Patent Publication Number: US-9428198-B2

Title: Monorail vehicle apparatus with gravity-augmented contact load

Description:
RELATED APPLICATIONS 
     This application is a continuation of presently allowed, U.S. patent application Ser. No. 14/550,960 filed on Nov. 22, 2014 which is a continuation-in-part of U.S. patent application Ser. No. 13/772,156 filed on Feb. 20, 2013 now U.S. Pat. No. 8,939,085. Each of these applications is incorporated herein in its entirety. 
    
    
     FIELD OF THE INVENTION 
     This application is related to monorail vehicle apparatus and methods for augmenting the normal load in monorail vehicles, and more precisely to augmenting the load between the drive wheel of such monorail vehicle and the traction surface through appropriate placement of the center of gravity of the monorail vehicle. 
     BACKGROUND ART 
     There are many types of vehicles designed to travel on several or on just one guide rail. Typically, such vehicles have one or more drive wheels that propel them along the guide rail. To accomplish this, a certain amount of torque has to be applied to the drive wheel or wheels engaged with the rail by a drive mechanism. In this way the state of motion of the vehicle can be controlled, e.g., motion at constant velocity or rapid acceleration as required by the application. 
     The drive force that is delivered by any drive wheels engaged with a guide rail is limited by traction. Consequently, since acceleration requires a certain amount of drive force and faster acceleration requires more force, the permissible acceleration is limited by traction. In many situations the drive force is applied by one traction wheel while others are provided for stability and control (e.g., idler wheels). Therefore it is usually the friction between the drive wheel and the bearing surface of the rail on which the drive wheel rolls that presents the limiting factor on maximum available drive force. 
     In a general configuration, for instance in a car, the center of gravity is balanced between the vehicle&#39;s wheels. A number of solutions exist to increase the normal contact load on traction wheels in such cases, including foils and springs. In fact, the prior art teaches that these solutions can also be applied in vehicles traveling on guide rails, including monorail vehicles traveling along just one rail. 
     For example, U.S. Pat. No. 5,069,141 to Ohara et al. discloses an overhead conveyor that provides increased reactive force and traction to a drive wheel on ascending rail sections. The conveyor engages the upper side of the track or rail. Its various means for creating a reactive force are positioned to engage the underside of the track to improve frictional forces during ascendancy. More precisely, the weight of the unit is employed to create the reactional force while guide rollers are resiliently biased by either separate springs or by making the guide rollers themselves resilient. Ohara&#39;s teachings are applicable to monorail type conveyors that convey articles along a path defined by the guide rail. 
     Another solution to monorail vehicles addressing stability and hill climbing capability with the aid of springs can be found in the teachings of U.S. Pat. No. 4,044,688 to Kita. Here a monorail transport apparatus travels while holding the monorail from above and below and uses a driving belt in conjunction with an auxiliary wheel. The apparatus deploys a compression spring to accomplish the intended objectives including increased traveling stability irrespective of the sinuousity of the monorail. 
     Still other solutions use hydraulics. For example, U.S. Pat. No. 5,372,072 to Hamy teaches a transportation system in which the vehicle is coupled to a track by a bogie whose wheels are mounted on mutually articulated frames. These frames are forcibly urged to pivot with the aid of hydraulic rams. In other words, Hamy teaches to achieve wheel contact load, and consequently maximum driving force, with the aid of certain types of hydraulics. 
     In contrast to the above references, some prior art solutions teach acting on the wheels of monorail vehicles without the use of springs or hydraulic elements. Rather, they teach to take advantage of the vehicle&#39;s own weight. For example, U.S. Pat. No. 3,935,822 to Kaufmann teaches a monorail trolley designed to travel on a monorail and having a truck in which the center of gravity of both the loaded and empty trolley truck is displaced with respect to the points of contact between the rail and the supporting wheel and the counter-wheel. This causes both wheels to engage firmly and adhere to the rail. Kaufmann&#39;s design accommodates rapid and easy placement of the truck on the monorail and permits the trolley to move up and down grades. He also teaches adjustments in the placement of the center of gravity without the use of springs or hydraulics. 
     There are many other prior art teachings that use the center of gravity of a monorail vehicle to achieve their objectives. The reader is referred here to U.S. Pat. Nos. 4,690,064 and 6,321,657 both to Owen as well as U.S. Pat. No. 7,650,843 to Minges and the many additional references cited therein. 
     Unfortunately, none of the prior art teachings, whether using springs, hydraulic elements or just the placement of the vehicle&#39;s center of mass are compatible with large increases in contact load on drive wheels of monorail vehicles that are light, low-cost and yet provide for periods of rapid acceleration along the guide rail as the vehicle transports itself between docking stations. Furthermore, the prior art does not address monorail vehicles that exhibit such desirable features and performance characteristics while being confined to travel along a low-grade (e.g., stock) rail that exhibits a substantial profile variation. 
     OBJECTS OF THE INVENTION 
     In view of the prior art limitations, it is an object of the invention to provide for monorail vehicle apparatus and methods that permit high accelerations by a monorail vehicle that is light and low-cost. More precisely, it is an object of the invention to reach these objectives by providing a constraint point with idler wheels to prevent lift-off while increasing the load on the drive wheel not only by the mass of the vehicle itself, but also by a moment established about a pivot point. 
     It is another object of the invention to provide for monorail vehicles and method that achieve such increased drive wheel loads without the use of additional springs or hydraulic elements, thus allowing the vehicle to be light weight and low-cost. 
     Still other objects and advantages of the invention will become apparent upon reading the detailed description in conjunction with the drawing figures. 
     SUMMARY OF THE INVENTION 
     Several advantageous aspects of the invention are secured by a monorail vehicle apparatus with a gravity-augmented normal load on a drive wheel. This goal is achieved by a judicious placement of a center of gravity of a monorail vehicle belonging to the apparatus. 
     The apparatus has a rail with a bearing surface and a contact surface that are non-parallel to the gravity vector. The vehicle has a structure that defines a pivot location against the bearing surface of the guide rail. Furthermore, the vehicle engages with the rail on the bearing surface and the contact surface. 
