Nonlinear Control Techniques for a Dual RR Robot MEEN 655: Design of Nonlinear Control Systems

Taimoor Daud Khan

Mechanical Engineering Department

Texas A&m University

Abstract—This paper details the different nonlinear control

methods for a parallel RR robot. The system is a four degrees

of freedom (DOF) redundant robot that is used to study the

fine manipulations of a finger. Such mechanisms make

popular grasping end effectors for industrial robots. The

research effort focuses on developing nonlinear controllers for

trajectory tracking problems. To this end the information

found in robotics and control literature is used to study the

performance of the system with different nonlinear

controllers. The paper concludes with a comment on the

performance and optimization of control using the results

obtained in our study.

Keywords—Redundant Robot, Parallel RR Robot, Nonlinear

Controllers

I. INTRODUCTION

The fine manipulations of human fingers are of particular interest to engineers who work with rehabilitation robots and grasping mechanisms. Robots performing similar motions tend to be redundant especially since the human fingers in themselves are redundant manipulators [1]. Each finger can be viewed as a separate manipulator and as such robotic devices capturing such fine manipulations tend to be parallel robots [2].

A simple mechanism that mimics the motion of human fingers is a dual RR robot as shown in Fig.1. The dual RR robot is a simplified model of two human fingers and have only two revolute joints for each finger instead of three. From the literature on biomechanics one can reasonably ascertain that the dual RR robot captures the abduction/adduction motion of the human thumb, and the flexion/extension motion of the human index finger. The kinematic redundancy coupled with multiple end effectors for such robots add to the challenge of controlling them. The nonlinear nature of the equations of motion of such robotic devices motivates this study to explore nonlinear control techniques to optimize the tracking performance.

There have been many other works to control such redundant parallel robots. One method utilizes the inertial properties of the links along with an augmented jacobian model [3]. Another method reduces a parallel robot to a five bar linkage mechanism [4]. This imposes holonomic constraints on the system. An adaptive nonlinear tracking control algorithm for a kinematically redundant algorithm was proposed by E. Tatliciaglo [5]. A Lyapunov type control for robots was proposed by Zergeroglu [7]. This controller was designed to utilize self-motions for sub-tracking problems. Earlier, nonlinear feedback linearization control for robots was proposed by Gilbert [7]. Another popular nonlinear control technique is the sliding mode robust controller for robotic manipulators [8]. Gravitational compensation along with classical PD control is also a popular control algorithm for some robots [9].

This paper evaluates the performance of a few latter

(a) (b)

Fig.1 Human thumb and index finger with the

corresponding RR robotic fingers

nonlinear control techniques on a dual RR robot. The

problem is reduced to that of tracking a circular trajectory in

the task space. The kinematics section makes use of a five

bar linkage mechanism for constraint matrix mapping. This

is followed by the dynamic formulation of the system which

is done by decoupling the system into two RR robots. The

next section formulates the nonlinear control laws. This is

followed by the simulation and the results section. The

paper then concludes with some comments on the

simulation results and with a note on further optimizing the

performance of such a system.

II. KINEMATICS

The kinematics of the dual RR robot are realized by simplifying the model to that of a fiver bar closed linkage mechanism. The robot is mathematically modelled when it is grasping a rigid object, the schematic of the system is shown in Fig.2.

To get the holonomic constraints of the system we write the closed loop equations of the linkage mechanism as follows

Where

A constraint is said to be holonomic if it restricts the

motion of the system to a smooth hypersurface in the

unconstrained configuration space Q. The corresponding

velocity constraints is given by Pfaffian constraint as shown

below

Fig.2 The Dual RR robot reprsentation as a 5 bar linkage

Where

The dual RR robot system under consideration has only

two actuated joints. The constraint mapping of dependent

and independent coordinates is then given by the following

relation

Using the above relation we derive the following

relations between the complete joint velocity vector and the

independent velocity vector

III. DYNAMICS

The dynamics of the system were realized by decoupling the system into two RR robots and formulating the equations of motion for each of them. For this purpose the Denavit Hartenberg convention was used and the DH parameters were used with the mathematical package Robotica in Mathematica to get the equations of motion. The DH parameters for a planar RR manipulator are shown in Table1.

Table 1 DH Parameters

n d a

1 0 0

2 0 0

The Euler Lagrange equations of motion for a RR robot is given by

Where τ is a 2x1 joint torque vector, M is the 2x2 mass matrix, C is the 2x2 Christoffel symbol matrix, G is the 2x1 gravity matrix and q is the joint angle.

With the following arbitrary choice of robot parameters

,

,

we get the following matrices (the parameters are not comparable to human fingers and were chosen only for simulation purposes)

The equations of the motion for the complete system are formed by considering the constraint forces due to the holonomic constraints. We can write the constraint force as

Where h represents the holonomic constraints and γ is the

vector of relative magnitudes of constraint forces. The scalar

elements of γ are called the Lagrangian multipliers. These

constraint forces can be viewed as acting normally to the

constraint surface. Using d’Alemberts’s principle which

states the forces that are generated due to constraint do not

work on the system we write our final equation of motion

for the complete system as

Differentiating eq.3 with respect to time and equating

the joint acceleration by using eq.7 we get the following

formula to compute the Lagrangian multipliers

IV. CONTROLLERS

This section formulates four different control laws for the dual RR robot. We define the error terms as follows

A. PD Control Law

This is the classical trajectory tracking PD control law and is mathematically formulated as

where and are the proportional and derivative gains

respectively. These gains are tuned to achieve some desired

performance. One popular method of tuning the gains is the

use of Ziegler-Nichols method. The control is dependent on

joint position and velocities and a control applied at one

joint is independent of the other. The gains therefore are

usually diagonal matrices. The equation of motion then

becomes

The stability of the solution when using PD control law can be proven by taking a Lyapunov function candidate as

where U is the potential energy of the system. By doing

some manipulation on the system it can be shown that the

derivative of this Lyapunov candidate function is

which shows that the system is stable for some appropriate

value of the proportional gain controller.

