Impedance Controller Tuned by Particle Swarm Optimization for Robotic Arms

Abstract

This paper presents an efficient and fast method for fine tuning the controller parameters of robot manipulators in constrained motion. The stability of the robotic system is proved using a Lyapunov-based impedance approach whereas the optimal design of the controller parameters are tuned, in offline, by a Particle Swarm Optimization (PSO) algorithm. For designing the PSO method, different index performances are considered in both joint and Cartesian spaces. A 3DOF manipulator constrained to a circular trajectory is finally used to validate the performances of the proposed approach. The simulation results show the stability and the performances of the proposed approach.

1. Introduction

In recent time, programming robot arms for maximizing their capabilities remains a challenging task [1]. Fine tuning of control parameters is particularly a time consuming and an expensive procedure [2, 3]. Indeed, control parameters must take account of coupling effects between multiple-joints. Furthermore, tuning procedure becomes a very complex problem when the robotic arm must operate in a constrained environment with high-accuracy performance and high speed. In fact, in such case, there is an increasing requirement for stable position/force control strategies [4, 5, 6]. Furthermore, not only position and speed parameters of the controllers must be tuned but also force controller parameters.

In many cases, using Artificial intelligence (AI) techniques is a good alternative to improve controller performances for robotic arms [7]. A wide range of AI techniques as neural network [8] and fuzzy logic [9] have been widely applied to proper tuning controller parameters of robot manipulators. Successful applications can be found in [10, 11, 12].

Alternatively, a wide range of new (AI) techniques, have submerged [13]. These novel techniques are biological-inspired optimization methods. They iteratively use random elements to transfer one candidate solution into a new better solution with regard to a given fitness function.

The main nature-inspired optimization approaches are Genetic Algorithms (GA), Simulated Annealing (SA), Tabu Search, Ant Colony Optimization (ACO) and Particle Swarm Optimization (PSO) algorithms.

Some relevant applications of such strategies in robotic field can be found in [14, 15, 16, 17]. Among the variety of evolution algorithms, Particle Swarm Optimization (PSO) algorithm [18, 19] seems to be one of the most promising techniques [20]. Particularly, PSO algorithms have less computational complexities and gives better performances [21].

PSO algorithms were successfully applied in several robotic applications, “For example, see [22, 23, 24, 25]”. Particularly, for tuning the controller parameters of robot arms in non constraint environment, several papers were recently proposed, “For example, see [26, 27]”. However, for the constrained motion case study, very few papers were devoted to this topic [28, 29].

Based on the lack of results in this framework, this paper proposes an efficient and fast method for fine tuning the controller parameters of robot manipulators in constrained motion using PSO intelligence. The stability of the constrained robotic system is proved using a Lyapunov-based impedance approach whereas the optimal design of the controller parameters are tuned, in offline, by the PSO algorithm using different index performances in both joint and Cartesian spaces.

The remainder of this paper is organized as follows. In Section 2, the stability conditions using the Lyapunov based impedance controller for constrained robotic systems are presented. In section 3, PSO algorithm is exposed. The case study of fine tuning the controller parameters for a 3DOF robotic system constrained to a circular profile are developed in section 4 where the performances of the proposed strategy are shown by simulation results.

2. Impedance control

Let consider the dynamics of a serial-chain n-link constrained robotic arm with n degrees of freedom described, in the joint space, by: $$\mathrm{M}(\mathrm{\theta })\ddot{\mathrm{\theta }}+\mathrm{H}(\mathrm{\theta },\dot{\mathrm{\theta }})+\mathrm{G}(\mathrm{\theta })=\mathrm{U}-{\mathrm{U}}_{\mathrm{F}}$$ (1)

where θ is the n × 1 vector of joint displacements, θ is the n × 1 vector of joint velocities, $\ddot{\mathrm{\theta }}$ is the n × 1 vector of joint accelerations, U is the n × 1 vector of torque inputs and U_F is the n×1 vector of torques exerted by the contact generalized forces of the manipulator on the environment, M(θ) is the n × n symmetric positive definite inertia matrix, $\mathrm{H}(\mathrm{\theta },\dot{\mathrm{\theta }})$ is the n × 1 vector of centripetal and Coriolis torques and G(θ) is the n × 1 vector of gravitational torques.

The Cartesian position of the end-effector of the constrained robotic arm is characterized by a vector X ∊ R^p where p is the task space dimension. Cartesian position is function of the joint displacements such that: $$\mathrm{X}=\mathrm{h}(\mathrm{\theta })$$ (2)

where h(θ): Rⁿ → R^p denotes the direct kinematics. The time derivative of the direct kinematic model (2) yields the following differential kinematic model: $$\dot{\mathrm{X}}=\mathrm{J}(\mathrm{\theta })\dot{\mathrm{\theta }}$$ (3)

Where $\ddot{\mathrm{X}}$ is the p × 1 Cartesian velocity vector and J(θ) ∊ R^pxn is the so-called analytical Jacobian matrix.

The vector of torques due to the contact generalized forces exerted by the manipulator on the environment can be written as: $${\mathrm{U}}_{\mathrm{F}}=\mathrm{J}{(\mathrm{\theta })}^{\mathrm{T}}\mathrm{F}$$ (4)

The time derivative of the differential kinematic model (3) can be written as: $$\ddot{\mathrm{X}}=\dot{\mathrm{J}}(\mathrm{\theta })\dot{\mathrm{\theta }}+\mathrm{J}(\mathrm{\theta })\ddot{\mathrm{\theta }}$$ (5)

where $\ddot{\mathrm{X}}$ is the $\mathrm{p}\times 1$ Cartesian acceleration vector and j(θ) is defined by: $$\dot{\mathrm{J}}(\mathrm{\theta })=\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}\mathrm{J}\left(\mathrm{\theta }\right)$$ (6)

Given a desired Cartesian position of the end-effector X_d ∊ R^p and a desired contact force F_d ∊ R^p, the control problem aim to ensure: $$\underset{\mathrm{t}\to {\mathrm{t}}_{\mathrm{f}}}{\lim }\mathrm{X}-{\mathrm{X}}_{\mathrm{d}}=0$$ (7)

and $$\underset{\mathrm{t}\to {\mathrm{t}}_{\mathrm{f}}}{\lim }\mathrm{F}-{\mathrm{F}}_{\mathrm{d}}=0$$ (8)

The last problem will be solved under the following assumptions:

A ₁ : The entire vectors of force F, Cartesian position X, Cartesian velocity $\dot{\mathrm{X}}$ and Cartesian acceleration $\ddot{\mathrm{X}}$ are measured.

