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Case Studies in Mechanical Systems and Signal

Processing

journal homepage: www.elsevier.com/locate/csmssp

Design, simulation and comparison of controllers for a redundant robot

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Claudio Urrea*, John Kern

Grupo de Automática, Departamento de Ingeniería Eléctrica, Universidad de Santiago de Chile, Chile

ARTICLE INFO

ABSTRACT

Article history:

Received 15 September 2015

Received in revised form 18 November 2015

Accepted 9 December 2015

Available online 22 December 2015

Keywords:

Controllers

Robots

Redundant manipulators Dynamic model Simulation

The simulation tools are the foundation for the design of robot systems, for the application of robots in complex environments and for the development of new control strategies and algorithms. Because of this, the design, simulation and comparison of the performance of controllers applied to a redundant robot with five degrees of freedom (DOF) are presented in this paper. Through homogeneous transformation matrices the inverse kinematic model of the redundant robot is obtained. Six controllers are prepared to test the robot's dynamic model: hyperbolic sine-cosine; computed torque; sliding hyperbolic mode; control with learning; and adaptive. A simulation environment is developed by means of the MatLab/ Simulink software, which allows analyzing the dynamic performance of the robot and of the designed controllers. This simulation environment is used to carry out different tests of the redundant manipulator model together with each controller as they are made to follow a trajectory in space. The results, obtained through a simulation environment, are represented by comparative curves and RMS indices of the joint and Cartesian errors, and they show that the redundant manipulator model follows the test trajectory with less pronounced maximum errors using the adaptive controller than the other controllers, with more homogeneous motions of the manipulator. The largest joint and Cartesian errors generated when testing the robot model, both in terms of maximum and RMS values, occurred when the computed torque controller is used. The results with that controller are obtained by executing three iterations for learning, because with more iterations the variations were not important.

© 2015 The Authors. Published by Elsevier Ltd. All rights reserved.

1. Introduction

The use of industrial robots, since its beginnings more than 60 years ago, has made it possible to increase productivity and improve the quality of manufactured products, becoming massified and extending rapidly to various fields of application such as the automotive, plastics, food, lumber, agricultural, aeronautics, railways, energy industries, and the aerospace industry (National Aeronautics and Space Administration: Curiosity in 2012), among others [1,2]. This extensive range of applications has therefore required flexibilizing the work space of the robots, a characteristic that can be achieved by increasing their degrees of freedom, i.e., providing them with redundancy. However, all these activities would not be possible without an adequate design of the robot and of its technical control. Fulfilling this requires the knowledge and study of a mathematical model and of a certain class of "intelligence" that can direct the manipulator to perform the assigned tasks.

* Corresponding author at: Departamento de Ingeniería, Eléctrica Universidad de Santiago de Chile, Av. Ecuador, 3519, Estación Central Santiago, Chile. E-mail address: claudio.urrea@usach.cl (C. Urrea).

http://dx.doi.org/10.1016/j.csmssp.2015.12.001

2351-9886/© 2015 The Authors. Published by Elsevier Ltd. All rights reserved.

Fig. 1. Scheme of a robotized manipulator with rotational and prismatic redundance.

Using the basic laws of physics that govern the robot's dynamics, it is possible to derive a mathematical model that represents its behavior, and through appropriate programming tools, develop an environmental simulation to subject it to different tests such as, for example, following trajectories [3-7]. Because the simulation tools are the foundation for the design of robot systems, for the application of robots in complex environments and for the development of new control strategies and algorithms, this paper takes up the modeling and control of a redundant robot with five DOF that is tested by making it follow a test trajectory composed of a spiral in Cartesian space. Six controllers are made to test the model: hyperbolic sine-cosine; computed torque; sliding hyperbolic mode; control with learning; and adaptive. A simulator is developed by means of MatLab/Simulink software on which the redundant robot model is executed together with each controller. This analysis also includes the dynamics of the actuators. The results are shown by means of comparative curves and RMS indices of the joint and Cartesian errors.

