A Provably Constrained Neural Control Architecture With Prescribed Performance for Fault-Tolerant Redundant Manipulators

In this paper, a novel neural control architecture is proposed and investigated for resolving redundancy in trajectory tracking applications for manipulators with joint velocity constraints. First, a nonlinear invertible map is invoked to transform the constrained system state into a set of unconstrained variables, which allows the proposed framework to realize solutions that rigorously adhere to the specified bound constraints. Next, a quadratic program (QP) architecture is synthesized by incorporating suitably prescribed performance constraints to ensure that the resulting system error achieves exponential convergence to the ground truth while also ensuring that the system states evolve along trajectories with good transient and steady-state behavior. Thus, in contrast with previous approaches that do not rigorously guarantee the satisfaction of the bound constraints in the transient phase and/or the steady-state, the proposed scheme ensures that these constraints are rigorously satisfied while achieving prescribed performance both during the transient phase and in the steady-state. The novelty of the proposed scheme lies in the fusion of prescribed performance constraints with the state and input constraints within the QP framework, which offers the important advantage of higher computational efficiency compared to leading alternative designs. A detailed theoretical analysis is undertaken to prove the global stability and convergence of the proposed scheme. Simulation and experimental results with the KUKA LBR IIWA 14 R820 manipulator are used to verify the efficacy of the proposed scheme in accomplishing trajectory tracking for the fault-free and fault-tolerant cases with multiple joint failures. Finally, detailed performance comparison studies with leading alternative designs are further used to illustrate the advantages of the proposed scheme.


I. INTRODUCTION
A redundant robotic manipulator refers to a manipulator 22 which possesses more degrees of freedom than it is required 23 to complete a task objective. The presence of these additional 24 degrees of freedom leads to multiple sets of solutions for the 25 same problem statement, thus allowing for the enforcement of 26 additional sets of constraints on the manipulator in addition 27 to the primary task. A fundamental problem in robotics is 28 the problem of redundancy resolution for trajectory tracking 29 applications using redundant manipulators. It refers to the 30 The associate editor coordinating the review of this manuscript and approving it for publication was Yangmin Li . computation of the joint poses required to achieve desired 31 end-effector path in the Cartesian space. Due to the nonlinear 32 nature of the mapping between the joint pose and end-effector 33 positions, redundancy resolution via inverse kinematics is 34 often quite difficult. Hence, redundancy resolution is usu-35 ally performed at the velocity level, which can be repre-36 sented in the form of a time-varying underdetermined linear 37 equation [1], [2]. 38 The zeroing neural network (ZNN) represents a special 39 class of recurrent neural networks that is primarily aimed at 40 finding the zeros of a time-varying system represented by 41 a set of linear or nonlinear equations. ZNNs were first pro-42 posed for finding the solution of the time-varying Sylvester's 43 redundancy resolution applications in robotics. 84 To overcome this drawback, a novel quadratic program 85 (QP)-based neural control architecture is proposed in this 86 paper, which incorporates prescribed performance constraints 87 with state and input constraints at the joint velocity level.  The proposed model further incorporates suitably prescribed 96 performance constraints, which guarantee desired tran-97 sient and steady-state performance characteristics, and have 98 been shown to deliver prescribed performance for various 99 dynamical systems [19], [20], [21], [22]. Prescribed perfor-100 mance control ensures that the system forces the tracking 101 error to converge to a small residual set with a convergence 102 rate greater than or equal to some predefined value in such 103 a manner that the overshoot does not exceed a predefined 104 limit. The proposed model uses a convex optimization frame-105 work to enforce both the bounds on the state and the input, 106 and the prescribed performance constraints simultaneously. 107 Combining these approaches and incorporating them within a 108 quadratic program further ensures that the solutions obtained 109 from the proposed model are optimal in accomplishing the 110 task objective. In particular, the optimization framework is 111 designed with the aim of enforcing bounds on the mathe-112 matical range of the solution, which is in contrast with the 113 ZNN-based optimization model proposed in [23], which does 114 not impose any bounds on the system state/input. Moreover, 115 the proposed framework is used to impose explicit expo-116 nentially convergent dynamic bounds on the residual error 117 as opposed to the study in [16], where only static bounds 118 are imposed on the tracking error and thus cannot guaran-119 tee desired transient and steady-state performance. Theoret-120 ical analysis is undertaken to demonstrate the stability and 121 convergence properties of the proposed scheme. Complex-122 ity analysis is also undertaken to demonstrate the compu-123 tational efficiency of the proposed model as compared to 124 leading alternative designs such as [14], [24]. Simulation 125 and experimental studies are carried out with a KUKA LBR 126 IIWA 14 R820 to show the efficacy of the proposed models 127 for path tracking applications in redundant manipulators with 128 bounds on the joint velocities. The suitability of the proposed 129 model for fault-tolerant trajectory tracking is then demon-130 strated in simulation and experiments in the presence of mul-131 tiple joint failures. Note that this approach differs from the 132 previous approaches proposed for fault tolerance such as [24], 133 in the sense that it incorporates limits on the state/input of 134 the system. Further, the presence of prescribed performance 135 constraints and the inclusion of formal guarantees for rigor-136 ous adhesion to bound constraints distinguishes it from the 137 fault-tolerant approaches proposed in [25], [26], and [27]. 138 To the best of the authors' knowledge, no other study has 139 focused on the constrained resolution of redundant manipu-140 lators that delivers prescribed transient and steady-state per-141 formance of the tracking error within an optimal framework. 142 The main contributions of this paper are summarized 143 below. 144 i) A novel QP-based neural control framework is intro-145 duced for redundancy resolution with bound constraints at the 146 joint velocity level. To the best of the authors' knowledge, this 147 is the only framework that incorporates both prescribed per-148 formance and joint velocity constraints within an optimiza-149 tion framework for redundancy resolution in fault-tolerant 150 trajectory tracking applications.

