US20120239199A1 - Coordinated joint motion control system - Google Patents

Abstract

A coordinated joint control system for controlling a coordinated joint motion system, e.g. an articulated arm of a hydraulic excavator blends automation of routine tasks with real-time human supervisory trajectory correction and selection. One embodiment employs a differential control architecture utilizing an inverse Jacobian. Modelling of the desired trajectory of the end effector in system space can be avoided. The invention includes image generation and matching systems.

Description

FIELD OF THE INVENTION
• The present invention relates to coordinated joint motion control systems, for example, excavating equipment, robotic and semi-robotic arms and certain three-dimensional image generators such as anatomic simulators and to control systems and methods therefor.
BACKGROUND OF THE INVENTION
• The mining, construction and other industries, are increasingly employing automation and robotics to enhance the efficiency of material processing operations, such as excavation and mining activities, using powered equipment, often with articulated arms having independent joint connections between the links. Some machines have built-in mechanical means to coordinate joint motions.
• Many tasks require a human operator to coordinate the movement of several machine links by simultaneous control of a corresponding number of joysticks or other control devices. One example is the control of earth moving equipment such as an excavator employing a bucket as a tool. It can be a difficult, skilled task, often requiring years of experience effectively to control the machine so as to move the tool along a desired path with an efficient trajectory. The task demands continuous concentration and careful adjustment by the operator of multiple links so as to effectively control their motion.
• Roberts, J. M. & Corke, P. I.1997. in “Automation of underground truck haulage.”Fourth International Symposium on Mine Mechanization and Automationhave described automated load-dump haulage roadheaders. Kogler, P., Melrose, R., Stratmann, T. & Sifferlinger, N. A.1997. “Further approaches in automation on roadheaders/bolter miners in production and development.”Fourth International Symposium on Mine Mechanization and Automation: A6-11-A6-18. have described partially automated dragline stripping, production drilling, rock bolting, and shotcreting.
• Of particular exemplary, but non-exclusive, interest for automation are large mining front end shovels. Such excavating machines may perform several hundred shovel cycles per day, each cycle including filling the machine's bucket, hoisting and swinging the load over a haul truck, dumping the load, and returning to the digging front. The term “bucket” is often used generically in the art to include a “shovel” and other tools with which an excavator arm may be equipped. Stentz, A., Bares, J., Singh, S. & Rowe, P. 1999. inA robotic excavator for autonomous truck loading. Autonomous Robots 7: 175-186. have proposed using sensory adjustments to vary the digging and loading points with a view to rendering fully robotic the repetitive components of the operations of a front end shovel. To integrate such fully automated or robotic components into the complex operations performed by mining and construction equipment, may require human supervisory control.
• Employment of a human operator, even in a supervisory capacity, is contrary to traditional industrial robotics practice, for example for assembly and paint spraying operations and the like in automobile manufacture, where the absence of a real-time human supervisor is an important advantage of robots. Unlike the “cookie cutter” complex but repeated operations common in manufacturing robotics the flexibility and diversity of operations required in mining, construction and the like may justify or require the partial or even full-time attention of a human operator even when an automated machine is utilized. In general, robotics systems are designed to perform large numbers of iterations of a procedure or suite of procedures or selection of one or more procedures from an available suite, each of which procedures or suite of procedures is familiar to the robotic system. In contrast, an ability to adapt to unfamiliar terrain, environment or circumstances is a prerequisite of mining and construction operations. For these reasons alone, known robotics methods may not be suitable for automation of mining and construction equipment and the like.
• Furthermore, conventional robotic equipment is generally unsuited to a mining and construction environment. Industrial robots employ precision engineering with fine-tuned valves and mechanical controls and sensitive hydraulics that require scrupulously clean oil. In a typical modern factory having a climate- and dust-controlled, indoor environment, these conditions can usually be met without undue difficulty. A typical mining or construction site offers quite the opposite conditions, presenting a hostile environment to industrial robotics. The air is typically dust and dirt laden. The equipment, as robust as it is, operates to relatively crude tolerances and commonly functions satisfactorily under mining conditions. Accordingly, severe problems may be encountered in adapting the principles of industrial robotics for operation of mining or construction machinery or the like.
• Perreira et al disclose in U.S. Pat. No. 4,763,276 a robot control method intended to locate a robot at a desired position and orientation (pose) wherein an anticipated pose of a robot is predicted, compared with a desired pose and a correcting command signal is employed to place the robot.
• Chan et al disclose in U.S. Pat. No. 4,893,254 a manipulator arm position sensing method wherein smoother operation of an the movement of the end point of an articulated arm to a preselected target is obtained by computer calculation of joint angle changes using an iterative pseudo inverse Jacobian having a damping factor. Though presumed to be useful for their intended purposes, neither the Perreira et al. nor the Chan et al. methods is suitable for controlling excavation or construction machinery in a manner capable of automating repeated operations and permitting flexible machine operation in a changing or diverse environment.
• As shown inFIG. 1, Alami, R., Chatila, R., Fleury, S., Ghallab, M. & Ingrand, F. 1998. in “An architecture for autonomy.”The International Journal of Robotics Research17(4): 315-337. have proposed a control architecture for mobile robots which reportedly integrates human supervisory direction into an automated robotic machine. Referring toFIG. 1, ahuman supervisor10 observes the motion and position of anend effector12, and provides mission guidance, to atrajectory planner14.Trajectory planner14 selects a software control protocol corresponding with the mission guidance instruction and provides power and motion control to the machine actuators, for example hydraulic pistons and cylinders to execute the mission guidance instruction. The traectory is planned either in Cartesian or joint space. The software control protocol iterates every step needed, including every actuator adjustment, for the end effector to execute the desired trajectory.
• The term “end effector” is used in the context of the invention herein, to reference the ultimate point or object component of the machine or system that is manipulated by the machine or system, for example a tool such as a front end loader shovel or drill, or an object picture in an imaging device.
• An example of such an integrated, human controlled partially robotic operation is that of a three-boom robotic tunneling drilling machine each boom of which collars, corrects the alignment and drills a hole.
• Ideally, a supervisor, who is effectively a robotic operator, “manually” checks and corrects the collaring of each pre-programmed drill hole using the machine's joysticks or other manual controls, but employs built-in automation to complete the drilling of each hole. If effective, the collaring and manual correction of the alignment and determination of the coordinates of the next hole can be made while the two other booms are drilling automatically. This is a typical example of sequentially applying operator adjustment and robotic automated control of a predetermined trajectory element.
• One known, robot control scheme suitable for control of a joint-based, continuous-path end effector is illustrated inFIG. 2. Referring toFIG. 2, the control scheme shown employs aCartesian trajectory generator16 and the control architecture allows the motion to be executed at a desired speed and the actual movement to be adjusted relative to the basic trajectory. For this control scheme, the desired trajectory is assumed to be predefined, either as a preprogrammed or as a user-recorded pattern.
• Operation of one of the levers or controls16 of the excavator'smanual control unit18 provides a Cartesian position control signal to atrajectory generator20.Trajectory generator20 utilizes atime division base22 to generate control signals comprising desired joint position, velocity and acceleration signals for moving a particular machine link through a desired trajectory.Trajectory generator20 employs forward kinematics, using a model to determine where the end effector should be at a future point in time. The control signals are supplied to ajoint control system24 which provides an appropriate power signal torobot actuators26. Real-time integration of the operation of any or all levers16 in each of three Cartesian directions yields machine kinematic configuration and an output which determines the actual Cartesianend effector position30 over time. Feedback control is provided by position sensors (not shown) at the robot actuators which provide joint position and velocity signals to thejoint control system24 enabling internal position error correction.
• Mathematically, the three-dimensional trajectory of the end effector can be described by employing a surrogate variable for the definition of the trajectory as functions of three Cartesian position control signals, without specifying velocity. The motion velocity can be given as the tangential velocity along the path. The incremental arc length of the path can be related to the partial displacements in the three Cartesian directions to provide the relationship between time and the surrogate variable for a given motion. This relationship may be assumed to be solved by using an appropriate initial condition for the motion. Further simplification can be obtained for the special case where the selected parameter is identical with the arc length of the path. In this case, the variation of the surrogate parameter over time can readily be determined.
• Variations in the tool orientation may be described by defining the tool direction with an ordered set of absolute-angle or Euler-angle rotations, one for each degree-of-freedom of the automated robotic machine, in a comparable manner to the description of the translational motions and displacements using a surrogate paramater. If the variation of the parameter with time is based on the position definition, the angular velocities can be defined and no further velocity relationship is needed. Otherwise, a further equation is required to specify the angular velocity, for example by prescribing the rate of the equivalent angular rotation as a function of the surrogate parameter. The position and angular velocities can then be defined in real time. For further purposes, the angular velocities of the ordered rotations with time can be determined based on the foregoing approach, using manipulator kinematics formulations, such as described e.g., by Craig, J. J. (1989) “Introduction to robotics: mechanics and control.”2^ndEd. Addison-Wesley, Reading, Mass.
• This conventional control system is suitable for continuous path robotic control and in effect models the desired trajectory of the end effector in system space and applies power signals to the machine actuators of an amplitude and duration calculated to achieve the desired trajectory in real space. However, it is deterministic, the end effector can only travel on a preprogrammed path, the control does not permit human supervisory participation, and cannot adapt to new circumstances, such as novel changes in the end effector working environment.
• A completely manual control based only on a human operator is shown inFIG. 3 for comparison, as another conventional system. Danko, the inventor herein, in “Coordinated Motion Control.”Presentation to Sandwick-Tamrock Personnel, Tampere, Finlandin 2000 (“Danko 2000a” hereinafter) described a modification of the conventional robot control scheme shown inFIG. 2, for human supervisory control, which modification is schematically illustrated inFIG. 3. In this example, in order to show for comparison the complexity of prior art approaches, human supervisory control of the end effector position and orientation is accomplished by adjusting the motion velocity components of the joint or machine actuators and integrating the individual components with respect to time much as is done in the manual supervisory control scheme illustrated inFIG. 1. The main component of the motion control is the manual guidance of the tangential and angular velocities along the pre-determined path. Other control inputs can be used to modify the trajectory. Translational and rotational trajectory modifications can be accomplished by adding position correction terms to the absolute position. The solution example, however, requires the combination of ordinary robotic control with a supervisory adjustment which further increases the inherent complexity of a robotic system.
• The foregoing description of background art may include insights, discoveries, understandings or disclosures, or associations together of disclosures, that were not known to the relevant art prior to the present invention but which were provided by the invention. Some such contributions of the invention may have been specifically pointed out herein, whereas other such contributions of the invention will be apparent from their context.
BRIEF SUMMARY OF THE INVENTION
• The present invention solves the problem of providing a control system for controlling a coordinated joint motion system which provides a coordinated joint motion system which is easy to operate and flexible in the tasks it can undertake.
• In one aspect, the invention provides a coordinated joint control system, for controlling a coordinated joint motion system the coordinated joint motion system comprising:
• • a) support;
  • b) multiple links;
  • c) multiple joints, optionally revolute or prismatic joints, connecting the links one to another and to a support, each joint permitting relative movement between the connected members;
  • d) multiple actuators to effect said relative movement between the connected members, the multiple actuators being controlled by the coordinate joint control system; and
  • e) an end effector supported by the jointed links for movement relative to the support;
    wherein the coordinated joint motion system is capable of execution of an automated end effector trajectory without human intervention wherein the coordinated joint control system comprises an operator interface enabling a human supervisor to change the end effector motion or position during execution of the automated trajectory.
• The system can comprise an internal feedback loop to determine a mathematical model of the coordinated joint motion system and provide a model-based forward predictor for directly controlling the joint actuators, optionally by employing a differential control architecture. The internal feedback loop can generate a differential inverse kinematics model of the machine configuration for a given end effector position and may comprise an inverse Jacobean matrix relating the joint-space variables as a vector, to an input vector of Cartesian variables.
• Preferably, the control system enables a human supervisor to employ velocity control to adjust and correct the end effector position. The operator interface can comprise a control box employing at least one manually movable member to input control signals and optionally a computer interface for selection of a trajectory and the manually movable member can comprise multiple joysticks. The control system can distribute a control signal from a single joystick to multiple joint actuators.
• The coordinated joint motion system comprises a mining or construction machine, optionally an excavator and the multiple links comprise a boom revolutely connected to the machine, an arm revolutely connected to the boom and a tool, optionally a bucket, revolutely connected to the arm.
• In another aspect the invention provides a coordinated joint control imaging system for imaging a coordinated joint motion system, the coordinated joint motion system comprising:
• • a) support;
  • b) multiple links;
  • c) multiple joints, optionally revolute or prismatic joints, connecting the links one to another and to a support, each joint permitting relative movement between the connected members;
  • d) multiple actuators to effect said relative movement between the connected members, the multiple actuators being controlled by the coordinate joint control system; and
  • e) an end effector supported by the jointed links for movement relative to the support;
    wherein the imaging system comprises an internal feedback loop to determine a mathematical model of the coordinated joint motion system and provide a a model-based forward predictor for directly controlling the joint actuators, optionally by employing a differential control architecture. method for controlling a coordinated joint motion system, the coordinated joint motion system comprising:
  • a) support;
  • b) multiple links;
  • c) multiple joints, optionally revolute or prismatic joints, connecting the links one to another and to a support, each joint permitting relative movement between the connected members;
  • d) multiple actuators to effect said relative movement between the connected members, the multiple actuators being controlled by the coordinate joint control system; and
  • e) an end effector supported by the jointed links for movement relative to the support;
    the method comprising execution of an automated end effector trajectory without human intervention a human supervisor changing the end effector motion or position during execution of the automated trajectory.
• The new differential control architecture of the invention provides a simple control system while permitting integration of operator adjustments into the system. In one embodiment, hardware required to implement the invention, may comprise a manual electronic electro-hydraulically controlled machine with the addition of joint position sensors and a control computer. Internal, position control feedback loops may not be necessary to implement the inventive control system are needed as they are in conventional robotic equipment. However, position control feedback loops may be employed, if already present in a particular machine, or if available or desired, for added precision. Nor are internal velocity feedback loops required, in preferred embodiments, but their utilization e.g. in the form of application method electro-hydraulic servo valves, may increase precision.
• While the invention is not bound by any particular theory, the new solution can be considered as blending, preferably continuously and in real time, operator machine control with automatic control by re-distributing the operator's velocity control components into machine joint velocity components according to a selected and/or predetermined and/or optimized trajectory characteristic, given e.g. in the form of a differential relationship between partial motion components either in joint or Cartesian (real-world) co-ordinates or partial velocities.
• Preferred embodiments of the present invention are capable of providing novel control systems and methods for coordinated joint motion systems which enable repeated tasks to be carried out automatically by the machine but which permit a human supervisor or operator to make corrections or adjustments, if necessary, and preferably also to select, import and/or create one or more predefined trajectories for the machine to execute.
• One particular preferred embodiment provides a hydraulically powered excavating machine embodying a preferred inventive control system which is easy to operate and flexible in the tasks it can undertake.
BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING
• Some embodiments of the invention, and of making and using the invention, as well as the best mode contemplated of carrying out the invention, if not described above, are described in detail below, by way of example, with reference to the accompanying drawings, in which like reference characters designate the same or similar elements throughout the several views, and in which:
• FIG. 1 is a block flow diagram showing, in simplified form, a prior art high-level control schematic of an automated robotic machine employing a human supervisor;
• FIG. 2 is a block flow diagram showing, in simplified form, a prior art joint-based, continuous-path robot control scheme;
• FIG. 3 is a block flow diagram showing, in simplified form, a prior art modification of the robot control scheme shown inFIG. 2 for human operator control;
• FIG. 4 is a side elevation of a powered shovel embodying a coordinated joint motion control system according to the invention;
• FIG. 5 is a schematic view of a motion control system according to the invention for the powered shovel shown inFIG. 4;
• FIG. 6 is a block flow diagram of a continuous differential control architecture according to an embodiment of the invention for operating the motion control system illustrated inFIG. 5;
• FIG. 7 is a schematic view illustrating the creation of a family of two-dimensional trajectories for use by the differential control architecture illustrated inFIG. 6;
• FIG. 8 is a view similar toFIG. 7 of another parallel trajectory family that may be employed in the invention;
• FIG. 9 is a time-lapse schematic depiction, in graphic form, of a first cut in a simulated prior art excavation operation employing a manually controlled front shovel;
• FIG. 10 is a view similar toFIG. 9 of second and third cuts in the simulated prior art excavation operation;
• FIG. 11 is a time-lapse schematic depiction, in graphic form, of a simulated three-cut excavation operation employing an embodiment of differential control architecture according to the invention;
• FIG. 12 is a schematic view of one example of a desirable excavation trajectory;
• FIG. 13 is a view similar toFIG. 11 of a first cut employing the desirable excavation trajectory illustrated inFIG. 12;
• FIG. 14 is a view similar toFIG. 11 of a third cut employing the desirable excavation trajectory illustrated inFIG. 12;
• FIG. 15 is a schematic view illustrating virtual configuration of an articulated machine such as an excavator using an embodiment of differential control architecture according to the invention;
• FIG. 16 is a schematic view of another embodiment of differential control architecture according to the invention, which architecture is useful for virtual machine configuration;
• FIG. 17 is a schematic view illustrating the application of an embodiment of the differential control architecture of the invention to simulation of the motion of a coordinated joint system, the particular example illustrated being an anatomical coordinated joint system, namely the human right arm; and
