Abstract:
A mobile base movable relative to a surface is disclosed. The base includes a main housing having at least two wheels pivotably and rotatably mounted to the housing with each wheel having a steering axis and a rotation axis. The steering and rotation axes are nonintersecting and are offset by a known caster distance. Further, included are drive means for rotating the wheels to roll along the surface, steering means for pivoting the wheels and changing their heading with respect to the surface, and controller means for reading an input vector from a host processor. The input vector is a three dimensional force torque vector.

Description:
This is a continuation-in-part of international application number PCT/US97/15605, filed on Sep. 5, 1997, which claims a priority of U.S. provisional application Ser. No. 60/025,406, filed on Sep. 6, 1996. 

   FIELD OF THE INVENTION 
   The present invention relates generally to manned or unmanned omni-directional, normal-wheeled vehicles. More precisely, this invention involves a method and apparatus for controlling the motion of mobile base used as a vehicle. 
   BACKGROUND OF THE INVENTION 
   Mobile bases with intended applications in robotics and industrial automation require a great deal of movement flexibility in order to be fully utilized. Current robot mobile bases often employ a “synchro-drive” mechanism—a complex set of gears and pulleys that constrains the wheels to steer and to translate simultaneously. Since steering and translation movement is fully decoupled in this system, “steering in place” is possible. (Contrast this maneuverability with that of a car.) While this allows a good deal of maneuverability, its mechanical complexity makes it difficult and expensive to manufacture. Additionally, this type of mobile base has limited movement due to its fixed orientation (i.e. its inability to rotate). 
   In general, a rigid body constrained to move in a plane (i.e. a mobile base moving on the floor) has three degrees of freedom (DOFs) such as, movement in the x direction, movement in the y direction and rotation. Combining these three DOFs results in movement in any direction while simultaneously rotating. 
   A holonomic mobile base, has the ability move in this manner and change its motion at any time. By constructing a mobile base out of wheels that each have two degrees of freedom (one for steering, one for translation) all three DOFs are possible under the proper control, and complex mechanisms found in synchro-drive mobile bases are no longer needed. Additionally, these 2-DOF wheels (2-DOFWs) can be easily integrated and manufactured as modular wheel assemblies. A complete description of one example of a robot base and modular wheel assembly is provided by U.S. patent application Ser. No. 09/134,241, by Holmberg et al., incorporated herein by reference. 
   It is also possible to increase the “caster” of a 2-DOFW by moving the translation axis behind the steering axis ( FIG. 1 ) and making it resemble a caster wheel. Unlike a 2-DOFW wheel with no caster (i.e. intersecting steering and translation axes), a mobile base constructed with caster 2-DOFWs is fully holonomic under the proper control. 
   Various problems arise, however, when trying to control a mobile base constrained to three DOFs with more than three controllable DOFs (e.g. a base with four 2-DOFWs has eight DOFs). A base with this many DOFs under improper control will certainly result in undesired motion and motor axes that “fight” each other. (i.e. consider two wheels facing opposite directions playing tug-of-war.) 
   This in turn creates wheel slippage, increased tire wear, increased power consumption, and more frequent mechanical problems. 
   It is considered important in robotics to be able to accurately assess the motion of the mobile base either through direct measurement or through estimation. This motion estimation can also be “summed-up” over time to create a “dead reckoned” position estimate of the mobile base with respect to fixed coordinates, which is also useful for autonomous tasks such as navigation. That is, consider a robot that wishes to navigate to a location (room) of which it knows the x-y coordinates. Motion estimation is also responsible for proper control of the mobile base, as will be described below. A control algorithm which minimizes wheel slippage will also allow for maximum motion estimation accuracy. 
   SUMMARY OF THE INVENTION 
   The present invention overcomes the deficiencies in the prior art described above, and provides motion control for a mobile robot base that is more accurate and more maneuverable Referring to FIG.  3  and the Detailed Description of the Preferred Embodiments, an overview of the inventive control sequence is provided in the following paragraph. 
   A supervisory controller reads the input vector from a host processor, and maps the input vector to the desired axis motion vector (i.e. the desired motion of each axis) by using the equations in Section  2  below. It then predicts if the axes are capable of the desired motion by calculating their control envelopes (i.e. the motion possible within one control cycle Δt.) as described in Section  3 . If all axes are capable of the desired motion within one control cycle the desired axis motion vector is passed to the low-level controller. If one or more axes are incapable of the desired motion, a modified axis motion vector is calculated as described in Section  4  and passed to the low-level controller. The modified axis motion vector lies within the control envelopes of all  2 N axes while minimizing control error (i.e. the difference between the commanded input vector and the actual base motion.) The control algorithm then estimates the motion of the mobile base since the last control cycle using the technique described in Section  5 . The estimated motion is then used to update the position and orientation [x B ,y B ,ψ B ] T  of the base coordinate frame in the fixed world coordinate frame as described also in Section  5 . The updated position and orientation is then made available for the host processor to read. The control algorithm then repeats the whole process by beginning another control cycle. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1A  is a perspective view showing a mobile base wheel with no caster offset. 
       FIG. 1B  is a perspective view showing a mobile base wheel with a caster offset. 
       FIG. 2  is a schematic diagram showing the flow of command information which controls the motion of the mobile base. 
       FIG. 3  is a schematic diagram showing the steps of a method for controlling the motion of the mobile base. 
       FIG. 4  is a perspective view schematically showing the layout of the mobile base wheels and the base and world coordinate systems. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   Section l—Overview of the Control Algorithm 
   Referring to Table 1, a description of notations and variables used herein is provided. 
   