     In accordance with the invention, the monorail vehicle is mounted on the rail such that its center of gravity has a rear longitudinal offset r rl  from the pivot location. The center of gravity produces a moment N ap  about the pivot location. This moment N ap  is resisted by the contact force with the contact surface of the monorail vehicle at a constraint point on the contact surface. The constraint point is located at a front longitudinal offset r fl  from the pivot location. Since the contact surface is not parallel to the gravity vector, the contact force adds to the forces resisted by the monorail vehicle on the bearing surface. In other words, the moment N ap  contributes to the load on any actual engagement element of the monorail vehicle, e.g., the drive wheel engaged with the bearing surface of the rail at the pivot location. The value of the resultant normal load is typically much beyond a standard load generated by the mass of the monorail vehicle alone. 
     It should be noted that the force amplification of normal load on the drive wheel is not affected by which end of the monorail vehicle is designated as front and rear. The rear offset of the center of gravity described above is merely a choice made for purposes of the description. Anyone skilled in the art will recognize that front and rear can be swapped in any embodiment according to the invention. 
     In the preferred embodiment, the monorail vehicle has at least one wheel to move along the rail. Preferably, the vehicle has drive wheel engaged with the bearing surface for propelling the monorail vehicle along the rail. In this preferred embodiment, the vehicle has one or more idler wheels that engage the contact surface of the rail. Alternatively, both the vehicle has drive wheels for propelling the monorail vehicle along both the bearing and contact surfaces of the rail. In still other embodiments, the wheel engaged with the bearing surface can be an idler wheel and the wheel engaged with the contact surface can be a drive wheel. 
     In addition to rear longitudinal offset r rl  from the pivot location, the center of gravity can have a lateral offset r lat  defined from a rail centerline along which the rail extends. Similarly, the center of gravity can have a vertical offset r vert  from the rail centerline. 
     The vertical offset r vert  can be selected to achieve a number of performance requirements. For instance, if vertical offset r vert  is negative, i.e., it defines a location below the pivot point, the monorail vehicle will be more resistant to losing contact in spite of imposed displacements or external forces. Additionally, especially for a vehicle that frequently accelerates or decelerates, a nonzero r vert  will increase or decrease the loads on certain wheels depending on vehicle motion. It will also allow the peak traction to be tuned for acceleration or for braking, as the application demands. For example, a negative r vert  will result in higher normal loads and more available traction when the vehicle is slowing down than when it is accelerating; this may be desirable in some applications. 
     In many cases the bearing surface and the constraint surface of the rail are geometrically opposite each other, e.g., they are the top and bottom surfaces of the rail for square and rectangular cross-sections. Furthermore, in order to ensure proper localization of the monorail vehicle an alignment datum can be provided for locating the bogie at any of the docking locations along the rail. 
     Some applications extend to methods for propelling the monorail vehicle along the rail with increased drive wheel normal load. That goal is accomplished by properly mounting the vehicle on the rail to augment the preload through the placement of the vehicle&#39;s center of gravity. In certain embodiments, the rail can be non-featured and have a certain cross-section defined along a rail centerline (parallel with the X-axis or longitudinal axis). 
     The elements of the apparatus and steps of the methods claimed by the invention do not necessarily require assemblies with wheels to engage with the rail. As such in certain embodiments, the monorail vehicle may just have a hollow cross-section to slide over the guide rail within the spirit of the invention. Additionally, such an embodiment may encapsulate a drive wheel on the bearing surface to define a pivot point and idler wheel or wheels on the contact surface to define a constraint point according to the teachings. Yet, other variations may just have protuberances on the vehicle that make contact with the rail to define a pivot point on the bearing surface and a constraint point on the contact surface. 
     The details of the invention, including its preferred embodiments, are presented in the below detailed description with reference to the appended drawing figures. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWING FIGURES 
         FIG. 1  is a partial isometric view of a monorail vehicle apparatus according to the invention. 
         FIG. 2  is a partial elevation view of the monorail vehicle apparatus of  FIG. 1  showing the pivot location and lift-off constraint on the rail that supports the monorail vehicle. 
         FIG. 3  is a partial isometric view of the monorail vehicle apparatus of  FIG. 1  illustrating the degrees of freedom in the placement of the center of gravity of the monorail vehicle. 
         FIG. 4  is a partial isometric view of another monorail vehicle apparatus according to the invention. 
         FIG. 5  is a partial elevation view of the monorail vehicle apparatus of  FIG. 4  showing the details of application of the drive force by a drive wheel traveling on the contact surface. 
         FIG. 6A  is an isometric view of a single second assembly equipped with a number of idler wheels. 
         FIG. 6B  is an isometric view of a structure deploying the second assembly of  FIG. 6A  in conjunction with a first assembly also equipped with additional idler wheels. 
         FIG. 6C  is an isometric view illustrating how the structure of  FIG. 6B  is mounted on a guide rail. 
         FIG. 6D  is an isometric view illustrating mounted structure of  FIG. 6C  along with a chassis of a monorail vehicle deploying the structure to achieve gravity-augmented drive wheel preload in accordance with the invention. 
         FIG. 7  are cross-sectional views of suitable rails for monorail vehicles and methods of the present invention. 
         FIG. 8  is a perspective view of a monorail vehicle apparatus deployed to adjust mechanisms at docking locations in an outdoor environment. 
         FIG. 9  is a partial isometric view of the monorail vehicle apparatus according to the invention that does not use any additional structures or assemblies to slide over the guide rail. 
         FIG. 10  shows the center of gravity and the various offsets of the monorail vehicle of the embodiment illustrated in  FIG. 9 . 
         FIG. 11  is partial elevation view of a variation of the monorail vehicle of  FIG. 9  that encapsulates a drive wheel and idler wheels. 
     
