy (7) with two additional adaptive control actions, that is, the adaptive integral action u I (t) in (8c), and the adaptive switching control action u N (t) computed either as in (19), for the EMRAC-UV strategy, or in (20), in the case of the EMRAC-EW solution. The adaptive integral control action improves the tracking of the reference model to unmodeled biases in the plant dynamics, for example constant disturbances, while the adaptive switching control action increases the robustness of the closed-loop tracking performance with respect to rapid varying bounded disturbances.
• The additional adaptive and integral control actions of the EMRAC, that is, u I (t) and u N (t), make this strategy more computational demanding with respect to traditional MRAC algorithms. However, implementations of EMRAC solutions in ROS, investigated in this paper, suggest that the two extra control actions do not jeopardize real-time implementability of EMRAC algorithms.
• For engineering control problems, nominal plant models can be used to design the reference dynamics such that conditions (3) hold. A possible approach is to select as a reference model the nominal plant model controlled via feedback control actions (e.g., LQR strategies, see References 12,33). Furthermore, the range of variations of the plant parameters and disturbances can be utilized to select the weights of the -modification. 34 • As for the multi-input MRAC strategies presented in Reference 18, the assumption on the existence of the matrix S in (3) replaces the condition of the knowledge of the sign ofΦ R for the single-input EMRAC. 33 However, when considering the canonical form of the model of robotic manipulators, the matrix S can be set as the identity matrix (see also Section 4).
• Compared to classical multi-input MRAC strategies, 18 in addition to the adaptive switching control action and the adaptive integral control action, the EMRAC algorithm augments the integral adaptive mechanism of the adaptive gains in (11) with (i) a proportional adaptive mechanism to improve the closed-loop tracking dynamics 35 and (ii) a -modification strategy, which is used, together with (24), for guaranteeing the ultimate boundedness of the closed-loop tracking error dynamics in presence of unmatched disturbances and unmodeled dynamics. Furthermore, for each component of the regressor (33), there is an integral adaptive weight, a proportional adaptive weight and leakage factor, thus allowing tailoring these control parameter for each entry of the regressor based on the control application.
• The novel adaptive laws for the gains of the switching control actions, that is, (21) and (22), guarantee that the dynamics of the gains Φ Nj , with j = 0, 1, … , n u , and their derivatives are bounded for any y e -trajectory as h 0 (||y e || Ω ) and h j (|y ej |) in (23) are bounded and the -modification strategy is adopted (see also Section 3.1).
• If n u = 1, both the EMRAC-UV and the EMRAC-EW algorithms reduce to the single-input EMRAC, 33 thus confirming that they are consistent extensions of the EMRAC family to the multi-input case. However, by using the novel adaptive mechanism for the switching control action in (21) and (22), when n u = 1 the ultimate bound computed as in (30) is smaller than that presented in Reference 16 for single-input systems. Furthermore, the adaptive mechanisms (21) and (22) reduce to the one presented for the single-input EMRAC by selecting j = 1, j = 1 j = 0 in (23).

PROOF OF THE MAIN THEOREMS
The analysis of the closed-loop system is based on the Lyapunov theory for smooth and nonsmooth dynamic systems and Barbalat's lemma to prove convergence of the tracking error. In what follows, the ultimate boundedness of Φ N0 and Φ N is first proven in Section 3.1. Then a quadratic auxiliary Lyapunov-like function is designed for the closed-loop error dynamics in Section 3.2. This Lyapunov-like function will be then used along with the theory for non-smooth dynamic systems presented in Appendix A to prove Theorem 1 and Theorem 2 (see Section 3.3, Section 3.4 and Section 3.5). Finally, the proof of Corollary 1 is presented in Section 3.6. The proof of the ultimate-boundedness of the closed-loop error dynamics requires the following lemma.
Lemma 1. The -modification strategy (15) and (16) guarantees Moreover Notice that, the proof of Lemma 1 follow identically to the proof of lemma 2 in Reference 34, thus and it is omitted for the sake of brevity.

