Microsoft Word - 28-Enebi_32436_autores_padronizar_artigooriginal.doc 1689 Original Article Biosci. J., Uberlândia, v. 32, n. 6, p. 1689-1702, Nov./Dec. 2016 DESIGN OF A ROBOTIC DEVICE ACTUATED BY CABLES FOR HUMAN LOWER LIMB REHABILITATION USING SELF-ADAPTIVE DIFFERENTIAL EVOLUTION AND ROBUST OPTIMIZATION DISPOSITIVO ROBÓTICO ATUADO POR CABOS PARA REABILITAÇÃO DO MEMBRO INFERIOR HUMANO UTILIZANDO EVOLUÇÃO DIFERENCIAL AUTO-ADAPTÁVEL E OTIMIZAÇÃO ROBUSTA Rogério Sales GONÇALVES 1 ; João Carlos Mendes CARVALHO 2 ; Fran Sérgio LOBATO 3 1. School of Mechanical Engineering (FEMEC), Federal University of Uberlândia, Uberlândia-MG – Brazil, rsgoncalves@ufu.br. 2. School of Mechanical Engineering (FEMEC), Federal University of Uberlândia, Brazil; 3. NUCOP – Laboratory of Modeling, Simulation, Control and Optimization, Federal University of Uberlândia, School of Chemical Engineering, Brazil. ABSTRACT: In engineering designed systems it is commonly considered that mathematical models, variables, and parameters are sufficiently reliable, i.e., there are no errors in modeling and estimation. However, the systems to be optimized can be sensitive to small changes in the designed variables causing significant changes in the objective function. Robust optimization is an approach for modeling optimization problems under uncertainty in which the modeler aims to find decisions that are optimal for the worst-case realization of the uncertainties within a given set of values. In this contribution, a self-adaptive heuristic optimization method, namely the Self-Adaptive Differential Evolution (SADE), is evaluated. Differently from the canonical Differential Evolution algorithm (DE), the SADE strategy is able to update the required parameters such as population size, crossover parameter, and perturbation rate, dynamically. This is done by considering a defined convergence rate on the evolution process of the algorithm in order to reduce the number of evaluations of the objective function. For illustration purposes, the SADE strategy is associated with the Mean Effective Concept (MEC) for insertion robustness, is applied to minimize forces applied in cables used for the rehabilitation of the human lower limbs by determining the positioning of motors. The results show that the methodology that was proposed (SADE+MEC) appears as an interesting strategy for the treatment of robust optimization problems. KEYWORDS: Robust Optimization. Self-Adaptive Differential Evolution. Mean Effective Concept. Rehabilitation. Robotic Device. INTRODUCTION Traditionally, during the engineering systems design it is considered that the result is not subject to the influence of small perturbations of design variables and/or parameters involved in the process. However, the global optimum solution can be sensitive to small perturbations of design variables vector. In this case, an optimal local solution may be less sensitive, which from a real point of view, is configured as a feasible and can be implemented in industry. The concept of robust optimization should be used in order to minimize this effect in engineering systems design. This approach is applied for modeling optimization problems under uncertainty, in which the modeler aims to find decisions that are optimal for the worst- case realization of the uncertainties within a given set of values. Robust optimization is defined as an approach that produces a solution which is not very sensitive to small changes in design variables (TAGUCHI, 1984). It is emphasized that a robust solution may not coincide with the nominal solution, i.e., a solution without robustness, as observed in Figure 1. In this context, the robustness characterizes an important tool to help getting a not very sensitive solution under certain conditions, when exposed to given conditions of uncertainty. In the literature, some studies considering the introduction of robustness in the mono and multi-objective optimization context can be found (DEB; GUPTA, 2006; SOUZA et al., 2015). Several of these studies require the introduction of new restrictions and/or new objectives, i.e., relations between the mean and the standard deviation of objective functions vector, and probability distribution functions for the design variables and/or objectives. As an alternative to these classical formulations, Deb and Gupta (2006) extended the Mean Effective Concept (MEC), originally proposed for mono-objective problems, to the multi- objective optimization context. In this approach, no additional restriction is inserted into the original problem. Thus, the problem is rewritten as a mean vector of original objectives. Received: 01/12/16 Accepted: 