Axial Insertion and Movement Along a Partially Constrained Path for Robotic Catheters and Other Uses

Abstract

Devices, systems, and methods are provided for control over automated movement of catheters and other elongate bodies. Fluid and/or pull-wire drive systems can be used to provide robotically coordinated lateral bending motions and a processor of the system can generate synchronized actuator drive signals to move the tool along an at least partially laterally constrained path, with the path optionally extending along the axis of a virtual model of the catheter that has been driven in silico into alignment with a target tissue adjacent an open workspace such as an open chamber of the heart.

Claims

1. A method for aligning a tool with a target tissue adjacent a workspace within a patient body, the method comprising: advancing an elongate structure axially along an access path within the patient body, a first segment of the elongate structure being configured to laterally bend in response to a first segment drive command, wherein the advancing is performed so that a first portion of the first segment is laterally unconstrained within the workspace and a second portion of the first segment is laterally constrained along the access path; determining, with a robotic controller, the first segment drive command from a desired tool position and a first portion axial length of the first portion; and transmitting the first segment drive command from robotic controller so that the unconstrained first portion of the first segment bends laterally and the tool moves toward the desired tool position.

2. The method of claim 1, wherein the desired tool position is defined by a desired tool path, and wherein the advancing, determining, and transmitting steps are repeated so that the tool advances along the tool path.

3. The method of claim 2, further comprising defining the path by determining an aligned pose of the elongate structure with the tool aligned with the target tissue, wherein the elongate structure has an axis and the path extends along the axis of the elongate structure in the aligned pose.

4. The method of claim 3, wherein the aligned pose is determined by driving a virtual model of the elongate structure with reference to a display showing an image of the target tissue, the driving of the virtual model including modeled lateral bending of the first segment and modeled axial advancement of the elongate structure, and superimposing images of the virtual model of the elongate structure on the image of the target tissue in the display.

5. The method of claim 4, further comprising validating a candidate path of the elongate structure along the axis of the elongate structure in the aligned pose by superimposing a virtual validation model of the elongate structure on the image of the target tissue in the display and moving the virtual validation model along the candidate path.

6. The method of claim 4, wherein the advancing of the elongate structure is measured with an axial sensor, the axial sensor generating axial sensor signals and the controller determining the first segment drive commands in response to the axial signals.

7. The method of claim 6, wherein the axial advancing is performed manually.

8. The method of claim 1, wherein the access path comprises a lumen of an outer sheath having a lateral bending stiffness, and wherein the first segment drive signal is determined using the lateral bending stiffness of outer sheath.

9. The method of claim 1, wherein a second segment of the elongate structure is configured to laterally bend in response to a second segment drive command, the second segment being axially offset from the first segment.

10. The method of claim 9, wherein the first segment is proximal of the second segment, and further comprising determining, with the robotic controller, the second segment drive command from the desired tool position so that the first and second drive commands together move the tool toward the desired tool position.

11. The method of claim 10, wherein the robotic controller determines the first and second segment drive commands with a first number of command degrees of freedom, and further comprising determining a constrained drive command with a second number of command degrees of freedom smaller than the first number when at least a portion of the first segment is constrained.

12. The method of claim 9, wherein the first segment is distal of the second segment, and further comprising determining, with the robotic controller, the first segment drive command in response to constrained lateral bending of the second segment.

13. A method for aligning a tool with a target tissue adjacent a workspace within a patient body, the method comprising: driving, with reference to a display, a virtual model of an elongate structure from an actual pose of the elongate structure to a desired pose, the display showing an image of the elongate structure, an image of the target tissue, and a superimposed image of the virtual model, the elongate structure having an axis; and advancing the elongate structure axially along a desired path within the patient body by transmitting drive commands to laterally bend the elongate body, the desired path being defined by the axis of the virtual model of the elongate body in the desired pose.

14. A method for aligning a tool with a target adjacent a workspace, the method comprising: driving a continuum robot per a first drive signal in response to a first command to move a distal end of the continuum robot toward the target, wherein the continuum robot has proximal and distal articulatable portions, and wherein the first signal is determined with a first number of degrees of freedom by a robotic controller in response to the proximal portion being laterally constrained; and driving the continuum robot per a second drive signal in response to a second command to move the distal end toward the target, wherein the second signal is determined with a second number of degrees of freedom by the robotic controller in response to the proximal portion being laterally unconstrained, the second number being larger than the first number.

15. A system for aligning a tool with a target tissue adjacent a workspace within a patient body, the system comprising: an elongate structure having a proximal end and a distal end with an axis therebetween, the elongate structure configured to advance axially along an access path within the patient body, a first segment of the elongate structure having a first segment length and being configured to laterally bend in response to a first segment drive command; and a robotic controller coupled to the first segment, the robotic controller configured to generate the first segment drive command from a desired tool position and a laterally constrained axial portion length so that the unconstrained first portion of the first segment bends laterally and the tool moves toward the desired tool position.

16. A system for aligning a tool with a target tissue adjacent a workspace within a patient body, the system comprising: an elongate structure having a proximal end and a distal end with an axis therebetween, the elongate structure configured to advance axially into a patient body; a simulation system configured to drive, with reference to a display, a virtual model of the elongate structure from an actual pose of the elongate structure to a desired pose, the display, in use, showing an image of the elongate structure and an image of the target tissue, the simulation system configured to superimpose an image of the virtual model on the display; and a robotic controller coupled with the elongate structure and configured to help advance the elongate structure axially along a desired path within the patient body by transmitting drive commands to laterally bend the elongate body, the desired path being defined by the axis of the virtual model of the elongate body in the desired pose.

17. A system for aligning a tool with a target adjacent a workspace, the system comprising: a continuum robot having a proximal end and a distal end with an axis therebetween, the continuum robot having axially offset proximal and distal articulatable portions and configured to advance axially from a lateral constraint into the workspace; a robotic controller coupled with the continuum robot, the controller having a first mode for use when the proximal portion is laterally constrained for transmitting to the continuum robot a first drive signal in response to a first command to move the distal end toward the target, the first signal having a first number of degrees of freedom; and the robotic controller having a second mode when the proximal portion is unconstrained for driving the continuum robot per a second drive signal in response to a second command to move the distal end toward the target, wherein the second signal has a second number of degrees of freedom, the second number being larger than the first number.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0020] FIG. 1 illustrates an interventional cardiologist performing a structural heart procedure with a robotic catheter system having a fluidic catheter driver slidably supported by a stand.

[0021] FIG. 2 is a simplified schematic illustration of components of a helical balloon assembly, showing how an extruded multi-lumen shaft can provide fluid to laterally aligned subsets of balloons within an articulation balloon array of a catheter.

[0022] FIGS. 3A-3C schematically illustrate helical balloon assemblies supported by flat springs and embedded in an elastomeric polymer matrix, and also show how selective inflation of subsets of the balloons can elongate and laterally articulate the assemblies.

[0023] FIG. 4 is a perspective view of a robotic catheter system in which a catheter is removably mounted on a driver assembly, and in which the driver assembly includes a driver encased in a sterile housing and supported by a stand.

[0024] FIG. 5 schematically illustrates a robotic catheter system and transmission of signals between the components thereof so that input from a user induces a desired articulation.

[0025] FIG. 6 is a high-level flow chart of an exemplary control system for the fluid-driven robotic catheter control systems described herein.

[0026] FIG. 7 is a flow chart showing an exemplary method and structure for use by the control systems described herein to solve for the inverse kinematics of the fluid-driven catheter structures.

[0027] FIG. 8 illustrates a relationship between reference frames at the base of the articulated segments and at the tip of the catheter, for use in the control systems described herein.

[0028] FIG. 9 illustrates Euler angles for determination of transformations between reference frames used in the control systems described herein.

[0029] FIGS. 10A-10C graphically illustrate coordinate frames, angle and length symbols representing variable values used by the control systems described herein.

[0030] FIG. 11 is a flow chart showing inverse kinematics for a segment as can be used to solve for lumen pressures in a control system of the fluid-driven catheter systems described herein.

[0031] FIG. 11A graphically illustrates segment and other component lengths of a catheter system, and shows how an outer sheath or other structure may laterally constrain a portion of a segment by increasing a spring constant of the portion of a segment disposed therein.

[0032] FIG. 12 graphically illustrates angles and values used in the user interface of the control systems described herein.

[0033] FIGS. 13A and 13B graphically illustrate angles and values used for communication of signals between the user interface and the robotic position control of the control systems described herein.

[0034] FIG. 14 schematically illustrate exemplary components of an input-output system for use in the robotic control and/or simulation systems described herein, and also show an image of a tool supported by a flexible body within an internal surgical site bordered by adjacent tissue.

[0035] FIGS. 15A and 15B schematically illustrate exemplary components of robotic control and/or simulation systems and show communications between those components.

[0036] FIGS. 16A-16D are screen prints of display images showing an in situ movement plan (or simulation thereof) and associated movement of a catheter-supported tool along a trajectory determined by disregarding one or more candidate intermediate input poses and a trajectory of a virtual catheter indicator.

[0037] FIGS. 17A-17D are screen prints of display images showing rotational and translational movements and associated indicia for use with a planar input device during movement of an actual or simulated robotic catheter.

[0038] FIGS. 18A-18C illustrate exemplary graphical indicia superimposed on an image of an actual or simulated robotic catheter to facilitate precise and predictable rotation, translation, and alignment with target tissues.

[0039] FIG. 19 is a functional block diagram of an exemplary fluid-driven structural heart therapy system having an augmented reality hybrid 2D/3D display for reference by a system user to position a therapeutic or diagnostic tool in an open chamber of a patient's beating heart.

[0040] FIGS. 20A and 20B are screen shots of an augmented reality display for use in the system of FIG. 19, showing a captured 2D image representing an actual tip of an articulated delivery system and adjacent tissue in FIG. 20A, and showing a 2D image of a virtual model of the articulated delivery system superimposed on the captured 2D tissue/tip image.

[0041] FIG. 21 is a screenshot showing a hybrid 2D/3D display for use in the system of FIG. 19, with the display presenting an image including a 3D virtual model of the articulated delivery system, and also presenting first and second 2D image planes, each having a 2D image of the virtual model projected thereon with the orientations of the 2D images relative to the model corresponding to the orientations of the planes on which they are projected, wherein the 2D image planes represent fluoroscopic image planes.

[0042] FIG. 22 is a screenshot showing another hybrid 2D/3D display presenting a 3D virtual image, a 2D fluoroscopic image plane, and first and second 2D ultrasound image slice planes, wherein the ultrasound planes are offset from the model.

[0043] FIG. 23 is a screenshot showing another hybrid 2D/3D display presenting a 3D virtual image of an articulated delivery system and an ultrasound transducer, a 2D fluoroscopic image plane having the articulated delivery system and transducer projected thereon, and first and second 2D ultrasound image slice planes of the transducer, and a 3D ultrasound image volume of the transducer so as to illustrate how image data of the fluoroscopic and ultrasound system can be registered and tracked.

[0044] FIG. 24 is a screenshot showing first and second 2D ultrasound image slice planes of a transducer, along with a 3D virtual image of an articulated delivery system and 2D virtual image sliced projected from the model to the 2D ultrasound image planes.

[0045] FIG. 25 is a screenshot showing yet another hybrid 2D/3D display showing a 3D virtual model of the articulated delivery system, offset of first and second 2D ultrasound image planes along their normals, projection of a phantom articulated delivery system onto the ultrasound image planes, and a 3D trajectory between the virtual 3D model and the 3D phantom.

[0046] FIGS. 26A-26C are screenshots showing a hybrid 2D/3D display showing 3D virtual models of articulated delivery systems and 2D images of the virtual models projected on 2D images, with both the 3D and 2D images including widgets adjacent the tip of the models, the widgets correlating to a constraint on the movement of the articulated delivery system.

[0047] FIGS. 26D-26G schematically illustrate geometric terms which may optionally be used to constrain motion of articulated devices as detailed in the associated text.

[0048] FIGS. 27A-27C are screenshots showing pages of a 6 degree of freedom input device, showing app pages and buttons for controlling the articulation system in different modes using a variety of alternative constraints.

[0049] FIGS. 28A-28C illustrate manual positioning of a 6 degree of freedom input device and corresponding movement of an image of a 3D virtual catheter within a 3D virtual workspace as shown on a 2D display, and also show spring-back of the view to a starting position and orientation when a view drive input button is released.

[0050] FIGS. 29A-29D illustrate optional image elements to be included in an exemplary hybrid 2D/3D display image.

[0051] FIGS. 30A-30C illustrate manual manipulation of the catheter body from outside the patient while driving the tip so as to inhibit resulting changes in the tip position.

[0052] FIG. 31 schematically illustrates a calculated trajectory that avoids a workspace boundary, along with a virtual trajectory verification catheter that moves along the trajectory to facilitate visual verification of safety prior to implementing a move of the actual catheter.

[0053] FIG. 32A illustrates a perspective view of a multi-segment articulated catheter (the steerable sleeve) supporting an instrument or tool (specifically a clip), with an instrument deployment catheter extending through the segments and a clip being supported by the deployment catheter distal of the segments; the clip, deployment catheter, and segments being introduced into a chamber of a heart or other workspace within a patient through a guide catheter.

[0054] FIGS. 32B, 32C, and 32D illustrates schematic views of an instrument, of a multi-segment articulated catheter and a guide catheter, and also identify locations along the catheter system and associated spring constants and catheter portion lengths.

[0055] FIG. 32E is a chart that defines in symbols and equations the relative positions and spring constants of sections of the catheter system for driving with different degrees of freedom in response to different portions of the catheter being laterally unconstrained and/or constrained.

DETAILED DESCRIPTION OF THE INVENTION

[0056] The improved devices, systems, and methods for controlling, image guidance of, inputting commands into, and simulating movement of powered and robotic devices will find a wide variety of uses. The elongate tool-supporting structures described herein will often be flexible, typically comprising catheters suitable for insertion in a patient body. Exemplary systems will be configured for insertion into the vascular system, the systems typically including a cardiac catheter and supporting a structural heart tool for repairing or replacing a valve of the heart, occluding an ostium or passage, or the like. Other cardiac catheter systems will be configured for diagnosis and/or treatment of congenital defects of the heart, or may comprise electrophysiology catheters configured for diagnosing or inhibiting arrhythmias (optionally by ablating a pattern of tissue bordering or near a heart chamber).