Ultimate boundedness of N0 and N
The ultimate boundedness, and thus the boundedness, of the adaptive gains Φ Nj , j = 0, 1, … , n u in (21) and (22) can be proved by using the theory of smooth nonlinear dynamics systems is 19 as follows. By selecting for each gain Φ Nj , j = 0, 1, … , n u the functions V N , W Na and W Nb as then As each h j , j = 0, 1, … , n u in (23) is bounded and by using the -modification functions in (24), after some algebraic manipulations the derivative of V N can be upper-bounded aṡ where H Nj = Nj ∕ Nj , Nj ∈ (0, 1) and W Nj Nj , j = 0, 1, … , n u . As the dynamics in (21) and (22) are smooth and (37) and (38) hold, it is possible to apply theorem 4.18, p. 172 in Reference 19, which guarantees the boundedness and the ultimate boundedness of the evolution of each Φ Nj , and the ultimate bound is W −1 Na (W Nb ( Ni )) = Ni , j = 0, 1, … , n w .

Closed-loop dynamics and the Lyapunov-like function candidate
The closed-loop error dynamics are obtained through (1)-(3), (7), and (11) and they arė with f being a smooth vector field for the dynamics of integral part of the adaptive gains e . The entries oḟe can be collected in the matrixΦ e ∈ R n u ×n w obtained by deriving Φ e in (26). After some algebraic manipulations, the dynamics of As the control action u N , computed either as (19) or (20), is discontinuous, the vector field of system (39a) is discontinuous, thus it is not possible to use the theory presented in Reference 19 to prove the ultimate boundedness of the closed-loop error system in (39). Consequently, the theory presented in Appendix A for Filippov systems is used, and the differential equation (39) is replaced with the corresponding differential inclusion wherex e is defined in (26) is the Filippov set valued map for the discontinuous closed-loop vector field, which is computed as being the set-valued vector function in Reference 36, where ⃗ y e is the unit vector in the direction of y e and  u is the closed unit sphere in R n u centered in the origin.
For the EMRAC-EW, with where For system (41), the Lyapunov-like function is selected as whereP is the strictly positive matrix defined in (27). The Lyapunov-like function (47) can be bounded as where can be rewritten as As the function V(x e ) is smooth, its generalized derivative can be computed aṡṼ = ∇V T K[ ] (see also Appendix A). By using (3), (12), (40), and (42), after some algebraic manipulations,̇Ṽ takes the following forṁṼ Now, as tr y T e SP −1 S T y e w T Γ w ≥ 0, ∀ y e ∈ R n u , and ∀ w ∈ R n w , (51c)Ṽ can be upper-bounded aṡṼ Furthermore, by using (17), (35), Lemma 1 and Φ −1 R = SP −1 , after some algebraic manipulations,̇Ṽ can be further upper-bounded aṡṼ where 2 in (28) is computed by using 2 ∞ =  2 ∞ .
In the case T B m ≥ 0, ∀t ≥ t 0 , with the same steps,̇Ṽ can be upper-bounded aṡṼ where 2 in (28) is computed by using 2

Proof of Theorem 1a
In the case of the EMRAC-UV, from (43) the term y T e SP −1 K[u N ] in (53) and (54) become and thus .
Hence, (53) can be further upper-bounded aṡṼ where ∈ (0, 1) and W 3 (x e ) ∈  is the positive function defined as Since (48) and (57)  ( ∞ ) as stated in Theorem 1a when 2 ∞ =  2 ∞ . Moreover, there exist a -class function Ψ ∶ R + × R + → R + , a time interval  and a function b ∈  ∞ such that the error dynamics are bounded as Consider now the case T B m > 0 ∀ t ≥ t 0 , along with the decomposition of the disturbance  in (6). Under these conditions, by using also (21), (25), there exists a time instant , for all t > t ⋆ 0 . Hence, by considering also (55),̇Ṽ in (54) can be upper bounded aṡṼ where ∈ (0, 1) and W 3 (x e ) ∈  is the function defined in (57). As (48) and (59) hold, similar to the previous case, Theorem 3 in Appendix A can be applied, thus the closed-loop error dynamics (39)-(41) are ultimate bounded, and the ultimate bound is W −1 1 (W 2 ( ( ∞ ))) = √ max (P) min (P) ( ∞ ) as stated in Theorem 1a when 2 ∞ = 2 ∞ . Moreover, there exists a -class function Ψ ⋆ ∶ R + × R + → R + , a time interval  ⋆ and a function ⋆ b ∈  ∞ such that the error dynamics are bounded as The boundedness of ‖ ‖x e ‖ ‖ , guaranteed either by (58) or (60), implies the boundedness of x e , e and Φ e . As x e is bounded (i.e., x e ∈  ∞ ) then y e is bounded, and because x m in bounded by assumption, then also x is bounded, while the boundedness of x I can be proven as in Reference 34. The boundedness of Φ e , x e , d, r imply that K X , K R , K I and K D are bounded. The boundedness of the signals x, x e , d, r, Φ N0 , K X , K R , K I , and K D implies that also u anḋx are bounded. Hence, the boundedness of all closed-loop signals remains proven when the EMRAC-UV is used, thus concluding the proof of Theorem 1a.