A ₂ : The Jacobian matrix J(θ) is assumed to be full ranked and bounded for all θ ∊ Rⁿ.

A ₃ : A desired impedance model for the contact point between the end-effector of the robotic arm and the constrained environment is imposed such that [30]: $${\mathrm{Z}}_{\mathrm{d}}=\frac{{\mathrm{F}}_{\mathrm{d}}-\mathrm{F}}{{\mathrm{X}}_{\mathrm{d}}-\mathrm{X}}={\mathrm{K}}_{\mathrm{d}}+{\mathrm{B}}_{\mathrm{d}}\mathrm{s}+{\mathrm{M}}_{\mathrm{d}}{\mathrm{s}}^2$$ (9)

where K_d, B_d, M_d ∊ R^pxp are stiffness, damping and inertia matrices, respectively, and s is the Laplace operator.

A ₄ : All control feedback gain matrices, used to solve the control problem are diagonal.

Theorem:

For desired stiffness, damping and inertia matrices K_d, B_d, M_d ∊ R^pxp and if there exist diagonal gain matrices K_p, K_v, K_f ∊ R^pxp such that the following conditions: $$\begin{array}{l}{\mathrm{K}}_{\mathrm{p}}+({\mathrm{I}}_{\mathrm{p}\times \mathrm{p}}+{\mathrm{K}}_{\mathrm{f}}){\mathrm{K}}_{\mathrm{d}}>0\\ {\mathrm{K}}_{\mathrm{v}}+({\mathrm{I}}_{\mathrm{p}\times \mathrm{p}}+{\mathrm{K}}_{\mathrm{f}}){\mathrm{B}}_{\mathrm{d}}>0\\ {\mathrm{M}}_{\mathrm{d}}=0\end{array}$$ (10)

or $$\begin{array}{l}{\mathrm{K}}_{\mathrm{p}}>0\\ {\mathrm{K}}_{\mathrm{v}}>0\\ {\mathrm{K}}_{\mathrm{f}}=-{\mathrm{I}}_{\mathrm{p}\times \mathrm{p}}\end{array}$$ (11)

are satisfied, then the robotic system described by the dynamical model (1) and the direct kinematic model (2) is asymptotically stable under the constrained force model: $$\mathrm{F}={\mathrm{F}}_{\mathrm{d}}-{\mathrm{K}}_{\mathrm{d}}({\mathrm{X}}_{\mathrm{d}}-\mathrm{X})-{\mathrm{B}}_{\mathrm{d}}({\dot{\mathrm{X}}}_{\mathrm{d}}-\dot{\mathrm{X}})-{\mathrm{M}}_{\mathrm{d}}({\ddot{\mathrm{X}}}_{\mathrm{d}}-\ddot{\mathrm{X}})$$ (12)

and the control law: $$\mathrm{U}=\mathrm{J}{\left(\mathrm{\theta }\right)}^{\mathrm{T}}[{\mathrm{K}}_{\mathrm{p}}({\mathrm{X}}_{\mathrm{d}}-\mathrm{X})+{\mathrm{K}}_{\mathrm{v}}({\dot{\mathrm{X}}}_{\mathrm{d}}-\dot{\mathrm{X}})+{\mathrm{K}}_{\mathrm{f}}({\mathrm{F}}_{\mathrm{d}}-\mathrm{F})+{\mathrm{F}}_{\mathrm{d}}]+\mathrm{G}$$ (13)

where K_p, K_v, K_f ∊ R^pxp are position, velocity and force gain matrices, respectively.

Proof:

Let $\mathrm{\Phi }∊{\mathrm{\Re }}^{\mathrm{n}}$ and $\mathrm{Y}(\mathrm{\Phi })∊{\mathrm{\Re }}^{\mathrm{p}}$ the error vectors defined in the joint and task space, respectively, by: $$\mathrm{\Phi }=\mathrm{\theta }-{\mathrm{\theta }}_{\mathrm{d}}$$ (14) $$\mathrm{Y}(\mathrm{\Phi })=\mathrm{X}(\mathrm{\theta })-{\mathrm{X}}_{\mathrm{d}}$$ (15)

Consider, now, the constrained robot system described by the dynamic model (1) for the force design (12) and the control law (13). Using the relations (14) and (15) we can write [31, 32]: $$\mathrm{M}(\mathrm{\Phi })\ddot{\mathrm{\Phi }}+\mathrm{H}(\mathrm{\Phi },\dot{\mathrm{\Phi }})+{\mathrm{J}}^{\mathrm{T}}(\mathrm{\Phi }){\mathrm{K}}_1\mathrm{Y}(\mathrm{\Phi })+{\mathrm{J}}^{\mathrm{T}}(\mathrm{\Phi }){\mathrm{K}}_2\dot{\mathrm{Y}}(\mathrm{\Phi })+{\mathrm{J}}^{\mathrm{T}}(\mathrm{\Phi }){\mathrm{K}}_3\ddot{\mathrm{Y}}(\mathrm{\Phi })=0$$ (16)

where $$\begin{array}{l}{\mathrm{K}}_1={\mathrm{K}}_{\mathrm{p}}+(\mathrm{I}+{\mathrm{K}}_{\mathrm{f}}){\mathrm{K}}_{\mathrm{d}}\\ {\mathrm{K}}_2={\mathrm{K}}_{\mathrm{v}}+(\mathrm{I}+{\mathrm{K}}_{\mathrm{f}}){\mathrm{B}}_{\mathrm{d}}\\ {\mathrm{K}}_3=(\mathrm{I}+{\mathrm{K}}_{\mathrm{f}}){\mathrm{M}}_{\mathrm{d}}\end{array}$$ (17)

