Parameterizing robust manipulator controllers under approximate inverse dynamics: A double‐Youla approach

We consider the goal of ensuring robust stability when a given manipulator feedback control law is modified online, for example, to safely improve the performance by a learning module. To this end, the factorization approach is applied to both the plant and controller models to characterize robustly stabilizing controllers for rigid‐body manipulators under approximate inverse dynamics control. Outer‐loop controllers to stabilize the nonlinear uncertain loop that results from approximate inverse dynamics are often derived by lumping uncertainty in a single term and subsequent analysis of the error system. Here, by contrast, the well‐known norm bounds of these uncertain dynamics are first recast into a generalized plant configuration that preserves the characteristic uncertainty structure. Then, the overall loop uncertainty is expressed with respect to the nominal outer‐loop feedback controller by means of an uncertain dual‐Youla operator. Therefore, using the dual‐Youla parameterization, we provide a novel way to rigorously quantify permissible perturbations of robot manipulator feedforward/feedback controllers. The method proposed in this paper does not constitute another robust control law for rigid‐body manipulators, but rather a characterization of a set of robustly stabilizing controllers. The resulting double‐Youla parameterization for the control of robot manipulators is amenable to numerous advanced design methods. The result is thoroughly discussed by a planar elbow manipulator and exemplified with a six‐degree‐of‐freedom robot scenario with varying payload.

On the one hand, if the complete dynamic model is available, feedback linearization can be applied. On the other hand, simple, completely model free controllers are ubiquitous, most prominently, the popular proportional-derivative (PD) controller. Both may be interpreted as two extremes on the spectrum of approximate inverse dynamics (AID) controllers: ideal linearization is theoretically achieved if the model is perfect; if, on the contrary, there simply is no dynamical model in the controller, only the PD action of the outer loop remains. In this article, we consider the following question. Given a robot manipulator under AID control, how and how much can the outer-loop controller be modified during operation without compromising stability? Answering this question ultimately allows to use for online control performance enhancement methods that are otherwise hard to employ in a strict robust stability framework, eg, black-box optimization or reinforcement learning.
It is intuitively clear that the worse the controller is informed about the dynamic characteristics of the controlled robot, the harder it will be to provide stability guarantees when the controller is modified during operation. This situation is depicted in Figure 1 and motivates this article: We provide a method to quantify this trade-off. We then characterize a set of robustly stabilizing controllers  R , ie, all controllers contained in  R stabilize the uncertain nonlinear inner loop resulting from a particular AID situation. Interestingly, not only the model accuracy but also the nominal outer-loop controller K influence the amount of apparent uncertainty in the loop. Only comparatively recently, Bascetta and Rocco 5 have reformulated the robust control of rigid manipulators to account for this issue, after it had not been considered by the most common robust robot control methods. [1][2][3][4] The new framework proposed in this article provides a unified perspective and constitutes a tool to explore freedom in controller (re)tuning, given a simplified model of the robot manipulator and a nominal outer-loop controller K.
Central to our method is a Q-parameterization (viz, Youla-Kučera parameterization due to the original developments in the works of Youla et al 6 and Kučera 7 ) of stabilizing controllers and the dual S-parameterization of plants stabilized by a controller. 8 The primary parameterization was already applied to the robust linear design of robot controllers around 30 years ago, 9 known in robotics as the stable factorization approach. 9,10 Our interest in extending this approach is due to the beneficial stability and robustness properties obtained by a Q-parameterization: Our recent work 11 suggests that such a parameterization is very helpful to leverage machine learning also in a feedback configuration, that is, in closed loop and on hardware. Inherently, the question is raised of how much feedback controller modification is admissible, given the effect of neglected uncertainties in the closed-loop system. The dual-Youla parameter S is needed to answer this question and there appears to be a gap in the literature of how to utilize the dual parameterization 12 for the control of robot manipulators. The following novel methodological contributions are given.
(i) A generalized plant description of the rigid-body robot manipulator under AID control is formulated, such that the sources of uncertainty remain separated and the outer-loop controller K is left undetermined. (ii) Next, a dual-Youla parameterization is derived to quantify the uncertainty occurring by the interplay of neglected manipulator dynamics in the inner loop and the outer-loop controller K. Opposed to the works of Bascetta and Rocco 5 and Spong and Vidyasagar, 9 our approach allows for a general control configuration and the sources of uncertainty are kept separated throughout. Hence, we propose a unified framework to rigorously assess the uncertainty arising in the closed-loop system. (iii) Our main result is a general parameterization of robustly stabilizing two-degree-of-freedom (DoF) (feedforward/ feedback) controllers for rigid-body manipulators under AID control that is amenable to a variety of control design

FIGURE 1
The goal of this article is to parameterize a set of robustly stabilizing feedback controllers for rigid-body robot manipulators. To this end, a general approximate inverse dynamics (AID) controller is considered in the inner loop that may range from a perfect inverse dynamics model to none at all. An outer-loop feedback controller K be given that stabilizes the loop but may not yet yield satisfactory performance. Therefore, in general, one can augment K by some stable parameter Q ∈  ⊆  ∞ . However, if the dynamic model of in the inner AID loop is imperfect, only a subset  ⊂  ∞ is admissible to preserve the stability of the overall loop. In this article, the set  and the controller parameterization K(Q) are characterized such that various advanced design methods can be used for robust controller performance enhancement [Colour figure can be viewed at wileyonlinelibrary.com] methods (learning, gain scheduling, etc). To this end, the worst-case uncertainty measure from (ii) is employed in a double-Youla parameterization, effectively tightening the set of nominally stabilizing controllers for the ideal double integrator system to the corresponding subset of robustly stabilizing controllers. (iv) The derivation offers several new developments of relevance to the broader control community beyond the scope of robotics. In particular: (a) A state-space realization for the dual-Youla parameter is derived for nondynamic metric-bounded uncertainty, given an arbitrary linear controller and a strictly proper plant. (b) We specialize the result to a double-Youla parameterization based on a static central controller and discuss a choice of the gain design parameter such that the results are further simplified.
The remainder of this article is structured as follows. We first introduce the methods underlying our work in Section 2.
The main result is given in Section 3, a general double-Youla parameterization for robot manipulators under AID control; the specialization using a static nominal controller is then reported in Section 4. In Section 5, the results are exemplified and thoroughly discussed by means of a planar elbow manipulator and a PUMA P560 robot model with six DoFs. The proofs and some technical details are provided in the Appendix.
Notation. Lowercase boldface letters are for vectors (x), matrices (M) are written in capital bold letters, sets () are calligraphic, transfer operators (G) that define input/output relations are in standard capital letters, and stands for a state-space realization. The symbol ≜ means equality by definition. The set of nonnegative real numbers is ℝ + 0 and the Euclidean space of vectors of dimension n is denoted ℝ n , with the 2-norm ||x|| = √ ∑ n i=1 |x i | 2 . The corresponding induced matrix norm of an n × n matrix M is written ||M||. The vertical concatenation of two column vectors We use the convention that if G(s) is a transfer matrix and u = u(t) is a Laplace transformable signal, then by Gu the signal (g * u)(t) is meant, where g(t) is the impulse response of G(s) and * denotes the convolution operator, with the technical distinction being given from the context. The space  ∞ denotes the set of all proper and real n × m rational stable transfer matrices G(s) and the associated  ∞ norm is ||G(s)|| ∞ ≜ sup ∈ℝ̄( G( )), whered enotes the largest singular value. If G is a nonlinear mapping, by ||G|| ∞ the induced  2 gain of G is meant. The feedback interconnection of two systems G 1 and G 2 through common input/output variables is written (G 1 , G 2 ). A lower linear fractional transformation of a partitioned operator G = The transformations are well defined if and only if the well-posedness condition is satisfied.

