US20210212777A1 - Inverse kinematic control systems for robotic surgical system - Google Patents

Inverse kinematic control systems for robotic surgical system Download PDF

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US20210212777A1
US20210212777A1 US16/081,773 US201716081773A US2021212777A1 US 20210212777 A1 US20210212777 A1 US 20210212777A1 US 201716081773 A US201716081773 A US 201716081773A US 2021212777 A1 US2021212777 A1 US 2021212777A1
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motion
tool
arm
pose
remote center
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Jiqi Cheng
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Covidien LP
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    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B34/00—Computer-aided surgery; Manipulators or robots specially adapted for use in surgery
    • A61B34/30—Surgical robots
    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B17/00—Surgical instruments, devices or methods, e.g. tourniquets
    • A61B17/28—Surgical forceps
    • A61B17/29—Forceps for use in minimally invasive surgery
    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B34/00—Computer-aided surgery; Manipulators or robots specially adapted for use in surgery
    • A61B34/30—Surgical robots
    • A61B34/37—Master-slave robots
    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B34/00—Computer-aided surgery; Manipulators or robots specially adapted for use in surgery
    • A61B34/70—Manipulators specially adapted for use in surgery
    • A61B34/74—Manipulators with manual electric input means
    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61B—DIAGNOSIS; SURGERY; IDENTIFICATION
    • A61B34/00—Computer-aided surgery; Manipulators or robots specially adapted for use in surgery
    • A61B34/70—Manipulators specially adapted for use in surgery
    • A61B34/76—Manipulators having means for providing feel, e.g. force or tactile feedback
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J13/00—Controls for manipulators
    • B25J13/02—Hand grip control means
    • B25J13/025—Hand grip control means comprising haptic means
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1679—Programme controls characterised by the tasks executed
    • B25J9/1689—Teleoperation
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1694—Programme controls characterised by use of sensors other than normal servo-feedback from position, speed or acceleration sensors, perception control, multi-sensor controlled systems, sensor fusion
    • B25J9/1697—Vision controlled systems
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1602—Programme controls characterised by the control system, structure, architecture
    • B25J9/1607—Calculation of inertia, jacobian matrixes and inverses
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B2219/00—Program-control systems
    • G05B2219/30—Nc systems
    • G05B2219/40—Robotics, robotics mapping to robotics vision
    • G05B2219/40495—Inverse kinematics model controls trajectory planning and servo system

Definitions

  • Robotic surgical systems such as teleoperative systems are used to perform minimally invasive surgical procedures that offer many benefits over traditional open surgery techniques, including less pain, shorter hospital stays, quicker return to normal activities, minimal scarring, reduced recovery time, and less injury to tissue.
  • Robotic surgical systems can have a number of robotic arms that move attached instruments or tools, such as an image capturing device, a stapler, an electrosurgical instrument, etc., in response to movement of input devices by a surgeon viewing images captured by the image capturing device of a surgical site.
  • instruments or tools such as an image capturing device, a stapler, an electrosurgical instrument, etc.
  • each of the tools is inserted through an opening, either natural or an incision, into the patient and positioned to manipulate tissue at a surgical site.
  • the openings are placed about the patient's body so that the surgical instruments may be used to cooperatively perform the surgical procedure and the image capturing device may view the surgical site.
  • the tools are manipulated in multiple degrees of freedom.
  • a singularity is created.
  • the operation of the tool may become unpredictable and the degrees of freedom of the tool may be decreased.
  • some systems employ hard stops.
  • Other systems allow the tool to reach a singularity and use a comparison of a speed of an input controller with a speed of movement of the tool to predict the movement of the tool.
  • This disclosure relates generally to correcting the pose of an arm and a tool of a robotic system to avoid singularities between joints and to maintain degrees of freedom of movement of the arm and the tool. Specifically, when a remote center of motion of the arm is within a boundary distance from an origin of a tool center-point frame, the remote center of motion is moved to the boundary distance while maintaining a position of a jaw axis of the tool and rotating the remote center of motion according to the rigid body kinematics. After the rotation, the tool center-point frame is expressed in the rotated remote center of motion frame, and treated as the corrected desired pose for inverse kinematics calculation.
