Patent Publication Number: US-9417632-B2

Title: Path planning autopilot

Description:
RELATED APPLICATIONS 
     This application is a division of U.S. Ser. No. 13/470,028 (“Path planning autopilot”) filed on May 11, 2012 and incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     The disclosure is generally related to automatic control systems for vehicles 
     BACKGROUND 
     Guiding a farm tractor, construction vehicle, mining truck, army tank or other large vehicle according to a precise, desired path often saves time and energy compared to sloppy operating. Precise farm tractor control, for example, benefits farmers, consumers and society as a whole. Farmers are able to work more efficiently, and spend less money on fertilizers and pesticides. Consumers enjoy lower prices for high quality produce, and precise use of farm land and farm chemicals reduces waste and excess runoff. 
     Autopilot systems that guide farm tractors, spray trucks, harvesters and the like with high accuracy and repeatability contribute to efficient land and chemical use. For example, fields are often sprayed with booms 90-feet-wide or even larger. When using such a wide boom it would seem prudent to allow a few feet of overlap from one spray swath to the next. However, the overlap may be reduced to just inches if the tractor or spray truck is equipped with a high performance autopilot. The savings accrued from not double-spraying swath edges add up quickly on large area farms. 
     Existing autopilot systems are adequate for keeping vehicles on a predefined path. These autopilots are based on feedback control techniques and they can make a large farm tractor, for example, follow a line within one or two inches cross-track error.  FIG. 1A  shows a vehicle following a path under feedback control; vehicle  105  follows path  110 . When the cross track error, XTE, is smaller than the vehicle wheelbase, b, the vehicle is in a small signal regime or “close” to the desired path. 
     Feedback autopilots do not perform as well when a vehicle is far off from a desired path. Common situations where this occurs include joining a path from far away or turning around at the end of a path to join a nearby path.  FIG. 1B  shows a vehicle with a possible path  125  that could result from using a small-signal feedback autopilot in a large-signal regime. Vehicle  115  is guided by a feedback autopilot toward path  120 . When the cross track error is larger than the vehicle wheelbase, the vehicle is in the large signal regime or “far” from the desired path. 
     With conventional feedback autopilots, tradeoffs exist between performance in the small signal regime and acceptable behavior in the large signal regime. For example, high feedback gain that keeps cross track error small in the small signal regime may result in steep approaches to a desired path and oscillation when joining the path as shown in  FIG. 1B . Conventional autopilots avoid undesirable large-signal behavior by invoking heuristic limits when large deviations from a desired path are encountered. These limits mean that large-signal guidance is not as efficient as it could be. 
     In the examples above wheelbase is used as an example of a characteristic length scale to divide small and large signal regimes. Other characteristic lengths may be used; e.g. distance travelled in a characteristic autopilot response time. More generally, the small signal regime refers to any situation where a vehicle autopilot behaves as a linear, time-invariant system. The large signal regime, on the other hand refers to situations where non-linear behavior, such as steering angle limits or steering angle rate limits, occurs. 
     What is needed is a vehicle autopilot that offers high performance guidance regardless of whether a vehicle is near or far from a desired path. The autopilot should not only keep a vehicle on a path, but also guide it efficiently to join a path from any starting point. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  shows a vehicle following a path under feedback control. 
         FIG. 1B  shows a vehicle with a possible path that could result from using a small-signal feedback autopilot in a large-signal regime. 
         FIG. 2A  shows a vehicle characterized by an arbitrary position (x, y), heading (φ)), and curvature (θ); and a desired path for the vehicle to follow. 
         FIG. 2B  shows a vehicle following a planned path to join a desired path. 
         FIG. 3  shows a turn-around path planned for a tractor towing an implement inside a field boundary. 
         FIG. 4  is a block diagram of a path planning autopilot including feedback and feed-forward controllers. 
         FIG. 5  is a block diagram of a feedback control system based on path planning algorithms. 
         FIG. 6  is a block diagram of a feedback and feed-forward control system in feedback mode. 
         FIG. 7  is a block diagram of a feedback and feed-forward control system in combined feedback and feed-forward mode. 
         FIG. 8  is a feedback map for a double integrator system subject to constraints. 
         FIG. 9  is a plot of a surface representing cross track distance travelled during turns from all possible initial conditions of heading and curvature subject to constraints. 
         FIG. 10A  shows an example path for a vehicle joining a straight line desired path. 
         FIG. 10B  shows u, θ and φ versus time for the path shown in  FIG. 10A . 
         FIG. 11  shows two example joining paths. 
         FIG. 12  is a first diagram of curvature versus time or distance used in an iterative path planning method. 
         FIG. 13  is a second diagram of curvature versus time or distance used in an iterative path planning method. 
         FIG. 14  is an example graph of final cross track error versus iteration number in an iterative path planning method. 
         FIGS. 15A-15D  illustrate turn-around paths. 
         FIGS. 16A and 16B  illustrate speed dependent curvature and curvature rate limits. 
         FIG. 17  illustrates an approach to a curved desired path. 
         FIGS. 18A and 18B  illustrate joining paths that are not necessarily optimal. 
     
