Abstract:
A lithography system and method for calculating an optimal discrete time trajectory for a movable device is described. A trajectory planner of the lithography system calculates an optimal discrete time trajectory subject to maximum velocity and maximum acceleration constraints. The trajectory planner begins by calculating a continuous time, three-segment trajectory for a reticle stage, a wafer stage or a framing blade, including a first phase for acceleration at the maximum acceleration to the maximum velocity, a second phase for travel at the maximum velocity and a third phase for deceleration at the negative maximum acceleration to a final velocity. Next, the trajectory planner converts said continuous time, three-segment trajectory to a discrete time trajectory. The time of execution of the resulting trajectory is at most three quanta greater than the time of execution of the continuous time trajectory. One advantage of the system is the reduction of scanning times of a lithography system. This advantage increases throughput and reduces manufacturing costs for a lithography system.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS  
       [0001]    This patent application is a continuation of Ser. No. 09/923,397 filed Aug. 8, 2001 (now U.S. Pat. No. ______ that issued ______), which is hereby incorporated by reference herein in its entirety.  
         [0002]    This patent application is related to the following commonly-owned U.S. Patent Application: U.S. patent application Ser. No. 09/757,622 (now U.S. Published Application No. 2003-0055855 A1, published Mar. 20, 2003), to Wiener, entitled “Method and System for Efficient and Accurate Filtering and Interpolation,” filed Jan. 11, 2001. The foregoing U.S. Patent Application is hereby incorporated by reference in it&#39;s entirety. 
     
    
     
       BACKGROUND OF THE INVENTION  
         [0003]    1. Field of the Invention  
           [0004]    This invention relates to trajectory planning. More specifically, this invention relates to trajectory planning for a lithography system.  
           [0005]    2. Background Art  
           [0006]    Many industrial processes involve motion along trajectories that are defined by precise positions at specific times. Photolithography is an example of such a process. In a photolithography process, an illumination source projects an illumination beam. The beam passes through, or is reflected off, a reticle to enable the transfer of a reticle image from the reticle to a substrate, such as a semiconductor wafer.  
           [0007]    Scanning techniques are employed in photolithography processes to project a reticle image onto a substrate. These scanning techniques involve moving a reticle across an illumination slot to allow the reticle image to be exposed onto a substrate that is simultaneously moving. Reticles and substrates are disposed on stages that are capable of motion in one or more dimensions.  
           [0008]    Long scanning times amount to greater manufacturing costs and less throughput. Therefore, it is desirable to reduce scanning times. In order to decrease the amount of time it takes to execute a trajectory, i.e, execution time, these systems aim to calculate optimal trajectories, i.e, trajectories requiring the least amount of time to execute. The calculation of an optimal continuous trajectory is well known in mathematics. However, a typical lithography system operates in discrete time, not continuous time. That is, a typical lithography system can only receive, process and react to commands in increments of time since the sampling rate of the command and control systems is limited. Further, a typical lithography system is restricted by maximum velocity and maximum acceleration constraints.  
           [0009]    Accordingly, what is needed is a system and method for calculating optimal discrete time trajectories for a lithography system.  
         BRIEF SUMMARY OF THE INVENTION  
         [0010]    The present invention relates to a lithography system and method for calculating an optimal discrete time trajectory.  
           [0011]    In an embodiment of the present invention, a trajectory planner of the lithography system calculates an optimal discrete time trajectory subject to maximum velocity and maximum acceleration constraints. The trajectory planner begins by calculating a continuous time, three-segment trajectory for a movable device, including a first phase for acceleration at the maximum acceleration to the maximum velocity, a second phase for travel at the maximum velocity and a third phase for deceleration at the negative maximum acceleration to a final velocity. The movable device can be a wafer stage, a reticle stage or a framing blade. Next, the trajectory planner converts said continuous time, three-segment trajectory to a discrete time trajectory. The time of execution of the resulting trajectory is at most three quanta greater than the time of execution of said continuous time, three-segment trajectory.  
           [0012]    The present invention reduces scanning times in a lithography system. This advantage increases throughput for a lithography system and reduces manufacturing costs.  
           [0013]    Further features and advantages of the invention as well as the structure and operation of various embodiments of the present invention are described in detail below with reference to the accompanying drawings. 
       