Proof of Theorem 1b
In the case = 0, by using (34), (43), and (55),̇Ṽ in (52) can be upper-bounded aṡṼ From (21) and (25), there exists a time instant t ⋆ 0 > t 0 such that for t > t ⋆ 0 , (61) can be further upper-bounded aṡṼ According to Appendix A, dV dt (x e (t), t)∈ a.eV (x e (t), t), thus Now, according to References 10,37, from (63) for any closed-loop x e -trajectory we have Since W(x e ) is a continuously differentiable positive-definite function, Barbalat's Lemma can be applied, and W(x e (t)) convergences to zero when t → +∞. 10,37 Consequently, the state tracking error x e (t) convergences to zero when t → +∞, thus concluding the proof of Theorem 1b.
Moreover, from (22) and (25) there exists a time instant t ⋆ 0 > t 0 such that Φ Nj > j∞ ∕c j , j = 1, … , n u for all t > t ⋆ 0 . Hence, in the case T B m > 0, ∀ t ≥ t 0 , and the decomposition (6) of the disturbance  is used, by using (65) when t > t ⋆ 0 ,Ṽ in (54) can be upper bounded aṡṼ where ∈ (0, 1) and W 3 (x e ) ∈  is the function defined in (57). As (66) provides the same upper-bound foṙṼ given by (59), the proof of the ultimate-boundedness of the closed-loop error dynamics (39)-(41) with ultimate bound given by From (22) and (25), there exists a time instant t ⋆ 0 such thaṫṼ can be further upper-bounded as in (62) for all t > t ⋆ . Hence, the convergence to zero of x e follow equivalently as shown in the proof of Theorem 1b in Section 3.4, thus also Theorem 2 remains proven.

Proof of Corollary 1
The Consequently, (40) still holds, and therefore the proofs of Theorems 1 and 2 follow identically.