[0057] Alternative applications may include use in steerable supports of image acquisition devices such as for trans-esophageal echocardiography (TEE), intra-coronary echocardiography (ICE), and other ultrasound techniques, endoscopy, and the like. The structures described herein will often find applications for diagnosing or treating the disease states of or adjacent to the cardiovascular system, the alimentary tract, the airways, the urogenital system, and/or other lumen systems of a patient body. Other medical tools making use of the articulation systems described herein may be configured for endoscopic procedures, or even for open surgical procedures, such as for supporting, moving and aligning image capture devices, other sensor systems, or energy delivery tools, for tissue retraction or support, for therapeutic tissue remodeling tools, or the like. Alternative elongate flexible bodies that include the articulation technologies described herein may find applications in industrial applications (such as for electronic device assembly or test equipment, for orienting and positioning image acquisition devices, or the like). Still further elongate articulatable devices embodying the techniques described herein may be configured for use in consumer products, for retail applications, for entertainment, or the like, and wherever it is desirable to provide simple articulated assemblies with one or more (preferably multiple) degrees of freedom without having to resort to complex rigid linkages.

[0058] Embodiments provided herein may use balloon-like structures to effect articulation of the elongate catheter or other body. The term “articulation balloon” may be used to refer to a component which expands on inflation with a fluid and is arranged so that on expansion the primary effect is to cause articulation of the elongate body. Note that this use of such a structure is contrasted with a conventional interventional balloon whose primary effect on expansion is to cause substantial radially outward expansion from the outer profile of the overall device, for example to dilate or occlude or anchor in a vessel in which the device is located. Independently, articulated medial structures described herein will often have an articulated distal portion, and an unarticulated proximal portion, which may significantly simplify initial advancement of the structure into a patient using standard catheterization techniques.

[0059] The robotic systems described herein will often include an input device, a driver, and an articulated catheter or other robotic manipulator supporting a diagnostic or therapeutic tool. The user will typically input commands into the input device, which will generate and transmit corresponding input command signals. The driver will generally provide both power for and articulation movement control over the tool. Hence, somewhat analogous to a motor driver, the driver structures described herein will receive the input command signals from the input device and will output drive signals to the tool-supporting articulated structure so as to effect robotic movement of an articulated feature of the tool (such as movement of one or more laterally deflectable segments of a catheter in multiple degrees of freedom). The drive signals may comprise fluidic commands, such as pressurized pneumatic or hydraulic flows transmitted from the driver to the tool-supporting catheter along a plurality of fluid channels. Optionally, the drive signals may comprise electromagnetic, optical, or other signals, preferably (although not necessarily) in combination with fluidic drive signals. Unlike many robotic systems, the robotic tool supporting structure will often (though not always) have a passively flexible portion between the articulated feature (typically disposed along a distal portion of a catheter or other tool manipulator) and the driver (typically coupled to a proximal end of the catheter or tool manipulator). The system will be driven while sufficient environmental forces are imposed against the tool or catheter to impose one or more bend along this passive proximal portion, the system often being configured for use with the bend(s) resiliently deflecting an axis of the catheter or other tool manipulator by 10 degrees or more, more than 20 degrees, or even more than 45 degrees.

[0060] The catheter bodies (and many of the other elongate flexible bodies that benefit from the inventions described herein) will often be described herein as having or defining an axis, such that the axis extends along the elongate length of the body. As the bodies are flexible, the local orientation of this axis may vary along the length of the body, and while the axis will often be a central axis defined at or near a center of a cross-section of the body, eccentric axes near an outer surface of the body might also be used. It should be understood, for example, that an elongate structure that extends “along an axis” may have its longest dimension extending in an orientation that has a significant axial component, but the length of that structure need not be precisely parallel to the axis. Similarly, an elongate structure that extends “primarily along the axis” and the like will generally have a length that extends along an orientation that has a greater axial component than components in other orientations orthogonal to the axis. Other orientations may be defined relative to the axis of the body, including orientations that are transvers to the axis (which will encompass orientation that generally extend across the axis, but need not be orthogonal to the axis), orientations that are lateral to the axis (which will encompass orientations that have a significant radial component relative to the axis), orientations that are circumferential relative to the axis (which will encompass orientations that extend around the axis), and the like. The orientations of surfaces may be described herein by reference to the normal of the surface extending away from the structure underlying the surface. As an example, in a simple, solid cylindrical body that has an axis that extends from a proximal end of the body to the distal end of the body, the distal-most end of the body may be described as being distally oriented, the proximal end may be described as being proximally oriented, and the curved outer surface of the cylinder between the proximal and distal ends may be described as being radially oriented. As another example, an elongate helical structure extending axially around the above cylindrical body, with the helical structure comprising a wire with a square cross section wrapped around the cylinder at a 20 degree angle, might be described herein as having two opposed axial surfaces (with one being primarily proximally oriented, one being primarily distally oriented). The outermost surface of that wire might be described as being oriented exactly radially outwardly, while the opposed inner surface of the wire might be described as being oriented radially inwardly, and so forth.

[0061] Referring first to FIG. 1, a system user U, such as an interventional cardiologist, uses a robotic catheter system 10 to perform a procedure in a heart H of a patient P. System 10 generally includes an articulated catheter 12, a driver assembly 14, and an input device 16. User U controls the position and orientation of a therapeutic or diagnostic tool mounted on a distal end of catheter 12 by entering movement commands into input 16, and optionally by sliding the catheter relative to a stand of the driver assembly, while viewing a distal end of the catheter and the surrounding tissue in a display D. As will be described below, user U may alternatively manually rotate the catheter body about its axis in some embodiments.

[0062] During use, catheter 12 extends distally from driver system 14 through a vascular access site S, optionally (though not necessarily) using an introducer sheath. A sterile field 18 encompasses access site S, catheter 12, and some or all of an outer surface of driver assembly 14. Driver assembly 14 will generally include components that power automated movement of the distal end of catheter 12 within patient P, with at least a portion of the power often being transmitted along the catheter body as a hydraulic or pneumatic fluid flow. To facilitate movement of a catheter-mounted therapeutic tool per the commands of user U, system 10 will typically include data processing circuitry, often including a processor within the driver assembly. Regarding that processor and the other data processing components of system 10, a wide variety of data processing architectures may be employed. The processor, associated pressure and/or position sensors of the driver assembly, and data input device 16, optionally together with any additional general purpose or proprietary computing device (such as a desktop PC, notebook PC, tablet, server, remote computing or interface device, or the like) will generally include a combination of data processing hardware and software, with the hardware including an input, an output (such as a sound generator, indicator lights, printer, and/or an image display), and one or more processor board(s). These components are included in a processor system capable of performing the transformations, kinematic analysis, and matrix processing functionality associated with generating the valve commands, along with the appropriate connectors, conductors, wireless telemetry, and the like. The processing capabilities may be centralized in a single processor board, or may be distributed among various components so that smaller volumes of higher-level data can be transmitted. The processor(s) will often include one or more memory or other form of volatile or non-volatile storage media, and the functionality used to perform the methods described herein will often include software or firmware embodied therein. The software will typically comprise machine-readable programming code or instructions embodied in non-volatile media and may be arranged in a wide variety of alternative code architectures, varying from a single monolithic code running on a single processor to a large number of specialized subroutines, classes, or objects being run in parallel on a number of separate processor sub-units.

[0063] Referring still to FIG. 1, along with display D, a simulation display SD may present an image of an articulated portion of a simulated or virtual catheter S12 with a receptacle for supporting a simulated therapeutic or diagnostic tool. The simulated image shown on the simulation display SD may optionally include a tissue image based on pre-treatment imaging, intra-treatment imaging, and/or a simplified virtual tissue model, or the virtual catheter may be displayed without tissue. Simulation display SD may have or be included in an associated computer 15, and the computer will preferably be couplable with a network and/or a cloud 17 so as to facilitate updating of the system, uploading of treatment and/or simulation data for use in data analytics, and the like. Computer 15 may have a wireless, wired, or optical connection with input device 16, a processor of driver assembly 14, display D, and/or cloud 17, with suitable wireless connections comprising a Bluetooth™ connection, a WiFi connection, or the like. Preferably, an orientation and other characteristics of simulated catheter S12 may be controlled by the user U via input device 16 or another input device of computer 15, and/or by software of the computer so as to present the simulated catheter to the user with an orientation corresponding to the orientation of the actual catheter as sensed by a remote imaging system (typically a fluoroscopic imaging system, an ultra-sound imaging system, a magnetic resonance imaging system (MM), or the like) incorporating display D and an image capture device 19. Optionally, computer 15 may superimpose an image of simulated catheter S12 on the tissue image shown by display D (instead of or in addition to displaying the simulated catheter on simulation display SD), preferably with the image of the simulated catheter being registered with the image of the tissue and/or with an image of the actual catheter structure in the surgical site. Still other alternatives may be provided, including presenting a simulation window showing simulated catheter SD on display D, including the simulation data processing capabilities of computer 15 in a processor of driver assembly 14 and/or input device 16 (with the input device optionally taking the form of a tablet that can be supported by or near driver assembly 14, incorporating the input device, computer, and one or both of displays D, SD into a workstation near the patient, shielded from the imaging system, and/or remote from the patient, or the like.

[0064] Referring now to FIG. 2, the components of, and fabrication method for production of, an exemplary balloon array assembly (sometimes referred to herein as a balloon string 32) can be understood. A multi-lumen shaft 34 will typically have between 3 and 18 lumens. The shaft can be formed by extrusion with a polymer such as a nylon, a polyurethane, a thermoplastic such as a Pebax™ thermoplastic or a polyether ether ketone (PEEK) thermoplastic, a polyethylene terephthalate (PET) polymer, a polytetrafluoroethylene (PTFE) polymer, or the like. A series of ports 36 are formed between the outer surface of shaft 36 and the lumens, and a continuous balloon tube 38 is slid over the shaft and ports, with the ports being disposed in large profile regions of the tube and the tube being sealed over the shaft along the small profile regions of the tube between ports to form a series of balloons. The balloon tube may be formed using a compliant, non-compliant, or semi-compliant balloon material such as a latex, a silicone, a nylon elastomer, a polyurethane, a nylon, a thermoplastic such as a Pebax™ thermoplastic or a polyether ether ketone (PEEK) thermoplastic, a polyethylene terephthalate (PET) polymer, a polytetrafluoroethylene (PTFE) polymer, or the like, with the large-profile regions preferably being blown sequentially or simultaneously to provide desired hoop strength. The ports can be formed by laser drilling or mechanical skiving of the multi-lumen shaft with a mandrel in the lumens. Each lumen of the shaft may be associated with between 3 and 50 balloons, typically from about 5 to about 30 balloons. The shaft balloon assembly 40 can be coiled to a helical balloon array of balloon string 32, with one subset of balloons 42a being aligned along one side of the helical axis 44, another subset of balloons 44b (typically offset from the first set by 120 degrees) aligned along another side, and a third set (shown schematically as deflated) along a third side. Alternative embodiments may have four subsets of balloons arranged in quadrature about axis 44, with 90 degrees between adjacent sets of balloons.

[0065] Referring now to FIGS. 3A, 3B, and 3C, an articulated segment assembly 50 has a plurality of helical balloon strings 32, 32′ arranged in a double helix configuration. A pair of flat springs 52 are interleaved between the balloon strings and can help axially compress the assembly and urge deflation of the balloons. As can be understood by a comparison of FIGS. 3A and 3B, inflation of subsets of the balloons surrounding the axis of segment 50 can induce axial elongation of the segment. As can be understood with reference to FIGS. 3A and 3C, selective inflation of a balloon subset 42a offset from the segment axis 44 along a common lateral bending orientation X induces lateral bending of the axis 44 away from the inflated balloons. Variable inflation of three or four subsets of balloons (via three or four channels of a single multi-lumen shaft, for example) can provide control over the articulation of segment 50 in three degrees of freedom, i.e., lateral bending in the +/−X orientation and the +/−Y orientation, and elongation in the +Z orientation. As noted above, each multilumen shaft of the balloon strings 32, 32′ may have more than three channels (with the exemplary shafts having 6 or 7 lumens), so that the total balloon array may include a series of independently articulatable segments (each having 3 or 4 dedicated lumens of one of the multi-lumen shafts, for example). Optionally, from 2 to 4 modular, axially sequential segments may each have an associated tri-lumen shaft, with the tri-lumen shaft extending axially in a loose helical coil through the lumen of any proximal segments to accommodate bending and elongation. The segments may each include a single helical balloon string/multilumen shaft assembly (rather than having a dual-helix configuration). Multi-lumen shafts for driving of distal segments may alternatively wind proximally around an outer surface of a proximal segment, or may be wound parallel and next to the multi-lumen shaft/balloon tube assemblies of the balloon array of the proximal segment(s).

[0066] Referring still to FIGS. 3A, 3B, and 3C, articulated segment 50 optionally includes a polymer matrix 54, with some or all of the outer surface of balloon strings 32, 32′ and flat springs 52 that are included in the segment being covered by the matrix. Matrix 54 may comprise, for example, a relatively soft elastomer to accommodate inflation of the balloons and associated articulation of the segment, with the matrix optionally helping to urge the balloons toward an at least nominally deflated state, and to urge the segment toward a straight, minimal length configuration. Alternatively (or in addition to a relatively soft matrix), a thin layer of relatively high-strength elastomer can be applied to the assembly (prior to, after, or instead of the soft matrix), optionally while the balloons are in an at least partially inflated state. Advantageously, matrix 54 can help maintain overall alignment of the balloon array and springs within the segment despite segment articulation and bending of the segment by environmental forces. Regardless of whether or not a matrix is included, an inner sheath may extend along the inner surface of the helical assembly, and an outer sheath may extend along an outer surface of the assembly, with the inner and/or outer sheaths optionally comprising a polymer reinforced with wire or a high-strength fiber in a coiled, braid, or other circumferential configuration to provide hoop strength while accommodating lateral bending (and preferably axial elongation as well). The inner and outer sheaths may be sealed together distal of the balloon assembly, forming an annular chamber with the balloon array disposed therein. A passage may extend proximally from the annular space around the balloons to the proximal end of the catheter to safely vent any escaping inflation media, or a vacuum may be drawn in the annular space and monitored electronically with a pressure sensor to inhibit inflation flow if the vacuum deteriorates.