     
       
             
             
           
         
             
               TABLE 1 
             
             
                 
             
           
           
             
               N 
               Number of wheels 
             
             
               [x Wi , y Wi ] T   
               Cartesian position of wheel attachment point of 
             
             
                 
               2-DOFW i to the base in base coordinates 
             
             
                 
               (m, m) 
             
             
               [x′ Wi , y′ Wi ] T   
               Cartesian position of wheel attachment point of 
             
             
                 
               2-DOFW i to the base in base coordinates as it 
             
             
                 
               has changed since the previous control cycle 
             
             
                 
               using the coordinates of the previous control 
             
             
                 
               cycle (m, m) 
             
             
               θ mWi   
               Measured steer angle of 2-DOFW i (rad) 
             
             
               θ dWi   
               Desired steer angle of 2-DOFW i (rad) 
             
             
               {overscore (m)} d   
               Input vector (IV), or desired mobile base 
             
             
                 
               motion in base coordinates read from host 
             
             
                 
               processor (m/s, m/s, rad/s), (m/s 2 , 
             
             
                 
               m/s 2 , rad/s 2 ), or (N, N, Nm) 
             
             
                 
               dependant on desired implementation 
             
             
               {overscore (m)} x  = [{dot over (x)}, {dot over (y)}, {dot over (ψ)}] T   
               Actual mobile base motion in base coordinates 
             
             
                 
               (m/s, m/s, rad/s) 
             
             
               {overscore (m)} e  = [{dot over (x)} e , {dot over (y)} e , {dot over (ψ)} e ] T   
               Estimated mobile base motion in base 
             
             
                 
               coordinates calculated as described in Section 5 
             
             
                 
               (m/s, m/s, rad/s) 
             
             
               Δm e  = [Δx e , Δy e , Δψ e ] T   
               Estimated change in mobile base position in 
             
             
                 
               base coordinates calculated as described in 
             
             
                 
               Section 5 (m, m, rad) 
             
             
               {overscore (m)} l (λ) 
               Possible mobile base motion decomposed along 
             
             
                 
               parametric line with parameter λ (m/s, 
             
             
                 
               m/s, rad/s) (Section 4) 
             
             
               {overscore (m)} m   
               Modified mobile base motion (m/s, m/s, rad/s) 
             
             
               {overscore (m)} a   
               Axis motion vector (AMV) of length 2N. 
             
             
                 
               (Type of wheel (caster or no caster) and 
             
             
                 
               type of low level controller determines units) 
             
             
               δ 
               Small increment much less than 1.0 (0.1 
             
             
                 
               nominal) (Section 4) 
             
             
               [{dot over (x)} dWi , {dot over (y)} dWi ] T   
               Desired velocity vector of attachment point of 
             
             
                 
               2-DOFW i to the base in base coordinates 
             
             
                 
               (m/s, m/s) 
             
             
               {dot over (μ)} dWi   
               Desired velocity magnitude of 2-DOFW i (m/s) 
             
             
               s dWi   
               Desired steering axis position for 
             
             
                 
               2-DOFW i (encoders) 
             
             
               s mWi   
               Measured steering axis position for 
             
             
                 
               2-DOFW i (encoders) 
             
             
               {grave over (s)} mWi   
               Measured steering axis position for 
             
             
                 
               2-DOFW i in the previous control 
             
             
                 
               cycle (encoders) 
             
             
               Δs mWi   
               Measured change in steering axis position 
             
             
                 
               for 2-DOFW i since the previous control 
             
             
                 
               cycle (m) 
             
             
               {dot over (s)} dWi   
               Desired steering axis velocity for 
             
             
                 
               2-DOFW i (encoders/s) 
             
             
               {dot over (s)} mWi   
               Measured steering axis velocity for 
             
             
                 
               2-DOFW i (encoders/s) 
             
             
               {dot over (s)} maxWi   
               Maximum steering axis velocity for 
             
             
                 
               2-DOFW i (encoders/s) 
             
             
               {umlaut over (s)} maxWi   
               Maximum steering axis acceleration for 
             
             
                 
               2-DOFW i (encoders/s 2 ) 
             
             
               s lWi   
               Lower-bound steering axis position for 
             
             
                 
               2-DOFW i within Δt time (encoders) 
             
             
               s uWi   
               Upper-bound steering axis position for 
             
             
                 
               2-DOFW i within Δt time (encoders) 
             
             
               {dot over (s)} lWi   
               Lower-bound steering axis velocity 
             
             
                 
               for 2-DOFW i within Δt time (encoders/s) 
             
             
               {dot over (s)} uWi   
               Upper bound steering axis velocity for 
             
             
                 
