; .:5 5 A fi \? 

16 6 

I 

Semiannual Progress Report on Real Time Control for 

NASA Robotic Gripper 

Carole A. Salter and'John S. Baras 

and 

Systems Research Center 

University of Maryland, College Park 

February 15, 1989 

Electrical Engineering Department Kfld,op ,# 

I 

(NASa-CR-184978) REAL TIHE COBTBOL FOR NASA N89-24608 

ROBOTIC GRIPPER Semiannual Progress Report (Haryland Univ.) 16 p CSCL 131 

Unclas 

G3/37 0209856 
1 Introduction 

Manipulating objects in space presents very different kinds of problems than it does 

on earth, in fact in many respects it is much more difficult. Accordingly, the design and 

control of space manipulators must reflect these differences. Reaction forces and torques 

normally damped by the earth, can create problems in space. For instance, an astronaut 

attempting to turn a valve on a relatively large space vehicle must be securely attached to 

the vehicle, otherwise, instead of turning the valve, the torque created will serve to turn 

the astronaut relative to the vehicle. This is by no means the only difficulty. Consider 

the following scenario: A master slave arm is being used to link beams for a large space 

structure. The sockets used to link the beams are transported on trays to the locations 

of the beam joints. Due to the zero gravity environment, the sockets must be fixed in 

place by some type of fastener to ensure they remain on the tray. Designing the end 

effector to release the fastener while maintaining a firm grasp on the socket will solve 

both the problems associated with torques and reaction forces, as well as the problem of 

minor perturbations dispersing the sockets in random directions. 

To accommodate these problems NASA developed the gripper/nut runner. The gripper/nut runner has two fingers which together have one degree of freedom. Similar to 

a vice, the width of the opening is the only variable. Centered between the fingers is 

a device for unscrewing nuts. Having one degree of freedom, this nut rmner will be 

actuated by wrist rotations. Pictures of the gripper, side, end, and top views, can be 

found in figures 1, 2, and 3 respectively. Still in the design phase, the nut runner is not 

present in the figures. 

This project’s aim is to develop control laws (in some optimal sense) for the gripper/nut runner end effector. Control laws for the gripper and nut runner portions of the 

end effector may be developed independently since these two systems are decoupled. A 

hybrid force/position controller will be used for both the gripper and nut runner. Developed here is the gripper controller only. Sensory data available to the controller is 

obtuned from an array of strain qauges [l] as well as a linear potentiometer. Applying 

well known optimal control theoretic principles, the control which minimizes the transition time between positions is obtained. In addition, a robust force control scheme is 

developed to contend with the strain gauge drift caused by extreme temperature variations encountered in space. 

2 Mat hematical Model 

For analytical purposes it is necessary to establish a mathematical model for the system. This model will include the actuator, transmission mechanism, and finger assembly. 

The finger actuator is a permanent-magnet DC motor. Models for these motors can 

be obtained from various sources. Used here is the model developed in hfodern Control 

Engineering 121. 

1 
. . ----------- 

Figure 1: Gripper (side view) 

The armature is modeled by a resistance R,, and inductance La, both in series with 

a voltage source eb representing the back emf generated by the rotor rotations. Applied 

to the armature terminals is the control voltage e,. 

Proportional to the armature current by a constant b;, the motor torque is given by 

Tm(t) = k&(t). (2.1) 

Also, the back emf is proportional to the rotor angular ve1ocit.y by constant kb, 

eb(t) = kbWm(t). 

The power in the motor armature can be expressed as 

or 

2 

ORIGINAL PAGE 

BLACK AND WHITE PHOTOGRAPH 
, 

.. .. 

ORfGfNAL PAGE 

BLACK AND WHITE PHOTOGRAPH 

Figure 2: Gripper (end view) 

yielding the relation __ 

* 

V 

A kb (- rad- s 

In all DC permanent magnet motors there exists a ripple torque caused by the nonuniformity of the magnetic field. Thus the torque constant ki is a function of the angular 

position of the rotor, and therefore periodic with period 2~. This periodic torque can 

effect the performance of the system, however, due to the high gear ratios being used, 

the effect is negligible. 