[0067] Referring now to FIG. 4, a proximal housing 62 of catheter 12 and the primary components of driver assembly 14 can be seen in more detail. Catheter 12 generally includes a catheter body 64 that extends from proximal housing 62 to an articulated distal portion 66 (see FIG. 1) along an axis 67, with the articulated distal portion preferably comprising a balloon array and the associated structures described above. Proximal housing 62 also contains first and second rotating latch receptacles 68a, 68b which allow a quick-disconnect removal and replacement of the catheter. The components of driver assembly 14 visible in FIG. 4 include a sterile housing 70 and a stand 72, with the stand supporting the sterile housing so that the sterile housing (and components of the driver assembly therein, including the driver) and catheter 12 can move axially along axis 67. Sterile housing 70 generally includes a lower housing 74 and a sterile junction having a sterile barrier 76. Sterile junction 76 releasably latches to lower housing 74 and includes a sterile barrier body that extends between catheter 12 and the driver contained within the sterile housing. Along with components that allow articulation fluid flow to pass through the sterile fluidic junction, the sterile barrier may also include one or more electrical connectors or contacts to facilitate data and/or electrical power transmission between the catheter and driver, such as for articulation feedback sensing, manual articulations sensing, or the like. The sterile housing 70 will often comprise a polymer such as an ABS plastic, a polycarbonate, acetal, polystyrene, polypropylene, or the like, and may be injection molded, blow molded, thermoformed, 3-D printed, or formed using still other techniques. Polymer sterile housings may be disposable after use on a single patient, may be sterilizable for use with a limited number of patients, or may be sterilizable indefinitely; alternative sterile housings may comprise metal for long-term repeated sterile processing. Stand 72 will often comprise a metal, such as a stainless steel, aluminum, or the like for repeated sterilizing and use.

[0068] Referring now to FIG. 5, components of a simulation system 101 that can be used for simulation, training, pre-treatment planning, and or treatment of a patent are schematically illustrated. Some or all of the components of system 101 may be used in addition to or instead of the clinical components of the system shown in FIG. 1. System 101 may optionally include an alternative catheter 112 and an alternative driver assembly 114, with the alternative catheter comprising a real and/or virtual catheter and the driver assembly comprising a real and/or virtual driver 114.

[0069] Alternative catheter 112 can be replaceably coupled with alternative driver assembly 114. When simulation system 101 is used for driving an actual catheter, the coupling may be performed using a quick-release engagement between an interface 113 on a proximal housing of the catheter and a catheter receptacle 103 of the driver assembly. An elongate body 105 of catheter 112 has a proximal/distal axis as described above and a distal receptacle 107 that is configured to support a therapeutic or diagnostic tool 109 such as a structural heart tool for repairing or replacing a valve of a heart. The tool receptacle may comprise an axial lumen for receiving the tool within or through the catheter body, a surface of the body to which the tool is permanently affixed, or the like. Alternative drive assembly 114 may be wireless coupled to a simulation computer 115 and/or a simulation input device 116, or cables may be used for transmission of data.

[0070] When alternative catheter 112 and alternative drive system 114 comprise virtual structures, they may be embodied as modules of software, firmware, and/or hardware. The modules may optionally be configured for performing articulation calculations modeling performance of some or all of the actual clinical components as described below, and/or may be embodied as a series of look-up tables to allow computer 115 to generate a display effectively representing the performance. The modules will optionally be embodied at least in-part in a non-volatile memory of a simulation-supporting alternative drive assembly 121a, but some or all of the simulation modules will preferably be embodied as software in non-volatile memories 121b, 121c of simulation computer 115 and/or simulation input device 116, respectively. Coupling of alternative virtual catheters and tools can be permed using menu options or the like. In some embodiments, selection of a virtual catheter may be facilitated by a signal generated in response to mounting of an actual catheter to an actual driver.

[0071] Simulation computer 115 preferably comprises an off-the-shelf notebook or desktop computer that can be coupled to cloud 17, optionally via an intranet, the internet, an ethernet, or the like, typically using a wireless router or a cable coupling the simulation computer to a server. Cloud 17 will preferably provide data communication between simulation computer 115 and a remote server, with the remote server also being in communication with a processor of other simulation computers 115 and/or one or more clinical drive assemblies 14. Simulation computer 115 may also comprise code with a virtual 3D workspace, the workspace optionally being generated using a proprietary or commercially available 3D development engine that can also be used for developing games and the like, such as Unity™ as commercialized by Unity Technologies. Suitable off-the-shelf computers may include any of a variety of operating systems (such as Windows from Microsoft, OS from Apple, Linex, or the like), along with a variety of additional proprietary and commercially available apps and programs.

[0072] Simulation input device 116 may comprise an off-the-shelf input device having a sensor system for measuring input commands in at least two degrees of freedom, preferably in 3 or more degrees of freedom, and in some cases 5, 6, or more degrees of freedom. Suitable off-the-shelf input devices include a mouse (optionally with a scroll wheel or the like to facilitate input in a 3.sup.rd degree of freedom), a tablet or phone having an X-Y touch screen (optionally with AR capabilities such as being compliant with ARCor from Google, ARKit from Apple, or the like to facilitate input of translation and/or rotation, along with multi-finger gestures such as pinching, rotation, and the like), a gamepad, a 3D mouse, a 3D stylus, or the like. Proprietary code may be loaded on the simulation input device (particularly when a phone, tablet, or other device having a touchscreen is used), with such input device code presenting menu options for inputting additional commands and changing modes of operation of the simulation or clinical system. A simulation input/output system 111 may be defined by the simulation input device 116 and the simulation display SD.

[0073] System Motion Equations

[0074] Referring now to FIG. 6, a system control flow chart 120 is shown that can be used by a processor of the robotic system to drive movement of an actual or virtual catheter. The following terms are used in flow chart 120 and/or in the analysis that follows. The following notation may be used in at least some of the equations herein:

Input

[0075] Q.sub.d Tip position desired

Measured

[0076] Pr.sub.0 Pressure for balloon arrays

Calculated

[0077] Pr.sub.d Pressure for balloon arrays as computed for joint, j.sub.d
j Joint space variables
j.sub.d Joint space computed for desired tip position, Q.sub.d
j.sub.n Joint space newly computed
+ Add increment to each variable one at a time
Q Tip position in world space
Q+ Tip positions by added increments to each variable one at a time to form Jacobian
Q.sub.n Tip position newly calculated

e Error

[0078] Seg FK Segment Forward Kinematics; pressures to joint space
Seg IK Segment Inverse Kinematics; joint space to pressures
FK System Forward Kinematics; joint space to world space (distal segments tip position)
IK System Inverse Kinematics; world space (distal segment tip position) to joint space
J Jacobian (numerically derived)

J.SUP.−1 .Inverse Jacobian

[0079] Referring now to FIG. 7, an inverse kinematics flow chart 122 is useful to understand how the processor can solve for joint angles and displacements. The calculations below reference the variables outlined in Table I. The input variables (α0, α1, α2, α3, α4, β1, β2, β3, β4, S0, S1, S2, S3, and S4) are used as input in these calculations. The calculations are arranged for articulated catheters having up to 4 articulated independently articulatable segments (segment Nos. 1-4), and a base segment (No. 0) can be used to accommodate a pose of the catheter body proximally of the most proximal articulated segment (the parameters for the base segment or segment No. 0 being treated as defined parameters in the calculations).

TABLE-US-00001 TABLE 1 Variable Matrix Segment no. n i 0 1 2 3 4 Angles Alpha α.sub.i α.sub.0 α.sub.1 α.sub.2 α.sub.3 α.sub.4 Input Beta β.sub.i β.sub.1 β.sub.2 β.sub.3 β.sub.4 Arc Length Arc S.sub.i S.sub.0 S.sub.1 S.sub.2 S.sub.3 S.sub.4 Arc Radii Radius r.sub.i r.sub.1 r.sub.2 r.sub.3 r.sub.4 Solve Arc P.sub.i x.sub.i 0 x.sub.1 x.sub.2 x.sub.3 x.sub.4 For End y.sub.i 0 y.sub.1 y.sub.2 Y.sub.3 y.sub.4 Points z.sub.i z.sub.0 z.sub.1 z.sub.2 z.sub.3 .sub.z4 Arc V.sub.i a.sub.i 0 a.sub.1 a.sub.2 a.sub.3 a.sub.4 Unit b.sub.i 0 b.sub.1 b.sub.2 b.sub.3 b.sub.4 Vectors c.sub.i 1 c.sub.1 c.sub.2 c.sub.3 c.sub.4

[0080] Startup Position:

[0081] Segments start by expanding the balloon array inflations to predetermined levels. The Segment is driven to predetermined and straight (or nearly straight) condition defined by an initial joint space vector j.sub.s, which accounts for all the Segments' initial states.

[0082] To move to a desired location, balloon array conditions are changed to locate segment angles and displacement. The first step is to determine the current and desired position vectors for the robot tip in the robot's base coordinate system, sometime referred to herein as world space. Note that world space may move relative to the patient and/or the operating room with manual repositioning of the catheter or the like, so that this world space may differ from a fixed room space or global world space.

[0083] User Input Commands/Tip Vector q:

[0084] Referring now to FIG. 8, The robot is driven by a user input vector q measured from a coordinate system attached to the distal end of the robot, or tip. This q vector is transformed into a desired world space vector Qa, where the current tip location in world space vector is Q.sub.c.

Q.sub.c=(X.sub.c,Y.sub.c,Z.sub.c,.sub.c,.sub.c)

The user input q represents a velocity (or small displacements) command. The tip coordinate system resides at the current tip position, and therefore the current q, or q.sub.c is always at the origin:

q.sub.c=(0,0, 0, 0, 0)

Since q.sub.c is zero the desired displacement, qa, is equivalent to the change in q or dq as shown here:

dq=q.sub.d—q.sub.c=q.sub.d

To simplify, dq is replaced simply with q to represent the desired change in position,

q=q.sub.d=(x.sub.T,y.sub.T,Z.sub.T,α.sub.T,β.sub.T),

where x.sub.T, y.sub.T, z.sub.T, α.sub.T and β.sub.T describes a change vector in tip space coordinates. q is then used by the current Transformation Matrix, T.sub.0Tc, to acquire the desired world coordinates, Q.sub.d.

Q.sub.d=T.sub.0Tc(q.sub.d)=(X.sub.d,Y.sub.d,Z.sub.d,.sub.d,.sub.d)

Where the Tip's current world coordinates are defined by Q.sub.c.

Q.sub.c=T.sub.0Tc(q.sub.c)=(X.sub.c,Y.sub.c,Z.sub.c,.sub.c,.sub.c)

Use Joint Space vector, J, contains the Segment angles and displacement information which is used to solve for the Transformation and Rotation matrix, T.sub.0T and R.sub.0T respectively. The Transformation and Rotation Matrix will be discussed below.

[0085] Current Catheter State or Pose

Q.SUB.c.:

[0086] The current world coordinate vector Q.sub.c is defined by the tip q vector with no displacement which is q.sub.c, and can be resolved as follows:

Q.sub.c=T.sub.0Tc(q.sub.c)=(X.sub.c,Y.sub.c,Z.sub.c,.sub.c,.sub.c)

The coordinates may be found by the following math,

(X.sub.c,Y.sub.c,Z.sub.c,1).sup.T=T.sub.0Tc.Math.(0,0,0,1).sup.T

a.sub.T=cos(0)*sin(0)=0

b.sub.T=sin(0)*sin(0)=0

c.sub.T=cos(0)=1

(A.sub.c,B.sub.c,C.sub.c,0).sup.T=T.sub.0Tc.Math.(0,0,1,0).sup.T

.sub.c=atan2(A.sub.c,B.sub.c)

.sub.c=atan2(C.sub.c,H.sub.c)

H.sub.c=(A.sub.c.sup.2B.sub.c.sup.2).sup.1/2

The range of Beta () may be limited if the rotation matrix is not used. This is due to use of the hypotenuse (H) quantity which removes the negative sign from one side of the atan2 formula as follows:

H=(A.sup.2+B.sup.2).sup.1/2

=atan2(C,H),0<<180

Q.SUB.d.:

[0087] The desired world coordinate vector Q.sub.d is defined by the tip q vector with desired displacement which is q.sub.d, and can be resolved as follows:

Q.sub.d=T.sub.0Tc(q.sub.d)=(X.sub.d,Y.sub.d,Z.sub.d,.sub.d,.sub.d)

The coordinates may be found by the following math,

(X.sub.d,Y.sub.d,Z.sub.d,1).sup.T=T.sub.0T.Math.(x.sub.T,y.sub.T,z.sub.T,1).sup.T

a.sub.T=cos(α.sub.T)*sin(β.sub.T); b.sub.T=sin(α.sub.T)*sin(β.sub.T); C.sub.T=COS(β.sub.T)

(A.sub.d,B.sub.d,C.sub.d,0).sup.T=T.sub.0T.Math.(a.sub.T,b.sub.T,c.sub.T,0).sup.T

=atan2(A.sub.d,B.sub.d)

.sub.d=atan2(C.sub.d,H.sub.d)

H.sub.d=(A.sub.d.sup.2+B.sub.d.sup.2).sup.1/2

The following alternative formula solves Beta (B) in all four quadrants for a full 360 degrees (as opposed to only two quadrans and 180 degrees) and for Gamma (Γ), the sixth and final coordinate to define the position in 3D space.