In order to evaluate the relationship between the dynamics of the constrained robotic system and its energy, we use Lagrange's equation described by [33]: $$\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}\left(\frac{\partial \mathrm{T}}{\partial \dot{\mathrm{\Phi }}}\right)-\frac{\partial \mathrm{T}}{\partial \mathrm{\Phi }}+\frac{\partial \mathrm{P}}{\partial \mathrm{\Phi }}+\frac{\partial \mathrm{D}}{\partial \dot{\mathrm{\Phi }}}=0$$ (18)

where $\mathrm{T}(\mathrm{\Phi },\dot{\mathrm{\Phi }})$ is the kinetic energy of the constrained robotic system defined by: $$\mathrm{T}(\mathrm{\Phi },\dot{\mathrm{\Phi }})=\frac{1}{2}{\dot{\mathrm{\Phi }}}^{\mathrm{T}}\mathrm{M}(\mathrm{\Phi })\dot{\mathrm{\Phi }}$$ (19)

and P(Φ), $\mathrm{D}(\mathrm{\Phi },\dot{\mathrm{\Phi }})$ are potential energy and dissipation function respectively. For the Lagrange equation (18) we can prove that [31]: $$\frac{\partial \mathrm{P}(\mathrm{\Phi })}{\partial \mathrm{\Phi }}={\mathrm{J}}^{\mathrm{T}}(\mathrm{\Phi }){\mathrm{K}}_1\mathrm{Y}(\mathrm{\Phi })$$ (20) $$\frac{\partial \mathrm{D}(\mathrm{\Phi },\dot{\mathrm{\Phi }})}{\partial \dot{\mathrm{\Phi }}}={\mathrm{J}}^{\mathrm{T}}(\mathrm{\Phi }){\mathrm{K}}_2\dot{\mathrm{Y}}(\mathrm{\Phi })+{\mathrm{J}}^{\mathrm{T}}(\mathrm{\Phi }){\mathrm{K}}_3\ddot{\mathrm{Y}}(\mathrm{\Phi })$$ (21) $$\mathrm{H}(\mathrm{\Phi },\dot{\mathrm{\Phi }})={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}\left(\frac{\mathrm{d}{\mathrm{\Phi }}_{\mathrm{i}}}{\mathrm{d}\mathrm{t}}\frac{\partial \mathrm{M}}{\partial {\mathrm{\Phi }}_{\mathrm{i}}}\right)}\frac{\dot{\mathrm{\Phi }}}{2}$$ (22)

Impose, now, to the system (16) to have a Lyapunov Hamiltonian function defined by [33]: $$\mathrm{V}(\mathrm{\Phi },\dot{\mathrm{\Phi }})=\mathrm{T}(\mathrm{\Phi },\dot{\mathrm{\Phi }})+\mathrm{P}(\mathrm{\Phi })-\mathrm{P}(0)$$ (23)

The error system (16) is asymptotically stable if $\mathrm{V}(\mathrm{\Phi },\dot{\mathrm{\Phi }})$ satisfies the following conditions [34]: $$\mathrm{V}(0,0)=0\text{ if }\mathrm{\Phi }=0,\dot{\mathrm{\Phi }}=0$$ (24) $$\mathrm{V}(\mathrm{\Phi },\dot{\mathrm{\Phi }})>0\text{ if }\mathrm{\Phi }\ne 0,\dot{\mathrm{\Phi }}\ne 0$$ (25) $$\dot{\mathrm{V}}(\mathrm{\Phi },\dot{\mathrm{\Phi }})<0\text{ if }\mathrm{\Phi }\ne 0,\dot{\mathrm{\Phi }}\ne 0$$ (26)

since T(0,0) = 0, then V(0,0) = 0 so the first Lyapunov condition (24) is verified. The second Lyapunov condition (25) will be verified if we prove that V_p (Φ) defined by: $${\mathrm{V}}_{\mathrm{p}}(\mathrm{\Phi })=\mathrm{P}(\mathrm{\Phi })-\mathrm{P}(0)$$ (27)

is positive definite [33] since the kinetic energy $\mathrm{T}(\mathrm{\Phi },\dot{\mathrm{\Phi }})$ is positive definite and V_p (0) = 0.

From (27) and (20) we can write: $$\left[\frac{\partial {\mathrm{V}}_{\mathrm{p}}(\mathrm{\Phi })}{\partial \mathrm{\Phi }}\right]=\left[\frac{\partial \mathrm{P}(\mathrm{\Phi })}{\partial \mathrm{\Phi }}\right]={\mathrm{J}}^{\mathrm{T}}(\mathrm{\Phi }){\mathrm{K}}_1\mathrm{Y}(\mathrm{\Phi })$$ (28)

then: $${\left[\frac{\partial {\mathrm{V}}_{\mathrm{p}}(\mathrm{\Phi })}{\partial \mathrm{\Phi }}\right]}_{\mathrm{\Phi }=0}={\mathrm{J}}^{\mathrm{T}}(0){\mathrm{K}}_1\mathrm{Y}(0)=0$$ (29)

Let consider the function W given by: $$\mathrm{W}=\frac{\partial }{\partial {\mathrm{\Phi }}^{\mathrm{T}}}\left(\frac{\partial {\mathrm{V}}_{\mathrm{p}}(\mathrm{\Phi })}{\partial \mathrm{\Phi }}\right)=\frac{\partial }{\partial {\mathrm{\Phi }}^{\mathrm{T}}}\left(\frac{\partial \mathrm{P}(\mathrm{\Phi })}{\partial \mathrm{\Phi }}\right)$$ (30)

Substituting (20) in (30) we have: $$\mathrm{W}=\frac{\partial }{\partial {\mathrm{\Phi }}^{\mathrm{T}}}\left({\mathrm{J}}^{\mathrm{T}}(\mathrm{\Phi }){\mathrm{K}}_1\mathrm{Y}(\mathrm{\Phi })\right)={\mathrm{w}}_{\mathrm{i}\mathrm{j}}+{\mathrm{J}}^{\mathrm{T}}(\mathrm{\Phi }){\mathrm{K}}_1\mathrm{J}(\mathrm{\Phi })$$ (31)

where: $${\mathrm{w}}_{\mathrm{i}\mathrm{j}}={\left(\frac{\partial {\mathrm{J}}_{\mathrm{i}}}{\partial {\mathrm{\Phi }}_{\mathrm{i}}}\right)}^{\mathrm{T}}{\mathrm{K}}_1\mathrm{Y}(\mathrm{\Phi })$$

At the equilibrium point we have: $${\left[\mathrm{W}\right]}_{\mathrm{\Phi }=0}={\mathrm{J}}^{\mathrm{T}}(0){\mathrm{K}}_1\mathrm{J}(0)$$ (32)

Based on (32), the function W is positive definite at the equilibrium point if K₁ is positive definite. Hence the second Lyapunov condition (25) is satisfied if K₁ is positive definite.

