Patent Publication Number: US-9850108-B2

Title: Movement system configured for moving a payload

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. patent application Ser. No. 13/664,947 filed on Oct. 31, 2012, which claims priority to U.S. Provisional Patent Application No. 61/555,825 filed on Nov. 4, 2011, which are hereby incorporated by reference in their entirety. 
    
    
     TECHNICAL FIELD 
     The present disclosure relates to a movement system that is configured for moving a mass along an X axis and a Y axis in response to articulation of a movement device. 
     BACKGROUND 
     Overhead bridge cranes are widely used to lift and relocate large payloads. Generally, the displacement in a pick and place operation involves three translational degrees of freedom and a rotational degree of freedom along a vertical axis. This set of motions, referred to as a Selective Compliance Assembly Robot Arm (“SCARA”) motions or “Schönflies” motions, is widely used in industry. A bridge crane allows motions along two horizontal axes. With appropriate joints, it is possible to add a vertical axis of translation and a vertical axis of rotation. A first motion along a horizontal axis is obtained by moving a bridge on fixed rails while the motion along the second horizontal axis is obtained by moving a trolley along the bridge, perpendicularly to the direction of the fixed rails. The translation along the vertical axis is obtained using a vertical sliding joint or by the use of a belt. The rotation along the vertical axis is obtained using a rotational pivot with a vertical axis. 
     There are partially motorized versions of overhead bridge cranes that are displaced manually along horizontal axes and rotated manually along the vertical axis by a human operator, but that include a motorized hoist in order to cope with gravity along the vertical direction. Also, some bridge cranes are displaced manually along all of the axes, but the weight of the payload is compensated for by a balancing device in order to ease the task of the operator. Such bridge cranes are sometimes referred to as assist devices. Balancing is often achieved by pressurized air systems. These systems need compressed air in order to maintain pressure or vacuum depending on the principle used which requires significant power. Also, because of the friction in the compressed air cylinders, the displacement is not very smooth and can even be bouncy. Balancing can be achieved using counterweights, which add significant inertia to the system. Although helpful and even necessary for the vertical motion, such systems attached to the trolley of a bridge crane add significant inertia regarding horizontal motion due to moving the mass of these systems. In the case of balancing systems based on counterweights, the mass added can be very large, even larger than the payload itself If the horizontal traveling speed is significant, the inertia added to the system becomes a major drawback. 
     There are also fully motorized versions of such bridge cranes that require powerful actuators, especially for the vertical axis of motion which has to support the weight of the payload. These actuators are generally attached to the trolley or bridge and are then in motion. The vertical translation actuator is sometimes attached to the bridge and linked to the trolley by a system similar to what is used in tower cranes. 
     SUMMARY 
     A movement system is configured for moving a payload. The movement system includes a bridge crane, a trolley, and a movement device. The bridge crane is configured for movement along an X axis. The trolley is movably attached to the bridge crane and is configured for movement along a Y axis, in perpendicular relationship to the X axis. The movement device depends from the trolley along a Z axis. The movement device includes a first four-bar mechanism, a second four-bar mechanism, and a sensor. The second four-bar mechanism is operatively connected to, and suspended from, the first four-bar mechanism. Each four-bar mechanism has a pair of kinematic links and a pair of base links. The pair of kinematic links extend in spaced and parallel relationship to one another. The pair of base links extend in spaced and parallel relationship to one another and are pivotally connected to ends of the pair of kinematic links to form a first, second, third, and fourth joint therebetween. The pair of kinematic links and the corresponding pair of base links form a parallelogram. A first axis extends through the first joint of the first four-bar linkage and the third joint of the second four-bar linkage. A second axis extends through the second joint of the first four-bar linkage and the fourth joint of the second four-bar linkage. A third axis extends through the third joint of the first four-bar linkage and the first joint of the second four-bar linkage. A fourth axis extends through the fourth joint of the first four-bar linkage and the second joint of the second four-bar linkage. The first, second, third, and fourth axis extend in parallel relationship to one another. The kinematic links are rotatable about the respective axes. The axes of the first four-bar mechanism are disposed in perpendicular relationship to the axes of the second four-bar mechanism. The sensor is operatively attached to one of the joints of one of the first and second four-bar mechanisms. The sensor is configured to measure an angle of rotation of the respective kinematic link about the respective axis. 
     A movement device depends from a trolley along a Z axis and is configured for moving along at least one of an X axis and a Y axis. The movement device includes a first four-bar mechanism, a second four-bar mechanism, and a sensor. The second four-bar mechanism is operatively connected to, and suspended from, the first four-bar mechanism. Each four-bar mechanism has a pair of kinematic links and a pair of base links. The pair of kinematic links extend in spaced and parallel relationship to one another. The pair of base links extend in spaced and parallel relationship to one another and are pivotally connected to ends of the pair of kinematic links to form a first, second, third, and fourth joint therebetween. The pair of kinematic links and the corresponding pair of base links form a parallelogram. A first axis extends through the first joint of the first four-bar linkage and the third joint of the second four-bar linkage. A second axis extends through the second joint of the first four-bar linkage and the fourth joint of the second four-bar linkage. A third axis extends through the third joint of the first four-bar linkage and the first joint of the second four-bar linkage. A fourth axis extends through the fourth joint of the first four-bar linkage and the second joint of the second four-bar linkage. The first, second, third, and fourth axis extend in parallel relationship to one another. The kinematic links are rotatable about the respective axes. The axes of the first four-bar mechanism are disposed in perpendicular relationship to the axes of the second four-bar mechanism. The sensor is operatively attached to one of the joints of one of the first and second four-bar mechanisms. The sensor is configured to measure an angle of rotation of the respective kinematic link about the respective axis. 
     A method of moving a movement device along at least one of an X axis and a Y axis includes providing a sensor configured to measure angle of rotation of at least one of a first and a second kinematic link about a respective axis of rotation. A force is imparted on at least one of the first and second kinematic links such that an angular displacement of at least one of the first and second kinematic links about the respective axis of rotation is achieved. The angular displacement of the at least one of the first and second kinematic links about the respective axis of rotation is determined. The movement device is moved along the at least one of the X axis and the Y axis in response to the determination of the angle of rotation of the at least one of the first and second kinematic links about the respective axis of rotation until first and second kinematic links are vertical. 
     The above features and advantages, and other features and advantages of the present disclosure, will be readily apparent from the following detailed description of the embodiment(s) and best mode(s) for carrying out the described invention when taken in connection with the accompanying drawings and appended claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic perspective view of a movement system including a movement device which is connected to a support structure; 
         FIG. 2  is a schematic perspective view of the movement device of  FIG. 1 , configured for moving a payload along an X axis and a Y axis; 
         FIG. 3  is another schematic perspective view of the movement device of  FIG. 1 , configured for moving a payload along an X axis and a Y axis; 
         FIG. 4  is a schematic perspective view of the movement device of  FIG. 3  having an articulated mechanism and the payload supported by the articulated mechanism; 
         FIG. 5  is a schematic block diagram of a high frequency oscillation scheme usable with the controller shown in  FIG. 1 ; and 
         FIG. 6  is a schematic block diagram of a control scheme usable with the controller shown in  FIG. 1 . 
     