    
    
     DETAILED DESCRIPTION 
     Introduction 
     The path planning autopilot described below provides more sophisticated vehicle guidance than is possible with a conventional autopilot.  FIG. 2A  shows a vehicle  205  in a configuration defined by an arbitrary position (x, y), heading (φ), and curvature (θ); and a desired path  210  for the vehicle to follow. A path planning autopilot creates an efficient joining path that takes the vehicle from an arbitrary initial configuration (y, φ, θ) to a desired path. The initial configuration may be near or far from the desired path. As an example,  FIG. 2B  shows a vehicle  215  following a planned path  220  to join a desired path  225 . 
     Unlike conventional autopilots, a path planning autopilot offers direct control over constraints such as maximum permitted steering angle or maximum steering angle rate. Such constraints may be specified as functions of vehicle speed. Path planning techniques may be implemented as a complementary feed-forward capability of a feedback autopilot or they may replace feedback all together. 
     Many advanced maneuvers are possible with a path planning autopilot. Paths may be joined at up to specified maximum angles such as a, the angle between dotted line  230  and desired path  225  in  FIG. 2B . A desired path may be joined in a specified direction, a capability that is used in turn-arounds.  FIG. 3  shows a turn-around path planned for a tractor  305  towing an implement  310  inside a field boundary  315 . Planned path  320  is designed to keep implement  310  more than an implement half-width, W, from the boundary. Planned path  320  joins desired path  325  in a direction opposite to the tractor&#39;s present heading. 
     System Overview 
     A path planning autopilot calculates an efficient path for a vehicle to follow. The autopilot guides the vehicle along this path using a pure feedback or a combined feed-forward and feedback control architecture. Two different approaches for calculating efficient paths are described. 
     A typical scenario that illustrates the utility of a path planning autopilot is joining a desired path. In farming, for example, a tractor operator may wish to join predefined, straight line for planting, spraying or harvesting. There is no requirement to join the line at any particular position. Rather, the operator would simply like to join the line in the least time within constraints of comfortable operation. 
     The joining path that guides a vehicle from its initial position to a desired line has continuous curvature, restricted to be less than a maximum curvature limit. This means that the joining path has no sharp corners and takes into account practical limits on how fast a vehicle can change its curvature and heading. 
     For wheeled vehicles, curvature is determined by wheel steering angle and wheelbase. Path planning autopilots may be used with truck-like vehicles having two steerable wheels and two or more fixed wheels, center hinge vehicles, tracked vehicles, tricycle vehicles, etc. Some vehicle control systems accept curvature as a direct input rather than steering angle. 
     Joining paths are calculated assuming constant vehicle speed. Joining paths may be recalculated whenever speed changes or they may be recalculated periodically regardless of whether or not vehicle speed has changed. 
     Joining paths are constructed as a series of straight line, circular and clothoid segments. Clothoids are used to join segments of different, constant curvature; e.g. to join a line and a circular arc. The curvature of a clothoid changes linearly with curve length. Clothoids are also called Euler spirals or Cornu spirals. Reasonable approximations to clothoids may also be used. 
       FIG. 4  is a block diagram of a path planning autopilot including feedback and feed-forward controllers. The autopilot includes a main unit  405  having a feedback controller, a feed-forward controller, a processor and memory. Functional blocks of the main unit may be implemented with general purpose microprocessors, application-specific integrated circuits, or a combination of the two. 
     The main unit receives input from GNSS (global navigational satellite system) receiver  410  and sensors  415 . The main unit sends output to actuators  420 . GNSS receiver  410  may use signals from any combination of Global Positioning System, Glonass, Beidou/Compass or similar satellite constellations. The receiver may use differential corrections provided by the Wide Area Augmentation System and/or real-time kinematic positioning techniques to improve accuracy. The receiver provides the main unit with regular position, velocity and time updates. 
     Sensors  415  include steering sensors that report vehicle steering angle and heading sensors that report vehicle heading to the main unit. Heading may be sensed from a combination of GNSS and inertial measurements. Steering angle may be sensed by a mechanical sensor placed on a steering linkage or by inertial measurements of a steering linkage. Sensors  415  may also be omitted if heading and curvature of vehicle track are estimated from GNSS measurements alone. 
     Actuators  420  include a steering actuator to steer a vehicle. A steering actuator may turn a vehicle&#39;s steering wheel or tiller, or it may control a steering mechanism directly. On some vehicles steering actuators operate hydraulic valves to control steering linkages without moving the operator&#39;s steering wheel. 
     A path planning autopilot may include many other functions, sensors and actuators. For simplicity, only some of those pertinent to path planning operations have been included in  FIG. 4 . A display and user input devices are examples of components omitted from  FIG. 4 . User input may be necessary for system tuning, for example. The system of  FIG. 4  may operate in different modes, three of which are illustrated in  FIGS. 5, 6 and 7 . 
     A path planning autopilot may use path planning techniques in a feedback control system or it may use path planning in a combined feedback/feed-forward control system.  FIG. 5  is a block diagram of a feedback control system based on path planning algorithms. In  FIG. 5 , feedback controller  505  controls vehicle  510  by sending curvature rate (u) or curvature (θ) command signals to it. Actual vehicle heading (φ) and cross track error (y) as measured by a GNSS receiver, heading sensor and/or other sensors are combined with desired curvature, heading and track (θ, φ, y) to form an error signal for controller  505 . 
     The method used by controller  505  to determine u or θ is described in detail below. At this point suffice it to say that the controller repeatedly calculates a joining path to guide the vehicle from its present position to a desired path, and generates control commands that cause the vehicle to follow the joining path. The controller is not based on conventional proportional-integral-derivative (PID) techniques. 
       FIGS. 6 and 7  are block diagrams of a feedback and feed-forward control system in feedback, and combined feedback and feed-forward modes, respectively. In  FIGS. 6 and 7 , feedback controller  605  controls vehicle  610  by sending curvature (θ) commands to it. (Alternatively, feedback controller  615  may send curvature rate (u) commands to the vehicle.) Actual vehicle heading (φ) and cross track error (y) as measured by a GNSS receiver, heading sensor and/or other sensors are combined with parameters of a desired path  620  to form an error signal for controller  605 . Feedback controller  605  may be a conventional PID controller unlike feedback controller  505 . 
     In  FIGS. 6 and 7 , solid arrows represent active signal paths, while dashed arrows represent signal paths that are turned off. Thus in  FIG. 6 , feedback controller uses an error signal derived from the difference between measured and desired paths to generate feedback control signal θ FB . This mode may be used when the vehicle is close to the desired path. 
     When the vehicle is far from a desired path, the control system switches to the mode illustrated in  FIG. 7 . A typical scenario in which a vehicle is far from a desired path is when an autopilot system is engaged before the vehicle is established on the path. A set of parallel swaths may have been created to guide a tractor over a farm field, for example, and the tractor operator may want the autopilot to guide the tractor to the nearest one. When this happens, path joining planner  615  calculates an efficient path from the vehicle&#39;s present position to the desired path. 
     Path joining planner  615  calculates a joining path to guide the vehicle to the desired path and provides a feed-forward control signal, OFF, which is combined with feedback control signal θ FB  to guide vehicle  610 . For purposes of feedback controller  605 , the joining path is the desired path until the original desired path  620  is reached. Thus feedback reference signal (φ, y) is derived from the path joining planner during that time. The path joining planner is the feed-forward mechanism described in connection with  FIG. 4 . Methods used by the feed-forward, path joining planner are described in detail below. 
     Alternatively, once path joining planner  615  calculates a joining path, that joining path continuing onto the original desired path (from the point where the joining path meets the original path) may become the new desired path. The original desired path before the point where the joining path meets it may then be discarded. 
     Autopilots having a path-planning, feedback and feed-forward control system like that of  FIGS. 6 and 7  may use a switch to alternate between modes. The mode is selected based on vehicle distance from desired path  620 ; near the path feedback control is used, far from it combined feed-forward and feedback control is used. Criteria to determine “near” and “far” may be based on characteristic lengths such as vehicle wheelbase or distance travelled in a characteristic response time, or whether or not steering angle or steering angle rate limits are encountered. 
     Alternatively, a control system may operate in the mode illustrated in  FIG. 7  continuously. When the vehicle is near the desired ( 620 ) path, output signals from the path joining planner are small and may have negligible practical influence on the feedback control system. Thus, the path joining planner may be kept online all the time. 
       FIGS. 5-7  show measured (φ, y) data being combined with desired path data and then sent to feedback controller  505  or  605 . However, measured data could alternatively be estimated by an observer, e.g. a Kalman filter, before being combined with desired path data. 
     Two different approaches for calculating efficient joining paths are described in detail below an integral approach and an iterative approach. Each approach is first described in terms of planning paths to join the straight line y=0. Extensions to joining curves are discussed later. Symbols used in the discussion below are defined in Table 1. 
                     TABLE 1                  Symbol definitions.                             (x, y)   position                       φ   heading           θ   curvature           u   curvature rate                        
These terms are used to describe vehicle configuration and characteristics of paths that vehicles follow.
 