Using the expression (23) we have: $$\frac{\mathrm{d}\mathrm{V}(\mathrm{\Phi })}{\mathrm{d}\mathrm{t}}=\frac{\mathrm{d}\mathrm{T}(\mathrm{\Phi },\dot{\mathrm{\Phi }})}{\mathrm{d}\mathrm{t}}+\frac{\mathrm{d}\mathrm{P}(\mathrm{\Phi })}{\mathrm{d}\mathrm{t}}$$ (33)

From equation (19) we can write: $$\begin{array}{l}\frac{\mathrm{d}\mathrm{T}(\mathrm{\Phi },\dot{\mathrm{\Phi }})}{\mathrm{d}\mathrm{t}}={\dot{\mathrm{\Phi }}}^{\mathrm{T}}\mathrm{M}(\mathrm{\Phi })\ddot{\mathrm{\Phi }}+{\dot{\mathrm{\Phi }}}^{\mathrm{T}}\frac{\mathrm{d}\mathrm{M}(\mathrm{\Phi })}{\mathrm{d}\mathrm{t}}\frac{\dot{\mathrm{\Phi }}}{2}\\ ={\dot{\mathrm{\Phi }}}^{\mathrm{T}}\mathrm{M}(\mathrm{\Phi })\ddot{\mathrm{\Phi }}+\mathrm{H}(\mathrm{\Phi },\dot{\mathrm{\Phi }})\end{array}$$ (34)

Furthermore, $$\frac{\mathrm{d}\mathrm{P}(\mathrm{\Phi })}{\mathrm{d}\mathrm{t}}={\dot{\mathrm{\Phi }}}^{\mathrm{T}}\frac{\partial \mathrm{P}(\mathrm{\Phi })}{\partial \mathrm{\Phi }}$$ (35)

Substituting (20) in (35) we have: $$\frac{\mathrm{d}\mathrm{P}\left(\mathrm{\Phi }\right)}{\mathrm{d}\mathrm{t}}={\dot{\mathrm{\Phi }}}^{\mathrm{T}}{\mathrm{J}}^{\mathrm{T}}(\mathrm{\Phi }){\mathrm{K}}_1\mathrm{Y}(\mathrm{\Phi })$$ (36)

Substituting (34) and (36) in (33) we obtain: $$\frac{\mathrm{d}\mathrm{V}\left(\mathrm{\Phi },\dot{\mathrm{\Phi }}\right)}{\mathrm{d}\mathrm{t}}={\dot{\mathrm{\Phi }}}^{\mathrm{T}}\mathrm{M}\left(\mathrm{\Phi }\right)\ddot{\mathrm{\Phi }}+{\dot{\mathrm{\Phi }}}^{\mathrm{T}}\mathrm{H}\left(\mathrm{\Phi },\dot{\mathrm{\Phi }}\right)+{\dot{\mathrm{\Phi }}}^{\mathrm{T}}{\mathrm{J}}^{\mathrm{T}}\left(\mathrm{\Phi }\right){\mathrm{K}}_1\mathrm{Y}\left(\mathrm{\Phi }\right)$$ (37)

From (16) we can write: $$\mathrm{M}\left(\mathrm{\Phi }\right)\ddot{\mathrm{\Phi }}+\mathrm{H}\left(\mathrm{\Phi },\dot{\mathrm{\Phi }}\right)+{\mathrm{J}}^{\mathrm{T}}\left(\mathrm{\Phi }\right){\mathrm{K}}_1\mathrm{Y}\left(\mathrm{\Phi }\right)=-({\mathrm{J}}^{\mathrm{T}}\left(\mathrm{\Phi }\right){\mathrm{K}}_2\dot{\mathrm{Y}}\left(\mathrm{\Phi }\right)+{\mathrm{J}}^{\mathrm{T}}\left(\mathrm{\Phi }\right){\mathrm{K}}_3\ddot{\mathrm{Y}}\left(\mathrm{\Phi }\right))$$ (38)

Substituting the second member of (38) in (37) gives: $$\frac{\mathrm{d}\mathrm{V}\left(\mathrm{\Phi },\dot{\mathrm{\Phi }}\right)}{\mathrm{d}\mathrm{t}}=-{\dot{\mathrm{\Phi }}}^{\mathrm{T}}({\mathrm{J}}^{\mathrm{T}}\left(\mathrm{\Phi }\right){\mathrm{K}}_2\dot{\mathrm{Y}}\left(\mathrm{\Phi }\right)+{\mathrm{J}}^{\mathrm{T}}\left(\mathrm{\Phi }\right){\mathrm{K}}_3\ddot{\mathrm{Y}}\left(\mathrm{\Phi }\right))$$ (39)

Using relations (3), (14) and (15) gives: $$\frac{\mathrm{d}\mathrm{V}\left(\mathrm{\Phi },\dot{\mathrm{\Phi }}\right)}{\mathrm{d}\mathrm{t}}=-{\dot{\mathrm{Y}}}^{\mathrm{T}}\left(\mathrm{\Phi }\right)({\mathrm{K}}_2\dot{\mathrm{Y}}\left(\mathrm{\Phi }\right)+{\mathrm{K}}_3\ddot{\mathrm{Y}}\left(\mathrm{\Phi }\right))$$ (40)

The third Lyapunov condition (26) is then verified if K₂ is positive definite and K₃ is null. Note that K₃ is null if I + K_f = 0 or M_d = 0. The control objectives (7) and (8) are then reached if the conditions (10) or (11) are satisfied. However, to ensure some desired performances, tuning the controller gain matrices K_p and K_v ∊ R^pxp remain a complex and time consuming procedure. This difficult problem will be solved in the next section.

3. PSO-Impedance Controller

3.1 Overview of PSO algorithm

The particle swarm optimization (PSO) method was firstly introduced by Kennedy and Eberhart [18]. It is a relatively new evolutionary algorithm that may be used to find optimal or near optimal solutions in big search space. The PSO method is developed from study on swarm such as fish schooling and bird flocking. It can be easily implemented, and has stable convergence characteristic with good computational efficiency.

During flight, each particle adjusts its position according to its own experience, and the experience of other particles, making use of the best position encounter by itself and its neighbors. The swarm direction of a particle is defined by the set of particles neighboring the particle and its history of experience. The best previous position of ith particle is recorded and represented as pbest_i. The best particle among all the particles in the group is represented as gbest [35].