2. Redundant robots

Redundant robots are those that have more degrees of freedom than those required to perform a given task [8-11]. In recent years special attention has been given to the study of redundant manipulators, and this redundancy has been considered as an important characteristic in the performance of tasks that require dexterity comparable to that of the human arm, such as, for example, in the space mission called Mars Science Laboratory (MSL), better known as Curiosity. Although most redundant manipulators do not have a sufficient number of degrees of freedom to carry out their main tasks, e.g., following the position and/or the orientation, it is known that its restricted manipulability results in a reduction of the work space1 due to the mechanical limitations of the joints and to the presence of obstacles in that space. This has led researchers to study the performance of the manipulators when more degrees of freedom are added (kinematic redundancy), allowing them to fulfill additional tasks defined by the user. Those tasks can be represented as kinematic functions, including not only the functions of kinematics that reflect some desirable properties of the manipulator's performance such as the characteristics of the joints and the evasion of obstacles, but can also be expanded to include measurements of the dynamic

1 Region of space where the manipulator can position its terminal effector (end of its wrist), that is determined by the robot's geometric configuration.

performance through the definition of functions in the robot's dynamic model, e.g., impact strength, control of inertia, etc. [12-15]. The robotized manipulator studied incorporates two additional degrees of freedom, giving it redundancy in its rotational motion, in its motion on the x-y plane, as well as in its prismatic motion along the z axis, as shown in Fig. 1.

3. Redundant manipulator with 5 DOF

Fig. 2 is a schematic diagram of the SCARA-type redundant manipulator showing its redundancy in its rotational motion, its motion on the x-y plane, as well as its prismatic motion along the z axis, as well as the distribution of the coordinate axes systems and the location of the centroids.

where q1,q2, q3, q4, q5 and l1, l2, l3, l4, l5, represent the generalized coordinates and the lengths of the links: first, second, third, fourth and fifth, respectively, and lc2, lc3 and lc4 are the lengths from the origins to the centroids of the corresponding second, third and fourth links. Now we make the corresponding calculations for the design of a kinematic model of the manipulator.

3.1. Kinematics

To get the kinematic model the standard method of Denavit-Hartenberg has been considered, whose parameters are indicated in Table 1. Then, using the homogeneous transformations given we get the direct kinematic model indicated by matrix Eq. (1):

c234 s234 0 hC2 t- I3C23 f /4c234

s234 -c234 0 /2s2 f /3S23 + /4s234

0 0 -1 /1 d1 - /5 - d5

0 0 0 1

where s2 = sin02, s23 = sin(02 + 03), s234 = sin(02 + 03 + 04), c2 = cos02, c23 = cos(02 + 03), and c234 = cos(02 + 03 + 04). Getting the inverse kinematics of a redundant robot requires looking at different methods and selecting the most adequate one according to the considerations of the model. In his case, the three rotary degrees of freedom that govern the motion of the robot on the x-y plane, we would get multiple solutions. That is why, for simplicity, the condition 04 = 03 is set. In accordance with this, and after the adequate simplifications, we get the inverse kinematic model expressed by Eqs. (2) and (3), where z and d1 are known.

03 = p ± arccosfal2 + /4)/4/2/4 ± ((4/2/4 - /32)(2/2/4 - I22 - k2) + 4/2/4(x2 + y2))V4/2/^ (2)

Fig. 2. Scheme of a redundant manipulator of the SCARA type.

Table 1

Assignment of Denavit-Hartenberg parameters. Joint i 9j dj at

1 0° li+ di 0 0°

2 02 0 I2 0°

3 03 0 I3 0°

4 04 0 I4 180°

5 0° l5 + d5 0 0°

02 = -2arctan( (x ± (x2 + y2 - s2 (4^/4C3 + /3) + /(y + /3S3 + ^2.3) andd5 = /1 + di - /5 - z (3)

3.2. Dynamics

Keeping in mind the characteristics of the manipulator presented so far, we get its dynamic model. For that purpose it is possible to make approximations through second order systems [16] or to develop a complete model, as achieved in [17]. In this work the Lagrange-Euler formulation that is based on the principle of the conservation of energy [18-20] is used; for which it is necessary to determine the kinetic and potential energy of the manipulator, the Lagrangian2 and then substitute in the Lagrange-Euler equation [21-25]. We get the dynamic model of the redundant robotized manipulator, which can be expressed by means of Eqs. (4) through (10), M is the inertia matrix (with dimension n x n), C is the centrifugal and Coriolis forces vector (with dimension n x 1), G is the gravitational force vector (with dimension n x 1), and F ([F11 F12 F13 F14 F15]T) is the friction forces vector (with dimension n x 1).