APPLICATION OF MULTI-INPUT EMRAC SOLUTIONS TO SPACE ROBOTIC ARMS
The free-flying space robot system is the spacecraft platform whose attitude is controlled during the manipulation. 24,38 When the position and orientation of the base-spacecraft are kept fixed by the AOCS 39 during the postcapturing phase of a noncooperative micro-satellite target, the model of a free-flying robotic manipulator with n links is expressed as where q ∈ R n is the vector of the joint variables, ∈ R n is the vector of the torques and forces provided to each joint (control variables), (q) ∈ R n×n is the inertial matrix, (q,q) ∈ R n×n is a matrix such that (q,q)q is the vector of Coriolis and centrifugal torques and forces,  v ∈ R n×n is the matrix of the damping coefficients,  p (q) ∈ R 3×n is the submatrix of the geometric Jacobian of the manipulator relating the joint velocities to the end-effector linear velocity. The force f p ∈ R 3 is the reaction disturbance force provided by the AOCS of the noncooperative target, and is modeled in accordance with an ideal cold gas thruster on the target spacecraft. Specifically, the AOCS measures the first-or second-order derivatives of the attitude or orbit drifts, and feedbacks them for attitude or orbit control. 40 For this study it is assumed that the target spacecraft uses the cold gas thruster as AOCS actuator, as it is a common solution for micro and small spacecrafts. 41 Furthermore, it is assumed that (i) the thruster control constant , which depends on the fuel used and the mechanical design of the nozzle of the thruster, 42 is a positive constant; (ii) the target is rigidly moved with the end-effector of the service robotic arm during the postcapturing phase, thus the acceleration drift of the target measured by the AOCS sensor is that of the end-effector (i.e.,p); (iii) the reaction forces are sensitive to the second-order derivative of the longitudinal orbital change as this variation is usually used by AOCS to compensate for external forces causing target orbital changes; and (iv) the residual orbital station-keeping and hypothetical anti-capture controls are activated during the entire postcapture servicing phase and provide uncooperative forces proportional to the second derivatives of orbital changes through the thruster control constant to resist the manipulation action. Hence, the following simplified model is used to account for the effect of the AOCS onto the manipulator dynamics According to Reference 43, when a full-state FL is used for compensating the term (q,q)q in (68) to make the closed-loop system behave as an n-dimensional decoupled mass-spring-damper system with stiffness K ∈ R n×n and damping matrix Θ ∈ R n×n , the resulting plant to control takes the form in (1) with = 0, d = 0, E = 0, n u = n, n x = 2n and where(q) and(q,q) are estimates of the matrices (q,q) and (q), respectively. Similarly to the robust control design in Reference 43, as the joint positions are confined in a finite set, and saturations exist on the maximum velocities and accelerations of the motors, then the resulting nonmeasurable disturbance is bounded. Notice that for system (70a) the interaction force f p and friction have not been compensated. Moreover, the matrices K and Θ are diagonal. For the robotic manipulator, EMRAC solutions can be designed for imposing reference trajectories in the joint space, despite the imperfect compensation of the robot nonlinearites, parameter uncertainties and disturbances.
The reference model for imposing a reference trajectory denoted as q R (t) takes the form in (2), where E m = 0 and Hence, the matching conditions (3) are satisfied and S =  n .
In what follows, the EMRAC solutions that are used for correcting the imperfect FL strategy and imposing trajectories in the joint space are referred to as EMRAC-FL strategies. Moreover, to further test the robustness of the novel EMRAC algorithms, EMRAC solutions with no feedback linearization (NFL) control action are also tested when the reference model is set as in (71). These EMRAC solutions are named as EMRAC-NFL strategies.

NUMERICAL RESULTS
The multi-input EMRAC solutions are tested for a threelink anthropomorphic arm whose fundamental parameters are reported in Table 1.
The simulation scenario is set to emulate the working conditions of a noncooperative interaction between a micro-satellite and the robotic arm in the post-capturing phase (see also Section 4). For this case study, the micro-satellite body is modeled as an unknown load at the end-effector, with cubic shape of side 1 m and mass m load = 100 kg (see also Figure 1 for a schematic of the manipulator within the simulation environment, notice that the servicing spacecraft model in Figure 1 is based on SpaceX dragon for illustration purposes). The satellite thrusters try to counteract the trajectory imposed by the space robotic arm, thus generating the disturbance f p in (69) (or equivalently the disturbance 2 in (70b)). Furthermore, as the mass of the micro-satellite is supposed to be unknown, the resulting FL control action provides an imperfect compensation (i.e., ≠  and ≠ ), thus generating the disturbance term 1 in (70b).
The path to be imposed to the end-effector position in the operational space is shown Figure 2A. The position on the path (i.e., the trajectory) is obtained by using two fifth-order interpolating polynomials. The resulting reference trajectory, denoted as p R (t), lasts 250 s, and consists of five submaneuvers, which are listed in Table 2.
The trajectory in the operational space is converted into the trajectory in the joint space by using the second-order closed-loop inverse kinematic algorithm presented in Reference 43. The resulting reference joint positions and speeds are depicted in Figure 2B and C, respectively.