  • a method of using inverse kinematics to control a robotic system includes receiving an input pose from a user interface to move an arm of the robotic system, calculating a remote center of motion for a desired pose from the input pose in a tool center-point frame, checking when the desire pose needs correction, correcting the desired pose of the arm, and moving the arm to the desired pose in response to the input pose.
  • the arm of the robotic system includes a tool having a jaw disposed at an end of the arm.
  • Checking when the desired pose needs correction includes verifying that the remote center of motion is at or beyond a boundary distance in the desired pose. Correcting the desired pose of the arm occurs when the remote center of motion is within the boundary distance.
  • Correcting the desired pose of the arm may include moving the remote center of motion to the boundary distance.
  • determining the boundary distance occurs as a function of a maximum joint angler of a pitch joint defined between the tool and the distance between the origin of the tool center-point and the pitch joint.
  • the maximum joint angle of the pitch joint may be about 75°.
  • Determining the boundary distance may include taking the sum of the distance between the origin of the tool center-point and the pitch joint and the cosine of the maximum joint angle of the pitch point. The boundary distance may be taken along the jaw axis.
  • the method includes providing feedback when the remote center of motion approaches the boundary distance in the desired pose.
  • Providing feedback when the remote center of motion in the desired pose approaches the boundary distance may include increasing the feedback as the remote center of motion approaches the boundary distance.
  • Increasing the feedback may include linearly or exponentially increasing the feedback.
  • increasing the feedback may include linearly increasing the feedback as the remote center of motion approaches the boundary distance and exponentially increasing the feedback as the remote center of motion crosses the boundary distance.
  • the method includes determining a check angle that is defined between the jaw axis and a vector between an origin of the tool center-point frame and the remote center of motion when the remote center of motion is within the boundary distance.
  • the method may include providing feedback when the check angle is below a predefined extreme angle.
  • a robotic surgical system in another aspect of the present disclosure, includes a processing unit, a user interface, and a robotic system.
  • the user interface is in communication with the processing unit and includes an input handle.
  • the robotic system is in communication with the processing unit and including an arm and a tool supported at an end of the arm.
  • the arm defines a remote center of motion and the tool defines a tool center-point frame.
  • the arm and the tool are configured to move to a desired pose in response to an input pose of the input handle.
  • the processing unit is configured to verify when the remote center of motion is within the boundary distance in the desired pose.
  • the processing unit is configured to correct the desired pose when the remote center of motion is within the boundary distance.
  • the processing unit is configured to verify that a check angle defined between the jaw axis and a vector defined between an origin of the tool center-point frame and the remote center of motion is below a predefined extreme angle when the remote center of motion is within the boundary distance in the desired pose.
  • the user interface may be configured to provide feedback to a clinician when the check angle is below the predefined extreme angle.
  • the input handle may be configured to provide force feedback to a clinician when the remote center of motion approaches the boundary distance in the desired pose.
  • FIG. 1 is a schematic illustration of a user interface and a robotic system of a robotic surgical system in accordance with the present disclosure
  • FIG. 2 is a side view of an exemplary arm and tool of the robotic system of FIG. 1 ;
  • FIG. 3 is a model of the arm and tool of FIG. 2 in a first case
  • FIG. 4 is a model of the arm and tool of FIG. 2 in a second case
  • FIG. 5 is a model of the arm and tool of FIG. 2 in a third case
  • FIG. 6 is the model of FIG. 3 illustrating a calculation of a joint angle ⁇ 6 ;
  • FIG. 7 is the model of FIG. 6 illustrating a calculation of a joint angle ⁇ 5 ;
  • FIG. 8 is the model of FIG. 7 illustrating a calculation of joint angles ⁇ 1 and ⁇ 2 ;
  • FIG. 9 is the model of FIG. 8 illustrating a calculation of a joint angle ⁇ 3 ;
  • FIG. 10 is a plan view of the ⁇ circumflex over (x) ⁇ 6 , ⁇ 6 plane of the model of FIG. 3 ;
  • FIG. 11 is a model illustrating determination of u 0 ;
  • FIG. 12 is the plan view of FIG. 10 including a boundary line B;
  • FIG. 13 is a model illustrating a projection method of calculating a rotation of a trocar point O to a rotated trocar point M;
  • FIG. 14 is a calculation of a rotation angle ⁇ from the trocar point O to the rotated trocar point M of FIG. 13 ;
  • FIG. 15 is a calculation of an orientation matrix for the rotation of the Base Frame ⁇ circumflex over (x) ⁇ 0 , ⁇ 0 , ⁇ circumflex over (z) ⁇ 0 ;
  • FIG. 16 is a model illustrating detection of an extreme condition
  • FIG. 17 is a model of a long jawed instrument with the arm of the robotic system of FIG. 1 in accordance with the present disclosure
  • FIG. 18 is a calculation of a rotation angle ⁇ in a Frame 6 of the trocar point O to the rotated trocar point M of FIG. 17 ;
  • FIG. 19 is a model illustrating determination of u 0 for the long jawed instrument of FIG. 17 ;
  • FIG. 20 is a projection of an End Effector Frame of FIG. 17 to the rotated trocar point M
  • FIG. 21 is a calculation of an orientation matrix for the rotation of the Frame 6;
  • FIG. 22 is a model illustrating detection of an extreme condition for the long jawed instrument of FIG. 17 .