    
    
     BRIEF DESCRIPTION OF THE FIGURES  
       [0014]    The features and advantages of the present invention will become more apparent from the detailed description set forth below when taken in conjunction with the drawings in which like reference numbers indicate identical or functionally similar elements. Additionally, the left-most digit of a reference number identifies the drawing in which the reference number first appears.  
         [0015]    [0015]FIG. 1 is a block diagram illustrating the architecture of a lithography system, in an embodiment of the present invention, showing connectivity among the various components.  
         [0016]    [0016]FIG. 2 is a flowchart depicting an embodiment of the operation and control flow of a lithography system of the present invention.  
         [0017]    [0017]FIG. 3A is a graph illustrating a continuous, three-segment trajectory subject to no constraints, in an embodiment of the present invention.  
         [0018]    [0018]FIG. 3B is a graph illustrating a continuous, three-segment trajectory subject to velocity and acceleration constraints, in an embodiment of the present invention.  
         [0019]    [0019]FIG. 4 is a flowchart depicting an embodiment of the operation and control flow of the trajectory calculation process of the present invention.  
         [0020]    [0020]FIG. 5 is a graph illustrating an alternative, continuous, three-segment trajectory subject to velocity and acceleration constraints, in an embodiment of the present invention.  
         [0021]    [0021]FIG. 6 is a block diagram illustrating the system architecture of an embodiment of a computer system and computer program product, showing connectivity among the various components, that may be used to implement the present invention.  
         [0022]    [0022]FIGS. 7A-7P are a collection of graphic surfaces representing the values of multi-dimensional variables.  
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
     Table of Contents  
       [0023]    system and method for DISCRETE TIME TRAJECTORY PLANNing for LITHOGRAPHY  
         [0024]    CROSS-REFERENCE TO RELATED APPLICATIONS  
         [0025]    BACKGROUND OF THE INVENTION  
         [0026]    Field of the Invention  
         [0027]    Background Art  
         [0028]    BRIEF SUMMARY OF THE INVENTION  
         [0029]    BRIEF DESCRIPTION OF THE FIGURES  
         [0030]    DETAILED DESCRIPTION OF THE INVENTION  
         [0031]    DETAILED DESCRIPTION OF THE INVENTION  
         [0032]    I. Overview  
         [0033]    A. Definition  
         [0034]    B. General Considerations  
         [0035]    II. Lithography System Architecture  
         [0036]    III. Trajectory Planning  
         [0037]    A. Optimal Trajectories  
         [0038]    B. Trajectory Calculation  
         [0039]    1. Calculation of a Continuous Trajectory  
         [0040]    2. Quantization.  
         [0041]    3. Special Cases  
         [0042]    IV. Computer System  
         [0043]    V. Conclusion  
         [0044]    system and method for DISCRETE TIME TRAJECTORY PLANNing for LITHOGRAPHY  
         [0045]    ABSTRACT OF THE DISCLOSURE  
       DETAILED DESCRIPTION OF THE INVENTION  
       [0046]    I. Overview  
         [0047]    The present invention relates to a lithography system and method for calculating an optimal discrete time trajectory. The present invention allows for a more efficient and timely production of semiconductors.  
         [0048]    A. Definition  
         [0049]    The following definitions are provided for illustrative purposes only. Alternative definitions for the listed terms will be apparent to the persons skilled in the relevant art(s) based on the discussion contained herein, and fall within the scope and spirit of embodiments of the invention.  
         [0050]    The term “trajectory” or “profile” refers to a path that is traveled by a projectile or a movable device.  
         [0051]    The term “quantum” refers to one unit of measurement. With reference to a trajectory, a quantum is one time increment. A time increment is also referred to as a “quantization interval.” 
         [0052]    The term “quantize” refers to the conversion of a continuous function or curve into a discrete function or curve. In the present invention, a continuous curve is quantized, or converted, to a discrete curve in accordance with the sampling rate of the command and control system of a lithography system.  
         [0053]    B. General Considerations  
         [0054]    The present invention is described in terms of the examples contained herein. This is for convenience only and is not intended to limit the application of the present invention. In fact, after reading the following description, it will be apparent to one skilled in the relevant art(s) how to implement the following invention in alternative embodiments.  
         [0055]    II. Lithography System Architecture  
         [0056]    [0056]FIG. 1 shows a block diagram illustrating the architecture of a lithography system  100 , in an embodiment of the present invention, showing connectivity among the various components. System  100  includes a trajectory planner  102 , a control system  104  and a movable device  106 . Movable device  106  can be any device that can be moved during the scanning process in a lithography tool, such as a reticle stage or a wafer stage. Trajectory planner  102  designs trajectories in accordance with user-specified requirements. This design process results in the generation of commands that define a trajectory for a movable device in a lithography tool. Trajectory planner  102  is implemented as a computer program product that is executed by a double precision computer. However, trajectory planner  102  can also be implemented in hardware and/or firmware. Trajectory planner  102  designs these trajectories in an off-line process. Thus, in the case of a lithography system, commands can be generated prior to commencement of substrate stage or reticle stage scanning operations. Control system  102  is responsible for implementing the trajectories designed by trajectory planner  102 . Control system  102  can be any lithography control system that is known to one of ordinary skill in the art.  
         [0057]    [0057]FIG. 2 shows a flowchart depicting an embodiment of the operation and control flow  200  of a lithography system of the present invention. FIG. 2 shows how a trajectory is calculated and implemented in a lithography system, such as system  100 . Control flow  200  begins at step  202 , with control passing immediately to step  204 .  
         [0058]    In step  204 , trajectory planner  102  calculates the intended trajectory of movable device  106 . The method by which trajectory planner  102  calculates the intended trajectory is described in greater detail below.  
         [0059]    In step  206 , control system  104  implements the trajectory calculated in step  204  above.  
         [0060]    In step  208 , movable device  106  travels the trajectory calculated in step  204  and implemented by control system  104  in step  206 .  
         [0061]    In step  210 , flow  500  ceases.  
         [0062]    III. Trajectory Planning  
         [0063]    A. Optimal Trajectories  
         [0064]    [0064]FIG. 3A shows a graph  300  illustrating a continuous two-phase trajectory. Graph  300  is a velocity versus time graph showing the optimal trajectory for an object moving from a first position to a second position, along one axis. There are no velocity or acceleration constraints shown in graph  300  other than the fact that acceleration is finite.  