Integral Approach
 
     The time evolution of a continuous curvature trajectory in the x-y plane is described by: 
     
       
         
           
             
               
                 
                   
                     
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     A vehicle following a trajectory defined by (1) has heading φ, curvature θ, and curvature rate u. The vehicle&#39;s velocity is v. 
     The problem to be solved is Find a path that leads a vehicle onto a target line in the least amount of time, subject to constraints. The target line to be joined is the x axis (y=0) and x position may be dropped from consideration entirely. Since (1) is invariant with respect to time, the initial time may be set to t=0. 
     A formal statement of the optimal control problem is Find a piecewise continuous solution for u for all times from t=0 to t=t f  such that constraints (2)-(6) below are met and the final time, t f , is minimized. 
     
       
         
           
             
               
                 
                   
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     Here θ max  is the maximum allowed curvature and μ is the maximum allowed rate of change of curvature. μ normalizes u such that |u|≦1. (Note μ&gt;0) Equations (2) and (3) represent initial conditions; equation (4) represents final conditions. Equation (5) represents the dynamics of the problem, while equation (6) is a constraint on curvature. 
     Physically, φ=φ±n·2π, so φ f =n·2π where n is an integer. In some cases, heading may be restricted. For example, one may require that a vehicle approach a target line on a heading no greater than a maximum angle away from the line. In that case |φ(t)|≦φ max  where 0&lt;φ max ≦π. 
     The problem just described is complicated. Rather than solve it directly, it is easier to consider a simpler problem Calculate a path that leads a vehicle from its initial heading and curvature to some desired final heading and zero curvature. In other words, ignore the y coordinate or cross track error. The statement of the control problem now becomes Find a piecewise continuous solution for u (−1≦u≦1) for all times from t=0 to t=t f  such that constraints (7)-(10) below are met and the final time, t f , is minimized. 
     
       
         
           
             
               
                 
                   
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     Here φ f ≠0 represents any desired final heading. 
     This problem is an example of the so-called “double integrator” problem, solutions to which are well known. The optimal input, u(t), is 0, 1 or −1 depending on the values of φ and θ. Let Δφ=φ−φ f  and 
                 ϕ   c     =       v   ·     θ   max   2         2   ⁢   μ         ,         
and define:
 
     
       
         
           
             
               
                 
                   
                     
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     which may be combined as: 
     
       
         
           
             
               
                 
                   
                     
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     The optimal input, u(t)=ū(Δφ(t), θ(t)) is then: 
     
       
         
           
             
               
                 
                   
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                         
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                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ϕ 
                         
                         = 
                         0 
                       
                     
                     
                       
                         θ 
                         = 
                         0 
                       
                     
                   
                   
                     
                       0 
                     
                     
                       for 
                     
                     
                       
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ϕ 
                         
                         &gt; 
                         
                           ϕ 
                           c 
                         
                       
                     
                     
                       
                         θ 
                         = 
                         
                           - 
                           
                             θ 
                             max 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ϕ 
                         
                         &lt; 
                         