  • the term “clinician” refers to a doctor, a nurse, or any other care provider and may include support personnel.
  • proximal refers to the portion of the device or component thereof that is closest to the clinician and the term “distal” refers to the portion of the device or component thereof that is farthest from the clinician.
  • This disclosure relates generally to correcting the pose of an arm and a tool of a robotic system to avoid singularities between joints and to maintain degrees of freedom of movement of the arm and the tool. Specifically, when a remote center of motion of the arm is within a boundary distance from an origin of a tool center-point frame, the remote center of motion is moved to the boundary distance while maintaining a position of a jaw axis of the tool and rotating the remote center of motion according to the rigid body kinematics. After the rotation, the tool center-point frame is expressed in the rotated remote center of motion frame, and treated as the corrected desired pose for inverse kinematics calculation.
  • a robotic surgical system 1 in accordance with the present disclosure is shown generally as a robotic system 10 , a processing unit 30 , and a user interface 40 .
  • the robotic system 10 generally includes linkages or arms 12 and a robot base 18 .
  • the arms 12 moveably support an end effector or tool 20 which is configured to act on tissue.
  • the arms 12 each have an end 14 that supports an end effector or tool 20 which is configured to act on tissue.
  • the ends 14 of the arms 12 may include an imaging device 16 for imaging a surgical site.
  • the user interface 40 is in communication with robot base 18 through the processing unit 30 .
  • the user interface 40 includes a display device 44 which is configured to display three-dimensional images.
  • the display device 44 displays three-dimensional images of the surgical site which may include data captured by imaging devices 16 positioned on the ends 14 of the arms 12 and/or include data captured by imaging devices that are positioned about the surgical theater (e.g., an imaging device positioned within the surgical site, an imaging device positioned adjacent the patient, imaging device 56 positioned at a distal end of an imaging linkage or arm 52 ).
  • the imaging devices e.g., imaging devices 16 , 56
  • the imaging devices may capture visual images, infra-red images, ultrasound images, X-ray images, thermal images, and/or any other known real-time images of the surgical site.
  • the imaging devices transmit captured imaging data to the processing unit 30 which creates three-dimensional images of the surgical site in real-time from the imaging data and transmits the three-dimensional images to the display device 44 for display.
  • the user interface 40 also includes input handles 42 which are supported on control arms 43 which allow a clinician to manipulate the robotic system 10 (e.g., move the arms 12 , the ends 14 of the arms 12 , and/or the tools 20 ).
  • Each of the input handles 42 is in communication with the processing unit 30 to transmit control signals thereto and to receive feedback signals therefrom. Additionally or alternatively, each of the input handles 42 may include input devices (not shown) which allow the surgeon to manipulate (e.g., clamp, grasp, fire, open, close, rotate, thrust, slice, etc.) the tools 20 supported at the ends 14 of the arms 12 .
  • Each of the input handles 42 is moveable through a predefined workspace to move the ends 14 of the arms 12 within a surgical site.
  • the three-dimensional images on the display device 44 are orientated such that movement of the input handle 42 moves the ends 14 of the arms 12 as viewed on the display device 44 .
  • the orientation of the three-dimensional images on the display device may be mirrored or rotated relative to view from above the patient.
  • the size of the three-dimensional images on the display device 44 may be scaled to be larger or smaller than the actual structures of the surgical site permitting a clinician to have a better view of structures within the surgical site.