    
    
     DETAILED DESCRIPTION 
     Referring to the drawings, wherein like reference numbers refer to like components, a movement system  10  configured for moving a payload  12  in a plurality of directions is shown at  10  in  FIG. 1 . The movement system  10  is mounted to a stationary support structure  14  that is configured to support the movement system  10  and the payload  12 . The support structure  14  includes, but is not limited to a pair of parallel rails  16  or runway tracks. 
     The movement system  10  includes a bridge crane  18 , a trolley  20 , and a movement device  22 . The bridge crane  18  is a structure that includes at least one girder  30  that spans the pair of parallel rails  16 . The bridge crane  18  is adapted to carry the payload  12  along a Y axis  19 . The trolley  20  is movably attached to girders  30  of the bridge crane  18  such that the trolley  20  is adapted to carry the payload  12  along an X axis  17 , in generally perpendicular relationship to the Y axis  19 . The movement device  22  is operatively attached to the trolley  20 . A Z axis  21  extends in a vertical direction, with respect to the ground, and is defined between the intersection of the X axis  17  and the Y axis  19 . 
     The movement device  22  includes four-bar mechanisms  24  and is configured to be a two degree-of-freedom articulated mechanism (X and Y). A two degree-of-freedom articulated mechanism is shown in  FIGS. 1 and 3 . The articulated mechanism includes four-bar mechanisms  24 . Additionally, the movement device  22  may be configured to allow the center of mass  26  of the payload  12  to be offset from a center line  25  of the movement device  22 . 
     With reference to  FIGS. 2 and 3 , the movement device  22  includes a first four-bar mechanism  24   a  and a second four-bar mechanism  24   b  which is operatively connected to, and is suspended from, the first four-bar mechanism  24   a . Each four-bar mechanism  24  includes a pair of four-bar linkages  32 , i.e., a first four-bar linkage  32   a  and a second four-bar linkage  32   b , which are rigid. Each four-bar linkage  32  includes a pair of kinematic links  34 , i.e., a first kinematic link  34   a  and a second kinematic link  34   b , and a pair of base links  36 , i.e., a first base link  36   a  and a second base link  36   b . The first base link  36   a  and the second base link  36   b  are disposed in spaced and parallel relationship to one another. Opposing ends  38  of the first kinematic link  34   a  are pivotally connected to ends  38  of the first and second base link  36   a ,  36   b  to form a respective first joint  40  and second joint  42  therebetween. The second kinematic link  34   b  is disposed in spaced and parallel relationship to the first kinematic link  34   a  and opposing ends  38  of the second kinematic link  34   b  are pivotally connected to ends  38  of the first and second base link  36   a ,  36   b  to form a respective third joint  44  and fourth joint  46  therebetween. Accordingly, each four-bar linkage  32  forms a parallelogram. 
     The first four-bar linkage  32   a  and the second four-bar linkage  32   b  of each of the first and second four-bar mechanisms  24   a ,  24   b  are disposed in spaced and generally parallel relationship to one another such that the first kinematic link  34   a  of the first four-bar linkage  32   a  is disposed in spaced and generally parallel relationship to the second kinematic link  34   b  of the second four-bar linkage  32   b  and the second kinematic link  34   b  of the first four-bar linkage  32   a  is disposed in spaced and generally parallel relationship to the first kinematic link  34   a  of the second four-bar linkage  32   b . Additionally, the first base link  36   a  and the second base link  36   b  of the first four-bar linkage  32   a  are disposed in spaced and generally parallel relationship to a corresponding first base link  36   a  and second base link  36   b  of the second four-bar linkage  32   b.    
     A first axis  48  extends through the first joint  40  of the first four-bar linkage  32   a  and the third joint  44  of the second four-bar linkage  32   b . A second axis  50  extends through the second joint  42  of the first four-bar linkage  32   a  and the fourth joint  46  of the second four-bar linkage  32   b.  A third axis  52  extends through the third joint  44  of the first four-bar linkage  32   a  and the first joint  40  of the second four-bar linkage  32   b.  A fourth axis  54  extends through the fourth joint  46  of the first four-bar linkage  32   a  and the second joint  42  of the second four-bar linkage  32   b.  The first axis  48 , second axis  50 , third axis  52 , and fourth axis  54  extend in spaced and generally parallel relationship to one another for each of the four-bar mechanisms  24   a,    24   b.  Additionally, the first axis  48 , second axis  50 , third axis  52 , and fourth axis  54  of the first four-bar mechanism  24   a  are generally perpendicular to the first axis  48 , second axis  50 , third axis  52 , and fourth axis  54  of the second four-bar mechanism  24   b.    
     Referring to  FIGS. 1-3 , each four-bar mechanism  24  includes a first connection link  56  and a second connection link  58 . The first connection link  56  rigidly connects the first kinematic link  34   a  of the first four-bar linkage  32   a  and the second kinematic link  34   b  of the second four-bar linkage  32   b.  The second connection link  58  rigidly connects the second kinematic link  34   b  of the first four-bar linkage  32   a  and the first kinematic link  34   a  of the second four-bar linkage  32   b.  The rigid connections mean that the first kinematic link  34   a  of the first four-bar linkage  32   a  and the second kinematic link  34   b  of the second four-bar linkage  32   b  rotate in unison about the respective first and second axes. Likewise, the second kinematic link  34   b  of the first four-bar linkage  32   a  and the first kinematic link  34   a  of the second four-bar linkage  32   b  rotate in unison about the respective third and fourth axes. The first and second four-bar linkages  32   a,    32   b  and the first and second connection links  56 ,  58  are used for each four-bar mechanism  24  such that each four-bar mechanism  24  can sufficiently support required forces, moments, and torques. Roller bearings may also be disposed in the joints  40 ,  42 ,  44 ,  46  in order to reduce friction. 