• FIG. 18 is a block flow diagram of another embodiment of differential control architecture according to the invention suitable for effecting the simulation illustrated inFIG. 17.
DETAILED DESCRIPTION OF THE INVENTION
• The coordinated joint motion control system of the invention and the control architecture embodied therein are here described, for illustrative purposes only and without limitation, in its application to the control of aparticular machine100, namely mechanical excavator, more specifically a front end shovel loader. It will, however be understood that the control system and architecture have a wide variety of applications, some of which are described or referenced herein, including for example, to robotic arms and other articulated mechanical systems capable of performing a diversity of functions having repetitive elements as well as to other systems such as anatomic simulators.
• The invention provides a motion control process for controlling the movement of an apparatus or machine employing manually operated multi-dimensional joysticks and a novel differential joint control architecture. Pursuant to the novel joint control architecture the control signal generated by an individual joystick is not dedicated to an individual actuator but can be distributed to multiple actuators. Thus, one or more of the joystick signals may comprise number of joint motion parameters for the multiple actuators. For example, the joint motion parameters may be differentially based on the instantaneous inverse or pseudo-inverse of the Jacobian of the machine or apparatus.
• The invention includes a control apparatus or machine, particularly but not exclusively an apparatus or machine employing a coordinated joint or articulated arm system, which apparatus or machine embodies the inventive motion control process to control movement of the apparatus or machine. The invention further extends to electronic systems for rendering graphical, three dimensional image in space, and optionally also in time, of coordinated joint systems, for example anatomical systems such as the human arm, illustrating its articulation, and to the rendered images on screen, in print, in electronic storage or in other media.
• The distribution of joystick control can be accomplished dynamically during motion based on a pre-programmed trajectory family, or pre-defined coordinate system, or a pre-defined control kinematics. If the Jacobean is non-singular, the inverse is calculated. If the joint parameters are over-determined relative to the end-effectors' Cartesian trajectory coordinate parameters, the least-square fit solution of the set of over-determined equations replaces of the inverse as a pseudo inverse. If the set is under-determined, additional constraints are added to aid the inverse solution. For the sake of simplicity of discussion, the existence of the inverse Jacobean is assumed. The coordinated control is differential and represented by the Jacobean of the pre-defined desired control kinematics. Other components of the coordinated control are the desired starting point and the velocity component of the motion.
• Coordinated joint parameters control greatly simplifies the control of machine tools or graphical objects through a desired task e.g., for navigating them through a course between obstacles. Artificial intelligence can be used to obtain the optimum trajectory family for typical, repetitive tasks. Sensory input can be used to obtain the optimum coordinate system for using as a pre-defined system. Global positioning system can use sensory input for coordinate system definition. A computer frame main space layout can be a pre-defined coordinate system. A surveyed underground or surface mine coordinates can also be entered as a pre-defined coordinate system. A fully autonomous target trajectory control system can utilize the differential coordinated joint manipulation. Such an application may arise in fitting a multi-link human body motion to captured images.
• Many tasks require performing coordinated movement of several machine links with the simultaneous control of the corresponding joysticks or other control signal. One example is the control of earth moving equipment such as an excavator with a bucket as a tool. It is difficult to control the tool along a desired path that demands continuous concentration and motion adjustment of multiple links from the operator. A new control architecture is advantageous to provide real-time operator support using programmable, automated coordination of the motion of the machine joints, while still retaining full manual control, if needed. The new control architecture allows re-defining the machine inherent motion kinematics characteristic into a task-specific kinematics that fits a desired task optimally. The definition of the task-specific kinematics can be re-programmed. The control computer can be trained for recognising the desired coordinated movement pattern.
• During execution, the machine movement follows the trained path while the operator still retains the master control for the overall trajectory parameters and the motion velocity. In addition to convenience and manual flexibility, the new control architecture allows for the optimisation of the control of the machine during operation. An example is provided, based on robotic simulation and analysis, to illustrate the operator-assisted control of a front shovel.
• The new architecture is based on the recognition that there is no need to generate an integrated path in the control computer model-space and then differentiate the trajectory for joint variable control during real-time machine control. The differentiating control architecture, shown inFIG. 1, does not use joint position control loops. A main, Cartesian position control loop is applied to each motion component through visual feedback to the supervisory operator.
• The main position control signal can be control velocity, as a most common example characteristic to hydraulic power machinery. Through velocity control, the operator adjusts and corrects the tool position. An internal feedback loop is applied to generate a differential inverse kinematics model of the machine configuration for a given position. This model is the inverse Jacobean, a sensitivity matrix that relates the joint-space θ variables as a vector, to an input vector of Cartesian variables x_i.
• The architecture uses joint position variables only for the determination of the Jacobean, i.e., the matrix of the partial derivatives for a given position. The internal feedback, therefore, is accomplished through the determination of the mathematical model of the machine, and not through a signal. Consequently, the new differential control architecture realizes a model-based forward predictor through which the joint actuators are directly controlled
• Referring toFIG. 4, the powered shovel shown, in this case a front end shovel loader comprises a tracked motive under carrier or ground-engagingsystem101 supporting arotatable turret102.Rotatable turret102 has acab103 to house a human supervisory operator and supports one end of an articulatedpositioning unit104 for manipulating a tool or other end effector supported at the other end ofpositioning unit104.
• Positioning unit104 comprises aboom105 articulated to turret102 for rotation about apivot106, thereby providing a boom joint. One end of alever arm108 is articulated to boom104 for rotation about apivot110, thereby providing an arm joint, while the other end is articulated to and supports a tool, in this case a shovel orbucket112, for rotation about apivot114, thereby providing a tool joint, such rotations all being about axes perpendicular to the plane of the paper. While the example illustrated in the drawings comprises revolute joints having one or multiple degrees of freedom, other joints may be utilized, for example prismatic joints.
• The tilt or orientation ofshovel112 is preferably also variable, for whichpurpose shovel112 can be mounted with two further degrees of rotational freedom about axes in the plane of the paper by means (not shown). Such further shovel rotational means may comprise, for example, a first pivotal attachment to the distal end ofarm108 providing rotation about a longitudinal arm axis extending throughpivots110 and114 and a second axis perpendicular thereto, also extending throughpivot114.
• Piston-and-cylinderhydraulic actuators116,118 and120 control the motion and position ofboom105,arm108 and shovel orbucket112, respectively. Each actuator116-120 is pivotally secured at its ends andcylinder120 is connected to shovel112 through an articulatedlink122 to enhance its leverage. One or more leveling, guide or ancillary cylinders such as levelingcylinder124 can be provided if desired.
• Referring toFIG. 5, other actuators such as126 can optionally also be provided for moving and positioning other elements or systems, forexample turret102, and ground-engagingsystem101, as will be further described hereinbelow.
• In the motion control system illustrated inFIG. 5, each actuator116-120 and126 is operated by its own, local hydraulic control unit130-136, respectively, which control unit receives a signal from ajoint control coordinator138band controls its associatedactuator116,118,120 or126 to provide a suitable output, namely a desired position and/or velocity of the associated articulatedelement104,108,112 or128, determined by thejoint control coordinator138bbased on the forward input from the differentialcontrol kinematics generator138aand the feedback input from the machine differentialkinematics feedback generator138c.
• Each articulatedelement104,108,112 or128 is provided with a transducer sensor140-146 to detect preferably at least the position of the respective articulatedelement104,108,112 or128 with respect to its supporting element, and report the sensed position, preferably as a continuous or time-divided signal, to the machine differentialkinematics feedback generator138c.
• Acontrol box140 comprises a suitable number of manually operable control members orjoysticks142 which can be moved, pressurized or otherwise manipulated by ahuman supervisor144 to theboom104 and/orarm108 and/or shovel112 and/or other elements according to a desired trajectory.Joysticks142 may be the same or different and may comprise any convenient and suitable manually operable input member such as a pivotable lever, wheel, slide, key, button or set of keys or buttons or comprise a pointing device such as a mouse, trackball or the like. However, in the mining and construction industries, pivoted levers are common. Conventional powered shovels and other such articulated hydraulic machinery employ at least one joystick for each powered joint. Pursuant to the present invention, this number may be varied, as will become apparent hereinbelow. Additional joysticks may be provided for operation ofrotatable turret102 and, possibly also for the drive to ground-engagingsystem101 to move the powered shovel around, or to or from, a work site, or for other desired purposes.
• The signal outputs fromcontrol box140 are supplied to the differentialcontrol kinematics generator138athat is provided with suitable data storage means, such as a hard drive, optical drive and the like. Suitable software and/or programs, as are known in the art of control computers or other appropriate art, for implementing the processes of the invention may be stored on such data storage means, or may be remotely accessed bycentral computer138.
• Preferably, but optionally, the differentialcontrol kinematics generator138ais also be provided with conventional input output devices such as mouse, keyboard, touch pad or touch screen, monitor printer data transfer means such as radio wave input/output, removable data storage media, and so on, enablinghuman supervisor144 to interface with the unit and mediate the operation of, for example, positioningunit104, if desired. Such mediation or intervention byhuman supervisor144 enablessupervisor144 to change the response of the control system to one or more inputs received bycontrol box140 to modify a programmed response or to choose a desired response from multiple possible preprogrammed responses, as will be further described hereinbelow.
• The differentialcontrol kinematics generator138a,joint control coordinator138b, and machine differentialkinematics feedback generator138care preferably realized in a commoncentral computer138.
• Preferably also, thecentral computer138 provides a visual display which can include schematic representations of the real machine's configuration and virtual machine configurations that can be employed byhuman supervisor144 as well preferably as available calculated and learned trajectories which can be made available for selection byhuman supervisor144. Optionally also,machine100 may be equipped with cameras providing one or more views of the tool and the work area, which views can be displayed by the computer, and optionally integrated with the control programs to enhance the control system.
• Referring toFIGS. 5-6, employing the control architecture illustrated inFIG. 6,human supervisor144 usescomputer138 to input, or select a predetermined shovel trajectory, for processing bytrajectory generator146.Trajectory generator146 employs adifferential time base148 to generate a differential trajectory with transformation which outputs to differentialinverse kinematics model150 which is generated based on aJacobian generator152. Differentialinverse kinematics model150 provides a value, series of values or continuous signal for each machine joint to be moved; which may be the double differential with respect to time of the particular joint angle θ.
• Employing anintegrator154, a matrix of angular acceleration and velocity signals for each joint to be moved is applied to thejoint control system154 inFIG. 6.Control system154 preferably comprises relevant processing components ofcentral computer138 and control units130-136. Operation of control units130-136 imparts suitable positioning and velocity to actuators116-120 and126, providing a desiredmachine kinematics configuration156 to themachine100.
• Sensors140-146 read the positions and optionally the velocities of actuators116-120 and126 which values or signals are applied toJacobian generator152 both directly and after forward kinematics processing158.
• Human supervisor144 observes the machine, notably the performance of the end effector,shovel112, but also the configuration of theboom104 and the articulatedarm108, and can employvisual feedback160 to controltrajectory generator146, or to apply tool trajectoryadjustments using joysticks142 or to select a different or modified trajectory usingcentral computer138.
• An optionalartificial intelligence component162 receiving input from machine actuators116-120,126 andhuman supervisor144 can be used to provide to trajectory generator146 a desirable or optimum trajectory family for typical, repetitive tasks.
• As used herein, the term “trajectory” refers to a series of motion coordinates of an end effector, e.g. a tool or shovel112, which represent a number of consecutive motion positions or a locus of the motion of the end effector within a time frame; the term “machine kinematics” refers to the relationship between the actuators' positions or movements and the end effector's position or movement;
• As described above, the signal generated by anindividual joystick142 can be distributed to more than one actuator116-120 and126. Preferably, the distribution of the joystick control is accomplished dynamically, while the system is in motion, using a pre-programmed trajectory family, or pre-defined coordinate system, or a pre-defined control kinematics.
• Employing the novel control architecture provided by the invention, it is possible provide efficient control of coordinated joint motion without generating an integrated path in the control computer model-space and then differentiating the trajectory for joint variable control during real-time machine control.
• Thus, the control system illustrated inFIG. 6 employing differentiating control architecture, does not use joint position control loops. Instead, as provided by the invention, a main, Cartesian position control loop can be applied to each motion component through visual feedback from thehuman supervisor144.
• Although not an inherent or required component in the inventive control architecture, position feedback loops may optionally be employed for linearization, backlash compensation and/or to achieve added stability.
• While modified control architectures based upon other control parameters will be, or become, apparent to those skilled in the art, in a preferred embodiment of the invention, the main position control signal is control velocity, as a common example characteristic of hydraulic power machinery. Using velocity control, applied byjoysticks142, thehuman supervisor144 can adjust and correct the tool position. An internal feedback loop is applied via sensors140-146,forward kinematics158 andJacobian generator152 to generate differentialinverse kinematics model150 of the machine configuration for a given tool position. In other words, for a given tool position a model is generated of how the machine should be configured, which model specifies parameters such as the angular position and velocity of each joint. Pursuant to one preferred embodiment of the invention, this model can be an inverse, or pseudo inverse, Jacobian, a sensitivity matrix that relates the joint-space6 variables as a vector, to an input vector of Cartesian variables x_i.
• A “Jacobian” or “Jacobian determinant” is a determinant of the matrix whose ith row lists all the first order partial derivatives of a function f_i(x₁, x₂, . . . , x_n), i=1, 2, . . . , n, of real variables x_i. A Jacobian is useful, inter alia, for effecting transformations between polar coordinates and Cartesian coordinates.
• In management of the control of coordinated joint systems such as described herein, the mathematical conversion from tool workspace to joint space should be made efficiently to permit real time operation of the controlled articulated arm or manipulator. Solution of the inverse kinematics problem required to effect the transformation can conveniently use an iterative method employing a Jacobian.
• The Jacobian is a linear relationship between various possible changes in joint space and various possible changes in workspace. General approaches are known for calculating an inverse Jacobian for coordinated joint systems and such methods may be used in the present invention as will be apparent to those skilled in the art. One approach suitable for application in the embodiment of the invention illustrated inFIG. 4 on will now be described, by way of example.
• Some basic relationships for the variables and derivations are:
• $\begin{array}{cc}\left[d{\theta }_i\right]=J^{-1}\left[{\mathrm{dx}}_i\right]\left[{\stackrel{.}{\theta }}_i\right]=J^{-1}\left[{\stackrel{.}{x}}_i\right]\left[{\ddot{\theta }}_i\right]=J^{-1}\left[{\ddot{x}}_i\right]& \left(1\right)\end{array}$
• where “x_i” is the Cartesian acceleration in direction x_i; θ_tis the joint-space acceleration in direction θ_i; and differentials are with respect to time increment “dt”.
• The control architecture can use joint position variables alone for the determination of the Jacobian, providing the following possible matrix of the partial derivatives for a given position:
• $\begin{array}{cc}J\left(i,j\right)=\left\{\frac{\partial x_i}{\partial {\theta }_j}\right\}i=1\dots N,j=1\dots N& \left(2\right)\end{array}$
• Internal feedback, can accordingly be accomplished by determination of the mathematical model of the machine, rather than from a signal. Thus, the novel differential control architecture provided by the invention comprises a model-based forward predictor, differentialinverse kinematics model150, through which the joint actuators116-120,126 are directly controlled.
• Position control of the tool can be based on the Cartesian acceleration components, {umlaut over (x)}_i=d{dot over (x)}_i/dt evaluated from the {dot over (x)}_i(t) functions using a sufficiently fine differential time base, dt, for example in the range of from about 1 microsecond to about 100 milliseconds, preferably from about 1 to about 10 milliseconds. These variables can be determined from the combination of the a priori trajectory and thesupervisor144's corrections input. Assuming that a predefined trajectory is given in the parametric form of x_i=f_i(g) with g chosen to be the arc length of the path s, {dot over (x)}_ican be determined as follows:
• $\begin{array}{cc}{\stackrel{.}{x}}_i=\frac{\partial f_i}{\partial g}\frac{g}{t}=\frac{\partial f_i}{\partial s}v\left(t\right),i=1\dots N& \left(3\right)\end{array}$
• wherein v(t) is the motion velocity controlled by thesupervisor144. The ∂f_i/∂s derivatives are determined by the predefined trajectory. Since differentiation eliminates any additive constants in the parametric trajectory, the movement can be originated from any starting point. Consequently, a trajectory family or families in the form of one or more sets of parallel trajectory curves can readily be generated bytrajectory generator146 by applying Equation (3).
• Parallel shifting of the trajectory during motion can be accomplished by super-imposing the control velocities of Equation (3) and the modification velocity component {dot over (θ)}_i^T, controlled by thesupervisor144 in machine joint space:
• $\begin{array}{cc}{\stackrel{.}{x}}_i=\frac{\partial f_i}{\partial s}v\left(t\right)+{\stackrel{.}{\theta }}_i^T,i=1\dots N& \left(4\right)\end{array}$
• The relationship between the {dot over (x)}_l^Tcorrection control and the {dot over (θ)}_lT machine joint control components defines a correction control kinematics that can be expressed with the introduction of the control kinematics Jacobian matrix, Jc, as follows using bracketed vector notation:
• 
  [{dot over (x)}_i^T]=J_c[{dot over (θ)}_i^T]  (5)
• Such implementation is illustrated inFIG. 5. Expressing [{dot over (θ)}_i^T] from Equation (5) and substituting it into Equation (4) for all components gives:
• $\begin{array}{cc}\left[{\stackrel{.}{x}}_i\right]=\left[\frac{\partial f_i}{\partial s}\left(s\right)\right]v\left(t\right)+J_c^{-1}\left[{\stackrel{.}{x}}_i^T\right]& \left(6\right)\end{array}$
• Equation (6) describes differential trajectory generation with coordinate transformation such, for example, as is illustrated inFIG. 7.
• Some examples of the physical meaning of the terms on the right-hand side ofEquation 6 are as follows:
• $\left[\frac{\partial f_i}{\partial s}\left(s\right)\right]$
• • is the predefined trajectory that is available to thehuman supervisor144 to copy into the motion to create a desired trajectory. This term is a function of the movement along the trajectory through s. If desired, thehuman supervisor144 can “re-wind” s and start at the beginning for a new motion.
  • v(t): is the real-time automated execution velocity along the pre-defined trajectory, and is controlled by thehuman supervisor144. If v=0, the automatedrobotic machine100 becomes fully manual.
  • J_c⁻¹: is the coordinate transformation feature for manual correction or fully manual control. Several J_c⁻¹choices can be provided for selection. If J_c⁻¹is the unit matrix, the automatedrobotic machine100 is controlled directly in the joint space, acting like a plain manual machine.
  • [{dot over (x)}_i^T
    