  • the tools 20 are moved within the surgical site as detailed below.
  • movement of the tools 20 may also include movement of the ends 14 of the arms 12 which support the tools 20 .
  • the end 14 of an arm 12 of the robotic system 10 is moveable about a Remote Center of Motion (RCM) 22 in four Degrees of Freedom (DOF) or joints.
  • the tool 20 is pivotal in two DOF about a first tool joint 24 and a second tool joint 26 , respectively.
  • the arm 12 of the robotic system 10 defines a frame x 0 , y 0 , z 0 (the Base Frame) positioned at the RCM.
  • the first DOF or yaw joint of the arm 12 is represented by the frame x 1 , y 1 , z 1 (Frame 1) with the position of the arm 12 about the yaw joint represented as joint angle ⁇ 1 .
  • the second DOF or pitch joint of the arm 12 is represented by the frame x 2 , y 2 , z 2 (Frame 2) with the position of the arm 12 about the pitch joint represented as joint angle ⁇ 2 .
  • the third DOF or roll joint of the arm 12 is represented by the frame x 3 , y 3 , z 3 (Frame 3) with the position of the arm 12 about the roll joint represented as joint angle ⁇ 3 .
  • the fourth DOF or linear joint is represented by the frame x 4 , y 4 , z 4 (Frame 4) and with the position of the first tool joint 24 from the RCM 22 represented as distance d 4 .
  • Movement of the tool 20 about the first tool joint 24 is defined in the fifth DOF or tool pitch joint represented in the frame x 5 , y 5 , z 5 (Frame 5) with the position of the tool pitch joint represented as joint angle ⁇ 5 .
  • Movement of the tool 20 about the second tool joint 26 is defined in the sixth DOF or tool yaw joint in the frame x 6 , y 6 , z 6 (Frame 6) with the position in the tool yaw joint represented as joint angle ⁇ 6 .
  • the orientation of the end effector 29 is represented by the ⁇ circumflex over (x) ⁇ 6 axis.
  • DH Denavit-Hartenberg
  • the RCM 22 is represented by trocar point “O” and is the point at which the end 14 of the arm 12 or the tool 20 passes through a trocar (not shown) in to the body cavity of the patient.
  • the tool pitch joint 24 is represented by point “P” and the tool yaw joint 26 is represented by yaw point “T”.
  • a yoke 25 ( FIG. 2 ) of the tool 20 may be represented by the line “PT”.
  • a frame of a jaw 29 of the tool 20 is represented by the Frame 6 and may be referred to as the tool-center-point frame (TCP frame) with the jaw direction positioned along the ⁇ circumflex over (x) ⁇ 6 axis.
  • TCP frame tool-center-point frame
  • the yaw point “T” may be referred to as the “TCP” point herein.
  • the jaw direction ⁇ right arrow over (T) ⁇ x 6 is known as the direction of the ⁇ circumflex over (x) ⁇ 6 axis.
  • the joint angle ⁇ 6 can be determined by placing the following two constraints on the pitch point “P”.
  • the pitch point “P” is in a yaw plane “YP” defined by rotation of the yoke “PT” around the ⁇ circumflex over (z) ⁇ 6 axis, shown as the dashed circle “YP” in FIG. 3 , with both the yoke PT and the jaw direction ⁇ right arrow over (T) ⁇ x 6 lying in the yaw plane “YP”.
  • the pitch point “P” when joint angle ⁇ 6 changes, the pitch point “P” will be in the yaw plane “YP” a distance a 6 from the yaw point T. Second, the pitch point “P” is also in a pitch plane “PP” which is perpendicular to the yaw plane “YP” defined by the trocar point “O” and the ⁇ circumflex over (z) ⁇ 6 axis.
  • the pitch plane “PP” is uniquely defined by the trocar point “O” and the 26 axis and intersects the yaw plane “YP” at the pitch point “P” on a proximal side (i.e., closer to the trocar point “O”) of the yaw point “T”.
  • the joint angle ⁇ 6 and the joint angle ⁇ 5 are in a range of about ⁇ /2 to about ⁇ /2. As used herein, angles are expressed in radians where it is equal to 180°.