Rigid-body manipulator dynamics and tracking control
We consider a rigid-body robot manipulator with n links, described by the standard Euler-Lagrange model 1-3
where q(t) ∈ ℝ n is the vector of generalized coordinates (representing joint positions), ∈ ℝ n is the input vector of generalized force (torque), M(q) ∈ ℝ n×n , M(q) ≻ 0 is the inertia matrix, and n ∈ ℝ n is a vector that summarizes the vector of Coriolis and centrifugal terms C(q(t), .

q(t))
. q(t) ∈ ℝ n , the friction terms f ( . q(t)) ∈ ℝ n , and the gravitational terms g(q(t)) ∈ ℝ n . It is assumed that the input disturbance dist (t) is a Lebesgue measurable function. The state of the system is denoted x(t) = col(q(t), . q(t)). Given a desired path in joint space q d (t) ∈  2 , q d , . q d ,q d ∈  ∞ (ℝ + , ℝ n ), the tracking error is formed as e(t) = col(e 1 (t), e 2 (t)) = col(q d (t) − q(t), . q d (t) − . q(t)). The measurement available for feedback is assumed aŝ q(t) = q + w 1 (t),̂. q(t) = . q(t) + w 2 (t), where w(t) represents measurement uncertainty. The goal of tracking control is to find a controller generating (t) such that e(t) vanishes. For the sake of readability, we henceforth drop the dependency on time t where it is unambiguously clear from the context.

Control of rigid manipulators using AID
The parameters of (1) are in pratice not known exactly. The approximate or realistic inverse dynamics control 2,3 therefore attempts to cancel the nonlinearities of (1) by feedback linearization based on the measured quantities and an available model of the dynamic parameters, denoted by( ·): The available model parameters may have been obtained by estimations (eg, due to unknown load) or approximations (unknown dynamic model, simplified models, etc). The error quantities are denoted using(·) as The dynamics given by application of the control law (3) to the system (1) is referred to as inner loop, and the vector u ∈ ℝ n is the new control input that is to be determined by an outer loop. Clearly, ifM ≡ 0,ñ ≡ 0, (3) achieves a perfect feedback linearization and the minimum-phase nonlinear robot equations (1) are turned into a set of double integrators. Thus, a common choice to stabilize the resulting system is a linear state feedback with positive definite gain matrices then effectively acts as a PD controller.
In the realistic situation, however, the control law (3) inserted into (1) yields the following perturbed double integrator: = M(q) −1ñ (q,̂. q, q, Robust AID controller design thus amounts to selecting u in the outer loop that rejects the inverse additive disturbance , subject to being robust w.r.t. the multiplicative input uncertainty M caused by an inaccurate inertia matrix estimateM. Hence, controller design in the following refers to the outer-loop controller generating u.

Manipulator norm bounds
A number of norm bounds are routinely assumed for the system (1)-(2) as summarized in Appendix A. Moreover, the accuracy of the dynamical model is assumed to fulfill the following properties.
1. There exists a constant > 0 such that, for all q ∈ ℝ n , For the analysis in this paper, it is not necessarily required that < 1 as in the works of Bascetta and Rocco 5 and Spong and Vidyasagar. 9 It will be shown, however, that robust performance enhancement is feasible only if < 1.

Controller parameterization by factorization
In this section, we provide a concise summary of the plant/controller factorization 10 approach central to this article. The resulting Q-parameterization is ubiquitously used in control theory, eg, in robust control, 15,16 computer-aided design, 17 gain scheduling, 18,19 model predictive, 20 hybrid, 21 and adaptive [22][23][24] control. For a general introductory exposition, the reader is referred to the work of Anderson. 8 The factorization approach has been introduced to the robust control of robots in the work of Spong and Vidyasagar 9 in a one-DoF controller design and in the work of Sugie et al 25 for a two-DoF controller design, and is nowadays covered in textbooks 4,26 as well. Proceeding in state space, our notation is along the lines of the work of Tay et al. 22 Consider the system G with state x G ∈ ℝ n x , such that exogenous signals w ∈ ℝ n w (reference signals, disturbances, noise) and control inputs u ∈ ℝ n u are mapped to the outputs y ∈ ℝ n y (measurements available for control) and the performance signals z ∈ ℝ n z (signals to be controlled): ] .
Let G 11 , G 12 , G 21 ∈  ∞ , ie, be rational proper asymptotically stable transfer function matrices. Then, internal stability of the system is given if and only if the feedback loop that is formed by G 0 = G 22 (s) = C y (sI − A) −1 B u + D yu and a controller K 0 is well-posed and internally stable ( Figure 2A).

Youla parameterization
To parameterize all stabilizing controllers, G 0 and K 0 are written in terms of double coprime factorizations such that the left( ·) and right (·) factors have no unstable pole-zero cancellations, ie, chosen to satisfy the double Bezout identity The following result is summarized, eg, from Th. 12.17 in the work of Zhou et al. 15 Proposition 1 (Q-parameterization). Given coprime factorizations (12) of the plant and the controller that fulfill (13), the set  of stabilizing controllers for G 0 can be characterized in terms of arbitrary stable Q as for a left stable linear fractional form. Reformulating from the definition, by the Bezout identity, we also have is invertible and the generator system J can be implemented as Proof. See the work of Zhou et al. 15 In words, by a Q-parameterization shown in Figure 2B, every linear internally stabilizing controller K can be implemented as a lower fractional transformation K(Q) =  ( J, Q). That is, some central system J, where J 11 is the nominal controller K 0 , is interconnected with another stable filter Q ∈  ∞ such that col(u, r) = [J] col(y, s), s = Qr. (17) Note that the relationship between K and Q is bijective and that the nominal controller K 0 is recovered by Q = 0.

Dual-Youla parameterization
The dual parameterization is most known for its use in closed-loop system identification in the Hansen scheme. 27 In addition, parameterizations of the dual form play a key role to reduce conservatism in the robust design of advanced motion control systems; see the work of Oomen. 28 The dual parameterization characterizes 8,22 in terms of a stable parameter system S all systems that are stabilized by one particular given controller.
Proposition 2 (S-parameterization). Given coprime factorizations (12) of the nominal plant G and the controller K 0 that fulfill (13), the set of all proper plants stabilizable by K 0 is characterized in terms of an arbitrary stable operator S as Proof. The coprime factors of G(S) and of K 0 satisfy the Bezout identity; hence, the result is dual to Proposition 1. A detailed derivation is spelled out in Chap. 3.4 in the work of Tay et al. 22 The relation between any S and the corresponding G(S) is bijective, and the nominal system G 0 is recovered by S = 0. Note that S ∉  ∞ implies that the pair (G(S), K 0 ) is not stabilizing.