                           - 
                           
                             ϕ 
                             c 
                           
                         
                       
                     
                     
                       
                         θ 
                         = 
                         
                           θ 
                           max 
                         
                       
                     
                   
                   
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                   
                   
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ϕ 
                         
                         ≥ 
                         
                           ϕ 
                           c 
                         
                       
                     
                     
                       
                         θ 
                         ≠ 
                         
                           - 
                           
                             θ 
                             max 
                           
                         
                       
                     
                   
                   
                     
                       1 
                     
                     
                       for 
                     
                     
                       
                         0 
                         ≤ 
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ϕ 
                         
                         ≤ 
                         
                           ϕ 
                           c 
                         
                       
                     
                     
                       
                         θ 
                         &gt; 
                         
                           
                             f 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ϕ 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                         
                           - 
                           
                               
                           
                           ⁢ 
                           
                             ϕ 
                             c 
                           
                         
                         ≤ 
                         Δϕ 
                         &lt; 
                         0 
                       
                     
                     
                       
                         θ 
                         ≥ 
                         
                           
                             f 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ϕ 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                   
                   
                     
                       
                         - 
                         1 
                       
                     
                     
                       for 
                     
                     
                       
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ϕ 
                         
                         ≤ 
                         
                           ϕ 
                           c 
                         
                       
                     
                     
                       
                         θ 
                         ≠ 
                         
                           θ 
                           max 
                         
                       
                     
                   
                   
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                         
                           - 
                           
                             ϕ 
                             c 
                           
                         
                         ≤ 
                         Δϕ 
                         ≤ 
                         0 
                       
                     
                     
                       
                         θ 
                         &lt; 
                         
                           
                             f 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ϕ 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                       
                     
                     
                       
                           
                       
                     
                     
                       
                         0 
                         &lt; 
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ϕ 
                         
                         ≤ 
                         
                           ϕ 
                           c 
                         
                       
                     
                     
                       
                         θ 
                         ≤ 
                         
                           
                             f 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ϕ 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
       FIG. 8  is a feedback map for the double integrator system subject to constraints given above. The feedback map shows two example trajectories in the (Δφ, θ) plane. The first starts at A and ends at 0, while the second starts at E and ends at 0. Beginning at A, curvature (θ) increases until the maximum curvature limit is reached at C. The curvature changes sign at B. From C to D the curvature is maximum and constant. The curvature decreases from D to 0. The second example trajectory starts at E and ends at 0. Beginning at E, curvature decreases until F. Curvature then increases until 0 is reached. 
     When optimal inputs (14) are applied to a vehicle obeying equation (1), the vehicle executes optimal turns. Following the optimal (Δφ, θ) plane trajectories, from A to 0 or from E to 0, of  FIG. 8  leads to optimal turns in the (x, y) plane. The optimal (Δφ, θ) plane trajectory from A to 0 leads to an optimal turn in the (x, y) plane constructed from clothoids (from A to B, B to C and D to 0) and a circular arc (from C to D). The optimal (Δφ, θ) plane trajectory from E to 0 leads to an optimal turn in the (x, y) plane constructed from two clothoids (from E to F and F to 0). 
     Thus, an optimal turn is defined as a path obtained by applying optimal double-integrator inputs (14) to continuous curvature equation (1). Optimal turns consist of (i) a clothoid, (ii) a circular arc followed by a clothoid, (iii) a clothoid followed by a clothoid, or (iv) a clothoid followed by a circular arc followed by a clothoid. In theory optimal turns guide a vehicle to a target configuration (e.g. zero heading and curvature) in minimum time. In practice optimal turns may at least be used to create an efficient joining path from an initial configuration to a target configuration while obeying heading, curvature, curvature rate or other constraints. 
     The approach described above may now be extended to account for cross track error. The control problem statement becomes Find a piecewise continuous solution for u (−1≦u≦1) for all times from t=0 to t=t f  such that constraints (15)-(18) below are met and the final time, t f , is minimized. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             y 
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             ϕ 
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             θ 
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             y 
                             0 
                           
                         
                       
                       
                         
                           
                             ϕ 
                             0 
                           
                         
                       
                       
                         
                           
                             θ 
                             0 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             y 
                             ⁡ 
                             
                               ( 
                               
                                 t 
                                 f 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             ϕ 
                             ⁡ 
                             
                               ( 
                               
                                 t 
                                 f 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             θ 
                             ⁡ 
                             
                               ( 
                               
                                 t 
                                 f 
                               
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             y 
                             f 
                           
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ⅆ 
                       
                         ⅆ 
                         t 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               y 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               ϕ 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               θ 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               v 
                               · 
                               sin 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ϕ 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             v 
                             · 
                             
                               θ 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             μ 
                             · 
                             
                               u 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                     θ 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ∈ 
                   
                     { 
                     
                       θ 
                       | 
                       
                         
                           - 
                           
                             θ 
                             max 
                           
                         
                         ≤ 
                         θ 
                         ≤ 
                         
                           θ 
                           max 
                         
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     The target line is the x-axis and φ f =0. This problem and its solutions are invariant with respect to y so any trajectory may be shifted in y such that y f =0. Equivalently, if y 0  is picked correctly, then y f =0. That value of y 0  that results in y f =0 is defined as  y (φ 0 , θ 0 ). 
     Or, said another way starting from initial conditions of heading, curvature and cross track error (φ 0 , θ 0 , y 0 ), an optimal turn to zero heading and zero curvature will end on the x-axis if y 0 = y (φ 0 , θ 0 ).  y  is found by integrating ∫y(t)dt backwards until a desired heading and curvature are reached for all permissible values of φ and θ: 
     
       
         
           
             
               
                 
                   
                     y 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       y 
                       0 
                     