Q.SUB.c.:

[0088] Use J.sub.c (current Joint Space variables) is used to solve for the current T.sub.0Tc and R.sub.0Tc

(X.sub.c,Y.sub.c,Z.sub.c,1).sup.T=T.sub.0Tc.Math.(0,0,0,1).sup.T

(a.sub.Tx,b.sub.Tx,c.sub.Tx).sup.T=R.sub.0Tc×(1,0,0).sup.T

(a.sub.Ty,b.sub.Ty,c.sub.Ty).sup.T=R.sub.0Tc×(0,1,0).sup.T

(a.sub.Tz,b.sub.Tz,C.sub.Tz).sup.T=R.sub.0Tc×(0,0,1).sup.T

.sub.c=atan2(a.sub.Tz, b.sub.Tz)

if |a.sub.Tz|<min; than a.sub.Tz=(a.sub.Tz/|a.sub.Tz|)*min

.sub.c=atan2(c.sub.αz,a.sub.αz)

a.sub.αz=cos α*a.sub.Tz+sin α*b.sub.Tz

C.sub.αz=C.sub.Tz

if |c.sub.αz|<min; than c.sub.αz=(c.sub.αz/|c.sub.αz|)*min

Γ.sub.c=atan2(a.sub.βx,b.sub.βx)

a.sub.βx=cos α*cos β*a.sub.Tx+sin α*cos β*b.sub.Tx−sin β*c.sub.Tx

b.sub.βx=−sin α*a.sub.Tx+cos α*b.sub.Tx

if |a.sub.βx|<min; than a.sub.βx=(a.sub.βx/|a.sub.βx|)*min

Q.SUB.d.:

[0089]
(X.sub.d,Y.sub.d,Z.sub.d,1).sup.T=T.sub.0Tc.Math.(x.sub.d,y.sub.d,z.sub.d,1).sup.T

Use Q.sub.d with Inverse Jacobian to solve for Segment angles and displacements, J.sub.d (desired Joint Space variables).
Use J.sub.d to solve for the new T.sub.0Td and R.sub.0Td

(X.sub.d,Y.sub.d,Z.sub.d,1).sup.T=T.sub.0Td.Math.(0,0,0,1).sup.T

(a.sub.Tx,b.sub.Tx,c.sub.Tx).sup.T=R.sub.0Td×(1,0,0).sup.T

(a.sub.Ty,b.sub.Ty,c.sub.Ty).sup.T=R.sub.0Td×(0,1,0).sup.T

(a.sub.Tz,b.sub.Tz,c.sub.Tx=R.sub.0Td×(0,0,1).sup.T

.sub.d=atan2(a.sub.Tz,b.sub.Tz)

if |a.sub.Tz|<min; than a.sub.Tz=(a.sub.Tz/|a.sub.Tz|)*min

.sub.d=atan2(c.sub.αz,a.sub.αz)

a.sub.αz=cos α*a.sub.Tz+sin α*b.sub.Tz

c.sub.αz=c.sub.Tz

if |c.sub.αz|<min; than c.sub.αz=(c.sub.αz/|c.sub.αz|)*min

Γ.sub.d=atan2(a.sub.βx,b.sub.βx)

a.sub.βx=cos α*cos β*a.sub.Tx+sin α*cos β*b.sub.Tx−sin β*c.sub.Tx

b.sub.βx=−sin α*a.sub.Tx+cos α*b.sub.Tx

if |a.sub.βx|<min; than a.sub.βx=(a.sub.βx/|a.sub.βx|*min

[0090] Basis for Alternative Formulas:

[0091] Solve for base coordinate axis unit vectors

(a.sub.Tx,b.sub.Tx,c.sub.Tx).sup.T=R.sub.0T×(1,0,0).sup.T

(a.sub.Ty,b.sub.Ty,c.sub.Ty).sup.T=R.sub.0T×(0,1,0).sup.T

(a.sub.Tz,b.sub.Tz,c.sub.Tx=R.sub.0T×(0,0,1)T

[0092] Referring now to FIG. 9, the Robot angles are resolved as follows:

α.sub.T=atan2(a.sub.Tz,b.sub.Tz)

The rotation matrix for alpha (α) is the following:

[00001]$R_{\alpha }=\left(\begin{array}{ccc}\cos \alpha & -s\mathrm{in}\alpha & 0\\ \sin \alpha & \cos \alpha & 0\\ 0& 0& 1\end{array}\right)$

The inverse of this rotation matrix is the transpose.

[00002]$R_{\alpha }^{-1}=R_{\alpha }^T=\left(\begin{array}{ccc}\cos \alpha & \sin \alpha & 0\\ -s\mathrm{in}\alpha & \cos \alpha & 0\\ 0& 0& 1\end{array}\right)$${\left(a_{\alpha z},b_{\alpha z},c_{\alpha z}\right)}^T=R_{\alpha }^T\times {\left(a_{Tz},b_{Tz},c_{Tz}\right)}^T$

Applying the rotation R.sub.α.sup.T removes the b.sub.α component and aligns the beta (β) angle within X′-Z′ plane. This allows full circumferential angle determination of beta (β).

β.sub.T=atan2(c.sub.αz,a.sub.αz)

a.sub.αz=cos α*a.sub.Tz+sin α*b.sub.Tz

b.sub.αz=−sin α*a.sub.Tz+cos α*b.sub.Tz=0

c.sub.αz=c.sub.Tz

To find gamma (γ) use alpha (α) and beta (β) to create a rotation matrix as follows:

[00003]$R_{\alpha \beta }=\left(\begin{array}{ccc}\cos \alpha & -s\mathrm{in}\alpha & 0\\ \sin \alpha & \cos \alpha & 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}\cos \beta & 0& \sin \beta \\ 0& 1& 0\\ -s\mathrm{in}\beta & 0& \cos \beta \end{array}\right)$$R_{\alpha \beta }=\left(\begin{array}{ccc}c\alpha *c\beta & -s\alpha & c\alpha *s\beta \\ s\alpha *c\beta & c\alpha & s\alpha *s\beta \\ -s\beta & 0& c\beta \end{array}\right)$

Remove the alpha (α) and beta (β) from the Y axis vector to solve for gamma (γ). This can be done by inverting the rotation matrix and multiply by Y axis unit vector. The inverse of this rotation matrix is the transpose.

[00004]$R_{\mathrm{\alpha \beta }}^{-1}=R_{\mathrm{\alpha \beta }}^T=\left(\begin{array}{ccc}c\alpha *c\beta & s\alpha *c\beta & -s\beta \\ -s\alpha & c\alpha & 0\\ c\alpha *s\beta & s\alpha *s\beta & c\beta \end{array}\right)$${\left(a_{\beta x},b_{\beta x},c_{\beta x}\right)}^T=R_{zy}^T\times {\left(a_{Tx},b_{Tx},c_{Tx}\right)}^T$

This new Roll vector should have a zero in the c.sub.Tr positions placing the vector on a Tip coordinate X-Y plane with values for a.sub.Tr and b.sub.Tr. Use these two values to determine gamma (γ) as follows:

γ.sub.T=atan2(a.sub.βx,b.sub.βx)

a.sub.βx=cos α*cos β*a.sub.Tx+sin α*cosβ*b.sub.Txx−sin β*c.sub.Tx

b.sub.βx=−sin α*a.sub.Tx+cos α*b.sub.Tx

c.sub.βx=cos α*sin β*a.sub.Tx+sin α*sin β*b.sub.Tx+cos β*c.sub.Tx=0

[0093] Limiting Input Commands to Facilitate Solution:

[0094] As discussed in the previous section, this user input vector q is used to find the desired world space vector Q.sub.d using the Transformation Matrix T.sub.0T. The Q.sub.d vector finds the desired coordinate values for X.sub.d, Y.sub.d, Z.sub.d, .sub.d, and .sub.d.

[0095] Due to coordinate frame limitations, beta () should be greater than zero and less than 180 degrees. As beta approaches these limits, the Inverse Kinematics solver may become unstable. This instability is remediated by assigning a maximum and minimum value for beta that is higher than zero and lower than 180 degrees. How close to the limits depends on multiple variables and it is best to validate stability with assigned limits. For example, a suitable beta minimum may be 0.01 degrees and maximum 178 degrees.

[0096] The optimization scheme used to solve for the joint vector j (through the Inverse Kinematics) may become unstable with large changes in position and angles. While large displacements and orientation changes will often resolve, there are times when it may not. Limiting the position and angles change will help maintain mathematical stability. For large q changes, dividing the path into multiple steps may be helpful, optionally with a Trajectory planner. For reference, the maximum displacement per command may be set for 3.464 mm and maximum angle set for 2 degrees. The displacement is defined by the following:

Displacement=(x.sub.T.sup.2+y.sub.T.sup.2+z.sub.T.sup.2).sup.1/2<2 mm
Angle=β.sub.T<2 degrees

[0097] Segment Rotational Matrix:

[0098] Referring now to FIG. 10A, solve rotation matrix

for an arc of a cord (no axial twist).

[00005]$R=\mathrm{Joint}\mathrm{Rotational}\mathrm{matrix}.$$R=R_z\times R_y\times R_z^T$$R_z=\left(\begin{array}{ccc}\cos \left(\alpha \right)& -\sin \left(\alpha \right)& 0\\ \sin \left(\alpha \right)& \cos \left(\alpha \right)& 0\\ 0& 0& 1\end{array}\right)\mathrm{about}Z\mathrm{axis}$$R_y=\left(\begin{array}{ccc}\cos \left(\beta \right)& 0& \sin \left(\beta \right)\\ 0& 1& 0\\ -\sin \left(\beta \right)& 0& \cos \left(\beta \right)\end{array}\right)\mathrm{about}Y\mathrm{axis}$$R_{zy}=R_z\times R_y=\left(\begin{array}{ccc}\cos \left(\alpha \right)& -\sin \left(\alpha \right)& 0\\ \sin \left(\alpha \right)& \cos \left(\alpha \right)& 0\\ 0& 0& 1\end{array}\right)\times \left(\begin{array}{ccc}\cos \left(\beta \right)& 0& \sin \left(\beta \right)\\ 0& 1& 0\\ -\sin \left(\beta \right)& 0& \cos \left(\beta \right)\end{array}\right)$$R=R_{zy}\times R_z^T=\left(\begin{array}{ccc}c\alpha *c\beta & -s\alpha & c\alpha *s\beta \\ s\alpha *c\beta & c\alpha & s\alpha *s\beta \\ -s\beta & 0& c\beta \end{array}\right)\times \left(\begin{array}{ccc}c\alpha & s\alpha & 0\\ -s\alpha & c\alpha & 0\\ 0& 0& 1\end{array}\right)$$R=\left(\begin{array}{ccc}c{\alpha }^2*c\beta +s{\alpha }^2& s\alpha *c\alpha *\left(c\beta -1\right)& c\alpha *s\beta \\ s\alpha *c\alpha *\left(c\beta -1\right)& s{\alpha }^2*c\beta +c{\alpha }^2& s\alpha *s\beta \\ -s\beta *c\alpha & -s\beta *s\alpha & c\beta \end{array}\right)$

[0099] Segment Position Matrix:

[0100] Referring now to FIGS. 10B and 10C, solve position matrix for the cord

[00006]$P=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$$P=\mathrm{point}\mathrm{in}\mathrm{space}\mathrm{relative}\mathrm{to}\mathrm{reference}\mathrm{frame}$$r=S/\beta$$x=r*c\alpha *\left[1-c\beta \right]$$x=\left(S/\beta \right)*c\alpha *\left[1-c\beta \right]$$y=\left(S/\beta \right)*c\alpha *\left[1-c\beta \right]$$z=\left(S/\beta \right)*s\beta$$P=\left(\begin{array}{c}\left(S/\beta \right)*c\alpha *\left[1-c\beta \right]\\ \left(S/\beta \right)*s\alpha *\left[1-c\beta \right]\\ \left(S/\beta \right)*s\beta \end{array}\right)$

[0101] Segment Transformation Matrix:

[0102] Combine rotation and position matrix into a transformation matrix.

[00007]$T=\left(\begin{array}{cc}R& P\\ 0& 1\end{array}\right)$$T=\left(\begin{array}{cccc}c{\alpha }^2*c\beta +s{\alpha }^2& s\alpha *c\alpha *\left(1-c\beta \right)& c\alpha *s\beta & \left(S/\beta \right)*c\alpha *\left[1-c\beta \right]\\ s\alpha *c\alpha *\left(1-c\beta \right)& s{\alpha }^2*c\beta +c{\alpha }^2& s\alpha *s\beta & \left(S/\beta \right)*s\alpha *\left[1-c\beta \right]\\ -c\alpha *s\beta & -s\alpha *s\beta & c\beta & \left(S/\beta \right)*s\beta \\ 0& 0& 0& 1\end{array}\right)$

Compound Lateral Stiffness Transformation Matrix:

[0103] For segments with an instruments tip passing through, or having instruments within with changing lateral stiffness, or when the segment is transitioning out the distal end of a guide sheath, the Transformation Matrix requires partial Transformation matrix within the length of the Segment. In the drawing to the right these partial Transformational Matrices are identified within the length of the Segment with prime marks (as seen in the drawing to the right) illustrated here is a segment with three distinct sections, S′, S″ and S′″ which are all sub sections of one segment S. Note that the sum of the lengths of S primes equals the total length of the segment.