Applying D’Alembert’s law at the motor output, and Kirchhoff’s voltage law to the 

circuit model of the motor, the equations for the motor are as follows: 

J,5&) = T,(t) - T&) - B+,(t) 

~,f,(t) = ea(t) - Rai,(t) - eb(t) 

(2.6) 

(2.7) 

where the coulomb friction, Tj, is included in the load torque, TL(~). 

Selected for the actuation, the TRW 5A540-10 hfhl Planetary Gearmotor is manufactured with gears inside the motor housing. The gearing has the effect of decreasing 

3 
' 

ORIGINAL PAGE 

BLACK APiD WHITE PHOTOGRAPH 

definition 

armature resistance 

armature inductance 

torque constant 

back emf constant 

rotor inertia 

viscous friction 

coulomb friction 

Figure 3: Gripper (top view) 

the angular velocity and increasing the torque. The gear ratio is 27.94 and the minimum 

efficiency is 0.6. A summary of the other constants and their values for this motor can 

be found in Table 2.1. Motor variables are summarized in Table 2.2. 

s 

value 

25.2 52 

7.2 mH 

0.0247 N - m/A 

0.0247 V/rud/s 

3.67 x io-' bg - m2 

0.0 N - m/rud/s 

0.0028 N - m 

constant 

Ra 

La 

bi 

Jm 

Brn 

Tf 

1.b 

4 
a R 

variable 

e,(t) 

ia(t) 

eb(t) 

wm(t) 

Tm(t) 

TL(~) 

e 

a 

definition 

control voltage 

armature current 

back emf 

angular velocity 

motor torque 

load torque 

Figure 4: Motor Model 

+ t 

Figure 5: Motor with Internal Gears * 

A problem with the motor model developed above is that it gives the equations for 

torque and angular velocity without internal gears. These equations must be modified 

to account for gear effects. 

Ideal gears are initially assumed in the derivation. A gear is termed ideal if it has no 

5 
L 

constant 

k; = ~Nki 

&, = Nkb 

Jm = N2Jm 

Bm = N2Bm 

moment of inertia, no stored energy, no friction, and a perfect meshing of teeth. 

definition value 

torque constant 

back emf constant 0.6901 V/rad/s 

rotor inertia 2.9 x kg - m2 

0.4141 N - m/A 

viscous friction 

Let fc(-) denote the contact force where the gears mesh and N = q/r2 the gear ratio. 

From equation (2.6), with TL(~) = rlfc(t), the equation for the motor and attached gear 

is 

JmGn(t) = - rlfc(t) - Bmwm(2) (2.8) 

also 

Combining these two equations, and using the geometrical equality om(t) = No,(t), 

gives 

N2JmGo(t) = NTm(t) - T,(t) - N2B,wo(t). (2.10) 

Of course, in reality the gears are not ideal. To help account for this, an efficiency term, 

c, is included in the equation as follows 

N2J,G,(t) = cNT,(t) - T,(t) - N2B,w,(t) (2.11) 

~ ~~ 

Table 2.3 Equivalent Motor Constants 

By using the equivalent constants defined in Table 2.3 the adapted motor equations 

become 

JmLjo(t) = Tm(t) - TL(~) - Bmw,(t) (2.12) 

~,l,(t) = ea(t) - Rai,(t) - &(t) (2.13) 

T,(t) = Icii,(t) (2.14) 

eb(t) = zbLd,(t) (2.15) 

The transmission mechanism used is an acme screw. The acme screw serves to transform rotary motion of the motor to linear motion needed to move the fingers. A screw 

with a small lead angle, t = 1.6 mm, was used so that larger finger tip forces could be 

developed while using a smaller motor. As a trade-off, the maximum velocity attained 

during opening and closing is lower. Another benefit of having a small lead angle is that 

the screw is self locking under reasonable load conditions. This is an important feature 

since it will allow the end effector to hold objects without using power. This property 

helps to eliminate problems created because of poor heat dissipation in space. 