B. PD Control Law with Gravity Compensation

This controller is similar to the PD control law but it has an additional Gravitational term in the control law. The control law can be rewritten as

This control law is used to improve the performance of the system by cancelling the effect of gravity. If the G matrix in eq. 13 is the same as in eq.7 the equation of motion reduces down to

Using an appropriate Lyapunov function candidate like the one shown below we can show that its derivative is less than or equal to zero.

This only guarantees the stability of the equilibrium

point so we invoke the LaSalle’s theorem and it can be

easily deduced that the equilibrium is globally

asymptotically stable.

C. Feedback Linearization Control Law

This nonlinear control law is also known as the

computed torque controller. This control law makes use of

output to input linearization with respect to state variables

and attempts to come up with a control law such that

nonlinear terms in the state equations are cancelled. The

nonlinear feedback control law is given as

Where u is the new control input and, and are

estimates of the corresponding mass, Christoffel and

gravitational matrices. If one has perfect knowledge of the

system then the corresponding term on both sides cancel out

and the control law reduces to

Since in reality this is not the case, we always have some

uncertainty in the system. Formulating trajectory tracking

dynamics in the task space and using jacobian relations as

follows we get a final control law

The computed torque controller can also be used to

formulate a decoupled force tracking problem that is to

regulate the Lagrange multipliers. An integral controller can

be used for this purpose as shown below

This formulation was not used in the simulations but was

introduced just as a concept for decoupled control laws in

task space. In simulations we assume a perfect knowledge of

the Lagrangian multipliers and hence the terms cancel out

on both sides.

The closed loop equation for this control law can be

written as

The linear autonomous system equation shown above

guarantees the stability of the system if and are

designed to be positive definite. This implies that as t tends

to infinity the error converges to zero.

D. Sliding Mode control and Computed Torque Control

The uncertainties due to feedback linearization motivates

us to find a new robust controller and hence we use the

sliding mode controller in conjunction with the computed

torque control. The modifies control law is given as

where , is the sliding manifold,

P is a positive definite matrix and is a positive sliding

gain matrix. For the system if this combine control law is

applied then equilibrium is asymptotically stable. This can

be shown by taking a Lyapunov function and performing the

following analysis

where are small positive values. The above analysis shows

that . From this we conclude that

.

V. SIMULATIONS AND RESULTS

The aforementioned control laws were implemented in

MATLAB/SIMULINK to track a circular trajectory. Fig.3

shows the 5 bar linkage mechanism being simulated and the

desired trajectory in Cartesian space. Fig 4 shows the

Simulink block diagram that was used for the

implementation of the control algorithms. The force control

block feeds out the exact Lagrangian multipliers. It was

formulated to do force control but was not implemented

with the integral control as shown earlier due to the time

constraint.

Fig.3. Dual RR robot tracking a circular trajectory

Fig.4. Simulink Implementation of feedback linearization

control

The PD control results are shown in Fig.5. Trajectory

tracking for each of the four joint angles is shown and the

corresponding error is also displayed. This show a

suboptimal performance. The gain values used for this were

.

(a)

(b)

(c)

(d)

Fig.5. Trajectory Tracking and Error results for joint

variables with PD control

The PD control with graviataional compensation results

are shown in Fig.6. Trajectory tracking for each of the four

joint angles is shown and the corresponding error is also

displayed. The gains used with this controller were

.

We see an improvement in tracking and the errors are lower

when compared to the results for the PD controller.

The results for feedback linearization are shown in Fig.7.

The performance exceeds to that of PD control with gravity

compensation. This is because of the inclusion of mass

matrices and Christoffel symbols in the control law. The

gains for this controller were chosen as

.

The results for the sliding mode control are shown in

Fig.8. This is indicative of an optimal performance as the

error is very low for all joint tracking problem. The

performance by the sliding mode control is shown to be

better than all three previous controller. The values of the

proportional and derivative gains were kept the same at that

in feedback linearization control. The sliding mode gain

matrix and associated parameter values used were as follows

(a)

(b)

(c)

(d)

Fig.6. Trajectory Tracking and Error results for joint

variables with Gravity Compensation PD Control

(a)

(b)

(c)

(d)

Fig.7. Trajectory Tracking and Error results for joint

variables with Feedback Linearization Control

(a) error is of magnitude 10^-3

(b) error is of magnitude 10^-4

(c) error is of magnitude 10^-3

(d)

Fig.8. Trajectory Tracking and Error results for joint

variables with Sliding Control

VI. CONCLUSION

Based on the results we conclude that the performance

of sliding mode controller with feedback linearization

exceed that of the other controllers. Although it is seen in

the results that error does not always converge to zero for

joint tracking problem, it is very small. The reason that the

error did not converge to zero might be associated with the

suboptimal gains. It is recommended that a known gain

tuning approach be implemented instead of arbitrarily

tuning them. Future works should also take into account the

uncertainty in the Lagrangian parameters and implement the

integral control as formulated in feedback linearization

control section.

ACKNOWLEDGMENT

The author would like acknowledge the support of Dr. Prabhakar Pagilla at the Mechanical Engineering department at Texas A&M University. Dr. Pagilla’s contribution as a mentor and a guide for this work are truly appreciated.

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