               2-DOFW i within Δt time (encoders/s) 
             
             
               t mWi   
               Measured translation axis position for 
             
             
                 
               2-DOFW i (encoders) 
             
             
               {grave over (t)} mWi   
               Measured translation axis position for 
             
             
                 
               2-DOFW i in the previous control cycle 
             
             
                 
               (encoders) 
             
             
               Δt mWi   
               Measured change in translation axis position 
             
             
                 
               for 2-DOFW i since the previous control 
             
             
                 
               cycle (m) 
             
             
               {dot over (t)} dWi   
               Desired translation axis velocity for 2-DOFW i 
             
             
                 
               (encoders/s) 
             
             
               {dot over (t)} mWi   
               Measured translation axis velocity for 
             
             
                 
               2-DOFW i (encoders/s) 
             
             
               {dot over (t)} maxWi   
               Maximum translation axis velocity for 
             
             
                 
               2-DOFW i (encoders/s) 
             
             
               {umlaut over (t)} maxWi   
               Maximum translation axis acceleration for 
             
             
                 
               2-DOFW i (encoders/s 2 ) 
             
             
               {dot over (t)} lWi   
               Lower-bound translation axis velocity for 
             
             
                 
               2-DOFW i within Δt time (encoders/s) 
             
             
               {dot over (t)} uWi   
               Upper-bound translation axis velocity for 
             
             
                 
               2-DOFW i within Δt time (encoders/s) 
             
             
               r Wi   
               Radius of 2-DOFW i (m) 
             
             
               σ Wi   
               Steering axis encoder pitch for 2-DOFW i 
             
             
                 
               (encoders/rad) 
             
             
               τ Wi   
               Translation axis encoder pitch for 2-DOFW i 
             
             
                 
               (encoders/rad) 
             
             
               c Wi   
               Amount of caster for 2-DOFW i (m) 
             
             
               Δt 
               Time elapsed during a control cycle (s) 
             
             
               [x eB , y eB , ψ eB ] T   
               Estimated position and orientation of the base 
             
             
                 
               coordinate frame (i.e. coordinates that 
             
             
                 
               are fixed with respect to the mobile base 
             
             
                 
               and move with the mobile base) in fixed 
             
             
                 
               world coordinates (i.e. coordinates that 
             
             
                 
               are fixed with respect to the floor on 
             
             
                 
               which the mobile base moves) (m, m, rad) 
             
             
               C 
               Constraint matrix which maps mobile base 
             
             
                 
               motion to axis speeds [2N × 3] 
             
             
               C f   #   
               The Force Projection Matrix which is 
             
             
                 
               the particular generalized inverse of the C 
             
             
                 
               matrix which is used to map the desired motion 
             
             
                 
               vector to the actuator control torques 
             
             
                 
               [3 × 2N] 
             
             
               C x   #   
               The Velocity Estimation Matrix which is the 
             
             
                 
               particular generalized inverse of the 
             
             
                 
               C matrix which is used to 
             
             
                 
               map actuator speeds to base speeds [3 × 2N] 
             
             
               Λ 
               The base mass matrix [3 × 3] 
             
             
               μ 
               The base centripetal, coriolis, and gravity 
             
             
                 
               vector [3 × 1] 
             
             
               γ dWi   
               Desired steering axis torque for 2-DOFW i 
             
             
                 
               (N · m) 
             
             
               γ lWi   
               Lower-bound steering axis torque for 
             
             
                 
               2-DOFW i (N · m) 
             
             
               γ uWi   
               Upper-bound steering axis torque for 
             
             
                 
               2-DOFW i (N · m) 
             
             
               ρ dWi   
               Desired translation axis torque for 2-DOFW i 
             
             
                 
               (N · m) 
             
             
               ρ lWi   
               Lower-bound translation axis torque for 
             
             
                 
               2-DOFW i (N · m) 
             
             
               ρ uWi   
               Upper-bound translation axis torque for 
             
             
                 
               2-DOFW i (N · m) 
             
             
                 
             
           
        
       
     
   