                     + 
                     
                       
                         ∫ 
                         0 
                         t 
                       
                       ⁢ 
                       
                         
                           v 
                           · 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 ϕ 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           t 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   
                     ϕ 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       ϕ 
                       0 
                     
                     + 
                     
                       
                         ∫ 
                         0 
                         t 
                       
                       ⁢ 
                       
                         
                           v 
                           · 
                           
                             θ 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           t 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   
                     θ 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       θ 
                       0 
                     
                     + 
                     
                       
                         ∫ 
                         0 
                         t 
                       
                       ⁢ 
                       
                         
                           μ 
                           · 
                           
                             u 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           t 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     where appropriate u(t) is determined from (14). 
       FIG. 9  is a plot of y= y (φ, θ) for all allowed φ and θ. In  FIG. 9 , v=1, μ=0.6 and θ max =50 degrees. All solutions for y(t f )=φ(t f )=θ(t f )=0 lie on the y= y (φ, θ) surface plotted. In particular, trajectories similar to those of  FIG. 8  are plotted as a dark curve on the surface. 
     The solution of the whole problem, including cross track error, may now be specified for two cases with and without heading constraints. When there are no heading constraints: 
                                       u   _     ⁡     (       ϕ   -     ϕ   f       ,   θ     )               if   ⁢           ⁢     y   ⁡     (   t   )         =       y   _     ⁡     (     ϕ   ,   θ     )                     u   ⁡     (       ϕ   ⁡     (   t   )       ,     θ   ⁡     (   t   )       ,     y   ⁡     (   t   )         )       =             u   _     ⁡     (       ϕ   -     ϕ   f     -     π   2       ,   θ     )               if   ⁢           ⁢     y   ⁡     (   t   )         &gt;       y   _     ⁡     (     ϕ   ,   θ     )                                 u   _     ⁡     (       ϕ   -     ϕ   f     +     π   2       ,   θ     )               if   ⁢           ⁢     y   ⁡     (   t   )         &lt;       y   _     ⁡     (     ϕ   ,   θ     )                     (   22   )               
where the choice of φ f =2πn (n being an integer) determines whether the first turn is to the right or left. In practice both possibilities may be calculated and whichever resulting trajectory is shorter may be used.
 
     Solution (22) prescribes an optimal turn onto the target line if y(t)= y (φ, θ). If y(t)≠ y (φ, θ) then an optimal turn is made toward a line perpendicular to the target line; i.e. to a line heading 
             ±       π   2     .           
A turn toward the perpendicular line, or subsequent travel along it, yields a trajectory that intersects with
 
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           
                             y 
                             _ 
                           
                           ⁡ 
                           
                             ( 
                             
                               ϕ 
                               , 
                               θ 
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         ϕ 
                       
                     
                     
                       
                         θ 
                       
                     
                   
                   ] 
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     hence a solution that reaches the final desired condition is always found. 
     When heading constraints are present: 
     
       
         
           
             
               
                 
                   
                     
                       
                           
                       
                     
                     
                       
                         
                           u 
                           _ 
                         
                         ⁡ 
                         
                           ( 
                           
                             ϕ 
                             , 
                             θ 
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           if 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             y 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         = 
                         
                           
                             y 
                             _ 
                           
                           ⁡ 
                           
                             ( 
                             
                               ϕ 
                               , 
                               θ 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           u 
                           ⁡ 
                           
                             ( 
                             
                               
                                 ϕ 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               , 
                               
                                 θ 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               , 
                               
                                 y 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                       
                     
                     
                       
                         
                           u 
                           _ 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               ϕ 
                               - 
                               
                                 min 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       ϕ 
                                       max 
                                     
                                     , 
                                     
                                       π 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                             , 
                             θ 
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           if 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             y 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         &gt; 
                         
                           
                             y 
                             _ 
                           
                           ⁡ 
                           
                             ( 
                             
                               ϕ 
                               , 
                               θ 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                       
                     
                     
                       
                         
                           u 
                           _ 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               ϕ 
                               + 
                               
                                 min 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       ϕ 
                                       max 
                                     
                                     , 
                                     
                                       π 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                             , 
                             θ 
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           if 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             y 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         &lt; 
                         
                           
                             y 
                             _ 
                           
                           ⁡ 
                           
                             ( 
                             
                               ϕ 
                               , 
                               θ 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     Solution (24) prescribes an optimal turn onto the target line if y(t)= y (φ, θ). If y(t)≠ y (φ, θ) then an optimal turn is made toward a line inclined at an angle min(φ max , π/2) to the target line. (min(x, y) equals x or y, whichever is least.) The turn toward the line, or subsequent travel along it, yields a trajectory that intersects with 
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           
                             y 
                             _ 
                           
                           ⁡ 
                           
                             ( 
                             
                               ϕ 
                               , 
                               θ 
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         ϕ 
                       
                     
                     
                       
                         θ 
                       
                     
                   
                   ] 
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     hence a solution that reaches the final desired condition is always found. 
     Solution (22) or (24) may be used in feedback controller  505  in  FIG. 5 . 
       FIG. 10A  shows an example path for a vehicle joining a straight line desired path while  FIG. 10B  shows u, θ and φ versus time for the path shown in  FIG. 10A . In  FIG. 10A , x-axis (y=0)  1005  is the target line to be joined. A joining path is shown that leads a vehicle from an initial condition (y 0 =2.5, φ 0 =0.3, θ 0 =0) to the target line. The joining path is composed of clothoid  1010 , circular arc  1015 , clothoid  1020 , straight line  1025 , clothoid  1030 , circular arc  1035  and clothoid  1040 . In  FIG. 10B , discontinuous line  1045  (for example) represents input u, piecewise continuous line  1050  represents curvature θ, smooth line  1055  represents heading φ, dotted lines  1060  represent a curvature limit, and dot-dash lines  1065  represent a heading limit. An optimal solution for u has only three possible values 1, 0 or −1. The slope of curvature θ is flat during circular and straight line segments; the slope of θ during clothoid segments reflects the maximum allowed steering angle rate. Heading  1055  varies smoothly; its maximum slope reflects the maximum allowed steering angle. 
     Joining paths may be computed by solving (1) numerically and choosing u according to (22) or (24) at each time step. Alternatively (19), (20) and (21) may be integrated analytically and the x coordinate may be found from: 
     
