amir robust report

UNIVERSITY OF BRITISH COLUMBIA

MECHANICAL ENGINEERING DEPARTMENT

SIMON FRASER UNIVERSITY

THE SCHOOL OF ENGINEERING SCIENCE

Multivariable Feedback Control Robust Controller Design for a

Robotic Hand Exoskeleton

Supervised by: Dr. Ryozo Nagamune

Dr. Carlo Menon

Amirreza Ziai M. A. Sc. Candidate

Simon Fraser University The School of Engineering Science

[email protected]

December 2009

of 20

Introduction

Intelligent and sophisticated robots are emerging in every aspect of our lives. In the past decade,

researchers have been looking for ways to incorporate robots with humans. Majority of these

efforts has had the goal of benefiting human motion or medical rehabilitation in mind. Wearable

robots, whether designed for rehabilitation, power augmentation or replicating the human

motion, are designed based on the same concepts.

More advanced robotic systems that are capable of assisting patients with their daily living tasks

(such as stroke or nerve disorder patients who are dependent on other) are sought after. Robots

that can intelligently detect the intention of patient and assist them with movements, making

them independent of others.

A robotic exoskeleton capable of providing torque to a patient's hand has been designed and

developed in Menrva lab (The school of engineering science, Simon Fraser University). Since

the device is intended to be used with human being, it is very crucial to control it in a precise and

safe manner, avoiding further injury to the patient.

Dynamics of the robotic exoskeleton and the patient's hand are coupled. Human involvement in

this system introduces a set of uncertainties. Every patient has a different hand size and weight

that in turn changes the inertia of the system as a whole. In order to mitigate the effects of the

uncertainties introduced by the patient and to have a general controller, designing a robust

controller was considered.

Modeling

Human wrist possesses three degrees of freedom as depicted in figure 1:

Flexion and extension about the wrist joint

Radial and Ulnar deviation

Pronation and supination

of 20

Figure 1- Human wrist degrees of freedom

http://www.revolutionarytennis.com/step12-5contact.html

The exoskeleton developed in the Menrva lab is capable of providing motion to the

flexion/extension and ulnar/radial deviation degrees of freedom. A linear actuator is used to

control each degree of freedom. The linear movement of the actuator is translated through a

linkage to the back of the hand, which in turn exerts a torque about the wrist joint as illustrated in

figure 2.

Figure 2- Force provided by the actuator creates torque about the wrist joint [1]

of 20

Sub-systems for both degree of freedom are identical and actuators operate independently of

each other. In both cases the actuator, linkages and the hand construct a four-bars mechanism

with identical actuator stroke length and linkage lengths.

Figure 3- CAD drawing of the exoskeleton with the hand [1]

In order to determine the operational range of the exoskeleton the kinematics of the system are

analyzed.

of 20

Figure 4- Kinematic model for both sub-systems [1]

L- Actuator stroke length

A- Actuator distance to the wrist joint

B- Distance of the center of the hand to the wrist joint

C- Distance of the center of the hand to the upper link connection point

D- Length of the upper link

E- Distance of the wrist joint to the upper link connection point

F- Distance of the connection point of the actuator to the wrist joint

H- Distance from the actuator to the wrist joint

of 20

The actuator selected for the system has a maximum stroke length of 10cm. Since each sub-

system has only a single degree of freedom, given a certain extension of the actuator the output

angle can be calculated using the trigonometric relationships previously derived. The actuator

stroke length was swept between 0cm and 10cm to determine the operational range in figure 5.

Figure 5- Exoskeleton operational range

The operational range of the exoskeleton is for both sub-systems.

Computing the value of stroke length given the output angular displacement involved solving a

highly-nonlinear trigonometric equation. In order to avoid the difficulties introduced, a cubic fit

to the above graph is used:

The torque exerted around the wrist joint by the actuator ( ) depends on the configuration of

the system and the input actuator force ( ).

of 20

Calculation of the value of can also introduce difficulties in modeling. Since this value is not

drastically changing, it can be considered as an uncertainty in the system. Figure 6 shows how

the value of " " changes with changing the wrist angle.

Figure 6- Values of "X" for the operational range of the exoskeleton

(20% uncertainty)

Dynamics of the horizontal sub-system where gravity does not affect the output are governed by

the following differential equation:

Where is the moment of inertia of the moving part parts of the exoskeleton coupled with the

hand, is the damping coefficient of the system and is the torsion spring. In reality evaluating

these values, especially the moment of inertia is a very difficult task. A few trial-and-error

iterations were carried out with the actual device to try and mimic a system with similar output in

Simulink.

of 20

Dynamics of the actuator were also taken into account. The actuator is unable to follow the input

current instantly and acts as a first-order lag system. The time constant of the actuator system

was also measured experimentally. The parameter " " equals " ".

The table below list all the uncertain parameters:

Parameter Nominal Value Variance Uncertainty A 20%

B 6%

J 6%

K 7%

T 3%

The transfer function of the system can be rearranged in the following way:

of 20

Figure 7- Block diagram of the system

System uncertainties can be represented using the LFT form:

I(s)

(s)

I(s) F(s) (s)

Actuator Subsystem

Exoskeleton + Hand Subsystem

of 20

Figure 8- Block diagram of the system with the LFT blocks

I(s)

(s)

of 20

Step response and Bode diagram of the uncertain system are presented in figure 9.