   Referring to  FIGS. 1A and 1B , a mobile base constructed according to the present invention includes as many two degrees of freedom wheels (2-DOFWs) as deemed necessary (N≧2, where N is the number of 2-DOFWs). All of the wheels are mounted on the mobile base which is a rigid platform (depicted in FIG.  4 .). Each 2-DOFW has two independent axes, one for steering and one for translation.  FIG. 1A  shows an example of a 2-DOFW with no caster, i.e. c Wi  =0, such that the steering and translation axes intersect.  FIG. 1B  shows an example of a 2-DOFW having a caster offset, i.e. c Wi  ≠0. In other words, the steering axis is offset from the translation axis by c Wi . Each of the 2-DOFWs on the mobile base can have a different amount of caster if necessary. 
   Referring to  FIG. 2 , a schematic diagram shows the flow of command information which controls the motion of the mobile base. Since each of the N wheels has two axes (i.e. steering and translation), the mobile base has a total of  2 N axes. Controlling the  2 N axes of the mobile base are  2 N servo amplifiers connected to a set of low-level controllers that perform closed-loop, high servo-rate control of all axis positions. Positional feedback of each axis is provided by an accurate encoding scheme. A supervisory controller interfaced to the low-level controllers coordinates all  2 N axes by sending position updates to the low-level controllers at each discrete control cycle. The position updates are calculated by the control algorithm which takes into account the mobile base geometry, motor dynamics, and a 3-DOF input vector sent from a host processor interfaced to the supervisory controller. The 3-DOF input vector completely describes the desired velocity-based motion of the mobile base, which is constrained to move within three DOFs as described previously. A possible input vector, for example, consists of an x-velocity, y-velocity, and rotational velocity with respect to the center of the mobile base, or alternatively, angle, magnitude, and rotational velocity with respect to a random fixed point (i.e. the representation is arbitrary as long as the axes of the input vector are independent.) 
   The control algorithm is optimal in that it controls the  2 N axes such that the mobile base moves as commanded by the input vector as accurately and as quickly as possible within the physical limits of the motors. 
   For example, if a motor is commanded beyond what it is physically capable of (i.e. it is commanded beyond its “saturation point”) while other motors are commanded to within their physical limits, wheel slippage occurs. The control algorithm is able to predict this situation and correct it before it occurs. It accomplishes this by anticipating velocity or torque and saturation points with working models of each motor axis. Thus, the control algorithm simultaneously minimizes wheel slippage and minimizes the difference between desired motion specified by the input vector and actual mobile base motion (motion error). 
   Section  2 —Mapping the Input Vector to the Desired Axis Motion Vector 
   This determines the axis motion vector (AMV), {right arrow over (m)} a , which is the motion required at each 2-DOFW and the corresponding motion at each axis such that the mobile base moves according to the commanded input vector (i.e. [{dot over (x)} d ,{dot over (y)} d ,{dot over (ψ)} d ] T ). Thus, the mapping is from  3  to  2N . It is accomplished by first calculating the desired velocity to each wheel attachment point as below. The desired wheel velocity for each wheel is expressed as a 2-vector [{dot over (x)} dWi ,{dot over (y)} dWi ] T  in base coordinates (FIG.  4 ). (∀i: 1≦i≦N)(i.e. for all wheels i):
 
{dot over (x)} dWi ={dot over (x)} d −y wi {dot over (ψ)} d  
 
{dot over (y)} dWi ={dot over (y)} d +x wi {dot over (ψ)} d  
 
where the [x Wi , y Wi ] T  are the Cartesian coordinates of the wheel attachment point of each 2-DOFW i to the base in base coordinates (m, m).
 
   The steering angle (θ mWi ) for each 2-DOFW is measured based on the raw measured encoder value of the steering axis (s mWi ) and the encoder pitch (σ Wi ): 
           θ   mWi     =       s   mWi       σ   Wi         ⁢               
 
and will be used in many of the following calculations.
 
   These desired wheel velocities are then mapped to the two axes of the 2-DOFW. For 2-DOFWs with no caster (c Wi =0) the desired steering axis position and desired translation axis velocity [s dWi , {dot over (t)} dWi ] are calculated for each 2-DOFW as follows:
 
θ dWi =arctan 2({dot over (y)} dWi ,{dot over (x)} dWi ) 
 
{dot over (μ)} dWi =√{square root over({dot over (x)})} dWi   2 +{dot over (y)} dWi   2 
 
     s dWi =σ Wi θ dWi                t   .     dWi     =         τ   Wi       r   Wi       ⁢       μ   .     dWi         ⁢               
 with
 {right arrow over (m)} a =[s dW1 ,{dot over (t)} dW1 , . . . ,s dWi ,{dot over (t)} dWi , . . . ,s dWN ,{dot over (t)} dWN ] T    
where
     θ dWi  is the desired steering angle for 2-DOFW i,   σ Wi  is the steering axis “encoder pitch” for 2-DOFW i expressed in encoders per radian, and   {dot over (μ)} dWi  is the desired velocity magnitude for wheel i   
   When using a velocity based low level controller for 2-DOFWs with caster (c Wi ≠0) the desired steering axis velocity and desired translation axis velocity [{dot over (s)} dWi ,{dot over (t)} dWi ] are calculated for each 2-DOFW as below. 
           s   .     dWi     =           σ   Wi       c   Wi       ⁢     (           y   .     dWi     ⁢     cos   ⁡     (     θ   mWi     )         -         x   .     dWi     ⁢     sin   ⁡     (     θ   mWi     )           )       -     ψ   .           
             t   .     dWi     =         τ   Wi       r   Wi       ⁢     (           x   .     dWi     ⁢     cos   ⁡     (     θ   mWi     )         +         y   .     dWi     ⁢     sin   ⁡     (     θ   mWi     )           )         ⁢               
 
with
 
{right arrow over (m)} a =[{dot over (s)} dW1 ,{dot over (t)} dW1 , . . . ,{dot over (s)} dWi ,{dot over (t)} dWi , . . . ,{dot over (s)} dWN ,{dot over (t)} dWN ] T  
 
where
     c Wi  is the amount of caster offset in meters,   τ Wi  is the encoder pitch for 2DOFW i expressed in encoders per radians, and   γ Wi  is the radius of the 2DOFW i   