Double-Youla parameterization
Combining the above parameterizations of plant G(S) and controller K(Q), the double-Youla parameterization 22 is obtained. It describes the set of loops formed by a nominal plant/controller interconnection and perturbations of the closed loop by systems S and Q.
Proposition 3 (Double-Youla parameterization). Let (G 0 , K 0 ) be a stabilizing nominal plant/controller pair, let G(S) be the class of plants described by (18)- (19), and let K(Q) be the class of controllers (14)- (15). Then, (G(S), K(Q)) is well-posed and internally stable if and only if the feedback system (Q, S) is well-posed and internally stable, ie, there exists Proof. See Th. 2.1 in the work of Tay et al. 29 Let us emphasize in particular that the transfer function matrix from the input s to the output r in the closed loop of Figure 2B is the parameter S. Thus, to establish robust stability under both plant and controller uncertainty, the nominal plant/controller interconnection has to be stable, as well as the loop (Q, S). For a more detailed exposition, the reader is referred to the work of Tay et al. 22 Proposition 3 allows to characterize the set  of maximally allowed perturbations Q ∈  of the controller K subject to robust stability of the closed loop for some set  of permissible models S ∈ . Proposition 4 (Stable plant/controller perturbations). Consider a set of plants  and a set of controllers  given by Then, all plants in  are stabilized by all controllers in  if and only if S · Q < 1.
Proof. This follows directly from application of the small-gain theorem 30 on the (Q, S) loop; see pp. 166f, 224 in the work of Tay et al. 22 Early investigations of such parameterizations were motivated by adaptive robust performance enhancement. 29 The advantages of the double-Youla framework are its inherent handling of uncertainty, the variety of design methods nowadays documented in the literature on the basis of the Youla parameter, and finally its nonconservatism compared to using other robustness metrics such as the Vinnicombe gap metric. 31 Therefore, recent work exploits the double parameterization approach, eg, to establish robust stability for systematic coupling controller design 32 or switching control for uncertain systems. 33 A double-Youla parameterization that is particularly suitable for implementation on robotic systems will be derived in Section 4.

Robust AID control
Before we proceed to characterize robust AID controllers via the double-Youla parameterization, a brief overview of the related work is given. The robust control of robot manipulators has been subject to research for decades, and a wealth of methods have been developed, 1-4 based on a multitude of underlying approaches, eg, linear multivariable, passivity-based, sliding mode. 34,35 We therefore review selected work paradigmatically by ascending dynamic model demand.

Static and model free.
It has long been known that a high-gain PD controller applied directly to the nonlinear rigid-body system robustly yields uniform ultimate boundedness of the tracking error. 36,37 Design methods for gain selection are nonetheless being investigated today 38 and a high-gain PD controller can be suitable even for fast dynamic manipulation tasks. 39

Based on crude approximation.
Even if not based on inverse dynamics, robust manipulator control methods typically employ a nominal inertia model, eg, using a disturbance observer 40 or model-free time delay control. 41 That is, only the most dominant part of the manipulator dynamics is approximated and the rest is treated as an uncertain disturbance input. [42][43][44] Taking only a diagonal estimate of the inertia matrix, the problem is handled as a linear decoupled system subject to the disturbances induced by neglected cross-coupling terms. This way, it is always possible to achieve < 1 in (9). In the authors' experience, 11 this approach constitutes a viable trade-off to build a parameterization-based robustly stable learning control system that noticeably exploits domain knowledge without a detailed dynamical model of the robot manipulator at hand.

Nonlinear model-based.
Taking a nonlinear dynamical model of the robot manipulator allows to accomplish a better approximate feedback linearization. Nonetheless, the model (6) is always imprecise to a certain extent, ie, (7) and (8) do not vanish. Classic robust manipulator control methods 1-4 therefore design the outer-loop controller to ensure robust stability of the overall loop. Recent research also considers performance or optimality criteria, eg,  ∞ optimality, 45 time-domain bounds, 46 or orbital stabilization around the desired trajectory. 47 Most relevant for the developments in this article are the classic linear multivariable design 9 and the Lyapunov-based robust manipulator control designs 5 reviewed in more detail next.

The linear multivariable approach
A double-Youla parameterization allows to consider perturbations of both the controller and the plant models. The early robust manipulator control methods, in contrast, only exploit the characterization of all stabilizing compensators for the linear unperturbed model. Spong and Vidyasagar 9 were able to show that one can always find a linear control law stabilizing the nonlinear loop resulting from AID if the dynamic model of the robot satisfies Assumptions 1 and 2 and < 1 in (9). To this end, consider the control where the additional term is supposed to increase the robustness against the uncertainties due to the approximate controller (3). Inserting (20) and (3) into (1), the resulting error dynamics read where the vector is induced by the uncertainty resulting from the approximate model in the closed loop: Hence, (q d , q, . q, ) is a nonlinear term that cannot simply be rejected as if it were an external disturbance. Instead, under the assumptions (9) and (10) with < 1, it is shown in the work of Spong and Vidyasagar 9 that, for some constants b and , the bound || || 30 Then, if ( 1 + 3 ) < 1, the control , the tracking error e, and the uncertainty are bounded. A sequence of Youla parameters Q k is thus designed such that 1 → 0 and 3 → 1 for k → ∞. The resulting robust controller K k is finally given from (15) and is generally a high-gain dynamic compensator.

Lyapunov-based method
Another popular approach to design a robust control input u for (3) is based on Lyapunov's second method. [1][2][3][4] It is usually assumed that a PD controller with feedforward acceleration compensation and an additional input term is employed in the outer loop, which results in the error system Similar to (21)- (22), the uncertainty in the closed loop is lumped in a single term and one needs to select an appropriate control Lyapunov function to design so as to suppress the destabilizing effects of . For the details and assumptions underlying the classic approach, the reader is referred to Chap. 6.5.3 of the work of Sciavicco and Siciliano, 2 the work of Spong et al, 3 and Chap. 5.2 of the work of Lewis et al. 4

Drawbacks of established approaches
The robot control methods based on the approaches of Sections 2.5.1 and 2.5.2 have a number of drawbacks in common. First, all of the AID uncertainty is lumped in a single quantity . However, it is clear from (6) that an inaccurate inertia model results in multiplicative input uncertainty, whereas the neglected manipulator nonlinearities induce the inverse additive disturbance . Therefore, the structure of (6) is dismissed when the analysis starts with a single term . Second, an outer-loop PD controller is commonly 1-4 applied to stabilize (6) prior to the robustness analysis. Therefore, in all subsequent developments, one has to work with an error system (24) where the gains K P , K D of the controller occur in the dynamic matrix A. These aspects entail unfavorable consequences. The additional input must be carefully constructed to suppress the internal nonlinear disturbance , and it remains prohibitive, in general, to adapt on the fly by data-driven methods. Furthermore, to design , a scalar bound || || ≤ (||e||) is required, which in the classic robust manipulator control 1-4 depends on the gains ||[K P , K D ]|| of the nominal PD controller. Thus, as pointed out in the works of Bascetta and Rocco 5 and Abdallah et al, 34 higher gains in the controller require stronger robustifying inputs to correct for seemingly higher uncertainty. In contrast, the practitioner's approach of using a high-gain PD controller is widely known to work robustly in practice. For the Lyapunov-based design, this discrepancy has been resolved by Bascetta and Rocco in the revised design. 5 We will return to these works by comparing to our novel parameterization in Section 5.3.