    : correction or manual motion control velocities guided by the original joysticks.
• Referring now toFIG. 7, there is shown an a priori orpredefined trajectory170 in a base coordinatesystem172 extending in orthogonal directions x₂and x₂which trajectory has a sinuous shape. An end effector position A is located a distance s from a home position H. Also shown, in broken lines, is aparallel trajectory174 in a transformed coordinatesystem176, which is illustrative of how a family of trajectories parallel topredefined trajectory170. Parallel trajectories such as174 may be generated by modifying the two-dimensional translational trajectory vectors ofpredefined trajectory170 by adding position correction terms x₁^Tand x₂^Tto the absolute position coordinates in Equation (6) resulting in displacement of end effector position A to a new position B. Three-dimensional and rotational embodiments of the parallel trajectory may be generated by adding corresponding third dimension and rotational corrections. A family of trajectories may be generated by varying the values of the corrections. The particular coordinate transformation employed may be selected bysupervisor144 and represents a parallel, similarity coordination between the individual control directions in addition to the one defined by the basic trajectory. The a priori orpredefined trajectory170 and its derivatives such asparallel trajectory174 define a trajectory family.
• InFIG. 8, the parallel trajectory family shown comprises a curvilinear digging trajectory family in an orthonormal s-n coordinate system. This family and other families of parallel trajectories that will be apparent to those skilled in the art may be employed in the invention to provide more or better choices tosupervisor144.
• Preferably, a range of available trajectory families and their parameters is graphically displayed tosupervisor144 for selection and implementation.
• The new optimized trajectory is recorded and added to a menu of trajectory selection available tosupervisor144. Software programming associates with each trajectory a corresponding implementation of the machine kinematics to control the automatedrobotic machine100 to perform the trajectory. Preferably, in the software defined kinematics, the original functions ofjoysticks142 are re-defined to distribute their outputs among actuators116-120 and126 for greater efficiency. If desired, eachjoystick142 may instruct the coordinated movement of several of actuators116-120 and126, depending on the need to simplify the control of a given task.
• Once a given trajectory has been input to the system and processed,supervisor144 can activate the appropriate machine kinematics corresponding with the new trajectory to perform the relevant task by selecting the new trajectory from a menu or visualization in graphical display, or by other suitable activation means.
• Another preferred embodiment of the invention includesartificial intelligence component162 integrated into the control architecture to identify one or more typical repetitive elements of the machine's movements. In one example of the employment ofartificial intelligence component162, themachine100 goes through a teaching period where multiple predefined trajectories are input to the systems and stored bycentral computer138. Suitable trajectories can be manually input bysupervisor144 or could be electronically supplied from simulations or experience with or training of other similar machines.
• Artificial intelligence component162 of the control architecture can then review all the predefined trajectories and determine a best possible or most suitable common trajectory of all the input trajectories for a particular machine task e.g. “fill bucket” or “dump load”, or trajectory e.g. “high swing”. The selected common trajectory can be determined for each relevant link. Theartificial intelligence component162 can determine the optimum parameters for the differential trajectory generation with transformation units. Thesupervisor144 is then notified by theartificial intelligence component162 that a satisfactory common trajectory has been successful determined. From this point, thesupervisor144 can execute an optimized trajectory which can be further adjusted real-time in an ever-changing uncontrolled task environment with the new control architecture of the invention.
• An exemplary computer program simulating the application of the inventive differential control architecture to the control of an excavator, using a coordinate system transformation is given in Appendix 1 to this specification. In the Appendix 1 program three machine actuators are coordinated by one control signal from a simulated joystick. The operation of the program will be readily understood by one skilled in the art. Accordingly, no further explanation is provided here. Subroutine functions used in the main program are given inAppendix 4. The appendices employ MATLAB “The Language of Technical Computing, The Mathworks Inc., Version 6.1, 2001. MATLAB is an enginnering computational and visualization package. It is also a programming environment in which a computer program script (in its specific sytax) can be executed to make (1) calculations, (2) graphs-images, (3) to access data from computer interface such as a joystick, etc. All the appendix scripts are in MATLAB symbolic language. The joystick drivers came from a third-party supplier, interfacing a common, 3-dimensional game joystick to MATLAB.
• For the simulated bucket manipulation operations illustrated inFIGS. 9-14, a large mining front shovel, for example a Caterpillar model E650, is chosen as the automatedrobotic machine100 on which to demonstrate the application of the differential control architecture of the invention. Three operating constraints (referenced “the three specified operating constraints” hereinafter) are used in comparison exercises between fully manual control and control using the inventive differential control method: (1) to fill three buckets from one base position, (2) to reduce the complexity of the manual control by reducing the number of joints simultaneously moved, and (3) to produce an identical excavation profile at the end of the third bucket fill to the initial profile. The last requirement is useful for emulating continuous shovel operation with a three-bucket load per machine advance excavating on a repeating pattern.
• For comparison, as illustrated inFIG. 9, a conventional manual use of themachine100 is graphically emulated using a simple graphical imaging tool for robotics by Danko (2000b) “Robotics Teaching support tools” or “Introduction to Robotics,” a 3-credit graduate course taught at the University of Technology, Helsinki, Finland.
• FIG. 9 shows the automatedrobotic machine100 executing an indicated first cut with time-lapse graphics showing the moving machine link positions at discrete time intervals. As depicted, the three specified operating constraints can only be satisfied by executing a steep, highly unrealistic excavation profile whereinboom105 is extended forwards and upwardly andarm108 is close to the horizontal throughout its pivoting movement. As may be seen, in executing the first cut, botharm108 andbucket112 are moved; which requires operator coordination of both movements. Also shown is a previous machine position, indicated in dotted lines, where the indicated previous excavation profile was executed.
• As shown inFIG. 10, pursuant to this prior art method, the second bucket loading also needs coordination of the movements of two “links”arm108 andbucket112, while the third, finishing cut can be made by controllingarm108 only, i.e. by swingingarm108 aboutpivot110 without changing the orientation ofbucket112 relative to arm108 aboutpivot114.
• FIG. 11 illustrates that excavation along a different, more practical profile, for example a slope at an angle close to the angle-of-repose of a muck pile, requires the coordinated movements of all three joints atpivots106,110 and114. Employment of the differential control architecture for the execution of these complex control requirements provides a satisfactory outcome wherein themachine100 can rake along a pre-defined slope angle in three layers from one base position, with simultaneous bucket angle adjustment, as shown inFIG. 11.
• Long and shallow raking is not advantageous, and a different digging strategy can be used to accomplish the three constraints defined previously while excavating along a slope. One possible desirable or even optimal predetermined trajectory for excavating along a slope is shown inFIG. 12 where f is the distance of horizontal excavation advance; s is the arc length; B is the bucket edge; H is the starting point and θ_bis the bucket angle. The trajectory parameters illustrated inFIG. 12 may be generated by adjustment of an a priori trajectory, by thehuman supervisor142. One suitable computer algorithm for effecting theFIG. 12 trajectory is given inAppendix 2, by way of example.
• InFIG. 12, the excavation profile is a spline arc with a horizontal beginning at H and an end tangential to the slope sg. The slope grade sg and the excavation advance f are variable parameters. The parametric trajectory is defined by the derivatives with respect to the motion along the arc, and is therefore independent of the starting position. Therefore, the same basic trajectory can be used to emulate the first and the second bucket fills, as illustrated by the excavation trajectories shown inFIGS. 13 and 14.
• As indicated by the multiple positions ofboom105,arm108 andbucket112, inFIGS. 13-14, all three cuts require coordination of three machine links, yet they are easily controlled byhuman supervisor144, employing the inventive control architecture. The third bucket fill employs a somewhat different, basic trajectory, a “cleaning trajectory”, so as to finish with a horizontal bench surface, followed by a raking motion along the slope until the height of the second cut is reached.
• Each of the parallel, inclined, straight cutting trajectories illustrated inFIGS. 11-14 can be created using only onejoystick control142, simultaneously coordinated to the boom, arm and bucket actuators116-120. Simulated, straight-line trajectories are shown inFIG. 11. Shift of the cutting line normal to the slope can be accomplished by positioning the bucket cutting edge to a starting point by another joystick. Although this second joystick remains active during the control of the movement along each cut, it requires no action from the super-operator for one cut along the slope.
• Referring toFIG. 15, the differential control architecture of the invention can be employed to enablehuman supervisor144 to configuremachine100 as a virtual machine, so that, for example, a small machine will follow joystick control according to the kinematics characteristics of a large machine.
• FIG. 15 shows in solid line the relativelyshorter boom105,arm108 andbucket112 of asmall machine100. Subtending the same beginning and end points, and shown in broken lines is a large virtual machine comprising a relatively longer boom105′,arm108′ andbucket112′.Human supervisor144 can define the virtual machine and obtain the motion kinematics for a large machine enablinghuman supervisor144 to select the virtual machine and have, for example, the small real machine move like a large machine. The real machine responds in motion to one ormore joysticks100 as though it were a large machine.
• Human supervisor144 can directly control the virtual angles θ₁^T, θ₂^Tand θ₃^Twithjoysticks142. The real machine moves according to the virtual machine kinematics, represented by the control kinematics inverse Jacobian, J_c⁻¹.
• One suitable differential control architecture for virtual configuration of an articulated machine in the manner described with reference toFIG. 15 is shown inFIG. 16, in which ρ_jdenotes the user-selected parameters of the virtual machine kinematic configuration. Other symbols having meanings consistent with or parallel to their usages elsewhere herein. The operation of the differential control architecture illustrated inFIG. 16 and modifications thereof or other suitable architectures for controlling an articulated machine for virtual operation will be, or will become, apparent to those skilled in the art from the disclosure herein.
• Two useful virtual machine scripts are included in the accompanying Appendix 5. These two scripts can control images based on real joystick signals according to the inventive control architecture. Script Demo1 controls the machine image movement in x-y Cartesian coordinates while the x-y coordinate system can be user-rotated between +/−45 degrees. Script Demo2 controls the machine image according to a pre-defined differential cutting trajectory.
• The graphical examples ofFIGS. 11-14 show that the inventive control architecture application can be used not only for effective and efficient control of an articulated machine, but also for creating related 3D images, for example, as shown, a complete schematic of an excavator executing coordinated joint control. One example of the implementation of the differential control architecture of the invention to generate a sequence of computer images representing an excavator employing three actuators coordinated by one control signal, of a simulated joystick, appears in Appendix 3 herein.
• In the foregoing examples, including those employingartificial intelligence component162, thehuman supervisor144 is, in most cases, an important element of the machine's control architecture. Thehuman supervisor144 guides themachine100 based on visual feedback and determines the preferred trajectory for a given task.
• It will however be apparent to one of ordinary skill in the art that the differential, coordinated joint motion control system of the invention may be applied to control problems that do not require a human operator or human supervisor as an element in a control loop. Such an application may arise when a multi-joint machine is required to follow a particular trajectory or trajectories generated by measurement and/or a mathematical model or models. The inventive concepts relating to the distribution of joint control velocity components from asingle joystick142 to multiple, or all, joint actuators can, pursuant to the present invention, be applied, with suitable adaptation to other problems involving the control, characterization, representation or simulation of coordinated joint systems.
• One example is the graphical matching problem of a captured, filmed, digitized, or otherwise scanned image to a physical model, for example an anatomical model or subject. Another problem, some solutions of which have been described hereinabove, is that of providing a graphical representation of a multi-linked machine.
• In the particular example of gait analysis of the human body, the trajectories to be followed are the measured, and recorded paths of body points or markers, and the end effectors are the corresponding points on the multi-linked biomechanical model of the human body. Matching the model to the measured trajectory with the use of the inventive, coordinated, joint motion control system can be advantageous and provide new solutions to evaluating joint parameters related to motion characteristics, and to evaluating muscle forces and torques in the human body. Such matching methods and systems may be utilized for research, teaching, rehabilitation, athletics training, prosthetics design and other purposes as will be apparent to those skilled in the art.
• In such matching and representation applications, thehuman supervisor144's visual feedback can be replaced by an automated error evaluator that calculates the difference between the measured or desired trajectory points and those calculated from a biomechanical forward kinematics model. The error evaluator also reduces this difference by correcting the joint actuator velocities coordinately in a manner analogous to the way the inventive control architecture supports control by thehuman supervisor144 in the above-described excavator examples.
• One example of such an automated image-matching application is illustrated inFIGS. 17 and 18.FIG. 17 depicts the example of captured, or measured, [X_i^c] and modeled [X_i] trajectories of part of a human right arm depicted as a three-link revolute machine.
• FIG. 18 is a schematic chart of one embodiment of the application of the inventive joint velocity control coordination system for efficiently achieving a best fit of a modeled trajectory to a captured trajectory. The illustrated method is intended to be iterative for each time interval until the best square fit between the desired trajectory, which is to say the measured or captured trajectory and the system executed or modeled trajectory is achieved. While the performance of the method is believed clear fromFIG. 18 when read in conjunction withFIG. 17, in light of the description herein, nevertheless, the control concept will now be described in more detail to illustrate the application of the inventive differential control principles in this coordinated joint manipulation example.
• The trajectory for each X_ipoint is guided by the captured trajectory X_i^c. The velocity components {dot over (x)}_i{dot over (y)}_iand ż_ican be calculated from the X_i^cvector for each time interval by numerical derivation. Cubic spline or other suitable smoothing of the trajectories can be used to achieve reliable derivatives. Cubic spline smoothing of the trajectory function in arc-length parametric form, X_i=F_i(s_i), can also be used, if desired. The control velocity is the sum of two components, the first term, for following the captured trajectory, and the second term, for correcting the matching error, ΔX_i^E(t):
• $\begin{array}{cc}{\stackrel{.}{X}}_i=\frac{\partial F_i}{\partial s_i}\left(s_i\right)·\frac{s}{t}+X_i^E,i=1\dots N& \left(7\right)\end{array}$
• The matching error correction velocity vector {dot over (X)}_i^Ecan be iteratively calculated from finite differences (or higher-order differential schematics if deemed necessary) for each time step which is to say for each consecutive frame, as follows:
• $\begin{array}{cc}{\stackrel{.}{X}}_i^E\left(t\right)=\frac{X_i^c\left(t\right)-X_i\left(t\right)}{\Delta t}=\frac{\Delta X_i^E\left(t\right)}{\Delta t}& \left(8\right)\end{array}$
• The control velocity vectors in Equation 7 are preferably distributed to the individual joints in terms of joints angular velocity components. This can be accomplished by applying the differential joint control coordination control concept based on a differential inverse kinematics model of the human body. It is convenient to assume that the trajectory coordinate data set over-determinates the set of joint angles. Therefore, a least-square-fit solution can be obtained for the joint motion velocities:
• 
  [θ_i^c]=[J_i]\{[X_i^c]+[X_i^E]}  (9)
• A further optimization loop is included inFIG. 18 to permit a further reduction in the multi-dimensional distance error, measured in root-mean-square norm, by varying the [θ_i] set. Fast convergence is achieved since every consecutive [X_i] set, being an element of the time-series of the positions, uses the previous set as a good initial approximate position.
• As illustrated by the foregoing examples the inventive systems and methods enable the design of efficient computational algorithm equipment applicable to problems associated with coordinated joint systems or related or equivalent systems, as will be apparent to those skilled in the art. The invention also provides a new level of operator control that can be called human supervisory control wherein the operator is relieved of repetitive and challenging manipulative tasks but serves a high level function selecting monitoring and adjusting automated machine performance.
• A comparison is given below between a machine operator control, as it is currently known, and ahuman supervisor144 control that becomes possible with the inventive, differential control architecture.
• Conventional articulated arm machines suffer from a number of drawbacks. A manual operator is required to control the machine. Conventional machines usually cannot be adapted to a given task-type without making hardware adjustment(s) and often no task-type adjustment is possible. Tool movement trajectory is usually directed point-to-point using multiple control joysticks or lever arms each of which is rigidly pre-defined to control only one corresponding machine actuator. The control of a complex trajectory may require the use of all joysticks simultaneously. Repeated tasks are always new tasks. Repetition increases the operator's skills, and the efficiency of task execution increases through the skills of the operator. Automated repeatability of a trajectory is not usually supported. Remote control with camera vision is difficult to execute since visually coordinated joystick movement is needed. Visual feedback is key to trajectory control; in poor visibility, the task is problematic.
• In contrast, the present invention provideshuman supervisor144 control of an automatedrobotic machine100 which with the inventive control architecture can be adapted to one or more given task-types by selecting from pre-programmed software-defined machine kinematics. New machine kinematics can also be created by training the machine to a new kinematics in a new coordinate system.
• Tool movement trajectory cant be directed point-to-point using one or more control joysticks or lever arms that are dynamically defined to control a number of machine actuators. The control of the same trajectory may be accomplished using only one joystick. Repeated tasks are built into the new control system's trajectory generator. The efficiency of task execution increases through the refinements of the control parameters either by the operator or the artificial intelligence component of the control system. possible to repeat a pre-defined trajectory. Remote control with the new architecture provides better control since part of the trajectory control is supported by a priori trajectory characteristics. Remote control in poor visibility employing one or more cameras other viewing systems is a feasible embodiment of the invention and can be safe and efficient to execute. Optionally, wireless remote communication may be used to transmit the control signals.
• An automated machine with a traditional robotic control architecture requires high-precision components to be used in the embedded, joint position control loops. Another disadvantage is that complex algorithms are needed to combine the operator control components with the pre-defined trajectory feature of a conventional, robotic equipment with trajectory planner. In contrast the differential control architecture of the present invention, in preferred embodiments can provide a number of advantageous features.
• Safety of control the manual and the pre-programmed motion control components can be blended together continuously and always under the guidance of thehuman supervisor144. Preferred inventive differential control schematics can apply simple velocity-controlled actuators that are similar to those of an electro-hydraulic, manual machine equipped with additional joint position sensors. The additional inventive control architecture can be implemented by modifying the control joystick circuits and the human supervisor's user interface often without significant other modifications.
• The inventive control architecture can accept a selection of parametric a priori trajectories. The inventive system provides flexibility. In addition to trajectory selection or parametric modification, re-programming of the motion can be made through motion modification control executed in a transformed or joint space coordinate system. Operator support optimization and computational assistance can be incorporated in the determination and selection of a basic trajectory, representing a motor schema of the motion. In addition the invention may favorably impact the performance of model-based evaluation of time-seriesed captured images.
INDUSTRIAL APPLICABILITY
• The present invention is particularly, but not exclusively, suitable for application in the mining and construction and image matching and generation industries providing a flexible control of equipment in an unstructured environment. In repetitive but slightly modulated tasks, the invention can provide relief to an operator by supporting automatic control regarding repetitive trajectory elements.
• The entire disclosure of each patent and patent application cross-referenced or referenced herein and of each non-patent publication referenced herein is hereby incorporated herein by reference thereto, as though wholly set forth herein. Each document incorporated by reference in any of the foregoing patents, patent applications or non-patent publications is also incorporated herein in its entirety by reference thereto.
• While illustrative embodiments of the invention have been described above, it is, of course, understood that various modifications will be apparent to those of ordinary skill in the relevant art, or may become apparent as the art develops. Many such modifications are contemplated as being within the spirit and scope of the invention.
APPENDIXES 1-5