Used to link the motion of the fingers, the rack and pinion gear’s effect is to move 

the fingers at the same velocity in opposite directions. The model, which is developed 

6 
Figure 6: System Diagram 

for the force and velocity of one of the fingers, is effected only by the efficiency of this 

gear. 

The finger assembly is modeled as a rigid body since operating forces are far below the 

magnitude necessary to result in significant deflections of the aluminum body. Because 

the motion is strictly linear the moving parts in the finger assembly are treated as a point 

mass located at the nut. Table 2.4 lists the moving parts and their weights. 

Since the errors due to backlash caused by the gears (the motor's internal gears, 

the acme screw, and the rack and pinion) will be within desired positioning accuracy 

of f0.25cm, they can be ignored in the model development. Position control will be 

employed only to achieve an approximate desired finger opening; force control will be used 

upon encountering an object. Therefore, a course positioning scheme will be sufficient. 

weight 

Table 2.4 Weight of Moving Gripper Parts 

The last step in the model development is to relate output force at the fingers and 

output velocity of the fingers to the input armature voltage. 

Assume initially that the screw is ideal. For an ideal screw, the translation from 

angular velocity of the motor output shaft to linear velocity of the fingers is 

e 

and the transformation from torque to force is 

2n 

h(t) = 2;;W"(t) 

C(t) = TW) 

(2.16) 

(2.17) 

7 
where Tc(.)and I?,(.) denote the coupling torque and force respectively between the screw 

and the nut. Then developing the equations as before 

J;n30(t) = Tm(t) - Tc(t) - Bmoo(t) 

FL(t) = Fc(t) - rnVL(t) 

(2.18) 

(2.19) 

Combining these equations and applying the screw equations (2.16) and (2.17) yields 

(2.20) 

As before efficiency terms, p and p, are incorporated into the model to account for 

the losses in the non-ideal screw and rack and pinion gears respectively. For an acme 

screw the efficiency is approximately 0.4, and for the rack and pinion 0.7. 

(2.21) 

Included in the load force, FL(-), is the friction due to bearing preload which is 

1.362 N. This is arrived at by estimating the preload force at 4.45 N (k: 10 Zbs), and the 

coefficient of friction for the roller bearings at 0.3. 

Taking the Laplace transform of equations (2.13)-(2.16) and (2.21), after some algebraic manipulation yields 

Found in appendix A are the magnitude Bode plots of the model transfer function for 

position, velocity and force. Obtained from NASA by measuring the gripper response to 

a series of incremented sine waves, the true bode plot for position are found in appendix 

B. The accuracy of the true Bode plot for frequency greater than 8 rad/s is extremely 

poor due to backlash in the system. This will not effect performance since the fingers 

will not be expected to make rapid changes in direction. The actual bode plot reveals 

that for the range of operation the system only second order dynamics are evident. This 

implies that the third order dynamics cause by the nonzero inductance can be ignored 

and thus it can be assumed that the inductance is zero. The magnitude Bode plots of 

the transfer function for position, and velocity with inductance equal to zero have been 

superimposed on the third order model plots found in appendix A. 

8 
, 

3 State Space Realization 

From the transfer function for the velocity of the fingers comes the position transfer 

function (assuming L, = 0): 

Rewritten to simplify notation as 

\ This transfer function describes a linear time invariant system which has two inputs, 

armature voltage, e,(-), and load torque, FL(-), one output, finger position, P(.). The 

minimal realization has two states, finger position, P(.), and finger velocity, I"'(.) In 

state space form, the minimal system is written as 

where 

*=(i -'e3> 

0 .=(E", -c,> 

C=(l 0). 

Note that since this realization is minimal, it is both controllable and observable. 

(3.4) 

4 Optimal Position Controller (Bang Bang Control) 

It is desirable to determine the control which moves the fingers from the initial state 

(position, velocity), xo = ( zo 0 )', to the desired final state, xi = ( zf 0 )', in minimum 

time. Such problems have been extensively studied in optimization theory. One solution 

to the minimum time problem follows from the well known Pontryagin Minimum Principle 

(PMP). The necessary conditions derived from the PMP for linear time invariant systems 

are detailed below [3]. 