; C =[0 C12 C13 C14 0]T and G = [(m1 + m2 + m3 + m4 + m5)gz 000 -m5gz ]T

M11 = m1 + m2 + m3 + m4 + m5,M15 = M51 = -M55 = -m5 and M12 = M21 = M13 = M31 = M14 = M41 = M25

= M52 = M35 = M53 = M45 = M54 = 0 (5)

M22 = l2c2m2 + (l2 + l2c3 + 2/2/C3C^ m3 + Çl^ + l2 + + 2/2/3C3 + 2/3/c4C4 + /2/C4 + C34)) m4

+ ... (l2 + l2 + l\ + 2/2/3C3 + 2(I3I4C4 + l2l4C34))m5 + /2zz + hzz + I4zzandM44 = l24m4 + l2m5 =

M23 = M32 = (l23 + b^) m3 + (l3 + l24 + ld3C3 + 2/3/c4C4 + /2/C4 + C34) m4

+/3zz + I4zz... (l2l4C34 + l3 + l4 + ld3C3 + ^Uc^)m5 + andM34 = M43 = (l^ + /3/c4C^m4 + (l2 + l3l4C4)m5 + /4zz

M24 = M42 = (4 + /3/c4C4 + /2/C4C3^ m4 + (J2 + l3l4C4 + l214C34) m5 + ^4zz and

M33 = l23m3 + (l\ + l24 + 2/3/C4C^ m4 + (l2 + l^ + d3l4C^ m5 + hzz + I4zz

C12 = - ^3-03 + 2l2s3 "0203 J (lc3m3 + l3m4 + ^5) + ^2 s34-^3 - 2 l3s4 + l2s34 ) 03004 ) ( lC4m4 + l4m5)

+ ... (l2s34 -02003 - (042 + l2s34 + ^4)) (Ic4m4 + ^5)

2 Scalar function that is defined as the difference between the kinetic energy and the potential energy of a mechanical system.

Fig. 3. Schematic of a servo motor coupled with a robotized manipulator as the load.

C31 = (lc3m3 + hm4 + ¡3m5)/253 • 6»2 + ^34 • 6»2 - (2(62 + 63)+64) ^¿^ (lc4m4 + 14ms) and

C41 = (lc4m4 + kmsX /3s4 + ^34) 6 2 + + 21384 •6263^ (lc4m4 + l4m5) (10)

where s3 = sin 63, s4 = sin 64, c3 = cos 63 c4 = cos 64, s34 = sin(63 + 64), c34 = cos(63 + 64); m1, m2, m3, m4 and m5 represent the masses of the first, second, third, fourth and fifth links, respectively; l2zz, l3zz and l4zz indicate the moments of inertia of the second, third and fourth links with respect to the first z axis of its joint, respectively.

4. Actuator dynamics

The actuators considered in this study correspond to analogic servo motors. Fig. 3 shows a schematic of a servo motor coupled with a robotized manipulator as the load [26]. These systems are constituted by a dc motor, a set of gears to reduce the rotational speed and increase the torque on its drive shaft, a potentiometer connected to that output shaft, which is used to know the position, and a feedback control circuit that converts an input signal of the PWM (Pulse-Width Modulation) type to voltage, comparing it with the fed back position and then amplifying it and activating an H bridge to cause a turn at a given speed [5]. Fig. 4 shows a block diagram of an analogic servo motor connected with a load consisting of a robot.