The PSO concept consists of, at each time step, changing the velocity (or acceleration) of each particle i toward its pbest and the gbest position. For the ith particle, a new velocity and position are updated such that [19]: $$\begin{array}{c}{\mathrm{V}}_{\mathrm{i}}^{(\mathrm{k}+1)}={\mathrm{w}}^{(\mathrm{k})}\cdot {\mathrm{V}}_{\mathrm{i}}^{(\mathrm{k})}+{\mathrm{c}}_1\cdot \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}()\cdot (\mathrm{p}\mathrm{b}\mathrm{e}\mathrm{s}{\mathrm{t}}_{\mathrm{i}}-{\mathrm{p}}_{\mathrm{i}}^{(\mathrm{k})})\\ +{\mathrm{c}}_2\cdot \mathrm{R}\mathrm{a}\mathrm{n}\mathrm{d}()\cdot (\mathrm{g}\mathrm{b}\mathrm{e}\mathrm{s}{\mathrm{t}}^{(\mathrm{k})}-{\mathrm{p}}_{\mathrm{i}}^{(\mathrm{k})})\end{array}$$ (41) $${\mathrm{p}}_{\mathrm{i}}^{(\mathrm{k}+1)}={\mathrm{p}}_{\mathrm{i}}^{(\mathrm{k})}+{\mathrm{V}}_{\mathrm{i}}^{(\mathrm{k}+1)}$$ (42)

for k = 1,2,…, N and i = 1,2,…, M and where:

M Number of particles in a group

N Maximum number of iterations

${\mathrm{V}}_{\mathrm{i}}^{(\mathrm{k})}$ Velocity of ith particle at iteration k

${\mathrm{p}}_{\mathrm{i}}^{(\mathrm{k})}$ Position of ith individual at iteration k

$\mathrm{p}\mathrm{b}\mathrm{e}\mathrm{s}{\mathrm{t}}_{\mathrm{i}}^{(\mathrm{k})}$ Best position of ith individual at iteration k

gbest^(k) Best position of the group until iteration k

w^(k) Inertia weight factor at iteration k

c₁,c₂ Acceleration factors

rand () Random numbers between 0 and 1

Rand () Random numbers between 0 and 1

In general, the inertia weight is calculated according to the following equation: $${\mathrm{w}}^{(\mathrm{k})}=\frac{{\mathrm{w}}_{\max }-{\mathrm{w}}_{\min }}{\mathrm{N}}\cdot \mathrm{k}$$ (43)

The design steps for implementing the PSO algorithm are given as follow:

1. Initialize a population array of particles with random positions and velocities on d-dimensional space.

2. For each particle, evaluate the desired optimization fitness function.

3. Compare particle's fitness evaluation with its pbest_i. If current value is better than pbest_i, then set pbest_i equal to the current value.

4. Identify the particle in the neighborhood with the best success so far, and assign it to gbest.

5. Change the velocity and position of the particle according to equations (41) and (42) respectively.

6. If a criterion is met (usually a sufficiently good fitness or a maximum number of iterations), go to 7 else go to 2.

7. The particle that generates the latest gbest is an optimal controller parameter.

3.2. PSO-Controller tuning

To design the impedance controller (13), only the stability condition (11) will be adopted in this paper to ensure that M_d ≠ 0_p×p. In this case, the force matrix gain is well defined as K_f = -I_p×p. Taking on account of the assumption A₄, the member of each individual of the swarm will be composed of the diagonal elements of the gain matrices K_p, K_v ∊ R^pxp. Consequently, whatever the number of degrees of freedom for the constrained robotic arm, the dimension of the individual member will be always 4 for a planar robot and 6 for a three dimensional robot.

Fig. 1 illustrates the block diagram of optimal design of the proposed Lyapunov-based impedance controller for constrained robotic arms using PSO algorithm.

Figure 1.

Impedance controller tuned with PSO

3.3. Performance indexes

A fundamental step in applying PSO algorithm remains in choosing the cost function or performance index which is used to evaluate fitness of each particle. In this paper the following cost functions:

1. Mean of Root Squared Error (MRSE)

2. Mean of Absolute Magnitude Error (MAE)

3. Mean of Time-weighted Magnitude Error (MTE)

Based on (7) and (8) we consider furthermore a multiobjectif optimization problem to be solved where the two objectives to be reached are really in conflict. Furthermore the multiobjectif problem will be solved under the stability constrained conditions (11).

We first try to solve the problem in the joint space by considering the following multiobjectif cost functions: $$\mathrm{M}\mathrm{R}\mathrm{S}\mathrm{E}{ }_{\mathrm{\theta }}=\frac{1}{\mathrm{N}}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{N}}\sqrt{{\mathrm{e}}_{\mathrm{k}}^{\mathrm{T}}\cdot {\mathrm{e}}_{\mathrm{k}}}}+{\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{N}}\sqrt{{\mathrm{F}}_{\mathrm{k}}^{\mathrm{T}}\cdot {\mathrm{F}}_{\mathrm{k}}}}\right)$$ (44) $$\mathrm{M}\mathrm{A}{\mathrm{E}}_{\mathrm{\theta }}=\frac{1}{\mathrm{N}}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{N}} {\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}\left|{\mathrm{e}}_{\mathrm{i},\mathrm{k}}\right|}}+{\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{N}} {\displaystyle \sum _{\mathrm{j}=1}^{\mathrm{p}}\left|{\mathrm{F}}_{\mathrm{j},\mathrm{k}}\right|}}\right)$$ (45) $$\mathrm{M}\mathrm{T}\mathrm{E}{ }_{\mathrm{\theta }}=\frac{1}{\mathrm{N}}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{N}}\mathrm{t}\cdot {\mathrm{e}}_{\mathrm{k}}^{\mathrm{T}}\cdot {\mathrm{e}}_{\mathrm{k}}}+{\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{N}}\mathrm{t}\cdot {\mathrm{F}}_{\mathrm{k}}^{\mathrm{T}}\cdot {\mathrm{F}}_{\mathrm{k}}}\right)$$ (46)

where:

${\mathrm{F}}_{\mathrm{k}}={\left[{\mathrm{F}}_{\mathrm{d}}(\mathrm{k})-\mathrm{F}(\mathrm{k})\right]}^{\mathrm{T}}∊{\mathrm{\Re }}^{\mathrm{p}}$

${\mathrm{e}}_{\mathrm{k}}={\left[\begin{array}{ccc}{\mathrm{\theta }}_{1,\mathrm{d}}(\mathrm{k})-{\mathrm{\theta }}_1(\mathrm{k})& \cdots & {\mathrm{\theta }}_{\mathrm{n},\mathrm{d}}(\mathrm{k})-{\mathrm{\theta }}_{\mathrm{n}}(\mathrm{k})\end{array}\right]}^{\mathrm{T}}∊{\mathrm{\Re }}^{\mathrm{n}}$

${\mathrm{e}}_{\mathrm{i},\mathrm{k}}=\left[{\mathrm{\theta }}_{\mathrm{i},\mathrm{d}}(\mathrm{k})-{\mathrm{\theta }}_{\mathrm{i}}(\mathrm{k})\right]∊\mathrm{\Re },\mathrm{i}=1,\cdots ,\mathrm{n}$

${\mathrm{F}}_{\mathrm{j},\mathrm{k}}=\left[{\mathrm{F}}_{\mathrm{j},\mathrm{d}}(\mathrm{k})-{\mathrm{F}}_{\mathrm{j}}(\mathrm{k})\right]∊\mathrm{\Re },\mathrm{j}=1,\cdots ,\mathrm{p}$

θ_i,d(k) design the ith element of desired angular displacement vector. F_j,d(k) and F_j (k) design the jth element of the desired force vector and the jth element of the force vector, respectively.