     The first four-bar mechanism  24   a  is operatively attached to the trolley  20 . More specifically, the first four-bar mechanism  24   a  depends from the trolley  20 . The second four-bar mechanism  24   b  depends from the first four-bar mechanism  24   a.  More specifically, the second four-bar mechanism  24   b  depends from the first four-bar mechanism  24   a  such that the first axis  48 , second axis  50 , third axis  52 , and fourth axis  54  of the first four-bar mechanism  24   a  are in generally perpendicular relationship to the first axis  48 , second axis  50 , third axis  52 , and fourth axis  54  of the second four-bar mechanism  24   b.    
     Referring to  FIGS. 2 and 3 , a pair of tubes  60  extend from the second four-bar mechanism  24   b,  along the X axis  17 . The payload  12  is suspended from at least one of these tubes  60  and is offset from the Z axis  21 . 
     Referring to  FIG. 4 , an articulated joint  61  may extend from one or both of the tubes  60  and further extend in an X and/or Y direction which is further offset from the Z axis  21 . The payload  12  may extend from the articulated joint  61  at an attachment point  84 . The payload  12  may be offset from the attachment point  84 . 
     During operation, an oscillation frequency of the movement device  22  is a function of a length L of the kinematic links  34 , but not on a position of the center of mass  26  of the payload  12 , with respect to the Z axis  21 . Shorted kinematic link  34  lengths L may be used to save space, while longer kinematic link  34  lengths L may be used to reduce the oscillation natural frequency. 
     The movement device  22  includes a cart  62  and a controller  63 . The cart  62  is configured for moving the bridge crane  18  and/or the trolley  20  along the respective X axis  17  and Y axis  19  in response to the application of a force F to the payload  12 . As the force F is applied to the payload  12  a direction along the X axis  17  and/or the Y axis  19 , the kinematic links  34  of the first and/or second four-bar mechanism  24   a,    24   b  rotate about the respective axes. Sensors  64  are operatively connected to at least one joint of each of the first and second four-bar mechanisms  24   a,    24   b.  These sensors  64  measure an angle of rotation θ 1  and θ 2  of the kinematic links  34  about the respective axes. The sensor  64  may include an encoder  66  and a Hall effect sensor  68  operatively disposed along the respective axis. While only one sensor  64  may be used per axis, signals from the combination of the encoder  66  and the Hall effect sensor  68  can be combined by using data fusion to obtain improved signal quality over using a single sensor  64 . Additionally, using two signals provides redundancy such that signals from both sensors  64  can be compared to one another to detect any signal problems. Additionally, the Hall effect sensor  68  provides an absolute signal, whereas the encoder  66  offers a precise signal. It should be appreciated that other sensors  64  may also be used. Absolute encoders, potentiometers or linear accelerometers (used as inclinometers) could be used as the position sensor. A gyroscope could be used to obtain the angular velocity while an accelerometer could be used to obtain angular acceleration. Accelerometers or gyroscopes placed on slotted parts could also help determine different dynamical effects. Photointeruptors could also be used at strategic places. Finally, the above signals can be derived/integrated to obtain corresponding signals. 
     The angular displacement and angular velocity estimations are obtained from the Kalman state estimation. Each signal, i.e., from the encoder  66  and the Hall effect sensors  68 , are independently Kalman filtered and then combined in proportion of their Kalman covariance matrix corresponding state value. 
     In order to be desensitized to small angle measurement precision errors, a deadband on the angle may be used. The deadband is an area of a sign range where no action on the system occurs. The movement device  22  may also be excited by small amplitude, high frequency unmodeled dynamics or it may be difficult for the control to manage high frequency oscillations. During oscillations, when the kinematic links  34  are close to a vertical position, since the angle measurement often changes sign, it becomes difficult to suppress the oscillations. One method of suppressing these oscillations is to increase the angle deadband. An algorithm, shown as an oscillation logic block  70  in  FIG. 5 , is provided to compensate for high frequency oscillations, while keeping precision and performance to keep the kinematic links  34  vertical. For a small deadband, θ db1  is still used to cope with precision errors of the angle measurements. Two other angles are defined, θ db2  and θ db3 . The signal θ p0  is determined in a deadband block  72  and expressed as follows: 
               θ     db   ⁢           ⁢   1       =     {             0           if   -     θ     db   ⁢           ⁢   1         &lt;   θ   &lt;     θ     db   ⁢           ⁢   1                   θ   -     θ     db   ⁢           ⁢   1                 if   ⁢           ⁢   θ     &gt;     θ     db   ⁢           ⁢   1                   θ   =     θ     db   ⁢           ⁢   1                 if   ⁢           ⁢   θ     &lt;     -     θ     db   ⁢           ⁢   1                 ⁢     
     ⁢     θ     p   ⁢           ⁢   0         =     {         0           if   -     θ     db   ⁢           ⁢   1         &lt;   θ   &lt;     θ     db   ⁢           ⁢   1                   θ   -     θ     db   ⁢           ⁢   1                 if   ⁢           ⁢   θ     &gt;     θ     db   ⁢           ⁢   1                   θ   +     θ     db   ⁢           ⁢   1                 if   ⁢           ⁢   θ     &lt;     -     θ     db   ⁢           ⁢   1                             
and the signal θ p1  is determined in a deadband and saturation block  74  and expressed as follows:
 