• APPENDIX 1
  Differential control architecture example to generate a sequence of
  computer images representing an excavator. Three actuators are
  coordinated by one control signal, of a simulated joystick. The
  coordinated control is based on coordinate transformation.

  %Figure 6.

  %dynamic trajectory planning

  %machine makes linear rakeing: cannot advance slices since it reaches

  envelope

  clear all

  close

  frame=[0 16 −1 15]; %figure frame size

  bl=39*.1789;

  sl=25*.1789;

  bm=[0 b1;0 0];

  ds=[0 s1;0 0];

  bt=[0  1  6.5   11   16  19   14.5  10 1 0;

  0  −9  −12  −10 −7.5   0   −3   −1.5 4.5 0];

  bt=bt*.1789;

  bct=bt(1,6)−bt(1,1);     %bucket opening size

  s1=ones(size(bt));

  s2=ones(size(ds));

  cab=[NaN  0  12  14.88  16.98  17.75  16.98  14.88   12   4

  4   6  0

  NaN  0   0  0.77   2.88   5.75   8.63   10.73  11.5  13

  34  44  44];

  cab=cab*.1789;

  bmss(1,:)=s2(1,:)*bm(1,2); %stick-size vector

  bmss(2,:)=s2(2,:)*bm(2,2); %stick-size

  bmbs(1,:)=s1(1,:)*bm(1,2); %bucket-size vector

  bmbs(2,:)=s1(2,:)*bm(2,2); %bucket-size

  dsbs(1,:)=s1(1,:)*ds(1,2); %bucket-size

  dsbs(2,:)=s1(2,:)*ds(2,2); %bucket-size

  bx=0;

  by=26*.1789;

  %control

  %home

  x=4.2;y=0;dh=−120; %horizontal vx constant

  %x=12;y=2.1468;dh=0; %verical vy constant

  d=dh;

  u=90;

  pf=1;

  inv_gr1 % inverse kinematic provides home joint coordinates

  hold on %keeps picture

  ah=a;

  bh=b;

  aint=ah;

  bint=bh;

  dint=dh;

  %dynamic trajectory planning

  prof=[(0:9)/10 ones(1,100) (9:−1:0)/10];

  abd=[ ];

  L=7.62;  %25 ft

  vav=mean(prof);

  dta=L/vav/120;

  for i=1:120

  vx=0.8*prof(i); %m/s

  vy=0.8*prof(i);

  dt=dta/.8;

  a=aint;

  b=bint;

  dd=.82;

  d=dint;  %bucket absolute angle

  pf=1; %pf=rem(i,10); %plots at every time pf==0

  pf=rem(i,10);if i==1;pf=0;end;

  fwd_gr1

  %Machine Jacobian JM

  JM=flipud([eds−bbm eds−ebm]);JM(1,:)=−JM(1,:);

  vbx=−(ebt(2,1)−eds(2,1))*dd*pi/180/dt;

  vby=(ebt(1,1)−eds(1,1))*dd*pi/180/dt;

  dang=inv(JM)*[vx−vbx;vy−vby]*180/pi*dt;

  aint=aint+dang(1,1);

  bint=bint+dang(2,1);

  dint=dint+dd;

  abd=[abd [dang;dd]];

  plot(ebt(1,1),ebt(2,1),‘k’) %plots bucket edge

  axis(frame)

  figure(gcf)

  pause(.01)

  end

  plot(cab(1,:),cab(2,:),‘k:’)

  xlabel(‘Horizontal Distance [m]’)

  ylabel(‘Elevation [m]’)

  %FORWARD BOUNDARY

  by=26*.1789;

  %control

  %home

  x=4.2+1.7;y=0;dh=−120; %horizontal vx constant

  %x=12;y=2.1468;dh=0; %verical vy constant

  d=dh;

  u=90;

  pf=1;

  inv_gr1 % inverse kinematic provides home joint coordinates

  hold on %keeps picture

  ah=a;

  bh=b;

  aint=ah;

  bint=bh;

  dint=dh;

  %dynamic trajectory planning

  prof=[(0:9)/10 ones(1,100) (9:−1:0)/10];

  abd=[ ];

  L=7.62;

  vav=mean(prof);

  dta=L/vav/120;

  for i=1:120

  vx=0.8*prof(i); %m/s

  vy=0.8*prof(i);

  dt=dta/.8;

  a=aint;

  b=bint;

  dd=.82;

  d=dint; %bucket absolute angle

  pf=rem(i,10);if i==1;pf=0;end; %plots at every time pf==0

  fwd_gr1

  %Machine Jacobian JM

  JM=flipud([eds−bbm eds−ebm]);JM(1,:)=−JM(1,:);

  vbx=−(ebt(2,1)−eds(2,1))*dd*pi/180/dt;

  vby=(ebt(1,1)−eds(1,1))*dd*pi/180/dt;

  dang=inv(JM)*[vx−vbx;vy−vby]*180/pi*dt;

  aint=aint+dang(1,1);

  bint=bint+dang(2,1);

  dint=dint+dd;

  abd=[abd [dang;dd]];

  plot(ebt(1,1),ebt(2,1),‘k’) %plots bucket edge

  axis(frame)

  figure(gcf)

  pause(.01)

  end

  %plot(cab(1,:),cab(2,:),‘k:’)

  xlabel(‘Horizontal Distance [m]’)

  ylabel(‘Elevation [m]’)

  %CUT UPPER CORNER

  plot(cab(1,:),cab(2,:),‘k:’)

  hold off

• APPENDIX 2
  Differential control architecture example to generate a sequence of
  computer images representing an excavator. Three actuators are
  coordinated by one control signal, of a simulated joystick.
  The control is based on a defined parametric trajectory. The
  first cut is simulated. The same parametric trajectory can
  be used for the second cut.