S=S′+S″+S′″

[0104] To resolve the full segment Transformation Matrix, partial transformation matrix is required for each subsection (with different lateral stiffness) of the segment. The subsection Transformation Matrix are mathematically similar to the homogenous (lateral stiffness) Segment Transformation above, but with partial values as indicated below such that T′ can be resolved as follows:

[00008]$\left(\begin{array}{cccc}c{\alpha }^{\mathrm{\prime 2}}*c{\beta }^{\prime }+s{\alpha }^{\mathrm{\prime 2}}& s{\alpha }^{\prime }*c{\alpha }^{\prime }*\left(1-c{\beta }^{\prime }\right)& c{\alpha }^{\prime }*s{\beta }^{\prime }& \left(S^{\prime }/{\beta }^{\prime }\right)*c{\alpha }^{\prime }*\left[1-c{\beta }^{\prime }\right]\\ s{\alpha }^{\prime }*c{\alpha }^{\prime }*\left(1-c{\beta }^{\prime }\right)& s{\alpha }^{\mathrm{\prime 2}}*c{\beta }^{\prime }+c{\alpha }^{\mathrm{\prime 2}}& s{\alpha }^{\prime }*s{\beta }^{\prime }& \left(S^{\prime }/{\beta }^{\prime }\right)*s{\alpha }^{\prime }*\left[1-c{\beta }^{\prime }\right]\\ -c{\alpha }^{\prime }*s{\beta }^{\prime }& -s{\alpha }^{\prime }*s{\beta }^{\prime }& c{\beta }^{\prime }& \left(S^{\prime }/{\beta }^{\prime }\right)*s{\beta }^{\prime }\\ 0& 0& 0& 1\end{array}\right)$

These Segment subsection Matrix are multiplied to resolve the full segment Transformation Matrix.

T=T′×T″×T′″× . . . T.sup.n′

[0105] Note that the first subsection Rotation Matrix orients the lateral bend direction and subsequent subsection Rotation Matrix bend on the same plan (though at a different rates). Therefore, the first partial segment Rotation Matrix sets the direction and the rest have an alpha with a value of zero.

α′=α; α″=α′″=α.sup.n′=0

[0106] Rotational Matrix Generalized:

[0107] R.sub.i indicates the rotation of a reference frame attached to the tip of segment “i” relative to a reference frame attached to its base (or the tip of Segment “i-1”).

R.sub.i=R(α.sub.i,β.sub.i,S.sub.i); i=0,1,2, . . . n;

i=0, sensor readings to input by manual (rotation & axial motion) actuation of the catheter proximal of the first segment.
i=1, is the most proximal segment.
i=n, the most distal segment (which is 2 for a two-segment catheter).

[0108] System Position Generalized:

[0109] P.sub.i indicates the origin of the distal end of segment “i” relative to a reference frame attached to its base (or the tip of Segment “i-1”).

[00009]$P=\left(\begin{array}{c}{x}_i\\ y_i\\ z_i\end{array}\right)$

[0110] Continuum Translation Matrix:

[0111] T.sub.i is the transformation matrix from a frame at the distal end of segment “i” to a frame attached to its base (or the tip of Segment “i-1”).

[00010]$T_i=\left(\begin{array}{cc}{R}_i& P_i\\ 0& 1\end{array}\right),\mathrm{for}i\mathrm{from}1\mathrm{to}n.$$T_i=\left(\begin{array}{cccc}c{\alpha }_i^2*c\beta i+s{\alpha }_i^2& s{\alpha }_i*c{\alpha }_i*\left(1-c{\beta }_i\right)& c{\alpha }_i*s{\beta }_i& \left(S_i/{\beta }_i\right)*c{\alpha }_i*\left[1-c{\beta }_i\right]\\ s{\alpha }_i*c{\alpha }_i*\left(1-c{\beta }_i\right)& s{\alpha }_i^2*c{\beta }_i+c{\alpha }_i^2& s{\alpha }_i*s{\beta }_i& \left(S_i/{\beta }_i\right)*s{\alpha }_i*\left[1-c{\beta }_i\right]\\ -c{\alpha }_i*s{\beta }_i& -s{\alpha }_i*s{\beta }_i& c{\beta }_i& \left(S_i/{\beta }_i\right)*s{\beta }_i\\ 0& 0& 0& 1\end{array}\right)$

T.sub.w is the transformation matrix from the most distal segment's tip reference frame to the world reference frame which is located proximal to the manually (versus fluidically) driven joints.

T.sub.w=T.sub.0×T.sub.1×T.sub.2× . . . T.sup.n

Use this matrix to solve the Forward Kinematics for current tip position P.sub.w and axial unit vector V.sub.wz.

P.sub.w=T.sub.w*(0,0,0,1)

P.sub.w=(x.sub.w,y.sub.w,z.sub.w)

V.sub.wz=T.sub.w*(0,0,1,0)

Vwz=(a.sub.wz,b.sub.wz,c.sub.wz)

[0112] Combine for World Space Tip Position:

[0113] Solve tip world space Q.sub.W, by combining P.sub.w and V.sub.wz as follows.

Q.sub.W=(X.sub.W,Y.sub.W,Z.sub.W,α.sub.w,β.sub.w)

X.sub.W=x.sub.w=x.sub.n

Y.sub.W=y.sub.w=y.sub.n

Z.sub.W=z.sub.w=z.sub.n

Solve for Segment Angles:

[0114]
α.sub.w=atan2(a.sub.wz,b.sub.wz)

if |a.sub.wz|<min; than a.sub.wz=(a.sub.wz/|a.sub.wz|)*min

Current β.SUB.w

[0115]
β.sub.w=atan2(c.sub.wz,ab.sub.wz)

ab.sub.wz=(a.sub.wz.sup.2+b.sub.wz.sup.2).sup.1/2

if |c.sub.wz|<min; than c.sub.wz=(c.sub.wz/|c.sub.wz|*min

New β.sub.w and γ.sub.w

β.sub.w=atan2(c.sub.wz,a.sub.αz)

a.sub.αz=cos α*a.sub.wz+sin α*b.sub.wz

if |c.sub.wz|<min; than c.sub.wz=(c.sub.wz/|c.sub.wz|)*min

γ.sub.w=atan2(aβx,b.sub.βx)

a.sub.βx=cos α*cos β*a.sub.wx+sin α*cos β*b.sub.wx−sin β*c.sub.wx

b.sub.βx=−sin α*a.sub.wx+cos α*b.sub.wx

if |a.sub.βx|<min; than a.sub.βx=(a.sub.βx/|a.sub.βx|)*min

Convert Q.sub.W to Q.sub.J for use with Jacobian

B.sub.Wx=β.sub.w*cos(α.sub.w)

B.sub.Wy=β.sub.w*sin(α.sub.w)

Q.sub.J=(X.sub.W,Y.sub.W,Z.sub.W,β.sub.Wx,β.sub.Wy)

[0116] Numerical Jacobian:

[0117] To solve the unique Q.sub.J for a deviation in joint variable (α.sub.i, β.sub.i, S.sub.i), one at a time, deviate each variable in every Segment by using the Transformation matrix. Then combine the resultant Q.sub.J vectors to form a numeric Jacobian. By using small single variable deviations from the current joint space, a localized Jacobean can be obtained. This Jacobean can be used in several ways to iteratively find a solution for segment joint variables to a desired world space position, Q.sub.d. Preferable the Jacobean is invertible in which case the difference vector between the current and desired world position can be multiplied by the inverse Jacobian J.sup.−1 to iteratively approach a correct joint space variables. Repeat this process until the Forward Kinematics calculates a position vector equivalent to Q.sub.d within a minimum error. Check Joint Space results (α.sub.i, β.sub.i, S.sub.i for i=0, 1, n) for accuracy, solvability, workspace limitations and errors.

J.sup.−1*J=I

α.sub.i>=−360° & α.sub.i<=360°

β.sub.i>β.sub.min & β.sub.i<β.sub.max

β.sub.i≈0° or 180° (or β.sub.max)

β.sub.w≈0° or 180°

S.sub.i>S.sub.min; S.sub.i<S.sub.max

[0118] A more stable Jacobian may optionally be achieved by altering α.sub.i, β.sub.i, S.sub.i, to be βx.sub.1, β.sub.yi, S.sub.i, where:

β.sub.xi=β.sub.i*cos(α.sub.i)

β.sub.yi=β.sub.i*sin(α.sub.i).

[0119] To convert back:

α.sub.i=atan2(β.sub.xi,β.sub.yi)

β.sub.i=(β.sub.xi.sup.2+β.sub.yi.sup.2).sup.1/2

[0120] Note that α.sub.i defines the direction that the segment is pointing in the world X-Y plane and when pointing in the direction of the Z axis, α.sub.i becomes undefined. Also, β.sub.i, being the middle angle in a gimble sequence of angles, may be limited to a range of 180° and always positive, which works well for the Pi covert back equation.

[0121] β.sub.xi and β.sub.yi values are always definable and using radians range between positive and negative Pi. Unlike α.sub.i and Pi, βx.sub.i and βy.sub.i both equal zero and us defined when the segment points in the Z direction. In addition, α.sub.i has a numerical step when transitioning from 359 to zero degrees which is more challenging to manage that the smooth transitions of βx.sub.i and βy.sub.i.

Target and Achievable:

[0122] With more input variables than target variables, the pseudo Jacobean will solve for a solution through the least squares fit. There may be more than one solution and any given solution target may be achieved while being beyond the input variable range of motion. Using several techniques workspace can be expanded and often a target may be achieved within the limits of the input variables range. There are at least three techniques that can be used within the interactive equations which are herein referred to as Scaling, Shifting, and Skewing, and these or other techniques can be used alone or in any combinations of two, three, or more.

[0123] The input for the robot can be used to change the pressure. The pressure range may have a minimum and a maximum, optionally 0 to 500 PSI or more. When a numerical solution produces a set of pressures that have some lumens outside the range, the solution may not be achievable.

[0124] Changing equal amounts of pressure within each of the three lumens of a Segment may cause a displacement change, while retaining orientation. Therefore, if 10 PSI is removed from Pr.sub.1A, Pr.sub.1B, and Pr.sub.1C, the orientation should remain the same and only the position will change. Shifting changes the pressure within a Segment by equal amounts in order to bring all three pressures within range. The consequence is a translation along an axis between the Segments origin and the Segments tip.

[0125] When a Segments pressure range between the lumens is larger than the available pressure range of the source, a process of Scaling occurs to proportionally reduce the three-lumen pressure so they will fit, after Shifting, within the minimum and maximum pressures available.

[0126] With a segment lumen pressures may be scaled and/or shifted; the Jacobean tends to achieve a solution that meets the Target Q variables but may also remain slightly shifted outside the pressures range. Skewing nudges the Segment towards being within the range available pressure while maintaining an accurate target output. A small multiple bias the Segment towards either higher or lower pressure. This skewing multiple (ζ) attempts to shift a segment towards being inside the pressure boundary while moving neighboring segments to compensate. The result is a tendency for the segments to gravitate towards an achievable solution. The multiple is only applied when the three-lumen pressures bandwidth is within the available pressure range, and when one segment is outside the pressure range, or two segments are out in complementary directions. Complementary directions in a two Segment system would have one Segment below minimum pressure and the other Segment above maximum pressure. The skewing multiple (ζ) may be applied to modify pressure, or to the segment tri-lengths. Both cases were evaluated and applying the multiple to the tri-lengths (post Shifting and Scaling) appears to optimize a bit quicker. The multiple ζ works well with the Segment tri-lengths (S) at about 2% (1.02 and 0.098), and with the pressure (Pr) at 9% (1.09 and 0.091).

TABLE-US-00002 S.sub.1A = S.sub.1A * ζ S.sub.2A = S.sub.2A/ζ S.sub.1B = S.sub.1B * ζ S.sub.2B = S.sub.2B/ζ S.sub.1C = S.sub.1C * ζ S.sub.2C = S.sub.2C/ζ Skew = .98 Test1 = IF( SEG.sub.1Pr.sub.MAX & SEG.sub.2PR.sub.MAX > Pr.sub.MAX, 1,    IF( SEG.sub.1Pr.sub.MIN & SEG.sub.2PR.sub.MIN < Pr.sub.MIN, −1, 0 )    ) Test2 = IF( Test1 = 0,    IF( SEG.sub.1Pr.sub.MAX OR SEG.sub.2PR.sub.MAX > Pr.sub.MAX, 1,    IF( SEG.sub.1Pr.sub.MIN OR SEG.sub.2PR.sub.MIN < Pr.sub.MIN, −1, 0 )    ), 0) ζ = IF(Test2 = 1, Skew, IF(Test2 = −1, 1/Skew, 1))

[0127] Referring now to FIG. 11, a lumen pressure flow chart 120 is shown that can be used by a processor of the robotic system to determine pressures associated with the desired movement or position of an actual or virtual catheter.

[0128] Solving for Segment Balloon Array End Coordinates:

[0129] A balloon array is a set of fluidically connected balloons along one side of a segment. Find end coordinates for each balloon array within a segment base frame. Assume balloon arrays are spaced at 120 degrees apart around the cordial axis, that the first is located on the X axis, and that the array balloons remain axially aligned through the length of the segment.

r.sub.A=radius of balloon center in balloon array from cordial axis
θ=angular period of balloons within segment (120° for a 3 array segment)
θ.sub.A=0; θ.sub.B=120; θ.sub.C=240
Arc Start points for balloon arrays:

[00011]$P_{0A}=\left(\begin{array}{c}{r}_A*\cos \left({\theta }_A\right)\\ r_A*\sin \left({\theta }_A\right)\\ 0\\ 1\end{array}\right),$$P_{0B}=\left(\begin{array}{c}{r}_A*\cos \left({\theta }_B\right)\\ r_A*\sin \left({\theta }_B\right)\\ 0\\ 1\end{array}\right),$$P_{0C}=\left(\begin{array}{c}{r}_A*\cos \left({\theta }_C\right)\\ r_A*\sin \left({\theta }_C\right)\\ 0\\ 1\end{array}\right)$

Arc unit vectors for axial orientation of balloon arrays are as follows:

[00012]$V_{0A}=\left(\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right),V_{0B}=\left(\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right),V_{0C}=\left(\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right),$

Use local (for one Segment) Transformation Matrix to find Balloon Array distal end coordinates.