    
    
     DETAILED DESCRIPTION 
     The figures and the following descriptions relate to preferred embodiments of the present invention by way of illustration only. It should be noted that alternative embodiments of the structures and methods disclosed herein will be readily recognized as viable options that can be employed without departing from the principles of the claimed invention. 
     Reference will now be made to several embodiments of the present invention, examples of which are illustrated in the accompanying figures. Similar or like reference numbers are used to indicate similar or like functionality wherever practicable. The figures depict embodiments of the present invention for purposes of illustration only. One skilled in the art will readily recognize from the following description that alternative embodiments of the structures and methods illustrated herein may be employed without departing from the principles of the invention described herein. 
     The present invention will be best understood by first reviewing the embodiment of a monorail vehicle apparatus  100  as shown in the isometric view afforded by  FIG. 1 . A monorail vehicle  102  belonging to apparatus  100  travels along a non-featured rail  104  that is supported on one or more posts or mechanical supports  105 . To understand the mechanics of the travel of monorail vehicle  102  we first review the definitions of relevant parameters in an appropriate coordinate system  106 . We also note that monorail vehicle  102  is not shown in full in  FIG. 1 . In fact, a substantial portion of monorail vehicle  102  is cut-away in this view for clarity. 
     It is convenient that coordinate system  106  be Cartesian with its X-axis, also referred to as the longitudinal axis by some skilled artisans, being parallel to a rail centerline  108  along which non-featured rail  104  extends. Both, rail centerline  108  and X-axis are also parallel to a displacement arrow  110  indicating the possible directions of travel of monorail vehicle  102 . It should be noted that arrow  110  shows that vehicle  102  can travel in either direction. In other words, vehicle  102  can travel in the positive or negative direction along the X-axis as defined in coordinate system  106 . Furthermore, coordinate system  106  is right-handed, and its Y- and Z-axes define a plane orthogonal to the direction of travel of vehicle  102 . 
     In addition to linear movement along any combination of the three axes (X,Y,Z) defined by coordinate system  106 , monorail vehicle  102  can also rotate. A total of three rotations are available to vehicle  102 , namely about X-axis, about Y-axis and about Z-axis. These rotations are indicated explicitly in  FIG. 1  by their corresponding names, specifically: roll, pitch and yaw. Although many conventions exist for defining three non-commuting rotations available to rigid bodies in three-dimensional space, the present one agrees with conventions familiar to those skilled in the art of mechanical engineering of suspensions. 
     In total, monorail vehicle  102  thus has six degrees of freedom; three translational ones along the directions defined by the axes (X,Y,Z) and three rotational ones (roll, pitch, yaw). The translational degrees of freedom are also referred to in the art as longitudinal translation along rail  104  (X-axis), lateral translation (Y-axis) and vertical translation (Z-axis). 
     Non-featured rail  104  has a rectangular cross-section  112 . Furthermore, top surface  114  of rail  104  is chosen to be the bearing surface and the geometrically opposite bottom surface  116  of rail  104  is chosen to be the contact surface. Note that bearing surface  114  and contact surface  116  are non-parallel, and indeed orthogonal (perpendicular) to a vector F g  denoting the force of gravity acting on monorail vehicle  102 . 
     Monorail vehicle  102  engages rail  104  such that it can travel along rail  104  in either direction, as already indicated by arrow  110 . The vehicle has a structure  118  that defines a pivot location  220  against bearing surface  114  of rail  104 . An axis through pivot location  122  and perpendicular to the X-Z plane can be used to sum the moments about pivot location  122 . In fact, such a pitch axis  124  through pivot location  122  is drawn in  FIG. 1  for clarity. 
     The monorail vehicle  102  includes a first assembly  126  for engaging rail  104  at pivot location  122 . First assembly  126  can have any number of first assembly wheels to engage rail  104 . In the present embodiment, first assembly  126  has just one wheel  128 , which is also a drive wheel that engages rail  104  on bearing surface  114 . Drive wheel  128  is connected to a drive mechanism  130  for moving or displacing vehicle  102  along rail  104  in either direction along the X-axis, as also indicated by displacement arrow  110 . 
     Although a person skilled in the art will recognize that any suitable drive mechanism  130  may be used, the present embodiment deploys a motor  132  with a shaft  134  on which drive wheel  128  is mounted. Thus, motor  132  can apply a corresponding torque to rotate shaft  134  about a rotation axis  136  and thereby drive wheel  128  that is engaged with top or bearing surface  114  of rail  104 . In this manner, motor  132  can use drive wheel  128  to propel vehicle  102  along the positive or negative longitudinal direction as defined by the X-axis of coordinate system  106 . 
     Further, the monorail vehicle  102  has a second assembly  138  for engaging rail  104  on its contact surface  116 . Second assembly  138  is designed to engage on contact surface  116  in such a way that it produces a contact force F c , explained in more detail in reference to  FIG. 2 , at a front longitudinal offset r fl  from pivot location  122 . More precisely, second assembly  138  engages contact surface with two second assembly wheels  140 A,  140 B that are constrained directly by contact surface  116  to prevent bogie  118  from pivoting about pitch axis  124 . 
     We now refer to  FIG. 2  where monorail vehicle apparatus  100  is shown in a partial elevation view. Here, pivot location  122  and contact force F c  against bottom or contact surface  116  of rail  104  are shown explicitly. More precisely, contact force F c  obtains a constraint point  142  between idler wheels  140 A,  140 B (note that only idler wheel  140 A is visible in  FIG. 2 ) of second assembly  138  and contact surface  116  at front longitudinal offset r fl  from pivot location  122 . 
     In accordance with the invention, monorail vehicle  102  is designed for producing a gravity-augmented normal load on drive wheel  128  and on idler wheels  140 A,  140 B. This objective is achieved by a judicious placement of a center of gravity  144  of vehicle  102 . Specifically, vehicle  102  has its center of gravity  144  offset longitudinally by r rl  from pivot location  122 . Such placement of center of gravity  144  produces a moment N ap  about pivot location  122  or rather about pitch axis  124  and thus generates the desired gravity-augmented preload at pivot location  122  and at constraint point  142 . As the value of rear longitudinal offset r rl  increases, the normal load can be increased much beyond a standard normal load generated by the mass of monorail vehicle  102  alone. 
     We now motivate the requirement for a large normal load F p  that is generated in accordance with the invention. F p  is a force parallel with gravity vector F g  shown acting on center of gravity  144 . Furthermore, the force of normal load F p  is experienced by drive wheel  128  of first assembly  126 . As the mass of monorail vehicle  102  increases, a drive force F d  (indicated by its vector in  FIG. 2 ) needed to accelerate it increases proportionately. Under ideal conditions, based on Newton&#39;s Second Law, the acceleration a mv  of monorail vehicle  102  of mass m mv  achieved by the application of drive force F d  would be given by: 
     