  %Figure 8.

  %dynamic trajectory planning

  %machine makes linear rakeing: cannot advance slices since it reaches envelope

  clear all

  close

  frame=[0 16 −1 15]; %figure frame size

  b1=39*.1789;

  s1=25*.1789;

  bm=[0 b1;0 0];

  ds=[0 s1;0 0];

  bt=[0  1  6.5 11  16  19 14.5   10  1 0;

  0 −9 −12  −10 −7.5  0  −3  −1.5 4.5 0];

  bt=bt*.1789;

  bct=bt(1,6)−bt(1,1);    %bucket opening size

  s1=ones(size(bt));

  s2=ones(size(ds));

  cab=[NaN 0 12 14.88 16.98 17.75 16.98 14.88  12  4  4  6 0

  NaN 0  0 0.77  2.88  5.75  8.63  10.73 11.5 13 34 44 44];

  cab=cab*.1789;

  bmss(1,:)=s2(1,:)*bm(1,2); %stick-size vector

  bmss(2,:)=s2(2,:)*bm(2,2); %stick-size

  bmbs(1,:)=s1(1,:)*bm(1,2); %bucket-size vector

  bmbs(2,:)=s1(2,:)*bm(2,2); %bucket-size

  dsbs(1,:)=s1(1,:)*ds(1,2); %bucket-size

  dsbs(2,:)=s1(2,:)*ds(2,2); %bucket-size

  bx=0;

  by=26*.1789;

  %control

  %home

  x=4.2;y=0;dh=−120; %horizontal vx constant

  %x=12;y=2.1468;dh=0; %verical vy constant

  d=dh;

  u=90;

  pf=1;

  inv_gr1 % inverse kinematic provides home joint coordinates

  hold on %keeps picture

  ah=a;

  bh=b;

  aint=ah;

  bint=bh;

  dint=dh;

  %dynamic trajectory planning

  prof=[(0:9)/10 ones(1,100) (9:−1:0)/10];

  abd=[ ];

  L=7.62; %25 ft

  vav=mean(prof);

  dta=L/vav/120;

  for i=1:120

  vx=0.8*prof(i); %m/s

  vy=0.8*prof(i);

  dt=dta/.8;

  a=aint;

  b=bint;

  dd=.82;

  d=dint; %bucket absolute angle

  pf=1; %pf=rem(i,10); %plots at every time pf==0

  % pf=rem(i,10);if i==1;pf=0;end;

  fwd_gr1

  %Machine Jacobian JM

  JM=flipud([eds−bbm eds−ebm]);JM(1,:)=−JM(1,:);

  vbx=−(ebt(2,1)−eds(2,1))*dd*pi/180/dt;

  vby=(ebt(1,1)−eds(1,1))*dd*pi/180/dt;

  dang=inv(JM)*[vx−vbx;vy−vby]*180/pi*dt;

  aint=aint+dang(1,1);

  bint=bint+dang(2,1);

  dint=dint+dd;

  abd=[abd [dang;dd]];

  plot(ebt(1,1),ebt(2,1),‘k’) %plots bucket edge

  axis(frame)

  figure(gcf)

  pause(.01)

  end

  plot(cab(1,:),cab(2,:),‘k:’)

  xlabel(‘Horizontal Distance [m]’)

  ylabel(‘Elevation [m]’)

  %FORWARD BOUNDARY

  by=26*.1789;

  %control

  %home

  x=4.2+1.7;y=0;dh=−120; %horizontal vx constant

  %x=12;y=2.1468;dh=0; %verical vy constant

  d=dh;

  u=90;

  pf=1;

  inv_gr1 % inverse kinematic provides home joint coordinates

  hold on %keeps picture

  ah=a;

  bh=b;

  aint=ah;

  bint=bh;

  dint=dh;

  %dynamic trajectory planning

  prof=[(0:9)/10 ones(1,100) (9:−1:0)/10];

  abd=[ ];

  L=7.62;

  vav=mean(prof);

  dta=L/vav/120;

  for i=1:120

  vx=0.8*prof(i); %m/s

  vy=0.8*prof(i);

  dt=dta/.8;

  a=aint;

  b=bint;

  dd=.82;

  d=dint; %bucket absolute angle

  pf=1; %pf=rem(i,10);if i==1;pf=0;end; %plots at every time pf==0

  fwd_gr1

  %Machine Jacobian JM

  JM=flipud([eds−bbm eds−ebm]);JM(1,:)=−JM(1,:);

  vbx=−(ebt(2,1)−eds(2,1))*dd*pi/180/dt;

  vby=(ebt(1,1)−eds(1,1))*dd*pi/180/dt;

  dang=inv(JM)*[vx−vbx;vy−vby]*180/pi*dt;

  aint=aint+dang(1,1);

  bint=bint+dang(2,1);

  dint=dint+dd;

  abd=[abd [dang;dd]];

  plot(ebt(1,1),ebt(2,1),‘k’) %plots bucket edge

  axis(frame)

  figure(gcf)

  pause(.01)

  end

  %plot(cab(1,:),cab(2,:),‘k:’)

  xlabel(‘Horizontal Distance [m]’)

  ylabel(‘Elevation [m]’)

  %CUT UPPER CORNER

  %DEFINE CUTTING TRAJECTORY

  %constants

  f=1.7; %forward advance

  sg=1; %slope grade 45 degrees

  dh=−90; %home bucket position

  dhr=dh*pi/180;

  %x=4.2;y=0; %hone coordinates

  x=8.75;y=4.6;

  d=dh;

  u=90;

  pf=0;

  inv_gr1 % inverse kinematic provides home joint coordinates

  ah=a;

  bh=b;

  pf=0; %pf=rem(i,10);if i==1;pf=0;end; %plots at every time pf==0

  fwd_gr1

  axis(frame)

  xh=ebt(1,1); %home coordinate of x

  yh=ebt(2,1); %home coordinate of x

  k=−bct*(sin(dh*pi/180)+cos(atan(sg)));

  h=f+k/sg;

  c3=sg/h{circumflex over ( )}2−2*k/h{circumflex over ( )}3;

  c2=3*k/h{circumflex over ( )}2−sg/h;

  hold on

  %variables

  aint=ah;

  bint=bh;

  prof=[(0:9)/10 ones(1,100) (9:−1:0)/10];

  abd=[ ];

  L=25; %degrees here

  vav=mean(prof);

  dta=L/vav/120;

  d1o=dh;

  for i=1:30

  dd=.06*prof(i); %degree/sec

  dt=dta;

  X=ebt(1,1)−xh;

  Y=ebt(2,1)−yh;

  if (X{circumflex over ( )}2+Y{circumflex over ( )}2)==0

  xpy=1;

  else

  xpy=X/(X{circumflex over ( )}2+Y{circumflex over ( )}2){circumflex over ( )}.5;

  end

  if X<=h

  vx1=(dd/dt)*h/(atan(sg)−dh−90)*xpy;

  vy1=(3*c3*X{circumflex over ( )}2+2*c2*X)*vx1;

  l1=(h−bct*(sin(atan(sg))−cos(dhr)));

  xdiff=X*(1−l1/h)+bct*cos(dhr);

  ydiff=bct*sin(dhr)+Y;

  d1=180/pi*atan2(ydiff,xdiff);

  else

  vx1=(dd/dt)*h/(atan(sg)−dh−90)*xpy;

  vy1=vx1*sg;

  xdiff=X*(1−l1/h)+bct*cos(dhr);

  ydiff=bct*sin(dhr)+Y;

  d1=180/pi*atan2(ydiff,xdiff)*(X/h){circumflex over ( )}2; %reduced angular bucket adjustment

  %d1=−(90−180/pi*atan(sg));

  end

  dd1=d1−d1o; %angular incremental change

  d1o=d1;

  JM=flipud([eds−bbm eds−ebm]);JM(1,:)=−JM(1,:);

  abd=[abd [vx1;vy1;d1]];

  vbx=−(ebt(2,1)−eds(2,1))*dd1*pi/180/dt;

  vby=(ebt(1,1)−eds(1,1))*dd1*pi/180/dt;

  dang=inv(JM)*[vx1−vbx;vy1−vby]*180/pi*dt;

  aint=aint+dang(1,1);

  bint=bint+dang(2,1);

  %dint=dint+dd;

  a=aint;

  b=bint;

  d=d1;

  pf=0; %pf=rem(i,10);if i==1;pf=0;end; %plots at every time pf==0

  fwd_gr1

  figure(gcf)

  pause(.1)

  end

  plot(cab(1,:),cab(2,:),‘k;’)

  hold off %pap8.m

  %dynamic trajectory planning

  %machine makes linear rakeing: cannot advance slices since it reaches envelope

  clear all

  close

  frame=[0 16 −1 15]; %figure frame size

  b1=39*.1789;

  s1=25*.1789;

  bm=[0 b1;0 0];

  ds=[0 s1;0 0];

  bt=[0  1 6.5  11  16  19  14.5  10 1  0;

  0 −9 −12  −10 −7.5  0  −3  −1.5 4.5  0];

  bt=bt*.1789;

  bct=bt(1,6)−bt(1,1);    %bucket opening size

  s1=ones(size(bt));

  s2=ones(size(ds));

  cab=[NaN 0 12 14.88 16.98 17.75 16.98 14.88  12  4  4  6 0

  NaN 0  0 0.77  2.88  5.75  8.63  10.73 11.5 13 34 44 44];

  cab=cab*.1789;

  bmss(1,:)=s2(1,:)*bm(1,2); %stick-size vector

  bmss(2,:)=s2(2,:)*bm(2,2); %stick-size

  bmbs(1,:)=s1(1,:)*bm(1,2); %bucket-size vector

  bmbs(2,:)=s1(2,:)*bm(2,2); %bucket-size

  dsbs(1,:)=s1(1,:)*ds(1,2); %bucket-size

  dsbs(2,:)=s1(2,:)*ds(2,2); %bucket-size

  bx=0;

  by=26*.1789;

  %control

  %home

  x=4.2;y=0;dh=−120; %horizontal vx constant

  %x=12;y=2.1468;dh=0; %verical vy constant

  d=dh;

  u=90;

  pf=1;

  inv_gr1 % inverse kinematic provides home joint coordinates

  hold on %keeps picture

  ah=a;

  bh=b;

  aint=ah;

  bint=bh;

  dint=dh;

  %dynamic trajectory planning

  prof=[(0:9)/10 ones(1,100) (9:−1:0)/10];

  abd=[ ];

  L=7.62; %25 ft

  vav=mean(prof);

  dta=L/vav/120;

  for i=1:120

  vx=0.8*prof(i); %m/s

  vy=0.8*prof(i);

  dt=dta/.8;

  a=aint;

  b=bint;

  dd=.82;

  d=dint; %bucket absolute angle

  pf=1; %pf=rem(i,10); %plots at every time pf==0

  % pf=rem(i,10);if i==1;pf=0;end;

  fwd_gr1

  %Machine Jacobian JM

  JM=flipud([eds−bbm eds−ebm]);JM(1,:)=−JM(1,:);

  vbx=−(ebt(2,1)−eds(2,1))*dd*pi/180/dt;

  vby=(ebt(1,1)−eds(1,1))*dd*pi/180/dt;

  dang=inv(JM)*[vx−vbx;vy−vby]*180/pi*dt;

  aint=aint+dang(1,1);

  bint=bint+dang(2,1);

  dint=dint+dd;

  abd=[abd [dang;dd]];

  plot(ebt(1,1),ebt(2,1),‘k’) %plots bucket edge

  axis(frame)

  figure(gcf)

  pause(.01)

  end

  plot(cab(1,:),cab(2,:),‘k:’)

  xlabel(‘Horizontal Distance [m]’)

  ylabel(‘Elevation [m]’)

  %FORWARD BOUNDARY

  by=26*.1789;

  %control

  %home

  x=4.2+1.7;y=0;dh=−120; %horizontal vx constant

  %x=12;y=2.1468;dh=0; %verical vy constant

  d=dh;

  u=90;

  pf=1;

  inv_gr1 % inverse kinematic provides home joint coordinates

  hold on %keeps picture

  ah=a;

  bh=b;

  aint=ah;

  bint=bh;

  dint=dh;

  %dynamic trajectory planning

  prof=[(0:9)/10 ones(1,100) (9:−1:0)/10];

  abd=[ ];

  L=7.62;

  vav=mean(prof);

  dta=L/vav/120;

  for i=1:120

  vx=0.8*prof(i); %m/s

  vy=0.8*prof(i);

  dt=dta/.8;

  a=aint;

  b=bint;

  dd=.82;

  d=dint; %bucket absolute angle

  pf=1; %pf=rem(i,10);if i==l;pf=0;end; %plots at every time pf==0

  fwd_gr1

  %Machine Jacobian JM

  JM=flipud([eds−bbm eds−ebm]);JM(1,:)=−JM(1,:);

  vbx=−(ebt(2,1)−eds(2,1))*dd*pi/180/dt;

  vby=(ebt(1,1)−eds(1,1))*dd*pi/180/dt;

  dang=inv(JM)*[vx−vbx;vy−vby]*180/pi*dt;

  aint=aint+dang(1,1);

  bint=bint+dang(2,1);

  dint=dint+dd;

  abd=[abd [dang;dd]];

  plot(ebt(1,1),ebt(2,1),‘k’) %plots bucket edge

  axis(frame)

  figure(gcf)

  pause(.01)

  end

  %plot(cab(1,:),cab(2,:),‘k:’)

  xlabel(‘Horizontal Distance [m]’)

  ylabel(‘Elevation [m]’)

  %CUT UPPER CORNER

  %DEFINE CUTTING TRAJECTORY

  %constants

  f=1.7; %forward advance

  sg=1; %slope grade 45 degrees

  dh=−90; %home bucket position

  dhr=dh*pi/180;