The optimization problem will be also solved in the Cartesian space by considering the following multiobjectif cost functions: $$\mathrm{M}\mathrm{R}\mathrm{S}{\mathrm{E}}_{\mathrm{x}}=\frac{1}{\mathrm{N}}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{N}}\sqrt{{\mathrm{E}}_{\mathrm{k}}^{\mathrm{T}}\cdot {\mathrm{E}}_{\mathrm{k}}}}+{\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{N}}\sqrt{{\mathrm{F}}_{\mathrm{k}}^{\mathrm{T}}\cdot {\mathrm{F}}_{\mathrm{k}}}}\right)$$ (47) $$\mathrm{M}\mathrm{A}{\mathrm{E}}_{\mathrm{x}}=\frac{1}{\mathrm{N}}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{N}} {\displaystyle \sum _{\mathrm{j}=1}^{\mathrm{p}}\left|{\mathrm{E}}_{\mathrm{j},\mathrm{k}}\right|}}+{\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{N}} {\displaystyle \sum _{\mathrm{j}=1}^{\mathrm{p}}\left|{\mathrm{F}}_{\mathrm{j},\mathrm{k}}\right|}}\right)$$ (48) $$\mathrm{M}\mathrm{T}{\mathrm{E}}_{\mathrm{x}}=\frac{1}{\mathrm{N}}\left({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{N}}\mathrm{t}\cdot {\mathrm{E}}_{\mathrm{k}}^{\mathrm{T}}\cdot {\mathrm{E}}_{\mathrm{k}}}+{\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{N}}\mathrm{t}\cdot {\mathrm{F}}_{\mathrm{k}}^{\mathrm{T}}\cdot {\mathrm{F}}_{\mathrm{k}}}\right)$$ (49)

where:

${\mathrm{E}}_{\mathrm{k}}={\left[{\mathrm{X}}_{\mathrm{d}}(\mathrm{k})-\mathrm{X}(\mathrm{k})\right]}^{\mathrm{T}}∊{\mathrm{\Re }}^{\mathrm{p}}$

${\mathrm{E}}_{\mathrm{j},\mathrm{k}}=\left[{\mathrm{X}}_{\mathrm{j},\mathrm{d}}(\mathrm{k})-{\mathrm{X}}_{\mathrm{j}}(\mathrm{k})\right]∊\mathrm{\Re },\mathrm{j}=1,\cdots ,\mathrm{p}$

X_j(k) and X_j,d(k) design the jth element of Cartesian displacement vector and the desired Cartesian displacement vector, respectively.

4. Application to a 3DOF constrained robotic arm

To verify the stability and performances of the proposed controller tuned by PSO intelligence, a 3DOF robotic arm constrained to circular trajectory is considered (see Fig. 2).

Figure 2.

3DOF manipulator at constrained circular motion

4.1. The 3DOF constrained robotic arm

The dynamical model (1), kinematic model (2) and differential kinematic models (3) of the 3DOF constrained robotic arm are defined, respectively, by: $$\mathrm{M}(\mathrm{\theta })=\left[\begin{array}{ccc}{\mathrm{I}}_1+{\mathrm{m}}_1{\mathrm{k}}_1^2+{\mathrm{m}}_2{\mathrm{L}}_1^2+{\mathrm{m}}_3{\mathrm{L}}_1^2& \mathrm{a}\cdot {\mathrm{c}}_{12}& \mathrm{b}\cdot {\mathrm{c}}_{13}\\ \mathrm{a}\cdot {\mathrm{c}}_{12}& {\mathrm{I}}_2+{\mathrm{m}}_2{\mathrm{k}}_2^2+{\mathrm{m}}_3{\mathrm{L}}_2^2& \mathrm{c}\cdot {\mathrm{c}}_{23}\\ \mathrm{b}\cdot {\mathrm{c}}_{13}& \mathrm{c}\cdot {\mathrm{c}}_{23}& {\mathrm{I}}_3+{\mathrm{m}}_3{\mathrm{k}}_3^2\end{array}\right]$$ $$\mathrm{H}\left(\mathrm{\theta },\dot{\mathrm{\theta }}\right)=\left[\begin{array}{ccc}0& \mathrm{a}\cdot {\mathrm{s}}_{12}& \mathrm{d}\cdot {\mathrm{s}}_{12}\\ -\mathrm{a}\cdot {\mathrm{s}}_{12}& 0& \mathrm{c}\cdot {\mathrm{s}}_{23}\\ -\mathrm{d}\cdot {\mathrm{s}}_{12}& -\mathrm{c}\cdot \sin {\mathrm{s}}_{23}& 0\end{array}\right]\left[\begin{array}{c}{\dot{\mathrm{\theta }}}_1^2\\ {\dot{\mathrm{\theta }}}_2^2\\ {\dot{\mathrm{\theta }}}_3^2\end{array}\right]$$ $$\mathrm{G}(\mathrm{\theta })=\mathrm{g}\left[\begin{array}{c}({\mathrm{m}}_1{\mathrm{k}}_1+{\mathrm{m}}_2{\mathrm{L}}_1+{\mathrm{m}}_3{\mathrm{L}}_1)\cdot {\mathrm{c}}_1\\ ({\mathrm{m}}_2{\mathrm{k}}_2+{\mathrm{m}}_3{\mathrm{L}}_2)\cdot {\mathrm{c}}_2\\ {\mathrm{m}}_2{\mathrm{k}}_3\cdot {\mathrm{c}}_3\end{array}\right]$$ $$\mathrm{h}\left(\mathrm{\theta }\right)=\left[\begin{array}{c}{\mathrm{L}}_1\cdot {\mathrm{c}}_1+{\mathrm{L}}_2\cdot {\mathrm{c}}_2+{\mathrm{L}}_3\cdot {\mathrm{c}}_3\\ {\mathrm{L}}_1\cdot {\mathrm{s}}_1+{\mathrm{L}}_2\cdot {\mathrm{s}}_2+{\mathrm{L}}_3\cdot {\mathrm{s}}_3\end{array}\right]$$ $$\mathrm{J}(\mathrm{\theta })=\left[\begin{array}{ccc}-{\mathrm{L}}_1\cdot {\mathrm{s}}_1& -{\mathrm{L}}_2\cdot {\mathrm{s}}_2& -{\mathrm{L}}_3\cdot {\mathrm{s}}_3\\ {\mathrm{L}}_1\cdot {\mathrm{c}}_1& {\mathrm{L}}_2\cdot {\mathrm{c}}_2& {\mathrm{L}}_3\cdot {\mathrm{c}}_3\end{array}\right]$$

where:

