Design of Prisoner’s dilemma based fuzzy logic computed torque controller with Lyapunov synthesis linguistic model for PUMA-560 robot manipulator

Abstract

The robot manipulators are highly nonlinear, time varying and one of the important challenges in the field of robotics is the effective control of manipulators. This paper presents a technique to extract the rules of Fuzzy Logic Computed Torque Controller for PUMA-560 robot arm with uncertainties. Fuzzy Logic Controllers are placed at the input of the PD Controller to make the gains adaptive. Prisoner’s dilemma is employed to systematically tune the gains of the controller. The interrelations between inputs and outputs of a Fuzzy Linguistic Model are assigned using payoff matrix through Prisoner’s Dilemma. The difficulty in designing of fuzzy controllers is the extraction of the rule base. The extraction of fuzzy control rules requires good understanding of the plant and control theory. The present paper utilizes Fuzzy Lyapunov Synthesis to constitute the rule base assuming that minimal knowledge about the plant to be controlled. Simulation results prove the effective performance of the proposed controller in minimizing the error in joint angles when compared to Proportional Derivative Computed Torque Controller (PD-CTC), normal Fuzzy Logic Controller (FLC) and that of the reference signal.

Keywords

PUMA-560 robot manipulator fuzzy controller Prisoner’s dilemma Lyapunov synthesis

1 Introduction

The robotic arm which is an important section of robotics such as PUMA-560 is widely used in welding, mechanical handling, assembling, grinding, painting and other industrial applications.

These applications require trajectory generation, path planning and controller design. The robot manipulators for industrial applications are usually controlled by linear PID controllers, but its dynamic functions are, nonlinear with strong coupling betweenjoints (low gear ratio), structure and unstructured uncertainty and multi- inputs multi- outputs (MIMO). The design of linear controller becomes very difficult especially if the velocity and acceleration of robot manipulator will be high and also when the ratio between joints gear will be small. However, conventional methods are model dependent [7], requires information about the system nonlinearities, uncertainties, time varying parameters of a nonlinear system. Literature shows [12] that many controllers like Hα, LQG and input shaping as well as singular perturbation, feedback linearization, manifolds and output redefinition techniques have been used for controlling purpose with a known model of the system. Hence, to eliminate the above mentioned problems the researcher has to choose nonlinear controllers for physical systems.

Computed torque controller (CTC) is a powerful nonlinear controller which is widely used in control of robot manipulator. The concept of CTC is based on Feedback linearization which computes the required arm torques by applying a nonlinear control theory. CTC is a model dependant control scheme, hence, it shows good performance when all dynamic and physical parameters of the robot manipulator are known and poor performance when dynamic parameters are varied. The dynamic and physical parameters of robot manipulator are time variant, the control research has to go towards an intelligent controller to solve the major issues like model dependency and stability problem.

Many researchers have suggested that Fuzzy Logic Controllers have potential for robust stabilization control in the phase of physical uncertainties and dynamic variations in robot manipulator. Fuzzy Logic in CTC eliminates a nonlinear part in pure computed torque controller. Fuzzy control systems are rule based systems in which a set of fuzzy rules represents a control decision mechanism to adjust the effect of uncertainties caused by the systems. In general FLCs are suitable for plants that cannot be described precisely by a mathematical formulation and chaotic systems.