Link
Length (m) Mass (kg) Diameter (m) The three-link-anthropomorphic arm within the Gazebo-robot operating system simulation environment.  The matrices of the reference model (2) have been chosen as in (71) with K = 10 −3 diag(6.7, 2, 1) and Θ = 10 −2 diag(16.8, 9, 6.2), while the adaptive weights have been selected as a trade-off between convergence time and reactivity of the control actions. Furthermore, similarly to References 12,33, in order to avoid unwanted chattering phenomena, the discontinuous terms in (19) and (20) have been smoothed as y e ‖y e ‖ = y e ‖y e ‖ + 0 , and sgn(y ej ) = y ej |y ej | + j , j = 1, 2, 3,
To challenge the ability of the closed-loop system to adjust to unknown working conditions, the adaptive gains of the four EMRAC solutions have been initialized to zero. Hence, no preliminary knowledge of the plant is used to initialise the EMRAC adaptive mechanisms.
In addition to the four EMRAC solutions, four benchmark controllers have been designed and implemented. The benchmark controllers are (i) a PID controller embedded in ROS (denoted as ROS-PID) and (ii) three full-state FL-based strategies, augmented with a proportional-integral controller (PI-FL), a proportional derivative controller (PD-FL), or the robust strategy in Reference 44 (ROBUST-FL).
The feedback gains of the PI-FL strategy have been selected through a trial-and-error approach, with the aim to preserve closed-loop stability and minimize the tracking error. The proportional (K P ) and derivative (K D ) gains of the PD-FL control action have been chosen to have ideal linear closed-loop dynamics given by the eigenvalues of the matrix A m in (71), thus they have been set as K P = K and K D = Θ, respectively. For the same reason, for the ROBUST-FL controller, the proportional feedback gain K rb weighting the state tracking error has been selected as K rb = [K Θ], while a heuristic approach has been used to tune (i) the gain that modulates the magnitude of the robust sliding mode-based control action (indicated as rb ) and (ii) the threshold modulating the boundary layer within which the tracking error is allowed to vary (indicated as rb ). Specifically, rb and rb have been heuristically selected respectively as the largest value and the smallest value which allow to avoid chattering in the control action.
For the implementation in the ROS environment detailed in Appendix B, the EMRAC control solutions and the benchmark controllers have been discretized with a sampling time T s = 5 ms, with the reference model and the required

Closed-loop dynamics
The norm of the closed-loop tracking error in the joint space provided by the four EMRAC solutions is depicted in Figure 4. The norm of the joint speed error never exceeds 0.01 rad/s as shown in Figure 4B. Furthermore, Figure 4A shows that the norm of the joint position error is always below 0.04 rad, except for the EMRAC-UV-NFL, which, however, remains below 0.075 rad. When t > t ⋆ = 35 s, Figure 4A also shows that the norm of the residual joint position error is much smaller than the magnitude of each reference joint position in Figure 2B. Hence, better tracking of the reference trajectory in the operational space is expected when the time exceeds the threshold t ⋆ . This is confirmed by Figure 5A , which shows that when t > t ⋆ the norm of the position tracking error remains below 10 cm. The highest values of the position tracking errors are at the beginning of the maneuvers (i.e., within the INIT sub-maneuvers) due to the zero initialisation of the adaptive gains of the EMRAC solutions. Indeed, during the initial part of the maneuvers, the control gains are not tuned for precisely compensating the unmodeled dynamics and the disturbance f p . Furthermore, the effect of the zero initialization conditions on the tracking error is more severe for the EMRAC-NFL solutions, where also the initial contribution of the nominal FL action is excluded. However, as time evolves, the control gains adapt to the actual operating condition, resulting in better tracking of the reference trajectory in the operational space. Specifically, when the initialisation maneuver is completed (i.e., t > 55 s), the residual position error in the operational space remains below 3.5 cm for the EMRAC-NFL solutions, while for the EMRAC-FL strategies the 3.5 cm threshold is exceeded only for a time interval of about 10 s at the end of the first-LAP maneuver (see also Figure 5A). Furthermore, the residual speed error is below 0.020 m/s for all the adaptive solutions, as shown in Figure 5B. As the tracking errors provided by the EMRAC solutions for t > t ⋆ are negligible when compared to the magnitude of the reference trajectory, a precise tracking of the reference path is obtained as shown in Figure 6.  Figure 7A,B shows the bounded evolution of the norm of the integral part of the adaptive gains for the continuous control action and discontinuous control action, respectively, while the proportional part of the adaptive gains is bounded as the residual tracking error in Figure 4 is bounded. Consequently, Figures 7 confirms the ability of the -modification strategies embedded within the EMRAC adaptive laws to prevent an unbounded drift of the adaptive control gains also in presence of persistent disturbances that could ultimately jeopardize the tracking of the reference trajectory. 45 Without the -modification strategy in (24), the adaptive gains of the discontinuous control actions diverge, thus resulting in chattering in the control action, which could damage the robot actuators or induce closed-loop instability.
As the control gains and the state variables are bounded, the torque demanded at each joint is bounded, as depicted in Figure 8A , where the norm of the torque is shown for each EMRAC solution. The demanded torques coincide with the control action for the EMRAC-NFL solutions. For the EMRAC-FL solutions, the control action is the required acceleration to each joint, which is also bounded, see Figure 8B. As the unit of measurement of the control action for the EMRAC-FL solutions is different from that of the EMRAC-NFL solutions, the order of magnitude of the corresponding adaptive control gains varies significantly, as depicted in Figure 7.

