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Nonlinear Control Techniques for a Dual RR...
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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.
REFERENCE
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