05/10/16 1690 Design of a robotic device… GONÇALVES, R. S.; CARVALHO, J. C. M.; LOBATO, F. S. Biosci. J., Uberlândia, v. 32, n. 6, p. 1689-1702, Nov./Dec. 2016 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 Robust Nominal f( x) x Figure 1. Nominal solution versus robust solution (Taguchi, 1984; Deb and Gupta, 2006). Traditionally, the engineering systems design has been obtained by using Deterministic Optimization Methods. In the last years, Non- Deterministic Optimization Methods have been used to solve this kind of problem. During the solution of optimization problems, the input parameters of any evolutionary algorithm are kept constant (i.e., population size, crossover parameter, and perturbation rate, among others). In this case, the computational implementation of the algorithms is simplified. However, the variation of the population size inherent to real biological systems (important aspect of biological evolution) is disregarded. Intuitively, it is interesting to expand the population size during the first generations (i.e., assume its maximum value) due to the high diversity faced at this stage. This aspect offers the opportunity to the individuals to explore the design space accordingly. On the other hand, from the optimization point of view, in the end of the evolutionary process, the natural tendency of the population is to become homogeneous, which implies unnecessary evaluations of the objective function and, consequently, the increase of computational cost, when the population size is kept constant. To overcome this disadvantage, in this work the Self-Adaptive Differential Evolution algorithm (SADE), proposed by Cavalini et al. (2015), is considered with an optimization strategy. In this evolutionary approach, the population size, crossover parameter, and perturbation rate are dynamically updated during the convergence process in the Differential Evolution algorithm (DE). Basically, the SADE strategy reduces the number of objective function evaluations based on the definition of a convergence rate in order to evaluate the homogeneity of the population in the evolutionary process (CAVALINI et al., 2015). In the engineering systems design context, the development of robotic devices to be applied in the rehabilitation process of human lower limbs is configured as an important application due to the large number of people with lower limb problems caused by stroke and/or accidents. In general, human beings have always tried to perform tasks making less effort as possible. In the last years, machineries and equipments were developed to simplify the execution of multiple tasks, reducing the time needed to fulfill them, improving the patient’s quality of life and safety. With the advancement of technology, it was possible to apply robots also in health, using them for surgeries, prosthetics and structures to assist in rehabilitation. In this contribution, the SADE algorithm associated with the MEC is used to optimize the motors positioning in a parallel robotic device actuated by cables for human lower limb rehabilitation used to assist patients and health professionals during rehabilitation sessions. The results obtained by proposed methodology (SADE+MEC) are compared with those obtained by using the canonical DE algorithm. MATERIAL AND METHODS Robust optimization Mean effective concept (Deb and Gupta, 2006): For the minimization of an objective function (f(x)), a solution x* is called a robust solution if it is the global minimum of the mean effective function feff(x), defined with respect to a δ-neighborhood as follows: ( ) 1 ( )= min ( ) eff y x f x fdy x δδ ∈ϒ ϒ ∫ (1) where ϒδ is the δ-neighborhood of the solution x and |ϒδ| is the hypervolume of the neighborhood. 1691 Design of a robotic device… GONÇALVES, R. S.; CARVALHO, J. C. M.; LOBATO, F. S. Biosci. J., Uberlândia, v. 32, n. 6, p. 1689-1702, Nov./Dec. 2016 Basically, a finite set of solutions H can be generated randomly using the latin hypercube for the evaluation of integral given by Eq. (1). In this case, defining the δ-neighborhood with respect to design variables vector, N solutions are generated employing the latin hypercube, with the integral evaluated numerically. It should be mentioned that the aspect increases the computational cost due to the number of integral evaluations necessary to evaluate the objective function (DEB; GUPTA, 2006). Robotic devices actuated by cables for lower limb human rehabilitation The parallel cable-driven manipulator consists on a base and a moving platform which are connected by multiple cables that can extend or retract (see Figure 2). Then, a cable-based manipulator can move the end-effector