Evaluation of the closed-loop performance via KPIs
To quantitatively compare the EMRAC solutions to the benchmark controllers, the root mean square error (RMSE) and the maximum error (ME) are used as KPIs for measuring the closed-loop tracking performance. By denoting as a closed-loop vector signal of interest , the corresponding RMSE and the ME are computed as where R is the reference value for the signal , while t i and t f are the initial and final time instants, respectively, delimiting the time interval on which the KPIs are computed. The control effort is measured by the integral of the norm of the demanded torque normalized with time (I || || ), and the maximum norm of the torque (M || || ) defined as The KPIs in (73) and (74) are computed for each submaneuver in Table 2. Table 3 shows the tracking performance KPIs for the joint position and speed, and the KPIs for the control effort, while Table 4 shows the tracking performance KPIs for the position and speed of the end-effector. As the PD with FL and the ROS-PID become unstable while performing the second lap of the circumference, it was not possible to compute the KPIs for the second-LAP and second-STST submaneuvers.
The following remarks are based on Table 3 and Table 4.
• All EMRAC solutions outperform the PD controller for the sub-maneuvers where the KPIs for the PD algorithm can be computed (i.e., over the time intervals before the closed-loop stability is lost). For instance, compared to the PD controller, the EMRAC solutions reduce the RMSE ||q|| and RMSE ||q|| by at least four times and 2.5 times during the INIT-maneuver, respectively. The reduction of the KPIs becomes even more significant over the first-STST and first-LAP submaneuvers. Specifically, for these submaneuvers, the adaptive solutions reduce, for instance, the RMSE ||q|| and RMSE ||q|| by at least 70 and 23 times, respectively (see the results for EMRAC-UV-NFL and EMRAC-EW-FL, computed over the first-STST maneuver) which result in a reduction of at least 24 and 29 times of the RMSE ||p|| and RMSE ||ṗ|| , respectively.
• All EMRAC solutions provide better tracking performance compared to the built-in ROS-PID controller over all sub-maneuvers before the ROS-PID solution becomes unstable. For instance, over the INIT maneuver, the adaptive solutions reduce the RMSE ||ṗ|| by a factor that ranges from 2.5 times and up to nine times. Furthermore, in the first-STST maneuver, the adaptive solutions provide a reduction of the ME ||q|| and ME ||q|| in excess of 14 and 50 times, respectively, which corresponds to a reduction of the ME ||p|| and ME ||ṗ|| by at least 28 and 60 times, respectively.
• Over the INIT maneuver, the KPIs measuring the position tracking performance (i.e., RMSE || || and ME || || with = {q, p}) provided by the ROBUST-FL and PI-FL controllers are smaller than those obtained with the adaptive solutions. The reduced tracking performance of the adaptive solutions along the first submaneuver is due to the zero initialization of the adaptive gains, which generates larger initial errors over the first 35 s, referred to as t ⋆ in the remainder (see also Figures 4A and 5A), and results in larger KPIs when the INIT maneuver is completed. The effect of the zero initialization on the KPIs is more severe for the EMRAC-NFL strategies, which, compared to the EMRAC-FL, are not equipped with the nominal FL control action.
• When the INIT maneuver is completed, the EMRAC adaptive gains have been self-adjusted to the actual plant dynamics, thus the residual tracking error and consequently the KPIs measuring the tracking performance shrink drastically. Furthermore, for the remaining submaneuvers (i.e., first-STST, first/second-LAP, second-STST), the KPIs measuring the tracking performance provided by the EMRAC solutions are always smaller compared to those obtained with the benchmark controllers (the ROBUST-FL strategy and the PI-FL controller). For instance, compared to the ROBUST-FL strategy, the EMRAC strategies reduce the RMSE ||p|| by a factor that ranges (i) from 2.8 to 8.9 times in the case of the first-LAP maneuver, and (ii) from 5.2 to about 10 times over the second-LAP. Similarly, when compared to the PI-FL controller, the EMRAC strategies reduce the RMSE ||p|| of a factor that ranges (i) from 2.6 to 8 times over the first-LAP maneuver, and (ii) from to 8.6 to 16.2 times over the second-LAP maneuver. A similar trend can be noted when comparing the KPIs tracking performance in the joint space of the adaptive solutions to those of the PI-FL and ROBUST-FL strategies over the first-LAP and second-LAP .