The dynamic model of the servo motors considered has been developed by the authors in Ref. [26] and is given by Eqs. (11) and (12):

tL = ((ka/Ra)AkskpVi -Jmq/n)/n - ((kakb/Ra + Bm)q/n + ka/RaAkspq + /eC(q/n))=n (11)

fecit)=Fec1tanh(kq)(1 + sgn(ij))/2 + Fec2tanh(kij)(1 - sgn(ij))/2 (12)

Fig. 4. Block diagram of a servo motor coupled with a robotized manipulator as the load.

where n represents the gear ratio (n1/n2), ka is the motor's torque constant, Ra is the armature resistance, A is the current amplifier gain (H bridge), ks is the sensitivity of the comparator, kp is the total gain of the PWM conversion (kp1 x kp2), vi is the input voltage to the servo motor, Jm is the moment of inertia of the motor, kb is the inverse electromotive force constant, Bm is the viscous friction of the motor, p is the gain of the position potentiometer, and k is the gain of the slope of the tan h function used to increase or reduce the slope of the curve as it crosses zero.

5. Controllers

Below we present a summary of the controllers considered for the evaluation of the robot model together with its actuators and their corresponding performance.

5.1. Hyperbolic sine-cosine controller

This controller, which was presented in Ref. [27], is composed of a proportional part based on hyperbolic sine and cosine functions, a derivative part based on a hyperbolic sine, and a gravity compensation, as shown in Eq. (13):

t = Kpsinh(e)cosh(e) - Kvsinh(q) + G(q) (13)

where Kp = diag(Kp1, Kp2.....Kpn) and Kv = diag(Kv1, Kv2.....Kvn) are proportional and derivative, diagonal and positive definite

(with dimension n x n) gain matrices, respectively.

5.2. Sliding mode control

Sliding mode control (SMC) systems correspond to a particular type of variable structure control (VSC) systems that have the characteristic of changing structure, by means of some law, in order to satisfy desired characteristics [28]. The SMC consists in defining a control law that, commuting at high frequency, succeeds in taking the state of a system to a surface called a sliding surface, and once there, keep it in the face of possible external perturbations [30]. The control law corresponds to:

t = -K • sgn(s) (14)

where K =diag(K1, K2.....Kn) is a definite positive diagonal matrix (with dimension n x n). The sliding surface is given by:

s = W•(q - qd) + (q - qd) (15)

where W = diag(W1, W2.....Wn) corresponds a definite positive diagonal matrix (with dimension n x n)

5.3. Computed torque control

The following algorithm uses the computed torque control, which consists in applying a torque with the purpose of compensating for the centrifugal and Coriolis; gravitational; and frictional effects, as shown in Eq. (16) [36]:

t = M(q) (qd + Kve + Kpe) + C (q, q) + G(q) + F(q) (16)

where M expresses the estimation of the inertia matrix (with dimension n x n), C is the estimation of the centrifugal and Coriolis forces vector (with dimension n x 1), G is the estimation of the gravitational force vector (with dimension n x 1), F is the estimation of the frictional force vector (with dimension n x 1), qd is the desired acceleration vector of the joints (with dimension n x 1) and e represents the velocity vector.

5.4. Sliding hyperbolic mode control

One of the advantages of the sliding mode control is its invariance when facing parametric uncertainties and external perturbations. However, the high commutation frequencies that characterize this control cannot be implemented [28], and it also incorporates the "chattering" vibration phenomenon in the actuators, which must be avoided in many physical systems such as servo control systems, structure vibration control systems, and robotized systems [29-33]. For that reason a modification of the classical SMC is introduced through the hyperbolic tangent function, with the purpose of reducing its characteristic abrupt commutation, as indicated in Eq. (17):

t = -K • tanh(a • s) (17)

where a = diag(a1, a2.....an) corresponds to a definite positive diagonal matrix (with dimension n x n).

Table 2

Parameters considered in the manipulator.