  %x=4.2;y=0; %hone coordinates

  x=8.75;y=4.6;

  d=dh;

  u=90;

  pf=0;

  inv_gr1 % inverse kinematic provides home joint coordinates

  ah=a;

  bh=b;

  pf=0; %pf=rem(i,10);if i==1;pf=0;end; %plots at every time pf==0

  fwd=gr1

  axis(frame)

  xh=ebt(1,1); %home coordinate of x

  yh=ebt(2,1); %home coordinate of x

  k=−bct*(sin(dh*pi/180)+cos(atan(sg)));

  h=f+k/sg;

  c3=sg/h{circumflex over ( )}2−2*k/h{circumflex over ( )}3;

  c2=3*k/h{circumflex over ( )}2−sg/h;

  hold on

  %variables

  aint=ah;

  bint=bh;

  prof=[(0:9)/10 ones(1,100) (9:−1:0)/10];

  abd=[ ];

  L=25; %degrees here

  vav=mean(prof);

  dta=L/vav/120;

  d1o=dh;

  for i=1:30

  dd=.06*prof(i); %degree/sec

  dt=dta;

  X=ebt(1,1)−xh;

  Y=ebt(2,1)−yh;

  if (X{circumflex over ( )}2+Y{circumflex over ( )}2)==0

  xpy=1;

  else

  xpy=X/(X{circumflex over ( )}2+Y{circumflex over ( )}2){circumflex over ( )}.5;

  end

  if X<=h

  vx1=(dd/dt)*h/(atan(sg)−dh−90)*xpy;

  vy1=(3*c3*X{circumflex over ( )}2+2*c2*X)*vx1;

  l1=h−bct*(sin(atan(sg))−cos(dhr)));

  xdiff=X*(1−l1/h)+bct*cos(dhr);

  ydiff=bct*sin(dhr)+Y;

  d1=180/pi*atan2(ydiff,xdiff);

  else

  vx1=(dd/dt)*h/(atan(sg)−dh−90)*xpy;

  vy1=vx1*sg;

  xdiff=X*(1−l1/h)+bct*cos(dhr);

  ydiff=bct*sin(dhr)+Y;

  d1=180/pi*atan2(ydiff,xdiff)*(X/h){circumflex over ( )}2; %reduced angular bucket adjustment

  %d1=−(90−180/pi*atan(sg));

  end

  dd1=d1−d1o; %angular incremental change

  d1o=d1;

  JM=flipud([eds−bbm eds−ebm]);JM(1,:)=−JM(1,:);

  abd=[abd [vx1;vy1;d1]];

  vbx=−(ebt(2,1)−eds(2,1))*dd1*pi/180/dt;

  vby=(ebt(1,1)−eds(1,1))*dd1*pi/180/dt;

  dang=inv(JM)*[vx1−vbx;vy1−vby]*180/pi*dt;

  aint=aint+dang(1,1);

  bint=bint+dang(2,1);

  %dint=dint+dd;

  a=aint;

  b=bint;

  d=d1;

  pf=0; %pf=rem(i,10);if i==1;pf=0;end; %plots at every time pf==0

  fwd_gr1

  figure(gcf)

  pause(.1)

  end

  plot(cab(1,:),cab(2,:),‘k:’)

  hold off\\\\\

• APPENDIX 3
  Differential control architecture example to generate a sequence of computer images representing an excavator.
  Three actuators are coordinated by one control signal, of a simulated joystick.
  The control is based on a pre-defined trajectory for cutting a sharp corner.

  %Figure 9.

  %pap8.m

  %dynamic trajectory planning

  %machine makes linear rakeing: cannot advance slices since it reaches envelope

  clear all

  close

  frame=[0 16 −1 15]; %figure frame size

  b1=39*.1789;

  s1=25*.1789;

  bm=[0 b1;0 0];

  ds=[0 s1;0 0];

  bt=[0	1	6.5	11	16	19	14.5	10	1  0;
  0	−9	−12	−10	−7.5	0	−3	−1.5	4.5  0];

  bt=bt*.1789;
  bct=bt(1,6)−bt(1,1);	%bucket opening size
  s1=ones(size(bt));
  s2=ones(size(ds));

  cab=[NaN 0	12	14.88	16.98	17.75	16.98	14.88	12	4	4	6	0
  NaN 0	0	0.77	2.88	5.75	8.63	10.73	11.5	13	34	44	44];

  cab=cab*.1789;

  bmss(1,:)=s2(1,:)*bm(1,2);  %stick-size vector

  bmss(2,:)=s2(2,:)*bm(2,2);  %stick-size

  bmbs(1,:)=s1(1,:)*bm(1,2);  %bucket-size vector

  bmbs(2,:)=s1(2,:)*bm(2,2);  %bucket-size

  dsbs(1,:)=s1(1,:)*ds(1,2);   %bucket-size

  dsbs(2,:)=s1(2,:)*ds(2,2);   %bucket-size

  bx=0;

  by=26*.1789;

  %control

  %home

  x=4.2;y=0;dh=−120; %horizontal vx constant

  %x=12;y=2.1468;dh=0; %verical vy constant

  d=dh;

  u=90;

  pf=1;

  inv_gr1  % inverse kinematic provides home joint coordinates

  hold on %keeps picture

  ah=a;

  bh=b;

  aint=ah;

  bint=bh;

  dint=dh;

  %dynamic trajectory planning

  prof=[(0:9)/10 ones(1,100) (9:−1:0)/10];

  abd=[ ];

  L=7.62;  %25 ft

  vav=mean(prof);

  dta=L/vav/120;

  for i=1:120

  vx=0.8*prof(i); %m/s

  vy=0.8*prof(i);

  dt=dta/.8;

  a=aint;

  b=bint;

  dd=.82;

  d=dint;  %bucket absolute angle

  pf=1;  %pf=rem(i,10); %plots at every time pf==0

  % pf=rem(i,10);if i==1;pf=0;end;

  fwd_gr1

  %Machine Jacobian JM

  JM=flipud([eds−bbm eds−ebm]);JM(1,:)=−JM(1,:);

  vbx=−(ebt(2,1)−eds(2,1))*dd*pi/180/dt;

  vby=(ebt(1,1)−eds(1,1))*dd*pi/180/dt;

  dang=inv(JM)*[vx−vbx;vy−vby]*180/pi*dt;

  aint=aint+dang(1,1);

  bint=bint+dang(2,1);

  dint=dint+dd;

  abd=[abd [dang;dd]];

  plot(ebt(1,1),ebt(2,1),‘k’) %plots bucket edge

  axis(frame)

  figure(gcf)

  pause(.01)

  end

  plot(cab(1,:),cab(2,:),‘k:’)

  xlabel(‘Horizontal Distance [m]’)

  ylabel(‘Elevation [m]’)

  %FORWARD BOUNDARY

  by=26*.1789;

  %control

  %home

  x=4.2+1.7;y=0;dh=−120; %horizontal vx constant

  %x=12;y=2.1468;dh=0; %verical vy constant

  d=dh;

  u=90;

  pf=1;

  inv_gr1  % inverse kinematic provides home joint coordinates

  hold on %keeps picture

  ah=a;

  bh=b;

  aint=ah;

  bint=bh;

  dint=dh;

  %dynamic trajectory planning

  prof=[(0:9)/10 ones(1,100) (9:−1:0)/10];

  abd=[ ];

  L=7.62;

  vav=mean(prof);

  dta=L/vav/120;

  for i=1:120

  vx=0.8*prof(i); %m/s

  vy=0.8*prof(i);

  dt=dta/.8;

  a=aint;

  b=bint;

  dd=.82;

  d=dint;  %bucket absolute angle

  pf=1;  %pf=rem(i,10);if i==1;pf=0;end; %plots at every time pf==0

  fwd_gr1

  %Machine Jacobian JM

  JM=flipud([eds−bbm eds−ebm]);JM(1,:)=−JM(1,:);

  vbx=−(ebt(2,1)−eds(2,1))*dd*pi/180/dt;

  vby=(ebt(1,1)−eds(1,1))*dd*pi/180/dt;

  dang=inv(JM)*[vx−vbx;vy−vby]*180/pi*dt;

  aint=aint+dang(1,1);

  bint=bint+dang(2,1);

  dint=dint+dd;

  abd=[abd [dang;dd]];

  plot(ebt(1,1),ebt(2,1),‘k’) %plots bucket edge

  axis(frame)

  figure(gcf)

  pause(.01)

  end

  %plot(cab(1,:),cab(2,:),‘k:’)

  xlabel(‘Horizontal Distance [m]’)

  ylabel(‘Elevation [m]’)

  %CUT UPPER CORNER

  %DEFINE CUTTING TRAJECTORY

  %constants

  f=1.7;  %forward advance

  sg=1;  %slope grade 45 degrees

  dh=−90;  %home bucket position

  dhr=dh*pi/180;

  %x=4.2;y=0; %hone coordinates

  x=8.75;y=4.6;

  %comment 1: use predefined parametric trajectory. Use approximate analytical solution for %

  derivative df/ds in Equation (6) for the parametric trajectory in Figure 7.

  d=dh;

  u=90;

  pf=0;

  inv_gr1  % inverse kinematic provides home joint coordinates

  ah=a;

  bh=b;

  pf=0; %pf=rem(i,10);if i==1;pf=0;end; %plots at every time pf==0

  fwd_gr1

  axis(frame)

  xh=ebt(1,1);  %home coordinate of x

  yh=ebt(2,1);  %home coordinate of x

  k=−bct*(sin(dh*pi/180)+cos(atan(sg)));

  h=f+k/sg;

  c3=sg/h{circumflex over ( )}2−2*k/h{circumflex over ( )}3;

  c2=3*k/h{circumflex over ( )}2−sg/h;

  hold on

  %variables

  aint=ah;

  bint=bh;

  prof=[(0:9)/10 ones(1,100) (9:−1:0)/10];

  abd=[ ];

  L=25;  %degrees here

  vav=mean(prof);

  dta=L/vav/120;

  d1o=dh;

  for i=1:30

  dd=.06*prof(i); %degree/sec

  dt=dta;

  X=ebt(1,1)−xh;

  Y=ebt(2,1)−yh;

  if (X{circumflex over ( )}2+Y{circumflex over ( )}2)==0

  xpy=1;

  else

  xpy=X/(X{circumflex over ( )}2+Y{circumflex over ( )}2){circumflex over ( )}.5;

  end

  if X<=h

  vx1=(dd/dt)*h/(atan(sg)−dh−90)*xpy;

  vy1=(3*c3*X{circumflex over ( )}2+2*c2*X)*vx1;

  	l1=(h−bct*(sin(atan(sg))−cos(dhr)));
  	xdiff=X*(1−l1/h)+bct*cos(dhr);
  	ydiff=bct*sin(dhr)+Y;

  d1=180/pi*atan2(ydiff,xdiff);

  else

  vx1=(dd/dt)*h/(atan(sg)−dh−90)*xpy;

  vy1=vx1*sg;

  xdiff=X*(1−l1/h)+bct*cos(dhr);

  ydiff=bct*sin(dhr)+Y;

  d1=180/pi*atan2(ydiff,xdiff)*(X/h){circumflex over ( )}2;  %reduced angular bucket adjustment

  %d1=−(90−180/pi*atan(sg));

  end

  dd1=d1−d1o; %angular incremental change

  d1o=d1;

  JM=flipud([eds−bbm eds−ebm]);JM(1,:)=−JM(1,:);

  abd=[abd [vx1;vy1;d1]];

  vbx=−(ebt(2,1)−eds(2,1))*dd1*pi/180/dt;

  vby=(ebt(1,1)−eds(1,1))*dd1*pi/180/dt;

  dang=inv(JM)*[vx1−vbx;vy1−vby]*180/pi*dt;

  aint=aint+dang(1,1);

  bint=bint+dang(2,1);

  %dint=dint+dd;

  a=aint;

  b=bint;

  d=d1;

  pf=1; %pf=rem(i,10);if i==1;pf=0;end; %plots at every time pf==0

  fwd_gr1

  plot(ebt(1,1),ebt(2,1),‘k’) %plots bucket edge

  figure(gcf)

  pause(.1)

  end

  %CUT middle

  %DEFINE CUTTING TRAJECTORY

  %constants

  f=1.7;  %forward advance

  sg=1;  %slope grade 45 degrees

  dh=−90; %home bucket position

  dhr=dh*pi/180;

  %x=4.2;y=0; %hone coordinates

  x=8.75−2.35;y=4.6−2.35; %4/1.7=2.35 -- a bucketful of material

  d=dh;

  u=90;

  pf=0;

  inv_gr1  % inverse kinematic provides home joint coordinates

  %see comment 1

  ah=a;

  bh=b;

  pf=0; %pf=rem(i,10);if i==1;pf=0;end; %plots at every time pf==0

  fwd_gr1

  axis(frame)

  xh=ebt(1,1);  %home coordinate of x

  yh=ebt(2,1);  %home coordinate of x

  k=−bct*(sin(dh*pi/180)+cos(atan(sg)));

  h=f+k/sg;

  c3=sg/h{circumflex over ( )}2−2*k/h{circumflex over ( )}3;

  c2=3*k/h{circumflex over ( )}2−sg/h;

  hold on

  %variables

  aint=ah;

  bint=bh;

  prof=[(0:9)/10 ones(1,100) (9:−1:0)/10];

  abd=[ ];

  L=25;  %degrees here

  vav=mean(prof);

  dta=L/vav/120;

  d1o=dh;

  for i=1:30

  dd=.06*prof(i); %degree/sec

  dt=dta;

  X=ebt(1,1)−xh;

  Y=ebt(2,1)−yh;

  if (X{circumflex over ( )}2+Y{circumflex over ( )}2)==0

  xpy=1;

  else

  xpy=X/(X{circumflex over ( )}2+Y{circumflex over ( )}2){circumflex over ( )}.5;

  end

  if X<=h

  vx1=(dd/dt)*h/(atan(sg)−dh−90)*xpy;

  vy1=(3*c3*X{circumflex over ( )}2+2*c2*X)*vx1;

  	l1=(h−bct*(sin(atan(sg))−cos(dhr)));
  	xdiff=X*(1−l1/h)+bct*cos(dhr);
  	ydiff=bct*sin(dhr)+Y;

  d1=180/pi*atan2(ydiff,xdiff);

  else

  vx1=(dd/dt)*h/(atan(sg)−dh−90)*xpy;

  vy1=vx1*sg;

  xdiff=X*(1−l1/h)+bct*cos(dhr);

  ydiff=bct*sin(dhr)+Y;

  d1=180/pi*atan2(ydiff,xdiff)*(X/h){circumflex over ( )}2;  %reduced angular bucket adjustment

  %d1=−(90−180/pi*atan(sg));

  end

  dd1=d1−d1o; %angular incremental change

  d1o=d1;

  JM=flipud([eds−bbm eds−ebm]);JM(1,:)=−JM(1,:);

  abd=[abd [vx1;vy1;d1]];

  vbx=−(ebt(2,1)−eds(2,1))*dd1*pi/180/dt;

  vby=(ebt(1,1)−eds(1,1))*dd1*pi/180/dt;

  dang=inv(JM)*[vx1−vbx;vy1−vby]*180/pi*dt;

  aint=aint+dang(1,1);

  bint=bint+dang(2,1);