MAIN RESULT
Our main goal is to shape a set of admissible controller perturbations such that numerous advanced design methods are applicable for enhancement of the AID-based controllers in a strict robust stability framework. To circumvent the drawbacks summarized in Section 2.5.3, we keep the uncertainty separated throughout instead of lumping the effects of (7)-(8) into a single additive term. As a sideline, we derive bounds on the perturbations to (6) before any outer-loop controller comes into place. To eventually keep the influence of the outer-loop controller transparent as well, we describe the uncertainty by means of the dual-Youla parameter. Therefore, in the framework proposed next, by construction, the uncertainty due to AID remains clearly distinguishable from the gains of the outer-loop controller.

Problem statement
It is unfortunately not obvious to find in the literature 1-4,26,48 a suitable generalized plant 15,49 description for AID, as most methods work with error dynamics (21) or (24). Therefore, the first problem we tackle is how the standard robotic bounds (9)-(10) translate into a generalized setup ( Figure 3A) without lumping the effects of (7)-(8) into a single additive term.
Problem 1 (Generalized plant formulation of AID). Given the system (1) and the control law (3) with Assumptions 1 and 2 fulfilled, reformulate the AID (6) and the corresponding bounds as a system (25) such that the uncertainty (7) and (8) acting on the nominal double integratorq = u is described by a linear fractional transformation as shown in Figure 3A. That is, the uncertainty is captured by an unknown norm-bounded Δ operating on the signals z Δ to yield the input perturbation w Δ , In the next step, we analyze how these bounds affect the closed-loop system when the nominal outer-loop controller is applied. The main result is a parameterization of robustly stabilizing controllers for the rigid-body manipulator under AID control. Once Problem 2 is solved, the solution to the following problem is immediate.
Problem 3 (Double-Youla for robust approximate inverse dynamics). For a given robot under AID control as in Problems 1 and 2, characterize in terms of the parameter set  a subset of robustly stabilizing controllers  R ⊂ K().

Generalized plant formulation of uncertainty bounds
We first tackle Problem 1 to obtain an uncertainty formulation in the form of the generalized plant depicted in Figure 3A.
For convenience, let us restate (6) in the formq where external disturbances w dist ≜ M −1 (q) dist , and the vector of internal model uncertainty w U

q,q,̂. q).
Subsequently, the dependency on joint positions and velocities is dropped for brevity in notation. With Assumption 3 the summand w dist is bounded by ||w dist ||  ∞ ⩽ M u C dist and depending neither on state nor on control; w dist is therefore considered as external disturbance input. Note that by (9)-(10), both w M Δ and w ψ Δ are nondynamic, time-varying nonlinear uncertainties; it is therefore possible to use a norm-bounded uncertainty description (26), where the upper bound on the induced matrix norm ||Δ|| ≜ max z Δ (t)≠0 ||Δz Δ (t)|| ||z Δ (t)|| ≤ is satisfied for all times t (Sec. 2.3.1 in the work of Petersen and Tempo 50 ). This corresponds to the maximum amplification over all input directions z Δ and is key to solving Problem 1.

Theorem 1 (Structured uncertainty under AID).
Under the conditions of Problem 1, the perturbed double integrator system (6) can be conservatively reformulated as a G − Δ structure depicted in Figure 4 The bounds M and are scalars given by (B3).
Proof. The derivation is given in Appendix B.1.
Remark 1 (Uncertainty separation). To obtain such a description, one could also work with w U Δ = Δ U z Δ instead of collecting the terms w M Δ , w ψ Δ of (27) separately in the vector w Δ . Then, however, the result is unstructured with an overly conservative bound on the uncertainty matrix .
By adopting the uncertainty structure (28), the separation of the two uncertainty sources is preserved because the summation of u, w M Δ , and w ψ Δ in (27) is captured by the signal interconnection in the generalized plant G, as depicted in Figure 4. Therefore, this structure is carried into the dual-Youla operator in the sequel.

Dual-Youla uncertainty characterization
We now calculate a realization of the uncertain dual-Youla parameter S when the nominal controller K 0 is applied to stabilize the uncertain plant  u (G, Δ). Recall that in (18)- (19) all systems stabilized by K 0 are parameterized as .

FIGURE 4 Generalized plant interconnection of a robot
manipulator under approximate inverse dynamics control with uncertainties "pulled out." 15 This figure only depicts the perturbed double integrator system and the uncertainty, ie, the G-Δ structure suitable for subsequent analyses. The precise configuration of the control and performance channels y, z depends on the manipulator control goal at hand, eg, tracking or impedance control. A tracking control example is given in Section 5 with the corresponding state-space realization in Appendix C Consequently, if G yu (Δ) =  u (G, Δ) is robustly stabilized by K 0 , ie, G yu (Δ) ∈  for all Δ ∈  Δ , then equivalently there exists an uncertain S ∈  Δ ⊆ . Key to our approach is the following explicit connection of the open-loop uncertainty description in terms of Δ and in terms of the S obtained in the closed loop with K 0 applied (Th. 3.4 in the work of Niemann,12 and the work of Niemann and Stoustrup 51 ): In (29), T Δ refers to the mapping from the inputs col(w Δ , s) to the outputs col(z Δ , r). In operator notation, T Δ is expressed as Remark 2. As Niemann 12 points out, "S depends only on the uncertain block Δ and the coprime factors." For the purpose of this paper, however, we would like to emphasize the fact that (29) and (30) also depend on the interconnection used in the generalized plant (25). This is an important feature of this article: in contrast to the robust control approaches summarized in Section 2.5, by the derivation in Section 3.2, we keep the inertia-induced uncertainty w M Δ completely separated from the nonlinear inputs w ψ Δ , thus reducing the conservatism of the resulting S. With the generalized plant of Figure 4 and the bounded perturbation set  Δ from (28), we obtain the following result.
Theorem 2 (Realization of dual-Youla uncertainty set). Consider the system (1) with the AID controller (3) such that Assumptions 1 and 2 are fulfilled. According to Section 3.2, the resulting system dynamics is conservatively covered by the linear, stabilizable, and detectable plant (25) with a state-space realization (C1) and the uncertainty structure (28). Let a nominal linear controller be applied to the outer loop, with the pairs (A K , B K ) stabilizable and (A K , C K ) detectable. Let the state-feedback gains F G and F K be designed such that A 11 + B 13 F G and A K + B K F K are stable. Then, the set of uncertain dual-Youla operators is given by Proof. The proof is given in Appendix B.2.
From the realization (33), it becomes apparent that S is much more involved than the diagonal uncertainty structure of Δ. Hence, standard robustness tools 52 such as analysis 53 or worst-case gain assessment 54 cannot be directly applied.
To solve Problem 2, it is nonetheless required to find an estimatêS of the worst-case gain To this end, we propose to adopt a randomized approach 55 allowing for the following probabilistic worst-case assessment.
Remark 3 (Realization of dual-Youla parameter for norm-bounded memoryless uncertainty). The state-space realization for the dual-Youla parameter S according to (33) is a fairly general result by itself: It covers, under the usual assumptions for double-Youla parameterizations, the linear generalized plant interconnection of the form (C1) subject to nondynamic (memoryless) norm-bounded uncertainty Δ, while allowing for any linear central controller K 0 in the Q-parameterization. Thus, the result extends 12 in that, due to the symbolic derivation, the realization of S given in (33) is explicit in the state-space matrices and hence suitable for numeric computation of the  ∞ -norm. By contrast, in our simulation study reported in Section 5, a straightforward * implementation by (29) quickly becomes inaccurate, presumably because of the numeric inversion that easily yields badly conditioned systems.