The objective of the present paper is to design a two input Fuzzy Logic Controller (FLC), which is similar to a PD controller based on its analytical structure and to systematically tune its parameters. FLCs are placed at the input of the PD Controller to make the gains adaptive. Deriving an analytical structure for the FLC gives its mathematical structure which systemizes the design and eliminates a nonlinear part in pure computed torque controller. It is necessary to systematically tune the parameters used in FLC to get the desired and optimum response under all dynamic variations and uncertainties considered in the present work. These parameters are commonly determined by trial and error method which is rather time consuming and does not guarantee an optimal control. The present work concentrates on proper tuning of FLC parameters using Prisoner’s Dilemma. In this paper Fuzzy sets are introduced to Prisoner’s Dilemma model by generalizing the payoffs to invoke fuzzy goal sets and to deem the strategies as fuzzy sets. In Prisoner’s Dilemma, the degree of cooperativeness of a strategy would seem a plausible way to proceed. A combination of these two ways results in a fuzzy game where each strategy has a membership degree in the cooperativeness subset and each strategy combination of players leads to a non-fuzzy outcome with its associated membership degrees in the players’ goal sets [5, 6]. This is the basic idea to combine fuzzy logic and prisoner’s dilemma. The design of PD like FLC is transformed as an uncertainty problem which is optimized by Prisoner’s dilemma. The paper proposes an effort to assess the optimal gains of Proportional-plus-Derivative (PD) like Fuzzy Logic controller by updating payoff matrix of players as inputs. The paper proposes a new Prisoner’s dilemma based optimal Fuzzy logic Computed Torque Controller (FLCTC) is derived to minimize the error in joint angles of PUMA-560 robot manipulator compared to the reference signal. PUMA-560 robot manipulator is a nonlinear system whose parameters are time invariant. Fuzzy Logic in applied to eliminate a nonlinear part in traditional computed torque controller whose parameters are tuned using Prisoner’s dilemma. This improves the efficacy of the proposed controller, uncertainties such as variation in inertia and gravitational constants are introduced in dynamic system of PUMA-560 robot manipulator. In order to achieve an optimal rule base for a FLCTC to show good performance during any range of uncertainties, the work in the present paper has been extended to constitute new rule base of FLCTC using Fuzzy Lyapunov Synthesis.

2 Formulation of PUMA-560 robot manipulator

Figure 1 shows the plant model of PUMA-560 robot manipulator [1]. The behavior of robot manipulators can be analyzed by having their dynamic model. The dynamic modeling illustrates the relationship between the motion of the joints, velocity and accelerations to torque of the manipulator and the resulting motion of the rigid bodies which form the robot. It also explains about dynamic variations such as inertia constants, Coriolis and centrifugal torques.

Most modern manipulators consist of a set of rigid links connected together by a set of joints. The dynamic equation of a nonlinear PUMA-560 robot manipulator [1 –4] is given in (1) considering three links among the total six links, such that q₄ = q₅ = q₆ = 0. $τ = A (q) \ddot{q} + B (q) [\dot{q} \dot{q}] + C (q) [q^{2}] + G (q)$ (1)

Where

q: n×1 position vector,

A (q): n×n inertia matrix of the manipulator,

G (q): n×1 vector of gravity terms

τ: n×1 vector of torques

B(q): n×n(n-1)/2 matrix of Coriolis torques

C(q): n×n matrix of Centrifugal torques

$\ddot{q}$ : n vector of acceleration

$[\dot{q} \dot{q}]$ and [q²] are notation for n(n-1)/2 vector of velocity products and the n-vector of squared velocities respectively.

Where $\begin{matrix} [\dot{q} \dot{q}] = [{\dot{q}}_{1} {\dot{q}}_{2}, {\dot{q}}_{1} {\dot{q}}_{3} \dots {\dot{q}}_{1} {\dot{q}}_{n}, {\dot{q}}_{n - 1} {\dot{q}}_{n}, q_{n - 2}, \dots]^{T} \\ [q^{2}] = [{\dot{q}}_{1}^{2}, {\dot{q}}_{2}^{2} \dots {\dot{q}}_{n}^{2}]^{T} \end{matrix}$ The above model of the robot arm is derived by generating the kinetic energy matrix and gravity vector symbolic elements by performing the summation of Lagrange’s nonlinear formulation [5]. These elements are simplified by combining inertia constants that multiply common variable expressions. The Centrifugal and Coriolis matrix elements are then calculated in terms of partial derivatives of kinetic energy, and then reduced using four relations that hold the partial derivatives.

3 Prisoner’s dilemma based optimal FLCTC

The main objective of the controller is to sense information from robot manipulator to improve the systems performance by achieving a small tracking error. CTC is used to compensate dynamic equation of robot manipulator tracking response in uncertain environment.

3.1 Design of PD-CTC

The design of CTC is based on feedback linearization. Assume that the desired motion trajectory for the manipulator is q (t). Define the tracking error as

e (t) = q_{d} (t) - q_{a} (t)

(2)

where e(t) is error of the plant, q_d (t) is desired d input variable, that in our system is desired displacement, q_a (t) is actual displacement. The required arm torques are computed as

τ = A (q) ({\ddot{q}}_{d} - U) + N (\dot{q}, q)

(3)

Where, $N (\dot{q}, q)$ is the vector of nonlinearity term. This is a nonlinear feedback control law that guarantees tracking of desired trajectory. Proportional-plus-derivative (PD) feedback for U(t) results in the PD-computed torque controller is chosen as (4) and the resulting error dynamics is shown in (5) $τ = A (q) ({\ddot{q}}_{d} + K_{d} \dot{e} + K_{p} e) + N (\dot{q}, q)$ (4) $({\ddot{q}}_{d} + K_{d} \dot{e} + K_{p} e) = 0$ (5)

Where K_d and K_p are the controller gains. As robot manipulators are time variant systems, a systematic tuning of the controller gains guarantees convergence of the tracking error to zero.