  • the pitch plane “PP” is uniquely defined by the trocar point “O” and the ⁇ circumflex over (z) ⁇ 6 axis and intersects the yaw plane “YP” at the pitch point “P” on a distal side (i.e., away to the trocar point “O”) of the yaw point “T”.
  • the joint angle ⁇ 6 and the joint angle ⁇ 5 are outside of a range of about ⁇ /2 to about ⁇ /2.
  • the 26 axis is directed towards the trocar point “O” such that the trocar point “O” and the ⁇ circumflex over (z) ⁇ 6 axis fail to uniquely define the pitch plane “PP”.
  • a singularity exists which makes it difficult to determine a range of the joint angle ⁇ 6 and the joint angle ⁇ 5 .
  • the third case can be modeled as described in greater detail below.
  • the homogenous transform of the “TCP” Frame in the Base Frame can be expressed as:
  • the Base Frame is expressed in the TCP Frame as:
  • the projection of the trocar point “O” in the ⁇ circumflex over (x) ⁇ 6 , ⁇ 6 plane is point “O′” and the projection of point “O′” on the ⁇ 6 axis is “O′′”.
  • Equation 6 Taking into account the relative sign and quarter location, as detailed above, in Equation 6:
  • ⁇ 6 a tan 2( v′, ⁇ u ′) (7)
  • the joint angle ⁇ 5 is given by:
  • ⁇ 5 a tan 2( w′, ⁇ u ′ cos ⁇ 6 +v ′ sin ⁇ 6 ⁇ a 6 ) (8)
  • a rotation matrix chain can be expressed as:
  • 6 5 R can be expressed as:
  • 5 4 R can be expressed as:
  • the projection of the pitch point “P” in the ⁇ circumflex over (x) ⁇ 0 , ⁇ 0 plane is point “P′” and the projection of the point “P′” on the ⁇ circumflex over (x) ⁇ 0 axis is “P′′”.
  • the joint angle ⁇ 1 can be expressed as:
  • ⁇ 1 a tan 2( ⁇ R 23 , ⁇ R 13 ) (16)
  • joint angle ⁇ 2 can be expressed as:
  • ⁇ 2 a tan 2( R 33 , ⁇ square root over ( R 13 2 + R 23 2 ) ⁇ ) (17)
  • a ⁇ circumflex over (z) ⁇ 0 ′ axis is created at the pitch point “P” that is parallel to the ⁇ circumflex over (z) ⁇ 0 axis, the projection point z 0′ in the plane ⁇ circumflex over (x) ⁇ 4 , ⁇ 4 is point z 0′′ .
  • the joint angle ⁇ 3 can be expressed as:
  • ⁇ 3 a tan 2( ⁇ R 32 , R 31 ) (18)
  • the joint angle ⁇ 6 is calculated in a manner similar to the first case detailed above and then Equation 20 is used to determine the joint angle ⁇ 6′ for the second case.
  • the rest of the joint angles ⁇ 1 , ⁇ 2 , ⁇ 3 , and ⁇ 5 as well as the distance d 4 can be determined using the above method.
  • the third case can be detected when
  • the previous joint angle ⁇ 6 can be used.
  • the inverse kinematics solution provides mathematically precise mapping between an input pose and joint angles of the surgical robot 10 .
  • other considerations such as joint limits and joint speed limits are not taken into account.
  • the joint limits of joint angles ⁇ 5 and ⁇ 6 are usually in the range of about ⁇ /2 to about ⁇ /2.
  • the mechanical constraints of the surgical robot 10 prevent cases 2 and 3, detailed above, from being physically realizable since one of the joint angles will be greater than ⁇ /2.
  • the poses presented in cases 2 and 3 may be commanded by the input device 42 because the input device 42 and the surgical robot 10 have different kinematic constructions and working spaces.
  • case 3 is a singularity where the inverse kinematic solution is not unique.
  • a small rotation of the input pose along the axis x 6 would cause a dramatic change in joint angle ⁇ 3 .
  • Such a dramatic change may violate a joint speed limit and an acceleration limit of the roll joint of the arm 12 represented by joint angle ⁇ 3 . It will be appreciated that such sudden motions are undesirable in clinical settings.
  • the joint angle ⁇ 5 is restricted in a range of about ⁇ 5 ⁇ /12 to about 5 ⁇ /12 and the joint angle ⁇ 6 is restricted in a range of about ⁇ /2 to about ⁇ /2.