EW-FL EW-NFL UV-FL UV-NFL ROBUST-FL PI-FL PD-FL ROS-PID
• For the PI-FL strategy, the RMSE || || indicators, with = {p,ṗ, q,q}, have an increasing trend over the last four maneuvers, which might mark the onset of unstable dynamics. For the ROBUST-FL algorithm, the magnitude of the KPIs measuring the tracking performance mainly depends on the trajectory to follow (e.g., the RMSE || || , with = {p, q}, for the maneuver first-STST and second-STST are similar). However, for the EMRAC solutions, the mag-TA B L E 4 Performance indicators computed in the operational space (RMSE ||p|| (m), RMSE ||ṗ|| (m/s), ME ||p|| (m), ME ||ṗ|| (m/s)). nitude of the KPIs could decrease for the same maneuver, because of the adaptation process. 12 For this case study, the reduction of the tracking errors caused by the gain adaptation is significant for the STST maneuvers. For instance, when second-STST is compared to first-STST, the RMSE ||p|| reduces by 5.4 times and 25.6 times for the EMRAC-EW-FL and the EMRAC-UV-NFL, respectively.

EW-FL EW-NFL UV-FL UV-NFL ROBUST-FL PI-FL PD-FL ROS-PID
• The KPIs measuring the control effort (i.e., I || || and M || || ) show that the magnitude of the control action required by the EMRAC strategies matches the one demanded by the benchmark controllers that complete the entire maneuver (i.e., the ROBUST-FL controller and the PI-FL strategy).

CONCLUSIONS
This paper has presented two extensions of the EMRAC strategy to multi-input plants. The novel multi-input EMRAC strategies, named EMRAC-UV and EMRAC-EW, include the -modification strategy to systematically bound all the adaptive gains also for not vanishing perturbations, but differ based on how the adaptive switching control action is computed. For both strategies, a proof of the global uniform ultimate boundedness of the closed-loop error dynamics has been derived, based on the Lyapunov theory for Filippov systems. The effectiveness of the multi-input EMRAC control framework has been numerically evaluated by designing and implementing, in a ROS-based simulation environment, four EMRAC solutions for controlling space robotic manipulators during the postcapturing phase, in case of unknown and noncooperative targets without any initial knowledge of the operating conditions. The simulation analysis has confirmed the ability of the adaptive closed-loop system to adjust to the unknown working conditions, thus providing low residual tracking errors. The EMRAC solutions have been compared with four benchmark controllers, that is, an embedded ROS PID controller, and PD, PI and robust controllers equipped with a full-state FL strategy. An extensive quantitative analysis carried out through the use of KPIs defined in the joint space and operational space has confirmed that the closed-loop tracking performance of the EMRAC solutions outperforms those of the benchmark controllers after an initialization maneuver. Future work will investigate the experimental assessment of the proposed multi-input EMRAC schemes.