       
         
           
             
               
                 
                   
                     x 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       x 
                       0 
                     
                     + 
                     
                       
                         ∫ 
                         0 
                         t 
                       
                       ⁢ 
                       
                         
                           v 
                           · 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 ϕ 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           t 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     The following steps may be used to compute joining paths to target lines starting from an initial vehicle configuration [y 0  φ 0  θ 0 ]:
         (1) (a) If y 0 = y (φ 0 , θ 0 ), then go to step (5).    (b) If y 0 ≠ y (φ 0 , θ 0 ) and φ 0 =−sgn(φ 0 )·φ max  (or       

                 ϕ   0     =       -     sgn   ⁡     (     ϕ   0     )         ·     π   2         )         
and θ 0 =0, then go to step (3).
          (c) Otherwise, continue to step (2).   (2) Compute an optimal turn onto a line inclined at an angle to the target line by ±φ max  or       

             ±       π   2     .           
Call the time at which this part of the trajectory ends t 1 . Then:
          (a) If y(t 1 )= y (φ(t 1 ), θ(t 1 )), then go to step (5).    (b) If y 0     y (φ 0 , θ 0 ) and y(t 1 )   y (φ(t 1 ), θ(t 1 )), then go to step (4).    (c) Otherwise, continue to step (3).   (3) Add a straight line until time t 3  where t 3  satisfies y(t 3 )= y (φ(t 3 ), θ(t 3 )). Go to step (5).   (4) Find t 2 , t 0 &lt;t 2 &lt;t 1 , where t 2  satisfies y(t 2 )= y (φ(t 2 ), θ(t 2 )). Discard the trajectory computed so far for all times t&gt;t 2 . Continue to step (5).   (5) Compute an optimal turn onto the target line.       

     Expressed without mathematical symbols these steps are, starting from any initial vehicle cross track error, heading and steering angle:
         (1) (a) If the vehicle is at the right distance away from the target line to start the final turn, then go to step (5).    (b) If not, and if the heading is at the maximum allowed angle to the target line (or perpendicular to the target line) then go to step (3).    (c) Otherwise, continue to step (2).   (2) Compute an optimal turn onto a line inclined at the maximum allowed angle to the target line or perpendicular to it.    (d) If this turn ends at the right distance away from the target line to start the final turn, then go to step (5).    (e) If, at some point during this turn, the right distance away from the target line to start the final turn is reached, then go to step (4).    (f) Otherwise, continue to step (3).   (3) Add a straight line proceeding in the direction of the current heading ending at the right distance away from the target line to start the final turn; then go to step (5).   (4) Find the point at which the right distance was reached. Continuing from this point, go to step (5).   (5) Compute an optimal turn onto the target line.       

     Joining paths computed by steps (1)-(5) above include one or more clothoids and may also include circular arcs and straight lines. An example joining path structure is clothoid-line-clothoid-arc-clothoid. 22 other combinations of clothoids, arcs and lines are possible. In general, a joining path from an arbitrary initial condition to a target line consists of (i) an optimal turn, (ii) a line followed by an optimal turn, (iii) an optimal turn followed by an optimal turn, or (iv) an optimal turn followed by a line followed by an optimal turn. Note that the line specified in steps (2) and (3) may have zero length in which case one optimal turn is followed directly by another. 
       FIG. 11  shows two example joining paths. In  FIG. 11 , x-axis  1105  is the target line. The longer joining path consists of an optimal turn  1110  to heading 
                 -   π     4     ,         
a straight line  1115 , and an optimal turn  1120  onto the target line. The shorter path shows how, following steps (4) and (5) above, an optimal turn  1125  to heading
 
               -   π     4         
got cut off at  y  and an optimal turn  1130  to the target line was added from that point. In both cases
 
               -   π     4         
is a maximum allowed approach heading (φ max ) specified by a vehicle owner or operator.
 