[00013]$T=\left(\begin{array}{cccc}c{\alpha }^2*c\beta +s{\alpha }^2& s\alpha *c\alpha *\left(1-c\beta \right)& c\alpha *s\beta & \left(S/\beta \right)*c\alpha *\left[1-c\beta \right]\\ s\alpha *c\alpha *\left(1-c\beta \right)& s{\alpha }^2*c\beta +c{\alpha }^2& s\alpha *s\beta & \left(S/\beta \right)*s\alpha *\left[1-c\beta \right]\\ -c\alpha *s\beta & -s\alpha *s\beta & c\beta & \left(S/\beta \right)*s\beta \\ 0& 0& 0& 1\end{array}\right)$

For segments with multiple lateral stiffnesses: T=T′×T″×T′″× . . . T.sup.n′

P.sub.i1=T×P.sub.0

P.sub.1A=T×P.sub.0A

P.sub.1B=T×P.sub.0B

P.sub.1C=T×P.sub.0C

For each Balloon Array set the origin at the starting points to normalize distal endpoint coordinates dP. This is helpful for solving for the Balloon Array arc, S, which follows in the Find Array Arc Lengths section below.

dP.sub.i=P.sub.i1−P.sub.i0=(x.sub.0,y.sub.0,z.sub.0)Segment Centerline Cord

dP.sub.A=P.sub.A1−P.sub.A0=(x.sub.A,y.sub.A,z.sub.A) Balloon Cord A

dP.sub.B=P.sub.B1−P.sub.B0=(x.sub.B,y.sub.B,z.sub.B) Balloon Cord B

dP.sub.C=P.sub.C1−P.sub.C0=(x.sub.C,y.sub.C,z.sub.C) Balloon Cord C

All cordial orientation vectors are equivalent.

V=(a.sub.iz,b.sub.iz,c.sub.iz), for i from 1 to n.

[0130] Find Array Arc Lengths:

[0131] Referring again to FIGS. 10B and 10C, solve for general Balloon Array cord lengths and radii as follows.

(α, β, S), α represent the bend direction (about z axis starting at x axis), β the bend amount (off the z axis), and S the length of an arc anchored to the origin of a reference frame.
(x, y, z) is the coordinate location of a point at the end of the arc.
r is the balloon array cord radius.

[00014]$h={\left(x^2+y^2\right)}^{1/2}$$r=\left(h^2+z^2\right)/\left(2*h\right)$$r-h=\left(z^2-h^2\right)/\left(2*h\right)$$\alpha =\mathrm{atan}2\left(x,y\right)$$\beta =\mathrm{atan}2\left(z,r-h\right)$$S=r*\beta$$\left(\begin{array}{c}\alpha \\ \beta \\ S\end{array}\right)=\left(\begin{array}{c}\begin{array}{c}\mathrm{atan}2\left(x,y\right)\\ \mathrm{atan}2\left(r-h,z\right)\end{array}\\ r*\mathrm{atan}2\left(r-h,z\right)\end{array}\right)$$S_i=\left(h_i^2+z_i^2\right)/\left(2*h_i\right)*\beta ,h_i={\left(x_i^2+y_i^2\right)}^{1/2}$

i (segment center cord)
Solve S for cords A, B, C, (segment center cord)
Note that segment center cord (S, β, α) is already determined.

When β>β.SUB.min

[0132]
S.sub.a=(h.sub.A.sup.2+z.sub.A.sup.2)/(2*h.sub.A)*β, h.sub.A=(x.sub.A.sup.2+y.sub.A.sup.2).sup.1/2

S.sub.B=(h.sub.B.sup.2+z.sub.B.sup.2)/(2*h.sub.B)*β, h.sub.B=(x.sub.B.sup.2+y.sub.B.sup.2).sup.1/2

S.sub.C=(h.sub.C.sup.2+z.sub.C.sup.2)/(.sup.2*h.sub.C)*β, h.sub.C=(x.sub.C.sup.2+y.sub.C.sup.2).sup.1/2

When β=<β.sub.min

S.sub.A=S.sub.B=S.sub.C=S.sub.i

[0133] For segments with multiple lateral stiffness' where T=T′×T″×T′″× . . . T.sup.n′, The process above is resolved for each segment partial section and the sub-Segment lengths (S′, S″, . . . S′.sup.n) are summed for each cord length (A, B, and C). Similar math is used for the subsection lengths. [0134] When β′>β.sub.min (may not be needed)

S′.sub.A=(h′.sub.A.sup.2+z′.sub.A.sup.2)/(2*h′.sub.A)*β′, h′.sub.A=(x′.sub.A.sup.2+y′.sub.A.sup.2).sup.1/2

S′.sub.B=(h′.sub.B.sup.2+z′.sub.B.sup.2)/(2*h′.sub.B)*β′, h′.sub.B=(x′.sub.B.sup.2+y′.sub.B.sup.2).sup.1/2

S′.sub.C=(h′.sub.C.sup.2+z′.sub.C.sup.2)/(2*h′.sub.C)*β′, h′.sub.C=(x′.sub.C.sup.2+y′.sub.C.sup.2).sup.1/2 [0135] When β′=<β.sub.min (may not be needed)

S′.sub.A=S′.sub.B=S′.sub.C=S′.sub.i

[0136] Repeat for each subsegment section and sum partial lengths

S.sub.A=S.sub.A′+S.sub.A″+ . . . S.sub.A.sup.n′

S.sub.B=S.sub.B′S.sub.B″+ . . . S.sub.B.sup.n′

S.sub.C=S.sub.C′+S.sub.C″+ . . . S.sub.C.sup.n′

[0137] Segment Internal Load Conditions:

[0138] Segment spring force may be proportional to a spring rate with extension.

F.sub.S=K.sub.F*S.sub.i+F.sub.0

Where F is the sum of the balloon forces, K.sub.F is the spring constant, and F.sub.0 is the offset force.

F.sub.0=F.sub.preload−K.sub.F*S.sub.min

Where F.sub.preload is the preload force at the minimum segment length S.sub.min.
Sum balloon array forces as follows:

F.sub.Pr=F.sub.A+F.sub.B+F.sub.C=A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)

F.sub.S=F.sub.Pr

S.sub.i=(A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0)/K.sub.F

Segment spring torque proportional to a spring rate with bend angle.

M.sub.S=K.sub.M*β+M.sub.0

β=(M.sub.S−M.sub.0)/K.sub.M

Where M is the internal moment applied to the Segment, K.sub.M is a angular spring constant can, and M.sub.0 is the preload moment (compensation for no load bend).
Sum of balloon array torques:

[00015]$r_A=\mathrm{radius}\mathrm{of}\mathrm{balloon}\mathrm{center}\mathrm{in}\mathrm{balloon}\mathrm{array}\mathrm{from}\mathrm{cordial}\mathrm{axis}$$\theta =\mathrm{angular}\mathrm{period}\mathrm{of}\mathrm{balloons}\mathrm{within}\mathrm{segment}\left(120°\mathrm{for}a3\mathrm{array}\mathrm{setment}\right)$${\theta }_A=\mathrm{\theta °};{\theta }_B=120°;{\theta }_C=240°$$\begin{array}{c}{M}_x=F_A*r_A*\sin \left({\theta }_A\right)+F_B*r_A*\sin \left({\theta }_B\right)+F_C*r_A*\sin \left({\theta }_C\right)\\ =\left(3^{1/2}/2\right)*A*r_A*\left({\mathrm{Pr}}_B-{\mathrm{Pr}}_C\right)\end{array}$$\begin{array}{c}{M}_y=-F_A*r_A*\cos \left({\theta }_A\right)-F_B*r_A*\cos \left({\theta }_B\right)-F_C*r_A*\cos \left({\theta }_C\right)\\ =\left(1/2\right)*A*r_A*\left({\mathrm{Pr}}_B+{\mathrm{Pr}}_C-2*{\mathrm{Pr}}_A\right)\end{array}$$\begin{array}{c}{M}_s={\left(M_x^2+M_y^2\right)}^{1/2}\\ =({\left(\left(3^{1/2}/2\right)*A*r_A*\left({\mathrm{Pr}}_B-{\mathrm{Pr}}_C\right)\right)}^2+(\left(1/2\right)*A*r_A*({\mathrm{Pr}}_B+{\mathrm{Pr}}_C-\\ {{2*{\mathrm{Pr}}_A))}^2)}^{1/2}\\ =\left(A/2\right)*r_A*{\left(3*{\left({\mathrm{Pr}}_B-{\mathrm{Pr}}_C\right)}^2+{\left({\mathrm{Pr}}_B+{\mathrm{Pr}}_C-2*{\mathrm{Pr}}_A\right)}^2\right)}^{1/2}\\ =\left(A/2\right)*r_A*(\left(3*{\mathrm{Pr}}_B^2-6*{\mathrm{Pr}}_B*{\mathrm{Pr}}_C+3*{\mathrm{Pr}}_C^2\right)+\\ ({\mathrm{Pr}}_B^2+{\mathrm{Pr}}_B*{\mathrm{Pr}}_C-2*{\mathrm{Pr}}_A*{\mathrm{Pr}}_B+{\mathrm{Pr}}_B*{\mathrm{Pr}}_C+{\mathrm{Pr}}_C^2-\\ {2*{\mathrm{Pr}}_A*{\mathrm{Pr}}_C-2*{\mathrm{Pr}}_A*{\mathrm{Pr}}_B-2*{\mathrm{Pr}}_A*{\mathrm{Pr}}_C+4*{\mathrm{Pr}}_A^2))}^{1/2}\\ =\left(A/2\right)*r_A*(4*{\mathrm{Pr}}_B^2-4*{\mathrm{Pr}}_B*{\mathrm{Pr}}_C+4*{\mathrm{Pr}}_C^2-4*{\mathrm{Pr}}_A*{\mathrm{Pr}}_C+\\ {4*{\mathrm{Pr}}_A^2)}^{1/2}\\ =A*r_A*{\left({\mathrm{Pr}}_B^2-{\mathrm{Pr}}_B*{\mathrm{Pr}}_C+{\mathrm{Pr}}_C^2-{\mathrm{Pr}}_A*{\mathrm{Pr}}_B-{\mathrm{Pr}}_A*{\mathrm{Pr}}_C+{\mathrm{Pr}}_A^2\right)}^{1/2}\\ =A*r_A*({\mathrm{Pr}}_A^2+{\mathrm{Pr}}_B^2+{\mathrm{Pr}}_C^2-{\mathrm{Pr}}_A*{\mathrm{Pr}}_B-{\mathrm{Pr}}_B*{\mathrm{Pr}}_C-{\mathrm{Pr}}_C-\\ {{\mathrm{Pr}}_C*{\mathrm{Pr}}_A)}^{1/2}\end{array}$$M_{\mathrm{Pr}}=M_s$$\beta =\left[A*r_A*{\left({\mathrm{Pr}}_A^2+{\mathrm{Pr}}_B^2+{\mathrm{Pr}}_C^2-{\mathrm{Pr}}_A*{\mathrm{Pr}}_B-{\mathrm{Pr}}_B*{\mathrm{Pr}}_C-{\mathrm{Pr}}_C*{\mathrm{Pr}}_A\right)}^{1/2}-M_0\right]/K_M$

Moment direction angle (α):

[00016]$M_x=\left(3^{1/2}/2\right)*A*r_A*\left({\mathrm{Pr}}_B-{\mathrm{Pr}}_C\right)$$M_y=\left(1/2\right)*A*r_A*\left({\mathrm{Pr}}_B+{\mathrm{Pr}}_C-2*{\mathrm{Pr}}_A\right)$$\cos \left(\alpha \right)=M_y/M_S$$\sin \left(\alpha \right)=-M_x/M_S$$\beta =\left(M_S-M_0\right)/K_M$$\begin{array}{c}{\beta }_x=\left[\left(M_S-M_0\right)/K_M\right]*\cos \left(\alpha \right)\\ =\left[\left(M_S-M_0\right)/K_M\right]*M_y/M_S\\ =\left[1-\left(M_0/M_S\right)\right]*M_y/K_M\\ =[1-(M_0/A*r_A*({\mathrm{Pr}}_A^2+{\mathrm{Pr}}_B^2+{\mathrm{Pr}}_C^2-{\mathrm{Pr}}_A*{\mathrm{Pr}}_B-{\mathrm{Pr}}_B*{\mathrm{Pr}}_C-\\ {{\mathrm{Pr}}_C*{\mathrm{Pr}}_A)}^{1/2})]*\left(1/2\right)*A*r_A*\left({\mathrm{Pr}}_B+{\mathrm{Pr}}_C-2*{\mathrm{Pr}}_A\right)/K_M\end{array}$$\begin{array}{c}{\beta }_y=-\left[\left(M_S-M_0\right)/K_M\right]*\sin \left(\alpha \right)\\ =-\left[\left(M_S-M_0\right)/K_M\right]*M_x/M_S\\ =-\left[1-\left(M_0/M_S\right)\right]*M_x/K_M\\ =-[1-(M_0/A*r_A*({\mathrm{Pr}}_A^2+{\mathrm{Pr}}_B^2+{\mathrm{Pr}}_C^2-{\mathrm{Pr}}_A*{\mathrm{Pr}}_B-{\mathrm{Pr}}_B*{\mathrm{Pr}}_C-\\ {{\mathrm{Pr}}_C-{\mathrm{Pr}}_C*{\mathrm{Pr}}_A)}^{1/2}))]*\left(3^{1/2}/2\right)*A*r_A*\left({\mathrm{Pr}}_B-{\mathrm{Pr}}_C\right)/K_M\end{array}$

With M.SUB.0.=0,

[0139]
β.sub.x=(1/2)*A*r.sub.A*(Pr.sub.B+Pr.sub.C−2*Pr.sub.A)/K.sub.M

β.sub.y=−(3.sup.1/2/2)*A*r.sub.A*(Pr.sub.B−Pr.sub.C)/K.sub.M

α=atan2(β.sub.x, β.sub.y); Note the minus sign for Mx is applied for matching direction.