3.2 Design of Prisoner’s dilemma based FLCTC

The basic structure of FLC [10, 11] is as shown in Fig. 2.

3.2.1 Fuzzy sets associated with FLCTC design

In the present formulation, the structure of the proposed FLCTC model is as shown in the Fig. (2). In proposed design, two variables $e, \dot{e}$ are used as input signals. The coefficients K_p, K_d which are called scaling factors, transform the scaled real values to the required value in decision limit. The output signal coefficient K_u is injected to the summing point. The normalized inputs of the proposed controller are A1 and A2 which are equal to $K_{p} e, K_{d} \dot{e}$ respectively. The scaling factors K_p, K_d are tuned using the Prisoner’s dilemma. The two similar fuzzy sets defining the two inputs of the proposed FLCTC are givenby: $K_{p} e = K_{d} \dot{e} = {NB, NM, NS, Z, PS, PM, PB}$

The triangular membership functions are considered and partitioned within the Universe of Discourse in the range [–6, +6] for the inputs and outputs. For instance, the mathematical model of membership function is given as follows: ${\begin{matrix} μ_{ZF} = - (7 / 20) x + 1, if 0 < x < 2.85 \\ μ_{ZF} = (7 / 20) x + 1, if - 2 . 85 < x < 0 \\ μ_{ZF} = 0 \end{matrix}$ (6)

Otherwise ${\begin{matrix} μ_{PS} = - (7 / 20) x + 1, if 0 < x < 2.85 \\ μ_{PS} = (7 / 20) x + 1, if - 2.85 < x < 0 \\ μ_{PS} = 0 \end{matrix}$ (7)

3.2.2 IF-Then rules

The decisions in fuzzy logic-based approach are made by forming series of rules which relate the inputs to outputs by IF–THEN statements. In the proposed case the number of control rules to cover all the possible combinations of the seven membership functions of each input variable is 7×7 (i.e.49)

3.2.3 Defuzzification

In this paper Centroid defuzzification method is adopted to calculate the output. The equation represents the final output U by computing the centroid of the area of the possibility distribution.

U = \frac{\sum_{i, j = 1}^{n} {MF value of the input \times correspoding output}}{\sum_{i, j = 1}^{n} {MF value of input}}

4 Prisoner’s dilemma

The Prisoner’s dilemma is one of the classical games. Various solutions have been derived by applying Prisoner’s Dilemma in ecosystems, social sciences, economic sciences, control strategies etc. The interactions of the players in the Prisoner’s Dilemma are generally described by a 2×2 payoff matrix of player A as in (8), $\begin{matrix} \begin{matrix} C & D \end{matrix} \\ P_{A} = \begin{matrix} C \\ D \end{matrix} (\begin{matrix} R & S \\ T & P \end{matrix}) \end{matrix}$ (8)

C &D, cooperation and defection are two strategies that can be selected by each player in each round, and T > R > P > S. For e.g. two (A1&A2) players are chosen to play, the entries of the payoff matrix are interpreted as follows:

If A1 and A2 choose cooperation, then both gets reward R;

If A1 and A2choose defection, then both deserves punishment P;

If A1 or A2defects and A2 or A1 cooperates, then A1 or A2 (defector) obtains payoff T (Temptation); while A2 or A1 (sucker) the sucker gets payoff S.