  %dint=dint+dd;

  a=aint;

  b=bint;

  d=d1;

  pf=1; %pf=rem(i,10);if i==1;pf=0;end; %plots at every time pf==0

  fwd_gr1

  plot(ebt(1,1),ebt(2,1),‘k’) %plots bucket edge

  figure(gcf)

  pause(.1)

  end

  %CUT bottom

  %DEFINE CUTTING TRAJECTORY

  %constants

  f=1.7;  %forward advance

  sg=1;  %slope grade 45 degrees

  dh=−120; %home bucket position

  dhr=dh*pi/180;

  %x=4.2;y=0;  %home coordinates

  x=4.2;y=0;  %4/1.7=2.35 -- a bucketful of material

  d=dh;

  u=90;

  pf=0;

  inv_gr1  % inverse kinematic provides home joint coordinates

  %see comment 1

  ah=a;

  bh=b;

  pf=0; %pf=rem(i,10);if i==1;pf=0;end; %plots at every time pf==0

  fwd_gr1

  axis(frame)

  xh=ebt(1,1);  %home coordinate of x

  yh=ebt(2,1);  %home coordinate of x

  k=−bct*(sin(dh*pi/180)+cos(atan(sg)));

  h=f+k/sg;

  c3=sg/h{circumflex over ( )}2−2*k/h{circumflex over ( )}3;

  c2=3*k/h{circumflex over ( )}2−sg/h;

  hold on

  %variables

  aint=ah;

  bint=bh;

  prof=[(0:9)/10 ones(1,100) (9:−1:0)/10];

  abd=[ ];

  L=25;  %degrees here

  vav=mean(prof);

  dta=L/vav/120;

  d1o=dh;

  for i=1:100

  dd=prof(i); %degree/sec

  dt=dta;

  X=ebt(1,1)−xh;

  Y=ebt(2,1)−yh;

  if (X{circumflex over ( )}2+Y{circumflex over ( )}2)==0

  xpy=1;

  else

  xpy=X/(X{circumflex over ( )}2+Y{circumflex over ( )}2){circumflex over ( )}.5;

  end

  if X<=f

  vx1=(dd/dt)*f/(atan(sg)−dh−90)*xpy;

  vy1=0;

  d1=dh;

  else

  vx1=(dd/dt)*f/(atan(sg)−dh−90)*xpy;

  vy1=vx1*sg;

  xdiff=X*(1−l1/h)+bct*cos(dhr);

  ydiff=bct*sin(dhr)+Y;

  d1=d1+dd*180/pi*dt/20;  %integrated angular bucket adjustment

  %d1=−(90−180/pi*atan(sg));

  end

  dd1=d1−d1o; %angular incremental change

  d1o=d1;

  JM=flipud([eds−bbm eds−ebm]);JM(1,:)=−JM(1,:);

  abd=[abd [vx1;vy1;d1]];

  vbx=−(ebt(2,1)−eds(2,1))*dd1*pi/180/dt;

  vby=(ebt(1,1)−eds(1,1))*dd1*pi/180/dt;

  dang=inv(JM)*[vx1−vbx;vy1−vby]*180/pi*dt;

  aint=aint+dang(1,1);

  bint=bint+dang(2,1);

  %dint=dint+dd;

  a=aint;

  b=bint;

  d=d1;

  pf=rem(i,5);if i==1;pf=0;end; %plots at every time pf==0

  fwd_gr1

  figure(gcf)

  pause(.1)

  end

  plot(cab(1,:),cab(2,:),‘k:’)

  hold off

• APPENDIX 4
  Subroutine functions used in the Matlab programs in Appendices 1 to 3

  	%inv_gr1.m
  	de=d*pi/180;
  	%wrist coordinates
  	xe=x−bct*cos(de);
  	ye=y−bct*sin(de);
  	%solve inverse
  	u2=0;v2=0; %u - alpha
  	err=1;
  	%for j=1:1000
  	while err>1e−12
  	v=(atan2(((b1*sin(u)−(ye−by)),(b1*cos(u)−(xe−bx))));
  	u=(atan2(((ye−by)+sl*sin(v)),((xe−bx)+sl*cos(v))));
  	err=abs(u−u2)+abs(v−v2);
  	u2=u;v2=v;
  	end
  	a=u*180/pi;
  	b=(v−u)*180/pi;
  	%boom rotation
  	%a=45;
  	al=a*pi/180;
  	rbm=[cos(al) −sin(al); sin(al) cos(al)];
  	%dipperstick rotation
  	%b=45;
  	be=(−180+b)*pi/180;
  	rds=[cos(be) −sin(be); sin(be) cos(be)];
  	%bucket rotation
  	g=180+d−a−b;
  	%g=20;
  	ga=g*pi/180;
  	rbt=[cos(ga) −sin(ga); sin(ga) cos(ga)];
  	%shape
  	shbm=rbm*bm;
  	shds=rbm*rds*ds+rbm*bmss;
  	shbt=rbm*rds*rbt*bt+rbm*bmbs+rbm*rds*dsbs;
  	%add boom base shift
  	shbm(1,:)=shbm(1,:)+bx;
  	shbm(2,:)=shbm(2,:)+by;
  	shds(1,:)=shds(1,:)+bx;
  	shds(2,:)=shds(2,:)+by;
  	shbt(1,:)=shbt(1,:)+bx;
  	shbt(2,:)=shbt(2,:)+by;
  	[m1 n1]=size(shbm);
  	[m2 n2]=size(shds);
  	[m3 n3]=size(shbt);
  	%beginning of boom
  	bbm=shbm(:,1);
  	%end of boom
  	ebm=shbm(:,n1);
  	%end of stick
  	eds=shds(:,n2);
  	%end of bucket
  	ebt=shbt(:,6);
  	%add boom base shift
  	if pf==0
  	plot(bbm(1,1),bbm(2,1),‘ko’,shbm(1,:),shbm(2,:),‘k’,ebm(1,1),
  	ebm(2,1),‘ko’,...shds(1,:),shds(2,:),‘k’,eds(1,1),eds(2,1),‘ko’,...
  	shbt(1,:),shbt(2,:),‘k’)
  	axis(frame);
  	figure(gcf)
  	end
  	%fwd_gr1.m
  	de=d*pi/180;
  	%boom rotation
  	%a=45;
  	al=a*pi/180;
  	rbm=[cos(al) −sin(al); sin(al) cos(al)];
  	%dipperstick rotation
  	%b=45;
  	be=(−180+b)*pi/180;
  	rds=[cos(be) −sin(be); sin(be) cos(be)];
  	%bucket rotation
  	g=180+d−a−b;
  	%g=20;
  	ga=g*pi/180;
  	rbt=[cos(ga) −sin(ga); sin(ga) cos(ga)];
  	%shape
  	shbm=rbm*bm;
  	shds=rbm*rds*ds+rbm*bmss;
  	shbt=rbm*rds*rbt*bt+rbm*bmbs+rbm*rds*dsbs;
  	%add boom base shift
  	shbm(1,:)=shbm(1,:)+bx;
  	shbm(2,:)=shbm(2,:)+by;
  	shds(1,:)=shds(1,:)+bx;
  	shds(2,:)=shds(2,:)+by;
  	shbt(1,:)=shbt(1,:)+bx;
  	shbt(2,:)=shbt(2,:)+by;
  	[m1 n1]=size(shbm);
  	[m2 n2]=size(shds);
  	[m3 n3]=size(shbt);
  	%beginning of boom
  	bbm=shbm(:,1);
  	%end of boom
  	ebm=shbm(:,n1);
  	%end of stick
  	eds=shds(:,n2);
  	%end of bucket
  	ebt=shbt(:,6);
  	%add boom base shift
  	if pf==0
  	plot(bbm(1,1),bbm(2,1),‘ko’,shbm(1,:),shbm(2,:),‘k’,ebm(1,1),
  	ebm(2,1),‘ko’,...shds(1,:),shds(2,:),‘k’,eds(1,1),eds(2,1),‘ko’,...
  	shbt(1,:),shbt(2,:),‘k’)
  	axis(frame);
  	figure(gcf)
  	end