         [0065]    Graph  300  shows an object beginning at time x 0  at a first position and velocity. At time x 0 , the object is at zero acceleration. Then, the object accelerates at maximum acceleration up to time x 1 . At time x 1 , the first segment of the trajectory ends and the second segment of the trajectory begins. Then, the object decelerates at minimum acceleration (the negative of the previous, maximum acceleration) until the object is located at the desired second position and velocity at time x 2 . At time x 2 , the object is at zero acceleration.  
         [0066]    It should be noted that, if the initial velocity at x 0  is the same as the desired velocity at x 2 , then x 1  is substantially at the midpoint between times x 0  and x 2  and at the midpoint between the first position and the second position. This is because 1) the absolute value of the maximum acceleration of the first segment is identical to the absolute value of the deceleration of the second segment and 2) the starting condition (at rest) is the same as the end condition. Thus, the trajectory is symmetrical.  
         [0067]    [0067]FIG. 3B shows a graph  350  illustrating a continuous, three-segment trajectory. Graph  350  is a velocity versus time graph showing the optimal trajectory for an object moving from a first position to a second position, along one axis, subject to constraints. The trajectory of graph  350  is subject to velocity and acceleration constraints. That is, the trajectory of graph  350  has a maximum velocity and a maximum acceleration.  
         [0068]    Graph  350  shows an object beginning at time x 0  at the first position and velocity. At time x 0 , the object is at zero acceleration. Then, the object accelerates at the maximum acceleration until the maximum velocity is reached at time x 1 . At time x 1 , the first segment of the trajectory ends and the second segment of the trajectory begins. Then, the object travels at the maximum velocity and zero acceleration until time x 2 . At time x 2 , the second segment of the trajectory ends and the third segment of the trajectory begins. Then, the object decelerates at the negative maximum acceleration (the negative of the previous, maximum acceleration) until the object reaches the desired second position and velocity at time x 3 . At time x 3 , the object is at zero acceleration.  
         [0069]    It should noted that the first segment of the trajectory is substantially congruent to the third segment of the trajectory. This is because 1) the absolute value of the acceleration of the first segment is identical to the absolute value of the deceleration of the third segment and 2) the starting condition is the same as the end condition. Thus, the trajectory is symmetrical.  
         [0070]    B. Trajectory Calculation  
         [0071]    [0071]FIG. 4 shows a flowchart depicting an embodiment of the operation and control flow  400  of the trajectory calculation process of the present invention. FIG. 4 shows how a trajectory is calculated by a trajectory planner, such as trajectory planner  102 , in a lithography system, such as system  100 . Flow  400  describes in greater detail how a trajectory is planned, as show in step  202  of FIG. 2. Control flow  400  begins at step  402 , with control passing immediately to step  404 .  
         [0072]    In step  404 , trajectory planner  102  calculates a continuous, three segment trajectory subject to velocity and acceleration constraints. A continuous, three segment trajectory subject to velocity and acceleration constraints is described in greater detail above (see FIG. 3B). The calculation of a continuous, three segment trajectory subject to velocity and acceleration constraints is described in greater detail below.  
         [0073]    In step  406 , the trajectory calculated in step  404  above is quantized. Quantization of a continuous function or curve is described in greater detail below.  
         [0074]    In step  408 , flow  400  ceases.  
         [0075]    1. Calculation of a Continuous Trajectory  
         [0076]    This section describes in greater detail the calculation of a continuous, three segment trajectory subject to velocity and acceleration constraints, as performed by trajectory planner  102  in step  404  above (see FIG. 4).  
         [0077]    [0077]FIG. 5 shows a graph  500  illustrating a continuous, three-segment trajectory, similar to graph  350 . Graph  500  is a velocity versus time graph showing the optimal trajectory for an object moving from a first position to a second position, subject to constraints. The trajectory of graph  500  is subject to velocity and acceleration constraints. A continuous, three-segment trajectory subject to constraints is described in greater detail above in FIG. 3B. In FIG. 5:  
         [0078]    v 1  is the initial velocity of the object  
         [0079]    v 2  is the second velocity of the object  
         [0080]    v 3  is the third velocity of the object  
         [0081]    v 4  is the fourth or end velocity of the object  
         [0082]    t 1  is the duration of the first segment of the trajectory  
         [0083]    t 2  is the duration of the second segment of the trajectory  
         [0084]    t 3  is the duration of the third segment of the trajectory  
         [0085]    Also associated with FIG. 5 (but not shown):  
         [0086]    s 0  is the initial position of the object (associated with the beginning of the first segment of the trajectory)  
         [0087]    s 1  is the second position of the object (associated with the beginning of the second segment of the trajectory)  
         [0088]    s 2  is the third position of the object (associated with the beginning of the third segment of the trajectory)  
         [0089]    s 3  is the fourth position of the object (associated with the end of the third segment of the trajectory)  
         [0090]    a 1  is the first acceleration of the object (associated with the first segment of the trajectory)  
         [0091]    a 2  is the second acceleration of the object (associated with the second segment of the trajectory  
         [0092]    a 3  is the third acceleration of the object (associated with the third segment of the trajectory  
         [0093]    v max  is the maximum velocity allowed for the object, i.e., a velocity constraint  
         [0094]    a max  is the maximum acceleration allowed for the object, i.e., an acceleration constraint  
         [0095]    d is the total desired displacement of the object, i.e., d=s 1 +s 2 +s 3    
         [0096]    Because FIG. 5 represents an “optimal” continuous trajectory, we assume that:  
         [0097]    a 2 =0  
         [0098]    v 2 =v 3    
         [0099]    a 1= a max    
         [0100]    a 3 =−a max    
         [0101]    As a result of the velocity and acceleration constraints, we see that:  
         [0102]    v i  is less than or equal to the maximum velocity, i.e. |v i |≦v max    
         [0103]    a i  is less than or equal to the maximum acceleration, i.e. |a i |≦a max    
         [0104]    In light of the above, we see that:  
                       v   2     =         a   1          t   1       +     v   1                     v   3     =         a   2          t   2       +     v   2                     v   4     =         a   3          t   3       +     v   3               }           (   1   )                         s   1     =         1   2          a   1          t   1   2       +       v   1          t   1                       s   2     =         1   2          a   2          t   2   2       +       v   2          t   2                       s   3     =         1   2          a   3          t   3   2       +       v   3          t   3                 }           (   2   )                               
 