     
       
         
           
             
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     The signal θ p0  then corresponds to the input angle signal above θ db1  while θ p1  corresponds to the input signal between θ db2  and θ db3 . In order to remove the high frequency oscillations from θ p1 , this signal is further processed. While a low pass filter could be used, phase delays may result, causing system instability. The absolute signal of θ p1  is determined in an absolute logic block  76  and then the absolute signal passes through a rate limiter block  78 . The rising limit is low and the falling limit is high, such that it takes time for the output signal to increase, filtering high frequency oscillations. However, the signal of the θ p1  can return to zero rapidly, avoiding a phase shift. This signal is then multiplied by the sign of θ p1  stored in a sign block  82 . The resulting signal, can then optionally be slightly filtered with a usual low pass filter at a low pass block  80 , resulting in the signal θ p2 . Although, θ p0  and θ p2  can be used individually in the control, they can also be grouped as:
 
θ pf =θ p0 +θ p2  
 
     In the following, the equations of motion are first obtained with a complete model called coupled motion. Then, with simplifications, a simplified model is obtained. With reference to  FIG. 2 , the following velocities are obtained:
 
 {dot over (X)}   p   ={dot over (X)}   c   +L  cos θ 1  {dot over (θ)}  1   −l   4  {dot over (Ø)}
 
 {dot over (Y)}   p   ={dot over (Y)}   c   +L  cos θ 2  {dot over (θ)}  2   −l   3  {dot over (Ø)}
 
Ż p   =Ż   c   +L  sin θ 1  {dot over (θ)}  1    +L  sin θ 2  {dot over (θ)}  2  Ø
 
{dot over (Ø)}  p ={dot over (Ø)}  c +{dot over (Ø)}  e  
 
where X p , Y p  and Z p  are the payload  12  center of mass position in fixed coordinates (the X axis  17  is aligned with the tubes  60 ), X C , Y C , Z C  are the cart  62  coordinates in fixed coordinates, φ C  is the mechanism rotation about the vertical axis and φ e  is the payload  12  rotation about the end-effector axis. φ p  is the total translation of φ e  plus φ c . The potential energy is provided as follows:
 
 V=mgL (cos θ 1 +cos θ 2 )− Z   c  
 
where m is the payload  12  mass and the kinetic energy is expressed as:
 
             T   =         1   2     ⁢     M   x     ⁢       X   .     c   2       +       1   2     ⁢     M   y     ⁢       Y   .     c   2       +       1   2     ⁢     M   z     ⁢       Z   .     c   2       +       1   2     ⁢     m   ⁡     (         X   .     p   2     +       Y   .     p   2     +       Z   .     p   2       )                 
where M X  is the cart  62  mass in the X direction and M Y  the cart  62  mass in the Y direction and M Z  is the cart  62  mass in the Z direction. One should note that masses of the kinematic links  34  were neglected. The equations of motion are obtained from the previous two equations and the Lagrange method as follows:
 