${\mathrm{c}}_{12}=\cos ({\mathrm{\theta }}_1-{\mathrm{\theta }}_2)$ , ${\mathrm{c}}_{13}=\cos ({\mathrm{\theta }}_1-{\mathrm{\theta }}_3)$ , ${\mathrm{c}}_{23}=\cos ({\mathrm{\theta }}_2-{\mathrm{\theta }}_3)$

${\mathrm{s}}_{12}=\sin ({\mathrm{\theta }}_1-{\mathrm{\theta }}_2)$ , ${\mathrm{s}}_{23}=\sin ({\mathrm{\theta }}_2-{\mathrm{\theta }}_3)$

${\mathrm{c}}_1=\cos {\mathrm{\theta }}_1$ , ${\mathrm{c}}_2=\cos {\mathrm{\theta }}_2$ , ${\mathrm{c}}_3=\cos {\mathrm{\theta }}_3$

${\mathrm{s}}_1=\sin {\mathrm{\theta }}_1$ , ${\mathrm{s}}_2=\sin {\mathrm{\theta }}_2$ , ${\mathrm{s}}_3=\sin {\mathrm{\theta }}_3$

$\begin{array}{l}\mathrm{a}={\mathrm{m}}_2{\mathrm{L}}_1{\mathrm{k}}_2+{\mathrm{m}}_3{\mathrm{L}}_1{\mathrm{L}}_2\\ \mathrm{b}={\mathrm{m}}_3{\mathrm{L}}_1{\mathrm{k}}_2\\ \mathrm{c}={\mathrm{m}}_3{\mathrm{L}}_2{\mathrm{k}}_3\\ \mathrm{d}={\mathrm{m}}_3{\mathrm{L}}_1{\mathrm{k}}_3\end{array}$

The parameters m_i, L_i, k_i and I_i, (i = 1,2,3), design mass, length, position of gravity center and moment of inertia of the links of the 3DOF constrained arm.

4.2 Trajectory generation

The desired Cartesian displacement, speed and acceleration of the constrained robotic system are used as inputs of the control laws (13). It is essential then to generate desired trajectories. Assume that joint motion must begin and end regularly such that: $$\begin{array}{l}{\mathrm{\theta }}_{\mathrm{i}}({\mathrm{t}}_0)={\mathrm{\theta }}_{\mathrm{i}0};{\dot{\mathrm{\theta }}}_{\mathrm{i}}({\mathrm{t}}_0)=0\\ {\mathrm{\theta }}_{\mathrm{i}}({\mathrm{t}}_{\mathrm{f}})={\mathrm{\theta }}_{\mathrm{i}\mathrm{d}};{\dot{\mathrm{\theta }}}_{\mathrm{i}}({\mathrm{t}}_{\mathrm{f}})=0\end{array}$$

where t₀ and t_f design initial and terminal times. Reference joint trajectories are chosen as: $$\begin{array}{l}{\mathrm{\theta }}_{\mathrm{i},\mathrm{d}}(\mathrm{t})={\mathrm{a}}_{\mathrm{i}0}+{\mathrm{a}}_{\mathrm{i}1}\mathrm{t}+{\mathrm{a}}_{\mathrm{i}2}{\mathrm{t}}^2+{\mathrm{a}}_{\mathrm{i}3}{\mathrm{t}}^3\hfill \\ {\dot{\mathrm{\theta }}}_{\mathrm{i},\mathrm{d}}(\mathrm{t})={\mathrm{a}}_{\mathrm{i}1}+2{\mathrm{a}}_{\mathrm{i}2}\mathrm{t}+3{\mathrm{a}}_{\mathrm{i}3}{\mathrm{t}}^2\hfill \end{array}$$

where: $$\left[\begin{array}{c}{\mathrm{a}}_{\mathrm{i}0}\\ {\mathrm{a}}_{\mathrm{i}1}\\ {\mathrm{a}}_{\mathrm{i}2}\\ {\mathrm{a}}_{\mathrm{i}3}\end{array}\right]={\left[\begin{array}{cccc}1& {\mathrm{t}}_0& {\mathrm{t}}_0^2& {\mathrm{t}}_0^3\\ 1& {\mathrm{t}}_{\mathrm{f}}& {\mathrm{t}}_{\mathrm{f}}^2& {\mathrm{t}}_{\mathrm{f}}^3\\ 0& 1& 2{\mathrm{t}}_0& 3{\mathrm{t}}_0^2\\ 0& 1& 2{\mathrm{t}}_{\mathrm{f}}& 3{\mathrm{t}}_{\mathrm{f}}^2\end{array}\right]}^{-1}\left[\begin{array}{c}{\mathrm{\theta }}_{\mathrm{i}0}\\ {\mathrm{\theta }}_{\mathrm{i}\mathrm{d}}\\ 0\\ 0\end{array}\right]$$

In the Cartesian space, we impose a circular motion. To realize this profile, we will introduce a sinusoidal signal on each axis. So, the axis references correspond to a harmony excitation in quadratic phase. Therefore, for a circle radius R, the desired end-effector motion of the constrained robotic system will be defined by: $${\mathrm{X}}_{\mathrm{d}}=\left[\begin{array}{c}\mathrm{R}\cos ({\mathrm{a}}_{30}+{\mathrm{a}}_{31}\mathrm{t}+{\mathrm{a}}_{32}{\mathrm{t}}^2+{\mathrm{a}}_{33}{\mathrm{t}}^3)\\ \mathrm{R}\sin ({\mathrm{a}}_{30}+{\mathrm{a}}_{31}\mathrm{t}+{\mathrm{a}}_{32}{\mathrm{t}}^2+{\mathrm{a}}_{33}{\mathrm{t}}^3)\end{array}\right]$$

4.3. Simulation Results

Simulation results are conducted solving the dynamical robotic model (1) using relations (2), (3), (4), (5) and (6) for the control law (13) and the force design (12). The parameter data of the robotic system are given by Table 1.