Characterization of robustly stabilizing controllers
With S according to (29) and (33), respectively, the control loop can now be described as shown in Figure 3B. The loop T(S) is stabilized but uncertain and the controller is determined by some Q ∈ . Writing out the matrix transfer operator The following sufficient condition is established to ensure the overall stability of the scheme.
*Direct computation of S by formula (29) using MATLAB state-space objects ss yields hinfnorm(S,1e-6)=inf although S is stable.
Lemma 1 (Double Youla under exogenous disturbance and plant uncertainty). Under the previous assumptions, for all Δ ∈  Δ , there exists correspondingly an S ∈  Δ . Denote by  the set of finite-gain stable, admissible parameters Q ∈ . Referring to Figure 3, T zw (S, Q) is robustly stable if both of the following conditions hold.
Proof. Please refer to Appendix B.3.
By the condition (ii), it is immediate to characterize a set of controller parameters  that preserves robust stability and is suitable for a wide range of performance enhancement methods.

If̂S is finite, K 0 is robustly stabilizing. A subset of all robustly stabilizing controllers is then given by
and every controller K ∈  R stabilizes the nonlinear system (6)- (8).
Proof. According to Theorem 1, the uncertain nonlinear dynamics (6)-(8) are conservatively covered by the uncertainty structure  Δ from (28). As by Theorem 2,  Δ corresponds to  Δ when the nominal controller K 0 is applied. With Proposition 5,̂S is the worst-case gain over all  Δ by letting p → 1, → 0. The result is then a direct consequence of Lemma 1.
On the one hand, the first condition (i) of Lemma 1, ie, robust stability of the uncertain loop with only the nominal controller applied, may seem quite restrictive. On the other hand, the stability of the (Q, S) loop is ensured by a small-gain argument; thus, it also guarantees internal stability when Q is a nonlinear or time-varying stable operator with an appropriately defined stability notion. 30 Hence, due to the double-Youla parameterization, a variety of advanced methods can be used for manipulator control design in the proposed rigorous robust stability framework (cf Figure 1).
Remark 4 (Model complexity trade-off). Theorem 3 provides a way to quantify the trade-off between the accuracy of the available manipulator model and the amount of controller enhancement permissible without sacrificing robust stability of the system: A very imprecise robot model will yield large S and consequently  R → {K 0 } as  → ∅. As for the other extreme, a perfect model allows for a true feedback linearization; hence, S = 0, and bŷS → 0, Theorem 3 recovers the set of all controllers that stabilize a double integrator.
Remark 5 (Robust stability under nominal control). If the nominal controller K 0 does not stabilize the system under all perturbations Δ ∈  Δ , then S = ∞, ie, there is some unstable S(Δ). In such cases, it may via Q still be possible to adaptively stabilize the (Q, S) loop of the double-Youla parameterization. 29 We do not pursue such approaches further but restrict attention to a more robust setting: here, stabilization under all perturbations Δ ∈  Δ by means of all controllers (36) is the asset allowing for straightforward online performance enhancement.

SPECIAL CASE: STATIC NOMINAL CONTROL
In this section, we specialize the main result to the very commonly used controller of the PD-type (5) or (23) as initial controller K 0 . To this end, under the previous assumptions, a Q-parameterization can be constructed as follows.
Proposition 6 (Central J for static nominal control 11 ). A realization of the central system J to build a Q-parameterization around a static controller for the system (C1) is Proof. See the work of Friedrich and Buss. 11 A block diagram of (37) is depicted in Figure 5.
Remark 6 (Components of control input). Comparing to Section 2.5, the overall control signal corresponds exactly to the standard ansatz for the outer loop (23), but the additional signal u is generated differently from the "robust-control term" 5 , precisely model based in the Q-parameterization generated from (37).
Note that the central system J inherits with (37) a separation structure that is very useful for implementation on hardware: The two-port system (37) is given directly by the static nominal controller D 0 and some dynamic augmentation to generate the signal r. Referring to Figure 5, the augmentation J can be further simplified by choosing the coprime factor stabilizing gain F G as By this choice, only the filtered output s is added to the control input u = u nom + u a + s (37) with (38) = u nom + s. Thus, the proposed special case is particularly simple to implement: Opposed to the standard Youla parameterization, 15 no observer-based central controller is required. Instead, robot manipulators that are already driven by a PD controller can be augmented to generate the parameterization.
To quantify the uncertainty in the closed loop bŷS, an expression for the corresponding dual-Youla parameter is derived as well. Proof. The derivation is analogous to the full case of Theorem 2, replacing the coprime factors with those for a static controller. 11 Alternatively, the given realization of S can be obtained from (33) by appropriately removing obsolete rows and columns for undefined A K , B K , C K , and taking D K = D 0 .

Summary of the proposed robust manipulator control framework
The steps to employ the control design in practice are summarized as follows.
Steps 1 and 2 are standard, whereas steps 3 -5 exploit the novel parameterization based on the double-Youla approach.

DISCUSSION AND ILLUSTRATIVE STUDY
First, let us discuss the implications of the main result by the familiar planar elbow manipulator example. Next, the practical utilization of the proposed framework is illustrated by means of a robotic manipulator with six rotational DoFs under varying payload. We will finally compare the novel robust stability framework to the existing ones and outline how a variety of control design methods can be used in the control approach put forward in this article.

Discussion of worst-case dual-Youla uncertainty measure
The dual-Youla characterization according to Proposition 5 of our main result provides a new perspective to uncertainty quantification in AID manipulator control.