  • One method of achieving this is in a brutal-force method by clamping the solution provided by the inverse kinematics solution.
  • the brutal-force method may cause discontinuity in the inverse kinematics solution which is not desirable in a clinical setting.
  • a method may be used to correct the input pose after it crosses from a desirable solution space to an undesirable solution space.
  • the joint limits and the joint speed limits are acceptable and in the undesirable solution space at least one of the joint limits or the joint speed limits are exceeded or undesirable.
  • the corrected input pose can be solved by using the inverse kinematics solution detailed above having deniable joint angles.
  • Such a method may avoid discontinuities in the joint angles when the boundary between the desirable and undesirable solution spaces is crossed.
  • such a method can realize the position component of the input pose by only correcting an orientation component of the TCP frame to avoid unintended motion of the TCP frame from being introduced into the inverse kinematic solution.
  • a boundary plane that separates a desirable and undesirable solution space may be defined by locating the trocar point on a spherical surface. A boundary plane then divides the spherical surface into two sections. If the trocar point is located on one side of the boundary (e.g., the right side), the input pose does not require correction. If no correction is required, the inverse kinematics solutions are desirable and within the joint limits and the joint speed limits.
  • the input pose is projected to the boundary plane.
  • the input pose is corrected by rotating the TCP frame to a projection point located on the boundary plane according to rigid body kinematics and expressing the corrected TCP frame in a rotated base frame for the inverse kinematics solution.
  • the inverse kinematics solution for the corrected TCP frame will be desirable and within the joint limits and the joint speed limits.
  • the projection point “O′” is positioned beyond, to the right as shown, of a dotted boundary line “B” with a dimension u o from the ⁇ 6 axis which is greater than the length a 6 of the yoke “PT” which requires that:
  • the coordinate u 0 is chosen to meet the physical limits of joint angle ⁇ 5 and joint angle ⁇ 6 to avoid a singular condition.
  • the projection point “O′” may be positioned on either side or on the boundary line “B”.
  • a distance r′′ is defined between the projection point “O” and pitch point “P”, such that
  • ⁇ 5 , max is a desirable limit (i.e., mechanical constraint) on joint angle ⁇ 5 (e.g., 75° or 5 ⁇ /12).
  • the projection point “O′” lies on the left side of the boundary line “B”
  • the projection point “O′” is systematically projected to the boundary line “B”.
  • the yaw point “T” is rotated according to the projection rule, as detailed below, such that after the rotation of the yaw point “T”, a new pose of the “TCP” Frame is realized.
  • the trocar point “O” and the yaw point “T” are connected such that a distance between the yaw point “T” and the trocar point “O” is distance r 0 .
  • the distance r 0 is equal to ⁇ square root over (u′ 2 +v′ 2 +w′ 2 ) ⁇ .
  • the vector ⁇ right arrow over (T) ⁇ O is rotated in the “TCP” Frame around the yaw point “T” such that the trocar point “O” will travel on a sphere centered at the yaw point “T”.
  • the projection point “O′” lies on the left side of the boundary line “B” as illustrated in FIG. 11
  • the yaw point “T” will rotate until the trocar point “O” is rotated to the point “M” such that the projection point “O′” lies on the boundary line “B”.
  • the rotation angle “ ⁇ ” of the yaw point “T” can be determined as:
  • the ⁇ circumflex over (x) ⁇ 7′ axis is aligned with the vector ⁇ right arrow over (T) ⁇ M and the ⁇ circumflex over (z) ⁇ 7′ axis is along the direction of ⁇ right arrow over (T) ⁇ M ⁇ right arrow over (T) ⁇ O.
  • the Base Frame becomes frame x 0′ , y 0′ , and z 0′ (Rotated Base Frame).
  • the Base Frame in Frame 7 is equal to the Rotated Base Frame in Frame 7′ such that:
  • 0′ 6 R is from the homogenous transform 0′ 6 T in Equation 31.
  • the robotic system 10 may decouple or stop the mapping between the input device 43 and the arm 12 of the surgical robot 10 . Similar to the determination of u 0 in Equations 23 and 24, the distance between trocar point “O” and pitch point “P” can be expressed as:
  • checking if the input pose needs correction is accomplished by choosing u 0 in Equation 22 as a function of the yaw point “T” and a maximum joint angle ⁇ 5, max .