by changing the cables’ lengths while preventing any cables to become slack. Therefore, feasible tasks are limited due to main static or dynamic characteristics of the cables because they can only pull the end-effector but cannot push it (CANNELLA et al., 2008; HILLER et al., 2009). These structures have characteristics that make them suitable for rehabilitation purposes. They have large workspace, which may be adapted to different patients and different training. The mechanical structure is easy to assembly and disassembly, which makes it easier to transport, and can be reconfigured in order to perform different therapies. In the clinical point of view, the use of cables instead of rigid links makes patients feel less constrained, which is important to help them accept the technology. These characteristics make the cable-based parallel manipulators ideal for rehabilitation. The drawbacks related to the use of a cable-driven parallel structure are the physical nature of cables that can only pull and not push and the fact that the workspace evaluation turns into dependent forces and can have a complex and irregular shape (HILLER et al., 2009). The cable-driven parallel manipulator, proposed in this paper, can be assembled from two to six cables arranged in a rigid structure (fixed platform) having a moving platform (splint), Figure 2(a). Figure 2(b) shows the prototype built at the Laboratory of Robotics and Automation at the Federal University of Uberlândia. Figure 2(a) shows the elements of the cable-based parallel manipulator, consisting on sets formed by 24 Volts x 45 Nm DC motor, encoder with 500 pulses per revolution and pulley. In this first step towards implementation of graphic simulations and experimental tests, a 1.80 m anthropometric wooden puppet was used to simulate a human body, Figure 2(b). The equipment works by using the "teaching by showing", repeating movements predetermined by the therapist to the patient. Therefore, it is necessary to perform the control of actuators in two distinct steps: a step called "teaching", in which the therapist "teaches" the movements to be performed by the machine, and another step of "playing", in which the machine runs the predetermined movements. The control of the actuators was conducted using three PIC18F4550 controllers that communicate with each other by the I2C interface. One is used as master, so it can command the others as slaves, and promote communication with a PC using a USB port. In the "teaching" step, in which the movements will be taught to the machine, the acquisition of position data and speed of each motor shaft is done through digital encoders. Each movement performed by the therapist in the splint in which the patient's lower limb is positioned, Figure 2(a), should cause a shift in the axis of the motors. However, just as cables are used to attach the splint to the axis of the actuators, the movement of the therapist could just loosen the cables, without any movement on the shaft of the engine was triggered. Thus, there must be a control loop that maintains the tension of the cables at this stage so as to cause the movement of the actuator when the therapist moves the splint. The signal from load cells attached to the cables will be used as a control variable, and the PWM signal of PIC microcontrollers promote the rotation of the actuators. Position data and angular velocity of each actuator are obtained by digital encoders, and saved to be replayed during the stage of playing (GONCALVES et al., 2015). The "playing" step, in which the structure that was proposed repeats the movements "taught" by the therapist, is done through the closed-loop control of speed and position of the actuators shafts, using PIC microcontrollers. This occurs via PWM control in order to guarantee the reproduction of movements executed in the "teaching" step (GONCALVES et al., 2013). To ensure the safe operation, emergency buttons are installed and the maximum allowable forces acting on the cables are set to prevent injuries involving patients. A graphical interface for PC was developed to control in which step of the operation the structure should run. The kinematic model of cable-driven parallel robots is obtained similarly to the model obtained from traditional parallel structures (CÔTÉ, 1692 Design of a robotic device… GONÇALVES, R. S.; CARVALHO, J. C. M.; LOBATO, F. S. Biosci. J., Uberlândia, v. 32, n. 6, p. 1689-1702, Nov./Dec. 2016 2003). The inverse kinematic problem consists in finding the cables lengths (ρi) as function of the end- effector pose. The forward kinematic problem consists of finding the end-effector poses for a given set of cables