       
         
           
             
               
                 
                   
                     a 
                     mv 
                   
                   = 
                   
                     
                       F 
                       d 
                     
                     
                       m 
                       mv 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     1 
                   
                   ) 
                 
               
             
           
         
       
     
     In practice, however, rolling friction μ places an upper limit on drive force F d  that can be applied to a drive wheel. That is because the available drive force F d  is limited by the force of friction F r  at impending slip between drive wheel  128  and rail  114 , and more precisely between drive wheel  128  and bearing surface  114 . The maximum drive force F dmax  for a prior art vehicle on a horizontal guide rail in which no moment N ap  is used for increasing normal load is thus limited to:
 
 F   r   =F   dmax   =μm   mv   a   g   (Eq. 2)
 
where a g  is the Earth&#39;s gravitational acceleration that produces a downward force on any drive wheel. Consequently, when wishing to apply a large drive force F d , the selection of materials for prior art drive wheels becomes limited to high-friction substances to obtain a high coefficient of rolling friction μ. Unfortunately, high-friction substances frequently have the undesirable properties of high wear, high rolling friction, adhesion and high deformation. Typical prior art solutions involve the use of foils and springs to increase the load on the traction wheel. Such solutions are dependent on vehicle dynamics or require additional mechanisms that add weight and complexity to the vehicle.
 
     We now present the mathematical expressions that demonstrate the relationship between the location of center of gravity  144  of vehicle  102  and its static and dynamic behavior. We start by defining a reference frame that travels with vehicle  102  and has its origin at pivot location  122 . For simplicity, we adopt the following conventions to allow several vector quantities to be treated as scalars by taking the three-degree-of-freedom equations of motion and constraining them to motion along rail  104 ; this simplifies their unit directions a priori. Thus, vectors a g , F g , F p  and N ap  are assumed to have the directions shown in  FIG. 1  and will be treated as scalars. Negative values indicate that the direction is opposite of that shown in  FIG. 1 . Vectors F c  and r fl  will be similarly treated using the directions illustrated in  FIG. 2 . Offset vectors r rl , r vert  and r lat  of center of mass  144  will be treated as scalars by assuming the directions shown in  FIG. 3 . Lastly, vehicle acceleration vector a mv  is assumed to act in the positive x-direction according to coordinate system  106 . Without complete mathematical rigor, which will be clear from the context to one skilled in the art, we may use the same symbol to denote either the vector or the scalar quantity. 
     By placing center of gravity  144  of vehicle  102  at a longitudinal offset r rl  from pivot location  122  where drive wheel  128  contacts bearing surface  114 , and by providing constrained idler wheels  140 A,  140 B in second assembly  138  normal load F p  on drive wheel  128  is no longer limited by the mass m mv  of vehicle  102 . This is shown by simplifying the equations that result from performing a static balance of the forces in the vertical direction and a static moment balance about pitch axis  124  that passes through pivot point  122 . It is seen that normal load F p  on drive wheel  128  can be increased by manipulating the value of rear longitudinal offset r rl  of center of gravity  144  from pivot location  122 . We note that as shown with the orientation of wheels in  FIG. 1 , it is necessary that r rl /r fl  be non-negative so vehicle  102  does not flip off rail  104 . A conventional monorail vehicle would have both wheels on top of the rail and r rl /r fl  would be non-positive. 
     To better understand the result of increasing rear longitudinal offset r rl , we now review the forces acting on vehicle  102  constructed in accordance with the invention. This means vehicle  102  is travelling in a straight line at a constant velocity on a horizontal section of rail  104 . Gravitational force F g  acts on center of gravity  144  of vehicle  102  and is given by:
 
 F   g   =m   mv   a   g   (Eq. 3)
 
     The vector corresponding to this force is indicated in  FIGS. 1 &amp; 2 . Normally, load F p  on drive wheel  128  is limited to at most the gravitational force F g , as we saw above. In apparatus  100  of the invention, however, rear longitudinal offset r rl  of center of gravity  144  creates moment N ap  about pitch axis  124  that is expressed by:
 
 N   ap   =m   mv   a   g   r   rl   =F   g   r   rl   (Eq. 4)
 
     Under these conditions the value of rear longitudinal offset r rl  can be increased to achieve a large moment N ap . 
     With N ap  taken into account, we sum the moments around pitch axis  124 . The result gives:
 
Sum of the Moments about 124=( m   mv   *a   g   *r   rl )−( F   c   *r   fl )
 
     We can solve for contact force F c  on idler wheels  140  at point of contact  142  for the constant velocity case as follows: 
     
       
         
           
             
               
                 
                   
                     F 
                     c 
                   
                   = 
                   
                     
                       
                         m 
                         mv 
                       
                       * 
                       
                         a 
                         g 
                       
                       * 
                       
                         r 
                         rl 
                       
                     
                     
                       r 
                       fl 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     5 
                   
                   ) 
                 
               
             
           
         
       
     
     With F c  known, we can now sum the forces in the z-direction (along the vertical or Z-axis of coordinate system  106 ) on vehicle  102 . In particular:
 
Sum of the Forces in  Z=F   p   −F   c −( m   mv   *a   g )
 
     Setting this sum equal to 0, since vehicle  102  is not free to translate along Z-axis and solving for load F p  on drive wheel  128  we obtain: 
     
       
         
           
             
               
                 
                   
                     F 
                     P 
                   
                   = 
                   
                     
                       m 
                       mv 
                     
                     * 
                     
                       a 
                       g 
                     
                     * 
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               r 
                               rl 
                             
                             
                               r 
                               fl 
                             
                           
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     6 
                   
                   ) 
                 
               
             
           
         
       
     
     The loading on drive wheel  128  is governed by the factor of 
               1   +       r   rl       r   fl         ,         
and since
 
               r   rl       r   fl           
is nonnegative, this factor is clearly greater than one. This permits increasing the normal force F p  on the drive wheel  128  to a theoretically arbitrary limit. It will be clear to a skilled artisan that suitable modifications to these expressions using trigonometric relations allow this analysis to be generalized to a guide rail having a non-zero inclination angle (non-horizontal rail).
 