         [0105]    If we combine equations (1) and (2), then:  
               t   1     =     -         v   1     -       1   /   2              2        v   4   2       +     2        v   1   2       +     4      d                   a   max                 a   max                 (     3      a     )                               
 
         [0106]    Or, if t 1  is negative:  
               t   1     =     -         v   1     +       1   /   2              2        v   4   2       +     2        v   1   2       +     4      d                   a   max                 a   max                 (     3      b     )                               
 
         [0107]    Similarly, for t 3 :  
               t   3     =         -     v   4       +       1   /   2              2        v   4   2       +     2        v   1   2       +     4      d                   a   max                 a   max               (     4      a     )                 t   3     =         -     v   4       -       1   /   2              2        v   4   2       +     2        v   1   2       +     4      d                   a   max                 a   max               (     4      b     )                               
 
         [0108]    Thus, the “optimal” v 2  velocity can be calculated as:  
           v   2optimal   =v   1   +a   1   t   1   (5)  
         [0109]    If the absolute value of v 2optimal  is greater than the absolute value of maximum velocity v max , then, in order to meet the velocity constraint, let:  
                       v   2     =     v   max                   t   1     =         v   max     -     v   1         a   1                     t   3     =     -         v   4     -     v   max         a   max                       v   3     =     v   max                   t   2     =       1   2            (         -   2          v   max   2       +     v   1   2     +     v   4   2     +     2      d                   a   max         )         v   max          a   max                   }           (   6   )                               
 