4.1 Problem Statement 

Given a completely controllable dynamical system 

x(t) = Ax(t)+ Bu(t) 

9 
such that the components ul(t), uz(t), . . .ur(t) of u(t) are constrained in magnitude by 

the relation 

luj(t)l 5 Mj j = 192,. . (4.2) 

A and that at the initial time to = 0, the initial state of the system is xo = x(t0) , find a 

control u*(.) that transfers the system from xo to xf in minimum time. This control is 

called the time optimal control. 

4.2 Necessary Conditions 

Assume that a time optimal control, u*(t), exists that transfers the initial state xo to 

xi. Let x*(t) denote the trajectory of the system corresponding to u*(t), originating at 

xo at to = 0, and hitting XI in the minimum time, T*. Then there exists a corresponding 

costate vector p*(t) such that: 

1. x*(t) and p*(t) are solutions of the canonical equations 

aH 

X*(t) = -(x*(t), ~*(t), p*(t)) = Ax*(t) + Bu*(t) bP* (4.3) 

aH 

P*(t) = -=(x*(t), ~*(t),p*(t)) = -A'p*(t) (4.4) 

with boundary conditions 

x*(O) = xo, x*(T*) = Xf. (4.5) 

Called the Hamiltonian function, 

for linear time invariant systems. 

2. The relation 

*(x*(t), u*(t), P*(t)) 5 H(x*(t), u(0, P*(t)) (4.7) 

(B'P'W, u*(t)) 5 (B'PV), W) (4.8) 

uJ(t) = -Msgn(bSp*(t)), (4.9) 

holds for all admissible u(t) and for t E [O,T*]. From equation (4.7), this implies * 

holds for all admissible u(t) and for t E [0, T*]. Clearly, < B'p*(t), u(t) > is 

minimized by 

where bj denotes the j'h column of B. 

3. The relation 

ab", u*(t), P*W = 0 

holds for all t E [O,T*]. 

(4.10) 

Thus it has been shown that if an optimal control exists, it may only take on values 

fMj Vj. The second input, F'(-), is depend on the particular task and thus cannot be 

set optimally. For the first input, e,(.), the set of feasible controls is practically limited 

by the maximum armature voltage, 24 V. 

10 
4.3 Switching Curves 

The necessary conditions do not contain any explicit information regarding either the 

initial costate, p*(O), or the terminal costate, p*(T*). However, from equation (4.6) and 

(4.10) it follows that the costate, p*(t), must be a nonzero vector. That is 

p*(t) # 0 Vt E [O,T*]. (4.11) 

The switching times of the control voltage can be found by finding the time at which 

b{p*(t) = ~lpz(t) = 0. (4.12) 

Solving equation (4.4) for the costate yields: 

p;(t) = p;(o) = constant 

p;(t) = (-3P;(o) + P*(O)) exPC3 - -3P;(o) 1 1 

C C 

(4.13) 

(4.14) 

Since p;(.) is constant, and p;(.) is monotonic in t, the control voltage changes signs at 

most once. Without information on the initial or final costate, the switching time cannot 

be determined from these equations. 

Disregarding the question of determining the initial or final costate, investigate which 

among the feasible controls just obtained steers the given initial state to the final state. 

Assumed here is that the load force will be essentially constant and equal to the internal 

frictional force. This will be the general case when operating in the position control 

mode. Solving equation (4.1) yields 

clu(2) - C2FL ClU(t) - C2FL 3*( 1 - exp - c3t) C Xl(t) = 20 + 31 + 

C 

clu(t) - c2Ft 

z&) = - 3( 1 - exp - c3t) C 

(4.15) 

(4.16) 

Bygubstituting for ti(-) in equations (4.15) and (4.16), find which of the feasible controls 

achieves the desired goal. This control is the optimal control. 

Substituting the model parameters from section 1, it can be seen that the switching 

time is within the position error margins of f.25cm. 