α=atan2((Pr.sub.B+Pr.sub.C−2*Pr.sub.A),−1.73205*(Pr.sub.B−Pr.sub.C))

if(|Pr.sub.B−Pr.sub.C|<min)&(|Pr.sub.B+Pr.sub.C−2*Pr.sub.A|<min); than α=0

Solve for P.sub.A, P.sub.B, and P.sub.C using the following three equations and a Jacobian. Pressures should meet these conditions to solve.

|Pr.sub.A−Pr.sub.B|+|Pr.sub.B−Pr.sub.C|+|Pr.sub.C−Pr.sub.A|<MIN.sub.DIfference; related to |β.sub.Calc|>β.sub.min

Pr.sub.A,Pr.sub.B,Pr.sub.C>Min.sub.Pressure

Setting up Input α.SUB.Desired .for Jacobian

[0140]
IF(α.sub.Desired>180°,α.sub.Desired−360°, IF(α.sub.Desired<−180°,α.sub.Desired+360°,α.sub.Desired))

Segment or joint position may be found with lumen pressures as follows:

S.sub.Calc=[A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0]/K.sub.F

β.sub.x=(1/2)*A*r.sub.A*(Pr.sub.B+Pr.sub.C−2*Pr.sub.A)/K.sub.M

β.sub.y−(3.sup.1/2/2)*A*r.sub.A*(Pr.sub.B−Pr.sub.C)/K.sub.M

and:

β.sub.Calc=[A*r.sub.A*(Pr.sub.A.sup.2+Pn.sub.B.sup.2+Pr.sub.C.sup.2−Pr.sub.A*Pr.sub.B−Pr.sub.B*Pr.sub.C−Pr.sub.C*Pr.sub.A).sup.1/2]/K.sub.M

α.sub.Calc=atan2((Pr.sub.B+Pr.sub.C−2*Pr.sub.A),−3.sup.1/2*(Pr.sub.B−Pr.sub.C))

IF(|α.sub.Calc|>45° AND|α.sub.Calc|<315°,

Note that:

IFα.sub.Calc>0°,IF(α.sub.Desired<0°,α.sub.Desired+360°,α.sub.Desired),

IF(α.sub.Desired>0°,α.sub.Desired−360°,α.sub.Desired)),α.sub.Desired)

[0141] For segments with multiple lateral stiffness lengths, we can solve for a resultant bend constant by adding the contributions of the subsegments. To avoid dividing by a zero length solve for the inverse of the bend constant.

dK.sub.M′.sup.−1=(S′/S)*(1/K.sub.M′)

ΣK.sub.M.sup.−1=(1/S)*[(S′/K.sub.M′)+(S″/K.sub.M″)+ . . . (Sn′/K.sub.M.sup.n′)]

ΣK.sub.M=S/[(S′/K.sub.M′)+(S″/K.sub.M″)+ . . . (Sn′/K.sub.M.sup.n′)]

[0142] The Lumen pressures for segments with multiple lateral stiffnesses can be solved as follows:

S.sub.Calc=[A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0]/K.sub.F

β.sub.x=(1/2)*A*r.sub.A*(Pr.sub.C+Pr.sub.B−2*Pr.sub.A)/ΣK.sub.M

β.sub.y=(3.sup.1/2/2)*A*r.sub.A*(Pr.sub.C−Pr.sub.B)/ΣK.sub.M

[0143] Calculating Segment Multiple Section Spring Constants

[0144] To solve the K.sub.M.sup.n′ constants above we can start with the component bend constants. Referring to FIG. 11A, bend constants are assigned to various components of the Catheter Assembly. As shown the Guide and Catheter overlap and the Instrument Body and runs through the Segments, Catheter and Guide. The lateral bend spring constants (K.sub.M.sup.n′) is a combination of these component bend constants and a change in lateral stiffness is associated with a change of components along the length of a segment.

[0145] For instance, for Segment 1 K.sub.M′ with the Guide overlapping would have the following

K.sub.M′=1/[(1/K.sub.M.S1)+(1/K.sub.M.Guide)]

[0146] Add the Instrument Body through it and K.sub.M′ becomes:

K.sub.M′=1/[(1/K.sub.M.S1)+(1/K.sub.M.Guide)+(1/K.sub.M.Instr-B)]

[0147] If the Guide ends and Instrument Body continues mid-way along S1, K.sub.M″ is as follows:

K.sub.M″=1/[(1/K.sub.M.S1)+(1/K.sub.M.Instr-B)]

[0148] If more distal along S1 the Instrument Body becomes the Instrument Tip, K.sub.M′″ is added and can be found as follows:

K.sub.M′″=1/[(1/K.sub.M.S1)+(1/K.sub.M.Instr-T)]

[0149] If prior to the end of S1 the Instrument Tip ends a fourth constant is introduced as, KM″″ and how with nothing passing through this portion of the Segment becomes as follows:

K.sub.M′″=1/[(1/K.sub.M.S1)]=K.sub.M.S1

[0150] Note that each KM represents the spring constant for the whole length of a Segment. To calculate the relative bend for each length the constants are normalized with the length relative to the Segment S value as indicated above.

Find lumen pressures from joint variables:

[00017]${\beta }_y=-\left(3^{1/2}/2\right)*A*r_A*\left({\mathrm{Pr}}_B-{\mathrm{Pr}}_C\right)/K_M$${\mathrm{Pr}}_C={\mathrm{Pr}}_B+2*\left({\beta }_y/3^{1/2}\right)*K_M/\left(A*r_A\right)$${\beta }_x=\left(1/2\right)*A*r_A*\left({\mathrm{Pr}}_B+{\mathrm{Pr}}_C-2*{\mathrm{Pr}}_A\right)/K_M$$2*{\beta }_x*K_M/\left(A*r_A\right)={\mathrm{Pr}}_B+{\mathrm{Pr}}_B+2*\left({\beta }_y/3^{1/2}\right)*K_M/\left(A*r_A\right)-2*{\mathrm{Pr}}_A$$2*{\beta }_x*K_M/\left(A*r_A\right)+2*{\mathrm{Pr}}_A=2*{\mathrm{Pr}}_B+2*\left({\beta }_y/3^{1/2}\right)*K_M/\left(A*r_A\right)$${\mathrm{Pr}}_B={\mathrm{Pr}}_A+{\beta }_x*K_M/\left(A*r_A\right)-\left({\beta }_y/3^{1/2}\right)*K_M/\left(A*r_A\right)$${\mathrm{Pr}}_B={\mathrm{Pr}}_A+\left({\beta }_x-{\beta }_y/3^{1/2}\right)*K_M/\left(A*r_A\right)$$S_{\mathrm{Calc}}=\left[A*\left({\mathrm{Pr}}_A+{\mathrm{Pr}}_B+{\mathrm{Pr}}_C\right)-F_0\right]/K_F$$\left(S_{\mathrm{Calc}}*K_F+F_0\right)/A={\mathrm{Pr}}_A+{\mathrm{Pr}}_B+{\mathrm{Pr}}_B+2*\left({\beta }_y/3^{1/2}\right)*K_M/\left(A*r_A\right)$$\left(S_{\mathrm{Calc}}*K_F+F_0\right)/A={\mathrm{Pr}}_A+2*\left[{\mathrm{Pr}}_A+\left({\beta }_x-{\beta }_y/3^{1/2}\right)*K_M/\left(A*r_A\right)\right]+2*\left({\beta }_y/3^{1/2}\right)*K_M/\left(A*r_A\right)$$\left(S_{\mathrm{Calc}}*K_F+F_0\right)/A=3*{\mathrm{Pr}}_A+\left[2*\left({\beta }_x-{\beta }_y/3^{1/2}\right)+2*\left({\beta }_y/3^{1/2}\right)\right]*K_M/\left(A*r_A\right)$$\left(S_{\mathrm{Calc}}*K_F+F_0\right)/A=3*{\mathrm{Pr}}_A+2*{\beta }_x*K_M/\left(A*r_A\right)$${\mathrm{Pr}}_A=\left(S_{\mathrm{Calc}}*K_F+F_0\right)/\left(3*A\right)-2*{\beta }_x*K_M/\left(3*A*r_A\right)$$\begin{array}{c}{\mathrm{Pr}}_B={\mathrm{Pr}}_A+\left({\beta }_x-{\beta }_y/3^{1/2}\right)*K_M/\left(A*r_A\right)\\ =\left(S_{\mathrm{Calc}}*K_F+F_0\right)/\left(3*A\right)-2*{\beta }_x*K_M/\left(3*A*r_A\right)+\underset{\_}{3*}\\ \left({\beta }_x-{\beta }_y/3^{1/2}\right)*K_M/\left(\underset{\_}{3*}A*r_A\right)\end{array}$${\mathrm{Pr}}_B=\left(S_{\mathrm{Calc}}*K_F+F_0\right)/\left(3*A\right)+\left({\beta }_x-3*{\beta }_y/3^{1/2}\right)*K_M/\left(3*A*r_A\right)$${\mathrm{Pr}}_C={\mathrm{Pr}}_B+2*\left({\beta }_y/3^{1/2}\right)*K_M/\left(A*r_A\right)$${\mathrm{Pr}}_C=\left(S_{\mathrm{Calc}}*K_F+F_0\right)/\left(3*A\right)+\left({\beta }_x-3*{\beta }_y/3^{1/2}\right)*K_M/\left(3*A*r_A\right)+\underset{\_}{6*}\left({\beta }_y/3^{1/2}\right)*K_M/\left(\underset{\_}{3*}A*r_A\right)$${\mathrm{Pr}}_C=\left(S_{\mathrm{Calc}}*K_F+F_0\right)/\left(3*A\right)+\left({\beta }_x+3*{\beta }_y/3^{1/2}\right)*K_M/\left(3*A*r_A\right)$

[0151] Lumen Pressures from Joint Variables

Pr.sub.A=(S.sub.Calc*K.sub.F+F.sub.0)/(3*A)−2*β.sub.x*K.sub.M/(3*A*rA)

Pr.sub.B=(S.sub.Calc*K.sub.F+F.sub.0)/(3*A)+(β.sub.x−3*β.sub.y/3.sup.1/2)*K.sub.M/(3*A*r.sub.A)

Pr.sub.C=(S.sub.Calc*K.sub.F+F.sub.0)/(3*A)+(β.sub.x+3*β.sub.y/3.sup.1/2)*K.sub.M/(3*A*r.sub.A)

For segments with multiple lateral stiffness lengths, solve for a resultant bend constant by adding the contributions of the subsegments. The sub constants are scaled by the section's relative length to the whole segment. To avoid dividing by a zero length solve for the inverse of the bend constant.

dK.sub.M′.sup.−1=(S′/S)*(1/K.sub.M′)

ΣK.sub.M.sup.−1=(1/S)*[(S′/K.sub.M′)+(S″/K.sub.M″)+ . . . (Sn′/K.sub.M.sup.n′)]

ΣK.sub.M=S/[(S′/K.sub.M′)+(S″/K.sub.M″)+ . . . (Sn′/K.sub.M.sup.n′)]

[0152] The Lumen pressures for segments with multiple lateral stiffnesses can be solved as follows:

Pr.sub.A=(S.sub.Calc*K.sub.F+F.sub.0)/(3*A)−2*β.sub.x*ΣK.sub.M/(3*A*r.sub.A)

Pr.sub.B=(S.sub.Calc*K.sub.F+F.sub.0)/(3*A)+(β.sub.x−3*β.sub.y/3.sup.1/2)*ΣK.sub.M/(3*A*r.sub.A)

Pr.sub.C=(S.sub.Calc*K.sub.F+F.sub.0)/(3*A)+(β.sub.x+3*β.sub.y/3.sup.1/2)*ΣK.sub.M/(3*A*r.sub.A)

[0153] Find joint variables with Segment Lengths

S.sub.Calc=[A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0]/K.sub.FL.sub.A=3*A*Pr.sub.A/K.sub.F; Pr.sub.A=K.sub.F*L.sub.A/(3*A) L.sub.B=3*A*Pr.sub.B/K.sub.F; Pr.sub.B=K.sub.F*L.sub.B/(3*A) L.sub.C=3*A*Pr.sub.C/K.sub.F; Pr.sub.C=K.sub.F*L.sub.C/(3*A) L.sub.OFFSET=−F.sub.0/K.sub.F

S.sub.Calc=[L.sub.A+L.sub.B+L.sub.C)/3+L.sub.OFFSET

β.sub.x=(1/2)*A*r.sub.A*(Pr.sub.B+Pr.sub.C−2*Pr.sub.A)/K.sub.M

β.sub.y=−(3.sup.1/2/2)*A*r.sub.A*(Pr.sub.B−Pr.sub.C)/K.sub.ML=r*β=r*[(3/2)*r*F/K.sub.M]=(3/2)*r.sup.2*(A*Pr)/K.sub.ML.sub.A=(3/2)*r.sub.A.sup.2*A*Pr.sub.A/K.sub.M; Pr.sub.A=(2/3)*L.sub.A*K.sub.M/(r.sub.A.sup.2*A) L.sub.B=(3/2)*r.sub.A.sup.2*A*Pr.sub.B/K.sub.M; Pr.sub.B=(2/3)*L.sub.B*K.sub.M/(r.sub.A.sup.2*A) L.sub.C=(3/2)*r.sub.A.sup.2*A*Pr.sub.C/K.sub.M; Pr.sub.C=(2/3)*L.sub.C*K.sub.M/(r.sub.A.sup.2*A)

β.sub.x=(1/3)*(L.sub.C+L.sub.B−2*L.sub.A)/r.sub.A

β.sub.y=(3.sup.1/2/3)*(L.sub.C−L.sub.B)/r.sub.A

[0154] Joint Variables from Segment Lengths

S.sub.Calc=[S.sub.A+S.sub.B+S.sub.C)/3

β.sub.x=(1/3)*(S.sub.C+S.sub.B−2*S.sub.A)/r.sub.A

β.sub.y=(3.sup.1/2/3)*(S.sub.C−S.sub.B)/r.sub.A

[0155] Find Segment Lengths from Joint Variables.

β.sub.y=(3.sup.1/2/3)*(L.sub.C−L.sub.B)/r.sub.A L.sub.C=L.sub.B+(3/3.sup.1/2)*r.sub.A*β.sub.y  (1)

β.sub.x=(1/3)*(L.sub.C+L.sub.B−2*L.sub.A)/r.sub.AL.sub.C=−L.sub.B+2*L.sub.A+3*r.sub.A*β.sub.x   (2)

[0156] Combine (1) and (2).