151
ii) The proposed model relies on an invertible nonlinear 152 map to transform the constrained system state into a new 153 set of unconstrained variables. This transformation formally 154 guarantees that the proposed model will rigorously satisfy the 155 joint constraints both during the transient phase and in the 156 steady-state, which is in contrast with most previous studies.

178
In this section, the problem of redundancy resolution for sys-179 tems involving inverse kinematics with bound constraints on 180 the joint velocity is introduced. The problem of redundancy 181 resolution can be stated as follows. Given a desired path 182 r d ∈ R m of the end-effector in the task space, the required 183 joint angles θ ∈ R n in the joint space have to be determined.

184
The solution to this problem can be achieved by solving the 185 forward kinematics of the manipulator, which is given as

197
where J(θ ) ∈ R m×n represents the Jacobian matrix,θ(t) ∈ 198 R n×1 represents the joint velocity, andṙ d ∈ R m×1 represents 199 the desired end-effector velocity.θ − andθ + represent the where θ − i and θ + i represent 208 the limits on the angular position of the i th joint, and κ p > 0 is 209 a scaling factor for the joint limits.

211
In this section, a quadratic program-based ZNN embedded 212 with performance constraints is proposed for the solution of 213 system (2) subject to bound constraints (3) along with the sub-214 sequent theoretical analysis to demonstrate the convergence 215 properties of the proposed scheme. For the proposed scheme, 216 both the system constraints and the prescribed performance 217 constraints are included in the formulation via a nonlinear 218 mapping, and a QP framework is subsequently introduced to 219 impose both of them simultaneously. In this subsection, a nonlinear transformation is introduced 223 to impose the constraints on the state of the system. Subse-224 quently, prescribed performance constraints (PPCs) are incor-225 porated into a ZNN framework for better transient and steady-226 state performance.