   When using a torque based low level controller for 2-DOFWs with caster (c Wi ≠0) the desired steering axis torque and desired translation axis torque [γ dWi ,ρ dWi ] T  are calculated for each 2-DOFW as below. First, gather coefficients of the base velocities from the previous equations used in the development of the velocity controller into the constraint matrix, C. The constraint matrix is defined by the ideal kinematic relationship:
 
{right arrow over (m)} a =C{right arrow over (m)} x  
 
where {right arrow over (m)} a  is the motion axis vector and {right arrow over (m)} x  is the actual mobile base motion. The constraint matrix for the preferred embodiment is: 
         C   =     [             -       σ   W1       c   W1         ⁢     sin   ⁡     (     θ   mWi     )                   σ   W1       c   W1       ⁢     cos   ⁡     (     θ   mW1     )                     σ   W1       c   W1       ⁢     (         x   W1     ⁢     cos   ⁡     (     θ   mW1     )         +       y   W1     ⁢     sin   ⁡     (     θ   mW1     )           )       -   1                   τ   W1       r   W1       ⁢     cos   ⁡     (     θ   mW1     )                   τ   W1       r   W1       ⁢     sin   ⁡     (     θ   mW1     )                   τ   W1       r   W1       ⁢     (         x   W1     ⁢     sin   ⁡     (     θ   mW1     )         -       y   W1     ⁢     cos   ⁡     (     θ   mW1     )           )               ⋮       ⋮       ⋮               -       σ   W1       c   Wi         ⁢     sin   ⁡     (     θ   mWi     )                   σ   W1       c   Wi       ⁢     cos   ⁡     (     θ   mWi     )                     σ   W1       c   Wi       ⁢     (         x   Wi     ⁢     cos   ⁡     (     θ   mWi     )         +       y   Wi     ⁢     sin   ⁡     (     θ   mWi     )           )       -   1                   τ   W1       r   Wi       ⁢     cos   ⁡     (     θ   mWi     )                   τ   W1       r   Wi       ⁢     sin   ⁡     (     θ   mWi     )                   τ   W1       r   Wi       ⁢     (         x   Wi     ⁢     sin   ⁡     (     θ   mWi     )         -       y   Wi     ⁢     cos   ⁡     (     θ   mWi     )           )               ⋮       ⋮       ⋮               -       σ   W1       c   WN         ⁢     sin   ⁡     (     θ   mWN     )                   σ   W1       c   WN       ⁢     cos   ⁡     (     θ   mWN     )                     σ   W1       c   Wi       ⁢     (         x   WN     ⁢     cos   ⁡     (     θ   mWN     )         +       y   WN     ⁢     sin   ⁡     (     θ   mWN     )           )       -   1                   τ   W1       r   WN       ⁢     cos   ⁡     (     θ   mWN     )                   τ   W1       r   WN       ⁢     sin   ⁡     (     θ   mWi     )                   τ   W1       r   WN       ⁢     (         x   WN     ⁢     sin   ⁡     (     θ   mWN     )         -       y   Wi     ⁢     cos   ⁡     (     θ   mWN     )           )             ]       ⁢               
 
   Calculate, C f   # , the force projection matrix, a generalized left inverse of C. Any generalized left inverse will work. By judicious choice of a particular C f   # various behaviors can be implemented. An example which is particularly useful is:
 
C f   # =(C T C) −1 C T  
 
which minimizes, in a least squares way, the axis torques. The axis motion vector containing the desired steering axis torque and desired translation axis torque [γ dWi ,ρ dWi ] T  is then:
 
{right arrow over (m)} a =C f   #     T   {right arrow over (m)} d  
 
where {right arrow over (m)} d  is the 3-DOF force torque input vector. For control of a base with a known dynamic model, one can dynamically decouple the undesired forces by calculating the axis torques with the following expression:
 
{right arrow over (m)} a =C f   #     T   (Λ{right arrow over (m)} d +μ) 
 
where {right arrow over (m)} d  is the 3 DOF acceleration input vector and
 
{right arrow over (m)} a =[γ dW1 ,ρ dW1 , . . . ,γ dWi ,ρ dWi , . . . ,γ dWN ,ρ dWN ] T  
 
are the axis torques.
 
   When controlling the base in 3 DOF it is desirable to eliminate or greatly reduce the undesired motion of the vehicle due to the dynamic effects of the various motions of the 2-DOFWs. This can be accomplished by using Λ and μ in the above equations as found from the dynamic model of the base. It is well understood by those skilled in the art that the dynamic model, that is defined by the dynamic equations of a system (base) can be found and written as:
 
F=Λ{umlaut over (x)}+μ
 
where F is the 3 DOF linear and rotational force on the base, Λ is the mass matrix, {umlaut over (x)} is the acceleration of the base in 3DOF, and μ is the centripetal, coriolis, and gravity vector. It is possible to control a base with unknown dynamics by using an estimate of the dynamic parameters such as: 
           Λ   =     [         1       0       0           0       1       0           0       0       1         ]       ,     μ   =     [         0           0           0         ]         ⁢               
 
but the base will not produce the desired motion as closely as when the values of the dynamic parameters for the actual base are used.
 