Example setup
As in the reference book, 4 we use a planar elbow manipulator with two rotational DoFs to illustrate the control scheme and discuss our results. For the description and detailed physical parameters of the particular manipulator, the reader is referred to the work of Schill and Buss. 39 The resulting model is given by (all in SI units)

q. (40b)
A preliminary analysis yields the following numeric values of the bounds (A1) for this manipulator: To analyze the interplay of the available model knowledge and the outer-loop controller w.r.t. the uncertainty S(Δ), the most commonly used controllers of type (3) are summarized in Table C1; the abbreviations PFL, AID, SID, DS, GC, and NID correspondingly refer to the inner-loop controllers in the following. The parameters used in the simulation study and the achievable uncertainty bounds are also given in Table C1.
Remark 7 (Two-DoF control). We employ a two-DoF control scheme in the sequel, ie, the feedforward and feedback gains can be tuned independently. By selecting y = col(q,̂. q, q d , e,q d ), the following developments hold for the general class of two-DoF controllers. We use the state-space realization as given by (C1)-(C2) in Appendix C, corresponding to a tracking control configuration of Figure 4.
To keep the following discussion simple, we restrict most of our attention to the specialization of the main result from Section 4. The central controller is based on static u 0 = K 0 y with and the Q-parameterization is generated from (37). Consequently, S is calculated according to Corollary 1 unless stated otherwise. The stabilizing gains F G , F K are design parameters in the construction of the stabilizing factorization (B4) of the plant and controllers and affect S . In this section, no F K is needed because of the static nominal controller, and unless stated differently, F G is calculated by (38). Note that, by construction, S is a dynamical operator mapping s ∈ ℝ n u to r ∈ ℝ n y ; hence, in this example with two joints and n y = 10 measurements according to Remark 7, S is a 10 × 2 uncertain system. With S from (34) describing the worst-case uncertainty in the closed-loop system given a specific outer-loop controller, the following qualitative properties should be captured. By inspection of (33), all these aspects influence the expression for S(Δ); therefore, each will be discussed separately. To obtain̂S, the procedure from Table 1  Remark 8 (Signal uncertainty for revolute manipulators). The example manipulator consists only of revolute joints; thus, a constant bound on the gravity error can be assumed. In other words, it is known exactly that ||ñ|| does not depend on joint positions and consequently, the uncertainty on nonlinearities can be taken as a matrix Ψ ∈ ℝ n×(n+1) .

Influence of AID parameters
We first investigate how the accuracy of the approximate model influences the worst-case gain S . To this end, for now, assume that the outer-loop controller is a PD feedback controller for critical damping in the nominal loop, ie, it is defined by (42) with First, consider the PFL case of Table C1, ie, M = Ψ = 0. Clearly, in this case, there is no uncertainty as the underlying feedback linearization uses a perfect model of the manipulator dynamics. The uncertainty set consequently degenerates to  Δ = {0} and from (33) immediately S = 0 follows. As expected from Remark 4, the nominal decoupled double integrators describe the resulting loop perfectly.
In the robust control approaches reviewed in Section 2.5, the accuracy of the inertia model plays a crucial role as measured by the value of from (9): Both Spong and Vidyasagar 9 and Bascetta and Rocco 5 need < 1 as a prerequisite. We therefore begin by investigating the influence of M on the worst-case gain S while = 0. The calculations were carried out over a fine grid on 0 < M ≤ 1.1. The results are shown in Figure 6A. The uncertainty characterized by the worst-case ||S(Δ)|| ∞ is very low for 0 < M < 0.7. We can then observe a rapid increase for 0.7 < M < 0.85 and̂S = ∞, ie, unstable S, for values greater than M ≈ 0.85. In other words, robust stability of the inner AID loop using the PD controller with the gains (43) is lost for inertia uncertainty greater in norm than 0.85, given thatñ = 0.
Similarly, the influence of the norm of the neglected nonlinearitiesñ can be evaluated. To this end, we have calculated S over a grid of 0 < Ψ ≤ 110 with M = 0. As depicted in Figure 6B, for low values of 0 ≤ Ψ < 50,̂S stays small with a sudden increase up to instability of S for Ψ > 63. Comparing with Table C1, we find that, for the gains (43), robust stability cannot be concluded for the inner-loop controller types NID, GC, DS, and SID, while viable robustness is obtained under the AID law.
We also performed a parameter sweep calculation over a grid of both M and Ψ , as shown in Figure 7. It can be observed that there is a relatively clear boundary between model accuracy that allows for robust controller enhancement and too inaccurate modeling that results in̂S = ∞. Note that these values are depending on the controller gains and were obtained † Implementations of suitable methods 55 for uniform sampling over norm-bounded real matrices are readily available. 56 (A) (B)   Table C1. Here, only the AID allows for robustly stable controller enhancement by some parameter Q. AID, approximate inverse dynamics; DS, diagonal scaling; SID, simplified inverse dynamics using controller (43). Figures 6 and 7 demonstrate how the main result allows to quantify the worst-case uncertainty set under a given AID situation.

Central feedback gains.
A high-gain PD controller is working robustly in practice to control the rigid manipulator. 37 It is also clear from (33) that the P-and D-gains of the nominal controller influence the uncertainty expressed by the dual-Youla parameter. Hence, we calculated the values of̂S over a grid of P/D combinations for the central gain (42). As shown in Figure 8, the evaluation was performed over a dense grid with values of 0 < K p < 5·10 4 and 0 < K d < 10 4 . Three different parameter sweeps were performed, with the uncertainty levels M and Ψ set corresponding to the inner-loop controllers AID, SID, and DS from Table C1. In accordance to intuition, one can observe that the uncertainty in the closed loop decreases with increasing gains and increasing model accuracy. Furthermore, a minimum D-gain of approximately K d > 10 is needed to conclude robust stability although the model employed in the AID controller is relatively good. Note that even if the model had been perfect, some K d > 0 would have been required as lead compensation for the double integrator plant. In accordance to the literature, 37 large PD gains are required for robustification in case of limited model knowledge.

Feedforward term.
The previous analysis only considered a controller feeding back position and velocity errors. Here, the effect of adding feedforward terms is examined, ie, K ff ≠ 0 in (42). In the robust robot controllers of other works, 2,5,9 the outer-loop controller (20) and (23) includes a summandq d . In the ideal case then, q = ∫ ∫q d dt 2 = q d even without feedback. In the methods reviewed in Section 2.5, the robust control term also depends on the maximum norm ||q d ||  ∞ of the desired trajectory acceleration. In the framework put forward in this article, on the contrary, the uncertainty measure S does not change when the feedforward gains are included in (42) by some K ff ≠ 0. This may  Table C1. Configurations wherêS is not finite are marked by x for visibility. The GC and NID controllers yield unstable S everywhere on the depicted grid and are hence excluded from the graph. The PFL case is also not depicted aŝS = 0, ∀K d , K p > 0. AID, approximate inverse dynamics; DS, diagonal scaling; GC, gravity compensation; NID, no inverse dynamics; PFL, perfect feedback linearization; SID, simplified inverse dynamics [Colour figure can be viewed at wileyonlinelibrary.com] seem counterintuitive, but it is a simple consequence of the clear structure obtained by the generalized plant: Due to the Q-parameterization, subsequent controller design is based on an internal model of the plant; hence, the structure is carried into all controllers parameterized by Q (cf Remark 2). More formally, a two-DoF controller can be thought of as a single controller K = [K ff , K fb ] in feedback connection with an augmented plant [0, G ⊤ ] ⊤ . 22(p32f) Hence, all two-DoF controllers stabilizing G are obtained, 22(pp49f,53) as all one-DoF controllers for [0, G ⊤ ] ⊤ . Consequently, the plant in feedforward control channels is known exactly: It is a zero operator. The control u cannot affect feedforward signals. Naturally, the associated uncertainty with this plant is zero, as are the corresponding entries in S = . Indeed, in our example system with the measurements according to Remark 7, calculating the uncertain operator S(Δ) confirms that its realization always has the form There is accordingly no uncertainty associated with the last six components of y, ie, q d , It is a clear advantage of the proposed parameterization that the loop uncertainty measure S can be made independent from quantities such as q d ,q d by the two-DoF design according to Remark 7. Note that Figure 8 is obtained irrespectively of K ff because ||S|| ∞ = ||S fb || ∞ .