     Note that for the shorter path in  FIG. 11 , and other paths like it, instead of computing a complete turn according to step (2) and then finding where it crosses  y  according to step (4), one could check if  y  is reached at each point in as the turn is computed. Other, small procedural variations are no doubt possible. 
     Joining paths may be computed using the integral approach by the path joining planner of  FIGS. 6 and 7 . 
     Iterative Approach 
     An alternative approach to computing joining paths is based on consideration of curvature versus time (or distance) diagrams such as those shown in  FIGS. 12 and 13 . In this approach a turn toward a target line is initiated and allowed to progress for a certain amount of time. The consequences of the turn are then computed, the consequences being whether or not a turn in the opposite direction can be constructed that results in a trajectory that ends on the target line with zero heading and curvature. If a satisfactory turn is not achieved, the procedure is repeated with the initial turn allowed to progress for another time increment. This iterative approach is most conveniently implemented as code for a microprocessor. It does not readily admit to analytic solution, unlike the integral approach. 
       FIG. 12  is a first diagram of curvature versus time or distance used in an iterative path planning method. In  FIG. 12  time is used as the abscissa; speed is assumed constant, so distance could be used instead. The situation depicted is one in which an initial, positive turn has been allowed to progress until point B, at which point the turn is unwound in the opposite direction. Steering angle limits are included in the curvature limit ±θ max  as steering angle and curvature are related by wheelbase in a truck-like vehicle and by similar parameters in other vehicle types. Steering angle rate limits are included in the maximum slope of θ(t). The shaded area under the θ(t) curve (i.e. the time integral of θ(t)) is proportional to heading change; see also (20) above. 
     Considering  FIG. 12  now in detail: At time t=0, curvature θ=0, heading φ=φ i , and cross track error y=y i . A positive turn is begun at maximum steering angle rate until θ reaches θ max . θ remains at θ max  until point B. For each point B, the heading at point C is tested to see if it exceeds a maximum heading constraint. If it has, point B has been incremented too far and the previous increment must be used instead. In  FIG. 12 , a maximum heading constraint is not reached and curvature is steadily decreased from point B until it reaches the minimum limit, —θ max .  FIG. 13 , discussed below, shows a case where a maximum heading limit is reached. 
     For each point B, the part of the diagram in  FIG. 12  representing negative curvature, i.e. the part to the right of point C is then constructed such that it has enough area to make the final heading φ f =0. In other words, the sum of the initial heading φ i , the accumulated positive heading change (shaded area denoted “+”), and the accumulated negative heading change (shaded area denoted “−”) equals zero. 
     Next the final cross track error y f  is computed. If y f =0, then the problem is solved; if not, then point B is incremented one time step and the procedure just outline is repeated. 
       FIG. 13  is a second diagram of curvature versus time or distance used in an iterative path planning method.  FIG. 13  is similar to  FIG. 12 . In  FIG. 13 , however, the initial curvature is not zero so the starting condition is (θ≠0, φ i , y i ). The curvature does not reach θ max , but heading does reach its maximum and this leads to the zero curvature (constant heading) line before point B. Point B is incremented in time until a negative turn can be constructed that results in the final condition (θ f =0, φ f =0, y f =0). 
     The iterative method does not provide a mechanism to decide a priori whether an initial positive or negative (i.e. left or right) turn leads to a shorter joining path. Results obtained from an initial positive turn from a given starting condition may be compared to results obtained from an initial negative turn from the same condition. Whichever initial turn leads to the shorter joining path may then be used in practice. 
     The iterative method proceeds until the final cross track error is zero or at least minimized to within a range that can be handled by a conventional feedback autopilot. Cross track error less than the vehicle wheelbase or some other characteristic length may be close enough.  FIG. 14  is an example graph of final cross track error versus iteration number. As illustrated in  FIG. 14 , the final cross track error, y f , may initially increase with further iterations of point B. However, once the cross track error becomes a decreasing function of B iteration number, an estimate of how far B must be incremented to achieve acceptable final cross track error may be obtained without trying every intermediate B iteration. Gradient descent methods may be used to estimate the optimum B, for example. 
     Paths generated with the iterative method may be used in the path joining planner of  FIGS. 6 and 7 . 
     Extensions 
     Joining paths computed by either the integral or iterative approach may be used in advanced maneuvers. Velocity depending constraints may be included. Either approach may be extended to join desired paths that are not straight lines. Finally, non-optimal joining paths may sometimes be desirable. 
     A 180 degree turn is an example of a path planning maneuver that has particular applicability in farming. Plowing, fertilizer, seeding and other operations are often performed along a set of parallel rows spanning a field. Upon reaching the end of a row, a farm vehicle turns around and joins the next row. A path planning autopilot can create a joining path from one row to another using the integral or iterative approaches described above. 
       FIGS. 15A-15D  illustrate turn-around paths. In each of  FIGS. 15A-15D  a curved joining path connects two straight line paths, e.g. field rows. The joining path has been computed using methods described above. In  FIG. 15A  two straight line paths (y=0 and y=2) are just the right distance apart that a maximum curvature joining path connects them without overshoot. In  FIG. 15B  the straight line paths are farther apart so the joining path includes a perpendicular, straight line section. In  FIG. 15C  the straight line paths are too close to be joined without overshoot. This path may be followed in either direction so that the overshoot crosses either the starting or ending straight line as desired. In  FIG. 15D  the joining path overshoots both straight line paths equally. This joining path is constructed by starting at half way point where y=half the distance between straight line paths, φ=−π/2, and θ=θ max  and computing a joining path to the line y=0. This generates half of the joining path shown in the figure; the other half is just the mirror image. 
     Once any of the joining paths of  FIGS. 15A-15D  are computed, the distance required for the path in the x direction is known. This information may be used to start the turnaround at an appropriate point such that a vehicle (or an implement towed by it) stays within a boundary as shown in  FIG. 3   
     Turnarounds are an illustrative example of the utility of path planning. There is, of course, no requirement that the straight lines to be joined are parallel. The path planning approaches described above are general and accommodate any initial cross track error, heading and steering angle condition. 
     The integral and iterative path planning approaches assume constant vehicle speed. If the speed changes, a new joining path may be computed to take the change into account. For any given vehicle speed maximum limits may be placed on parameters such as curvature and curvature rate. For a typical farm vehicle, these limits correspond to maximum allowed steering angle and steering angle rate.  FIGS. 16A and 16B  illustrate speed dependent curvature and curvature rate limits. 