Parameter Link 1 Link 2 Link 3 Link 4 Link 5 Units

l 0.524 0.2 0.2 0.2 0.14 [m]

lc - 0.0229 0.0229 0.0229 - [m]

m 1.228 1.023 1.023 1.023 0.5114 [kg]

Izz - 0.0058 0.0058 0.0058 - [kgm2]

Fv 0.03 0.025 0.025 0.025 0.02 [Nms/rad]

F 1 eca 0.05 0.05 0.05 0.05 0.03 [Nm]

Fecb -0.05 -0.05 -0.05 -0.05 -0.03 [Nm]

Table 3

Parameters considered in the servo motors.

Parameter Servo 1 Servos 2-3-4 Servo 5 Units

Ra 1.6 1.6 1.6 [V]

La 0.0048 0.0048 0.0048 [H]

Jm 0.007 0.007 0.007 [kgm2]

Bm 0.01413 0.01313 0.01208 [Nms/rad]

ka 0.35 0.35 0.35 [Nm/A]

kb 0.04 0.04 0.04 [vs/rad]

F eca 0.05 0.05 0.03 [Nm]

Fecb -0.05 -0.05 -0.03 [Nm]

n 1/600 1/561.6 1/561.6 [Times]

A 15 15 15 [Times]

ks 10 10 10 [Times]

kp 1 1 1 [Times]

P 1 1 1 [Times]

Table 4

Gains considered in the controllers (part 1).

Controller Constants

Hyperbolic sliding mode K.....K5 W1.....W5 01.....05

1.2, 1.8, 1.8, 8, 10, 10, 200, 200,180, 160, 300

1.8, 1.56 10,10

Learning Kp1.....Kp5 Kv1.....Kv5 M1.....M5

200, 120, 160, 120, 60 60, 80, 80, 1, 1, 1,

80, 40 1, 1

Adaptive inertia Kv1.....Kv5 11.....15 g1.....g 5

20, 26, 24, 10, 20, 18, 0.4,1.2,1,

22, 20 18, 8 1, 0.6

Table 5

Gains considered in the controllers (part 2).

Controller Constants

Sinh-Cosh Kp1.....Kp5 Kv1.....Kv5

400, 300, 200,100, 100 5, 4, 3, 2, 2

Sliding mode K1.....K5 W1.....W5

0.74, 1.45, 1.4, 1.35,1.54 10, 10, 10, 10, 10

Computed Kp1.....Kp5 Kv1.....Kv5

torque 400, 600, 700, 800,100 120, 100, 60, 50, 40

Joint error (sinh-cosh)

012345678

Time (s)

Fig. 5. Joint trajectory error using the hyperbolic sine-cosine controller.

5.5. Contro/ with Learning

Control with learning is based on the correction of the control system through successive repetitions of the operations in order to compensate for the model's uncertainties. In this way, a first control torque is generated, estimating that one part of the model is known, and a second control torque is generated from a model that is adjusted by means of a learning law, in successive repetitions of the same operation. A control scheme that considers Proportional and Derivative (PD) terms, and terms dependent on the known model, are shown in Eq. (18) [34]:

t = M (q)(qd + Kve + Kpe) + C (q, q) + G (q) + yk (18)

Joint error (sm)

•31-1-1-'-1-1-1-1-1

012345678

Time (s)

Fig. 6. Joint trajectory error using the sliding mode controller.

Joint error (sm-tanh)

•3'-1-1-1-1-'-1-1-1

012345678

Time (s)

Fig. 7. Joint trajectory error using the sliding mode hyperbolic controller. where yk expresses the torque obtained by successive learning repetitions of the motion (k = 1, 2, . . . ). 5.6. Adaptive control

Adaptive control has the purpose of getting the correct performance of the robotized system in spite of the many uncertainties related to different aspects of the manipulator, e.g., the flexibility of the links and joints, external perturbations, the dynamics of the actuators, friction at the joints, the noise of sensors, and in other not modeled dynamic behaviors. In that control the parameters are variables that are estimated online and are adjusted through a mechanism based on the system's measurements [30]. The adaptive control considered is based on a law of control presented in Refs. [18,35,36], for which it is necessary to define an auxiliary error signal r = Ae + e and its derivative r = Ae + e with respect to time, where A = diag(11,

1......1n) corresponds to a definite positive diagonal matrix (with dimension n x n). When r and r are combined properly, we

t = Y(-)w - M(q)r - Vm(q, q)r (19)

Joint error (tor-calc)

012345678

Time (s)

Fig. 8. Error of the joint trajectory using the computed torque controller.