     In practice, the normal load F p  on drive wheel  128  is limited by a number of factors. First, moment N ap  produces stresses in vehicle  102  that require management. Additionally, a large normal load F D  can produce high rolling friction, increased wear and high deformation of drive wheel  128 . A person skilled in the art will understand the trade-offs between these loads and the advantages of loading drive wheel  128 . 
     Second, front longitudinal offset r fl  is limited by requirements on the performance of monorail vehicle  102 . Many vehicles must retain accurate location while resisting wear. The pitching of vehicle  102  on bearing surface  114  of rail  104  caused by the wear of wheels  140 A and  140 B can be described by: 
     
       
         
           
             
               Induced 
               ⁢ 
               
                   
               
               ⁢ 
               Pitch 
             
             = 
             
               
                 tan 
                 
                   - 
                   1 
                 
               
               ⁢ 
               
                 
                   ( 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Wheel 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     104 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     B 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     radius 
                   
                 
                 
                   r 
                   fl 
                 
               
             
           
         
       
     
     Further, the vibrational mode of vehicle  102  in pitch is a function of front longitudinal offset r fl . Assuming the pitch stiffness is dominated by the wheel, rather than chassis compliance, a larger r fl  will create a stiffer mechanism. 
     Third, rear longitudinal offset r rl  is also limited by requirements on the performance of apparatus  100 . By the requirement of apparatus  100 , the mass m mv  of monorail vehicle  102  is supported by a cantilevered portion of the chassis having of length equal to r rl . Vehicle  102  can thus be modeled as a cantilever beam with a mass; with its center of gravity  144  attached to the end of the beam. Vehicular strength and stiffness requirements dictate that r rl  cannot be arbitrarily increased. 
     For example, supposing that wheel compliance is negligible and the vehicle chassis is modeled as a compliant beam of uniform cross-section. The natural frequency of apparatus  100 , and in particular of vehicle  102  mounted on rail  104  can then be calculated as: 
     
       
         
           
             
               ω 
               nat 
             
             = 
             
               
                 
                   3 
                   * 
                   E 
                   * 
                   I 
                 
                 
                   
                     r 
                     rl 
                     2 
                   
                   * 
                   
                     ( 
                     
                       
                         r 
                         rl 
                       
                       + 
                       
                         r 
                         fl 
                       
                     
                     ) 
                   
                   * 
                   
                     m 
                     mv 
                   
                 
               
             
           
         
       
     
     Where E is the Young&#39;s Modulus of the structure of vehicle  102  and I is the area moment of inertia of the structure of the vehicle  102 . We therefore see that, for a given structural cross-section, r rl  is limited by a minimum natural frequency of the mechanical system represented by vehicle  102  mounted on rail  104  and cannot be arbitrarily increased. 
       FIG. 3  is a partial isometric view of monorail vehicle apparatus  100  that illustrates the full freedom in the placement of center of gravity  144  of vehicle  102  within a volume  146 . In this drawing we see that in addition to rear longitudinal offset r rl  from pivot location  122 , center of gravity  144  can have a lateral offset r lat  in the Y-Z plane along the Y-axis as defined in coordinate system  106 . Lateral offset r lat  is defined from rail centerline  108  along which rail  104  extends. This degree of freedom in the placement of center of gravity  144  can be useful when vehicle  102  is not symmetric in its lateral weight distribution and for other engineering reasons. 
     Similarly, center of gravity  144  can have a vertical offset r vert  from rail centerline  108 . Vertical offset r vert  is also in the Y-Z plane and along the Z-axis as defined in coordinate system  106 . Vertical offset r vert  is defined from pivot location  122 . 
     In principle, vertical offset r vert  can be set above rail centerline  108  or below it. With vertical offset r vert  above rail centerline  108  (direction shown in  FIG. 3 , and thus a positive scalar value), a displacement of center of gravity  144  in roll will create a contributing moment that exacerbates the displacement. By contrast, with r vert  set below pivot  122 , displacement of center of gravity  144  in roll will create an opposing moment. Any lateral or longitudinal forces, such as centrifugal forces due to centripetal acceleration a c  when monorail vehicle  102  travels along a curve in rail  104  will tend to displace center of gravity  144 . 
     In this application, r vert  has additional implications. The above example of loads at pivot location  122  where drive wheel  128  contacts bearing surface  114  assumed constant velocity. With acceleration in a straight path included, and using D&#39;Alembert&#39;s Principle of inertial forces to perform force and moment balances that sum to zero, the term for moment N ap  is different, namely:
 
Sum of the Moments= N   ap −( F   c   *r   fl )=0
 
where:
 
 N   ap   =m   mv   *a   g   *r   rl   −m   mv   *a   mv   *r   vert  
 
     Following this equation through, the expression for the normal load F p  on drive wheel  128  is: 
     
       
         
           
             
               F 
               P 
             
             = 
             
               
                 
                   m 
                   mv 
                 
                 * 
                 
                   a 
                   g 
                 
                 * 
                 
                   ( 
                   
                     1 
                     + 
                     
                       
                         r 
                         rl 
                       
                       
                         r 
                         fl 
                       
                     
                   
                   ) 
                 
               
               + 
               
                 
                   
                     
                       m 
                       mv 
                     
                     * 
                     
                       a 
                       mv 
                     
                     * 
                     
                       r 
                       vert 
                     
                   
                   
                     r 
                     fl 
                   
                 
                 . 
               