     In  FIGS. 16 and 16B , curvature θ and curvature rate dθ/dt, respectively, are plotted as solid curves versus velocity v. These curves show an example of how an autopilot may be tuned. As vehicle speed increases, curvature and curvature rate decrease. Dashed lines (“max”) represent maximum limits in each figure. The shape of the curvature and curvature rate versus velocity curves may be adjusted to suit vehicle operator or autopilot manufacturer preferences. 
     Curvature vs. velocity and curvature rate vs. velocity may be tuned in several different ways. One way is to drive a vehicle through a series of test maneuvers to discover the vehicle&#39;s maximum capabilities at various speeds. The autopilot may record a path followed by the vehicle, including speed, and derive maximum curvature and curvature rate from recorded data. A second way is for the autopilot to accept user input to specify a certain make and model of vehicle with known capabilities. A third way is to ask a vehicle operator to drive the vehicle as aggressively as he would ever want the autopilot to drive it. The autopilot may then record a path followed by the vehicle while the operator is driving and use recorded data to derive maximum operator-desired curvature and curvature rate. Some operators do not realize the full turning capabilities of their vehicles. Matching autopilot performance to operator comfort level helps prevent frightening the operator when he lets the autopilot take over vehicle control. 
     So far path planning techniques have been described in terms of creating joining paths to straight lines. It is also possible to join curved paths, however. The overall strategy for joining a curve is to use path planning to guide a vehicle close enough to the curve for feedback techniques to take over. First a joining path to the nearest tangent of the curve is created. As the joining path approaches the curve the nearest tangent changes; a new joining path to the new nearest tangent is then created. The process of creating and following joining paths to a nearest tangent may be repeated to construct an overall joining path that ends close enough to the curved path that feedback techniques may be used to guide a vehicle the rest of the way. 
       FIG. 17  illustrates an approach to a curved desired path.  FIG. 17  fixes  1705  and  1710  are indicated by triangles. Curve  1715  is the desired path to be joined. The tangent to the desired path closest to fix  1705  is labeled “ 1 ” while the tangent to the desired path closest to fix  1710  is labeled “ 2 ”. Dashed lines connect each fix to their respective tangent. Curve  1720  is a joining path from the first fix to the first tangent “ 1 ”, while curve  1725  is a joining path from the second fix to the second tangent “ 2 ”. Dotted curves on either side of curved path  1715  indicate maximum cross track error considered “close” to the curved path. A vehicle closer to the curved path than either of these two dotted curves may be guided onto the curved path with a small signal feedback autopilot. Farther away, a large signal, feed-forward autopilot, e.g. based on path planning techniques, may be used. 
     Constructing a joining path to curved desired path  1715  proceeds as follows. Starting at fix  1705 , tangent “ 1 ” to the nearest point on curve  1715  is computed. Path planning techniques, such as integral or iterative approaches, are used to compute joining path  1720  to tangent “ 1 ”. Upon reaching fix  1710 , tangent “ 2 ” to the now nearest point on curve  1715  is computed. Path planning techniques are used to compute joining path  1725  to tangent “ 2 ”. This process may be repeated as many times as needed to construct a smooth joining path. 
     After the joining path has been calculated the vehicle follows it under the guidance of an autopilot such as described in connection with  FIGS. 4-7 . Once the vehicle is close to the curved path it is guided the rest of the way by a feedback autopilot. Dashed curve  1730  is the track that results from feedback guidance to curve  1715 . 
     When or how often to recalculate a new tangent to the current nearest point on the curved desired path may be decided based on how far away from the path the vehicle starts and/or the degree of curvature of the path. A new tangent may also be calculated at regular intervals regardless of distance or path curvature. 
     As previously mentioned, vehicle speed changes, whether joining a curved or straight desired path, may be accommodated by recalculating a joining path. High-speed maneuvers take more space than low-speed maneuvers, however, and a path planning autopilot may take this into account. When planning a maneuver near a boundary, such as a turnaround at the edge of a farm field, a path planning autopilot may compute a joining path for the current speed and also a higher speed. If currently operating at 50% of the maximum vehicle speed, for example, an additional path appropriate for 90% of the maximum vehicle speed may also be computed. This way the autopilot can provide a margin in case the vehicle operator decides to speed up. Anticipating higher speeds helps prevent becoming stuck in maneuvers that lead to undesirable high-speed joining paths. A path planning autopilot may also be restricted by specifying the maximum curvature rate μ to be less than that prescribed by vehicle absolute limits. Smaller values of μ lead to less sharp turns. Paths may be recalculated whenever heading, curvature or curvature rate limits change. 
     There are some cases when optimal, least-time joining paths may not be desired.  FIGS. 18A and 18B  illustrate joining paths that are not necessarily optimal. In  FIGS. 18A and 18B , the desired path is the line y=0. In  FIG. 18A , a joining path starts from approximately (0, −1.8); in  FIG. 18B , a joining path starts from approximately (0, 4). 
     The joining path in  FIG. 18A  is one that maintains initial heading until getting close enough to the target line (e.g. reaching  y ) to begin an optimal final turn. This path is not necessarily optimal because a steeper approach to the target line may be allowed. The joining path in  FIG. 18B  is one that maintains initial curvature until getting close enough to the target line (e.g. reaching  y ) to begin an optimal final turn. This path is not necessarily optimal because a greater curvature may be allowed. Thus the joining paths of  FIGS. 18A and 18B  begin with a (possibly) non-optimal and end with an optimal final turn. 
     Non-optimal paths such as those illustrated in  FIGS. 18A and 18B  may be desirable from the point of view of operator comfort. In both cases, the non-optimal paths begin by guiding the vehicle to continue doing what it is already doing no heading change in the case  FIG. 18A ; no steering angle change in the case of  FIG. 18B . This behavior provides a smooth, non-startling transition from operator control to autopilot control of a vehicle. Of course, strategies that lead to non-optimal paths are not desirable if the vehicle does not eventually reach a position (e.g.  y ) from which an optimal final turn may be made. Maintaining an initial heading, for example, does not work if that heading takes the vehicle away from the target path. 
     A path planning autopilot and methods for computing joining paths have been described. The autopilot provides efficient vehicle guidance even when a vehicle is far from a desired path. The autopilot also offers direct control over parameters such as steering angle rate without relying on heuristic limits. 
     The above description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the principles defined herein may be applied to other embodiments without departing from the scope of the disclosure. Thus, the disclosure is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.