Figure 9- Step response and Bode diagram of the uncertain system

of 20

Controller Design

It is desirable to design a controller that satisfies the following condition:

Robust stability

Robust performance

Low-order

Figure 10- Closed loop system with the controller [2]

The weighting functions are chosen to reject disturbances, track reference signal and constraint

control-effort over different frequency ranges. ensures good disturbance rejection and

reference tracking while constraints the control input. Low frequency unit disturbance (up to

1rad/s) is rejected to a order of or less.

and controllers were designed with Matlab. The following table summarizes the properties

of each of the two controllers:

Upper bound 13.9973 Lower bound 4.9285 4.9273

Destabilizing frequency 0.1066 Model uncertainty tolerance 493% 493%

Controller order 5 23 (full) 4 (reduced) Sensitivity to A 100% 103% Sensitivity to B 0% 0% Sensitivity to J 6% 20%

of 20

Sensitivity to K 5% 26% Sensitivity to T 1% 1%

Both controllers are robustly stable. The full controller was of order 23. Figure 11 depicts the

frequency characteristics of the full and reduced controllers.

Figure 11- Frequency characteristics of the full and reduced order controllers

The controller does not satisfy robust performance as shown in figure 12.

of 20

Figure 12 - Nominal and robust performance for the controller

Parameters resulting in the largest peak in the sensitivity function are computed as follows:

Figure 13 illustrates the closed loop response to a step reference signal for both controllers.

Figure 13 - Closed loop step response

of 20

Reference tracking for both controllers, given the dynamics of the actuator, is satisfactory. The

designed µ controller is faster and introduces bigger overshoots compared to the H∞ controller.

Figure 13 shows rejection of a disturbance input. Effects of the disturbance on the system output

is diminished and wiped out in two seconds.

Controller effort is shown in figure 14. Each linear actuator is capable of delivering at most 50N

of push/pull force. The controller effort for tracking a 15° reference signal is less than 10N that is

within the feasible range.

It is also very important for this system to track a sinusoidal input (back and forth motion of the

hand for rehabilitation purposes) with frequencies no more than 0.5rad/s (period of 12.6s). Figure

15 depicts how the closed loop system tracks a sinusoidal input with a frequency of 0.5rad/s with

the presence of a sinusoidal disturbance of twice the frequency. Results are very satisfactory for

both controllers.

Figure 13 - Closed loop disturbance rejection

of 20

Figure 14 - Controller effort

Figure 15 - Sinusoidal referencce signal tracking

of 20

Due to the presence of a torsion spring in the system, wrist angle is always returned to zero

degrees when there's no torque exerted on the patient's hand. For instance, if the system is

released from a deflected wrist angle of 15°, it'll take about 1 seconds for the torsion spring to

return the system to resting position which is 0°. On the other hand, dynamics of the actuator

dictate a two seconds delay to compensate for the same amount of deflection. As shown in figure

16, it'll take the controller about two seconds to cancel out the effect of spring and return the

system back to the reference signal when the initial and reference signal are both at 15°. Within

these 2 seconds the output will be reduced to 6° in less than 500ms.

Figure 16 - Closed loop response with initial condition equal the reference signal

Although this is an interesting observation, there's no necessity to try and further reduce the time

it takes for the controller to compensate for the spring action. In reality because of the presence

of the spring, there would be no situation where the initial condition is anywhere but in the

resting position (0° wrist angle).

of 20

The linear actuator used in this device is fed with 12V electrical potential and has a 1:210

gearing ratio. Figure 17 shows the relationship between the input current and output force of the

actuator in steady state. Since there's a linear relationship between the two, output of the

controller that is in Newtons can be easily converted to the input current provided to the actuator

using a simple multiplication and addition:

Figure 17 - Steady-state force relationship for the linear actuator

of 20

Conclusions & Future Work

Mechanics of a robotic hand exoskeleton were analyzed. Dynamics of the system consisting of

the linear actuator, exoskeleton and the patient's hand were modeled with uncertainties involved

in the application of the device.

Two controllers, based on the performance specifications imposed by two weighting functions

were designed. While both controllers guarantee robust stability, only the µ controller guarantees

robust performance. Closed loop time response characteristics of both controller were

satisfactory. The µ controller provides faster transient time responses with bigger overshoots

while the H∞ controller results in a more sluggish response with no overshoot in most cases.

Both controllers are of relative low order making their physical implementation feasible.

Next step to this project would be to implement the controller using a microcontroller in order to

control the device in real-time. The ultimate goal of this project would be to design a controller

that can satisfy robust stability and performance for a wide range of users with different hand

weights and dimensions.

References

[1] Zhen Gang Xiao, " CO-OP Work Report for ENSC 196", Menrva Lab, The School of

Engineering Science, Simon Fraser University, Summer 2009

[2] D.-W. Gu, P. Hr. Petkov and M. M. Konstantinov, "Robust Control Design with MATLAB",

Springer

amir robust report

Documents