248
Introducing the tracking error (t) = f (θ) − r d , and using 249 a non-negative constant k ≥ 0, the system error is defined as 250 (6) 253 where −1 (χ) : R n → R n represents the inverse mapping 254 for the transformation (θ). To improve tracking perfor-255 VOLUME 10, 2022 mance, especially in the presence of multiple joint failures, 256 we consider the inclusion of the performance constraints to 257 impose bounds on the system error (6) through the following 258 transformation [21]  consider the component-wise transformation by using (7) as and [24].

289
To account for the bounds on the error function, a vector 290 g(χ) is defined as To solve this optimization problem, the Lagrangian function 298 is expressed as [32] represents the 301 Lagrange's multipliers. The design methodology used in [23] 302 is adopted to formulate the desired ZNN model. The aug-303 mented state vector for the system is given as For the solution of problem (10), a ZNN model is formulated 308 to drive the components of h(y(t), t) to zero as where (·) : R m+n → R m+n represents a vector array 311 of monotonically increasing odd activation functions, and 312 γ ≥ 0 is a constant that scales the convergence rate of ZNN. 313 The total derivative of h(y(t), t) can be expressed as The ZNN model (14) can be reformulated using (15) as where † represents the pseudoinverse. The neuronal form of 321 the proposed model (16) can be written as + 9mn + m + 2n multiplications, m + 2n nonlin-334 ear operations, and 3n integrator operations. While as the 335 VP-ZNN model proposed in [24] requires 3(m 2 + n 2 + m 2 a ) + 336 6(mn + m a n + m a m) additions/subtractions, 3(m 2 + n 2 + 337 m 2 a ) + 6(mn + m a n + m a m) + m + n + m a multiplications, 338 m + n + m a nonlinear operations, and m + n + m a integrator 339 operations, where m a is the number of faulty joints. Despite 340  (10) is convex with respect to the state variablesθ(t).

353
Proof: The function f (χ) can be expanded as

357
where δ ij represents the Kronecker delta function. Note that From (19) and (20), it can be concluded that ∂ 2 f 1, .., n. Hence, the Hessian ∂ 2 f ∂θ∂θ is a diagonal matrix with 365 positive diagonal elements and hence is positive definite. 366 Thus, the function f is convex with respect toθ(t) [33].
The ij th element of the Hessian ∂ 2 g k ∂η∂η ∈ R m×m is then given 374 as where δ ijk represents the Kronecker delta function. Sinceχ k 377 is uniformly bounded and thus finite withχ k ∈ (−∞, ∞), 378 henceη k ∈ (0, 1). Proceeding in the same manner as 379 Lemma 1, it can be shown that ∂ 2 g k Thus, the Hessian ∂ 2 g i ∂θ ∂θ ∈ R n×n is given as which represents a positive semi-definite matrix. Hence, the 383 constraint functions g(χ) is convex with respect toθ(t) [33]. is defined as

394
The time-derivative of the Lyapunov function is given as

481
The evolution of tracking errors is shown by Figs. 4b and 5b. 482 These tracking errors eventually converge to an order of 483 10 −5 m. As is evident from the simulations, an additional 484      tracking errors in the absence of PPCs still significantly 543 exceed the corresponding values in the presence of PPCs for 544 the case with multiple joint failures. However, the apparent 545 difference is not as pronounced as before due to the presence 546 of nonzero initial tracking errors which contribute signifi-547 cantly to the mean tracking error values. To glean a deeper 548 insight into the effect of the PPCs for the case with nonzero 549 initial tracking errors, the corresponding simulation results 550 for trajectory tracking in the absence of PPCs are shown in 551 Figs. 10 -13. It can be seen (Figs. 10b -13b) that the absence 552 of prescribed performance constraints consistently results in 553 a significant overshoot and a non-exponential decay, espe-554 cially for the fault-tolerant case, thus leading to significant 555 degradation of the transient performance. Further, as can be 556 seen from Figs. 10c -13c, absence of PPCs leads to a slower 557 decay rate and larger steady-state errors which results in a 558 poorer steady-state performance. The performance difference 559 is drastic in some cases, as seen in Fig. 12a, where the 560 traced path shape is severely distorted. The superior tracking 561 error obtained in the presence of performance constraints 562 is due to the fact that PPCs force the residual error of the 563 system to exponentially converge to a small residual set at 564 a decay rate equal to or faster than a predefined rate, forcing 565 the tracking error to decay faster and achieve lower steady-566 state values. Thus, the inclusion of prescribed performance 567 constraints significantly improves both the transient as well 568 as the steady-state performance of the proposed scheme (16). 569