         [0110]    where v max  in this case has the appropriate sign. That is, if the three-segment trajectory involves an initial negative acceleration, then v max  must be negative. Further, if the three-segment trajectory involves an initial positive acceleration, then v max  must be positive.  
         [0111]    If the absolute value of v 2optimal  is not greater than the maximum velocity v max , then t 1  is given by equations (3a) or (3b) and t 3  is given by equations (4a) or (4b). As a result, t 2 =0 and v 2  is given by equation (5).  
         [0112]    The following conditions should be noted. First, equations (6) define the ideal trajectories for a velocity and acceleration constrained problem. Second, notice that the closer v 1  and v 4  are to v max , the smaller t 1  and t 3  become. Third, note that both t 1  and/or t 3  exist, both are positive and both are real if the element in the square root of equations (3a)/(3b) and (4a)/(4b) adheres to the following inequalities:  
         (ν 1   2 +ν 4   2 )&gt;2 da   max    
         a max &gt;0  (7)  
         [0113]    It should be noted that if a max &lt;0, usually equations (3b) and (4b) are used to obtain the proper result. Lastly, t 3  and t 2  may be negligible if:  
         ν 4 ≅ν 1   +t   1   a   max  and        d   ≅     s   1     ≅         1   2          a     a                 max            t   1   2       +       v   1          t   1                                 
         [0114]    Furthermore, t 1  and t 3  might be negligible if:  
         v 1 ≅v 2 ≅v 3 ≅v 4    
         [0115]    So far, the velocity and acceleration constraints have been easily met by the choice of a 1  and a 3  and equations (5) and (6). At this point, however, t 1 , t 2  and t 3  are real numbers, not necessarily integer multiples of a quantization interval. The following section deals with an additional constraint—the quantization of t 1 , t 2  and t 3 .  
         [0116]    2. Quantization.  
         [0117]    This section describes in greater detail the quantization of the continuous, three-segment trajectory defined above, as performed by trajectory planner  102  in step  406  above (see FIG. 4).  
         [0118]    In this section, a new constraint is added to the continuous, three-segment trajectory defined above:  
         [0119]    t i  is an integer multiple of a quantization interval T, i.e.,  
           t   i   =k*T (i=1, 2, 3)  
         [0120]    In order to meet this constraint, t 1 , t 2  and t 3  must be quantized, i.e., converted to an integer multiple of the quantization interval T without violating the maximum velocity and acceleration constraints.  
         [0121]    Assume that instead of using t 1  to t 3  calculated in the previous section, a set of times t 1calc  to t 3calc  are chosen as follows:  
         [0122]    if (t 1 ≧T and t 2 &gt;T and t 3 ≧T), then let:  
         t 1calc =t 1 ;  
         t 2calc =t 2 ;  
         t 3calc =t 3 ;  
         [0123]    if (t 1 ≧T and t 3 ≧T and t 2 &lt;T), then let:  
           t   1calc   =t   1 +( t   2 /2);  
         t 2calc =0;  
           t   3calc   =t   3 +( t   2 /2);  
         [0124]    if (t 3 ≧T and t 1 &lt;T and t 2 ≧T), then let:  
           t   1calc   =t   1   +t   2 ;  
         t 2calc =0;  
         t 3calc =t 3 ;  
         [0125]    if (t 1≧T and t   3 &lt;T and t 2 ≧T), then let:  
         t 1calc =t 1 .  (8a)  
         [0126]    Notice that the following two cases are not covered:  
         [0127]    (t 3 ≧T and t 1 &lt;T and t 2 &lt;T)  
         [0128]    (t 1 ≧T and t 3 &lt;T and t 2 &lt;T)  
         [0129]    Both of these cases indicate that applying a single acceleration (or deceleration) of the maximum magnitude “almost” covers the total desired displacement d, and that the remaining required adjustments in position and/or velocity are smaller than the ones that can be performed during a multiple of the quantization interval T. The most straightforward way to deal with these cases is to accelerate or decelerate in order to meet the target velocity; then generate a new set of trajectories in order to compensate for the displacement error. The reason for doing this will become obvious in the following.  
         [0130]    For the cases described in equations (8a), the new accelerations a 1calc  and a 3calc  can be calculated from a modified version of (1) and (2):  
                       v     2      calc       =         a     1      calc            t     1      calc         +     v   1                     v     3      calc       =         a   2          t     2      calc         +     v     2      calc                       v   4     =         a     3      calc            t     3      calc         +     v     3      calc                 }           (     8      b     )                         s     1      calc       =         1   2          a     1      calc            t     1      calc     2       +       v   1          t     1      calc                         s     2      calc       =         1   2          a   2          t     2      calc     2       +       v     2      calc            t     2      calc                         s     3      calc       =         1   2          a     3      calc            t     3      calc     2       +       v     3      calc            t     3      calc                   }           (     8      c     )                               
 