Table 1.

Physical parameters of the robotic arm

Link i	mi(Kg)	Li(m)	ki(m)	Ii(Kgm−2)
1	1.960	0.321	0.140	0.016
2	1.120	0.253	0.109	0.006
3	0.420	0.187	0.095	0.001

Desired circular motion is obtained by imposing θ_i0 = [0 0 0]^T and θ_id = [π π π]^T as initial and final joint positions, t₀ = 0 and t_f = 1s as initial and final times and R = 0.76 m for the circle Radius. The desired circular trajectory is then given by: $${\mathrm{X}}_{\mathrm{d}}=\left[\begin{array}{c}0.76\cos (3\mathrm{\pi }{\mathrm{t}}^2-2\mathrm{\pi }{\mathrm{t}}^3)\\ 0.76\sin (3\mathrm{\pi }{\mathrm{t}}^2-2\mathrm{\pi }{\mathrm{t}}^3)\end{array}\right]$$

PSO intelligence is applied using the variables given by Table 2. To carry out simulation results, two case studies for desired contact force, stiffness, damping and inertia parameters were assessed (see table 3). Fig. 3 and Fig. 4 show the convergence characteristics of cost functions for the two cases with respect to index performances (44)–(49).

Figure 3.

Evolution of fitness functions: case1

Figure 4.

Evolution of fitness functions: case2

Table 2.

Variables of the PSO algorithm

Designation	Variable	Value
Number of particles in a group	M	30
Maximum number of iterations	N	50
Minimum inertia weight factor	wmin	0.4
Maximum inertia weight factor	wmax	0.9
Search interval for kp	$\left[\begin{array}{ll}{\mathrm{k}}_{{\mathrm{p}}_{\text{min}}}\hfill & {\mathrm{k}}_{{\mathrm{p}}_{\text{max}}}\hfill \end{array}\right]$	[100 900]
Search interval for kv	$\left[\begin{array}{ll}{\mathrm{k}}_{{\mathrm{v}}_{\text{min}}}\hfill & {\mathrm{k}}_{{\mathrm{v}}_{\text{max}}}\hfill \end{array}\right]$	[30 500]
Search interval velocity of kp	$\left[\begin{array}{ll}0\hfill & {\mathrm{k}}_{{\mathrm{p}}_{\text{max}}}/2\hfill \end{array}\right]$	[0 450]
Search interval velocity of kv	$\left[\begin{array}{ll}0\hfill & {\mathrm{k}}_{{\mathrm{v}}_{\text{max}}}/2\hfill \end{array}\right]$	[0 250]
acceleration constants	c1 c2	2

Table 3.

Case study parameters

	Case study 1	Case study 2
Fd	[6 0]T	[6 6]T
Md	diag[0.02 0.02]	diag[0.05 0.05]
Bd	diag[5 5]	diag[10 10]
Kd	diag[20 20]	diag10 10]

Best cost functions and their corresponding controller gain parameters were reported in Tables 4 and 5 using the performance indexes (44)–(46) and in Tables 6 and 7 using the performance indexes (47)–(49). It is clear that optimization procedure with respect to the Cartesian space errors gives better results.

Table 4.

PSO-best controller tuning in the joint space: case1

	MRSEθ	MAEθ	MTEθ
kp1	799.3	640.9	757.8
kp2	839.6	727.2	322.4
kv1	358	307.2	500
kv2	297.2	319.5	290.6
Processing time (s)	1961	1965	2093
Best fitness value	0.0369	0.0408	0.0374

Table 5.

PSO-best controller tuning in the joint space: case2

	MRSEθ	MAEθ	MTEθ
kp1	691	646.8	746.6
kp2	860.2	699.6	380.9
kv1	450.1	464.8	451.4
kv2	272.6	458.8	438.3
Processing time (s)	2973	2958	3210
Best fitness value	0.0457	0.0458	0.0582

Table 6.

PSO-best controller tuning in the Cartesian space: case1

	MRSEX	MAEX	MTEX
kp1	255	566.1	551
kp2	801.5	572.4	391.4
kv1	500	269.3	200.5
kv2	367.4	222.2	324.5
Processing time (s)	1988	1960	2052
Best fitness value	0.0286	0.0292	0.0355

Table 7.

PSO-best controller tuning in the Cartesian space: case2

	MRSEX	MAEX	MTEX
kp1	821.3	493.3	900
kp2	792.1	487.2	620.5
kv1	434.2	331.5	485.2
kv2	364.5	356.2	436.3
Processing time (s)	2969	2966	3305
Best fitness value	0.0342	0.0341	0.0543

Best solutions are reported in Table 8 and will be used in the following to illustrate the performances of the PSO-impedance controllers. Fig. 5 and Fig. 6 show the convergence of the PSO algorithm to these solutions with respect to the number of iterations. Fig. 7 and Fig. 8 show the evolution of joint positions and velocities using best controller gains whereas Fig. 9 and Fig. 10 illustrate the tracking performances of the desired forces and the smooth profile of control laws.

Figure 5.

Convergence of controller parameters: case1

Figure 6.

Convergence of controller parameters: case2

Figure 7.

Evolution of joint position and velocity: case1

Figure 8.

Evolution of joint position and velocity: case2

Figure 9.

Contact force response and control laws: case1

Figure 10.

Contact force response and control laws: case2

Table 8.

Case study controller parameters

	kp1	kp2	kv1	kv2
Case study 1	255	801.5	500	367.4
Case study 2	493.3	487.2	331.5	356.2

Position and force errors are depicted in Fig. 11 and Fig. 12, respectively. Results prove the tracking performances. Fig. 13 shows the evolution of the constrained robotic arm to a circular profile.

Figure 11.

Position and force errors: case 1

Figure 12.

Position and force errors: case 2

Figure 13.

Robotic arm constrained to a circular profile

6. Conclusion

This paper proposes a Lypunov-based impedance controller tuned by a multiobjectif Particle Swarm Intelligence. In order to examine effects of the cost functions on the controller parameter optimization, different performance indexes were used in both join and Cartesian spaces. The PSO-impedance controller was tested on a 3DOF robot constrained to a circular trajectory. The simulation results prove the stability and the performances of the offline tuned controller.