  • the maximum joint angle ⁇ 5, max may be selected as 5 ⁇ /12 (i.e., 75°).
  • the tool 20 ( FIG. 2 ) has two DOF in addition to a grasping function. It is contemplated that other tools may be used with this inverse kinematic model by manipulating constraints of the joint angles as detailed below.
  • Equation 23 can be modified as follows:
  • Equation 23 can be modified as follows:
  • Equation 23 can be modified such that:
  • Equations 37-39 are applied after the correction Equations 25-34 have been applied. Thus, two rounds of corrections are required.
  • a frame x e , y e , z e (End Effector Frame) is defined orientated similar to Frame 6 and offset a distance a 7 from Frame 6.
  • the End Effector Frame is defined adjacent a tip of the long jaws 29 along a centerline of the long jaws 29 . It is contemplated that the End Effector Frame may be on the centerline of the long jaws 29 , in a middle of the long jaws 29 , or any point along or between the long jaws 29 .
  • Equation 3 Equation 3
  • u′ is constrained according to Equation 22. If u′>u 0 , where u 0 can be dynamically constrained by using Equation 40, the trocar point “O” is projected to rotated trocar point “M” as shown in FIG. 20 . The coordinates of the rotated trocar point “M” is calculated according to Equation 25.
  • the distance between trocar point “O” and rotated trocar point “M” is:
  • r 0 ′ ⁇ square root over ( r 0 2 ⁇ u′ 2 +( u′+a 7 ) 2 ) ⁇ (46)
  • r 0 ′ ⁇ square root over ( r 0 2 ⁇ u 0 2 +( ⁇ u 0 ⁇ a 7 ) 2 ) ⁇ (47)
  • the rotation angle ⁇ ′ can be expressed as:
  • ⁇ ′ a ⁇ ⁇ cos ( l 2 - r 0 ′2 - r 0 ′′2 2 ⁇ r 0 ′ ⁇ r 0 ′′ ) ( 48 )
  • Equation 31 remains accurate; however, Equation 32 changes to the following:
  • Equation 33 becomes the following:
  • e 6 T is the End Effector Frame in the TCP frame and is expressed as:
  • the Rotated TCP Frame can be expressed in the Base Frame as:
  • the tool 20 when the input pose 6′ 0 T positions the trocar point “O” on the left side of the boundary line “B” the tool 20 would be functioning in an undesired configuration (e.g., with a possible decrease in DOF or the desired pose of the tool 20 may not be reachable in light of the current pose of the arm 12 or the desired pose of the tool 20 may be close to a singularity).
  • it may be desirable to correct the pose of the arm 12 using the projection method, as detailed above.
  • the user interface 40 may provide feedback to a clinician interfacing with the input handles 42 .
  • the feedback may be visual, audible, or haptic.
  • One form of haptic feedback is force feedback that may inform the clinician of the current status of the configuration of the arm 12 and the tool 20 .
  • the force feedback would provide a tactile force through the input handle 42 as the boundary line “B” is approached and/or crossed.
  • the tactile force may have a first feedback force as the boundary line B is approached and a second stronger feedback force as the boundary line “B” is crossed.
  • the feedback forces can be generated by a feedback torque of a force feedback system (not explicitly shown). With reference to FIG. 15 , the feedback torque can be expressed in the Base Frame as:
  • f( ⁇ ) is a scalar function that determines the magnitude of the feedback torque ⁇ circumflex over (t) ⁇ and the cross product of ⁇ right arrow over (T) ⁇ M and ⁇ right arrow over (T) ⁇ O determines the direction of the feedback torque ⁇ circumflex over (t) ⁇ .
  • the f( ⁇ ) may be a linear function or may be an exponential function that is limited by a maximum torque.
  • the maximum torque may be set by torque that is realizable by the force feedback system and/or may be set by torque that is physiologically meaningful to a clinician interfacing with the input handles 42 .
  • Equation 42 the feedback torque ⁇ circumflex over (t) ⁇ is expressed in the Base Frame and that when displayed or applied by the input handles 42 of the user interface 40 , the feedback torque ⁇ circumflex over (t) ⁇ would be expressed in the proper frame of the user interface 40 .

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