
                               INTRODUCTION

     Servo motors are widely used in the field of motion control in
     factory automation. The control target can be position, speed, or
     force, among others. For this example application we take force
     as the control variable.
     
     In order to implement force control, we need to know the
     compliance (response) of the controlled object to force. The
     feedback gain in the control loop changes as a function of
     compliance.
     
     Grasping a range of objects from, for example, a soft tennis
     ball to a hard steel ball using conventional servo control is
     extremely difficult. The traditional control model does not
     handle a variety of objects with differing material
     characteristics very well. The system can become unstable. Fuzzy
     logic, with its inherent flexibility, can be employed
     effectively as an alternative in this situation.
     
     
     FUZZY FORCE CONTROLLER
     
     Control Objective
     
     Grasp objects of various compliance, ranging, for example, from a
     soft tennis ball to a hard steel ball with a constant force.
     
     
     Control System
     
     The control block diagram is shown in Figure 1. Output force
     applied to the object is measured by a sensor and compared
     against a reference force to obtain the difference. A control
     gain Kg is applied to diminish this force difference. This gain
     also varies as a function of the compliance of the grasped
     object. Thus, control gain Kg is affected by two
     factors: (1) the compliance of the object and (2) the difference
     between a reference force and the measured force.  Branches
     coming off the error (e) node and speed (v) node of the above
     diagram are expansions of those nodes, and represent variables
     to be used to determine the control gain.  They do not represent
     additional control paths.  We can write the control gain and
     diagram its components as shown in Figure 2 below.
     
     
     Ks (compliance component) is a function of Ke. Kf (force
     component) is a function of error e and its time
     derivative ‚. Both can be inferred by fuzzy logic. 
     
     The compliance Ke is determined by injecting a
     speed command v into the servo motor and
     measuring the output force f. Compliance is
     expressed as follows: 
     
     Ke = df/dx = (df/dt)/(dx/dt) = âf/v
     
     We obtain 
     
     Ke = (fk-f(k-1))/v(k-1)
     
     Compliance can be thought of as the change in force (df) required
     for a given deformation (dx) of an object. For example, a tennis
     ball has a large compliance because the force needed to initiate
     deformation is small, but increases significantly as the
     deformation process proceeds. The change in force from initiation
     to termination is large. At the other extreme is the steel ball,
     which has small compliance. Although the force required to
     initiate deformation is large, the force to continue deformation
     does not change significantly.  Consequently, the change from
     initiating to terminating force is small.
     
     It is known that the control gain Kg is the
     reciprocal of the compliance Ke, so Ks can be inferred from Ke
     by the following fuzzy rules: 
     
     If Ke is small then Ks is large 
     If Ke is large then Ks is  small
     
     These two rules make up the fuzzy inference unit A which connects
     Ke with Ks.
     
     
     Definition of Input/Out Variables for Unit B 
     
     Now let us consider fuzzy inference unit B, inferring Kf from e 
     and ‚. The two inputs into Unit B are error e and its time derivative
     ‚. e is the difference between a reference force and the
     applied output force. Labels and membership functions for e and
     ‚ are defined as shown in Figure 3a, 3b respectively. Figure 3c shows 
     the labels and membership functions for Kf.
     
     
     FIU Source Code of Unit B
     
     The following is the source code of Unit B written in FIDE's
     Fuzzy Inference Language (FIL). Note that in the definition of
     input variable Error, the value of P_VerySmall is given as  (@-3,
     0, @0, 1, @50, 0), and that of N_VerySmall is  (@-50, 0, 
     @0, 1, @3, 0). We use -3 and 3 instead of -1 and 1
     respectively because the data range of Error must be accommodated
     in a resolution of 8 bits. This means the smallest interval of
     Error is 600/256 = 3. The membership functions of these
     fuzzy sets are shown in Figure 3a, 3b, and 3c as we have seen.
     
     $ FILENAME:	motor/motor1.fil 
     $ DATE: 	08/12/1992 
     $ UPDATE:	08/14/1992
     
     $ Two inputs, one output, to determine control gain
     $ INPUT(S):	Error, Derivative(_of_Error)
     $ OUTPUT(S):	Gain
     
     $ FIU HEADER
     
     fiu tvfi (min max) *8;
     