   Thus, by concatenating all N wheel motion pairs described above, we create a desired axis motion vector which is  2 N in length. 
   Section  3 —Calculating the Control Envelopes 
   The control envelope for an axis describes the possible motion an axis can perform within a fixed time. For a 2-DOFW i with no caster (c Wi =0) we must calculate the steering axis position lower and upper bounds (s lWi  and s uWi ) and the translation axis velocity lower and upper bounds ({dot over (t)} lWi  and {dot over (t)} uWi ). First, calculating the steering position lower bound, we assume that the axis should move based on a value proportional to the error (s dWi -s mWi ) scaled by gain value (k p ) minus a small tolerance (e s ) or based on the maximum (negative) acceleration possible for the steering axis ({umlaut over (s)} max Wi ) (the measured position and velocity (s mWi  and {dot over (s)} mWi ) are required to calculate the position as a result of maximum acceleration), whichever is greater:
 
s lWi =max(s mWi +k p (s dWi −s mWi )−e s ,s mWi +{dot over (s)} mWi Δt−½{umlaut over (s)} max Wi Δt 2 ) 
 
   Calculating the upper bound, we similarly assume the axis should move based on a value proportional to the error scaled by a gain value plus a small tolerance or based on the maximum (positive) acceleration, whichever is lesser.
 
s uWi =min(s mWi +k p (s dWi −s mWi )+e s ,s mWi +{dot over (s)} mWi Δt+½{umlaut over (s)} max Wi Δt 2 ) 
 
   Calculating the lower bound of the translation axis velocity, we assume the control envelope is determined by either maximum (negative) acceleration, or by the maximum possible (negative) velocity of the translation axis ({dot over (t)} max Wi ), whichever is greater.
 
{dot over (t)} lWi =max({dot over (t)} mWi −{umlaut over (t)} max Wi Δt, −{dot over (t)} max Wi ) 
 
   Calculating the upper bound, we similarly assume the control envelope is determined by either the maximum (positive) acceleration, or by the maximum possible (positive) velocity of the translation axis, whichever is lesser.
 
{dot over (t)} uWi =min({dot over (t)} mWi +{umlaut over (t)} max Wi Δt, {dot over (t)} max Wi ) 
 
   Once these values have been determined, the desired motion [s dWi ,{dot over (t)} dWi ] lies within the control envelope for Δt time duration if and only if
 
s dWi ≧s lWi  and s dWi ≦s mWi  
 
or 
 
s dWi ≦s uWi  and s dWi ≧s mWi  
 
   For a 2-DOFW i with caster (c Wi ≠0) when a velocity based low level controller is used, we must calculate the steering axis velocity lower and upper bounds (s lWi and s uWi ) and translation axis velocity lower and upper bounds ({dot over (t)} lWi  and {dot over (t)} uWi ) Calculating the lower bound of the steering axis velocity, we assume the control envelope is determined by either maximum (negative) acceleration ({umlaut over (s)} max Wi ), or by the maximum possible (negative) velocity of the translation axis ({dot over (s)} max Wi ), whichever is greater.
 
{dot over (s)} lWi =max({dot over (s)} mWi −{umlaut over (s)} max Wi Δt, −{dot over (s)} max Wi ) 
 
   Calculating the upper bound, we similarly assume the control envelope is determined by either the maximum (positive) acceleration, or by the maximum possible (positive) velocity of the steering axis, whichever is lesser.
 
{dot over (s)} uWi =min({dot over (s)} mWi +{umlaut over (s)} max Wi Δt, {dot over (s)} max Wi ) 
 
   Calculating the translation velocity lower and upper bounds is the same for 2-DOFWs with caster.
 
{dot over (t)} lWi =max({dot over (t)} mWi −{umlaut over (t)} max Wi Δt, −{dot over (t)} max Wi ) 
 
{dot over (t)} uWi =min({dot over (t)} mWi +{umlaut over (t)} max Wi Δt, {dot over (t)} max Wi ) 
 
   Once these values have been determined, the desired motion [{dot over (s)} dWi ,{dot over (t)} dWi ] lies within the control envelope for Δt time duration if and only if
 
{dot over (s)} lWi ≦{dot over (s)} dWi ≦{dot over (s)} uWi  
 
and 
 
{dot over (t)} lWi ≦{dot over (t)} dWi ≦{dot over (t)} uWi  
 
   For a 2-DOFW i with caster (c Wi ≠0), when a torque based low level controller is used, we must determine if the motion input vector ({right arrow over (m)} d ) is within the upper bound of the actuator torque (γ uWi ,ρ uWi ) and the lower bound of the actuator torque (γ lWi ,ρ lWi ):
 