CONFLICT OF INTEREST STATEMENT
We confirm that no one on the author list has a conflict of interest to disclose.

DATA AVAILABILITY STATEMENT
Data available upon request to the authors.

APPENDIX A. ULTIMATE BOUNDEDNESS OF NON-SMOOTH DYNAMIC SYSTEMS
This appendix provides details about the theory of nonsmooth dynamic systems, which has been used to prove the ultimate boundedness of the closed-loop error system when the multi-input EMRAC solutions are used. The theory can be applied to nonsmooth time-varying systems of the form wherex ∈ R n is the state of the system and ∶ R × R n → R n is a discontinuous vector field.
supports C++ code generation. The simulation framework is completed with two ad hoc in-house C++ and Python codes whose uses are detailed below. The robotic arm within the simulator is modeled through the Unified Robotic Description (URDF) format. The micro-satellite attached to the end-effector is modeled in the URDF file as an additional fixed link. A picture of Gazebo when the URDF is loaded is shown in Figure 1.
The non-cooperative target disturbance in (69) is emulated by using the IMU plugin sensor, 49 the apply_body_wrench service, 50 and the in-house Python script. The IMU plugin is a build-in feature provided in Gazebo. It measures and publishes the acceleration information of a selected 'link'. In the simulation, the IMU plugin is added to the target. The in-house Python script is used to subscribe the acceleration information of the target and subsequently calculate the reaction forces generated by the target in (69). Then the apply_body_wrench ROS service 50 is recalled to apply the attitude disturbance forces to the target.
The four EMRAC solutions and the benchmark controllers (except for the ROS-PID controller) in Sections 4 and 5 are implemented in Matlab/Simulink and connected to the ROS environment through the ROS Publish and Subscribe blocks (available in the MatLab ROS toolbox). The Publish block uses the node of the Simulink model to create a ROS publisher for a specific topic. The input of this block is a Simulink nonvirtual bus that corresponds to the specified ROS message type and publishes it to the ROS network. For the specific simulation framework, one Publish block was used for each of the three joint torques, creating the topics named /space_arm/joint_i_position_controller/command (i = 1, 2, 3) and providing the torque values (computed by the controller) using the std_msgs/Float64 message type. The block is responsible for converting the Msg input from a Simulink bus signal to a ROS message and publishing it at each sampling time. In code generation, the input is a C++ ROS message. 51 The Subscribe block uses the Simulink model node to create a ROS subscriber for a specific topic, takes a specified ROS message type as input, and provides a corresponding Simulink nonvirtual bus as output. The URDF model publishes the joint states values to the /space_arm/joint_states topic using sensor_msgs/JointState message type, thus, a Subscribe block was used to feed the manipulator's joint position and velocities to the controller. The block is responsible, on each simulation step, for checking if a new message is available on the specific topic and, if it is, retrieving the message and converting it to a Simulink bus signal. 51 The combined use of these two blocks allows for providing the controller output torques to the URDF manipulator which, in return, feeds the joint states signals to the control algorithm, therefore enabling to close the control loop. An additional "To File" block is added in the Simulink model in order to collect all the simulation data and store them into a MAT-file. The "To File" block is compatible with C++ code generation and has minimal memory overhead during simulation, which means that it will not affect the performance of the controller.
The Simulink code of each controller is converted into a C++ code by using Simulink Coder and Embedded Coder. Furthermore, to finalize the deployment of the code of the controller into the simulation framework, the in-house C++ code has been developed which subscribes to the /space_arm/joint_i_position_controller/command topic and applies the controller' input torques values to the URDF manipulator joints using the EffortJointInterface hardware interface. Finally, after the simulations in Gazebo, a MAT-file with all the simulation data will be created.