         [0131]    Note that:  
         
       d=s 
       1 
       +s 
       2 
       +s 
       3 
       =s 
       1calc 
       +s 
       2calc 
       +s 
       3calc  
     
         [0132]    Then, we solve for a 1calc  and a 3calc :  
                     a     1      calc       =       (         -   2          v   1          t     1      calc         -     2        t     2      calc            v   1       -       t     3      calc            v   1       -       t     3      calc            v   4       +     2      d       )         t     1      calc            (       t     1      calc       +     2        t     2      calc         +     t     3      calc         )                       a     3      calc       =       (         v   1          t     1      calc         +     2        t     3      calc            v   4       -     2      d     +       v   4          t     1      calc         +     2        v   4          t     2      calc           )         (       t     1      calc       +     2        t     2      calc         +     t     3      calc         )          t     3      calc                         (     8      d     )                               
 
         [0133]    Now let t 1q , t 2q  and t 3q  be the quantized counterpart of t 1calc , t 2calc  and t 3calc , i.e.:  
           t   iq   =n   i   *T ; i=1, 2, 3; n=0, 1, 2, . . .   (9)  
         [0134]    Notice that t iq  can also be represented in terms of t icalc  as:  
           t   iq   =t   icalc   +dt   i   T ; i=1, 2, 3; (0≦dt i &lt;1)  (10)  
         [0135]    Where  
           dt   i   =t   iq   −t   icalc   =n   i   *T−t   icalc   (11)  
         [0136]    Also notice that if t i  is rounded towards infinity, then  
               ceil        (   x   )       =     {         x         only                 if                 x                 is                 an                 integer             y       else                   (   12   )                               
 
         [0137]    where y is equal to truncating the decimal part of x+1.  
         [0138]    Then (10) can be expressed as:  
           t   iq   =t   icalc   +dt   i   T ; i=1, 2, 3  (13a)  
           t   icalc   =K   i   T  (K i =0, 1, 2, . . . )  (13b)  
           t   iq =( K   i   +dt   1 ) T ; i=1, 2, 3  (13c)  
         [0139]    Where K i  and d i  are identically equal to zero only if t icalc  is equal to zero.  
         [0140]    Notice from equations (8a), (9), (10), (11) and (13) that:  
           t   1q   +t   2q   +t   3q   &lt;t   1calc   +t   2calc   +t   3calc +3 T≦t   1   +t   2   +t   3 +3 T   (14)  
         [0141]    Thus, equation (14) shows that the quantized execution time of the trajectory (t 1q +t 2q +t 3q ) is at most three quanta (3T) longer than the execution time described by the continuous, optimal solution (t 1 +t 2 +t 3 ).  
         [0142]    Given equation (14), the goal of quantization is to recalculate a 1  and a 3  as to still satisfy the velocity and acceleration constraints. Let the values of these new accelerations be a 1q  and a 3q , respectively. Then, equations (1) and (2) can be re written as:  
                       v     2      q       =         a     1      q            t     1      q         +     v   1                     v     3      q       =         a   2          t     2      q         +     v     2      q                       v   4     =         a     3      q            t     3      q         +     v     3      q                 }           (   13   )                         s     1      q       =         1   2          a     1      q            t     1      q     2       +       v   1          t     1      q                         s     2      q       =         1   2          a   2          t     2      q     2       +       v     2      q            t     2      q                         s     3      q       =         1   2          a     3      q            t     3      q     2       +       v     3      q            t     3      q                   }           (   14   )                               
 
         [0143]    (Note that a iq  are not quantized.)  
         [0144]    Note that a 2 =0 in equations (8a), (8b), (13) and (14). Then, we solve equations (8b), (8c), (13) and (14) for a 1q  and a 3q . Note that v 2q =v 2calc,  in order to guarantee the velocity constraint, and that s 1q +s 2q +s 3q =s 1calc +s 2calc +s 3calc =s 1 +s 2 +s 3 =d. Then, (8d) is substituted for a 1calc  and a 3calc . Subsequently:  
                     a     1      q       =     -           t     3      q            v   4       -     2      d     +       (       t     3      q       +     2        t     1      q         +     2        t     2      q           )          v   1             t     1      q            (       t     3      q       +     t     1      q       +     2        t     2      q           )                           a     3      q       =           (       2        t     3      q         +     t     1      q       +     2        t     2      q           )          v   4       +       v   1          t     1      q         -     2      d           t     3      q            (       t     3      q       +     t     1      q       +     2        t     2      q           )                                    (   15   )                               
 