             
           
         
       
     
     It is clear that for r vert  set below pivot location  122  (negative scalar according to the vector convention established in  FIG. 3 ), a negative acceleration a mv  will produce a larger normal load F p  on drive wheel  128  at pivot location  122  where it contacts rail  104 . Alternatively, if r vert  is positive, a positive acceleration will produce a larger load F p  on drive wheel  128  at its contact point with rail  104 —i.e., at pivot location  122 . This is particularly helpful in applications where one direction of agility is more valuable than another. For example, if vehicle  102  must stop much faster than accelerate to achieve certain stopping distances, e.g., in order to comply with safety concerns, selecting a negative r vert  will allow vehicle  102  to achieve such short stopping distances without unnecessarily loading drive wheel  128  in normal operation. 
     For example, for a 50 kg vehicle  102  with a friction coefficient of about 0.3 seeking to achieve about 0.5 g acceleration, drive wheel  128  must be loaded to approximately 735 N (i.e., F p =735 N). With a standard vehicle, these agility parameters would not be achievable as the total available force from the mass of the vehicle is only 500 N. In accordance with the present invention, a designer can then select rear longitudinal offset r rl  to be 0.25 m and front longitudinal offset r fl  to be 0.5 m. This would correspond to a normal load F p  on drive wheel  128  of 735 N and thus permit vehicle  102  to achieve high agility requirements. 
     Further, suppose that vehicle  102  exhibiting the above parameters and offsets has to come to a complete stop from a speed of 8 m/s in less than 1 second for safety reasons. This would require an acceleration of 0.81 g and a normal load F p  on drive wheel  128  equal to about 1,200 N. A designer would want to avoid unnecessarily loading drive wheel  128  and could therefore select an r vert  so that braking would contribute to normal load F p  on drive wheel  128 . In this case, if the designer were to select r vert  of −0.6 m, then vehicle  102  would experience a normal force of 1,215 N on drive wheel  218  during braking, ceteris paribus. This permits vehicle  102  to achieve its braking parameters without unduly loading drive wheel  128  in normal operation. 
     In reviewing monorail vehicle apparatus  100  it is important to note, that since contact force F p  on drive wheel  128  rolling along top bearing surface  114  also benefits from the standard force of weight m mv a g  it is preferable that it roll along top surface  114  rather than bottom contact surface  116 . However, given a sufficiently large moment N ap , it is possible to provide one or more drive wheels that travel on bottom contact surface  116 . 
       FIG. 4  is an isometric view that illustrates a monorail vehicle apparatus  200  in which a monorail vehicle  202  traveling along rail  104  has a first assembly  204  with idler wheels  206 A,  206 B and a second assembly  208  with a drive wheel  210 . The drive mechanism associated with drive wheel  210  is not shown in  FIG. 4 . Persons skilled in the art will appreciate that a suitable drive mechanism can deploy any known motor. Drive mechanisms with a remote motor mounted in the main body of vehicle  202  and a belt drive for transmitting its torque to drive wheel  210  in order to minimize the mass of second assembly  208  are preferred. 
     A structure  212  connecting first and second assemblies  204 ,  208  with the main body of vehicle  202  establishes a pivot location  214  against bearing surface  114  of rail  104 . It is at pivot location  214  that idler wheels  206 A,  206 B belonging to first assembly  204  contact bearing surface  114 . More precisely, idler wheels  206 A,  206 B contact bearing surface  114  along a pitch axis  216  defined through pivot location  214 . 
     Referring now to  FIG. 5 , which shows a partial elevation view of monorail vehicle  202  of  FIG. 4 , we see that a moment N ap  is created about pitch axis  216  by the placement of center of gravity  218  of vehicle  202  at a rear longitudinal offset r rl  from pivot location  214 . Meanwhile, drive wheel  210  of second assembly  208  engages with bottom or contact surface  116  of rail  104  at a constraint point  220 . Constraint point  220  is located at a front longitudinal offset r fl  from pivot location  214 . 
     In this embodiment, load force F p  acts on idler wheels  206  (only idler wheel  206 B visible in  FIG. 5 ) at pivot location  214 . Contact force F c  acts on drive wheel  210  at constraint point  220 . Because contact force F c  is created by moment N ap  and is not augmented by the force of weight of vehicle  202 , drive force F d  that can be applied to drive wheel  210  in this embodiment is lower than in the preferred embodiment described above. Thus, vehicle  202  will generally not achieve the levels of agility attained by vehicle  102 . 
     In another embodiment, however, vehicle  202  may deploy one or more drive wheels in the place of idler wheels  206 A,  206 B. Clearly, when using drive wheels engaged with both top surface  114  and bottom surface  116  of rail  104  very high levels of agility can be achieved. In fact, both first and second assemblies  204 ,  208  can in general use any suitable combination of one or more drive wheels and one or more idler wheels. The idler wheels may include wheels that roll along surfaces of rail  104  other than bearing surface  114  and contact surface  116 . For example, idler wheels can be arranged to travel on side surfaces of rail  104  that are generally parallel with the gravity vector. 
       FIG. 6A  is an isometric view of an exemplary second assembly  300  that deploys a single idler wheel  302  for engaging a contact surface of a rail. Assembly  300  also has one idler wheel  304  for engaging one side surface of a rail and two idler wheels  306 A,  306 B for engaging the other side surface of a rail. In practical applications, assemblies with additional idler wheels are desirable since they help in stabilizing the monorail vehicle and constraining the rotational degrees of freedom (e.g., yaw and roll). 
       FIG. 6B  is an isometric portion of a structure  308  deploying second assembly  300  in conjunction with a first assembly  310 . First assembly  310  has a drive wheel  312  powered by a drive mechanism  314  that includes a motor  316 . In addition, first assembly  310  also has one idler wheel  318  for engaging one side surface of a rail and two idler wheels  320 A,  320 B for engaging the other side surface of a rail. 
       FIG. 6C  is an isometric view illustrating how structure  308  is mounted on a guide rail  322  that has a rectangular cross-section. Note that drive wheel  312  of first assembly  310  engages against a top surface of rail  322 , which is the bearing surface in this case. Idler wheel  302  of second assembly  300  engages against a bottom surface of rail  322 , which is the contact surface. The remaining idler wheels of assemblies  300 ,  310  engage the side surfaces of rail  322  to stabilize any monorail vehicle deploying structure  308 . 
     A center of gravity  324  of such monorail vehicle and its location with respect to assemblies  300 ,  310  is shown in  FIG. 6C  for reference. Note that besides the rear longitudinal offset (not expressly shown in  FIG. 6C ) center of gravity  324  can additionally exhibit a lateral and/or a vertical offset, as previously discussed. 
     An additional advantageous aspect of the invention involves the manner in which assemblies  300 ,  310  are mounted on structure  308 . Specifically, first assembly  310  and second assembly  300  support mutual rotation to provide for travel of any monorail vehicle using structure  308  along curves in rail  322 . Corresponding axes of rotation  326 ,  328  of first and second assemblies  310 ,  300  are indicated along with arrows indicating the possible rotations. 
       FIG. 6D  is an isometric view illustrating structure  308  attached to a chassis  330  of a monorail vehicle. The cover of monorail vehicle as well as its parts are not expressly shown in  FIG. 6D  for reasons of clarity. Because of the advantageous design and mutual rotation capability of first and second assemblies  310 ,  300  the monorail vehicle using structure  308  not only achieves normal load on drive wheel  312  exceeding that obtained by the force of weight alone, but also can move along curves in rail  322  that have a small radius of curvature. The rotation capacity of assemblies  310 ,  300  allow the monorail vehicle to navigate tight turns having a turning radius at least as small as the wheel base between the two rotating assemblies. 
     Those skilled in the art will recognize that the shape of curved monorail  322 , the manner in which a straight section of rail  322  blends with a turn, and the desired velocity of the monorail vehicle as it navigates through a turn all impact the loads that turning applies to the vehicle. It should also be recognized that provisions must be made to ensure that the rotating assemblies have a stable yaw equilibrium in all operational locations on monorail  322  to keep the assembly aligned with the tangent vector to monorail  322 . Among many possible options available to the designer, such stability could be provided by springs that generate a restoring force to bias the assembly to return to center. Another alternative is to incorporate multiple wheels into the rotating assembly to thereby provide alignment of the assembly to the tangent vector of monorail  322 . 
     The apparatus and method of invention are compatible with guide rails that are non-featured and have various cross-sections. In fact, a monorail vehicle with gravity-augmented normal load according to the invention can travel even along a low-grade stock rail that exhibits substantial profile variation. 