5 Hardware Implementation 

Chosen to implement the controller in real time, the hlac I1 has been equipped with a 

MacAdios board for D/A and A/D conversions. To boost the power in the control signal, 

an amplifier was built using the Burr-Brown high power operational amplifier (Figure 

9). The decision to use the Mac I1 was based on the need for a design environment 

that would facilitate rapid design and testing of control algorithms. In addition the 

mouse/icon oriented environment allows new users to come up to speed with a minimum 

of training. 

11 
Figure 7: Hardware Flow Diagram 

The time optimal position controller has been implimented. The controller can position to within f.25 cm of the desired position without overshoot. The top velocity can 

be set arbitrarily between the maximum value, approximately 1.25 cm/s, and 0 cm/s. 

Since fingers equipped with the full array of strain gauges have not yet been delivered 

by NASA, the force controller currently running receives force information from only one 

strain gauge on each finger and thus is not able to compute the grip points or finger 

torques. Any finger force between 0 N and approximately 6.0 N can be maintained. 

Addressing the problem of hitting an unexpected obstacle, the hybrid force/position 

controller stops if a prespecified force is achieved during positioning. 

5.1 Future Plans 

Impeded by disagreements in the initial design environment and delays in delivering 

the mechanical hardware, progress is slow and there is still much work to be done. The 

major focuses are detailed below. 

* 

1. Upon receipt of the fingers equipped with the complete array of strain gauges, an 

algorithm which determines grip points an finger torques will be tested. 

2. Once the control algorithms have beei perfected, they will be programmed onto a 

dedicated signal processing chip and the h4ac I1 will be used only for higher level 

processing, leaving the low level calculations and commands to the signal processing 

chip. 

References 

[l] J. Vranish, “Wheatstone bridge,”. Details the force sensor design. 

[a] K. Ogata, Modern Control Engineering. Elglewood Cliffs, NJ: Prentice Ilall, 1970. 

12 
Figure 8: Amplifier Diagram 

[3] M. Athans and P. Falb, OPTIMAL CONTROLS: An Intduction to the Theory and 

It’s Applications. New York, NY: McGraw-Hill, 1966. 

[4] C. Close and D. F’rederic, Modeling and Analysis of Dynamic Systems. Boston, MA: 

Houghton Mifflin Company, 1978. 

[5] C. Chen, Linear Systems Theory and Design. New York, NY: Holt, Reinhard and 

Winston, 1984. 

[6] Electrocraft Corporation, ed., DC Motors, Speed Controls, Servo Systems. Elmsfor, 

NY: Pergamon Press, fifth ed., 1980. 

13 
A Model Bode Plots: position, velocity, force 

-50 

n 

rn 

0,-100 

a 

? 

5 

C all 

(d 

E 

a v 

-150 

0 

-50 

n 

rn 

23- 100 

0) 

a 

3 -150 ..-I 

5 

g -200 

-250 

I I I IIIII I I I I1111 I I11l111 I I111111 I I I I Ill1 I I I I1 Ill1 I I I I I Ill 

- - - - - - - - - - - 

- 

- 

- 

- 

- with inductance - - - - - - - - - - without inductance - 

- I I I I1111 I I I I I Ill 1 I I I I IIII I I I I I Ill I I111111 

Position Transfer Function 

I I I I I Ill1 1 I I 1 I Ill[ I I I I I Ill I I I I I Ill I 1 I I I Ill] I I I I I Ill 1 I I I1111- 

7 - 

- - 

- 

- 

- 

- 

- 

IO0 I I I1l111 I I 111111 I I I I1 Ill I I I IIIII I I I I I Ill I 1 1 I I Ill I I I IIIII 
m 

m 

N 

, I ! 

I I 1 

! I 1 j 1 i 

i i I i i 

I 

I 

--r' ! - 

__-_ 1 I I 

I 

I 

1 I 

I 

a 

N 

I 

d 

m 

m 

Ln 

U 

I 

cn 

Ln 

0 

N 

7 

b 

b 

00 

a 

'5' 

I