Therefore with respect to the above, every player should tends to defect because it will get more total payoff irrespective of the strategy of its opponent (for T > R and P > S). Hence, one only gets the penalty P. Distinctly, when they both choose cooperation (2R > T + S), they will get higher total payoffs in the long run. This is the dilemma. Based on the literature, a rescaled form about this matrix where T >1, R = 1, P = S = 0 is generally adopted. The strategies of Prisoner’s dilemma can be updated using MAX Payoff strategy updating Technique, where, a player in each site plays against its neighbors including itself. Each individual implement the strategy of the player who gains the highest total payoff. Hence, the total payoff P_i of ith player is calculated as follows: $P_{i} = \sum_{j \in ψ_{i}}^{j} X_{i}^{T} {Ax}_{j}$ (9)

Where Ψ_i is the neighborhood of ith player including itself, A denotes the payoff matrix (8), (.) ^T denotes the transpose. And X_i, x_j satisfy the following requirements; If ith player chooses defection, x_i = (-1 1) ^T; if ith chooses cooperation, x_i = (-1 1) ^T. Sometimes a player doesn’t know the exact payoff of its opponents and even its own, in this case it is difficult to decide its updating strategy. Fuzzy Logic is a good approach to deal such approximate uncertainty. The present work adopts a fuzzy linguistic rule model based strategy updating scheme. And we obtain a series of reasonable simulation results 49-fuzzy-rule base. This paper presents an approach to derive the optimal parameters of Prisoner’s Dilemma based FLCTC. The problem is defined as a tournament between two player’s payoffs as fuzzy inputs, the winning possibility as fuzzy outputs. Through fuzzy reasoning, it is possible to get the possibility of one player forcing its strategy on its opponent in the strategy update.

4.1 Updating payoff to obtain optimal parameters of Prisoner’s Dilemma based FLCTC

Figure 3 shows the flow chart to update the payoff matrix of Prisoner’s Dilemma based FLCTC; the updated strategy is introduced as inputs to the fuzzy controller. The payoff matrix is derived by optimizing the (7) to reduce the error. K_P and K_d are the controller gains that are derived by updating payoff matrix to acquire the desired response. These gains are introduced as scaling factor of fuzzy logic CTC. Hence a new Prisoner’s dilemma based Prisoner’s Dilemma based FLCTC is derived and proposed.

4.2 A new Fuzzy rule base for an updated Prisoner’s dilemma based FLCTC

PUMA-560 arm is a robot manipulator with 3 links and six degree of freedom. The above designed Prisoner’s dilemma based FLCTC is tested for step and ramp inputs applying to PUMA-560. The efficacy of the proposed technique is tested by introducing uncertainties i.e., variations in inertia constants as shown in APPENDIX. In order to achieve an optimal rule base for a FLCTC to show good performance during any range of uncertainties, the original rule base of FLCTC is modified using Fuzzy Lyapunov Synthesis.

4.2.1 Lyapunov Synthesis in deriving the new rule base

Consider the motion of a link in PUMA-560 robot arm. Let the state variables are x₁ = e (angle) and $x_{2} = \dot{e}$ (angular velocity). The system’s dynamic equations are described as follows

$F_{1} (x) {\begin{matrix} {\dot{x}}_{1} = x_{2} = F_{1} (x) \\ {\dot{x}}_{2} = f (x_{1}, x_{2}) + g (x_{1}, x_{2}) u = F_{2} (x) \end{matrix}$

Where $\begin{matrix} f (x_{1}, x_{2}) = \frac{9.8 {Sinx}_{1} - \frac{{mlx}_{2}^{2} {Cosx}_{1} {Sinx}_{1}}{m_{c} + m}}{1 (\frac{4}{3} - \frac{{mCos}^{2} x_{1}}{m_{c} + m})} and \\ g (x_{1}, x_{2}) = \frac{\frac{{Cosx}_{1} {Sinx}_{1}}{m_{c} + m}}{l (\frac{4}{3} - \frac{{mCos}^{2} x_{1}}{m_{c} + m})} \end{matrix}$

Where m is the mass of the link, l is the pole’s length, and u is the applied force or torque for control. The fuzzy control rules of FLCTC are commonly applied to control the motion of the link in PUMA-560, are obtained heuristically. Assume that the model of a nonlinear system is unknown. However, based on the physical intuition and the experience of balancing the motion of a link, the interpretation is based on the following information.

The state variables are described as ${\dot{x}}_{1} = x_{2}$

The angular acceleration is proportional to the force or torque (u) applied for the motion of the link.

u is inversely proportional to e and ė.

Consider a Lyapunov “candidate function” [18, 19] as $V (x_{1}, x_{2}) = \frac{1}{2} (x_{1}^{2} + x_{2}^{2})$ . Differentiating V gives $\dot{V} = x_{1} {\dot{x}}_{1} + x_{2} {\dot{x}}_{2}$ .