γ lWi ≦γ dWi ≦γ uWi  
 
ρ lWi ≦ρ dWi ≦ρ uWi  
 
   Thus we can determine for a given increment in time (Δt) the possible motion for each axis (control envelope) of each 2-DOFW, and for a given axis motion vector, whether it lies within the control envelopes. 
   Section  4 —Calculating the Modified Axis Motion Vector 
   We describe an algorithm that can be applied when the desired axis motion vector mapped from the input vector does not lie within the control envelopes of all axes. Given a desired base motion input vector ({right arrow over (m)} d ) and a current estimated base motion vector ({right arrow over (m)} e ) (calculated in Section  5 ) we can determine a modified motion vector that is closest to the desired input vector while staying within the control envelopes of all axes. 
   When a velocity based low level controller is used, we begin by defining a parametric line m l (λ) as a function of λ, which varies between 0 and 1:
 
{right arrow over (m)} l (λ)=λ({right arrow over (m)} d −{right arrow over (m)} e )+{right arrow over (m)} e  
 
0≦λ≦1 
 
   Evaluating m l (0) results in me, which is the current base motion vector. Evaluating m l (1) results in {right arrow over (m)} d , which is the desired input vector. Thus, we define an algorithm that finds a value of λ whose m l (λ) mapping is closest to the desired input vector and lies within the control envelopes when mapped to the axis motion vector. The basic idea is to increment λ by a small amount (δ) until the mapping of m l (λ) to the axis motion vector lies within the control envelopes. This becomes the modified axis motion vector. 
   When a torque based low level controller is used, we again begin by defining a parametric line m l (λ) as a function of λ, which varies between 0 and 1, where now:
 
{right arrow over (m)} l (λ)=λ{right arrow over (m)} d  
 
0≦λ≦1 
 
   Evaluating m l (0) results in zero torque command such that the base continues its current base motion vector. Evaluating m l (1) results in {right arrow over (m)} d , which is the desired input vector. Thus, we define an algorithm that finds a value of λ whose m l (λ) mapping is closest to the desired input vector and lies within the control envelopes when mapped to the axis motion vector. The basic idea is to increment λ by a small amount (δ) until the mapping of m l (λ) to the axis motion vector lies within the control envelopes. This becomes the modified axis motion vector.
     Using: the appropriate equation for m l (λ) from above, and with 0≦δ&lt;&lt;1:
       begin procedure
 
λ=δ
           while {right arrow over (m)} l (λ) mapped to axis motion vector (Section  2 ) lies within control envelopes of all axes of each 2-DOFW (Section  3 ) do
 
λ=λ=δ
   end
 
{right arrow over (m)} m ={right arrow over (m)} l (λ) 
   return axis motion vector mapping of {right arrow over (m)} m      
           end   
       

   Note, there are many possible ways to implement the same algorithm. This method is presented because of its simplicity. 
   Section  5 —Estimating the Motion of the Mobile Base 
   Here, we describe how to estimate the motion of the base during the discrete time interval Δt. First, we calculate the new apparent wheel positions with respect to the base coordinates. These new coordinates are expressed as x Wi ′ and y Wi ′ for wheel: i. For wheels with no caster (c Wi ≠0), we begin by calculating the measured steering angle as in Section 1: 
           θ   mWi     =       s   mWi       σ   Wi         ⁢               
 
We proceed by calculating the measured translation distance (Δt mWi ) since the beginning of the previous control cycle:
 
Δt mWi =(t mWi −′t mWi ) 
 
Here, we introduce ′t mWi  which is the measured translation axis position at the beginning of the previous control cycle expressed in encoders. The apparent wheel positions are then simply the wheel positions in base coordinates (x Wi  and y Wi ) plus the calculated motion: 
           x   Wi   ′     =       x   Wi     +     Δ   ⁢           ⁢     t   mWi     ⁢       r   Wi       τ   Wi       ⁢     cos   ⁡     (     θ   mWi     )             ⁢               
         y   Wi   ′     =       y   Wi     +     Δ   ⁢           ⁢     t   mWi     ⁢       r   Wi       τ   Wi       ⁢     sin   ⁡     (     θ   mWi     )               
 
   For wheels with caster (c Wi ≠0), we similarly calculate the steering angle, θ mWi , and the apparent wheel motion as a result of the translation axis, Δt mWi . However, because of the caster offset, steering motion results in apparent wheel motion as well. Here we introduce ′s mWi  which is the measured steering axis position at the beginning of the previous control cycle expressed in encoders:
 
Δs mWi =(s mWi −′s mWi ) 
 
The apparent wheel motion is then calculated by a rotation with respect to the steer angle: 
         x   Wi   ′     =       x   Wi     +     Δ   ⁢           ⁢     t   mWi     ⁢       r   Wi       τ   Wi       ⁢     cos   ⁡     (     θ   mWi     )         -     Δ   ⁢           ⁢     s   mWi     ⁢       c   Wi       σ   Wi       ⁢     sin   ⁡     (     θ   mWi     )               
           y   Wi   ′     =       y   Wi     +     Δ   ⁢           ⁢     s   mWi     ⁢       c   Wi       σ   Wi       ⁢     sin   ⁡     (     θ   mWi     )         +     Δ   ⁢           ⁢     t   mWi     ⁢       r   Wi       τ   Wi       ⁢     sin   ⁡     (     θ   mWi     )             ⁢               
 