       FIG. 7  illustrates several suitable rails and their cross-sections along rail centerlines. Specifically, a rail  350  has a square cross-section  352  and can be used in the same way as previously discussed rails with rectangular cross-sections. Another suitable rail  354  has a rectangular cross-section  356 . Note that in the case of rail  354  all side surfaces are non-parallel to the gravity vector when mounted in the orientation shown. Triangular cross-section  356 , however, is not widely available and therefore it is desirable to use rectangular cross-section instead. 
     Another desirable rail  358  with circular cross-section  360  is also shown. Note that in the case of rail  358  additional mechanisms are required to constrain roll about longitudinal axis (X-axis). Still another possible rail  362  has a desirable closed cross-section afforded by its hexagonal cross-section  264 . Based on these non-exhaustive examples a person skilled in the art will recognize that there are many other suitable cross-sections that are compatible with the apparatus and methods of the present invention. 
       FIG. 7  shows in order of decreasing desirability two other possible cross-sections that can be used in non-featured rails deployed in monorail vehicle apparatus of the invention. Specifically, rails  366  or  370  with I cross-section  368  or T cross-section  372  may not be as desirable. Normally, rails  366 ,  370  with I and T cross-sections  368 ,  372  are easy to obtain and offer features that a vehicle could grasp rendering them popular with monorails. However, in apparatus with long unsupported spans of guide rail, such cross-sections are not as desirable due to their low torsional stiffness and resulting susceptibility to low frequency mechanical resonance modes. 
       FIG. 8  offers a perspective view of a monorail vehicle apparatus  400  deployed in accordance with the method of invention in an outdoor environment  402 . Apparatus  400  uses a low-cost, non-featured rail  404  made of steel and having a rectangular cross-section  406 . Rail  404  is suspended above the ground on posts  408  and has provisions  410  such as alignment data or other arrangements generally indicated on rail  404  for accurate positioning of a monorail vehicle  412  traveling on it. 
     Provisions  410  correspond to the locations of associated docking stations and are designed to accurately locate vehicle  412  at each one. Mechanical adjustment interfaces  420  for changing the orientation of corresponding solar panels  422  are present at each docking station. Further, vehicle  412  has a robotic component  414  for engaging with the interfaces  420  and performing adjustments to the orientation of solar panels  422 . 
     In accordance with the invention, vehicle  412  is agile and can accelerate and decelerate rapidly. Hence, it can move rapidly between adjustment interfaces  420  on relatively long unsupported spans of low-cost rail  404  with rectangular cross-section  406  exhibiting substantial profile variation (as may be further exacerbated by conditions in outdoor environment  402 , such as thermal gradients). These advantageous aspects of the invention thus permit rapid and low-cost operation of a solar farm while implementing frequent adjustments in response to changing insolation conditions. 
       FIG. 9  shows another preferred embodiment of the present invention that does not require first and second assemblies. In other words, the monorail vehicle  502  of the present invention comprises a hollow cross section that simply slides over guide rail  104  of our previous embodiments. 
       FIG. 10  is a partial isometric view of monorail vehicle apparatus  500  of  FIG. 9  that illustrates the full freedom in the placement of center of gravity  544  of vehicle  502  within volume  546  according to above teachings. The drawing shows pivot location  522  on bearing surface  114  and constraint point  542  on contact surface  116 . Note while pivot location  522  and constraint point  542  may appear to be in the body of monorail vehicle  502  in this three dimensional view, they are intended to be on the top or bearing surface  114  and on the bottom or contact surface  116  respectively of rail  104  where monorail vehicle  502  defines its pivot location and constraint according to preceding explanation. The drawing also shows the rear longitudinal offset r rl  from pivot location  522  and lateral offset r lat  from center of gravity  544  in the Y-Z plane and along the Y-axis as defined in coordinate system  106 . Lateral offset r lat  is defined from rail centerline  108  along which rail  104  extends. As in previous embodiments, this degree of freedom in the placement of center of gravity  544  can be useful when vehicle  502  is not symmetric in its lateral weight distribution and for other engineering reasons. 
     Similarly, center of gravity  544  has a vertical offset r vert  from rail centerline  108 . Vertical offset r vert  is also in the Y-Z plane and along the Z-axis as defined in coordinate system  106 . In principle, vertical offset r vert  can be set above rail centerline  108  or below it with the corresponding pros and cons taught above. 
     While the principles of the instant invention fully apply to embodiments where there are no other attachments or assemblies facilitating the mounting of monorail vehicle  502  over guide rail  104  and there are conceivable applications of such embodiments within the scope of the invention, a variety of practical applications will require monorail vehicle  502  to have wheels to counter friction and facilitate its motion along guide rail  104 . Alternatively, referring still to  FIG. 10 , it is conceivable for the instant invention to merely have protuberances or other suitable features for defining pivot location  522  and constraint point  542  on bearing surface  114  and contact surface  116  respectively. Such features will reduce friction as monorail vehicle  502  translates along guide rail  104  as will be apparent to people of skill. 
       FIG. 11  shows a partial elevation view of a similar embodiment of monorail vehicle apparatus  500  having a monorail vehicle  502  that has wheels to overcome friction and facilitate its motion along guide rail  104 . Specifically, monorail vehicle  502  has a drive wheel  528  against bearing surface  114  to propel it along guide rail  104  and idler wheels  540 A,  540 B (note that only idler wheel  540 A is visible in  FIG. 11 ) against contact surface  116 . Here, pivot location  522  and contact force F c  against bottom or contact surface  116  of rail  104  are shown explicitly. More precisely, contact force F c  obtains a constraint point  542  between idler wheels  540 A,  540 B and contact surface  116  at front longitudinal offset r fl  from pivot location  522 . Note the motor or drive mechanism responsible for translating monorail vehicle along rail  104  by rotating drive wheel  528  around rotation axis  536  is not shown in  FIG. 11 . Note also that alternate drive mechanisms for propelling monorail vehicle  502  in this embodiment are entirely possible within the scope of the invention and are not delved into detail further. Finally also note, that such an embodiment of the present invention may encapsulate additional idler and drive wheels against either bearing surface  114 , contact surface  116  or both, to provide requisite propulsion and stability to monorail vehicle  502 . 
     In accordance with the invention, monorail vehicle  502  is designed for producing a gravity-augmented normal load on drive wheel  528  and on idler wheels  540 A,  540 B. This objective is achieved by a judicious placement of center of gravity  544  of vehicle  502 . Specifically, vehicle  502  has its center of gravity  544  offset longitudinally by r rl  from pivot location  522 . Such placement of center of gravity  544  produces a moment N ap  about pivot location  522  or rather about pitch axis  524  and thus generates the desired gravity-augmented preload at pivot location  522  and at constraint point  542 . As the value of rear longitudinal offset r rl  increases, the normal load can be increased much beyond a standard normal load generated by the mass of monorail vehicle  502  alone. 
     Let us look at the requirement for a large normal load F p  that is generated in accordance with the invention. F p  is a force parallel with gravity vector F g  shown acting on center of gravity  544 . Furthermore, the force of normal load F p  is experienced by drive wheel  528  contained in monorail vehicle  502 . As the mass of monorail vehicle  502  increases, a drive force F d  (indicated by its vector in  FIG. 11 ) needed to accelerate it increases proportionately. Under ideal conditions, based on Newton&#39;s Second Law, the acceleration a mv  of monorail vehicle  502  of mass m mv  achieved by the application of drive force F d  is governed by Eq. 1 as explained above. 
     Further as explained above, in practice, rolling friction μ places an upper limit on drive force F d  that can be applied to a drive wheel. That is because the available drive force F d  is limited by the force of friction F r  at impending slip between drive wheel  528  and rail  104 , and more precisely between drive wheel  128  and bearing surface  114 . 
     As per above teachings, by placing center of gravity  544  of vehicle  502  at a longitudinal offset r rl  from pivot location  522  where drive wheel  528  contacts bearing surface  114 , and by providing constrained idler wheels  540 A,  540 B, normal load F p  on drive wheel  128  is no longer limited by the mass m mv  of vehicle  502 . This was taught above by simplifying the equations that result from performing a static balance of the forces in the vertical direction and a static moment balance about pitch axis  524  that passes through pivot point  522 . It is seen that normal load F p  on drive wheel  528  can be increased by manipulating the value of rear longitudinal offset r rl  of center of gravity  544  from pivot location  522 . As such, per the above teachings, we are directly led to the computation of gravitational force F g  (Eq. 3), moment N ap  (Eq. 4), contact force F c  (Eq. 5) and load F p  on drive wheel  528  (Eq. 6). 
     As explained earlier in reference to  FIG. 1-3 , the loading on drive wheel  128  is governed by a factor of 
               1   +       r   rl       r   fl         ,         
and since
 