     $ DEFINITION OF INPUT VARIABLE(S)
     
     invar Error " " : -300 () 300 [
     	P_Large        (@100, 0,  @200,  1,  @300, 1),
     	P_Medium       (@50,  0,  @100,  1,  @200, 0),
     	P_Small        (@0,   0,  @50,   1,  @100, 0),
     	P_VerySmall    (@-3,  0,  @0,    1,  @50,  0),
     	N_VerySmall    (@-50, 0,  @0,    1,  @3,   0),
     	N_Small        (@-100,0,  @-50,  1,  @0,   0),
     	N_Medium       (@-200,0,  @-100, 1,  @-50, 0),
     	N_Large        (@-300,1,  @-200, 1,  @-100,0)
     	]; 
     
     invar Derivative " " : -30 () 30 [
     	P_Large        (@10,  0,  @20,  1,  @30, 1),
     	P_Medium       (@5,   0,  @10,  1,  @20, 0),
     	P_Small        (@0,   0,  @5,   1,  @10, 0),
     	P_VerySmall    (@-1,  0,  @0,   1,  @5,   0),
     	N_VerySmall    (@1,   0,  @0,   1,  @-5,  0),
     	N_Small        (@0,   0,  @-5,  1,  @-10,0),
     	N_Medium       (@-5,  0,  @-10, 1,  @-20,0),
     	N_Large        (@-10, 0,  @-20, 1,  @-30,1)
     	]; 
     
     $ DEFINITION OF OUTPUT VARIABLE(S)
     
     outvar Gain " " : -2 () 2 * (
     	P_Large        = 2.00,
     	P_Medium       = 1.00,
     	P_Small        = 0.50,
     	P_VerySmall    = 0.25,
     	Zero           = 0.00,
     	N_VerySmall    = -0.25,
     	N_Medium       = -1.00
      	);
     
     
     $ RULES
     
     if Error is N_Large  and Derivative is P_Large then Gain is P_Medium;
     if Error is N_Large  and Derivative is P_Medium then Gain is P_Medium;
     if Error is N_Large  and Derivative is P_Small then Gain is P_Medium;
     if Error is N_Large  and Derivative is P_VerySmall then Gain is P_Medium;
     if Error is N_Large  and Derivative is N_VerySmall then Gain is P_Medium;
     if Error is N_Large  and Derivative is N_Small then Gain is P_Medium;
     if Error is N_Large  and Derivative is N_Medium then Gain is P_Small;
     if Error is N_Large  and Derivative is N_Large then Gain is P_Small;
     
     if Error is N_Medium and Derivative is P_Large then Gain is P_Medium;
     if Error is N_Medium and Derivative is P_Medium then Gain is P_Medium;
     if Error is N_Medium and Derivative is P_Small then Gain is P_Medium;
     if Error is N_Medium and Derivative is P_VerySmall then Gain is P_Medium;
     if Error is N_Medium and Derivative is N_VerySmall then Gain is P_Medium;
     if Error is N_Medium and Derivative is N_Small then Gain is P_Medium;
     if Error is N_Medium and Derivative is N_Medium then Gain is P_Small;
     if Error is N_Medium and Derivative is N_Large then Gain is Zero;
     
     if Error is N_Small  and Derivative is P_Large then Gain is P_Medium;
     if Error is N_Small  and Derivative is P_Medium then Gain is P_Medium;
     if Error is N_Small  and Derivative is P_Small then Gain is P_Medium;
     if Error is N_Small  and Derivative is P_VerySmall then Gain is P_Medium;
     if Error is N_Small  and Derivative is N_VerySmall then Gain is P_Medium;
     if Error is N_Small  and Derivative is N_Small then Gain is P_Small;
     if Error is N_Small  and Derivative is N_Medium then Gain is P_VerySmall;
     if Error is N_Small  and Derivative is N_Large then Gain is N_VerySmall;
     
     if Error is N_VerySmall  and Derivative is P_Large then Gain is P_Medium;
     if Error is N_VerySmall  and Derivative is P_Medium then Gain is 
     P_Medium;
     if Error is N_VerySmall  and Derivative is P_Small then Gain is P_Medium;
     if Error is N_VerySmall  and Derivative is P_VerySmall then Gain is 
     P_Medium;
     if Error is N_VerySmall  and Derivative is N_VerySmall then Gain is 
     P_Large;
     if Error is N_VerySmall  and Derivative is N_Small then Gain is 
     P_VerySmall;
     if Error is N_VerySmall  and Derivative is N_Medium then Gain is 
     N_VerySmall;
     if Error is N_VerySmall  and Derivative is N_Large then Gain is 
     N_Medium;
     