We then use the apparent wheel motion to calculate the amount of base rotation Δψ e  since the last control cycle. We do this by calculating the rotation for each possible wheel pair and averaging the results:
 
     M=C(N,2)          Δψ   e     =         1   M     ⁢       ∑     ∀     {     j   ,   k     }         ⁢     arctan   ⁢           ⁢   2   ⁢     (         y   Wk   ′     -     y   Wj   ′       ,       x   Wk   ′     -     x   Wj   ′         )           ⁢           -     arctan   ⁢           ⁢   2   ⁢     (         y   Wk     -     y   Wj       ,       x   Wk     -     x   Wj         )               
   The above equation is summed over all possible wheel pairs j, k. That is, there are C(N,2) (N choose 2 combinatorial) possible wheel combinations. 
   Since the position of a mobile base having only two wheels (i.e. N=2) can be estimated with feedback from only three of its four motion axes (the fourth one being redundant), there are many other ways to estimate the motion of the base. The preceding formula can be used for bases having only two wheels, but there will only be one wheel pair to “average.” As the number of wheels used on a mobile base goes up (i.e. as N increases), the better the above estimation algorithm gets. 
   We now calculate the change in x and y coordinates (Δx e  and Δy e ) by evaluating the following equations: 
         Δ   ⁢           ⁢     x   e       =       1   N     ⁢       ∑     i   =   1     N     ⁢           ⁢     (       x   Wi   ′     +       y   Wi     ⁢     Δψ   e         )             
           Δ   ⁢           ⁢     y   e       =       1   N     ⁢       ∑     i   =   1     N     ⁢           ⁢     (       y   Wi   ′     -       x   Wi     ⁢     Δψ   e         )           ⁢               
 
The estimated base motion vector can be calculated by dividing by the time increment Δt:
 
{right arrow over (m)} e =1/Δt [Δx e ,Δy e ,Δψ e ] T  
 
   A similar method to calculate mi, which directly averages the measured readings with a more compact notation, uses the matrix, C, introduced above:
 
{right arrow over (m)} e =1/Δt C x   #[Δs   mW1 ,Δt mWi , . . . ,Δs mWi ,Δt mWi , . . . ,Δs mWN ,Δt mWN ] T  
 
where C x   # , the velocity estimation matrix, is a generalized left inverse of C. Any generalized left inverse will work. By judicious choice of a particular C x   #  various behaviors can be implemented. An example which is particularly useful is:
 
C x   # =(C T C) −1 C T  
 
which minimizes, in a least squares way, the difference in the measured motion of the axis sensors and the motion of axis sensors on an ideal model of the PCV which perfectly obeys the velocity relationship described by C.
 
   The final step in motion estimation is determining the “summed-up” position of the mobile base in fixed world coordinates. This is accomplished by adding the rotation angle change Δψ eB  to the existing angle estimate(ψ eB ). Similarly X eB  and y eB  are calculated by adding to the existing estimates the rotated change in x and y coordinates with respect to ψ eB :
 
ψ eB =ψ eB +Δψ e  
 
x eB =x eB +Δx e  cos(ψ eB )−Δy e  sin(ψ eB ) 
 
y eB =y eB +Δy e  cos(ψ eB )−Δx e  sin(ψ eB ) 
 
Section  6 —Summary
 
   The present invention, as described above, provides a method and apparatus for controlling the motion of a mobile base with increased accuracy and maneuverability. In its preferred embodiment, the present invention is used on a mobile robot base having three wheels each with a predetermined amount of caster. The mobile robot is controlled by an off-board host processor (as shown in  FIG. 2. ) The host processor sends command signals and receives motion feedback from an onboard supervisory controller by radio, cable, infrared, or similar type of link. The motion control signals are mapped to axis control signals by the supervisory controller and sent to six low-level controllers. Each of the six low level controllers corresponds to either a steering axis or translation axis for one of the three wheels. Each of the low level controllers in turn sends an axis control signal to an associated servo amplifier, which provides the proper voltage and current to drive the respective axis motor. Each of the six motors has an encoder, which provides motor position feedback to both the low-level controller and the supervisory controller. 
   The host processor and supervisory controller are preferably microprocessors that are commonly used for embedded control. Propriety software code is written preferably in C programming language to implement the inventive control method on the microprocessors and low level controllers. 
   The inventive method and apparatus can be utilized with other configurations (not shown), such as on drive systems for forklifts or automated guided vehicles (AGV&#39;s.) Also, the mobile base described above can be inverted with the positions of the mobile base and the surface it rolls on transposed. In other words, two degree of freedom wheels can be mounted pointed upward on a stationary base, and can translate and rotate a horizontal surface resting on the wheels. In another possible application (not shown), multiple bases, each having single or multiple wheels, can be pivotably linked together in a snake-fashion to form a non-rigid base which is controlled by the inventive method. 
   The above descriptions and drawings are for illustrative purposes only, and are not exhaustive of possible alternate embodiments of the invention. It is to be understood that the present invention is not limited to the sole embodiments described above and illustrated herein, but encompasses any and all variations falling within the scope of the appended claims.