               r   rl       r   fl           
is nonnegative, this factor is clearly greater than one. This permits increasing the normal force F p  on the drive wheel  128  to a theoretically arbitrary limit. However, the normal load F p  on drive wheel  128  is generally limited by a number of practical factors as previously explained. It will be clear to a skilled artisan that suitable modifications to the above expressions using trigonometric relations allow this analysis to be generalized to a guide rail having a non-zero inclination angle (non-horizontal rail).
 
     As in previous embodiments, it is also entirely conceivable in this embodiment to have the drive wheel propelling monorail vehicle  502  on contact surface  116  instead of bearing surface  114 , or drive wheels propelling the vehicle on both surfaces, within the scope of the invention. Furthermore, the present embodiment will also function on a low-grade stock rail that exhibits substantial profile variation or lack of smoothness of surface. Such low-grade stock rail, whose surface finish does not require highly sophisticated manufacturing processes is inexpensive to produce and easier to obtain than the rails of prior art whose surface characteristic need to be more refined. This opens up the instant invention to a variety of additional industrial applications, including the operation of a mobile robot to align the orientation of solar panels in a solar farm (refer to  FIG. 8  and associated explanation). 
     In view of the above teaching, a person skilled in the art will recognize that the apparatus and method of invention can be embodied in many different ways in addition to those described without departing from the spirit of the invention. Therefore, the scope of the invention should be judged in view of the appended claims and their legal equivalents.