Illustrative example: P560 with varying payload
With the previous example, only the novel dual-Youla perspective on AID uncertainty was discussed. In this section, the utilization of the complete double-Youla parameterization is illustrated by means of a multi-DoF robot system under varying payload. To this end, we consider a PUMA P560 manipulator with six DoF, the dynamic model being publicly available from the work of Corke. 57 The task is to track a fast reference trajectory for each joint while the payload m p applied in 10 cm distance to the end-effector is uncertain within m p ∈ [0.5, 1.5] kg. The trajectory is chosen to cross areas of the state-space where nonlinearities and inertial interactions are strong; precisely, we use Therefore, an outer-loop controller K 0 is designed to robustify the loop. We use the proportional gains K p = 900 and derivative gains K d = 2 √ K p = 60 in each joint. 2 Next, one needs to obtain numeric values for the bounds (9), (10), and (A1a). By a simulation as in the work of Rocco, 13 we find = 0.3438 (worst case with m p = 0.5), 0 = 5.4028, 1 = 4.3689, M u = 5.7032. 3 While the previous bounds are only due to the manipulator and uncertain feedback linearization, we now construct the novel generalized plant setup proposed in Theorem 1. Just as in the previous example, we use the two-DoF design structure given in Appendix C. Now, the uncertainties are conceptually pulled out as shown in Figure 4. To keep the conservatism reasonable, some characteristics of the manipulator are considered. In particular, • all six DoFs of the P560 are rotational (cf Remark 8); • the Coriolis and centripetal effects of the P560 are to a very large extent determined from . q 3 ; • Friction is independent of payload, hence compensated by the inverse dynamics controller.
Accordingly, the uncertainty structure (28) is described by matrices M ∈ ℝ 6×6 and Ψ ∈ ℝ 3×(3+1) . From the values obtained in step 2 , we have as of (B3) the associated norm bounds M = 0.3438 and Ψ = 39.627. 4 Next, the realization of the uncertain dual-Youla operator (39) is calculated. We then assign the probability levels p = 99.9% and = 10 −4 to employ Algorithm 1 from Table 1, using 10 4 samples. The worst estimate iŝS = 0.1137. 5 The detailed uncertainty quantification of the dual-Youla parameter S by 1 -4 is worthwhile once the Q-parameterization is used to enhance performance. That is, the nominal PD controller is firstly augmented, as shown in Figure 5. According to Theorem 3, the parameterization now allows to search over robustly stabilizing controllers simply by choosing a (possibly time-varying) finite-gain  2 stable parameter system Q. By the value of There is a multitude of design methods for the parameter Q, and the reader is referred to the literature. [15][16][17]19,21,22,24 Controller design in Q is beyond the scope of this article, yet in general, the less restricted ||Q|| ∞ needs to be, the more design freedom there is for any such method. Let us briefly illustrate this trade-off. To this end, some controllers Q are "designed" by randomly sampling stable dynamic systems of maximum order 10. We run 20 times 50 nonlinear simulations of the closed loop over T = 10 seconds, each time increasing the allowed threshold of the loop norm ||Q|| ∞ ·̂S in a logarithmic range of 0.05, 0.1, 0.25, 0.75, … , 500, 10 3 , 10 4 . To account for the uncertain load, a random value 0.5 ≤ m p ≤ 1.5 is assigned to the payload in each simulation run. The initial state is q 0 = [0, ∕4, − ∕2, 0, 0, 0, 0 1×6 ] ⊤ . A discrete-time formulation of the controller is used for implementation. The sampling time is constant with t s = 5 ms, and the solutions are obtained by a third-order Runge-Kutta method (Bogacki-Shampine). To keep the comparison irrespective of feedforward control, only a feedback controller augmentation is used in this example, ie, Q ff = 0 6×18 in all simulations.
The result of these simulations is summarized in Figure 9, depicting the 2 -norm of the six-dimensional error signals. Controllers on the left side of the vertical dashed line in Figure 9 are within the set  R from (36). It can be observed that none of the controllers in  R leads to a significant increase in the error norm. Leaving the set of admissible robust controllers, the error norms rapidly increase up to practically useless control behavior and instability of the simulated loops starting from approximately ||Q|| ∞ > 20∕̂S. Compared to Q = 0, 46 of the 548 controllers K ∉  R yield better performance; however, 211 controllers K ∉  R lead to instability. This study shows how the parameterization of Theorem 3 is indeed useful: It allows to search exclusively over robustly stabilizing controllers. Robust performance can subsequently be obtained by suitable design of Q subject to ||Q|| ∞ < 1∕̂S. Approximately half of the LTI systems Q actually improve the performance of the overall control system in this study, albeit being only randomly sampled. This is depicted in Figure 9B, which shows a magnification of the area marked by the rectangle in the lower left corner of Figure 9A.

Comparison to related work
Single primary vs double-Youla parameterization.
The robust control of Section 2.5.1 derived in the classic frequency domain Youla parameterization always yields a high-gain dynamic compensator. 9 Here, the parameterization is in state-space and it can be based on both dynamic and static central K 0 . Moreover, the conventional approach is lumping all uncertainties in a single term (22) of internal feedback disturbances that must be suppressed. In contrast, the structure (28) proposed in this article more accurately reflects that of AID uncertainty (6)- (8). It may be less evident that the purpose of the parameterizations is quite different. In the method of Spong and Vidyasagar, 9 the central controller does not ensure robust stability; robustness is obtained by careful design of Q ("The choice of [Q] is not easy to see" 9 ). As the primary Youla parameterization only yields stabilizing controllers for the unperturbed plant, ie, the nominal double integrator, in the work of Spong and Vidyasagar, 9 the design of the filter system Q must ensure that the internal disturbance is suppressed and not destabilizing. In this paper, we do not report another robustification tool but rather characterize a whole set of robustly stabilizing AID controllers (36) such that the control performance can be enhanced online (cf Figure 1 and Remark 5). Such enhancement is possible using time-varying or switching schemes, including adaptive, 23,29 learning, 11 model predictive, 20 hybrid, 21 and gain scheduling 18,19 approaches. Most of these references only focus on the design aspect of the system Q, ie, design for a nominal model within the set of stabilizing controllers. By the contributions in this article, such a design can be systematically tightened to the subset of robustly stabilizing controllers for robot manipulators. Finally, the traditional approach dictates the design of the system Q and is therefore limited to robust trajectory tracking control. The double-Youla framework reported in this article, in general, does not make this restriction because all derivations refer to the generalized plant (25).