 F   X   =M   x   {umlaut over (X)}   c   +m ( {umlaut over (X)}   c   −L  sin θ 1  {dot over (θ)}  1   2   +L  cos θ 1  {umlaut over (θ)}  1   −l   4  {umlaut over (Ø)} )
 
 F   Y   =M   y   Ÿ   c   +m ( Ÿ   c   −L  sin θ 2  {dot over (θ)}  2   2   +L  cos θ 2  {umlaut over (θ)}  2   +l   3 {umlaut over (Ø)} )
 
 F   Z   =+M   z   {umlaut over (Z)}   c   +m ( {umlaut over (Z)}   c   +L  cos θ 1  {dot over (θ)}  1   2   +L  sin θ 1  {umlaut over (θ)}  1   +L  cos θ 2  {dot over (θ)}  2   2   +L  sin θ 2  {umlaut over (θ)}  2   +g )
 
 F   θ1 =0 =mL ( {umlaut over (X)}   c  cos θ 1   −l   4  cos θ 1    {umlaut over (Ø)} +{umlaut over (Z)}   c  sin θ+ L{umlaut over (θ)}    1   +L  sin θ 1  cos θ 2  {dot over (θ)}  2   2   +L  sin θ 1   +L  sin θ 1  sin θ 2  {umlaut over (θ)}  2 +mg sin θ 1 )
 
 F   β1 =0 =mL ( Ÿ   c  cos θ 2    +l   3  cos θ 2  {umlaut over (Ø)} + {umlaut over (Z)}   c  sin θ 2   +L{umlaut over (θ)}    2   +L  sin θ 2  cos θ 1  {dot over (θ)}   1   2   +L  sin θ 1  sin θ 2  {umlaut over (θ)}  1    +mg  sin θ 2 )
 
     One should note that similar equations could be found with the other angle representation as (θ 2 , β 2 ). Additionally, the coupling between angles θ 1  and θ 2  is negligible for relatively small angles and angular velocities. Thus, motion along the X axis  17  and Y axis  19  will be treated separately, as described below. 
     Referring to  FIG. 4 , with only one degree-of-freedom where θ refers to θ 1  or θ 2 , while the other angle remains fixed, and a small rotation rate, equations of motion are as follows:
 
 F =( M+m ) {umlaut over (x)}+m{umlaut over (θ)} L  cos θ− mL {dot over (θ)}  2  sin θ+ m{umlaut over (L)}  sin θ+2 mθ{dot over (L)}  cos θ
 
τ=0=( {umlaut over (x)}  cos θ+ g  sin θ+ L {umlaut over (θ)} +2 {dot over (L)}{dot over (θ)} ) mL  
 
which can be simplified to the pendulum equations for constant link lengths L of the kinematic links  34  as follows:
 
 F =( M+m ) {umlaut over (x)}+m{umlaut over (θ)} L  cos θ− mL{dot over (θ)}    2  sin θ+ m{umlaut over (L)}  sin θ+2 mθ{dot over (L)}  cos θ
 
τ=0=( {umlaut over (x)}  cos θ+ g  sin θ+ L {umlaut over (θ)}) mL  
 
where M is the mass of the cart  62  and m is the mass of the payload  12 . Assuming small angles and a slowly varying vertical translation and neglecting {dot over (θ)}  2 , the equations can be approximated as follows:
 
 F =( M+m ){umlaut over (x)}+ m{umlaut over (θ)}L  
 
0 ={umlaut over (x)}+gθ+L{umlaut over (θ)} 
 
     The movement mechanism may be operated in a cooperation mode. It is possible to manage an offset of the center of mass  26  of the payload  12  from the central line  25 . In  FIGS. 2 and 3 , the offset is from the movement device  22  and in  FIG. 4 , the offset is from the attachment point  84 , allowing the operator  28  to operate the movement device  22  by placing their hands  31  directly on the payload  12 . The movement mechanism allows the operator  28  to impart an angle θ 1  and θ 2  to the movement device  22 , i.e., the first four-bar mechanism  24   a  and the second four-bar mechanism  24   b,  by pushing the payload  12 , and this angle θ 1  and θ 2  is measured by the sensors  64 . The operator  28  is permitted to place their hands  31  directly on the payload  12  because the angles θ 1  and θ 2  imparted to the links of the first four-bar mechanism  24   a  and the second four-bar mechanism  24   b,  which are measured by the sensors  64 , are done above the payload  12 . The control system moves the cart  62  in response to the angle θ 1  and θ 2  measured by the sensors  64  to keep the kinematic links  34  vertical. Thus, the cart  62  moves in the direction desired by the operator  28 , while controlling any sway of the kinematic links  34 , resulting in assistance to the operator  28 . Additionally, since the controller  63  insures that the kinematic links  34  remain vertical, the operator  28  is not required to manually stop the load, since the control system manages itself to stop the payload  12 . An autonomous mode, where the payload  12  position is prescribed, while reducing links sway, may also be desired. 
     More specifically, the angle θ 1  and θ 2  is imparted by the kinematic links  34  of the first and/or second four-bar mechanisms  24   a,    24   b  pivoting about the axes in response to the operator  28  pushing on the mechanism. An objective of the control system is to move the overhead cart  62 , in response to the imparted angles θ 1  and θ 2  to keep the kinematic links  34  vertical. Thus, the cart  62  moves in the direction imparted by the operator  28  to the payload  12 , while controlling swaying of the kinematic links  34 . Additionally, since the controller  63  ensures that the kinematic links  34  remain vertical, the operator  28  is not required to stop the load. More specifically, the control system functions to stop the cart  62 , and the associated payload  12 . 
     The force F required for an operator  28  to move the payload  12  would be reduced because a measure of the imparted angle(s) θ 1  and θ 2  of the kinematic links  34  about the respective axes can be precisely and accurately measured. This results in a system that moves along the corresponding X axis  17  and/or Y axis  19 . 
     The controller  63  includes a control block  86 , shown in  FIG. 6 , which is configured to operate for cooperative motion or autonomous motion. The cart  62  acceleration will be considered as the input. The payload  12  and cart  62  mass do not need to be known. The following equations are obtained in a Laplace domain as follows:
 