Using the interpretation that, the angular acceleration is proportional to the force or torque (u) applied for the motion of the link, ${\dot{x}}_{2} = u$ . Therefore, substituting ${\dot{x}}_{2} = u$ . $\dot{V} = x_{1} {\dot{x}}_{1} + x_{2} u = x_{1} x_{2} + x_{2} u = x_{2} (x_{1} + u)$

Theorem 2 (Zhou and Ruan, 2002): If V (x) is a Lyapunov function and the linguistic value LV (V (x)) = Negative, where Supp (Negative)⊂ [- ∞ , 0], then the fuzzy controller designed by fuzzy Lyapunov synthesis is locally stable. Furthermore, if Supp (Negative) ⊂ (- ∞ , 0), then the stability is asymptotic.

According to Lyapunov Synthesis [20] and theorem 2 the fuzzy linguistic control rules are systematically obtained as shown in Table 1. For example, if x₁= Positive and x₂= Positive, from our heuristics, u should be Negative Big to ensure x₁ + u= (Positive –Negative Big) = Negative, and hence x₂ (x₁ + u)= (Positive-Negative) = Negative, that is LV $(\dot{V} (x))$ = Negative. From Theorem 2, if Supp (Negative) ⊂ [–8, 0], then the FLCTC designed by the fuzzy Lyapunov synthesis approach is locally stable. Hence a new Prisoner’s dilemma based FLCTC is defined with new fuzzy rule base shown in Table 1, based on Fuzzy Lyuapnov Synthesis is proposed as Prisoner’s dilemma based Lyapunov FLCTC.

5 Results and discussions

The proposed Prisoner’s dilemma based Fuzzy logic applied to PUMA 560 robot manipulator has been tested for both step and ramp inputs and compared with PD-CTC, FLCTC, reference signal and Fuzzy Lyapunov with and without uncertainties. The performance of the proposed controller is tested by incorporating at each joint of PUMA-560 robot manipulator without and with uncertainties such as inertial and gravitational constants as shown in APPENDIX. The results presented in this paper prove the effective performance of the proposed controller. From Figs. 4–16, it can be observed that the errors in the theta values are minimized with Fuzzy Lyapunov controller when both ramp and step input is given when compared to other controllers. Table 2 shows the comparative numerical data analysis in error in theta of link1, link2, link3 of PUMA-560.

6 Conclusions

In this paper a novel technique to constitute a new rule base of a Fuzzy Logic Computed toque Controller to make the controller model free. A Lyapunov “candidate function”, V is chosen to determine the fuzzy control rules so that V is a Lyapunov function. Prisoner’s Dilemma is employed to systematically tune the Fuzzy scaling factors to define it as an optimal controller for the introduced uncertainties. The paper proposes a new Prisoner’s Dilemma Based Fuzzy Logic Computed Torque Controller with Lyapunov Synthesis Linguistic Model forPUMA-560 Robot Manipulator. Hence, it is concluded that the proposed controller in this paper guarantees the effective performance of the proposed controller in minimizing the error in joint angles compared to other controllers and Reference signal i.e., ramp and step signals.

Footnotes

Appendix

Inertial Uncertainties

I₁ = 1.43±0.05	I₂ = 1.75±0.07
I₃ = 1.38±0.05	I₄ = 0.0333±0.02
I₅ = 0.372±0.031	I₆ = 0.333±0.016
I₇ = 0.298±0.029	I₈ = –0.134±0.014
I₉ = 0.0238±0.012	I₁₀ = –0.0213±0.0022
I₁₁ = –0.0142±0.007	I₁₂ = –0.011±0.0011
I₁₃ = –0.00379±0.0009	I₁₄ = 0.00164±0.0003
I₁₅ = 0.00125±0.0003	I₁₆ = 0.00124±0.0003
I₁₇ = 0.000642±0.0003	I₁₈ = 0.000431±0.000013
I₁₉ = 0.0003±0.0014	I₂₀ = –0.000201±0.0008
I₂₁ = –0.0001±0.0006	I₂₂ = –0.000058±0.015
I₂₃ = 0.00004±0.00002	I_m1 = 1.14±0.27
I_m2 = 4.71±0.54	I_m3 = 0.827±0.093
I_m4 = 0.2±0.016	I_m5 = 0.179±0.014
I_m6 = 0.193±0.0016

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