Comparison to Lyapunov-based designs.
By the revised robust control design, Bascetta and Rocco 5 essentially undo the step in the Lyapunov-based designs (Section 2.5.2) of lumping the closed-loop uncertainty into a single term in (24). Their method consists of two steps: First, the nominal PD controller is designed to ensure global asymptotic stability of the error system under perturbation with any admissible matrix || M || ≤ . Second, the asymptotic performance under the influence of the neglected termsñ is recovered by an additional feedback term = f(e) obtained via a quadratic Lyapunov function. Our requirement of (35) that robust stability is ensured by the nominal controller is therefore similar to the first step of Bascetta and Rocco. 5 While the methodological approach is quite different, the additional input u in this paper corresponds to the term in the work of Bascetta and Rocco, 5 as emphasized in Remark 6. The parameterization of Theorem 3 thus constitutes a viable alternative to the Lyapunov-based methods. The general drawback of a factorization approach is that only uniform ultimate boundedness is ensured as long as the linear bound (10) holds. In turn, the double-Youla parameterization brings some advantages. First, the Lyapunov approaches to robust manipulator control typically result in high-frequency additional control signals . Here, in contrast, the characteristics of u depend exclusively on the design of Q. For example, the controllers of Section 5.2 generate smooth signals because Q is LTI. Second, Bascetta and Rocco 5 only deal with robust tracking control. The novel double-Youla parameterization, however, is derived for a generalized plant structure (25) and an arbitrary LTI controller K 0 . It therefore complements the Lyapunov-based approaches in terms of versatility, as (25) allows to represent via z many more control objectives in the design step of Q.

Recent approaches.
Let us finally recapitulate the distinctive features of the presented double-Youla approach compared to recent literature on robust manipulator control. The conservatism is determined by the accuracy of the model bounds and nominal outer-loop compensator gains. Helwa et al 58 consequently propose a learning-based adjustment of the uncertainty bounds in the Lyapunov-based robust design. Nonetheless, the robustification term is a switching signal of potentially high frequency, and the overall design is tailored towards tracking control; one could then also work with a robust-adaptive scheme. 59 In addition, the work of Kim et al 46 is related in that the L 1 robustness bounds therein are as well derived by consideration of a generalized plant interconnection. However, Kim et al 46 start from the error system (24), and the disturbance signal is only split into exogenous and internal components. The uncertainty structure used in the work of Kim et al 46 hence does not distinguish between inertia and signal uncertainty, resulting in a conservative design (cf Remark 1). In all these works, the analysis is conducted regardless of the nominal outer-loop PD controller that has considerable influence on robustness. The dual-Youla measure of uncertainty proposed in this article, in turn, provides a general approach to systematically quantify this influence.

CONCLUSION
We provide for the first time a double-Youla parameterization for robust control of rigid body manipulators, with the dual parameterization being the key tool to quantify the uncertainty of the control loop. Using a static nominal controller, the theory specializes to a handy structure suitable for practical implementations. In summary, the proposed methodology constitutes a control strategy that allows to apply numerous advanced design methods for enhancement of inverse dynamics-based feedback controllers in a strict robust stability framework.
We emphasize that the parameterization generally entails controllers based on the internal model principle. Clearly, also the disturbance observer compensation designs are a specific implementation of the internal model principle. 60 Just as the linear robust internal-loop compensator structure from the work of Kim et al 60 has been generalized to the nonlinear setting, 45 a nonlinear flavor of our method might be developed in future work: With suitable extensions, 61 one could take advantage of the available manipulator model directly in terms of a parameterization. It would also be worthwhile to develop a dual-Youla uncertainty characterization for the robust manipulator control with a disturbance observer-based inner loop compensation. 40 Another important research direction is to include the analysis of flexible robots 62

STANDARD ROBOTIC NORM BOUNDS
The reader is referred to other works 2,4,9 for details and an in-depth interpretation of the following assumptions.
Assumption 2 (Manipulator dynamic bounds 2,4 ). When the robot arm (1) is revolute, there exist known positive constants M l , M u , C u , F u , and g u such that, for all (t, q, . q) ∈ ℝ t × ℝ n × ℝ n , the matrices M(q), C(q, . q) and the vectors f ( . q) and g(q) satisfy the following inequalities: The workspace of the manipulator is bounded ||q||  ∞ ≤ q max ; hence, the inertia matrix M(q) is invertible for all q. Finally, || . q||  ∞ ≤ v max holds due to the power limitation.
1. The input disturbance is bounded from above by a known constant C dist such that || dist ||  ∞ ≤ C dist < ∞.

PROOFS
The following simple auxiliary inequality is introduced.
of the plant. Then, for the general controller (31), a coprime factorization can be obtained directly in state-space by adopting the formulae given in the works of Tay et al 29 and Ishihara and Sales 64 Next, an expression for T Δ from (30) has to be calculated, which we repeat here for convenience: ] .
Let us also explicitly state the following systems determined from (C1) as The steps to obtain a compact expression of T Δ,11 are the insertion of the coprime factors U 0 andM 0 from (B4) and algebraic simplification, subsequent application of a state similarity transformation such that = Tx with Finally, plugging the above realizations of T Δ,12 , (I − T Δ,11 Δ) −1 and T Δ,21 into (29) and a number of straightforward but tedious simplifications yield the result (33).

B.3 Proof of Lemma 1
Proof. To obtain a robustly stable loop, the stability condition on (Q, S) from Proposition 4 is not sufficient yet because it only refers to the controlled channel; in Figure 3B, however, the prestabilized yet uncertain loop T(S) is subject to exogenous inputs w. Hence, to ensure robust closed-loop internal stability, it is required that the pair ) be stable 22(p80) for all allowed perturbations. To this end, it is sufficient to show 22(p32) that T 11 (S), T 12 (S), T 21 (S) ∈  ∞ as S ∈  ∞ , and (Q, T 22 (S)) = (Q, S) should be a stabilizing loop by construction. Denoting with P(S) the transfer matrix between col(w, u) and col(z, y) for the plant determined by S, the uncertain operator T(S) is given by 22(p79) T(S) = By the first condition (i), consider that (35) is necessary and sufficient for the stability of the nominal uncertain loop. Then, for all Δ ∈  Δ , there exists a stable dual-Youla operator S ∈  ∞ whose coprime factors satisfỹ M(S), M(S) ∈  ∞ by construction and consequently S Δ ⊂  ∞ . Using (29) to calculate S, by the factorization, the plant under control P(S) in (B6) is precisely that of the uncertain plant  u (G, Δ) from the controller's point of view. Thus, with (35) also P 11 (S), P 12 (S), P 21 (S) ∈  ∞ and consequently T(S) ∈  ∞ ∀Δ ∈  Δ . Alternatively, the requirement (i) could also be proved by coprimeness of the factors involved (App. B in the work of Thomsen et al 20 ). The requirement (ii) follows directly from the small-gain condition of Proposition 4: S is an uncertain stable time-varying operator and the only assumption about the plug-in controller Q is that ||Q|| ∞ ≤ Q for some finite Q . The small-gain theorem is therefore necessary (Th. 9.1 in the work of Zhou et al 15 ) to ensure that all S ∈  Δ are stabilized by all Q ∈ Q.