 {umlaut over (X)} ( s )+ g θ( s )+ s   2   L θ( s )=0
 
The state-space representation is as follows:
 
 {dot over (x)}   S   =A   S     x     S   +B   S   u   S  
 
 y   S   =C   S     x     S   +D   S   u   S  
 
where y S  the output vector,  x   S  is the state vector, us is the input scalar, A S  is an n×n state matrix, B S  is an n×m input matrix, C S  is a p×n output matrix, D S  is a p×m feed through matrix and where n is the number of states, m is the number of inputs and p is the number of outputs. Here,  x   S =[x {dot over (x)} θ {dot over (θ)} ] T  and u S ={umlaut over (x)}, with
 
     
       
         
           
             
               A 
               s 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           
                             - 
                             g 
                           
                           L 
                         
                       
                       
                         0 
                       
                     
                   
                   ] 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   B 
                   s 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       0 
                     
                   
                   
                     
                       1 
                     
                   
                   
                     
                       0 
                     
                   
                   
                     
                       
                         
                           - 
                           1 
                         
                         L 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     The above equation, obtained from the Laplace domain, is used, where u={umlaut over (x)}, the control law is u S =K R e, where: 
                 K   R     ⁢   e     =         [           K   x           K   v           -     K   θ             -     K     θ   ⁢           ⁢   p               ]     ⁢           ⁢   and   ⁢           ⁢   e     =     [           x   d           -   x                 x   .     d           -     x   .                 θ   d           -   θ                 θ   .     d           -     θ   .             ]             
where {dot over (x)} d , θ d , and {dot over (θ)}  d  equal zero.
 
     Referring again to the control logic block of  FIG. 6 , the input, u S , is the acceleration of the cart  62 , and because controlling acceleration is not practical, velocity control is used in the cooperation mode and position control is used in the autonomous mode. The output of the latter lower level controller block  88  is shown as u 2  in  FIG. 6 . 
     In the cooperation mode, the state space controller block  90  output of  FIG. 6  is obtained as a discrete velocity with a zero-order-hold integration, as follows:
 
 {umlaut over (x)}   d(k)   =u=K   r   e  
 
 {dot over (x)}   d(k)   ={dot over (x)}   d(k−1)   +{umlaut over (x)}   d(k)   T   S  
 
Likewise, in the autonomous mode, the state space controller block  90  output of  FIG. 6  is obtained as a position by integrating once more, as follows:
 
 x   d(k)   =x   d(k−1)   +{dot over (x)}   d(k-1)   T   S +0.5 {umlaut over (x)}   d(k)   T   S   2  
 
     It should be appreciated that the measured velocity could be used in the preceding equations, instead of the last time step desired value. 
     One should note that the measured velocity could be used in the preceding equations instead of the last time step desired value. This integration method is used to achieve acceleration control in an admittance control scheme. The desired acceleration is then obtained by using velocity or position control, which is more practical. It is also possible to additionally use computed torque control using the previous force equations. Although the payload  12  and cart  62  mass would then be required, an approximation is sufficient since feedback control is also used. Additionally, the payload  12  and cart  62  mass are not required in order to adapt the state space controller block  90  gains to varying parameters. Additionally, a limit and saturation block  92  may be used for virtual walls and to limit velocity and acceleration of the cart  62 . 
     In the cooperation mode, since there is no reference position, K x  is set to zero. The control gain K θp , i.e., gain on the angular velocity signal, can be optionally used, depending on the angle derivative signal quality. An adaptive controller  63 , based on pole placement and state space control may be used. The pole of the system may be obtained by:
 
det[ sI−A+BK   r ]
 
leading to the equation:
 
                   s   3     ⁢   L     +       s   2     ⁡     (         K   θ     ⁢   p     +       K   v     ⁢   L       )       +     s   ⁡     (     g   +     K   θ       )       +       K   v     ⁢   g       L         
where K θ  and K θp  are assumed negative.
 