L.sub.B+(3/3.sup.1/2)*r.sub.A*β.sub.y=−L.sub.B+2*L.sub.A+3*r.sub.A*β.sub.x L.sub.A=L.sub.B−(3/2)*r.sub.A*β.sub.x+(1/2)*(3/3.sup.1/2)*r.sub.A*β.sub.y   (4)

S.sub.Calc=[L.sub.A+L.sub.B+L.sub.C)/3+L.sub.OFFSETL.sub.A=−L.sub.B−L.sub.C+3*(S.sub.Calc−L.sub.OFFSET)

[0157] Insert (1)

L.sub.A=−L.sub.B−{L.sub.B+(3/3.sup.1/2)*r.sub.A*β.sub.y}+3*(S.sub.Calc−L.sub.OFFSET) L.sub.A=−2*L.sub.B—(3/3.sup.1/2)*r.sub.A*β.sub.y+3*(S.sub.Calc−L.sub.OFFSET)   (5)

[0158] Combine (4) and (5)

L.sub.B—(3/2)*r.sub.A*β.sub.x+(1/2)*(3/3.sup.1/2)*r.sub.A*β.sub.y=−2*L.sub.B−(3/3.sup.1/2)*r.sub.A*β.sub.y+3*(S.sub.Calc−L.sub.OFFSET)

L.sub.B+2*L.sub.B=−(3/3.sup.1/2)*r.sub.A*β.sub.y−(1/2)*(3/3.sup.1/2)*r.sub.A*β.sub.y+(3/2)*r.sub.A*β.sub.x+3*(S.sub.Calc−L.sub.OFFSET)

3*L.sub.B=−(3/2)*(3/3.sup.1/2)*r.sub.A*β.sub.y+(3/2)*r.sub.A*β.sub.x+3*(S.sub.Calc−L.sub.OFFSET)

L.sub.B=S.sub.Calc+(1/2)*r.sub.A*β.sub.x−(1/2)*(3/3.sup.1/2)*r.sub.A*β.sub.y−L.sub.OFFSET   (6)

[0159] Insert (6) into (1)

L.sub.C=(1/2)*r.sub.A*β.sub.x−(1/2)*(3/3.sup.1/2)*r.sub.A*β.sub.y+S.sub.Calc−L.sub.OFFSET+(3/3.sup.1/2)*r.sub.A*β.sub.y

L.sub.C=S.sub.Calc+(1/2)*r.sub.A*β.sub.x+(1/2)*(3/3.sup.1/2)*r.sub.A*β.sub.y−L.sub.OFFSET   (7)

[0160] Insert (6) into (4)

L.sub.A=(1/2)*r.sub.A*β.sub.x−(1/2)*(3/3.sup.1/2)*r.sub.A*β.sub.y+S.sub.Calc−L.sub.OFFSET−(3/2)*r.sub.A*β.sub.x+(1/2)*(3/3.sup.1/2)*r.sub.A*β.sub.y

L.sub.A=S.sub.Calc−r.sub.A*β.sub.x−L.sub.OFFSET   (8)

[0161] Segment Balloon Array Lengths from Joint Variables

L.sub.A=S.sub.Calc−r.sub.A*β.sub.x−L.sub.OFFSET

L.sub.B=S.sub.Calc+(1/2)*r.sub.A*β.sub.x−(1/2)*(3/3.sup.1/2)*r.sub.A*β.sub.y−L.sub.OFFSET

L.sub.C=S.sub.Calc+(1/2)*r.sub.A*β.sub.x+(1/2)*(3/3.sup.1/2)*r.sub.A*β.sub.y−L.sub.OFFSET

[0162] Segment Lengths from Joint Variables

S.sub.A=S.sub.Calc−r.sub.A*β.sub.x

S.sub.B=S.sub.Calc+(1/2)*r.sub.A*β.sub.x−(1/2)*(3/3.sup.1/2)*r.sub.A*β.sub.y

S.sub.C=S.sub.Calc+(1/2)*r.sub.A*β.sub.x+(1/2)*(3/3.sup.1/2)*r.sub.A*β.sub.y

[0163] Find Segment Lengths from Lumen Pressures (Using Spring Pressure Math)

S=[S.sub.A+S.sub.B+S.sub.C)/3

S.sub.Calc=[A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0]/K.sub.FS.sub.A+S.sub.B+S.sub.C=3*[A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0]/K.sub.F   (9)

β.sub.x=(1/3)*(S.sub.C+S.sub.B—2*S.sub.A)/r.sub.A

β.sub.x=(1/2)*A*r.sub.A*(Pr.sub.C+Pr.sub.B−2*Pr.sub.A)/K.sub.M (1/3)*(S.sub.C+S.sub.B−2*S.sub.A)/r.sub.A=(1/2)*A*r.sub.A*(Pr.sub.C+Pr.sub.B−2*Pr.sub.A)/K.sub.MS.sub.C+S.sub.B−2*S.sub.A=(3/2)*A*r.sub.A.sup.2*(Pr.sub.C+Pr.sub.B−2*Pr.sub.A)/K.sub.M   (10)

β.sub.y=(3.sup.1/2/3)*(S.sub.C−S.sub.B)/r.sub.A

β.sub.y=(3.sup.1/2/2)*A*r.sub.A*(Pr.sub.C−Pr.sub.B)/K.sub.M (3.sup.1/2/3)*(S.sub.C−S.sub.B)/r.sub.A=(3.sup.1/2/2)*A*r.sub.A*(Pr.sub.C−Pr.sub.B)/K.sub.MS.sub.C−S.sub.B=(3/2)*A*r.sub.A.sup.2*(Pr.sub.C−Pr.sub.B)/K.sub.M   (11)

[0164] Add (10) and (11)

2*S.sub.C−2*S.sub.A=(3/2)*A*r.sub.A.sup.2*[(Pr.sub.C+Pr.sub.B−2*Pr.sub.A)+(Pr.sub.C−Pr.sub.B)]/K.sub.M

S.sub.C−S.sub.A=(3/2)*A*r.sub.A.sup.2*(Pr.sub.C−Pr.sub.A)/K.sub.M   (12)

[0165] Add (9) and (11)

S.sub.A+2*S.sub.C=3*[A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0]/K.sub.F+(3/2)*A*r.sub.A.sup.2*(Pr.sub.C−Pr.sub.B)/K.sub.M  (13)

[0166] Add (12) and (13)

3*S.sub.C=3*[A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0]/K.sub.F+(3/2)*A*r.sub.A.sup.2*(Pr.sub.C−Pr.sub.B)/K.sub.M+(3/2)*A*r.sub.A.sup.2*(Pr.sub.C−Pr.sub.A)/K.sub.M

S.sub.C=[A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0]/K.sub.F+(1/2)*A*r.sub.A.sup.2*(2*Pr.sub.C−Pr.sub.A−Pr.sub.B)/K.sub.M  (14)

[0167] Substitute (14) into (11)

S.sub.B=S.sub.C−(3/2)*A*r.sub.A.sup.2*(Pr.sub.C−Pr.sub.B)/K.sub.M

S.sub.B=[A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0]/K.sub.F+(1/2)*A*r.sub.A.sup.2*(2*Pr.sub.C−Pr.sub.A−Pr.sub.B)/K.sub.M−(3/2)*A*r.sub.A.sup.2*(Pr.sub.C−Pr.sub.B)/K.sub.M

S.sub.B=[A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0]/K.sub.F+(1/2)*A*r.sub.A.sup.2*(2*Pr.sub.C−Pr.sub.A−Pr.sub.B)−3*(Pr.sub.C−Pr.sub.B)/K.sub.M

S.sub.B=[A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0]/K.sub.F+(1/2)*A*r.sub.A.sup.2*(2*Pr.sub.B−Pr.sub.C−Pr.sub.A)/K.sub.M  (15)

[0168] Substitute (14) into (12)

S.sub.A=S.sub.C−(3/2)*A*r.sub.A.sup.2*(Pr.sub.C−Pr.sub.A)/K.sub.M

S.sub.A=[A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0]/K.sub.F+(1/2)*A*r.sub.A.sup.2*(2*Pr.sub.C−Pr.sub.B−Pr.sub.A)/K.sub.M −(3/2)*A*r.sub.A.sup.2*(Pr.sub.C−Pr.sub.A)/K.sub.M

S.sub.A=[A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0]/K.sub.F+(1/2)*A*r.sub.A.sup.2*[(2*Pr.sub.C−Pr.sub.B−Pr.sub.A)−3*(Pr.sub.C−Pr.sub.A)]/K.sub.M

S.sub.A=[A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0]/K.sub.F(1/2)*A*r.sub.A.sup.2*(2*Pr.sub.A−Pr.sub.B−Pr.sub.C)/K.sub.M  (16)

[0169] Segment Lengths from Lumen Pressure

S.sub.A=[A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0]/K.sub.F+(1/2)*A*r.sub.A.sup.2*(2*Pr.sub.A−Pr.sub.B−Pr.sub.C)/K.sub.M

S.sub.B=[A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0]/K.sub.F+(1/2)*A*r.sub.A.sup.2*(2*Pr.sub.B−Pr.sub.C−Pr.sub.A)/K.sub.M

S.sub.C=[A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0]/K.sub.F+(1/2)*A*r.sub.A.sup.2*(2*Pr.sub.C−Pr.sub.A−Pr.sub.B)/K.sub.M

[0170] For segments with multiple lateral stiffness lengths, solve for a resultant bend constant by adding the contributions of the subsegments. The sub constants are scaled by the section's relative length to the whole segment. To avoid dividing by a zero length solve for the inverse of the bend constant.

dK.sub.M′.sup.−1=(S′/S)*(1/K.sub.M′)

ΣK.sub.M.sup.−1=(1/S)*[(S′/K.sub.M′)+(S″/K.sub.M″)+ . . . (Sn′/K.sub.M.sup.n′)]

ΣK.sub.M=S/ [(S′/K.sub.M′)+(S″/K.sub.M″)+ . . . (Sn′/K.sub.M.sup.n′)]

[0171] The Segment lengths for segments with multiple lateral stiffnesses can be solved as follows:

S.sub.A=[A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0]/K.sub.F+(1/2)*A*r.sub.A.sup.2*(2*Pr.sub.A−Pr.sub.B−Pr.sub.C)/ΣK.sub.M

S.sub.B=[A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0]/K.sub.F+(1/2)*A*r.sub.A.sup.2*(2*Pr.sub.B−Pr.sub.C−Pr.sub.A)/ΣK.sub.M

S.sub.C=[A*(Pr.sub.A+Pr.sub.B+Pr.sub.C)−F.sub.0]/K.sub.F+(1/2)*A*r.sub.A.sup.2*(2*Pr.sub.C−Pr.sub.A−Pr.sub.B)/ΣK.sub.M

[0172] Find Lumen Pressures from Segment Lengths (Using Spring Pressure Math)

[0173] From (9), (10), and (11)

Pr.sub.A+Pr.sub.B+Pr.sub.C=[(1/3)*K.sub.F*(S.sub.A+S.sub.B+S.sub.C)+F.sub.0]/A   (17)

Pr.sub.C+Pr.sub.B−2*Pr.sub.A=(2/3)*[K.sub.M/(A*r.sub.A.sup.2)]*(S.sub.C+S.sub.B−2*S.sub.A)   (18)

Pr.sub.C−Pr.sub.B=(2/3)*[K.sub.M/(A*r.sub.A.sup.2)]*(S.sub.C−S.sub.B)  (19)

[0174] Add (18) and (19)

(Pr.sub.C+Pr.sub.B−2*Pr.sub.A)+(Pr.sub.C−Pr.sub.B)=(2/3)*[K.sub.M/(A*r.sub.A.sup.2)]*(S.sub.C+S.sub.B−2*S.sub.A)+(2/3)*[K.sub.M/(A*r.sub.A.sup.2)]*(S.sub.C−S.sub.B)

Pr.sub.C−Pr.sub.A=(2/3)*[K.sub.M/(A*r.sub.A.sup.2)]*(S.sub.C−S.sub.A)   (20)

[0175] Subtract (17) from (18)

(Pr.sub.C+Pr.sub.B−2*Pr.sub.A)−(Pr.sub.A+Pr.sub.B+Pr.sub.C)=(2/3)*[K.sub.M/(A*r.sub.A.sup.2)]*(S.sub.C+S.sub.B−2*S.sub.A)−[(1/3)*K.sub.F*(S.sub.A+S.sub.B+S.sub.C)+F.sub.0]/A

−3*Pr.sub.A=(2/3)*[K.sub.M/(A*r.sub.A.sup.2)]*(S.sub.C+S.sub.B−2*S.sub.A)−[(1/3)*K.sub.F*(S.sub.A+S.sub.B+S.sub.C)+F.sub.0]/A

Pr.sub.A=(1/3)*[(1/3)*K.sub.F*(S.sub.A+S.sub.B+S.sub.C)+F.sub.0]/A+(1/3)*(2/3)*[K.sub.M/(A*r.sub.A.sup.2)]*(2*S.sub.A−S.sub.C−S.sub.B)   (21)

[0176] Substitute (21) into (20)

Pr.sub.C=Pr.sub.A+(2/3)*[K.sub.M/(A*r.sub.A.sup.2)]*(S.sub.C−S.sub.A)

Pr.sub.C=(1/3)*[(1/3)*K.sub.F*(S.sub.A+S.sub.B+S.sub.C)+F.sub.0]/A+(1/3)*(2/3)*[K.sub.M/(A*r.sub.A.sup.2)]*(2*S.sub.A−S.sub.C−S.sub.B)+(2/3)*[K.sub.M/(A*r.sub.A.sup.2)]*(S.sub.C−S.sub.A)

Pr.sub.C=(1/3)*[(1/3)*K.sub.F*(S.sub.A+S.sub.B+S.sub.C)+F.sub.0]/A+(1/3)*(2/3)*[K.sub.M/(A*r.sub.A.sup.2)]*(2*S.sub.A−S.sub.C−S.sub.B)+(1/3)*(2/3)*[K.sub.M/(A*r.sub.A.sup.2)]*(3*S.sub.C−3*S.sub.A)