         [0145]    Equations (15), are easy to implement but do not shed any light on the relation between the quantization interval and the magnitude of a 1q  and a 3q .  
         [0146]    In an alternative approach, we solve (8b), (8c) and (14) for a 1q  and a 3q . Note that v 2q =v 2calc , in order to guarantee the velocity constraint, and that s 1q +s 2q +s 3q =s 1calc +s 2calc +s 3calc =s 1 +s 2 +s 3 =d. However, (8d) is not substituted for a 1calc  and a 3calc . Subsequently:  
         a     1      q       =     -                 t     3      q            a     1      calc            t     1      calc         +       t     3      q            t     3      calc            a     3      calc         -       a     1      calc            t     1      calc     2       -                 2        v   1          t     1      calc         -     2        t     2      calc            a     1      calc            t     1      calc         -     2        t     2      calc            v   1       -       a     3      calc            t     3      calc     2                   t     1      q            (       t     3      q       +     t     1      q       +     2        t     2      q           )                                      -   2          t     3      calc            a     1      calc            t     1      calc         -     2        t     3      calc            v   1       +     2        v   1          t     1      q         +     2        t     2      q            v   1       +     2        t     3      q            v   1         _               a     3      q       =               2        t     3      q            a     1      calc            t     1      calc         +     2        t     3      q            t     3      calc            a     3      calc         -       a     1      calc            t     1      calc     2       -                 2        v   1          t     1      calc         -     2        t     2      calc            a     1      calc            t     1      calc         -     2        t     2      calc            v   1       -       a     3      calc            t     3      calc     2                   (       t     3      q       +     t     1      q       +     2        t     2      q           )          t     3      q                                      -   2          t     3      calc            a     1      calc            t     1      calc         -     2        t     3      calc            v   1       +     2        v   1          t     1      q         +     2        t     2      q            v   1       +     2        t     3      q            v   1       +     _                                a     1      calc            t     1      calc            t     1      q         +     2        a     1      calc            t     1      calc            t     2      q         +       t     3      calc            a     3      calc            t     1      q         +     2        t     3      calc            a     3      calc            t     2      q           _                           
 
         [0147]    Clearly these equations are too complicated as to shed any light on the behavior of a 1q  an a 3q  as a function of the calculated times, however they can be simplified to:  
                     a     1      q       =     -           A   1          a     1      calc         +       B   1          a     3      calc         +       C   1          v   1             t     1      q            (       t     3      q       +     t     1      q       +     2        t     2      q           )                           a     3      q       =           A   3          a     1      calc         +       B   3          a     3      calc         +       C   3          v   1             t     3      q            (       t     3      q       +     t     1      q       +     2        t     2      q           )                                    (   16   )                               
 