     The transfer function from angle θ to an angle initial condition θ 0  is as follows: 
     
       
         
           
             
               
                 
                   θ 
                   0 
                 
                 ⁡ 
                 
                   ( 
                   
                     s 
                     + 
                     
                       K 
                       v 
                     
                   
                   ) 
                 
               
               ⁢ 
               
                 L 
                 s 
               
             
             
               
                 
                   s 
                   3 
                 
                 ⁢ 
                 L 
               
               + 
               
                 
                   s 
                   2 
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       
                         K 
                         θ 
                       
                       ⁢ 
                       p 
                     
                     + 
                     
                       
                         K 
                         v 
                       
                       ⁢ 
                       L 
                     
                   
                   ) 
                 
               
               + 
               
                 s 
                 ⁡ 
                 
                   ( 
                   
                     g 
                     + 
                     
                       K 
                       θ 
                     
                   
                   ) 
                 
               
               + 
               
                 
                   K 
                   v 
                 
                 ⁢ 
                 g 
               
             
           
         
       
     
     The poles may be placed to the following:
 
( s+p   1 )( s   2 +2ζ 1 ω n1 +ω n1   2 )
 
     In a first method, Kν and K θ  are used, which leads to the following: 
                     K   v     =         p   1     +     2   ⁢     ζ   1     ⁢     w     n   ⁢           ⁢   1       ⁢     p   1     ⁢     
     ⁢     g   L       +       K   θ     L       =         w     n   ⁢           ⁢   1     2     +     2   ⁢     ζ   1     ⁢     w     n   ⁢           ⁢   1       ⁢     
     ⁢         K   v     ⁢   g     L         =       p   1     ⁢     w     n   ⁢           ⁢   1     2                   
and then, the following are used:
 
               p   1     =       2   ⁢   g   ⁢           ⁢     ζ   1     ⁢     2     n   ⁢           ⁢   1             -   g     +       w     n   ⁢           ⁢   1     2     ⁢   L                       K   v     =         p   1     ⁢     w     n   ⁢           ⁢   1     2     ⁢   L     g                   K   θ     =       (       w     n   ⁢           ⁢   1     2     -     g   L     +     2   ⁢   ζ   ⁢           ⁢     w     n   ⁢           ⁢   1       ⁢     p   1         )     ⁢   L           
where
 
               ω     n   ⁢           ⁢   1       ≥       g   L             
and are ζ design parameters. The control gains are thus obtained. The transfer function zero influences the response, but without practical effect, since it is relatively high, ω n1  is chosen very close to
 
                 g   L       ,         
but not too close to avoid numerical problems.
 
     Referring again to  FIG. 3 , the control scheme is then used with these gains to manage the cooperation with the operator  28 , while stabilizing the movement device  22 . 
     In a second method, Kν, K θ , and K θp  are used, which leads to the following: 
     
       
         
           
             
               
                 K 
                 v 
               
               + 
               
                 
                   K 
                   
                     θ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     p 
                   
                 
                 L 
               
             
             = 
             
               
                 p 
                 1 
               
               + 
               
                 2 
                 ⁢ 
                 
                   ζ 
                   1 
                 
                 ⁢ 
                 
                   w 
                   
                     n 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 g 
                 L 
               
               + 
               
                 
                   K 
                   θ 
                 
                 L 
               
             
             = 
             
               
                 w 
                 
                   n 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
                 2 
               
               + 
               
                 2 
                 ⁢ 
                 
                   ζ 
                   1 
                 
                 ⁢ 
                 
                   w 
                   
                     n 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                 
                 ⁢ 
                 
                   p 
                   1 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   K 
                   v 
                 
                 ⁢ 
                 g 
               
               L 
             
             = 
             
               
                 p 
                 1 
               
               ⁢ 
               
                 w 
                 
                   n 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
                 2 
               
             
           
         
       
     
     The second method allows the poles to remain constant. Using the gain K θp  allows the cart  62  to move in regards to the angle and angular velocity. The following is then obtained: 
               p   1     =       -     g   ⁡     (       K     θ   ⁢           ⁢   p       -     2   ⁢   ζ   ⁢           ⁢     w     n   ⁢           ⁢   1     2     ⁢   L       )           L   ⁡     (       -   g     +       w     n   ⁢           ⁢   1     2     ⁢   L       )                       K   v     =           p   1     ⁢     w     n   ⁢           ⁢   1     2     ⁢   L     g     ⁢           
     ⁢       K   θ     =       (       w     n   ⁢           ⁢   1     2     -     g   L     +     2   ⁢   ζ   ⁢           ⁢     w     n   ⁢           ⁢   1       ⁢     p   1         )     ⁢   L                 
where
 
                 ω     n   ⁢           ⁢   1       ≥       g   L         ,         
ζ, and K θp  are design parameters. The control gains are thus obtained. The transfer function zero influences the response, but without practical effect since it is relatively high, ω n1  is chosen very close to
 
                 g   L       ,         
but not too close to avoid numerical problems.
 
     Referring again to  FIG. 3 , the control scheme is then used with these gains to manage the cooperation with the operator  28 , while stabilizing the movement device  22 . 
     Neglected terms from the complete model as {dot over (L)}, {dot over (β)} , {dot over (θ)}  2  and viscous friction can be compensated for, for example, with gains K θ  and K θp  by considering the terms constant over a time step, similarly as with the lengths L of the kinematic links  34 . 
     Control gains may also be heuristically modified from the computed gains. Additionally, control gains on θ p0  and θ p2  and their derivatives may be different from each other. 
     In the autonomous mode, K x  is used to control the cart  62  position. The control gain K θp  can be optionally used. An adaptive controller  63  based on pole placement and state space control using K θp  is provided. Similar to the cooperation mode, the system poles are: 
                   s   4     ⁢   L     +       s   3     ⁡     (         K   θ     ⁢   p     +       K   v     ⁢   L       )       +       s   2     ⁡     (     g   +     K   θ     +       K   x     ⁢   L       )       +     s   ⁡     (       K   v     ⁢   g     )       +       K   x     ⁢   g       L         
where K θ  and K θp  are assumed to be negative.
 