Joint error (learning)

- r _____

"*2 e3 e, 5

w 0 -2 -4 -6

012345678

Time (s)

Fig. 9. Joint trajectory error using the controller with learning.

where Y(-) represents a regression matrix (with dimension n x n), f is a parameter vector (with dimension n x 1) and Vm is the centrifugal and Coriolis forces matrix.

6. Simulation environment

The six control laws mentioned above, together with the dynamic model of the SCARA type redundant manipulator and the actuator dynamics, are executed in a simulation structure carried out using MatLab/Simulink programming tools. The values of the parameters considered in the manipulator are shown in Table 2. Table 3 shows the set of values of the parameters used for each actuator. Tables 4 and 5 show the set of values of the gains used for each type of controller.

7. Results

After developing the manipulator model and the simulation environment, incorporating the actuator dynamics, and establishing the control laws to be used, a test trajectory in space was determined to make the manipulator model follow it,

2 3 4 5 Time (s)

Fig. 10. Error of the joint trajectory using the adaptive controller.

Controller type

Fig. 11. Performance index corresponding to the joint's trajectory.

and then study the results as a function of the performance of each controller. The test trajectory in Cartesian space corresponds to a spiral.

Figs. 5 and 6 show the graphs of the joint trajectory (e1.....e5) errors using the sinh-cosh controller and the sliding

mode controller, respectively. Figs. 7 and 8 show the curves of the errors obtained from the desired and real joint trajectories using the hyperbolic sliding mode controller and the computed torque controller, respectively.

Figs. 9 and 10 show the curves of the errors obtained from the desired and real joint trajectories using the controller with learning and the adaptive controller, respectively.

tor-calc

Controller type

Fig. 12. Performance index corresponding to the Cartesian trajectory.

Fig. 11 and Fig. 12 show the Cartesian and joint RMS errors, respectively, according to Eq. (20), where ei represents the joint as well as the Cartesian errors of the trajectory, and n is the number of data.

8. Conclusion

The use of a robotics simulator for development of a robotics control program is highly recommended regardless of whether an actual robot is available or not. Because of this, a kinematic and dynamic model of a redundant robot with five degrees of freedom using the methods of Denavit-Hartenberg and Lagrange-Euler, respectively, was developed. Six controllers were made hyperbolic sine-cosine; computed torque; sliding hyperbolic mode; control with learning; and adaptive. A simulator was made using the MatLab/Simulink software. The tests of the manipulator model were presented, including the dynamics of the actuators and together with each controller, by following a test trajectory composed of a spiral in the Cartesian space.

The results, obtained through a simulation environment, were represented by comparative curves and RMS indices of the joint and Cartesian errors, and they showed that the redundant manipulator model followed the test trajectory with less pronounced maximum errors using the adaptive controller than the other controllers, with more homogeneous motions of the manipulator.

It was seen that the largest joint and Cartesian errors generated when testing the robot model, both in terms of maximum and RMS values, occurred when the computed torque controller was used. It should be mentioned that the results with that controller were obtained by executing three iterations for learning, because with more iterations the variations were not important. Therefore the best performance results of the robotized manipulator model were achieved using the adaptive controller, as shown in Fig. H and Fig. 12.

It is important to point out that the hyperbolic sliding mode controller presents a lower simulation complexity due to the simplicity of its control law and because it does not require the second derivative of the joint position, and this situation can be determining if high performance processors are not available.

9. Future work

From the performance achieved through the simulation tests, the redundant robot model together with its actuators, and the different control laws discussed, a new stage begins in the study and analysis of redundant robotized manipulators consisting in the practical implementation of real industrial type robots, their actuators and their controllers by means of the development of the necessary hardware.

Acknowledgements

This work was supported by Proyectos Basales y Vicerrectoría de Investigación, Desarrollo e Innovación of the Universidad de Santiago de Chile, Chile.

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