         [0148]    [0148]FIGS. 7A-7P show a collection of graphic surfaces representing the values of variables A 1 , B 1  and C 1  versus varying values of t 1  and t 3 , as described in equations (16).  
         [0149]    3. Special Cases  
         [0150]    Equations (15), (16) and (8d) show the effect of small displacements, i.e., cases where (t 3 ≧T and t 1 &lt;T and t 2 &lt;T) or (t 1 ≧T and t 3 &lt;T and t 2 &lt;T). Considering equations (16), if t 1calc  varies from 0.001T to 0.9999T, and t 3 calc varies from T to 4T, then B 3  is, in general, greater than or equal to the value of the denominator (see FIGS. 7A-7P). However, the value of A 3  will be subtracted from the value of the denominator due to the fact that the sign of a 1calc  is opposite to the sign of a 3calc . Thus, no real conclusion can be achieved. In contrast, the values of B 1  and A 1  will add up, in most cases, to a value greater than the denominator.  
         [0151]    Assuming that the magnitude of a 3calc  is “close” to the maximum value allowed (this will typically be the case), then a 1q  will be greater than the maximum allowed acceleration. As seen by the behavior of C 1 , if v 1  is not zero, it can reduce or increase the value of the quantized acceleration.  
         [0152]    A similar analysis of the case when t 3calc  varies from 0.001T to 0.9999T and t 1calc  varies from T to 4T shows that a 3q  that will be greater than the maximum value allowed, as shown in FIGS. 7A-7P. From the previous analysis, it is clear that the algorithm is well behaved for all values when at least two time intervals are greater than the quantization interval.  
         [0153]    In short, if the desired displacement and the initial and/or final velocity violate the three-segment routine constraints, a four-segment trajectory is used instead. The new trajectory will accelerate or decelerate at maximum acceleration until the appropriate velocity is zero. The trajectory then includes a three-segment trajectory to compensate for the remaining displacement.  
         [0154]    Another special case occurs when the maximum velocity constraint is violated by a small amount due to quantization, i.e., when t 1q  is slightly larger than one quantization interval, and v 1  is close to v max . In order to avoid the throughput impact of generating a four-segment trajectory in this case, the three-segment routine is reevaluated while setting v max  to a percentage of its value.  
         [0155]    IV. Computer System  
         [0156]    The functions performed by trajectory planner  102 , which is capable of calculating trajectories as shown in control flow  400 , are preferably implemented in software. Alternatively, the same may be implemented using hardware or a combination of hardware and software.  
         [0157]    In an embodiment of the present invention, trajectory planner  102  comprises a computer system which may be connected to a network. An example of such a computer system  600  is shown in FIG. 6. The computer system  600  represents any single or multiprocessor computer. Single-threaded and multi-threaded computers can be used. Unified or distributed memory systems can be used.  
         [0158]    The computer system  600  includes one or more processors, such as processor  604 . One or more processors  604  can execute software implementing the operations described in flow  400 . Each processor  604  is connected to a communication bus  602  (e.g., cross-bar or network). Various software embodiments are described in terms of this exemplary computer system. After reading this description, it will become apparent to a person skilled in the relevant art how to implement the invention using other computer systems and/or computer architectures.  
         [0159]    Computer system  600  also includes a main memory  606 , preferably random access memory (RAM), and can also include a secondary memory  608 . The secondary memory  608  can include, for example, a hard disk drive  610  and/or a removable storage drive  612 , representing a floppy disk drive, a magnetic tape drive, an optical disk drive, etc. The removable storage drive  612  reads from and/or writes to a removable storage unit  614  in a well known manner. Removable storage unit  614  represents a floppy disk, magnetic tape, optical disk, etc., which is read by and written to by removable storage drive  612 . As will be appreciated, the removable storage unit  614  includes a computer usable storage medium having stored therein computer software and/or data.  
         [0160]    In alternative embodiments, secondary memory  608  can include other means for allowing computer programs or other instructions to be loaded into computer system  600 . Such means can include, for example, a removable storage unit  622  and an interface  620 . Examples can include a program cartridge and cartridge interface (such as that found in video game devices), a removable memory chip (such as an EPROM, or PROM) and associated socket, and other removable storage units  622  and interfaces  620  which allow software and data to be transferred from the removable storage unit  622  to computer system  600 .  
         [0161]    Computer system  600  can also include a communications interface  624 . Communications interface  624  allows software and data to be transferred between computer system  600  and external devices via communications path  626 . Examples of communications interface  620  can include a modem, a network interface (such as Ethernet card), a communications port, etc. Software and data transferred via communications interface  624  are in the form of signals which can be electronic, electromagnetic, optical or other signals capable of being received by communications interface  624 , via communications path  626 . Note that communications interface  624  provides a means by which computer system  600  can interface to a network such as the Internet.  
         [0162]    The present invention can be implemented using software running (that is, executing) in an environment similar to that described above with respect to FIG. 6. In this document, the term “computer program product” is used to generally refer to removable storage unit  614 , a hard disk installed in hard disk drive  610 , or a carrier wave carrying software over a communication path  626  (wireless link or cable) to communication interface  624 . A computer useable medium can include magnetic media, optical media, or other recordable media, or media that transmits a carrier wave. These computer program products are means for providing software to computer system  600 .  
         [0163]    Computer programs (also called computer control logic) are stored in main memory  606  and/or secondary memory  608 . Computer programs can also be received via communications interface  624 . Such computer programs, when executed, enable the computer system  600  to perform the features of the present invention as discussed herein. In particular, the computer programs, when executed, enable the processor  604  to perform the features of the present invention. Accordingly, such computer programs represent controllers of the computer system  600 .  
         [0164]    In an embodiment where the invention is implemented using software, the software can be stored in a computer program product and loaded into computer system  600  using removable storage drive  612 , hard drive  610 , or communications interface  624 . Alternatively, the computer program product can be downloaded to computer system  600  over communications path  624 . The control logic (software), when executed by the one or more processors  604 , causes the processor(s)  604  to perform the functions of the invention as described herein.  
         [0165]    In another embodiment, the invention is implemented primarily in firmware and/or hardware using, for example, hardware components such as application specific integrated circuits (ASICs). Implementation of a hardware state machine so as to perform the functions described herein will be apparent to persons skilled in the relevant art(s).  
         [0166]    V. Conclusion  
         [0167]    While various embodiments of the present invention have been described above, it should be understood that they have been presented by way of example, and not limitation. It will be apparent to persons skilled in the relevant art(s) that various changes in form and detail can be made therein without departing from the spirit and scope of the invention. Thus the present invention should not be limited by any of the above-described exemplary embodiments, but should be defined only in accordance with the following claims and their equivalents.