     There is a compromise between the cart  62  position trajectory and the kinematic links  34  oscillations cancellation. In regards to the equations, this is due to the transfer function zeros. 
     Pole placement is used using the characteristic equation:
 
( s+p   1 ) 2 ( s   2 +2◯ 1 ω n1 +ω n1   2 )
 
     Equaling the previous equations for the system poles and pole placement provides: 
                 2   ⁢     ζ   1     ⁢     w     n   ⁢           ⁢   1         +     2   ⁢     p   1         =       K   v     +       K     θ   ⁢           ⁢   p       L                       w     n   ⁢           ⁢   1     2     +     4   ⁢     ζ   1     ⁢     w     n   ⁢           ⁢   1       ⁢     p   1   2         =         K   θ     L     +     K   x     +     g   L                       2   ⁢     w     n   ⁢           ⁢   1     2     ⁢     p   1       +     2   ⁢     ζ   1     ⁢     w     n   ⁢           ⁢   1       ⁢     p   1   2         =         K   v     ⁢   g     L                     w     n   ⁢           ⁢   1       ⁢     p   1   2       =           K   x     ⁢   g     L                 
and then the following are used:
 
               K   x     =         w     n   ⁢           ⁢   1     2     ⁢     p   1   2     ⁢   L     g                   K   v     =       2   ⁢     w     n   ⁢           ⁢   1       ⁢     p   1     ⁢     L   ⁡     (       w     n   ⁢           ⁢   1       +       ζ   1     ⁢     p   1         )         g                   K   θ     =       (       w     n   ⁢           ⁢   1     2     +     4   ⁢   ζ   ⁢           ⁢     w     n   ⁢           ⁢   1       ⁢     p   1       +     2   ⁢     p   1       -     K   x     -     g   L       )     ⁢   L                   K     θ   p       =       (       2   ⁢     ζ   1     ⁢     w     n   ⁢           ⁢   1         +     2   ⁢     p   1       -     K   v       )     ⁢   L               
where
 
               ω     n   ⁢           ⁢   1       ≥       g   L             
and ζ are design parameters and p 1  is heuristically chosen to be equal to ω n1  as to lie on the same circle as the other poles. It is a design choice to use two complex poles and two equal real poles as other choices are possible. The state space controller  63  gains to adapt are thus obtained. The transfer function zero influence the response but without practical effect since it is relatively high. ω n1  is chosen very close to
 
                 g   L       ,         
but not too close to avoid numerical problems.
 
     One should note that the operator  28  can still push the payload  12  in autonomous mode. The cart  62  position will move in the direction desired by the operator  28 , while being attracted to its reference position and cancelling oscillations of the movement device  22 . Depending on the control gains, it will be more or less easy to move the cart  62  away from its reference position. Referring to  FIG. 6 , the control block  86  will then be used with these gains to manage autonomous and cooperation with the operator  28 , while stabilizing the movement device  22 . 
     Neglected terms from the complete model as {dot over (L)}, {dot over (β)} , {dot over (θ)}  2  and viscous friction can be compensated for, for example, with gains K θ  and K θp  by considering the terms constant over a time step, similarly as with the lengths L of the kinematic links  34 . 
     Control gains can also be heuristically modified from the computed gains. Additionally, control gains on θ p0  and θ p2  and their derivatives can be different from one another. 
     When switching between the modes, i.e., cooperation mode, autonomous mode, stopping, and the like, rude acceleration and jerk profile may be required. The most frequent abrupt profile happens when switching modes when the angles θ 1  and θ 2  of the kinematic links  34  are non-zero. “Bumpless” transfer or smooth transfer between modes may be achieved. In one embodiment, the last control input is memorized or observed. In another embodiment, the measured velocity is memorized when the mode switch happens. In the cooperation mode, the output bumpless velocity is as follows:
 
ν DesBumpl =α bt ν mem +(1−α bt )ν des  
 
     The variable α bt  is reinitialized at 1 when a mode switch happens and is then multiplied by b bt  at each time step. At first ν Desbumpl  is then equal to the measured velocity (ν mem ) and after some time, depending on parameter b bt , α bt  goes to 0 and ν DesBumpl  to ν des . b bt  should be defined as a parameter to be chosen by the designer. The goal is to go from the present velocity as the mode switch moment (ν mem ) to the desired velocity (ν des ) in a smooth filtered way. For the autonomous mode, the desired position is first reset to the measured position and the desired bumpless velocity is integrated to obtain a new desired position respecting this velocity. Further smoothing may also be possible by considering the acceleration in the mode switch. 
     It should also be appreciated that the movement device  22  may be configured such that the payload  12  may include an end effector which is slidable, relative to the four-bar mechanisms  24   a ,  24   b  and which also allows the payload to be rotated, as indicated at  94  in  FIG. 1 . Movement in a vertical direction may be accomplished between the movement device  22  and the trolley  20  or between the movement device  22  and the end effector. More specifically, the end effector may include a slidable and rotatable mechanism such that the payload  12  could be translated on the four-bar mechanisms  24   a ,  24   b  or rotated about  94 . 
     While the best modes for carrying out the disclosure have been described in detail, those familiar with the art to which this disclosure relates will recognize various alternative designs and embodiments for practicing the disclosure within the scope of the appended claims.