     if Error is P_VerySmall  and Derivative is P_Large then Gain is 
     N_Medium;
     if Error is P_VerySmall  and Derivative is P_Medium then Gain is 
     N_VerySmall;
     if Error is P_VerySmall  and Derivative is P_Small then Gain is 
     P_VerySmall;
     if Error is P_VerySmall  and Derivative is P_VerySmall then Gain is 
     P_Large;
     if Error is P_VerySmall  and Derivative is N_VerySmall then Gain is 
     P_Medium;
     if Error is P_VerySmall  and Derivative is N_Small then Gain is 
     P_Medium;
     if Error is P_VerySmall  and Derivative is N_Medium then Gain is 
     P_Medium;
     if Error is P_VerySmall  and Derivative is N_Large then Gain is 
     P_Medium;
     
     if Error is P_Small  and Derivative is P_Large then Gain is N_VerySmall;
     if Error is P_Small  and Derivative is P_Medium then Gain is 
     P_VerySmall;
     if Error is P_Small  and Derivative is P_Small then Gain is P_Small;
     if Error is P_Small  and Derivative is P_VerySmall then Gain is 
     P_Medium;
     if Error is P_Small  and Derivative is N_VerySmall then Gain is 
     P_Medium;
     if Error is P_Small  and Derivative is N_Small then Gain is P_Medium;
     if Error is P_Small  and Derivative is N_Medium then Gain is P_Medium;
     if Error is P_Small  and Derivative is N_Large then Gain is P_Medium;
     
     if Error is P_Medium  and Derivative is P_Large then Gain is Zero;
     if Error is P_Medium  and Derivative is P_Medium then Gain is P_Small;
     if Error is P_Medium  and Derivative is P_Small then Gain is P_Medium;
     if Error is P_Medium  and Derivative is P_VerySmall then Gain is 
     P_Medium;
     if Error is P_Medium  and Derivative is N_VerySmall then Gain is 
     P_Medium;
     if Error is P_Medium  and Derivative is N_Small then Gain is P_Medium;
     if Error is P_Medium  and Derivative is N_Medium then Gain is P_Medium;
     if Error is P_Medium  and Derivative is N_Large then Gain is P_Medium;
     
     if Error is P_Medium  and Derivative is P_Large then Gain is P_Small;
     if Error is P_Medium  and Derivative is P_Medium then Gain is P_Small;
     if Error is P_Medium  and Derivative is P_Small then Gain is P_Medium;
     if Error is P_Medium  and Derivative is P_VerySmall then Gain is 
     P_Medium;
     if Error is P_Medium  and Derivative is N_VerySmall then Gain is 
     P_Medium;
     if Error is P_Medium  and Derivative is N_Small then Gain is P_Medium;
     if Error is P_Medium  and Derivative is N_Medium then Gain is P_Medium;
     if Error is P_Medium  and Derivative is N_Large then Gain is P_Medium
     
     end
     
     
     Input/Output Response
     
     Figure 4 shows the response surface of the FIU defined above.
     This surface is obtained by using the Analyzer tool provided in
     FIDE. 
     
     
     COMMENTS
     
     Through experimentation, we can obtain a set of rules to infer
     compliance Ke from speed v, and the measured force f.  The rules 
     are in essence as follows: 
     
     If v is large and â is small, then Ke is very small
     If v is large and â is medium,then Ke is small
     If v is large and â is large, then Ke is medium
     If v is small and â is small, then Ke is medium
     If v is small and â is medium,then Ke is large
     If v is small and â is large, then Ke is very large
     
     The label names used here give an intuitive sense of how the
     rules apply. However, even though label names are the same for
     different variables, the fuzzy sets associated with these labels
     may be different. For speed v, the label large
     may be a fuzzy set as shown in Figure 5a, and for compliance
     Ke, label large could be another fuzzy set as shown in Figure 5b. 
     
     
     The ranges of these variables can be determined by experiment on
     the devices and objects of interest. For example, compliance data
     gathered from a soft tennis ball and a hard steel ball can be
     used to define large and small labels respectively for variable
     Ke.
     
     If we use an FIU to infer compliance Ke, the control gain
     function now becomes three FIUs and an operations block (FOU) as
     shown in Figure 6. The FOU implements Kg = Ks . Kf . Using
     Fide's Composer capability, these four blocks can be combined
     into a single system for analysis and simulation purposes.
    
     (Weijing Zhang, Applications Engineer, Aptronix Inc.)
     

     
                  For Further Information Please Contact:
    
                          Aptronix Incorporated
                      2150 North First Street #300
                           San Jose, CA 95131
                           Tel (408) 428-1888
                           Fax (408) 428-1884
                   FuzzyNet (408) 428-1883 data 8/N/1
    
    

                        Aptronix Company Overview
    
    Headquartered in San Jose, California, Aptronix develops and
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                         Servo Motor Force Control
     
                    FIDE Application Note 003-140892		
                           Aptronix Inc., 1992 

         
     
     
     