570
This subsection includes the performance comparison studies 571 for the proposed model (16) with the ZNN model [14] and 572 the VP-ZNN model [24]. This is followed by a qualitative 573 comparison study with previous related studies to show the 574 novelty of present work.

575
The performance comparison studies are only undertaken 576 for scenarios with zero initial tracking error as the ZNN 577 model in [14] implements trajectory tracking at the velocity 578 level which cannot compensate for nonzero initial track-579 ing errors. Figs. 14 and 15 present the trajectory track-580 ing results obtained with the ZNN model [14] and the 581 VP-ZNN model [24] for the fault-free case respectively. 582 To illustrate the benefits of the proposed schemes for robotic 583 trajectory tracking applications, a comparison of the average 584 tracking error (29) and the average control effort required 585 VOLUME 10, 2022       in [24]. The marginally lower computational effort required 601 by the VP-ZNN scheme in [24] is realized at the cost of . This deviation from the desired path occurs 615 due to saturation of statesθ 1 (t) andθ 2 (t) (Fig. 14c) corre-616 sponding to the computed solution by the ZNN model [14]. 617 As is apparent from Fig. 14b, this situation leads to a sudden 618 surge in errors which leads to deviation from the desired path. 619 A similar scenario can be seen for the VP-ZNN proposed 620 in [24], where statesθ 2 (t) andθ 4 (t) show violation of bounds 621 as the scheme lacks any structure for enforcing velocity 622 constraints (Fig. 15c). Again, due to saturation limits, these 623 desired velocities (which violate the bounds constraints for 624 this controller) cannot be achieved, leading to input satura-625 tion, as seen in Fig. 15c. Again, as before, there is a surge 626 in tracking error, as seen in Fig. 15b, which leads to a sig-627 nificant deviation from the desired path. In contrast, as seen 628 in the previous section, the proposed model (16)   input constraints, as well as prescribe performance which 651 separates it from the previously mentioned studies. As seen 652 from this comparison, the proposed scheme (16) is the only 653 scheme that combines both the state and input constraints as 654 well as prescribed performance constraints in a QP frame-655 work to synthesize an optimal input for trajectory tracking 656 applications both in a fault-free and a fault-tolerant setting.    Moreover, the proposed model is shown to realize zero 716 terminal joint velocities obtained by embedding the non-717 linear mapping for imposing state constraints within the 718 optimization framework. Finally, physical experiments are 719 carried out to further substantiate the efficacy of the proposed 720 scheme.

721
A possible direction for future work is the inclusion of 722 obstacle avoidance capabilities in the proposed models to 723 accomplish obstacle-aware trajectory tracking with bounds 724 on the system state and control input. Moreover, incorpo-725 rating a state observer for estimating the system's Jacobian 726 online, which could potentially lead to a platform-agnostic 727 observer-controller framework, would find wide suitabil-728 ity in constrained trajectory tracking applications. Finally, 729 the efficacy of the proposed scheme for trajectory tracking 730 applications in redundant robotic manipulators alludes to 731 the suitability of these schemes to other constrained robotic 732 applications such as visual target tracking and visual servo 733 control. 734 VOLUME 10, 2022