lengths. For the kinematic model, the parameters used are shown in Fig. 2(c). The kinematic variables are the cables’ lengths. The equilibrium equations for forces and moments acting on each cable can be given by 1F J W−= (2) where vector F represents the cable tension, which consists on forces that must be done by actuators, W is the vector of external forces and moments applied to the system, which are the limb and the splint weight and, J is the Jacobian matrix of the structure. More details about the Jacobian calculus can be found in (CARVALHO et al., 2013; BARBOSA, 2013). (a) (b) (c) (d) Figure 2. (a) Scheme of the parallel structure proposed; (b) Prototype built; (c) Kinematic parameters; (d) gait simulation with the proposed structure. During the rehabilitation movements it is possible that at least one cable has no traction, leading to a region of non-movement control in the structure, since cables cannot push the lower limb but only pull it. To avoid this, it is necessary that positive forces exist in the cables throughout the movement, otherwise it is not possible to move the limb in all positions required for the rehabilitation session. Thus, in this paper, the optimization approach was used to determine the best points to fix the motor in a fixed platform, Figure 2(a), according to the type of movement performed, through the following objective function (OF): 1693 Design of a robotic device… GONÇALVES, R. S.; CARVALHO, J. C. M.; LOBATO, F. S. Biosci. J., Uberlândia, v. 32, n. 6, p. 1689-1702, Nov./Dec. 2016 ( )( ) 2 m i n = m a x 0 , ( 0 . 5 )i i O F F ∈ Γ − −∑ (3) where Fi is the compression forces set. Intuitively, this function seeks to minimize the number of negative or very low force on cables throughout the movement (less than 0.5 N). For this case, the design variables are the positions of the motors in the structure (xi, i = 1,..., 6), as presented in Figure 2 (a). Self-adaptive differential evolution The DE algorithm is an optimization technique that belongs to the family of evolutionary computation, which differs from other evolutionary algorithms in the mutation and recombination schemes. Basically, DE executes its mutation operation by adding a weighted difference vector between two individuals to a third individual. Then, the mutated individuals will perform discrete crossover and greedy selection with the corresponding individuals from the last generation to produce the offspring. The key control parameters for DE are the population size (NP), the crossover constant (CR), and the so-called weight (Fw). The canonical pseudo-code of DE algorithm is presented in Figure 3, in which P is the population of the current generation, and P’ is the population to be constructed for the next generation, C[i] is the candidate solution with population index i, C[i][j] is the j-th entry in the solution vector of C[i], and r is a random number between 0 and 1. Differential Evolution Initialize and evaluate population P while (not done) { for (i = 0; i < N; i++) { Create candidate C[i] Evaluate C[i] if (C[i] is better than P[i]) P0[i] = C[i] else P0[i] = P[i]} and P = P0 } Create candidate C[i] Randomly select parents P[i1], P[i2], and P[i3] where i, i1, i2, and i3 are different. Create initial candidate C’[i] = P[i1] + Fw×(P[i2]-P[i3]). Create final candidate C[i] by crossing over the genes of P[i] and C’[i] as follows: for (j = 0; j < N; j++) { if (r 1 the variance is decreased and premature convergence situations can be avoided. ( )( ) ( )( ) 1Var x g Var x g γ + = (11) The controlling idea is based on the parameter Fw, such that the recombination applied in the generation g compensates the effect of the previous application of recombination and selection. 1696 Design of a robotic device… GONÇALVES, R. S.; CARVALHO, J. C. M.; LOBATO, F. S. Biosci. J., Uberlândia, v. 32, n. 6, p. 1689-1702, Nov./Dec. 2016 For this aim, Eq. (12) and Eq. (13) have to be solved. 2 2 21 2 w CR CR F CR c NP NP + − + = (12) where c is defined as ( )( ) ( )( ) 1Var x g c Var x g γ + ≡ (13) Equation (12) can be solved with respect to Fw. Thus, 1 if 0 2 o th e rw is e w w m in F N P C R F η η  ≥ ≡    (14) where ( ) ( )1 2NP c CR CRη ≡ − + − and Fwmin is the minimal value of Fw. A sufficient condition for increasing the population variance by recombination is that Fw>(1/NP) 0.5. Thus, Fwmin=(1/NP) 0.5 should be used. An upper bound for Fw can also be imposed as suggested by Storn et al. (2005). In this case, Fwmax=2. By solving Eq. (14) with respect to CR, one obtains the following adaptation rule for CR: ( ) ( ) ( ) 22 21 1 1 if 0 otherwise w w min NP F NP F NP c c CR CR  − × − + × − − − ≥ ≡   (15) with CRmin=0.01