• APPENDIX 5
  Demo 1
  %pap1_new.m
  %dynamic trajectory planning
  %basic swing
  clear all
  close all
  axis equal
  hMatStick = InitMatStick;   %Create the Joystick handle
  joy0=GetMatStick(hMatStick); %Get the Joystick middle position
  frame=[0 16 −1 15]; %figure frame size
  figure(1)
  set(1,‘renderer’,‘opengl’)
  axis(frame)
  bl=39*.1789;
  sl=25*.1789;
  bm=[0 bl;0 0];
  ds=[0 sl;0 0];
  bt=[0 1 6.5 11 16 19  14.5 10 1 0;
  0 −9 −12  −10 −7.5 0  −3 −1.5 4.5 0];
  bt=bt*.1789;
  bct=bt(1,6)−bt(1,1);      %bucket opening size
  s1=ones(size(bt));
  s2=ones(size(ds));
  cab=[NaN 0 12 14.88 16.98 17.75 16.98 14.88 12 4 4 6 0
  NaN 0 0 0.77 2.88 5.75 8.63 10.73 11.5 13 34 44 44
  ];
  cab=cab*.1789;
  bmss(1,:)=s2(1,:)*bm(1,2); %stick-size vector
  bmss(2,:)=s2(2,:)*bm(2,2); %stick-size
  bmbs(1,:)=s1(1,:)*bm(1,2); %bucket-size vector
  bmbs(2,:)=s1(2,:)*bm(2,2); %bucket-size
  dsbs(1,:)=s1(1,:)*ds(1,2); %bucket-size
  dsbs(2,:)=s1(2,:)*ds(2,2); %bucket-size
  bx=0;
  by=26*.1789;
  %control
  %home
  %Prepare 45 degree-angle slope as forward boundary
  x=4.2+1.7;y=0;dh=−120; %horizontal vx constant
  %x=12;y=2.1468;dh=0; %verical vy constant
  d=dh;
  u=90;
  pf=0;
  inv_gr1 % inverse kinematic provides home joint coordinates
  ah=a;
  bh=b;
  aint=ah;
  bint=bh;
  dint=dh;
  EB1=[ ];
  prof=[(0:9)/10 ones(1,100) (9:−1:0)/10];
  L=7.62;
  vav=mean(prof);
  dta=L/vav/120;
  for i=1:120
  vx=0.8*prof(i); %m/s
  vy=0.8*prof(i);
  dt=dta/.8;
  a=aint;
  b=bint;
  dd=.82;
  d=dint; %bucket absolute angle
  pf=1; %pf=rem(i,10);if i==1;pf=0;end; %plots at every time pf==0
  fwd_gr1
  %Machine Jacobian JM
  JM=flipud([eds−bbm eds−ebm]);JM(1,:)=−JM(1,:);
  vbx=−(ebt(2,1)−eds(2,1))*dd*pi/180/dt;
  vby=(ebt(1,1)−eds(1,1))*dd*pi/180/dt;
  dang=inv(JM)*[vx−vbx;vy−vby]*180/pi*dt;
  aint=aint+dang(1,1);
  bint=bint+dang(2,1);
  dint=dint+dd;
  EB1=[EB1 ebt];
  end
  %Original trajectory plot preparation
  x=5.5;y=0;dh=−90; %horizontal vx constant
  %x=12;y=2.1468;dh=0; %verical vy constant
  d=dh;
  u=90;
  pf=0; %first frame
  inv_gr1 % inverse kinematic provides home joint coordinates
  ah=a;
  bh=b;
  aint=ah;
  bint=bh;
  dint=dh;
  prof=[(0:9)/10 ones(1,100) (9:−1:0)/10];
  L=90+18.5; %angle here
  vav=mean(prof);
  dta=L/vav/120;
  %trajectory plot
  EB=[ ];
  for i=1:120
  vx=0.8*prof(i); %m/s
  vy=0.8*prof(i);
  dt=dta/.8;
  %a=aint;
  b=bint;
  %dd=.4;
  d=dint; %bucket absolute angle
  pf=1;%pf=rem(i,10); %plots at every time pf==0
  fwd_gr1
  aint=aint; %+dang(1,1);
  bint=bint+vy*dt; %+dang(2,1);
  dint=dint+vy*dt; %+dd;
  EB=[EB ebt];
  end
  hold on
  %re-wind bucket to bottom position
  d=dh;
  u=90;
  pf=0; %first frame
  inv_gr1 % inverse kinematic provides home joint coordinates
  aint=a;
  bint=b;
  dint=d;
  type=0;
  holdf=0;
  joy=GetMatStick(hMatStick);  %Get the Joystick middle position
  %manual control in machine kinematics
  while ( joy(4)~=16)
  if (joy(4)==0)
  holdf=0;
  end
  if (joy(4)==1 & holdf==0)
  type=~type;
  holdf=1;
  end
  %vx=0.8*prof(i); %m/s
  %vy=0.8*prof(i);
  joy=GetMatStick(hMatStick);   %Get the Joystick middle position
  ii=[1; 2; 3; 5];
  djoy=joy−joy0; %Get the joystick movement
  djoy(ii)=djoy(ii).*(abs(djoy(ii))>2045/1); %makes a 2045-unit
  joystick deadzone
  vx=−0.0003/5*djoy(1);
  vy=−0.0003/5*djoy(2);
  vz=−0.0003/5*djoy(5);
  disp([vx vy joy′])
  dt=dta/.8;
  a=aint;
  b=bint;
  %dd=.4;
  dd=(vx{circumflex over ( )}2+vy{circumflex over ( )}2){circumflex over ( )}.5;
  d=dint; %bucket absolute angle
  pf=0; %rem(i,10); %plots at every time pf==0
  fwd_gr1
  if type==0
  %original kinematics
  aint=aint+vx*dt; %+dang(1,1);
  bint=bint+vy*dt; %+dang(2,1);
  dint=dint+vz*dt; %+dd;
  else
  a2=djoy(3)/65280*90;
  s=a2*pi/180;
  R=[cos(s) −sin(s);sin(s) cos(s)];
  v1=R*[vx;vy];
  vx=v1(1);
  vy=v1(2);
  slow=.1;
  %dd=1;
  %transformed kinematics
  %Machine Jacobian JM
  JM=flipud([eds−bbm eds−ebm]);JM(1,:)=−JM(1,:);
  vbx=−(ebt(2,1)−eds(2,1))*dd*pi/180/dt;
  vby=(ebt(1,1)−eds(1,1))*dd*pi/180/dt;
  dang=slow*inv(JM)*[vx−vbx;vy−vby]*180/pi*dt;
  aint=aint+dang(1,1);
  bint=bint+dang(2,1);
  %dd is replaced here with 1
  %dint=dint+dd*vz;
  dint=dint+1*vz;
  %abd=[abd [dang;dd]];
  end
  hold on
  plot(cab(1,:),cab(2,:),‘k’,EB(1,:),EB(2,:),‘:’,EB1(1,:),EB1(2,:),‘:’)
  %plot(cab(1,:),cab(2,:),‘k’,EB(1,:),EB(2,:))
  xlabel(‘Horizontal Distance [m]’)
  ylabel(‘Elevation [m]’)
  if type==0
  title(‘Original’)
  else
  title([‘Transformed 1 (angle=‘ num2str(a2) ’)’])
  end
  axis equal
  hold off
  pause(.01)
  end
  hold off
  Demo 2
  %pap1_new.m
  %dynamic trajectory planning
  %basic swing
  clear all
  close all
  axis equal
  hMatStick = InitMatStick; %Create the Joystick handle
  joy0=GetMatStick(hMatStick);  %Get the Joystick middle position
  frame=[0 16 −1 15]; %figure frame size
  figure(1)
  set(1,‘renderer’,‘opengl’)
  axis(frame)
  bl=39*.1789;
  sl=25*.1789;
  bm=[0 bl;0 0];
  ds=[0 sl;0 0];
  bt=[0 1 6.5 11 16 19  14.5 10 1 0;
  0 −9 −12 −10 −7.5 0  −3 −1.5 4.5 0];
  bt=bt*.1789;
  bct=bt(1,6)−bt(1,1);   %bucket opening size
  s1=ones(size(bt));
  s2=ones(size(ds));
  cab=[NaN 0 12 14.88 16.98 17.75 16.98 14.88 12  4 4 6 0
  NaN 0 0 0.77  2.88  5.75  8.63  10.73 11.5 13 34 44 44
  ];
  cab=cab*.1789;
  bmss(1,:)=s2(1,:)*bm(1,2); %stick-size vector
  bmss(2,:)=s2(2,:)*bm(2,2); %stick-size
  bmbs(1,:)=s1(1,:)*bm(1,2); %bucket-size vector
  bmbs(2,:)=s1(2,:)*bm(2,2); %bucket-size
  dsbs(1,:)=s1(1,:)*ds(1,2); %bucket-size
  dsbs(2,:)=s1(2,:)*ds(2,2); %bucket-size
  bx=0;
  by=26*.1789;
  %control
  %home
  %Prepare 45 degree-angle slope as forward boundary
  x=4.2+1.7;y=0;dh=−120; %horizontal vx constant
  %x=12;y=2.1468;dh=0; %verical vy constant
  d=dh;
  u=90;
  pf=0;
  inv_gr1 % inverse kinematic provides home joint coordinates
  ah=a;
  bh=b;
  aint=ah;
  bint=bh;
  dint=dh;
  EB1=[ ];
  prof=[(0:9)/10 ones(1,100) (9:−1:0)/10];
  L=7.62;
  vav=mean(prof);
  dta=L/vav/120;
  for i=1:120
  vx=0.8*prof(i); %m/s
  vy=0.8*prof(i);
  dt=dta/.8;
  a=aint;
  b=bint;
  dd=.82;
  d=dint; %bucket absolute angle
  pf=1; %pf=rem(i,10);if i==1;pf=0;end; %plots at every time pf==0
  fwd_gr1
  %Machine Jacobian JM
  JM=flipud([eds−bbm eds−ebm]);JM(1,:)=−JM(1,:);
  vbx=−(ebt(2,1)−eds(2,1))*dd*pi/180/dt;
  vby=(ebt(1,1)−eds(1,1))*dd*pi/180/dt;
  dang=inv(JM)*[vx−vbx;vy−vby]*180/pi*dt;
  aint=aint+dang(1,1);
  bint=bint+dang(2,1);
  dint=dint+dd;
  EB1=[EB1 ebt];
  end
  %Prepare 45 degree-angle slope as starting boundary
  x=4.2;y=0;dh=−120; %horizontal vx constant
  %x=12;y=2.1468;dh=0; %verical vy constant
  d=dh;
  u=90;
  pf=0;
  inv_gr1 % inverse kinematic provides home joint coordinates
  ah=a;
  bh=b;
  aint=ah;
  bint=bh;
  dint=dh;
  EB2=[ ];
  prof=[(0:9)/10 ones(1,100) (9:−1:0)/10];
  L=7.62;
  vav=mean(prof);
  dta=L/vav/120;
  for i=1:120
  vx=0.8*prof(i); %m/s
  vy=0.8*prof(i);
  dt=dta/.8;
  a=aint;
  b=bint;
  dd=.82;
  d=dint; %bucket absolute angle
  pf=1; %pf=rem(i,10);if i==1;pf=0;end; %plots at every time pf==0
  fwd_gr1
  %Machine Jacobian JM
  JM=flipud([eds−bbm eds−ebm]);JM(1,:)=−JM(1,:);
  vbx=−(ebt(2,1)−eds(2,1))*dd*pi/180/dt;
  vby=(ebt(1,1)−eds(1,1))*dd*pi/180/dt;
  dang=inv(JM)*[vx−vbx;vy−vby]*180/pi*dt;
  aint=aint+dang(1,1);
  bint=bint+dang(2,1);
  dint=dint+dd;
  EB2=[EB2 ebt];
  end
  %Original trajectory plot preparation
  x=5.5;y=0;dh=−90; %horizontal vx constant
  %x=12;y=2.1468;dh=0; %verical vy constant
  d=dh;
  u=90;
  pf=0; %first frame
  inv_gr1 % inverse kinematic provides home joint coordinates
  ah=a;
  bh=b;
  aint=ah;
  bint=bh;
  dint=dh;
  prof=[(0:9)/10 ones(1,100) (9:−1:0)/10];
  L=90+18.5;  %angle here
  vav=mean(prof);
  dta=L/vav/120;
  %trajectory plot
  EB=[ ];
  for i=1:120
  vx=0.8*prof(i); %m/s
  vy=0.8*prof(i);
  dt=dta/.8;
  %a=aint;
  b=bint;
  %dd=.4;
  d=dint; %bucket absolute angle
  pf=1;%pf=rem(i,10); %plots at every time pf==0
  fwd_gr1
  aint=aint; %+dang(1,1);
  bint=bint+vy*dt; %+dang(2,1);
  dint=dint+vy*dt; %+dd;
  EB=[EB ebt];
  end
  hold on
  %re-wind bucket to bottom position
  d=dh;
  u=90;
  pf=0; %first frame
  inv_gr1 % inverse kinematic provides home joint coordinates
  aint=a;
  bint=b;
  dint=d;
  home0=ebt;
  type=0;
  holdf=0;
  joy=GetMatStick(hMatStick);  %Get the Joystick middle position
  %manual control in machine kinematics
  while ( joy(4)~=16)
  if (joy(4)==0)
  holdf=0;
  end
  if (joy(4)==1 & holdf==0)
  type=~type;
  holdf=1;
  end
  %vx=0.8*prof(i); %m/s
  %vy=0.8*prof(i);
  joy=GetMatStick(hMatStick);  %Get the Joystick middle position
  ii=[1; 2; 3; 5];
  djoy=joy−joy0; %Get the joystick movement
  djoy(ii)=djoy(ii).*(abs(djoy(ii))>2045/1); %makes a 2045-unit
  joystick deadzone
  vx=−0.0003/5*djoy(1);
  vy=−0.0003/5*djoy(2);
  vz=−0.0003/5*djoy(5);
  disp([vx vy joy′])
  %  dt=dta/.8;
  %  a=aint;
  %  b=bint;
  %  %dd=.4;
  %  dd=(vx{circumflex over ( )}2+vy{circumflex over ( )}2){circumflex over ( )}.5;
  d=dint; %bucket absolute angle
  pf=0; %rem(i,10); %plots at every time pf==0
  fwd_gr1
  if type==0
  L=90+18.5; %angle here
  vav=mean(prof);
  dta=L/vav/120;
  dt=dta/.8;
  a=aint;
  b=bint;
  %dd=.4;
  dd=(vx{circumflex over ( )}2+vy{circumflex over ( )}2){circumflex over ( )}.5;
  d=dint; %bucket absolute angle
  pf=0; %rem(i,10); %plots at every time pf==0
  fwd_gr1
  %original kinematics
  aint=aint+vx*dt; %+dang(1,1);
  bint=bint+vy*dt; %+dang(2,1);
  dint=dint+vz*dt; %+dd;
  home=ebt;
  once=1;
  else
  if once
  %constants
  f=1.7; %forward advance
  sg=1; %slope grade 45 degrees
  dh=−90; %home bucket position
  dhr=dh*pi/180;
  x=home(1);y=home(2); %hone coordinates
  % x=8.75−2.35;y=4.6−2.35; %4/1.7=2.35 -- a bucketful of material
  %
  % d=dh;
  % u=90;
  % pf=0;
  inv_gr1 % inverse kinematic provides home joint coordinates
  ah=a;
  bh=b;
  pf=0; %pf=rem(i,10);if i==1;pf=0;end; %plots at every time pf==0
  fwd_gr1
  axis(frame)
  xh=ebt(1,1); %home coordinate of x
  yh=ebt(2,1); %home coordinate of x
  k=−bct*(sin(dh*pi/180)+cos(atan(sg)));
  h=f+k/sg;
  c3=sg/h{circumflex over ( )}2−2*k/h{circumflex over ( )}3;
  c2=3*k/h{circumflex over ( )}2−sg/h;
  hold on
  %variables
  aint=ah;
  bint=bh;
  prof=[(0:9)/10 ones(1,100) (9:−1:0)/10];
  abd=[ ];
  L=25; %degrees here
  vav=mean(prof);
  dta=L/vav/120;
  d1o=dh;
  once=0;
  end
  %CUT middle
  %DEFINE CUTTING TRAJECTORY
  %do once
  %  for i=1:30
  dd=0.06*vy;
  %   dd=.06*prof(i); %degree/sec
  dt=dta;
  X=ebt(1,1)−xh;
  Y=ebt(2,1)−yh;
  if (X{circumflex over ( )}2+Y{circumflex over ( )}2)==0
  xpy=1;
  else
  xpy=X/(X{circumflex over ( )}2+Y{circumflex over ( )}2){circumflex over ( )}.5;
  end
  if X<=h
  vx1=(dd/dt)*h/(atan(sg)−dh−90)*xpy;
  vy1=(3*c3*X{circumflex over ( )}2+2*c2*X)*vx1;
  l1=(h−bct*(sin(atan(sg))−cos(dhr)));
  xdiff=X*(1−l1/h)+bct*cos(dhr);
  ydiff=bct*sin(dhr)+Y;
  d1=180/pi*atan2(ydiff,xdiff);
  else
  vx1=(dd/dt)*h/(atan(sg)−dh−90)*xpy;
  vy1=vx1*sg;
  xdiff=X*(1−l1/h)+bct*cos(dhr);
  ydiff=bct*sin(dhr)+Y;
  d1=180/pi*atan2(ydiff,xdiff)*(X/h){circumflex over ( )}2; %reduced angular bucket
  adjustment
  %d1=−(90−180/pi*atan(sg));
  end
  dd1=d1−d1o; %angular incremental change
  d1o=d1;
  JM=flipud([eds−bbm eds−ebm]);JM(1,:)=−JM(1,:);
  vbx=−(ebt(2,1)−eds(2,1))*dd1*pi/180/dt;
  vby=(ebt(1,1)−eds(1,1))*dd1*pi/180/dt;
  dang=inv(JM)*[vx1−vbx;vy1−vby]*180/pi*dt*(abs(djoy(2))>2045);
  aint=aint+dang(1,1);
  bint=bint+dang(2,1);
  %dint=dint+dd;
  a=aint;
  b=bint;
  d=d1;
  %dint=d;
  pf=0; %pf=rem(i,10);if i==1;pf=0;end; %plots at every time pf==0
  a=mod(a,360);
  b=mod(b,360);
  d=mod(d,360);
  if a>=60, a=60; end
  if a<=−60, a=−60; end
  if b>=179, b=179; end
  if b<=1, b=1; end
  %  if d>=90, d=90; end
  %  if d<=−90, d=−90; end
  fwd_gr1
  axis equal
  end
  %  end
  hold on
  plot(cab(1,:),cab(2,:),‘k’,EB(1,:),EB(2,:),‘:’,EB1(1,:),EB1(2,:),‘:’,
  EB2(1,:),EB2(2,:),‘:’)
  %plot(cab(1,:),cab(2,:),‘k’,EB(1,:),EB(2,:))
  xlabel(‘Horizontal Distance [m]’)
  ylabel(‘Elevation [m]’)
  if type==0
  title(‘Original’)
  else
  %title([‘Transformed 1 (angle=‘ num2str(a2) ’)’])
  title([‘Transformed 1 (angle=‘ ’)’])
  end
  axis equal
  hold off
  pause(.01)
  end
  hold off

Claims (10)

1-32. (canceled)

33. A tangible computer-readable storage medium storing computer-executable instructions for causing a computer system programmed thereby to perform a method comprising:

outputting to a display device a menu of virtual kinematics configurations, each virtual kinematics configuration comprising a plurality of trajectory paths of a user-manipulable implement, each trajectory path comprising at least two dimensions;

calculating velocity components for at least one of the plurality of trajectory paths using a model-based forward predictor;

modifying the velocity components based on signals received from an input device to provide modified velocity components;

generating a position-dependent inverse kinematics model of the modified velocity components; and

based on the position-dependent inverse kinematics model, outputting the modified velocity components to at least one of a plurality of control connections associated with a plurality of joints of the user-manipulable implement.

34. The computer readable storage medium ofclaim 33, the method further comprising receiving signals from an input device regarding a virtual kinematics configuration selected by a user.

35. The computer readable storage medium ofclaim 33, the method further comprising receiving signals from an input device for modifying the velocity components.

36. A computer-implemented method of operating an apparatus comprising a user-manipulable implement comprising a plurality of joints, the method comprising:

outputting to a display device a menu of virtual kinematics configurations, each virtual kinematics configuration comprising a plurality of trajectory paths of a user-manipulable implement, each trajectory path comprising at least two dimensions;

calculating velocity components for at least one of the plurality of trajectory paths using a model-based forward predictor;

modifying the velocity components based on signals received from an input device to provide modified velocity components;

generating a position-dependent inverse kinematics model of the modified velocity components; and

based on the position-dependent inverse kinematics model, outputting the modified velocity components to at least one of a plurality of control connections associated with a plurality of joints of the user-manipulable implement.

37. A method comprising:

storing on a computer-readable storage medium computer-executable instructions for causing a computer system programmed thereby to perform acts comprising:

outputting to a display device a menu of virtual kinematics configurations, each virtual kinematics configuration comprising a plurality of trajectory paths of a user-manipulable implement, each trajectory path comprising at least two dimensions;

calculating velocity components for at least one of the plurality of trajectory paths using a model-based forward predictor;

modifying the velocity components based on signals received from an input device to provide modified velocity components;

generating a position-dependent inverse kinematics model of the modified velocity components; and

based on the position-dependent inverse kinematics model, outputting the modified velocity components to at least one of a plurality of control connections associated with a plurality of joints of the user-manipulable implement.

38. A system comprising:

a processor;

the tangible computer readable storage medium ofclaim 33.

39. The system ofclaim 38, further comprising a display.

40. The system ofclaim 38, further comprising an input device, the input device configured to generate signals allowing an operator to select a virtual kinematics configuration from the displayed menu.

41. The system ofclaim 38, further comprising an input device, the input device configured to generate signals modifying the velocity components.

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